Conjugate heat transfer with rarefaction in parallel plates microchannel

Superlattices and Microstructures 60 (2013) 370–388

Contents lists available at SciVerse ScienceDirect

Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / s u p e r l a t t i c e s

Conjugate heat transfer with rarefaction inparallel plates microchannel

0749-6036/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.spmi.2013.05.016

⇑ Corresponding author. Tel.: +213 34 50 14 00; fax: +213 34 50 35 35.E-mail address: [email protected] (Y. Kabar).

Yassine Kabar a,⇑, Rachid Bessaïh b, Mourad Rebay c

a Université de Jijel, Laboratoire d’Energétique Appliquée et des Matériaux, Faculté des Sciences et de la Technologie, BP. 96, 18000Jijel, Algeriab Université Constantine I, Laboratoire d’Energétique Appliquée et de Pollution LEAP, Faculté des Sciences de la Technologie, Routede Ain el Bey, 25000 Constantine, Algeriac Université de Reims Champagne-Ardenne, GRESPI/Laboratoire de Thermomécanique, Faculté des Sciences, BP. 1039, 51687Reims, France

a r t i c l e i n f o

Article history:Received 19 February 2013Received in revised form 29 April 2013Accepted 8 May 2013Available online 22 May 2013

Keywords:Conjugate heat transferAxial conductionSlip-flowMicrochannel

a b s t r a c t

In this paper, we study numerically the effects of axial wall con-duction and rarefaction in parallel plates microchannel. The simul-taneously developing laminar flow with a constant heat flux (H2)boundary condition will also be considered. The finite volumemethod is used to solve the two-dimensional Navier–Stokes andenergy equations, with slip velocity and temperature jump bound-ary implemented at the fluid/solid interface. The results obtainedby our computer code are compared to the analytical results foundin the literature. For different Knudsen number Kn, thermal con-ductivity ratio K and dimensionless thickness E, the influence ofaxial conduction is demonstrated for Kn = 0, especially for largevalues of K and E. Concerning slip-flow, the effect of axial conduc-tion proved to be negligible for all values of K and E.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The use of micro-channels for removal heat is undertaking a great interest in various industrialfields, such as electronics, micro heat exchangers and bio-engineering. This type of cooling offers highperformances in heat transfer. However, to conceive and manufacture such channels, it is necessary tounderstand and characterize flows as well as heat transfer in micro-scale heat transfer problems. It isalso important to examine the interaction of convective heat transfer in the fluid with conduction in

http://crossmark.dyndns.org/dialog/?doi=10.1016/j.spmi.2013.05.016&domain=pdf

http://dx.doi.org/10.1016/j.spmi.2013.05.016

mailto:[email protected]

http://dx.doi.org/10.1016/j.spmi.2013.05.016

http://www.sciencedirect.com/science/journal/07496036

http://www.elsevier.com/locate/superlattices

Nomenclature

a coefficient in the discretization equationb source term in the discretization equationCp specific heat (J/kg K)Dh hydraulic diameter (m)E dimensionless thickness of the plateH height of microchannel (m)h heat transfer coefficient (W/m2 K)K thermal conductivity ratio K = ks/kf

k thermal conductivity (W/m K)Kn Knudsen number Kn = k/Dh

L length (m)Nu Nusselt numberNux local Nusselt numberP dimensionless pressurep pressure (N m�1)Pe Péclet number Pe = Pr�RePo Poiseuille number Po = f RePr Prandtl numberq0 heat flux (W/m2)q dimensionless heat flux at the interfaceq⁄ normalized heat flux q⁄ = q/qm at the interfaceRe Reynolds number Re = qumDh/lT temperature (K)U, V dimensionless velocityu, v velocity components in Cartesian coordinates (m s�1)X⁄ dimensionless axial coordinate X⁄ = x/PeDh

X+ dimensionless axial coordinate X+ = x/ReDh

x, y Cartesian coordinates (m)Y dimensionless coordinateb ratio of accommodation coefficientc heat capacity ratioD thickness of the plate (m)h dimensionless temperaturek molecular mean-free-path (m)l viscosity dynamic (kg/m s)q density (kg/m3)s shear stress (N/m2)rT thermal accommodation coefficient (rT = 1)rv momentum accommodation coefficient (rv = 1)/ independent variable

Subscriptsb Bulkf FluidN, S, E, W neighboring grid pointsin inletint interfacem means solidslip slipw wall

Y. Kabar et al. / Superlattices and Microstructures 60 (2013) 370–388 371

372 Y. Kabar et al. / Superlattices and Microstructures 60 (2013) 370–388

the walls of the microchannel. For example, the thickness of the wall becomes greater than thehydraulic diameter of the passage section of the fluid in some cases, the conjugate effects also becomemore significant. This phenomenon has received a special attention during the recently past years inthe modeling of heat transfer in micro-channels, and a large number of scientists worked on.

