CHAPTER 1
INTRODUCTION
1.1 Introduction and Background
Porous media has a wide variety of industrial applications such as electronic cooling,
geothermal systems and many others. Modeling of non-Darcian transport through porous
media has been the subject of interest in recent times, due to its practical applications,
which includes electronic air-cooling in microelectronics. The demand for high execution
speed and memory capacity for modern computers has led to a high circuit density per
unit chip resulting in higher fluxes at the chip level to the order of 10 - 25 W/cm2 for
DIPs and PGA respectively (Mahalingam and Berg, 1984; Mahalingam, 1985). For
reliable operation these units need to be maintained at a safe level of temperature (1400
C). Natural and forced convection remove only a small amount of heat around 0.001
W/cm2 0 C by natural convection, 0.01W/cm2 0C in forced air and 0.1 W/ cm2 0C in forced
liquid (Simons, 1984).
While, the primary mode of removal of heat from the package surface is by
convection, the thermal resistance between the coolant and the surface plays a key factor
and therefore a large heat transfer coefficient or large contact surface for heat extraction
is pursued for package cooling.
Studies (Koh and Colony, 1974; Koh and Stevens, 1975) in forced convection in a
porous channel by using a Darcy flow model have shown that under a constant heat flux
boundary condition at the boundary the temperature at the wall and the temperature
difference between wall and coolant can be drastically reduced by the introduction of
porous media in the channel. Also, investigators (Kuo and Tien, 1988; Hunt and Tien,
1
1988) have utilized a foam material to enhance liquid forced convection cooling and their
results show an increase of two to four times in heat transfer as compared with that of
laminar slug-flow in a clear duct. Many other researchers have investigated the flow and
heat transfer through constant porosity (Poulikakos and Kazmierczak, 1998) and variable
porosity categories (Renken and Poulikakos, 1988; Hsai, Cheng and Chen, 1992).
Thermal dispersion caused by the presence of solid matrix plays a key role in
heat transfer augmentation. A number of investigations have been carried out (Hunt and
Tien, 1988, Hsiao et al., 1992, Hsu and Cheng, 1990) to study the effect of thermal
dispersion in porous medium. Later investigations (Cheng and Vortmeyer, 1988; Amiri
and Vafai, 1998) have revealed the effect of transverse thermal dispersion in forced
convection and their studies show that effect of transverse dispersion is much more
important than that of longitudinal dispersion.
The use of local thermal equilibrium (LTE) is used widely to analyze the transport
process through the porous media. In such a packed bed operated under steady state
conditions, a difference in the local temperatures between the fluid and the particle may
exist, but the overall solid and fluid temperature profiles are considered to be identical to
each other. In estimating the overall steady-state temperature profile, the heterogeneous
packed bed may be considered to be a homogenous single phase. The temperature
profiles in the bed are then predicted in terms of the effective thermal conductivities and
wall heat transfer coefficients. Several studies have been conducted using the assumption
of local thermal equilibrium.
The above assumption is not valid when there is a substantial temperature
difference between the solid and the fluid phases or the ratio of the thermal conductivities
2
is large. Dixon and Criswell (1979) investigated the problem of LTNE between the two
phases and were the first to obtain a fluid to solid heat transfer correlation. Other
researchers (Vafai and Amiri, 1998, Hwang et al., 1994) have obtained their own
correlations. Several studies (Amiri and Vafai, 1998, Amiri et al, 1995, Hwang and
Chao, 1994, Kuznestov, 1997) employed the two-equation model to study forced
convection in porous medium.
1.2 Objectives
The main objective of this work is to numerically analyze using FLUENT 5.5 of a
study of non-Darcian forced convection in an asymmetric heated, constant porosity
sintered porous channel (Hwang and Chao, 1994; Hwang, Wu and Chao, 1995). Also,
1. to develop the model and to analyze the wall temperature distribution; to extend
the model to a higher thermal conductivity ratio material namely Copper and
analyze the wall temperature distribution;
2. analyze the model (for both the materials) for two heat transfer correlations
(Hwang et al., 1994, Amiri and Vafai, 1998) considering local thermal nonequilibrium (LTNE) between the two phases and their respective fluid to solid
specific area correlations;
3. to investigate the wall temperature dependence on variants like particle Reynolds
Number (Rep), Heat transfer correlations (HTC), particle diameter (Dp) and solid
to fluid thermal conductivity ratio in the constant porosity category.
3
CHAPTER 2
INTRODUCTION TO POROUS MEDIA AND NAVIER STOKES EQUATIONS
2.1 Definition
Porous medium can be defined as a material consisting of a solid matrix with an
interconnected void. The solid matrix is either rigid (non-consolidated) or it undergoes
slight deformation (consolidated). The transport in the porous media is achieved by the
flow of fluid in the voids that are interconnected.
In single-phase flow, a single fluid saturates the voids and in two-phase flow a
liquid and a gas share the void space. The phases are chemically homogenous, separated
by a physical boundary. In nature there are many materials that can be defined as porous
medium and some examples of these are: wood, the human lung, Soils, etc.
2.2 Mechanics of Fluid Flow in Porous Media
The flow parameters in the porous medium will clearly be irregular at the pore
scale (microscopic) level and therefore are investigated at the macroscopic level. The
quantities of interest are measured over areas that cross many pores and such quantities at
the macroscopic level are known to change in a regular fashion with respect to space and
time. Treating the medium at macroscopic level and obtaining the governing equations as
a continuum model achieve the laws governing the flow in porous media.
The phases in disjoint domains are replaced by a continuum, which fills the entire
domain for each phase. For each point in the flow domain, representative elementary
volumes (REV) are chosen and phase variables are averaged over the REV’s. These
averaged values are called the macroscopic values. The length scale of the REV is much
4
larger than the pore scale, but considerably smaller than the length scale of the flow
domain.
2.3 Porosity (  )
The porosity (  ) of a porous medium is defined as the fraction of the total
volume that is occupied by the void spaces i.e.; the total void volume divided by the total
volume occupied by the solid matrix and void volumes. (1-  ) represents the fraction that
is occupied by the solid. In defining the porosity we are assuming that all the void spaces
are interconnected but in reality some of them may be connected to only one other pore
(dead end) or not connected to any other pore (isolated) and in this case we need to define
effective porosity which is the volume fraction of the inter connected pores. In nonconsolidated media like, particles that are loosely packed the effective porosity and the
porosity are equal. For natural media the porosity does not exceed 0.6 .For packed bed of
spheres, it varies from 0.245 to 0.476.
2.4 Solid Matrix Structures
Based on the simplicity of the matrix structures we can classify it as follows:
1. Very long straight cylinders (circular and other cross sections).
2. Spheres (and other three dimensional particles).
3. Short and long fibers (circular and other cross sections)
4. Rocks, silicon gels, coal, fabrics, etc., have complex topology and in
general very difficult to analyze.
2.5 Stokes Flow and Darcy Equation
The first experiments of calculating the bulk resistance to flow of an
incompressible fluid through a solid matrix was measured by Darcy (1856). He
5
conducted his experiment on non-consolidated (loosely packed particles), uniform, rigid
and isotropic solid matrix. The macroscopic flow was steady, one-dimensional, and
.
driven by gravity. The mass flow rate m was measured and the filtration velocity or
seepage velocity was determined by dividing the mass flow rate by the product of the
fluid density and the cross-section area A of the channel i.e.;
.
m
uD 
A
[2.1]
By applying a volumetric force balance to this flow, he discovered that the
viscosity of the fluid and the permeability of the solid matrix characterize the bulk
resistance to the fluid flow such that
- P 

uD
k
[2.2]
The permeability of the solid phase k is independent of the nature of the fluid but
it depends on the geometry of the medium. It has the dimensions (length) 2 and is called
the specific permeability or intrinsic permeability of the medium. In case of single-phase
flow it is termed as permeability. The unit for permeability is Darcy. One unit of Darcy is
equivalent 9.87×10-13 m2. The Darcy model has been examined extensively and it is not
followed for liquid flows at high velocities and for gas at very low and very high
velocities.
In three dimensions, Eq. [2.2] is generalized as
P 

v
K
[2.3]
6
where K is now a second order tensor. For an isotropic medium the pressure gradient p
and the velocity vector v are parallel and permeability is a scalar .In this case Eq. [2.3]
reduces to
P 

v
k
[2.4]
Typical values of permeability are 10-9 to 10-22 for clean sand 1.910-12 to 2.41016
for spherical packing (well shaken).
2.6 Permeability
The permeability K of the porous medium is a measure of the flow conductance
of the matrix. Various models are used for the determining the relationship between
permeability and the matrix property parameters (porosity and other structure variables).
The most widely used among them are the Capillary models and the Hydraulic radius
model. In case of packed bed of spheres the Hydraulic radius model is most widely used
for determining the relationship between the porosity and the permeability of the
medium. . The effective particle diameter (Dp) is determined and using the hydraulic
radius theory of Carmen-Kozenzy which leads to the relationship:
K
D2p 3
[2.5]
180(1 )2
where the constant 180 is obtained by seeking the best fit with experimental data.
The Carmen-Kozenzy equation gives satisfactory results for particles that are of spherical
shape and whose diameters fall within a narrow range and is widely used.
7
2.7 Equation of Continuity
The conservation of mass applied to the porous media as a continuum model;
based on the R.E.V concept is defined as follows:

f
 .(f v)  0
t
[2.6]
The equation is obtained by considering the mass flux into a representative elementary
volume to the increase of the fluid within the volume.
2.8 Momentum equation:
For low filtration velocities the Darcy equation (Eq. [2.4]) can be used for each
R.E.V in the domain.
Darcy’s equation is linear to the filtration velocity and is valid when the filtration velocity
is small or the Rep (Reynolds number based on the particle diameter) is of the order of
unity or smaller. As u D increases the form drag due to the solid is considerable as
compared to the surface drag due to friction and transition to non-linear drag is quite
smooth in the range of Rep 2 to 20.Thus the Darcy equation does not hold true and the
equation is replaced by an modification of the Darcy equation to take into account form
drag.
The modified equation also known as the Darcy Forchheimer equation is as
shown below:
P 

v  CF K 0.5 f | v | v
K
[2.7]
where CF is the dimensionless form drag constant or the Ergun coefficient
8
and the term CF K 0.5 f | v | v is also called the Forchheimer term and is proportional to
the square of the filtration velocity and hence the name quadratic drag. The value of C F
varies with the nature of the porous medium and is not a constant. The deviation of the
Darcy law begins when ReK0.5 (Reynolds Number based on the square root of
permeability) is greater than or equal to 0.2.
2.9 Energy Equation for Porous Medium
The energy equation of an isotropic porous medium neglecting viscous
dissipation, radiative effects and work done by pressure changes can be defined for an
elemental volume of the medium as follows:
for solid phase,
s
(1 )(c)s
 (1 ).(k s Ts )  (1 )q ''' ,
[2.8]
t
and for the fluid phase
T
[2.9]
 (cp )f f  (cp )f u D .Tf .(k f Tf )  q f'''
t
where the subscripts s and f refer to the solid and fluid phases respectively. c is the
specific heat of the solid and c p is the specific heat at constant pressure of the fluid; k is
the thermal conductivity,  is the porosity of the medium and q ''' [W/m3 ] is the heat
production per unit volume.
Considering local thermal equilibrium (LTE) between the solid and the fluid
phases i.e.; where the solid and the fluid phase temperature are equal (Ts = Tf = T) and
setting the same in equation 2 and 2 we get:
(c) m
T
 (c)f u D .T  .(k m T) q '''m
t
where
9
[2.10]
(c)m  (1 )(c)s   (c P )f ,
k m  (1 )k s   k f ,
q '''m  (1 )qs'''  q '''f
is the overall heat capacity per unit volume, overall thermal conductivity, and overall heat
production per unit volume of the medium.
2.10 Overall Thermal conductivity of the porous medium:
When the heat conduction between the two phases is in parallel, then the overall
thermal conductivity is the weighted arithmetic mean of the conductivities of the two
phases i.e.;
k e  (1 )k s  k f
[2.11]
If the heat conduction takes place in series, with all the heat flux passing through
both the solid and the fluid, then the overall thermal conductivity is the Harmonic mean
of the two conductivities.
1 1  


