and Nanoscale Thermal Flows in Porous Structures

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A. H. Meghdadi Isfahani

Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad 8514143131, Iran e-mails: [email protected];

[email protected]

Parametric Study of Rarefaction Effects on Micro- and Nanoscale Thermal Flows in Porous

Structures

Hydrodynamics and heat transfer in micro/nano channels filled with porous media for different porosities and Knudsen numbers, Kn, ranging from 0.1 to 10, are considered.

The performance of standard lattice Boltzmann method (LBM) is confined to the microscale flows with a Knudsen number less than 0.1. Therefore, by considering the rarefaction effect on the viscosity and thermal conductivity, a modified thermal LBM is used, which is able to extend the ability of LBM to simulate wide range of Knudsen flow regimes. The present study reports the effects of the Knudsen number and porosity on the flow rate, permeability, and mean Nusselt number. The Knudsen’s minimum effect for micro/nano channels filled with porous media was observed. In addition to the porosity and Knudsen number, the obstacle sizes have important role in the heat transfer, so that enhanced heat transfer is observed when the obstacle sizes decrease. For the same porosity and Knudsen number, the inline porous structure has the highest heat transfer performance.[DOI: 10.1115/1.4036525]

Keywords: micro/nano flows, porous media, lattice Boltzmann method, convection

1 Introduction

Recent technological developments have made it possible to design various micro/nano devices such as micro/nano channels, nozzles, and pumps, where fluid flow and heat transfer are involved. For example, an electroosmotic micropump was fabri- cated by Ai et al. [1] by using a thin nanoporous anodic Alumi- num oxide membrane with the pore sizes about 200 nm. Also, the photoelectrical wet etching method was used by Jiang et al. [2] to fabricate a nanochannel embedded with the nanoporous structures to fabricate light-emitting diodes (LEDs). In another work, nanoporous silicon with average pore size less than 20 nm is produced by electrochemical etching method [3]. Furthermore, Ohba [4]

introduced porous materials with average pore sizes from 0.7 nm to 2.9 nm, which can be fitted into nano channels by using electrochemical methods.

In micro/nano devices, porous media can be used for microfil- tration, fractionation, catalysis and microbiology related applications [5]. For example, micropacked beds or sintered metal fibers can be used in microstructured reactors for catalytic reaction [6,7]

and the filter medium prepared from commercially available glass fibers of 70–140lm thickness is suitable for applications in bio filtration systems [8]. Also, charged porous media structures have been employed in microdevices to magnify the pumping [9,10], mixing [11,12], and separating [13] effects.

Micro-heat-exchangers can be applied in many important fields:

micro-electronics, aviation and aerospace, medical treatment, bio- logical engineering, materials sciences, cooling of high temperature superconductors, thermal control of film deposition, cooling of powerful laser mirrors, and other applications where light- weight, small heat exchangers are required. Compared with conventional heat exchangers, the main advantage of the micro-heat exchangers is their extremely high heat transfer area per unit volume [14]. Previous results have shown that flow through porous media can greatly enhance the convection heat transfer [15,16].

The use of porous media in the micro/nano heat exchangers can verify this phenomenon. The microporous heat exchangers have better heat transfer performance than the microchannel heat exchangers [17,18].

Conventional numerical methods using the Navier–Stokes equations fail to predict some aspects of micro/nano flows such as nonlinear pressure distribution, increase mass flow rate, slip flow and temperature jump at the solid boundaries. This is because of the fact that when the mean free path,k, of the molecules becomes comparable to the characteristic length of the flow domain, the continuum flow model (Navier–Stokes equations) breaks down and the Knudsen number, defined as the ratio of the molecular mean free path to the characteristic length of the system, increases. For Kn<10³the continuity assumption with no slip boundary conditions is valid. For 10³<Kn<10¹ (slip flow regime) the velocity and temperature of the gas near the wall are no longer equal to the wall velocity and temperature, respectively, and for Kn>10¹(transition and free molecular flow regime) the continuity assumption is under question [19].

This necessitates the development of new computational methods which depend on the kinetic theory that are both accurate and computationally efficient. Recently, the lattice Boltzmann method (LBM) has received considerable attention to simulate slip flow regimes in microchannels [20–22] and microporous media [23]

but only a few papers can be mentioned for the use of LBM in transition regime [24–30]. Though the LBM depends on the Boltzmann equation, which is valid in the whole rarefaction regimes, the performance of standard LBM is confined to the microscale flows with Kn, less than 0.1. To cover a wide range of the flow regimes, in the previous works [31,32], a new relaxation time formulation, named as modified LBM, by considering the rarefaction effect on the viscosity and thermal conductivity is proposed. The modified LBM was also utilized in some researches.

For example, Zhuo and Zhong [33] simulated the microchannel gas flows with a wide range of Knudsen numbers covering from the slip regime up to the entire transition regime via the D₂Q₉ LBM. For all test cases conducted in this study, the results are found to be in quite good agreement with the solutions of the linearized Boltzmann equation, the results of the direct simulation

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 8, 2016; final manuscript received March 11, 2017; published online May 9, 2017. Assoc. Editor:

Peter Vadasz.

