University of Wollongong Thesis Collections
University of Wollongong Thesis Collection
University of Wollongong
Year 
Turbulent plumes generated by a
horizontal area source of buoyancy
Apichart Chaengbamrung
University of Wollongong
Chaengbamrung, Apichart, Turbulent plumes generated by a horizontal area source
of buoyancy, PhD thesis, Faculty of Engineering, University of Wollongong, 2005.
http://ro.uow.edu.theses/445
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Turbulent Plumes Generated by a
Horizontal Area Source of Buoyancy
A thesis submitted in fulfilment of the
requirements for the award of the degree
PhD of Engineering
(Mechanical)
from
University of Wollongong
by
Apichart Chaengbamrung, B.Eng, M. Eng
Faculty of Engineering
2005
Thesis Certification
Thesis Certification
CERTIFICATION
I, Apichart Chaengbamrung, declare that this thesis, submitted in fulfilment of the
requirements for the award of Doctor of Philosophy (Mechanical Engineering), in the
Faculty of Engineering, University of Wollongong, is wholly my own work unless
otherwise referenced or acknowledged. The document has not been submitted for
qualifications at any other academic institution.
Apichart Chaengbamrung
February 2005
ii
Abstract
Abstract
Plumes generated from hot surfaces may contain fumes and other contaminants that
constitute major health and environmental hazards. Current design techniques of
contaminated plume control for area sources of heat have limited applicability, and
provide little information on plume characteristics such as vertical velocity distribution,
density distribution, etc. in the zone near the source.
This study is an experimental and computational (CFD) investigation into the
fundamental processes of plume generation and dispersion from a hot area source, such
as a hot metal bath, blast furnaces etc. The main aims are to provide quantitative data
and theoretical models that will enable engineers/designers to greatly improve the
efficiency of exhaust systems and reduce exposure of workers and the community to
harmful contaminants.
The main aim of this project was to conduct a fundamental investigation into the
complex processes of plume generation from a hot surface of finite size and its
subsequent dispersion. Particular objectives included:
1.
Use of experimental techniques to investigate plume characteristics such as
velocity and density/temperature fields.
2.
Computational fluid dynamics (CFD) analysis of velocity and temperature fields.
3.
Validation of the CFD simulations, using analytical solutions and specific
experiments of increasing complexity.
The experimental work was conducted in two distinct programs. In the first part, saline
plumes generated from a uniform area source descending into quiescent water were
studied. This represented a ‘cold flow’ analogy of thermal plumes ascending in
iii
Abstract
quiescent air. A 1.4m cubed environmental tank was designed, built and commissioned
in which experiments were carried out. Experiments are described in which the velocity
and salt concentration profiles in the plume were measured. Two-dimensional velocity
profiles were determined using a PTV (Particle Tracking Velocimetry) technique. The
plume shape was recorded on video tape and also analysed using the DigImage PTV
analysis software. A special conductivity probe with digital position control was used to
measure the concentration of salt solution (or plume density distribution) both inside
and outside the plume at different levels within a 1.4m cubed environmental tank. In the
CFD simulations, two geometries were used to determine plume characteristics and
were compared with experimental results.
In the second program, thermal plumes generated from a uniform temperature area
source were investigated inside a glass enclosure. Both vertical and horizontal velocity
profiles were measured using the LDV (Laser Doppler Velocimetry) technique. Fine
thermocouples were used to measure the temperature distribution of in the plume inside
the enclosure. CFD analysis of confined and unconfined thermal plumes was carried
out.
Saline plume experiments showed that the plume flow from an area source can be
modelled as two regions, one from the area source to the plume neck and the second
region beyond the neck. The saline plume radius was measured using the Shadowgraph
and the Particle Tracking Velocimetry (PTV) technique. The plume radius determined
by the Shadowgraph technique and image analysis was greater than the plume radius
determined by the vertical velocity analysis by approximately 50%.
It was found that the Gaussian equation is suitable as a model of the vertical velocity
and density distributions where the plume reaches the self-similarity region but in the
region near an area source, a proposed modification to the Gaussian equation showed a
better fit than the standard Gaussian equation. The centreline vertical velocity as a
function of height by experiment showed two distinct regions and this supports the work
of Colomer, et al. (1999). In the first region, the present experimental results showed
the same pattern as the model of Colomer, et al. but in the second region the model of
iv
Abstract
Colomer, et al. is in doubt because the value of their centreline velocity increases with
distance away from the area source, which is in contrast to the present experimental
results and contradicts to a decreasing velocity that one would expect on dimensional
grounds.
In the experimental study of thermal plumes in an enclosure, the spread of the plume
vertical velocity radius was found to be wider than the plume temperature radius. The
Gaussian equation also provided a good fit to the vertical velocity and temperature data
in the far-field region. The modified Gaussian equation was used to fit the data in the
near-field region.
In the numerical study, it was found that the choice of the value of constants in the
turbulence model had the effect on the results of simulation. For example, when the
value of Cε 3 and the value of turbulent Prandtl number (Prt) were set to 0.6 and 0.65,
respectively the numerical results showed a good match with experimental results of
both saline plumes and thermal plumes.
Plumes from the ‘hot’ processes such as metallurgical operations (e.g. foundries,
furnace tapping, charging), are often generated from area sources and such plumes
frequently do not reach a self-similar state due to the space over the ‘hot’ processes is
not high enough. Therefore, the main aim of the present study was to determine plume
characteristics, such as temperature or density distribution, velocity profile and plume
width in the near-field region where the plume profiles are not self-similar.
The results of this study provide new information on the fundamental behaviour of
plumes from area sources and will assist in the improvement of the design procedures
for industrial ventilation systems and other building ventilation system applications.
v
Acknowledgements
Acknowledgments
I am very much pleased to acknowledge the university staff and some people who
deserve special mention for their assistance and support in the development of this work
in various ways. Without these people, this work would not have been successful.
Firstly, I would like to express my sincere and grateful thanks to my supervisor, A/Prof.
Paul Cooper, for his imparted knowledge, kind assistance, useful advice, valuable
comments and regular suggestions throughout the completion of this thesis.
Secondly, I am very pleased to thank my co-supervisors, A/Prof. Peter Wypych and Dr.
Ajit Godbole, for their helpful support and important suggestions throughout the
progression of this work. Thirdly, I am very grateful to thank Dr. Buyuug Kosasih for
his kind assistance in providing the essential instructions for using the equipment in the
Laser Laboratory.
Fourthly, I would like to thank Prof. Kiet Tieu for his permission in using the Laser
Laboratory. I also would like to thank Mr. Stuart Rodd (Senior Technical Officer), Mr.
Keith Maywald (Senior Technical Officer) and Mr. Martin Morillas (Senior Technical
Officer) for their technical assistance during the implementation of my experiment.
Finally, thanks to my sponsors, the Royal Thai Government Scholarship, Faculty of
Engineering of Kasetsart University Scholarship and Faculty of Engineering of
University of Wollongong Scholarship who financially supported the author throughout
this study. Without their support, the author would never have been able to undertake
this study. And on a personal note, the most important thanks to my family members,
especially my dear father for their love, support and morale.
(Apichart Chaengbamrung)
February 2005
vi
Table of Contents
Table of Contents
Thesis Certification…………………………………………………………………….ii
Abstract………………………………………………………………………………...iii
Acknowledgments……………………………………………………………………...vi
Table of Contents……………………………………………………………………...vii
List of Figures…………………………………………………………………………xii
List of Tables ………………………………………………………………………...xxv
Nomenclature……………………………………………………………………….xxvii
List of Publications………………………………………………………………...xxxiii
Chapter 1 Introduction……………………………………………………………….2
1.1
Background…………………………………………………………………....2
1.2
Thesis Contents……………………………………………………………......7
1.3
Objectives…………………………………………………………………......7
1.4
Thesis Outline…………………………………………………………............8
Chapter 2 Literature Review………………………………………………………..10
2.1
Plume Analysis………………………………………………………………10
2.2.1
Plumes from Point Sources of Buoyancy in a Uniform Environment............10
2.2.2
Plumes from Area Sources of Buoyancy in a Uniform Environment…….....19
2.2.3
Plume Virtual Origin………………………………………………………...30
2.3
Numerical Simulation of Plume Flows……………………………………...36
2.3.1
Turbulence Modelling…….………………………………...……………….36
2.3.2
Turbulence Models Used in Previous Plume Numerical Studies.......……….37
2.4
Experimental Techniques……………………………………………...…….45
2.4.1
Temperature Measurement………………………………………………......45
2.4.2
Flow Measurement……………………………………………………….....48
2.4.2.1
Velocity Measurement………………………………………………….........48
2.4.2.2
Flow Visualisation……………………………………………………….......50
2.4.3
Direct and Indirect Density Measurement…………………………………...52
2.5
Summary……………………………………………………………………..53
vii
Table of Contents
Chapter 3 Experimental Facilities, Techniques and Procedures…………………56
3.1
Saline Plume Experimental Facilities, Techniques and Procedure………….57
3.1.1
Saline Plume Experimental Facilities………………………………………..57
3.1.1.1
Experimental Tanks………………………………………………………….58
3.1.1.2
Area Sources for Saline Plume………………………………………………62
3.1.1.3
Traversing Mechanism………………………………………………………63
3.1.1.4
Constant-Head Salt Solution Supply………………………………………...64
3.1.1.5
Conductivity Probe…………………………………………………………..65
3.1.1.6
Refractometer………………………………………………………………..68
3.1.1.7
Densitometer………………………………………………………………...68
3.1.1.8
Velocity Field Determination for Saline Plume……………………………..68
3.1.1.9
DigImage System…………………………………………………………....69
3.1.2
Saline Plume Data Collection Techniques…………………………………..71
3.1.2.1
Shadowgraph Technique…………………………………………………….72
3.1.2.2
Particle-Tracking Velocimetry Technique…………………………………..73
3.1.3
Saline Plume Data Collection Procedure………………………………….....75
3.1.3.1
Determination of Saline Plume Shape and Density Field…………………....75
3.1.3.2
Determination of Velocity Field…………………………………………......76
3.2
Thermal Plume Experimental Facilities and Procedures…………………….77
3.2.1
Thermal Plume Experimental Facilities……………………………………..78
3.2.1.1
Heat Source ……………………………………………………………….....78
3.2.1.2
Glass Enclosure and Base……………………………………………………80
3.2.1.3
Thermocouples……………………………………………………………....80
3.2.1.4
Smoke and Smoke Generator………………………………………………..83
3.2.1.5
LDV System…………………………………………………………………84
3.2.2
Experimental Procedure for Thermal Plume Study………………………….86
Chapter 4 Computational Modelling……………………………………………….89
4.1
Governing Equations………………………………………………………...90
4.2
Turbulence Modelling……………………………………………………….94
4.3
The Finite Volume Discretisation……………………………………………99
4.4
Implementation of Boundary Conditions and Source Terms………………100
viii
Table of Contents
4.5
The SIMPLEST Solution Algorithm Technique…………………………...104
4.6
PHOENICS Software………………………………………………………105
Chapter 5 Saline Plume Experimental Results…………………………………...108
5.1
Plume Shape………………………………………………………………..108
5.2
Velocity Distribution……………………………………………………….112
5.3
Density Distribution………………………………………………………..122
Chapter 6 Saline Plume Numerical Models and Results ………………………...127
6.1
Steady-State Numerical Model for a Saline Plume from an Area Source….127
6.2
Transient Numerical Model for Axisymmetric Saline Plume……………...131
6.2.1
Effect of Number of Time Steps……………………………………………132
6.2.2
Effect of Number of Sweeps……………………………………………….134
6.2.3
Grid Dependence…………………………………………………………...134
6.2.4
Comparison of Numerical Results between Saline Plume in Circular Tank and
Rectangular Tank…………………………………………………………..135
6.3
Transient Numerical Model and Results for Saline Plume in a Rectangular
Enclosure…………………………………………………………………...138
6.3.1
Velocity Distribution……………………………………………………….139
6.3.2
Density Distribution………………………………………………………..141
Chapter 7 Thermal Plume Experimental Results………………………………..145
7.1
Experimental Results for 150oC Source……………………………………146
7.1.1
Temperature Distribution…………………………………………………..150
7.1.2
Vertical Velocity Distribution……………………………………………...154
7.1.3
Horizontal Velocity Distribution…………………………………………...157
7.1.4
Turbulence Intensity………………………………………………………..159
7.2
Experimental Results for 200oC Source……………………………………160
7.2.1
Temperature Distribution…………………………………………………..161
7.2.2
Vertical Velocity Distribution……………………………………………...163
7.2.3
Horizontal Velocity Distribution…………………………………………...166
7.2.4
Turbulence Intensity………………………………………………………..167
ix
Table of Contents
Chapter 8 Thermal Plume Numerical Models and Results……………………...170
8.1
Effect of Grid Refinement………………………………………………….171
8.2
The Effect of Turbulence Modelling Parameter, Cε3, on Numerical Results of
Thermal Plume….………………………………………………………….173
8.3
Numerical Results for 150oC Heated Source………………………………176
8.3.1
Vertical Velocity Distribution……………………………………………...178
8.3.2
Temperature Distribution…………………………………………………..180
8.3.3
Horizontal Velocity Distribution…………………………………………...182
8.4
Numerical Result for 200oC Heated Source………………………………..182
8.4.1
Vertical Velocity Distribution……………………………………………...183
8.4.2
Temperature Distribution…………………………………………………..184
8.4.3
Horizontal Velocity Distribution…………………………………………...185
8.5
Numerical Result for 200oC Heated Source without Enclosure……………186
8.5.1
Vertical Velocity Distribution……………………………………………...188
8.5.2
Temperature Distribution…………………………………………………..188
8.5.3
Horizontal Velocity Distribution…………………………………………...190
Chapter 9 Discussion……………………………………………………………….192
9.1
Saline Plume………………………………………………………………..192
9.1.1
Experimental Results of Saline Plume……………………………………..192
9.1.1.1
The Effect of Enclosure on Vertical Velocity Distribution………………...192
9.1.1.2
The Modified Gaussian Equation vs. the Standard Gaussian Equation for
Near-Field Flow……………………………………………………………194
9.1.1.3
Comparison of Plume Radius Measured by Shadowgraph and PTV
Technique…………………………………………………………………..196
9.1.1.4
Comparison of Dneck and zneck………………………………………………197
9.1.1.5
Comparison of the Virtual Origin Location, zv……………………………..200
9.1.1.6
Plume Centreline Vertical Velocity………………………………………...203
9.1.2
Comparison of Saline Plume Experimental and Numerical Results……….205
9.1.2.1
Comparison of Velocity Distribution………………………………………205
9.1.2.2
Density Distribution………………………………………………………..209
9.2
Thermal Plume……………………………………………………………..210
x
Table of Contents
9.2.1
Experimental Results of Thermal Plume…………………………………...210
9.2.1.1
Plume Vertical Velocity Radius, bv , vs Plume Temperature Radius, bT…210
9.2.2
Comparison between Experimental and Numerical Results……………….210
9.2.2.1
Comparison of Vertical Velocity Distribution……………………………..212
9.2.2.2
Comparison of Horizontal Velocity Distribution…………………………..212
9.2.2.3
Comparison of Temperature Distribution………………………………….216
9.2.2.4
Comparison of Plume Radius and Centreline Values of Experimental and
Numerical Results for the Thermal Plume…………………………………217
9.2.3
Effect of Enclosure on Plume Structure……………………………………220
9.3
Comparison of the Thermal and Saline Plumes……………………………222
Chapter 10 Conclusions and Recommendations…………………………………225
10.1
Conclusions………………………………………………………………...225
10.2
Recommendations………………………………………………………….229
References…………………………………………………………………………….232
Appendix I
Experimental Tank Drawing……………………………………...247
Appendix II
Traversing Mechanism…………………………………………….251
Appendix III
Conductivity Probe Design and Drawing………………………...253
Appendix IV
Thermocouple Calibrations……………………………………….255
Appendix V
Table of Properties………………………………………………...257
Appendix VI
Densitometer……………………………………………………….263
Appendix VII
Conductivity Monitor……………………………………………..265
xi
List of Figures
List of Figures
Figure
Description
Page
1.1
Cover and extraction system for a ladle carrying molten iron in
3
automotive industry.
1.2
Hot plume during “tapping” of a blast furnace in the steel industry.
4
1.3
Location of virtual origin of plume from area source recommended by
4
ACGIH (2001).
1.4
Shadowgraph image of a saline plume from area source (virtual origin
5
located by intersection of extended plume edge lines).
1.5
Location of virtual origin of plume from area source. (The Industrial
6
Ventilation Design Guidebook, Goodfellow and Tähti, 2001).
2.1
Relationship between Top-hat and Gaussian profiles.
13
2.2
Velocity vector plots in a plane normal to the horizontal source at
21
different source temperature, D = 30 mm. (Chiari and Guglielmini,
1998)
2.3
Percent relation velocity fluctuations (Chiari and Guglielmini, 1998)
22
2.4
A vertical cross-section through the plume, indicating resultant velocity
23
vectors at difference times. (Colomer, et al., 1999)
2.5
Regions of saline plume identified by Colomer, et al. (1999) showing
24
plume neck width, L0.
2.6
Velocity vector after the flow has become quasi-stationary (at t = 80s).
25
(Colomer, et al., 1999).
2.7
A front-view streak photograph of flow field around the plume under
25
quasi-steady conditions. (Colomer, et al., 1999).
2.8
Plume width vs. distance z. (Colomer, et al., 1999).
xii
26
List of Figures
2.9
The non-dimensional centreline mean vertical velocity of Colomer, et
26
al. (1999) as a function of non-dimensional height.
2.10
Predicted shape of a converging-diverging plume from an area source.
29
(Fannelop and Webber, 2003).
2.11
Dimensionless centreline concentration and velocity as a function of
(
z ′ = ( z − z 0 ) L where L = F 2 g
)
(
15
and w′ = w0 U where U = g 2 F
)
15
29
.
(Fannelop and Webber, 2003).
2.12
The plume width from concentration and vertical profile as a function
(
of z ′ = (z − z 0 ) L where L = F 2 g
)
15
30
. (Fannelop and Webber, 2003).
2.13
The conical correction for the virtual point.
32
2.14
Schematic of the Schlieren flow visualisation (Holman, 1994).
51
2.15
Schematic of the Mach-Zender interferometer (Holman, 1994).
51
3.1
Schematic summarising the present experimental study of turbulent
56
plumes from an area source.
3.2
Schematic of the apparatus for saline plume experiment.
57
3.3
Estimation of water tank size required for saline plume experiment.
59
3.4
Result of numerical stress calculation for water tank.
60
3.5
Design drawings of environmental fluid dynamics tank.
61
3.6
Photograph of tank.
62
3.7
Cross-section drawings of the area source.
62
3.8
Photograph of the saline plume area source.
63
3.9
Schematic of traversing mechanism and controller.
64
3.10
Photograph of the conductivity probe in the large tank.
65
3.11
Cross-section of the present conductivity probe.
66
3.12
Schematic of the conductivity probe and conductivity meter
67
xiii
List of Figures
arrangement.
3.13
Typical calibration graph for the conductivity probe.
68
3.14
The shadowgraph technique arrangement.
72
3.15
Image of grid in the experimental tank for position mapping.
74
3.16
The particle tracking experimental arrangement.
75
3.17
View of thermal plume experimental arrangement.
79
3.18
Drawing of heat source and floor of thermal plume experiment.
79
3.19
Cross-section of heat source and floor of thermal plume experiment.
80
3.20
Thermocouple locations in and around the heat source.
81
3.21
Thermocouple bank.
82
3.22
Typical calibration graph (thermocouple no.5).
83
3.23
Forward-Scattering LDV.
84
3.24
Laser generator.
85
3.25
Laser transmitter and receiver probes.
86
4.1
Schematic of control volume and their faces.
99
4.2
Diagrammatic representation of the PHOENICS software.
106
5.1
The period used for averaging profile images in DigImage.
110
5.2
Example of time-averaged and enhanced image of saline plume shape.
111
Test no.4, B0 = 10.50×10-4 m2/s3, M0 = 4.46×10-9 m4/s2 and Q0 =
6.22×10-6 m3/s area source (at 60s over 30s averaging time).
5.3
Plume edge of test no.4, B0 = 10.50×10-4 m2/s3, M0 = 4.46×10-9 m4/s2
112
and Q0 = 6.22×10-6 m3/s source conditions (at 60s over 30s averaging
time).
5.4
Example of particles detected by DigImage during PTV processing.
113
5.5
The particle path detected by DigImage during velocity analysing
114
process at 60s over 10s period (Test no.5, B0 = 1.60×10-4 m2/s3, M0 =
xiv
List of Figures
4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s source conditions).
5.6
Plume flow from the area source at each time (average over one
115
second). (Test no. 5, B0 = 1.60×10-4 m2/s3, M0 = 4.33×10-10 m4/s2 and
Q0 = 1.94×10-6 m3/s source conditions).
5.7
Average velocity field of saline plume of test no.5, B0 = 1.60×10-4
116
m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s source conditions
(at 60s over 30s averaging time).
5.8
Vertical velocity profile of saline plume at ~0.04m below the area
117
source (test no.5, B0 = 1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 =
1.94×10-6 m3/s (at 60s over 30s averaging time, n = 2.13).
5.9
Vertical velocity profiles in saline plume at different levels showing
118
best fit to experimental data using Equation 5.2 (test no.5, B0 =
1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s at 60s
over 30s averaging time).
5.10
Mean horizontal velocity distribution in saline plume at different levels
119
(test no.5, B0 = 1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 =
1.94×10-6 m3/s at 60s over 30s averaging time).
5.11
Variations of wC and bv with position in saline plume (test no.5, B0 =
120
1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s at 60s
over 30s averaging time).
5.12
Variation of centreline vertical velocity (wC) with heights of plume
121
under different source conditions.
5.13
Variation of plume velocity radius (bv) with heights of plume under
121
different source conditions.
5.14
Variation of shape index (n) with heights of plume under different
122
source conditions.
5.15
Density distribution in saline plume at 0.10m below source ( B0 =
xv
123
List of Figures
1.07×10-4 m2/s3, M 0 = 1.54×10-9 m4/s2 and Q0 = 10.04×10-6 m3/s source
condition (test no.9)).
5.16
Density distribution of saline plume at different levels ( B0 = 1.07×10-4
124
m2/s3, M 0 = 1.54×10-9 m4/s2 and Q0 = 10.04×10-6 m3/s source condition
(test no.9), symbols shows experimental results and line best fits).
5.17
Density distribution in saline plume at different levels ( B0 = 1.66×10-4
124
m2/s3, M 0 = 3.73×10-9 m4/s2 and Q0 = 16.25×10-6 m3/s source condition
(test no.10), symbols shows experimental results and line best fits).
5.18
Density distribution in saline plume at different levels ( B0 = 9.56×10-5
125
m2/s3, M 0 = 3.04×10-10 m4/s2 and Q0 = 4.64×10-6 m3/s source condition
(test no.11), symbols shows experimental results and line best fits).
5.19
Density distribution in saline plume at different levels ( B0 = 2.15×10-4
125
m2/s3, M 0 = 1.54×10-9 m4/s2 and Q0 = 10.44×10-6 m3/s source condition
(test no.12), symbols shows experimental results and line best fits).
6.1
The 1×42×100 cells and computational domain for steady-state saline
128
plume simulation.
6.2
Velocity vector and density contour of steady-state saline plume
129
simulation ( Cε 3 = 0.6 and Prt = 1.0).
6.3
Effect of Cε 3 on centreline velocity in saline plume (Prt = 1.0).
130
6.4
Effect of Prt on saline plume vertical velocity at 0.06m below an area
131
source ( Cε 3 =1.0).
6.5
The cylindrical polar geometry of saline plume simulation.
132
6.6
The effect of time step on vertical velocity distribution of plume.
133
(r = 0.00m, z = 0.10m below an area source at 60s after plume was
released).
xvi
List of Figures
6.7
The effect of number of sweeps on vertical velocity at r = 0.00m and z
133
= 0.10m (at t = 60s).
6.8
The effect of cells on vertical velocity at r = 0.08 and z = 0.10m (at t =
135
60s).
6.9
Computational domain for numerical study of saline plume in
136
rectangular tank.
6.10
The 32×32×60 cells of rectangular tank geometry.
136
6.11
Comparison of centreline vertical velocity between two models of
137
tanks (at 60s).
6.12
Comparison of vertical velocity distribution between two models of
137
tanks (at 60s).
6.13
Comparison of density distribution between two models of tanks (at
138
60s).
6.14
Velocity vector field 60s after plume was released (0.276m radius and
140
0.5m height cylindrical tank). The source conditions of test no.5, w0 =
2.24×10-4 m/s and 10% by weight salt concentration.
6.15
Vertical velocity profile in saline plume 60s after plume was released
140
(the source conditions of test no.5, w0 = 2.24×10-4 m/s and 10% by
weight salt concentration).
6.16
Horizontal velocity profiles in saline plume 60s after plume was
141
released (the source conditions of test no.5, w0 = 2.24×10-4 m/s and
10% by weight salt concentration).
6.17
Geometry model and the 1×35×100 numerical cells of large
142
computational circular tank.
6.18
Density contour in saline plume 120s after plume was released
(0.7841m radius and 1.20m height cylindrical tank). The source
conditions of test no. 9, w0 = 1.48×10-4 m/s, 10% by weight of salt
concentration.
xvii
143
List of Figures
6.19
CFD results of density profiles in saline plume 120s after plume was
143
released (the source conditions of test no. 9).
7.1
Position of thermocouple used to measure temperature inside an
146
enclosure.
7.2
Development of quasi-steady thermal conditions inside the enclosure
147
before plume velocity and temperature distribution data were collected.
7.3
Surface temperature distribution in 150oC hot plate.
148
7.4
Temperature distribution of thermal plume from 150oC hot plate at
149
0.10m above the floor for three separate experiments to illustrate
repeatability of results.
7.5
Vertical velocity distribution of thermal plume from 150oC hot plate at
149
0.10m above the floor for three separate experiments to illustrate
repeatability of results.
7.6
Horizontal velocity distribution of thermal plume from 150oC hot plate
150
at 0.10m above the floor for three separate experiments to illustrate
repeatability of results.
7.7
The comparison of two fitting-equations with temperature distribution
151
o
data at 0.02m (z/D = 0.1064) above the floor from 150 C hot plate.
7.8
The comparison of bT from two fitting equations.
152
7.9
Temperature distributions at different elevations in thermal plume
153
above 150oC hot plate.
7.10
The graph of ∆TC and bT of thermal plume above 150oC hot plume.
154
7.11
Vertical velocity distribution at 0.10m above floor from 150oC hot
155
plate compared with equation 5.2, bV = 0.0547m.
7.12
Vertical velocity distribution in thermal plume from 150oC hot plate.
156
7.13
The graph of wC and bV of thermal plume above 150oC hot plate.
156
xviii
List of Figures
7.14
The comparison graph of bT and bV of thermal plume above 150oC hot
157
plate.
7.15
Horizontal velocity distribution of thermal plume at 0.004m above the
158
hot plate with 150oC.
7.16
Horizontal velocity distribution of thermal plume above 150oC hot
158
plate.
7.17
Vertical velocity turbulence intensity at different levels from 150oC hot
159
plate.
7.18
Horizontal velocity turbulent intensity at different levels from 150oC
160
hot plate.
7.19
Surface temperature distribution of 200oC hot plate.
161
7.20
Temperature distribution at 0.10m above the floor above 200oC hot
162
plate compared with equation 7.2.
7.21
Temperature distribution of thermal plume above 200oC hot plate.
162
7.22
∆TC and bT of thermal plume above 200oC hot plate.
163
7.23
Vertical velocity distribution of thermal plume above 200oC hot plate.
164
7.24
Graph of wC and bV of thermal plume above 200oC hot plate.
165
7.25
Comparison of bT and bV of thermal plume above 200oC hot plate.
165
7.26
Horizontal velocity distribution of thermal plume from 200oC hot plate
166
at z = 0.004m.
7.27
Horizontal velocity distribution of thermal plume above 200oC hot
166
plate.
7.28
Vertical velocity turbulence intensity at different levels from 200oC hot
167
plate.
7.29
Horizontal velocity turbulence intensity at different levels from 200oC
hot plate.
xix
168
List of Figures
8.1
Computation domain of thermal plume simulation.
172
8.2
The comparison of vertical velocity at middle of plume from different
172
number of cells (at z = 0.10m above hot plate).
8.3
The comparison of temperature at middle of plume from different
173
number of cells (at z = 0.10m above hot plate).
8.4
Comparison of vertical velocity distribution at 0.1m above the floor
174
when the value of Cε 3 is changed.
8.5
Contour plots of GENB and GENK of the results of the thermal plume
175
0
simulation. (Temperature of hot plate = 200 C, Probe located at z =
0.01m, r = 0.00m).
8.6
Contour plots of GENB and GENK of the results of saline plume
176
simulation (Test no. 5).
8.7
The computation domain and 69×69×91 computational mesh of
177
thermal plume simulation.
8.8
Velocity vectors and temperature contour of thermal plume from
177
~150oC hot plate.
8.9
Numerical surface temperature distributions in 150oC hot plate
178
comparing with experimental data.
8.10
Vertical velocity distribution in thermal plume from ~150oC hot plate.
179
8.11
Vertical velocity distribution of thermal plume from ~150oC hot plate
179
at 0.10m above hot plate compared with equation 7.3, n = 1.55.
8.12
Temperature distribution of thermal plume from ~150oC hot plate.
180
8.13
Comparison of bT and bV deduced from CFD simulations of thermal
181
plume from ~150oC source.
8.14
Horizontal velocity distribution of thermal plume from ~150oC hot
182
plate.
8.15
Numerical surface temperature distributions in 200oC hot plate
xx
183
List of Figures
comparing with experimental data.
8.16
Vertical velocity distribution in thermal plume from ~200oC hot plate.
184
8.17
Temperature distribution of thermal plume from ~200oC hot plate.
184
8.18
Comparison of bT and bV deduced from numerical simulation of
185
thermal plume from ~200oC source.
8.19
Horizontal velocity distribution of thermal plume from ~200oC hot
186
plate.
8.20
Geometry and 69×69×91 grids of thermal plume without an enclosure.
187
8.21
Vertical velocity vectors and temperature contour of thermal plume
187
from ~200oC hot plate without an enclosure.
8.22
Plume vertical velocity distributions from ~200oC hot plate without an
188
enclosure.
8.23
Temperature distribution in thermal plume from ~200oC hot plate
189
without an enclosure.
8.24
Comparison of bT and bV of thermal plume from ~200oC hot plate
190
without an enclosure.
8.25
Horizontal velocity distribution of thermal plume from ~200oC without
190
an enclosure.
9.1
Average velocity filed of saline plume test no.5 at 60s over 30s
193
averaging time.
9.2
Average vertical velocity distribution of saline plume (Test no. 5 at
193
0.02m below an area source (over 60s with 30s averaging time).
9.3
Experimental and curve-fit velocity profiles of saline plume.
194
9.4
Value of the shape index, n, in Equation 5.2 for vertical velocity
195
profiles as a function of distance from the area source for saline plume.
9.5
Vertical velocity plume radius, bV, deduced from the two equations.
(Test no. 5).
xxi
196
List of Figures
9.6
Comparison of the plume radius estimated by two methods.
197
9.7
Comparison of Dneck from experimental data compared with results of
199
Colomer, et al. (1999).
9.8
Comparison of zneck from experimental data compared with results of
199
Colomer, et al. (1999).
9.9
Estimation of the virtual origin location from two experimental
200
methods.
9.10
Virtual origin location from different methods compared with
202
experimental results for saline plumes.
9.11
Dimensionless centreline vertical velocity of a saline plume as a
203
function of dimensionless distance from source using the scaling of
Colomer, et al. (1999).
9.12
Dimensionless centreline vertical velocity of a saline plume as a
205
function of dimensionless distance from source using the scaling of
Fannelop and Webber (2003).
9.13
Comparison of experimental and numerical results for vertical velocity
207
distribution at different levels of saline plume of test no.5.
9.14
Comparison of experimental and numerical results for horizontal
208
velocity at different levels of saline plume of test no. 5.
9.15
Screw head in the area source.
209
9.16
Comparison of experimental and numerical results for density
211
distribution in a saline plume below an area source of test no. 9.
9.17
The comparison graph of bT and bV of thermal plume above 150oC and
211
o
200 C hot plate.
9.18
Comparison of vertical velocity distribution of thermal plume from
212
~150oC hot plate.
9.19
Comparison of vertical velocity distribution of thermal plume from
xxii
213
List of Figures
~200oC hot plate.
9.20
Comparison of horizontal velocity distribution of thermal plume at
213
0.004m above ~150oC hot plate.
9.21
Comparison of horizontal velocity distribution of thermal plume at
214
0.020m above ~150oC hot plate.
9.22
Comparison of horizontal velocity distribution of thermal plume at
214
0.100m above ~150oC hot plate.
9.23
Comparison of horizontal velocity distribution of thermal plume at
215
0.004m above ~200oC hot plate.
9.24
Comparison of horizontal velocity distribution of thermal plume at
215
0.020m above ~200oC hot plate.
9.25
Comparison of horizontal velocity distribution of thermal plume at
216
0.100m above ~200oC hot plate.
9.26
Comparison of temperature distribution of thermal plume from ~150oC
216
hot plate.
9.27
Comparison of temperature distribution of thermal plume from ~200oC
217
hot plate.
9.28
Comparison of vertical velocity radius, bV, of thermal plume from
218
150oC and 200oC hot plate.
9.29
Comparison of temperature radius, bT, of thermal plume from 150oC
219
and 200oC hot plate.
9.30
Comparison of wC of thermal plume from 150oC and 200oC hot plate.
220
9.31
Comparison of ∆TC of thermal plume from 150oC and 200oC hot plate.
220
9.32
Comparison of plume vertical velocity radius, bV, of thermal plume
221
with and without an enclosure above ~200oC hot plate.
9.33
Comparison of plume temperature radius, bT, of thermal plume with
and without an enclosure above ~200oC hot plate.
xxiii
222
List of Figures
9.34
Comparison of dimensionless plume vertical velocity radius, bv/D0, vs.
223
dimensionless distance, z/D0, between thermal and saline plume.
A-I 1
Tank drawing.
247
A-I 2
Tank base.
248
A-I 3
Position of tank frame.
248
A-I 4
The 0.5×0.5×0.5 m3 water tank.
250
A-II 1
The drawing of the present traversing mechanism used for saline plume
251
experiments.
A-III 1
Drawing of original conductivity probe.
253
A-III 2
Drawing of modified conductivity probe.
254
A-VI 1
Densitometer.
263
A-VII 1
The conductivity meter.
265
xxiv
List of Tables
List of Tables
Table
Description
Page
2.1
Summary of some round axisymmetric plume research.
31
2.2
Advantages and disadvantages of different turbulence models
39
2.3
Range and polynomial coefficients for difference thermocouples.
47
5.1
The conditions of all experimental tests
109
5.2
Values of establishment time (te) for plumes to reach quasi-steady
110
state as recommended by Colomer, et al. (1999).
5.3
Centreline velocity, wC, vertical velocity radius, bV, and shape
119
index, n, at different heights of plume of test no.5, B0 = 1.60×10-4
m2/s3, M 0 = 4.33×10-10 m4/s2 source conditions (at 60s with 30s
averaging time).
6.1
Number of cells used for each numerical model.
134
7.1
The experimental conditions of the thermal plume experiments.
145
7.2
Values of ∆TC , bT and n of thermal plume above 150oC hot plate.
153
7.3
Values of wC , bV and n of thermal plume above 150oC hot plate.
155
7.4
Values of ∆TC , bT and n of thermal plume above 200oC hot plate.
163
7.5
Values of wC , bV and n of thermal plume above 200oC hot plate.
164
8.1
Meshes used in grid refinement studies.
171
8.2
Values of wC , bV and n of thermal plume from ~150oC hot plate at
180
different levels.
8.3
Values of ∆TC , bT and n of thermal plume from ~150oC hot plate at
different levels.
xxv
181
List of Tables
8.4
Values of wC , bV and n of thermal plume from ~200oC hot plate at
183
different levels.
8.5
Values of ∆TC , bT and n of thermal plume from ~200oC hot plate at
185
different levels.
8.6
Values of wC , bV and n of thermal plume from ~200oC hot plate
188
without an enclosure.
8.7
Values of ∆TC , bT and n of thermal plume from ~200oC hot plate
189
without an enclosure.
9.1
Standard Deviation of two equations after fitting with experimental
195
data.
9.2
Values of Dneck and zneck compared with calculated value of
198
Colomer, et al. (1999).
9.3
The value of virtual origin location, zv, from different methods.
201
A-I 1
Material details of 1.4×1.4×1.4m3 water tank.
249
A-I 2
The 0.5×0.5×0.5m3 water tank.
250
A-II 1
Material details of traversing mechanism.
252
A-IV 1
The liner equation of all thermocouple calibrations.
255
A-V 2
Properties of air
260
A-VI 1
The technical data of the present densitometer.
264
A-VII 1
The technical data of the conductivity meter.
265
xxvi
Nomenclature
Nomenclature
English Symbols
Capital Letters
A
Area
m2
Ai
Area of a cell face
m2
AW
Constant value for time average vertical velocity (eq. 2.11)
AT
Constant value for time average temperature (eq. 2.12)
B
Buoyancy flux per unit area
m2/s3
B0
Buoyancy flux per unit area at source
m2/s3
BW
Constant value for time average vertical velocity (eq. 2.11)
BT
Constant value for time average temperature (eq. 2.12)
CP
Plume invariants (eq. 2.31)
Cε1
Constant in k-ε turbulent model (eq. 4.37)
Cε2
Constant in k-ε turbulent model (eq. 4.37)
Cε3
Constant in k-ε turbulent model (eq. 4.37)
Cµ
Constant value in k-ε turbulent model (eq. 4.38)
C1
Percentage by weight of salt solution
D
Diameter
m
Dc
Plume diameter at xc
m
Ds
Source Diameter
m
E
A constant depending on the roughness (eq. 4.46)
F
Buoyancy flux
F
Mean mixture fraction
Fg
Gravitational force
Fr
Froude number
F0
Buoyancy flux at source
G
N2
H
Depth of confined region
m
L
Line source width
m
Lm
Jet length
m4/s3
kg.m/s2
m4/s3
xxvii
Nomenclature
L0
Necking diameter
m
M
Specific momentum flux
m4/s2
M0
Momentum flux at source
m4/s2
N
Buoyancy frequency of the ambient
Pr
Prandtl number
Q
Specific mass flux (or volume flux)
m3/s
Q0
Volume flux at source
m3/s
R
Radius of confined region
m
R
Universal ideal gas constant
Ra
Rayleigh number
R0
Source Richardson number
RP
Plume Richardson number
Sbc
Boundary condition source term
Sg
Source term of buoyancy
Sm
Mass flow boundary condition term
SΦ
Source term of Φ
T
Temperature
T
A geometrical multiplier
TC
o
C
Centreline temperature of plume
o
C
TW
Wall temperature
o
C
T ′2
Mean square temperature fluctuation
V
Volume
m3
Vm
Mass flux = ρin×uin
kg/(m2.s)
∆V
Volume of a cell
m3
W
Mean square vorticity fluctuation
Lowercase Letters
a
Constant value
an
Polynomial coefficient at n order
b
Plume radius
m
bav
the width-averaged buoyancy (in equation 2.55)
(m/s3)1/3
xxviii
Nomenclature
bG
Gaussian plume radius
m
bT
Plume temperature radius
m
btop
Top-hat plume radius
m
c
Constant in equation 2.43
c
The value in percent to which the mean amplitude of the
profile has decreased to c% of peak value (eq. 2.66)
%
cp
The specific heat of the fluid at constant pressure
kJ/(kg.oC)
g
Gravitational acceleration
m/s2
g′
Reduced gravity
m/s2
g G′
Gaussian reduced gravity
m/s2
′
g top
Top-hat reduced gravity
m/s2
h
Total enthalpy
kJ/kg
k
Turbulent kinetic energy
kJ/kg
k
Thermal conductivity
W/(m.K)
k
1 η
km
Constant in equation 2.29
kq
Constant in equation 2.29
p
Pressure
Pa
pref
Reference density
kg/m3
q
Heat flux
W/(m2.s)
r
Radius from centre of plume
m
t
Time
s
te
A quasi-steady state time
s
y
Distance from centre of plume in case of plume from line
source
y+
Non-dimensional distance from the wall
y P+
Non-dimensional distance from the wall to node P
∆yP
Distance of the near wall node P to the solid surface
u
Velocity vector
u
Velocity in x component
m
m/s
xxix
Nomenclature
uin
Inlet velocity
m/s
uP
Velocity at the grid node
m/s
uτ
Friction velocity
m/s
v
Velocity in y component
m/s
ve
Lateral entrainment velocity
m/s
w
Velocity in z direction
m/s
w
Time averaged vertical velocity
m/s
wG
Gaussian vertical velocity
m/s
wtop
Top-hat vertical velocity
m/s
z
Vertical distance from a source
m
zavs
Asymptotic virtual origin
m
zc
Location of necking from a circular area source
m
z0
Position of the first front
m
zv
Location of virtual point source
m
Greek Symbols
α
Entrainment constant
β
Volumetric coefficient of thermal expansion
γ
Intermittency factor
δij
Kronecker delta
ε
Rate of dissipation of turbulent kinetic energy
ζ
z/ H
η
ρ ρ∞
κ
Von Karman’s constant
λ
Second viscosity
λ
Universal constant for forced plume
µ
The molecular vicosity
kg/(m.s)
µt
Eddy viscosity
kg/(m.s)
ν
Laminar kinematic viscosity
m2/s
νt
Turbulent kinematic viscosity
m2/s
ρ
Density
kg/m3
K-1
kg/(m.s)
xxx
Nomenclature
ρ∞
Ambient density
kg/m3
ρin
Inlet density
kg/m3
ρref
Reference Density
kg/m3
∆ρ
Density difference at same level
kg/m3
σ
Laminar Prandtl number/Schmidt number
σh
The effective Prandtl number for the diffusion of heat
σt
Turbulent Prandtl number/Schmidt number
σk
Prandtl number value of k
σε
Prandtl number value of ε
τ
The viscous stress
kg.m/s2
τw
The wall shear stress
kg.m/s2
φ
Angle of the spread of plume
degree
Г
Source parameter
ГФ
Diffusion coefficient
Ф
Fluid property (eq. 4.18)
Ф
(Γ − 1)
Γ (eq. 2.78)
Superscripts
_
Time average
/
Fluctuation components
Subscripts
ambient
ambient
c
Centreline value
G
Gaussian profile
i
The component in i direction, i = 1, 2, 3
j
The component in j direction, j = 1, 2, 3
k
The component in k direction, k = 1, 2, 3
top
Top-hat profile
xxxi
Nomenclature
Abbreviations
DNS
Direct Numerical Simulation
FEM
Finite Element Method
LES
Large Eddy Simulation
LDV
Laser Doppler Velocimetry
PIV
Particle Image Velocimetry
PTV
Particle Tracking Velocimetry
RTD
Resistance Temperature Detector
xxxii
Introduction
List of Publications
During the course of this PhD study, various papers were published by the author as a
result of the study. They are listed as follows:
1. Chaengbamrung, A., Cooper, P., Wypych, P. and Godbole, A., Theoretical and
Experimental Investigation of Descending Salt Plume from a Circular Source,
Seventh Australian Natural Convection Workshop, 21-22 July 2003, University of
Sydney, Australia
2. Chaengbamrung, A., Cooper, P., Wypych, P. and Godbole, A., Theoretical and
Experimental Investigation of Thermal Plumes in Air from a Uniform-Temperature
Circular Heat Source, Seventh Australian Natural Convection Workshop, 21-22 July
2003, University of Sydney, Australia.
3. Chaengbamrung, A., Cooper, P., Wypych, P. and Godbole, A., Theoretical and
Experimental Investigation of Plume from a Circular Distributed Source, The 7th
International Symposium on Ventilation for Contaminant Control, August 5-8 2003,
University of Hokkaido, Japan
4. Chaengbamrung, A., Godbole, A., Cooper, P. and Wypych, P., Experimental and
Numerical Investigation of Hot Process Plumes for Local Exhaust Ventilation, 8th
International Conference on Bulk Materials Storage, Handling and Transportation,
July 2004, University of Wollongong, NSW, Australia
5. Godbole, A., Cooper, P., Wypych, P. and Chaengbamrung, A., Theoretical
Investigation of Generation and Dispersion of Fume from Hot Metal Baths into an
Inert Atmosphere, 8th International Conference on Bulk Materials Storage, Handling
and Transportation, July 2004, University of Wollongong, NSW, Australia
xxxiii
Chapter 1
Introduction
Introduction
Chapter 1
Introduction
1.1
Background
Natural Convection is one of the most important classes of fluid flow and heat transfer
and one of the most prevalent. It is buoyancy-induced, arising due to a body force such
as gravity, acting on density variations existing within a fluid. A plume is one type of
natural convection flow that is generated by a continuous source of buoyancy and is a
feature of many natural and artificial phenomena such as eruption of volcanoes, fires in
forests, metal-making processes, natural ventilation in workplaces, electronic cooling,
and so on.
The plume was first studied quantitatively by Schmidt (1941), who observed that the
flow of hot air rising from a small source occupies a conical region above the source
when the flow is turbulent. He studied the mean velocity and temperature profiles in
both two- and three-dimensional plumes. Morton, et al. (1956) proposed the
subsequently well-known theory of plume flow. After their work, many researchers
such as Caulfield and Woods (1998), Rooney and Linden (1998), Elicer-Cortés (1998),
Hunt and Kaye (2001) modified that theory using different assumptions for various
situations.
All these studies consider plume characteristics such as temperature distribution,
velocity distribution, etc. in the “self-similar” (or far-field) region only. The location of
the virtual origin, from which the far-field plume profile of the same characteristics as
that which would evolve from a pure source of buoyancy, was also considered in this
work.
In the flow region close to the source, where the plume profiles cannot be assumed to be
self-similar, relatively little research work has been carried out to date. One example is
that of Colomer, et al. (1999) who studied descending saline plumes from an area
source. They presented two correlations for the velocity distribution in the plume: one
2
Introduction
for the region between the source and the plume “neck”, and the other for the region
beyond.
In ‘hot’ processes such as metallurgical operations (e.g. foundries, furnace tapping,
charging), exposure of hot metal to the atmosphere can generate a contaminant plume
that disperses the contaminants such as metal fume into the surroundings and can
constitute major health and environmental hazards (Figures 1.1 and 1.2).
The design of fume control systems for such situations to date has been hampered by a
lack of quantitative information on temperature and velocity fields in plumes above hot
metal processes. The limited guidance available, such as that in the Industrial
Ventilation Manual by American Conference of Governmental Industrial Hygienists
(ACGIH, 2001), typically has very simple algorithms for the calculation of air flow
above a hot object (Figure 1.3), leading to questionable recommendations.
Please see print copy for Figure 1.1
Figure 1.1
Cover and extraction system for a ladle carrying molten iron in the
automotive industry (Cooper, 2001).
3
Introduction
Please see print copy for Figure 1.2
Figure 1.2
Hot plume during “tapping” of a blast furnace in the steel industry
(Cooper, 2001).
Please see print copy for Figure 1.3
Figure 1.3
Location of virtual origin of plume from area source
recommended by ACGIH (2001).
4
Introduction
In Figure 1.3,
Ds :
diameter of heat source.
Dc :
Plume diameter at hood face (Dc ≈ 0.5Xc0.88).
Y:
distance from the process surface to the hood face.
Z:
distance from the actual area source to the hypothetical point source
(Z = (2Ds)1.138).
Xc :
(Y+Z), the distance from the hypothetical point source to the hood face.
But in reality, plumes from area sources such as the surface of molten metal always
exhibit the phenomenon of ‘necking’ because of limited ambient flow entrainment near
the surface of an area source. This is shown in Figure 1.4, for the case of a saline plume
descending into a body of still water, studied as part of the present work.
Virtual
Origin
Width of area source
Entrainment
Figure 1.4
Entrainment
Shadowgraph image of a saline plume from an area source (virtual origin
located by intersection of extended plume edge lines).
5
Introduction
Therefore, the estimation of effective plume width and virtual origin location, and also
velocity distribution, etc. as recommended in the Industrial Ventilation Manual by
ACGIH (2001) is not completely correct because it does not include consideration of
the necking in the plume flow from large sources.
The effect of necking is included for determination of the virtual origin of the plume in
the Industrial Ventilation Design Guidebook (Goodfellow and Tähti, 2001). Here,
reference is made to the research work of Morton et al. (1956) that suggested that the
virtual source be located at z v = 1.7 to 2.1D below the real source as shown in Figure
1.5. However, this does not include a measure of the strength of the source that may
have an effect on the virtual origin. Therefore, one of the objectives of the present work
was to study the effect of source conditions on the virtual origin of plumes from area
sources.
Please see print copy for Figure 1.5
Figure 1.5
Location of virtual origin of plume from area source.
(The Industrial Ventilation Design Guidebook, Goodfellow and Tähti, 2001).
In addition, an important assumption of the classical plume theory of Morton et al.
(1956) is the Boussinesq approximation. This approximation is not valid in the case of
a strong source of buoyancy, especially in gases. In the case of the exposure of hot
6
Introduction
metal baths to the atmosphere, the exposed area is finite. Therefore, classical plume
theory and many modified plume theories that use the self-similarity and Boussinesq
approximations may not be appropriate in the near-field regions of strong plumes,
particularly in gases.
Therefore, the main aim of the present investigation was to determine plume
characteristics such as temperature or density distribution, velocity profile and plume
width in the near-field region where the plume profiles are not self-similar. It is hoped
that the results of these studies will provide new information on the fundamental
behaviour of plumes from area sources and assist in the improvement of the design
procedures for industrial ventilation systems.
1.2
Thesis Contents
In this thesis, the turbulent buoyancy-driven plume from an area source was studied
using both experimental and numerical methods. In the experimental investigation,
special facilities were designed and built to collect the data for velocity profiles, density
profiles (in case of saline plumes) and temperature distribution (in case of thermal
plumes). The data was analysed in order to find out the main characteristics of the
plume such as necking diameter and location, plume width, and temperature or density
profiles. All characteristics of plumes that were deduced from the experimental data
were also compared with the results of numerical analyses carried out using the
PHOENICS Computational Fluid Dynamics (CFD) package.
1.3
Objectives
This study was a fundamental investigation into the structure of natural convection
plumes generated by area sources of buoyancy. Particular objectives included:
1.
Development of specific experimental apparatus and methodologies for
measurement of characteristics of saline and thermal plumes from area sources.
2.
Computational Fluid Dynamics (CFD) simulations of the velocity and density or
temperature fields of plumes above area sources of buoyancy.
3.
Validation of the CFD simulations.
7
Introduction
4.
Comparison of the experimental and numerical results with previously theoretical
models and also development for determining the validity of these models in the
near-field of an area source of buoyancy.
1.4
Thesis Outline
This thesis consists of nine chapters. In Chapter 2, previous theoretical and
experimental research is reviewed. Chapter 3 describes the experimental facilities,
measurement techniques and experimental procedures used in this study for saline
plume and thermal plume experiments. In Chapter 4, the necessary governing equations
that were used to simulate the flow of plumes from area sources are discussed, including
the boundary conditions, turbulence modelling and solution techniques. In Chapter 5,
the results of the experimental study of saline plumes are presented and discussed
including vertical velocity, horizontal velocity and density profiles. In the case of
necking, the minimum diameter of the plume, the results of necking diameters and
necking locations from experiments are compared with those of other researchers. In
Chapter 6, the results of the experimental and numerical studies of saline plumes are
compared and presented. In Chapter 7, the results of the experimental study of thermal
plumes are shown for two source conditions, 150oC and 200oC. Chapter 8 includes the
results of numerical simulations of thermal plumes in air, in terms of vertical velocity,
temperature distribution, etc., along with comparisons with experimental results. The
discussions of all results of both experimental and numerical study are presented in
Chapter 9. Finally, conclusions and recommendations for further study are presented in
Chapter 10.
8
Chapter 2
Literature Review
Literature Review
Chapter 2
Literature Review
In this chapter, previous research that forms the background to the present project is
reviewed. Firstly, earlier attempts at understanding the characteristics of plumes such as
velocity profile, temperature profile or density profile are reviewed. Secondly,
numerical analysis research work on buoyancy-driven flows and plumes is summarized.
This includes a discussion of turbulence modelling techniques for buoyancy driven
flows generally and plumes in particular. Finally, a review of experimental techniques
appropriate for the present project is presented.
2.1
Plume Analysis
Plumes represent one particular case of natural convection flow where the fluid motion
is set up as a result of buoyancy forces. The analytical study of plumes was initiated by
Schmidt (1941). After this work, many researchers investigated the phenomena of
plumes. Many methods and many assumptions were used to predict plume
characteristics such as plume width, velocity profile, density profile, etc. In this section,
the main outcomes of previous studies of plumes are summarised.
2.2.1
Plumes from Point Sources of Buoyancy in a Uniform Environment
The first quantitative plume study was the research work of Schmidt (1941) who
observed that the plume of hot air rising from a small source occupies a conical region
when the flow is turbulent. He used mixing-length hypotheses to obtain expressions for
the mean velocity and temperature profiles for both plane (two-dimensional) and round
(three-dimensional) plumes. The plume was further studied by Rouse et al. (1952). The
aim of their study was to find the mean velocity and temperature profiles in both
axisymmetric and two dimensional (line source) plumes in a uniform environment.
They suggested from dimensional analysis that:
10
Literature Review
For the case of line source:
w
F L
(
1 3 −1 3
0
(F
 y 
z
) = f 
g′
L
)z
2 3 −2 3
0
−1
(2.1)
= f  y 
 z
(2.2)
For the case of point sources:
w
F z
(
13 13
0
g′
(F z
and
r
) = f ( z)
2 3 −5 3
0
(2.3)
r
) = f ( z)
(2.4)
where F0 is buoyancy flux at source
w is time average vertical velocity
g ′ is reduced gravity
r is radial distance from the centre of a plume in the case of a plume from a
point source
y is distance from centre of the plume in the case of a plume from a line source
z is vertical distance from a source
and
L is line source width
In order to estimate the constants in the above equations, experimental measurements
using vane anemometers and thermocouples were used. After fitting the data to the
above equations, it was found that:
For a line source:
2