ke
ks kf
[2.12]
where ke = effective thermal conductivity of the porous medium.
The assumption of Local thermal equilibrium is not valid when there is a
substantial temperature difference between the two phases or when the thermal
conductivities of the two phases are very different. In this case of local thermal nonequilibrium (LTNE) between the two phases, the energy equation is replaced by the
following two equations:
10
for solid phase,
(1 )(c)s
Ts
 (1 ).(k s Ts ) (1 )q s''' h(Tf  Ts ),
t
[2.13]
and for the fluid phase,
 (c p )f
T
 (c p )f u D .Tf  .(k f Tf )  q f'''  h(Ts  Tf ) ,
t
[2.14]
where
h is the heat transfer coefficient.
2.11 Thermal Dispersion
In case of forced convection in the porous medium, there would be significant
heat transfer due to hydrodynamic mixing of interstitial fluid at the pore level called
thermal dispersion. In addition to the molecular diffusion of heat, there is a mixing due to
nature and geometry of the porous medium. Hydrodynamic Mixing can occur due to any
of the following reasons:
1. Mixing due to obstructions,
2. Recirculation caused by local regions of reduced pressure arising from flow
restrictions,
3. Within a flow channel mixing can occur between fluid particles moving at
different velocities,
4. Eddies that form when the flow becomes turbulent and many more.
5. Thus, dispersion is a very complex phenomenon and plays a very important role
in forced convection in packed columns. The steep temperature gradients that
exist near heated or cooled walls or attributed to not only the channeling effect but
11
also to the thermal dispersion effect. A number of investigations were done to
consider the effect of thermal dispersion on fluid flow (Hsu and Cheng 1990) and
their results show that transverse dispersion is much more important than the
longitudinal dispersion (Alazami and Vafai, 2000, p.303).
12
CHAPTER 3
POROUS MEDIUM AS HEAT EXCHANGER
3.1 Introduction
Porous media has diverse engineering applications, which include thermal insulation,
packed bed heat exchanger, geothermal industries and electronic cooling.
Of particular interest is the innovative use of a porous matrix as potential heat
exchanger in electronic cooling. This report is on the use of porous medium as an
electronic cooling device. The problem of forced convection or the non-darcian transport
in porous media is most relevant to packed bed of spheres or solid matrix heat
exchangers. A packed bed of spheres as a potential heat exchanger is analyzed here.
The assumption of Local thermal equilibrium is widely used in analyzing the
transport process in porous media. In such a packed bed operated under steady state
conditions, a difference in the local temperatures between the fluid and the particle may
exist, but the overall solid and fluid temperature profiles are considered to be identical to
each other. In estimating the overall steady-state temperature profile, the heterogeneous
packed bed may be considered to be a homogenous single phase. The temperature
profiles in the bed are then predicted in terms of the effective thermal conductivities and
wall heat transfer coefficients.
However the assumption of local thermal equilibrium does not hold true when
there is internal heat generation or when there is a substantial difference between the
solid and fluid temperatures (or the difference between the two thermal conductivities are
very significant). A porous media provides a solid-air contact area many more than the
duct surface area. Using this concept the experimental/numerical results of high
13
performance porous channel using Sintered beads (Hwang & Chao, 1994; Hwang, Wu &
Chao, 1995) are evaluated numerically using FLUENT code. Furthermore, the model was
extended using the same configuration for higher thermal conductivity ratio material
namely Copper beads too. The porous channel of dimensions 511 cm, is made up of
sintered bronze/Copper beads of two different diameters Dp = 0.72 and 1.59 mm.
3.2 Forced Convection in Porous Media
Forced convection in porous materials is known to increase the heat transfer
coefficient three to four times as compared to slug flow in a clear duct. The thermal
dispersion caused by the presence of solid matrix plays a key role in this heat transfer
augmentation.
A porous medium will provide solid–air contact many times greater than the duct
surface area. The total heat transfer rate will be increased by several orders in spite of the
lower heat transfer coefficient on the solid-air contact surface.
There are two modes of forced convection in a porous media one considering a
Local thermal equilibrium between the two phases and the second considering a local
thermal non equilibrium between the solid and the fluid phases, which are appropriately
named as LTE and LTNE respectively.
The effect of thermal dispersion is also very important and it has been proven that
the effect of transverse dispersion is of more significance than the longitudinal dispersion
(Amiri and Vafai, 2000). Since there exists a large difference in the thermal
conductivities of the two phases, the LTNE conditions are used for modeling the flow.
14
3.3 Formulation of the Problem
A parallel plate channel filled with spheres which is homogenous and isotropic is
modeled using the configuration shown below:
q’’
U
H = 1cm
T
L = 5cm
Fig. 3.1: Schematic diagram of the model.
3.4 Assumptions
The assumptions for the present investigation are as follows:
1. The flow is steady and incompressible.
2. The properties of the fluid and the porous media are assumed constant.
3. The thermal properties of the fluid vary but the properties of the porous matrix
are assumed to be constant.
4. Fully developed flow is introduced at the inlet.
3.5 Governing Equations for the Model
3.5.1 Momentum Equation
The momentum equation for the model is given by the equation below:
0
 F


V f
[ V  .  V ]   2  V    P 
K

K
15
[3.1]
where K is the permeability of the porous media and F is the inertial coefficient. The first
and the second term on the RHS are the momentum source terms for the porous media.
The equation neglects the convective term as the flow is fully developed.
3.5.2 Energy Equation
Due to high solid to fluid conductivity ratio, the temperature difference between the
solid and air is not small and hence the LTNE equations are applied here.
The Fluid phase equation under steady state conditions is:
(Cp )f  V  .  Tf  .{k feff .  Tf }  h sf a sf ( Ts    Tf )
[3.2]
and the solid phase energy equation under steady state condition is:
.{k seff .  Ts }  h sf a sf ( Ts    Tf )  0
where
k feff = Effective thermal conductivity of fluid (  ×Kf).
k seff = k s + k d ,
k s = Thermal conductivity of the solid phase ((1-  ) ×Ks)
k d = Dispersion conductivity of the fluid.
h sf = Internal heat transfer coefficient
a sf = Fluid to solid specific area
16
[3.3]
3.6 Heat Transfer Coefficient and Fluid to Solid Specific Area Correlations
Two models are used for finding the heat transfer correlation and the solid specific
area correlations as described below:
1. The first correlation is a general correlation, which can be used for all Particle
Reynolds Number (Amiri and Vafai, 1998):
hsf  kf (2  1.1Pr 0.333 Re0.6 ) / dp
[3.4]
and the area correlation for this model is given by,
a sf  6(1 ) / d p
[3.5]
2. The second correlation (Hwang and Chao, 1994) used in the model is as follows:
h sf  0.0040(k f / d v )  Prf0.333  Re1.35 (Re d  100)
[3.6]
h sf  0.0156  (k f / d v )  Prf0.333  Re1.04 (Red  100)
[3.7]
a sf  20.346  (1 ) 2 / d p
[3.8]
and
3.7 Nusselt Number Correlations:
The Nusselt Number at each axial location of the flow is defined by the following
correlation:
Nu fx 
hDe
kf
[3.9]
h=
heat transfer correlation
[3.10]
where
q = heat input per unit area
Tw = Temperature at Wall
17
Tfb = Fluid bulk temperature
3.8 Thermal Dispersion Conductivity Correlation
Thermal dispersion plays a particularly important role in forced convection as discussed
previously in section 2.10. Cheng and his colleagues [4] assumed that the local transverse
thermal dispersion kd is given by
 U 
[3.11]
k d  k f Dt Ped l 

U
 m
where
Dp
, Peclet number is in terms of the mean seepage velocity Um, the
Ped = Um
f
particle diameter Dp and fluid thermal diffusivity  f , while Dt is a constant and l is a
dimensionless dispersive length normalized with respect to Dp .
The dispersive length is modeled by a wall function of the Van Driest type:
 Y 
1
l  1  exp 
0  Y  H
2
 Dp 
 (H  Y)  1
l  1  exp 
 HYH
 Dp  2
[3.12]
Replacing  f with
kf
in Ped and substituting in Eq. [3.11] we get,
C p f
k d  D t (c p ) f D p Ul
[3.13]
where
U=
l
u
2
 v2 
= Van Driest type wall function as defined above.
18
 = Empirical constant = 1.5
Dp = Diameter of the particle
Dt = Thermal dispersion constant = 0.375
Y = y coordinate
19
CHAPTER 4
NUMERICAL MODELING OF POROUS MEDIA IN FLUENT
4.1 Introduction
The porous media model in FLUENT can be used for a wide variety of problems,
including flows through packed beds, filter papers, perforated plates, flow
distributors, and tube banks. In this model, we define a cell zone in which the porous
media model is applied and the pressure loss in the flow is determined via user inputs.
Heat transfer through the medium can also be represented, and is subject to the
assumption of thermal equilibrium between the medium and the fluid flow.
A 1D simplification of the porous media model, termed the ``porous jump,'' can
be used to model a thin membrane with known velocity/pressure-drop characteristics.
The porous jump model is applied to a face zone, not to a cell zone, and should be used
(instead of the full porous media model) whenever possible because it is more robust and
yields better convergence.
4.2 Limitations of Porous Media Model in FLUENT
The porous media model defined in FLUENT incorporates an empirically determined
flow resistance in a region of the model defined as ``porous''. In essence, the porous
media model is nothing more than an added momentum sink added to the governing
momentum equations.
20
1. The fluid does not accelerate as it moves through the medium, since the volume
blockage, which is present physically, is not represented in the model. This may
have a significant impact in transient flows since it implies that the transit time for
flow through the medium is not correctly represented by FLUENT.
2. The effect of the porous medium on the turbulence field is only approximated.
3. The model solves the energy equation based on the LTE model for energy.
4.3 Momentum Equations for Porous Media in FLUENT
Porous media are modeled by the addition of a momentum source term to the
standard fluid flow equations. The source term is composed of two parts, a viscous loss
term (Darcy), and an inertial loss term:
3
3
1
Si   Dijv j   Cij  | v j | v j
2
j1
j1
[4.1]
where Si is the source term for the ith (x, y) momentum equation, and D and C are
prescribed matrices. This momentum sink contributes to the pressure gradient in the
porous cell, creating a pressure drop that is proportional to the fluid velocity (or velocity
squared) in the cell.
In case of simple homogeneous porous media Eq. [4.1] reduces to:
Si 

v i  C2  0.5   | v i | v i
K
[4.2]
where K is the permeability and C2 is the inertial resistance factor.
21
The permeability and the inertial resistance coefficient for each cell zone are
computed as a material property and then used in equation as source terms for the
momentum equation in the x and y directions.
4.4 Energy equation for porous media in FLUENT
The energy equation that FLUENT solves, makes modifications to the conduction
flux and the transient terms only of the standard energy equation:


