Taipei University of Technology, Taipei, Taiwan, Republic of China

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1 This article was downloaded by:[2008 National Taipei University of Technology] On: 18 June 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Online Publication Date: 01 January 2008 Numerical Heat Transfer, Part A: Applications An International Journal of Computation and Methodology Publication details, including instructions for authors and subscription information: Numerical Study of Heat Transfer of a Porous-Block-Mounted Heat Source Subjected to Pulsating Channel Flow Yueh-Liang Yen a ; Po-Chuan Huang b ; Chao-Fu Yang b ; Yen-Jen Chen b a Department of Pharmacy, National Taiwan University Hospital, Taipei, Taiwan, Republic of China b Department of Energy and Refrigerating Air-Conditioning Engineering, National Taipei University of Technology, Taipei, Taiwan, Republic of China To cite this Article: Yen, Yueh-Liang, Huang, Po-Chuan, Yang, Chao-Fu and Chen, Yen-Jen (2008) 'Numerical Study of Heat Transfer of a Porous-Block-Mounted Heat Source Subjected to Pulsating Channel Flow', Numerical Heat Transfer, Part A: Applications, 54:4, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Numerical Heat Transfer, Part A, 54: , 2008 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / NUMERICAL STUDY OF HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED HEAT SOURCE SUBJECTED TO PULSATING CHANNEL FLOW Yueh-Liang Yen 1, Po-Chuan Huang 2, Chao-Fu Yang 2, and Yen-Jen Chen 2 1 Department of Pharmacy, National Taiwan University Hospital, Taipei, Taiwan, Republic of China 2 Department 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 compactness, heat generation from integrated chips is excessive. For this reason, efficient removal of excessive heat has been a crucial requirement for the reliable operation of sophisticated electronics. In response to these demands, various highly effective cooling techniques have been used in the past to obtain heat transfer enhancement with a minimum of frictional losses, including the traditional methods of natural and forced convective cooling. Among the heat transfer enhancement schemes, one of the promising techniques is the use of a porous material subjected to flow pulsation. The porous medium has emerged as an effective passive cooling enhancer because 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 another active augmenting method, because of the hydrodynamic instability in a shear layer, which substantially increases lateral flow mixing and hence augments the convective thermal transport in the direction normal to the heated surface. Received 30 August 2007; accepted 1 April 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 E and 97-D 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. pchuang@ntut.edu.tw 426

3 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 427 A oscillating amplitude of axial inlet velocity C p specific heat at constant pressure, J=kg K Da Darcy number (¼K=R 2 ) f dimensional forcing frequency, Hz F function used in expressing inertia terms h convective heat transfer coefficient, W=m 2 K H p height of porous block, m k thermal conductivity, W=mK K permeability of the porous medium, m 2 L i length of channel upstream from the first porous block, m L o length of channel downstream from the second porous block, m L t total length of channel, m Nu m cycle-space average Nusselt number ½¼ð R s R W 0 0 Nu x;t dx dtþ=swš Nu x cycle-averaged local Nusselt number ¼ R s 0 Nu x;t dt =s Nu x,t local instantaneous Nusselt number ¼hðx; tþr=k f P pressure, N=m 2 Pe Pelect number (¼u o R=a) q 00 uniform heat flux from each heat source, W=m 2 R height of channel, m Rc eff effective heat capacity ratio ½¼ðpC p Þ eff =ðpc p Þ f Š Re Reynolds number (¼u o R=n) St dimensionless pulsating frequent, Strouhal number (¼fR=u o ) t time, s NOMENCLATURE T temperature, K u x-component velocity, m=s u i inlet pulsating velocity, m=s u o 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=qC p ), m 2 =s a eff effective thermal diffusivity (¼k eff =q f c p;f ), m 2 =s d boundary-layer thickness, m e porosity of the porous medium k eff effective thermal conductivity ratio (¼k f =k eff ) m dynamic viscosity, kg=ms n kinematic viscosity, m 2 =s n vorticity q density, kg=m 3 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 Thermal convection in fluid-saturated porous media has been of continuing interest because of its relevance in a broad range of engineering applications such as thermal insulation, heat exchangers, geothermal energy systems, enhanced oil recovery, heat pipe technology, industrial furnaces, cooling of electronic equipment, etc. A comprehensive review of the existing studies on these topics can be found in Nield 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 transfer enhancement using different configurations of porous obstacles in a channel. Khanafer and Vafai [4] studied the production and regulation of an isothermal surface 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 a channel flow. The effects of flow inertia, variable porosity, and solid boundary were included. They reported that for the blocked ratio Hp ¼ 0:5, the thermal