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Monte Carlo (DSMC) and IP-DSMC methods. Also, Liou and Lin [34] performed numerical simulations on the pressure-driven rarefied flow through channels with a sudden contraction–expansion of 2:1:2 using D2Q9and D3Q19lattice Boltzmann methods covering the slip and transition flow regimes. The numerically computed results are found to be in reasonably good agreement with the experimental ones. In another work, Li et al. [35] used this effective relaxation time with a multiple relaxation time LBM to account for the rarefaction effect on gas viscosity. The results, including the velocity profile, the nonlinear pressure distribution along the channel, and the mass flow rate, are in good agreement with the solution of the linearized Boltzmann equation, the DSMC results, and the experimental results over a broad range of Knud- sen numbers. Kalarakis et al. [36] employed a single relaxation time model with this modified D2Q9lattice Boltzmann method to simulate flow between parallel plates and flow in porous media.

They tested the modified LB method against the DSMC method in the simple case of a straight channel. It was found that the velocity profiles were reproduced with reasonable accuracy for Knudsen numbers in the range of 0.1–10. In addition, the permeability predictions of the two methods for computer-aided recon- structions of porous media are in very good agreement with each other over the entire transition flow regime. They concluded that the modified D2Q9LB method is suitable for prediction of the gas permeability in porous media over the entire slip and transition flow regimes.

Previous researches [37,38] demonstrate the remarkable ability of LBM for modeling flow features in complicated geometries such as porous media. Therefore, the aim of the present article is to use the modified LBM to simulate gas flows in two- dimensional micro/nano porous geometries in order to investigate the effect of various parameters such as porosity and Knudsen number on the volume flow rate, permeability, pressure drop, and Nusselt number.

2 Theory

2.1 Flow Through Porous Media.For relatively low Reyn- olds number flows between parallel plates with large aspect ratio, the inertial effects in the momentum equation can be neglected.

Under such conditions, the momentum equation along the stream- wise direction is reduced to the following form

d²u dy²¼1

l dp

dx (1)

where u is the axial velocity along the x direction, l is the dynamic viscosity of the fluid andpis the pressure. By using the second-order slip boundary conditions, this equation results in the following analytical solution for the corresponding volumetric flow rate [19]:

q¼ h² 2l

dp dx

1

6þC1Knþ2C2Kn²

(2) wherehis the channel height andC1andC2are the first-order and second-order slip coefficients, respectively.

For creeping flows, moving with a slow steady velocity (Re<1), the volumetric flow rate per unit cross-sectional area of porous medium,q, obtains through the Darcy’s law

q¼ k l

dp

dx (3)

wherekis the permeability. In comparison with Eq.(2), the following formula can be derived fork:

k¼h²

121þ6C1Knþ12C2Kn²

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Thus, for micro-/nanoscale flows, the permeability depends only on the structure of the medium and the Knudsen number and is independent of the fluid viscosity.

The mean Knudsen number is defined as Kn¼ l

hp ffiffiffiffiffiffiffiffiffi pRT 2 r

(5) where R is the gas constant,T is the temperature, and pis the mean pressure of the channel inlet and outlet, ðpinþpoutÞ=2.

Thus, the permeability can be written as k¼h²

12 1þ6C1

l h

ffiffiffiffiffiffiffiffiffi pRT 2

r 1

pþ12C2

l² h²

pRT 2

1 p²

! (6)

For Kn1 the second-order correction term in Eq.(4)can be neglected. Therefore, the gas permeability is a linear function of 1=p. This is consistent with the Klinkenberg results [39]. The effect of 1=p²accounts for deviation from a straight line from the curve relatingkand 1=p, especially for large Knudsen numbers.

In the present study, the permeability is calculated based on the flow rate and pressure drop. For the micro-/nanoscale flows, the rarefaction effects lead to a nonlinear pressure drop resembling an incompressible flow [19]. Therefore, the gas permeability should be evaluated using the solution of compressible gas flows [40]

k¼ 2lLq_op_o

p_i²p_o² (7)

wherep_i,p_o,q_o, andLare the inlet pressure, outlet pressure, outlet velocity, and channel length, respectively. Then the Darcy number can be estimated using the following relation:

Da¼ k

h² (8)

2.2 Lattice Boltzmann Method.The continuum Boltzmann equation is a fundamental model for rarefied gases in the kinetic theory [41,42]

@_tfþ ðn:rÞf ¼Qðf;f⁰Þ (9) wherefðt;x;nÞis the probability of finding a molecule at position xwith molecular velocity ofnat timet, andQðf;f⁰Þis a nonlinear collision integral term withf andf⁰being the pre- and postcolli- sion probability distribution functions, respectively.

In this article, the two-distribution thermal lattice Boltzmann model based on the work of He et al. [43] is used, which utilizes two different distribution functions—one for the velocity field,f, and the other for the internal energy field,g

f_iðxþc_iDt;tþDtÞ f_iðx;tÞ ¼ 1

s_fþ0:5f_iðx;tÞ f_i^eqðx;tÞ (10) g_iðxþc_iDt;tþDtÞ g_iðx;tÞ ¼ 1

s_gþ0:5g_iðx;tÞ g_i^eqðx;tÞ (11) where s_f and s_g are the relaxation times for the number and energy density distribution functions, respectively. The new varia- blesfandg, which are introduced in order to have an explicit scheme, are defined as

f¼fþ 1

2s_fðff^eqÞ (12)

g¼gþ 1

2s_gðgg^eqÞ (13)