 
T 

( f h f  (1 )s h s ) 
 f u i h f    k eff   
t
x i
x i 
x i 
x i
u
 ik i   Sfh  (1 )Ssh
x k
Dp
 h J   Dt 
j j
j
[4.3]
where hf is the fluid enthalpy, hs is the solid medium enthalpy,  is the porosity of
the medium, keff is the effective thermal conductivity of the medium, S fh is the fluid
enthalpy source term and Ssh is the solid enthalpy source term.
4.5 Effective Thermal Conductivity of Porous Medium
The effective thermal conductivity in the porous medium is computed by
FLUENT as the volume average of the fluid conductivity and the solid conductivity:
k eff  k f  1  ks
[4.4]
When this simple volume averaging is not desirable, perhaps due to the effects of
the medium geometry, the effective conductivity can be computed via a user-defined
22
function. In all cases, however, the effective conductivity is treated as an isotropic
property of the medium.
In the porous medium, the conduction flux uses an effective conductivity and the
transient term includes the thermal inertia of the solid region on the medium. Also
FLUENT uses a single equation for the computation of the energy equation considering
local thermal equilibrium between the solid and fluid phases. The above assumption hold
true for the following conditions:
1.
When the variation in temperature between the two phases is not much.
2.
When there is no internal heat generation.
Hence, in order to model the local thermal non-equilibrium condition (LTNE) due
to variations in the stagnant thermal conductivities of the solid and fluid phases, we need
to make use of two equations, one for the solid phase and the other for the fluid phase.
For the fluid phase the energy equation is:
(Cp )f  V  .  Tf  .{k feff .  Tf }  h sf a sf ( Ts    Tf )
[4.5]
and for the solid phase the energy equation is:
.{k seff .  Ts }  h sf a sf ( Ts    Tf )  0
[4.6]
These two equations need to be incorporated in FLUENT to make use of the LTNE
condition and the same are modeled using UDFs in FLUENT.
4.6 Effective Thermal Conductivity of Porous Medium Considering LTNE Between
the Two Phases
23
The effective thermal conductivities in the porous medium for both the solid
phase and the fluid phase’s kseff and kfeff respectively are given by the following
correlations:
k seff = (1 -  )k s .
[4.7]
k feff =  k f + k d
[4.8]
where, ks kf are the solid and fluid thermal conductivities respectively and  is the
porosity and kd is the dispersion conductivity.
4.7 Convective Heat Transfer Coefficient and Interstitial Area Correlations:
The two correlations for convective heat transfer coefficient (hsf) and interstitial
area (asf) as described in equations 3.4 through 3.8 are used for modeling the convective
heat transfer coefficient and fluid to solid area correlation.
4.8 Developing the Model in FLUENT
The porous media model defined in FLUENT doesn’t effectively make use of the
set of equations that govern the porous media transport equations considering LTNE
between the two phases. Hence, in order to make use of the transport equations
effectively the momentum and the energy equations for the porous media model are
defined by making use of User Defined Functions.
4.9 User defined Functions (UDF) in FLUENT
24
User-defined functions (UDFs) are functions that one can write to enhance the
standard features of FLUENT. UDFs are written in the C programming language and
have two different modes of execution: interpreted and compiled. Interpreted UDFs are
simpler to use but have coding and speed limitations. Compiled UDFs execute much
more quickly and have no coding limitations, but require more effort to set up and use.
We can use UDFs to customize:
1. Boundary conditions
2. Source terms
3. Property definitions (except specific heat)
4. Surface and volume reaction rates
5. User-defined scalar transport equations
6. Discrete phase model (e.g., body force, drag, source terms)
7. Algebraic slip mixture model (slip velocity and particle size)
8. Solution initialization
9. Wall heat fluxes
10. Post processing using user-defined scalars
UDFs assign values to individual cells and cell faces in Boundary and fluid zones. In
UDF zones are referred to as threads. A looping macro is used for accessing individual
cells in a thread. Thread and variable references are passed automatically to the UDF
when assigned to the boundary in GUI. The values that are returned by the UDF should
be in SI units. Basic Steps for using UDFs in FLUENT are as follows:
25
1. Create a file containing the source code
2. Start the solver and read in the Case or data file.
3. Interpret or compile the UDF
4. Assign the UDF to the appropriate variable and zone in the BC panel.
5. Set the UDF update frequency in the Iterate panel.
6. Run the calculation
4.10 Macros
Macros are predefined functions. They allow the definition of the UDF
functionality and the name of the function. It allows access to field variables, cell
information etc.
The macros are defined in the header files and one of the most important header
files is the “udf.h” header file. Each file must contain this header file and is included in
the file using the #include preprocessor macro. Other header files may be included
depending on the need for it. Some of the header files that are typically used in the
program are “mem.h” for field variable access and “metric.h” for cell geometry data.
Any UDF that are written must begin with a DEFINE macro; there are eleven
general-purpose macros and ten discrete phase modeling (DPM) macros that are available
for the user. For the case of porous media modeling, most of the eleven general-purpose
macros were made use of. Some the general-purpose macros and their definitions can be
found in Appendix.
26
4.11 Looping Macros
There are specialized variables that are used in looping macros and these are:
1. cell_t c defines a cell.
2. face_t f defines a face.
3. Thread *t pointer to a thread.
4. Domain *d pointer to a collection of all threads.
Some of the looping macros that were used in the UDFs and there descriptions are given
below:
1. Thread_loop_c (t,d){ } loop that steps through all the cells in the
domain.
2. thread_loop_f (t,d){ } loop that steps through all the faces in the
domain.
3. begin_c_loop(c,t) { } end_c_loop(c,t) loops through all the cells in the
thread.
4. begin_f_loop(c, t) { } end_f_loop(c,t) loops through all the faces in
the thread.
Code enclosed in the {…} are executed in the loop. Apart from these there are other
macros like Geometry and time macros, Cell field variable macros, Face field variable
macros.
27
4.12 User Defined Scalars
FLUENT can solve the generic transport equations for any user-defined scalars.
FLUENT can solve the transport equation for any arbitrary, user-defined scalar (UDS) in
the same way that it solves the transport equation for a scalar such as species mass
fraction. Extra scalar transport equations may be needed in certain types of applications
for example; in this case the solid phase temperature is solved with the help of a userdefined scalar. For an arbitrary scalar k, FLUENT solves one of three equations,
depending on the method used to compute the convective flux:
1. If convective flux is not to be computed, FLUENT will solve the equation:


x i
 k 
 k
  Sk k  1...n
 x i 
[4.9]
where k and Sk are the diffusion coefficient and source term supplied by the user for
each of the N scalar equations.
2. If convective flux is to be computed with mass flow rate, FLUENT will solve the
equation:
k 
 
 u i k  k
  Sk k  1...n
x i 
x i 
[4.10]
3. It is also possible to specify a user-defined function to be used in the computation
of convective flux. In this case, FLUENT solves the equation:
28

x i

k 
 Fi k  k
  Sk k  1...n
x i 

[4.11]
where, Fi is the user-defined flux.
One important feature in FLUENT is that the user-defined scalars are solved only in fluid
cells, not in solid cells.
4.14 Problem Solving Strategy:
FLUENT solvers are based on the finite volume method. The domain is
discretized into finite set of control volumes or cells and the governing transport
equations are solved for each cell.
The general transport equation for mass, momentum and energy is shown below:

dV   V.dA 
A
t v

A
.dA   SdV
[4.12]
v
i.e.
Unsteady term + convection term = diffusion term + Source term
where  = 1 for continuity u for x-momentum v for y-momentum and h (CpdT) for
energy.
FLUENT solves these equations by discretizing into algebraic equations, and all
equations are solved for the solution of the flow field.
29
The governing equations of the porous media model considering LTNE
conditions are as follows:
The momentum equation under steady state condition:
0
 F


