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Effect of Axial Heat Conduction and Internal Heat

Generation on the Effectiveness of Counter Flow

Microchannel Heat Exchangers

B. Mathew1 and H. Hegab

2

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

The influence of axial heat conduction and internal heat generation (viscous dissipation)

on the effectiveness of a counter flow microchannel heat exchanger is analyzed in this paper.

The ends of the wall separating the hot and cold fluid are kept isothermal, i.e. non-adiabatic,

thereby leading to thermal interaction between the heat exchanger and its surroundings. A

thermal model of this particular heat exchanger consists of three one dimensional governing

equations. This system of equations is solved using finite difference method. The hot and

cold fluid effectiveness is found to depend on parameters such as NTU, axial heat conduction

parameter, end wall temperatures and internal heat generation parameter. Increase in axial

heat conduction parameter of a heat exchanger subjected internal heat generation can either

increase or decrease the effectiveness of the fluids depending on the temperature of the end

walls. The effect of internal heat generation in a counter flow microchannel heat exchanger

with axial heat conduction is to always degrade and improve the effectiveness of the hot and

cold fluid, respectively.

Nomenclature

A = area

C = heat capacity

Cp = specific heat

h = heat transfer coefficient

L = length

MCHX = microchannel heat exchanger

NTU = number of transfer units

P = pressure

p = perimeter

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

ε = effectiveness

θ = nondimensional temperature

µ = viscosity

ψ = internal heat generation parameter

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

71272; Email: bma017@latech.edu, Ph: +1-318-514-9618 2Associate Professor, Mechanical Engineering Program, College of Engineering and Science, LA Tech, Ruston, LA

71272; Email: hhegab@latech.edu, 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 2010-4767

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

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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 fluidic devices that employ channels with hydraulic diameter

smaller than 1 mm [1]. The advantages of using such small channels in a MCHX include 1) enhanced heat

transfer coefficient, 2) increased surface area density, and 3) compactness [2]. Present day MCHXs can be made

using materials such as silicon, glass, alumina, silicon carbide, and PDMS [3]. The fabrication method depends on

the material used for fabricating the MCHX. Some of the commonly used fabrication methods include dry and wet

chemical etching, mechanical micromachining, laser etching, hot embossing, and LIGA [4]. Depending on the

fabrication method the channel profile can be square, rectangular, circular, triangular or trapezoidal. In certain

situations, in order to achieve high through put, a layer of hot and cold fluid channels are alternately stacked over

one other.

The current design of MCHX is done using the conventional ε-NTU relationship [5]. However, these equations

are based on assumptions which may not always be valid at the microscale [1]. Thus the validity of such

assumptions at the microscale has to be checked before using the existing equations for design purposes. Two

assumptions, among the several assumptions that are made for formulating the conventional ε-NTU equations, need

to be reevaluated. These are 1) the insignificance of axial heat conduction, and 2) absence of internal heat

generation. Regarding axial heat conduction, even though neglected in the original formulation of the ε-NTU

relationships its influence on the effectiveness of the fluids in macro and microscale heat exchangers have been

extensively studied by several researchers in the past [6, 7]. Unlike in a parallel flow heat exchanger, axial heat

conduction exists in a counter flow heat exchanger because of the temperature gradient existing in the wall

separating the fluids [7]. Moreover, due to the short length of the MCHX the temperature gradient in it would be

higher than that in a macroscale heat exchanger under the same heat duty. Thus axial heat conduction is a more

severe problem in MCHX than in its conventional counterpart. In addition, due to the fact that the ends of the wall

separating the fluids are either part of the manifold or the substrate in which it is fabricated. Thus there is always

thermal interaction between the MCHX and its surroundings which is absent in a macroscale heat exchanger.

Therefore this additional heat transfer path aggravates the existing problem of axial heat conduction.

Regarding internal heat generation, it is the heat generated in the channels due to the conversion of flow work

into heat [8]. Flow work is the work needed to push a certain quantity of liquid against the frictional force (assuming

that no other forces act on the fluid under consideration) existing between the fluid and the wall. It is a function of

volumetric flow rate and the pressure drop between the inlet and outlet of the channel [8]. Pressure drop, for a

specific volumetric flow rate, is inversely proportional to the hydraulic diameter of the channel. For flow in

microchannels the flow work is not a negligible quantity because the pressure drop is significantly high in these

channels for a specific flow rate. On the other hand, in macroscale channels flow work is small due to the fact that

channels are big. Thus it is important to consider the effect of the conversion of flow work into heat while designing

MCHXs. The heat generated inside the channels due to the conversion of mechanical work into heat is also referred

to as viscous dissipation in literature [9].

This paper analyzes of the thermal performance of a counter flow MCHX (MCHXCF) subjected to axial heat

conduction (coupled with heat transfer) and internal heat generation. Counter flow arrangement has the best thermal

performance among the three flow arrangements and thus it is studied in this paper.

