Numerical Heat Transfer, Part A: Applicationsrx.mc.ntu.edu.tw/alumni/epaper/23/03.pdf · heat...

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Numerical Heat Transfer, Part A:ApplicationsAn International Journal of Computation andMethodologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713657973

Numerical Study of Heat Transfer of aPorous-Block-Mounted Heat Source Subjected toPulsating Channel FlowYueh-Liang Yen a; Po-Chuan Huang b; Chao-Fu Yang b; Yen-Jen Chen ba Department of Pharmacy, National Taiwan University Hospital, Taipei, Taiwan,Republic of Chinab Department of Energy and Refrigerating Air-Conditioning Engineering, NationalTaipei University of Technology, Taipei, Taiwan, Republic of China

Online Publication Date: 01 January 2008

To cite this Article: Yen, Yueh-Liang, Huang, Po-Chuan, Yang, Chao-Fu and Chen, Yen-Jen (2008) 'Numerical Studyof Heat Transfer of a Porous-Block-Mounted Heat Source Subjected to Pulsating Channel Flow', Numerical HeatTransfer, Part A: Applications, 54:4, 426 — 449

To link to this article: DOI: 10.1080/10407780802164496URL: http://dx.doi.org/10.1080/10407780802164496

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NUMERICAL STUDY OF HEAT TRANSFEROF A POROUS-BLOCK-MOUNTED HEAT SOURCESUBJECTED TO PULSATING CHANNEL FLOW

Yueh-Liang Yen1, Po-Chuan Huang2, Chao-Fu Yang2, andYen-Jen Chen2

1Department of Pharmacy, National Taiwan University Hospital, Taipei,Taiwan, Republic of China2Department of Energy and Refrigerating Air-Conditioning Engineering,National Taipei University of Technology, Taipei, Taiwan, Republic of China

A numerical analysis was conducted to investigate the convective characteristics of

pulsating flow through a channel with a porous-block-mounted heat source. Comprehensive

time-dependent flow and temperature data are calculated and averaged over a pulsation

cycle in a periodic steady state. The impacts of the Darcy number, pulsating frequency

and amplitude, and porous blockage ratio are documented in detail. The results indicate that

the periodic alteration in the structure of recirculation flows, caused by both porous block

and flow pulsation, has a direct impact on the flow behavior in the vicinity of the porous

block and on the heat transfer rate from the heater.

INTRODUCTION

With today’s rapid advances in high-density electronic packaging for compact-ness, heat generation from integrated chips is excessive. For this reason, efficientremoval of excessive heat has been a crucial requirement for the reliable operationof sophisticated electronics. In response to these demands, various highly effectivecooling techniques have been used in the past to obtain heat transfer enhancementwith a minimum of frictional losses, including the traditional methods of naturaland forced convective cooling. Among the heat transfer enhancement schemes,one of the promising techniques is the use of a porous material subjected to flow pul-sation. The porous medium has emerged as an effective passive cooling enhancerbecause of its large surface area-to-volume ratio and intense mixing of fluid flow.The forced pulsation of incoming fluid at the entrance of the channel is anotheractive augmenting method, because of the hydrodynamic instability in a shear layer,which substantially increases lateral flow mixing and hence augments the convectivethermal transport in the direction normal to the heated surface.

Received 30 August 2007; accepted 1 April 2008.

This work was supported by the R.O.C. National Science Council and the R.O.C. Ministry of

Economic Affairs, Bureau of Energy, under contracts NSC 95-2221-E-027-113 and 97-D0137-3.

Address correspondence to Po-Chuan Huang, Department of Energy and Refrigerating Air-

Conditioning Engineering, National Taipei University of Technology, 1 Sec. 3, Chung-Hsiao E. Rd.,

Taipei 106, Taiwan, Republic of China. E-mail: [email protected]

426

Numerical Heat Transfer, Part A, 54: 426–449, 2008

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780802164496

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Thermal convection in fluid-saturated porous media has been of continuinginterest because of its relevance in a broad range of engineering applications suchas thermal insulation, heat exchangers, geothermal energy systems, enhanced oilrecovery, heat pipe technology, industrial furnaces, cooling of electronic equipment,etc. A comprehensive review of the existing studies on these topics can be found inNield and Bejan [1]. Related to the thermal control application, Huang and Vafai[2, 3] analyzed the effect of the presence of porous blocks on the heat transferenhancement using different configurations of porous obstacles in a channel.Khanafer and Vafai [4] studied the production and regulation of an isothermal sur-face utilizing porous inserts for the thermal control applications of electric devices.Fu et al. [5] dealt with heat transfer from a porous-block-mounted heat plate in achannel flow. The effects of flow inertia, variable porosity, and solid boundarywere included. They reported that for the blocked ratio H�p ¼ 0:5, the thermal

NOMENCLATURE

A oscillating amplitude of axial inlet

velocity

Cp specific heat at constant pressure,

J=kg K

Da Darcy number (¼K=R2)

f dimensional forcing frequency, Hz

F function used in expressing inertia terms

h convective heat transfer coefficient,

W=m2K

Hp height of porous block, m

k thermal conductivity, W=m K

K permeability of the porous medium, m2

Li length of channel upstream from the

first porous block, m

Lo length of channel downstream from the

second porous block, m

Lt total length of channel, m

Num cycle-space average Nusselt number

½¼ðR s

0

RW0 Nux;t dx dtÞ=sW �

Nux cycle-averaged local Nusselt number

¼R s

0 Nux;t dt� �

=s� �

Nux, t local instantaneous Nusselt number

¼h x; tð ÞR=kf

� �P pressure, N=m2

Pe Pelect number (¼uoR=a)

q00 uniform heat flux from each heat

source, W=m2

R height of channel, m

Rceff effective heat capacity ratio

½¼ðpCpÞeff=ðpCpÞf �Re Reynolds number (¼uoR=n)

St dimensionless pulsating frequent,

Strouhal number (¼ fR=uo)

t time, s

T temperature, K

u x-component velocity, m=s

ui inlet pulsating velocity, m=s

uo cycle-averaged velocity of the inlet flow,

m=s

v y-component velocity, m=s

V velocity vector, m=s

W width of heat source or porous block, m

x, y Cartesian coordinates, m

a thermal diffusivity (¼k=qCp), m2=s

aeff effective thermal diffusivity

(¼keff=qf cp;f ), m2=s

d boundary-layer thickness, m

e porosity of the porous medium

keff effective thermal conductivity ratio

(¼kf =keff )

m dynamic viscosity, kg=m s

n kinematic viscosity, m2=s

n vorticity

q density, kg=m3

s oscillatory period for a cycle

u stream function

x angular velocity, 1=s

Subscripts

eff effective

f fluid

i inlet

p porous

s nonpulsating component

x local

Superscript� dimensionless quantity

HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 427

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performance is enhanced by higher porosity and porous particle diameter. However,the result is opposite for H�p ¼ 1. Huang et al. [6] investigated forced-convective heattransfer from multiple heated blocks in a channel by porous covers and found that therecirculation caused by the porous-covering block significantly augments the heattransfer rate on both the top and right faces of the second and subsequent blocks.

