Heat and Fluid Flow of Gases in Porous Media with Micropores: Slip Flow Regime

16 Heat and Fluid Flow o Gases in Porous Media with Micropores: Slip Flow Regime Moghtada Mobedi, Murat Barisik, and A. Nakayama CONTENTS 16.1 Introduction...407 16. VAM Motion and Heat Transer Equations...409 16.3 Heat and Fluid Flow in Microscale Channels... 411 16.4 The Considered Two-Dimensional Porous Media... 411 16.5 Governing Equations, Boundary Conditions, and Solution Method... 41 16.5.1 Calculation o Permeability... 413 16.5. Calculation o Interacial Heat Transer Coeicient... 414 16.6 Results and Discussion... 416 16.6.1 Permeability... 416 16.6. Interacial Convective Heat Transer... 418 16.7 Conclusion...40 Acknowledgment... 4 Reerences... 4 16.1 INTRODUCTION Heat and luid low in porous media have ound wide applications ranging rom nature to industry. The low o air into a human lung, low o water in soil, luid low in underground geothermal ields, low o crude oil in underground reservoirs, and even air low between tree trunks in orests are some examples o transport in a porous environment. The low o hot air or drying in ixed packed beds, the use o metal oams to enhance heat transer, and luid low in membranes are examples o industrial applications. Due to these extensive uses, the number o studies that ocuses on porous transport ield has increased exponentially in recent years. The mechanism o heat and luid transport through the pores is very complex such that determining the velocity, pressure, and temperature ields is a challenging task. There are two main theoretical approaches to analyze heat and luid low in a porous structure known as pore scale method (PSM) and volume-averaged method (VAM). In PSM, the conservation o mass and momentum is solved or the luid lowing through the voids o the porous medium. The energy equation can be solved or both solid and luid low in the pores to ind temperature ield both in solid and luid regions. The PSM yields accurate distributions or velocity, pressure, and temperature. However, its practical application on porous media, which are generally heterogeneous with numerous pores and voids, is troublesome. This diiculty with PSM leads researchers to develop and employ VAM. VAM is developed based on volume average theory. The results o VAM may not provide an exact view or temperature, pressure, and velocity in the porous media since spaced-averaged values are used. However, the method is practical and the accuracy o results is generally in acceptable ranges. Porous media consists o solid and luid phases considered as a continuum domain and the heat and 407

