Y. S. Muzychka
Professor
Mem. ASME
Faculty of Engineering and Applied Science,
Memorial University of Newfoundland,
St. John’s, NL, A1B 3X5, Canada
E. J. Walsh
Senior Research Fellow
Mem. ASME
P. Walsh
Senior Research Fellow
Mem. ASME
Stokes Research Institute,
University of Limerick,
Castletroy, County Limerick, Ireland
1
Heat Transfer Enhancement Using
Laminar Gas-Liquid Segmented
Plug Flows
Heat transfer enhancement using segmented nonboiling gas-liquid flow is examined.
Segmentation results in a two phase flow of liquid/gas having a constant liquid fraction;
i.e., no phase change occurs. In this flow configuration, enhanced heat transfer occurs as
a result of a shorter effective thermal length due to internal fluid circulation in the liquid
plugs. A simple theory for laminar segmented flows is developed based on scaled Graetz
theory and comparisons made with existing published data from the literature and new
experimental data obtained in a companion study. The proposed model is valid for an
isothermal tube wall provided that the axial residence time of the flow is such that
dimensionless tube length L쐓 ⬍ 0.1. 关DOI: 10.1115/1.4002807兴
Keywords: heat transfer, segmented flow, two phase flow, thermal enhancement, Graetz
flow, plug flow, Poiseuille flow, Taylor flow
Introduction
This present study examines the potential of heat transfer enhancement using uniformly segmented fluid streams. Such
streams are frequently referred to as Taylor flows 关1兴 after Taylor’s original study 关2兴 of a simple two phase plug flow, Bolus
flow 关3兴 in one of the earliest such studies involving heat transfer
in a laminar two phase plug flow, and segmented flows 关4兴. Segmented flows find many applications in mass transfer systems
where enhanced radial transport is desired while axial dispersion
is minimized. A segmented fluid stream may be easily produced
by using plugs of solid, liquid, or gas introduced into a liquid
stream, denoted here as the carrier fluid, to which heat is to be
transferred. In these cases, a two phase flow stream of alternating
elements of segmenting media and carrier fluid are produced. The
easiest such system to work with is the two phase gas-liquid system.
A distinction is made between the traditional two phase plug 共or
slug兲 flow and segmented flow for the following reasons. In a
traditional two phase plug flow, the liquid plugs may not all be of
uniform length, shape, and distribution, and as such, the fluid
stream is much more complex and less amenable to theoretical
analysis. In a uniformly segmented flow, both the liquid plugs in
the carrier fluid and segmenting media have constant lengths and
uniform spacing or frequency. Further, the segmenting media may
or may not contribute significantly to the overall heat transfer rate.
In this present study, we are considering only uniformly distributed gas-liquid segmented flows, as shown in Fig. 1.
Segmented flows, while simple in composition, have many issues that must be considered, which affect heat transfer rates.
These include
•
•
•
•
•
•
•
•
thermal boundary condition
type of velocity distribution
plug length
plug circulation
liquid fraction
liquid properties
film evaporation
duct shape
Contributed by the Heat Transfer Division of ASME for publication in the JOURHEAT TRANSFER. Manuscript received May 6, 2009; final manuscript received
September 25, 2010; published online January 11, 2011. Assoc. Editor: S. A. Sherif.
NAL OF
Journal of Heat Transfer
Early studies in two phase plug flow hypothesized that enhancement in heat transfer rates in laminar flows could be obtained by splitting up the coolant stream into a series of shorter
elements or plugs 关3–8兴. These early studies in two phase flow
heat transfer considered thermal enhancement using gas or solid
plugs as a segmenting medium. It has long been known that
within the liquid plugs, internal circulations arise as a result of the
gas-liquid, solid-liquid, or liquid-liquid interfaces 关1,2,4–8兴. It has
been hypothesized that thermal enhancement occurs due to two
effects. One is a result of the internal circulation in the liquid
plugs, leading to greater radial transport of heat or mass, while the
other mechanism results from the increased velocity that liquid
plugs experience as a result of the reduced liquid fraction for a
constant mass flow rate. The latter will be shown to be unlikely
using simple heat transfer theory. This leaves only plug circulation
effects and/or general plug velocity distribution as the primary
effect contributing to enhanced heat transfer.
A number of these early works have also attempted to predict
the two phase plug flow data using the single phase Graetz solution for a tube 关3–9兴. Each of these studies failed to successfully
show any correlation with the Graetz theory but showed increased
heat transfer rates for segmented flows having the same mass flow
rate as compared with a continuous flow stream.
While these studies carefully examined the effect of void fraction and plug length on heat transfer enhancement using carefully
planned experiments, they failed to consider plug length as an
actual scaling parameter in heat transfer predictions. The data
from these and other studies are re-analyzed using different scaling principles in the present work.
Three early studies considered in this work are those of Horvath
et al. 关4兴, Oliver and Young-Hoon 关6,7兴, and Vrentas et al. 关8兴. In
all three studies, the authors made attempts to relate single phase
heat transfer theory to either predict or show the level of enhancement in Nusselt data versus some reference parameter that was
usually taken as the Reynolds number, the Graetz number, or the
Peclet number of the actual flow. In all cases when the Graetz–
Leveque model was compared with experimental data, the authors
used the total heated tube length in their data reduction for defining the dimensionless thermal length and invariably used the total
or wetted surface area of the tube when determining Nusselt numbers. It is this inconsistency in addition to the definitions of Nusselt number and Reynolds number that leads to the ultimate failure
in correlating these carefully obtained experimental plug flow data
using the classic Graetz flow theory.
Copyright © 2011 by ASME
APRIL 2011, Vol. 133 / 041902-1
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Gas - Liquid Flow
L/d = 1
L/d = 4
Gas
L/d = 6
Liquid
L/d = 10
L
Fig. 1 Gas-liquid plug flow for different liquid slug lengths
2
Literature Review
One of the earliest studies on laminar plug flow heat transfer is
the study of Prothero and Burton 关3兴. In a one of a series of
studies, they compared the heat transfer rate of single phase Graetz flow with that of a segmented gas-liquid stream. While Prothero and Burton 关3兴 were able to show enhancement over the
single phase flow, they did not assess the effect of slug length.
Rather, Prothero and Burton 关3兴 only considered the effect of void
fraction in a limited sense. Their data will be presented in this
study in a general plot in a later section but cannot be rereduced
according to the approach suggested in this paper due to the lack
of information on actual liquid plug lengths.