The work undertaken by Maranzana et al. [1] concerned the influence of axial conduction in thewalls of mini or micro-channels. The authors introduced a criterion in order to quantify the effectof axial conduction in the wall of micro-channels. Li et al. [2] undertook an experimental and numer-ical analysis on the heat transfer characterization of a liquid laminar flow of a micro-channel. Theyfound that the influence of the axial conduction is more pronounced for the small Reynolds numbersand the high thickness of the wall. Nonino et al. [3] analyzed the effects of axial heat conduction in thesolid walls of microchannels of circular cross-sections. They demonstrated the effects of wall thick-ness, thermal conductivity ratio and length of the microtube on the local Nusselt number. However,Rahimi and Mehryar [4] studied numerically conjugate heat transfer in microtube. The results showthat the axial heat conduction in the duct wall due to wall thermal conductivity and its thicknesscause a reduction of local Nusselt number at the entrance region, and also a deviation in the local Nus-selt number at the ending region of the microtube. Concerning Kabar et al. [5], they analyzed the heattransfer in the dynamic and thermal developing zone in a micro-channel, by taking into account theeffects of combined axial conduction in the fluid and the wall, and the viscous dissipation. They provedthat viscous heating of the fluid and the conduction effect in the wall can significantly influence theheat transfer in the micro-channel heat sink. Zhang et al. [6] numerically simulated the effects of wallaxial heat conduction in a conjugate heat transfer problem in simultaneously developing laminar flowand heat transfer in straight thick wall of circular tube, with constant outside wall temperature. Theyfound in this case that the heat transfer highly dependent on the thermal conductivities ratio. Re-cently, Avcı et al. [7] numerically investigated laminar conjugate heat transfer of a liquid flow inthe entrance region of a microtube. They examined in details the effects of thermal conductivity ratio,diameter ratio, channel length and viscous dissipation on the Nusselt number as well as on the tem-perature and the interface heat flux distribution and they found that the effects of the thickness of thewalls and the ratio of thermal conductivities are important for short microtube.

Since the dimensions of channels constituting miniaturized cooling systems mounted in electroniccomponents are of order of micron and the working fluid is a gas, the mean free path of molecules gasis the order of hydraulic diameter of these channels, and the rarefaction effect is not negligible. Theflow regime depends on the degree of the rarefaction, which is quantified by the Knudsen number.For small values of Kn (<0.01), the flow is considered as continuum flow. However, for0.01 < Kn < 0.1 the flow is in the slip-flow regime, for 0.1 < Kn < 10 the flow is in the transition flowregime, and for large values of Kn (>10) the flow is considered as free-molecular flow. In slip-flow re-gime, the conventional continuum momentum and energy equations can be used with slip velocityand temperature jump boundary conditions at the walls.

The convective heat transfer of slip-flow was studied few years ago. Larrodé et al. [8] showed, inslip-flow forced convection, that the effect of jump temperature at the wall which was neglected inthe previous studies of Barron et al. [9,10], is essential in the analysis of heat transfer, but Yu andAmeel [11] found that the rarefaction reduces the Nusselt number. Such a conclusion was reachedby Mikhailov and Cotta [12], and Chen [13] through an analytical analysis on laminar flow of a gasin a microchannel formed by two parallel plates neglecting viscous dissipation, and considering theconstant thermo-physical properties. They obtained a Nusselt number in an established state lowerthan the classical value of 7.54 giving an idea of the influence of rarefaction on heat transfer. Jeongand Jeong [14] analyzed the Graetz problem in a parallel plate microchannel subjected to constanttemperatures of walls, taking into account the effects of rarefaction, viscous dissipation and axial con-duction. The Nusselt number decreases when the Knudsen numbers Kn increases. Barbaros et al. [15]studied numerically the fully developed flow of a gas in microchannels, taking into account the viscousdissipation and neglecting the axial conduction. The authors have shown the effects of rarefaction andviscous dissipation on the Nusselt number. Another numerical study was conducted by Van Rij et al.[16] on the effect of viscous dissipation, axial conduction, and rarefaction of forced convection in arectangular microchannel. The authors concluded that these effects are significant on the simulta-neous development of hydrodynamic and thermal boundary layer, and the Nusselt number depends


mainly on the degree of rarefaction. Madhawa-Hettiarachchi et al. [17] numerically simulatedthree-dimensional slip-flow and heat transfer in rectangular microchannels for thermally andsimultaneously developing flows, including the effect of axial conduction. They explained the imple-mentation of the temperature-jump and studied the influence of Péclet number and rarefaction on theNusselt number for thermally and simultaneously developing flows.