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f^eq and g^eq are the equilibrium distribution functions approxi- mated as Maxwellian form given by Eqs.(14)and(15)

f_i^eqðx;tÞ ¼q:w_i 1þu:ci

RTþðu:c_iÞ² 2ðRTÞ²u:u

2RT

!

w0¼4

9;wi¼1;2;3;4¼1

9;wi¼5;6;7;8¼ 1 36

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g^eq_i ðx;tÞ

¼

w0qe3ðu:uÞ

2c² i¼0

g^eq_a ¼w0qe 1:5þ1:5c_i:u

c² þ4:5ðci:uÞ²

c⁴ 1:5ðu:uÞ c²

i¼1;2;3;4 g^eq_a ¼w0qe 3þ6ci:u

c² þ4:5ðc_i:uÞ²

c⁴ 1:5ðu:uÞ c²

i¼5;6;7;8 8>

>>

><

>>

:

(15) In the present work, the nine velocity 2D model (D2Q9), is used to discrete momentum space [44]. In this model, the discrete velocity fieldc_i¼ ðc_ix;c_iyÞis given by

ci¼ 0;0 ð Þ; i¼0

cos i1 2 p

;sin i1 2 p

c; i¼1; :::;4 ffiffiffi2

p cos 2i9

4 p

;sin 2i9

4 p

c; i¼5; :::;8 8>

>>

><

>>

:

(16) where c¼Dx=Dt: Dxand Dtare lattice spacing and time step, respectively. The flow parameters are then calculated in terms of the particle distribution functionf_iandg_iby

q¼X

i

f_i; qu¼X

i

c_if_i; qe¼qRT¼X

i

g_i

t¼c²_ss_f c_s²¼RT¼c²

3 (17)

wheretis the kinematic viscosity.

In Refs. [31] and [32], we proposed an effective relaxation time which can simulate flow features for wide range of Knudsen numbers, including the transition regime, as follows:

seff¼ Kn

1þ2:2KnH (18)

s_g¼Prs_eff (19)

where Pr is the Prandtl number andHis the channel width in lattice units. In spite of the standard LBM, the velocity and temperature distributions, the volumetric flow rate and the local Nusselt number obtained from this modified LBM agree well with the other numerical and empirical results in a wide range of Knudsen numbers.

3 Results

3.1 Hydrodynamics.Isothermal Poiseuille flows in micro/

nano channels filled with porous media with different Knudsen numbers are simulated by using Eqs. (10), (14), and (18). For boundary conditions on the wall surfaces, the bounce-back boundary condition is coupled with specular reflection using a reflection coefficient r¼0.7 as presented by Succi [45]. It should also be noted that this method is formulated in terms off, notfand one

needs to convert betweenfandf for using the boundary conditions. To achieve a nonlinear pressure drop, inlet and outlet pressures are imposed using the method proposed by Lim et al. [46].

At the inlet and outlet nodes, the unknown velocitiesuandvare computed from interior fluid nodes according to the extrapolation method. After the extrapolation, the unknown distribution functions are computed according to the equilibrium distribution functions.

Although the Bhatnagar–Gross–Krook (BGK) model is used extensively with specular reflective boundary condition (SRBC), some researchers have noticed that using the BGK model with the bounce back boundary condition and its combinations such as SRBC leads to certain defects in prediction of the slip velocity at the walls especially for high Kn flows, so that the slip velocity is a numerical artifact of the BGK model rather than a physical effect because of its dependence on the grid resolution [47]. This defi- ciency is viscosity dependent as a consequence of the viscosity dependence of the bounce back type boundary conditions with the BGK model, which leads to a severe problem for simulating flow through porous media because the permeability becomes viscosity dependent, while it should be a characteristic of the physical prop- erties of porous medium alone [48]. Using the new relaxation time relation (Eq.(18)) causes the values ofs_f ands_gto be decreased which improves the accuracy of the boundary conditions signifi- cantly. On the other hand, in Ref. [31] we showed that by using the new relaxation time, the form of the slip boundary condition used in the proposed LBM will be equivalent to the higher-order boundary conditions used elsewhere, and consequently, higher Knudsen number flows can be simulated. Similar explanations can be developed for the inlet and outlet boundary conditions.

The simulated 2D porous media are similar to the structures shown in Fig.1. Disordered solid blocks are packed between two parallel plates. Porosityerepresents the fraction void volume in the porous structure and it is defined ase¼ ðVVsÞ=V, whereV is the total volume, andVsis the volume of the solid blocks.

Figure 2shows the volumetric flow rate through micro/nano channel without solid blocks between the two parallel plates (e¼1) normalized byDph²=Lðp ffiffiffiffiffiffi

RT

p Þ. The results of kinetic gas theory, linearized Boltzmann, and standard LBM are also presented. It can be seen from the figure that despite the standard LBM, the present LBM predicts the flow rate in wide range of Knudsen numbers accurately.

The calculated permeability under mesh refinement is shown in Table 1 for e¼0:825, p¼1:0510⁵Pa and Kno¼0:0236, which indicates that the results converge for the lattice 240600.

Tables2and3present the permeability of the porous structure withe¼1 ande¼0:825 calculated by the modified LBM as well as the results obtained by using the standard LBM [49].