II. Literature Review

This section is dedicated to the review of several articles that deal with topics similar to that being analyzed in

this paper. Some of the articles included in this section deal with the individual effect of axial heat conduction and

internal heat generation and the few others are on the combined influence of these two effects. The articles dealt

with in this section are solely on two fluid heat exchangers. Mathew and Hegab [10] theoretically analyzed the effect

of axial heat conduction in a MCHXCF in which the ends of the wall separating the fluids is in thermal contact with

its surroundings. The end walls were maintained at constant temperatures. Analytical equations for estimating the

M

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temperature of the fluids and the wall are developed as part of this work. The effectiveness of the fluids is found to

depend also on axial heat conduction, and end wall temperatures in addition to its dependence on NTU. Using this

model they studied the effectiveness of the fluids for a few different cases. The effectiveness of the fluids increased

with increase in axial heat conduction parameter when the temperature at the end wall at the hot and cold fluid side

are kept at the inlet temperature of the hot and cold fluid, respectively. On the other hand, the effectiveness of hot

fluid increased while that of the cold fluid decreased when the temperature of the end wall at the inlet side of the hot

fluid is reduced 20% below the inlet temperature of the hot fluid without changing the temperature of the other end

wall. The influence of axial heat conduction on the effectiveness of the fluids depends on the end wall temperatures

unlike in a MCHXCF with insulated end walls. Mathew and Hegab [11] also studied the effect of external heat

transfer and internal heat generation on the effectiveness of the fluids of a MCHXCF. The external heat transfer

refers to the heat transfer between the fluids and an external heat source which in this paper is the ambient. The

effectiveness is determined to depend on NTU, internal heat generation parameter, ambient temperatures, and the

thermal resistance between the ambient and the fluids. In the presence of internal heat generation the hot and cold

fluid effectiveness decreased and increased, respectively. On the other hand, the sole effect of external heat transfer

can either increase or decrease the effectiveness of the fluids depending on the ambient temperature. If the ambient

temperature is lower than that inlet temperature of the fluids then the effectiveness of the hot and cold fluid will

increase and decrease, respectively. The opposite trend in the effectiveness of the fluids is observed if the ambient

temperatures are above the inlet temperature of the fluids. The presence of external heating of the fluids of a

MCHXCF subjected to internal heat generation will aggravate the degradation of the hot fluid effectiveness while

improving that of the cold fluid. On the other hand, external cooling will reduce the negative impact of internal heat

generation on the effectiveness of the hot fluid. However, external cooling will definitely reduce the improvement

brought about by internal heat generation on the effectiveness of the cold fluid. Mathew and co-researchers [12, 13]

considered the influence of axial heat conduction and external heat transfer on the effectiveness of the fluids of a

MCHXCF. In [12] the effect of external heat flux on the effectiveness is studied while in [13] the influence of

external heat transfer with the ambient is analyzed. The ends of the wall separating the MCHXCF are taken to be

insulated in both these studies. For both the cases the effectiveness of the fluids is found to depend on NTU, axial

heat conduction parameter. For the study in [12] the effectiveness of the fluids is observed to depend on the

nondimensional external heat transfer parameter along with the above mentioned parameters. While, in addition to

NTU and axial heat conduction parameter, the effectiveness of the fluids considered in [13] depends on ambient

temperature and the thermal resistance between the ambient and the fluids. The presence of the axial heat

conduction alone brings about an equal reduction in the effectiveness of the fluids. However, with increase in

external heat transfer the effectiveness of the fluids can either increase or decrease depending on the whether heat is

being gained or lost by the individual fluids. Recently Mathew and Hegab [14] conducted theoretical investigation

on MCHXCF subjected to external heat flux. Analytical equations were developed for both balanced and unbalanced

flow conditions of the MCHXCF. In the presence of external heating the effectiveness of the hot and cold fluid

decreased and increased, respectively. The trends in effectiveness of the fluids are reversed in the presence of

external cooling. Under unbalanced flow conditions, the effectiveness of the fluids is found to depend on the fluid

with the lowest heat capacity. On comparing the two unbalanced flow conditions, the effectiveness of fluids is better

when the hot fluid has the lowest heat capacity among the two fluids. They introduced a new concept called

performance factor for accessing the relative change in effectiveness of the fluids in the presence of external heat

transfer. From the few articles reviewed here it is clear that there have not been any previous attempts at analyzing

the effect of axial heat conduction and internal heat generation on the thermal performance of a MCHXCF. Therefore

this is dealt with in this paper.

III. Theoretical Model

This section deals with the development of the thermal model of the MCHXCF described in the previous section.