4 428 Y.-L. YEN ET AL. performance is enhanced by higher porosity and porous particle diameter. However, the result is opposite for Hp ¼ 1. Huang et al. [6] investigated forced-convective heat transfer from multiple heated blocks in a channel by porous covers and found that the recirculation caused by the porous-covering block significantly augments the heat transfer rate on both the top and right faces of the second and subsequent blocks. Recently, with the advent of high-performance electronic devices, there has been a growing need to achieve augmented heat transfer from fully=partially porous channel flow. One such effort has been directed to exploring the use of coupling a porous heat sink with pulsating flow. Here, a pulsating channel flow is that an oscillating component superposed on the mean flow in a confined passage can enhance the axial transfer of energy because of the large oscillating temperature gradients in the direction normal to the heated wall. Pulsating flow is frequent encountered in natural systems (human respiratory and vascular systems) and engineering system (exhaust and intake manifolds of internal combustion engines, thermoacoustic coolers, Stirling engines, etc.). For the problem of forced pulsating convection in a channel 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 effect of oscillating boundary conditions on the transport phenomena was investigated with the packed wall insulated. Kim et al. [8] simulated forced pulsating flow in a fully porous channel. Their results showed that the effect of pulsation on heat transfer between the channel wall and the fluid is more pronounced in the case of small pulsating frequency and large pulsating amplitude. Khodadadi [9] analyzed a fully developed oscillatory flow through a porous-medium channel bounded by two impermeable parallel plates. It was indicated that the velocity profiles exhibit maxima next to the wall. Paek et al. [10] treated experimentally the pulsating flow through a porous duct, showing that the heat transport from the porous material decreases as pulsating frequency decreases at given amplitude and is decreased when the 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 oscillating flow and found that the length-average Nusselt number for oscillating flow is higher than that for steady flow. Most of these studies are related to the aspect of forcedpulsation convection over the full porous system; however, little is known about the problem of combining forced pulsation in a fluid=porous composite system. Guo et al. [12] reported the pulsating flow and heat transfer in a partially porous pipe and indicated that the maximum effective thermal diffusivity was gained by pulsating flow through a pipe partially filled with porous medium rather than the limiting case of no porous medium or full filling of porous medium. In addition, the forced pulsation in a fully=partially porous channel with discrete heat sources was of special interest because of its applications in the enhanced cooling of electronics. A literature survey reveals that no published reports have dealt with the issue of associated heat transport. The main motivation of the present study is to explore the effects of both heat transfer enhancement factors by flow pulsation and fiber porous blocks on the convective cooling of an electronic device. In this study, a numerical analysis was carried out to investigate the flow field and heat transfer characteristics on a porous-block-mounted heat source subjected to both steady and pulsating channel flow. Through the use of a stream function vorticity transformation, solution of the coupled governing equations for the

5 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 429 porous=fluid composite system is obtained using the control-volume method. The basic interaction phenomena between the porous substrate and the fluid region, as well as the action of pulsation on the transport process, are scrutinized. Detailed numerical results are obtained to describe the effects of various governing parameters defined in the problem, such as the permeability of the porous block, the frequency and the amplitude of pulsation, and the porous blockage ratio. In addition, the results are also compared with those obtained for a steady, nonpulsating flow. It is shown that specific choices of descriptive parameters can exert a significant influence in the cooling of the heat source. MATHEMATICAL FORMULATION Consider a pulsating flow in a channel with an isolated, heated strip source, as shown in Figure 1a. The channel height and total length are R and L t, respectively, and both channel walls are insulated. The heat source dissipates a uniform heat flux q 00 over its length W. A porous-block heat sink with height H p and width W p is mounted on the heat source. At the channel inlet, a pulsating flow u i with a uniform temperature T i is imposed, in which u i ¼ u o ½1 þ A sinðxtþš, where A and x are the pulsating amplitude and frequency, respectively. The flow is assumed to be unsteady, incompressible, and two-dimensional. In addition, the thermophysical properties of the fluid and the porous matrix are assumed to be constant, and the fluid-saturated porous medium is considered to be homogeneous, isotropic, nondeformable, and in local thermodynamic equilibrium with the fluid. Possible channeling near the wall is neglected in the present study because fibrous media are considered, for which the porosity and permeability are relatively constant even close to the wall [13]. The effective viscosity of the porous medium is equal to the viscosity of the fluid. In this work, 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.