These results agree very well for flows with Kn less than 0.1, which represents the validation of our results, while by increasing Kn, the deviation between the results of standard LBM [49] and modified LBM increases, which is because of the fact that the standard LBM is accurate only for slip flow regime with Knudsen numbers less than 0.1.

Figure3shows the volume flow rate as a function of the pressure difference between the inlet and outlet for various porosities and outlet Knudsen numbers, Kno. The figure presents that the values of velocities is small as the Mach number is less than 0.3.

Since the Reynolds number is small, the curves are linear, which is in agreement with Darcy low. By increasing the porosity, the obstacles decreases and the flow rate increases. However, for porous structures with relatively small porosities, the flow rate is affected to a less degree than those having higher levels of porosities. According to the figure, despite the fact that the porosity of the inline structure is less than that of the staggered one, the flow rate is higher. This phenomenon is related to the less tortuosity of the flow path in the inline porous structure.

The effect of Knudsen number on the volume flow rate at mean pressure p¼1:0510⁵Pa is shown in Fig. 4. Increasing Kn causes the increase in velocity slip, which leads to increase in

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flow rate. On the other hand, increase in Kn causes the increase in the collisions of the molecules with the walls. As a result, the effective viscosity increases, which leads to the decrease in flow rate. The net result is that by increasing the Knudsen number, the flow rate decreases initially and has a minimum value about Kn 0.1–0.2, and then it increases. This phenomenon is called Knud- sen’s minimum effect. For the pressure-driven flows in nonporous channels (e¼1), this effect occurs at Kn¼1. In Ref. [31], we showed that despite the standard LBM, the new LBM is able to capture the Knudsen’s minimum effect at Kn¼1 for nonporous channels. In the porous structures, the molecular collisions with the walls are more important with respect to the nonporous structures. Therefore, the increase in the effective viscosity is higher and the Knudsen’s minimum effect occurs at lower Knudsen numbers. Thus, the Knudsen’s minimum effect depends on the porosity and the structure of the porous media.

The average gas permeability is calculated from Eq. (7). The characteristics of Darcy number, Da, are shown in Fig.5for various Kn ande. As previously mentioned, Da is a linear function of the reciprocal mean pressure, 1=p which is consistent with the Klinkenberg effect. Because of the gas slippage on the solid boundaries, the friction drag reduces. Thus, the permeability and Darcy number increase with increasing the Knudsen number.

From the figure, we can see that Da decreases evidently as the mean pressure increases, especially for relatively high Knudsen numbers. The reason is that according to the relation Kn¼ Kn_op_o=pfor the fixed outlet conditions, by increasing the mean pressure, the Knudsen number and the rarefaction effects decrease, which results in larger flow resistance and lower gas permeability. On the other hand, by increasing Kn_othe reduction of the Knudsen number from the outlet to the inlet of the channel increases, so the reduction of the permeability increases. There- fore, the slope of the straight line relating the gas permeability to reciprocal mean pressure decreases.

Regarding the figure, it can be found that as the porosity decreases, the Knudsen number influence on the gas permeability becomes more evident. For example, for the porous structure with e¼0:881, the ratio of the Darcy numbers for two cases, Kn_o¼ 10 and Kn_o¼0:0236, is 14.54 while it is 19.77 for the porous Fig. 1 The simulated porous structure: (a)e50:881, (b)e50:861, (c)e50:825, (d)e50:732,

(e) inlinee50:877, and (f) staggerede50:897

Fig. 2 Volumetric flow rate as a function of exit

Table 1 Convergence under mesh refinement Lattice 40100 120300 240600 4801200 K(m²) 4:7410¹⁵ 3:6610¹⁵ 3:1910¹⁵ 3:2210¹⁵

Table 2 The permeability of the porous structure withe51 at mean pressurep51:05310⁵Pa

Output Knudsen Standard LBM [43] Present work Kno¼0:0236 1:2110¹³ðm²Þ 1:20210¹³ðm²Þ Kno¼0:055 1:4810¹³ðm²Þ 1:4610¹³ðm²Þ Kno¼0:1 1:810¹³ðm²Þ 1:6910¹³ðm²Þ

Table 3 The permeability of the porous structure withe50:825 at mean pressurep51:05310 ⁵Pa

Output Knudsen Standard LBM [43] Present work Kn_o¼0:0236 3:2610¹⁵ðm²Þ 3:1910¹⁵ðm²Þ Kno¼0:055 6:5210¹⁵ðm²Þ 5:7210¹⁵ðm²Þ Kno¼0:1 10:9710¹⁵ðm²Þ 9:4210¹⁵ðm²Þ

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structure with e¼0:732. This characteristic can be explained regarding the fact that: as the porosity decreases, generally the total surface area of the solid blocks increases and the slippage effect on the solid surfaces is more dominant, which leads to lower flow resistance through the porous structure.