Oliver and Wright 关5兴 conducted a series of measurements to
investigate the effect of plug flow on heat transfer and friction in
laminar flow. They surmised that the internal circulation would
increase the heat transfer coefficient significantly and therefore,
one would not be able to use the Graetz–Leveque theory. They
attributed the increase in heat transfer coefficient to both the effects of internal circulation and increased liquid velocity that results at constant mass flow rates due to the void fraction. They
concluded that the effect due to void fraction is independent of
plug length but that circulation effects would be strongest for
shorter plugs. They reserved the assessment of plug length for a
later study since the apparatus they used to collect data could not
control liquid plug lengths very well. As part of their study, Oliver
and Wright 关5兴 developed simple correlations based on their experimental data and modification of the Graetz–Leveque model.
Hughmark 关9兴 proposed a simple correlation using a modified
Graetz–Leveque theory. However, his modifications were based
on the experimental observations of Oliver and Wright 关5兴 and
contained no new insights. Further, no rational basis was given for
the proposed modification. The proposed model was reported as
冉 冊冉 冊
hTPD冑␣L
ṁC p
= 1.75
k
␣LkL
1/3
␮
␮w
0.14
共1兲
Oliver and Young-Hoon 关6,7兴 developed an experimental facility
to produce liquid plugs of constant length using gas as a segmenting medium. This study was motivated by the earlier study of
Oliver and Wright 关5兴, which observed increased enhancement for
Newtonian and non-Newtonian flows. Oliver and Young-Hoon
关6,7兴 conducted similar experiments as Oliver and Wright 关5兴,
except that plug length and liquid fraction were carefully controlled. They obtained data for a segmented fluid stream of constant liquid mass flow rate that contained liquid plugs of uniform
length and distribution. They reported data for two different liquid
flow rates and for plugs varying in length from 1 in. to 20 in. in a
1/4 in. 共2.54 cm to 50.8 cm in a 6.35 mm兲 diameter tube. They
could also control the rate of gas flow while maintaining the plug
size to study the effect of gas stream velocity and hence, plug
velocity. Heat transfer data were reported as a Nusselt number for
the liquid phase versus a combined or two phase flow Graetz
041902-2 / Vol. 133, APRIL 2011
number. This was the first reported study to assess the effect of
liquid plug length and liquid fraction on heat transfer.
Horvath et al. 关4兴 obtained mass transfer data for liquid plugs of
varying length for four Reynolds numbers using gas as a segmenting medium. They reported their data as a Nusselt number for the
liquid plug phase only versus plug length at various Reynolds
numbers for a stream having a constant liquid fraction of 0.5. The
effect of plug velocity was studied by varying the two phase flow
Reynolds number. Horvath et al. 关4兴 also examined the effect of
dimensionless tube length for very short plugs, but these data
cannot be used since the liquid fraction varies from point to point
and is not reported. However, they showed that dimensionless
tube length 共L / D兲 has a small effect on the overall heat transfer
rate. We will delve into this issue in a later section.
Vrentas et al. 关8兴 used solid steel spheres as a segmenting medium in a stream of silicone oil. They reported liquid phase Nusselt data as a function of the stream Peclet number for three plug
lengths. They also considered two different Prandtl numbers. The
primary distinguishing feature of this work is that the liquid plug
ends cannot be assumed to be approximately adiabatic as in a
gas-liquid flow as the authors report that the steel spheres heat up
to the wall temperature and thus provide another heat transfer
path. This issue becomes more pronounced for shorter plugs.
However, we will make use of some of the data reported by Vrentas et al. 关8兴 for comparative purposes to illustrate the general
scaling property.
More recent studies using a numerical solution were undertaken
by Lakehal et al. 关10兴 and Narayanan and Lakehal 关11兴. They
re-examined the potential of nonboiling two phase plug flows as a
means of heat transfer enhancement and proposed a simple model
that, while modestly predicting their data, does not adequately
address all the issues of plug length outlined above. Further, they
combined the simple laminar fully developed flow Nusselt number with a turbulent flow Nusselt behavior despite the liquid flow
Reynolds numbers falling in the laminar regime,
4/5 0.4
NuD ⬇ 3.67 + 0.022ReD
Pr
共2兲
where ReD = UGD / ␯ is based on the gas phase velocity.
Finally, Young and Mohensi 关12兴, Mohseni and Baird 关13兴, and
Baird and Mohseni 关14兴 obtained numerical results for plug flows
in a plane channel. They examined the heat transfer characteristics
for a single droplet translating in a confined channel. They have
coined the term digitized heat transfer to describe a discrete flow
of droplets or plugs. The results reported 关12–14兴 are primarily
concerned with low Peclet number flows using liquid metals as a
proposed coolant. The reported behavior in Refs. 关12–14兴 is also
the characteristic of the experimental data of the present authors
关15兴. Thus, while the initial interest in Refs. 关12–14兴 was heat
transfer from microfluidic drops, all of the analysis that is presented is also applicable for laminar segmented flows in tubes and
finite channels.
Transactions of the ASME
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Table 1 Summary of experimental and numerical data for tubes
Parameter
Ls / D
L/D
D 共mm兲
ReD
Pr
PeD
Ls+
Ls쐓
L쐓
␣L
N
Type
Wall
Oliver and Young-Hoon 关6兴
Horvath et al. 关4兴
Vrentas et al. 关8兴
Narayanan and Lakehal 关11兴
Walsh et al. 关15兴
1.2–74
144
6.35
977–1362
10
9770–13,620
0.00087–0.069
0.000077–0.0061
0.011–0.015
0.61–0.85
1–100
Gas-liquid
Isothermal
1.6–20
260
2.32
30–220
1700
51,000–374,000
0.0042–0.73
0.0000025–0.00043
0.00069–0.0051
0.5
6–80
Gas-liquid
Isothermal
10.5–61
128
9.5
0.7–15.8
1000, 10,000
4210–17,800
3.85–31.2
0.00083–0.0093
0.0073–0.030
0.913–0.984
2–12
Solid-liquid
Isothermal
1.92–6.17
40
1
1396–2135
5.5
7678–11,742
0.0014–0.0029
0.00025–0.00052
0.0034–0.0052
0.520–0.795
2–10
Gas-liquid
Isothermal
0.67–28.9
50
1.5
70–565
5.5
388–3106
0.0012–0.17
0.00022–0.031
0.016–0.13
0.11–0.8
1–37
Gas-liquid
Isoflux
Additional studies in the literature, such as Refs. 关16–18兴, also
considered two phase plug flows; however, only the references
discussed above have considered plug length and/or plug circulation at some level of detail. Due to the manner in which each of
these studies presents Nusselt data, a direct comparison with
single phase flow theory was not appropriately undertaken. These
data will be re-examined and replotted using more appropriate
definitions for the Nusselt number and dimensionless plug length,
which coincide with the physics of a Graetz flow. The details of
each data set are summarized in Table 1. These represent data for
a circular tube. With the exception of Narayanan and Lakehal
关11兴, all data are experimental. Additional new data obtained by
the authors will also be considered 关15兴. These data are obtained
for the isoflux condition, but for this study, they have been presented as equivalent isothermal data using the mean wall temperature. In general, this is a reasonable approach as integration of the
Graetz–Leveque equation for isoflux conditions yields an expression that is within 7% of the result for an isothermal tube wall.