6 430 Y.-L. YEN ET AL. in the porous matrix to incorporate the viscous and inertial effects and by the unsteady Navier-Stokes equation in the fluid domain, and the thermal field by the energy equation. Then, an efficient alternative method for combining the two sets of governing equations for the fluid and porous regions into one set of conservation equations is used to model the whole fluid=porous composite system as a single domain governed by one set of conservations, the solution of which satisfies the matching conditions at the fluid=porous interfaces. The above-mentioned resulting time-dependent momentum and energy equations in terms of dimensionless variables are as follows [14]: e qf qf qf þ u þ v qt qx qy ¼ e Re r2 f þ Su r 2 u ¼ f Rc qt qt qt þ u þ v qt qx qy ¼r 1 k Pe rt where e denotes the porosity; u and f are the stream function and vorticity, respectively, which are related to the fluid velocity components u and v by u ¼ qu qy v ¼ qu qx f ¼ qv qu qx qy The dimensionless parameters in Eqs. (1) (3) are the Reynolds number Re, the heat capacity ratio Rc, the thermal conductivity ratio k, and the Pelect number Pe. S u is the source term, which can be considered as that contributing to the vorticity generation due to the presence of the rectangular porous block. Then, the nondimensional parameters in the fluid region are Rc f ¼ qc p f ¼ 1 qc p f Re f ¼ u or n f Pe f ¼ u or a f ð1þ ð2þ ð3þ ð4þ k f ¼ k f k f ¼ 1 S u ¼ 0 e ¼ 1 ð5þ And in the porous region the nondimensional parameters are Rc eff ¼ qc p qc p eff f Re eff ¼ u or n eff Pe eff ¼ u or a eff k eff ¼ k f k eff ð6aþ Su ¼ 1 Re eff Da n Fe2 pffiffiffiffiffiffi Da V n p Fe2 ffiffiffiffiffiffi Da v q V qx u q V! qy ð6bþ where the Darcy number, Da ¼ K=R 2, is related to the permeability of the porous medium, e denotes the porosity, F is the inertia coefficient of the porous medium,

7 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 431 and k eff is the effective thermal conductivity of fluid-saturated porous medium. The dimensionless variables appearing in the equations above are defined as u ¼ u u i R x ¼ x y ¼ y R R! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j V j¼ u 2 þ v 2 f ¼ Rf u i u ¼ u u o W p ¼ W p R T ¼ T T o q 00 R=k f v ¼ v u o t ¼ tu o R H p ¼ H p R St ¼ fr u o The associated dimensionless boundary conditions necessary to complete the formulation of the present problem are as follows. 1. At the inlet (x ¼ 0, 0 < y < 1, and t > 0), u ¼ 1 þ A sinð2p St t Þ v ¼ 0 T ¼ 0 u ¼ y f ¼ 0 ð9þ where St ¼ fr=u o is the dimensionless pulsating frequency parameter (Strouhal number). 2. At the outlet (x ¼ L t,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 ð7þ ð8þ ð10þ 3. At the bottom channel wall (0 5 x 5 L t, y ¼ 0, and t > 0), u ¼ 0 v ¼ 0 u ¼ 0 f ¼ q2 u qy 2 ð11þ qt qt ¼ 0 (at insulated area) qy ¼ 1 (at heat source area) qy ð12þ 4. At the top channel wall (0 5 x 5 L t, y ¼ 1, and t > 0), u ¼ 0 v ¼ 0 u ¼ 1 f ¼ q2 u qy 2 qt qy ¼ 0 ð13þ In addition, the continuities of the velocity, pressure, stress, temperature, and heat flux are satisfied at the porous=fluid interface [6]. To evaluate the effects of both flow pulsation and porous block on the heat transfer rate at the heat source, the local instantaneous Nusselt number along the

8 432 Y.-L. YEN ET AL. surface of the heat source is evaluated as Nu x; t ¼ hðx; tþr k f k eff R ¼ k f ðt w T o Þ qt qy ¼ 1 y¼0 k eff Tw where T w ¼ (T w T i )=(q 00 R=k f ) is the dimensionless heater surface temperature. Then the corresponding local Nusselt number in a time average over one cycle of pulsation is calculated as Nu x ¼ 1 s Z s 0 Nu x;t dt qt qy The cycle-space averaged Nusselt number over a heat source is defined as Nu m ¼ 1 Z s Z w Nu x;t dx dt sw 0 where W is the overall exposed length of the heat source. Noted that the definition of Nusselt number based on the conductivity of the fluid permits a direct comparison for a heat source with and without a porous block. NUMERICAL METHOD To obtain the solution of the preceding system of equations, the region of interest is overlaid with a variable grid system, Figure 1b. Applying the first-order fully implicit scheme for the time derivatives, central differencing for the diffusion terms, and second upwind differencing for the convective terms, the transient finitedifference form of the vorticity transport, stream function, and energy equations were derived by control-volume integration of these differential equations over discrete cells surrounding mesh points. The transient finite-difference equations were solved by the extrapolated-jacobi scheme [15]. In this work, convergence was considered to have been achieved when the absolute value of the relative error on each mesh point between two successive iterations was found to be less than In most cases, steady periodic solutions were obtained after cycles of pulsation. The time resolution was such that one pulsating period was divided into 60 time steps during the first 5 10 cycles, and into 120 time steps for later cycles. In addition, the interface between the porous medium and the fluid space requires special consideration. This is due to the sharp change of thermophysical properties, such as permeability, porosity, and thermal conductivity, across the interface. The harmonic mean formulation suggested by Patankar [16] was used to handle these discontinuous characteristics in the porous=fluid interface. It has been found that this approximation provides good agreement with experimental data [17]. In this study, the computational domain was chosen to be larger than the physical domain to eliminate the entrance and exit effects and to satisfy continuity at the exit. A systematic set of numerical experiments was performed to ensure that the use of a fully developed velocity profile for the outflow boundary condition had no detectable effect on the flow solution within the physical domain. 0 ð14þ ð15þ ð16þ