The most important relations for calculation the pressure drop of a fluid flowing through a packed bed of solids are the Ergun (Eq.(20)) [50] and the Carman–Kozeny (Eq.(21)) correlations [51]

Dp L

D_p q_iu²_i

e³ 1e

ð Þ¼150lð1eÞ

q_iu_iD_p þ1:75 (20)

Dp L

Dp

q_iu²_i e³ 1e

ð Þ¼180lð1eÞ

q_iu_iD_p (21) where q_i and u_i represent the inlet mean density and velocity, respectively.D_pis the equivalent diameter of the particles. First, the value of the equivalent diameter of the body,D_p, should be determined. It should be noted that the Ergun and Carman–

Kozeny relations are able to calculate the pressure drop for continuum incompressible flows with Kn0:001:To calculateD_pwe used the method presented in Ref. [49]. A continuum incompressible flow with Kn¼0:001 and Dp¼0:110⁵ðPaÞ was Fig. 3 The volume flow rate versus the pressure gradient. Labelse50:877ðLÞande50:897ðSÞ

represent the results of inline and staggered porous structures, respectively.

Fig. 4 Knudsen minimum effect

Fig. 5 The Knudsen number influence on Darcy number

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considered andu_iwas determined by using the lattice Boltzmann method. Table 4 shows the equivalent diameter values, determined by comparing the calculated results with the Carman–

Kozeny equation.

Figure6compares the calculated results of pressure drops with those of the Ergun and Carman–Kozeny equations for various Knudsen number in slip and transitional flow regimes.

Because of the velocity slip and the rarefaction effects, the dimensionless pressure drop is lower than that of the Ergun and Carman–Kozeny equations at lower Reynolds numbers,qu_iD_p=

½lð1eÞ. However, the discrepancies between the results become lower and eventually the dimensionless pressure drop exceed the results obtained by Ergun equation at larger Reynolds numbers. As the inlet pressure increases, Re increases and the mean Knudsen number of the inlet and outlet decreases. That is, the mean Knudsen number discrepancies are small at higher inlet pressures, which results in the quite close pressure drops.

Note that the Ergun and Carman–Kozeny equations are limited to nearly incompressible flow with a linear pressure distribution along the channel while for the range of Knudsen number used in this work, the compressibility effect is important and causes a nonlinear pressure distribution. The stronger the rarefaction effects, the smaller the deviation from the linear pressure distribution. Therefore, for highest Reynolds numbers, which corresponds to the lowest Knudsen numbers, the pressure drop exceeds the

Table 4 The equivalent diameter of the porous structures

e 0.881 0.861 0.805 0.780 0.742 0.732 Inline Staggered DpðlmÞ 0.17 0.16 0.18 0.19 0.19 0.20 1.2 7.6

Fig. 6 Nondimensional pressure drop versus Reynolds number

Fig. 7 Pressure drops considering the compressibility versus Reynolds number

Fig. 8 Nusselt number obtained from the DSMC and the new LBM (Pr52/3)

Fig. 9 The heat transfer control volume

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continuum Ergun equation. Thus, considering the compressibility effects and comparing Eq.(7)with Eq.(3), we rewrite the ordinate as shown in Fig.7. From Fig.7, we can see that the pressure drops are below the continuum Ergun equation.

3.2 Heat Transfer.In this section, we consider a channel withe¼1 to validate the modified thermal LBM. The developing thermal flow in a micro/nano channel with different porosities is considered. Equations(7)and(8)are solved by using the effective relaxation times presented in Eqs. (13) and (14). A pressure- driven flow with the uniform inlet temperatureTi¼1 is imposed at the inlet of the channel. The channel outlet is fixed at atmos- pheric pressure with the fully developed temperature profile. It is assumed that the channel walls are heated uniformly with a constant temperature T_w¼10. In this section, the diffuse scattering boundary condition (DSBC) [52,53] is imposed on the solid walls to predict the slip velocity and temperature jump.

Figure 8compares the values of Nusselt number, Nu, at the fully developed region of the channel with e¼1 obtained from the modified LBM with the results based on the DSMC method [54]. It can be seen from the figure that the results, have good agreement for Kn0:2.

For the porous structure shown in Fig.9with the inlet and outlet temperaturesTiandTothe heat flux of the fluid is

q¼mc_ _pðT_oT_iÞ (22)

wherem_ andc_pare the mass flow rate and specific heat capacity.

The outlet temperature is obtained by the extrapolation method.

The average Nusselt number can be calculated as Fig. 10 The Knudsen number effect on the mean temperature

along the channel forp_i=p_o51:1

Fig. 11 The effect of Knudsen number at the channel outlet on the average Nusselt number

Fig. 12 The porosity effect on the mean Nusselt number

Fig. 13 The porosity effect on the mean temperature distribution along the channel

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Nu¼mc_ _p2H Tð _oT_iÞ

cA Tð _iT_wÞ (23)

wherecis the gas thermal conductivity andAis the total surface area of the solid surfaces.

For the present study, the flowing fluid is air with Pr¼0:7 and c_p¼1:0035ðkJ=kg KÞ. To study the Knudsen number effects on the temperature distribution, variation of the mean temperature along the channel is plotted in Fig. 10. In the entrance region, increasing Kn causes the temperature jump to be increased lead- ing to less amount of heat transfer from the wall to the fluid.

Hence, the thermal entrance length is increased as Knudsen number increases. On the other hand, increasing Kn increases Nu due to increase in mass flow rate. It can be concluded from the figure that at the entrance region of the channel, the temperature jump is dominant factor in determining the Nusselt number. In other words, in the entrance region, the decrease of Nusselt number due to the increase of temperature jump dominates the increase of Nusselt number due to the increase of mass flow rate.