 −32 y
2

z
1 3 −1 3 
0
w = 1.80 F L e
g ′ = 2.6 F
23
0
−2 3
L




2

 − 41 y
2

z
−1 
z e
(2.5)




(2.6)
The value of lateral entrainment velocity, ve, was inferred to be
ve = ±0.28(− F0 L )
11
1
3
(2.7)
Literature Review
For a point source:
2

 −96 r
2

z
1 3 −1 3 
0




2

 −71 r
2

z
−5 3 




w = 4.7 F z
g ′ = 11F02 3 z
e
e
(2.8)
(2.9)
The value of lateral entrainment velocity, ve, was inferred as
ve = ±0.041F01 3 z 2 3 r
(2.10)
Batchelor (1954) also studied plumes arising from a point source. He suggested that the
velocity and reduced gravity distributions depend on height z, buoyancy flux F0, and the
radial distance r :
w = AW F z
13
0
−1 3
e
r
− BW  
z
2
r
− BT  
2 3 −5 3
z
0
g ′ = AT F
z
(2.11)
2
e
(2.12)
where AW , AT , BW , BT are constants.
Perhaps the most important piece of research on the fundamental characteristics of the
plume was that by Morton, Taylor and Turner (1956). Their analysis was based on three
main assumptions:
• The entrainment velocity at a particular altitude at the edge of the plume is directly
proportional to the vertical velocity at the same altitude with a constant of
proportionality, called “the entrainment constant”, α (or ventrain = α vvertical)
• Profiles of vertical velocity and buoyancy force in horizontal sections are similar,
and
• The Boussinesq approximation is valid. In this case, the largest local variations of
density the inside plume are negligible compared with density difference between
inside and outside the plume at the same level.
Morton, et al. (1956) modelled the evolution of the specific mass flux (or volume flux),
Q, the specific momentum flux, M and buoyancy flu,x F with height, z, using the
equations of conservation of mass, momentum, buoyancy for a plume above a “point
12
Literature Review
source” of buoyancy (with finite buoyancy flux, F0, but zero volume and momentum
fluxes), all assumptions of Morton, Taylor and Turner (1956) and assumption of “tophat profiles for plume flows as shown below:
The top-hat values above can be converted to Gaussian values by using the following
relationships:
btop = 2 bG , wtop =
wG
g′
′ = G , α top = 2α G
, g top
2
2
(2.13)
In order to avoid the confusion about top-hat and Gaussian profile, the variables w, b,
g ′ and α without subscribe “top” from here are the variables for Gaussian profile.
wG
Gaussian
Top-hat
wG /2
bG
r
btop
Figure 2.1
Relationship between Top-hat and Gaussian profiles.
Therefore, the relationship of the specific mass flux (or volume flux), Q, the specific
momentum flux M and buoyancy flux F with height were defined as:
dQ
= 2α M 1 2
dz
(2.14)
dM FQ
=
dz
M
(2.15)
dF
= −QN 2
dz
(2.16)
13
Literature Review
where Q ≡ b 2 w , M ≡ b 2 w 2 , F ≡ g ′ b 2 w , N is buoyancy frequency of the ambient and
b is the plume width.
N 2 (z ) ≡ −
ρ −ρ
g dρ o

and reduced gravity g ′ ≡ g  ∞
ρ1 dz
 ρ0 
(2.17)
Here, ρ∞ is the ambient density, ρ is the density inside the plume and ρ0 is the density
outside the plume at the same level.
In the case of N = 0 (uniform ambient), the buoyancy flux in a plume is constant at all
heights. Therefore, using the top-hat profile, Morton, et al. estimated the plume width,
′ , as:
btop, top-hat vertical velocity, wtop, and top-hat reduced gravity, g top
btop =
6α top
5
13
z , wtop
5 Fo  9
5 9

 −1 3
′ =
=
 α F0 
 α top F0  z and g top
6α  10
6α top  10


−1 3
z −5 3
(2.18)
By examining the evolution of plumes from sources with finite fluxes of buoyancy,
volume and momentum (F0, Q0, M0), Morton (1959a) continued his research using
Gaussian profiles for time-averaged vertical velocity and time-averaged buoyancy.
He quoted the research work of Rouse et al. (1952) that suggested a greater lateral
spread of heat compared to that of vertical momentum. Therefore, he suggested that the
entrainment coefficient, α, measures the rate of flow into a forced plume with the
velocity profile characterised by the plume width, b, and with an associated buoyancy
profile characterised by λb where α and λ are universal constants for forced plumes.
Therefore, the plume can be assumed to have a Gaussian profile of time-averaged mean
vertical velocity and reduced gravity:
(
g ′ = g ′ exp(− r
)
(2.19)
λ2 b 2 )
(2.20)
w = wC exp − r 2 b 2
C
2
Morton (1959a) also classified “forced” plumes into three categories using the ‘source
parameter’, Г, where
a)
“Forced” plume when 0 < Γ < 1
14
Literature Review
b)
“Pure” plume when Γ = 1
c)
“Lazy” plume when Γ > 1,
(
)
−5
Γ = 2 9 2 5 α −1 1 + λ2 F0 M 0 Q0
where
2
(2.21)
This issue is discussed further in section 2.2.2 below.
George, Alpert and Tamanini (1978) studied axisymmetric turbulent buoyant plumes.
They used two-wire probes to measure the temperature and velocity fields in the plume
in a uniform ambient. The velocity and temperature distributions were estimated to be
given by:
w = 3.4 F z
13
0
−1 3
g ′ = 9.1 F02 3 z
e
r
−55  
z
2
r
− 65  
−5 3
z
(2.22)
2
e
(2.23)
Baines and Turner (1969) studied the effect of continuous convection from small
sources of buoyancy on the properties of a uniform environment when the region was
bounded. They coined the term “the first front” for the front of buoyant fluid which first
reaches the upper boundary (or bottom in case of dense plume in low density ambient)
and begins to descend. They found that the time taken by the first front reach to a given
height, z0, is given by:

5  5π 
− 2 3 −1 3  H 
2
t=

 R H F0  
4α  18α 
 z 0 
13
23

− 1

(2.24)
where R is the radius of the confined region,
H is the depth of confined region,
z0 is the position of the front,
and
t is the time.
In a non-dimensional form:
13
[
]
5
τ = 5  ζ o −2 3 − 1
8
15
(2.25)
Literature Review
where
2
13
4
z
H F
and τ = 1 3 α 4 3   04 3 t
ζ =
π
H
R H
(2.26)
List (1982) presented a summary on turbulent jets and plumes. In the case of turbulent
thermal plumes, if a plume source is thermal the specific buoyancy flux is defined as:
F = gB q (ρ c p )
(2.27)
where B is the volumetric coefficient of thermal expansion,
q is the heat flux,
c p is the specific heat of the fluid at constant pressure.
If the plume rises from a source with mass flux then the specific buoyancy flux is
F = g (ρ ∞ − ρ ) Q ρ
(2.28)
List (1982) also quoted that for fully developed plumes it can be shown that the total
specific momentum is given by
M = km F 2 3 z 4 3
(2.29)
Q = kq F 1 3 z 5 3
(2.30)
and the volume flux by
where km and kq are constants.
Therefore, these results can be used to derive the plume Richardson number, Rp
where
Rp = Q F 1 2 M 5 4
(2.31)
List (1982) quoted from the research work of Rouse et al. (1952) that the values of Cp
and Rp for a pure plume in the fully developed region are 0.25 and 0.56 respectively.
The rates of entrainment into the plumes were defined as
For round plume
dQ dz = (5 3) C p M 1 2
16
(2.32)
Literature Review
dQ dz = (C p M z )
12
For plane plume
(2.33)
Baines (1983) continued Baines and Turner (1968) work on “filling box theory” and
proposed a technique for the direct measurement of the volume flux in a plume as a
function of source condition and height. This technique used the location of the
interface in a filling box experiment to define the volume flux in the plume. In his work,
Baines (1983) plotted Q 3 5 F0−1 5 vs. z. He found that the value of the Gaussian
entrainment coefficient, α, for a pure saline plume was 0.074 and the Froude number of
a pure plume is constant along the height and is given by:
Fr 2 =
5 2
M52
= 1 2 = 6.74
2
Q F0 8π α
(2.34)
Papanicolaou and List (1987, 1988) studied round turbulent buoyant jets. They
suggested the use of a dimensionless parameter that defined the degree of jet-like or
plume-like character of the initial flow, the Richardson number, R0, defined as:
Q0 F01 2
R0 =
M 05 4
(2.35)
The initial R0 varies from very small for jet-like flows to unity for plume like flows.
Therefore, the value of R0 at the source is one of main parameters used to check the
condition of the source. In this research, they also suggested using the jet length, Lm, to
check whether the buoyant flow is jet-like or plume-like. They suggested that the flow
is plume like when the value of z/Lm > 5,
where
Lm = 2 −3 2 α −1 2
M 03 4
F01 2
(2.36)
In the fully developed plume region, z/Lm > 5, Papanicolaou and List (1987, 1988)
suggested that the vertical velocity and concentration profiles are given by:
w = 3.85 F z
13
0
−1 3
17
e
r
−90  
z
2
(2.37)
Literature Review
g
∆ρ
ρ
= g ′ = 14.3 F02 3 z
r
−80  
−5 3
z
2
e
(2.38)
They also suggested the value of the entrainment coefficient is 0.0875 in the plume
region.
Caulfield and Wood (1991) extended the Morton, et al. (1956) model. In the case of
plumes in uniform environments, they suggested a special constant, c, to define plume
types:
c ≡ 1−
a)
“Forced” plume when c < 0
b)
“Pure” plume when c = 0
c)
“Lazy” plume when c > 0
8α M 05 2
2
5 F0 Q0
(2.39)
The constant c is related to the source parameter, Г, defined by Morton (1959) by:
c = 1−1 Γ
(2.40)
Caulfield and Wood (1991) also suggested that the spread of the plume depended on the
entrainment coefficient, α. For a pure plume, tan (φ ) = 6α/5 while for a forced plume
2α > tan (φ ) > 6α/5 and for a lazy plume tan (φ ) < 6α/5 where φ is the plume half angle
at the virtual source.
Shabbir and George (1994) reported their experimental study of a round turbulent
plume in a uniform ambient. They found that:
w = 3.4 F z
13
0
−1 3
g ′ = 9.4 F02 3 z
e
r
−58  
z
2
r
−68  
−5 3
z
e
(2.41)
2
(2.42)
Dai, Tseng and Faeth (1994, 1995a and 1995b) studied round buoyant turbulent plumes
in the fully developed region. They found that:
18
Literature Review
g ′ = 12.6 F02 3 z
r
−125  
−5 3
z
2
e
(2.43)
In the above research work, a consistently used assumption was the Boussinesq
approximation: the largest local variations of density ∆ρ inside the plume are small
compared with density ρ outside the plume at the same level. In many practical
situations such as fires or industrial processes, the Boussinesq approximation is not
valid because (∆ρ/ρ) > 0.1, the limit for validity of the Boussinesq approximation
(Malin, 2003). Therefore, in order to study plumes from industrial processes, studies of
non-Boussinesq plumes are important. Rooney and Linden (1996, 1997) studied nonBoussinesq plumes in a uniform environment. They presented a mathematical analysis
of plumes without the Boussinesq approximation. Their similarity solution for the nonBoussinesq case is
13
wtop
5 9 
=   π −1 3 α −2 3 F 1 3 z −1 3
6  10 
btop
6α
=
5
5 9
′ =  
g top
6  10 
 ρ 
z  
 ρ0 
(2.44)
−1 2
(2.45)
−1 3
π −2 3α −4 3 F 2 3 z −5 3
(2.46)
where
F=
2
btop
wtop ∆ρ g
ρ0
(2.47)
The value of the entrainment velocity, ve, in the non-Boussinesq case can be defined as:
5 6α
ve = ×
3 5
2.2.2
12
 ρ 
  wtop
 ρ0 
(2.48)
Plumes from Area Sources of Buoyancy in a Uniform Environment
This section presents a summary of previous work on plumes generated from area
sources of buoyancy.
19
Literature Review
Kotsovinos (1985) studied the structure of the mean and fluctuating temperature field in
a turbulent, vertical round plume in a uniform ambient. The focus of his study was on
the transition of jet-like flow to plume-like flow. However, he did not measure fluid
velocities. He suggested that the parameter often known as the “jet length”,
Lm = M 03 4 F01 2 , could be used to define of the position where the flow changes from
jet-like to plume-like (where M0 and F0 are momentum and buoyancy fluxes
respectively).
In the region where z << M 03 4 F01 2 , buoyancy induced momentum is relatively small
compared to the initial momentum of the source and the flow is jet-like. In the region
where z >> M 03 4 F01 2 , the flow is plume-like since the buoyancy-induced momentum
dominates. He proposed that the temperature distribution equation is a function of the
plume width, bt (z):
2

 r  

T ( z , r ) = TC exp − (ln 2) 


 bt ( z )  

(2.49)
From his experimental results, Kotsovinos (1985) found that
(
T ( z , r ) = TC exp − 69(r z )
where
2
)
(2.50)
bt (z) ≈ 0.1z
He also suggested that a pure thermal plume generated from a heated disk of diameter D
reaches self-similarity at z D ≈ 24 , and that any round buoyant jet becomes a fully
developed, self-preserved, plume at a distance zF01 2 M 03 4 > 14 or at zR01 2 D > 12.5
where R0 is the source Richardson number defined by:
R0 =
Q02 F0
M 05 2
(2.51)
Elicer-Cortés (1998) studied the temperature field in a pure thermal plume. He
suggested that for z / D > 2.15 , the mean centreline temperature varied as z −5 3 but for
z / D < 2.15 , the mean centreline temperature did not follow the z −5 3 law. He also
20
Literature Review
stated that for z D > 1.5, the thermal field achieved a state of self-similarity. The halfwidth of temperature bht (z) was linear in the self-similarity zone and the spread rate was
dbT dz = 0.096.
Chiari and Guglielmini (1998) investigated the thermal plume from an area source.
They used fluorocarbon liquid FC-72 as their working fluid. Laser Doppler Velocimetry
was used for velocity measurement in their work. The results of velocity vectors for
different Rayleigh numbers are shown in Figure 2.2.
Please see print copy for Figure 2.2
Figure 2.2
Velocity vector plots in a plane normal to the horizontal source at
different source temperature, D = 30 mm. (Chiari and Guglielmini, 1998).
The results for turbulence intensity given as percent relative velocity fluctuations as a
function of r of their work are shown in Figure 2.3.
21
Literature Review
Please see print copy for Figure 2.3
Figure 2.3
Percent relative velocity fluctuations.
(Chiari and Guglielmini, 1998).
Colomer, Boubnov and Fernando (1999) studied saline plumes from an isolated area
source. They found that:
i) The velocity vector of plume from area source from starting time as shown in Figure
2.4 below.
22
Literature Review
Figure 2.4
A vertical cross-section through a saline plume, indicating
resultant velocity vectors at difference times:
(a) t = 20s and (B0 D 2 ) t = 3.82 , (b) 40 and 7.64, (c) 60 and 11.46. The external
13
parameters are D = 19.1cm and B0 = 2.54 cm2s-3. (Colomer, et al. 1999)
From Figure 2.4, it can be seen that the entrainment flow first appeared near the
perimeter of the area source, but with time, it spreads along the entire plume.
ii) The plume achieved a quasi-steady state at a time, te, that is a function of diameter of
source, D, and the buoyancy flux per area, B0:
t e ≈ 1.8(D 2 B0 )
13
(2.52)
After the plume reaches a quasi-steady state, the flow field of the plume can be divided
into two regions as shown in Figure 2.5 below.
23
Literature Review
Please see print copy for Figure 2.5
Figure 2.5
Regions of saline plume identified by Colomer, et al. (1999)
showing plume neck width, L0.
Figure 2.6 shows the velocity vector of the plume measured and analysed by Digimage
when the plume reached a quasi-steady defined by Equation 2.52. Figure 2.7 shows a
streak photograph of the flow field of the plume from the area source of Colomer, et al.
(1999). Because of difficulties in image processing, the particle tracks within the plume
were not presented.
Figure 2.8 shows the plume width for different values of buoyancy flux and Colomer, et
al. claimed that the buoyancy flux has an insignificant effect on Dneck and zneck of a
plume from the area source. They suggested that:
Dneck = (0.55±0.05)D
(2.53)
zneck = (0.28±0.13)D
(2.54)
and
24
Literature Review
Please see print copy for Figure 2.6
Figure 2.6
Velocity vector after the flow has become quasi-stationary (at t = 80s).
D = 19.1cm, B0 = 0.53 cm2s-3. (Colomer, et al. 1999)
Please see print copy for Figure 2.7
Figure 2.7
A front-view streak photograph of flow field around the plume under
quasi-steady conditions. D = 19.1 cm and B0 = 2.54 cm2s-3. (Colomer, et al. 1999).
25
Literature Review
Please see print copy for Figure 2.8
Figure 2.8
Plume width vs distance z. D = 19.1 cm and B0, ● = 2.71; ○ = 2.63;
+ = 0.75; × = 6.28; ■ = 4.43; □ = 7.80; ▲ = 2.19; ∆ = 1.35 cm2s-3.
(Colomer, et al. 1999).
Please see print copy for Figure 2.9
Figure 2.9
The non-dimensional centreline mean vertical velocity of Colomer, et al.
(1999) as a function of non-dimensional height.
26
Literature Review
Colomer, et al. (1999) suggested that in region I, 0 < z < zc where zc ≈ 0.28D, the
horizontal and vertical velocities and the plume width are functions of D, B0 and z. The
average maximum horizontal velocity, ve, was found to be
ve ≈ 0.66(B0 D )
13
(2.55)
The centreline vertical velocity, wC , was found to be
wC ≈ 2.7(B0 D )
13
(z D )
(2.56)
The equation of the width-averaged buoyancy was found to be
(
bav ≈ 10 B02 D
)
13
(2.57)
For region II, z > zc, the equations of centreline vertical velocity and the width-averaged
buoyancy are not dependent on the diameter of source, D. They were found to be
wC ≈ 1.2(B0 z )
13
and
z D > 0 .3
for
bav ≈ 3.7(B02 z )
13
for
z D > 0 .4
(2.58)
(2.59)
Recently, Fannelop and Webber (2003) developed a theoretical model of buoyant
plumes from area sources. They suggested a power law solution in terms of height to
describe a converging-diverging flow above an area source:
Q = η w b2 ,
M = η w 2 b 2 and F = (1 − η ) w b 2
(2.60)
where η is ρ/ρα. They suggested the relation for z as a function of M to be
z = (1 ( gF ))∫ (M Q ) dM
(2.61)
where Q = Q(M) depends on the value of an arbitrary function of the density ratio, k.
For one source only k = 1 η used by Morton et al., 1956, and Taylor, 1958 or k = 1
used by Ricou and Spalding, 1961.
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Literature Review
In the case of k = 1, Fannelop and Webber (2003) showed that
 8α  5 2
Q = Q02 + 
M
 5 gF 
(2.62)
In the case of k = 1 η , they showed that
[(Q + 3 F 2)
]
 3F 2
Q(Q + F ) − (Q0 + 3 F 2) Q0 (Q0 + F ) − 
 2

  Q+
 ln 
  Q +
  0
(Q + F )   8α  5 2
=
 M (2.63)
(Q0 + F )   5gF 
Fannelop and Webber (2003) also found that at the point of maximum velocity in the
plume from an area source is given by
2α k M 5 2
gF Q 2
=1 .
(2.64)
zw
At the point of minimum radius or at the “neck” of the plume:
2α (1 − η ) k M 5 2
gF
Q2
=1 .
(2.65)
zb
From the equations above, it is clear that the maximum velocity in the plume from an
area source always occurs above the neck of plume.
Figure 2.10 to 2.12 shows the predicted converging-diverging plume from an area
source using Equation 2.59 of source condition b = 5 m, w = 0.2 ms-1, η = 0.6. Figure
2.11 to Figure 2.12 show the comparison of non-dimensional concentration and velocity
profiles with experimental data of Fannelop and Webber (2003). Fannelop and Webber
(2003) did define w′ in their paper, however, it has been subsequently established
(Webber, 2005) that they used w′ = w U .
28
Literature Review
Please see print copy for Figure 2.10
Figure 2.10
Predicted shape of a converging-diverging plume from an area source
(source condition b = 5m, w = 0.2ms-1, η = 0.6 and used the Morton-Taylor-Turner
entrainment model with α = 0.1) (Fannelop and Webber, 2003).
Please see print copy for Figure 2.11
Figure 2.11
Dimensionless centreline concentration and velocity as a function of
z ′ = ( z − z 0) L where L = (F 2 g )
15
and w′ = w0 U where U = (g 2 F ) . (Fannelop and
15
Webber, 2003).
29
Literature Review
Please see print copy for Figure 2.12
Figure 2.12
The plume width from concentration and vertical velocity profile as a
function of z ′ = ( z − z 0 ) L where L = (F 2 g ) . (Fannelop and Webber, 2003).
15
The table 2.1 is a summary of some of the main results of the previous studies on
plumes in a uniform ambient.
2.2.3
Plume Virtual Origin
The virtual origin of a plume of finite area source with buoyancy flux, specific volume
flux and specific momentum flux (F0, Q0, M0) is the position of an idealized plume
point source with finite buoyancy flux but zero initial fluxes of volume and momentum.
The concept of a virtual origin of the plume is very useful to determine other plume
characteristics such as plume width, vertical velocity distribution, temperature
distribution, etc. From previous research work, the location of the virtual origin can be
modelled using a variety of different methods. However, these methods can be
categorised in terms of four main techniques. These are:
30
Literature Review
Table 2.1
Name & Year
Rouse et al.
(1952)
Summary of some round axisymmetric plume research.
Methods
Exp.
2 