9 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 433 A grid independence test showed that there is only a very small difference (less than 1%) in the space-time averaged Nusselt number among the solutions for (81 81), (94 93), ( ), and ( ) grid distributions. Also, the time step was reduced until a further reduction did not significantly affect the results on amplitude and frequency. Therefore, a grid system was adopted for the present work. In addition, special attention was given to the spatial mesh points in the boundary layer, since the boundary-layer thickness d=r for the classical oscillatory flow on a flat plate can be estimated as follows [18]: d 2n 1=2 d x R 1 ðst ReÞ 1=2 Thus the dimensionless boundary-layer thickness becomes smaller as St and=or Re increases. Spatial grids were clustered to resolve the region of this thin boundary layer for high-frequent pulsation. The mathematical model and the numerical scheme were validated by comparing the present numerical results with three relevant limiting cases available in the literature. This was achieved by making the necessary adjustments of our model to reduce it to a system equivalent to the simplified available cases. The relevant studies for our case correspond to the following problems: (1) a steady, nonpulsating forced 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, H s ¼ 0.25, W s ¼ 0.25, k s =k f ¼ 10, L i ¼ 2.0, L o ¼ 8.0 at Re f ¼ðu o R=n f Þ¼500; (2) a steady forced convection from an isolated heat source in a channel with a porous block attached to the upper surface wall vertically above the heating zone (i.e., St ¼ 0, H p ¼ 0.5, W p ¼ 1.0 for Re f ¼ 500, Da ¼ , F ¼ 0.55, e ¼ 0.9, k eff =k f ¼ 1, Pr ¼ 0.72, L i ¼ 5.5, L o ¼ 15); (3) a forced pulsating flow in a channel filled with fluid-saturated porous media, that is, H p ¼ 1, W p!1. For the first case, the results agree to better than 2.1% with data provided by Young and Vafai [19] and Cess and Shaffer [20] for streamlines and time-averaged local Nusselt number Nu x of a solid block for the steady, nonpulsating channel flow over a heated block, as shown in Figures 2a and 2b. The results for the second case are within less than 1% agreement with the data reported by Sung et al. [21] for both streamlines and isotherms, as shown in Figures 2c 2d. The third validation was to compare with the study of Kim et al. [8] for Da ¼ 10 4,Re¼50, Pr ¼ 0.7, F ¼ 0.057, e ¼ 0:6, A ¼ 0.75, St ¼ and Comparisons between the profiles of normalized time-dependent fluctuation u t s ¼ u t u s of velocity u, where u t is the total instantaneous velocity and u s denotes the nonpulsating steady part, calculated by Kim et al. [8], and the current analysis show discrepancies less than 1.5% as shown in Figures 2e and 2f. ð17þ 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, W p ¼ 1, F ¼ 0.057, e ¼ 0:6, k eff ¼ 1, L i ¼ 3, and L o ¼ 9. In this study, emphasis is placed on the effects of Darcy number ( Da ), pulsation frequency (0.6 St 1.8), pulsation

10 434 Y.-L. YEN ET AL. amplitude (0 A 0.7), and the porous-block aspect ratio (0.1 H p 0.5) on the flow and heat transfer characteristics. To illustrate the results of flow and temperature fields near the porous-block-mounted strip heat source clearly, only this region and its vicinity are presented. However, it should be noted that the computational domain included a much larger region than what is displayed in the subsequent figures. Furthermore, for the sake of brevity, only the main features and characteristics of some of the results are discussed; the corresponding figures are not presented. Steady Flow For comparison purposes, representative velocity and temperature fields in a channel 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].