Figure 11shows the effect of Knudsen number at the channel outlet on the average Nusselt number. The increase of Kn has two opposite effects on the Nusselt number. The first effect is that it increases the temperature jump. Therefore, the Nusselt number decreases. The second effect is that it leads to increase the slip velocity at the wall, since larger amount of fluid is moved near the wall. As a result, larger amount of fluid absorbs heat from the wall, so Nusselt number increases. It can be noted that by increasing the Knudsen number, the Nusselt number decreases initially and has a minimum value around Kn¼0.1, and then it increases.

The effect of Kn on the Nusselt number is similar to its effect on the mass flow rate (Fig.4). It can be concluded that for the current conditions, the effect of mass flow rate on the Nusselt number is more significant as compared with the effect of temperature jump.

Figure 12 shows the effect of porosity on the mean Nusselt number. At fixed Knudsen numbers and pressure ratios, there are two opposite trends which the porosity affects the Nusselt number.

Any increase in porosity would result in an increase in mass flow rate causing the Nusselt number to increase. On the other hand, with increasing porosity the net surface of the solid obstacles decreases, which leads to decrease in heat transfer area and Nus- selt number. The figure shows that the effect of mass flow rate on the Nusselt number dominates the effect of heat transfer area.

This in turn causes an increase in Nu as porosity increases.

The porosity effect on the mean temperature distribution along the channel length is shown in Fig.13. The curve shows that with

decreasing the porosity, the contact surface between the solid particles and fluid increases, which leads to the increase in mean temperature.

Figure14shows the effect of particle size on the heat transfer.

The symbols L and S indicate the inline and staggered porous structures, respectively. The figure indicates that while the porosity of three structures is relatively close together, an enhanced heat transfer is observed for the inline and staggered porous structures. The reason is that the solid blocks in the inline and staggered structures are smaller, so the heat transfer area is larger. It can be concluded that as the obstacle sizes in the porous media decreases, the thermal performance of the porous media increases.

We also conclude that the inline configuration has the highest heat transfer rate and lower pressure drop.

4 Conclusion

A lattice Boltzmann model is presented for the case of flow in micro/nano channels filled with porous media which can take into account the influence of gas rarefactions for wide range of flow regimes. Different porosities and Knudsen numbers are used and the effects of each parameter on the hydrodynamics and heat transfer are considered and discussed. The following conclusions can be drawn from the results:

(1) The Knudsen’s minimum effect occurs at Kn¼0.1 because of the tortuosity effects.

(2) For the same porous structure, Da increases with the increase of Knudsen number, but the rarefaction influence is more evident for low-porosity structures.

(3) Because of the velocity slip on solid boundaries, the dimensionless pressure drop is lower than that of the Ergun and Carman–Kozeny equations.

(4) The effect of Kn on the Nusselt number is similar to the Knudsen’s minimum effect. There is a minimum in the curve Nu versus Kn.

(5) For the same porosity, the inline porous structure with the smaller obstacles has the most efficient heat transfer.

References

[1] Ai, Y., Yalcin, S. E., Gu, D., Baysal, O., Baumgart, H., Qian, S., and Beskok, A., 2010, “A Low-Voltage Nano-Porous Electroosmotic Pump,”J. Colloid Interface Sci.,350(2), pp. 465–470.

[2] Jiang, R. H., Lin, C. F., Yang, C. C., Fan, F. H., Huang, Y. C., Tseng, W.-P., Cheng, P. F., Wu, K. C., and Wang, J. H., 2012, “InGaN Light-Emitting Diode With a Nanoporous/Air-Channel Structure,”Appl. Phys. Express,6(1), p. 012103.

[3] Kondrashova, D., Lauerer, A., Mehlhorn, D., Jobic, H., Feldhoff, A., Thommes, M., Chakraborty, D., Gommes, C., Zecevic, J., Jongh, P., Bunde, A., K€arger, J., and Valiullin, R., 2017, “Scale-Dependent Diffusion Anisotropy in Nanoporous Silicon,”Sci. Rep.,7, p. 40207.

[4] Ohba, T., 2016, “Limited Quantum Helium Transportation Through Nano- Channels by Quantum Fluctuation,”Sci. Rep.,6(1), p. 28992.

[5] Jeong, N., Choi, D. H., and Lin, C. L., 2006, “Prediction of Darcy-Forchheimer Drag for Micro-Porous Structures of Complex Geometry Using the Lattice Boltzmann Method,”J. Micromech. Microeng.,16(10), pp. 2240–2250.

[6] Yuranov, I., Renken, A., and Kiwi-Minsker, L., 2005, “Zeolite/Sintered Metal Fibers Composites as Effective Structured Catalyst,”Appl. Catal.,281(55), pp.

55–60.

[7] Kiwi-Minsker, L., and Renken, A., 2005, “Microstructured Reactors for Cata- lytic Reactions,”Catal. Today,110(2), pp. 2–14.

[8] Assis, O. B. G., and Claro, L. C., 2003, “Immobilized Lysozyme Protein on Fibrous Medium: Preliminary Results for Microfilteration Applications,”Elec- tron. J. Biotechnol.,6(2), epub.

[9] Brask, A., Goranovic, G., Jensen, M. J., and Bruus, H., 2005, “A Novel Electro- Osmotic Pump Design for Nonconducting Liquids: Theoretical Analysis of Flow Rate–Pressure Characteristics and Stability,”J. Micromech. Microeng., 15(4), pp. 883–891.

