Laminar forced convection in a heat generating bi-disperse porous medium channel

International Journal of Heat and Mass Transfer 54 (2011) 636–644

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Laminar forced convection in a heat generating bi-disperse porous medium channel Arunn Narasimhan ⇑, B.V.K. Reddy Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e

i n f o

Article history: Received 27 July 2010 Received in revised form 30 August 2010 Accepted 31 August 2010 Available online 18 October 2010 Keywords: Forced convection Mono-disperse Bi-disperse Porous medium Electronics cooling Heat generation

a b s t r a c t Thermal management of heat generating electronics using the Bi-Disperse Porous Medium (BDPM) approach is investigated. The BDPM channel comprises heat generating micro-porous square blocks separated by macro-pore gaps. Laminar forced convection cooling fluid of Pr = 0.7 saturates both the microand macro-pores. Bi-dispersion effect is induced by varying the porous block permeability DaI and external permeability DaE through variation in number of blocks N2. For fixed Re, when 105 6 DaI 6 102, the heat transfer Nu is enhanced four times (from 200 to 800) while the pressure drop Dp* reduces almost eightfold. For DaI < 105, Nu decreases quickly to reach a minimum at the Mono-Disperse Porous Medium (MDPM) limit (DaI ? 0). Compared to N2 = 1 case, Nu for BDPM configuration is high when N2  1, i.e., the micro-porous blocks are many and well distributed. The pumping power increase is very small for the entire range of N2. Distributing heat generating electronics using the BDPM approach is shown to provide a viable method of thermo-hydraulic performance enhancement v. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction With progressive miniaturisation and associated higher heat generation, thermal management of electronic components is a pertinent problem as discussed in detail in recent review articles [1,2]. Spatial re-distribution of electronic components based on principles enables cooling by all forms of convection. Parallel plate channels containing distributed heated solid blocks mimicking electronics and cooled by forced convection, is studied using direct (non-porous medium approach) numerical simulations. These include configurations with uniform and random distribution of heated solid blocks [3–7]. Use of porous medium as inserts enhancing forced convection is another well-explored option [8]. In [9], laminar forced convection through a porous block mounted on a heated wall was studied using numerical methods. In [10], the flow field and thermal characteristics of external laminar forced convection flow over a porous block array were investigated numerically. Enhanced heat transfer in a channel using porous blocks was reported in [11]. Apart from the direct use of porous media, a porous medium approach – treating distribution of solids as porous media – for investigating thermal management of electronics has also been in practice due to faster computations than direct simulations. Some earlier research [12,13] and recent studies [14–17] have modelled the array of pin–fin heat sinks cooling electronics as Mono⇑ Corresponding author. Tel.: +91 44 22574696. E-mail addresses: [email protected] (A. Narasimhan), [email protected] (B.V.K. Reddy). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.08.022

Disperse Porous Medium (MDPM) channel using porous medium approach. The thermo-hydraulic performance of such a MDPM channel depends on the inlet fluid flow rate, thermal properties of fluid and solid matrix, permeability, inertial coefficient and porosity of the porous medium. In contrast to MDPM, a Bi-Disperse Porous Medium (BDPM) [18] is a macro-pored porous medium with its solid matrix replaced by a micro-porous medium. In a BDPM, the same fluid permeates the micro- and macro-pores, which are usually connected. Similar to heat exchangers enhanced by porous media [19,20], forced convection cooling of electronics can also be modeled using a BDPM approach. Recently, analytical studies for forced convection in BDPM channels were performed for thermally developing [21,22] and developed regions [23,24]. In [25], coupled conduction in plane slabs bounding the BDPM channel was studied. The local Nusselt number depended strongly on the BDPM thermal conductivity ratio and only moderately on the velocity ratio in the two bi-disperse pore scales. Heat generation effects under forced convection were not investigated in these studies. Modeling distributed heat generating electronics as a BDPM channel, influence of bi-dispersion effects on the forced convection thermo-hydraulic performance is investigated using numerical simulations in the present work. The BDPM domain considered is made of square heat generating micro-porous blocks separated by uniform macro-gaps. Bi-dispersion effect is induced by varying separately, the permeability of the porous blocks DaI and the permeability of the BDPM channel through the number of porous blocks N2.