Figure 1 represents a schematic of the differential element of such a MCHXCF. Three governing equations make up

the thermal model. Each of the governing equations represents one of the entities (hot fluid, cold fluid and wall) of

the MCHXCF. These equations are formulated by considering these entities as a continuum. Therefore the equations

and the results provided in this paper are suitable even for a macroscale (conventional) heat exchanger. However,

most often the scenario presented in this paper will occur only in MCHXs. In order to ease the development of the

thermal model few assumptions are made and these are provided below.

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0wθ

)( cwsc dAh θθ −

+ Xd

Xd

dC h

hh

θθ

+ Xd

Xd

dC c

cc

θθccC θ

hhCθ

1wθ

)( whshdAh θθ − Pν

Pν

+ XdXd

dPPν

+ XdXd

dPPν

X Xd

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

this study.

1. The MCHXCF is

operating under

steady state

conditions, i.e. fluid

temperatures are

independent of time.

2. The temperature of

each fluid at every

cross section is

considered to be a

function of just the

axial coordinate.

3. Both the fluids are

assumed to be fully

developed

(hydrodynamically and thermally) at the entrance of the MCHXCF.

4. Among the several forces that can possibly act on the fluid volume under consideration, only frictional force is

assumed to exist in this study.

5. The effect of transverse thermal resistance of the wall separating the fluids on the overall heat transfer

coefficient is considered inconsequential due to the negligible thickness of this wall.

6. Neither the hot nor the cold fluid undergoes phase change in the MCHXCF.

7. Velocity of the fluids on the microchannel wall is assumed to be zero, i.e. slip does not exist.

8. Influence of effects such as external heating/cooling, non-uniform flow distribution and longitudinal heat

conduction (in the fluids) on the temperature of the fluids are neglected.

9. The thermophysical properties of the fluids are assumed constant between the inlet and outlet of the MCHXCF.

Based on these assumptions three governing equations are developed for the MCHXCF studied in this paper. The

governing equations are developed by simplifying the energy equation of the fluids and the wall. The set of

governing equations is shown in Eq. (1), Eq. (2) and Eq. (3). Equation (1) represents the governing equation of the

hot fluid while Eq. (2) is that of the cold fluid and Eq. 3 is the governing equation of the wall. Regarding Eq. (1) and

Eq. (2), the first term in these equations represent the axial variation of the fluid temperature. The second term in

these equations accounts for the local heat transfer between the individual fluid and the wall. The term on the right

hand side of these equations considers the effect of heat generated internally per unit length of the microchannel, i.e.

over every cross section of the microchannel, on the local temperature of the fluids. Regarding the third governing

equation, the first term represents the axial variation in heat conduction through the wall separating the fluids. The

second term of this equation represents the local heat transfer between the hot fluid and the wall. The third term in

Eq. 3 accounts for local heat transfer between the wall and the cold fluid.

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)

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It is mentioned earlier that the second term of Eq. (1) and Eq. (2) represents the local heat transfer between the

individual fluids and the wall. The coefficient of these terms is the ratio of the convective thermal resistance per unit

length to the heat capacity of the fluid. Once these equations are normalized the coefficient of the second term on the

right hand side of Eq. (1) and (2) represent the NTU associated with the hot and cold fluid, respectively [14, 15].

The NTU associated with the hot and cold fluid are represented as NTUh and NTUc, respectively [14, 15]. In the

conventional ε-NTU equations, the term NTU represents the ratio of the overall heat transfer coefficient to the

minimum heat capacity [5]. In order to make sure that the terms used in this thermal model is same as that used in

the ε-NTU equation it is important to find a relationship between these NTU terms. For most microfluidic

applications the flow regime is laminar and thus the heat transfer coefficient is a function of Nusselt number,

hydraulic diameter of the channel and the thermal conductivity of the fluid. Therefore, as the channels associated

with the hot and cold fluid are equal in size the heat transfer coefficient associated with the both fluids are equal.

Moreover when the MCHXCF is operated under balanced flow conditions the heat capacities of both fluids are equal.

Thus based on these two conditions it is possible to write the NTU associated with each of the fluids in terms of the

NTU based on the overall heat transfer coefficient. This relationship is shown in Eq. (4).

NTUNTUNTUC

hA

hA

C

Ah

C

Ah

C

NTUNTUNTUch

s

ssc

c

sh

h

ch

22111

min

min ===⇒=+=+= (4)

It is also mentioned in the previous section that internal heat generation considered in this paper is same as

viscous dissipation. In the following section the equivalency between internally generated heat due to flow work and

viscous dissipation function (Φ) in the standard energy equation of any Newtonian fluid is proved. When the flow is

fully developed the velocity in the y- and z- direction are zero and the velocity in the x-direction is function just the

y- and z- coordinate. Using this information the viscous dissipation function which is part of the 3D energy equation

of a fluid can be written as shown in Eq. (5) [16]:

( )

∂+

∂=∇⋅∇=

22

dz

u

dy

uuu µµφ (5)

Morini [17] estimated the viscous dissipation function to be the product of friction factor for internal flows.

