Heat transfer engineering an international journal, tập 31, số 3, 2010

Heat Transfer Engineering, 31(3):159–167, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903304335

Similarities and Differences Between
Flow Boiling in Microchannels
and Pool Boiling
SATISH G. KANDLIKAR
Mechanical Engineering Department, Rochester Institute of Technology, Rochester, New York, USA

Recent literature indicates that under certain conditions the heat transfer coefficient during flow boiling in microchannels
is quite similar to that under pool boiling conditions. This is rather unexpected, as microchannels are believed to provide
significant heat transfer enhancement under single-phase as well as flow boiling conditions. This article explores the
underlying heat transfer mechanisms and illustrates the similarities and differences between the two processes. Formation
of elongated bubbles and their passage over the microchannel walls have similarities to the bubble ebullition cycle in pool
boiling. During the passage of elongated bubbles, the longer duration between two successive liquid slugs leads to wall
dryout and a critical heat flux that may be lower than that under pool boiling conditions. A clear understanding of these
phenomena will help in overcoming these limiting factors and in developing strategies for enhancing heat transfer during
flow boiling in microchannels.

INTRODUCTION
The nucleation criterion developed by Hsu [1] has been successful in predicting the onset of nucleation in pool boiling as
well as in flow boiling. The criterion was also shown to be quite
accurate for flow boiling in microchannels by a number of investigators, including Zhang et al. [2] and Kandlikar et al. [3].
The high single-phase heat transfer coefficient value prior to nucleation in flow boiling leads to nucleation cavity diameters that
are smaller than those in pool boiling. This link between pool
boiling and flow boiling is an important factor in comparing the
two boiling modes.
In the quest for improved heat removal rates, in general, pool

boiling is considered to be more efficient (higher heat transfer
coefficient) than single-phase liquid flow, while flow boiling
provides the highest heat transfer coefficients. However, recent
data obtained with enhanced single-phase flow channels and
flow boiling in microchannels indicate that this may not be necessarily true with the current status of these two modes of heat
transfer. Following the definition of Kandlikar and Grande [4],
microchannels are defined as channels with hydraulic diameter

Address correspondence to Professor Satish G. Kandlikar, Mechanical Engineering Department, Rochester Institute of Technology, Rochester, NY 14623,
USA. E-mail:

(or the smallest flow passage width of a channel) between 10 µm
and 200 µm.
Table 1 shows a comparison of heat transfer coefficients and
heat fluxes for four cases: single-phase flow in plain microchannels, single-phase flow in enhanced microchannels, pool boiling,
and saturated flow boiling in plain and enhanced (with reentrant
cavities) microchannels. Due to the pressure drop constraints,
the flow in microchannels is generally in laminar flow regime.
The single-phase heat transfer coefficients are therefore calculated for laminar flow (including the entrance region effect).
The plain microchannels are unable to meet the high heat flux
cooling requirement of 1000 W/cm2 (10 MW/m2 ). However,
the microchannels enhanced with short offset strip fins provide
a very high heat transfer coefficient. In a practical system with
this geometry, Colgan et al. [5] employed multiple-inlet/-outlet
regions with a flow length of only 2 mm through the microchannels. This configuration holds the most promise in meeting the
future chip cooling challenges. Results with single-phase flow
[5, 6], pool boiling [7, 8], and flow boiling in microchannels
under stable and unstable conditions [9, 10] are used in the
comparison presented in Table 1.
The pool boiling mode at macroscale offers an efficient mode

of heat transfer. The saturated flow boiling heat transfer with
plain microchannels [9] and that with reentrant cavities [10]
both provide an improvement over plain microchannels, but fall
substantially below the desired values. Although these values

159


160

S. G. KANDLIKAR

Table 1 Comparison of heat transfer coefficients with water under different
modes
Heat transfer mode
Single-phase flow in plain 100- to
200-µm square microchannels
Single-phase flow in enhanced
microchannels, 48 µm × 256 µm,
offset strip fins, 500 µm, Colgan et al.
[5], Steinke and Kandlikar [6]
Pool boiling, flat surface, fully developed
boiling, Nukiyama [7], microdrilled
surface, Das et al. [8]
Saturated flow boiling in Dh ∼ 207 µm
rectangular microchannels, Steinke
and Kandlikar [9], with reentrant
cavities, Kuo and Peles [10]

h, kW/m2◦ C

q , MW/m2

10–15

0.1–0.2

>500

5–10

∼30–300

1.2–4.8

30–80

2–3.5

are higher than those under pool boiling, the increase is not
significant. In fact, employing enhanced pool boiling surfaces
can improve the performance by a factor of 2–4, e.g., by microdrilling the heater surface with holes at 5–10 mm pitch as
reported by Das et al. [8].
In comparison, a set of parallel microchannels with or without reentrant cavities yields much lower performance as compared to single-phase flow in enhanced microchannels or pool
boiling on enhanced surfaces. This has been a major concern in
developing flow boiling systems to meet the needs in electronic
cooling applications.
There have been a number of papers published exploring the
effect of diameter on flow boiling heat transfer. A comparison
of two data sets obtained by Kenning and Cooper [11] for a

9.6 mm diameter circular tube and by Steinke and Kandlikar [9]
for a 207 µm hydraulic diameter rectangular channel is shown
in Figure 1. Both data sets are obtained at close to atmospheric
pressure and the Boiling numbers are around 1.5 × 10−4 in
both cases. The effect of diameter on the ratio of two-phase
to liquid-only single-phase heat transfer coefficients is depicted

Figure 1 Effect of channel hydraulic diameter on the ratio hT P / hLO for Bo
≈ 1.5 × 104 during flow boiling of water near atmospheric pressure.

heat transfer engineering

in Figure 1. It is seen that this ratio is reduced considerably
from a value of 9.5 for the 9.6 mm-diameter tube to 4 for
the 207 µm channel. Further, considering the fact that hLO
in the microchannel corresponds to laminar flow conditions, a
compelling argument can be made for the dramatic reduction in
heat transfer coefficient for the smaller diameter tube.
It has been suggested by a number of authors, including
Lazarek and Black [12], that flow boiling in narrow channels
can be predicted reasonably well with a pool boiling correlation. Kew and Cornwell [13] compared the flow boiling data in
narrow channels with an established pool boiling correlation by
Cooper [14] with some degree of success. The other correlations
that were similarly successful had heat flux as the primary variable. This realization, brought about by the success of the pool
boiling correlations in predicting the flow boiling in microchannels, is really the precursor to the present article. This similarity
is further explored using the available literature on experimental
data and theoretical models. The discussion is focused on water
as the working fluid, but by no means is this study intended to be
restrictive in this regard. The broad availability of experimental
data with water makes it possible to present a more comprehensive comparison between the pool boiling and microchannel

flow boiling.

NUCLEATE BOILING AND CONVECTIVE BOILING
CONTRIBUTIONS
The contributions from nucleate boiling and convective boiling during flow boiling are well recognized. The nucleate boiling
contribution is dependent on the heat flux, in a manner similar to
the pool boiling with an exponent of around 0.7. The convective
boiling component is independent of the heat flux and varies
with the mass flux. For the conventional large diameter tubes,
the mass flux dependence was identified with an exponent of
0.8, which is in agreement with the turbulent single-phase flow
relationship. A flow boiling map proposed by Kandlikar [15]
showed these contributions with Boiling number Bo and density ratio, ρL /ρG , as parameters. The map was developed with
hT P / hLO versus x using the Kandlikar [16] correlation. The
map was instrumental in explaining the different dependencies
observed in the two-phase heat transfer data as a function of
quality. The nucleate boiling component is adversely affected
with an increase in quality, while the convective boiling term
increases with quality due to the higher specific volume of vapor
being produced. The relative contributions from these components are governed by Bo and ρL /ρG . A higher density ratio
causes a larger increase in the overall flow velocity upon vaporization, leading to a greater increase in the heat transfer
coefficient, while a low value of density ratio causes the convective contribution to increase only moderately. A combination
of low Bo and high ρL /ρG causes hT P /hLO to increase with an
increase in x, while a combination of high Bo and low ρL /ρG
causes hT P /hLO to decrease with an increase in x.
vol. 31 no. 3 2010


S. G. KANDLIKAR
Table 2 Comparison of two-phase flow structures in the two boiling modes

Flow boiling

Pool boiling

Bubble inception as the nucleation
criterion is met for specific
cavities under single-phase liquid
flow.
Elongated bubble covering the
channel walls.
Liquid slug being pushed between
the two consecutive elongated
bubbles.
Liquid slugs are intensely mixed
with vapor in a churn flow.

Bubble inception as the nucleation
criterion is met for specific
cavities under natural convection
with liquid.
Growing bubbles covering the
heater surface.
Liquid circulation around the
nucleating bubbles as a result of
the individual bubble ebullition
cycles.
Liquid surrounding bubbles
(undergoing ebullition cycles) is
intensely mixed with vapor at
high heat fluxes under fully

developed boiling conditions.

These trends, as described by Kandlikar [15], are further
affected by laminar flow occurring in small-diameter channels.
Depending on the single-phase liquid Reynolds number, the
flow may be in the laminar region, where the single-phase liquid
heat transfer coefficient under fully developed flow conditions
is independent of the mass flux. This is one of the reasons why
the two-phase heat transfer coefficient is dramatically altered in
microscale channels.
Another effect of the small channel dimensions arises due
to the changes occurring in the flow patterns. The nucleating
vapor bubbles are confined in the small channels and grow as
elongated bubbles, forming alternate liquid slugs and elongated
bubbles. The two-phase flow structures during flow boiling resemble the respective pool boiling characteristics as shown in
Table 2.
The single-phase heat transfer in microchannels is generally
under laminar flow conditions due to the pressure drop limitations and the small channel dimensions. As pointed out earlier,
the convective contribution from the single-phase liquid flow
needs to be considered using the laminar flow equation. The dependence of the convective contribution is thus altered from the
conventional channel trends since the Nusselt number in laminar
flow is independent of the flow rate. These effects are accounted
for in the flow boiling correlation proposed by Kandlikar and
Balasubramanian [17]. The correlation is rewritten in terms of
the density ratio and Boiling number as follows:
For 400 ≤ ReLO ≤ 1600:
hT P
= larger of
hLO

hT P ,N BD
hLO
hT P ,CBD
hLO

161

hT P ,CBD
= 1.136(ρL /ρG )0.45 x 0.72 (1 − x)0.0.08
hLO
+ 667.2 Bo0.7 (1 − x)0.8 FF l

(3)

The single-phase heat transfer coefficient hLO is calculated from
the laminar flow equation instead of the Gnielinski [18] correlation for the turbulent region. In the region of Re from 1600 to
3000, a linear interpolation is recommended.
For the low Reynolds number range 100 ≤ ReLO < 400, the
heat transfer coefficient is found to be always nucleate boiling
dominant (NBD). Thus:
hT P
hT P ,NBD
=
= 0.6683(ρL /ρG )0.1 x 0.16 (1 − x)0.64
hLO
hLO
+ 1058.0Bo0.7 (1 − x)0.8 FF l

(4)

In the range Re < 100, the convective component in the above
NBD term is reduced further and hT P depends on the nucleate
boiling component alone:
hT P
= 1058.0Bo0.7 (1 − x)0.8 FF l
hLO

(5)

Equations (1)–(5) are used to generate a flow boiling map
for microchannels. Three values of density ratio, 10, 100, and
1000, and two values of Bo∗ , 10−4 and 10−3 , are used to generate the plots. The modified Boiling number Bo∗ is defined as
follows:
Bo∗ = Bo × (FF l )1/0.7

(6)

Figures 2–4 show the plots generated for different Re ranges.
Figure 2 shows the variation of the ratio hT P / hLO with x
with different values of Bo for 400 ≤ ReLO ≤ 1600. This
plot is same as the one for large-diameter tubes, but the actual
heat transfer coefficient will be different since the single-phase

(1)

where
hT P ,NBD
= 0.6683 (ρL /ρG )0.1 x 0.16 (1 − x)0.64
hLO
+ 1058.0 Bo0.7 (1 − x)0.8 FF l

(2)

heat transfer engineering

Figure 2 Flow boiling map for microchannels in the range 400 ≤ ReLO ≤
1600, Bo∗ = Bo × (FF l )1/0.7 .

vol. 31 no. 3 2010


162

S. G. KANDLIKAR

HEAT TRANSFER PROCESSES DURING POOL
AND FLOW BOILING
Factors Responsible for Heat Transfer Degradation in
Microchannels

Figure 3 Flow boiling map for microchannels in the range 100 ≤ ReLO <
400, Bo∗ = Bo × (FF l )1/0.7 .

coefficient hLO will be derived from the laminar flow
equations.
Figure 3 shows the flow boiling map for 100 ≤ ReLO < 400.
Here the nucleate boiling component begins to play a major
role as seen by the continuous decrease in h with x throughout
the range. In other words, the increased flow velocity at higher
x does not provide the expected benefits in terms of improved

convective heat transfer.
Figure 4 shows the flow boiling map for very low values of
ReLO < 100. The convective component is completely blocked
off; the density ratio has no effect on h. Here the suppression
effects are overriding and the heat transfer exhibits similar characteristics as in nucleate boiling with the increased suppression
effects at higher qualities.