In the present work, the main objective is to examine the effects of axial conduction on fluid andsolid-wall, rarefaction, simultaneous development of both dynamics and thermal boundary layers. Nostudies which investigate these effects simultaneously have been never reported in the literature. Theintroduction of the slip velocity and temperature jump at the fluid/solid interface is shown in details.

This paper is organized as follows. Section 2 presents the problem description. Section 3 discussesthe numerical method and techniques, used for the computation, the grid independence study, andthe code validation. Section 4 discusses the results. Finally, a conclusion is given.

2. Analyse

The problem under consideration, shown in Fig. 1, is a two-dimensional channel of height H andlength L. The thickness of the wall is taken equal to D. At the fluid/solid wall interface, first-order slipflow and temperature jump boundary conditions are introduced. The first boundary conditions reflect-ing a first-order velocity slip and temperature jump at the wall have been established by Maxwell in1879 and by Smoluchowski in 1898. They are respectively [18], as follows:

uslip � uw ¼2� rv

rvk@u@n

��w

ð1Þ

Tslip � Tw ¼2� rT

rT

2cðcþ 1Þ

kcpl

k@T@y

ð2Þ

where rv and rT are the tangential momentum and temperature accommodation coefficients, respec-tively. They describe the interaction of the fluid molecules with the wall. These coefficients vary fromnear zero to unity for specular and diffuse reflections, respectively. Their values depend on the type ofgas, type of solid, surface roughness, gas temperature, gas pressure, etc. They are determined exper-imentally. For most of the gas solid couples used in engineering applications, values of accommoda-tion coefficients are close to the unity.

The following dimensionless variables are introduced:

Fig. 1. Geometry of problem and boundary conditions.


X ¼ xH

; Y ¼ yH

; U ¼ uuin

; V ¼ vuin

; P ¼ ppu2

in

; h ¼ ðT � TinÞkf

H � q0ð3Þ

By neglecting the viscous dissipation, the dimensionless equations for the two-dimensionallaminar and steady state forced convection, Newtonian fluid and incompressible flow with constantproperties; this assumption is justified as the microchannel can be a part of a heat exchanger withmultiple microchannels mounted on the electronic components; the operating temperature ofelectronic components is less than 400 K, are written in dimensionless form, as follows:

@U@Xþ @V@Y¼ 0 ð4Þ

U@U@Xþ V

@U@Y¼ � @P

@Xþ 2

Re@2U

@X2 þ@2U

@Y2

!ð5Þ

U@V@Xþ V

@V@Y¼ � @P

@Yþ 2

Re@2V

@X2 þ@2V

@Y2

!ð6Þ

The energy equation in the fluid region is:

U@h@Xþ V

@h@Y¼ 2

Pe@2h

@X2 þ@2h

@Y2

!ð7Þ

and in the solid region:

@2h

@X2 þ@2h

@Y2

!¼ 0 ð8Þ

The governing Eqs. (4)–(8) are subject to the following dimensionless boundary conditions:

At X ¼ 0 U ¼ 1;V ¼ 0 and h ¼ 0 ð9Þ

At X ¼ LH¼ 25

@U@X¼ @V@X¼ @h@X¼ 0 ð10Þ

At Y ¼ 0:5þ E U ¼ V ¼ 0 and@h@Y¼ 1

2Kð11Þ

At Y ¼ 0@h@Y¼ @U@Y¼ 0 and V ¼ 0 ð12Þ

3. Numerical method and code validation

The governing Eqs. (4)–(8), with the associated boundary conditions (9)–(12), are solved using afinite volume technique [19]. They have been implemented in a FORTRAN code. The numerical proce-dure called SIMPLER [19] is used to handle the pressure–velocity coupling. Convection fluxes in theseequations are evaluated by the power-law scheme, and diffusion fluxes by the central differencesscheme. The discretized algebraic equations are solved by the line-by-line tri-diagonal matrixalgorithm (TDMA). Evaluation of thermal conductivities at the interface of two neighbouring nodesis carried out using a harmonic function [19]. The convergence criterion used is defined as follows:

max/nþ1 � /n

/n

�� 6 e ð13Þ

where / denotes one of the main variables U, V, h and n is the iteration index. The criterion is satisfiedwhen the relative error between two successive iterations in all nodes is lower than e = 10�7 for the

Fig. 2. Control volume: fluid/wall interface.


dimensionless temperatures (hs and hf), and e = 10�4 for the dimensionless velocities (U and V).Calculations were carried out on a Linux Cluster PC with 24 Go DDR3.