Using the identity provided by Morini [17], i.e. Eq. (6), it is possible to include the effect of axial pressure gradient

in Eq. (5). The momentum equation in the x-direction for fully developed flow can be written as shown in Eq. (7).

The simplified momentum equation is used for rewriting the first term on the right hand side of Eq. (6) as shown in

Eq. (8).

222 5.0 uuuuu ∇+∇−=∇⋅∇ (6)

dx

dPu

µ12 =∇ (7)

( )

∇+−=∇+∇−=⇒

25.0

22222 u

dx

dPuuuu

µµµφ (8)

The pressure gradient in Eq. (8) is function of just the axial coordinate. Equation (8) represents the viscous

dissipation function at an arbitrary point within the volume of the fluid contained in the microchannel. This is

because the viscous dissipation function is developed from the 3D energy equation of the fluid. However, it can be

seen that Eq. (1), Eq. (2) and Eq. (3) are one dimensional. Therefore it is necessary to calculate the viscous

dissipation function that is averaged over the cross section of the microchannel for use in Eq. (1) and Eq. (2). The

mathematical technique of averaging the viscous dissipation function over the cross section of the microchannel is

shown in Eq. (9).

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( ) dydzu

dydzdx

dPudydzuudydz

AAAA

∫∫∫∫∫∫∫∫∇

+−=∇⋅∇=2

22

µµφ (9)

The second term on the right hand side of Eq. (9) is calculated to be zero by Morini [17]. Using this particular

simplification provided by Morini [17] the value of the viscous dissipation function averaged over the cross section

can be equated to the product of volumetric flow rate and pressure gradient as shown in Eq. (10).

dx

dPdxdy

dx

dpdxdy

AA

νφ −=−= ∫∫∫∫ (10)

dydz∫∫φ is the viscous dissipation function per unit length of a microchannel. In Eq. (10) it has been proved

that the average value of viscous dissipation function is numerically equal to the product of volumetric flow rate and

axial pressure gradient. This is the quantity that appears on the right hand side of Eq. (1) and Eq. (2). Moreover, for

fully developed flow the average value of viscous dissipation function is a constant since volumetric flow rate and

axial pressure gradient are both constants. In addition, the product of volumetric flow rate and axial pressure

gradient represents the pumping power per unit length and thus the viscous dissipation averaged over the area is

equal to pumping power per unit length.

In a MCHXCF the hot and cold fluid flows in opposite direction. In this paper the hot fluid is taken to flow in the

positive x-direction. Thus the axial velocity, u, is positive for the hot fluid. On the other hand, the axial velocity is

negative for the cold fluid since it is flowing in the negative x-direction. Equations (5) – (10) are formulated for

positive values of axial velocity. Therefore Eq. (10) is valid only for the hot fluid. The pressure gradient is negative

for the hot fluid. On the other hand the pressure gradient is positive for the cold fluid since the flow is in the

negative x-direction, i.e. the axial velocity is negative for the cold fluid. Thus the equivalent of Eq. (10) for the cold

fluid is provided in Eq. (11).

dx

dPdxdy

A

νφ =∫∫ (11)

Equations (1) – (3) are rewritten using the formulation provided in Eq. (4). In addition certain terms are used for

nondimensionalizing these equations. These terms are provided below. X is the nondimensional axial coordinate. λ

represents the axial heat conduction parameter. This term is the ratio calorific thermal resistance to the axial

conduction thermal resistance. Ψ is the internal heat generation parameter. The pumping power is

nondimensionalized with respect to the maximum heat transfer possible in a MCHXCF. The nondimensional

governing equations of the thermal model are provided in Eq. (12), Eq. (13) and Eq. (14).

L

xX = ,

LC

Ak crw

min

=λ , )(min cihi TTC

Ldx

dP

−

−=

νψ

Hot fluid:

ψθθθ

=−+ )(2 whh NTU

Xd

d

(12)

Cold fluid:

ψθθθ

−=−+ )(2 wcc NTU

Xd

d

(13)

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Wall:

02242

2

=++

− chw NTUNTUNTU

Xd

dθθθλ

(14)

The governing equations represent a system of equations that need to be solved simultaneously. In order to solve

these equations four boundary conditions are needed. One boundary condition is needed per fluid since the

governing equation of the fluids is first order. The boundary condition associated with the hot and cold fluid

governing equation is the inlet temperature. The inlet temperature of these fluids is known a priori and they are used

as the boundary condition. Equations (15) and (16) provide the boundary condition of the hot and cold fluid,

respectively.