Recently, with the advent of high-performance electronic devices, there hasbeen a growing need to achieve augmented heat transfer from fully=partially porouschannel flow. One such effort has been directed to exploring the use of coupling aporous heat sink with pulsating flow. Here, a pulsating channel flow is that an oscil-lating component superposed on the mean flow in a confined passage can enhancethe axial transfer of energy because of the large oscillating temperature gradientsin the direction normal to the heated wall. Pulsating flow is frequent encounteredin natural systems (human respiratory and vascular systems) and engineering system(exhaust and intake manifolds of internal combustion engines, thermoacoustic cool-ers, Stirling engines, etc.). For the problem of forced pulsating convection in a chan-nel with a fluid-saturated porous material occupying the passage, Sozen and Vafai [7]conducted a numerical study of compressible flow through a packed bed. The effectof oscillating boundary conditions on the transport phenomena was investigatedwith the packed wall insulated. Kim et al. [8] simulated forced pulsating flow in afully porous channel. Their results showed that the effect of pulsation on heat trans-fer between the channel wall and the fluid is more pronounced in the case of smallpulsating frequency and large pulsating amplitude. Khodadadi [9] analyzed a fullydeveloped oscillatory flow through a porous-medium channel bounded by twoimpermeable parallel plates. It was indicated that the velocity profiles exhibit max-ima next to the wall. Paek et al. [10] treated experimentally the pulsating flowthrough a porous duct, showing that the heat transport from the porous materialdecreases as pulsating frequency decreases at given amplitude and is decreased whenthe pulsating amplitude is large (>1) enough to cause a backward flow. Fu et al. [11]explored experimentally the heat transfer of a porous channel subjected to oscillatingflow and found that the length-average Nusselt number for oscillating flow is higherthan that for steady flow. Most of these studies are related to the aspect of forced-pulsation convection over the full porous system; however, little is known about theproblem of combining forced pulsation in a fluid=porous composite system. Guo etal. [12] reported the pulsating flow and heat transfer in a partially porous pipe andindicated that the maximum effective thermal diffusivity was gained by pulsatingflow through a pipe partially filled with porous medium rather than the limiting caseof no porous medium or full filling of porous medium. In addition, the forced pul-sation in a fully=partially porous channel with discrete heat sources was of specialinterest because of its applications in the enhanced cooling of electronics. A literaturesurvey reveals that no published reports have dealt with the issue of associated heattransport. The main motivation of the present study is to explore the effects of bothheat transfer enhancement factors by flow pulsation and fiber porous blocks on theconvective cooling of an electronic device.

In this study, a numerical analysis was carried out to investigate the flow fieldand heat transfer characteristics on a porous-block-mounted heat source subjectedto both steady and pulsating channel flow. Through the use of a stream function–vorticity transformation, solution of the coupled governing equations for the

428 Y.-L. YEN ET AL.

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porous=fluid composite system is obtained using the control-volume method. Thebasic interaction phenomena between the porous substrate and the fluid region, aswell as the action of pulsation on the transport process, are scrutinized. Detailednumerical results are obtained to describe the effects of various governing para-meters defined in the problem, such as the permeability of the porous block, thefrequency and the amplitude of pulsation, and the porous blockage ratio. Inaddition, the results are also compared with those obtained for a steady, nonpulsat-ing flow. It is shown that specific choices of descriptive parameters can exert asignificant influence in the cooling of the heat source.

MATHEMATICAL FORMULATION

Consider a pulsating flow in a channel with an isolated, heated strip source, asshown in Figure 1a. The channel height and total length are R and Lt, respectively,and both channel walls are insulated. The heat source dissipates a uniform heat fluxq00 over its length W. A porous-block heat sink with height Hp and width Wp ismounted on the heat source. At the channel inlet, a pulsating flow ui with a uniformtemperature Ti is imposed, in which ui ¼ uo½1þ A sinðxtÞ�, where A and x are thepulsating amplitude and frequency, respectively. The flow is assumed to be unsteady,incompressible, and two-dimensional. In addition, the thermophysical properties ofthe fluid and the porous matrix are assumed to be constant, and the fluid-saturatedporous medium is considered to be homogeneous, isotropic, nondeformable, and inlocal thermodynamic equilibrium with the fluid. Possible channeling near the wall isneglected in the present study because fibrous media are considered, for which theporosity and permeability are relatively constant even close to the wall [13]. Theeffective viscosity of the porous medium is equal to the viscosity of the fluid. In thiswork, the flow is modeled by the transient Darcy-Brinkman-Forchheimer equation

Figure 1. Present configuration. (a) Schematic diagram of the problem and the corresponding coordinate

systems. (b) Typical nonuniform grid system for the whole computational domain.