The evolution of local Nusselt number obtained by the three mesh considered in this study(200 � 272, 300 � 272 and 300 � 377 nodes) is shown in Fig. 3. It is found that the difference betweenthe curves does not exceed 2% for the best case. The grid corresponding to 300 � 272 nodes is there-fore adopted for all numerical simulations, in order to optimize the CPU time and the cost ofcomputations.

3.1. Implementation of the jump temperature condition at the fluid/solid interface

For a rarefied flow, we introduce the dimensionless boundary conditions slip velocity and jumptemperature on the fluid/solid interface, defined as follows:

Uslip � Uw ¼ 2Kn2� rv

rv

@U@Y

��w

ð14Þ

hslip � hw ¼ 2KKn2� rT

rT

2cðcþ 1Þ

1Pr@h@Y

��w

ð15Þ

The problem arises when introducing the jump temperature condition on the wall. For this reason,we adopt the method presented by Madhawa Hettiarachchi et al. [17], a method adapted to our casei.e. the presence of a solid wall. The discretized energy equation can be written in the followingalgebraic form:

Fig. 3. The variations of local Nusselt number Nux for different grid sizes.

Fig. 4a. Comparison between the present work and the analytical solution obtained by Van Rij et al. [16] for the velocityprofiles.

Fig. 4b.Nux alo


aphp ¼ aEhE þ aWhW þ aNhN þ aShS þ b ð16Þ

Eq. (16) can also be rewritten in the following form:

aNðhN � hpÞ þ aSðhS � hpÞ þ aEðhE � hpÞ þ aWðhW � hpÞ þ b ¼ 0 ð17Þ

We write the jump temperature condition in a dimensionless form of the fluid/solid interface,which is represented in Fig. 2.

10-3 10-2 10-1 100

X*

0

2

4

6

8

10

12

14

16

18

20

Nux

Kn = 0.0 Jeong et al.Kn = 0.08 Jeong et al.Present

8.238.23

5.72

Comparison between the present work and the results obtained by Jeong and Jeong [14] for the local Nusselt numberng the microchannel.

YY


hn � hw ¼2cKKnðcþ 1ÞPr

@h@Y

��Y¼H

ð18Þ

After some manipulations, we obtain:

hp � hw ¼ 1þ 2cKKnðcþ 1ÞPr

� �ðhp � hNÞ ð19Þ

and we replace the coefficient aN at the interface by a0N in the discretized energy equation:

a0N ¼aN

1þ 2cKKnðcþ1ÞPr

� � ð20Þ

3.2. Code validation

The numerical code developed for this study has already been validated with the analyticalsolution of velocity profile in fully developed region of parallel plates microchannel proposed byVan Rij et al. [16]:

(a) X=0.1 )b( X=0.7

(c) X=1.6 )d( X=25

0 0.25 0.5 0.75 1 1.25 1.5

0 0.25 0.5 0.75 1 1.25 1.5 0 0.25 0.5 0.75 1 1.25 1.5

0 0.25 0.5 0.75 1 1.25 1.5U

0

0.1

0.2

0.3

0.4

0.5

Kn = 0.00Kn = 0.04Kn = 0.10

U

0

0.1

0.2

0.3

0.4

0.5

Y

Kn = 0.00Kn = 0.04Kn = 0.10

U

0

0.1

0.2

0.3

0.4

0.5

Kn = 0.00Kn = 0.04Kn = 0.10

U

0

0.1

0.2

0.3

0.4

0.5

Y

Kn = 0.00Kn = 0.04Kn = 0.10

Fig. 5. Development of axial velocity profiles along the microchannel for various values of X at Re = 100.


uðy=HÞum

¼ uslip

umþ 6 1� uslip

um

� �yH� y2

H2

� �ð21Þ

uslip

um¼ 1� 1

1þ 12rvKn2� � ð22Þ

Fig. 4a shows the comparison between the present work and the analytical solution obtained byVan Rij et al. [16] for the velocity profiles. For different Knudsen numbers, these curves are almostidentical to those calculated from analytical relations (21) and (22). The error does not exceed 1%.