10=

=Xhθ (15)

01=

=Xcθ (16)

The governing equation of the wall is second order and thus two boundary conditions are needed. The boundary

conditions consist of the temperature at the ends of the wall separating the fluids. Such boundary conditions are

assumed because these end walls are either part of the manifold or the substrate in which the MCHXCF is fabricated.

The boundary conditions of the wall are presented in Eq. (17) and Eq. (18).

inwXw ,0θθ =

= (17)

outwXw ,1θθ =

= (18)

The governing equations subjected to the above mentioned boundary conditions are numerically solved using

finite difference method. The computational domain consists of the hot fluid, cold fluid and the wall. The number of

nodes in each of the entities that make up the computational domain is kept same. The total number of nodes in the

computational domain is 3n+3 since the nodes in each of the entity is n+1. However, the temperature at 4 nodes is

known beforehand and thus the temperature at only 3n-1 nodes need be determined. The first order differential term

in Eq. (12) and Eq. (13) is discretized using central difference scheme for all interior nodes. For the hot fluid, the

differential term at its exit is discretized using backward different scheme. On the other hand, this term for the cold

fluid at its exit is discretized using forward difference scheme. The error associated with the discretization of the

first order term, irrespective of the scheme, is second order in nature. The second order term of Eq. (14) is

discretized using central difference scheme. For the wall the temperature at the inlet and exit section is not

calculated since they are already known. Only the temperature at the interior nodes is determined. The error

associated with this scheme is also second order in nature. For obtaining results that are independent of the grid size,

calculations are performed for three different node lengths. For each set of iterations the node length is reduced by

half. For all calculations performed in this paper the first iteration is performed at a node length of 1×10-3

. The

second and third set iterations are performed with node length of 5×10-4

and 2.5×10-4

. The temperature of the fluids

obtained from the third set of iterations is used for calculating the effectiveness and heat transfer between the fluids.

The effectiveness of the fluids can be determined once the temperatures of the fluids are numerically determined.

For calculating the effectiveness only the outlet temperature of the fluid is required. The inlet temperature of the

fluids is not needed because of the unique way in which the temperatures are nondimensionalized. The equations

needed for determining the effectiveness are provided in Eqs. (19) and (20).

1,,min

,,1

)(

)(=

−=−

−=

Xh

icih

ohihh

hTTC

TTCθε (19)

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0,,min

,,

)(

)(=

=−

−=

Xc

icih

icocc

cTTC

TTCθε (20)

Using the local temperatures of the fluids it is possible to determine the heat transfer between the individual

fluids and the wall. It is also possible to determine the heat transfer between the MCHXCF and its surroundings

through the end walls. The general equation for determining the heat transfer between the hot fluid and the wall is

provided in Eq. (21). The temperature of the hot fluid and wall obtained using node length of 2.5×10-4

is used for

calculating the heat transfer between the hot fluid and the wall.

∫ ∫ −==1

0

1

0

,, )(2 XdNTUdQQ whwhwh θθ (21)

The integration in Eq. (21) can be performed using any numerical quadrature technique. Composite Simpsons

Method is used for calculating the integral of the difference in the temperature of the hot fluid and the wall [18]. The

error associated with this method is O(4

X∆ ). An equation similar to that presented in Eq. (21) is used for

determining the heat transfer between the wall and the cold fluid. Eq. (22) represents the general equation for

determining the heat transfer between the wall and the cold fluid.

∫ ∫ −==1

0

1

0

,, )(2 XdNTUdQQ cwcwcw θθ (22)

It is also possible to quantify the thermal interaction between the MCHXCF and its surroundings through the end

walls. The heat transfer through the end walls can be determined using one dimensional Fourier’s heat conduction

equation. The equation necessary for determining the heat transfer between the MCHXCF and its surroundings

through the end wall at the inlet of the hot fluid is shown in Eq. (23).

0

0,

=

−=X

wcond

Xd

dQ

θλ (23)

The wall temperature gradient at this section, i.e. at the inlet of the hot fluid, can be determined using numerical

techniques. The three point formula is used for determining this temperature gradient. Three points inside the

computational domain are used for this calculation. The error associated with this method is O(3

X∆ ). Equation

(24) is used for determining the heat transfer between the MCHXCF and its surroundings through the end wall at the

entrance of the cold fluid. Positive values of Qcond,0 represent that heat transfer is from the surroundings to the

MCHXCF and vice versa. While positive values of Qcond,1 represent that the heat transfer is from the MCHXCF to its

surroundings and vice versa.

1

1,

=

−=X

wcond

Xd

dQ

θλ (24)

It is also important to calculate the net heat accumulated/lost from the MCHXCF due to the heat transfer between

end walls and the surroundings. Eq. (25) represents the net heat accumulated/lost from the MCHXCF. If Qcond,net is

positive it represents that the heat transferred out of the MCHXCF through the end wall at the inlet section of the cold

fluid is greater than that entering the MCHXCF through the one at the inlet section of the hot fluid. Qcond,net is

negative if the heat entering the MCHXCF through the end wall at the hot fluid inlet section is greater than that

leaving through the other end wall.