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Nomenclature cF d d* D D* Dh Da H k K L L* n

inertial coefficient, dimensionless size of the electronic component, m size of the electronic component (d/H), dimensionless porous block size, m porous block size (D/H), dimensionless hydraulic diameter of the channel (4WH/2(W + H)), m Darcy number (K/H2), dimensionless depth of the channel, m thermal conductivity, Wm1 K1 permeability of the porous medium, m2 length of the channel, m length of the channel (L/H), dimensionless number of electronic components in the first column of n  n block configuration number of porous blocks in the first column of N  N porous blocks array Nusselt number, Eq. (12), dimensionless pressure, Nm2 pressure, dimensionless pressure drop ðpL  p2L Þ, Eq. (11), dimensionless Prandtl number (m/a) volumetric heat generation, W m3 Reynolds number based on hydraulic diameter of the channel (uinDh/m) Reynolds number based on depth of the channel (uinH/m) temperature, K velocity components along the x and y-axes, respectively, ms1 velocity components along the x and y-axes, respectively, dimensionless width of the channel, m

N Nu p p* Dp* Pr 000 Q Re ReH T u, v u*, v* W

2. Geometry and numerical model The BDPM channel, modeled as a two-dimensional porous block array, is shown in Fig. 1a. Each porous block of size D*  D* can be viewed as an array of n  n regularly arranged square electronic components, each electronic component having dimensions of d*  d* as shown in Fig. 1b. In this case, n = 5. The heat generating porous block in a flow channel of size W*  L* is considered an independent unit, disconnected from neighbouring blocks. The blocks are distributed uniformly in the channel. The generated heat is removed by the fluid flowing through micro- and macro-pores of the BDPM channel. The fluid flow and heat transfer behavior are studied using the two-dimensional versions of the mass, momentum (modified Navier–Stokes) and energy conservation equations. The following assumptions are made: The steady forced convection flow is incompressible and laminar. The Newtonian fluid retains its phase and its thermo-physical properties are invariant. The porous blocks are homogeneous and isotropic with constant thermo-physical properties. The volumetric heat generation in the porous blocks is uniform. The convecting fluid and the porous matrix are in local thermal equilibrium. The system of governing partial differential equations are nondimensionalized using the following terms

x x ¼ ; H 

p ¼

y y ¼ ; H 

ðp  pref Þ ; qu2in

u ; u ¼ uin 

Pr ¼

v



¼

v uin

;

h¼

T  T in 2 Q 000 s H =kf

;

keff m KI quin H ; DaI ¼ 2 ; ReH ¼ ; c¼ ; a l kf H ð1Þ

W* x, y x*, y*

width of the channel (W/H), dimensionless coordinates, m coordinates, dimensionless

Greek symbols thermal diffusivity, m2 s1 d macro-pore channel width, dimensionless c thermal conductivity ratio (keff/kf), dimensionless / volumetric porosity, dimensionless l dynamic viscosity, kgm1 s1 m kinematic viscosity, m2 s1 h temperature, dimensionless q density, kg m3 w* stream function (u* = ow*/@y* and v* =  ow*/@x*), dimensionless v performance parameter, Eq. (14), dimensionless

a

Subscripts cf clear fluid eff effective E external, macro-porous medium f fluid gen generation in inlet I internal, micro-porous medium max maximum out outlet pm porous media s solid, surface ref reference

where pref and Tin are the reference pressure and temperature and are taken as the atmosphere pressure and inlet temperature respectively. Using the above terms, the nondimensional governing equations for mass, momentum in the x* and y* directions and heat transport in porous media are written as

@u @ v  þ ¼0 @x @y

ð2Þ !