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in the porous matrix to incorporate the viscous and inertial effects and by theunsteady Navier-Stokes equation in the fluid domain, and the thermal field by theenergy equation. Then, an efficient alternative method for combining the two setsof governing equations for the fluid and porous regions into one set of conservationequations is used to model the whole fluid=porous composite system as a singledomain governed by one set of conservations, the solution of which satisfies thematching conditions at the fluid=porous interfaces. The above-mentioned resultingtime-dependent momentum and energy equations in terms of dimensionless variablesare as follows [14]:

eqf�

qt�þ u�

qf�

qx�þ v�

qf�

qy�¼ e

Rer2f� þ S�u ð1Þ

r2u� ¼ �f� ð2Þ

RcqT�

qt�þ u�

qT�

qx�þ v�

qT�

qy�¼ r � 1

k PerT�

� �ð3Þ

where e denotes the porosity; u and f are the stream function and vorticity, respect-ively, which are related to the fluid velocity components u and v by

u ¼ quqy

v ¼ � quqx

f ¼ qv

qx� qu

qyð4Þ

The dimensionless parameters in Eqs. (1)–(3) are the Reynolds number Re, the heatcapacity ratio Rc, the thermal conductivity ratio k, and the Pelect number Pe. Su

� isthe source term, which can be considered as that contributing to the vorticity gener-ation due to the presence of the rectangular porous block. Then, the nondimensionalparameters in the fluid region are

Rcf ¼qCp

� �f

qCp

� �f

¼ 1 Ref ¼uoR

nfPef ¼

uoR

afkf ¼

kf

kf¼ 1 S�u ¼ 0 e¼ 1

ð5Þ

And in the porous region the nondimensional parameters are

Rceff ¼qCp

� �eff

qCp

� �f

Reeff ¼uoR

neffPeeff ¼

uoR

aeffkeff ¼

kf

keffð6aÞ

S�u ¼ �1

Reeff Dan� � Fe2ffiffiffiffiffiffiffi

Dap V

�� n� � Fe2ffiffiffiffiffiffiffiDap v�

q V��

qx�� u�

q V��

qy�

!ð6bÞ

where the Darcy number, Da ¼ K=R2, is related to the permeability of the porousmedium, e denotes the porosity, F is the inertia coefficient of the porous medium,


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and keff is the effective thermal conductivity of fluid-saturated porous medium. Thedimensionless variables appearing in the equations above are defined as

x� ¼ x

Ry� ¼ y

Ru� ¼ u

uov� ¼ v

uo

jV��!j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu�2 þ v�2

pW �

p ¼Wp

RH�p ¼

Hp

Rð7Þ

u� ¼ uuiR

f� ¼ Rfui

T� ¼ T � To

q00R=kft� ¼ tuo

RSt ¼ fR

uoð8Þ

The associated dimensionless boundary conditions necessary to complete theformulation of the present problem are as follows.

1. At the inlet (x� ¼ 0, 0 < y� < 1, and t� > 0),

u� ¼ 1þ A sinð2p St t�Þ v� ¼ 0 T� ¼ 0 u� ¼ y� f� ¼ 0 ð9Þ

where St ¼ fR=uo is the dimensionless pulsating frequency parameter (Strouhalnumber).

2. At the outlet (x� ¼ Lt�, 0 < y� < 1, and t� > 0),

qu�

qx�¼ 0

qv�

qx�¼ 0

qu�

qx�¼ 0

qf�

qx�¼ 0

qT�

qx�¼ 0 ð10Þ

3. At the bottom channel wall (05 x�5Lt�, y� ¼ 0, and t� > 0),

u� ¼ 0 v� ¼ 0 u� ¼ 0 f� ¼ � q2u�

qy�2 ð11Þ

qT�

qy�¼ 0 (at insulated area)

qT�

qy�¼ �1 (at heat source area) ð12Þ

4. At the top channel wall (05 x�5Lt�, y� ¼ 1, and t� > 0),

u� ¼ 0 v� ¼ 0 u� ¼ 1 f� ¼ � q2u�

qy�2

qT�

qy�¼ 0 ð13Þ

In addition, the continuities of the velocity, pressure, stress, temperature, and heatflux are satisfied at the porous=fluid interface [6].

To evaluate the effects of both flow pulsation and porous block on the heattransfer rate at the heat source, the local instantaneous Nusselt number along the


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surface of the heat source is evaluated as

Nux; t ¼hðx; tÞR

kf¼ � keff R

kf ðTw � ToÞqT

qy

��y¼0

¼ � 1

keff T�w

qT�

qy�ð14Þ

where Tw� ¼ (Tw� Ti)=(q00R=kf) is the dimensionless heater surface temperature.

Then the corresponding local Nusselt number in a time average over one cycle ofpulsation is calculated as

Nux ¼1

s

Z s

0

Nux;t dt ð15Þ

The cycle-space averaged Nusselt number over a heat source is defined as

Num ¼1

sW

Z s

0

Z w

0

Nux;t dx dt ð16Þ

where W is the overall exposed length of the heat source. Noted that the definition ofNusselt number based on the conductivity of the fluid permits a direct comparisonfor a heat source with and without a porous block.

NUMERICAL METHOD

To obtain the solution of the preceding system of equations, the region of inter-est is overlaid with a variable grid system, Figure 1b. Applying the first-order fullyimplicit scheme for the time derivatives, central differencing for the diffusion terms,and second upwind differencing for the convective terms, the transient finite-difference form of the vorticity transport, stream function, and energy equationswere derived by control-volume integration of these differential equations overdiscrete cells surrounding mesh points. The transient finite-difference equations weresolved by the extrapolated-Jacobi scheme [15]. In this work, convergence was con-sidered to have been achieved when the absolute value of the relative error on eachmesh point between two successive iterations was found to be less than 10�6. In mostcases, steady periodic solutions were obtained after 15–45 cycles of pulsation. Thetime resolution was such that one pulsating period was divided into 60 time stepsduring the first 5–10 cycles, and into 120 time steps for later cycles.

In addition, the interface between the porous medium and the fluid spacerequires special consideration. This is due to the sharp change of thermophysicalproperties, such as permeability, porosity, and thermal conductivity, across the inter-face. The harmonic mean formulation suggested by Patankar [16] was used to handlethese discontinuous characteristics in the porous=fluid interface. It has been foundthat this approximation provides good agreement with experimental data [17]. In thisstudy, the computational domain was chosen to be larger than the physical domainto eliminate the entrance and exit effects and to satisfy continuity at the exit. Asystematic set of numerical experiments was performed to ensure that the use of afully developed velocity profile for the outflow boundary condition had no detect-able effect on the flow solution within the physical domain.


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A grid independence test showed that there is only a very small difference (lessthan 1%) in the space-time averaged Nusselt number among the solutions for(81� 81), (94� 93), (113� 113), and (123� 123) grid distributions. Also, the timestep was reduced until a further reduction did not significantly affect the resultson amplitude and frequency. Therefore, a 94� 93 grid system was adopted for thepresent work. In addition, special attention was given to the spatial mesh pointsin the boundary layer, since the boundary-layer thickness d=R for the classical oscil-latory flow on a flat plate can be estimated as follows [18]:

d � 2nx

� �1=2 dR� 1

ðSt ReÞ1=2ð17Þ

Thus the dimensionless boundary-layer thickness becomes smaller as St and=or Reincreases. Spatial grids were clustered to resolve the region of this thin boundarylayer for high-frequent pulsation.