The flow boiling maps depicted in Figures 2–4 are based on
the experimental data and provide a visual tool to illustrate the
effects of flow parameters on heat transfer coefficient. As the
ReLO is reduced, it is seen that the nucleate boiling becomes
the dominant mode, with its decreasing trend in h versus x.
Further decreases in Reynolds number cause the heat transfer to
deteriorate, with the elimination of the convective contribution
term in Eq. (5).
The boiling instabilities experienced in microchannels are
another major cause for heat transfer reduction. These instabilities occur at lower mass fluxes as the inertia of the incoming
liquid is insufficient to prevent the liquid from rushing back.
The reasons for instabilities and methods for preventing them
have been discussed in a number of publications, including Serizawa et al. [19], Steinke and Kandlikar [9], Hetsroni et al. [20],
Kandlikar et al. [3], and Kuo and Peles [21]. As a result of the
instabilities, the walls of the microchannels remain exposed to
the expanding vapor bubble, creating local dry patches on the
wall and causing heat transfer deterioration.
Another method to avoid the instabilities is to change the
operating conditions with increased mass fluxes. Dong et al.
[22] conducted experiments with R-141b in 60 µm × 200 µm
parallel rectangular microchannels for mass fluxes of 400 to
980 kg/m2 -s. Boiling was initiated within the channels with
subcooled liquid inlet. Pressure drop oscillations were not observed and stable boiling was attained. The stable results obtained under such conditions were shown to agree quite well with

the Kandlikar and Balasubramanian [17] correlation, whereas
the unstable data observed in the Steinke and Kandlikar correlation showed a marked deterioration with increasing quality as shown in Figure 5. The results of Dong et al. [22] are
shown in Figure 6. Although a higher mass flux is beneficial for
heat transfer, the resulting pressure drop could be prohibitively
large.
Similarities Between Pool Boiling and Microchannel Flow
Boiling Mechanisms

Figure 4 Flow boiling map for microchannels in the range ReLO < 100,
Bo∗ = Bo × (FF l )1/0.7 .

heat transfer engineering

Some of the recent publications provide an insight into the
reasons for this dramatic reduction in h with x, even under
stable conditions. Using the elongated bubble flow pattern description, Kandlikar [23] pointed out the similarities between the
microchannel flow boiling and pool boiling heat transfer. As a
bubble grows, the downstream interface represents the receding
liquid–vapor interface of a growing nucleating bubble, whereas
the upstream interface of the elongated bubble is similar to the
advancing liquid–vapor interface of a nucleating bubble as its
base shrinks and the bubble begins to depart from the heated
vol. 31 no. 3 2010


S. G. KANDLIKAR

163

Figure 5 Flow boiling results from Steinke and Kandlikar [9] for water,

showing dramatic reduction in heat transfer performance at increased qualities
due to instabilities; q is heat flux, W/m2 .

surface in pool boiling. Figure 7a depicts the respective interfaces as elongated bubbles are formed in a microchannel. These
two interfaces were experimentally studied in a moving meniscus on a heated surface by Kandlikar et al. [24] and numerically
by Mukherjee and Kandlikar [25]. Their studies showed the
important roles played by transient conduction as the liquid interface advances over the heater surface. The microconvection
caused by the liquid flow behind the advancing liquid interface
for a moving meniscus is shown in Figure 7b, and during a nucleate boiling bubble ebullition cycle is shown in Figure 7c. The
receding interface provides a phase change surface where the
liquid superheat is dissipated and cooled liquid becomes avail-

Figure 7 Elongated bubbles in microchannels presenting advancing and receding interfaces in (a) that are similar to interface movements of a moving
meniscus (b) and a nucleating bubble during a bubble ebullition cycle in pool
boiling shown in (c).

Figure 6 Experimental data for flow boiling of R-141b by Dong et al. [22]
and predictions from Kandlikar and Balasubramanian [17] at G = 500 kg/m2 s
under stable operation, FF l = 1.8, Bo* = [q/(Ghfg )] × FF l = 1.2 × 10−3
(lower line) and 1.6 × 10−3 (upper line).

able for the transient conduction process. The advancing and receding interfaces seen around an elongated bubble are shown in
Figure 7a.
In the model proposed by Jacobi and Thome [26], the
heat transfer in the liquid slug region is assumed to be by
laminar steady-state convection, and its contribution is quite
small compared to that from microlayer evaporation. However,
the numerical simulation and the results from transient conduction model described by Mukherjee and Kandlikar [25]
and Kandlikar et al. [24] indicate that transient conduction
and microconvection modes contribute significantly in the

evaporating meniscus geometry. Mukherjee and Kandlikar [27]
simulated the bubble growth and elongated flow pattern development in microchannels and concluded that the transient conduction and the subsequent convection behind theevaporating
liquid–vapor interface were the major contributors to the total heat transfer process in microchannel flow boiling as
well.

heat transfer engineering

vol. 31 no. 3 2010


164

S. G. KANDLIKAR

Role of Microlayer Evaporation during Elongated Bubble
Flow Pattern
Comparing pool boiling and the microchannel flow boiling
processes, the three distinct modes of heat transfer that can be
identified in both cases are:
1. Transient conduction heat transfer resulting from the motion
of the liquid–vapor interface over the heated surface. The heat
transfer is enhanced due to the cooler liquid being brought
in contact with the heater surface as a result of interface
movement.
2. Microconvection heat transfer resulting from the increased
convection from the interface movement. It is combined with
the transient conduction contribution effect described above,
since it is difficult to identify and isolate their individual
effects.
3. Microlayer evaporation resulting from the evaporation of a

thin layer of liquid left on the heater by the receding liquid–
vapor interface.
Relative contributions from these three mechanisms have
been a topic of intense research in pool boiling. Myers et al.
[28] used silicon chips with heaters and sensors to determine
the localized heat fluxes and surface temperatures around nucleating bubbles. Figure 8 shows the relative contributions from
these three mechanisms for water. It can be seen that the transient conduction/microlayer convection together are the largest
contributor to the total heat flux during a bubble ebullition cycle.
The microlayer contribution was seen to be quite small, around
20%. These results are in agreement with the numerical work
by Son et al. [29]. Recent work by Moghaddam and Kiger [30]
showed similar results for FC-72.
The microlayer contribution has received considerable attention in recent flow boiling studies in microchannels. It is

Figure 8 Relative contributions from different mechanisms during pool boiling. Redrawn using data from Myers et al. [28].

Figure 9 Equivalent convective coefficient for films under a steady-state conduction model.

very difficult to measure the microlayer thickness in the microchannel flows. Calculating from the experimental data, a
film thickness on the order of 10–20 µm has been estimated by
Jacobi and Thome [26] from a parametric study. The initial film


S. G. KANDLIKAR

165

Table 3 Similarities and differences between pool boiling and microchannel flow boiling processes
Process

Similarities

Pool boiling

Microchannel flow boiling

Nucleation

The same nucleation criterion by Hsu [1]
is applicable for both processes.

Bubble growth

Transient conduction and
microconvention heat transfer
processes are similar in the liquid slug.
At higher flow rates, the two-phase
flow characteristics of large-diameter
tubes appear and the microchannel
flow boiling becomes distinctly
different from pool boiling.
Role of microlayer evaporation is
relatively limited in both cases,
accounting for only 20–25% of the
total heat transfer.

Nucleation cavities and bubble departure
sizes are larger. The low h in single-phase
flow prior to nucleation allows nucleation
at lower wall superheats.

The bulk liquid is not highly superheated
prior to onset of nucleation, causing the
bubbles to grow predominantly near the
heater surface.

Nucleation cavities and bubble departure
sizes are smaller. The high h value during
single-phase liquid flow prior to
nucleation introduces large wall superheat.
Bulk liquid also reaches a high degree of
superheat causing explosive bubble
growth following nucleation.

The microlayer thickness is on the order of
1–3 µm, Koffman and Plesset [34]. The
high frequency in the bubble ebullition
cycle limits the occurrence, extent, and
duration of dry patches from microlayer
depletion from evaporation at high heat
fluxes.
Smaller bubbles coalesce prior to CHF as the
liquid interface retracts away.

The microlayer under bubbles in flow
boiling are thicker and impede heat
transfer. The lower frequency of elongated
bubble passage allows longer time for
microchannel wall dryout.

Microlayer

Critical heat flux
(CHF)

CHF condition results from the inability
of the advancing liquid front to rewet
the dry patches.

Heat transfer
enhancement

Altering nucleation characteristics will
provide significant heat transfer
enhancements in both cases.

Providing early nucleation by introducing
cavities of right sizes and geometries is
successfully implemented in pool boiling.

Comparison Between the Pool Boiling and Microchannel
Flow Boiling Processes
The underlying heat transfer mechanisms in the two processes have many similarities, with transient conduction, microconvection, and microlayer evaporation playing similar roles
in both. The essential differences between the two processes
emerge from the presence of strong inertia forces in the bulk
flow and the large shear stress present at the wall. These forces
affect the nucleation and other flow characteristics directly. Heat
transfer processes are also affected.
Table 3 summarizes the main features that are common between the two processes. The role of gravity is critical in pool
boiling, but this effect is negligible in microchannels, where the
interface motion is mainly governed by evaporation momentum

and inertia forces. Although the forces are different, the resulting interface movement leads to similarities in the underlying
heat transfer mechanisms in the two cases. The effect on critical
heat flux is also described, and some enhancement strategies are
outlined. Avoiding microlayer dryout and avoiding or delaying
the elongated bubble formation are seen as ways to improve the
heat transfer and CHF in microchannels. Microbubbles seem to
have great promise in improving the heat transfer. They may be
generated in microchannels by using localized heating elements
heat transfer engineering

The dry patches formed during long duration
of elongated bubble flow are heated to a
high temperature before the arrival of the
liquid front, leading to the CHF condition.
New ideas need to be developed.
Microbubble generation to avoid or delay
formation of large elongated bubbles may
lead to higher heat transfer rates. Local
heating elements driven by pulsed
currents, vibrations, or dissolved gases
may be used to generate the microbubbles.

that are supplied with pulsed electric supply. Introducing vibrations using piezoelectric elements is also seen as a promising
technique to generate microbubbles. Although dissolved gases
will also lead to generation of microbubbles, their overall effect
on the interfacial heat transfer and system performance needs to
be investigated. Experimental results from Steinke and Kandlikar [36] indicate an increase in the subcooled flow boiling heat
transfer at the nucleation, but the heat transfer was reduced as
the bubbles formed a thin insulating layer. Effective removal
of bubbles is important. Further research on these topics is

warranted.