[10] Bazant, M. Z., and Squires, T. M., 2004, “Induced-Charge Electrokinetic Phenom- ena: Theory and Microfluidic Applications,”Phys. Rev. Lett.,92(6), p. 066101.

[11] Oddy, M. H., Santiago, J. G., and Michelsen, J. C., 2001, “Electrokinetic Insta- bility Micromixing,”Anal. Chem.,73(24), pp. 5822–5832.

[12] Biddiss, E., Erickson, D., and Li, D. Q., 2004, “Heterogeneous Surface Charge Enhanced Micromixing for Electrokinetic Flows,”Anal. Chem.,7(11), pp. 3208–3213.

[13] Wong, P. K., Wang, J. T., Deval, J. H., and Ho, C. M., 2004, “Electrokinetics in Micro Devices for Biotechnology Applications,”IEEE/ASME Trans. Mecha- tronics,9(2), pp. 366–376.

Fig. 14 The particle size effect of the mean temperature distribution along the channel

(9)

[14] Jiang, P. X., Fan, M. H., Si, G. S., and Ren, Z. P., 2001, “Thermal-Hydraulic Performance of Small Scale Micro-Channel and Porous-Media Heat- Exchangers,”Int. J. Heat Mass Transfer,44(5), pp. 1039–1051.

[15] Mahdavi, M., Saffar-Avval, M., Tiari, S., and Mansoori, Z., 2014, “Entropy Generation and Heat Transfer Numerical Analysis in Pipes Partially Filled With Porous Medium,”Int. J. Heat Mass Transfer,79, pp. 496–506.

[16] Rong, F., Zhang, W., Shi, B., and Guo, Z., 2014, “Numerical Study of Heat Transfer Enhancement in a Pipe Filled With Porous Media by Axisymmetric TLB Model Based on GPU,”Int. J. Heat Mass Transfer,70, pp. 1040–1049.

[17] Buonomo, B., Manca, O., and Lauriat, G., 2014, “Forced Convection in Micro- Channels Filled With Porous Media in Local Thermal Non-Equilibrium Conditions,”Int. J. Therm. Sci.,77, pp. 206–222.

[18] Shokouhmand, H., Meghdadi Isfahani, A. H., and Shirani, E., 2010, “Friction and Heat Transfer Coefficient in Micro and Nano Channels Filled With Porous Media for Wide Range of Knudsen Number,”Int. Commun. Heat Mass Trans- fer,37(7), pp. 890–894.

[19] Karniadakis, G., Beskok, A., and Aluru, N., 2005,Microflows and Nanoflows Fundamentals and Simulation, Springer, New York.

[20] Abolfazli Esfahani, J., and Norouzi, A., 2014, “Two Relaxation Time Lattice Boltzmann Model for Rarefied Gas Flows,”Physica A,393, pp. 51–61.

[21] Liu, X., and Guo, Z., 2013, “A Lattice Boltzmann Study of Gas Flows in a Long Micro-Channel,”Comput. Math. Appl.,65(2), pp. 186–193.

[22] Gokaltun, S., and Dulikravich, G. S., 2014, “Lattice Boltzmann Method for Rarefied Channel Flows With Heat Transfer,”Int. J. Heat Mass Transfer,78, pp. 796–804.

[23] Islam, M. S., Caulkin, R., Jia, X., Fairweather, M., and Williams, R. A., 2012,

“Prediction of the Permeability of Packed Beds of Non-Spherical Particles,”

Comput. Aided Chem. Eng.,30, pp. 1088–1092.

[24] Lopez, P., and Bayazitoglu, Y., 2013, “An Extended Thermal Lattice Boltz- mann Model for Transition Flow,”Int. J. Heat Mass Transfer,65, pp. 374–380.

[25] Chikatamarla, S. S., and Karlin, I. V., 2006, “Entropy and Galilean Invariance of Lattice Boltzmann Theories,”Phys. Rev. Lett.,97(19), p. 190601.

[26] Ansumali, S., Karlin, I. V., Arcidiacono, S., Abbas, A., and Prasianakis, N. I., 2007, “Hydrodynamics Beyond Navier–Stokes: Exact Solution to the Lattice Boltzmann Hierarchy,”Phys. Rev. Lett.,98(12), p. 124502.

[27] Kim, S. H., Pitsch, H. P., and Boyd, I. D., 2008, “Accuracy of Higher-Order Lattice Boltzmann Methods for Microscale Flows With Finite Knudsen Numbers,”J. Comput. Phys.,227(19), pp. 8655–8671.

[28] Zhang, Y. H., Gu, X. J., Barber, R. W., and Emerson, D. R., 2006, “Capturing Knudsen Layer Phenomena Using a Lattice Boltzmann Model,”Phys. Rev. E., 74(4), p. 046704.

[29] Tang, G. H., Zhang, Y. H., and Emerson, D. R., 2008, “Lattice Boltzmann Models for Nonequilibrium Gas Flows,”Phys. Rev. E.,77(4), p. 046701.

[30] Tang, G. H., Zhang, Y. H., Gu, X. J., and Emerson, D. R., 2008, “Lattice Boltz- mann Modeling Knudsen Layer Effect in Non-Equilibrium Flows,”EPL,83(4), p. 40008.