 −96 r 
2 

z
1 3 −1 3 

0
w = 4.7 F z
Morton, et al.
(1956)
wtop
(Top hat)
George, et al.
(1978)
Exp.
Exp.
(1988)
Shabbir and
George (1994)
Exp.
g ′ = 9.1F
23
0
e
2 

 −90 r 

z 2 
1 3 −1 3 
0
z
g ′ = 14.3F
2 

 −58 r 
z 2 
1 3 −1 3 
0
g ′ = 9.4 F
23
0
e
Exp.
Linden
Theo.
(1996,1997)
wtop =
0.153
e
z
z
g ′ = 12.6 F02 3 z
(1994)
Rooney and
(top hat)
2 

 −65 r 
z 2 
−5 3 
Dai, Tseng
and Faeth
0.093
2 

 −80 r 

z 2 
2 3 −5 3 
0
e
w = 3.4 F z
0.041
e
5F 9α 
1 3 −5 3
′ = 0 
g top
 F0 z
6α  10 
2 

 −55 r 
z 2 
1 3 −1 3 
0
w = 3.85F z
z
13
5  9α 
1 3 −1 3
=

 F0 z
6α  10 
Papanicolaou
and List
g ′ = 11F
e
w = 3.4 F z
Coefficient
2 

 −71 r 
2 

z
2 3 −5 3 

0
13
Theo.
Entrainment
g′
Velocity Distribution
e
2 

 −68 r 
z 2 
−5 3 
e
2 

 −125 r 

z 2 
−5 3 
e
13
non-Boussinesq
−1 3
5 9 
−1 3 − 2 3 1 3 −1 3
5
9


π
α
F
z
−
2
3
−
4
3
2
3
−
5
3
 
0.2
′ =   π α F z
g top
6  10 
6  10 
(top hat)
Region I
Colomer, et al.
(1999)
wc ≈ 2.7(Bo D )
13
Exp.
(z D )
Region II
wc ≈ 1.2(Bo z )
13
Provided general integral formulation but with no explicitly relation for velocity etc. versus z.
z=
Fannelop
and Webber
Theo.
[
(2003)
1
M
dM
∫
(gF ) Q(M )
 8α  5 2
where k = 1
Q = Q02 + 
M
 5 gF 
Or (Q + 3F / 2 ) Q(Q + F ) − (Q0 + 3F / 2 ) Q0 (Q0 + F )
−
3F 2  Q +
In 
2
 Q0 +
(Q + F )   8α  5 2
M where k = 1
=
(Q0 + F )   5 gF 
31
]
η
Literature Review
Corrections based on empirical measurement
In this method, the virtual origin is located by iteratively fitting experimental results to a
given function such as (z vs. F01 4 G −3 8 (note: G = N 2 ) of Morton, et al. (1956)). This is
carried out graphically. Similarly, Baines (1983) plotted z vs. (Q (c1 F 1 3 )) and w3z vs.
35
w3 as suggested by Bachelor (1954).
A conical source correction
This method was also suggested by Morton, Taylor and Turner (1956). They used the
radius of the nozzle as the effective radius of the plume (see Figure 2.3):
D = 2b − ln(c)
(2.66)
where D is the diameter of the area source, b is the radius of the plume, c is the value in
percent to which the mean amplitude of the profile has decreased to c% of the peak
value.
The location of the virtual point source then can be determined from:
zv
5
=
D 12α − ln(c)
(2.67)
D
An area source
zv (c)
Figure 2.13
The conical correction for the virtual point.
Morton, et al. (1956) also suggested that the value of c be equal to 0.01.
32
Literature Review
A source correction based on the initial properties of F0, M0 and Q0
Morton (1959a, b) suggested the term “forced plume” for a plume generated by a source
which has finite fluxes of buoyancy, momentum and mass. His analysis showed that in
the case of a uniform environment, the behaviour of a forced plume with a source of
buoyancy, mass and momentum can be matched to that from a virtual point source of
buoyancy only using the following two steps:
1) Replacing the real forced plume source (F0, M0, Q0) at z = 0 with a virtual point
source “forced” plume of modified strength (F0, γM0, 0) located at z = zv.
In this case, Morton (1959a) proposed the equation:
z v Lm = −101 2 γ
32
1 γ
sgn F0 ∫
sgn γ
t 5 − sgn γ
−1 2 3
t dt
(2.68)
where γ 5 = 1 − Γ and
Γ=
5Q02 F0
4αM 05 2
Lm = 2
−2 3
α
(2.69)
−1 2
M 03 4
F01 2
(2.70)
From the above solution, he classified the forced plume into four types:
a)
For 0 < Γ < 1, the value of zv can be calculated by:
1γ
z v Lm = −101 2 γ 3 2 ∫
1
b)
(t
)
−1
5
−1 2 3
t dt
(2.71)
For Γ = 1, a forced plume behaves as a purely buoyant plume (F0, 0, 0) at z = zv.
No next step is required:
z v Lm = −2.108
c)
For Γ > 1, the value of zv can be calculated by:
z v Lm = 3.162 γ
d)
(2.72)
∫ (t
32 1 γ
−1
)
t dt
(1 − t )
t dt
5
+1
−1 2 3
(2.73)
For Γ < 1, the value of zv can be calculated by:
1
z v Lm = −3.162 γ 3 2 ∫
1γ
5 −1 2 3
(2.74)
2) If γ ≠ 0 (or the case of Γ = 1), replacing force plume of modified strength (F0, γM0, 0)
at z = zv with a pure buoyancy source plume (F0, 0, 0) at z = zv + zavs.
33
Literature Review
In this step, the position of zavs can be found from:
z avs Lm = 101 2 sgn F0
∫
v
sgn γM 0
t 5 − sgn γM 0
−1 2
t 3 dt
(2.75)
where dw dz = v
In 2001, Hunt and Kaye had modified Morton’s model (1959a). he suggested one step
model for replacing two step model of Motor to estimate the virtual origin location of a
lazy plume (Г > 0.5), i.e. a plume where the source has less momentum than in a pure
plume of the same radius and buoyancy flux. They used a modified source parameter, Г,
that was defined as:
Γ=
5Q02 F0
4αM 05 2
Or in terms of the actual physical fluxes, Qˆ 0 , Fˆ0 , Mˆ 0 .
Γ=
 Q02 F0 
A5 2 g 0′


Γ
≈
or
M 02
2 7 2 α π 1 2  M 05 2 
5
(2.76)
The position of their dimensionless asymptotic virtual origin was defined as:
*
z avs
= Γ −1 5 (1 − δ )
*
=
z avs
where
where
δ=
(2.77)
z avs
5  Q0 


6α  M 01 2 
n

3
9 2
11 3
3 ∞ 
Φn
(1 + 5( j − 1)) (2.78)
Φ+
Φ +
Φ + ... = ∑  n −1
∏
35
425
1125
5 n=1  5 n!(10n − 3) j =1

Φ=
( Γ − 1)
Γ
(2.79)
For Γ > 88, the approximation of δ = 0.147 leads to results accurate to within 5%. Then
z avs
6α M 01 2
= 0.853 Γ −1 5
5 Q0
For Γ = 1,
34
(2.80)
Literature Review
z avs
6α M 01 2
=1
5 Q0
(2.81)
In the above studies, the positions of the virtual origin are determined for a plume
source with finite fluxes of buoyancy, momentum and mass. In case of a pure thermal
plume from a finite area source, it has no finite momentum and mass fluxes therefore it
is necessary to adapt the previous equation in order to find out the virtual point of plume
from this kind of an area source.
In 1998, Elicer-Cortés studied thermal plumes from a hot area source. He found that the
position of the virtual origin, z, depended only on the diameter of the area source, D :
z D = 0.19
(2.82)
The “Industrial Ventilation Manual” of ACGIH (1998) suggested that the position of
the virtual origin can be found from:
z = (2 D )
1.138
(2.83)
The ASHRAE Application Handbook (1999) recommended using the method of
Skistad (1994) to find the position of virtual origin. Skistad suggested the “maximum”
and “minimum” cases of estimation. In the maximum case, the real source is replaced
by a point source passes through the top edge of the real source. The minimum case is
when the diameter of the vena contrata of the plume is about 80% of upper surface
diameter and is located approximately 1/3 diameter above the source. He also suggested
that the maximum case is suitable for low-temperature sources and the minimum case is
suitable for high-temperature sources.
The “Industrial Ventilation Design Guidebook” of Goodfellow and Tähti (2001) quoted
the research work of Morton et al. (1956) and suggested that the position of virtual
origin is a function of the diameter of the hot surface:
z v ≈ 1.7 to 2.1D
Depending on the maximum (2.1D) or minimum (1.7D) case is used.
35
(2.84)
Literature Review
2.3
Numerical Simulation of Plume Flows
The Computational Fluid Dynamics (CFD) technique applied to plume flow is
summarised in this section. This technique is very useful and powerful. It has been used
to study and design many industrial processes. In industrial ventilation systems such as
ventilation in hot metal processes, ventilation in welding areas where plume flow is
involved, the design processes of these ventilation systems are complicated. Empirical
models may not be accurate enough to predict the plume flow and design of the
ventilation system if the workplace is complicated and if it has multiple sources of flow.
The CFD work can overcome some limitations of empirical theory but it is necessary to
validate CFD results against analytical or experimental results.
Application of CFD to plume flow requires the identification of suitable models e.g.
turbulence models that can be implemented and also give acceptable accurate results.
Therefore, in this section, previous CFD studies on plume flow are reviewed.
In the following sub-sections, previously used turbulence models are explained and
grouped. Secondly, a literature review CFD studies of plume behaviour is presented and
discussed. Finally, the turbulence model used in the present research is explained.
2.3.1
Turbulence Modelling
Turbulence is a complex physical phenomenon. The earliest attempts to study
turbulence involved developing a mathematical description of turbulent stresses. Further
progress was made possible by a discovery by Prandtl about the “boundary layer” in
1904. Prandtl introduced the mixing length model that can be used for computing the
“eddy viscosity” in terms of the “mixing length”. The mixing-length hypothesis formed
the basis of virtually all turbulence modelling research for the next twenty years.
By definition, an “n-equation” turbulence model requires n additional transport
equations to be solved simultaneously along with those for the conservation of mass,
momentum and energy. Thus, we can describe the types of turbulence model in terms of
the number of additional transport equations as:
36
Literature Review
1.
Zero-equation models such as The Prandtl mixing-length model. Although, zero
equation models are easy to implement and cheap in terms of computing
resources, they are completely incapable of describing flows with separation and
recirculation. They are rarely used by present researchers.
2.
One-equation models such as Prandtl energy with prescribed length scale. Like
zero equation models in cheap in terms of computing resources but can not
describe complicated flow.
3.
Two-equation models include the standard k-ε model, the Chen-Kim modified k-
ε model, Lam-Bremhorst k-ε model, RNG-derived k-ε model.
Two-equation models are widely used in industrial flow simulations.
4.
Second-order closure models such as the Reynolds-stress turbulence model. This
type of model is the most general of all classical turbulence models with very
accurate calculation of mean flow properties but consume very large amounts of
computing resources and are not as widely validated as the two-equation models.
5.
Advanced CFD models such as large eddy simulation (LES) and Direct
Numerical Simulation (DNS). LES and DNS are at present at the research stage
and they require large computing resources and are not yet suitable as general
purpose models for industrial flow simulations.
A comparison of different turbulence models is shown in Table 2.2 below:
2.3.2
Turbulence Models Used in Previous Plume Numerical Studies
The first turbulence modelling for a plume was devised by Schmidt (1941) when he
studied the turbulent thermal plume from a small source. He used a mixing-length-type
hypothesis to obtain expressions for the mean velocity and temperature profiles for both
plane and round geometries and compared the results with measurement.
Trent and Welty (1973) studied momentum jets and forced plumes using a finite
difference numerical model with the Prandtl eddy diffusivity model for turbulence, the
37
Literature Review
Boussinesq approximation and assuming a fully turbulent flow. In the case of a forced
plume discharged from a nozzle, they used the Froude number at source, Fr0, to define
the source conditions. The case studies of their work considered Froude numbers of 46,
52 and 1000.
Their results of centreline temperature and vertical velocity showed good agreement at
downstream locations but poor agreement closer to the discharge source. In addition, at
downstream points, in the self-similarity region, the centreline temperature closely
approached the experimentally verified the -5/3 log-log scale slope indicative of purely
buoyant flows and also centreline vertical velocity closely attained the -1/3 slope.
At the downstream locations, they proposed the following equations for centreline
vertical velocity and temperature:
13
 4F 
TC = 8 0  z −5 3
 3 
(2.85)
13
92  3  −1 3