3
Graetz Flow Problems
We begin with a quick review of the Graetz flow theory and
some useful alternatives for representing dimensionless heat transfer rates. As with external forced convection, a heat transfer coefficient is typically defined. Unlike external flows, where the
stream temperature remains constant, i.e., T⬁, the temperature difference between the wall and the moving stream does not remain
constant over the length of a duct or a channel, even in the case of
constant wall temperature. In a duct or channel flow, we may
define this wall to stream temperature difference in a number of
different ways. These include
•
•
•
wall to bulk mean, Tw − Tm
wall to inlet, Tw − Ti
mean wall to inlet, T̄w − Ti
The correct choice should be based on the application. For example, in single fluid problems such as heat sinks, the best and
easiest approach is to use the wall to inlet temperature difference,
Tw − Ti. However, in two fluid problems such as heat exchangers,
the better choice is most often the wall to bulk temperature difference. The most frequently used form for defining the local heat
transfer coefficient in an internal flow for constant wall temperature has traditionally been in terms of the bulk temperature, Tm
= Tm共z兲, where
Journal of Heat Transfer
冕冕
冕冕
␳c puTdA
Tm =
共3兲
␳c pudA
The local heat flux, qz, is often related to a local heat transfer
coefficient, hz, by means of some defined characteristic temperature difference in the local flow,
qz = hz共Tw − Tm兲
共4兲
where Tw − Tm is the local wall to bulk temperature difference. In
a duct where the prescribed wall temperature remains constant,
the heat flux varies due to changes in the bulk temperature. In a
duct where the prescribed wall flux remains constant, the wall
temperature varies. In these applications, the above equation is
written as
qw = hz共Tw,z − Tm兲
共5兲
and is utilized for determining the local wall temperature.
Care must be taken when considering this issue, as either the
wall may be maintained at a constant temperature or the wall heat
flux may be maintained at a constant value but not both at the
same time. In a real application, the true boundary condition may
lie somewhere in between. Aggregate analysis using a mean wall
temperature in flux specified problems and a mean flux in temperature specified problems often yields results within a few percent of each other.
We may define a dimensionless local or mean heat transfer
coefficient or Nusselt number for the two special cases of constant
wall temperature and constant wall heat flux. These will be treated
separately. Traditionally, the heat transfer rate is nondimensionalized using a Nusselt number defined as
NuD =
qD
hD
=
k共Tw − Tm兲
k
共6兲
For the purposes of this study, we are primarily interested in dimensionless heat transfer rates using the wall to inlet temperature
difference. Expressions differ slightly based on the thermal
boundary condition prescribed. These are addressed below.
3.1 Constant Wall Temperature. If the duct wall is maintained at a uniform constant temperature Tw, then we define
NuD =
q zD
k共Tw − Tm兲
共7兲
which is based on the wall to bulk mean fluid temperature. Alternatively, we could also define
APRIL 2011, Vol. 133 / 041902-3
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
NuD =
q zD
k共Tw − Ti兲
This definition proves useful in the boundary layer region for
developing asymptotic solutions where the bulk temperature
scales to the inlet fluid temperature, i.e., Tm ⬃ Ti.
If a mean Nusselt number is desired, then we must integrate
along the duct length
NuD =
1
L쐓
冕
L쐓
NuDdz쐓 =
0
1
⌬Tw = 쐓
L
共8兲
q̄D
k⌬Tln
共9兲
However, in the case of a Nusselt number defined on the basis of
the wall to inlet temperature difference, such as in the case of
single fluid systems, i.e., heat sinks, it is more convenient to designate the dimensionless mean wall flux simply as:
q̄D
q쐓 =
k共Tw − Ti兲
共10兲
NuD =
q wD
k共Tw,z − Tm兲
共11兲
0
q쐓 =
4
共12兲
in the thermal boundary layer region, where the bulk temperatures
scales according to Tm ⬃ Ti.
It is clear that if a uniform constant heat flux is specified, then
the local Nusselt number is utilized for the purpose of predicting
the wall to bulk fluid temperature rise or the wall temperature
distribution once Tm is known. In other words,
Tw,z = Tm +
q wD
kNuD
共13兲
In problems where the wall flux is prescribed, the concept of the
heat transfer coefficient is limited. Thus, given an expression for
the relationship of Nu as a function of dimensionless duct position
zⴱ = z / DPeD, one can easily obtain the local wall or wall to bulk
temperature difference. In this regard, in single fluid heat exchange systems, where the wall temperature is the parameter of
interest, it is more appropriate to consider a dimensionless local
wall temperature
⌬Tw쐓 =
共Tw,z − Ti兲k
1
=
Nu
q wD
共14兲
since the principal solution variable of interest will be the temperature field, Tw,z.
Frequently, sources of Nu data and models report a mean Nusselt for the constant flux wall condition. In these cases, the mean
heat transfer coefficient is utilized to find the mean wall to bulk
temperature difference. This is only useful in the boundary layer
or thermal entrance region where the wall to bulk temperature
difference varies since in a fully developed flow, Tw,z − Tm is constant. If one desires the mean wall temperature or mean wall to
bulk temperature difference, then we must obtain the integral relationship
041902-4 / Vol. 133, APRIL 2011
0
dz쐓
NuD
共15兲
q wD
共16兲
k⌬Tw
Heat Transfer Characteristics
Given the special scaling characteristics of the boundary layer
and fully developed regions, it is convenient to examine several
asymptotic formulations that can be used in a simple modeling
approach. These include
• fully developed flows
• plug flow limit 共Pr→ 0兲
• Poiseuille flow limit 共Pr→ ⬁兲
Using the asymptotic characteristics of slug and Poiseuille
flows for the constant wall boundary condition 关19,20兴 and the
Churchill–Usagi asymptotic correlation method 关21,22兴, the authors have developed the following expressions for thermally developing Graetz flows.