The mathematical model and the numerical scheme were validated by compar-ing the present numerical results with three relevant limiting cases available in theliterature. This was achieved by making the necessary adjustments of our modelto reduce it to a system equivalent to the simplified available cases. The relevant stu-dies for our case correspond to the following problems: (1) a steady, nonpulsatingforced flow in a channel with a heated solid block at uniform temperature, that is,Da! 0, and St ¼ 0 for Pr ¼ 0.7, Hs

� ¼ 0.25, Ws� ¼ 0.25, ks=kf ¼ 10, Li

� ¼ 2.0,Lo� ¼ 8.0 at Ref ¼ ðuoR=nf Þ ¼ 500; (2) a steady forced convection from an isolated

heat source in a channel with a porous block attached to the upper surface wall ver-tically above the heating zone (i.e., St ¼ 0, Hp

� ¼ 0.5, Wp� ¼ 1.0 for Ref ¼ 500,

Da ¼ 1� 10�2, F ¼ 0.55, e ¼ 0.9, keff=kf ¼ 1, Pr ¼ 0.72, Li� ¼ 5.5, Lo

� ¼ 15); (3) aforced pulsating flow in a channel filled with fluid-saturated porous media, that is,Hp� ¼ 1, Wp

� !1. For the first case, the results agree to better than 2.1% with dataprovided by Young and Vafai [19] and Cess and Shaffer [20] for streamlines andtime-averaged local Nusselt number Nux of a solid block for the steady, nonpulsat-ing channel flow over a heated block, as shown in Figures 2a and 2b. The results forthe second case are within less than 1% agreement with the data reported by Sunget al. [21] for both streamlines and isotherms, as shown in Figures 2c–2d. The thirdvalidation was to compare with the study of Kim et al. [8] for Da ¼ 10� 4, Re ¼ 50,Pr ¼ 0.7, F ¼ 0.057, e ¼ 0:6, A ¼ 0.75, St ¼ 0.006 and 0.16. Comparisons betweenthe profiles of normalized time-dependent fluctuation u�t�s ¼ ut

� � us� of velocity

u�, where ut� is the total instantaneous velocity and us

� denotes the nonpulsatingsteady part, calculated by Kim et al. [8], and the current analysis show discrepanciesless than 1.5% as shown in Figures 2e and 2f.

RESULTS AND DISCUSSION

The fixed input parameters that were utilized in the simulation were Pr ¼ 0.7(the air is used as the cooling fluid), Re ¼ 250, Wp

� ¼ 1, F ¼ 0.057, e ¼ 0:6,keff ¼ 1, Li

� ¼ 3, and Lo� ¼ 9. In this study, emphasis is placed on the effects of Darcy

number (1� 10�5�Da� 1� 10�3), pulsation frequency (0.6� St� 1.8), pulsation


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amplitude (0�A� 0.7), and the porous-block aspect ratio (0.1�Hp� � 0.5) on the

flow and heat transfer characteristics. To illustrate the results of flow and temperaturefields near the porous-block-mounted strip heat source clearly, only this region and itsvicinity are presented. However, it should be noted that the computational domainincluded a much larger region than what is displayed in the subsequent figures. Fur-thermore, for the sake of brevity, only the main features and characteristics of some ofthe results are discussed; the corresponding figures are not presented.

Steady Flow

For comparison purposes, representative velocity and temperature fields in achannel with a porous-block-mounted heat source at Re ¼ 250, Pr ¼ 0.7, e ¼ 0.6,

Figure 2. Results compared with other works: (a), (b) streamlines and time-averaged local Nusselt

number for the steady, nonpulsating flow compared with Young and Vafai [19] and Cess and Shaffer

[20]; (c), (d) streamlines and isotherms compared with Sung et al. [21]; (e), (f) profiles of time-dependent

fluctuation of u� compared with Kim et al. [8].


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F ¼ 0.057, keff ¼ 1, Hp� ¼ 0.3, Li

� ¼ 3, and Lo� ¼ 9 for a steady, nonpulsating flow

(St ¼ 0) at different Darcy numbers Da ¼ 1� 10�5, 4� 10�5, 1� 10�4, 5� 10�4,1� 10�3, and 1 (no porous block) are presented in Figure 3. The flow fields dis-played in Figure 3a reveal that as Darcy number decreases from 1 to 1� 10�4,the depth of streamlines penetrating into the porous block becomes less pronounced,

Figure 2. Continued.


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and a recirculation zone behind the porous block is gradually formed. As Da isdecreased further to 4� 10�5, the core flow creates three vortex effects: two recircu-lating cells, both upstream and downstream of the porous block, and a relativelylarge anticlockwise eddy zone on the smooth upper plate surface corresponding tothe reattached region on the bottom plate. The height of the recirculations behindthe porous block is higher than that of the porous block. The above complicatedflow-field change within the channel is the net result of four interrelated effects:(1) a penetrating effect pertaining to the porous medium, (2) a blowing effect causedby porous media transversely displacing the fluid from the porous region into thefluid region, (3) a suction effect caused by the pressure drop behind the porous block,resulting in a reattached flow, (4) the effect of boundary-layer separation. It shouldbe noted that the presence of the porous block creates various interesting effects forcontrolling the flow field while augmenting the heat transfer. The temperature fieldsshown in Figure 3b, corresponding to the above flow fields show that as Dadecreases, the extent of distortion of isotherms becomes more pronounced, andthe thermal boundary-layer thickness increases over the porous block. This is dueto the effect of the larger bulk frictional resistance that the flow encounters at smallervalues of Darcy number, which in turn causes a larger blowing effect through theporous block.