 z
wC =
2  4 F0 
(2.86)
Trent and Welty (1973) also concluded that although a constant value of momentum
and thermal diffusivity would permit accurate computation in the case of a momentum
jet, a variable coefficient was needed for buoyant plumes or large errors would result.
Madni and Pletcher (1977) used a finite-difference calculation with a mixing-length
model to solve the equations of conservation of mass, momentum and energy for a
buoyant turbulent forced plume. In the case of a uniform environment, the results
showed that the centreline temperature value of their model was in good agreement with
experimental results for z/D > 5.
38
Literature Review
Table 2.2
Advantages and disadvantages of different turbulence models (Versteeg
and Malasekera, 1999, CHAM, 2003).
Please see print copy for Table 2.2
39
Literature Review
The mixing-length model was modified continuously until the two-equation model was
proposed. It was proved by many researches that although the mixing-length model of
turbulence was easy to implement and needed shorter computation times but it always
gave unacceptable accuracy when it was used for simulation of flows with separation
and recirculation (CHAM, 2003, Versteeg and Malakasekara, 1999). Therefore, the
two-equation model was developed in order to be more general-purpose turbulence
models. The most popular two-equation model was the k − ε turbulence model. For
turbulent buoyant plume study, use of the standard k − ε model showed 40%
overprediction in the growth rate in the case of the axisymmetric jet (Dewan, Arakeri
and Srininasan, 1997). In addition, the research work of Nam and Bill (1993) showed
that the standard k − ε model overpredicted the centreline velocity and temperature and
consequently underpredicted the plume width when this model was used for a buoyant
plume. Therefore, many attempts were made improve the accuracy of k − ε model as
shown below. Launder and Spalding (1974) modified the standard k − ε model of
turbulence by adding the buoyancy effect term in k and ε equations. The results of this
adding, it had the extra constant value, Cε 3 , in the standard k − ε model (see chapter 4
for more details). They suggested that the Cε 3 had big improvement on numerical
results but needed to adjustment appropriately. CHAM (2003) suggested that the value
of Cε 3 should set depending on flow situation for example, It should be close to zero for
stably-stratified flow, and to 1.0 for unstably-stratified flows.
Chen and Nikitopoulos (1979) used the k − ε − T ′ 2 model ( T ′ 2 is the mean square
temperature fluctuation) of Chen and Rodi (1975) to predict the near field
characteristics of buoyant jets discharging into a stagnant uniform environment. They
divided the flow discharge from the nozzle into two zones, the zone of flow
establishment (ZFE) and the zone of established flow (ZEF). They stated that the zone
of flow establishment extends to about 10 diameters from the source. In their results,
they showed that the location of the virtual origin depends on the source mean profiles
(flat or triangle), the turbulence level and the buoyant force but they did not propose a
quantitative/mathematical the relationship between them. It was shown, in the case of a
40
Literature Review
buoyant jet, the normalised entrainment velocity starts increasing at a constant rate in
the zone of established flow.
Chen and Chen (1979) modified the k − ε − T ′ 2 turbulence model for predicting the
decay of vertical buoyant jets. They divided the flow discharged from the nozzle into
two zones: 1) the zone of flow establishment and 2) the zone of established flow.
They mentioned that the zone of flow establishment is the zone where the flow
undergoes a change from the flat source profile to a self preserving profile and is
approximated well by the Gaussian distribution function. They also divided the zone of
established flow into three regions: 1) non-buoyant region, 2) intermediate region and 3)
plume region. For round buoyant jets, they suggested that the centreline vertical
velocity is a function of x′ , where
x ′ = Fr 1 2 (ρ ρ ∞ )
−1 4
(z D )
(2.87)
where Fr is source Froude number = ρ w02 gD(ρ ∞ − ρ )
In the non-buoyant region, x′ ≤ 0.5 ;
(wC
w0 )Fr 1 2 (ρ ρ ∞ )
−1 4
= 7 x ′ −1
(2.88)
where wC is mean vertical velocity.
In the intermediate region, 0.5 ≤ x′ ≤ 5 ;
(wC
w0 )Fr 1 2 (ρ ρ ∞ )
−1 4
= 0 .2 x ′ − 4 5
(2.89)
(wC
w0 )Fr 1 2 (ρ ρ ∞ )
−1 4
= 4.4 x ′ −1 3
(2.90)
In the plume region, 5 < x′ ;
In 1992, Cho and Chung showed that the k − ε turbulence model cannot be used for
general-purpose simulations because accurate predictions of the k − ε turbulence model
required adjustment of the model constants according to the flow geometry as well as
the flow type. Therefore, they modified the k − ε turbulence model by adding the
41
Literature Review
intermittency factor, γ. They also called their model the k − ε − γ turbulence model.
They compared the computational results of different turbulence models. In the case of
a round jet emerging in stagnant surroundings, the computational results given by the
k − ε − γ turbulence model, the
k − ε turbulence model of Hanjalic and Launder
(1980), the conditional Reynolds stress model of Byggstoyl and Kollmann (1986), the
k − ε turbulence model of Pope (1978) were compared with experimental data. The
results of the mean velocity, the intermittency, the Reynolds shear stress profile and the
turbulent kinetic energy profiles given by the k − ε − γ turbulence model gave better
agreement with experimental results than others.
The k − ε − T ′ 2
model of Chen, et al. (1979) showed good agreement with
measurement but the velocity and thermal growth rate results for round jets and plumes
were still overpredicted, as shown by Dewan, et al. (1997), Kalita, et al. (2000).
Therefore, Dewan, et al. and Kalita, et al. proposed the k − ε − T ′ 2 − γ turbulence
model for plane and axisymmetric plumes. They combined the k − ε − γ ( γ is the
‘intermittency’) of Cho and Chung (1992) and the k − ε − T ′ 2 model of Chen, et al.
(1979). They stated that the standard k − ε turbulence model and Reynolds stress
model seriously overpredict the velocity and thermal growth rates of round jets and
plumes unless empirical corrections are used to match the predictions with
measurements. Without changing the model constants or employing empirical
corrections, they presented the k − ε − T ′ 2 − γ turbulence model for buoyant flows,
especially buoyant jets and plumes. They compared the numerical results of the
k −ε −γ
model,
the
k − ε − T ′2 − γ
model,
the
k − ε − T ′2
model,
the
k − ω − T ′ 2 model (ω is the mean square vorticity fluctuation), and the Reynolds stress
model in the case of the self-similar axisymmetric plume with experimental results of
Shabbir and George (1994). Their results showed that the prediction of all mean and
turbulence quantities for both the axisymmetric plume and plane plume using the
k − ε − T ′ 2 − γ model are better than the k − ε − γ model, the k − ε − T ′ 2 model, the
k − ω − T ′ 2 model, and the Reynolds stress model.
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Literature Review
The modified k − ε models use an additional term in the main model but as mentioned
above, a way of improving the numerical results of the standard k − ε model is to
change model constants appropriately. However, identifying suitable values of constants
of the standard k − ε model for studying plume flow presents a problem.
In the case of a buoyant thermal plume, Nam and Bill (1993) showed improvement in
computational results by changing the k − ε model constants. They modified the
standard k − ε model using different values of two constants: C D (the constant in the
effective turbulent viscosity equation) and σ h (the effective Prandtl number for the
diffusion of heat). They changed these values from 0.09 and 1.0 to 0.18 and 0.85,
respectively. Their comparisons of the results between the modified and the standard
k − ε model showed significant improvement when the modified k − ε model was
employed for thermal plume simulation.
Another two-equation model is the k-ω turbulence model proposed by Spalding in 1972.
This model is not as popular as the k-ε model because all secondary variables other than
ε required a near wall correction term (Malin and Spalding, 1984). Therefore, Malin and
Spalding (1984) tried to eliminate the need for the near-wall correction term in the k- ω
model. They suggested the k − ω − T ′ 2 model for prediction of turbulent jets and
plumes where ω is the mean square vorticity fluctuation. They used this model in four
case studies: 1) turbulent jets in a uniform ambient, 2) turbulent plumes in a uniform
ambient, 3) turbulent forced plumes in a uniform ambient and 4) turbulent forced plume
in a stably stratified ambient. In the case of axisymmetric turbulent plumes in a uniform
ambient, they showed that the spreading rates of velocity and temperature profile agree
fairly well with experimental data and showed more significant improvement in the
prediction of spread rate than the standard k- ω model. The computed entrainment
coefficient from this model was in good agreement with experimental results in the case
of a plane plume but the entrainment coefficient was overperdicted by about 8% in the
case of an axisymmetric plume.
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Literature Review
The other turbulence models that are completely different from the zero-equation model
and the two-equation model are Direct Numerical Simulation (DNS) and Large-Eddy
Simulation (LES), which have also been used for studying the plume.
Bastiaans, Rindt, Nieuwstadt, and Steenhoven (2000) used Direct Numerical Simulation
(DNS) and Large-Eddy Simulation (LES) to simulate the free convection flow induced
by a line heat source in a confined geometry. Both 2D and 3D flows were simulated.
The results of LES were compared with DNS results but not with experimental results.
It was found that the most of their LES simulation results were similar to the DNS
simulations.
Zhou, Luo, and Williams (2001) used LES to simulate a spatially developing round
turbulent buoyant forced plume. Comparisons were made between LES results,
experimental measurements and plume theory for a forced plume at moderate Reynolds
number. In their work, the Reynolds number of the plume is set at 1273, based on the
inflow mean velocity, viscosity, and source diameter. The injection fluid in this work
was heated air. They applied an azimuthal disturbance combined with a high level of
forcing in their model in order to get results consistent to the experimental results of
Shabbir and George (1994). They found that LES with an azimuthal forcing showed
much better comparison results with experiment than LES with axisymmetric forcing.
The result for the half-width of the plume showed that the half-width of velocity is
wider than the half-width for temperature in the far-field. This means that the velocity
spread is faster than the temperature spread. They found that the entrainment coefficient
increases rapidly to reach a peak value of 0.12 at about four source diameters and
decreases to a steady value of 0.09.
Worthy (2003) studied a buoyant plume using LES. The objective of his study was
investigation of the behaviour and characteristics of the different LES turbulence
models for a buoyant plume. Both static and dynamic LES models were tested. He
found that the suitable LES turbulence model for buoyant plume is the dynamic mixed
Smagorinsky/Bardina stress model with the full implementation.
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Literature Review
2.4
Experimental Techniques
In this section, the experimental techniques used for plume study are reviewed. The
main variables of interest in this study are velocity, density or salt concentration, and
plume geometry for saline plumes. In case of thermal plumes, velocity and temperature
are measured.
2.4.1
Temperature Measurement
In this sub-section, temperature measurement devices are explained. All of these
devices use the changes in properties influenced by temperature itself: 1) the physical
state, 2) dimensions, 3) electrical properties and 4) radiation properties. Accordingly
temperature measurement devices can be classified as follows:
1.
Expansion Thermometers
This kind of thermometer uses the changes in physical dimension when the temperature
changes. A well-known expansion thermometer is the bimetallic thermometer. It
measures temperature by means of the differential thermal expansion of two metals. The
materials used in a bimetallic thermometer are a copper alloy, a high thermal expansion
coefficient material, and nickel steel, a low thermal expansion coefficient material. The
accuracy of bimetallic thermometers is about ±2oC to ±5oC and the upper temperature
limit is about 300oC (Tse and Morse, (1989)).
2.
Filled Thermometers
Filled thermometers work on the expansion principle if completely filled with fluid. The
mercury-in-glass thermometer is the one of the well-known filled thermometers. Due to
the greatly different coefficient of expansion of mercury (about eight times that of glass)
the mercury rise up the capillary in the glass stem indicates temperature to acceptable
accuracy.
Certified mercury-in-glass thermometers are widely used as standard thermometers for
calibration systems and one was also in calibration work for thermocouples in this
study.
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Literature Review
Normally, the maximum accuracy of this type of thermometer is about ±0.01oC for the
range 0 to 150oC and ±1oC for the range 300 to 500oC.
3.
Thermocouples
The thermocouple is the most popular device used for temperature measurement. A
thermocouple consists of two metallic wires A and B depending on the types of
thermocouple:
Type E:
Chromel vs. Constantan (Copper-Nickel alloy)
Type J:
Iron vs. Constantan
Type K:
Chromel vs. Alumel (Nickel-Aluminum alloy)
Type T:
Copper vs. Constantan
Type S:
Platinum/10% Rhodium vs. Platinum
Type R:
Platinum/13% Rhodium vs. Platinum
The calculation of temperature from value of thermocouple voltage of difference type is
defined by a ninth-order polynomial:
T = a0 + a1 x + a2 x 2 + ... + a9 x 9
(2.91)
where T is temperature (oC) x is the thermocouple voltage (volts) with reference
junction at 0oC and an are polynomial coefficients.
The temperature for different range thermocouples and the corresponding polynomial
coefficients are shown in Table 2.3 (Holman, 1994).
In this study, K-type thermocouples with LabTech software were used to measure the
temperature of thermal plume and also temperature of the hot area source and the
ambient.
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Literature Review
Table 2.3
Range and polynomial coefficients for difference thermocouples
(Holman, 1994).
Please see print copy for Table 2.3
4.
Resistance Temperature Detectors (RTDs)
This type of temperature measurement device uses the phenomenon of electrical
resistance of metal that increases with temperature. Common materials used for RTDs
are Platinum (from -260 to 1000oC), Copper (from -200 to 260oC), Nickle and Balco
(70%Ni/30%Fe) (from -100 to 230oC) and Tungsten (from -100 to 2500oC). The range
of temperature measurement and accuracy of RTDs also depend on manufacturers and
construction of the RTDs.
5.
Thermistors
A thermistor is also a resistive element. It is used for temperature sensing and also in
electronics components and control systems. A thermistor has the smallest temperature
47
Literature Review
range, about -100 to 150oC, but has very high sensitivity (about ten times that of the
RTDs).
In the research work of Kotsovinos (1985), he used 0.3 mm thermistors that were
insulated and mounted in a stainless-steel tube. The time constant for these thermistors
was determined by quickly moving the thermistor from air to water. He found that his
temperature probe required 25 ms to obtain 63% of the final reading and his probe had
an accuracy of temperature measurement of 0.01oC.
6.
Pyrometers or Radiation Thermometers
Pyrometers are non-contact temperature measurement devices. They measure the
electromagnetic radiation emitted from a body and convert it to a temperature reading.
Pyrometers are applicable for wide temperature ranges. They are used to measure
temperature under difficult conditions in the working area such as strong chemical
environments, high pressure and very high temperature. The disadvantage of this
instrument is that it can measure only the surface average temperature of a solid body.
2.4.2
Flow Measurement
In this study, plume shape, plume necking, horizontal and vertical velocity are the main
flow characteristics that are measured and recorded. In this section, two aspects of the
method and technique of flow measurement are described: velocity measurement and
flow visualisation.
2.4.2.1 Velocity Measurement
1.
Vane anemometer
A very simple flow velocity measurement device for low flow speed is the vane
anemometer, consisting of a windmill if the flow direction is known. It can be used
down to a speed limited by the fiction of the bearing. This anemometer was used by
Rouse, et al. (1952) in order to find out velocity distribution in a plume.
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Literature Review
2.
Hot-wire anemometer
A hot-wire anemometer based on the convective heat transfer from a heated sensor to its
surrounding fluid. Hot-wire anemometers are used mainly to measure rapidly
fluctuating velocities in gases and liquids. The probe of the hot-wire anemometer
consists of a fine wire sensor made of Tungsten, Platinum or Platinum alloys, the wire
support, a probe body and electrical leads. Hot-wire anemometer have been used for
transient flow with very low time constant. Two types of electrical compensation are
used: (1) a constant current and (2) a constant temperature arrangement in order to
balance voltage of bridge circuit to maintain temperature of probe.
A modified hot-wire probe used in plume study was a two-wire probe of George, et al.
(1998) who used a constant current anemometer for temperature measurement and a
constant temperature anemometer for velocity measurement.
3.
Laser Doppler Anemometer (LDA)
The LDA is a device that offers non-obtrusive flow measurement with a very high
accuracy. In this system, the flow must carry some type of small particles to scatter the
light. It is clear that the LDA measures the velocity of the scattering particles. But if
they are sufficiently small, the slip velocity between particles and fluid will be small,
and thus an adequate indication of fluid velocity can be obtained.
Many researchers have used LDA for both vertical and horizontal velocity
measurement: eg. Ramaprin and Chandradekhara (1989), Chiari and Guglielmini
(1998), because it is a velocity measurement device that does not disturb the flow of
interest.
4.
Particle Image Velocimetry (PIV)
The Partical Image Velocimetry (PIV) is the one of powerful velocity measurement
techniques. The whole flow field can be analysed in order to determine the velocity
vectors. In this method, two sheets of laser light shine though the flow area in quick
succession. The images of particles in the illuminated sheets are recorded. The velocity
vector of fluid flow is deduced from the two images.
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Literature Review
5.
Particle Tracking Velocimetry (PTV)
This method was used for the present study, where the motion of neutrally buoyant
particles carried by the flow are recorded and analysed in order to find out particle
velocity. Because the particles are sufficiently small and also have the same density of
experimental fluid the velocity and direction of particles can closely approximate to that
of the flow of fluid in the field of interest. More details of this technique are described
in Chapter 3. Colomer, et al. (1999) used this technique to find out the vertical velocity
of the saline plume from an area source.
2.4.2.2 Flow Visualisation
1.
Shadowgraph Technique
The Shadowgraph technique is a method for direct viewing of flow phenomena. When
the parallel light rays enter the flow field, in the regions where there is no density
gradient, the light rays pass straight with no deflection, but in the regions with density
gradients, and hence gradients in refractive index, the light rays are deflected. The net
effect of those deflected and un-deflected rays will form bright and dark areas on the
screen. This technique is very simple, requiring only the naked eye, local lighting and
some image recorder. This method was used in many research works such as those by
Linden et al. (1990). This technique was used for flow visualisation of plume including
plume shape and plume necking in the present study.
2.
Schlieren Method
The Schlieren method involves an optical arrangement which indicates density
gradients. A schematic of the Schlieren flow visualisation technique is shown in figure
2.14.
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Literature Review
Please see print copy for Figure 2.14
Figure 2.14
Schematic of the Schlieren flow visualisation (Holman, 1994).
a) Optical Arrangement b) Detail of knife edge.
The Schlieren method has been used extensively for location of shock waves and
complicated boundary-layer phenomena in supersonic flow systems. Noto et al. (1999)
used the Schlieren technique for plume visualisation.
3.
The Mach-Zehnder Interferometer
The Mach-Zehnder interferometer is the most precise instrument for density field
visualisation. A schematic of it is shown in Figure 2.15.
Please see print copy for Figure 2.15
Figure 2.15
Schematic of the Mach-Zehnder interferometer (Holman, 1994).
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Literature Review
The Mach-Zehnder interferometer gives a direct quantitative indication of density
changes. It is used in a wide range of flow conditions ranging from low-speed (free
convection) to supersonic flow.
Yousef, et al. (1982) used the Mach-Zehnder interferometer to obtain the temperature
field in a plume above a heated surface.
2.4.3
Direct and Indirect Density Measurement
1.
Conductivity Probe (for saline plume)
A conductivity probe may be used to measure the electrical conductivity of salt
solution. The value of electrical conductivity of salt solution can be converted to the
value of density of salt solution. This kind of conductivity probe was developed at the
University of Cambridge, UK (Holford, 1997). A modified version of this conductivity
probe was used for the saline plume study in this project. The details of the present
conductivity probe are explained in Chapter 3.
Wong and et al. (2001) measured density profiles using a commericial conductivity
probe (MSCI 5201) with temperature compensation, using simultaneous thermistor
probe (GB38P12) measuring for density , in order to obtain the density distribution
within the plume.
2.
Light-Induced Fluorescence and Laser-Induced Iodine Fluorescence (LIF)
The light-induced fluorescence technique uses the light shining through a thin slit and
through the flow (such as plume) laced with a fluorescent dye such as sodium
fluorescein in order to observe the flow within the light sheet. This technique enables
estimation of the mean concentration of the dye. This technique was used in the work of
Kaye and Linden (2004), for example.
In the case of laser-induced iodine fluorescence technique, the light is replaced by a
laser source. The LIF signal is produced by a laser beam such as from an argon-ion
laser. The LIF has been used to determine mixture fraction (or density distribution) of
the plume (Dai et al., 1994) who studied the structure of round buoyant turbulent
plumes using laser-induced fluorescence (LIF). In their system, Mixture fractions were
measured with 2% maximum error for a carbon dioxide plume and 0.2% for a sulphur
52
Literature Review
hexafluoride plume. In self-similarly region, they suggested the relationship between
mean mixture fraction and density as:
f = (π Fr0 4 )
23
(ρ 0
ρ ∞ )(( z − z v ) d )−5 3 F (r (z − z v ))
(2.92)
where F (r ( z − z v )) is the appropriate scaled radius profile function of mean mixture
fraction, which becomes a universal function in the self-similar region.
3.
A Dye Attenuation Technique
This technique was used by Kaye and Linden (2004) in order to find the effect of the
coalescence of two co-flowing axisymmetric turbulent plumes. This technique uses the
strength of light passing through a dyed region as a function of the initial light intensity,
local dye concentration and the region depth. For more detail of this technique see Kaye
and Linden (2004).
2.5
Summary
In this chapter, previously reported research work on the plume was summarised,
including theoretical, experimental and numerical techniques. Many turbulence models
have been used in past numerical simulations of plume phenomena. The objective of a
present study was not to develop a new turbulence model for plumes from an area
source but rather to review these models and apply the most suitable turbulence model
for validation against the present experimental work and for calculation of industrial
plume characteristics.
Therefore, from details of all the turbulence models that are available in PHOENICS
and also from recommendations (Nam and Bill, 1993), it was found that the most
suitable, general and easy to apply turbulence model to study the characteristics of
plume from an area source is the standard k-ε model with adjustment of model
constants. The values of turbulent Prandtl no. in the standard k-ε model were to be
determined using experimental data in order to obtain a good comparison. Furthermore,
Launder and Spalding (1974), CHAM, 2003, suggests that Cε 3 in the equation of the
rate of dissipation of turbulent kinetic energy has a large impact in numerical results of
53
Literature Review
plumes that are affected by buoyancy force. Therefore changing of Cε 3 is done to obtain
a good comparison with experimental data.
54
Chapter 3
Experimental Facilities, Techniques
and Procedures
Experimental Facilities, Techniques and Procedures
Chapter 3
Experimental Facilities, Techniques and Procedures
In order to study the development of a plume from an area source, two types of
experiment were designed and carried out:
1
Descending saline plume in water;
2
Ascending thermal plume in air.
Experimental Study
of Turbulent Plumes
Descending Saline
Plume in Water
Ascending Thermal
Plume in Air
Velocity Distribution
Experiments
(Laser Doppler)
Plume Shape
Experiments
(Shadowgraph)
Figure 3.1
Density Distribution
Experiments
(Conductivity
Measurement)
Temperature Distribution
Experiments
(Thermocouple)
Velocity Distribution
Experiments
(Particle Tracking)
Schematic summarising the present experimental study of turbulent
plumes from an area source.
56
Experimental Facilities, Techniques and Procedures
3.1
Saline Plume Experimental Facilities, Techniques and Procedure
In the first part of the experimental study, the saline plume from an area source was
studied. In this section, the saline plume experimental facilities are described. A
visualisation technique the Shadowgraph technique and the velocity measurement
technique, Particle Tracking Velocimetry (PTV), are explained and then the data
collection processes for plume shape experiments, density and velocity distribution
experiments are described.
3.1.1
Saline Plume Experimental Facilities
Firstly, experiments designed to generate a descending salt solution plume in water
were conducted to study ‘lazy’ plumes in uniform, unstratified environments. These
experiments were separated into three parts to determine: 1) plume shape; 2) density
distribution; and 3) velocity distribution.
Header tank
Traversing
mechanism
Plume
source
Rotameter
Needle
valve
Filter
Pump
Figure 3.2
Tank
Storage
tank
Schematic of the apparatus for saline plume experiment
(not to scale).
Here, the evolution of the plume was recorded using video footage which was analysed
using the DigImage software package (Dalziel, 1993). The main aim of these
57
Experimental Facilities, Techniques and Procedures
experiments was to clearly visualize the plume shape and plume necking for different
source strengths.
A conductivity probe was simultaneously used to measure density (or salt
concentration) across the plume width at different levels and for different source
strengths.
A Particle Tracking Velocimetry (PTV) method was used to determine the velocity field
in and around the plumes. In this experiment, the working fluid was ‘seeded’ with
neutrally buoyant particles that were convected passively by the plume flow. The
motion of these particles was tracked using a video recorder. The video footage was
analysed using the Particle Tracking Velocimetry (PTV) option in the DigImage
software in order to determine the velocity profiles across the plume at different
distances below the source.
3.1.1.1 Experimental Tanks
In order to study the saline plumes from an area source, provision of a suitable large
water tank was necessary. Ideally, the tank should be large enough to minimise the
effect of the confining walls on the shape, density and velocity distribution in the
plume.
At the start of this project, there was no large environmental fluid dynamics tank
available at the University of Wollongong. Thus, as part of this thesis, a large glass tank
had to be designed and constructed first.
The required tank size was estimated using the point-source plume theory from ACGIH,
2001, as shown in Figure 3.3.
58
Experimental Facilities, Techniques and Procedures
Ds = 0.3 m.
Dc
Assume Y = 1.4 m.
From Z = (2Ds)1.138 then Z = 0.56m
Y
Then Xc = 1.96m therefore Dc = 0.9m
Xc
Ds
Z
Figure 3.3
Estimation of water tank size required for saline plume experiment.
From the calculation above, it was decided that a tank width of 1.4m was sufficient to
carry out plume shape experiments with the already available 300mm diameter source.
A cubic tank (1.4m x 1.4m x 1.4m) was designed and fabricated with a steel frame and
toughened glass walls. The tank had a large water storage capacity (max ≈ 2,744 kg),
and the possibility of large hydrostatic pressures on the floor and walls (max ≈ 13,734
Pa g on the floor, max average 6,687 Pa g on the vertical walls, assuming a full tank).
The strength of the tank frame and the thickness of the glass walls were calculated using
recommendations from Australian standards for design of glass tanks (AS 1288, 1994)
and the Glass Engineering Handbook (McLellan and Shand, 1984). The strength of the
tank frame and glass also were computationally checked using the Finite Element
Method (FEM) analysis program MSC/NASTRAN by a member of the academic staff
at the University of Wollongong (Remennikov, 2001). For more details, see Appendix I.
In the preliminary design, the tank was to have three sides of 15 mm thick glass and one
side of 3 mm steel plate. The calculation results showed that the 15 mm glass was
strong enough with a safety factor of about 2.2 but that the steel plate side showed an
unacceptably high deformation. In the actual construction, it would have been difficult
to make firm joints between the glass pane and the steel plate.
59
Experimental Facilities, Techniques and Procedures
Please see print copy for Figure 3.4
Figure 3.4
Result of numerical stress calculation for water tank (Remennikov,
2001).
An additional problem would have been corrosion of steel in salt water. Therefore, the
steel side was replaced with 15mm thick glass. The frame of the tank was made from
50mm square-section hollow steel tube coated with anti-corrosion paint. On the tank
floor, two holes, 50 mm in diameter, were drilled for the drainage system.
Some design drawings of the tank and a picture of the real tank are shown in Figures 3.5
and 3.6, respectively.
The saline plume source and the conductivity probe traversing mechanism were
mounted on a frame at the top of the tank. The level of the area source was adjusted up
and down using threaded rod.
In the study of velocity distribution of the saline plume, the seed particles had to be
large enough in number to capture the details of the flow. Therefore, the larger tank
(1.4m x 1.4m x 1.4m) was not suitable because it would have involved using an
60
Experimental Facilities, Techniques and Procedures
excessively large number of particles, which are difficult to recover after the
experiment.
Plan view
Front view
Figure 3.5
Side view
Design drawings of environmental fluid dynamics tank.
Therefore, a second smaller tank was used for determining the velocity field. This new
tank (0.46m x 0.46m x 0.46m), was designed to be geometrically similar to the larger
tank. Compatible with these smaller dimensions, a 105mm diameter area source was
made from an existing area source (Overton, 1993).
61
Experimental Facilities, Techniques and Procedures
Traversing
mechanism
Plume
source
Conductivity
probe
Glass
Figure 3.6
Photograph of tank.
3.1.1.2 Area Sources for Saline Plume
The saline plume area sources were an important part of the saline plume experimental
program. The prerequisite was a constant flow rate per unit area. A drawing of the area
source of saline plume is shown in figure 3.7.
Flow distribution
baffle
Salt solution
inlet
O-ring
Support rod
300 mm
Porous sheet
Figure 3.7
Cross-section drawing of the area source.
62
Experimental Facilities, Techniques and Procedures
Salt solution inlet
Flow distribution baffle
O-ring
Figure 3.8
Photograph of the saline plume area source.
The two area sources were made from plastic (side wall) and Perspex sheets (top). The
source areas (bottom) were made from 6 mm thick fine grade sintered PTFE porous
sheet, in order to allow uniform flow rate of the salt solution. The flow rate from the
area source was visually checked for uniformity before starting each experimental run
by feeding fresh water with dye through the porous sheet. After every experimental run,
the porous sheet was thoroughly washed with fresh water.
The diameters of the area sources were 300mm and 105mm. The 300mm diameter area
source was used to study the density distribution in the plume in the large tank. The
smaller area source in the smaller tank was used to study the plume shape and velocity
distribution in the plume.
3.1.1.3 Traversing Mechanism
The conductivity probe was required to be moved to the desired positions in order to
measure density distribution in the saline plume. Therefore a traversing mechanism was
required. The author designed and commissioned a traversing mechanism based on two
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Experimental Facilities, Techniques and Procedures
principles: 1) easy to make and operate and 2) computer controlled. The main
component in this traversing mechanism was the stepper motor. The body of the
traversing mechanism was made from aluminium, non-corrosive in salt solution. The
movement of the conductivity probe was controlled by rotating a stainless steel threaded
rod. At the end of this rod, the stepper motor was attached in order to control the
number of rotations. Two stepper motors were used to control the movement of the
conductivity probe in two directions, the vertical and horizontal, independently. The
YA-CNC software with position control box was used to control the two stepper
motors. A drawing of the traversing mechanism system is shown in Figure 3.9. More
details of the traversing mechanism are provided in Appendix II.
Stepper
motors
Aluminium
rod
Position
control box
Pentium I computer
with YA-CNC
software
Figure 3.9
Conductivity
probe support
Stainless
steel thread
Schematic of traversing mechanism and controller (not to scale).
3.1.1.4 Constant-Head Salt Solution Supply
The salt solution through the area source was supplied continuously from a constant
head tank situated approximately 3m above the experimental tank to facilitate a constant
flow during the experiments. The salt solution was continuously pumped up to the
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Experimental Facilities, Techniques and Procedures
header tank from a separate reservoir (see Figure 3.2). A return drain to the reservoir
maintained the salt solution level in the header tank.
3.1.1.5 Conductivity Probe
A special conductivity probe (Figure 3.10) was designed and built for density
measurements. The design of this probe was based on that of the probes used in the
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
(Holford, 1997, see Appendix III).
Conductivity
probe
Figure 3.10
Photograph of conductivity probe in the large tank.
The conductivity probe works on the principle that salt solution is an electrolyte whose
conductivity increases monotonically with salt concentration. The probe was
constructed from two concentric stainless steel tubes of 3 mm and 8 mm outside
diameter, separated by an air gap. Saline solution was sucked up the inner tube through
a narrow orifice 0.5 mm in diameter and 7.0 mm long in the conical Nylon probe tip.
65
Experimental Facilities, Techniques and Procedures
Plastic
tube
“Silastic”
Air gap
Nylon tip
500
20
7
10
Silastic
Electrical
connection
Figure 3.11 Cross-section of the present conductivity probe.
When a voltage difference was applied between the inner and outer tubes, electric
current would flow between the two tubes through the salt solution between them. The
electrical resistance of the probe was almost entirely that due to the small volume of salt
solution within the probe tip orifice.
The conductivity probe was connected to a commercial conductivity meter, AIC PM2
series. Figure 3.12 shows a schematic of the conductivity probe experimental
arrangement.
The measurement range used with this AIC PM2 series conductivity meter was 00.0099.99 µs/cm with 0.01 µs/cm resolution. The sampling rate of this meter was 1
sample/sec. The calibration procedure for the conductivity probe is explained next.
Conductivity Probe Calibration
The conductivity probe measured electrical conductivity. Because the electrical
conductivity of salt solution varies with density, the output of the conductivity probe
could be converted into a density value. It was therefore necessary to calibrate the
device.
66
Experimental Facilities, Techniques and Procedures
Traversing
Mechanism
Inlet pipe
Area
Source
Conductivity meter
AIC PM2 series
Conductivity
probe
Pentium 2 computer
with DAS 16 card
EXP 32 card
Figure 3.12
Schematic of the conductivity probe and conductivity meter arrangement
(not to scale).
The conductivity probe was calibrated using a density meter (Anton Paar model DMA
35N). The electrical conductivity and density of samples of salt solution varying from
0% to 20% by weight were measured by the conductivity probe and the density meter,
respectively, and the results correlated. Figure 3.13 shows the calibration curve of the
conductivity probe.
Each point on this graph resulted from averaging 20 measured data points that were
taken for each value of salt solution. The conversion equation for conductivity is:
Density (kg/m3) = 8.69110×10-3 X3 - 7.987143×10-2 X2 + 5.75797 X + 9.97505×102
Where X = conductivity value measured from conductivity probe (µs/cm).
The conductivity probe calibration process was repeated twice after using for a period
of time in order to check the repeatability of the conductivity probe measurements. The
change in the measured value of conductivity was ±2% (see data of calibration data2 in
Figure 3.13).
67
Experimental Facilities, Techniques and Procedures
1160
y = 8.69110E-03x3 - 7.97143E-02x2 + 5.757597E+00x + 9.97505E+02
R2 = 9.98503E-01
1140
Density (kg/m 3)
1120
1100
Calibration Data1
1080
Calibration Data2
Poly. (Calibration Data1)
1060
1040
1020
1000
980
0
2
4
6
8
10
12
14
16
18
20
Conductivity (µs/cm )
Figure 3.13
Typical calibration graph for the conductivity probe.
3.1.1.6 Refractometer
A digital refractometer (ATAGO Paletle, PR-32) which uses the variation of refractive
index of the salt solution with concentration, and hence density, was used as an
additional device for density measurement of the saline water samples. The
refractometer measured density to within ±0.001 % Brix or ±0.5 % by weight of salt.
3.1.1.7 Densitometer
An Anton-Paar (Model DMA 35N) density meter was used for direct measurement of
the density of the salt solution and also the density of water in the tank before starting
the experiment. This was also used for calibration of the conductivity probe. The
measured density values are accurate to within ± 0.001 g/cm3 as claimed by the
manufacturer (Appendix VI).
3.1.1.8 Velocity Field Determination for Saline Plume
The velocity field in the evolving plume was determined using a particle tracking
technique. For this, the working fluid was ‘seeded’ with neutrally buoyant particles
68
Experimental Facilities, Techniques and Procedures
whose motion was recorded on video tape. The seed particles had to have special
properties. They must be: 1) insoluble in water; 2) small enough to have a minimal
effect on the flow; 3) be of nearly the same density as water in order to remain
suspended. Particles of Pliolite, a white plastic used in reflective paints, were found to
be suitable. However, before they could be used in the experiment, they had to be sized
and their density determined.
First, a large enough number of particles of average size ~300 µm (suggested by
Colomer, et al, 1999) was separated from the wide range of sizes available, using a
sieve. Next, the selected particles were washed with clean water and then rinsed with
detergent solution, before introducing them into the water. The particles that either
floated up to the surface and or sank to the floor were removed. Only those particles that
remained suspended in the water were collected and used in the experimental water
tank.
Video footage of the evolving plume was analysed frame-by-frame using the DigImage
software to determine the plume shape and velocity field.
3.1.1.9 DigImage System
The DigImage system was developed at the Department of Mathematics and Theoretical
Physics, University of Cambridge, specifically for analysis of geophysical fluid
dynamics experiments (Dalziel, 1992, 1993). DigImage includes standard imageprocessing modules such as filtering, contouring, image enhancement, particle tracking,
PIV, optical thickness, refractive index analysis, LIF enhancement, time series
generation, etc. These capabilities enable greatly enhanced visualisations and allow
previously qualitative experiments to yield accurate quantitative data.
The DigImage system consisted of the following main hardware components:
1)
Pentium II Computer with DigImage software and the frame grabber board, DT2862;
2)
High quality Colour video monitor, SONY PVM-1450QM;
3)
SVHS Video Recorder, PANASONIC AG-7350;
69
Experimental Facilities, Techniques and Procedures
and for image recording
4)
CCD camera, SONY XC-77RR;
5)
Camera adaptor, SONY DC-77RR.
The two main DigImage features used were Particle-tracking velocimetry, the
command for deducing the velocity distribution in the plume, and time-averaging, the
command for averaging a series of plume images to determine plume shape.
Two dimensional particle tracking velocimetry in DigImage
There were three processes used for tracking the movement of particles:
1)
Image capture. DigImage uses Super VHS video tape to record a single view of
an illuminated region of an experiment. During the tracking phase, the video tape is
replayed and the images are captured by digitising the video signal.
2)
Particle location. DigImage employs two methods of locating particles, one
which uses all the information, and the other which summarises some of the
information. The location of a particle is determined using both these methods, the
results being compared to ensure consistency. A ‘particle’ in DigImage is defined as an
area of the enhanced image satisfying a number of criteria, based on the intensity, size
and shape. The most basic criterion is the intensity which is used to identify potential
particles or blobs within an image.
3)
Particle matching. Once all the particles in an image have been found, they must
be related back to the previous image to determine which particle image corresponds to
which physical particle.
Time-averaging command in DigImage
In this command, the plume image can be averaged over the duration of a particular
experiment. This command is specifically designed to efficiently compute image time
average over medium to long times. A temporal arithmetic average of a large number of
70
Experimental Facilities, Techniques and Procedures
frames is produced by utilising as many buffers as required to store the accumulated
total.
For further details about these commands and others in DigImage see Dalziel (1992,
1993).
3.1.2
Saline Plume Data Collection Techniques
In the saline plume study, the above techniques were used determine the main
characteristics of saline plume such as velocity distribution, density or temperature
distribution, plume shape etc. The shadowgraph technique was used to determine plume
shape and plume necking position and the Particle Tracking Velocimetry (PTV)
technique was used to determine the vertical and horizontal velocity distribution. A
special conductivity probe was used to determine density distribution.
An objective of this research was to study plumes in a quiescent environment. Some
time was needed for the water in the experimental tank to become still after introducing
the area source into the tank. When the area source was put into the experimental tank,
some diffusion between salt solution and fresh water occurred at the porous membrane,
changing the salt solution density inside the area source.
Many attempts were made to reduce the diffusion between the source salt solution and
fresh water before an experiment was started such as using a gate to block the surface of
the porous membrane or slowly filling water in the tank in order to raise the water level
up to the area source. But these two attempts still allowed significant water movement
inside the experimental tank.
The best method to reduce the initial diffusion effect and the water movement inside the
experimental tank before starting the experiment was gently putting the area source into
the experimental tank with as little water disturbance as possible and then waiting for
movement in the water inside the experimental tank to die out (after about 5 minutes).
The experimental procedures for saline plume study will be explained in section 3.1.3.
71
Experimental Facilities, Techniques and Procedures
3.1.2.1 Shadowgraph Technique
This technique was used to visualize the plume shape, plume ‘necking’ and also for
locating the virtual origin point of the plume. The shadowgraph technique consists of
three main components: a light source, shadow screen and video recording system
(Figure 3.14).
The image of the plume was projected onto the shadow screen (translucent tracing
paper) attached to the glass wall on the opposite side of the light source. The evolution
of the plume was recorded using a SONY SC-77RR CCD camera and an SVHS video
recorder (PANASONIC AG-7350).
Individual frames of the video footage were
analysed using the DigImage software in order to determine the average plume shape
over a 30 second duration. Use of the DigImage software is explained in section 3.1.1.9.
Plume source
Translucent paper
CCD camera,
SONY SC-77RR
Light source
Camera adaptor,
SONY DC-77RR
Figure 3.14
TV and
SVHS Video recorder,
Panasonic AG-7350
The shadowgraph technique arrangement (not to scale).
The beam emitted by the light source was not a precisely parallel beam, so that the
shadow of the plume presents a slightly distorted picture of the plume position and
72
Experimental Facilities, Techniques and Procedures
outline. This distortion had to be compensated for. This was done using the image of a
‘position scale’ (2-D grid drawn on a transparent Perspex sheet) that was suspended in
the field of view before starting each experimental test (see Figure 3.15). This made it
possible to set the coordinates of the shadow image accurately using the coordinate
mapping function in DigImage, that is to correlate the exact x-y coordinates in
millimetres to the x-y pixel position of the image.
3.1.2.2 Particle-Tracking Velocimetry Technique
The Particle Tracking Velocimetry (PTV) technique was the main method used to
determine the velocity distribution in the saline plume. This method determines the
velocity of small, neutrally buoyant ‘seed’ particles that are passively carried by the
flow. Pliolite particles were used as seed particles (average size 300 µm, also used by
Colomer et al, 1999), and their motion in a thin illuminated slice of the flow field
passing through its axis of symmetry was recorded on video tape (Figure 3.16). The
video footage was analysed using the DigImage software. In this technique, a light sheet
approximately 2mm wide was generated and passed though the mid-plane of the water
tank. Because Pliolite particles are white, a black background was used in order to
generate clear images of the Pliolite particles.
73
Experimental Facilities, Techniques and Procedures
(a)
(b)
Figure 3.15
Image of grid in the experimental tank for position mapping.
(a) Image from DigImage sequence of Shadowgraph experiment.
(b) Image from DigImage sequence of PTV experiment.
74
Experimental Facilities, Techniques and Procedures
Plume source
Light source
Slit in black
plastic sheet
TV and SVHS
Video recorder
Figure 3.16
3.1.3
Light sheet
Camera adaptor,
SONY DC-77RR
CCD camera, SONY
SC-77RR
The particle tracking experimental arrangement (not to scale).
Saline Plume Data Collection Procedure
Two experimental procedures for saline plume study are described in this section. For
plume shape and density distribution experiments, the larger tank and the shadowgraph
technique were used. The small tank and the PTV technique were used for velocity
distribution experiments.
3.1.3.1 Determination of Saline Plume Shape and Density Field
In these experiments, the 1.4m cubic tank was used. The steps were as follows:
1.
Preparation of source salt solution of the desired density.
2.
Filling experimental tank with fresh water.
3.
Put the area source into the experimental tank.
4.
Use the computer-controlled traversing mechanism to set the conductivity probe at
its initial position and set the step movements of the conductivity probe.
5.
Set up the light source and video recording system for shadowgraph technique.
75
Experimental Facilities, Techniques and Procedures
6.
Suspend the reference grid in the tank and record the position scale for use in
coordinate mapping.
7.
Remove the reference grid and the area source.
8.
Set the salt solution flow valve to the desired flow rate.
9.
Wait for water in experimental tank reach a quiescent state.
10. Gently put the area source back into the experimental tank making as small a
disturbance of water inside the tank as possible and then wait for another 5 minutes.
11. Start recording using the SVHS video recorder.
12. Open the valve to release plume source flow, at the same time a rod was moved
across the field of view to mark t = 0 on the video recording.
13. After 5 minutes, the density measuring process was started. The computer program
YA-CNC controlled the position of the conductivity probe tip and the densities
across different levels below the area source were measured.
14. Analysis of the video footage using DigImage in order to determine plume shape
and plume neck position/diameter.
3.1.3.2 Determination of Velocity Field
The
velocity
field
in
a
saline
plume
in
a
quiescent
environment
was
determined/obtained using the particle tracking velocimetry capability of DigImage.
The small water tank was used for this purpose. The experimental steps were as follows:
1.
Prepare the source salt solution of the desired density.
2.
Fill the experimental tank.
3.
Introduce a selected batch of Pliolite particles into the tank and then put the area
source without salt solution into tank.
4.
Set up the light source and video recording system as required for the particle
tracking velocimetry technique.
5.
Suspend the reference grid in the tank and record the scale for later use for
coordinate mapping and then wait for about 30 minutes to let the water become
quiescent.
6.
Fill the area source reservoir with the salt solution and set up the valve to the
desired source flow rate.
7.
Gently put the area source back into the tank and wait for 5 minutes.
76
Experimental Facilities, Techniques and Procedures
8.
Starting video-recording.
9.
Open the valve to release plume flow and mark start of video.
10. Analyse the video footage using DigImage in order to determine the velocity field
in the illuminated slice.
3.2
Thermal Plume Experimental Facilities and Procedures
A second experimental apparatus for investigating thermal plumes in air was designed
and fabricated. In these experiments, a horizontal circular heated plate at uniform
temperature acted as an area source for the plume.
In case of the salt solution plume, the relative density difference between the plume
fluid and the ambient fluid was sufficiently small that the Boussinesq approximation
was valid. This was not so for the thermal plumes in air since for the test conditions
(∆ρ/ρ) > 0.1 which is considered to be the limit for validity of the Boussinesq
approximation (Malin, 2003). In these experiments two source temperatures were used,
150 ºC and 200 ºC, giving values of ∆ρ/ρ equal to 0.31 and 0.38, respectively.
In order to reproduce conditions in the industrial workplace, ideally, the thermal plume
should not be artificially confined. However, for practical reasons this was not possible
since:
•
Natural convection flows are extremely sensitive to their surrounding
environment. It was necessary to confine the flow in an enclosure to ensure that
the plume was not affected by drafts in the surroundings caused by airconditioning, movement of people, etc.
•
For these delicate natural convection velocity measurements, it was necessary to
use a non-obtrusive technique. This required ‘seeding’ of the air above the heat
source. The enclosure above the heat source served to confine the seed particles,
preventing their escape and possible deposition on the sensitive and expensive
optical instruments in the laboratory, and reduced the risk of inhalation by the
laboratory personnel.
77
Experimental Facilities, Techniques and Procedures
Thus, a confined thermal plume experiment was set up as shown in Figure 3.17.
The velocity field and the temperature field were measured to determine various plume
characteristics such as plume width, plume necking, etc. Fine thermocouples were used
to measure the temperature field. For the velocity field, two techniques of velocity
measurement, the Laser Doppler Velocimetry, LDV, and the Particle Image
Velocimetry, PIV, were trialled. It was found that the PIV technique was not successful
since the velocity in the plume flow was too small and the resolution of the laser and
camera system were not sufficient to record the movement of the seeding particles.
After much initial development of the experimental methodology the velocity in the
plume was successfully measured using Laser Doppler Velocimetry.
3.2.1
Thermal Plume Experimental Facilities
The main experimental facilities for the thermal plume tests are presented below.
3.2.1.1 Heat Source
A cast iron plate, 0.188m in diameter, was used as the heat source. The temperature of
the hot plate was controlled using temperature controller, EUROTHERM Model 91e, to
within ± 2ºC. A 10 mm thick, 0.188m diameter copper disc was placed on the hot plate
to ensure that the area source surface was flush with the surrounding floor and was at a
uniform temperature.
The hot plate and copper disc were joined using six bolts. Temperature measurements
and infrared imaging using a thermographic thermometer showed that the thermal
contact between the copper plate and the hot plate was good, without the need for
additional material such as thermal cement. Small gaps between the copper disc and the
fibre cement floor were filled with Kaowool strips.
78
Experimental Facilities, Techniques and Procedures
Forward-scattering
receiver on Manual
Traverse
Thermocouple Bank
Computer-controlled
Traverse for Transmitter
Glass
Enclosure
Laser
Source
Laser
Transmitter
Traverse
Control
Box
Heat
Source
Fibre-Cement Base
with cutout
Figure 3.17
Kaowool
Insulation
View of thermal plume experimental arrangement.
Fibre-cement sheet
Copper disc
Kaowool strip
Kaowool sheet
Figure 3.18
Drawing of heat source and floor of thermal plume experiment.
79
Experimental Facilities, Techniques and Procedures
Fibre-cement sheet
Copper disc
Kaowool sheet
Figure 3.19
Heater with
temperature controller
Cross-section of heat source and floor of thermal plume experiment.
3.2.1.2 Glass Enclosure and Base
The enclosure was designed to be compatible with the requirements and constraints in
the laboratory (such as available space, width of laboratory access, maximum available
focal length of LDV lenses, etc.). The enclosure was designed as a box of height 1.0m
and square base of side 0.76m. The vertical walls were made of toughened optical glass,
10 mm thick. The ceiling was made of 10 mm Perspex, with suitable sealable holes as
inlets for the seed particles, and also to accommodate vertical supports for the
thermocouple bank.
The glass box stood on a large square base, made of layers of insulating material
(Kaowool and Fibre-Cement sheet). The base had a centrally located opening, 0.188 m
in diameter, to accommodate the heat source. The entire assembly was supported on a
specially fabricated robust steel frame. The heat source and plume enclosure are shown
in Figures 3.17, 3.18 and 3.19.
3.2.1.3 Thermocouples
A total of 32 fine thermocouples (Type K, 30 gauge) were used for temperature
measurement. Of these, six were inserted into and held in place in 1.0 mm diameter
holes on the underside of the hot plate. Another five were positioned to monitor the
temperature in the ‘insulating’ base surrounding the hot plate, as shown in Figure 3.20.
One thermocouple monitored the ambient temperature in the laboratory, and another
80
Experimental Facilities, Techniques and Procedures
five measured the temperature at different heights and 10mm away from the vertical
walls inside the glass enclosure.
A bank of 15 thermocouples, 20 mm apart and with tips protruding out of thin 100 mm
long stainless steel tubes, was used to measure the temperature in the plume (Figure
3.21). The entire assembly was supported by two long steel rods and was moveable
up/down by regular intervals to allow temperature measurement at different vertical
distances from the source. Each thermocouple was individually calibrated using a
NATA certified calibration thermometer.
Copper plate
Positions of thermocouple
Copper plate
Fibre-cement sheet
Fibre-cement sheet
Figure 3.20
Dimension in mm.
Thermocouple locations in and around the heat source.
Thermocouple calibration
All thermocouples used in the thermal plume experiments were calibrated against a
standard NATA-certified calibration thermometer. The temperature ranges for
calibration depended on the range of temperature that the thermocouples were used for:
10oC - 100oC for plume temperature measurement and 10oC - 200oC for heat source
surface temperature measurement. For thermocouple calibration, all the thermocouples
and the standard thermometer were immersed in a thermal bath filled with either water
81
Experimental Facilities, Techniques and Procedures
or vegetable oil, depending upon the desired temperature range. A temperature regulator
was used to control the temperature of the liquid in the bath.
Thermocouples
10 cm
Transmitter
probe
Figure 3.21
Thermocouple bank.
The output from each thermocouple was recorded using LabTech software. The
temperature inside the thermal bath was set at 15oC and increased in 5oC steps in order
to measure the temperature read by the thermocouples and the standard thermometer.
Then, calibration graphs of each thermocouple were produced and linear calibration
equations were deduced. These were used in LabTech software for converting the
signals from thermocouples to actual temperature values. The data of calibration in the
graph below resulted from averaging data over 10 seconds with sampling rate of five
samples/second.
A typical calibration graph for a thermocouple is shown in Figure 3.22. Calibration
equations for all thermocouples are shown in Appendix IV.
82
Real Temperature (
o
C)
Experimental Facilities, Techniques and Procedures
100
90
80
70
60
50
40
30
20
10
0
No.5
y = 1.05009E+00x - 9.59813E-01
R2 = 9.99448E-01
20
30
40
50
60
70
80
90
100
Temperature from Thermocouple (o C)
Figure 3.22
Typical calibration graph (Thermocouple no.5).
3.2.1.4 Smoke and Smoke Generator
After many trial experiments with different seeding particles and following many
unsuccessful
attempts
to
measure
the
plume
velocities
with
the
large
transmitter/receiver in back scattering mode, the author was successful in achieving
velocity measurement in the forward scattering mode. It was found that the ‘seeding’
particles produced by a commercially available smoke generator were adequate for the
purpose. The smoke was produced by electrically heating a liquid (commercial name
“Fog Juice”) in the smoke generator. Separate ‘wet’ and ‘dry’ tests using a Malvern
Particle Analyser showed that the average particle size was about 4-5 µm, and which
appeared to decrease with time. This suggested that the particles had a liquid envelope
that tended to evaporate away, leaving a more solid core.
The smoke was conveyed to the test section (glass enclosure above the heat source) by
plastic tubing, and the inlet hole sealed. It was found that the smoke concentration
inside the enclosure had to be optimum to expedite velocity data collection. A period of
about 25 minutes was allowed to elapse to ensure that the random smoke movement
initiated during the inflow of the smoke had died down before the LDV measurements
commenced.
83
Experimental Facilities, Techniques and Procedures
3.2.1.5 LDV System
A Laser Doppler Velocimetry system (LDV) is a velocity measuring methodology that
does not disturb the flow when the measuring process is in progress. A 2-component
LDV technique was used here. Four laser beams, two green and two blue, were focused
onto a single point in the flow field. The natural thickness of the beams resulted in the
creation of an egg-shaped ‘measuring volume’ at the ‘point’ of intersection. Light
scattered from the particles passed to the receiver and was translated into a velocity
value by the associated DataView software. It was assumed that the plume flow was
nearly axisymmetric (with a negligibly small circumferential velocity component), so
that a 2-component measurement was sufficient.
After many trials it was found that, for these measurements in a gaseous medium, it was
necessary to set up the signal receiver in the ‘forward-scattering’ mode. In liquid media,
the ‘backward-scattered’ signal is strong enough to be picked up by the receiver built
into the transmitter itself. Despite repeated attempts, this did not work in the present
experiment. This is due to the fact that the strength of an electromagnetic signal
scattered from a small particle depends on the angle of scattering. It is the weakest in
the direction of the incident beam, and thus indistinguishable from random ‘noise’. The
scattered signal is much stronger in the direction opposite to the incident beam. In the
present experiment, the receiver axis was inclined at an angle of about 10° to the
incident beam. Figure 3.23 represents schematically the LDV technique as used in the
experiment.
Receiver for
forward-scattered
signal
Screen blocks
direct beams
Laser
Transmitter
Seeded
Flow
Figure 3.23
Green beams : Horizontal Component
Blue beams : Vertical Component
Forward-scattering LDV.
For velocity measurements, it was possible to locate the measuring volume as close as 4
mm from the floor of the enclosure. At these very short distances (4 mm – 16 mm) from
84
Experimental Facilities, Techniques and Procedures
the floor, only the horizontal velocity component could be measured, as one of the blue
beams required for measuring the vertical component was blocked by the floorboard
itself. The shortest distance where both velocity components could be measured was 16
mm above the source.
A non-intrusive velocity probe, consisting of a 2-component, Class IIIB, Argon-ion
LDV system was used for velocity measurements (Figures 3.23 to Figure 3.25). The
laser beam transmitter was mounted on a computer-controlled, three-axis traverse
system. Four laser beams, two blue and two green, were focused on the measuring
volume by means of a 500 mm focal length lens to measure the horizontal and vertical
components of the velocity. The Doppler signal from the measuring volume was
received by a receiver situated on the far side of the box, in the forward-scattering mode
(Figure 3.17). Black cardboard screens (not shown in Figure 3.17) were appropriately
placed to block out any stray laser beams from the environment as much as possible.
Laser
generator
Laser
splitter
Figure 3.24
Laser generator.
85
Experimental Facilities, Techniques and Procedures
Laser receiver
probe
Figure 3.25
3.2.2
1.
Laser transmitter
probe
Laser transmitter and receiver probes.
Experimental Procedure for Thermal Plume Study
Switch on the heat source at least 2 hours before commencement of data recording.
This allowed a near-steady-state to be reached for the buoyancy-driven flow
established inside the enclosure, with the heated air in the plume rising in the
middle, impinging on the ceiling, spreading out laterally, cascading down the cooler
vertical walls before getting caught up in the ascending plume again. It is assumed
that the heat transfer through the glass walls makes it possible for the flow to
achieve the quasi-steady-state.
2.
Introduce the seed particles (smoke) into the enclosure. A period of about 30
minutes was allowed to elapse to ensure that the random smoke movement initiated
during the inflow of the smoke died down, and the smoke particles followed the
buoyancy-driven flow in the plume reasonably faithfully.
3.
Start the velocity data collection using the LDV system. The positions of measuring
points were controlled by the computer-controlled traversing mechanism.
86
Experimental Facilities, Techniques and Procedures
4.
After finishing velocity measurement, the temperature measurement was started.
The position of the measuring point was controlled by moving the thermocouple bank to
desired levels manually.
87
Chapter 4
Computational Modelling
Computational Modelling
Chapter 4
Computational Modelling
A major part of this research project was the numerical study of plumes developing
from area sources. The computational work complemented the experimental
investigation since there were limitations to the practical application of the experimental
work which are described below.
In the thermal plume study, ideally, the area source of the thermal plume is only the hot
copper plate set at the desired uniform temperature. In reality heat conduction from the
hot copper plate to the surrounding floor could not be avoided. This means that the area
of the source of thermal plume may be increased. In addition, the plume from an area
source in a uniform ambient should be unconfined. In the experiment, an enclosure was
necessary to confine the smoke that was used for LDV measurement and eliminate
drafts. The dimensions of the enclosure were dictated by the limitations of the
experimental facility as explained in Chapter 3.
In the saline plume experiment, a lot of water was used, especially with the large tank
(about 3 m3 per test) which was a problem due to drought and water restrictions at the
time of the present work. Water from each experimental test could not be reused
because it was contaminated with salt solution. For salt concentration or density
measurement, a lot of time was required to measure plume density distribution of each
level. Therefore, these measurements were carried out at only five elevations for each
experiment. This was really inadequate for precise determination of plume width
development by density distribution.
Some of the above practical limitations can be overcome using in numerical modelling.
Also, validated numerical modelling allows a greater and easier control over the flow
parameters than physical experiments.
89
Computational Modelling
This chapter describes the mathematical model used in the present work and its
implementation using the PHOENICS CFD software. The information has been sourced
from a number of texts, papers and other sources including: CHAM (2003), Verteeg and
Malalasekera (1999), Patankar and Spalding (1972), Malin (2003), Spalding (1980), etc.
4.1
Governing Equations
The plume from an area source was simulated using the governing equations of fluid
flow that represent the conservation laws of physics applied to a fluid as a continuum:
•
The mass of a fluid is conserved;
•
The rate of change of momentum equals the sum of the forces acting on
a fluid particle;
•
Energy is conserved;
In the case of three dimensional, unsteady flows, the equation of mass continuity is:
∂ρ ∂ (ρu ) ∂ (ρv ) ∂ (ρw)
+
+
+
=0
∂t
∂x
∂y
∂z
(4.1)
alternatively this can be written in tensor form as:
∂ρ ∂ (ρu j )
+
=0
∂t
∂x j
(4.2)
The momentum equation comes from the Newton’s second law that states that the rate
of change of momentum of a fluid particle equals the sum of the forces on the particle.
In three dimensional, unsteady flow, we can write this equation for the x, y and z
directions.
x-momentum:
y-momentum:
∂ (− p + τ xx ) ∂τ yx ∂τ zx
∂ ( ρu )
+ div(ρuu ) =
+
+
+ S Mx
∂t
∂x
∂y
∂z
(4.3)
∂τ xy ∂ (− p + τ yy ) ∂τ zy
∂ ( ρv )
+ div(ρvu ) =
+
+
+ S My
∂t
∂x
∂y
∂z
(4.4)
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Computational Modelling
z-momentum:
∂τ yz ∂ (− p + τ zz )
∂τ
∂ ( ρw )
+ div(ρwu ) = xz +
+
+ S Mz
∂t
∂x
∂y
∂z
(4.5)
In a Newtonian fluid, the viscous stresses are assumed to be proportional to the rates of
deformation, then the values of stress in above equations are defined as;
τ xx = 2µ
∂w
∂v
∂u
+ λ div u , τ yy = 2µ
+ λ div u , τ zz = 2µ
+ λ div u
∂y
∂z
∂x
(4.6)
 ∂u ∂v 
 ∂v ∂w 
 ∂u ∂w 
 (4.7)
and τ xy = τ yx = µ  +  , τ xz = τ zx = µ 
+
 , τ yz = τ zy = µ  +
 ∂z ∂x 
 ∂y ∂x 
 ∂z ∂y 
where µ is the dynamic molecular viscosity, λ is the second viscosity.
Therefore, the momentum equations can be written as:
x-momentum:
ρ
∂p ∂ 
Du
∂u
 ∂   ∂u ∂v  ∂   ∂u ∂w 
= − +  2µ
+ λ div u  +  µ  +  +  µ  +
 + S Mx (4.8)
Dt
∂x ∂x 
∂x
 ∂y   ∂y ∂x  ∂z   ∂z ∂x 
y-momentum:
ρ
 ∂
∂p ∂   ∂u ∂v  ∂ 
Dv
∂v
= − +  µ  +  +  2µ
+ λ div u  +
Dt
∂y ∂x   ∂y ∂x  ∂y 
∂y
 ∂z
  ∂v ∂w 
 + S My (4.9)
 µ  +
∂
z
∂
y



z-momentum:
ρ
∂p ∂   ∂u ∂w  ∂   ∂v ∂w  ∂ 
∂w
Dw

 +  2µ
= − + µ  +
+ λ div u  + S Mz (4.10)
 +  µ  +
∂z ∂x   ∂z ∂x  ∂y   ∂z ∂y  ∂z 
∂z
Dt

The momentum equation can be written in tensor form when gravity is the only body
force as:
∂ρu i ∂ρu i u j
∂p
∂
+
=−
+ ρg i +
∂t
∂x j
∂x i
∂x i
where δij is the Kronecker delta.
91
  ∂u i ∂u j
+
 µ 
  ∂x j ∂x i

∂u 
 + λδ ij k 

∂x k 

(4.11)
Computational Modelling
The source terms, S Mx , S My and S Mz , include contributions due to body forces only. In
the case of the body force being due to gravity only, S Mx , S My = 0 and S Mz = -Fg.
The energy equation is derived from the first law of thermodynamics which states that:
Rate of increase
of energy of fluid
particle
=
Net rate of heat
added to fluid
particle
+
Net rate of work
done on fluid
particle
The energy equation can be written in term of specific enthalpy value as:
∂p  ∂ (uτ xx ) ∂ (uτ yx ) ∂ (uτ zx ) ∂ (vτ xy ) ∂ (vτ yy ) ∂ (vτ zy )
∂ ( ρh )
+
+
+
+
+
+
+ div[ρhu ] = div(k grad T ) +
∂z
∂y
∂x
∂y
∂z
∂t  ∂x
∂t
+
∂ (vτ xy ) ∂ (vτ yy ) ∂ (vτ zy ) ∂ (wτ xz ) ∂ (wτ yz ) ∂ (wτ zz )
+
+
+
+
+
 + S h (4.12)
∂x
∂y
∂z 
∂x
∂y
∂z
where h = specific enthalpy, k = thermal conductivity.
div (k grad T ) =
∂  ∂T  ∂  ∂T  ∂  ∂T 
 + k