In the case of a slug flow, i.e., uniform velocity profile, one
obtains for Nu and q쐓 the following:
Nu =
q wD
k共Tw,z − Ti兲
冕
L쐓
These formulations involving Eqs. 共10兲 and 共16兲 will prove much
more useful in the analysis of single fluid systems since the parameter of interest is defined explicitly in the nondimensional formulation.
or, alternatively, as
NuD =
q wD 쐓 q wD
dz = 쐓
kNuD
kL
Using the above result, we may define a dimensionless heat transfer rate similar to Eq. 共10兲 as
in order to avoid confusion with the traditional definition of the
Nusselt number. Since, in these applications, it is the total heat
transfer rate related to the wall temperature and duct geometry
that is of interest, not the actual heat transfer coefficient.
3.2 Constant Wall Flux. In other applications where electric
resistance heating is used and/or we have a low conductivity duct
wall, it may be more realistic to assume a constant heat flux
boundary condition. If we maintain a constant flux at the wall, qw,
then we may define a Nusselt number as
冕
L쐓
and
q쐓 =
冋冉 冑 冊
1.128
L
2
+ 5.782
쐓
册
1/2
冋冉 冑 冊 冉 冊 册
1.128
L
−2
+
쐓
1
4L쐓
共17兲
−2 −1/2
共18兲
In the case of a Poiseuille flow, one obtains for Nu and q쐓 the
following:
Nu =
and
q쐓 =
冋冉 冊
1.614
L쐓1/3
5
+ 3.655
册
1/5
冋冉 冊 冉 冊 册
1.614
L쐓1/3
−3/2
+
1
4L쐓
共19兲
−3/2 −2/3
共20兲
In all cases, the dimensionless thermal duct length is defined as
L쐓 =
L/D
kL
=
PeD ␳C pUD2
共21兲
The above expressions are used for comparing the available published data that have been used to assess plug length. They represent the theoretical solutions for the Graetz–Nusselt class of duct
problems. The short duct asymptote, i.e., the first term in each of
the latter two expressions, is also known as the Graetz–Leveque
asymptote. The above equations are used in Sec. 5 to compare the
experimental and numerical data of Table 1, assuming that the
characteristic contact length 共or plug length兲 is used to define L쐓,
actual local mean plug velocity U considering ␣L when required,
and actual thermal contact area ␲DL␣L, are used to define the
dimensionless heat transfer rate.
5
Enhancement Mechanisms
The hydrodynamics and heat transfer characteristics of plug
flows varies depending on the type of segmenting medium, i.e.,
gas, liquid, or solid, and the surface characteristics. The principal
Transactions of the ASME
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
1.2
1
(a)
0.8
Q*
Flow
L
0.6
0.4
0.2
Segmented Slug / Continuous Slug
Segmented Poiseuille / Continuous Poiseuille
0 -4
10
(b)
10
-3
10
-2
Flow
10
L*
-1
10
0
10
1
Fig. 4 Q쐓 for segmented slug and Poiseuille Graetz flows having the same ṁ and no secondary flow mechanism
L
Fig. 2 Internal liquid plug circulation: „a… hydrophobic surface
and „b… hydrophyllic surface
mechanism for thermal enhancement is widely accepted to be
fluid circulation, which is a result of interfacial effects, see Fig. 2.
It has also been suggested that higher local fluid velocity for the
same liquid feed rate in a gas-liquid flow also contributes to a
higher heat transfer coefficient 关5兴. It can easily be shown that in
a laminar Graetz flow, this increase in heat transfer coefficient is
more than offset by the reduction in contact area and overall no
gains in heat transfer result.
In the absence of a secondary mechanism, i.e., interfacial effects, a fluid stream ideally would merely be divided into N segments such that a boundary layer is formed within each isolated
segment and the stream would have a liquid fraction ␣L = 1.
Regardless of whether the flow is Poiseuille or slug, the thermal
boundary layer characteristics would in effect remain the same, as
shown in Figs. 3共a兲 and 3共b兲. Since each segment would initiate a
boundary layer on entering the tube, and as each segment translates downstream, its boundary layer grows in such a manner that
the ensemble has the same boundary layer profile as a continuous
stream.
Thus, with no secondary mechanism 共interfacial effect兲 to
change the local velocity profile, segmentation alone will not result in higher heat transfer coefficients. Next, if one assumes an
ideal model whereby voids are created, as shown in Fig. 3共c兲, then
the heat transfer characteristic will still be based on the actual tube
length, not the segment or plug length, but with a reduced contact
area. Further, if the stream has a liquid fraction less than one for
the same mass flow rate, a reduced heat transfer rate would be
observed due to a reduced liquid contact area despite a higher
average heat transfer coefficient. In this case, the Nusselt number
would merely rescale based on introducing the void fraction effects on surface area and mass flow. Further, it can be shown that
the increase in local velocity at constant mass flow has no effect
on enhancing the overall heat transfer rate, as the reduction in
surface area outweighs an increase in heat transfer coefficient.
We begin with the following simple models for ideal thermally
developing plug flow, Eq. 共18兲, and ideal thermally developing
(a)
U
(b)
U
(c)
2U
Fig. 3 Ideal plug flow heat transfer
Journal of Heat Transfer
Poiseuille flow, Eq. 共20兲, in a tube with a constant wall temperature condition, i.e., Graetz flows, based on the inlet temperature
difference Tw − Ti. The wall to inlet temperature difference forms
are utilized since the wall to mean temperature difference approach would yield different values of the log mean temperature
difference 共LMTD兲, and hence, direct comparison could not be
made for a fixed temperature difference.
We can define Q for each case while considering the liquid
fraction ␣L for a stream having the same mass flow rate ṁ, i.e.,
L 쐓 → ␣ LL 쐓
共22兲
since the stream velocity becomes U → U / ␣L. Thus, it can easily
be seen that increasing U for the same mass flow rate with the
introduction of gas voids leads to a smaller effective L쐓 and hence
a larger effective h. But one must also consider the reduced liquid
contact area, i.e.,
Q s = ␣ LQ c共 ␣ LL 쐓兲
共23兲
To make a direct comparison with the continuous Graetz flow, we
will also define a dimensionless heat transfer ratio Q쐓 = Qs / Qc,
where Qs denotes the segmented flow heat transfer rate and Qc
denotes the continuous flow heat transfer rate. This leads to the
following expression for segmented slug flow relative to continuous slug flow denoted as 共s-s兲:
␣L
쐓
=
Qs-s
冋冉 冑 冊 冉 冊 册
冋冉 冑 冊 冉 冊 册
1.128
␣ LL 쐓
1.128
L쐓
and
␣L
Q쐓p-p =
−2
+
1
4 ␣ LL 쐓
+
1
4L쐓
−2
−2 −1/2
−2 −1/2
冋冉 冊 冉 冊 册
冋冉 冊 冉 冊 册
1.614 −3/2
1
+
共␣LL쐓兲1/3
4 ␣ LL 쐓
1.614 −3/2
1 −3/2
+
쐓1/3
L
4L쐓
共24兲
−3/2 −2/3
−2/3
共25兲
for segmented Poiseuille flow relative to continuous Poiseuille
flow denoted as 共p-p兲.