Pulsating Flow

By inducing pulsation, the above stable and steady flow pattern can be desta-bilized, which results in the strong interaction of the bulk fluid flow with the heatersurface and thus enhances thermal transport. The influence of pulsation is now man-ifested. Figure 4 illustrates the flow and temperature fields over one pulsating cycleat a periodic-steady state with six successive phase angles of xt ¼ 0, p=3, 2 p=3, p,4p=3, and 5 p=3 for Re ¼ 250, Pr ¼ 0.7, A ¼ 0.5, St ¼ 0.6, e ¼ 0:6, keff ¼ 1,Hp� ¼ 0.3, Li

� ¼ 3, and Lo� ¼ 9 at three different Darcy numbers Da ¼ 1� 10�3,

1� 10�4, and 4� 10�5, respectively. When the x-component velocity u� is plottedas a function of time at a monitoring point (x ¼ Li

� þWp�=2, y ¼ Hp

�=2) for the casewith Da ¼ 4� 10�5, it exhibits a time-asymptotic periodic-steady behavior afterabout 15 cycles of pulsation. The phase diagrams of u� versus v� and u� versus T�

at the same monitoring point display a simple closed loop, which clearly indicatesthat the flow and thermal fields are in a high time-periodic regime (for brevity, thesephase diagrams are not presented here). The same well-closed loops are also foundfor other cases of Darcy numbers. It can be seen in Figures 4a–4c, that for smallerDarcy number, Da ¼ 4� 10�5, two recirculating cells, one in front of the porousblock on the bottom wall and another behind the porous block on the upper wall,shrink and expand cyclically as the result of external forcing pulsation. Each tempor-ary flow pattern is the overall result of four competing effects of penetrating, blow-ing, suction, and boundary-layer separation, as mentioned earlier. This periodicalternation of flow pattern contributes significantly to the bulk mixing of fluids inthe porous-block region. The interaction of the core flow with recirculations, causedby both the porous block and flow pulsation, can significantly augment the heattransfer rate from the heat source face if the downstream recirculation zone onthe upper wall extends transversely to closer to the bottom wall. As Da increases,


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Figure 3. (a) Streamlines ðDu� ¼ 0:2 for 0 < u� < 1Þ and (b) isotherms ðDT� ¼ 0:1 for 0 < DT� < 1Þ for

a steady, nonpulsating flow (A ¼ 0) through a channel containing a heat source mounted with a porous

block at different Darcy numbers.


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Figure 4. (a)–(c) Isnstantaneous streamlines ðDu� ¼ 0:2 for 0 < u� < 1Þ and (d)–(f ) isotherm (DT� ¼ 0:1

for 0 < DT� < 1) for a pulsating flow (A ¼ 0.5, St ¼ 0.6) through a channel containing a heat source

mounted with a porous block at different Darcy numbers.


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the downstream recirculation becomes smaller and finally disappears, and theupstream recirculation moves downstream. This is due to the relatively smallerdrag force that the flow experiences in the porous region, which in turn acceleratesthe core flow through the porous block and confines the development of recircula-tion zones in the transverse direction. Figures 4d–4f show the impact of pulsationon the thermal field. Comparison of Figures 4d–4f with Figure 3b indicates thatthe thermal field under a pulsating flow presents a periodic oscillation of the ther-mal boundary-layer thickness. The thermal boundary-layer thickness descends dur-ing the acceleration phase of the cycle (xt ¼ 0 to p=2 and 3 p=2 to 2 p), and risesduring the deceleration phase of the cycle ðxt ¼ p=2 to 3 p=2Þ. This is becausewhen the flow velocity is low, the ratio of fluid residence time over the heat sourceplate to the heat diffusion time is high, allowing more heat to diffuse per unit volu-metric flow. This leads to higher flow temperatures and less steep temperature gra-dients at the wall. The depth of heat penetration into the fluid increases at thosetimes. Decreasing that ratio results in lower flow temperatures and greater tem-perature gradients when the flow velocity is high. Comparison of the isothermalvariation in Figures 5d–5f shows that for smaller Darcy number (Da ¼ 4� 10�5),the instantaneous thickness of the thermal boundary layer in the rear part of theheat source becomes smaller because the transverse growth of the downstreamrecirculation zone pushes the core flow to reattach the trailing edge of the heatsource. This brings about higher heat transport from the heat source to thecore flow.

The relationship between local cycle-averaged Nusselt number Nux and Darcynumber Da is shown in Figure 5a. For the case of steady, nonpulsating flow (St ¼ 0)without a porous block (Da!1), a large Nusselt number Nux occurs at the leadingedge of the heat source, where the thermal boundary layer begins to grow, and thenNusselt number declines toward the downstream edge due to boundary-layergrowth. For the case of steady, nonpulsating (St ¼ 0) flow with a porous block, ata larger value of Da (Da ¼ 1� 10�3), Nux is largest at the leading edge and thendecreases rapidly to a local minimum value. Near the trailing edge of the heat source,Nux increases slightly. This can be explained by noting that as the flow penetrates theporous block, a thermal boundary layer starts to develop at the left corner. Underthe blowing effect caused by the porous matrix attached to that surface, the thicknessof the thermal boundary layer grows quickly. Downstream of the heat source face,boundary-layer separation occurs, resulting in an increase in the convective energytransport, again due to the fluid mixing. The heat source has a smaller Nux valueat the leading edge than the pure flat heat source because of the impact of the coreflow as it penetrates the porous block with a relatively small vortex shedding at thefront part of the porous block and a higher temperature gradient at the leading edgeof heat source. In addition, as Da decreases from 1� 10�3 to 4� 10�5, the distri-bution curve of local Nusselt number with peak value of Nux appearing at the lead-ing edge of the heat source gradually transforms to that with the peak value at thetrailing edge of the heat source. The heat transfer in the rear part of the heat source ishigher due to the increased convection, aided by higher velocities in the recirculationeddy. For the case of pulsation flow with a porous block, the variation tendency ofNux versus Da is the same as that in the case of a steady, nonpulsation flow with aporous block. The larger heat transfer occurring at the rear part of heat source


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surface is caused by the oscillating reattachment of the core flow on the heat sourcesurface, resulting in a smaller cycle-average temperature gradients.

In order to obtain an overall measure of heat transport characteristics in thepresent study, the influence of both flow pulsation and the porous-block heat sinkon the heat transfer enhancement factor Num=(Num)non-s and (Num)s=(Num)non-s,which gives the cycle-space-averaged Nusselt number Num and the steady, nonpul-sating averaged Nusselt number (Num)s over a heat source normalized by the corre-sponding steady, nonpulsation, nonporous-block value (Num)non-s, respectively, iscalculated. Figure 5b exhibits the effect of Da on Num=(Num)non-s and(Num)s=(Num)non-s. Here, the abscissa is expressed in log scale to show clearly theeffect of Da in the range 1� 10�2 to 4� 10�5. It is clear from Figure 5b that inthe calculation range of Da, there exists a critcal Darcy number (aboutDa ¼ 1� 10�4) corresponding to the smallest values of Num=(Num)non-s and(Num)s=(Num)non-s beyond which both heat transfer enhancement factors increase.The value of heat transfer enhancement factor for pulsating flow is higher than thatfor steady, nonpulsating flow because of the larger cycle-space averaged temperaturegradient near the heat source surface. For Da ¼ 4� 10�5, the cycle-space averageNusselt number of pulsating flow is about 1.20 times that of nonpulsating flowand about 1.25 times that of nonpulsating flow over a nonporous-block heater.