@u @u @p / @ 2 u @ 2 u þ v   ¼ /2I  þ I þ @x @y @x ReH @x2 @y2 " # ffi /2I cF /I1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v 2 þ u2 u þ pffiffiffiffiffiffiffi  ReH DaI DaI !   @v  @v @p /I @ 2 v  @ 2 v  u þ v   ¼ /2I þ þ @x @y @y ReH @x2 @y2 " # 1=2 ffi /2I cF /I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ pffiffiffiffiffiffiffiffi v þ u v   ReH DaI DaI ! 2 2 c @ h @ h ð1  /I Þ  @h  @h þ þv u ¼ þ ; @x @y ReH Pr @x2 @y2 ReH Pr

u

ð3Þ

ð4Þ ð5Þ

where c(keff/kf) is the ratio of effective thermal conductivity of the porous medium to the thermal conductivity of a flowing fluid. Inside the macro-pore region, the Eqs. (3) and (4) are solved by setting /I = 1 and DaI = infinity. Eq. (5) is solved by setting keff = kf. The associated boundary conditions for Eqs. (2)–(5) with respect to the geometry shown in Fig. 1a are

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Fig. 1. Schematic of (a) physical and computational configuration with 5  5 porous block array, (b) porous block with 5  5 pin–fins and (c) mesh.

At the inlet

x ¼ 0 :

u ¼ 1;

v ¼ 0

and h ¼ 0:

ð6Þ

At the exit

x ¼ ðLin þ L þ Lout Þ :

@ 2 u @2h ¼ v  ¼ 2 ¼ 0; 2 @x @x

W :

u ¼ v  ¼ 0 and

@h ¼ 0: @y

ð7Þ

ð8Þ

Sudden changes in thermophysical properties such as permeability, porosity and thermal conductively at the fluid/porous interface, are treated using harmonic mean formulation [26]. At the interface between the macro-pores and the porous blocks the following conditions are imposed

ucf ¼ upm ; hcf ¼ hpm ;

pcf ¼ ppm ;  @h    @n cf

  leff @u  @u  ¼ and @n cf lcf @n pm  @h  ¼ c  ; @n

1 Dp ¼  W 

where Lin is the upstream length and Lout is the downstream length of the BDPM channel (Fig. 1a). Sufficient buffer lengths are provided for the up and down streams of the BDPM channel. An upstream length Lin ¼ L has been provided such that flow is uniform (uin, Tin) and there is no thermal ’back’ diffusion at the inlet. An outflow boundary condition is used at the exit of the flow domain. To meet this criterion, an additional buffer length Lout ¼ 20L is provided down stream of the BDPM channel. At the upper and lower walls

y ¼ 0;

The pressure drop and heat transfer rate are predicted in terms of nondimensional pressure drop Dp* and Nusselt number Nu. These are evaluated as

ð9Þ ð10Þ

pm

where n denotes the direction normal to the corresponding wall of each porous block.

"Z

W





p dy jx ¼L 

0

Z 0

#

W 



p dy jx ¼2L

ð11Þ

and 2

Nu ¼

RNi¼1

R D R D 0

0

2





Nux y dx dy 2

ðN D Þ

;

ð12Þ

where

Nux y ¼

2 Q 000 1 s H ¼ : ðT x;y  T in Þkf hx y

ð13Þ

The performance parameter of a uniformly heat generating BDPM channel, v, is defined as the maximum ratio of the dissipated heat to pumping power and is written as

v¼

Nu ; Dp Re3

ð14Þ

where, Re is the Reynolds number based on the hydraulic diameter of the channel (Re = ReH  (Dh/H)). For a given fluid (Pr = 0.7), BDPM channel volume (W*  L*  1) and temperature difference, the dissipated heat is proportional to Nu which is dependent on the area (W*  L*) of the BDPM channel. The pumping power is proportional to the D p*Re3. Further, for the BDPM geometry of W* = L* in Fig. 1a, once the macro-pore volume fraction /E and the number of porous blocks N2 are known, individual block size D* and macro-pore width d can be determined uniquely by using the relations,