0,1,, condcondnetcond QQQ −= (25)

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-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5NTU

ε

λ= ψ = 0.0 λ= 0.0, ψ = 0.1 λ = 0.1, ψ = 0.0

Conventional ε-NTU relationship [5] (λ = 0.0, ψ = 0.0)

Mathew and Hegab [13] (Qh = Qc = 0.1)

Mathew and Hegab [10] (λ = 0.1, θw,0 = 1.2, θw,1 = 0.1)

cold fluid

hot fluid

Figure 2. ε-NTU relationship of a balanced flow

MCHXPF (θw,0 = 1.2 and θw,1 = 0.1).

On inspecting the governing equations, i.e. Eq. (12), Eq. (13) and Eq. (14); the influence of the shape of the

channel is not directly evident. The effect of channel shape is influences the heat transfer coefficient, surface area,

and pressure drop. These parameters are in turn used in determining parameters such as NTU and internal heat

generation parameter. Thus the shape of the channel and mass flow rate decides NTU and internal heat generation

parameter. However, it is has to be understood that for a particular NTU and internal heat generation parameter the

effectiveness is independent of the channel shape.

The equations developed in this paper assume the three entities to be continuum. This puts certain limitation on

the applicability of this model in terms of the hydraulic diameter of the microchannel. With reduction in channel size

certain assumptions that are fundamental to the field of continuum mechanics becomes questionable. One such

assumption is that the flow velocity on the wall is zero. In addition, the phenomenon of rarefaction in gases is not

accounted for in the theories formulated based on the principles of continuum mechanics. An easy way of checking

the validity of the models developed using ideas of continuum mechanics in the microscale is by using the criterion

based on Knudesen number (kn). If kn < 10-3

the theories based on continuum mechanics is valid. Based on this

criterion when air is used as the fluid the model developed in this paper is valid only when the hydraulic diameter of

the channel is greater than 70 micrometers. Similarly, if hydrogen is used as the fluid then the theories developed in

this paper is not valid for channels with hydraulic diameter smaller than 125 micrometers. For liquids, the existence

of no-slip flow condition has been experimentally validated even for channels with dimensions as small as 500 µm

by 5 µm. Thus the theory developed in this model is valid for most practical situations when liquids are used.

However in the cases of gases the lower limit on the hydraulic diameter depends on the type of gases. So care has to

be taken especially when gases are used.

IV. Results and Discussions

In this section the thermal model developed in the

previous section is used to study the effect of axial heat

conduction and internal heat generation on the

effectiveness of the fluids. The parameters in the thermal

model consist of NTU, axial heat conduction parameter

and internal heat generation parameter. The effect of

each of these parameters is studied in this section. Prior

to using this model for performing the above mentioned

study it is important to validate the model against some

known model. The validation of this model is performed

using the conventional ε-NTU relationship, and two

other models previously developed by Mathew and

Hegab [10, 13]. Figure 2 represents the comparison

between the present model and these models. For

comparing the present model and the conventional ε-

NTU relationship the parameters representing the effect

of axial heat conduction and internal heat generation, i.e.

λ and ψ, are set to zero. The end wall temperatures are

kept at θw,0 = 1.2 and θw,1 = 0.1. The solid line represents the results calculated from the current model and the data

points represent the effectiveness as obtained from the conventional ε-NTU relationship. There is good match

between the two models. This figure also shows the comparison between the model developed in this paper with the

previous model of Mathew and Hegab [10] that deals with axial heat conduction coupled with heat transfer. For this

comparison the internal heat generation parameter is set to zero. The axial heat conduction parameter is kept at 0.1

and the end walls temperatures are maintained at 1.0 (hot fluid inlet section) and 0.0 (cold fluid inlet section). The

solid line corresponds to the effectiveness of the fluids calculated from the present model and the data points

represent the effectiveness of the fluids determined from the model developed by Mathew and Hegab [10]. The third

comparison is between the present model and the model developed by Mathew and Hegab [13] of a MCHXCF

subjected to external heat flux. In the governing equations, Eq. (1) and Eq. (2), the term on the right hand side

represents the internal heat generation per unit length. This term is independent of the axial distance, x. In the model

previously developed by Mathew and Hegab [13] the fluids of the MCHXCF are subjected to constant axial heat

transfer. Thus these two models are mathematically equivalent when axial heat conduction is eliminated in the

model developed in this paper. In order to compare the two models the internal heat generation parameter and

external heat transfer parameters of [13] are kept at 0.15. Results of both the models are same indicating the match

American Institute of Aeronautics and Astronautics

10

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5NTU

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

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5NTU

Qco

nd

,net

λ = 0.05, ψ = 0.15

λ = 0.10, ψ = 0.15

λ = 0.25, ψ = 0.15

Figure 3. ε-NTU relationship of a balanced flow

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

Figure 4. Qcond,net of a balanced flow MCHXPF (ψ

= 0.15, θw,0 = 1.2 and θw,1 = 0.1).

between them. These comparisons validate the present model.