Effect of Pulsating Amplitude A

Figure 6 displays the effect of A on the variation of both flow and temperaturefields over a periodic-steady pulsating cycle for Re ¼ 250, Pr ¼ 0.7, St ¼ 0.6,

Figure 5. Effects of Darcy number on (a) cycle-averaged Nusselt number and (b) heat transfer enhance-

ment factor.


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e ¼ 0:6, F ¼ 0.057, keff ¼ 1, Da ¼ 4� 10�5, Hp� ¼ 0.3, Wp

� ¼ 1.0, Li� ¼ 3, and

Lo� ¼ 9 at A ¼ 0.3, 0.5, and 0.7. Based on the inlet pulsating velocity in Eq. (9),

the larger the pulsating amplitude A is, the higher the flow deceleration becomes dur-ing the flow pulsation reversal (xt ¼ p to 2 p). This leads to production of strongerdownstream recirculation zones on the upper plate surface due to the lower flowmomentum and the thinner thickness of the temporal thermal boundary layer onthe bottom plate surface due to the reattachment of the larger amount of core fluidto the rear part of the heat source surface. It is seen from Figure 6b that Nux

increases with increased A because of the smaller oscillation temperature gradientsnear the heat source surface. Therefore, with an increase in A from 0.1 to 0.9, thegain in Num=(Num)non-s increases from 0.95 to 1.8, as shown in Figure 6c. Inaddition, when the value of A is beyond 0.4, the Num of the pulsating flow is largerthan that of nonpulsating flow.

Effect of Pulsating Frequency St

The effect of variations in the pulsating frequency or Strouhal number isdepicted in Figure 7 for Re ¼ 250, Pr ¼ 0.7, e ¼ 0:6, F ¼ 0.057, keff ¼ 1,Da ¼ 4� 10�5, A ¼ 0.5, Wp

� ¼ 1.0, Hp� ¼ 0.3, Li

� ¼ 3, and Lo� ¼ 9 with St ¼ 0.6,

1.0, and 1.8. The flow fields during a pulsating cycle with a phase angle incrementof p=3 reveal that as St increases from 0.6 to 1.0, the upper recirculation zone behindthe porous block becomes smaller and moves upstream. This gives rise to more fluidreattaching to the rear part of the heat source surface. As St is increased further to1.8, the upper downstream recirculation zone gradually disappears at each timeinstant. Because of the decreasing downward motion of the core flow, the amountof fluid reattaching to the heat source plate is reduced. The instantaneous thicknessof the thermal boundary layers decreases slightly with increase of St from 0.4 to 1.0,while it increases slightly with increase of St from 1.0 to 1.8 (the corresponding tem-perature fields are not shown). The effects of St on Nux are shown in Figure 7b. Itcan be seen that an increase in St from 0.4 to 1.8 results in a slight increase in a Nux,until an optimal St=heat transfer rate (around St ¼ 1.0) is reached, and thendecreases afterward. As expected, when St increases, the gain in Num=(Num)s gradu-ally increases to a maximum (about 1.28) around St ¼ 1, as displayed in Figure 7c,and decreases afterward. In addition, in this study range the heat transfer enhance-ment factor of the pulsating flow is always larger that of the steady flow.

Effect of Porous Blockage Ratio Hp�

Figure 8 presents the changes in Nux, Num=(Num)non-s, and (Num)s=(Num)non-s asthe porous block height Hp

� increases from 0.1 to 0.5 for Re ¼ 800, Da ¼ 4� 10�5,A ¼ 0.5, St ¼ 0.6, Wp

� ¼ 1.0, e ¼ 0:6; F ¼ 0.057, and keff ¼ 1. It can be seen in Figure8a that the distortions for streamlines become more conspicuous in each time instant asthe porous block height increases from 0.1 to 0.5. The size and strength of the upstreamand downstream recirculation zones of porous block increase with increasing Hp

�. This isdue to the relative increase in the height of the porous block, which in turn offers a higherdegree of obstruction and a larger blowing action to the flow for larger values of Hp

�. Thisleads to a stronger suction effect caused by the pressure drop behind the porous block,


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resulting in a stronger reattached flow. Meanwhile, the thickness of the instantaneousthermal boundary layer decreases with increasing Hp

� from 0.3 to 0.5, but increases asHp� increases from 0.1 to 0.3 (figures not presented). As expected, Nux increases with

increasing Hp� from 0.3 to 0.5, shown in Figure 8b, due to the existence of a larger tem-

perature gradient near the heater surface for the larger Hp�. Therefore, the gain in

Figure 6. Effect of pulsating amplitude A on (a) variation of instantaneous streamlines (Du ¼ 0:2 for

0 < u� < 1), (b) cycle-averaged Nusselt number, and (c) heat transfer enhancement factor during a

periodic-steady cycle.


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Num=(Num)non-s is more substantial for larger Hp�, as shown in Figure 8c. However, as

Hp� decreases from 0.3 to 0.1, Nux slightly increases inversely because of the smaller

blowing effect resulting in a larger transient temperature gradient near the heater surface.For the steady, nonpulsating flow, the increase trend of heat transfer enhancement factor(Num)s=(Num)non-s versus Hp

� is steeper than that in the pulsating-flow case, especially inHp� from 0.3 to 0.5. In other words, when the value of Hp

� is less than 0.5, the heat transferenhancement factor of the heater Num=(Num)non-s > (Num)s=(Num)non-s, while the value

Figure 7. Effect of pulsating frequency St on (a) variation of instantaneous streamlines (Du� ¼ 0:2 for

0 < u� < 1Þ, (b) cycle-averaged Nusselt number, and (c) heat transfer enhancement factor during a



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of Hp� is larger than 0.5, Num=(Num)non-s < (Num)s=(Num)non-s. This is due to the larger

cycle-space average temperature gradient existing on the heater surface for the steadyflow than for the pulsating flow as Hp

� > 0.5.