V f
[ V  .  V ]   2  V    P 
K

K
[4.13]
It is the standard momentum equation with addition of a momentum source term to the
standard fluid flow equations. The source term is composed of two parts, a viscous loss
term (Darcy), and an inertial loss term as discussed previously. These source terms are
added in to the standard flow equation of FLUENT as x and y momentum source terms
developed using UDF macro DEFINE_SOURCE (name, cell, thread, ds, index).
4.14 The Energy Equation:
The energy equation for the fluid phase is as follows:
(Cp )f  V  .  Tf  .{k feff .  Tf }  h sf a sf ( Ts    Tf )
{Convective term}
{Diffusion term}
[4.14]
{Source term}
The convective term and the diffusion term are automatically taken care of by the
FLUENT solver, but the diffusion term consists an effective thermal conductivity which
needs to be modified and the source term need to be incorporated into the model. The
effective thermal conductivity for the fluid phase (kfeff) is incorporated in the model using
30
the function macro: DEFINE_PROPERTY (name, cell, thread, index) and placed as a
UDF in the thermal conductivity panel of the material panel.
And the source term is incorporated in the model using the DEFINE_SOURCE
(name, cell, thread, dS, index) macro and added as a user defined source term for the
standard energy equation for the fluid phase.
The following relation gives the solid phase energy equation:
.{k seff .  Ts }  h sf a sf ( Ts    Tf )  0
{Diffusion term}
[4.15]
{Source term}
In order to model this equation a user defined scalar needs to be incorporated to
standard flow equations for the solid phase. The user-defined scalar (UDS) represents the
solid phase temperature. The diffusion and the source term are modeled using the
DEFINE_DIFFUSIVITY (name, thread, cell, index) and DEFINE_SOURCE (name,
thread, cell, index) user defined functions. These macros are added as Diffusivity term in
the material panel and UDS energy source in the fluid panel of FLUENT.
The UDF’s have been developed in C language and compiled the same into a
library to be used in the individual case files. A listing of the source code is given in
Appendix-D.
31
Chapter 5
Mesh Generation
5.1 Introduction
In order to build a computational solution, the spatial domain of interest must first be
discretized to form a grid. A mesh is the discretization of the domain into small shapes
called elements. A structured mesh is usually a warped grid of boxes, while an
unstructured mesh is typically a triangulation. On this grid the problem of interest is
numerically solved using any of the numerical codes, and the results are combined to get
the field solution. There are four distinct operations in CFD which are: creating the
surface geometry definition, discretizing the flow domain (grid generation), computing
the solution to the equations of fluid dynamics (Flow solution) and Postprocessing.
5.2 Computational Steps in CFD
The computational steps traditionally taken for Computational Fluid Dynamics
(CFD), Structural Analysis, and other simulation disciplines (or when these are used in
design) are:
1. Surface Generation
The surfaces of the object(s) are generated usually from a CAD system. This
creates the starting point for the analysis and is what is used for manufacturing.
2. Grid Generation
These surfaces are used (with possibly a bounded outer domain) to create the
volume of interest. Both steps 1 and 2 are developed using GAMBIT. Gambit is a
32
preprocessing tool that sets up the geometry and generates grid for Fluent. It has
both structured and unstructured grid capability. It provides a single unified
interface for all Fluent's preprocessing methods. GAMBIT provides a concise and
powerful set of solid modeling-based geometry tools for generating meshes and
grids to facilitate CFD simulations. Different CFD problems require different
mesh types, and GAMBIT gives all the options that are necessary in a single
package. It is a part of Fluent software.
3. Flow Solver or Simulation
The solver takes as input the grid generated by the second step (and
information about how to apply conditions at the bounds of the domain). Because
of the different styles of gridding, the solver is usually written with ability to use
only one of the volume discretization methods. In fact there are no standard file
types, so most solvers are written in close cooperation with the grid generator. For
fluids, the solver usually simulates either the Euler or Navier-Stokes equations in
an iterative manner, storing the results either at the nodes in the mesh or in the
element centers. The output of the solver is a file that contains the solution.
4.
Post-processing Visualization
After the solution procedure has successfully completed, the output from
the grid generator and the simulation are displayed and examined in a graphical
manner by the fourth step in this process. Usually a workstation with a 3D
graphics adapter is used to quickly render the output from data extraction
techniques. The tools (such as iso-surfacing, geometric cuts and streamlines)
allow the examination of the volumetric data produced by the solver. Even for
33
steady-state solutions, much time is usually required to scan, poke and probe the
data in order to understand the physics in the flow field. Both steps 3 and 5 are
done using FLUENT solver. FLUENT solvers are based on the finite volume
method.
Fig. 5.1 shows the integration of the computational steps in CFD.
CAD  GRID  GENERATION  SOLVING  VISUALIZE
Fig. 5.1: Computational Steps in CFD
Both Gambit and FLUENT are part of the FLUENT suite of CFD software.
Fig.5.2 shows the integration of FLUENT software codes.
GAMBIT
Geometry
or Mesh
Other CAD/CAE
Packages
Geometry setup
2D/3D mesh generation
Boundary
Mesh
2D/3D
mesh
FLUENT
PrePDF
PDF files
Mesh import and
adaptation
Phyisical models
B.C’s
Material
properties
Calculation
Post processimg
Boundary
and Volume
mesh
TGrid
Mesh
2D triangular mesh
etc.
Mesh
Fig. 5.2: The integration of FLUENT software code.
34
5.3 Mesh Components
Computational domain is defined by mesh that represents the fluid and solid regions
of interest. The mesh (2D or 3D) consists of the following components:
1. Cell = control volume into which domain is broken up
2. Face = boundary of a cell
3. Edge = boundary of a face
4. Node = grid point
5. Zone = grouping of nodes, faces, and/or cells
The individual components of the mesh are shown schematically in the figures 5.3 and
5.4.
node
cell
center
face
cell
Fig. 5.3: Simple 2D mesh
node
edge
cell
face
35
Fig. 5.4: Simple 3D mesh
5.4 2D Grid Generation
The rectangular channel of dimension 511 cm with one side heated and other
three insulated was modeled in Gambit as 2-D parallel plate channel of dimension 51
cm and the domain was discretized into structured grid elements for mesh generation. A
rectangular domain of above said dimensions was discretized using structured grid
generation tools of GAMBIT pre-processing tool to generate mesh of size 500 and 1000
cells. The interior of the domain was designated as fluid zone and the appropriate
boundary types were assigned to the extents of the domain. Grids of different mesh sizes
were developed to check for the grid independence as shown in the following figures.
Fig. 5.5: Grid of dimension 51 cm and size 1050 cells
36
Fig. 5.6: Grid of dimension 51 cm and size 10100 cells
Fig. 5.7: Grid of dimension 51 cm and size 10100 cells with boundary adaptation of
two cells on either side of the walls.
5.5 Grid Independence
Before starting of the numerical analysis an appropriate grid-size needs to be
incorporated for further analysis. Hence, the model of different grid sizes was imported
into FLUENT 5.5 and was subjected to numerical analysis.
37
Models having grid sizes 1050 cells, 10100 cells and 10100 cells with boundary
adaptation of two cells on either sides of the wall were subjected to the analysis. The
parameters that were used for the model were the same and plots of temperature profiles
were obtained for various grids. The graphs obtained are shown below from fig.5.8 to fig.
5.12. A comparison of the temperature profiles of various grids (fig. 5.11 - 5.12) shows
that there is no much variation between the temperature profiles. Hence, the higher order
grid with boundary adaptation (for better results at the boundary) was adopted for further
simulations.
Fig. 5.8: Temperature distribution for porous channel (Dp = 0.72 m.m) at 110 particle
Reynolds Number (inlet velocity = 3.35m/s) with input heat flux of 2.5 W/cm2 (Net heat
flux = 1.9 W/cm2) and E1 model for HTC. (Grid size: 1050 cells)
38
Fig. 5.9: Temperature distribution for porous channel (Dp = 0.72 m.m) at 110 particle
Reynolds Number (inlet velocity = 3.35m/s) with input heat flux of 2.5 W/cm2 (Net heat
flux = 1.9 W/cm2) and E1 model for HTC. (Grid size: 10100 cells)
Fig. 5.10: Temperature distribution for porous channel (Dp = 0.72 m.m) at 110 particle
Reynolds Number (inlet velocity = 3.35m/s) with input heat flux of 2.5 W/cm2 (Net heat
flux = 1.9 W/cm2) and E1 model for HTC. (Grid size: 10100 cells with boundary
adaptation)
39
Fig. 5.11: Variation of temperature profiles between two grids of size1050 cells and of
size 10100 cells.
Fig. 5.12: Variation of results between two grids of 1050 cells and size 10100 cells
with boundary adaptation.
40
CHAPTER 6
ANALYSIS
6.1 Introduction
The model was analyzed for heat transfer calculation for two different particle diameters
and for two models of solid to fluid heat transfer correlations and their respective fluid to
solid specific areas. The temperature profiles were obtained for heat fluxes of 0.8, 1.6, 2.4
and 3.2W/cm2 with air velocities ranging from 1.3 m/s to 6.437 m/s. The inlet pressure
(Operating pressure) was set to the prescribed value as described in the experimental
setup and the corresponding fully developed velocity profile was imposed on the inlet
(Hwang and Chao, 1994).
6.2 Model Parameters
The porous channel is made of two materials sintered bronze and copper beads;
with two different mean diameters of 0.72 mm and 1.69 mm. Subjecting the upper wall to
a constant heat flux (Hwang et.al, 1996, p.731) and varying the air velocity and inlet air
pressure between 1-3 atms the individual analysis are done. The physical parameters for
sintered bronze and copper beads are as shown in Table 6.1.
Table 6.1: Physical parameters for porous media model
SAMPLE
SINTERED BRONZE BEADS
COPPER BEADS
DAIMETER,
0.72
1.69
Dp [mm]
41
0.72
1.69
Table 6.1 continued…
Porosity, 
0.37
0.38
0.37
0.38
Permeability,
2.9
10.0
2.9
10.0
K1010 [m2]
(4.4)
(24.1)
(4.4)
(24.1)
Inertial
0.242
0.118
0.242
0.118
(0.636)
(0.610)
(0.636)
(0.610)
10.666
10.287
260.11
246.14
coefficient, F
Effective
Thermal
conductivity,
kseff [W/mK]
NOTE: The permeability and the inertial coefficient values are the original values got
from the experimental setup; the values in the bracket are the Ergun model values
Table 6.2 shows the fluid to solid heat transfer and their respective specific area
correlation.
Table 6.2: Different models of heat transfer and specific area correlation
Model
Heat Transfer correlation
Specific area correlation
h sf  0.0040(k f / d v )  Prf0.333  Re1.35 (Re d  100)
a sf  20.346  (1 ) 2 / d p
E1
h sf  0.0156  (k f / d v )  Prf0.333  Re1.04 (Red  100)
E2
hsf  kf (2  1.1Pr 0.333 Re0.6 ) / dp
42
a sf  6(1 ) / d p
6.3 Boundary Conditions
Once the mesh is defined, the next step is to assign boundary conditions to the
model. For the purposes of this study, a constant value of density was specified for the
entire model. FLUENT recommends velocity inlet condition at inflow and out let
boundary condition in order to simulate incompressible flow .The velocity distribution is
at the inlet is fully developed.
6.3.1 Fully Developed Inlet Velocity Profile
A parabolic velocity inlet profile was introduces at the inlet .The centerline
velocity was taken to be 1.5 times the average velocity through the medium. The average
velocity was calculated from the mass flux and inlet pressure by the following
correlation:
.
v avg
vc 
m

f
[6.1]
3
v avg
2
where v avg is the average velocity through the porous media and v c is the
centerline velocity at inlet.
The velocity distribution at the inlet was developed using the following correlation:
  y 2 
profile(y)  v c 1    
 H 