11 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 435 Figure 2. Continued. F ¼ 0.057, k eff ¼ 1, H p ¼ 0.3, L i ¼ 3, and L o ¼ 9 for a steady, nonpulsating flow (St ¼ 0) at different Darcy numbers Da ¼ ,4 10 5,1 10 4,5 10 4, , and 1 (no porous block) are presented in Figure 3. The flow fields displayed in Figure 3a reveal that as Darcy number decreases from 1 to , the depth of streamlines penetrating into the porous block becomes less pronounced,

12 436 Y.-L. YEN ET AL. and a recirculation zone behind the porous block is gradually formed. As Da is decreased further to , the core flow creates three vortex effects: two recirculating cells, both upstream and downstream of the porous block, and a relatively large anticlockwise eddy zone on the smooth upper plate surface corresponding to the reattached region on the bottom plate. The height of the recirculations behind the porous block is higher than that of the porous block. The above complicated flow-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 caused by porous media transversely displacing the fluid from the porous region into the fluid 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 should be noted that the presence of the porous block creates various interesting effects for controlling the flow field while augmenting the heat transfer. The temperature fields shown in Figure 3b, corresponding to the above flow fields show that as Da decreases, the extent of distortion of isotherms becomes more pronounced, and the thermal boundary-layer thickness increases over the porous block. This is due to the effect of the larger bulk frictional resistance that the flow encounters at smaller values of Darcy number, which in turn causes a larger blowing effect through the porous block. Pulsating Flow By inducing pulsation, the above stable and steady flow pattern can be destabilized, which results in the strong interaction of the bulk fluid flow with the heater surface and thus enhances thermal transport. The influence of pulsation is now manifested. Figure 4 illustrates the flow and temperature fields over one pulsating cycle at 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, k eff ¼ 1, H p ¼ 0.3, L i ¼ 3, and L o ¼ 9 at three different Darcy numbers Da ¼ , , and , respectively. When the x-component velocity u is plotted as a function of time at a monitoring point (x ¼ L i þ W p =2, y ¼ H p =2) for the case with Da ¼ , it exhibits a time-asymptotic periodic-steady behavior after about 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 indicates that the flow and thermal fields are in a high time-periodic regime (for brevity, these phase diagrams are not presented here). The same well-closed loops are also found for other cases of Darcy numbers. It can be seen in Figures 4a 4c, that for smaller Darcy number, Da ¼ , two recirculating cells, one in front of the porous block 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 temporary flow pattern is the overall result of four competing effects of penetrating, blowing, suction, and boundary-layer separation, as mentioned earlier. This periodic alternation of flow pattern contributes significantly to the bulk mixing of fluids in the porous-block region. The interaction of the core flow with recirculations, caused by both the porous block and flow pulsation, can significantly augment the heat transfer rate from the heat source face if the downstream recirculation zone on the upper wall extends transversely to closer to the bottom wall. As Da increases,

13 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 437 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.

14 438 Y.-L. YEN ET AL. 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.

15 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 439 the downstream recirculation becomes smaller and finally disappears, and the upstream recirculation moves downstream. This is due to the relatively smaller drag force that the flow experiences in the porous region, which in turn accelerates the core flow through the porous block and confines the development of recirculation zones in the transverse direction. Figures 4d 4f show the impact of pulsation on the thermal field. Comparison of Figures 4d 4f with Figure 3b indicates that the thermal field under a pulsating flow presents a periodic oscillation of the thermal boundary-layer thickness. The thermal boundary-layer thickness descends during the acceleration phase of the cycle (xt ¼ 0top=2 and3p=2 to2p), and rises during the deceleration phase of the cycle ðxt ¼ p=2 to3p=2þ. This is because when the flow velocity is low, the ratio of fluid residence time over the heat source plate to the heat diffusion time is high, allowing more heat to diffuse per unit volumetric flow. This leads to higher flow temperatures and less steep temperature gradients at the wall. The depth of heat penetration into the fluid increases at those times. Decreasing that ratio results in lower flow temperatures and greater temperature gradients when the flow velocity is high. Comparison of the isothermal variation in Figures 5d 5f shows that for smaller Darcy number (Da ¼ ), the instantaneous thickness of the thermal boundary layer in the rear part of the heat source becomes smaller because the transverse growth of the downstream recirculation zone pushes the core flow to reattach the trailing edge of the heat source. This brings about higher heat transport from the heat source to the core flow. The relationship between local cycle-averaged Nusselt number Nu x and Darcy number 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 Nu x occurs at the leading edge of the heat source, where the thermal boundary layer begins to grow, and then Nusselt number declines toward the downstream edge due to boundary-layer growth. For the case of steady, nonpulsating (St ¼ 0) flow with a porous block, at a larger value of Da (Da ¼ ), Nu x is largest at the leading edge and then decreases rapidly to a local minimum value. Near the trailing edge of the heat source, Nu x increases slightly. This can be explained by noting that as the flow penetrates the porous block, a thermal boundary layer starts to develop at the left corner. Under the blowing effect caused by the porous matrix attached to that surface, the thickness of the thermal boundary layer grows quickly. Downstream of the heat source face, boundary-layer separation occurs, resulting in an increase in the convective energy transport, again due to the fluid mixing. The heat source has a smaller Nu x value at the leading edge than the pure flat heat source because of the impact of the core flow as it penetrates the porous block with a relatively small vortex shedding at the front part of the porous block and a higher temperature gradient at the leading edge of heat source. In addition, as Da decreases from to , the distribution curve of local Nusselt number with peak value of Nu x appearing at the leading edge of the heat source gradually transforms to that with the peak value at the trailing edge of the heat source. The heat transfer in the rear part of the heat source is higher due to the increased convection, aided by higher velocities in the recirculation eddy. For the case of pulsation flow with a porous block, the variation tendency of Nu x versus Da is the same as that in the case of a steady, nonpulsation flow with a porous block. The larger heat transfer occurring at the rear part of heat source