[31] Homayoon, A., Meghdadi Isfahani, A. H., Shirani, E., and Ashrafizadeh, M., 2011, “A Novel Modified Lattice Boltzmann Method for Simulation of Gas Flows in Wide Range of Knudsen Number,”Int. Commun. Heat Mass Transfer, 38(6), pp. 827–832.

[32] Shokouhmand, H., and Meghdadi Isfahani, A. H., 2011, “An Improved Thermal Lattice Boltzmann Model for Rarefied Gas Flows in Wide Range of Knudsen Number,”Int. Commun. Heat Mass Transfer,38(10), pp. 1463–1469.

[33] Zhuo, C., and Zhong, C., 2013, “Filter-Matrix Lattice Boltzmann Model for Microchannel Gas Flows,”Phys. Rev. E,88(5), p. 053311.

[34] Liou, T.-M., and Lin, C.-T., 2013, “Study on Microchannel Flows With a Sud- den Contraction–Expansion at a Wide Range of Knudsen Number Using Lattice Boltzmann Method,”Microfluid. Nanofluid.,16(1), pp. 315–327.

[35] Li, Q., He, Y. L., Tang, G. H., and Tao, W. Q., 2011, “Lattice Boltzmann Mod- eling of Microchannel Flows in the Transition Flow Regime,”Microfluid.

Nanofluid.,10(3), pp. 607–618.

[36] Kalarakis, A. N., Michalis, V. K., Skouras, E. D., and Burganos, V. N., 2012,

“Mesoscopic Simulation of Rarefied Flow in Narrow Channels and Porous Media,”Transp. Porous Media,94(1), pp. 385–398.

[37] Jin, Y., Uth, M. F., and Kuznetsov, A. V., 2015, “Numerical Investigation of the Possibility of Macroscopic Turbulence in Porous Media: A Direct Numeri- cal Simulation Study,”J. Fluid Mech.,766, pp. 76–103.

[38] Uth, M. F., Jin, Y., and Kuznetsov, A. V., 2016, “A Direct Numerical Simula- tion Study on the Possibility of Macroscopic Turbulence in Porous Media:

Effects of Different Solid Matrix Geometries, Solid Boundaries, and Two Porosity Scales,”Phys. Fluids,28(6), p. 065101.

[39] Klinkenberg, L. J., 1941, “The Permeability of Porous Media to Liquids and Gases,” Drilling and Productions Practices, American Petroleum Institute, New York.

[40] Scheidegger, A. E., 1972,The Physics of Flow Through Porous Media, 3rd ed., University of Toronto Press, Toronto, ON, Canada.

[41] Peng, Y., 2005, “Thermal Lattice Boltzmann Two-Phase Flow Model for Fluid Dynamics,”Ph.D. thesis, University of Pittsburgh, Pittsburgh, PA.

[42] Cercignani, C., 1988,The Boltzmann Equations and Its Applications, Springer- Verlag, New York.

[43] He, X. Y., Chen, S. Y., and Doolen, G. D., 1998, “A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit,”J. Comput. Phys., 146(1), pp. 282–300.

[44] Succi, S., 2001,The Lattice Boltzmann Equation: for Fluid Dynamics and Beyond, Oxford University Press, Oxford, UK.

[45] Succi, S., 2002, “Mesoscopic Modeling of Slip Motion at Fluid-Solid Interfaces With Heterogeneous Catalysis,”J. Phys. Rev. Lett.,89(6), p. 064502.

[46] Lim, C. Y., Shu, C., Niu, X. D., and Chew, Y. T., 2002, “Application of Lattice Boltzmann Method to Simulate Microchannel Flows,”J. Phys. Fluids.,14(7), pp. 2299–2308.

[47] Verhaeghe, F., Luo, L. S., and Blanpain, B., 2009, “Lattice Boltzmann Modeling of Microchannel Flow in Slip Flow Regime,”J. Comput. Phys.,228(1), pp. 147–157.

[48] Pan, C., Luo, L. S., and Miller, C. T., 2006, “An Evaluation of Lattice Boltz- mann Schemes for Porous Medium Flow Simulation,” Comput. Fluids, 35(8–9), pp. 898–909.

[49] Tang, G. H., Tao, W. Q., and He, Y. L., 2005, “Gas Slippage Effect on Micro- scale Porous Flow Using the Lattice Boltzmann Method,”Phys. Rev. E,72(5), p. 056301.

[50] Ergun, S., 1952, “Fluid Flow Through Packed Columns,” Chem. Eng. Prog., 48(2), pp. 89–94.

[51] Bear, J., 1972,Dynamic of Fluids in Porous Media, Elsevier, Amsterdam, The Netherlands.

[52] Niu, X. D., Chew, Y. T., and Shu, C., 2004, “A Lattice Boltzmann BGK Model for Simulation of Micro Flows,”Europhys. Lett.,67(4), p. 600.

[53] Niu, X. D., Shu, C., and Chew, Y. T., 2007, “A Thermal Lattice Boltzmann Model With Diffuse Scattering Boundary Condition for Micro Thermal Flows,”

Comput. Fluids,36(2), pp. 273–281.

[54] Hadjiconstantinou, N. G., and Simek, O., 2002, “Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels,” ASME J. Heat Transfer, 124(2), pp. 356–364.