The evolution of local Nusselt number obtained by our code is also compared with that given byJeong and Jeong [14], who consider that the problem of Greatz for Knudsen numbers Kn = 0.0 andKn = 0.08 (with Dh = 4H, rather than Dh = 2H used here). Our implementation has been done withE = 0.0 (without influence of the plate) and b = 1.67 (typical value for air) and Pe = 100. The compar-ison shows a very good agreement between our numerical solution and those of Jeong and Jeong[14], except in the entrance region since the influence of dynamical and thermal developing regionsin our solution is clearly visible in the case of Kn = 0 (Fig. 4b). This difference decreases with increasingKn, because the rarefaction decreases the thermally establishment length.

4. Results and discussion

Consider a short microchannel, that length is equal to L/H = 25 and that the working fluid is air(Pr = 0.7). The study of the problem has been made according to the following parameters:0 6 Kn 6 0.1 (in this range of Knudsen number the maximum hydraulic diameter of microchannelis Dh = 65 lm, since for air kair = 0.65 nm under ambient condition), the thickness of the micro chan-nels is E = D/H = 0, 0.5,1.0,1.5 and 2, the Reynolds number Re = 100, and the thermal conductivity ratioK = ks/kf = 1,10,100 and 500. These values cover some materials used in the manufacture of micro-heatexchangers or used in the electronics industry such as glass and ceramic, FR-4, polymers PMMA,silicium germanium and steel.

10-3 10-2 10-1

X+

5

10

15

20

25

30

35

40

45

50

Po

Kn = 0.00Kn = 0.02Kn = 0.04Kn = 0.06Kn = 0.08Kn = 0.10

Fig. 6. Poiseuil number for different values of Kn.

(a) Kn = 0.0

(b) Kn = 0.06

(c) Kn = 0.1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

X*

6.00

6.50

7.00

7.50

8.00

8.50

9.00

9.50

10.00

Nux

No wallK = 1K = 10K = 100K = 500

Kn = 0.0

X*

6.00

6.10

6.20

6.30

6.40

6.50

6.60

6.70

6.80

6.90

7.00

Nux

No wallK = 1K = 10K = 100K = 500

Kn = 0.06

X*

5.00

5.25

5.50

5.75

6.00

Nux

No wallK = 1K = 10K = 100K = 500

Kn = 0.1

Fig. 7. Evolution of local Nusselt number Nux along the micro-channel for three values of Kn at E = 1 and K = 1, 10, 100, and 500.



4.1. Hydrodynamic field

The evolutions of velocity profiles along the microchannel are illustrated in Fig. 5a–d for differentKnudsen number Kn = 0.0, 0.04 and 0.01. At the entrance of the microchannel, the velocity profile isU = 1, boundary layer develop along the wall of micro-channel and the velocity is being redistributedgradually on account of the viscosity. At the exit (station X = 25), the fully developed no-slip laminarflow (Kn = 0.0) is obtained. The evolution of velocity profiles for slip-flow (Kn = 0.04 and 0.1) related inFig. 5 shows that slip velocity near the wall diminishes when Kn decreases unlike the velocity locatedin the center of microchannel, which decreases with increasing Kn. The final profiles (at theestablished state) is a flat parabolic profile. The velocity gradient and the friction coefficient on thewall decrease with increasing Kn number.

The Poiseuil number (Fig. 6) calculated for non-slip laminar according to the following relation:

Table 1The evo

X⁄

0.000.000.000.000.010.020.030.040.180.180.300.300.330.340.35

Po ¼ f Re ð23Þ

where f is the friction factor, defined by the following expression:

f ¼ sw12 qu2

in

ð24Þ

In established region, Po = 24 (value reported in the literature [20]). For K = 0.1 we obtain Po = 10.84.

4.2. Temperature field

In order to see the effect of axial conduction on the heat transfer in a short microchannel, we deter-mine the local Nusselt number according to the following relation:

Nu ¼ hDh

kf¼�ks

@T@y

ðTb � TwÞð25Þ

By introducing dimensionless quantities, the local Nusselt number is:

Nu ¼ K int�2 @h

@Y

ðhb � hwÞð26Þ

with

lution of the local Nusselt number along the microchannel E = 1.