 + k
k
∂x  ∂x  ∂y  ∂y  ∂z  ∂z 
(4.13)
or this can be written in tensor form.
∂(ρh ) ∂ (ρhu j )
∂  ∂T
+
=
k
∂t
∂x j
∂x j  ∂x j
 ∂p ∂ (u iτ ij )
+
+
 ∂t
∂x j

(4.14)
If a general variable Φ is a fluid property, such as salt concentration for example, the
conservation equation for the property Φ is given by:
∂(ρΦ )
+ div(ρΦu ) − div( Γ Φ grad Φ ) = SΦ
∂t
(4.15)
where ГΦ is a diffusion coefficient.
This equation is often used as the starting point for computational procedures in the
finite volume method. So that in the continuity equation:
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Computational Modelling
Φ =1
ΓΦ = 0
and
S Φ = 0 + boundary source ,
Γ Φ = ρ (ν t + ν l )
and
SΦ = -
In the momentum equation:
Φ = u , v, w
∂p
+ gravity + friction + ... ,
∂x k
And in the energy equation (in enthalpy form):
Φ=h
ν
ν 
Γ Φ = ρ  t + l 
σt σl 
and
SΦ = -
Dp
+ heat source + ... ,
Dt
where ν t = µ t ρ , ν = µ ρ are the turbulent and laminar kinematic viscosities and
σ t ,σ are the turbulent and laminar Prandtl/Schmidt Numbers.
Gravitational force, Fg, for momentum equation
In this work, the value of the gravitational force, Fg, can be represented in the
momentum equations in two ways.
Direct Statement or “Constant Density” Formulation
The gravitational force per unit volume, Fg, is defined:
Fg = ρ g
(4.16)
where ρ is the fluid density and g is the gravitational acceleration.
“Density Difference” or Reduced-Pressure Formulation
In this case, the gravitational force per unit volume, Fg, is given by:
Fg = (ρ − ρ ref ) g
Here, ρ ref is a reference density, which is a function of the reference pressure.
93
(4.17)
Computational Modelling
Because (ρ − ρ ref ) is used, the value of a reference pressure must be defined that will be
also used in term of ( p − pref ) in pressure term of the momentum equation instead. The
value of the reference pressure, pref , is defined by
grad ( pref ) + ρ ref g = 0
(4.18)
This formula was used in the case of saline water and thermal plume simulation in this
study.
Equations of State
In the case of a thermal plume in air, the ideal gas law was assumed. Therefore the
equation of state below was used:
p = ρ RT
(4.19)
In the case of the saline plume simulation, density, ρ, and viscosity, µ, of the saline
solution depends on salt concentration as follows:
ρ = 7.53348 × C1 + 9.96721×10 3
(4.20)
µ = 2.62129 ×10 −11 × C13 − 7.08646 ×10 −11 × C12 + 9.02391×10 −9 × C1 + 1.00451×10 −6
(4.21)
where C1 is the percentage by weight of salt solution.
These equations come from fitting a cubic polynomial to data from the table of
properties of salt solutions (Appendix V).
4.2
Turbulence Modelling
In this study of turbulent plume flow, the above equations (e.g. 4.2, 4.11 and 4.14) were
needed to describe the turbulent flows that have a statistically steady mean and were
94
Computational Modelling
solved by substituting the various velocity and scalar Reynolds decompositions. The
Reynolds decomposition of a variable Φ is defined as:
Φ( xi , t ) = Φ ( xi , t ) + Φ ′(xi , t )
(4.22)
where
1
Φ ( xi , t ) = Φ ( xi ) =
∆t
to + ∆t
∫ Φ(x , t )dt
(4.23)
i
to
therefore,
u i = u i + u i′ , p = p + p ′ , ρ = ρ + ρ ′
(4.24)
The vertical velocity root mean square value is defined as:
wrms = w =
2
∑ (w
T
T
− w)
2
(4.25)
N
The turbulent intensity is the root mean square of the fluctuations. For the x-component,
for example:
( )
′ = u′
u rms
2
12
1 T

=  ∫ u ′ 2 dt 
T 0

12
(4.26)
By substituting this description of variables in equations 4.2, 4.11, and 4.14, we obtain a
group of equations that can describe the effect of turbulent flow in a plume or other
‘buoyancy-driven flows’ such that:
(
)
∂
∂
+
ρ u j + ρ ′u ′j = 0
∂t ∂x j
(
)
(
)
(
(4.27)
)
∂p
∂
∂
∂
+ ρg i +
τ ij − u j ρ ′u i′ − ρ u i′u ′j − ρ ′u i′u ′j (4.28)
ρu i + ρ ′u i′ +
ρu i u j + u i ρ ′u ′j = −
∂x j
∂x i
∂t
∂x j
in which
 ∂u
∂u j 
∂u
 + λδ ij k

∂x k

τ ij = µ  i +
 ∂x j ∂x i
95
(4.29)
Computational Modelling
and
(
)
(
)
∂
∂
∂
ρh + ρ ′h ′ +
ρh u j + h ρ ′u ′j =
∂t
∂x j
∂x j
+
 ∂
 ∂
k
(
T + T ′) + ( p + p ′)
 ∂x
 ∂t
j


(
∂
u iτ ij + u i′τ ij′ − u j ρ ′h ′ − ρ h ′u ′j − ρ ′h ′u ′j
∂x j
)
(4.30)
In the case of laminar flow, the instantaneous continuity, momentum, and energy
equations form a closed set of five equations with five unknowns u, v, w, p and h. By
substituting the Reynolds decomposition for turbulent flow in the momentum equations,
results in six extra unknowns. In addition, extra unknowns u ′Φ ′ , v ′Φ ′ and w′Φ ′ are
also generated when substituting the Reynolds decomposition for any given variable Φ.
These extra unknowns that come from adding the Reynolds decomposition, leads to the
need for additional assumptions.
In this study the standard k-ε turbulence model with gravity correction was used to
study plumes from an area source. In this turbulence model, the values of the Reynolds
stress are linked to the mean rate of deformation through the following equation:
 ∂u i ∂u i
+
 ∂x
∂x i
j

τ ij = − ρ u i′u ′j = µ t 




(4.31)
where µt is eddy viscosity (Pa s).
Similarity, the turbulent transport of a scalar is taken to be proportional to the gradient
of the mean value of the transported quantity, such that:
− ρui′Φ ′ = Γ t
∂Φ
∂xi
(4.32)
where Γ t is the turbulent diffusivity and σ t = µt Γ t = a turbulent Prandtl/Schmidt
number.
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Computational Modelling
In the standard k-ε model (Launder and Spalding, 1974) two transport equations for the
turbulent kinetic energy, k, and the rate of dissipation of turbulent kinetic energy, ε, are
solved.
The equations of k and ε
Firstly, the turbulent kinetic energy, k, is defined as:
k=
1
2
(u ′
2
+ v ′ 2 + w′ 2
)
(4.33)
The concept of the transport equations for k or ε is summarised by the following.
Rate of
change of k
or ε
Transport of
+ k or ε by
convection
Transport of
k or ε by
diffusion
=
Rate of
Rate of
+ production of k - destruction
or ε
of k or ε
Therefore, the transport equation for turbulent kinetic energy k is:
ρ
∂k
∂k
∂
+ ρU j
=
∂t
∂x j ∂x j

µt
 µ +
σk

 ∂k 
∂U i

− ρε + S g
 + τ ij
∂x j
 ∂x j 
(4.34)
The rate of dissipation of turbulent kinetic energy, ε,
ρ
∂ε
∂ε
∂
+ ρU j
=
∂t
∂x j ∂x j

µt
 µ +
σε

 ∂ε 
ε ∂U i

− Cε 2 ρε 2 + Cε 3 S g
 + Cε 1 τ ij
k ∂x j
 ∂x j 
(4.35)
where Sg is a buoyancy source term discussed below.
The effects of gravity on the turbulence field may be accounted for by the introduction
of buoyancy source terms, Sg, into the transport equations for k and ε. For stably-
97
Computational Modelling
stratified flows, the user is advised that many workers recommend that such a source
term should be omitted from the ε equation, (CHAM, 2003).
In the case of a thermal plume, Sg is function of temperature that is defined as:
Sg =
− µt β g Ti
ρ σT
(4.36)
And in the case of a saline plume Sg is function of density that is defined as:
Sg =
− µt g ρi
ρ σ C1
(4.37)
Eddy viscosity can be defined as:
ρC µ k 2
µt =
ε
(4.38)
The k and ε equations have five empirical constants, C ε 1 , C ε 2 , C µ , σ k , and σ ε . In the
standard k-ε model, from experimental validation by previous researchers (e.g. Patankar
and Spalding (1972), Wilcox (1993), Worthy, et al. (2001), Spalding (1980), etc.) the
values of these constants are set to:
C ε 1 = 1.44 , C ε 2 = 1.92 , C µ = 0.09 , σ k = 1.0 , σ ε = 1.3
The constant Cε3 has been found to depend on the flow situation. It is thought that it
should be close to zero for stably-stratified flow, and to 1.0 for unstably-stratified flows
(CHAM, 2003).
This was an important issue for the present work since the plume flow near the area
source has both stably-stratified and unstably-stratified flow. Therefore the best value of
98
Computational Modelling
Cε3 for plume numerical simulation from the area source was an issue that was
investigated in some detail.
4.3
The Finite Volume Discretisation
From Equation 4.15,
∂ (ρΦ )
+ div(ρΦu ) − div( Γ Φ grad Φ ) = S
∂t
(4.15)
can be written in the form of;
∂ (ρΦ ) ∂ (ρuΦ ) ∂ (ρvΦ ) ∂ (ρwΦ ) ∂ 
∂Φ  ∂ 
∂Φ  ∂ 
∂Φ 
 −  Γ Φ
+
+
+
−  ΓΦ
 = S Φ (4.39)
 −  Γ Φ
∂t
∂x
∂y
∂z
∂x 
∂x  ∂y 
∂y  ∂z 
∂z 
North
High
West
East
P
z
y
x
South
Figure 4.1
Low
Schematic of control volume and their faces.
After integration, the finite volume equation has the form:
a pΦ p = a N Φ N + a S ΦS + a E ΦE + aW ΦW + a H ΦH + a LΦL + aT ΦT + S u (4.40)
where subscripts N, S, E, W, H, L refer to north, south, east, west, high, low and time,
respectively, and,
a P = a N + a S + a E + aW + a L + a H + aT + ∆F − S p
with
aT =
ρ T ∆V
∆t
99
(4.41)
Computational Modelling
The neighbour coefficients of this equation for the Hybrid differencing scheme are as
follows:
aW

F  

max  Fw ,  Dw + w , 0
2  


aE

F  

max − Fe ,  De − e , 0
2 


aS
 
F  
max  Fs ,  Ds + s , 0
2 
 
aN

F  

max − Fn ,  Dn − n , 0
2  


 
F
max  Fl ,  Dl + l
2
 
aL
 
, 0
 
aH

F  

max − Fh ,  Dh − h , 0
2  


∆F
Fe − Fw + Fn − Fs + Fh − Fl
where
Face
w
e
s
n
l
h
F
(ρu )w Aw (ρu )e Ae (ρu )s As (ρu )n An (ρu )l Al
(ρu )h Ah
D
Γw
Aw
δxWP
Γh
Ah
δz PH
Γe
Ae
δx PE
Γs
As
δy SP
Γn
An
δy PN
Γl
Al
δz LP
where Ai is the area of a cell face and ∆V is the volume of a cell.
4.4
Implementation of Boundary Conditions and Source Terms
In order to solve the differential equations, boundary conditions are needed. The
boundary condition, Sbc, is added to the right hand side of equation 4.18.
Therefore;
100
Computational Modelling
∂ (ρΦ )
+ div(ρΦu ) − div( Γ grad Φ ) = S + S bc
∂t
(4.42)
In the PHOENICS CFD software package the integral of the boundary source is
represented in linearized form;
S bc = a BC (Φ
−Φ
)
(4.43)
where ΦBC is the value of Φ at boundary, ΦP is the value of Φ at cell adjacent to the
boundary and a BC is the coefficient.
The finite-volume discretization of the differential equation thus yields, for each cell in
the domain, the following algebraic equation:
Φ =
where
∑a Φ
k
k
∑a Φ
k
+ a BC Φ
a p + a BC
(4.44)
= a N Φ N + a S ΦS + a E ΦE + aW ΦW + a H ΦH + a LΦL + aT ΦT
The types of boundary conditions used this study were:
Fixed value. The value of a BC is set to be very large (1×1030) therefore;
Φ =Φ
(4.45)
Inflows and outflows. All mass flow boundary conditions are introduced as linearized
sources (Sm) in equation 4.38.
S m = TC m (V m − Pp )
(4.46)
where T = a geometrical multiplier, Cm = the coefficient = 2×10-10, Vm = the mass flux
≈ ρ in u in and Pp = the pressure at node P.
101
Computational Modelling
At an inflow boundary, the mass flow is fixed irrespective of the internal pressure. The
sign convention is that inflows are positive and outflows are negative. A fixed outflow
rate can thus be fixed by setting a negative mass flow.
Fixed Pressure Boundary Condition (or Outlet Condition in PHOENICS). In the
case of a fixed pressure boundary condition, from equation 4.43, the pressure is fixed by
putting a large number for Cm (equal to 1×103) and the required pressure for Vm. The
direction of flow is determined for each cell by whether Pp>Vm, or Pp< Vm. The first
produces local outflow, the second local inflow.
Wall conditions. The wall is the most common boundary encountered in confined fluid
flow problems. In the case of turbulent flow, The inner layer is characterised by the
velocity scale uτ = τ w ρ , the “friction velocity” and a length scale ν u * . The nondimensional distance from the wall the first node is therefore:
y P+ =
∆y p
(ν u )
*
=
∆y p τ w
ν
ρ
(4.47)
where ∆y p is the distance of the near wall node P to the solid surface, τ w is the wall
shear stress.
A near wall flow is taken to be laminar if y + ≤ 11.63. If y + >11.63 the flow is turbulent
and the wall function approach is used. The exact value of y + = 11.63 is the
intersection of the linear and the log-law velocity profiles.
y+ =
1
κ
(
ln Ey +
)
(4.48)
where κ is von Karman’s constant = 0.41 and E is a constant that depends on the
roughness of the wall. For smooth walls E = 8.6.
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Computational Modelling
Linear sub-layer
The wall shear stress force is entered into the discretised u-momentum equation as a
source as:
τw = µ
up
(4.49)
∆y p
where up is the velocity at the grid node.
Heat transfer from a wall at fixed temperature, Tw, into the near-wall cell in the sublayer is calculated from:
qs = −
µ C p (T p − Tw )
Acell
σ
∆y p
(4.50)
where Acell is the wall area of the control volume.
Log-law wall functions
If the value of y + is greater than 11.63 node P is considered to be in the log-law region
of a turbulent boundary layer, then:
u+ =
(
U 1
+
= ln Ey p
uτ κ
)
(4.51)
2
k=
uτ
Cµ
(4.52)
2
u
ε= τ
κy
(4.53)
where U is the absolute value of the resultant velocity parallel to the wall at the first
grid node, C µ is a constant equal to 0.09 in the standard k-ε model.
Strictly the log-law should be applied to a point whose y + value is in the range 30 < y +
<130.
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Computational Modelling
The advice given with the PHOENICS package indicates that the boundary condition
for turbulent kinetic energy, k, assumes that the turbulence is in local equilibrium and
consequently, this set of wall functions is not really suitable under separated conditions,
as turbulent energy diffusion towards the wall is significant, leading to appreciable
departures from local equilibrium. However, in the present study separated flow
conditions do not occur and the wall function defined by equation 4.49 is implemented
in the momentum equations by way of source terms which takes the form (CHAM,
2003):
S mom = ρ s U (u w − u )
(4.54)
where s is the friction factor, u is the in-cell value of velocity and uw is the value of the
velocity at the wall.
General log-law wall function or Non-equilibrium log-law wall function
Launder and Spalding (1974) have proposed a generalisation of the log-law function to
non-equilibrium conditions. The form is:
U
where
k
uτ
2
y

 Est k 
ν 
= ln
 kst 




kst = k C µ
0.25
Est = ε C µ
0.25
(4.55)
(4.56)
(4.57)
These two kinds of log-law function are both provided in the PHOENICS CFD code.
4.5
The SIMPLEST Solution Algorithm Technique
The PHOENICS (v3.5) code used in the present study employed the SIMPLEST
algorithm. SIMPLEST stands for Semi-Implicit Method for Pressure-Linked Equations
ShorTened and is the most efficient derivative of the SIMPLE algorithm introduced by
104
Computational Modelling
Patankar and Spalding (1972). The main steps in the SIMPLEST algorithm (CHAM,
2003, Spalding, 1980 and Moukalled and Darwish, 2000) are:
•
Guess a pressure field.
•
Solve the momentum equations using this pressure field, thus obtaining
velocities which satisfy momentum, but not necessarily continuity.
•
Construct continuity errors for each cell: inflow - outflow.
•
Solve a pressure-correction equation. The coefficients are d(vel)/d(p), and the
sources are the continuity errors.
•
Adjust the pressure and velocity fields. Obtain velocities which satisfy
continuity, but not momentum.
•
Go back to step 2, and repeat with the new pressure field. Repeat until continuity
and momentum errors are acceptably small.
4.6
PHOENICS Software
In recent years, a number of commercial CFD packages have become available. The
one used in the present project was PHOENICS, a general-purpose CFD package that
can be used for simulation of fluid flow, heat transfer, mass transfer and associated
chemical reactions as well as stress analysis in solids. The simulations in PHOENICS
are based on the finite volume method and the PHOENICS program performs three
main functions. Those are:
Problem definition, called SATELLITE and VR (Virtual Reality) Editor, in which the
user prescribes the situation to be simulated.
Simulation, called EARTH, which is the main solver of the set of differential
equations.
Presentation, called PHOTON, VR-VIEWER, and AUTOPLOT, of the results of the
computation, by way of graphical displays, tables of numbers, and other means.
105
Computational Modelling
Please see print copy for Figure 4.2
Figure 4.2
Diagrammatic representation of the PHOENICS software (CHAM,
2003).
106
Chapter 5
Saline Plume Experimental Results
Saline Plume Experimental Results
Chapter 5
Saline Plume Experimental Results
This chapter presents the results of the experimental studies of saline plumes in a
quiescent environment. The results of velocity distribution, density distribution, plume
shape and plume necking are shown. Three main types of experimental tests were
conducted to determine:
•
Plume shape;
•
Velocity distribution;
•
Density distribution.
and
The source conditions for all tests are shown below in Table 5.1. In the case of the
density distribution experiments, the 0.3m diameter source was used.
5.1
Plume Shape
The shadowgraph technique was used to visualise the shape and location of necking of
the plume. The flow from the area source was varied from 1.98 mL/s to 6.64 mL/s with
salt concentration values of between 10% by weight and 20% by weight. This led to the
following range of source conditions in terms of buoyancy flux per unit area ( B0 ) and
total source momentum flux ( M 0 ): from B0 min = 1.6×10-4 m2/s3 and M 0 min = 4.5×10-10
m4/s2 to B0 max = 1.0×10-3 m2/s3 and M 0 max = 5.1×10-9 m4/s2.
The video footage of the plume released from the 105 mm diameter area source was
analysed using the DigImage software. The images of the plume were averaged after the
plume reached a quasi-steady state. Colomer, et al. (1999) have suggested that time (te)
when the plume reaches a quasi-steady state depends only on the diameter of the area
source and the source buoyancy flux as given by:
108
Saline Plume Experimental Results
(
t e = 1.8 D 2 B0
Table 5.1
)
13
(5.1)
The conditions of all experimental tests.
Test
Source
F0
B0
M0
diameter
× 10
-4
(m)
2
3
(m /s )
(m /s )
1.
0.105
1.61
Plume
2.
0.105
shape
3.
no.
×10
-6
4
3
Q0
×10
-9
4
2
×10
-6
Lm
Γ
7
×10-5
Gr*
×10
10
Ra*
×1012
(m /s )
3
(m /s)
×10
1.40
0.45
1.98
1.69
9.61
1.03
8.20
5.46
4.73
5.09
6.64
0.15
0.32
3.45
27.52
0.105
2.82
2.44
3.19
1.66
4.99
5.60
1.06
10.32
4.
0.105
10.50
9.09
4.46
6.22
0.36
0.21
3.98
38.59
5.
0.105
1.60
1.39
0.43
1.94
1.73
9.52
1.01
8.02
Velocity
6.
0.105
5.64
4.89
5.39
6.83
0.14
0.33
3.55
28.30
distribution
7.
0.105
2.94
2.55
0.34
1.72
4.73
5.75
1.09
10.61
8.
0.105
10.92
9.46
4.69
6.37
0.34
0.21
4.06
39.43
9.
0.300
1.07
7.53
1.54
10.04
11.80
0.10
44.35
353.40
Density
10.
0.300
1.66
11.71
3.73
16.25
4.88
0.16
69.00
549.88
distribution
11.
0.300
0.95
6.76
0.30
4.64
121.02
3.25
24.29
235.74
12.
0.300
2.15
15.21
1.54
10.44
23.90
7.31
54.70
530.93
(m)
The values of te from all experimental tests are shown below (Table 5.2).
It is seen that the maximum value of te shown in Table 5.2 was about 18 seconds.
Therefore, a time of 45 seconds after saline plume was released was considered for the
saline plumes reach to quasi-steady state. Then, in the image-averaging procedure, a 30
second centred at the 60 second mark was used (as shown in Figure 5.1).
109
Saline Plume Experimental Results
Table 5.2
Values of establishment time (te) for plumes to reach quasi-steady state
as recommended by Colomer et al. (1999).
Please see print copy for Table 5.2
Plume release
0s
Figure 5.1
Average time
(30s)
60s
120s
180s
240s
The period used for averaging plume profile images in DigImage.
The average plume shape realised for source conditions B0 = 10.50×10-4 m2/s3, M0 =
4.46×10-9 m4/s2 and Q0 = 6.22×10-6 m3/s is shown in Figures 5.2 and 5.3 below.
110
Saline Plume Experimental Results
Source diameter
(105mm)
Figure 5.2
Example of time-averaged and enhanced image of saline plume shape.
Test no.4, B0 = 10.50×10-4 m2/s3, M0 = 4.46×10-9 m4/s2 and Q0 = 6.22×10-6 m3/s area
source (at 60s over 30s averaging time).
The corresponding plume edge is shown in Figure 5.3.
Experimental values of Dneck, the minimum diameter of the plume, and zneck, the necking
location, are compared with the value calculated from the equations of Colomer, et al.
(1999) in Table 9.2.
111
Saline Plume Experimental Results
Radius (mm)
-100 -75 -50 -25 0
25
0
Dneck
-50
50
75
100
zneck
-100
z position(mm)
-150
-200
-250
-300
-350
-400
-450
-500
Figure 5.3
Plume edge of test no.4, B0 = 10.50×10-4 m2/s3, M 0 = 4.46×10-9 m4/s2
and Q0 = 6.22×10-6 m3/s source condition (at 60s over 30s averaging time).
From the above graph, the virtual origin of the plume can be located by extrapolating
the edges of the plume. The edges of the plume were located using Digimage, time
averaging of plume image mode. Such deduced values of estimated virtual origin
location (zv) from the graph are compared with previously published values in Table 9.3.
5.2
Velocity Distribution
The form of the velocity distribution within the plumes was studied using the particletracking velocimetry technique. The flow released from the area source was varied from
1.98 mL/s to 6.64 mL/s and the salt concentration was varied from 10% by weight to
20% by weight. The corresponding range of “Buoyancy Flux” and “Momentum Flux”
at an area source were B0 min = 1.61×10-4 m2/s3, M 0 min = 4.52×10-10 m4/s2 to B0 max =
1.05×10-3 m2/s3, M 0 max = 5.09×10-9 m4/s2.
112
Saline Plume Experimental Results
The velocity distribution within the plume for each area source condition was analysed
using the DigImage software. The values of velocity at each point were averaged over
30 seconds after the plume reached a quasi-steady state. From the values of te shown in
Table 5.2, one can assume that the plume flow reaches a quasi-steady after about 45
seconds after the flow was initiated.
Figure 5.4
Example of particles detected by Digimage during PTV processing.
Figure 5.4 shows an example of the instantaneous positions of particles as detected by
DigImage at a particular time.
The particle moving path over 10 seconds at 60 second released flows is shown in
Figure 5.5 below.
113
Saline Plume Experimental Results
Figure 5.5
Particle paths detected by DigImage during velocity analysis at 60s over a
10s period (Test no.5, B0 = 1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 =1.94×10-6
m3/s, source conditions).
The plume flows at each particular time are shown in Figure 5.6. The average velocity
shown below was averaged over one second at particular time.
Figure 5.7 shows a typical 2-D velocity field thus deduced at 60 second released plume
over 30s averaging time.
After this processing, the data was also analysed to determine the distributions of
vertical and horizontal components of velocity at different levels in the “near-field”
region.
114
Saline Plume Experimental Results
Previous researches (e.g. Rouse, et al. (1958), George, et al. (1958), etc.) suggested that
the vertical velocity distribution in the “far-field” region of the plume is best described
by the Gaussian curve. However, in the “near-field” region, the best-fitting curve must
be determined from experiments such as those described above.
Figure 5.6
a) at 1 second
b) at 3 seconds
c) at 5 seconds
c) at 10 seconds
Plume flow from the area source at each time (average over one second).
(Test no.5, B0 = 1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s source
conditions).
115
Saline Plume Experimental Results
Figure 5.7
Average velocity field of saline plume of test no.5, B0 = 1.60×10-4 m2/s3,
M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s source conditions (at 60s over 30s
averaging time).
In Gaussian curve description, the exponent is fixed for all levels far away from the area
source. For the “near-field” region a modified from of the Gaussian curve is proposed.
This is defined as:
w = wC e − (r bv )
n
(5.2)
where wC is the centreline vertical velocity, bv is the plume vertical velocity width and
n is the “shape index”.
Figure 5.8 shows an example of the use of the equation 5.2 comparing with
experimental data.
116
Saline Plume Experimental Results
0.06
Experiment
Experiment
Equation
5.2
Power fitting
Velocity (m/s)
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.08
Figure 5.8
-0.06
-0.04
-0.02
0.00
0.02
Radius (m)
0.04
0.06
0.08
Vertical velocity profile of saline plume at ~0.04m below the area source
(test no.5, B0 = 1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s (at 60s
over 30s averaging time, n =2.13 and bv = 0.0147 m).
It was found that the standard deviation of Power equation is less than that of the
Gaussian equation. However, the plume width from the Gaussian equation is not
significantly different from that from the Power (see Figure 7.8 and Figure 9.5).
Equation 5.2 proposed here should be suitable for a description of the near-field flow of
plume from an area source.
Figure 5.9 shows the vertical velocity profiles in a plume with source conditions, B0 =
1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s at different levels.
117
Saline Plume Experimental Results
0.06
0.010m
0.020m
0.040m
0.060m
0.080m
0.100m
0.200m
Vertical Velocity (m/s)
0.05
Velocity (m/s)
0.04
0.03
0.02
0.01
0.00
-0.01
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
Radius (m)
Figure 5.9
Vertical velocity profiles in saline plume at different levels showing best
fit to experimental data using Equation 5.2 (test no.5, B0 = 1.60×10-4 m2/s3, M 0 =
4.33×10-10 m4/s2 and Q0 =1.94×10-6 m3/s at 60s over 30s averaging time).
The horizontal velocity distribution in saline plumes with the same source conditions
are shown in Figure 5.10.
The values of centreline velocity, wC and velocity radius, bv , and shape index, n, at
different levels in a plume with this source condition are shown in Table 5.5 and Figure
5.11 below.
118
Saline Plume Experimental Results
Horizontal Velocity (m/s)
0.010
0.01m
0.02m
0.04m
0.06m
0.08m
0.10m
0.005
0.000
-0.005
-0.010
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12
Radius (m)
Figure 5.10
Mean horizontal velocity distribution in saline plume at different levels
(Test no.5, B0 = 1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s at 60s
over 30s averaging time).
Table 5.3
Centreline velocity, wC , vertical velocity radius, bv and shape index, n, at
different heights of plume of test no.5, B0 = 1.60×10-4 m2/s3, M 0 =
4.33×10-10 m4/s2 and Q0 =1.94×10-6 m3/s source condition (at 60s with
30s average time).
z (m)
0.01
0.02
0.04
0.06
0.08
0.10
0.12
0.15
0.20
wC (m/s) 0.0144 0.0233 0.0413 0.0489 0.0459 0.0530 0.0469 0.0400 0.0431
bv (m)
0.0208
0.0189
0.0147
0.0124
0.0138
0.0131
0.0161
0.0197
0.0223
n
4.53
3.48
2.13
1.75
1.80
1.68
1.74
1.99
1.95
119
0.07
0.035
0.06
wwc
bbv
c
v
0.030
0.05
0.025
0.04
0.020
0.03
0.015
0.02
0.010
0.01
0.005
0.00
0.00
0.05
0.10
0.15
bv (m)
wC (m/s)
Saline Plume Experimental Results
0.000
0.20
zz position (m)
Figure 5.11
Variations of wC and bv with position in saline plume (test no.5, B0 =
1.60×10-4 m2/s3, M 0 = 4.33×10-10 m4/s2 and Q0 = 1.94×10-6 m3/s at 60s over 30s
averaging time).
From the above graph, the values of zneck and Dneck are 0.070m and 0.025m,
respectively.
Similar tests were conducted under different conditions, test no.5 to test no.8 (see Table
5.1).
120
Saline Plume Experimental Results
0.09
0.08
0.07
wC (m/s)
0.06
0.05
0.04
0.03
0.02
test no.5
test no.6
test no.7
test no.8
0.01
0.00
0.00
Figure 5.12
0.05
0.10
z position (m)
0.15
0.20
Variation of centreline vertical velocity (wC) with heights of plume under
different source conditions (see table 5.1 and 5.2).
0.030
0.025
bv (m)
0.020
0.015
0.010
0.005
0.000
0.00
Figure 5.13
test no.5
test no.6
test no.7
test no.8
0.05
0.10
z position (m)
0.15
0.20
Variation of plume velocity radius (bv) with heights under different
source conditions (see table 5.1 and 5.2).
121
Saline Plume Experimental Results
7.0
test no.5
test no.6
test no.7
test no.8
6.0
5.0
n
4.0
3.0
2.0
1.0
0.0
0.00
0.05
0.10
0.15
0.20
z position (m)
Figure 5.14
Variation of shape index (n) with heights under different source
conditions.
5.3
Density Distribution
The large tank was used for saline plume density distribution tests. The 300mm
diameter area source was used to suit the spatial requirements of the special
conductivity probe to measure the density distributions in the plume. The density
measurement was started two minutes after the plume was released. The values of
density at each point were collected at the rate of 10 per second over 10 second periods
and were averaged in order to find out the average density at each point. The traversing
mechanism could move to measure the density data over only half the width of the
plume. In addition, the minimum time required for density measurement at one level
was about 3 minutes. Therefore density data could be collected at only four levels:
10mm, 50mm, 100mm and 200mm below the area source.
Density data at each level was fitted with the equation 5.3 below:
122
Saline Plume Experimental Results
ρ − ρ ∞ = ∆ρ M e
 r
−
 bd