Figure 8. Effect of porous blockage ratio Hp� on (a) variation of instantaneous streamlines (Du� ¼ 0:2 for

0 < u� < 1), (b) cycle-averaged Nusselt number, and (c) heat transfer enhancement factor during a



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Variations of Cycle-Average Local Surface Temperature Distribution

The local surface temperature of the heater is more important than the averagesurface temperature in the application of electronic cooling [22]. Figure 9 shows thevariations of cycle-average local temperature distribution along heat source surfacesfor various values of Da, A, St, and Hp

�. As seen in Figure 9, for the pure steady-flowcase, the surface temperature of the heater increases along the flow direction andapproaches a constant maximum value when the flow approaches the thermallydeveloped region. However, for the steady-flow or pulsating-flow case with porousblock, the heater surface temperature distribution curves are of convex shape, as aresult of the boundary layer reattaching on the rear parts of the heaters. Themaximum temperature point occurs at the center section of the heater surface, wherethe thickness of the thermal boundary layer approaches maximum.

It can be seen from Figure 9a (at Re ¼ 250, St ¼ 0.8, A ¼ 0.6, Wp� ¼ 1.0,

Hp� ¼ 0.3, and keff ¼ 1.0) that for steady flow with a porous block the maximum

temperature moves from the rear of the heater surface to the front of the heater sur-face as Da decreases from 1� 10�3 to 3� 10�5. The maximum temperature occurs atDa ¼ 1� 10�4. Below and above this critical value, the maximum temperature goesdown. Comparison of the cycle-average local surface temperature distribution Tx

�

for the pulsating-flow case with that for the steady-flow case in Figure 9a indicatesthat the cycle-average local surface temperature distribution for the pulsating-flowcase with a porous block is more uniform than that for the steady-flow case withor without a porous block. That is, the temperature difference between the maximumand minimum temperatures on the surface of the heat sources for the pure steady-flow case is higher than that for the pulsating-flow case with a porous block. Inaddition, the maximum temperatures Tx

� at the heater surface is smaller for thepulsating-flow case with a porous block than for the steady-flow case with or with-out a porous block. This is because the larger cycle-space averaged temperaturegradient near the heater surface leads to more heat removal.

The local surface temperature distribution uniformity of the heater increases aspulsating amplitude A (at Re� 250, Da ¼ 4� 10�5, St ¼ 0.8, Wp

� ¼ 1.0, Hp� ¼ 0.3,

and keff ¼ 1.0) increases, as seen in Figure 9b. As seen in Figure 9c, the surface tem-perature distribution uniformity of the heater increases slightly as St increases from0.4 to 1.0, while it decreases to a fixed uniform distribution as St increases from 1.0to 1.8 (for Re ¼ 250, Da ¼ 3� 10�5, A ¼ 0.6, Hp

� ¼ 0.3, Wp� ¼ 1.0, and keff ¼ 1).

The effect of Hp� on the heater surface temperature distribution uniformity is shown

in Figure 9d. It can be seen that an increase in Hp� from 0.1 to 0.3 (at Re ¼ 800,

Da ¼ 4� 10�5, A ¼ 0.5, St ¼ 0.6, and Wp� ¼ 1.0) results in a slight increase in the

heater surface temperature distribution uniformity until a maximum temperaturedifference is reached, and then it decreases as Hp

� increases from 0.3 to 0.5.

Pressure Drop Calculation

The overall pressure drop throughout the entire channel length is anotherimportant quantity, since this added pressure drop is the price one pays for the gainin heat transfer enhancement by both forced pulsation and porous medium. In thestream function–vorticity formulation, the pressure field is eliminated in obtaining


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the solution. However, the pressure field can be recovered from the converged streamfunction and vorticity fields. This is done by integrating the pressure gradient alongthe upper channel wall. The temporal pressure gradient in a periodic steady state isderived from the unsteady momentum equation using the no-slip boundary con-ditions on the solid wall. The total temporal pressure drop DP� along the upperchannel wall is then obtained from

DP� ¼Z Lt�

0

qP�

qx�

��y�¼1

dx� ¼ �Z L�t

0

qu�

qt�þ 1

Re

qn�

qy�

� ��y�¼1

dx� ð18Þ

Figure 9. Variations of cycle-average local temperature distribution along heat source surfaces for various

values of (a) Da, (b) A, (c) St, and (d) Hp�.


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The corresponding steady nonpulsating nonporous-block pressure drop (DP�)non-s is

ðDP�Þnon-s ¼Z L�t

0

qP�

qx�

��y�¼1

dx� ¼ �Z L�t

0

1

Re

qn�

qy�

��y�¼1

dx� ð19Þ

where pressure P� is nondimensionalized with respect to qu2o. The effects of Da, A,

St, and Hp� on the temporal pressure drop factor DP�=ðDP�Þnon�s, which gives the

overall pressure drop throughout the entire channel length, normalized by the corre-sponding steady nonpulsating nonporous-block value (DP�)non-s, is presented inFigure 10. In Figure 10a, the amplitude of the temporal pressure drop factor is lessaffected by a change in St (at Re ¼ 250, Da ¼ 3� 10�5, A ¼ 0.6, Hp

� ¼ 0.3, and

Figure 10. Temporal variations of pressure drop factor along the upper plate for various values of (a) St,

(b) A, (c) Da, and (d) Hp�.


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Wp� ¼ 1.0). However, the magnitude of DP�=ðDP�Þnon-s increases substantially as A

(at Re ¼ 250, Da ¼ 4� 10�5, St ¼ 0.8, Wp� ¼ 1.0, Hp

� ¼ 0.3) or Hp� (at Re ¼ 800,

Da ¼ 4� 10�5, A ¼ 0.5, St ¼ 0.6, and Wp� ¼ 1.0) increases and or as Da (at

Re ¼ 250, St ¼ 0.6, A ¼ 0.5, Wp� ¼ 1.0, and Hp

� ¼ 0.3) decreases, as seen in Figures10b–10d. This is because as the flow approaches the smaller passage formed bythe porous block and the upper surface of the channel, the fluid starts to accelerate,resulting in an increase in the pressure drop. The temporal pressure recovery behindeach porous block is not complete, because of the pressure loss in the recirculationzones. An increase in Hp

� or a decrease in Da provides larger bulk frictional resist-ance that the flow encounters in the channel, resulting in a significant increase invariations of temporal pressure drop. An increase in A also leads to a largerpressure loss because of the formation of larger recirculating zones in the channel,as stated previously. Therefore, the required pumping power to maintain a pulsatingflow increases with pulsation amplitude, Darcy number, and porous blockage ratio.The phase lead of the temporal pressure gradient DP�=ðDP�Þnon-s over the inletpulsating velocity for all cases studied here is around p=2. This indicates that theflow pulsation considered in this study is in a higher-frequency regime comparedto the oscillating flow inside a smooth duct [18]. In this classical oscillating flow,the phase lead of pressure drop over the inlet velocity approaches p=2 from zeroas x increases.

CONCLUSIONS

This article has presented a numerical simulation of forced-pulsating convec-tive flow in a parallel-plate channel with an isolated porous-block-mounted heatsource. The results can be summarized as follows.

1. For the nonpulsating, steady flow case, the heat transfer rate from a strip heatercan be enhanced by a fiber porous-block heat sink, depending on the consoli-dated result of four interrelated effects caused by the porous block: penetrating,blowing, suction, and boundary-layer separation.

2. For the pulsating-flow case, the steady and stable flow field is substantiallydestabilized by introducing pulsation and exhibits a periodic-changing flowpattern with cyclically expanding and shrinking alteration of the vortices. Thetemperature field is substantially affected in a similar way and presents a periodicoscillation of the thermal boundary-layer thickness near the heater surface. Thecycle-average local surface temperature distribution of the heater for thepulsating-flow case with a porous block is more uniform than that for the puresteady-flow case.

3. The heat transfer enhancement factor of the heater increases with the pulsationamplitude. However, the effects of the Darcy and Strouhal numbers and ofthe porous blockage ratio are not straightforward. There exists a critical valuefor which the heat transfer enhancement factor is minimum (for Darcynumber and blockage ratio number) or maximum (for Strouhal number). Belowand above this critical value, the heat transfer enhancement factor goes up ordrops off.


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REFERENCES

1. D. A. Nield and A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 1992.2. P. C. Huang and K. Vafai, Analysis of Forced Convection Enhancement in a Channel

Using Porous Blocks, AIAA J. Thermophys. Heat Transfer, vol. 8, pp. 563–573, 1994.3. P. C. Huang and K. Vafai, Internal Heat Transfer Augmentation in a Channel Using

an Alternate Set of Porous Cavity-Block Obstacles, Numer. Heat Transfer A, vol. 25,pp. 519–539, 1994.

4. K. Khanafer and K. Vafai, Isothermal Surface Production and Regulation for HighHeat Flux Applications Utilizing Porous Inserts, Int. J. Heat Mass Transfer, vol. 44,pp. 2933–2947, 2001.

5. W. S. Fu, H. C. Huang, and W. Y. Liou, Thermal Enhancement in Laminar ChannelFlow with a Porous Block, Int. J. Heat Mass Transfer, vol. 39, pp. 2165–2175, 1996.

6. P. C. Huang, C. F. Yang, J. J. Hwang, and M. T. Chiu, Enhancement of Forced-Convection Cooling of Multiple Heated Blocks in a Channel Using Porous Covers, Int.J. Heat Mass Transfer, vol. 48, pp. 647–664, 2005.

7. M. Sozen and K. Vafai, Analysis of Oscillating Compressible Flow through a Packed Bed,Int. J. Heat Mass Transfer, vol. 12, pp. 130–136, 1991.

8. S. Y. Kim, B. H. Kang, and J. M. Hyun, Heat Transfer from Pulsating Flow in ChannelFilled with Porous Media, Int. J. Heat Mass Transfer, vol. 37, pp. 2025–2033, 1994.

9. J. M. Khodadadi, Oscillatory Fluid Flow through a Porous Medium Channel Bounded byTwo Impermeable Parallel Plates, ASME J. Fluid Eng., vol. 113, pp. 509–511, 1991.

10. J. W. Paek, B. H. Kang, and J. M. Hyun, Transient Cool-down of a Porous Medium inPulsating Flow, Int. J. Heat Mass Transfer, vol. 42, pp. 3523–3527, 1999.

11. H. L. Fu, K. C. Leong, and C. Y. Liu, An Experimental Study of Heat Transfer of aPorous Channel Subjected to Oscillating Flow, ASME J. Heat Transfer, vol. 110,pp. 946–954, 2001.

12. Z. Guo, S. Y. Kim, and H. Y. Sung, Pulsating Flow and Heat Transfer in a Pipe PartiallyFilled with a Porous Medium, Int. J. Heat Mass Transfer, vol. 40, pp. 4209–4218, 1997.

13. M. L. Hunt and C. L. Tien, Effects of Thermal Dispersion on Forced Convection inFibrous Media, Int. J. Heat Mass Transfer, vol. 31, pp. 301–310, 1988.

14. A. Amiri, Analysis of Momentum and Energy Transfer in a Lid-Driven Cavity Filled witha Porous Medium, Int. J. Heat Mass Transfer, vol. 43, pp. 3513–3527, 2000.

15. J. Adams and J. Ortega, A Multicolor SOR Method for Parallel Computation, Proc. Int.Conf. on Parallel Procession, pp. 53–56, IEEE Computer Society Press, Piscataway, NJ, 1982.

16. S. W. Patankar, Numerical Heat Transfer and Fluid Flow, pp. 273–299, McGraw-Hill,New York, 1996.

17. G. Neale and W. Nader, Practical Significance of Brinkman’s Extension of Darcy’s LawCoupled Parallel Flows within a Channel and a Bounding Porous Medium, Can. J. Chem.Eng., vol. 52, pp. 475–478, 1974.

18. H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 1979.19. T. J. Young and K. Vafai, Convective Cooling of a Heated Obstacle in a Channel, Int.

J. Heat Mass Transfer, vol. 41, pp. 3131–3148, 1998.20. R. D. Cess and E. C. Shaffer, Heat Transfer to Laminar Flow between Parallel Plates with

a Prescribed Wall Heat Flux, Appl. Sci. Res., vol. A8, pp. 339–344, 1959.21. H. J. Sung, S. Y. Kim, and J. M. Hyun, Forced Convection from an Isolated Heat Source

in a Channel with Porous Medium, Int. J. Heat Fluid Flow, vol. 16, pp. 527–535, 1995.22. G. Hwang and C. Chao, Heat Transfer Measurement and Analysis for Sintered Porous

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