Nu

K = 10 K = 500

Kn = 0.0 Kn = 0.06 Kn = 0.1 Kn = 0.0 Kn = 0.06 Kn = 0.1

0 122.190 17.449 11.036 122.121 17.503 11.0561 36.531 14.674 9.904 33.585 14.606 9.8903 20.618 12.228 8.803 18.337 11.925 8.7126 15.420 10.649 8.024 13.529 10.205 7.8721 12.011 9.066 7.163 10.456 8.563 6.9662 9.895 7.655 6.296 8.628 7.260 6.1180 9.287 7.158 5.963 8.161 6.865 5.8253 8.807 6.734 5.666 7.879 6.567 5.5843 8.237 6.251 5.302 7.939 6.259 5.3085 8.238 6.251 5.302 7.938 6.258 5.3072 8.201 6.252 5.303 7.927 6.238 5.3004 8.198 6.252 5.302 7.919 6.235 5.2990 8.142 6.251 5.302 7.861 6.197 5.2797 8.049 6.251 5.302 7.803 6.146 5.2519 7.881 6.234 5.294 7.736 6.070 5.203


K int ¼kint

kfð27Þ

Kint is the thermal conductivity ratio at the interface, it is evaluated by a harmonic mean and hb isthe dimensionless bulk temperature defined as:

Fig. 8. Temperature distribution in a micro-channel for two values of K at E = 1.5 and Kn = 0.

F


hb ¼Z Y

0UhdY

Z Y

0UdY

�ð28Þ

4.2.1. Influence of the wall thermal conductivityThe influence of thermal conductivity ratios K on the axial variations of local Nusselt number Nux

along the microchannel is shown in Fig. 7a–c for K = 1,10, 100, and 500, and Kn = 0, 0.06 and 0.01. Forthe non-conjugate problem (no wall, E = 0) and non slip-fow (Kn = 0) (see Fig. 7a), the Nusselt numberdecreases and tends towards an asymptotic value around X⁄ = 0.1, which corresponds to the fullydeveloped flow characteristic (Nu = 8.23). For the conjugate case (E = 1), the local Nusselt number isreduced compared to the non-conjugate case (no wall), by increasing the thermal conductivity ratioK, as also shown in Table 1. The larger reductions of Nux are observed near the entrance and at the exitof the microchannel. This is in agreement with the results obtained by Nonino et al. [3], Rahimi andMehryar [4] and, Avcı et al. [7].

It is obvious that the axial wall conduction reduces the local Nusselt number, its effect is significantwhen the wall thermal conductivity ks increases because the thermal resistance of the wall decreases.It is clear from Fig. 8a and b, which represent the temperature distribution in the micorchannel for

(a) Kn = 0.0 (b) Kn = 0.06

(c) Kn = 0.1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

X*

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

q*

K = 1K = 10K = 100K = 500

E = 1, Kn = 0.06

X*

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

q*

K = 1K = 10K = 100K = 500

E = 1, Kn = 0.1

ig. 9. Evolutions of q and q⁄ along the micro-channel for three values of Kn at E = 1 and K = 1, 10, 100, and 500.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

Bul

k te

mpe

ratu

re

No wallK = 1K = 10K = 100K = 500

Kn = 0.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.00

0.25

0.50

0.75

1.00

1.25

Inte

rfac

ial t

empe

ratu

re

No wallK = 1K = 10K = 100K = 500

Kn = 0.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.00

0.01

0.02

0.03

0.04

0.05

Bul

k te

mpe

ratu

re

No wallK = 1K = 10K = 100K = 500

Kn = 0.06

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.00

0.01

0.02

0.03

0.04

0.05

Inte

rfac

ial t

empe

ratu

re

No wallK = 1K = 10K = 100K = 500

Kn = 0.06

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.00

0.01

0.02

0.03

0.04

0.05

Bul

k te

mpe

ratu

re

No wallK = 1K = 10K = 100K = 500

Kn = 0.1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.00

0.01

0.02

0.03

0.04

0.05

Inte

rfac

ial t

empe

ratu

re

No wallK = 1K = 10K = 100K = 500

Kn = 0.1

(a) (b)

(c) (d)

(e) (f)

Fig. 10. Variations of bulk and interfacial temperatures along the micro-channel for two values of Kn (0.06 and 0.1) at E = 1 andK = 1, 10, 100, and 500.



E = 1, K = 100 and 500 that increasing K the heating of fluid is better. At the exit of microchannel, thedimensionless temperature increases respectively from 0.88 to 1.18 for K = 100 and 500.