n
(5.3)
where ρ is the density at point (kg/m3), ρ∞ is reference density (kg/m3), r = radius (m),
bd = plume density radius (m) and n = shape index.
Experimental density data and the best fit curves of density distribution at 100mm
below the area source are shown in Figure 5.15.
3.5
Experiment
Equation 5.3
3.0
ρ-ρ
(kg/m33)
ρ-ρ∞0 (kg/m
2.5
2.0
1.5
1.0
0.5
0.0
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
Radius (m.)
Figure 5.15
Density distribution in saline plume at 0.10m below source
( B0 = 1.07×10-4 m2/s3, M 0 = 1.54×10-9 m4/s2 and Q0 = 10.04×10-6 m3/s source
condition (test no.9)).
Due to insufficient density measurements, the data was not used to deduce the changing
of plume width with plume height. The density distribution data was mainly used to
validate the saline plume simulation model.
The results of density distribution measurements for test no. 9 to test no. 12 (see Table
5.1) are shown below in Figure 5.16 to 5.19.
123
Saline Plume Experimental Results
4.0
3.5
10mm
50mm
100mm
200mm
ρ-ρ
(kg/m3)
ρ-ρ∞0 (kg/m3)
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
Radius (m)
Figure 5.16
Density distribution in saline plume at different levels
( B0 = 1.07×10-4 m2/s3, M 0 = 1.54×10-9 m4/s2 and Q0 = 10.04×10-6 m3/s source
condition (test no.9), symbols shows experimental results and line best fits).
4.0
3.5
10mm
50mm
100mm
200mm
33
ρ-ρ
∞0(kg/m
ρ-ρ
(kg/m) )
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
Radius (m)
Figure 5.17
Density distribution in saline plume at different levels
( B0 = 1.66×10-4 m2/s3, M 0 = 3.73×10-9 m4/s2 and Q0 = 16.25×10-6 m3/s source
condition (test no. 10), symbols shows experimental results and line best fits).
124
Saline Plume Experimental Results
4.0
3.5
10mm
50mm
100mm
200mm
33
ρ-ρ
ρ-ρ∞0(kg/m
(kg/m))
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
Radius (m)
Figure 5.18
Density distribution in saline plume at different levels
( B0 = 9.56×10-5 m2/s3, M 0 = 3.04×10-10 m4/s2 and Q0 = 4.64×10-6 m3/s source
condition (test no. 11), symbols shows experimental results and line best fits).
5.5
5.0
4.5
10mm
50mm
100mm
200mm
33
ρ-ρ
ρ-ρ∞0(kg/m
(kg/m))
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
Radius (m)
Figure 5.19
Density distribution in saline plume at different levels
( B0 = 2.15×10-4 m2/s3, M 0 = 1.54×10-9 m4/s2 and Q0 = 10.44×10-6 m3/s source
condition (test no.12), symbols shows experimental results and line best fits).
125
Chapter 6
Saline Plume Numerical Models and
Results
Saline Plume Numerical Models and Results
Chapter 6
Saline Plume Numerical Models and Results
The geometries and boundary conditions used in the saline plume simulation are
explained in this chapter. Various simulation types for saline plumes were investigated
and compared with experimental results. Determination of the optimal values of the
turbulent Prandtl numbers (Prt) for velocity and concentration and Cε 3 in the standard kε turbulence model was the main outcomes of this numerical work.
6.1
Steady-State Numerical Model for a Saline Plume from an Area Source
In the experimental study of saline plumes, an unsteady (filling box) situation was
investigated. Thus, in numerical study of saline plume, an unsteady CFD model really
should be used in order to compare with the experimental results.
However, for a full transient simulation, the time required for computation is the main
concern. Due to the excessive computation time, the effect of the turbulent Prandtl
numbers for velocity and concentration and the value of the empirical constant Cε 3 in
the standard k-ε could not be fully investigated using full transient model of the saline
plume. Therefore, the effects of the above parameters were investigated using a steadystate model.
In order to remove the effects of the filling box situation and concentrate on the
characteristics of the plume, the physical domain for the steady-state simulation was a
0.5m diameter × 1.25m long circular cylinder. The area source diameter was 0.105m
with 1.981×10-4 m/s source velocity and 20% by weight source salt concentration. This
axisymmetric geometry model with 1×42×100 cells is shown below. A slice of the
physical domain was chosen as the computational domain and had 1, 42 and 100 cells,
respectively in the x (circumferential), y (radial) and z (axial) directions.
127
Saline Plume Numerical Models and Results
Wall
Area Source
Fixed Pressure
condition
z
z
y
y
a) Geometry.
Figure 6.1
b) Computational cells.
The 1×42×100 cells and computational domain for steady-state saline
plume simulation.
128
Saline Plume Numerical Models and Results
Figure 6.2
Velocity vectors and density contours of a steady-state saline plume
simulation ( Cε 3 = 0.6 and Prt = 1.0).
The main aim of this work is the study of plume structure near an area source, where
the plume parameters profiles are not self-similar. The effect of Cε 3 on the near-field
flow could be investigated. The variations of the centreline vertical velocity in the
steady-state saline plume for different values of Cε 3 were compared with the results
reported by Colomer, et al. (1999). The comparison is shown in Figure 6.3 below.
The graph shows that the value of Cε 3 that gives a good match with the results of
centreline velocity value (Colomer, et al., 1999) is between 0.5- 0.7. A value of 0.6 for
Cε 3 will be used for the numerical simulation of the saline plume.
Nam and Bill (1993) suggested that changing the value of the turbulent Prandtl number
for diffusion of heat (Prt) can improve the numerical results for turbulent plume
simulations. Therefore, in this section, next the effect of the turbulent Prandtl numbers
for velocity and concentration on the numerical results will be investigated.
129
Saline Plume Numerical Models and Results
The effect of the turbulent Prandtl number for velocity and concentration on the results
of steady saline plume simulation is shown in Figure 6.4 below.
Centreline Velocity (m/s)
0.12
Colomer
Ce3=0.1
Ce3=0.3
Ce3=0.5
Ce3=0.7
Ce3=0.9
Ce3=1.1
Ce3=1.3
Ce3=1.5
0.10
0.08
0.06
0.04
0.02
0.00
0.0
0.1
0.2
0.3
Distance from
source
(m) (m)
Distance
from area
an area
source
Figure 6.3
Effect of Cε 3 on centreline velocity in saline plume (Prt = 1.0).
The value of the turbulent Prandtl numbers of velocity and concentration has an effect
on the plume shape and centreline velocity. For values of turbulent Prandtl no. less than
0.3, significant changes in plume width and centreline velocity are seen but for values
greater than 0.3, only centreline velocity is affected significantly.
From these observations, it can be concluded that, a suitable value for Cε 3 is 0.6 in
near-field region before the plume develops self-similarity. Changing the turbulent
Prandtl number of velocity and concentration between 0.4 and 1.0 can be used to reduce
the centreline value of vertical velocity to suit with experimental results.
In the remainder of the present work a value of Prt = 0.65 has been used. This gave the
best fit to experimental data. This value is somewhat lower than that used in some CFD
studies such as Wilcox (1993), Worthy, et al. (2001) and Nam and Bill (1993), but was
chosen to give a good agreement particular with the centreline velocity.
130
Saline Plume Numerical Models and Results
0.00
Prt=0.1
Prt=0.3
Prt=0.5
Prt=0.7
Prt=0.9
Prt=1.0
Vertical Velocity (m/s)
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
0.00
0.05
0.10
0.15
0.20
Radius (m)
Figure 6.4
Effect of Prt on saline plume vertical velocity at 0.06m below an area
source ( Cε 3 = 1.0).
6.2
Transient Numerical Model for Axisymmetric Saline Plume
In this section, an unsteady simulation of a saline plume in a circular tank is presented.
The main reason for this study was to arrive at an optimal computational time for
unsteady simulation. This was done by conducting grid dependence tests, time-step
dependence tests and number-of-sweep dependence tests.
The computational domain was cylindrical polar, with a radius of 0.27645m, which
yielded the same cross-section area as the small experimental tank (0.2401m2 in the case
of 0.5×0.5m2 tank). The computational domain used for this test is shown in Figure 6.5.
The source condition used in this test was 20% by weight of salt solution with 1.98×10-4
m/s source velocity.
131
Saline Plume Numerical Models and Results
Outlet Condition
Saline Plume
Source
Wall Condition
z
z
y
y
Figure 6.5
6.2.1
The cylindrical polar geometry of saline plume simulation.
Effect of Number of Time Steps
In this case, the 1×30×100 cells model of a saline plume in a cylindrical box was used
to determine the effect of time step size on the numerical results.
The evolution of the plume over 60 seconds was studied with time step varied from 60
to 960 corresponding the ∆t equals to 0.0625 to 1 second per step. The effect on a
typical “spot value”, the vertical velocity component at r = 0.0m, z = 0.10m is shown in
Figure 6.6.
From Figure 6.6, it was clear that a suitable value for the number of time steps was
about 480. The value of vertical velocity at that position decreased from 60 to about 250
number of steps and then increased and reached to steady at number of steps of 480.
Therefore, number of steps that was suitable for simulation for this saline plume should
more than 400. The 480 steps was used for this study corresponding to 60s duration
time.
132
Saline Plume Numerical Models and Results
-0.0314
r = 0.00m
-0.0316
Velocity (m/s)
-0.0318
-0.0320
-0.0322
-0.0324
-0.0326
60
240
420
~480
600
780
960
Number
of of
Time
Steps
Number
steps
Figure 6.6
The effect of time step size on vertical velocity distribution of plume.
(r = 0.00m, z =0.10m below an area source at 60s after plume was released).
-0.0302
r = 0.00m
Centreline
Velocity
Velocity
(m/s) (m/s)
-0.0304
-0.0306
-0.0308
-0.0310
-0.0312
-0.0314
-0.0316
-0.0318
-0.0320
-0.0322
0
50
100
150
NumberNumber
of Sweeps
per Time-Step
of sweeps
Figure 6.7
The effect of number of sweeps on vertical velocity at r = 0.00m and z =
0.10m (at t = 60s).
133
Saline Plume Numerical Models and Results
6.2.2
Effect of Number of Sweeps
The number of sweeps is the number of iterations per time step. In this test, 10, 25, 50,
75 and 100 sweeps were used to find out the effect on numerical results such as velocity
and density. The 1×30×100 geometry was used. The effect of changing the number of
sweeps on a spot value of the vertical velocity is shown in Figure 6.7.
It is seen that about 100 sweeps per time-step is adequate for the numerical simulation.
6.2.3
Grid Dependence
In order to find out a suitable mesh size, five different meshed were tested: Model I
(1×10×30), Model II (1×20×60), Model III (1×30×90), Model IV (1×40×120) and
Model V (1×50×150). The number of cells used for each model is shown in Table 6.1.
Table 6.1
Number of cells used for each numerical model.
Model No.
ny
nz
Total
I
10
30
300
II
20
60
1200
III
30
90
2700
IV
40
120
4800
V
50
150
7500
Due to the centre position of the first numerical cell was difference because the number
of cells used for each model was different, therefore it was not possible to compare the
value of centreline vertical velocity at the same position. Therefore, in order to find out
the suitable number of cell used for saline plume simulation, the spot values of vertical
velocity at about r = 0.08m and z = 0.10m was used as a measure of the griddependence.
134
Saline Plume Numerical Models and Results
0.0018
Velocity (m/s)
0.0016
0.0014
0.0012
0.0010
0.0008
0.0006
0
Figure 6.8
1000
2000
3000 4000 5000
Number of cells
6000
7000
8000
The effect of cells on vertical velocity at r = 0.08m and z = 0.10m (t =
60s).
From above graph, the value of vertical velocity reached to steady at ny about 35 and nz
about 100. Therefore, number of cell used in this numerical study was ny = 35 and nz =
100 suitable for study the saline plume in this study.
From all of the above testing, it could be concluded that ny = 35, nz = 100, 100 sweeps
per time-step and ∆t = 60/480 s would be optimal values for the saline plume
simulation.
6.2.4 Comparison of Numerical Results between Saline Plume in Circular Tank
and Rectangular Tank
All saline plume numerical results above were obtained using a circular cylinder tank
because it was easy to simulate due to easy geometry and save the computational
time.but the real experimental study, the environmental tanks were a box-shaped.
Therefore, in this section, the two geometries, circular and rectangular, was considered
numerically to study the effect of tank shape on saline plume characteristics.
135
Saline Plume Numerical Models and Results
The computation domain and mesh for a “quarter-box” simulation are shown in Figure
6.9 and 6.10, respectively.
Saline plume
source
Source support
plate (wall)
Figure 6.9
Ambient
pressure
condition
Tank wall
Computational domain for numerical study of saline plume in
rectangular tank.
The 32×32×60 cells of the rectangular tank geometry (Figure 6.10) corresponds to the
1×32×60 cells of polar circular cylinder tank geometry. The source conditions used for
this test were w0 = 1.981×10-4 m/s and 20% by weight source salt concentration.
Figure 6.10
The 32×32×60 cells of rectangular tank geometry.
136
Saline Plume Numerical Models and Results
The results of centreline vertical velocity, vertical velocity distribution at 0.10m below
an area source and density distribution at 0.10m below an area source of two models are
shown below.
0.00
Centreline Velocity (m/s)
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
-0.09
-0.10
0.00
Rectangular Tank
Circular Tank
0.10
0.20
0.30
0.40
0.50
Distance from
area
source
from an
area
source
(m)(m)
Figure 6.11
Comparison of centreline vertical velocity between two models of tank
(at 60s).
0.02
Vertical Velocity (m/s)
0.00
-0.02
Rectangular Tank
Circular Tank
-0.04
-0.06
-0.08
-0.10
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Radius (m)
Figure 6.12
Comparison of vertical velocity distribution between two models of tank
(at 60s).
137
Saline Plume Numerical Models and Results
1005
Rectagular Tank
Circular Tank
1003
3
Density (kg/m )
1004
1002
1001
1000
999
998
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Radius (m)
Figure 6.13
Comparison of density distribution between two models of tank (at 60s).
This results show that the simulation of the two environmental tanks gave almost
identical shapes of velocity profile and also density profile. The main difference in
comparison results is in the values of centreline velocity and density, but these
differences are less than 6% for velocity and 0.02% for density. This can be assumed
acceptable for the numerical results. Hence, the numerical results of saline plume in a
circular cylinder tank can be used to predict the results of saline plume in rectangular
tank for this study.
6.3
Transient Numerical Model and Results for Saline Plume in a Rectangular
Enclosure
From the above results, it was clear that the axisymmetric geometry could be used to
represent the results of saline plume simulation in a rectangular tank. Therefore in
numerical study, an axisymmetric geometry of circular cylinder tank was used to
simulate the development of the saline plume. The 1×35×100 cells model with at least
100 sweeps per time-step, 480 time-steps was used to simulate the evolution of the
saline plume filling the tank. In addition, a comparison of numerical and experimental
results was also presented.
138
Saline Plume Numerical Models and Results
Because two experimental tanks were used, two equivalent axisymmetric models of
circular cylinder tank will be used to compare the results.
The radii of the circular tanks representing the rectangular tanks were calculated from
the cross-section areas of the rectangular tanks. For the small rectangular tank
(0.5m×0.5m×0.5m, outer dimensions), a 0.27645m radius circular tank can be used in
the numerical model of saline plume. For the big rectangular tank ((1.4m×1.4m×1.4m,
outer dimensions), a 0.78414m radius circular tank can be used in the numerical model.
6.3.1
Velocity Distribution
In the experimental study, the small tank (0.5m×0.5m×0.5m, outer dimensions) was
used to study the velocity distribution in the saline plume. Correspondingly, the circular
tank of 0.27645m radius and 0.5 height was used for the numerical study.
The source conditions of test no. 5, w0 = 2.24×10-4 m/s and 10% by weight salt
concentration, were used for this simulation. The results of plume characteristics such
as plume shape, vertical velocity field are shown in Figure 6.14 and Figure 6.15. The
horizontal velocity profiles at different levels: 0.01m, 0.02m, 0.04m, 0.06m, 0.08m,
0.10m and 0.20m are shown in Figure 6.16.
139
Saline Plume Numerical Models and Results
Figure 6.14
Velocity vector field 60s after plume was released (0.276m radius and
0.5m height cylindrical tank). The source conditions of test no. 5, w0 = 2.24×10-4 m/s
and 10% by weight salt concentration.
0.01
Vertical Velocity (m/s)
0.00
-0.01
-0.02
-0.03
-0.04
0.01m
0.02m
0.04m
0.06m
0.08m
0.10m
0.20m
-0.05
-0.06
-0.07
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Radius (m)
Figure 6.15
Vertical velocity profile in saline plume 60s after plume was released
(the source conditions of test no. 5, w0 = 2.24×10-4 m/s and 10% by weight salt
concentration).
140
Saline Plume Numerical Models and Results
0.010
0.01m
0.02m
0.04m
0.06m
0.08m
0.10m
0.008
Horizontal Velocity (m/s)
0.006
0.004
0.002
0.000
-0.002
-0.004
-0.006
-0.008
-0.010
-0.08
Figure 6.16
-0.04
0.00
Radius (m)
0.04
0.08
Horizontal velocity profiles in saline plume 60s after plume was released
(the source conditions of test no. 5, w0 = 2.24×10-4 m/s and 10% by weight salt
concentration).
6.3.2
Density Distribution
The experimental data of density distribution was collected from the large
environmental tank (1.4m×1.4m×1.4m, outer dimensions). Therefore the corresponding
numerical model used for comparison is the numerical study of saline plume in the
0.7841m radius circular tank.
The source conditions of test no.9 (D0 = 0.30m, w0 = 1.48×10-4 m/s, 10% by weight of
salt concentration) was used in the numerical model.
The computational domain and mesh are shown in Figure 6.17.
141
Saline Plume Numerical Models and Results
r =0.7841m
1.4m
Outlet condition
Area source
Wall condition
z
z
y
y
Figure 6.17
Geometry model and the 1×35×100 numerical cells of large
computational circular tank.
Because the density data collection in experimental study started two minutes after the
plume was released, then in order to compare with experimental results, in numerical
work, the density distribution in the saline plume two minutes after the flow was
released was used for comparison with the experimental data.
The results of density contour and density distributions at some levels are shown below.
142
Saline Plume Numerical Models and Results
Figure 6.18
Density contour in saline plume 120s after plume was released (0.7841m
radius and 1.2m heights cylindrical tank). The source conditions of test no.9, w0 =
1.48×10-4 m/s, 10% by weight of salt concentration.
1007
0.01m
0.05m
0.10m
0.20m
1006
3
Density (kg/m )
1005
1004
1003
1002
1001
1000
999
998
997
0.000
Figure 6.19
0.025
0.050
0.075
0.100
Radius (m)
0.125
0.150
CFD results of density profiles in saline plume 120s after plume was
released (the source conditions of test no.9).
143
Chapter 7
Thermal Plume Experimental Results
Thermal Plume Experimental Results
Chapter 7
Thermal Plume Experimental Results
Thermal plumes generated by a hot surface at two different temperatures were studied
in a comprehensive series of experiments in order to determine plume characteristics
including velocity and temperature distributions, plume necking and plume width. The
conditions for these experiments are summarised in Table 7.1.
Table 7.1
The experimental conditions for the thermal plume experiments.
Test no.
Tambient (oC)
1
25.5
2
26.5
3
26.2
4
25.2
5
24.3
6
24.2
Average of
o
Tambient ( C)
Tsurface (oC)
Average of
o
24.5
Ra
7
× 107
5.47
3.80
5.45
3.79
153.5
5.46
3.80
198.9
5.71
3.96
5.74
3.98
5.74
3.98
Tsurface ( C)
150.8
26.1
Gr
152.8
200.9
200.7
152.4
200.2
× 10
Note: the length scale, L, in the equations of Gr and Ra number is the source diameter,
D.
The average values of Tambient and average values of Tsurface were used to set the surface
and ambient conditions for the numerical modelling. This is discussed in chapter 8. In
addition, the values of average of Tambient were also used in the data fitting process.
Velocity data at each elevation was collected using the LDV system with a 0.1 burst
threshold, 0.3125s sampling period and the data collection process was terminated when
4000 bits of data had been collected at each measurement point. The temperature
measurements using thermocouples and the LabTech software were monitored at a
100Hz sampling rate over 20 seconds. The experimental tests for each surface
145
Thermal Plume Experimental Results
temperature were repeated three times, on different days, in order to check experimental
repeatability and to confirm quasi-steady state of the flows in these experiments.
7.1
Experimental Results for 150oC Source
The first test in the thermal plume experiments was for a thermal plume from a 150oC
area source. Before data was taken a quasi-steady state had to be achieved. The heat
source inside the enclosure was switched on at least 2 hours before commencement of
plume velocity and temperature distribution data recording. Changes of temperature
inside the enclosure and in the external ambient temperature before plume velocity and
temperature data were collected are shown in Figure 7.2.
Tambient
no.19
20mm
no.18
20mm
no.17
20mm
no.16
20mm
Figure 7.1
Positions of thermocouple used to measure temperature inside an
enclosure.
146
Thermal Plume Experimental Results
34
no.16
no.17
no.18
no.19
Tambient
Temperature (oC)
32
30
28
26
24
22
20
0
3000
6000
9000
Time (s)
Figure 7.2
Development of quasi-steady thermal conditions inside the enclosure
before plume velocity and temperature distribution data were collected.
Figure 7.2 shows that after about 2 hours switching on the heat source, it can be
assumed that the quasi-steady state was achieved. Therefore all experimental data were
collected after 2 hours switching on the heat source including velocity distribution,
temperature distribution and also temperature distribution on the hot surface and the
insulation.
The temperature distribution on the hot surface and on the insulation next to the hot
plate was measured. The average temperature at each point on the surface of the hot
plate and insulation for the three tests are shown in Figure 7.3.
The results showed a uniform temperature condition (to within ±0.5oC) on the surface of
the hot copper plate. The temperature at the surface of the insulation around the hot
plate dramatically decreased with distance away from the edge of the hot copper plate.
147
Thermal Plume Experimental Results
In addition, the temperature at each measurement point in three tests shows almost the
same values.
Diameter of hot
plate (188mm)
200
test1
test2
test3
Temperature(oC)
150
100
50
0
-0.2
-0.15
Figure 7.3
-0.1
-0.05
0
0.05
Radius(m)
0.1
0.15
0.2
Surface temperature distribution in 150oC hot plate.
The main plume characteristics that were of interest in these tests were temperature,
vertical velocity and horizontal velocity distributions. The results of these distributions
at 0.10m above the floor are shown in Figures 7.4 to 7.6.
148
Thermal Plume Experimental Results
55
test1
test2
test3
Temperature (oC)
50
45
40
35
30
-0.15
Figure 7.4
-0.10
-0.05
25
0.00
Radius (m)
0.05
0.10
0.15
Temperature distribution of thermal plume from 150oC hot plate at
0.10m above the floor for three separate experiments to illustrate repeatability of results.
0.40
test1
test2
test3
Velocity
(m/s)
Vertical
Velocity
(m/s)
0.30
-0.15
Figure 7.5
0.20
0.10
0.00
-0.10
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distribution of thermal plume from 150oC hot plate at
0.10m above the floor for three separate experiments to illustrate repeatability of results.
149
Thermal Plume Experimental Results
0.06
test1
test2
test3
Velocity
(m/s) (m/s)
Horizontal
Velocity
0.04
0.02
0.00
-0.02
-0.04
-0.15
Figure 7.6
-0.10
-0.06
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Horizontal velocity distribution of thermal plume from 150oC hot plate at
0.10m above the floor for three separate experiments to illustrate repeatability of results.
From the graphs above, the values of temperature and velocity at each point of each test
provide evidence of the repeatability of the experiments to within ±3.0oC (±0.5% of the
overall temperature difference) for temperature and ±0.02m/s for velocity.
7.1.1
Temperature Distribution
In this section, the data of temperature profiles in the plume at different levels were
fitted with a suitable equation. Rouse, et al. (1952), George, et al. (1978) etc. have
recommended that in the plumes that reach self-similarity the temperature and vertical
velocity distributions can be modelled by the Gaussian profile. Therefore, at first, the
Gaussian equation was used to fit the temperature data. This equation is defined by:
∆T = ∆TC e
 r
−
 bT



2
(7.1)
where ∆T is distribution of temperature difference between temperature of plume and
ambient temperature at the same level, ∆TC is the centreline temperature difference, r is
radial distance from centre of the plume, bT is the plume “temperature radius”.
150
Thermal Plume Experimental Results
Investigation of the plume flow near an area source (ie. where the flow may not be fully
self-similar) was one of the main aims of this study. In Chapter 5, it is clear that
equation 5.2 is suitable for fitting vertical velocity and density (or temperature)
distribution. Therefore, the equation (7.2) was used to correlate all temperature
distribution at all levels.
∆T = ∆TC e
 r
−
 bT



n
(7.2)
where n is the empirical shape index.
The comparison between two equations after fitting with experimental data is shown in
Figure 7.7 below.
35
Experiment
Eq. 7.2
7.2, n = 2.1184
Gaussian
30
o
∆T ( C)
25
20
15
10
5
0
-5
-0.15
Figure 7.7
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
The comparison of two fitting-equations with temperature distribution
data at 0.02m (z/D = 0.1064) above the floor from 150oC hot plate.
A least squares best fit to the data showed that Equation 7.2 gave on slightly better
results than the Gaussian profile. Therefore, all of temperature and vertical velocity
distribution were fitted with the equation 7.2.
151
Thermal Plume Experimental Results
In order to show the effect of fitting equation on plume width, the value of plume width
from equations 7.1 and 7.2 were compared in Figure 7.8.
0.070
Eq. 7.2
Gaussian
plume width (m)
0.065
0.060
0.055
0.050
0.045
0.040
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Position above the area source (m)
Figure 7.8
The comparison of bT from two fitting equations.
(Plume from 150oC hot plate).
From Figures 7.8 above, it is seen that the Equation 7.2 has very small effect on the
value of bT but in Figure 7.7, the Equation 7.2 gives a better estimate of the near-field
temperature and vertical profile than the pure Gaussian distribution as shown in small
value of the standard deviation using the Equation 7.2 comparing with using the
Gaussian profile (see section 9.1.1.2 for more detail).
Temperature distributions were measured at elevations of 0.02, 0.04, 0.06, 0.08, 0.10,
0.20 and 0.30 m above the floor. After correlation of all temperature data with Equation
(7.2), the temperature distributions at difference levels are shown in Figure 7.9 and the
value of ∆TC and also bT at all levels are shown in Table 7.2.
152
Thermal Plume Experimental Results
30
0.02m
0.04m
0.06m
0.08m
0.10m
0.20m
0.30m
25
o
∆T ( C)
20
15
10
5
0
-5
-0.15
Figure 7.9
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Temperature distributions at different elevations in thermal plume above
150oC hot plate.
(Solid lines represent best fit of equation 5.2 to experimental data).
Values of ∆TC , bT and n of the plume above 150oC hot plate.
Table 7.2
Distance from the floor (m)
0.02
0.04
0.06
0.08
0.1
0.2
0.3
(z/D)
(0.106)
(0.213)
(0.319 )
(0.426)
( 0.532)
( 1.064)
( 1.596)
∆TC (oC)
28.57
26.19
25.24
23.57
22.21
15.95
10.73
bT (m)
0.0653
0.0568
0.0493
0.0468
0.0447
0.0435
0.0517
n
2.1184
2.0769
2.0936
2.1955
2.3381
1.8516
1.7370
The value of ∆TC , bT and n were plotted in order to find out the effect of distance above
the hot plate on ∆TC and bT . A graph of ∆TC and bT for the thermal plume from 150oC
hot plate is shown in Figure 7.10.
153
Thermal Plume Experimental Results
35
∆T
∆TC
bbT
T
0.06
25
0.05
20
0.04
15
0.03
10
0.02
5
0.01
0
0.00
0.35
bT (m)
∆TC (K)
(oC)
30
0.07
0
0.05
0.1
0.15
0.2
0.25
0.3
Position above hot plate (m)
Figure 7.10
The graph of ∆TC and bT of thermal plume above 150oC hot plate.
Plume temperature radius ( bT ) as a function of height shows that the necking of the
plume occurs at zTneck ≈ 0.14m above the hot plate and that the bTneck ≈ 0.041m. This is
indicated by the minimum value of bT in above graph of plume width.
7.1.2
Vertical Velocity Distribution
The vertical velocity distribution data of these experiments were fitted with Equation
(7.2):
w(z , r ) = wC e
 r
−
 bV
n



(7.2)
where wC is the centreline velocity at level z above the floor, bV is the plume “vertical
velocity radius”.
The data of vertical velocity distribution and best fitting curve of 0.10m levels above the
floor are shown in Figure 7.11.
154
Thermal Plume Experimental Results
0.40
0.35
0.30
w (m/s)
w (m)
0.25
0.20
0.15
0.10
0.05
0.00
-0.05
-0.15
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distribution at 0.10m above floor from 150oC hot plate
Figure 7.11
compared with equation 5.2, bv = 0.0547m.
The value of wC and also vertical velocity radius (bV) at levels 0.02, 0.04, 0.06, 0.08,
0.10, 0.20 and 0.30 m above the hot plate are shown in the Table 7.3.
Values of wC , bV and n of thermal plume above 150oC hot plate.
Table 7.3
Distance from the floor (m)
0.02
0.04
0.06
0.08
0.1
0.2
0.3
z/D
(0.106)
(0.213)
(0.319)
(0.426)
(0.532)
(1.064)
(1.596)
wC (m/s)
0.1257
0.1894
0.2676
0.3045
0.3392
0.4154
0.4019
bV (m)
0.0571
0.0583
0.0561
0.0541
0.0547
0.0558
0.0636
n
2.0645
2.5535
1.9956
2.7266
2.921
2.6905
2.6226
The vertical velocity distributions at difference levels are shown in Figure 7.12.
155
Thermal Plume Experimental Results
0.5
0.02m
0.04m
0.06m
0.08m
0.10m
0.20m
0.30m
0.4
w (m)
0.3
0.2
0.1
0.0
-0.1
-0.15
Figure 7.12
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distribution in thermal plume from 150oC hot plate
(D0 =0.188m).
(Solid lines represent best fit of equation 5.2 to experimental data).
The value of wC and also bV were plotted and shown in Figure 7.13.
0.50
wVc
C
bbv
V
0.45
0.40
0.070
0.065
0.30
0.060
0.25
0.20
0.055
bv (m)
wC (m/s)
0.35
0.15
0.10
0.050
0.05
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.045
0.35
Position above hot plate (m)
Figure 7.13
The graph of wC and bV of thermal plume above 150oC hot plate.
156
Thermal Plume Experimental Results
The results for plume velocity radius ( bV ) shows that the necking of the plume occurs
also at zVneck ≈ 0.14m above the hot plate, the same as the necking position using bT ,
and the bVneck ≈ 0.054m.
In order to compare the plume temperature radius (bT) and the plume vertical velocity
radius (bV), the value of bT and bV at difference level were plotted versus the distance
above the hot plate. The result is shown in Figure 7.14.
0.07
bbv
V
bbT
T
Plume Width (m)
0.06
0.05
0.04
0.03
0.02
0
Figure 7.14
0.05
0.1
0.15
0.2
0.25
Distance above the floor (m)
0.3
0.35
The comparison graph of bT and bV of thermal plume above 150oC hot
plate.
From Figure 7.13, the plume temperature radius (bT) is seen to be less than the plume
vertical velocity radius (bV) but both of bT and bV data show that the position of necking,
zneck, is approximately 0.14m above hot plate.
7.1.3
Horizontal Velocity Distribution
The horizontal velocity data in the plume above the 150oC hot plate were collected. The
lowest level at which the LDV system could measure was 0.004m above the hot plate.
Figure 7.15 shows the horizontal velocity distribution at 0.004m above the floor.
157
Thermal Plume Experimental Results
Horizontal Velocity (m/s)
0.12
test no.1
test no.2
test no.3
0.08
0.04
0.00
-0.04
-0.08
-0.12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 7.15
Horizontal velocity distribution of thermal plume at 0.004m above 150oC
hot plate.
Some of horizontal velocity distributions at different levels of Test no.1 are shown in
Figure 7.16.
Horizontal Velocity (m/s)
0.12
0.08
0.04
0.00
20 mm.
40 mm.
60 mm.
80 mm.
100 mm.
200 mm.
300 mm.
-0.04
-0.08
-0.12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 7.16
Horizontal velocity distribution of thermal plume above 150oC hot plate.
158
Thermal Plume Experimental Results
7.1.4 Turbulence Intensity
In this section, the turbulence intensity of both vertical and horizontal velocity of
thermal plume from 150oC hot area source is presented.
The turbulence intensity, u ′rms , is the root mean square of the fluctuation, u ′ , defined as:
( )
u ′rms = u ′
2
12
( )
12
1 T