[6.2]
43
where H is the height of the channel (0.01m) and y is the Y coordinates from the
center of the channel. The temperature of the fluid at inlet was maintained at 3000C and a
zero diffusion flux for the solid phase was incorporated at the inlet.
6.3.2 The Wall Boundary Conditions
The upper wall was subjected to a known constant heat flux (Hwang et al., 1996,
p.731) and the lower wall was kept insulated.
6.3.3 The Outlet Boundary Condition
Fluent suggests using outflow boundary condition with velocity inlet boundary
condition to simulate incompressible flow. An outflow boundary condition as defined in
FLUENT was imposed at the outlet. In this condition the diffusion flux of the flow
variables is set to zero in the principal direction of flow.
The zero diffusion flux condition applied at outflow cells means that the
conditions of the outflow plane are extrapolated from within the domain and have no
impact on the upstream flow. The extrapolation procedure used by FLUENT updates the
outflow velocity and pressure in a manner that is consistent with a fully developed flow
assumption, as noted below, when there is no area change at the outflow boundary.
The zero diffusion flux condition applied by FLUENT at outflow boundaries is
approached physically in fully developed flows. Fully developed flows are flows in
which the flow velocity profile (and/or profiles of other properties such as temperature) is
unchanging in the flow direction.
44
6.4 Solving the Simulation
The FLUENT 6.6 solver was used to solve the simulation. Using the UDFs that
were developed to do the simulation and plugging the appropriate UDFs in the Boundary
condition panel of the FLUENT solver, the simulations were conducted. The present
simulations were done on a grid of 1000 cells with boundary adaptation (two cells) at the
walls. The generalized heat transfer correlation (model E2) was used first to simulate the
results on different particle diameters and different particle Reynolds Number. Next the
results were simulated using the recommended heat transfer correlations (model E1)
described in the experiment (Hwang and Chow, 1994, p.461) on different particle
diameters and different Reynolds Number.
45
CHAPTER 7
RESULTS AND DISCUSSION
7.1 Introduction
The model was analyzed for two different particle materials (Sintered bronze beads and
Copper) each of two different particle diameters (0.72 and 1.59 mm) and was subjected to
two different heat transfer correlations (E1 and E2) and their respective fluid to solid
specific area correlations. The values of heat fluxes imposed at the upper wall are in
accordance with the numerical analysis (Hwang et al., 1995, p.731) conducted. The
model was extended to a higher thermal conductivity material namely copper with other
parameters same as that of SBB (Table 6.1) and the temperature profile was obtained.
The influence of particle Reynolds number (Rep), thermal conductivity of the material
and particle diameter (Dp) on temperature profile were analyzed and the results show
similarity with the results (Vafai and Alazami, 2000, p.319). The wall temperature
(heated section) profile for SBB matched with the results obtained with the experimental
results (Hwang and Chow, 1994, p.470) in the constant Reynolds Number region (X/De
>0.25 to X/De < 0.3).
7.2 Notation used for Case Files
SBBXXX = (Sintered bronze beads of particle diameter (0.72 or1.59))100
CUXXX = (Copper spheres of particle diameter (0.72 or 1.59)) 100
YYY Re = Particle Reynolds Number (Rep)
GEN = E2 model of Heat transfer correlation,
SPEC = E1 model of Heat transfer correlation
ZZZ = (heat flux at the boundary) 10
46
For example: SBB72110ReGEN24 indicates the case file is for sintered bronze beads of
particle diameter 0.72 mm with particle Reynolds Number 110. The heat transfer
correlation used is of type E1 with a heat flux at the boundary to be 2.4 W/cm2.
7.3 Results
The results obtained during the simulations were then plotted in terms of the wall
and porous zone temperatures. The temperature plots show the variation of temperature
along the principal direction of flow. The X-axis shows the dimensionless distance
parameter X/De where De is the hydraulic diameter (5H/3). The Y-axis of the temperature
plots shows the temperature in Kelvin.
The particle Reynolds Number (Rep) plots show the variation of particle Reynolds
Number (Rep) along the principal direction of flow w.r.t dimensionless distance
parameter (X/De).
7.3.1 Temperature and Rep plots for SBB of Dp 0.72mm.
Figs. 7.1 and 7.3 show the temperature distribution for sintered bronze beads of
diameter 0.72 mm at 110 particle Reynolds Number with 2.4 W/cm2 heat flux
imposed at the upper wall (The net heat flux being 1.9 W/ cm2) using E2 and E1
model of Heat transfer coefficient respectively. The particle Reynolds Number (Rep)
for the two case files are shown in figs 7.2 and 7.4. The E2 model (Fig.7.1) shows an
initial dip in wall temperature at inlet and then an increase in temperature along the
flow direction. In the case of E1 model (Fig. 7.3) the wall temperature increase
gradually from the inlet along the flow direction. This trend is followed for all of the
subsequent case files for SBB (Dp = 0.72 and 1.59 mm).
47
Fig. 7.1: Temperature distribution for porous channel (Dp = 0.72 mm) at 110 particle
Reynolds Number (inlet velocity = 3.34m/s) with input heat flux of 2.4 W/cm2 (Net heat
flux = 1.9 W/cm2) and E2 model for HTC. (Case file: SBB72110ReGEN24.cas)
Fig. 7.2: Plot of particle Reynolds Number. (Case file: SBB72110ReGEN24.cas).
48
Fig. 7.3: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34m/s) with input heat flux of 2.4 W/cm2 (Net heat
flux = 1.9 W/cm2) and E1 model for HTC. (Case file: SBB72110ReSPEC24.cas)
Fig. 7.4: Plot of particle Reynolds Number. (Case file: SBB72110ReSPEC24.cas)
49
The Particle Reynolds Number (Rep) plots (Fig. 7.2 and 7.4) show an initial scattering
(X/De < 0.25) due to the introduction of parabolic fully developed velocity profile at the
entrance. The exit section (X/De > 2.9) shows a drop in the particle Reynolds Number
(Rep) due to the outflow boundary condition imposed at the outlet i.e.; diffusive fluxes in
the principal direction of flow (X-direction) are set to zero. The rest of the case files both
for copper and SBB have similar pattern for the plots of particle Reynolds Number.
Fig. 7.5 and 7.6 shows the temperature distribution with an input heat flux of 3.2 W/cm2
(net heat flux of 2.4W/cm2) at particle Reynolds Number of 110 for E2 and E1 models of
HTC respectively.
Fig. 7.5: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34m/s) with input heat flux of 3.2 W/cm2 (Net heat
flux = 2.4 W/cm2) and E2 model for HTC. (Case file: SBB72110ReGEN32.cas)
50
Fig. 7.6: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34m/s) with input heat flux of 3.2 W/cm2 (Net heat
flux = 2.4 W/cm2) and E1 model for HTC. (Case file: SBB72110ReSPEC32.cas)
Fig. 7.7 and 7.8 show the temperature distribution of the porous channel at 75 particle
Reynolds Number and 2.4 W/ cm2 heat flux imposed at the upper wall for E2 and E1
models of HTC respectively. The E2 model shows a lower upper wall temperature profile
compared to the E1 model. Fig. 7.9 shows the particle Reynolds Number plot for the two
case files shown in Fig.7.7 and 7.8.
51
Fig. 7.7: Temperature distribution for porous channel (Dp = 0.72mm) at 75 particle
Reynolds Number (inlet velocity = 1.937m/s) with input heat flux of 2.4 W/cm2 (Net heat
flux = 1.7 W/cm2) and E2 model for HTC. (Case file: SBB7275ReGEN24.cas)
Fig. 7.8: Temperature distribution for porous channel (Dp = 0.72mm) at 75 particle
Reynolds Number (inlet velocity = 1.937m/s) with input heat flux of 2.4 W/cm2 and (Net
heat flux = 1.7 W/cm2) and E1 model for HTC. (Case file: SBB7275ReSPEC24.cas)
52
Fig. 7.9: Plot of particle Reynolds Number (SBB7275ReSPEC24.cas)
Fig. 7.10 and 7.11 show the wall temperature distribution for particle Reynolds Number
210 and heat flux of 2.4 W/cm2 imposed at the upper wall for E2 and E1 models of HTC
respectively. The temperature distributions match significantly for both the case files
depicted in Fig.7.10 and 7.11. Fig. 7.12 shows the particle Reynolds distribution for the
case files depicted in Fig. 7.10 and 7.11.A detailed discussion of wall temperature
profiles for SBB porous channel of particle diameter 0.72 mm are discussed in section
7.3.3.
53
Fig. 7.10: Temperature distribution for porous channel (Dp = 0.72mm) at 210 particle
Reynolds Number (inlet velocity = 7.4327m/s) with input heat flux of 2.4 W/cm2 and(Net
heat flux = 2.0 W/cm2) and E2 model for HTC. (Case file: SBB72175ReGEN24.cas)
Fig. 7.11: Temperature distribution for porous channel (Dp = 0.72mm) at 210 particle
Reynolds Number (inlet velocity = 7.4327m/s) with input heat flux of 2.4 W/cm2 and(Net
heat flux = 2.0 W/cm2) and E1 model for HTC. (Case file: SBB72175ReSPEC24.cas)
54
Fig. 7.12: Plot of particle Reynolds Number. (SBB72175ReGEN24.cas).
7.3.2 Temperature and Rep plots for SBB of Dp 1.59mm
Figs. 7.13 and 7.14 show the temperature plots obtained for SBB for particle diameter
1.59 mm for E2 and E1 model of HTC at 117 particle Reynolds Number and net heat flux
of 1.8 W/cm2
imposed at the upper wall. The immediate effect of increase in particle diameter using
E2 HTC can be seen in Fig 7.13, it shows a decrease in wall temperature profile with the
temperature difference near the upper wall (boundary) also to be small. The E1 model
tends to increase the wall temperature at the inlet as can be seen in Fig.7.14, thus
widening the temperature difference in upper wall temperatures. Fig. 7.15 shows the
Particle Reynolds Number plots for the case files depicted in Fig. 7.13 and fig 7.14.
55
Fig. 7.13: Temperature distribution for porous channel (Dp = 1.59mm) at 117 particle
Reynolds Number (inlet velocity = 1.55 m/s) with input heat flux of 2.4 W/cm2 (Net heat
flux = 1.8 W/cm2) and E2 model for HTC. (Case file: SBB159110ReGEN24.cas)
Fig. 7.14: Temperature distribution for porous channel (Dp = 1.59mm) at 117 particle
Reynolds Number (inlet velocity = 1.55 m/s) with input heat flux of 2.4 W/cm2 (Net heat
flux = 1.8 W/cm2) and E1 model for HTC. (Case file: SBB159110ReSPEC24.cas)
56
Fig. 7.15: Plot of particle Reynolds Number. (SBB159175ReGEN24.cas)
Similar plots of temperature profiles for different heat transfer correlations and particle
Reynolds number has been simulated and they can be found in Appendix B. The
comparisons of wall temperature profiles are in section 7.3.3.
7.3.3 Comparison of Wall temperature between E1 and E2 HTC for SBB
Figs. 7.16 shows the wall temperature distribution for particle diameter of 0.72 mm at
110 particle Reynolds Number and net heat flux imposed at the upper wall are 1.9 W/
cm2. The upper wall temperature profile obtained by E1 HTC shows a close resemblance
with experimental results of upper wall temperature profile (Hwang and Chao, 1994).
The lower wall temperature profile is the same for both the HTC.
57
Fig. 7.16: Wall temperature distribution for porous channel (Dp = 0.72mm) at 110
particle Reynolds Number (inlet velocity = 3.34m/s) with input heat flux of 2.4 W/cm2
(Net heat flux = 1.9 W/cm2) and E1, E2 model for HTC.
The temperature plot shows a variation of 7 to 10ºC between the upper wall temperature
profiles irrespective of the heat flux at the boundary.
Figs. 7.17 shows the wall temperature distribution for particle diameter of 0.72 mm at