The results of Eqs. 共24兲 and 共25兲 are plotted in Fig. 4. It is clear
that for a stream of constant mass flow rate, the effect of segmentation alone, i.e., introduction of voids with no interfacial effects
assumed, will not enhance the heat transfer rate, as suggested by
Oliver and Wright 关5兴. In fact, segmentation creates a large deficit
in heat transfer rate that must be overcome by the interfacial circulation effect. Next, if we consider that the effect of a change in
local velocity profile is a result of segmentation, i.e., the flow
becoming more uniform or less parabolic, then a similar assessment as Eqs. 共24兲 and 共25兲 yields
APRIL 2011, Vol. 133 / 041902-5
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
4
can therefore be manipulated through plug length and tube length.
Thus, given a tube with a characteristic thermal length L쐓 for a
continuous flow stream defined as
Segmented Slug / Continuous Poiseuille
3
L쐓 =
Q*
2
1
Ls쐓 =
10
共27兲
we will now define a new dimensionless thermal length for the
segmented flow as
0 -5
10
L/D
PeD
-4
10
-3
10
-2
10
L*
-1
10
0
10
1
PeD =
冋冉 冑 冊 冉 冊 册
冋冉 冊 冉 冊 册
쐓
Qs-p
=
−2
1.128
␣ LL
1.614
L쐓1/3
쐓
+
1
4 ␣ LL 쐓
+
1
4L쐓
−3/2
共26兲
assuming a segmented slug flow relative to a continuous Poiseuille flow, denoted as 共s-p兲. In this case, one obtains the results
in Fig. 5. It is clear that segmentation can lead to enhanced heat
transfer over the same continuous flow stream if and only if a
change in local velocity profile can be induced through segmentation, i.e., the interfacial effect. The change in local velocity profile has been demonstrated in a recent paper by King et al. 关23兴,
which shows that segmentation produces velocity profiles that lie
between Poiseuille and slug flows, which also depend strongly on
plug length.
In an actual segmented fluid stream, interfacial effects give rise
to internal circulations within the liquid plugs, as shown in Fig. 2.
The plug configuration shown in Fig. 2共b兲 is typical of those produced in gas-liquid interfaces. These circulations increase the radial transport of heat and in effect give rise to a renewal in thermal
boundary layer by continuously replenishing the heated fluid near
the wall with cooler fluid from the core. In these cases, the heat
transfer rate can thus be controlled by plug length, as circulation
effects become less pronounced for longer slugs. Therefore, the
only way to make the curves in Fig. 4 give rise to Q쐓 ⬎ 1 is to
effectively reduce the L쐓 of the segmented flow. This is analogous
to slotting of surfaces in compact heat exchangers utilizing fins.
This can only be achieved through a secondary mechanism such
as circulation, which reduces the characteristic thermal length of
the system since each plug is now, in essence, decoupled from the
behavior shown in Fig. 3共c兲, since the boundary layer characteristics no longer depend on axial position of the plug but rather on
the length of the plug.
The induced plug circulations give rise to an oscillatory behavior in local and mean Nusselt numbers, as reported in Refs.
关12–15兴. These oscillations in Nusselt number result from the local circulation effect and begin downstream at one plug length
from the inlet. After three to five oscillations, the effect is essentially dampened out and the flow reaches a new but higher fully
developed flow Nusselt number. Provided that a heated tube is of
sufficient length that several circulations occur over the flow
length, the thermal characteristics are such that a higher but constant Nusselt number will be observed. In the work of Horvath et
al. 关4兴, the authors observed a slight dependency on dimensionless
duct length L / D for this reason, as their experiments typically had
many plugs in the flow stream at any instant 共see Table 1兲.
The thermal boundary layer characteristics in a segmented flow
041902-6 / Vol. 133, APRIL 2011
共29兲
L
⬃ ␣ L␭
N
共30兲
where N is the plug count in the tube and ␭ is the pitch of the plug
train. Thus, we can obtain
−2 −1/2
−3/2 −2/3
4ṁ
␲␳D␣
But the plug length Ls can be related to the original tube length by
means of
Ls ⬃ ␣L
␣L
共28兲
where PeD is defined using the mass flow rate of the liquid stream,
쐓
Fig. 5 Q for segmented slug versus continuous Poiseuille
Graetz flows having the same ṁ and no secondary flow
mechanism
Ls/D
PeD/␣L
Ls쐓 =
␣L2 쐓
L
N
共31兲
Therefore, considering both the effect of liquid fraction ␣L and
plug count N, a reduction in the effective dimensionless thermal
length is achieved by means of the factor ␣L2 / N. Since ␣L ⬍ 1 and
N ⬎ 1 for a segmented flow, considerable increases in heat transfer
coefficient can be obtained. If we consider a continuous flow having a dimensionless thermal length of L쐓 ⬃ 0.1 close to the optimal
length for a single phase flow and develop a plug train having a
liquid fraction of 0.5 and only one plug, we obtain a theoretical
enhancement ratio of 1.55 for the same feed rate. This simple
example clearly illustrates the promise of gas-liquid segmented
flow. Further, even with the mean liquid velocity two times higher
for the same mass flow rate, the pressure drop in laminar flow is
still quite reasonable 关1,22兴. In general, when Ls / D ⬎ 15, the pressure drop is not appreciably changed over the single phase flow
stream as most of the pressure drop is due to liquid shear at the
tube wall.
6
Data Scaling and Analysis
Since heat transfer occurs primarily to the liquid stream and not
to the segmenting gas medium, both the Nusselt number and dimensionless plug length should use the thermophysical properties
of the liquid. Further, only the wetted heat transfer surface should
be used in rendering the Nusselt data and not the total tube area.
The independent parameter 共dimensionless thermal length兲 is defined on the basis of the plug length and plug characteristics and
not based on the tube length as done in previous studies. Using the
present convention, this yields the true heat transfer coefficient for
the liquid phase and also allows for a proper comparison with
single phase Graetz theory. Only the study of Vrentas et al. 关8兴
came close to this reduction scheme since the authors had accounted for all heat transfer to the liquid and defined the Peclet
number using the actual liquid velocity for the same mass flow
rate as used in their single phase experiments. However, the authors used the dimensionless plug length Ls / D as a characterizing
parameter to generate a family of curves but did not use it directly
as a Graetz flow parameter.