′ = w′ 2
=  ∫ u ′ 2 dt  or wrms
0
T

12
1 T

=  ∫ w′ 2 dt 
0
T

12
(4.26)
where u is velocity of x component and w is velocity of z component.
a) Vertical Turbulence Intensity
The data of vertical velocity turbulent intensity at different levels of thermal plume from
0.6
0.4
0.2
(m/s)
1.4
Vertical Turbulence Intensity
0.0
-0.15
1.2
-0.10
-0.05
0.00
0.05
0.10
0.15
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
-0.10
-0.05
0.00
0.05
a) at 0.02m
b) at 0.06m
0.8
0.6
0.4
0.2
-0.05
0.00
0.05
0.10
0.15
1.4
1.2
0.10
0.15
Exp1
Exp2
Exp3
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Radius (m)
c) at 0.10m
Figure 7.17
Exp1
Exp2
Exp3
Radius (m)
1.0
-0.10
1.2
Radius (m)
Exp1
Exp2
Exp3
0.0
-0.15
(m/s)
0.8
Vertical Turbulence Intensity
1.0
1.4
(m/s)
1.2
Exp1
Exp2
Exp3
Vertical Turbulence Intensity
(m/s)
1.4
Vertical Turbulence Intensity
150oC hot area source are shown in Figure 7.17 below.
d) at 0.20m
Vertical velocity turbulence intensity at different levels from 150oC hot
plate.
159
Thermal Plume Experimental Results
b) Horizontal turbulence intensity
Figure 7.18 shows the data of horizontal velocity turbulence intensity of thermal plume
1.4
Horizontal Turbulence Intensity (m/s)
Horizontal Turbulence Intensity (m/s)
from 150oC hot plate.
Exp1
Exp2
Exp3
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
1.4
Exp1
Exp2
Exp3
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
0.15
-0.10
-0.05
Radius (m)
1.4
Exp1
Exp2
Exp3
1.2
1.0
0.8
0.6
0.4
0.2
-0.10
-0.05
0.00
0.10
0.05
0.10
0.15
1.4
0.15
Exp1
Exp2
Exp3
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Radius (m)
c) at 0.10m
Figure 7.18
0.05
b) at 0.06m
Horizontal Turbulence Intensity (m/s)
Horizontal Turbulence Intensity (m/s)
a) at 0.02m
0.0
-0.15
0.00
Radius (m)
d) at 0.20m
Horizontal velocity turbulence intensity at different levels from 150oC
hot plate.
7.2
Experimental Results for 200oC Source
In order to determine the effect of source conditions on the thermal plume, the surface
temperature of the hot plate must be changed but because of limitation in temperature
measuring equipment available, which could not measure surface temperatures above
230oC. Therefore, the second test for thermal plume study was conducted from 200oC
hot plate. The same data collection setting and same levels of measurement were used
160
Thermal Plume Experimental Results
as in Section 7.1. The surface temperature distribution in these experiments is shown
below.
Diameter of hot
plate (188mm)
250
test4
test5
Temperature(oC)
200
test6
150
100
50
0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Radius(m)
Figure 7.19
7.2.1
Surface temperature distribution of 200oC hot plate.
Temperature Distribution
The data of temperature distribution in the thermal plume above the 200oC hot plate was
fitted with Equation 7.2. The results of temperature distribution at z = 0.10m are shown
in Figure 7.20.
161
Thermal Plume Experimental Results
35
Experiment
Eq. 7.2
30
∆T (oC)
25
20
15
10
5
0
-5
-0.15
Figure 7.20
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Temperature distribution at 0.10m above the floor above 200oC hot plate
compared with equation 7.2.
The temperature distributions at difference are shown below and also the value of ∆T
and also bT are shown in Figure 7.21 and Table 7.4.
45
0.02m
0.04m
0.06m
0.08m
0.10m
0.20m
0.30m
40
35
o
∆T ( C)
30
25
20
15
10
5
0
-5
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 7.21
Temperature distribution of thermal plume above the 200oC hot plate.
162
Thermal Plume Experimental Results
Value of ∆TC , bT and n of plume above the 200oC hot plate.
Table 7.4
Distance from the floor (m)
0.02
0.04
0.06
0.08
0.1
0.2
0.3
∆TC (K)
39.64
36.42
33.17
31.07
29.31
20.31
13.86
bT (m)
0.0644
0.0548
0.0508
0.0484
0.0457
0.0449
0.0530
n
1.9815
1.9048
2.0000
2.2337
2.0873
1.8122
1.7347
The values of ∆TC and bT are plotted and shown in Figure 7.22.
45
∆T
∆TC
bbT
T
40
0.07
0.06
35
∆TC (oC)
25
0.04
20
0.03
15
0.02
10
0.01
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.35
Position above hot plate (m)
Figure 7.22
7.2.2
∆TC and bT of thermal plume above 200oC hot plate.
Vertical Velocity Distribution
The vertical velocity distribution in this case is shown in Figure 7.23.
163
bT (m)
0.05
30
Thermal Plume Experimental Results
0.6
0.02m
0.04m
0.06m
0.08m
0.10m
0.20m
0.30m
0.5
0.4
w (m/s)
0.3
0.2
0.1
0.0
-0.1
-0.15
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distribution of thermal plume above 200oC hot plate
Figure 7.23
The value of wC and bV of thermal plume from ~200oC are shown below in Table 7.5.
Values of wC , bV and n of thermal plume above 200oC hot plate.
Table 7.5
Distance from the floor (m)
0.02
0.04
0.06
0.08
0.1
0.2
0.3
wC (m/s)
0.1508
0.2555
0.3196
0.3641
0.3930
0.4863
0.5213
bV (m)
0.0556
0.0622
0.0582
0.0559
0.0566
0.0555
0.0597
n
1.7606
2.4190
2.4210
2.3881
2.5837
2.4238
1.9450
164
0.6
0.070
0.5
0.065
0.4
0.060
0.3
0.055
0.2
0.050
0.1
0.0
0.00
Figure 7.24
wwc
C
bbv
V
0.05
0.10
0.15
0.20
0.25
Position above hot plate (m)
0.30
bv (m)
wC (m/s)
Thermal Plume Experimental Results
0.045
0.040
0.35
Graph of wC and bV of thermal plume above 200oC hot plate
The value of bT and bV were compared in Figure 7.25.
0.07
bbV
V
bbT
T
Plume width (m)
0.06
0.05
0.04
0.03
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Distance above the area source (m)
Figure 7.25
Comparison of bT and bV for thermal plume above 200oC hot plate.
165
Thermal Plume Experimental Results
7.2.3
Horizontal Velocity Distribution
The horizontal velocity distributions of this case are shown in Figures 7.26 and 7.27.
Figure 7.26 shows the horizontal velocity distribution at 0.004m level of three tests. The
values of horizontal velocity at different levels of Test no.4 are shown in Figure 7.27.
Horizontal velosity (m/s)
0.12
0.08
0.04
0.00
-0.04
test no.4
test no.5
test no.6
-0.08
-0.12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 7.26
Horizontal velocity distribution of thermal plume from 200oC hot plate at
HorizontalVelocity
Velocity
(m/s)
Horizontal
(m/s)
z = 0.004m.
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
-0.15
Figure 7.27
0.02m
0.04m
0.06m
0.08m
0.10m
0.20m
0.30m
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Horizontal velocity distribution of thermal plume above 200oC hot plate.
166
Thermal Plume Experimental Results
7.2.4 Turbulence Intensity
The distributions of velocity turbulence intensity of thermal plume from 200oC hot area
source are shown here separating to two sub-sections that are vertical turbulence
intensity and horizontal vertical turbulence intensity.
a) Vertical Turbulence Intensity
The distributions of vertical velocity turbulence intensity at different levels of thermal
plume from 200oC hot area source are shown in Figure 7.28 below. In Figure 7.28b,
only two experimental data are shown because the data of turbulence intensity at this
1.2
Exp1
Exp2
Exp3
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
Vertical Turbulence Intensity (m/s)
(m/s)
1.4
Vertical Turbulence Intensity
level was not collected in experimental Test no. 4.
1.4
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
0.15
Exp2
Exp3
1.2
-0.10
-0.05
1.0
0.8
0.6
0.4
0.2
-0.10
-0.05
0.00
0.05
0.10
(m/s)
Exp1
Exp2
Exp3
0.0
-0.15
1.4
1.2
0.15
0.8
0.6
0.4
0.2
0.0
-0.15
0.15
Exp1
Exp2
Exp3
1.0
Radius (m)
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
c) at 0.10m
Figure 7.28
0.10
b) at 0.06m
Vertical Turbulence Intensity
(m/s)
Vertical Turbulence Intensity
a) at 0.02m
1.2
0.05
Radius (m)
Radius (m)
1.4
0.00
d) at 0.20m
Vertical velocity turbulence intensity at different levels from 200oC hot
plate.
167
Thermal Plume Experimental Results
b) Horizontal Turbulence Intensity
The horizontal velocity turbulence intensity distributions of thermal plume from 200oC
1.4
Horizontal Turbulence Intensity (m/s)
Horizontal Turbulence Intensity (m/s)
hot area source are presented below.
Exp1
Exp2
Exp3
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
1.4
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
0.15
Exp2
Exp3
1.2
-0.10
-0.05
1.4
Exp1
Exp2
Exp3
1.2
1.0
0.8
0.6
0.4
0.2
-0.10
-0.05
0.00
0.10
0.05
0.10
1.4
0.15
1.0
0.8
0.6
0.4
0.2
0.0
-0.15
0.15
Exp1
Exp2
Exp3
1.2
Radius (m)
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
c) at 0.10m
Figure 7.29
0.05
b) at 0.06m
Horizontal Turbulence Intensity (m/s)
Horizontal Turbulence Intensity (m/s)
a) at 0.02m
0.0
-0.15
0.00
Radius (m)
Radius (m)
d) at 0.20m
Horizontal velocity turbulence intensity at different levels from 200oC
hot plate.
Because the values of temperature were set to 150oC and 200oC due to want to find out
the effect of the source strength on the thermal plume but from the results of two source
conditions, it shows small effect on flow pattern but magnitude of centreline vertical
velocity and temperature. Because the limitation of ability of instrument that can not
increase the source temperature greater than 200oC therefore, in this present work the
effect of the source strength on the thermal plume can not be done.
168
Chapter 8
Thermal Plume Numerical Models and
Results
Thermal Plume Numerical Models and Results
Chapter 8
Thermal Plume Numerical Models and Results
In the case of the numerical simulation of the thermal plume, although the flow of main
interest in this study is confined by the glass enclosure, the computational domain must
extend beyond the boundaries of the enclosure into the ambient at atmospheric pressure.
In this way, heat transfer through the glass walls, floor, etc., may be accounted for. This
allowed a full 3D, steady-state computation. The properties of glass and floor material
in the simulation model were set to be same value as the materials used in the
experiment.
All buoyancy-driven plume flows are characterized by periodic entrainment of the
ambient fluid into the plume (Bejan, 1984). However, in the present case, it is assumed
that after an initial period of time during which the plume develops inside the enclosure,
the flow reaches a quasi-steady state, provided the heat transfer across the walls of the
confining glass enclosure is accounted for. Simulation of such conjugate heat transfer
situations is possible in PHOENICS. This allowed a steady-state calculation to be
performed.
Experiments also validated the quasi-steady flow assumption: the experimental results
were found to be repeatable to within small errors (see chapter 7).
In the experiments, two test source temperatures were considered: 150ºC and 200º C.
From the ideal gas law, the fractional changes in density of air at different temperatures
and 1 atm pressure is (∆ρ ρ )150o C = 0.307 or 30% and (∆ρ ρ )200o C = 0.381 or 38.1%
compared to the density at 20oC.
Immediately above the heat source, the air in contact with it is at the same temperature
as the source itself, so that the fractional density changes calculated above are the
maximum possible in the flow field. This shows that for both the test conditions, (∆ρ/ρ)
170
Thermal Plume Numerical Models and Results
exceeds 0.1 which is considered to be the limit of applicability for the Boussinesq
approximation (e.g. Malin, 2003). Hence, in this case, the Boussinesq approximation is
not suitable. The more realistic ‘density difference’ model for the gravitational body
force was used in the simulations. The reference density was chosen as the density of air
depending on temperature of air surrounding the experimental enclosure.
The standard k-ε turbulence model with a buoyancy correction was used in this thermal
plume simulation. In order to improve the results of the simulation work, adjustments
were made to values of constants such as the turbulent Prandtl numbers for velocities
and temperature and Cε 3 in the standard k-ε model. Determination of these constants for
the present flow situation was one of the main objectives of this work.
8.1
Effect of Grid Refinement
Grid refinement is an important process required to find out the optimum number of
grids points needed for an accurate simulation. In this process, the simulation model of
thermal plume from 200oC hot plate (see figure 8.1) with difference five models of grids
in x,y and z were used. The value of number of cells in x-direction (nx), y-direction (ny)
and z-direction (nz) were set. All models used in this work are shown in Table 8.1
below.
Table 8.1
Meshes used in grid refinement studies.
Model No.
nx
ny
nz
Total cells
I
19
19
31
11191
II
45
45
61
123525
III
69
69
91
433251
IV
95
95
150
1353750
V
165
165
200
5445000
171
Thermal Plume Numerical Models and Results
Computational
domain boundary
Glass enclosure
Insulation Base
Figure 8.1
Heat Source
Computation domain for thermal plume simulation.
The effect of number of cells in x and z direction on the result of simulation is shown.
The comparison of velocity and temperature at r = 0.00m and z = 0.10m above the
0.50
0.50
0.45
0.45
Velocity (m/s)
Velocity (m/s)
heated plate are shown in Figure 8.2 and 8.3, respectively.
0.40
0.35
0.40
0.35
0.30
0.30
0.25
0.25
0
25
50
75
100
125
150
175
0
25
50
75
Number of cell in x direction
a)
Figure 8.2
100
125
150
175
200
Number of cell in z direction
b)
The comparison of vertical velocity at middle of plume from different
number of cell (at z = 0.10m above hot plate).
172
225
55
55
54
54
53
53
Temperature ( C)
52
o
o
Temperature ( C)
Thermal Plume Numerical Models and Results
51
50
49
52
51
50
49
48
48
47
47
46
46
0
25
50
75
100
125
150
175
0
25
50
75
125
150
175
200
Number of cell in z direction
Number of cell in x direction
a)
Figure 8.3
100
b)
The comparison of temperature at middle of plume from different
number of cell (at z = 0.10m above hot plate).
The centreline vertical velocity at z = 0.10m increases with number of cells in both x
and z directions and is stabilized after the number of cells increase to more than about
50 in the x direction and about 70 in the z direction. Similarly, the value of temperature
at the same position increases with number of cells and reach to stabilizes when the
number of cells equals about 50 in the x direction and 70 in the z direction.
From the above graph, it is quite clear that a suitable number of cells for the thermal
plume simulation is more than 50 in the x direction and more than 70 in the z direction
Because of small effect on numerical results, about 4.5% on velocity and about 1% on
temperature when comparing between model III and model V, the model III was finally
chosen for the thermal plume simulation.
8.2
The Effect of Turbulence Modelling Parameter, Cε 3 on Numerical Results of
Thermal Plume
In this section, the effect of turbulence modelling parameter, Cε 3 on numerical results of
thermal plume simulation is investigated. The value of Cε 3 was varied from 0.1 to 0.9.
A comparison of vertical velocity distribution at 0.10m above the floor of each setting
value of Cε 3 is shown below.
173
225
Thermal Plume Numerical Models and Results
0.5
0.1
0.3
0.5
0.7
0.9
test no.4
test no.5
test no.6
Vertical velocity (m/s)
0.4
0.3
0.2
0.1
0.0
-0.1
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 8.4
Comparison of vertical velocity distribution at 0.1m above the floor
when the value of Cε 3 is changed.
The graph shows that the value of Cε 3 has no significant effect on thermal plume
simulation results of this numerical geometry model. It is recommended by various
authors that Cε 3 should be close to zero for stably-stratified flow (dense below light),
and close to unity for unstably-stratified flows (light below dense) (e.g. CHAM, 2003).
In a buoyant plume in a confined region, the overall recirculating flow assumes the form
of a large Bernard cell. Horizontal (z) slabs of the flow field do not show any clear-cut
stable or unstable stratification. This may be why the value of Cε 3 has no effect.
One reason for the sensitivity of Cε 3 is that it is only significant when the turbulence
production terms due to buoyancy are significantly compared to shear production (see
Equations 4.34 to 4.37). The present author investigated this issue and found that the
value of the buoyant volumetric production rate (defined as GENB in PHOENICS) was
low (approximately 10 times lower) compared with the shear volumetric production rate
(GENK in PHOENICS) for the thermal plume. However, these terms were of a similar
174
Thermal Plume Numerical Models and Results
magnitude in some region of the saline plume case. For example, a spot value taken in
the boundary layer of the saline plume gave GENB ~ 5.66×10-5 m2/s and GENK ~
8.54×10-5 m2/s and in the thermal situation GENB ~ 3.72×10-4 m2/s and GENK ~
3.27×10-3 m2/s. Contour plot and spot values of GENB and GENK of both thermal and
saline plumes are shown in Figure 8.5 and 8.6, respectively.
Figure 8.5
Contour plots of GENB and GENK of the results of the thermal plume
simulation. (Temperature of hot plate = 200oC, Probe located at z = 0.01 m, r = 0.00 m).
175
Thermal Plume Numerical Models and Results
Figure 8.6 Contour plots of GENB and GENK of the results of the saline plume simulation (Test
no. 5). (Probe located at z = 0.01 m, r = 0.00 m).
8.3
Numerical Results for 150oC Heated Source
The copper plate (hot plate) was set to 152.4oC, which is the same as the average
experimental value of the temperature surface of the hot copper plate averaging from
three experiments. The average ambient temperature in the simulation is 26.1oC, which
came from averaging the three average ambient data from three experiments. Therefore,
the value of the Rayleigh number (Ra) is 3.08×107. The values of properties use to
calculate the value of Ra is the properties at average temperature.
The computational domain is shown in Figure 8.7. The mesh contains 69 x-cells, 69 ycells, 91 z-cells, non-uniformly distributed to allow dense cell population inside the
enclosure, in the plume and near the solid surfaces.
176
Thermal Plume Numerical Models and Results
Figure 8.7
The computational domain and 69×69×91 computational mesh for
thermal plume simulation.
The experimental data were used to validate the results of the numerical study. The
main modification to the standard k-ε turbulent model in this study was to the values of
the turbulent Prandtl numbers of the three velocity components and for temperature.
The best value of turbulent Prandtl number for velocities and temperature was found by
trial and error to be 0.65. Comparison of results from the experimental and numerical
studies is shown in Chapter 9. The results for the velocity and temperature fields are
shown below.
a) Velocity distribution
Figure 8.8
b) Temperature distribution
Velocity vectors and temperature contour of thermal plume from ~150oC
hot plate.
177
Thermal Plume Numerical Models and Results
The numerical result of temperature in the hot surface and insulation around the hot
plate compared with experimental data is shown in Figure 8.9.
200
test1
test2
test3
Simulation
Temperature(oC)
150
100
50
0
-0.2
-0.15
Figure 8.9
-0.1
-0.05
0
0.05
Radius(m)
0.1
0.15
0.2
Numerical surface temperature distributions in 150oC hot plate
comparing with experimental data.
In order to determine the plume width from the numerical results, the vertical velocity
and temperature distribution data from the numerical study at different levels were fitted
with Equations 7.2 and 7.3 below.
∆T = ∆TC e − (r bT )
(7.2)
w = wC e − (r bV )
(7.3)
n
n
8.3.1
Vertical Velocity Distribution
The vertical velocities at different levels determined from the numerical study are
shown in Figure 8.10.
178
Thermal Plume Numerical Models and Results
The result from numerical study was fitted with Equation 7.3 for the vertical velocity in
order to find out wC and bV. Figure 8.11 shows the simulated vertical velocity profile at
100mm, compared with the best fit of Equation 7.3.
0.6
10mm
50mm
100mm
150mm
200mm
250mm
300mm
Vertical
Velocity
(m/s)
Velocity
(m/s)
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.15
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distributions in thermal plume from ~150oC hot plate.
Figure 8.10
0.40
Simulation
Eq 7.3
0.35
Velocity (m/s)
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.15
Figure 8.11
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distribution of thermal plume from ~150oC hot plate at
100 mm above hot plate compared with equation 7.3, n = 1.55.
179
Thermal Plume Numerical Models and Results
The values of wC and bV at different levels are shown in Table 8.2 below.
Values of wC , bV and n of thermal plume from ~150oC hot plate at
Table 8.2
different levels.
Distance from the floor (m)
0.01
0.02
0.05
0.10
0.15
0.20
0.25
0.3
wC
0.105
0.155
0.275
0.377
0.443
0.491
0.520
0.552
bV
0.039
0.044
0.053
0.055
0.056
0.057
0.058
0.060
n
1.747
1.786
1.543
1.553
1.582
1.576
1.536
1.464
8.3.2
Temperature Distribution
The numerical results of temperature distribution from ~150oC hot plate at different
levels are shown below.
65
10mm
50mm
100mm
150mm
200mm
250mm
300mm
60
Temperature (oC)
55
50
45
40
35
30
25
20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 8.12
Temperature distribution of thermal plume from ~150oC hot plate.
As for the vertical velocity distribution, the temperature distribution data at each level
were fitted with Equation 7.2 for temperature.
180
Thermal Plume Numerical Models and Results
The value of ∆TC , bT and n are shown in Table 8.3.
Values of ∆TC , bT and n of thermal plume from ~150oC hot plate at
Table 8.3
difference levels.
Distance from the floor (m)
0.01
0.02
0.05
0.10
0.15
0.20
0.25
0.30
∆TC
35.19
31.83
24.52
18.66
14.81
12.12
10.60
9.00
bT
0.0722
0.0643
0.0544
0.0515
0.0521
0.0545
0.0569
0.0603
n
1.5469
1.5359
1.6097
1.6708
1.6894
1.6563
1.6241
1.5908
The value of bT and bV are compared in Figure 8.13.
0.075
bTbT
bV
bV
0.070
Plume width (m)
0.065
0.060
0.055
0.050
0.045
0.040
0.035
0.00
Figure 8.13
0.05
0.10
0.15
0.20
0.25
Position above the floor (m)
0.30
0.35
Comparison of bT and bV deduced from CFD simulations of thermal
plume from ~150oC source.
181
Thermal Plume Numerical Models and Results
8.3.3
Horizontal Velocity Distribution
The horizontal velocity distributions at different levels are shown in Figure 8.14.
0.15
10 mm
50 mm
100 mm
150 mm
200 mm
250 mm
300 mm
Horizontal
Velocity
Velocity
(m/s)(m/s)
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
Figure 8.14
8.4
-0.15
-0.10
-0.05
0.00
0.05
Radius (m)
0.10
0.15
0.20
Horizontal velocity distribution of thermal plume from ~150oC hot plate.
Numerical Result for 200oC Heated Source
In this case, the same computational model as used previously for the 150oC hot plate
was used. The source temperature was set to 200.2oC, same as the average
source/copper plate temperature in the experiments. The ambient temperature in this
simulation is 24.5oC. Therefore, the value of Rayleigh number (Ra) is 3.97×107.
The numerical result of temperature in the hot plate and insulation around it is shown in
Figure 8.15.
182
Thermal Plume Numerical Models and Results
250
test4
test5
test6
Temperature(oC)
200
Simulation
150
100
50
0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Radius(m)
Figure 8.15
Numerical surface temperature distributions in 200oC hot plate
comparing with experimental data.
8.4.1
Vertical Velocity Distribution
Figure 8.16 shows the vertical velocity distribution of thermal plume from ~200oC
source. The values of wC , bV and n are shown in Table 8.4 below.
Values of wC , bV and n of thermal plume from ~200oC hot plate at
Table 8.4
different levels.
Distance from the floor (m)
0.01
0.02
0.05
0.10
0.15
0.20
0.25
0.30
wC
0.1289
0.1870
0.3248
0.4424
0.5195
0.5756
0.6092
0.6467
bV
0.0384
0.0424
0.0503
0.0525
0.0532
0.0542
0.0554
0.0577
n
1.6417
1.7090
1.5394
1.5583
1.5925
1.5850
1.5425
1.4699
183
Thermal Plume Numerical Models and Results
0.7
10mm
50mm
100mm
150mm
200mm
250mm
300mm
Vertical velocity (m/s)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.15
Figure 8.16
8.4.2
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Vertical velocity distribution in thermal plume from ~200oC hot plate
Temperature Distribution
Temperature distributions at different levels and the values of ∆TC and bT are shown in
Figure 8.17 and Table 8.5, respectively.
80
10mm
50mm
100mm
150mm
200mm
250mm
300mm
Temperature (oC)
70
60
50
40
30
20
-0.15
Figure 8.17
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Temperature distribution of thermal plume from ~200oC hot plate.
184
Thermal Plume Numerical Models and Results
Values of ∆TC , bT and n of thermal plume from ~200oC hot plate at
Table 8.5
different levels.
Distance from the floor (m)
0.01
0.02
0.05
0.10
0.15
0.20
0.25
0.30
∆TC
48.12
43.48
33.41
25.54
20.38
16.75
14.67
12.49
bT
0.0692
0.0609
0.0509
0.0481
0.0490
0.0514
0.0536
0.0566
n
1.4953
1.4989
1.6016
1.6709
1.6704
1.6315
1.6058
1.5831
The values of bT and bV are compared in Figure 8.18.
0.065
bT
bV
Plume width (m)
0.060
0.055
0.050
0.045
0.040
0
Figure 8.18
0.05
0.1
0.15
0.2
0.25
Position above hot plate (m)
0.3
0.35
Comparison of bT and bV deduced from numerical simulations of thermal
plume from ~200oC source.
8.4.3
Horizontal Velocity Distribution
The horizontal velocities at different levels in this case are shown in Figure 8.19.
185
Thermal Plume Numerical Models and Results
Horizontal
Velocity
Velocity
(m/s) (m/s)
0.15
10 mm
50 mm
100 mm
150 mm
200 mm
250 mm
300 mm
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Radius (m)
Figure 8.19
8.5
Horizontal velocity distribution of thermal plume from ~200oC hot plate.
Numerical Result for 200oC Heated Source without Enclosure
In the numerical study of thermal plume in an enclosure, the enclosure has an effect on
plume characteristics such as the velocity and temperature distributions inside and
outside the plume.
For this reasons, results from numerical model of thermal plume within an enclosure
cannot be compared with previously proposed theoretical models. Therefore, a separate
numerical simulation of the thermal plume without enclosure was carried out. In this
case, same mesh of thermal plume with cover of model III (69×69×91cells) and settings
were used, but the cover was removed. The geometry and grids for this model are
shown below.
186
Thermal Plume Numerical Models and Results
Hot plate
Figure 8.20
Geometry and 69×69×91 grids of thermal plume without an enclosure.
The results of this simulation are shown below.
Figure 8.21
Vertical velocity vectors and temperature contour of thermal plume from
~200oC hot plate without an enclosure.
187
Thermal Plume Numerical Models and Results
8.5.1
Vertical Velocity Distribution
Vertical velocity distributions at different levels are shown in Figure 8.22 below.
1.0
10mm
50mm
100mm
150mm
200mm
250mm
300mm
Vertical
VerticalVelocity
velocity(m/s)
(m/s)
0.8
0.6
0.4
0.2
0.0
-0.2
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 8.22
Plume vertical velocity distributions from ~200oC hot plate without an
enclosure.
The value of wC , bV , and n that resulted from best fit of numerical results of vertical
velocity with Equation 7.3 are shown in Table 8.6.
Table 8.6
Values of wC , bV and n of thermal plume from ~200oC hot plate without
an enclosure.
Distance from
0.012
0.025
0.051
0.105
0.157
0.215
0.247
0.314
wC (m/s)
0.087
0.228
0.413
0.616
0.726
0.812
0.848
0.913
bV (m)
0.065
0.055
0.049
0.046
0.046
0.046
0.046
0.048
n
2.535
1.904
1.752
1.643
1.559
1.485
1.452
1.395
the floor (m)
8.5.2
Temperature Distribution
Figure 8.23 and Table 8.7 show the results of temperature distribution, ∆TC , bT and n.
188
Thermal Plume Numerical Models and Results
140
10mm
50mm
100mm
150mm
200mm
250mm
300mm
Temperature (oC)
120
100
80
60
40
20
-0.15
Figure 8.23
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Temperature distributions in thermal plume from ~200oC hot plate
without an enclosure.
In Figure 8.23, the temperature distribution of thermal plume at all levels show the same
pattern except at a height of 10mm that shows a kink in the radial temperature profile.
The possible reason is because of the temperature distribution between the hot copper
plate and the insulation around it. The temperature at the hot copper plate is hotter than
the temperature of insulation around it then the suddenly change in temperature
distribution on flat surface has the effect on the temperature distribution especially, at
the level near an area source.
Table 8.7
Values of ∆TC , bT and n of thermal plume from ~200oC hot plate
without an enclosure.
Distance from
0.012
0.025
0.051
0.105
0.157
0.215
0.247
0.314
∆TC (K)
93.87
81.97
67.21
50.73
41.30
34.13
31.20
26.38
bT (m)
0.072
0.058
0.047
0.041
0.042
0.044
0.045
0.048
n
1.701
1.568
1.594
1.649
1.632
1.605
1.594
1.578
the floor (m)
189
Thermal Plume Numerical Models and Results
The comparison of bT and bV for this case are shown in Figure 8.24. In Figure 8.25,
horizontal velocities at different levels are shown.
0.08
bT
bV
Plume width (m)
0.07
0.06
0.05
0.04
0.03
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Position above hot plate (m)
Comparison of bT and bV of thermal plume from ~200oC hot plate
Figure 8.24
without an enclosure.
8.5.3
Horizontal Velocity Distribution
0.25
10 mm
50 mm
100 mm
150 mm
200 mm
250 mm
300 mm
Velocity (m/s)
Horizontal Velocity (m/s)
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
Radius (m)
Figure 8.25
Horizontal velocity distribution of thermal plume from ~200oC without
an enclosure.
190
Chapter 9
Discussion
Discussion
Chapter 9
Discussion
In this chapter, the results of experimental and numerical investigations of both the
saline plume and the thermal plume are discussed and compared. Some of these results
are also compared with past work. This chapter is separated into two sections,
discussion on saline plume and discussion on thermal plumes respectively.
9.1
Saline Plume
In this section, the discussion on the results of experimental and numerical works for
saline plume are presented. In section 9.1.1, the results from experimental works using
shadowgraph technique and the PTV technique assisting by Digimage software are
discussed and compared. Section 9.1.2 shows the discussion on comparison between
experimental and numerical results of saline plume study.
9.1.1
Experimental Results of Saline Plume
9.1.1.1 The Effect of Enclosure on Vertical Velocity Distribution
The presence of the enclosure has a significant effect on the vertical velocity field at all
levels. At the levels near the area source for example, as seen in Figures 9.1 and 9.2, the
vertical velocity profile 0.02m below the area shows some positive velocity values that
would not occur had the plume been unconfined.
In the curve fitting process, these positive velocity values were not used. This
phenomenon is not seen to the same extent at levels are farther from the area source, for
example 0.1m below an area source (see Figure 9.3b). Figure 9.1 also shows a strong
upward flow of the ambient in the lower regions of the tank, as expected.
192
Discussion
0.02m level
Figure 9.1
Averaged velocity field in saline plume (test no.5, 30s averaging time
centred on 60s after start of experiment.
Vertical Velocity (m/s)
0.005
-0.10
-0.05
0.000
0.00
-0.005
-0.010
-0.015
0.05
0.10
Exp.
Gaussian Eq.
Equation 5.2
The vertical velocity profile due
to confining effect of the
-0.020
-0.025
Radius (m)
Figure 9.2
Averaged vertical velocity distribution of saline plume (Test no.5 at
0.02m below the area source (over 60s with 30s averaging time).
193
Discussion
9.1.1.2 The Modified Gaussian Equation vs. the Standard Gaussian Equation for
Near-Field Flow
One of the main aims of the present study was to determine the vertical velocity and
density (or temperature) distributions in the plume near the area source and hence to
propose a suitable equation that can be used as a best fit to these profiles.
In this study, new equations for the vertical velocity profile and density distribution for
the near-field region are presented. These are discussed in the following paragraphs.
Figure 9.3 shows a comparison between experimental data and the two curve-fit profiles
for saline plume experiments.
Standard Deviation from the two equations is compared in Table 9.1 below.
0.005
-0.05
-0.0050.00
0.05
-0.015
0.10
Exp.
Gaussian Eq.
Equation 5.2
-0.025
-0.035
-0.005
Vertical velocity (m/s)
Vertical Velocity (m/s)
-0.10
0.005
-0.015
Exp
Gaussian eq.
Equation 5.2
-0.025
-0.035
-0.045
-0.045
-0.055
-0.10
-0.055
Radius (m)
0.00
0.05
Radius (m)
(a) at 0.02m below
Figure 9.3
-0.05
(b) at 0.10m below
Experimental and curve-fit velocity profiles of saline plume.
From Figure 9.3 above, it is seen that the modified Gaussian equation showed better
agreement than the Gaussian equation, especially at levels near the area source (see
Figure 9.3a). The modified Gaussian equation does not show significant difference
compared with the Gaussian equation when used to fit with the experimental data of
saline plume vertical velocity at levels away from the area source.
194
0.10
Discussion
Table 9.1
z (m)
Standard Deviations of two equations after fitting with experimental data.
0.011
0.020
0.039
0.062
0.081
0.100
0.118
0.151
0.202
1.48×10-4
1.42×10-4
6.29×10-5
4.43×10-5
5.14×10-5
1.32×10-4
4.50×10-5
3.80×10-5
5.82×10-5
9.67×10-5
7.69×10-5
6.06×10-5
3.23×10-5
4.49×10-5
1.09×10-4
2.94×10-5
3.79×10-5
4.84×10-5
SD of the
Gaussian
eq.
SD of
equation
5.2
The standard deviation of the Gaussian equation is 1.42×10-4, compared to 7.69×10-5 for
the modified Gaussian equation for experimental data 0.02m below the area source, and
1.32×10-4 and 1.09×10-4 respectively for data 0.10m below the area source. This shows
clearly that at levels near the area source the modified Gaussian equation gave a closer
match to the experimental results.
Figure 9.4 shows the reason for the above observation. It is seen that the value of shape
index in Equation 5.2 is in the range 1.0 < n < 2.0 at distances away from the area
source.
7.0
test no.5
test no.6
test no.7
test no.8
6.0
5.0
n
4.0
3.0
2.0
1.0
0.0
0.00
0.05
0.10
0.15
0.20
z position (m)
Figure 9.4
Value of the shape index, n, in Equation 5.2 for vertical velocity profile
as a function of distance from the area source for saline plume.
195
Discussion
Vertical velocity radius, bV, (m)
Vertical velocity plume width (m)
0.024
0.022
0.020
0.018
0.016
0.014
0.012
0.010
0.00
Gaussian
Modified Gaussian
0.05
0.10
0.15
0.20
0.25
Distance from an area source (m)
Figure 9.5
Vertical velocity plume radius, bV, deduced from the two equations.
(Test no. 5).
Figure 9.5 shows a comparison between vertical velocity based plume radius, bV , from
the Gaussian equation and the modified Gaussian equation. It is seen that the values of
bV are not significantly different especially at positions far from the area source.
9.1.1.3 Comparison of Plume Radius Measured by Shadowgraph and PTV
Technique
In this section, the plume radius found experimentally using two methods for the saline
plume at the same source conditions are compared below.
196
Discussion
0
0
-20
-20
-40
ShadowGraph Exp
PTV Exp
-60
z position (mm)
z position (mm)
-40
-80
-100
-120
-140
-80
-100
-120
-140
-160
-160
-180
-180
-200
ShadowGraph Exp
PTV Exp
-60
-200
0
20
40
60
80
0
Radius (mm)
20
40
Radius (mm)
60
80
a) B0 = 1.61×10-4 m2/s3, M0 = 0.44×10-9 m4/s2
b) B0 = 10.50×10-4 m2/s3, M0 = 4.46×10-9 m4/s2
(Test no. 1 compared with test no. 5)
(Test no. 4 compared with test no. 8)
Figure 9.6 Comparison of the plume radius estimated by two methods.
(D0 = 0.105m)
The qualitative shape of the plume radius versus height is the same for both the
Shadowgraph and PTV technique. However, from Figure 9.6, it can be seen that from
the Shadowgraph technique is greater by approximately 40% to 50%. This means that
there will be a significant difference in the location of the virtual origin deduced from
the two methods, as discussed.
9.1.1.4 Comparison of Dneck and zneck
It is important to determine the effect, if any, of the source conditions on the plume
neck location and diameter.
Colomer, et al. (1999) stated that the values of Dneck and zneck depend on only the
diameter of the area source such that Equation 2.53 and Equation 2.54. In Figure 9.7
and 9.8, the experimental values of Dneck and zneck in the saline plume for different
source conditions are compared with the results of Colomer, et al. (1999). The
parameter that is used to define the strength of the source is the source parameter, Г,
recommended by Morton (1959a, b) and Hunt and Kaye (2001). The source parameter,
Г, is defined as:
197
Discussion
Γ=
5Q02 F0
4αM 05 2
(2.67)
where Q0 is the volume flux at source,
F0 is the buoyancy flux at source,
M0 is the momentum flux at source and
Α is the entrainment coefficient.
An important issue in this part of work was the values of zneck and Dneck of plume at
different source conditions. The experimental results of zneck and Dneck of plume at
different source conditions compared with work of Colomer, 1999, are summarised in
Table 9.2.
Table 9.2
Values of Dneck and zneck compared with calculated value of Colomer, et
al. (1999).
zneck/Dsource
zneck/Dsource
(Colomer)
(experiment)
(Colomer)
(m)
(m)
(m)
(m)
1.69
0.44
0.55
0.61
0.28
27.52
0.15
0.46
0.55
0.64
0.28
2.82
10.32
4.99
0.35
0.55
0.59
0.28
4
10.50
38.59
0.36
0.50
0.55
0.54
0.28
5
1.60
8.02
1.73
0.24
0.55
0.67
0.28
6
5.64
28.30
0.14
0.22
0.55
0.54
0.28
7
2.94
10.61
4.73
0.21
0.55
0.69
0.28
8
10.92
39.43
0.34
0.20
0.55
0.43
0.28
Bo× 10-4
Ra*
Γ
(m2/s3)
×1012
×107
1
1.61
8.20
2
5.46
3
Test no.
Dneck/Dsource
Dneck/Dsource
(experiment)
198
Discussion
0.60
D neck/D source
0.50
0.40
0.30
0.20
Equation 2.53
0.10
Plume shape experiment
Plume velocity experiment
0.00
0
1
2
3
7
Source parameter,
Γ x 10 Г × 10
Figure 9.7
4
5
-7
Comparison of Dneck from saline plume experimental data compared with
results of Colomer, et al (1999).
0.80
0.70
z neck /D source
0.60
0.50
0.40
0.30
0.20
Equation 2.54
Plume shape experiment
Plume velocity experiment
0.10
0.00
0
1
2
3
4
5
Γ x 107 Г × 10-7
Source parameter,
Figure 9.8
Comparison of zneck from experimental data compared with results of
Colomer, et al (1999).
199
Discussion
Expect visual estimation of plume width (from Test no. 1 to 4) is always grater than
plume width by vertical velocity profile (from Test no. 5 to 8).
From present experimental data available, it appears that the values of Dneck and zneck do
not depend significantly on the source parameter, Г, However, it is clear that the values
of Dneck from the present experimental results are lower than those suggested by
Colomer, et al., 1999 (Equation 2.53). On the other hand, the values of zneck from the
present experimental results are higher than those suggested by Colomer, et al., 1999
(Equation 2.54).
9.1.1.5 Comparison of the Virtual Origin Location, zv
Similarly, in some past researches, such as the “Conical Theory” suggested by Morton
(1956a), ACGIH (2001), Goodfellow and Tähti (2001) it is stated that the virtual origin
location, zv, depends only the diameter of the area source. But Morton (1971a) and Hunt
and Kaye (2001) suggested that it depends on the source strength also. Morton (1971a)
and Hunt and Kaye (2001) suggested a parameter that can be used to define the source
strength. They called it “the source parameter, Г ” given by Equation 2.69 in the present
work.
240
200
180
200
80
40
0
-40
-80
ShadowGraph Exp
PTV Exp
165
160
120 115
120
z position (mm)
z position (mm)
160
240
ShadowGraph Exp
PTV Exp
95
80
40
0
-40
-80
-120
-120
-160
-160
-200
-200
0
20
40
60
80
0
Radius (mm)
a) B0 = 1.61×10-4 m2/s3, M0 = 0.44×10-9 m4/s2
20
40
Radius (mm)
60
80
b) B0 = 10.50×10-4 m2/s3, M0 = 4.46×10-9 m4/s2
(Test no. 1 compared with test no. 5)
(Test no. 4 compared with test no. 8)
Figure 9.9 Estimation of the virtual origin location from two experimental methods.
(D0 = 0.105m)
200
Discussion
The location of zv from experimental data was estimated using the plume radius
predicted from the PTV experiments and the Shadowgarph experiments are shown in
Figure 9.9.
Table 9.3 and Figure 9.10 show the variation of zv with Г as proposed by previous
researchers, and compared with the present experiments. Note that for the same Г, by
the conical theory (Equation 2.65), in ACGIH (Equation 2.81), by Goodfellow and
Tähti (Equation 2.82) and by the Shadowgraph experiments are nearly the same.
The zv values deduced from the PTV experiments are lower than those calculated by the
conical theory, in ACGIH, by Goodfellow and Tähti and the Shadowgraph. The PTV
experimental results give zv much higher than that proposed by Hunt and Kaye, for the
same value of Г. Nevertheless, the virtual origin location from the PTV experiments as
shows a slight decrease with increasing as predicted by Hunt and Kaye’s equation
(Equation 2.80) but inadequate experimental data makes it difficult to determine the
form of the function.
Table 9.3
The value of virtual origin location, zv, from different methods.
zv (Conical)
Test
Ra*
Γ
no.
×1012
×107
zv
zv
zv + zavs
(ShadowGraph)
(PTV)
(G.Hunt,2001)
(m)
(m)
(m)
and
zv
(H. Goodfellow,
E. Tähti 2001)
zv
(ACGIH,2001)
(m)
(m)
1
8.20
1.69
0.180
0.115
0.046
0.179
0.169
2
27.52
0.15
0.185
0.130
0.075
0.179
0.169
3
10.32
4.99
0.155
0.090
0.037
0.179
0.169
4
38.59
0.36
0.165
0.095
0.063
0.179
0.169
201
Discussion
0.20
zv (m)
0.16
0.12
0.08
0.04
0.00
0
15
30
-6
Γ x10
Source parameter,
Г × 10
45
-6
ShadowGraph Exp
PTV Exp
Hunt and Kaye, 2001
Conical
ACGIH, 2001
Goodfellow, 2001
Figure 9.10
Virtual origin location from different methods compared with
experimental results for saline plumes.
An explanation of why the virtual origin location of Hunt and Kaye (2001) is very low
compared to present experimental result is because of the assumption of self-similarity
in the plume. Hunt and Kaye (2001) derived their equation based on the assumption of
self-similarity but in the near-field, the plume does not reach a self-similar state.
In the present experimental work, the virtual origin location of the plume was deduced
by extrapolating the plume radius back to a point where b = 0. However, it should be
note that even in the present experiments the plume may not have been fully developed
and the influence of the enclosure should not be ignored.
A powerful method to determine the virtual origin location of the plume is using a direct
measurement of plume volume flux, proposed by Baines (1983) but this was not
suitable for present work because it would have required an excessively large amount of
water for the present experimental setup.
202
Discussion
9.1.1.6 Plume Centreline Vertical Velocity
Colomer, et al. (1999) stated that the plume from an area source can be divided into two
regions. They also proposed that the value of centreline vertical velocity, wC , is a
function of the diameter of the area source, the buoyancy flux per unit area and the
distance from the area source.
The centreline vertical velocity calculated from Colomer, et al. (1999) (Equation 2.56
and 2.58) and the experimental data of centreline vertical velocity of are compared in
Figure 9.11.
2.5
wc/(B0D)1/3
2.0
1.5
1.0
0.5
Colomer's Region I
Colomer's Region II
0.0
0.0
Figure 9.11
0.5
1.0
z /D
1.5
2.0
Dimensionless centreline vertical velocity of a saline plume as a function
of dimensionless distance from source using the scaling of Colomer, et al.
Symbols represent experimental results and solid lines correlations by Colomer, et al.
(1999).
203
Discussion
From Figure 9.11, the centreline vertical velocity of both experimental tests and
Colomer, et al. (1999) showed two regions. In region I, the value of vertical centreline
velocity starts from zero and then increases sharply to the position above an area source.
The model of Colomer, et al. (1999) showed that the centreline vertical velocity
increases linearly (Equation 2.56) to the necking position and then continues to increase
with z1/3 (Equation 2.58). It means the velocity of the plume increase with the distance
from an area source but from the present experimental results, it is seen that the
centreline vertical velocity is a maximum in value at some distance above the area
source and after this distance, the value of vertical velocity deceases with the distance
from an area source that contrary to Colomer, et al.,. However, this present result is
similar to the theory of Fannelop and Webber (2003) (see Figure 2.11).
Another important result from experimental data was the position of the maximum
centreline vertical velocity and the position of necking, For example, for test no. 5, the
maximum centreline vertical velocity occurred at about 0.08m below the area source but
the position of necking occurred at about 0.07m. This supports the work of Fannelop
and Webber (2003) that stated that the maximum centreline vertical velocity occurs