110 particle Reynolds Number and net heat flux of 2.4 W/cm2 imposed at the upper wall.
The variation in upper wall temperature profiles between the two models of HTC is found
to be 7 to 10ºC.Due to the higher heat flux imposed at the upper wall, as is the case in
Fig.7.17, the wall temperature profiles show higher values than that of Fig.7.16.
.
58
Fig. 7.17: Wall temperature distribution for porous channel (Dp = 0.72mm) at 110
particle Reynolds Number (inlet velocity = 3.34m/s) with input heat flux of 3.2 W/cm2
(Net heat flux = 2.4 W/cm2) and E1, E2 model for HTC.
Fig. 7.18 shows the wall temperature distribution for the porous channel at 75 particle
Reynolds Number (Rep) and net heat flux of 1.7 W/cm2 imposed at the upper wall. The
variation in upper wall temperature profile between the two models of HTC is much more
significant (to the order of 12 to 15º C) between the upper wall temperature profiles. The
lower wall temperature profiles along the flow direction are the same for the both the
HTC.
Comparing Fig.7.18 with Fig.7.16, which have the same configurations except for the
particle Reynolds Number; we can deduce that the Reynolds Number plays a role in
variation of temperature profiles obtained by using the two HTC. The higher the particle
Reynolds Number the lower is the variation in wall temperature profiles between the two
HTC models. This is further substantiated by wall temperature profiles as shown in Fig.
7.19.
59
Fig. 7.18: Wall temperature distribution for porous channel (Dp = 0.72mm) at 75 particle
Reynolds Number (inlet velocity = 1.937m/s) with input heat flux of 2.4 W/cm2 and (Net
heat flux = 1.7 W/cm2) and E1, E2 model for HTC.
Fig. 7.19: Wall temperature distribution for porous channel (Dp = 0.72mm) at 210
particle Reynolds Number (inlet velocity = 7.4327m/s) with input heat flux of 2.4 W/cm2
and (Net heat flux = 2.0 W/cm2) and E1, E2 model for HTC.
60
Fig. 7.19 shows the wall temperature distribution for porous channel of SBB at 210
particle Reynolds Number and net heat flux of 2.0 W/cm2 imposed at the upper wall. The
variation in the upper wall temperature profiles between the two models of HTC (E1 and
E2) in this case is very less (to the order of 2 to 3º C).
The size of the particle used in the porous channel plays an important role in the variation
of temperature profiles obtained using the two HTC. Fig 7.20 shows the wall temperature
profile for the same configuration as of Fig.7.16 except for the increase in particle
diameter to1.59 mm. The variation in the temperature profile is much more significant in
this case (to the order of 19 to 21º C).
Fig. 7.20: Wall temperature distribution for porous channel (Dp = 1.59mm) at 117
particle Reynolds Number (inlet velocity = 1.55 m/s) with input heat flux of 2.4 W/cm2
and (Net heat flux = 1.8 W/cm2) and E1, E2 model for HTC.
Hence, larger particle diameter increases the variation in the temperature profiles
obtained by the two HTC.
61
Fig. 7.21: Wall temperature distribution for porous channel (Dp = 1.59mm) at 82 particle
Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 0.8W/cm2 and (Net
heat flux = 0.5 W/cm2) and E1, E2 model for HTC.
Fig. 7.21 shows the wall temperature distribution for SBB porous channel at 82 particles
Reynolds Number with a heat flux of 0.8 W/ cm2. Although the particle Reynolds number
is low (Rep = 82), the variation in temperature profiles is low (8 to 10C) due to the low
input of heat flux at the upper wall (Net heat flux = 0.5 W/cm2). Hence, it can be
concluded that heat flux imposed at the upper wall also plays a crucial role in variance of
temperature profiles among the models.
Fig. 7.22 and 7.23 show the variation in temperature profiles at 240 particle Reynolds
Number (Rep) and heat fluxes of 1.6 and 0.8 W/ cm2 imposed at the upper wall
respectively. The difference in upper wall temperature between the two models of HTC is
less (2 to 3C) in each of the plots of temperature profiles. The higher particle Reynolds
Number plays a crucial role in limiting the difference of temperature profiles between the
two models of HTC.
62
Fig. 7.22: Wall temperature distribution for porous channel (Dp = 1.59mm) at 240
particle Reynolds Number (inlet velocity = 3.301 m/s) with input heat flux of 1.6 W/cm2
(Net heat flux = 1.3 W/cm2) and E1, E2 model for HTC.
Fig. 7.23: Wall temperature distribution for porous channel (Dp = 1.59mm) at 240
particle Reynolds Number (inlet velocity = 3.301 m/s) with input heat flux of 0.8W/cm2
and (Net heat flux = 0.7 W/cm2) and E1, E2 model for HTC.
63
7.3.4 Discussion of results for Porous channel made of SBB
The E1 model of HTC correlation used gave an excellent match with the
experimental wall temperature profiles. The particle Reynolds number played a crucial
role in the temperature distribution of the porous channel. With higher Reynolds Number
(Rep > 110) the two heat transfer correlations (E1and E2) resulted in a temperature profile
very similar to each other with a difference of 2 to 4º C for various heat fluxes at the
boundary (fig.7.31, 7.32). For Reynolds Number less than 100(fig. 7.26 and 7.29) the
variations in the temperature profile were found to be large (10-15º C). Higher Particle
Reynolds Number (Rep) enhances the LTE and hence there is no much variation of the
temperature profiles between the two models. Where as lower Particle Reynolds Number
enhances the LTNE and hence there is a large variation in the temperature profiles
between the tow models (E1 and E2) of HTC. With the increase in particle diameter from
0.72 to 1.59 mm the variation of the temperature profile of the porous channel between
the two heat transfer models was much more significant. This is due to the fact that an
increase in the particle diameter enhances LTNE and therefore larger the variation in the
temperature profile in the two models. These results are in accordance to the conclusions
(Amiri and Alazami, 2000, p.319).
7.3.5 Temperature and Rep plots for Copper of Dp 0.72mm
Copper has a higher thermal conductivity than that of SBB. Although the thermal
conductivity of the solid phase has no relationship in hsf and asf , it plays a significant
role on the temperature profiles of the porous channel. The immediate effect (due to
higher thermal conductivity material), which can be seen in the subsequent plots, is to
give totally different plots for lower and upper wall temperature profiles for E1 and E2
64
model of HTC. The E1 model tends to show a sharp increase in temperature at the inlet
(0.2 < X/De < 0.6) for both the upper and lower wall temperature profiles. The E2 model
tends to give a gradual change in temperature for the upper wall with a little dip in
temperature at the inlet and for the lower wall it gives a sharp step increase in
temperature at inlet (X/De < 0.02) and then the temperature change is gradual along the
flow direction.
Figs. 7.24 and 7.25 show the temperature profile for E2 and E1 models of HTC
respectively at 110 Rep and net heat flux of 1.9W/cm2 imposed at the upper wall. Due to
the higher thermal conductivity material for the porous channel the deviation is very
prominent between the two models.
Fig. 7.24: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2 and (Net
heat flux = 1.9 W/cm2) and E2 model for HTC. (Case file: CU72110ReGEN24.cas)
65
The E2 model gives a large temperature difference between the wall and the boundary
temperature profiles, whereas the E1 model gives a comparatively less difference
between the wall and the boundary temperature. The temperature increase at the entrance
zone is gradual in the case of E2 model whereas for the E1 model increases rapidly at the
entrance region and attains a steady state (X/De > 0.30).
Fig. 7.25: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 1.9 W/cm2) and E1 model for HTC. (Case file: CU72110ReSPEC24.cas)
Similar plots of Temperature profiles for various particle Reynolds number(Rep) and heat
flux conditions imposed on the upper wall for porous channel made of Copper beads od
particle diameter (0.72 mm) were obtained and they can be seen in Appendix B. The
figures show the same characteristics of temperature distribution as described above.
Fig.7.26 shows the Reynolds number distribution for figs. 7.24 and 7.25.The analysis of
wall temperature profiles are discussed in detail in Section 7.3.7.
66
Fig. 7.26: Plot of particle Reynolds Number (Case file: CU72110ReGEN24.cas)
7.3.6 Temperature plots for Copper of Dp 1.59mm
The effect of increase in particle diameter in the case of Copper beads was also
analyzed and the effect of this increase can be seen by larger deviation of temperature
profiles for E1 and E2 models of HTC.
Figs. 7.27 and 7.28 show the temperature profiles for porous channel of particle diameter
1.59 mm at 110 particle Reynolds number (Rep) and net heat flux of 1.8 W/cm2 and E2
and E1 HTC. E2 model of HTC gives a step rise in temperature at lower wall and the
temperature difference between the wall and the boundary is small compared with the
same configuration for particle diameter of 0.79 mm.
67
Fig. 7.27: Temperature distribution for porous channel (Dp = 1.59mm) at 110 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 1.8 W/cm2) and E2 model for HTC. (Case file: CU159110ReGEN24.cas)
The E1 model (Fig.7.28) for the same configuration gives a more even distribution of
temperature profile. The temperature increases sharply at the entrance region (similar to
temperature profiles for particle diameter 0.72mm) and attains a constant temperature
after (X/De > 0.75). Figs. 7.29 and 7.30 show the temperature distribution for 82 particle
Reynolds number and 0.5 W/cm2 net heat flux at the boundary. The plots show a similar
trend as discussed above for figs 7.27 and 7.28.
68
Fig. 7.28: Temperature distribution for porous channel (Dp = 1.59mm) at 110 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 1.8 W/cm2) and E1 model for HTC. (Case file: CU159110SPEC24.cas)
Fig. 7.29: Temperature distribution for porous channel (Dp = 1.59mm) at 82 particle
Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 0.8W/cm2 (Net heat
flux = 0.5 W/cm2) and E2 model for HTC. (Case file: CU15982GEN08.cas)
69
Fig. 7.30: Temperature distribution for porous channel (Dp = 1.59mm) at 82 particle
Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 0.8W/cm2 (Net heat
flux = 0.5 W/cm2) and E1 model for HTC. (Case file: CU15982SPEC08.cas)
Similar plots of Temperature profiles for various particle Reynolds number (Rep) and
heat flux conditions imposed on the upper wall for porous channel made of Copper beads
of particle diameter (1.59 mm) were obtained and they can be seen in Appendix B.A
detail description of the wall temperature profiles for E1 and E2 models of HTC are
discussed in Section 7.3.7.
70
7.3.7 Comparison of Wall temperature between E1 and E2 HTC for Copper
Fig. 7.31: Wall temperature distribution for porous channel (Dp = 0.72mm) at 110
particle Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2
(Net heat flux = 1.9 W/cm2) and E1, E2 model for HTC. (Case file:
CU72110ReSPEC24.cas)
Fig.7.31 shows the wall temperature distribution for E1 and E2 models of HTC for
porous channel made of copper beads of particle diameter 0.72 mm at 110 particle
Reynolds Number (Rep). The wall temperature profiles show that the higher thermal
conductivity material drastically increases the variation of wall temperature profiles as
compared to SBB. The E1 model tends to sharply increase the wall temperature profile