CONCLUSIONS
The similarities between the pool boiling and microchannel
flow boiling processes are discussed. The roles of transient
conduction, microconvection, and microlayer evaporation
during elongated bubble flow patterns in microchannel flow
boiling are similar to those in pool boiling. Avoiding liquid
film dryout, and delaying the formation of elongated bubble
flow pattern by introducing microbubbles are proposed as some
of the ways to enhance the heat transfer and critical heat flux
(CHF). As the flow velocity increases, the microchannel flow
vol. 31 no. 3 2010


166

S. G. KANDLIKAR

boiling is expected to resemble the flow boiling in minichannels
and conventional-sized channels (>3 mm) with the presence of
churn flow pattern. The resulting high pressure drop needs to
be considered while operating under such high flow conditions.
Shorter flow lengths and improved header arrangements are
needed to alleviate the pressure drop limitations. Microbubbles
are seen as an effective way to improve heat transfer by avoiding
or delaying the formation of elongated bubble flow pattern.

NOMENCLATURE
Bo

Bo*
Dh
FF l
G
h
hfg
q
Re
x

Boiling number = q /(Ghfg ), dimensionless
modified Boiling number, defined by Eq. (6), dimensionless
hydraulic diameter, m
fluid surface parameter, dimensionless
mass velocity, kg/m2 -s
Heat transfer coefficient, W/m2 -K
latent heat of vaporization, J/kg
heat flux, W/m2
Reynolds number, dimensionless
vapor quality, dimensionless

[8]

[9]

[10]

[11]

density, kg/m3

[12]

Subscripts
CBD
G
L
LO
NBD
TP
V

[6]

[7]

Greek Symbols
ρ

[5]

[13]

convective boiling dominant
gas
liquid
entire flow as liquid
nucleate boiling dominant
two-phase
vapor

[14]

[15]

[16]

REFERENCES
[17]
[1] Hsu, Y. Y., On the Size Range of Active Nucleation Cavities on a
Heating Surface, Journal of Heat Transfer, vol. 84, pp. 207–216,
1962.
[2] Zhang, L., Wang, E. N., Goodson, K. E., and Kenny, T. W., Phase
Change Phenomena in Silicon Microchannels, International Journal of Heat and Mass Transfer, vol. 48, pp. 1572–1582, 2005.
[3] Kandlikar, S. G., Kuan, K., Willistein, D. A., and Borrelli, J.,
Stabilization of Flow Boiling in Microchannels Using Pressure
Drop Elements and Fabricated Nucleation Sites, Journal of Heat
Transfer, vol. 128, pp. 389–396, 2006.
[4] Kandlikar, S. G., and Grande, W., Evolution of Microchannel
Flow Passages—Thermohydraulic Performance and Fabrication

heat transfer engineering

[18]

[19]

[20]

Technology, Heat Transfer Engineering, vol. 24, no. 1, pp. 3–17,

2003.
Colgan, E. G., Furman, B., Gaynes, M., Graham, W., LaBianca,
N., Magerlein, J. H., Polastre, R. J., Rothwell, M. B., Bezama,
R. J., Choudhary, R., Marston, K., Toy, H., Wakil, J., Zitz, J., and
Schmidt R., A Practical Implementation of Silicon Microchannel
Coolers for High Power Chips, IEEE Transactions on Components
and Packaging Technologies, vol. 30, pp. 218–225, 2007.
Steinke, M. E., and Kandlikar, S. G., Single-Phase Liquid Heat
Transfer in Plain and Enhanced Microchannels, Proceedings of
the ASME 4th International Conference on Nanochannels, Microchannels and Minichannels, ICNMM2006, pp. 943–951, 2006.
Nukiyama, S., Translation—The Maximum and Minimum Values of Heat Q Transmitted from Metal to Boiling Water Under
Atmospheric Pressure, International Journal of Heat and Mass
Transfer, vol. 9, pp. 1419–1433, 1966. Original article in Japanese
Society of Mechanical Engineers, vol. 37, pp. 367–374, 1934.
Das, S. K., Das, P. K., and Saha, P., Nucleate Boiling of Water
from Plain and Structured Surfaces, Experimental Thermal and
Fluid Science, vol. 31, pp. 967–977, 2007.
Steinke, M. E., and Kandlikar, S. G., An Experimental Investigation of Flow Boiling Characteristics of Water in Parallel
Microchannels, Journal of Heat Transfer, vol. 126, 518–526,
2004.
Kuo, C.-J., and Peles, Y., Local Measurement of Flow Boiling in
Structured Surface Microchannels, International Journal of Heat
and Mass Transfer, vol. 50, pp. 4513–4526, 2007.
Kenning, D. B. R., and Cooper, M. G., Saturated Flow Boiling of
Water in Vertical Tubes, International Journal of Heat and Mass
Transfer, vol. 31, pp. 445–458, 1988.
Lazarek, G. M., and Black, S. H., Evaporative Heat Transfer,
Pressure Drop and Critical Heat Flux in a Small Vertical Tube
with R-113, International Journal of Heat and Mass Transfer,
vol. 25, pp. 945–960, 1982.

Kew, K., and Cornwell, K., Correlations for Prediction of Boiling Heat Transfer in Small Diameter Channels, Applied Thermal
Engineering, vol.17, pp. 705–715, 1997.
Cooper, M. G., Saturated Nucleate Pool Boiling—A Simple Correlation, 1st UK National Heat Transactions, Conference, IChemE
Symposium Series, vol. 86, no. 2, pp. 785–793, 1984.
Kandlikar, S. G., Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids inside
Circular Tubes, Journal of Heat Transfer, vol. 113, no. 1, pp.
190–200, 1991.
Kandlikar, S. G., A General Correlation for Two-Phase Flow Boiling Heat Transfer inside Horizontal and Vertical Tubes, Journal
of Heat Transfer, vol. 112, no. 1, pp. 21–228, 1990.
Kandlikar, S. G., and Balasubramanian, P., An Extension of the
Flow Boiling Correlation to Transition, Laminar and Deep Laminar Flows in Minichannels and Microchannels, Heat Transfer
Engineering, 25, no. 3, pp. 86–93, 2004.
Gnielinski, V., New Equations for Heat and Mass Transfer in
Turbulent Pipe and Channel Flow, International Chemical Engineering, vol. 16, pp. 359–368, 1976.
Serizawa, A., Feng, Z., and Kawara, Z., Two-Phase Flow in Microchannels, Experimental Thermal and Fluid Science, vol. 26,
pp. 703–714, 2002.
Hetsroni, G., Mosyak, A., Pogrebnyak, E., and Segal, Z., Explosive Boiling in Parallel Microchannels, International Journal of
Multiphase Flow, vol. 31, pp. 371–392, 2005.

vol. 31 no. 3 2010


S. G. KANDLIKAR
[21] Kuo, C.-J., and Peles, Y., Flow Boiling Instabilities in Microchannels and Means for Mitigation by Reentrant Cavities,
Journal of Heat Transfer, vol. 130, article 072402, 10 pp.,
2008.
[22] Dong, T., Yang, Z., Bi, Q., and Zhang, Y., Freon R141b Flow
Boiling in Silicon Microchannel Heat Sinks: Experimental Investigation, Heat Mass Transfer, vol. 44, pp. 315–324, 2008.
[23] Kandlikar, S. G., Scale Effects of Flow Boiling in Microchannels:
A Fundamental Perspective, Keynote paper presented at the 7th

International Conference on Boiling Heat Transfer, Florianopolis,
Brazil, May 2–7, 2009.
[24] Kandlikar, S. G., Kuan, W. K., and Mukherjee, A., Experimental
Study of Heat Transfer in an Evaporating Meniscus on a Moving
Heated Surface, Journal of Heat Transfer, vol. 127, pp. 244–252,
2005.
[25] Mukherjee, A., and Kandlikar, S. G., Numerical Study of an Evaporating Meniscus on a Moving Heated Surface, Journal of Heat
Transfer, vol. 128, pp. 1285–1292, 2006.
[26] Jacobi, A. M., and Thome, J. R., Heat Transfer Model for Evaporation of Elongated Bubble Flows in Microchannels. Journal of
Heat Transfer, vol. 124, pp. 1131–1136, 2002.
[27] Mukherjee, A., and Kandlikar, S. G., Numerical Simulation of
Growth of a Vapor Bubble During Flow Boiling in a Microchannel, Microfluidics and Nanofluidics, vol. 1, vol. 2, pp. 137–145,
2005.
[28] Myers, J. G., Yerramilli, V. K., Hussey, S. W., Yee, G. F., and
Kim, J., Time and Space Resolved Wall Temperature and Heat
Flux Measurements during Nucleate Boiling With Constant Heat
Flux Boundary Conditions, International Journal of Heat and
Mass Transfer, vol. 48, pp. 2429–2442, 2005.
[29] Son, G., Dhir, V. K., and Ramanujapu, N., Dynamics and Heat
Transfer Associated With a Single Bubble During Nucleate Boiling on a Horizontal Surface, Journal of Heat Transfer, vol. 121,
pp. 623–629, 1999.
[30] Moghaddam, S., and Kiger, K., Physical Mechanisms of Heat
Transfer during Single Bubble Nucleate Boiling of FC-72 Under
Saturation Conditions—I. Experimental Investig(al)sT Inaanismkad7ahB1erTand

167


Heat Transfer Engineering, 31(3):168–178, 2010
Copyright C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903304343

Performance of Counterflow
Microchannel Heat Exchangers
Subjected to External Heat Transfer
BOBBY MATHEW and HISHAM HEGAB
Mechanical Engineering Program, Louisiana Tech University, Ruston, Louisiana, USA

This article analyzes the effect of external heat transfer on the thermal performance of counterflow microchannel heat
exchangers. Equations for predicting the axial temperature and the effectiveness of both fluids as well as the heat transferred
between the fluids, while operating under external heating or cooling conditions, are provided in this article. External
heating may decrease and increase the effectiveness of the hot and cold fluids, respectively. External cooling may improve
and degrade the effectiveness of the hot and cold fluids, respectively. For unbalanced flows, the thermal performance of
the microchannel heat exchanger subjected to external heat transfer depends on the fluid with the lowest heat capacity. At
a particular number of transfer units (NTU), the effectiveness of both the fluids increased with decrease in heat capacity
ratio when the hot fluid had the lowest heat capacity. When the cold fluid had the lowest heat capacity, the effectiveness of
both fluids increased with decrease in heat capacity ratio at low values of NTU but at high values of NTU the effectiveness
increased with increase in heat capacity ratio. A term called the “performance factor” has been introduced in this article to
assess the relative change in effectiveness due to external heat transfer.