above the neck of the plume.
Unfortunately, there was little explanation about the length scale and velocity scale used
in Fannelop and Webber (2003). Despite communication directly with Webber (2005),
it appears that the length and velocity scale they used to compare their data does not
successfully collapse the experimental data of Colomer, et al. and that of the present
author. This suggests that their scaling may not be valid. A plot using the scaling of
Fannelop and Webber is shown in Figure 9.12. However, without the experimental raw
data used of Liedtke and Schatzmann (1997), it was very hard for the present author to
determine whatever the scaling used by Fannelop and Webber or that by Colomer, et al.
is the most suitable. Nevertheless, leaving aside concern about the length and velocity
scales used the shape of the Fannelop and Webber centreline velocity of the plume is
qualitatively the same as for the present data; especially in Region II where the
centreline vertical velocity is seen to decrease with distance away from the area source.
204
Discussion
1.4
1.2
1.0
w'
0.8
0.6
0.4
0.2
0.0
0
50
100
150
z /L
test no. 5
Colomer's model (Region I)
Liedtke and Schazmann's exp data
Figure 9.12
200
250
300
Fannelop' model
Colomer's model (Region II)
Dimensionless centreline vertical velocity of a saline plume as a function
of dimensionless distance from source using the scaling of Fannelop and Webber (2003).
9.1.2 Comparison of Saline Plume Experimental and Numerical Results
In this section, the results of numerical study of saline plume are compared with
experimental results. Test no. 5 (D0 = 0.105m, saline plume in small environmental
tank) and test no. 9 (D0 = 0.300m, saline plume in large environmental tank) were
simulated. The results of the numerical study are compared in terms of vertical velocity
distribution (of test no.5), horizontal velocity distribution (of test no.5) and density
distribution (of test no.9).
9.1.2.1 Comparison of Velocity Distribution
Comparisons between vertical and horizontal velocity distributions of experimental and
numerical study are shown in Figure 9.13 and 9.14. Test no.5, w0 = 2.24×10-4 m/s, 10%
by weight of salt concentration, was used for this numerical simulation conditions.
205
Discussion
The results of vertical velocity distribution show that at 0.01m and 0.02m below the
area source, the simulated plume width agrees well with the experimental data. But the
simulated centreline vertical velocity shows significant difference in the near-field
region, but at the levels far from the area source, the agreement is quite good for both
the centreline vertical velocity and the vertical velocity profile.
A possible physical reason for the large near-field discrepancy is the presence of the
screw head (about 10mm diameter) at the centre of the area source (see Figure 9.15).
The screw was needed to support the porous sheet in order to keep the sheet flat during
the experiments. At first the screw head was assumed that has no effect on the vertical
velocity, therefore it was not included in the simulations.
At levels below 0.02m from the source, the effect of screw head decreases, therefore the
spot values of centreline vertical velocity and also vertical velocity profile from the
numerical study agree well with the experimental data, for example, at 0.06m, 0.08m
and 0.10m that are shown in Figure 9.13.
On the other hand the discrepancy on the centreline in the near-field region may be due
to the failure of the numerical simulation to model the details of the boundary layer-type
flow on the porous plate, particularly with respect to turbulent entrainment.
Changing of value of Cε 3 in k-ε turbulent model has effect on the near-field region.
CHAM (2003) suggests that Cε 3 should be zero for stably-stratified flow and unit for
unstably-stratified flow but in this region, the flow combine two type of flow, stably and
unstably stratified then only one value of Cε 3 throughout the domain may not give a
good model of the flow in the critical near-field region.
206
0.005
0.005
-0.005
-0.005
-0.015
Vertical Velocity (m/s)
Vertical Velocity (m/s)
Discussion
Simulation
Experiment
-0.025
-0.035
-0.045
-0.055
-0.015
Simulation
Experiment
-0.025
-0.035
-0.045
-0.055
-0.065
-0.065
-0.075
0.000
0.025
0.050
-0.075
0.000
0.075
0.025
Radius (m)
(m/s)
(a) 0.01m below source
0.005
-0.005
-0.005
Simulation
Experiment
Vertical Velocity (m/s)
Vertical Velocity (m/s)
0.075
(b) 0.02m below source
0.005
-0.015
-0.025
-0.035
-0.045
-0.055
-0.065
Simulation
Experiment
-0.015
-0.025
-0.035
-0.045
-0.055
-0.065
-0.075
0.000
0.025
0.050
-0.075
0.000
0.075
0.025
Radius (m)
0.050
0.075
Radius (m)
(c) 0.04m below source
(d) 0.06m below source
0.005
0.005
-0.005
-0.005
Simulation
Experiment
-0.015
Vertical Velocity (m/s)
Vertical Velocity (m/s)
0.050
Radius (m)
-0.025
-0.035
-0.045
-0.055
-0.065
Simulation
Experiment
-0.015
-0.025
-0.035
-0.045
-0.055
-0.065
-0.075
0.000
0.025
0.050
0.075
-0.075
0.000
0.025
0.050
Radius (m)
Radius (m)
(e) 0.08m below source
(f) 0.10m below source
Figure 9.13
Comparison of experimental and numerical results for vertical velocity
distribution at different levels of saline plume of test no.5.
207
0.075
Discussion
0.015
Simulation
Experiment
0.010
Horizontal Velocity (m/s)
Horizontal Velocity (m/s)
0.015
0.005
0.000
-0.005
-0.010
-0.015
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.15
0.15
Simulation
Experiment
-0.10
-0.05
Radius (m)
(a) 0.01m below source
Horizontal Velocity (m/s)
Horizontal Velocity (m/s)
0.010
0.005
0.000
-0.005
-0.010
-0.05
0.00
0.05
0.10
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.15
0.15
-0.10
-0.05
0.010
0.005
0.000
-0.005
-0.010
0.15
0.05
0.10
0.15
Simulation
Experiment
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.15
-0.10
-0.05
Radius (m)
(e) 0.08m below source
Figure 9.14
0.10
0.015
Simulation
Experiment
0.00
0.05
(d) 0.06m below source
Horizontal Velocity (m/s)
Horizontal Velocity (m/s)
0.015
-0.05
0.00
Radius (m)
(c) 0.04m below source
-0.10
0.15
Simulation
Experiment
Radius (m)
-0.015
-0.15
0.10
0.015
Simulation
Experiment
-0.10
0.05
(b) 0.02m below source
0.015
-0.015
-0.15
0.00
Radius (m)
0.00
Radius (m)
0.05
0.10
(f) 0.10m below source
Comparison of experimental and numerical results for horizontal velocity
at different levels of saline plume of test no.5.
208
0.15
Discussion
In the case of horizontal velocity distribution (see Figure 9.14), a comparison of
horizontal velocity values and profiles are good agreement in the near-field regions, for
example at 0.01m and 0.02m, and also at levels away from the area source.
From Figure 9.14f, it can be seen that at 0.10m below an area source, the zero value
position of the horizontal velocity distribution from experiment was not in the middle of
the domain. This is probably due to meandering (buckling) of the saline plume due to
the effect of recirculation of the flow at the bottom of the experimental tank is dominant
at that averaging time in velocity analysing process.
Flow distribution
baffle
Salt solution
inlet
O-ring
Support rod
105 mm
Porous sheet
Screw head
Figure 9.15
Screw head in the area source.
9.1.2.2 Density Distribution
In the case of comparison of density distribution, source conditions of test no.9, w0 =
1.48×10-4 m/s, 10% by weight of salt concentration, were used for the numerical
simulation. Comparisons of density distribution at 0.01, 0.05, 0.10 and 0.20m below the
area source are shown in Figure 9.16.
The simulated density distributions in the saline plume show good agreement with the
experimental data, especially at levels away from the area source. At 0.01m below the
area source, the simulated value of centreline density higher than the experimental data,
but both plume width and density values away from the centre are in good agreement
with experimental data. The effect of the screw head may also have been significant
209
Discussion
dominated at the levels near the area source but decreased at levels away from the area
source, as shown by the good agreement at 0.1m and 0.2m.
9.2
Thermal Plume
This section presents a discussion of the results of the thermal plume both experimental
and numerical studies. First parts, the results from experimental work for thermal plume
are discussed including the comparison between plume vertical velocity radius, bV , and
plume temperature different radius, bT . Second parts, the comparison between
experimental and numerical works are discussed.
9.2.1
Experimental Results of Thermal Plume
9.2.1.1 Plume Vertical Velocity Radius, bV , vs Plume Temperature Different
Radius, bT
From the comparison graph between bT and bV as shown in Figure 9.17, from
experimental results, it was found that the vertical velocity spread out wider than the
temperature. The plume vertical velocity radius has the large effect on an enclosure that
can be seen at the position near an area source. It will be explained in section 9.2.3.
9.2.2 Comparison between Experimental and Numerical Results
The results from simulation were validated against experimental data for both source
temperature conditions. The main modification required in the simulations was
numerous trials with different values of turbulent Prandtl numbers for the velocity
components and temperature in the standard k-ε turbulent model. The value of turbulent
Prandtl number for u, v, w and temperature that gave the best agreement between
experimental and numerical data was 0.65 which is relatively compared low with
previous numerical studies by Nam and Bill (1993) who used a value of 0.85.
Comparisons between experimental and numerical data for vertical velocity distribution,
horizontal velocity distribution and temperature distribution for the two source
conditions, at 0.02m, 0.10m and 0.30m are shown in the following sections.
210
Discussion
1006
1006
Simulation
Experiment
1005
1004
1003
3
3
Density (kg/m )
1004
Density (kg/m )
Simulation
Experiment
1005
1002
1001
1000
1003
1002
1001
1000
999
999
998
998
997
0.00
0.05
0.10
Radius (m)
0.15
997
0.00
0.20
0.05
(a) 0.01m below source
1006
1006
1003
3
Density (kg/m )
1004
1003
1002
1001
1000
1002
1001
1000
999
999
998
998
0.10
0.15
997
0.00
0.20
Radius (m)
0.10
Radius (m)
(c) 0.10m below source
(d) 0.20m below source
Figure 9.16
0.20
Simulation
Experiment
1005
1004
0.05
0.15
0.20
Comparison of experimental and numerical results for density
distribution in a saline plume below an area source of test no.9.
0.07
0.07
bbV
V
bbT
T
Plume Width (m)
0.05
0.04
0.03
0.00
bbv
V
bbT
T
0.06
0.06
Plume width (m)
3
Density (kg/m )
1005
0.05
0.15
(b) 0.05m below source
Simulation
Experiment
997
0.00
0.10
Radius (m)
0.05
0.04
0.03
0.02
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0
Distance above the area source (m)
a) 150oC
Figure 9.17
0.05
0.1
0.15
0.2
0.25
Distance above the floor (m)
0.3
b) 200oC
The comparison graph of bT and bV of thermal plume above 150oC and
200oC hot plate.
211
0.35
Discussion
9.2.2.1 Comparison of Vertical Velocity Distribution
The comparisons of results of vertical velocity distribution in the thermal plume for
different source conditions are shown below.
Figures 9.18 and 9.19 show the comparisons of vertical velocity between experiment
and simulation for source temperatures of ~150oC and ~200oC respectively. The
comparisons show that the simulation results agree with experimental data within 10%.
9.2.2.2 Comparison of Horizontal Velocity Distribution
The comparisons of results of horizontal velocity in the thermal plume at different
levels are shown in Figure 9.20 to 9.25.
The comparisons below show that the simulated horizontal velocity profiles compare
well at the positions near the centre of plume (r = 0.00m) (within 2%). At a distant
radial location, the agreement is not as good as near the centre position. The values of
horizontal velocity from the simulation differ by a maximum of ±0.03 m/s from the
experimental results.
0.7
20mm
100mm
300mm
Exp.
Vertical
Velocity
Velocity
(m/s)(m/s)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.15
Figure 9.18
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of vertical velocity distribution of thermal plume from
~150oC hot plate (solid lines are numerical and symbols experimental results).
212
Discussion
0.7
20mm
100mm
300mm
Exp.
Vertical
Velocity
Velocity
(m/s)(m/s)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius (m)
Figure 9.19
Comparison of vertical velocity distribution of thermal plume from
~200oC hot plate (solid lines are numerical and symbols experimental results).
0.20
Simulation
Exp1
Exp2
Exp3
Horizontal velocity (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
Figure 9.20
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of horizontal velocity distribution of thermal plume at
0.004m above ~150oC hot plate.
213
Discussion
0.20
Simulation
Exp1
Exp2
Exp3
Horizontal velocity (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
Figure 9.21
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of horizontal velocity distribution of thermal plume at
0.020m above ~150oC hot plate.
0.20
Simulation
Exp1
Exp2
Exp3
Horizontal velocity (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
Figure 9.22
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of horizontal velocity distribution of thermal plume at
0.100m above ~150oC hot plate.
214
Discussion
0.20
Simulation
Exp4
Exp5
Exp6
horizontal velocity (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
Figure 9.23
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of horizontal velocity distribution of thermal plume at
0.004m above ~200oC hot plate.
0.20
Simulation
Exp4
Exp5
Exp6
Horizontal velocity (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
Figure 9.24
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of horizontal velocity distribution of thermal plume at
0.020m above ~200oC hot plate.
215
Discussion
0.20
Simulation
Exp4
Exp5
Exp6
Horizontal velocity (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.15
Figure 9.25
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of horizontal velocity distribution of thermal plume at
0.100m above ~200oC hot plate.
9.2.2.3 Comparison of Temperature Distribution
80
20mm
100mm
300mm
Exp.
o
Temperature ( C)
70
60
50
40
30
20
-0.15
Figure 9.26
-0.10
-0.05
0.00
Radius (m)
0.05
0.10
0.15
Comparison of temperature distribution of thermal plume from ~150oC
hot plate.
216
Discussion
80
20mm
100mm
300mm
Exp.
o
Temperature( C)
70
60
50
40
30
20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Radius(m)
Figure 9.27
Comparison of temperature distribution of thermal plume from ~200oC
hot plate.
From Figures 9.26 and 9.27, it is seen that the simulated temperature distributions at all
levels agree to within 5% when compared with experimental data.
9.2.2.4 Comparison of Plume Radius and Centreline Values of Experimental and
Numerical Results for the Thermal Plume
In this section, the value of plume vertical velocity radius, bV, plume temperature radius,
bT, centreline vertical velocity, wC and centreline temperature difference, ∆TC , from
experiment and simulation are compared.
a)
Plume Vertical Velocity Radius, bV
Experimental results indicate that the plume vertical velocity radius, bV , increases
dramatically in the near-field region. Therefore, the plume vertical velocity radius
decreases again indicating necking and the spreading out at some distance away from
the area source. The numerical results show the same pattern in the far-field region.
217
Discussion
0.08
0.08
Simulation
Experiment
0.07
0.07
Good agreement
bV (m)
bV (m)
0.06
0.05
0.04
0.03
0.00
Simulation
Experiment
Good agreement
0.06
0.05
0.04
0.05
0.10
0.15
0.20
0.25
Position above hot plate (m)
0.30
0.35
0.03
0.00
a) 150oC source
Figure 9.28
0.05
0.10
0.15
0.20
0.25
Position above hot plate (m)
0.30
b) 200oC source
Comparison of vertical velocity radius, bV , of thermal plume from 150oC
and 200oC hot plate.
At the region near the area source of the thermal plume, the plume vertical velocity
radius appears to shows increase with distance, z which did not happen in the case of
the saline plumes. A possible reason for this phenomenon maybe due to the nonadiabatic boundary condition of glass enclosure that give rise to a boundary layer on the
glass but in the case of the saline plume this circulation flow does not occur due to the
strongly stratified ambient and the zero flux condition on the vertical walls.
Another reason is the effect of the enclosure in this case. The effect of the enclosure on
thermal plume is presented in Section 9.2.3 below (See Figure 9.32).
b)
Plume Temperature Radius, bT
The simulated plume temperature radius, bT , is in very good agreement with
experimental results at positions near the area source.
218
0.35
Discussion
0.08
0.08
simulation
experiment
0.07
0.06
0.06
bT (m)
bT (m)
0.07
0.05
0.05
0.04
0.04
0.03
0.00
simulation
experiment
0.05
0.10
0.15
0.20
0.25
Position above hot plate (m)
0.30
0.35
0.03
0.00
0.05
a) 150oC source
Figure 9.29
0.10
0.15
0.20
0.25
Position above hot plate (m)
0.30
b) 200oC source
Comparison of temperature radius, bT , of thermal plume from 150oC and
200oC hot plate.
c)
Centreline Vertical Velocity, wC
The value of centreline vertical velocity, wC , increases exponentially from low value
(see in Figure 9.30). The difference between numerical and experimental results is very
small at position near the area source but increases at positions far from the area source.
d)
Centreline Temperature Difference, ∆TC
The centreline temperature difference, ∆TC , decreases with the distance from the area
source. The numerical results show a good agreement with experimental results for all
positions above the area source as shown in Figure 9.31.
219
0.35
Discussion
0.7
simulation
experiment
0.6
0.6
0.5
0.5
wC (m/s)
wC (m/s)
0.7
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.00
0.0
0.00
0.05
0.10
0.15
0.20
0.25
Position above hot plate (m)
0.30
0.35
simulation
experiment
0.05
a) 150oC source
Figure 9.30
0.30
0.35
b) 200oC source
Comparison of wC of thermal plume from 150oC and 200oC hot plate.
45
60
simulation
experiment
40
simulation
experiment
50
35
40
∆TC ( C)
30
25
o
o
∆TC ( C)
0.10
0.15
0.20
0.25
Position above hot plate (m)
20
30
20
15
10
10
5
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0
0.00
Position above hot plate (m)
9.2.3
0.10
0.15
0.20
0.25
0.30
Position above hot plate (m)
a) 150oC source
Figure 9.31
0.05
b) 200oC source
Comparison of ∆TC of thermal plume from 150oC and 200oC hot plate.
Effect of Enclosure on Plume Structure
The glass enclosure has a significant effect on plume shape, velocity distribution and
also density distribution. In this section, the numerical results of thermal plume of both
with cover and without cover are shown.
The simulated values of plume vertical velocity radius, bV , and plume temperature
different radius, bT in the enclosure and without the enclosure are shown in Figures
9.32 and 9.33 below.
220
0.35
Discussion
Figure 9.32 shows that the two variations show opposite trends in the near-field region.
In the far-field region, the confined plume radius is always greater.
The plume temperature radius, bT , of thermal plume with and without an enclosure
show same pattern. Without an enclosure, the temperature profile spreads out to a lesser
extent. This means an enclosure has a noticeable effect on the plume spread both in
terms of vertical velocity and temperature.
0.08
Without Uncover
enclosure
With Cover
enclosure
bV (m)
0.07
0.06
0.05
0.04
0.03
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Distance above the floor (m)
Figure 9.32
Comparison of plume vertical velocity radius, bV , of thermal plume with
and without an enclosure above ~200oC hot plate.
221
Discussion
0.08
WithoutUncover
enclosure
WithCover
enclosure
bT (m)
0.07
0.06
0.05
0.04
0.03
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Distance above the floor (m)
Figure 9.33
Comparison of plume temperature radius, bT , of thermal plume with and
without an enclosure above ~200oC hot plate.
From the figure above, it seems that the effect of the enclosure on plume vertical
velocity radius is significant but not to the same extent on the plume temperature radius.
This means that the ratio of the width of the enclosure, Le, and source diameter, D0, ≤
4.0 the effect of the enclosure on the near-field vertical velocity is significant but on the
near-field temperature, it is not so important.
9.3
Comparison of the Thermal and Saline Plumes
In this section, an attempt is made to compare the saline plume’s results with the
thermal plume’s results in order to find out the relationship between the saline and
thermal plumes but due to the boundary and source conditions of two types of flow
were somewhat different in the experiments. that are:
-
The ratio of the width of the enclosure, Le, to the source diameter, D0, was 4.76
and 4.04 for the saline and thermal plumes, respectively.
-
The type of flow, unsteady flow of the saline plume vs. steady flow of thermal
plume. In the case of the saline plume the density inside the environmental tank
changes with the time resulting in transient flow but in the case of the thermal
222
Discussion
plume the density inside the enclosure reaches a quasi-steady state and
recirculation flow inside the enclosure occurs.
-
The source parameter, Г, that had a finite value for the saline plume but was
infinite for the thermal plume in the present work.
Therefore the comparison of the saline plumes vs. thermal plumes is difficult using the
present data. However, given the above difference it is nevertheless of interest to
compare the dimensionless plume vertical velocity radius, bv/D0, vs. dimensionless
distance, z/D0, between the saline and thermal plumes as shown below.
0.40
0.35
0.30
b v/D
0.25
0.20
0.15
0.10
0.05
0.00
0.00
0.25
0.50
0.75
z /D
1.00
1.25
1.50
Exp of thermal plume within enclosure(200C)
(200oC)
o
Num of thermal plume within enclosure (200
(200C)
C)
o
C)
Num of thermal plume without enclosure (200
(200C)
Exp of saline plume (test no. 5)
Exp of saline plume (test no. 6)
Exp of saline plume (test no. 7)
Figure 9.34
Comparison of dimensionless plume vertical velocity radius, bv/D0, vs.
dimensionless distance, z/D0, between thermal and saline plume.
223
Chapter 10
Conclusions and Recommendations
Appendix VII. Conductivity Monitor
Chapter 10
Conclusions and Recommendations
10.1
Conclusions
In the present work, the structure of buoyancy-driven plumes generated from area
sources in enclosures was studied using experimental and numerical methods.
In the experimental section, two comprehensive and separate sets of experiments were
set up in order to investigate turbulent plume characteristics including plume vertical
velocity distribution, horizontal velocity distribution, vertical velocity radius,
temperature radius, plume necking, virtual origin location, etc. The present author
developed a range of experimental facilities to carry out the experimental program
including: a large environmental fluid dynamics tank, conductivity probe, computer
controlled probe traversing mechanism, thermal enclosure, LDV measurement system,
etc.
The major outcome in the numerical section of this work was the successful
development of numerical models of the saline and thermal plumes. Because the
configuration of each type of experiment was different, both transient and steady
numerical models were used. These were validated against the experimental results
including: vertical velocity distribution, density and temperature distribution, horizontal
velocity, plume radius and also plume centreline velocity. It was found that the k-ε
turbulent model with the gravitational correction is suitable to simulate the plume
generated from the area source and that improvement of the numerical results can be
optimised by adjustment changing Prt and Cε3 in the k-ε turbulent model.
The experimental and numerical results were compared with previous research such as
that by Morton (1959a), Hunt and Kaye (2001), Colomer, et al. (1999), Fannelop and
Webber (2003), etc. It is concluded that the field of the plumes from area sources still
requires much further investigation because in these research works and the present
225
Appendix VII. Conductivity Monitor
work, there is considerable disagreement as to suitable length scales and velocity scales
are as to suitable models of virtual origin location, position and diameter of the neck.
The conclusions that may be drawn from the experimental work can be summarised as
follows.
Saline plume experiments
1.
It was found that the Gaussian equation was a suitable model for the velocity and
density distributions in the plume in the far field. However, near the area source,
the profiles were much flatter (approaching a “top-hat” profile) which was
matched well with the modified Gaussian (Equation 5.2).
2.
At the Rayleigh numbers dealt with in the present work, the forms for Dneck, zneck
and zv are only very weakly dependent on source conditions.
3.
The plume vertical velocity radius measured by the PTV technique was always
significantly narrower than the plume shadowgraph image radius. The present
experimental data showed that the plume image radius is approximately 40%-50%
wider than that found by PTV.
4.
The plume generated from an area source can be separated into two regions as
reported by Colomer, et al. (1999) and others. The centreline vertical velocity
equations proposed by Colomer, et al. (1999) agree with the present experimental
results in region I but not in region II. The centreline vertical velocity in region II
was found to decrease with distance z, but Colomer, et al. (1999) reported that it
increased with the distance. The author believes this to be because Colomer, et al.
(1999) only present results for z/D ≤ 1 and at greater heights one would expect
vertical velocity to vary as w ~ z-1/3 as for a point source.
5.
The pattern of the centreline vertical velocity is in agreement with the centreline
vertical velocity of Fannelop and Webber (2003).
6.
The maximum vertical centreline velocity occurred at a position above the neck of
the plume. This confirms some of the theory developed by Fannelop and Webber
(2003).
7.
The virtual origin locations from the shadowgraph and the PTV experiments were
not the same. The virtual origin location from the shadowgraph experimental data
226
Appendix VII. Conductivity Monitor
is further from the actual source than that from the PTV experimental data
because the plume radius from the shadowgraph experimental data is wider than
from the PTV experimental data. The virtual origin location from the
shadowgraph experimental data showed good agreement with that suggested by
conical theory, ACGIH (2001), Goodfellow and Tähti (2001) who also suggest
that the virtual origin location of the plume is a function of only the source
diameter.
8.
The virtual origin location from the PTV experimental data was lower than that
suggested by conical theory, ACGIH (2001), Goodfellow and Tähti (2001) and
the shadowgraph experimental data but higher than the value given by Hunt and
Kaye (2001) who presented that the virtual origin of the plume is a function of the
source strength, Г.
9.
The virtual origin location from the PTV experimental data showed a slight
decrease in distance from the source with increasing Г as predicted by Morton
(1956a) and Hunt and Kaye (2001). However, the magnitude of the experimental
distance was significantly greater than that predicted by theory.
Thermal plume experiments
1.
Both Particle Image Velocimetry (PIV) and Laser Doppler Velocimetry (LDV)
measurement systems were trialled to measure the velocity distribution in the air
plume from a hot area source. It was found that the Laser Doppler Velocimetry
method was more suitable than the Particle Image Velocimetry for study of
thermal plumes in air.
2.
Following tests at each source condition, 150oC and 200oC, it was confirmed that
the experiments for determining vertical velocity, horizontal velocity and
temperature distribution were repeatable.
3.
In the far-field region, the plume temperature radius, bT, was consistently
narrower than the plume vertical velocity radius, bV by approximately 20%.
4.
At the region near the area source of the thermal plume, the plume vertical
velocity radius appears to increase with distance, z, which did not happen in the
case of the saline plumes. A possible reason for the phenomenon may be due to
227
Appendix VII. Conductivity Monitor
the non-adiabatic condition of the glass enclosure and the effect of enclosure on
flow near the source.
5.
In the near-field, the effect of the presence of the enclosure on plume vertical
velocity radius was found to be significant (for Le/D0 ≤ 4.0) but this was not the
case for plume temperature radius.
Numerical investigations. Two sets of numerical simulations were conducted to
analyse descending saline plumes and ascending thermal plumes. The following overall
conclusions can be drawn:
Saline plume simulations:
1.
The value of Cε 3 in the k-ε turbulence model with gravity correction has a
significant effect on the form of the numerical results. It was found that Cε 3 = 0.6
was suitable for modelling flow of the saline plume, as shown by the close
agreement with experimental results for both velocity and density distributions.
2.
Changing the value of the turbulent Prandtl number (Prt) for the velocity
components and salt concentration was found to improve numerical results, as
reported by others CFD researchers. In this study, it was found that the optimal
value of Prt for all velocity components (u, v and w) and salt concentration (C1)
was 0.65.
3.
Near the area source, the simulated centreline vertical velocity was significantly
higher than that measured experimentally. Plume shape and plume radius showed
good agreement between simulations and experiment. At levels away from the
area source, the centreline vertical velocity, vertical velocity distribution, plume
radius and density distribution showed good agreement with experimental results.
Thermal plume simulations
1.
Changing the value of Prt for the velocity components and temperature can be
used to improve numerical results. It was found that the value of Prt for all
velocity components (u, v and w) and temperature that gave the best fit to the
thermal plume experimental results was Prt = 0.65.
228
Appendix VII. Conductivity Monitor
2.
Following optimisation of the spatial and temporal grids, the results of vertical
velocity distribution, horizontal velocity distribution and temperature difference
profile from numerical work for both source conditions were in good agreement
with experimental work.
3.
The effect of the enclosure on the near-field vertical velocity profile was found to
be significant but its effect on the near-field temperature was not so important.
10.2
Recommendations
In this work the main characteristics of the plume such as plume vertical velocity
distribution, horizontal velocity distribution, plume necking, etc were studied. Despite
several attempts, entirely satisfactory results could not be obtained in some phases of
the project due to experimental and computational difficulties. Some unforseen
difficulties such as severe water restrictions, due to drought in NSW, also had an effect
on the number of saline plume experiments that could be carried out. Some
recommendations for future work are as follows.
Saline plume
1.
A bank of many conductivity probes should be developed and used for
simultaneous density measurements in an evolving plume. Alternatively, a quasisteady plume could be achieved by activating drains on the floor of the tank and
compensating water has to add into the tank to maintain the water level. Then
only one conductivity probe can be more easily used but a lot of water will be
consumed.
2.
A smaller-scale apparatus could be used to prevent excessive water usage, and
allow the use of the volume flux measurement method (Baines, 1983) for
estimating the virtual origin location and volume flux within the plume as a
function of height.
3.
It was found that counter diffusion between the salt solution and fresh water
across the porous sheet before the start of each experimental test led to some
uncertainty as to the starting condition of the saline plume. A removable thin sheet
of non-porous material could reduce the diffusion effect before starting plume
flow, although physically this will be difficult to construct.
229
Appendix VII. Conductivity Monitor
4.
The steady plume in the smaller environmental tank with a drain is recommended
when plume shape and plume density distribution are the main focus of the study
because this will eliminate the transient filling box situation and also reduce the
simulation time for the numerical study.
5.
To determine the velocity distribution, the filling box situation is still necessary
because seed particles are used for the Particle Tracking Velocimetry method.
Therefore a taller tank should be used to reduce the effect of recirculation near the
bottom of the tank and also increase the video recording time for velocity
analysis. For the 0.5m×0.5m bottom area of the tank, the tank height should be
double the tank width. The area source also must be suitable for that height and
width of the tank.
Thermal plume
1.
The surface temperature of the area source should be set to higher values (up to or
above 500oC possible) in order to determine the effect of source conditions on a
non-Boussinesq turbulent thermal plume. This will reproduce the situation above
hot molten metal baths more closely, for example.
2.
The traversing mechanisms of the laser transmitter and receiver probes in the
present study were controlled separately and manually. It was difficult to control
the positions of the two probes so that they focused on precisely the same
measurement volume. A support that can carry both probes and can be controlled
by one traversing mechanism is needed in the future. This will eliminate the
difficulty and also reduce the time required for each experiment.
3.
The size of the area source and an enclosure should be designed suitably to reduce
the effect of the enclosure on the thermal plume.
230
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245
Appendix I. Experimental Tank Design and Drawing
Appendix I
Experimental Tank Drawing
In this present study, two tanks were used. Firstly, the 1.4×1.4×1.4 m3 tank was
designed and constructed. Drawings of this tank are shown below.
8
2
9
14
1
15
12
13
10
3
4
11
5, 6 and 7
Figure A-I 1 Tank drawing.
247
Appendix I. Experimental Tank Design and Drawing
17
16
18
19
20
Figure A-I 2
Figure A-I 3
Tank base.
Position of tank frame.
248
Appendix I. Experimental Tank Design and Drawing
Material Details
Table A-I 1
Material details of 1.4×1.4×1.4m3 water tank.
No.
Material
Length
Quantity
Note
1.
Steel Hollow Square 50×50×3
1410
8
2.
Steel Hollow Rectangular 50×25×3
1360
8
3.
Steel Hollow Square 50×50×3
1335
2
4.
Steel Hollow Square 50×50×3
1375
2
5.
Glass 1362×1402×5
1
6.
Styrofoam 1475×1435×20
1
7.
Stainless Plate 1452×1403×5
1
8.
Steel Hollow Square 50×50×3
1535
2
45o cut
9.
Steel Hollow Square 50×50×3
1575
2
45o cut
10.
Steel L-Section 51×51×3
1458.16
2
45o cut
11.
Steel L-Section 51×51×3
1418.36
2
45o cut
12.
Steel L-Section 51×51×3
1402
2
45o cut
13.
Steel L-Section 51×51×3
1362.2
2
45o cut
14.
Steel L-Section 51×51×3
1359
4
15.
Steel L-Section 51×51×3
1331
4
16.
Steel Hollow Square 100×100×5
1510
2
45o cut
17.
Steel Hollow Square 100×100×5
1550
2
45o cut
18.
Steel Hollow Square 100×100×5
605
6
19.
Steel Hollow Square 100×100×5
150
4
20.
Steel Hollow Square 100×100×5
1350
1
Secondly, the 0.5×0.5×0.5 m3 tank was designed in order to study plume velocity
distribution. The drawing details of this tank are shown below.
249
Appendix I. Experimental Tank Design and Drawing
100mm
2
100mm
Top View
Front View
1
z
Figure A-I 4
The 0.5×0.5×0.5 m3 water tank.
Material Details
Table A-I 2
Material details of 0.5×0.5×0.5 water tank.
No.
Material
Length
Quantity
1.
Perspex
510×510×5
1
2.
Perspex
505×505×5
4
250
Note
Appendix II. Traversing Mechanism
Appendix II
Traversing Mechanism
Traversing mechanism for the saline plume experimental study was used to support and
move the conductivity probe to the measuring point. It was controlled using the stepper
motor and CNC software. The drawing of the present traversing mechanism is shown in
Figure A-II 1 below. The detail of material used for this traversing mechanism is
shown in Table A-II 1.
8
3
4
2
6
12
9
5
7
13
11
10
1
Figure A-II 1 The drawing of the present traversing mechanism used for saline plume
experiments.
251
Appendix II. Traversing Mechanism
Material Details
Table A-II 1
Material details of traversing mechanism.
No.
Item, Material
Quantity
1.
Vertical Thread supporter, Aluminium
1
2.
Moving supporter, Aluminium
2
3.
Vertical base supporter, Aluminium
1
4.
Horizontal thread supporter, Aluminium
2
5.
Moving slot component 1, Aluminium
2
6.
Moving slot component 2, Aluminium
4
7.
Conductivity probe supporter, Aluminium
1
8.
Horizontal support rod, Stainless steel
3
9.
Horizontal moving thread, Stainless steel
1
10.
Vertical support rod, Stainless steel
3
11.
Vertical moving rod, Stainless steel
1
12.
Aluminium bar, Aluminium
2
13.
Stepper Motor
2
252
Appendix III. Conductivity Probe Design and Drawing
Appendix III
Conductivity Probe Design and Drawing
The conductivity probe used in this study was modified from the conductivity probe
designed by Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, UK. The original design of conductivity probe is shown in figure A-III 1
below.
Please see print copy for Figure A-III 1
Figure A-III 1 Drawing of original conductivity probe (Holford, 1997).
At first, the same conductivity probe was needed for this study but in order to reduce the
effect of probe on the flow a smaller stainless steel tube, 8mm outside diameter and
3mm outside diameter were chosen. In the construction process, it was very difficult to
insert epoxy between two tubes therefore; between two tubes of new conductivity probe
it had only an air gap as shown in Figure A-III 2.
253
Appendix III. Conductivity Probe Design and Drawing
Figure A-III 2 Drawing of modified conductivity probe.
254
Appendix IV. Thermocouple Calibration
Appendix IV
Thermocouple Calibrations
All thermocouples used in the thermal plume experiments were calibrated against a
standard NATA-certified calibration thermometer. The temperature ranges for
calibration depended on the range of temperature that the thermocouples were used for
which were 10oC - 100oC for plume temperature measurement and 10oC - 200oC for
heat source surface temperature measurement. For thermocouple calibration, all the
thermocouples and the standard thermometer were immersed in a thermal bath filled
with either water or vegetable oil, depending upon the desired temperature range. A
temperature regulator was used to control the temperature of the liquid in the bath. The
output from each thermocouple was recorded using LabTech software. The temperature
inside the thermal bath was set at 15oC and increased every 5oC in increment in order to
measure temperature read by the thermocouples and the standard thermometer. Then,
calibration graphs of each thermocouple were produced and linear calibration equations
were deduced. These were used in LabTech software for converting the temperature
signal from thermocouples to actual temperature values. The data of calibration in the
graph below came from averaging data over 10 seconds with sampling rate equals to
five data per second.
Below table shows the linear equation of all thermocouples that were used in this study.
Table A-IV 1 The linear equation of all thermocouple calibrations.
(Tmea is measured temperature from thermocouples).
Thermocouple
Calibration equation
R2
1
1.05205×Tmea - 9.62066×10-1
9.99519×10-1
2
1.05537×Tmea – 1.11009
9.99464×10-1
3
1.04953×Tmea – 7.91720×10-1
9.99455×10-1
4
1.05455×Tmea – 1.01047
9.99505×10-1
5
1.05009×Tmea – 9.59813×10-1
9.99448×10-1
no.
255
Appendix IV. Thermocouple Calibration
6
1.05517×Tmea – 1.11616
9.99468×10-1
7
1.05044×Tmea – 8.77087×10-1
9.99419×10-1
8
1.05315×Tmea – 1.01949
9.99343×10-1
9
1.05442×Tmea – 9.09038×10-1
9.99462×10-1
10
1.05468×Tmea – 9.10947×10-1
9.99471×10-1
11
1.05359×Tmea – 8.03603×10-1
9.99352×10-1
12
1.05265×Tmea – 5.61343×10-1
9.99397×10-1
13
1.05216×Tmea – 5.56840×10-1
9.99422×10-1
14
1.05140×Tmea – 4.63111×10-1
9.99433×10-1
15
1.05164×Tmea – 4.22818×10-1
9.99365×10-1
16
1.02700×Tmea + 1.77572
9.99770×10-1
17
1.01395×Tmea + 2.54878
9.99778×10-1
18
1.02637×Tmea + 1.33086
9.99630×10-1
19
1.01820×Tmea + 2.25565
9.99824×10-1
20
1.01040×Tmea + 3.0094
9.99830×10-1
21
9.98204×10-1×Tmea + 1.21917
9.99790×10-1
22
9.98430×10-1×Tmea + 1.08638
9.99802×10-1
23
9.99286×10-1×Tmea+ 9.30087×10-1
9.99794×10-1
24
9.98680×10-1×Tmea+ 9.33863×10-1
9.99791×10-1
25
1.06072×Tmea – 6.01807×10-1
9.99346×10-1
26
1.00860×Tmea+ 1.11241
9.99722×10-1
27
9.93841×10-1×Tmea+ 1.93609
9.99764×10-1
28
9.93057×10-1×Tmea+ 2.46713
9.99784×10-1
29
1.00166×Tmea+ 8.72451×10-1
9.99762×10-1
30
1.00079×Tmea+ 8.79165×10-1
9.99791×10-1
31
9.88845×10-1×Tmea+ 1.28250
9.99790×10-1
32
9.90132×10-1×Tmea+ 1.36015
9.99773×10-1
256
Appendix V. Table of Properties
Appendix V
Table of Properties
A-V 1 Properties of Salt Solution
Please see print copy for Appendix A-V 1
257
Appendix V. Table of Properties
Please see print copy for Appendix A-V 1
258
Appendix V. Table of Properties
Please see print copy for Appendix A-V 1
259
Appendix V. Table of Properties
Table A-V 2 Properties of Air (Özişik, 1985)
Please see print copy for Appendix A-V 2
260
Appendix V. Table of Properties
A-V 3 Properties of Glass
Please see print copy for Appendix A-V 3
261
Appendix V. Table of Properties
Please see print copy for Appendix A-V 3
262
Appendix VI. Densitometer
Appendix VI
Densitometer
Densitometer is the main device used to measure the density data of salt solution for the
saline plume experiment and it was also used to calibrate the conductivity probe used
for plume density distribution measurement. Densitometer used in this present work is
Anton Paar, model DMA 35N. The drawing of this densitometer is shown in Figure AVI 1 below.
7
6
9
5
10
11
12
8
1
4
2
13
15
3
14
Figure A-VI 1 Densitometer
1. Display
2. Measuring cell
3. Filling tube
4. Screw plug
5. Built-in pump
6. Pump lock
7. Operating keys
8. Infrared interface connection
9. Data storage key
10. Screw for battery cover
263
Appendix VI. Densitometer
11. Battery cover
12. Warning plate
13. Type plate
14. DKD calibration number
15. Registration number and classification
The technical data of this densitometer is shown in Table A-VI 1.
Table A-VI 1 The technical data of the present densitometer.
Measuring range
Density
0 to 1.999 g/cm3
Temperature
0 to 40oC, 32 to 104oF
Viscosity
0 to approx. 1000 mPa.s
Uncertainty of measurement
Density
±0.001 g/cm3
Temperature
±0.2oC
Repeatability
Density
±0.0005 g/cm3
Temperature
±0.1oC
Resolution
Density
0.0001 g/cm3
Temperature
0.1oC or 0.1oF
Sample volume
Approx. 2 ml
Sample temperature
0 to 100oC
Ambient temperature
-10 to 40oC
Memory
1024 values
Interface
Infrared/RS 232
Power supply
2×1.5V Alkaline battery Micro LR03
Housing
Polypropylene or Stat-Kon NS
Dimensions
140×130×25mm
Weight
245 g
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Appendix VII. Conductivity Monitor
Appendix VII
Conductivity Monitor
The conductivity monitor, PM2 series of the Amalgamated Instrument Co Pty Ltd., was
used to display the value of densities that were measured from the conductivity probe.
Picture of this device is shown in Figure A-VII 1 below.
Figure A-VII 1
The conductivity Monitor
The technical data of this conductivity monitor is shown in Table A-VII 1.
Table A-VII 1 The technical data of the conductivity meter.
Measurement range
Range 1: 0.000 – 9.999 µS/cm
Resolution: 0.001 µS/cm
Range 2: 00.00 – 99.99 µS/cm
Resolution: 0.01 µS/cm
Range 3: 0.0 – 999.9 µS/cm
Resolution: 0.1 µS/cm
Range 4: 0 – 9999 µS/cm
Resolution: 1 µS/cm
Display type
5 digital 15 mm. LED
Display update
1 reading /second
Solution temperature measurement
-40 to 120oC
265
Appendix VII. Conductivity Monitor
Set-point accuracy
Range 0 - 9.999
0.001 µS/cm
Range 10.0 - 99.99
0.01 µS/cm
Range 100.0 - 999.9
0.1 µS/cm
Range 1000 - 9999
1 µS/cm
Power supply
220 - 250V (AC) 50/60 Hz
Housing
Extruded aluminium instrument case
Dimensions
222×91×44 mm.
Weight
1.0 kg.
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