The data of the four studies summarized in Table 1 are now
re-examined considering plug length and liquid fraction as the
controlling variables. In this present work, we have chosen to
redefine the dependent and independent variables so that all the
Transactions of the ASME
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
102
1
3
10
2
10
1
10
0
Oliver and Young-Hoon (1968) - A
Oliver and Young-Hoon (1968) - B
Oliver and Young-Hoon (1968) - C
Oliver and Young-Hoon (1968) - D
Graetz-Poiseuille Flow Theory
q*
q*
10
10
10
10
Oliver and Young-Hoon (1968) - A
Oliver and Young-Hoon (1968) - B
Oliver and Young-Hoon (1968) - C
Oliver and Young-Hoon (1968) - D
Horvath et al. (1973)
Vrentas et al. (1978)
Lakehal and Narayanan (2008)
Prothero and Burton (1961)
Graetz-Poiseuille Flow Theory
Graetz-Slug Flow Theory
0
-1
10
-3
10
-2
L*
10
10
-4
L*s
10
-3
10
-2
-1
Fig. 7 Data of Oliver and Young-Hoon †6,7‡ replotted considering plug length and liquid fraction
Fig. 6 All data plotted using nondimensional scaling consistent with total tube length when reducing data
Q/共␣LA兲 h̄D
=
k⌬Tln
k
NuD =
共32兲
based on actual wetted surface area and
q쐓 =
Q/共␣LA兲
1
关1 − exp共− 4NuDLs쐓兲兴
=
k共Tw − Ti兲 4Ls쐓
共33兲
which may easily be related to the mean Nusselt number and then
Ls쐓. The dimensionless thermal length Ls쐓 is now based on the local
liquid plug velocity and plug length, i.e.,
Ls쐓 =
Ls/D Ls/D
=
PeD UD/␣
共34兲
or, if written in terms of the liquid stream mass flow rate,
Ls쐓 =
␲ ␣ LL sk
4ṁC p
共35兲
Equations 共32兲 and 共35兲 reduce to their single phase flow counterparts when ␣L → 1. Thus, the effects of plug length and liquid
fraction can be properly addressed from the point of view of the
actual heat transfer coefficient, or put another way, the actual heat
transfer rate. This is quite important since in all the studies given
in Table 1, the Nusselt number is reported based on either the total
surface area or the wetted surface area, while the dimensionless
thermal length is always based on the true duct length. Both of
these issues have lead the published data showing no correlation
with single phase Graetz flows. To illustrate this issue, all of the
data from the studies summarized in Table 1 are plotted in Fig. 6.
In all cases, one can see that the effect of reducing slug length is
an increase in the nominal Nusselt number based on total tube
area when plotted against the dimensionless thermal length based
on total tube length. Figure 6 illustrates directly the level of enhancement that can be achieved with a segmented stream over a
continuous stream. However, in this form, no means of prediction
can be achieved. If one considers the carefully obtained data of
Horvath et al. 关4兴, it can be seen that as plug length decreases, the
nominal Nusselt number increases and in fact surpasses the Graetz
slug flow line.
In the study of Oliver and Young-Hoon 关6,7兴, the authors based
the Nusselt number on the total surface area of the tube and define
a two phase Graetz number using a velocity equal to the sum of
the liquid and gas velocities. Data were presented showing the
effect of dimensionless local plug velocity 共Graetz number兲 and
plug length on the Nusselt number. These data sets are denoted as
A and B in Fig. 6. Data were also presented showing Nusselt
Journal of Heat Transfer
103
Horvath et al. (1973)
Graetz-Poiseuille Flow Theory
102
q*
results may be easily compared with the simpler single phase
Graetz flow theory. Data are re-analyzed considering the following definitions:
number variation with plug length and liquid fraction for a constant mass flow rate. These data sets are denoted as C and D in
Fig. 6. Both sets of data have been reduced using the information
in the original paper along with the observations of Hughmark 关9兴
on liquid holdup calculations for laminar flows when the component gas and liquid flow rates are known. These data are presented
in Fig. 7 along with the simple Graetz flow theory. One can see
that a redistribution of the experimental data has occurred, with
the data now uniformly spread along the Graetz flow theory. The
data reported in Refs. 关6,7兴 for Nu versus Gz and Ls show much
less scatter.
In the study of Horvath et al. 关4兴 and Vrentas et al. 关8兴, the
authors based the Nusselt number on the wetted surface area,
␲DL␣L, but Horvath et al. 关4兴 defined the Reynolds number using
the sum of the phase velocities as was done by Oliver and YoungHoon 关6,7兴, whereas Vrentas et al. 关8兴 defined the Reynolds number using the local liquid plug velocity. In both studies, the Nusselt data are plotted versus the Reynolds number 关3兴 or the Peclet
number 关8兴 for various dimensionless plug lengths Ls / D. In this
form, the data clearly show increased Nusselt values with decreasing plug length. The data of these two studies have been reanalyzed and plotted versus Ls쐓. Figure 8 presents the data of Horvath et al. 关4兴 while the data of Vrentas et al. 关8兴 are given in Fig.
9. The data of Horvath et al. 关4兴 clearly demonstrate the scaling of
heat transfer data with dimensionless plug length. The data of
Vrentas et al. 关8兴 are somewhat higher for the larger Ls / D ratios.
This can be explained by the fact that in their experiments for
these plug lengths, typically, there were between two and four
liquid plugs. Additional data of Horvath et al. 关4兴 show that a
modest entrance effect occurs as tube length decreases for a given
flow. In other words, as tube length decreases, plug count N also
decreases. The available data and our own experimental observations 关15兴 indicate that if fewer than five plugs exist in the plug
10
1
0
10 -6
10
10-5
L*s
10-4
10-
Fig. 8 Data of Horvath et al. †4‡ replotted considering plug
length and liquid fraction
APRIL 2011, Vol. 133 / 041902-7
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
103
103
Vrentas et al. (1978) - L/D = 128
Vrentas et al. (1978) - Ls/D = 10.5
Vrentas et al. (1978) - Ls/D = 33.5
Vrentas et al. (1978) - Ls/D = 61
Lakehal and Narayanan (2008)
Graetz-Poiseuille Flow Theory
102
q*
q*
102
10
Oliver and Young-Hoon (1968) - A
Oliver and Young-Hoon (1968) - B
Oliver and Young-Hoon (1968) - C
Oliver and Young-Hoon (1968) - D
Horvath et al. (1973)
Vrentas et al. (1978)
Lakehal and Narayanan (2008)
Walsh et al. (2009)
Graetz-Poiseuille Flow Theory
1
0
10 -4
10
10
10-3
L*s
10-2
0
10 -6
10
10-1
Fig. 9 Data of Vrentas et al. †8‡ and Narayanan and Lakehal
†11‡ replotted considering plug length and liquid fraction
10-5
10-4
10-3
L*s
10-2
10-
Fig. 11 All data replotted considering plug length and liquid
fraction in the data reduction
7
train, higher heat transfer rates are observed over the case of plug
trains with many plugs in excess of 5. This would indicate that for
the isothermal boundary condition, a fully developed recirculating
heat transfer coefficient is obtained. It is this recirculating flow
characteristic that leads to the Ls / D scaling that is observed in the
present data sets.