INTRODUCTION
The conventional ε-NTU (number of transfer units) equations are based on several assumptions [1]. One among them
is that the thermal interaction in a two-fluid heat exchanger is
limited to that between the fluids. This is a valid assumption
as long as the heat exchangers are reasonably insulated from
their surroundings. However, thermal insulation of microchannel heat exchangers (MCHXs) by packaging them in materials
of low thermal conductivity or vacuum can significantly affect
their size and prevent them from being integrated with other
micro devices [2, 3]. The lack of proper insulation can cause the

fluids in a MCHX to thermally interact with their surroundings
due to reasons such as:
1. Low thermal resistance between the ambient and the individual fluids due to small wall thickness and high heat transfer
coefficient in the channels,
2. Proximity between the MCHX and other thermal microelectromechanical systems (MEMS) devices placed on the same
substrate,
Address correspondence to Professor Hisham Hegab, Mechanical Engineering Program, Louisiana Tech University, P.O. Box 10348, Ruston, LA 71272,
USA. E-mail:

3. High temperature differences between the ambient and the
fluids, as in microchannel heat exchanger reactors, microminiature refrigerators, and microcombustors [2–5].
The need for thermal isolation can be further understood by
analyzing the microchannel heat exchanger that was recently
developed by Hill et al. [6] for conducting chemical reactions.
Chemical reactions occurred in one set of channels while the
coolant was pumped through the other set of channels. This
microdevice interacted with the ambient, as it was not provided
with proper thermal isolation. Hill et al. [6] observed that 10 W
of heat was transferred to the coolant from the ambient while
operating this microdevice. The effect of external heat transfer
was not taken into consideration during the design stage, and due
to this the outlet temperature of the chemicals could be higher
than that estimated during designing. Based on these reasons
it is important to consider the effect of external heat transfer
while designing a MCHX. Thus, there is need for extending the
conventional ε–NTU relationship to account for the effect of external heat transfer on the effectiveness of a MCHX. Heat transfer between the fluids and the external heat source is referred
to as external heat transfer in this study. External heat transfer
considered in this article is the result of subjecting the fluids
of a MCHX to uniform axial heat flux. Therefore the external

168


B. MATHEW AND H. HEGAB

heat transfer is constant over the entire length of the MCHX. A
counterflow microchannel heat exchanger (MCHXCF ) has the
best performance, compared with other types of heat exchangers, for a specific value of NTU and thus it is often preferred over
other flow arrangements. Consequently, it is examined in this
article.
There are a few articles that have examined the effect of
external heat transfer on the thermal performance of heat exchangers. Hurd [7] developed an analytical expression for the
mean temperature difference between the fluid in the annular
section of a bayonet tube heat exchanger and the external fluid.
Bayonet tube heat exchangers are commonly used for recovering heat from the surroundings and thus their interaction with
the environment is necessary and is not considered a drawback.
Hurd [7] concluded that best recovery of heat (external heat
transfer) occurs when the external fluid and that in the annular section flow in opposite directions. Barron [8] formulated
equations for predicting the performance of counterflow heat
exchangers subjected to heat transfer between the ambient and
the individual fluids. He considered two cases; in each of these
cases only one of the fluids was externally heated/cooled by the
ambient. For both cases, he considered the effectiveness of the
hot fluid alone. He provided formulas for determining the axial
temperatures of the fluids, as well as the heat transferred between the fluids and the ambient. From both models, Barron [8]
observed that the hot fluid effectiveness degraded whenever the
ambient temperature was greater than the inlet temperature of
the fluids. Under extreme conditions of external heating, the effectiveness of the hot fluid became negative. Barron [8] defined
the heat capacity ratio with respect to the fluid that was being
heated/cooled; thus his models can only be used for balanced

flow or whenever the heated/cooled fluid has the lowest heat
capacity.
Chowdhury and Sarangi [9] developed analytical equations
for predicting the axial temperature of the fluids in a double pipe
counterflow heat exchanger subjected to external heat transfer.
In this model the fluid in the inner tube was free from external
heat transfer. They developed analytical equations for determining the axial temperature of the fluids in such a heat exchanger.
However, only the effectiveness of the fluid in the outer tube was
considered by them. They introduced the concept of effective
NTU, which is determined from the actual effectiveness. If the
external heat transfer causes degradation of effectiveness, then
the effective NTU would be lower than the NTU at which the
heat exchanger was originally designed to operate. In the absence of external heat transfer the effective NTU becomes same
as the NTU at which the heat exchanger was originally designed.
Comparison of both the NTUs can provide information about
the reduction in heat transfer surface area that occurs due to
external heating. However, the concept of effective NTU is useful only as long as the effectiveness is between zero and unity.
Chowdhury and Sarangi [9] used this model to analyze a case
where the hot and cold fluids were pumped through the inner and
outer tubes, respectively, of a heat exchanger subjected to external heating. They observed that the presence of external heating
heat transfer engineering

169

brought about considerable reduction in the effectiveness of the
hot fluid.
Ameel and Hewavithrana [10] developed a model for predicting the thermal performance of counterflow heat exchangers subjected to heat transfer with the ambient. Their model was
very similar to that of Barron [8] except that in their model
both fluids could be simultaneously subjected to external heat
transfer. They provided equations for estimating the temperature profile of the fluids and the heat exchanged between each

fluid and the ambient; however, they considered only the hot
fluid effectiveness. According to their model, the hot fluid effectiveness decreased whenever heat was transferred from the
ambient to the fluids. The degree of degradation of the hot fluid
effectiveness depended on the amount of heat transferred from
the ambient to the fluids. Moreover, they stressed the importance
of temperature cross in the design of heat exchangers subjected
to external heat transfer. They noticed that no improvement, for
a specific amount of heating and capacity ratio, in the effectiveness of the heat exchanger could be achieved by increasing
its heat transfer surface area once temperature cross had occurred. In their model the heat capacity ratio was defined with
respect to the cold fluid and thus their model is only relevant
for balanced flow or when the cold fluid has the minimum heat
capacity.
Peterson and Vanderhoff [11] computationally analyzed an
MCHXCF that experienced performance degradation due to radiation heat loss and axial heat conduction. This kind of situation
usually exists in an MCHXCF that is used in devices such as
microcombustors and microminiature refrigerators. The ends of
the wall separating the coolants were not insulated. Radiation
heat loss occurring between the ambient and the outer surface
of the MCHXCF was considered in their study. The axial heat
conduction through the entire MCHXCF structure was taken into
account in this model. Radiation heat transfer was accounted for
by using the concept of radiation heat transfer coefficient. The
authors presented their results in terms of heat loss (due to axial
heat conduction, radiation, and finite heat transfer surface area)
rather than in terms of effectiveness or fluid temperatures. Heat
loss between the MCHXCF and its surroundings was found to
decrease with initial increase in NTU, but it increased as NTU
was further raised. Peterson and Vanderhoff [11] suggested fabricating a MCHXCF using materials of low thermal conductivity
in order to reduce the effect of radiation heat loss and axial heat
conduction on its thermal performance.

Seetharamu et al. [12] applied the concept of a three-fluid
heat exchanger for predicting the thermal performance of a
two-fluid parallel-flow double-pipe heat exchanger subjected to
heating/cooling from the ambient. The fluid in the outermost
tube was considered to be ambient, and thus only the fluid in
the intermediate tube interacted with the ambient. Seetharamu
et al. [12] numerically analyzed this particular three-fluid heat
exchanger, and their predictions, for fluid temperature and effectiveness, were found to be in good agreement with the solutions
provided by earlier researchers [8]. They observed that when the
inlet temperatures of both fluids are above that of the ambient,
vol. 31 no. 3 2010


170

B. MATHEW AND H. HEGAB

reduction in the thermal resistance between the ambient and the
fluid in the intermediate tube improved and degraded the hot
and cold effectiveness, respectively.
Nellis and Pfotenhauer [13] theoretically analyzed a counterflow heat exchanger in which the fluids were subjected to
uniform external heat flux. They considered the effectiveness of
both fluids in their model. Using the model they observed that
the external heating of either of the fluids by the application
of uniform heat flux always decreased the hot fluid effectiveness [13]. Nellis and Pfotenhauer [13] defined the external heat
transfer with respect to the product of thermal conductance and
inlet temperature difference (UA(Thi− Tci )). This term does not
have any physical significance with respect to the design and
operation of heat exchangers. Moreover, Nellis and Pfotenhauer
[13] defined heat capacity ratio with respect to the hot fluid.

Thus, the NTU defined in their work will become the same as
the conventional NTU only for balanced flow or when the hot
fluid has the lowest heat capacity.
Mathew and Hegab [14] recently conducted experimental
studies on the thermal performance of a MCHXCF subjected to
uniform external heat flux. To the knowledge of the authors this
is the only article that has reported experimental studies on the
performance degradation of MCHXCF due to external uniform
heat flux heating. In their experiments both fluids were equally
heated by the external heat source. The NTU was varied between
0.35 and 1.4 and the corresponding hot fluid effectiveness, with
no external heating, ranged from 0.26 to 0.58 [14]. The hot fluid
effectiveness corresponding to 15% and 25% external heating
range from 0.23 to 0.5 and 0.21 to 0.44, respectively [14].
These experimental data show a drastic reduction in the hot
fluid effectiveness and provide further proof for the need for
extending the conventional ε-NTU relationship. An in depth
analysis of the many articstudies that have dealt with similar
topics can be found elsewhere [15].
A thermal model of an MCHXCF whose fluids are subjected
to external heat transfer has been developed in this article. In
this model the cause of external heat transfer is the uniform heat
flux that is applied to the fluids of the MCHXCF . The concept put
forward in this study is simpler than the ones already existing, as
the input parameters such as NTU and heat capacity ratio have
been defined in a conventional way. The external heat transfer
has been defined with respect to the maximum heat transfer
(qmax ) thermodynamically possible in a heat exchanger without
external heat transfer, i.e., qmax = Cmin (Th,i – Tc,i ), and thus
its impact on the thermal performance of an MCHXCF can be

easily understood. The theory provided in this study can be used
irrespective of the fluid that is being externally heated/cooled.
The same cannot be said about the analytical equations provided
by Barron [8], Ameel and Hewavithrana [10], and Nellis and
Pfotenhauer [13], as has been explained previously. In addition,
formulas for determining the axial temperature of the fluids have
also been provided in this article. Toward the end of this article
the authors have introduced a term, the “performance factor,”
to analyze the relative change in effectiveness of an MCHXCF
when subjected to external heat transfer.
heat transfer engineering

THEORETICAL MODEL
The following assumptions were made to develop a model
for the MCHXCF subjected to external heat transfer:
1. MCHXCF operates under steady-state conditions.
2. The temperatures of the fluids vary only in the axial direction.
3. No-slip boundary condition is assumed in the microchannels
(Kn < 0.001).
4. There is no phase change in either of the fluid streams.
5. Effects of longitudinal heat conduction in the fluid, viscous
dissipation and flow maldistribution are neglected.
6. Axial heat conduction through the wall separating the fluids
is neglected.
7. The thermophysical properties of the fluids are assumed to
be constant over the length of the MCHXCF .
8. The ends of the wall separating the fluids are considered to
be insulated.
Figure 1 represents a differential element of the MCHXCF
considered in this study. Equations (1) and (2) can be obtained

by applying the first law of thermodynamics to the individual
fluids.
dTh
dx + dqex
dx

(1)

dTc
dx + dqex + qc dA = Cc Tc
dx

(2)

Ch Th + qh dA = Ch Th +
Cc T c +

These equations are rearranged and dqex has been replaced
by UP(Th – Tc ) dx to obtain the governing equations:
Ch

dTh
+ U P (Th − Tc ) − qh P = 0
dx

(3)

Cc

dTc

+ U P (Th − Tc ) + qc P = 0
dx

(4)

Equations (3) and (4) in nondimensional form are:
dθh
+ CRh N T U (θh − θc ) − Qh CRh = 0
dZ

(5)

dθc
+ CRc N T U (θh − θc ) + Qc CRc = 0
dZ

(6)

Figure 1 Schematic representation of a differential element of the MCHXCF
studied in this article.

vol. 31 no. 3 2010


B. MATHEW AND H. HEGAB

The boundary conditions, in nondimensional form, are:
θh |Z=0 = 1

(7)

θc |Z=1 = 0

(8)