for upper and lower wall at the entrance zone (X/De <= 0.30) and then attain a steady
state. In the case of E2 model the wall temperature profile shows a initial dip in
temperature at the entrance and then increases gradually, the lower wall temperature
shows a step change in temperature at inlet and the temperature profile changes
71
gradually. The E2 model gives a lower temperature profile as that of the E1 model. The
temperature difference between the upper wall temperatures is 16 to 18C.
Fig. 7.32 shows the wall temperature profile for the same configuration as in Fig.7.31 but
at higher net heat flux at the upper wall (2.4 W/cm2). The variation in temperature
profiles between the two models for this configuration is the same as described above.
The temperature difference between the two models for the upper wall in this case
(Fig.7.32) is 24 to 26C. The larger difference in temperature in Fig. 7.32 as compared to
Fig.7.31 is due to the higher heat flux input at the wall.
Fig. 7.32: Wall temperature distribution for porous channel (Dp = 0.72mm) at 110
particle Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of
3.2W/cm2(Net heat flux = 2.4 W/cm2) and E1,E2 model for HTC. (Case file:
CU72110ReSPEC32.cas)
Fig 7.33 shows the wall temperature distribution for porous channel of diameter 0.72 mm
at 75 particle Reynolds number and net heat flux of 1.6 W/cm2. Although the nature of
the temperature profiles are the same as described previously the lower Reynolds number
72
(Rep = 82) increases the temperature variation between the two models.The temperature
difference of the upper wall for the two models of HTC is 24 to 26C.
Fig. 7.34 shows the temperature variation at 210 particle Reynolds number (Rep) for the
same amount of heat flux at the boundary (net heat flux = 2.0 W/cm2) as in Fig.7.33 but
higher Reynolds number. The higher Reynolds number model has an effect of decreasing
the variation between the two models .The variation between the upper wall temperatures
for fig.7.34 is of the order of 6 to 8 C whereas for the fig 7.33 the variation of upper wall
temperatures is of the order of 31 to 34 C. Hence, for a given particle diameter, the
particle Reynolds number is known to play significant role for the variation of
temperature profiles between the two models with higher Reynolds Number decreasing
the variation between the two models.
Fig. 7.33: Wall temperature distribution for porous channel (Dp = 0.72mm) at 75 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 1.6 W/cm2) and E1, E2 model for HTC. (Case file: CU7275ReSPEC24.cas)
73
Fig. 7.34: Wall temperature distribution for porous channel (Dp = 0.72mm) at 210
particle Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2
(Net heat flux = 2.0 W/cm2) and E1, E2 model for HTC. (Case file:
CU72175ReSPEC24.cas)
Fig. 7.35: Wall temperature distribution for porous channel (Dp = 1.59mm) at 110
particle Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 2.4W/cm2
(Net heat flux = 1.8 W/cm2) and E1, E2 model for HTC. (Case file:
CU159110SPEC24.cas)
74
Fig.7.35 shows the wall temperature distribution of porous channel of particle diameter of
1.59 mm at 110 particle Reynolds number (Rep). The figure shows a drastic variation in
temperature profiles between E1 and E2 models of HTC. Comparing fig.7.31 and 7.35
that have the same particle Reynolds number and heat flux boundary condition, the
increase in particle diameter (Dp = 0.72 mm to 1.59 mm) increases the variation between
the two models. The temperature difference between the upper wall profiles for E1 and
E2 model of HTC for Fig.7.35 is of the order of 35 to 37C as compared to 16C for
Fig.7.31.
The effect of particle Reynolds number on the temperature distribution can be analyzed
from Fig.7.36 and fig.7.37. Fig.7.36 shows the variation of the wall temperature profiles
where Rep = 82 and heat flux at the boundary is 0.8 W/cm2 (net heat flux = 0.5 W/cm2).
Although the heat flux at the boundary is comparatively small, the variation of upper wall
temperature profiles between the two models of HTC is found to be 17 to 18C. Fig.7.37
shows the wall temperature distribution for Rep = 240 and higher heat flux (net heat flux
= 1.3 W/cm2) at the boundary compared to that of Fig.7.36.The variation in the
temperature profiles at the upper wall is less as compared to that of Fig.7.35 (to the order
of 8 to 10C).
75
Fig. 7.36: Wall temperature distribution for porous channel (Dp = 1.59mm) at 82 particle
Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 0.8W/cm2 (Net heat
flux = 0.5 W/cm2) and E1, E2 model for HTC. (Case file: CU15982SPEC08.cas)
Fig. 7.37: Wall temperature distribution for porous channel (Dp = 1.59mm) at 240
particle Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 1.6W/cm2
(Net heat flux = 1.3W/cm2) and E1, E2 model for HTC. (Case file:
CU159240SPEC16.cas)
76
7.3.8 Discussion of Results for Copper Beads
The particle Reynolds number played a crucial role in the temperature distribution of the
porous channel. With higher Reynolds Number (Rep >= 210) the use of the two heat
transfer correlations (E1and E2) resulted in a temperature profiles little similar to each
other with a difference of 10 to 12ºC for various heat fluxes at the boundary (fig. 7.54).
For Reynolds Number less than 110 the variations in the temperature profile were found
to be large approximately 26 to 40ºC (fig.7.48 through 7.50). With the increase in particle
diameter from 0.72 to1.59 mm, the variation of the temperature profile of the porous
channel between the two heat transfer models was much more significant due to the
enhancement of LTNE.
Since, the ratio of thermal conductivity of Copper versus air is much more than that of
sintered bronze beads versus air the variation of wall temperatures is much more
significant in the case of Copper, but the temperature profiles are less compared to that of
SBB. Higher thermal conductivity ratio material is known to enhance LTNE and hence
the large variation in temperature profiles between the two heat transfer models E1 and
E2. These results adhere to the conclusions (Amiri and Alazami, 2000, p.319).
77
CHAPTER 8
CONCLUSION
The following conclusions can be drawn, in order of significance from the results
are as follows:
1. The material used in the porous channel plays a crucial role, with higher
thermal conductivity ratio material enhancing the LTNE condition and thus a
large variation in the temperature profile can be seen between the two models
of HTC.
2. The Particle diameter (Dp) is second significant factor in the analysis. The
larger the particle diameter, the larger is the variation in temperature profiles
between the two HTC. Larger particle diameter enhances LTNE and there
would be a large variation in the temperature profiles.
3. The particle Reynolds Number (Rep) is also crucial, with higher Reynolds
Number enhancing LTE and hence a decrease in the variation of temperatures
between the two models.
4. The heat flux is imposed at the upper wall is the last factor that needs to be
taken into consideration. The higher the heat flux at the upper wall, higher
would be the variation between the two models of HTC.
Thus, a material with low thermal conductivity ratio and small particle diameter (Dp)
would show a small variation in wall temperature profile provided the particle Reynolds
Number would be high (Rep > 150).
78
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80
APPENDICES
81
APPENDIX A: Temperature and Rep plots for SBB of Dp 1.59 mm.
Fig. A-1: Temperature distribution for porous channel (Dp = 1.59mm) at 82 particle
Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 0.8W/cm2 (Net heat
flux = 0.5 W/cm2) and E1 model for HTC. (Case file: SBB15982ReSPEC08.cas)
Fig. A-2: Temperature distribution for porous channel (Dp = 1.59mm) at 82 particle
Reynolds Number (inlet velocity = 1.15 m/s) with input heat flux of 0.8W/cm2 (Net heat
flux = 0.5 W/cm2) and E2 model for HTC. (Case file: SBB15982ReGEN08.cas)
82
Fig. A-3: Plot of particle Reynolds Number. (SBB15982ReGEN24.cas)
Fig. A-4: Temperature distribution for porous channel (Dp = 1.59mm) at 240 particle
Reynolds Number (inlet velocity = 3.301 m/s) with input heat flux of 1.7W/cm2 (Net heat
flux = 1.3 W/cm2) and E2 model for HTC. (Case file: SBB159240ReGEN17.cas)
83
Fig. A-5: Temperature distribution for porous channel (Dp = 1.59mm) at 240 particle
Reynolds Number (inlet velocity = 3.301 m/s) with input heat flux of 1.7W/cm2 and (Net
heat flux = 1.3 W/cm2) and E1 model for HTC. (Case file: SBB159240ReSPEC17.cas)
Fig.A-6: Plot of particle Reynolds Number (Case file: SBB159240ReGEN17.cas)
84
Fig. A-7: Temperature distribution for porous channel (Dp = 1.59mm) at 240 particle
Reynolds Number (inlet velocity = 3.301 m/s) with input heat flux of 0.8W/cm2 and (Net
heat flux = 0.7 W/cm2) and E1 model for HTC.(Case file: SBB159240ReSPEC08.cas)
Fig. A-8: Temperature distribution for porous channel (Dp = 1.59mm) at 240 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 0.8W/cm2 (Net heat
flux = 0.7 W/cm2) and E2 model for HTC. (Case file: SBB159240ReGEN08.cas)
85
APPENDIX B: Temperature and Rep plots for Copper of Dp 0.72 mm.
Fig. B-1: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 3.2W/cm2 (Net heat
flux = 2.4 W/cm2) and E2 model for HTC. (Case file: CU72110ReGEN32.cas)
Fig. B-2: Temperature distribution for porous channel (Dp = 0.72mm) at 110 particle
Reynolds Number (inlet velocity = 3.34 m/s) with input heat flux of 3.2W/cm2 (Net heat
flux = 2.4 W/cm2) and E1 model for HTC. (Case file: CU72110ReSPEC32.cas)
86
Fig.B-3: Temperature distribution for porous channel (Dp = 0.72mm) at 75 particle
Reynolds Number (inlet velocity = 1.937 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 1.7 W/cm2) and E2 model for HTC. (Case file: CU7275ReGEN24.cas)
Fig. B-4: Temperature distribution for porous channel (Dp = 0.72mm) at 75 particle
Reynolds Number (inlet velocity = 1.937 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 1.7 W/cm2) and E1 model for HTC. (Case file: CU7275ReSPEC24.cas)
87
Fig.B-5: Temperature distribution for porous channel (Dp = 0.72mm) at 210 particle
Reynolds Number (inlet velocity = 7.423 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 2.0 W/cm2) and E2 model for HTC. (Case file: CU72175ReGEN24.cas)
Fig. B-6: Temperature distribution for porous channel (Dp = 0.72mm) at 210 particle
Reynolds Number (inlet velocity = 7.423 m/s) with input heat flux of 2.4W/cm2 (Net heat
flux = 2.0 W/cm2) and E1 model for HTC. (Case file: CU72175ReSPEC24.cas)
88
Fig.B-7: Plot of particle Reynolds Number. (Case file: CU72210ReGEN24.cas)
89
APPENDIX C: Temperature and Rep plots for Copper of Dp 1.59 mm.
Fig.C-1: Temperature distribution for porous channel (Dp = 1.59mm) at 240 particle
Reynolds Number (inlet velocity = 7.418 m/s) with input heat flux of 1.7W/cm2 (Net heat
flux = 1.3W/cm2) and E2 model for HTC. (Case file: CU159240GEN17.cas)
Fig.C-2: Temperature distribution for porous channel (Dp = 1.59mm) at 240 particle
Reynolds Number (inlet velocity = 7.418 m/s) with input heat flux of 1.7W/cm2 and (Net
heat flux = 1.3W/cm2) and E2 model for HTC. (Case file: CU159240SPEC17.cas)
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APPENDIX-D: SOURCE-CODE
/* This program is developed to model porous media as constant porosity model
for a packed bed of sintered bronze beads/Copper beads along with heat transfer.
The solid-state temperature is achieved by using a user-defined scalar (UDS).