16 440 Y.-L. YEN ET AL. Figure 5. Effects of Darcy number on (a) cycle-averaged Nusselt number and (b) heat transfer enhancement factor. surface is caused by the oscillating reattachment of the core flow on the heat source surface, resulting in a smaller cycle-average temperature gradients. In order to obtain an overall measure of heat transport characteristics in the present study, the influence of both flow pulsation and the porous-block heat sink on the heat transfer enhancement factor Nu m =(Nu m ) non-s and (Nu m ) s =(Nu m ) non-s, which gives the cycle-space-averaged Nusselt number Nu m and the steady, nonpulsating averaged Nusselt number (Nu m ) s over a heat source normalized by the corresponding steady, nonpulsation, nonporous-block value (Nu m ) non-s, respectively, is calculated. Figure 5b exhibits the effect of Da on Nu m =(Nu m ) non-s and (Nu m ) s =(Nu m ) non-s. Here, the abscissa is expressed in log scale to show clearly the effect of Da in the range to It is clear from Figure 5b that in the calculation range of Da, there exists a critcal Darcy number (about Da ¼ ) corresponding to the smallest values of Nu m =(Nu m ) non-s and (Nu m ) s =(Nu m ) non-s beyond which both heat transfer enhancement factors increase. The value of heat transfer enhancement factor for pulsating flow is higher than that for steady, nonpulsating flow because of the larger cycle-space averaged temperature gradient near the heat source surface. For Da ¼ , the cycle-space average Nusselt number of pulsating flow is about 1.20 times that of nonpulsating flow and 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 temperature fields over a periodic-steady pulsating cycle for Re ¼ 250, Pr ¼ 0.7, St ¼ 0.6,

17 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 441 e ¼ 0:6, F ¼ 0.057, k eff ¼ 1, Da ¼ , H p ¼ 0.3, W p ¼ 1.0, L i ¼ 3, and L o ¼ 9atA 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 during the flow pulsation reversal (xt ¼ p to 2 p). This leads to production of stronger downstream recirculation zones on the upper plate surface due to the lower flow momentum and the thinner thickness of the temporal thermal boundary layer on the bottom plate surface due to the reattachment of the larger amount of core fluid to the rear part of the heat source surface. It is seen from Figure 6b that Nu x increases with increased A because of the smaller oscillation temperature gradients near the heat source surface. Therefore, with an increase in A from 0.1 to 0.9, the gain in Nu m =(Nu m ) non-s increases from 0.95 to 1.8, as shown in Figure 6c. In addition, when the value of A is beyond 0.4, the Nu m of the pulsating flow is larger than that of nonpulsating flow. Effect of Pulsating Frequency St The effect of variations in the pulsating frequency or Strouhal number is depicted in Figure 7 for Re ¼ 250, Pr ¼ 0.7, e ¼ 0:6, F ¼ 0.057, k eff ¼ 1, Da ¼ , A ¼ 0.5, W p ¼ 1.0, H p ¼ 0.3, L i ¼ 3, and L o ¼ 9 with St ¼ 0.6, 1.0, and 1.8. The flow fields during a pulsating cycle with a phase angle increment of p=3 reveal that as St increases from 0.6 to 1.0, the upper recirculation zone behind the porous block becomes smaller and moves upstream. This gives rise to more fluid reattaching to the rear part of the heat source surface. As St is increased further to 1.8, the upper downstream recirculation zone gradually disappears at each time instant. Because of the decreasing downward motion of the core flow, the amount of fluid reattaching to the heat source plate is reduced. The instantaneous thickness of 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 temperature fields are not shown). The effects of St on Nu x are shown in Figure 7b. It can be seen that an increase in St from 0.4 to 1.8 results in a slight increase in a Nu x, until an optimal St=heat transfer rate (around St ¼ 1.0) is reached, and then decreases afterward. As expected, when St increases, the gain in Nu m =(Nu m ) s gradually 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 enhancement factor of the pulsating flow is always larger that of the steady flow. Effect of Porous Blockage Ratio H p Figure 8 presents the changes in Nu x,nu m =(Nu m ) non-s,and(nu m ) s =(Nu m ) non-s as the porous block height H p increases from 0.1 to 0.5 for Re ¼ 800, Da ¼ , A ¼ 0.5, St ¼ 0.6, W p ¼ 1.0, e ¼ 0:6; F ¼ 0.057, and k eff ¼ 1. It can be seen in Figure 8a that the distortions for streamlines become more conspicuous in each time instant as the porous block height increases from 0.1 to 0.5. The size and strength of the upstream and downstream recirculation zones of porous block increase with increasing H p.thisis due to the relative increase in the height of the porous block, which in turn offers a higher degree of obstruction and a larger blowing action to the flow for larger values of H p.this leads to a stronger suction effect caused by the pressure drop behind the porous block,