Fig. 9a shows the variation of interfacial heat flux q at the inner wall of microchannel with X⁄ forKn = 0. For small K, the interfacial heat flux q decreases rapidly from the entry section and tends to-wards a constant value (q = 1). With increasing K, the interfacial heat flux decreased gradually fromthe entry to the exit. The effects of axial conduction in the wall of the channel are well highlighted.We can see the larger deviations between the curves with K = 1 and K = 500, localized near the en-trance and exit. At the entrance region, more heat enters the fluid while at the exit, an opposing effectis observed. This observation can be considered as an aiding effect of axial conduction. The axial var-iation of the dimensionless wall and bulk temperatures, corresponding to the cases showed in Fig. 7a,are reported in Fig. 10a and b. By increasing K (decrease in the thermal resistance of wall) the temper-ature gradient at the fluid/solid interface decreases at the end of microchannel.

The evolutions of temperature profiles for Kn = 0 and different conductivity ratios K = 1, 10 and 100at different axial locations X = 0.004, 0.358 and 25 are presented in Fig. 11a. Near microchannel entry(X = 0.004), for no-slip flow (Kn = 0), the temperature gradient in the fluid increases when K increases,we obtain a large interfacial heat flux, contrary to the exit (X = 25) where the heat flux inverselyproportional with K. For Kn = 0.1 (Fig. 11b), temperature gradients in the fluid are very close to allthe values of K.

The effects of rarefaction and axial conduction in the wall of microchannel on the local Nusseltnumber are shown in Fig. 7b for Kn = 0.06 and E = 1, the developments Nusselt curves for different ra-tios K are almost identical, except at the exit of microchannel where small decreases of local Nusseltnumber are observed when K increases. The difference between the various curves does not exceed3%. In Fig. 7c, for Kn = 0.1 the Nusselt curves become confused. The difference is less than 1%.

Since the increasing Knudsen number leads to increased temperature jump at the wall, as a result,the adjacent fluid to the wall does not feel the real temperature of the wall. The fluid is like insulated.By increasing K, the wall axial conduction is more significant so the temperature jump increases with

X = 0.004 X = 0.358 X = 25 (a) Kn = 0

X = 0.004 X = 0.358 X = 25

(b) Kn = 0.1

Fig. 11. Development of dimensionless temperature profiles along the micro-channel for different values of K = 1, 10, and 100.

Fig. 12. Temperature distribution in a micro-channel for two values of Kn at E = 2 and K = 100.


K (see Eq. (15)), consequently this lowers the heat transfer from the wall to fluid. This observation isshown in Fig. 12a and b, which represents the temperature distribution in the microchannel forrespectively E = 2, K = 100 and Kn = 0 and 0.1. The enhancement of fluid temperature through themicrochannel is insignificant for Kn = 0.1. The rarefaction reduces the effect of wall axial conduction.

(a) Kn=0.0

(b) Kn=0.1

0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

6.00

7.00

8.00

9.00

10.00

Nux

E = 0.0E = 0.5E = 1.0E = 1.5E = 2.0

K = 100, Kn = 0.0

0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

5.00

5.25

5.50

5.75

6.00

Nux

E = 0.0E = 0.5E = 1.0E = 1.5E = 2.0

K = 1, Kn = 0.1

Fig. 13. Evolution of local Nusselt number along the micro-channel for various values of E.


Thus, the interfacial normalized heat flux q⁄ curves schematically respectively in Fig. 9b and c forKn = 0.06 and 0.1, are confused for all values of K contrary to the case Kn = 0 (Fig. 9a). Also accordingto Maranzana et al. [1] the bulk fluid temperature is linear for all values of K (Fig. 11b and e),

4.2.2. Influence of the wall thicknessThe effects of wall thickness on the evolution of local Nusselt number along the microchannel are

shown in Fig. 13a for Kn = 0, K = 100 with different values of E. An increase in thickness wall E leads todecrease the local Nusselt number at the entry and exit of micro-channel. The curves of Nusselt

(a) Kn=0.0

(b) Kn=0.1

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35X*

0.50

0.75

1.00

1.25

1.50

q*

E = 0.0E = 0.5E = 1.0E = 1.5E = 2.0

K = 100, Kn = 0.1

Fig. 14. Evolutions of q and q⁄ along the micro-channel for various values of E.


number for different wall thicknesses become confused at the center X⁄ = 0.20. We observe that, byincreasing the thickness E, the local Nusselt number decreases.

According to Avcı et al. [7] the reductions in Nu when E increases, can be clarified by the back-heating mechanism due to the larger axial conduction in the solid. We can see that the increase in wallthickness decreases the thermal resistance of the solid. The thickness wall E exerts the same effect onthe local Nusselt number Nu as K.