Under the balanced flow condition, CRh and CRc would be
the same and numerically equal to unity. While operating the
MCHXCF under unbalanced flow condition, CRh and CRc depend on the heat capacities of the fluids. If the cold fluid has the
lowest heat capacity then CRc would be unity and CRh would
represent the conventional heat capacity ratio (CR ). Likewise,
when the hot fluid has the lowest heat capacity then CRh and
CRc would be numerically equal to unity and the conventional
heat capacity ratio, respectively. Q is the nondimensional external heat transfer. It can be noticed from Eqs. (5) and (6) that the
external heat transfer (q A) and the heat exchanged between the
fluids (qex ) are defined with respect to qmax . Both Qh and Qc are
positive if the fluids are externally heated and negative if they
are being externally cooled.
The mathematical procedure for solving these equations has
been provided by Wylie [16]. For balanced flow condition, Eqs.
(9) and (10) represent the hot and cold fluid temperature, respectively.
θh = a1 + a2 Z −
θc = a1 +
−

N T U · (Qh + Qc ) 2
Z
2

a2 − Qh
NT U

(9)

+ (a2 − Qh − Qc )Z

N T U · (Qh + Qc ) 2
Z
2

(10)

where
a1 = 1,
a2 =

−NT U + Qh + N T U · (Qh + Qc )(1 + 0.5N T U )
1 + NT U

Equations (11) and (12) represent the nondimensional axial
temperature of the fluids for unbalanced flow conditions.
θh =

CRc e−k1 − CRh e−k1 Z
CRc e−k1 − CRh
+ (e−k1 Z − 1)
−

k2
k1

k2
C NT U
k1 Rh

171

+

CRc −k1 Z
e
−1
CRh

−

k2
k1

+

N T U (CRc

e−k1

N T U (CRc

k2
k1

e−k1

+ CRh Qh
− CRh )

+ CRh Qh

(12)

CRh N T U

k1 = N T U · (CRh − CRc ),

k2 = CRh CRc N T U · (Qh + Qc )

The effectiveness of an MCHXCF , for all values of nondimensional external heat transfer parameters, is defined in the
same way as conventional effectiveness, i.e., as the ratio of actual heat transfer (qact ) to the maximum heat transfer possible
between the fluids (qmax ) [1]. In order to determine the actual
and maximum heat transfer, the heat capacities as well as the
inlet and outlet temperatures of the fluids are required. The
nondimensional outlet temperatures can be determined from
Eqs. (9)–(12). The equations for determining the effectiveness
of the fluids for both balanced and unbalanced flow conditions
are shown in Table 1.
Equations (9)–(12) cannot be directly used when NTU is zero
or infinite. In order to obtain the effectiveness when NTU is zero
or infinite, L’Hospital’s rule is applied to these equations. The
effectiveness of the fluids thus obtained is provided in Table 2. A
MCHXCF may never be operated at the extreme values of NTU;
however, these equations have been presented here to provide a
complete discussion of their thermal performance.

Equations (9)–(12) can be used for predicting the temperature
as well as effectiveness of the fluids even when there is no
external heating/cooling. The temperature or effectiveness of
the fluids for such a situation can be obtained by substituting
zero for Qh and Qc in the appropriate equations provided earlier.
Equations (13) and (14) represent the temperature of the hot and
cold fluid under balanced flow condition. The temperature of the
hot and cold fluid for unbalanced flow conditions is presented
in Eqs. (16) and (17), respectively. The effectiveness of the hot
and cold fluid is the same when external heat transfer to either
of the fluids is absent. Equation (15) represents the effectiveness
for balanced flow, while Eq. (18) represents that for unbalanced
flow. Equation (15) can be obtained from either Eq. (13) or (14).
Similarly, Eq. (18) can be derived from either Eq. (16) or (17).

+ CRh Qh
− CRh )

+

where

θh =
k2
k1

k2
k1

Z−

k2
C NT U
k1 Rh

1 + (1 − Z)N T U
1 + NT U

(13)

Table 1 Hot and cold fluid effectiveness for balanced and unbalanced flows
Effectiveness

Z

(11)

Fluids
Hot fluid

CRc e−k1 − CRc e−k1 Z
θc =
CRc e−k1 − CRh

Cold fluid

heat transfer engineering

Cmin = CRc

Cmin = Ch

εh = (1 − θh |Z=1 )

εh = (1 − θh |Z=1 )

εc = θc |Z=0

vol. 31 no. 3 2010

εc =

θ c |Z=0
CRc

Cmin = Cc
εh =

(1− θ h |Z=1 )
CRh

εc = θc |Z=0


172

B. MATHEW AND H. HEGAB
Table 2 ε-NTU relationships when NTU = 0 and NTU = ∞
Effectiveness
Balanced flow

NTU

Hot fluid

Unbalanced flow
Cold fluid

Hot fluid

Cold fluid

0

−Qh

+Qc

–Qh

+Qc

∞

1 – 0.5(Qh + Qc )

1 + 0.5(Qh + Qc )

1 (Cmin = Ch )

1 + Qh + Qc (Cmin = Ch )

1 − Qh – Qc (Cmin = Cc )

1 (Cmin = Cc )

θc =

(1 − Z)N T U
1 + NT U

(14)

NT U
1 + NT U

(15)

θh =

CRc e−k1 − CRh e−k1 Z
CRc e−k1 − CRh

(16)

θh =

CRc e−k1 − CRc e−k1 Z
CRc e−k1 − CRh

(17)

1 − e−k1
CRh − CRc e−k1

(18)

εh = εc =

εh = εc =

Shah and Sekulic [1] had developed equations for determining the axial temperature of the fluids in a counterflow heat
exchanger free of external heating/cooling; however, the flow
paths of the fluids in their heat exchanger were in a direction
opposite to that assumed in this work. Therefore, for the purpose
of comparison the nondimensional axial distance parameter in
the preceding equations was replaced by 1 – Z. Equations (13),
(14), (16), and (17) were then validated to be exactly same as
those formulated by Shah and Sekulic [1].
The heat exchanged between the fluids can be determined
from the equations that define their axial temperature. The general form for calculating the nondimensional heat transferred
between the fluids is given in Eq. (19).
1

Q∗ex

= NT U

(θh − θc )dZ

(19)

0

The heat transferred between the fluids has been nondimensionalized with respect to the maximum heat transfer (qmax )
possible in a MCHXCF without external heat transfer. Equations
(20) and (21) represent the heat transferred, in dimensionless
form, between the fluids for both balanced and unbalanced flow
conditions, respectively.
Q∗ex =

a1
− a2 + Qh
2

(20)

heat transfer engineering

Q∗ex =

1
CRh − CRc
(CRh N T U (1 − k2 )
k1 CRh k1 (CRc e−k1 − CRh )
− k2 − k1 CRh Qh ) + k2 + k1 CRh Qh

(21)

Viscous dissipation is another phenomenon that is almost
always neglected while designing heat exchangers [1]. Viscous

dissipation can be significant at high pumping power. In MCHXs
due to the small hydraulic diameter of the channels employed
the pumping power can be significantly greater than for their
conventional counterparts for the same flow rate. The effect
of viscous dissipation is to raise the temperature of the fluids
in the microchannels in a manner similar to that of subjecting
the fluids to a uniform external heat flux [17–19]. Thus the
solution developed here can be used for determining the thermal
performance of a MCHXCF with viscous dissipation as well,
even though it was not explicitly included in the governing
equations [19].

RESULTS AND DISCUSSION
For this section the thermal performance of a MCHXCF subjected to external heat transfer has been analyzed using the
equations already derived. These equations can be used for determining the hot and cold fluid effectiveness irrespective of the
amount of external heat transfer. For illustrative purposes, in
this article the case when both fluids are subjected to the equal
amounts of external heat transfer (Qh = Qc = 0, 0.25, 0.5, 0.75,
1) is analyzed. Toward the end of this article the authors have
introduced a term called the “performance factor” for assessing
the relative change in effectiveness of the fluids of a MCHXCF
due to external heat transfer.
The variation of hot and cold fluid effectiveness with respect
to NTU for balanced flow has been shown in Figure 2. The solid
line in this figure represents the effectiveness of both fluids when
external heating is absent. As seen from Figure 2, the effectiveness of the hot fluid decreased, for a particular value of NTU,
with increase in external heating. This is an expected trend;
with increase in the amount of external heating there occurs an
increase in the outlet temperature of the hot fluid and thus an observed reduction in its effectiveness. Further, the effectiveness
vol. 31 no. 3 2010

B. MATHEW AND H. HEGAB

173

Figure 2 ε–NTU relationship of a balanced flow MCHXCF (CRh = CRc ).

Figure 3 ε-NTU relationship of an unbalanced flow MCHXCF (Cmin = Ch ,
CRh = 1.0, CRc = 0.5).

of the hot fluid increased with increasing NTU for all levels of
external heating. This may be explained by the fact that raising
the NTU by decreasing the flow rate, for a specific Qh and Qc ,
is accompanied by a decrease in the heat flux that is applied
to fluids if the geometry of the MCHXCF is kept constant. This
can be understood from the equation defining Q. Thus there is
a decrease in the external heat transfer (qext ) to the individual
fluids. Moreover, in this scenario, there is an increase in the heat
transfer surface area per unit volume of the fluid. These two
effects combined lead to an improvement in the effectiveness of
the hot fluid as seen in Figure 2. The NTU can also be raised
by increasing the heat transfer surface area of the MCHXCF .
Even for this scenario the external heat flux to the individual
fluids decreases due to changes in the geometry of the channels.
However, the external heat transfer (qext ) remained the same as
Q was kept constant. Thus the hot fluid effectiveness improved
with increase in heat transfer surface as shown in Figure 2. It
can be noticed from Figure 2 that when NTU is zero, the hot
fluid effectiveness is numerically equal to the negative of the

nondimensional external heat transfer parameter (Qh ). This is
due to the fact that when NTU is zero there is no heat transfer
between the fluids, and consequently the heat supplied to the
hot fluid from the external source raises its outlet temperature.
At infinite NTU, it can be noticed from Table 2 that the effectiveness reaches a finite value. The effectiveness of the fluids in
an MCHXCF operating at infinite NTU and free from external
heating is unity. Whenever the fluids in an MCHXCF that is operating at very large NTU are externally heated, the externally
supplied heat is equally shared between the two fluids since
the thermal resistance between the hot and cold fluid is very
low. Therefore, the application of external heat degrades the hot
fluid effectiveness from unity to 1 – 0.5(Qh + Qc ). Moreover,
the addition of external heat to the hot fluid will cause the outlet
temperature to be higher than the inlet temperature of the cold
fluid and vice versa.
In Figure 2 the effectiveness of the cold fluid increased, for
a specific value of NTU, with increase in the external heating.
External heating raised the outlet temperature of the cold fluid
and thereby its effectiveness. Raising the NTU raised the effectiveness for a particular value of Qh and Qc . When NTU is
zero, the effectiveness of the cold fluid is numerically equal to

the nondimensional external heat added to it. If the MCHXCF is
operated at infinite NTU, then the effectiveness increases to a
value presented in Table 2. The reasons for these behaviors are
similar to those already explained for the hot fluid.
With regard to unbalanced flow an MCHXCF can be operated
in two ways, with either of the fluids as the one with the lowest
heat capacity. Under conditions of zero external heating the
thermal performance of an MCHXCF depends only on the heat
capacity ratio (Cmin /Cmax ) and NTU; it does not depend on the
fluid that has the minimum heat capacity [1]. However, when the

MCHXCF is subjected to external heat transfer this is not the case
as investigated in this section. Thus two cases are investigated:
(1) Ch = Cmin and Cc = Cmax , and (2) Cc = Cmin and Ch =
Cmax . However, the heat capacity ratio (Cr ) for both these cases
is kept at 0.5 for the results presented in this article.
The effectiveness of a MCHXCF for the first case, i.e., when
the hot fluid has the lowest heat capacity (CRh = 1.0 and CRc =
0.5), is shown in Figure 3. When the MCHXCF is free of external heating, the effectiveness of the fluids is shown by the solid
line in this figure. As seen from this figure, the effectiveness
of the fluids increased with NTU for a specific amount of external heating. When this MCHXCF was operated at zero NTU,
the effectiveness of the hot fluid was numerically equal to the
negative of the nondimensional external heat transfer parameter
(Qh ). Thus, the effectiveness of the fluids in a balanced flow and
the effectiveness in an unbalanced flow heat exchanger were

heat transfer engineering

Figure 4 Temperature profile of an unbalanced flow MCHXCF (Cmin = Ch ,
CRh = 1, CRc = 0.5, Qh = Qc = 0.5).

vol. 31 no. 3 2010


174

B. MATHEW AND H. HEGAB

Figure 5 ε-NTU relationship for an unbalanced flow MCHXCF (Cmin = Cc ,
CRh = 0.5, CRc = 1.0).

Figure 6 Temperature profile of an unbalanced flow MCHXCF (Cmin = Cc ,
CRh = 0.5, CRc = 1.0, Qh = Qc = 0.5).

the same when the MCHXCF was operated at zero NTU. The
reasons for these observed trends are similar to those already
mentioned for the balanced flow condition. When the MCHXCF
is operated at infinite NTU the effectiveness of the fluids reaches
a constant value, as mentioned in Table 2. Figure 4 is a graphical
representation of the temperature profile of the fluids for various
values of NTU with the nondimensional external heat transfer
parameters maintained at 0.5. From this figure it can be seen
that the hot fluid temperature is always greater than the cold
fluid temperature for all values of NTU. Thus, the heat transfer
is always from the hot to the cold fluid, irrespective of the NTU.
Therefore, when the MCHXCF is operated at infinite NTU the
hot fluid transfers heat, i.e., qmax + qext,h , to the cold fluid until
its outlet temperature becomes equal to the inlet temperature of
the cold fluid. Thus, the hot fluid effectiveness becomes equal
to unity. On the other hand, the cold fluid acts as the sink and
receives all the heat, qmax + qext,h + qext,c . Therefore, the cold
fluid effectiveness becomes equal to 1 + Qh + Qc . The temperature profile of the fluids is the same when NTU is infinite, as
seen in the figure.
In the second mode of operation associated with unbalanced
flow, the cold fluid has the lowest heat capacity among the fluids
in the MCHXCF (CRh = 0.5 and CRc = 1.0). Figure 5 represents
the ε-NTU relationship of the MCHXCF with respect to the hot
and cold fluid. The effectiveness of the fluids in a MCHXCF
without external heating is represented by the solid line of
Figure 5. Careful examination of Figures 3 and 5 show that
the ε-NTU relationship is dependent on the fluid that has the

lowest heat capacity. The only similarity in the ε-NTU relationship of the fluids occurs when NTU is zero. As NTU was raised
there was an initial increase in the hot and cold fluid effectiveness. However, it soon peaked and then started to decline with
further increase in NTU. This behavior can be understood from
the temperature profile of the fluids. Figure 6 represents the temperature profile of the fluids in the MCHXCF for NTU values of
1, 5, 10, 20, and ∞. The external heat transfer parameters of the
MCHXCF shown in Figure 6 are 0.5.
When NTU is 1, the hot fluid temperature is always higher
than that of the cold fluid. Thus, the hot fluid heats the cold fluid
throughout the length of the MCHXCF . Due to this the curve

representing the effectiveness has a positive slope at this NTU.
When the NTU is raised to 5, temperature cross is observed.
Between the inlet of the cold fluid and the location of temperature cross the hot fluid heats the cold fluid and beyond that point
the cold fluid heats the hot fluid. However, it can be seen from
Figure 5 that the effectiveness of the fluids has a positive slope at
this particular NTU. This means that the net heat transfer from
the hot fluid stream remained positive. On further increasing the
NTU, the point of temperature cross moved closer to the cold
fluid inlet. This behavior can be confirmed by comparing the
temperature profile of the fluids at NTU of 5, 10, and 20. From
the point of temperature cross to the inlet of the hot fluid, the
hot fluid gets heated by the cold fluid for certain values of NTU
that are shown in Figure 5. Between the location of temperature
cross and the inlet of the hot fluid the cold fluid takes the role
of the hot fluid and vice versa. As NTU is raised, the hot fluid
gets heated over a greater length of the MCHXCF . Thus, with
increase in NTU there was reduction in the net heat transferred
from the hot fluid, and this explains the negative slope in the
effectiveness of the fluids as observed in Figure 5. The change
from positive slope to negative slope just described can also be

observed in the curves representing the effectiveness when Qh =
Qc = 0.5, 0.75, and 1. The peak in the effectiveness occurred
when the net heat transfer from the hot fluid was the maximum.
The NTU at which the maximum effectiveness occurs is called
the critical NTU (NTUcritical ).
When NTU is raised to extremely high values, the effectiveness of the fluids reaches a constant value. At infinite NTU
temperature cross occurs in the immediate vicinity of the inlet
of the cold fluid due to the effect of temperature cross. Thus, the
hot fluid is heated by the cold fluid throughout the entire length
of the MCHXCF , except in a very small section near the cold
fluid inlet where the cold fluid is heated by the hot fluid. The exit
temperature of the hot fluid at infinite NTU has been marked in
Figure 6. Irrespective of the amount of heat transferred from the
hot to the cold fluid near the inlet of the cold fluid, all the heat
from the cold fluid is transferred to the hot fluid by the time the
cold fluid reaches its exit as the MCHXCF is operating at infinite
NTU. Thus, the effectiveness of the cold fluid becomes unity
and that of hot fluid become 1 – Qh – Qc .

heat transfer engineering

vol. 31 no. 3 2010


B. MATHEW AND H. HEGAB

Figure 7 Effect of heat capacity ratio on the effectiveness of fluids (Cmin =
Ch , CRh = 1, Qh = Qc = 0.25).

Figures 7 and 8 represent the effect of heat capacity ratio on

the effectiveness of the fluids in an MCHXCF in which the hot
fluid has the lowest heat capacity. In this figure CRc represents
Cr . In Figure 8 Cr is same as CRh as this figure contains the
ε-NTU relationship of a MCHXCF , in which the cold fluid has
the lowest heat capacity. For these figures both the nondimensional external heat transfer parameters (Qh and Qc ) were kept
at 0.25. In Figure 7 with decrease in heat capacity ratio the
effectiveness of both fluids increased. This trend is similar to
that observed in an MCHXCF free of external heating. In this
figure, the hot fluid effectiveness converges to a common value
when NTU is zero. This is because, as explained earlier, when
NTU is zero the hot fluid effectiveness is equal to –Qh which is
–0.25 for the cases presented in Figure 7. On the other hand, for
unbalanced flows, when NTU is infinite the effectiveness of the
hot fluid, irrespective of CRc , becomes equal to unity. Under the
balanced flow condition, the effectiveness of the hot fluid will
be 0.75 at infinite NTU. With regard to cold fluid effectiveness,
while operating under unbalanced flow conditions the curves in
Figure 7 converge to 0.25 and 1.5 when NTU is zero and
infinite, respectively. For balanced flow the cold fluid effectiveness is 1.25 when NTU = ∞. The reasons for these have
been already mentioned. The values of effectiveness, i.e., when
NTU = 0 or ∞, just mentioned were determined from Table 2.
Figure 8 represents the effectiveness of the fluids for several
values of CRh . For a particular CRh the effectiveness of the hot

Figure 8 Effect of heat capacity ratio on the effectiveness of fluids (Cmin =
Cc , CRc = 1, Qh = Qc = 0.25).

heat transfer engineering

175

and cold fluid initially increased, reached a maximum, and then
decreased with increase in NTU. This behavior is in contrast to
that observed in a MCHXCF without external heating/cooling.
The reversal in effectiveness with increase in NTU is due to the
temperature cross that occurs when the cold fluid has the lowest
heat capacity in an externally heated MCHXCF . As in Figure 7,
the effectiveness of the hot and cold fluid when NTU is zero
was –0.25 and 0.25, respectively. From Figure 8 it can also be
noticed that at high values of NTU the effectiveness of the hot
and cold fluid in an unbalanced flow MCHXCF converges toward 1 and 0.5, respectively. These values of effectiveness were
also calculated from Table 2.
Equations (9)–(12) can also be used when the fluids are externally cooled or when one of the fluids is heated and the other
is cooled using an external source. As mentioned earlier, Qh and
Qc are positive when the fluids are externally heated, and they
are negative when the fluids are externally cooled.
A performance factor is used to define the relative change
in the effectiveness of a MCHXCF subjected to external heating/cooling with respect to its effectiveness under zero external
heat transfer. Mathematically it can be represented by Eqs. (22)
and (23). Equations (22) and (23) represent the performance factors of the hot and cold fluids, respectively. Positive and negative
values of performance factor represent relative improvement and
degradation, respectively.
ρh =

εheating/cooling − εno−heating/cooling
εno−heating/cooling

(22)

ρc =

εheating/cooling − εno−heating/cooling
εno−heating/cooling

(23)

This same formulation has been called the longitudinal heat
conduction parameter and the degradation factor by Chiou [20]
and by Gupta and Atrey [21], respectively. They used these
terms mainly for defining the degradation in effectiveness of the
hot fluid alone. However, when heat exchangers are subjected to
external heat transfer the effectiveness of the fluids need not always deteriorate. For example, when both fluids of an MCHXCF
are subjected to external heating, the effectiveness with respect
to the cold fluid improves while that with respect to the hot fluid
degrades. An opposite situation would occur when the fluids are
cooled by an external source. A performance factor can be used
to define both the relative improvement and the degradation in
the effectiveness of an MCHXCF . Moreover, the performance
factor is presented here as a function of the NTU and nondimensional external heat transfer parameters. The authors of this
article feel that the main purpose of the performance factor
would be to simultaneously analyze several operating parameters. One purpose of such an analysis would be to determine
the best range of NTU values of an MCHXCF that may experience significant external heating and/or cooling. Conversely,
performance factors could also be used for determining the levels of external heating/cooling that may significantly impact the
performance of a given MCHXCF (known NTU).
vol. 31 no. 3 2010


176

B. MATHEW AND H. HEGAB

Figure 9 Performance factor of a balanced flow MCHXCF (CRc = CRh ,
Qh = Qc ): (a) hot fluid, (b) cold fluid.

The performance factor of the hot and cold fluids, for balanced flow, is shown in Figure 9a and b, respectively. For the
particular case analyzed here (external heating) the performance
factor of the hot fluid has negative values which represent deterioration in its effectiveness. It can be seen from Figure 9a that
at low values of NTU the degradation in thermal performance is
more than that at moderate and high values of NTU. This is attributed to the fact that the thermal performance of an MCHXCF
is low at small values of NTU because the heat transferred between the fluids is not significant. Therefore, the addition of heat
from an external source brings about greater deterioration in its
thermal performance and thus the observed trend in the performance factor. It should be noted that even as the NTU is raised
the deterioration in hot fluid performance still exists. However,
it is not as severe as it was at low values of NTU. This is the
result of the improvement in the heat transfer between the fluids
at high values of NTU, as explained previously. From Figure
9a it can be noticed that the lines separating the regions are not
smooth, in contrast to the curves in previous figures. This trend
is due to the fact the points (performance factors) lying on each
heat transfer engineering

Figure 10 Performance factor of an unbalanced flow MCHXCF (Cmin = Ch ,
CRh = 1, CRc = 0.5, Qh = Qc ): (a) hot fluid, (b) cold fluid.

of these lines do not belong to a specific Qh and Qc . This can
be better understood by analyzing the points laying on the line
separating the 0 to –1 and –1 to –2 regions. The performance
factor corresponding to NTU of 1 on this line is when Qh and
Qc are equal to ≈0.5. Similarly, at NTU of 3 the maximum
performance factor occurs when Qh and Qc are equal to ≈0.75.

Similar conclusions can be drawn about all the points lying on
this line and thus the lack of smoothness of the line.
For the cold fluid as shown in Figure 9b, the performance
factor is positive, which represents improvement in its effectiveness. At a specific value of NTU, with increase in external heating the outlet temperature of the cold fluid increases
further, resulting in an increase in the performance factor of the
cold fluid. Further, the performance factor had high values at
low NTU. The lines separating the regions of this figure are also
not smooth. The reasons for these trends are similar to those
already explained for the hot fluid performance factor.
When the hot fluid has the lowest heat capacity then the
performance factor of the fluids are represented in Figure 10a
vol. 31 no. 3 2010


B. MATHEW AND H. HEGAB

177

that of the cold fluid improved. This trend in the effectiveness
reversed when the fluids were cooled by the external heat source.
For an unbalanced flow MCHXCF the effectiveness of the fluids
depended on the fluid with the lowest heat capacity. Whenever
the hot fluid has the lowest heat capacity in an MCHXCF , the
effectiveness of the fluids always increases with increase in
NTU, irrespective of the degree of external heating. On the
other hand, the effectiveness of the fluids initially increases and
then decreases with increase in NTU if the cold fluid has the
lowest heat capacity in a MCHXCF . An additional advantage of
this theory would be in the design of na MCHXCF with viscous
heating. This is because the effect of viscous heating is similar

to that of subjecting the fluids to uniform heat flux. A parameter
called a performance factor was introduced to aid designers in
analyzing the effect of external heat transfer on the thermal
performance of MCHXs. Designers may graphically analyze
several operating parameters simultaneously using the concept
of the performance factor.

NOMENCLATURE
A
a1 , a2
C
CR
Cr

Figure 11 Performance factor of an unbalanced flow MCHXCF (Cmin = Cc ,
CRh = 0.5, CRc = 1, Qh = Qc ): (a) hot fluid, (b) cold fluid.

and b. Similar to the case of balanced flow, the performance
factor of the hot fluid was the lowest at low values of NTU.
Even for the cold fluid, the performance factor was the highest
at low values of NTU. The reasons for these trends are the same
as those already mentioned for balanced flow.
The performance factor of an MCHXCF under unbalanced
flow with the cold fluid having the minimum heat capacity conditions is shown in Figure 11a and b. The behavior of the fluids
is similar to that observed in earlier cases.

k1 , k2
Kn
L
MCHX

NTU
P
q
q
Q
Q∗
T
U
x
Z

heat transfer surface area (m2 )
constants used in Eqs. (9), (10), and (20)
heat capacity of individual fluid (W/K)
ratio of minimum heat capacity to the heat capacity
of the individual fluid, CR = CCmin
ratio of minimum heat capacity to the maximum heat
min
capacity, Cr = CCmax
terms used in Eqs. (11), (12), and (21)
Knudsen number
length of the MCHXCF (m)
microchannel heat exchanger
A
number of transfer units, N T U = CUmin
perimeter (m)
heat transfer (W)
heat flux (W/m2 )
nondimensional external heat transfer parameter,
Q = qqmaxA = Cmin (Tqh,iA−Tc,i )

nondimensional heat transfer, Q∗ = q/qmax =
q/Cmin (Th,i − Tc,i )
temperature of the fluid (◦ C)
overall heat transfer coefficient (W/K-m2 )
axial distance (m)
nondimensional axial distance, Z = Lx

CONCLUSIONS
From the analysis provided it is clear that the performance of
an externally heated/cooled MCHXCF may significantly deviate
from what is predicted by the conventional ε-NTU equations
that do not consider this effect. When subjected to external
heating the effectiveness of the hot flui always degraded while
heat transfer engineering

Greek Symbols
ε
ρ
θ

act
act
= Cmin (Tqh,i
effectiveness, Q = qqmax
−Tc,i )
performance factor
nondimensional fluid temperature, θ =

vol. 31 no. 3 2010

T −Tc,i
Th.i −Tc,i


178

B. MATHEW AND H. HEGAB

Subscripts
act
c
CF
ex
ext
h
i
min
max
o

actual
cold fluid
counterflow
exchanged between the fluids
external
hot fluid
inlet
minimum
maximum
outlet

[13]

[14]

[15]

[16]
[17]

REFERENCES
[1] Shah, R. K., and Sekulic, D. P., Fundamentals of Heat Exchanger
Design, John Wiley and Sons, Hoboken, NJ, pp. 100–101, 2003.
[2] Paugh, R. L., New Class of Microminiature Joule–Thompson
Refrigerator and Vacuum Package, Cryogenics, vol. 30, no. 12,
pp. 1079–1088, 1990.
[3] Peterson, R. B., High Temperature Microscale Reactor Analysis
Using a Counterflow Heat Exchanger Model, Microscale Thermophysical Engineering, vol. 3, no. 1, pp. 17–30, 1999.
[4] Little, W. A., Microminiature Refirgerator, Review of Scientific
Instruments, vol. 55, no. 5, pp. 661–680, 1984.
[5] Ronney, P. D., Analysis of Non Adiabatic Heat Recirculating
Combustors, Combustion and Flame, vol. 135, no. 4, pp. 421–
439, 2003.
[6] Hill, T. F., Velasquez-Garcia, L. F., Wilhite, B. A., Rawlins, W. T.,
Lee, S., Davis, S. J., Jensen, K. F., Epstein, A. H., and Livermore,
C., A MEMS Singlet Oxygen Generator—Part II: Experimental Exploration of the Performance Space, Journal of Microelectromechanical Systems, vol. 16, no. 6, pp. 1492–1505, 2007.
[7] Hurd, B. L., Mean Temperature Difference in the Field or Bayonet
Tube, Industrial and Engineering Chemistry, vol. 38, no. 12, pp.
1266–1271, 1946.
[8] Barron, R., Effect of Heat Transfer from Ambient on Cryogenic

Heat Exchanger Performance, Advances in Cryogenics, vol. 38,
pp. 265–272, 1984.
[9] Chowdhury, K., and Sarangi, S., Performance of Cryogenic Heat
Exchangers with Heat Leak From the Surroundings, Advances in
Cryogenics, vol. 38, pp. 273–280, 1984.
[10] Ameel, T. A., and Hewavithrana, L., Countercurrent Heat Exchangers with Both Fluids Subjected to External Heating, Heat
Transfer Engineering, vol. 20, no. 3, pp. 37–44, 1999.
[11] Peterson, R. B., and Vanderhoff, J. A., Analysis of a BayonetType Heat Counterflow Heat Exchanger With Axial Conduction
and Radiative Heat Loss, Numerical Heat Transfer, Part A, vol.
40, no. 3, pp. 203–219, 2001.
[12] Seetharamu, K. N., Quadir, G. A., Zainal, Z. A., and Krishnan,
G. M., FEM Analysis of Multifluid Heat Exchangers, Interna-

heat transfer engineering

[18]

[19]

[20]

[21]

tional Journal of Numerical Methods for Heat & Fluid Flow, vol.
14, no. 2, pp. 242–255, 2004.
Nellis, G. F., and Pfotenhauer, J. M., Effectiveness–NTU Relationship for a Counterflow Heat Exchanger Subjected to an External Heat Transfer, Journal of Heat Transfer, vol. 127, no. 9, pp.
1071–1073, 2005.
Mathew, B., and Hegab, H., External Heating Effects on the
Effectiveness–NTU Relationship of a Counterflow Microchannel Heat Exchanger, Proc. 2006 ASME International Mechanical
Engineering Congress and Exposition, Chicago, 2006.

Mathew, B., Performance Evaluation of Microchannel Heat Exchanger Subjected to External Heat Flux, M.S. Thesis, Louisiana
Tech University, Ruston, LA, pp. 39–97, 2007.
Wiley, C. R., Advanced Engineering Mathematics, McGraw-Hill,
New York, pp. 66–71, 1965.
Sahin, A. Z., Thermodynamic Design Optimization of a Heat Recuperator, International Communication in Heat and Mass Transfer, vol. 24, no. 7, pp. 1029–1038, 1997.
Murakami, Y., and Mikic, B. B., Parametric Investigation of Viscous Dissipation Effects on Optimized Air Cooling Microchanneled Heat Sinks, Heat Transfer Engineering, vol. 24, no.1, pp.
53–62, 2003.
Mathew, B., and Hegab, H., Effectiveness of Parallel Flow Microchannel Heat Exchangers with External Heat Transfer and Internal Heat Generation, Proc. ASME 2008 Summer Heat Transfer
Conference, Jacksonville, FL, 2008.
Chiou, J. P., The Effect of Longitudinal Heat Conduction on Crossflow Heat Exchanger, Journal of Heat Transfer, vol. 100, no. 2,
pp. 346–351, 1978.
Gupta, P., and Atrey, M. D., Performance Evaluation of Counter
Flow Heat Exchangers Considering the Effect of Heat In Leak
and Longitudinal Conduction for Low-Temperature Applications,
Cryogenics, vol. 40, no. 12, pp. 469–474, 2000.

Bobby Mathew is a Ph.D. student in Engineering at
Louisiana Tech University. In May 2007 he received
his M.S. degree in engineering at Louisiana Tech
University. His current research interests include microchannel heat exchangers, micro loop heat pipes
and microscale heat transfer and fluid mechanics.

Hisham Hegab is an associate professor of mechanical engineering at Louisiana Tech University and
serves as the program chair of Microsystems and
Nanosystems Engineering within the College of Engineering and Science. He received his Ph.D. in mechanical engineering in 1994 from the Georgia Institute of Technology. His current research interests
are in microscale heat transfer, microfluidic systems,
micro heat exchangers, and micro-/nanotechnology
education.

vol. 31 no. 3 2010

Heat Transfer Engineering, 31(3):179–192, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457630903304350

Heat Transfer and Pressure Drop
Under Dry and Humid Conditions
in Flat-Tube Heat Exchangers With
Plain Fins
´ 2 PER FAHLEN,
´ 3
CAROLINE HAGLUND STIGNOR,1 BENGT SUNDEN,
1
and SOFIA STENSSON
1

SP Technical Research Institute of Sweden, Energy Technology, Bor˚as, Sweden
Lund University, Lund Institute of Technology, Heat Transfer, Lund, Sweden
3
Chalmers University of Technology, Building Services Engineering, G¨oteborg, Sweden
2

Flat-tube heat exchangers could be an interesting alternative to make indirect cooling of display cabinets more energyefficient. This application involves low air velocities in combination with condensation of water vapor on the air side, so
plain fins could be suitable. Two different heat exchangers having flat tubes and plain fins on the air side were evaluated
experimentally. One of the heat exchangers had continuous plate fins, and the other had serpentine fins. The performances
during dry and wet test conditions were compared and related to theoretical predictions for different assumptions. The
influence of air velocity, air humidity, and inclination angle was investigated. The results show that, in most cases, the heat
transfer performance is somewhat reduced under wet conditions in comparison with dry test conditions, and that wet heat

transfer surfaces lead to an increased pressure drop. At the lower air velocity range that was investigated, the heat exchanger
having continuous plate fins drained better than the one with serpentine fins.

INTRODUCTION
Flat-Tube Heat Exchangers in Heating, Ventilation,
Air-Conditioning, and Refrigeration Applications
Flat-tube heat exchangers (FTHE), predominately with louvered fins, have been used for a long time in applications where
compactness and performance are important, such as in automotive applications. For heating, ventilation, air-conditioning,
and refrigeration (HVAC&R) applications, heat exchangers with
round tubes are still most frequently used in stationary applications. According to Webb [1], the flat-tube configuration has
some advantages over the round-tube heat exchangers. For example, it has better fin efficiency and a smaller wake region
behind the flat tubes. A limited range of operating conditions
Financial and material support from the Swedish Energy Agency, Hydro
Alunova, Grundfos, and Wilo is kindly acknowledged.
Address correspondence to Caroline Haglund Stignor, SP Technical Research Institute of Sweden, Energy Technology, PO Box 857, SE-501 15 Bor˚as,
Sweden. E-mail:

had been studied earlier, but in recent years the studied operating ranges have been widened to include HVAC&R applications,
i.e., operating conditions involving wet and frosted surfaces on
the air side. Jacobi and coworkers have performed extensive
studies of flat-tube heat exchangers with flat, wavy, strip, and
louvered fins under dry, wet, and frosting conditions. The studies involve literature studies, calculations, and experiments, and
they present their results in two reports [2, 3]. According to
the authors, the research presented in those reports constituted
probably the most comprehensive studies of flat-tube heat exchangers in HVAC&R applications at that date.

Indirectly Cooled Display Cabinets
In recent decades, the use of indirect cooling by means of
a secondary refrigerant (coolant) has become very frequent in
supermarkets, especially in some of the Nordic countries. This

system arrangement has recently attracted more attention, and
is getting more and more common in other parts of the world
as well [4, 5]. Traditionally, different kinds of tube coils with

179


180

C. H. STIGNOR ET AL.

aluminum fins on expanded copper tubes have been used as
cooling coils in display cabinets. The liquid flow regime is often
laminar with low heat transfer coefficients as a consequence, due
to the high viscosity of many secondary refrigerants. However,
good energy efficiency can be obtained even under laminar flow
conditions, if the geometry of the heat exchanger is adapted
to this kind of flow, e.g., with smaller hydraulic diameters on
the liquid side. For this reason, flat-tube heat exchangers are an
interesting alternative.
The display cabinet application is different from many other
heat exchanger and HVAC&R applications, since it involves low
air velocities (down to 0.3 m/s) in combination with condensation of water vapor and sometimes even frosting. However,
it should be possible to operate an energy-efficient indirectly
cooled cooling coil in a medium-temperature display cabinet
without frosting. The question is how to design the liquid and
air sides of the cooling coil/heat exchanger to achieve the best
performance and avoid frosting. In earlier research work, the
authors of the present article investigated heat transfer on the
liquid side of multiport extruded aluminum tubes [6]. They also

developed a preliminary model to predict the performance of
a flat-tube heat exchanger in a display cabinet application in a
parameter study, and found promising results for some designs
[7]. However, the air-side performance when condensation of
water vapor occurs on the fins had to be verified experimentally
for these results to be reliable. When fouling takes place on a
heat transfer surface, i.e., the surface is coated with deposits
originating from the flow systems, the heat transfer resistance
is found to increase. At the same time, the area available for
flow is decreased, which results in higher fluid velocity. This,
in combination with a rougher surface, leads to increased pressure drop [8]. Condensation of water vapor on a heat exchanger
surface might have a similar effect. However, since the condensate drains continuously, the influence of the condensate on
heat transfer and pressure drop depends on how much condensate is accumulated in the heat exchanger before steady state is
reached.

Selection of Appropriate Fin Design
A louvered fin geometry is often reported as superior to the
plain fins [2, 3]. Davenport [9] compared test results from louvered samples with results from samples with plain fins. It was
found that both the Colburn j factor and the friction factor, f ,
for the louvered samples were between 2 and 3.5 times greater
than for the plain samples, depending on louver geometry. The
results by Davenport [9] also illustrated the irrelevance of hydraulic diameter for the performance of louvered surfaces but
that the hydraulic diameter was relevant for the samples with
plain fins. For louvered surfaces, louver-pitch-based Re was recommended as given by the equation (ReLp = (µ · ρ · Lp )/µ).
Colburn j factors were presented down to ReLp = 100, but not
for lower values of Re. The reason was that the curves flattened
at low velocities, which was thought to be caused by boundary
heat transfer engineering

layer thickening on the louver surface that in turn changed the air

flow pattern through the fins of the heat exchanger. Therefore, j
curves should not be extrapolated to lower Re values since this
will almost certainly result in an overestimation. Achaichia and
Cowell [10] presented performance data for a range of plate and
flat tube and louver fin geometries. Their resulting Stanton number (St) curve demonstrated characteristics that were consistent
with the earlier findings by, for example, Davenport [9] that at
high Re the fluid flow is predominately parallel to the louvers,
but as Re is reduced the flow direction becomes increasingly
determined by the plate fins and the St-curves flattened.
In a display cabinet application, for which frosting can occur,
the fin pitch is often equal to or greater than 5 mm in order to
have long operating periods between the defrosts. However, if
frosting could be minimized or avoided the fin pitch could be
decreased. Nevertheless, due to the low air velocities and occurrence of dust and condensate, it is not desirable to have too
small fin pitches in the heat exchangers. A fin pitch of around
4 mm has therefore been the focus of investigations for this
study. In a source database presented by Jacobi et al. [3] the
louver pitch for most specimens ranges from 1 to 2.3 mm. For
such small dimensions, the flow in a display cabinet application
will most certainly be duct-orientated even if the fins are louvered, due to the low air velocities. In such a case, there is a risk
of the louver resulting only in an additional pressure drop over
the heat exchanger. In addition, condensed water might also
partially clog the louver gaps, which contributes to making the
flow duct-directed. Under such circumstances heat exchangers
with flat tubes and plain (flat) fins will be preferable.
Jacobi et al. [2] performed experiments with a flat-tube heat
exchanger with plain fins under dry, dehumidifying and frosting conditions. The heat exchanger had dimensions similar but
not identical to those of the MPET heat exchanger evaluated
in the present study (see Table 1), but the plain fins were continuous plate fins (no serpentine fins). The researchers found
that for dry test conditions the heat transfer and friction factor

performance could be predicted simply by duct-flow modeling.
They compared the performance of this heat exchanger with a
similar heat exchanger with wavy fins in a Reynolds number
region of 300 < Redh < 2100. They found the Colburn j factor
to be higher or similar for the plain fins in the lower Reynolds
number region (Re ≤ 1000) compared to the wavy fins under
both dry and wet conditions, while the opposite was true for
higher Reynolds number. Almost the same relations were found
for the friction factor, f . The Colburn j factor was not much
affected by the condensation of water vapor for either the plain
or the wavy fin geometry, while the friction factor was found to
increase by a factor of 2.2 to 2.8. Jacobi et al. [2] also carried
out experiments with a heat exchanger with louvered fins having a fin pitch of 5.08 mm and a louver pitch of 1.14 mm and
compared its performance for dry and wet conditions. The difference in performance was found to be very small which proves
that the flow is duct-oriented as was expected for such large fin
pitch. No heat transfer enhancement was therefore obtained by
boundary-layer restarting for either of the test conditions.
vol. 31 no. 3 2010


C. H. STIGNOR ET AL.

181

Table 1 Dimensions of the evaluated heat exchangers
Dimension

FFC

MPET-HE

250
460
152
10.0
19.0
3.61
0.11
7.7
19.0
4.8
13.6
1.9
13.0

250
458
118
23.1
73.0
3.95
0.203
19.1
45.0
6.3
45.0
3.20
44.2 (total),
1.42 (duct)
0.4

2.06
94.6
10

Height
Width
Depth
Tube pitch, transversal
Tube pitch, longitudinal
Fin pitch
Fin thickness
Fin length
Fin depth
Hydraulic diameter (air side)
Tube depth (outer width)
Tube inner height
Tube inner width

H
W
De
Tp.t
Tp,l
Fp
δ f in
Fl
Fd
dh,a
Td
Th,i

Tw,i

mm
mm
mm
mm
mm
mm
mm
mm
mm
mm
mm
mm
mm

Tube wall thickness
Hydraulic diameter (tube side)
Outer perimeter of tube
Number of tubes, transversal to
air flow
Number of fins, transversal to air
flow
Number of tubes, longitudinal to
air flow
Number of passes/heat
exchangers
Number of parallel vertical tube
rows in each pass

δ tube
dh,b
Ptube,o
Ntube,t

mm 0.2
mm 3.32
mm 31.8
(—) 25

Nf in,t

(—)

—

11

Ntube,l

(—)

8

2

n

(—)

4

2

Ntube,p (—)

2

1 tube (with 25
parallel channels)

Purpose of Study
On the basis of the results found in the literature, and from
earlier research by the authors presented earlier, flat-tube heat
exchangers with plain fins could be an interesting alternative for
indirect cooling of the air in display cabinets and other applications where dehumidifying conditions are combined with low
air velocities. However, it is of interest to study the performance
under dehumidifying conditions and even lower air velocities
than has been done in earlier investigations, with particular emphasis on investigating whether condensed water drains well
from serpentine plain fins or from continuous plate fins, and
whether and to what extent the humidity of the incoming air
influences the sensible heat transfer coefficient and the pressure
drop. Two different flat tube heat exchangers with plain fins
of the different types and horizontal flat tubes with different
depths have therefore been investigated experimentally. Their
geometrical data are presented in Table 1.
Figure 1 Schematic drawing and nomenclature of the “flat-fin-core” heat
exchanger, denoted FFC: (a) side view of cross section, (b) cross section of
staggered tube layout, (c) plate fin geometry.

EXPERIMENTS
Description of Tested Objects
Two different heat exchangers having flat tubes and plain
(flat) fins on the air side were evaluated experimentally. The
dimensions of the heat exchangers are presented in Table 1
heat transfer engineering

(see Figures 1 and 2 for description of nomenclature), while
Figure 3 shows the media flow arrangements of the heat exchangers. One of the heat exchanger is called “Flat-Fin-Core”
by its manufacturer and is therefore denoted FFC. It consists
vol. 31 no. 3 2010


182

C. H. STIGNOR ET AL.

Figure 3 Liquid circuitry of the FFC and the MPET-HE heat exchanger.

Figure 2 Schematic drawing and nomenclature of the multiport extruded
tube heat exchanger, denoted MPET-HE: (a) side view of cross section, (b)
serpentine fin geometry, (c) cross section of MPE tube.

of plain continuous plate fins on a tube bundle. The tube configuration of the FFC heat exchanger is between an in-line and
a staggered configuration, as can be seen in Figure 1. The flat
tubes are plain and empty. As can be seen, the liquid circuit is
arranged in such way that the heat exchanger consists of four
cross-flow heat exchangers connected in series, with an overall
counter(current) flow. The fins are made of copper and the tubes
are made of brass. The fins are connected to the tubes by soft

heat transfer engineering

soldering (tin). When punching the tube holes out of the plate
fins, a collar is left. Before the soldering, the tubes are coated
with tin. The heat exchanger is then put into an oven and inclined in such way that the melted tin fills the gap between the
tube and fin material and creates good thermal contact between
the tubes and the fins. The FFC type heat exchanger is normally
used in forest and agricultural machinery.
The other heat exchanger investigated in this work consists of
MULTIPort extruded tubes (MPE tubes) and folded serpentine
flat (plain) fins on the air side. It is denoted here as MPET-HE
(multiport extruded tube heat exchanger). As Figure 3b shows,
it consists of two cross-flow heat exchangers connected in series with an overall counter (current) flow. The cross section of
the MPE tubes of the heat exchanger can be seen in Figure 2c.
Both the fins and the tubes are of aluminum and the heat exchanger is put together by a controlled-atmosphere brazing process using a cladding material (AlSi). Due to the design of the
heat exchanger, with serpentine fins between flat tubes, it is possible to apply external pressure on the heat exchanger during the
brazing process, to ensure good thermal contact between the fins
and the tubes. This heat exchanger was uniquely constructed for
this research study.
As far as the air sides of the heat exchangers are concerned,
the main difference is that the FFC heat exchanger has continuous plain fins, while the MPET heat exchanger has serpentine
fins folded between the flat tubes. In addition, the tube depth
(or outer width)—i.e., the length of the air-side channels—is
much greater for the MPET heat exchanger than for the FFC
heat exchanger.
vol. 31 no. 3 2010