The effect of axial heat conduction in a MCHXCF that is subjected to internal heat generation is analyzed in Fig.

3. The axial heat conduction parameters are varied between 0.05 and 0.25. This means that the axial heat conduction

thermal resistance is between 5% and 2.5% of the

calorific resistance of the fluid under study. The internal

heat generation parameter is maintained at 0.15 implying

that the pumping power is 15% of the maximum heat that

can be transferred in a MCHXCF. The temperature of the

end wall (θw,0) at the inlet section is maintained at 20%

higher than the inlet temperature of the hot fluid. The

other end wall is maintained at a much lower

temperature, i.e. it is maintained at a temperature that is

90% lower than the inlet temperature of the hot fluid.

From this figure it can be seen that the effect of axial

heat conduction in a MCHXCF already subjected to

internal heat generation is to increase the effectiveness of

the fluids. The effectiveness of the fluids increased with

increase in axial heat conduction parameter. The

effectiveness of the hot fluid increased because of the

fact that the heat it transferred to the wall increased with

axial heat conduction in comparison with that in a

MCHXCF with just internal heat generation. With

increase in axial heat conduction parameter there is

further increase in the heat transferred to the wall for all

values of NTU. In addition at high values of NTU the net

heat conducted out of the MCHXCF is observed to be

positive indicating that more heat is conducted out of the

MCHXCF than that is conducted into it. This additional

heat that is conducted out of the MCHXCF has to have

originated from the hot fluid. This is shown in Fig. 4.

With regard to the cold fluid, its effectiveness increased

in the presence of axial heat conduction for all values of

NTU. This is because in the presence of axial heat

conduction the heat transferred from the wall to the cold

fluid increased over the NTU range shown in Fig. 3. The

heat transferred between the wall and the cold fluid

increased with increase in axial heat conduction

parameter. In addition the heat between the MCHXCF

and its surroundings also helped improved the cold fluid

effectiveness at low NTU values. From Fig. 4 it can be seen that Qcond,net for low values of NTU is negative. This

implies that heat from the surroundings accumulates in the MCHXCF. At high values of NTU the parameter Qcond,1 is

positive as seen from Fig. 4, however the increase in heat transfer between wall and the cold fluid due to increase in

NTU brought about the observed increase in its effectiveness. This additional heat that enters the MCHXCF is carried

away by the cold fluid thereby increasing its effectiveness. It can also be seen that for all values of NTU and axial

heat conduction parameter the effectiveness of the hot fluid is below that of the cold fluid. This is because of the

presence of internal heat generation. Internally generated heated, irrespective of axial heat conduction, degrades and

improved the effectiveness of the hot and cold, respectively. Thus the positive effect of axial heat conduction on the

effectiveness of the fluids is counter acted by internally generated heat and this is by the effectiveness of the hot

fluid is always below that of the cold fluid. The effectiveness of the hot fluid is numerically equal to the internal heat

generation parameter. For the hot fluid when NTU is zero there is no heat transfer with the cold fluid. This is either

due to very small heat transfer surface area or due to very high flow rate. Thus the thermal energy that is supplied to

it due to the conversion of pumping power into heat stays in it and rises its temperature. The total heat supplied to it

between the inlet and exit of the hot fluid is equal to the product of volumetric flow rate and total pressure which is

same as the pumping power. The nondimensionalized form of pumping power is equivalent to the internal heat

generation parameter. This also happens to be the hot fluid effectiveness due to the unique way in which the

American Institute of Aeronautics and Astronautics

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-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5NTU

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

Figure 5. ε-NTU relationship of a balanced flow

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

Figure 6. Qcond,net of a balanced flow MCHXCF (ψ

= 0.15, θw,0 = 0.8 and θw,1 = 0.2) Figure 7. ε-NTU relationship of a balanced flow

MCHXCF (λ= 0.1, θw,0 = 0.8 and θw,1 = 0.1)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5NTU

Qco

nd

,net

λ = 0.05, ψ = 0.15

λ = 0.10, ψ = 0.15

λ = 0.25, ψ = 0.15

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5NTU

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

nondimensionalization of internal heat generation is set up. Similarly, the effectiveness of the cold fluid when NTU

is zero is equal to the internal heat generation parameter.