Finally, Narayanan and Lakehal 关11兴 undertook a numerical
study of segmented flow in a minitube and define the Nusselt
number using the total surface area and the Reynolds number
using the gas velocity and liquid properties. The data have been
re-analyzed according to Eqs. 共32兲 and 共35兲 and have also been
plotted in Fig. 9 with the data of Vrentas et al. 关8兴. These comparisons are only for a qualitative scaling purpose.
Recent data obtained by the authors on electrically heated tubes
are presented in Fig. 10. The data that are obtained for an isoflux
boundary condition have been reduced according to Eq. 共16兲 to
make comparisons with the mean isothermal wall theory developed earlier. These data represent the fully developed flow or
limiting Nusselt numbers. Two types of data are presented. Those
with plug counts less than 3, and those with plug counts greater
than 7. In the latter, the integrated mean wall temperature is higher
than in the former case, and the data for q쐓 appear lower. All data
are now plotted in Fig. 11 along with the single phase Graetz
theory. Overall, the predictive ability of the single phase Graetz
flow theory for a gas-liquid segmented flow appears sound, particularly when compared with Fig. 6 共i.e., before rescaling L쐓兲.
The effect of plug length as a controlling variable is clearly being
demonstrated.
1
Predictive Heat Transfer Model
The proposed model is now summarized for a laminar gasliquid segmented flow. The dimensionless mean heat flux for an
isothermal wall based on wetted contact area can be determined
using
q쐓 =
冋冉 冊 冉 冊 册
1.614
−3/2
1
+
Ls쐓1/3
4Ls쐓
−3/2 −2/3
共36兲
where
q쐓 =
Q/共␣L␲DL兲D
k共Tw − Ti兲
共37兲
␲ ␣ LL sk
4ṁC p
共38兲
and
Ls쐓 =
Equation 共36兲 is applicable to segmented flow systems assuming
that N ⲏ 5, i.e., typically more than five plugs in the heated tube at
any given time. Further, given the nature of thermal entrance
length problems, the axial residence time of the plug flow should
provide for L쐓 ⱗ 0.1 since even a short liquid plug experiencing a
higher heat transfer coefficient will become thermally saturated
after this distance is achieved.
In a laminar Taylor 共plug兲 flow, it has been well documented
that the liquid fraction can be approximated very well using
␣L ⬇
Q̇l
Q̇l + Q̇g
共39兲
or if there is significant liquid film around the gas voids, the
Armand 关24兴 correlation
␣L ⬇ 0.83
103
Walsh (2009) - N < 3
Graetz-Poiseuille Flow Theory
Walsh (2009) - N > 7
q*
1
0
10 -4
10
10-3
L*s
10-2
共40兲
10-
Fig. 10 Data of Walsh et al. †15‡ replotted considering plug
length and liquid fraction
041902-8 / Vol. 133, APRIL 2011
Q̇l + Q̇g
may be used.
Since it is anticipated that most applications will involve liquid
cooled heat sinks, we limit ourselves to the dimensionless heat
transfer rate based on the inlet temperature difference Tw − Ti.
However, in a two fluid-type heat exchanger, Eqs. 共36兲 and 共37兲
can be replaced by Eq. 共19兲 with L쐓 replaced with Ls쐓 and Eq. 共32兲
in order to compute the heat transfer coefficient, h, based on the
wall to bulk temperature difference. If pressure drop is also desired, the reader is directed to another recent work of the authors
关25兴.
102
10
Q̇l
8
Summary and Conclusions
Nonboiling gas-liquid plug flow heat transfer was examined
from the point of view of plug length and plug circulation. Since
Transactions of the ASME
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
plug circulation is closely related to plug length, it was shown that
the effect of plug length on boundary layer renewal is strong, and
as such, predictive methods should utilize the actual plug length
when applying classic Graetz flow theory. Using the limited available data from the literature and some new data of the authors, we
have shown that better scaling of the dimensionless data is obtained when true wetted surface area and plug length are considered. It was shown that data redistribute themselves along the
Graetz–Poiseuille curve with a considerable reduction in scatter
and predictive error. Overall, root mean square 共rms兲 errors are
reduced from 120% to 24% for all data sets. In the case of four of
these carefully obtained data sets, the rms error is approximately
10–15%.
Acknowledgment
The authors acknowledge the financial support of the Natural
Sciences and Engineering Research Council of Canada 共NSERC兲
and Enterprise Ireland 共EI兲.
Nomenclature
A
Cp
D
Gz
h
k
L
Ls
L쐓
Ls쐓
ṁ
N
NuD
P
PeD
Pr
q̄
q쐓
Q
Q̇g
Q̇l
Q쐓
ReD
T
Ti
Tm
Tw
⌬T쐓
U
z
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
Greek Symbols
␣ ⫽
␣L ⫽
⑀ ⫽
␭ ⫽
␮ ⫽
␯ ⫽
␳ ⫽
flow area, m2
specific heat, J / kg K
diameter of circular duct, m
Graetz number 共⬅ṁC p / kL兲
heat transfer coefficient, W / m2 K
thermal conductivity, W / m K
length of tube, m
length of liquid plug, m
dimensionless length of tube 共⬅L / DPeD兲
dimensionless length of plug 共⬅Ls / DPeD兲
liquid mass flow rate, kg/s
number of plugs in tube
Nusselt number 共⬅hD / k兲
perimeter, m
Peclet number 共⬅UD / ␣兲
Prandtl number 共⬅␯ / ␣兲
mean heat flux, W / m2
dimensionless wall heat flux
heat transfer, W
gas flow rate, m3 / s
liquid flow rate, m3 / s
thermal enhancement ratio
Reynolds number 共⬅UD / ␯兲
temperature, K
inlet temperature, K
bulk temperature, K
wall temperature, K
dimensionless temperature
average liquid velocity, m/s
local axial coordinate, m
thermal diffusivity, m2 / s
liquid fraction 共⬅1 − ⑀兲
void fraction
plug pitch, m
dynamic viscosity, N s / m2
kinematic viscosity, m2 / s
fluid density, kg/ m3
Superscripts
쐓
⫽ dimensionless
共 · 兲 ⫽ mean value
Journal of Heat Transfer
Subscripts
c
D
i
m
pois
s
slug
w
z
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
continuous
based on diameter D
inlet
mixed mean
Poiseuille flow
segmented
slug flow
wall
local 共axial兲 variation
References
关1兴 Kreutzer, M. T., Kapteijn, F., Moulijin, J. A., Kleijn, C. R., and Heiszwolf, J.