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/E ¼ 1  N2 D2

639

ð15Þ

and

d¼

1  D N ; N

ð16Þ

where Eq. (15) is obtained by observing that the channel length or width has the same length as N macro-pore widths and N porous blocks. 3. Numerical solution procedure and grid independence study The finite volume formulation of Eqs. (2)–(5) and the associated boundary conditions described by Eqs. (6)–(10) are used for numerical simulations. The convective terms of Eqs. (2)–(5) are discretized using a hybrid scheme and the diffusion and pressure terms with the central difference scheme. The pressure and velocity coupling in Eqs. (2)–(4) is handled using SIMPLE algorithm [26]. The resulting algebraic equations are solved iteratively with ADI method using the tri-diagonal matrix algorithm. The convergence criteria (the difference between consecutive iterations at each cell centre) for the mass, momentum and energy equations are set as 106, 106 and 109 respectively. Grid independence tests under steady state are performed for various numbers of porous blocks (N2 = 1–81) (Table 1). For brevity, the non-uniform, orthogonal, cosine grid used in the control volume of the porous block is shown in Fig. 1c. Numerical simulations are performed with the grids marked bold for various numbers of porous blocks. A numerical parameter (Qout  Qin)/Qgen is defined to check the consistency of the steady state energy balance, and the results are found to validate as (Qout  Qin)/Qgen = 0.999.

Table 1 Grid independence study for the porous block arrays at Re = 1500, DaI = 103, Pr = 0.7, c = 100, /E = 0.64 and /I = 0.75 (values in bold face are grids chosen for further simulations). NN

Grid size

Nu

Percentage of error

11

33

55

77

99

Fig. 2. The validation of the numerical methods (a) the local Nusselt number variation along the top surface of the solid block [27] and (b) the local Nusselt number variation along the surface of the porous block [9].

351  31 641  121 881  241 1241  361 1581  481

805.33 769.25 746.95 752.73 751.52

4.69 2.99 0.77 0.16

349  29 597  57 753  113 1029  169 1325  225 1687  337

505.47 837.71 923.34 930.28 928.96 929.16

39.66 9.27 0.75 0.14 0.02

465  45 719  89 1137  179 1565  265 1833  353

568.09 949.82 1066.56 1083.12 1088.44

40.19 10.94 1.53 0.49

581  61 871  121 1191  181 1381  241 1711  361 1941  481

642.24 1073.26 1176.89 1212.31 1235.67 1240.84

40.16 8.8 2.9 1.8 0.41

667  77 933  153 1229  229 1545  305 1947  457

716.27 1193.0 1310.44 1351.65 1359.09

39.96 8.96 3.05 0.55

jNunew Nuold j  Nunew

100

The numerical code has been validated with published results for laminar convective cooling over (a) solid block [27] and (b) porous block [9] mounted to the wall as shown in Figs. 2a and b. 4. Results and discussion The numerical simulations are performed for fixed values of Pr = 0.7, c = 100, cF = 0.08, /I = 0.75, /E = 0.64, H = 25 mm and W = 75 mm. The influence of internal permeability of the porous blocks DaI (107 6 DaI 6 102) and the external permeability DaE of the BDPM channel (by varying number of porous blocks 1 6 N2 6 81) on the fluid flow and heat transfer characteristics of the Bi-Disperse Porous Medium (BDPM) channel are analysed in detail. 4.1. Effects of the porous block internal permeability (DaI) For Re = 1000, N2 = 25, /I = 0.75 and /E = 0.64, the effects of internal permeability DaI of the porous blocks on the streamlines and isotherms are shown in Fig. 3. The heat generating micro-porous blocks are shown as thin continuous wall squares. Streamlines and isotherms in Fig. 3a, are for DaI = 107, which can be considered as the Mono-Disperse Porous Medium (MDPM) limit where the blocks are solid (DaI ? 0). Convection is restricted mostly to the macro-pores, around the micro-porous blocks. From this

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Fig. 3. Streamlines and isotherms for 5  5 porous blocks at various internal permeability DaI values (a) 107 (b) 104 (c) 103 (d) 102 [Re = 1000, /E = 0.64 and /I = 0.75].

MDPM limit, as DaI increases, bi-dispersion effects appear. Fluid permeates the micro-pores as well, resulting in more uniform distribution of streamlines as seen in Fig. 3(b)–(d). Convection heat transfer increases significantly at DaI = 102. The local maximum temperature of the porous blocks decreases with an increase in DaI value. The flow disturbance (vortex zone) downstream of BDPM channel also decreases with an increase in DaI value. For parameters given in Fig. 3, the effect of DaI on temperature profiles h(y*) along the vertical sections of the 5  5 porous block array are shown in Fig. 4. Fig. 4(a) corresponds to x* = 1.5L* and Fig. 4(b), to x* = 2L* (see Fig. 1a). For Re = 1000, in Fig. 4(a), increas-

ing DaI increases bi-dispersivity (i.e. effect of bi-dispersion). Since /I = 0.75 is fixed, the heat generation magnitude remains same even when DaI is increased. Hence, the local maximum temperature of the heat generating micro-porous blocks decreases due to improved forced convection to the fluid that permeates the micro-pores. At the limit of DaI = 107 (i.e. as DaI ? 0), the micro-porous blocks behave almost as solids. The forced convection flow dominates the horizontal macro-gaps between two rows of micro-porous blocks. However, the fluid trapped in the successive macro-gaps between the micro-porous blocks along the flow