Figure 5 also represents the case of a MCHXCF

subjected to axial heat conduction and internal heat

generation. The temperature of the end wall at the inlet

section of the hot fluid is taken to be 20% lower than hot

fluid inlet temperature. The temperature of the end wall

at the inlet section of the cold fluid is maintained at 20%

of the inlet temperature of the hot fluid. The internal heat

generation parameter is maintained at 0.15 and the axial

heat conduction parameters are varied between 0.05 and

0.25. The effectiveness of the hot fluid in a MCHXCF

subjected to internal heat generation is observed to

increase in the presence of axial heat conduction for all

values of NTU. On the other hand, the effectiveness of

the cold fluid of the same MCHXCF decreases in the

presence of axial heat conduction parameter. Increase in

axial heat conduction parameter at a particular NTU

enhances the improvement and degradation in the

effectiveness of the hot and cold fluid, respectively. This

is because in this particular case the heat transferred out of the MCHXCF through the end wall at the cold fluid inlet

section is greater than that entering the MCHXCF through the other end wall. This is shown in Fig. 6. This additional

heat lost from the MCHXCF comes from the hot fluid and this enhances its effectiveness. On the other hand, the heat

that should have been gained by the cold fluid is carried out due to axial heat conduction and this degrades its

effectiveness. With increase in axial heat conduction parameter the net heat transferred out of the MCHXCF (Qcond,net)

increases for a particular NTU and thus the observed increase and decrease in the effectiveness of the hot and cold

fluid, respectively. Even for this case the effectiveness of the hot fluid is equal to the negative of the internal heat

generation parameter when NTU is zero. Similar to that observed in the previous case the effectiveness of the cold

fluid when NTU is zero is equal to the internal heat generation parameter.

The effect of internal generation on the effectiveness of the fluids of a MCHXCF subjected to axial heat

conduction is studied in Fig. 7. The internal heat generation parameter is varied between 0.0 and 0.2. When ψ = 0.0

the effect of internal heat generation on the effectiveness of the fluids is neglected. On the other hand, when ψ = 0.2

the heat generated in each of the fluids from pumping power is kept at 20% of the maximum heat transfer possible in

a MCHXCF. The temperatures of the end walls at the inlet section of the hot and cold fluid are maintained at 0.8 and

0.1, respectively. The axial heat conduction parameter is kept at 0.1. The effectiveness of the hot fluid of a MCHXCF

subjected to axial heat conduction decreases in the presence of internal heat generation. This is because with internal

American Institute of Aeronautics and Astronautics

12

heat generation heat is being added to the hot fluid which in turn decreases its effectiveness. With regard to the cold

fluid, its effectiveness increases in the presence of internal heat generation. The reason for this is that the internally

generated heat tends to heat the cold fluid thereby bringing about a positive effect on its effectiveness. The

effectiveness of the fluids increased with increase in NTU irrespective of axial heat conduction and internal heat

generation. This happens because of the improvement in the heat transfer between the individual fluids and the wall

that is associated with increase in NTU. With increase in internal heat generation the effectiveness of the hot and

cold fluid decreased and increased, respectively. With increase in internally generated heat the heat added to each of

the fluids increase. This increase in the heat added to the fluids has a positive and negative effect on the

effectiveness of the hot and cold fluid, respectively.

From the several cases studied in this paper it is clear that the effect of axial heat conduction and internal heat

generation cannot be neglected in a MCHXCF. Using the model developed in this paper it is possible to account for

the effect of axial heat conduction and internal heat generation on effectiveness of the hot and cold fluid of a

MCHXCF.

V. Conclusion

The effect of axial heat conduction and internal heat generation on the thermal performance of a counter flow

microchannel heat exchanger is dealt with in this paper. The effectiveness of the fluids is found to depend on NTU,

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

cases of a counter flow microchannel heat exchanger subjected to axial heat conduction and internal heat generation

are analyzed in this paper for understanding the influence of these effects on its performance. The studies have

shown that the presence of axial heat conduction in a counter flow microchannel heat exchanger already subjected to

internal heat generation improves the effectiveness of the fluids depending on the temperature of the end walls. If

the temperature of the end wall at the inlet section of the hot fluid is greater than the inlet temperature of the hot

fluid then there is increase in the effectiveness of both the fluids. If the temperature at this end wall is below that of

the inlet temperature of the hot fluid then the effectiveness of hot and cold fluid in comparison with their

effectiveness in a heat exchanger with just internal heat generation increases and decreases, respectively. Increase in

internal heat generation in a counter flow microchannel heat exchanger that is subjected to axial heat conduction

always brings about the degradation and improved in the hot and cold fluid effectiveness, respectively.

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10, 1999, pp. 811-819. 16White, F. M., Viscous Fluid Flow, McGraw Hill, New York, 2006. 17Morini, G. L., “Viscous Heating in Liquid Flows in Micro-Channels,” 2005, International Journal of Heat and Mass

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