The heat transfer correlations and fluid to solid specific area are modeled in
accordance with those given by E1 and E2 model.
*/
/*
Preprocessor directives and variable declarations for the model*/
/*
*****************************************************************
Preprocessor directives
*****************************************************************
*/
# include "udf.h"
# include "metric.h"
# include "sg.h"
/*
******************************
Declaration of variables and constants
******************************
*/
# define con1 1.5207e-11
# define con2 4.8574e-08
# define con3 1.0184e-04
# define con4 3.9333e-04
/*
****************************
Variables specific to porous media
****************************
*/
//# define K 2.9e-10 //permeability of porous media for dia = 0.72
# define K 10.0e-10 //permeabilty of porous media for dia = 1.59
//# define F 0.242 //Inertial coefficient for dia = 0.72
# define F 0.118 //Inertial coeeficient for for dia = 1.59
//# define Dp 0.00072 (Particle diameter)
#define Dp 0.00159 //(particle diameter)
//#define TKS 16.77 //Thermal conductivity of SBB for Dp = 0.00072
# define TKS 16.591 //Sinterd beads Dp 0.00159
#define vel 3.301 //center line velocity at the inlet
//# define zeta 0.37//(for Dp = 0.72mm)
# define zeta 0.38//(for Dp = 1.59mm)
/*
*******************
End of variable declaration
********************
*/
extern Domain *domain;
enum {TS};//user defined scalars
//FILE *fp;
//FILE *fp2;
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/*
************************************************************************
THE FOLLOWING MODULES ARE FOR CALCULATING THE X AND Y MOMENTUM SOURCE
TERMS,
ACCORDING TO THE CONSTANT POROSITY RELATIONS FOR THE X AND Y SOURCE TERMS
S(i) = (viscosity/K)*U(i) + (density)*(F/sqrt(K))*U(i)*|U(i)|
where,
i = x,y;
K = permeabilty of the medium
U = velocity in the ith direction
F = friction factor
The modules defined for these are:
1.void Material_Properties(cell_t c, Thread *t, real *C1, real *C2)
for calculating the material properties used only for variable porosity
2.DEFINE_SOURCE(srce_xmom,c,t,dS,eqn)
for calculating the X momentum source term
3.DEFINE_SOURCE(srce_ymom,c,t,dS,eqn)
for calculating the Y momentum source term
************************************************************************
*/
/*
1.
This module is used for variable porosity category.The Material properties of the
each individual cell namely Inertial term and the darcy term are calculated for each
cell. This module is called in the X and Y momentum source terms
*/
void Material_Properties(cell_t c, Thread *t, real *C1, real *C2)
{
real zeta3;
/* --- use this routine to return the permeability, alpha and the
co-efficient C2, for each location x,y,z in the domain. No
Modifications needed for 2D. For time dependent porous media
use the RP_Get_Real("flow time") function to find out t (secs) */
zeta3 = pow(zeta,3);
/* ---- Calculate local permeability = 1/C1 */
*C1=(150.0*pow((1.0-zeta),2.0))/(zeta3*pow(Dp,2.0));
//printf("%3.20f\n",(1/(*C1)));
/* ---- Calculate local C2 co-efficient */
*C2 = 1.75*(1.0-zeta)/(Dp*zeta3);
}
/*
2.
X_Momentum_Source
This routine returns the source term for the X-momentum term for each
92
Control volume in the domain. The local properties are obtained by calling Material_Properties. For
constant porosity category the material properties module is not used. THIS MODULE IS ADDED TO
THE X-MOMENTUM TERM IN THE FLUID PANEL SOURCE TERMS.
*/
DEFINE_SOURCE(srce_xmom,c,t,dS,eqn)
{
real C1, C2, constant1, constant2, Ux,Uy,source,dens,x[2],y,p,U;
real term,termg;
/* --- determine local properties */
Material_Properties(c,t,&C1, &C2);
/* --- determine constants 1,2 */
//direct correlation for darcy term
constant1 = C_MU_L(c,t) * (1/K);
constant2 = C_R(c,t)*(F/sqrt(K));
/* --- determine x-velocity */
Ux = C_U(c,t);
source = -(constant1*Ux + constant2 * fabs(Ux)*Ux);
dS[eqn] = -(constant1 + 2 * constant2 * fabs(Ux));
return source;
/*Uy = C_V(c,t);
C_UDSI(c,t,1) = sqrt(pow(Ux,2.)+pow(Uy,2.));
C_UDSI(c,t,2) = C_UDSI_G(c,t,1)[0];
C_UDSI(c,t,3) = C_UDSI_G(c,t,2)[0];
term = (C_MU_EFF(c,t)/zeta)*C_UDSI(c,t,3);
termg = (C_MU_EFF(c,t)/zeta)*C_UDSI_G(c,t,3)[0];*/
}
/*
3.
Y_Momentum_Source
This routine returns the source term for the Y-momentum term for each
control volume in the domain. The local properties are obtained by
calling Material_Properties.For the constant porosity category the
Material_Properties module is not used but the Permeability (K) and the
friction factor (F) are used. The actual values of Permeability (K) and friction
factor(K) are used here.
*/
DEFINE_SOURCE (srce_ymom, c, t, dS, eqn)
{
real C1, C2, constant1, constant2,Ux, Uy, source;
real term,termg;
real prod;
/* --- Determine local properties */
Material_Properties(c, t, &C1, &C2);
/* --- Determine constants 1,2 */
//direct correaltion for the darcy term
constant1 = C_MU_L(c,t) * (1/K);
//direct correlation for the inertial term
constant2 = C_R(c,t)*(F/sqrt(K));
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Ux = C_U(c,t);
Uy = C_V(c,t);
source = -(constant1*Uy + constant2 * fabs(Uy) * Uy);
dS[eqn] = -(constant1 + 2 * constant2 * fabs(Uy));
return source;
}
/*
************************************************************************
THE FOLLOWING MODULES ARE FOR MODELLING THE HEAT TRANSFER ACCORDING TO
THE LTNE ONDITIONS DEFINED BELOW:
-The solid phase energy equation is given by:
0 = h(loc)*A*(T(s) -T(f)) + ( K(s)* del.(del(T(s))).
i.e; 0 = source + diffusion
where,
T(s) is modeled as UDS,
h(loc) = heat transfer coeff between the solid and the fluid phases,
T(s) = solid temperature
T(f) - fluid temperature(air)
del = del operator
K(s) = effective solid thermal conductivity,
K(s) = (1-zeta)*TKS (Diffusivity of the medium)
where,
zeta = porosity.
TKS = thermal conductivity of the solid.
-The fluid phase energy equation is given by:
(PCp)U*d(T(f))/d(X) = h(loc)*A*(T(s)-T(f)) + (K(f) + K(t))*(del.(del(T(f))).
i.e; convection = source + diffusion
where,
T(f) =temperature of air
k(f) = thermal conductivity of air
K(t) = dispersion thermal conductivity given as a wall function
K(eff) = effective thermal conductivity of air
K(eff) = K(f) + K(t)
where,
K(t) is the solid thermal diffusion conductivity.
K(t) = 0.375*density*Cp*U*Dp*l;
where,
l is the wall function and U = sqrt(Ux^2 + Uy^2).
l = 1 - (exp((-(Y + 0.005))/(1.5*Dp)));
0.005 is the height of the channel.
**imp**----->The properties of solid phase are assumed to be constant.
The modules defined for the energy equation are below:
1.DEFINE_DIFFUSIVITY(uds_diffusivity, cell, thread, i)(for solid diffusivity)
2.DEFINE_PROPERTY(fluid_thermal_conductivity,cell,thread)(for effective conductivity of fluid)
94
3.real heat_t_coeff(cell_t c,Thread *t)(for h(loc) calculations)
4.real Area(cell_t c,Thread* t)( for interstial area calculations)
5.DEFINE_SOURCE(solid_energy,c,t,ds,eqn)(for solid energy calculations)
6.DEFINE_SOURCE(fluid_energy,c,t,ds,eqn)(for fluid energy calculations)
7.DEFINE_PROFILE(inlet_velocity_profile,t,i)(for inlet velocity profile)
8.float therm_conduct(float T)(for thermal conductivity of air t differnt temp.)
************************************************************************
*/
/*
1.
This function is for the solid-state diffusivity. The following relation is modeled according the solid-state
energy equation.
*/
DEFINE_DIFFUSIVITY (uds_diffusivity, cell, thread, i)
{
real source,Ts;
if (i == TS)//if solid state temperature
{
/*source = (1-porosity)* (Thermal conductivity of solid)*/
source = ((1-zeta)*TKS);
}
else source = 0.0;
return source;
}
/*
2.
This module is for calculating the effective thermal conductivity of the
fluid taking into account the thermal dispersion, which is modeled as a
Wall function.
*/
DEFINE_PROPERTY (fluid_thermal_conductivity,cell,thread)
{
real source,dis_con,zeta2,u,U,l,Y,x[ND_ND],v,X;
//omega = 1.5
//K = 3.36e-08;//permeabilty
zeta2 = pow(zeta,2);
u = pow(C_U(cell,thread),2.0);
v = pow(C_V(cell,thread),2.0);
U = sqrt(u + v);//for beads
C_CENTROID (x,cell,thread);
Y = x[1];
X = x[0];
l = 1 - (exp((-(Y + 0.005))/(1.5*Dp)));
//source = zeta*(C_K_EFF (cell,thread));
/*Thermal dispersion conductivity for Sintered bronze porous media*/
95
dis_con = 0.375*C_R(cell,thread)*C_CP(cell,thread)*U*Dp*l;
//dis_con = 0.025*C_R(cell,thread)*C_CP(cell,thread)*sqrt(K)*U;
//dis_con = 0.01*((1-zeta)/pow(zeta,2))*C_R(cell,thread)*C_CP(cell,thread)*U*Dp;
source = zeta*C_K_L(cell,thread) + dis_con;
return source;
}
/*
3.
This module calculates the Heat transfer coefficient h(loc) for both the solid
and the fluid phase based on the particle Reynolds Number Rep.
*/
real heat_t_coeff(cell_t c,Thread *t)
{
//FILE *t;
/*HTC = (kf*(2+1.1Pr^0.333*Re^0.6))/dp;*/
real Pr,Re,HTC,U,dv,Rep,Area,Ux;
//Interstial area between the solid and the Fluid
Area = 6.0*(1-zeta)/Dp;
//void diameter
dv = (4.0*zeta)/Area;
//Average velocity
U = sqrt(pow(C_U(c,t),2.) + pow(C_V(c,t),2.));
// modified Reynolds Number based on the void diameter
Re = (U*dv*C_R(c,t))/C_MU_EFF(c,t);
//Particle Reynolds Number based on Diamter of the particle(Dp).
Rep = (U*Dp*C_R(c,t))/C_MU_EFF(c,t);
//Fluid Prandtl Number
Pr = (C_MU_EFF(c,t)*C_CP(c,t))/C_K_EFF(c,t);
/*Use this part of the code for E2 model of HTC*/
HTC = (C_K_EFF(c,t)*(2.0 +(1.1*pow(Pr,0.333)*pow(Rep,0.6))))/Dp;
/*Use this part of code for E2 model of HTC
if(Rep < 100)
HTC=0.004*(C_K_EFF(c,t)/dv)*pow(Pr,0.3333)*pow(Re,1.35);
else
HTC =0.0156*(C_K_EFF(c,t)/dv)*pow(Pr,0.3333)*pow(Re,1.04);
*/
return HTC;
}
/*
4.
This module calculates the Interstial area between the solid and the fluid*/
96
real Area(cell_t c,Thread* t)
{
real area;
area = 6.0*(1-zeta)/Dp; //E2 model
//area = (20.346*(1-zeta)*pow(zeta,2))/Dp;//E1 model
return area;
}
/*
4.
This is the energy source term for the Solid phase(Sinterd Bronze beads).
This module is hooked to the UDS energy source term in the Fluid source terms panel*/
DEFINE_SOURCE (solid_energy,c,t,ds,eqn)
{
real source,Ts,HTC,area;
/*source = HTC*area*(TS-TF);*/
Ts = C_UDSI(c,t,TS);
/*Calculate the heat transfer coefficient term from the
heat_t_coeff(c,t) correlation*/
HTC = heat_t_coeff(c,t);
/*XCalculate the interstial area*/
area = Area(c,t);
/*Calculate the source term for the Solid state energy*/
source = -HTC*area*(Ts - C_T(c,t));
ds[eqn] = -HTC*area;
return source;}
/*
5.
This module is for the energy source term for the Fluid phase. This module is
attached to the fluid energy source term in the Fluid source term panel
*/
DEFINE_SOURCE(fluid_energy,c,t,ds,eqn)
{
real source,Ts,HTC,area;
/*source = HTCOEFF*area*(TS-TF);*/
Ts = C_UDSI(c,t,TS);
HTC = heat_t_coeff(c,t);
area = Area(c,t);
source = HTC*area*(Ts - C_T(c,t));
ds[eqn] = -HTC*area;
return source;
}
/*
6.
This module is used for developing fully developed inlet Y velocity
profile based on the input center line velocity = vel defined above. This
is hooked as velocity inlet profile at the inlet boundary
*/
DEFINE_PROFILE (inlet_velocity_profile,t,i)
{
97
real x[3];
real y;
face_t f;
begin_f_loop(f,t)
{
F_CENTROID(x,f,t);
y = x[1];
F_PROFILE(f,t,i) = vel - vel*pow((y/0.005),2);
}
end_f_loop(f,t)
}
/*
7.
This module is for finding the thermal conductivity of the air as
a function of temperature.Used for Nusselt number correlation
*/
float therm_conduct(float T)
{
float result;
/*k = 1.5207e-11*T^3 - 4.8574e-08*T^2 + 1.0184e-04*t-3.9333e-04*/
//printf("%3.3f %3.3f\n",pow(3.0,4),pow(2.5,2));
//return;
result =con1*pow(T,3)-con2*pow(T,2)+con3*T-con4;
//printf("The conductivity is:%3.5f",result);
return result;
}
98