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direction recirculates with higher magnitude. This local forced convection effect cools the micro-porous blocks more than that for higher DaI values. Hence in Fig. 4a the local micro-porous block temperature for DaI = 107 is much lower than that for other DaI values. However, this effect is not to be observed when only the fluid temperature is measured, as reported in Fig. 4b at the exitsection of the BDPM domain. Here, the fluid adjacent (downstream) to the micro-porous blocks record higher temperature when DaI = 107, as expected. The effects of DaI on the Nusselt number Nu (Eq. (12)) and the non-dimensional pressure drop Dp* (Eq. (11)) in the BDPM domain are shown in Fig. 5. The Dp* across the BDPM channel is calculated between vertical sections at x* = L* and x* = 2L* and the pressure at each section is the integral value. By the explanation of Figs. 3 and 4, an increment in DaI causes reduction in BDPM channel pressure drop and increase in the forced convection. Correspondingly, in Fig. 5, Nu is enhanced almost four times (from 200 to 800), as DaI is increased from 107 to 102 while the Dp* reduces almost eightfold. However, the variation in Nu and Dp* with DaI is not linear. For DaI < 105, Nu reduces quickly to reach a minimum at the MDPM limit (DaI ? 0), Fig. 5. Beyond DaI > 105, the enhancement in Nu and the corresponding reduction in Dp* is phenomenal. This can be attributed to the increased permeation of flow inside the micro-porous blocks due to the reduced local viscous drag. The BDPM

641

channel accommodates more uniform mass flow in a cross section, reducing Dp* and the block temperature h (thus increasing Nu). In Fig. 6, the reduction in hmax and increase in channel performance v with an increase in DaI is observed. When DaI > 105, steep rise of v and sharp fall of hmax is observed. This suggests, more permeable the porous block, better would be the cooling performance of the BDPM channel. It can be surmised that, when compared to a MDPM channel, the effect of bi-dispersion by increasing the DaI is to provide large heat transfer enhancement (4 times) and corresponding large reduction (8 times) in pressure-drop. Distributing heat generating electronics that comprise the micro-porous blocks by suitably choosing DaI provides thermo-hydraulic performance enhancement using BDPM approach. 4.2. Effects of number of porous blocks (N2) Finally, the effect of inducing the bi-dispersion effect by changing the number of blocks N2 is studied by keeping Re = 1000, /E = 0.64 and DaI = 103 as constants. The N2 values used are 1, 9, 25, 49 and 81 and the corresponding block sizes D* are calculated using Eq. (15). As /E is fixed, the block sizes progressively decrease with increase in N2 values.

Fig. 5. The effect of internal permeability DaI on Nusselt number (Nu) and pressure drop (Dp*) for 5  5 porous blocks [Re = 1000, /E = 0.64 and /I = 0.75].

Fig. 4. Temperature profiles h(y*) for 5  5 porous blocks at various internal permeability DaI values. (a) x* = 1.5L* (b) x* = 2L* [Re = 1000, /E = 0.64 and /I = 0.75].

Fig. 6. The effect of internal permeability DaI on performance parameter (v) and maximum temperature (hmax) for 5  5 porous blocks [Re = 1000, /E = 0.64 and /I = 0.75].

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Streamlines and isotherms for the different N2 values are shown in Fig. 7. The configuration can be viewed as follows: In Fig. 7, the heat generating electronics, for instance, is packed in such a way

that they crowd as a single micro-porous block in a channel flow.In the other figures (from b to e), the electronics are redistributed in smaller and smaller groups of greater number of micro-porous

Fig. 7. Streamlines and isotherms for various number of porous blocks N  N (a) 1  1 (b) 3  3 (c) 5  5 (d) 7  7 (e) 9  9 [Re = 1000, DaI = 103, /E = 0.64 and /I = 0.75].