18 442 Y.-L. YEN ET AL. 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. resulting in a stronger reattached flow. Meanwhile, the thickness of the instantaneous thermal boundary layer decreases with increasing H p from 0.3 to 0.5, but increases as H p increases from 0.1 to 0.3 (figures not presented). As expected, Nu x increases with increasing H p from 0.3 to 0.5, shown in Figure 8b, due to the existence of a larger temperature gradient near the heater surface for the larger H p. Therefore, the gain in

19 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 443 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 periodic-steady cycle. Nu m =(Nu m ) non-s is more substantial for larger H p, as shown in Figure 8c. However,as H p decreases from 0.3 to 0.1, Nu x 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 (Nu m ) s =(Nu m ) non-s versus H p is steeper than that in the pulsating-flow case, especially in H p from 0.3 to 0.5. In other words, when the value of H p is less than 0.5, the heat transfer enhancement factor of the heater Nu m =(Nu m ) non-s > (Nu m ) s =(Nu m ) non-s, while the value

20 444 Y.-L. YEN ET AL. Figure 8. Effect of porous blockage ratio H p 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. of H p is larger than 0.5, Nu m =(Nu m ) non-s < (Nu m ) s =(Nu m ) non-s. This is due to the larger cycle-space average temperature gradient existing on the heater surface for the steady flow than for the pulsating flow as H p > 0.5.

21 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 445 Variations of Cycle-Average Local Surface Temperature Distribution The local surface temperature of the heater is more important than the average surface temperature in the application of electronic cooling [22]. Figure 9 shows the variations of cycle-average local temperature distribution along heat source surfaces for various values of Da, A, St, and H p. As seen in Figure 9, for the pure steady-flow case, the surface temperature of the heater increases along the flow direction and approaches a constant maximum value when the flow approaches the thermally developed region. However, for the steady-flow or pulsating-flow case with porous block, the heater surface temperature distribution curves are of convex shape, as a result of the boundary layer reattaching on the rear parts of the heaters. The maximum temperature point occurs at the center section of the heater surface, where the 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, W p ¼ 1.0, H p ¼ 0.3, and k eff 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 surface as Da decreases from to The maximum temperature occurs at Da ¼ Below and above this critical value, the maximum temperature goes down. Comparison of the cycle-average local surface temperature distribution T x for the pulsating-flow case with that for the steady-flow case in Figure 9a indicates that the cycle-average local surface temperature distribution for the pulsating-flow case with a porous block is more uniform than that for the steady-flow case with or without a porous block. That is, the temperature difference between the maximum and minimum temperatures on the surface of the heat sources for the pure steadyflow case is higher than that for the pulsating-flow case with a porous block. In addition, the maximum temperatures T x at the heater surface is smaller for the pulsating-flow case with a porous block than for the steady-flow case with or without a porous block. This is because the larger cycle-space averaged temperature gradient near the heater surface leads to more heat removal. The local surface temperature distribution uniformity of the heater increases as pulsating amplitude A (at Re 250, Da ¼ ,St¼0.8, W p ¼ 1.0, H p ¼ 0.3, and k eff ¼ 1.0) increases, as seen in Figure 9b. As seen in Figure 9c, the surface temperature distribution uniformity of the heater increases slightly as St increases from 0.4 to 1.0, while it decreases to a fixed uniform distribution as St increases from 1.0 to 1.8 (for Re ¼ 250, Da ¼ , A ¼ 0.6, H p ¼ 0.3, W p ¼ 1.0, and k eff ¼ 1). The effect of H p on the heater surface temperature distribution uniformity is shown in Figure 9d. It can be seen that an increase in H p from 0.1 to 0.3 (at Re ¼ 800, Da ¼ , A ¼ 0.5, St ¼ 0.6, and W p ¼ 1.0) results in a slight increase in the heater surface temperature distribution uniformity until a maximum temperature difference is reached, and then it decreases as H p increases from 0.3 to 0.5. Pressure Drop Calculation The overall pressure drop throughout the entire channel length is another important quantity, since this added pressure drop is the price one pays for the gain in heat transfer enhancement by both forced pulsation and porous medium. In the stream function vorticity formulation, the pressure field is eliminated in obtaining