J., 2005, “Inertial and Interfacial Effects on Pressure Drop of Taylor Flow in
Capillaries,” AIChE J., 51共9兲, pp. 2428–2440.
关2兴 Taylor, G. I., 1961, “Deposition of a Viscous Fluid on the Wall of a Tube,” J.
Fluid Mech., 10, pp. 161–165.
关3兴 Prothero, J., and Burton, A. C., 1961, “The Physics of Blood Flow in Capillaries: 1 The Nature of the Motion,” Biophys. J., 1, pp. 565–579.
关4兴 Horvath, C., Solomon, B. A., and Engasser, J. M., 1973, “Measurement of
Radial Transport in Slug Flow Using Enzyme Tubes,” Ind. Eng. Chem. Fundam., 12共4兲, pp. 431–439.
关5兴 Oliver, D. R., and Wright, S. J., 1964, “Pressure Drop and Heat Transfer in
Gas-Liquid Slug Flow in Horizontal Tubes,” Br. Chem. Eng., 9, pp. 590–596.
关6兴 Oliver, D. R., and Young-Hoon, A., 1968, “Two Phase Non-Newtonian Flow:
Part 1 Pressure Drop and Holdup,” Trans. Inst. Chem. Eng., 46, pp. 106–115.
关7兴 Oliver, D. R., and Young-Hoon, A., 1968, “Two Phase Non-Newtonian Flow:
Part 2 Heat Transfer,” Trans. Inst. Chem. Eng., 46, pp. 116–122.
关8兴 Vrentas, J. S., Duda, J. L., and Lehmkuhl, G. D., 1978, “Characteristics of
Radial Transport in Solid-Liquid Slug Flow,” Ind. Eng. Chem. Fundam.,
17共1兲, pp. 39–45.
关9兴 Hughmark, G. A., 1965, “Holdup and Heat Transfer in Horizontal Slug GasLiquid Flow,” Chem. Eng. Sci., 20, pp. 1007–1010.
关10兴 Lakehal, D., Larrignon, G., and Narayanan, C., 2008, “Computational Heat
Transfer and Two Phase Flow Topology in Miniature Tubes,” Microfluid.
Nanofluid., 4共4兲, pp. 261–271.
关11兴 Narayanan, C., and Lakehal, D., 2008, “Two Phase Convective Heat Transfer
in Miniature Pipes Under Normal and Microgravity Conditions,” ASME J.
Heat Transfer, 130, p. 074502.
关12兴 Young, P., and Mohensi, K., 2008, “The Effect of Droplet Length on Nusselt
Numbers in Digitized Heat Transfer,” Proceedings of I-THERM, 2008, 11th
IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena
in Electronic Systems, pp. 352–359.
关13兴 Mohseni, K., and Baird, E. S., 2007, “Digitized Heat Transfer Using Electrowetting on Dielectric,” Nanoscale Microscale Thermophys. Eng., 11, pp.
99–108.
关14兴 Baird, E., and Mohseni, K., 2008, “Digitized Heat Transfer: A New Paradigm
for Thermal Management of Compact Micro Systems,” IEEE Trans. Compon.
Packag. Technol., 31共1兲, pp. 143–151.
关15兴 Walsh, P., Walsh, E., and Muzychka, Y. S., 2010, “Heat Transfer Model for
Gas–Liquid Slug Flows Under Constant Flux,” Int. J. Heat Mass Transfer,
53共15–16兲, pp. 3193–3201.
关16兴 Fries, D., Trachsel, F., and von Rohr, P. R., 2008, “Segmented Gas-Liquid
Flow Characterization in Rectangular Channels,” Int. J. Multiphase Flow, 34,
pp. 1108–1118.
关17兴 Yu, Z., Hemminger, O., and Fan, L. S., 2007, “Experiment and Lattice Boltzmann Simulation of Two Phase Gas-Liquid Flows in Microchannels,” Chem.
Eng. Sci., 62, pp. 7172–7183.
关18兴 Ua-Arayaporn, P., Fukagata, K., Kasagi, N., and Himeno, T., 2005, “Numerical Simulation of Gas-Liquid Two Phase Convective Heat Transfer in a Micro
Tube,” ECI International Conference on Heat Transfer and Fluid Flow in Microscale, Castelvecchio Pascoli, Italy, Sept. 25–30.
关19兴 Kakac, S., Aung, W., and Shah, R. K., 1987, Handbook of Single Phase Convective Heat Transfer, Wiley, New York.
关20兴 Bejan, A., 2005, Convection Heat Transfer, Wiley, New York.
关21兴 Churchill, S. W., and Usagi, R., 1972, “A General Expression for the Correlation of Rates of Transfer and Other Phenomena,” Am. Inst. Chem. Eng. Symp.
Ser., 18, pp. 1121–1128.
关22兴 Muzychka, Y. S., and Yovanovich, M. M., 2004, “Laminar Forced Convection
Heat Transfer in the Combined Entry Region of Non-Circular Ducts,” ASME
J. Heat Transfer, 126, pp. 54–62.
关23兴 King, C., Walsh, E. J., and Grimes, R., 2007, “PIV Measurements of Flow
Within Plugs in a Microchannel,” Microfluid. Nanofluid., 3共4兲, pp. 463–472.
关24兴 Wallis, G. B., 1969, One Dimensional Two Phase Flow, McGraw-Hill, New
York.
关25兴 Walsh, E., Muzychka, Y. S., Walsh, P., Egan, V., and Punch, J., 2009, “Pressure
Drop in Two Phase Slug/Bubble Flows in Mini Scale Capillaries,” Int. J.
Multiphase Flow, 35, pp. 879–884.
APRIL 2011, Vol. 133 / 041902-9
Downloaded 16 Mar 2011 to 134.153.184.170. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm