[American Institute of Aeronautics and Astronautics 10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference - Chicago, Illinois ()] 10th AIAA/ASME Joint Thermophysics and Heat

American Institute of Aeronautics and Astronautics

1

Parallel Flow Microchannel Heat Exchangers Subjected to

Axial Heat Conduction and Internal Heat Generation

B. Mathew1 and H. Hegab

2

College of Engineering and Science, Louisiana Tech University, Ruston, LA, 71272

This paper studies the effect of axial heat conduction and internal heat generation

(viscous dissipation) on the effectiveness of a parallel flow microchannel heat exchanger. The

ends of the wall separating the fluids are maintained at constant temperatures. This leads to

heat transfer between the heat exchanger and its surroundings. The thermal model

developed in this paper consists of three governing equations; one for each of the fluids and

one for the wall. This system of coupled equations is solved simultaneously using finite

difference method. The effectiveness of the fluids is found to depend on )TU, axial heat

conduction parameter, end wall temperatures and internal heat generation parameter. In

the presence of just internal heat generation the effectiveness of the hot and cold fluid

degrades and improves, respectively. For situations when the temperature of the end wall at

the inlet side is greater than that at the outlet side, increase in axial heat conduction

parameter of a heat exchanger subjected internal heat generation increases and decreases

the effectiveness of the hot and cold fluid, respectively. The effect of internal heat generation

in a parallel flow microchannel heat exchanger with axial heat conduction is to always

degrade and improve the effectiveness of the hot and cold fluid, respectively.

)omenclature

A = area

C = heat capacity

Cp = specific heat

h = heat transfer coefficient

L = length

MCHX = microchannel heat exchanger

TU = number of transfer units

P = pressure

Q = nondimensional external heat transfer parameter used in [10]

T = temperature

U = overall heat transfer coefficient

u = velocity in x-direction

x = axial coordinate

X = nondimensional axial coordinate

y = transverse coordinate (in the direction of channel width)

z = transverse coordinate (in the direction of channel depth)

υ = volumetric flow rate (m3sec

-1)

ε = effectiveness

θ = nondimensional temperature

µ = viscosity (Pa.sec)

ψ = internal heat generation parameter

1 Doctoral Student, Micro and Nano Systems Track, College of Engineering and Science, LA Tech, Ruston, LA

71272; Email: [email protected], Ph: +1-318-514-9618 2Associate Professor, Mechanical Engineering Program, College of Engineering and Science, LA Tech, Ruston, LA

71272; Email: [email protected], Ph: +1-318-257-3791, Fax: +1-318-257-4922

10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4893

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

Subscripts:

c = cold

CF = counter flow

cr = cross section

h = hot

PF = parallel flow

s = surface

w = wall

I. Introduction

icrochannel heat exchangers (MCHXs) are used in several fields of engineering. These devices employ

channels with hydraulic diameter smaller than 1 mm [1]. The use of channels with such small size has certain

advantages such as 1) enhanced heat transfer coefficient, 2) increased surface area density, and 3) compactness [2].

The major drawback of employing such small channels is pressure drop and fouling. MCHXs can be fabricated in

silicon, glass, ceramics, copper and stainless steel. The type of material selected for fabricating a MCHX depends on

the operating parameters, i.e. pressure and temperature. The fabrication method depends on the type of material

selected for fabricating the MCHX; some of the common methods include dry and wet chemical etching, mechanical

micromachining, sand blasting, and stereolithography [3]. MCHXs are used as components in several devices such

as micro fuel cells, microminiatue refrigerators, microcombustors, and micro reactors. Many of the devices that

employ a MCHX employ it for the purpose of recuperation of waste heat energy. In certain other devices MCHXs

are used for performing chemical reactions or for thermal management of chemical reactions.

The term heat exchanger has been conventionally used for referring to heat transfer devices with two fluids.

However, in recent times microchannel heat sinks which employ just one fluid are also addressed as heat exchangers

[4, 5]. Even though both these devices are heat exchangers in a broad sense there are differences in working

principle and application. The heat transfer is between two fluids in the heat transfer device that is conventionally

termed as heat exchanger. In a heat sink the heat transfer is between a fluid and a heated surface. The application to

which a two fluid heat transfer device is put to is for heating or cooling one of the fluids using the other fluid. On the

other hand, the single fluid heat transfer device is used for cooling of the heated surface with which it exchanges

heat. These specific differences should prohibit the use of same terminology for addressing the two different heat

transfer devices. As two-fluid heat transfer devices have been conventionally referred to as heat exchangers it would

only be logical to address even the microscale versions of such devices as heat exchangers. The single fluid heat

transfer device should be referred to as heat sinks even at the microscale level.

Most present day MCHXs are designed using the conventional ε-NTU equations [6]. However, as these

equations are based on certain assumptions it is important to reinvestigate the validity of these assumptions at the

microscale prior to employing these equations for design purposes. Two of the assumptions that need to be

investigated are: 1) insignificance of axial heat conduction and 2) lack of internal heat generation. In all macro-

sized parallel flow heat exchangers axial heat conduction is proven to be nonexistent or almost negligible [7]. This is

because the temperature of the wall separating the fluids is constant over the entire length of the heat exchanger. In

addition, the ends of the wall separating the fluids in a macroscale heat exchanger can be easily kept insulated and

this prevents heat transfer with its surroundings. However, in the case of MCHXs the ends of the wall are either part

of the manifold or the substrate due to manufacturing constraints. This leads to heat transfer between the MCHX and

its surroundings thereby leading to axial heat conduction through the wall separating the fluids. In addition, MCHXs

being small in length have lower conduction thermal resistance in the axial direction and this can aggravate the

problem of axial heat conduction. Thus while designing a MCHX it is important to consider the effect of axial heat

conduction, through the wall separating the fluids, on its effectiveness.

Internal heat generation refers to the situation where frictional pressure drop is converted into heat. For pushing a

certain quantity of fluid through a channel it is necessary to overcome the friction between the fluid and the wall.

The work needed to push fluid through a channel against friction is called flow work in literature. The power

corresponding to flow work is the product of volumetric flow rate and pressure drop. Pressure drop increases with

decrease in channel hydraulic diameter for a specific volumetric flow rate. Thus when the hydraulic diameter of a

channel is in the micrometer range the pressure drop experienced across its ends is extremely high for a specific

volumetric flow rate. Consequently, its effect on the effectiveness of the fluids has to be considered. This additional

heat source is not accounted for in the conventional ε-NTU relationship and thus it needs to be extended for use in

the design of MCHXs. The conversion of pumping power into heat is also referred to as viscous dissipation.

M


3

Figure 1. Schematic of the differential element of the MCHXPF considered in

this study.

XdX

0wθ

)( cwsc dAh θθ −

+ XdXd

dC h

hh

θθ

+ Xd

Xd

dC c

cc

θθccC θ

hhC θ

1wθ

)( whshdAh θθ −Pν

Pν

+ XdXd

dPPν

+ XdXd

dPPν

In this paper the influence of these effects with respect to a parallel flow MCHX (MCHXPF) is analyzed. A

counter flow MCHX (MCHXCF) is known to have better performance than a MCHXPF for the same operating

parameters. However, for certain applications such as those involving chemical reactions a MCHXPF has been

proved to be better than a MCHXCF and thus it is analyzed in this paper [8].

II. Literature Review

In this section, a few articles related to the effects analyzed in this paper are reviewed with respect to two fluid

heat exchangers. Mathew and Hegab [9] have theoretically analyzed the effect of external heat transfer and internal

heat generation on the thermal performance of a balanced flow MCHXPF. External heat transfer represents the heat

transfer between the fluids and the external ambient. While internal heat generation is used to take into account the

effect of viscous heating on the effectiveness of the fluids. When the ambient temperature is higher than the inlet

temperatures of the fluid, the presence of external heat transfer alone degraded and improved the effectiveness of the

hot and cold fluid, respectively. On the other hand, external heat transfer, when the ambient temperature is below the

inlet temperature of the fluids, increased and decreased the hot fluid and cold fluid effectiveness, respectively. The

effect of internal heat generation alone always decreased and increased the effectiveness of the hot and cold fluid,

respectively. The combined effect of external heat transfer and internal heat generation can improve or degrade the

effectiveness of the fluids depending on the amount of heat gained/lost to the fluids due to these effects. Mathew and

Hegab [10] analyzed the effect of uniform external heat flux on the thermal performance of a MCHXPF. They

developed analytical equations for both balanced and unbalanced flow conditions. The effect of external heating is

to degrade and improve the effectiveness of the hot and cold fluid, respectively. On the other hand external cooling

will increase and decrease the hot and cold fluid effectiveness, respectively. Under unbalanced flow conditions the

effectiveness of the fluids of a MCHXPF subjected to external heat flux is found to depend on the fluid with the

lowest heat capacity. Mathew and Hegab [11] studied the effect of axial heat conduction coupled with heat transfer,

through the ends of the wall separating the fluids, on the effectiveness of a balanced flow MCHXPF as well. The

ends of the wall separating the fluids are not kept insulated but rather maintained at fixed temperature. This leads to

heat transfer between the MCHXPF and its surroundings. The effectiveness is found to depend on NTU, axial heat

conduction parameter and the end wall temperatures. Reduction in the temperature of the wall at the inlet section of

fluids leads to improvement and reduction in the hot and cold fluid effectiveness, respectively. Axial heat

conduction is non-existent when the temperature of the both end walls is the average of the inlet temperature of the

fluids. From the few articles reviewed here it is clear that there is currently no work that deals with the effect of axial

heat conduction and internal heat generation on the performance of a MCHXPF.

III. Theoretical Model

The thermal model of the

MCHXPF studied in this paper

is developed in this section.

Figure 1 is a schematic

representation of the

MCHXPF analyzed in this

paper. The thermal model of

this MCHXPF consists of

governing equations which

are based on the principles of

continuum mechanics. Thus

the model developed in this

paper can be used for

macroscale heat exchangers

as well. However, the scenario

studied here would most often

be observed only in

microscale heat exchangers. Prior to developing the governing equations certain assumptions are made and they are

provided below.

1. The MCHXPF operates under steady state conditions.

2. The flow is taken to be hydrodynamically and thermally fully developed at the inlet of the MCHXPF.


4

3. The temperature of the fluids is assumed to be constant at every cross section along the direction of flow.

4. The contribution of wall thermal resistance in the overall thermal resistance is taken to be negligible.

5. There is no phase change of the fluids while flowing through the MCHXPF.

6. No-slip wall boundary condition is assumed on the walls of the microchannel.

7. External heat transfer, flow maldistribution and axial heat conduction in the fluids is considered negligible.

8. The thermophysical properties of the fluids are assumed constant between the inlet and outlet of the MCHXPF.

The governing equations developed for the MCHXPF are based on these assumptions and are provided in Eqs. (1)

– (3). The thermal model consists of three governing equations; one for each of the fluid and one for the wall. The

governing equations are obtained by performing an energy balance on each of three entities of the MCHXPF. The

first term of Eqs. (1) and (2) represent the axial variation in the temperature of the fluids. The second term of these

equations represents the heat transfer between the individual fluid and the wall. The term on the right hand side of

these equations accounts for internal heat generation. It is the heat generated internally per unit length of the

microchannel, i.e. over every cross section of the microchannel. Regarding the third governing equation, the first

term represents the variation in axial heat conduction through the wall separating the fluids. The second and third

term of this equation accounts for heat transfer between the hot fluid and wall and the wall and the cold fluid,

respectively.

Hot fluid:

dx

dP

CTT

C

ph

dx

dT

h

wh

h

shh ν−=−+ )(

(1)

Cold fluid:

dx

dP

CTT

C

ph

dx

dT

c

wc

c

scc ν−=−+ )(

(2)

Wall:

0)()(2

2

=−−−+ cwscwhshw

crw TTphTTphdx

TdAk

(3)

The coefficient of the second term of Eqs. (1) and (2) represent the NTU per unit length with respect to each of

the fluids. These terms represent the ratio of the convective thermal resistance per unit length to the heat capacity of

the fluid flowing the through the channel. Upon nondimensionalizing these equations the coefficients of the second

term of Eqs. (1) and (2) will represent the ratio of the convective thermal resistance to the heat capacity of the fluid,

i.e. the NTU respect to individual fluid. In literature the NTU associated with the hot and cold fluid are addressed as

NTUh and NTUc, respectively [12, 13]. In the conventional ε-NTU relationship the term NTU represents the ratio of

the overall thermal conductance to the lowest heat capacity among the two fluids in the heat exchanger. Therefore it

is importance to write the NTUh and NTUc in terms of NTU. In a device with microchannels the flow is almost

always laminar. Thus the flow rate does not have any influence on the heat transfer coefficient (h). The heat transfer

coefficients on either side of the wall depend only on Nusselt number, channel dimensions and the thermal

conductivity of the fluid. In most microfluidic devices with multiple channels the dimensions of the channels would

be kept constant. Moreover, for balanced flow case the heat capacity of both fluids are kept same. Thus for a specific

MCHXPF it is safe to assume NTUh and NTUc to be same. Thus NTUh and NTUc in terms of NTU can be

represented as shown in Eq. (4).

TU TU TUC

hA

hA

C

Ah

C

Ah

C

TU TU TUch

s

ssc

c

sh

h

ch

22111

===⇒=+=+= (4)

Earlier it was mentioned that internal heat generation which is taken to be equal to the power associated with

flow work is equivalent to viscous heat dissipation. For a fully developed internal flow the viscous dissipation at

every point in a 3D flow can be simplified and written as follows [14]:


5

( )

∂+

∂=∇⋅∇=

22

dz

u

dy

uuu µµφ (5)

Using the simplification provided by Morini [15], i.e. Eq. (6), it is possible to rewrite Eq. (5) as Eq. (8). Equation

(7) can be used to rewrite the first term on the right hand side of Eq. (6). Equation (7) is obtained by rearranging the

momentum equation in the x-direction for fully developed flow [14].

222 5.0 uuuuu ∇+∇−=∇⋅∇ (6)

dx

dP

z

u

y

uu

µ1

2

2

2

22 =

∂∂

+∂∂

=∇ (7)

∇+−=

∇+

∂∂

+∂∂

−=⇒ 2222

2

2

2

2

5.05.0 udx

dPuu

z

u

y

uu

µµµφ (8)

Φ is the viscous dissipation at any arbitrary point inside the microchannel. However, Eqs. (1) – (3) are one

dimensional, i.e. it accounts for variation only in the axial direction (x-direction). It implies that all parameters used

in these equations are averaged over the cross section area of the microchannel. Thus even Φ has to be averaged

over the cross section area of the microchannel as shown in Eq. (9) for use in Eqs. (1) – (3).

dydzudydzdx

dPudydz

z

u

y

udydz

AAAA

∫∫∫∫∫∫∫∫ ∇+−=

∂∂

+

∂∂

= 22

22

5.0µµφ (9)

Morini [15] was able to mathematically prove the second term in Eq. (9) to be equivalent to zero. Thus Eq. (9)

can be simplified as shown in Eq. (10).

dx

dPdydzu

dx

dpdydz

AA

νφ −=−= ∫∫∫∫ (10)

The parameter dydzA

∫∫φ represents viscous dissipation per unit length of the microchannel. This quantity is

same as the quantity represented by term on the right hand side of Eqs. (2) and (3). Thus pumping power associated

with flow work per unit length represents the viscous dissipation at every cross section in the microchannel of

interest. Since both volumetric flow rate and pressure gradient are constant for fully developed flow the internally

generated heat per unit length is also a constant.

In light of the information provided in Eq. (4), Eqs. (1) – (3) can be rewritten in terms of NTU. In addition these

equations need to be nondimensionalized and for this certain new terms are used and they are provided below. The

term X represents the nondimensional axial coordinate. The second term is the axial heat conduction parameter (λ)

and the third term is the internal heat generation parameter (ψ). ψ is the heat that is internally generated over the

entire length of the MCHXPF. The new set of governing equations based on these terms is represented in Eqs. (11) –

(13).

L

xX = ,

LC

Ak crw

min

=λ , )(min cihi TTC

Ldx

dP

−

−=

νψ


6

Hot fluid:

ψθθθ

=−+ )(2 whh TU

Xd

d

(11)

Cold fluid:

ψθθθ

=−+ )(2 wcc TU

Xd

d

(12)

Wall:

02242

2

=++

− chw TU TU TU

Xd

dθθθλ

(13)

The influence of microchannel profile is not directly obvious in the governing equations, Eqs (11) – (13).

Microchannel profile influences the heat transfer coefficient and the pressure drop. Heat transfer coefficient is

included in NTU and pressure drop is used in calculating the internal heat generation parameter. Thus the effect of

microchannel profile is indirectly included in Eqs. (11) – (13).

Four boundary conditions are needed to solve the set of governing Eqs. (11) – (13). One boundary condition is

needed for each of the governing equations representing the hot and cold fluid. The boundary conditions

corresponding to Eqs. (11) and (12) are their inlet temperature which is known a priori. Two boundary conditions

are needed for the equation representing the wall. It was mentioned earlier that the ends of this wall would be either

part of the manifold or the substrate in which it is fabricated. Thus the boundary conditions consist of the

temperature at the ends of the wall separating the fluids. These boundary conditions are mathematically presented in

Eqs. (14) – (17).

10=

=Xhθ (14)

00=

=Xcθ (15)

inwXw ,0θθ =

= (16)

outwXw ,1θθ =

= (17)

The governing equations, Eqs. (11) – (13), subjected to the boundary conditions shown in Eqs. (14) – (17), are

solved using the finite difference method. The computational domain consists of the two fluids and the wall

separating them. The first order term of Eqs. (11) and (12) is discretized using second order central difference

scheme for all nodes except the ones at the inlet and outlet. For the nodes at the outlet of the microchannels the

second order backward difference scheme is used. For the second order term of Eq. (13) the second order central

difference scheme is used at all the interior nodes. Nodes are uniformly spaced throughout the computational

domain. Three iterations at different node lengths are performed for a given set of operating parameters (NTU, λ,

ψ,θw,0, and θw,1) in order to ensure grid independency of the results. For all cases the first iteration is performed for a

node length of 0.001, the second one at 0.0005 and the third one at a node length of 0.00025. The temperature

corresponding to the solution of the governing equations with node size of 0.00025 is used for calculations of

effectiveness and heat transfer.

The effectiveness of the each fluid is determined from the inlet and outlet temperature of the fluids. The inlet

temperature of both the fluids is known a priori. Moreover, due to the way in which the temperatures are

nondimensionalized there is need for only the outlet temperature of the fluids for determining the effectiveness. The

effectiveness of each fluid is shown in Eqs. (18) and (19).


7

1,,min

,,1

)(

)(=

−=−

−=

Xh

icih

ohihh

hTTC

TTCθε (18)

1,,min

,,

)(

)(=

=−

−=

Xc

icih

icocc

cTTC

TTCθε (19)

Knowledge of the local temperature of the fluids and wall can be used for determining the heat transfer between

each fluid and the wall. The equations necessary for determining the two heat transfers are provided in Eq. (20) and

(21). Qh,w represents the heat transfer between the hot fluid and the wall while the heat transfer between the wall and

the cold fluid is represented by Qw,c. The integration in Eqs. (20) and (21) are performed using Composite Simpsons

Method [15]. The error associated with this method is proportional to the fourth power of the node length (4

X∆ ).

∫ ∫ −==1

0

1

0

,, )(2 Xd TUdQQ whwhwh θθ (20)

∫ ∫ −==1

0

1

0

,, )(2 Xd TUdQQ cwcwcw θθ (21)

The heat transfer through the ends of the wall separating the MCHXPF and its surroundings is also calculated in

this paper using 1D Fourier’s heat conduction equation and is presented in Eqs. (22) and (23). The wall temperature

gradient at the inlet and outlet section is numerically determined from the wall temperatures [15]. The 3 point

formula is used for determining the temperature gradient. The error associated with this method is proportional to3

X∆ . When Qcond,0 is positive it means that the heat transfer is from the surroundings to the MCHXPF. The

opposite is the case when Qcond,0 is negative. On the other hand, when Qcond,1 is positive it indicates heat transfer

from the MCHXPF to its surroundings. The heat transfer is from the surroundings to the MCHXPF when Qcond,1 is

negative.

0

0,

=

−=Z

wcond

Xd

dQ

θλ (22)

1

1,

=

−=Z

wcond

Xd

dQ

θλ (23)

The net heat transferred between the MCHXPF and its surroundings through the ends of the wall separating the

fluids can be determined using Eqs. (22) and (23). Qcond,net represents the heat that is either accumulated or lost from

the MCHXPF due to heat transfer through the ends of the wall separating the fluids. If Qcond,net is positive it indicates

accumulation of heat in the MCHXPF and a negative value of Qcond,net indicates the opposite. Qcond,net is

mathematically represented in Eq. (24).

0,1,, condcondnetcond QQQ −= (24)

It is mentioned at the beginning of this section that the theoretical model developed in this paper are based on the

principles of continuum mechanics. Thus there are some limitations on the size of the MCHXPF channel for which

these theories hold true. These limitations exist especially when gases are used. For gases the continuum theories

hold true only if the Knudsen number (Kn) is less than 10-3

[17]. If Kn > 10-3

, no-slip boundary condition, and

rarefaction has to be considered in the governing equations. Certain researchers have used Kn > 10-2

as the criterion

above which any gas can be considered as a continuum. For example when air is used as the fluid in a MCHXPF this


8

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

)TU

ε

λ= 0.0, ψ = 0.0 λ= 0.0, ψ = 0.1

Conventional ε-)TU relationship [6] (λ = 0.0, ψ = 0.0)

Mathew and Hegab [10] (λ = 0.0, Qh = Qc = 0.1)

cold fluid

hot fluid

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5)TU

ε

cold fluid

hot fluid

λ = 0.00, ψ = 0.00

λ = 0.00, ψ = 0.15

λ = 0.05, ψ = 0.15

λ = 0.10, ψ = 0.15

λ = 0.25, ψ = 0.15

Figure 2. ε-)TU relationship of a balanced flow

MCHXPF (λ = 0.0, θw,0 = 1.2 and θw,1 = 0.2).


MCHXPF (ψ = 0.15, θw,0 = 1.2 and θw,1 = 0.1).

criteria on Knudsen number will be satisfied only in channels with hydraulic diameter greater than 70 micrometers

based on the criterion that Kn > 10-3

[17]. Similarly, the channels of a MCHXPF cannot be made smaller than 200

micrometers if gaseous helium is employed [17]. Therefore there is no absolute value for the lower limit of the

hydraulic diameter of the channels employed in a MCHXPF. The lower limit on the hydraulic diameter has to be

determined on a case by case basis.

IV. Results and Discussions

The thermal model developed in this paper is used to study few cases associated with the thermal performance of

a MCHXPF with axial heat conduction and internal heat generation. The operating parameters especially axial heat

conduction parameter and internal heat generation parameter are randomly selected to obtain a general

understanding of the influence of these phenomena on the effectiveness of the fluids. Prior to this study the validity

of this thermal model is checked by comparing the results obtained from the model with that obtained using the ε-

NTU relationship for a specific NTU. For comparison purposes the axial heat conduction parameter (λ) and internal

heat generation parameter (ψ) are equated to zero in the governing equations. Figure 2 shows this comparison with

the solid line representing the effectiveness calculated from the present model and the data points showing the

effectiveness as determined from the conventional ε-

NTU equation. The end wall temperature the inlet

section is maintained at 1.2, i.e. 20% higher than the inlet

temperature of the hot fluid, and that at the exit section is

kept at 0.2 (80% lower than the inlet temperature of the

hot fluid). The data points exactly on top of the solid

lines indicating the validity of the model developed in

this paper. It was mentioned earlier that in the presence

of internal heat generation the fluids are uniformly

heated between the inlet and outlet of the microchannel.

This is similar to subjecting the fluids to uniform

external heat flux, a situation previously studied by the

Mathew and Hegab [10]. Thus as part of the validation

process the results obtained from this model is compared

with the results obtained from the analytical solutions

previously developed by Mathew and Hegab [10]. This

comparison is also shown in Fig. 2. The effectiveness

obtained from the previous model of Mathew and Hegab

[10] is shown as data points while the results of the present model are shown using dashed lines. In order to compare

the two models axial heat conduction parameter (λ) is set to zero and the internal heat generation parameter is set to

a desired value. In this paper it is set to 0.1, i.e. 10% of the maximum heat transfer possible in a MCHXPF. Up on

comparing the two models it is clear that the predictions of the present model is exactly equal to the predictions of

the previous model developed by Mathew and Hegab [10].

Figure 3 represents the hot and cold fluid

effectiveness of a MCHXPF with axial heat conduction

and internal heat generation. The internal heat generation

parameter is maintained at 0.15, i.e. it is taken to be 15%

of the maximum heat transfer that is possible in a

MCHXPF. The temperature of the end wall at the inlet

section is kept at 1.2, i.e. 20% higher than the inlet

temperature of the hot fluid. On the other hand the

temperature of the end wall at the outlet section of the

fluids is maintained at 0.1, i.e. 90% lower than the inlet

temperature of the hot fluid. The axial heat conduction

parameter is varied between 0.0 and 0.25. The

effectiveness of the hot fluid in the presence of just

internal heat generation is lower than that of a MCHXPF

free of axial heat conduction and internal heat

generation. This is because the addition of heat that is

generated due to viscous dissipation works to increase the


9

local temperature of the hot. Therefore the outlet temperature of such a MCHXPF will be higher than that of a

MCHXPF free of any additional phenomena. This leads to lower effectiveness for the MCHXPF with just internal

heat generation.

With increase in axial heat conduction parameter the effectiveness of the hot fluid increased for a particular NTU

even in the presence of internal heat generation. This can be understood by investigating the heat transfer between

the MCHXPF and its surroundings due to axial heat conduction. This heat transfer depends on the axial heat

conduction parameter and the temperature at the end walls. For the case studied in Fig. 3 the heat transfer through

the end wall at the outlet section is higher than that through the inlet section. Thus Qcond,net is a positive quantity.

This implies that more heat is getting transferred out of the MCHXPF through the end wall at the exit section than

that is entering the MCHXPF through that at the inlet section. This is shown Fig. 4 for various axial heat conduction

parameters (λ). The additional heat that is being transferred out of the MCHXPF comes from the hot fluid because it

is the only heat source in the MCHXPF. This has a positive effect on its effectiveness as seen in Fig. 3. However, the

internal heat generation always heats the hot fluid. Thus the presence of internal heat generation partially counteracts

the positive effect brought about by axial heat conduction on the effectiveness of the hot fluid.

The effectiveness of the hot fluid is equal to the negative of the internal heat generation parameter when NTU is

zero. This is because at this NTU there is no heat transfer from the hot fluid to the wall. This is because the heat

transfer surface area is either very small or the flow rate is very high for any heat transfer to take place. Thus the hot

fluid does not lose any heat rather it gains heat due to viscous dissipation. The amount of heat gained is equal to the

internal heat generation parameter which is defined with respect to the maximum heat transfer possible in a

MCHXPF. Thus its effectiveness is equal to the negative of the internal heat generation parameter.

The effectiveness of the cold fluid is equal to the internal heat generation parameter when the NTU is zero. This

situation is similar to that observed for the hot fluid. With regard to the cold fluid effectiveness shown in Fig. 3, its

effectiveness decreased in the presence of axial heat conduction and internal heat generation. This is due to the fact

that axial heat conduction will cause a portion of the heat lost by the hot fluid to transfer out of the MCHXPF rather

than being completely transferred to the cold fluid. Thus with increase in axial heat conduction parameter the

amount of heat gained by the cold fluid decreases and this leads to decrease in its effectiveness as shown in Fig. 3.

However, the presence of internal heat generation reduces the severity in the degradation of the cold fluid

effectiveness that would have occurred with just axial heat conduction.

Figure 5 represents a second case analyzed to understand the effect of axial heat conduction and internal heat

generation. In this case the end wall temperature at the inlet section is taken to be 0.8, i.e. 20% lower than the inlet

temperature of hot fluid inlet temperature. Other end wall temperature is maintained at 0.2 which is 80% lower than

the inlet temperature of the hot fluid inlet temperature. The axial heat conduction parameters are varied between 0.0

and 0.25. The internal heat generation parameter is kept at 0.15 even in this case.

Figure 4. Variation of Qcond,net with )TU for a

balanced flow MCHXPF (ψ = 0.15, θw,0 = 1.2 and θw,1

= 0.1).


MCHXPF (ψ = 0.15, θw,0 = 0.8 and θw,1 = 0.2).

Just like in the previous case the presence of axial heat conduction in a MCHXPF already subjected to internal

heat generation leads to improvement and degradation in the effectiveness of the hot and cold fluid, respectively. In

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5)TU

Qco

nd

,net

λ = 0.05, ψ = 0.15

λ = 0.10, ψ = 0.15

λ = 0.25, ψ = 0.15

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5)TU

ε

cold fluid

hot fluid

λ = 0.00, ψ = 0.00

λ = 0.00, ψ = 0.15

λ = 0.05, ψ = 0.15

λ = 0.10, ψ = 0.15

λ = 0.25, ψ = 0.15


10

the presence of just internal heat generation the effectiveness of the hot and cold fluid, with respect to that of a

MCHXPF free of axial heat conduction and internal heat generation, are lower and higher, respectively. The

effectiveness of the hot fluid gradually increased with increase in NTU while that of the cold fluid decreased over

the same NTU range. The effectiveness of the fluids when NTU is zero is numerically equal to the internal heat

generation parameter. These trends are similar to that already observed for the case presented in Fig. 3. Thus the

reasons for these observed trends are also same as that already explained.

Figure 7 represents the effect of internal heat generation in a MCHXPF that is subjected to axial heat conduction.

The axial heat conduction parameter is kept at 0.1. The end wall temperature at the inlet section is held at 1.1 and the

other end is held at 0.2. The internal heat generation parameters are varied between 0.0 and 0.2.

Figure 6. Variation of Qcond,net with )TU for a

balanced flow MCHXPF (ψ = 0.15, θw,0 = 0.8 and θw,1

= 0.2).


MCHXPF (λ = 0.1, θw,0 = 1.1 and θw,1 = 0.2).

The effect of axial heat conduction tends to improve the effectiveness of the hot fluid as mentioned earlier.

However, as already seen internal heat generation alone will always degrade the effectiveness of the hot fluid. Thus

when these two effects are occurring simultaneously the effectiveness of the hot fluid can increase or decrease

depending on the amount of heat lost/gained by the hot fluid due to these phenomena. Similar situation occurs even

for the cold fluid. For the cases analyzed in Fig. 7 the effectiveness of the cold fluid increased, reached a peak and

then decreased. This is because of the reversal in the influence of these additional effects on the effectiveness of the

fluids. In this case, i.e. Fig. 7, the heat supplied to the cold fluid by internal heat generation for a particular curve is

constant and equal to ψ, i.e. it is independent of NTU. On the other hand, with increase in NTU the heat transfer

between the hot fluid and the wall and that between the wall and cold fluid increases and decreases, respectively.

Therefore at low NTU internal heat generation has greater influence on the cold fluid effectiveness than axial heat

conduction. Thus the effectiveness of the cold fluid increases with NTU for low values of NTU. It can also be

observed that the ε-NTU curves at low NTU are very similar to that observed in the previous work of Mathew and

Hegab [10] which considered just the effect of uniform heat flux on the effectiveness of the fluids. At high values of

NTU there is reduction in the heat transfer between the wall and the cold fluid and this brings down the

effectiveness as seen in Fig. 7. For the hot fluid the effectiveness continuously increases over the NTU range of 0 to

5 because of the effect of internal heat generation is counteracted by the increase in heat transfer between the hot

fluid and the wall associated with increase in NTU.

V. Conclusion

This paper deals with the effect of axial heat conduction and internal heat generation on the effectiveness of a

parallel flow microchannel heat exchanger. The effectiveness of the fluids is found to depend not only on NTU but

also on internal heat generation parameter, axial heat conduction parameter and the temperature of the end walls.

With increase in axial heat conduction parameter, the effectiveness of the hot fluid of a parallel flow microchannel

heat exchanger with internal heat generation increased whenever the temperature of the end wall at the inlet side is

higher than that at the outlet side. For the same parallel flow microchannel heat exchanger the cold fluid

effectiveness decreased with increase in axial heat conduction parameter. The effect of internal heat generation in a

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3 4 5)TU

Qco

nd

,net

λ = 0.05, ψ = 0.15

λ = 0.10, ψ = 0.15

λ = 0.25, ψ = 0.15

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5)TU

ε

cold fluid

hot fluid

λ = 0.00, ψ = 0.00

λ = 0.10, ψ = 0.00

λ = 0.10, ψ = 0.05

λ = 0.10, ψ = 0.10

λ = 0.10, ψ = 0.15

λ = 0.10, ψ = 0.20


11

parallel flow microchannel heat exchanger with axial heat conduction is to always degrade the effectiveness of the

hot fluid while improving that of the cold fluid.

References 1Rosa, P., Karayiannis, and T. G., Collins, M. W., “Single Phase Heat Transfer in Microchannels: The Importance of Scaling

Effects,” Applied Thermal Engineering, Vol. 29, No. 17-18, 2009, pp. 3447-3468. 2Kutter, J. P., and Fintschenko, Y., 2005, Separation Methods in Microanalytical Systems, CRC Press, Boca Raton, FL, 2005. 3 Ashman. S., and Kandlikar, S. G., “A Review of Manufacturing Processes for Microchannel Heat Exchanger Fabrication,”

Fourth International Conference on anochannels, Microchannels and Minichannels, Limerick, Ireland, 2006, pp. 19-21. 4Choi, U. S., Rogers, C. S., and Mills, D. M., “High Performance Microchannel Heat Exchanger for Cooling High Heat Load

X-Ray Optical Components,” Winter Annual Meeting of ASME, 1992, pp. 83-89. 5Park, H., “A Microchannel Heat Exchanger Design for Microelectronics Cooling Correlating the Heat Transfer Rate in

Terms of Brinkman Number,” Microsystem Technologies, Vol. 15, No. 9, 2009, pp. 1373-1378. 6Shah, R. K., and Sekulic, D. P., Fundamentals of Heat Exchanger Design, John Wiley and Sons, New Jersey, 2003. 7Ranganayakulu, Ch., Seetharamu, K. N., and Sreevatsan, K. V., “The Effects of Longitudinal Heat Conduction in Compact

Plate-Fin and Tube-Fin Heat Exchanger Using a Finite Element Method,” International Journal of Heat and Mass Transfer, Vol.

40, No. 6, 1997, pp. 1261-1277. 8Deshmukh, S. R., and Vlachos, D. G., “Effect of Flow Configuration on the Operation of Coupled Combustor/Reformer

Microdevices for Hydrogen Production,” Chemical Engineering and Science, Vol. 60, No. 21, 2005, pp. 5718-5728. 9 Mathew, B., and Hegab, H., “Effectiveness of Parallel Flow Microchannel Heat Exchangers with External Heat Transfer

and Internal Heat Generation,” 2008 ASME Summer Heat Transfer Conference, Jacksonville, FL, 2008, pp. 175-184. 10Mathew, B., and Hegab, H., “Application of Effectiveness-NTU Relationship to Parallel Flow Microchannel Heat

Exchangers Subjected to External Heat Transfer,” International Journal of Thermal Science,

doi:10.1016/j.ijthermalsci.2009.06.014. 11Mathew, B., and Hegab, H., “Axial Heat Conduction in Parallel Flow Microchannel Heat Exchanger,” 2009 ASME

International Mechanical Engineering Congress and Exposition, Lake Buena Vista, FL, 2009. 12Chiou, J. P., “Effect of Longitudinal Heat Conduction on Crossflow Heat Exchanger,” Journal of Heat Transfer, Vol. 100,

No. 2, 1978, pp. 346-351. 13Venkatarathnam, G., and Narayanan, S. P., “Performance of a Counter Flow Microchannel Heat Exchanger with

Longitudinal Heat Conduction through the Wall Separating the Fluid Streams from the Environment”, Cryogenics, Vol.39, No.

10, 1999, pp. 811-819. 14White, F. M., Viscous Fluid Flow, McGraw Hill, New York, 2006. 15Morini, G. L., “Viscous Heating in Liquid Flows in Micro-Channels,” 2005, International Journal of Heat and Mass

Transfer, Vol. 48, No. 17, 2005, pp. 3637-3647. 16Burden, R. L., and Faires, J. D., umerical Analysis, Thomson Brooks/Cole, USA, 2005. 17Kandlikar, S. G., and Grande, W. J., “Evolution of Microchannel Flow Passages – Thermohydraulic Performance and

Fabrication Technology,” Heat Transfer Engineering, Vol. 24, No. 1, 2003, pp. 3-17.

Documents

[American Institute of Aeronautics and Astronautics 10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference - Chicago, Illinois ()] 10th AIAA/ASME Joint Thermophysics and Heat