22 446 Y.-L. YEN ET AL. 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) H p. the solution. However, the pressure field can be recovered from the converged stream function and vorticity fields. This is done by integrating the pressure gradient along the upper channel wall. The temporal pressure gradient in a periodic steady state is derived from the unsteady momentum equation using the no-slip boundary conditions on the solid wall. The total temporal pressure drop DP along the upper channel wall is then obtained from DP ¼ Z Lt 0 qp qx dx ¼ y ¼1 Z L t 0 qu qt þ 1 qn Re qy dx y ¼1 ð18þ

23 HEAT TRANSFER OF A POROUS-BLOCK-MOUNTED SOURCE 447 The corresponding steady nonpulsating nonporous-block pressure drop (DP ) non-s is ðdp Þ non-s ¼ Z L t 0 qp qx dx ¼ y ¼1 Z L t 0 1 qn Re qy dx ð19þ y ¼1 where pressure P is nondimensionalized with respect to qu 2 o. The effects of Da, A, St, and H p on the temporal pressure drop factor DP =ðdp Þ non s, which gives the overall pressure drop throughout the entire channel length, normalized by the corresponding steady nonpulsating nonporous-block value (DP ) non-s, is presented in Figure 10. In Figure 10a, the amplitude of the temporal pressure drop factor is less affected by a change in St (at Re ¼ 250, Da ¼ , A ¼ 0.6, H p 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) H p.

24 448 Y.-L. YEN ET AL. W p ¼ 1.0). However, the magnitude of DP =ðdp Þ non-s increases substantially as A (at Re ¼ 250, Da ¼ ,St¼ 0.8, W p ¼ 1.0, H p ¼ 0.3) or H p (at Re ¼ 800, Da ¼ , A ¼ 0.5, St ¼ 0.6, and W p ¼ 1.0) increases and or as Da (at Re ¼ 250, St ¼ 0.6, A ¼ 0.5, W p ¼ 1.0, and H p ¼ 0.3) decreases, as seen in Figures 10b 10d. This is because as the flow approaches the smaller passage formed by the 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 behind each porous block is not complete, because of the pressure loss in the recirculation zones. An increase in H p or a decrease in Da provides larger bulk frictional resistance that the flow encounters in the channel, resulting in a significant increase in variations of temporal pressure drop. An increase in A also leads to a larger pressure loss because of the formation of larger recirculating zones in the channel, as stated previously. Therefore, the required pumping power to maintain a pulsating flow increases with pulsation amplitude, Darcy number, and porous blockage ratio. The phase lead of the temporal pressure gradient DP =ðdp Þ non-s over the inlet pulsating velocity for all cases studied here is around p=2. This indicates that the flow pulsation considered in this study is in a higher-frequency regime compared to 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 zero as x increases. CONCLUSIONS This article has presented a numerical simulation of forced-pulsating convective flow in a parallel-plate channel with an isolated porous-block-mounted heat source. The results can be summarized as follows. 1. For the nonpulsating, steady flow case, the heat transfer rate from a strip heater can be enhanced by a fiber porous-block heat sink, depending on the consolidated 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 substantially destabilized by introducing pulsation and exhibits a periodic-changing flow pattern with cyclically expanding and shrinking alteration of the vortices. The temperature field is substantially affected in a similar way and presents a periodic oscillation of the thermal boundary-layer thickness near the heater surface. The cycle-average local surface temperature distribution of the heater for the pulsating-flow case with a porous block is more uniform than that for the pure steady-flow case. 3. The heat transfer enhancement factor of the heater increases with the pulsation amplitude. However, the effects of the Darcy and Strouhal numbers and of the porous blockage ratio are not straightforward. There exists a critical value for which the heat transfer enhancement factor is minimum (for Darcy number and blockage ratio number) or maximum (for Strouhal number). Below and above this critical value, the heat transfer enhancement factor goes up or drops off.

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