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643

Fig. 10. The effect of porous blocks N2 on performance parameter (v) and maximum temperature (hmax) [Re = 1000, DaI = 103, /E = 0.64 and /I = 0.75].

Fig. 8. Temperature profiles h(y*) at (a) x* = 1.5L* (b) x* = 1.5L* for various number of porous blocks N  N [Re = 1000, DaI = 103, /E = 0.64 and /I = 0.75].

pores, resulting in a uniform temperature distribution as observed in the isotherms in Fig. 7. The temperature profiles h(y*) at x* = 1.5 L* and x* = 2L* for various values of N2 are shown in Fig. 8. An increase in N2 resultsin a decrease in the local maximum temperature and an increase in the local minimum temperature. The temperature gradients in microporous blocks and macro-pores increase with N2 indicating improved heat transfer. For fixed values of DaI = 103, Re = 1000 and /E = 0.64 the effects of N2 on Nu and Dp*, and, v and hmax are shown in Figs. 9 and 10, respectively. The variation in Dp* with N2 is minor and is due to a small, single order change in effective permeability of the BDPM channel over the entire range of N2. However, there is considerable increase in the heat transfer rate Nu, and hence the performance v (in Fig. 10) of the BDPM channel with increase in N2 value. The maximum temperature hmax decreases as expected with increase in N2 as shown in Fig. 10. The effect of N2 on these parameters is more when N2 < 49. Increasing N2 > 49 results only in a minor increase in v, the BDPM channel performance parameter. It can be inferred that, in comparison to the N2 = 1 case, the BDPM configuration provides higher heat transfer enhancement when the micro-porous blocks are many and well distributed (N2  1). The pumping power remains almost unaltered (mildly increases from 2.5 to 2.9 for the entire range of N2). 5. Conclusions

Fig. 9. The effect of porous blocks N2 on Nusselt number (Nu) and pressure drop (Dp*) [Re = 1000, DaI = 103, /E = 0.64 and /I = 0.75].

blocks. From a to e, in Fig. 7, the magnitude of heat generation remains same, as /E and /I are fixed. From the streamlines, the fluid flow is inferred to become more uniform across the BDPM channel cross section, as N2 increases. Bidispersion causes fluid permeation through micro- and macro-

Using the Bi-Disperse Porous Medium (BDPM) approach, steady, laminar, forced convection cooling of heat generating electronics is investigated. In the numerical simulations, the BDPM channel is made of uniformly distributed square micro-porous blocks separated by macro-pore gaps. The effect of bi-dispersion is induced by varying the porous block permeability DaI separately and by varying the external permeability DaE through variation in number of blocks N2 in the channel. The variation in the heat transfer Nu and pressure drop D p* with DaI is found to be non-linear. For DaI < 105, the bi-dispersivity effect on Nu and Dp* is weaker and reaches minimum and maximum value respectively, at the Mono-Disperse Porous Medium (MDPM) limit (DaI ? 0). However, for DaI > 105, as local viscous drag reduces, the enhancement in Nu and the corresponding reduction in Dp* is phenomenal. The BDPM channel accommodates more uniform mass flow in a cross section, reducing Dp* and the block temperature h.

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When compared to a MDPM channel (i.e. DaI ? 0), the effect of bi-dispersion by increasing DaI provides large enhancement in heat transfer (4 times) and corresponding large pressure drop reduction (8 times). The increase in Dp* with N2 is minor over the entire range of N2. When N2 < 49, the degree of increase in Nu is significant resulting in improved cooling performance (large v values). However, further increase in heat transfer is not observed for N2 > 49, which results only in a minor increase in v. Compared to the N2 = 1 case, the BDPM configuration provides higher heat transfer enhancement when N2  1, i.e. the micro-porous blocks are many and well distributed. Using the BDPM approach, modeling heat generating electronics as micro-porous blocks separated by macro-gaps is advantageous. The geometry and distribution of such a BDPM channel can be suitably tuned through the bi-dispersion parameters (namely, DaI and N2) of such a BDPM channel to augment the heat transfer and reduce the pressure drop.

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