The effects of slip-flow and axial conduction in the wall of microchannel on the local Nusselt num-ber is given in Fig. 13b for Kn = 0.1. All curves are confused for all values of E along the microchannel.The effect of rarefaction is negligible for the various thicknesses E.


Fig. 14a and b shows the wall influence on the distribution flux q along X⁄. It is clear that for Kn = 0(Fig. 14a) and when E increases, the flux q is deviated from its nominal value (q = 1). For a rarefied flow(Fig. 14b), the influence of the wall on the distribution of normalized flux q⁄ at the interface isnegligible.

5. Conclusion

The numerical study of a slip-flow and conjugate heat transfer in parallel plates microchannel hasbeen carried out. The introduction of the boundary conditions of the temperature jump at the wall hasalso been presented. The main results obtained in this study can be summarized as follows:

� For non slip-flow Kn = 0, the wall thickness E increases considerably the effect of axial conductionof the solid and becomes more important for large values of E.� It was established that the conduction effect in the wall is more pronounced for solid with high

thermal conductivity, especially at the entrance of the channel, where as the effect of axial conduc-tion increases the rate of heat transfer in this area, by its effect against is negative about the heattransfer at the micro-channel exit.� For rarefied flows, the jump temperature causes a considerable decrease in the rate heat exchange.

The effect of axial conduction is negligible for any Knudsen number, thermal conductivity ratio andwall thickness.

Acknowledgments

Dr. Y. Kabar gratefully acknowledges the computer center (Centre de calcul de Champagne-Ardenne ROMEO, URCA Reims), for computer time on a Linux Cluster PC.

References

[1] G. Maranzana, I. Perry, D. Maillet, Int. J. Heat Mass Transfer 47 (2004) 3993–4004.[2] Z. Li, Y. He, G. Tang, W. Tao, Int. J. Heat Mass Transfer 50 (2007) 3447–3460.[3] C. Nonino, S. Savino, S. Del Giudice, L. Mansutti, Int. J. Heat Fluid Flow 30 (2009) 823–830.[4] M. Rahimi, R. Mehryar, Int. J. Therm. Sci. 59 (2012) 87–94.[5] Y. Kabar, M. Rebay, M. Kadja, C. Padet, Heat Transfer Res. 43 (2010) 247–263.[6] S.X. Zhang, Y.L. He, G. Lauriat, W.Q. Tao, Int. J. Heat Mass Transfer 53 (2010) 3977–3989.[7] M. Avcı, O. Aydın, M.E. Arıcı, Int. J. Heat Mass Transfer 55 (2012) 5302–5308.[8] F.E. Larrode, C. Housiadas, Y. Drossinos, Int. J. Heat Mass Transfer 43 (2000) 2669–2680.[9] R.F. Barron, X.M. Wang, R.O. Warrington, Int. Com. Heat Mass Transfer 23 (1996) 563–574.

[10] R.F. Barron, X.M. Wang, T.A. Ameel, R.O. Warrington, Int. J. Heat Mass Transfer 40 (1997) 1817–1823.[11] S. Yu, T.A. Ameel, Int. J. Heat Mass Transfer 44 (2001) 4225–4234.[12] M.D. Mikhailov, R.M. Cotta, Int. Commun. Heat Mass Transfer 32 (2005) 341–348.[13] C.H. Chen, Heat Mass Transfer 42 (2006) 853–860.[14] H.E. Jeong, J.T. Jeong, Int. J. Heat Mass Transfer 49 (2006) 2151–2157.[15] C. Barbaros, H. Yuncu, S. Kakaç, Int. J. Transp. Phenom. 8 (2006) 297–315.[16] J. Van Rij, T. Harman, T. Ameel, Int. J. Therm. Sci. 48 (2009) 271–281.[17] H.D. Madhawa Hettiarachchi, M. Golubovic, W.M. Worek, W.J. Minkowycz, Int. J. Heat Mass Transfer 51 (2008) 5088–5096.[18] M. Gad-el-Hak, MEMS: Introduction and Fundamentals, second ed., Taylor & Francis, New York, 2006.[19] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.[20] R.K. Shah, A.L. London, Laminar flow forced convection in ducts, in: Advanced Heat Transfer, Academic Press, New York,

1978.

http://refhub.elsevier.com/S0749-6036(13)00147-X/h0005




























Documents

Conjugate heat transfer with rarefaction in parallel plates microchannel