408 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 luid low equations are established or this imaginary continuum domain. Establishment o VAM governing equations requires the deinition o the volume average velocity, pressure, and temperatures as main dependent variables. Taking volume integral o the continuity, momentum and energy equations over a representative elementary volume (REV) o a porous structure results in additional unknown parameters such as permeability, Forchheimer coeicient, interacial heat transer coeicient, and thermal dispersion coeicients. These parameters are known as VAM transport parameters and the application o VAM requires the knowledge o values o VAM transport parameter. The values o permeability tensor depend on the geometrical parameters o the porous media. However, the values o Forchheimer, interacial convective heat transer coeicient, and thermal dispersion tensor depend on geometrical parameters, character o low in the pores, etc. It should be mentioned that, recently, Nakayama et al. [1] showed that in addition to the aorementioned macroscopic transport properties, thermal tortuosity also plays an important role in determining an accurate temperature ield in porous media. Consequently, a VAM transport parameter relating to thermal tortuosity should also be added to the aorementioned unknown transport parameters. The two methods or determining VAM transport parameters are the experimental and theoretical methods. Determination o permeability and thermal dispersion by experimental methods were discussed in the review studies o Sharma and Siginer [] and Ozgumus et al. [3] in detail. Pore scale simulation o heat and luid low or a REV o the porous media has become popular in recent years due to developments both in computer technology and computational methods. Multiple studies on pore scale simulation o heat and luid low in porous media can be ound in the literature. For heat and luid low in isotropic periodic porous media, examples such as the studies o Nakayama et al. [4] on D porous media with square rods, Saito and de Lemos [5] on the determination o interacial heat transer coeicient or D porous media with square rods, and Nakayma and Kuwahara [6] on the modeling o 3D PSM or a regular and periodic porous media can be given. For nonisotropic but periodic porous media structure, studies were conducted by Ozgumus and Mobedi [7] or D porous media with rectangular rods and Nakayama et al. [8] or D porous media with dierent directional cell. For real heterogeneous porous media that can be represented in computer environment by using methods such as tomography or MRI, the studies o Peszynska and Trykozko [9], Akolkar and Petrasch [10], and Ucar et al. [11] should be noted. The aorementioned studies were perormed to determine VAM transport properties or porous media o pore sizes greater than micron scales in which gas lows at standard conditions permit the use o continuum assumptions and constitutive laws. However, there exist many porous media with micron-scale pores (such as microiltration devices, packed beds with micron-scale particles, or micron-scale size iber porous structure) and low-pressure environments (vacuum absorption applications) where nonequilibrium gas behavior develops known as the rareaction. Most o the existing correlations suggested or the determination o VAM transport properties are developed without considering this rareaction eect and may not be valid or the porous media having pores in micron scale and/or operated under low pressures. For such cases, the Knudsen number deined as the ratio o mean ree path to the characteristic length is employed as a measure o the degree o rareaction. For a certain level o rareaction, Navier Stokes equations are ound to be valid with slip boundary conditions. Also known as the slip low regime, Maxwell deined that velocity slip and temperature jump conditions can incorporate the rareaction ormed as a result o small scale and/or low pressure. New correlations or diagrams or the determination o VAM transport properties should be investigated or the suggested correlations in literature should be revised as a unction o rareaction. Literature survey showed that the number o PSM studies examining the VAM transport parameters o porous media with microchannels is limited compared to studies perormed or macro channels. Most o the previous studies are on the determination o permeability rather than thermal VAM transport properties, such as the study o Jeong [1] in which lattice Boltzmann method was used to determine the permeability or microporous structures. For thermal VAM transport properties, the study o Vu et al. [13] on orced convection o air through networks o square rods or

Heat and Fluid Flow o Gases in Porous Media with Micropores 409 Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 cylinders can be given as an example. They ound permeability values or D porous media with staggered squared rods. Furthermore, Kuwahara et al. [14] studied the heat and luid low or D porous media with square rods and microscale channels and determined the values o permeability and tortuosity. Xu et al. [15] studied the rareied eects on internal heat transer coeicients in microporous media or sintered bronze porous media with average particle diameters rom 11 to 5 μm. The aim o the present work is to perorm a numerical study on D porous media consisting o square rods in micron size in order to determine VAM transport properties, including permeability and interacial heat transer coeicient in slip low regime. Brie inormation on the VAM motion and heat transer equations and the concept o heat and luid low in microscale channel with rareaction eect in slipping region is provided. Then, the obtained results or permeability and interacial heat transer coeicient are presented or a D porous medium. 16. VAM MOTION AND HEAT TRANSFER EQUATIONS A porous medium consists o solid particles and the voids between solid particles are shown in Figure 16.1. In order to obtain the pore scale temperature, velocity, and pressure ields, an REV is considered in the porous medium, then continuity, momentum and energy equations can be solved or the considered REV. I a steady-state single phase and incompressible luid low exists in the voids between particles, the governing equations can be expressed as Ñ V = 0 1 ( Ñ. V) V = - Ñ p+ nñ V r ( rcp) ( V. Ñ ) T = kñ T Ñ T = 0 s Fluid phase Solid phase (16.1) The solution o these equations with appropriate boundary conditions yields velocity, pressure, and the solid and luid phase temperature distributions or the REV o porous media. For an isotropic porous media, the obtained microscopic results are valid or the entire porous media. For nonisotropic or heterogeneous porous media, dierent REVs at dierent locations should be considered. For VAM, the porous medium consists o solid and voids in which the luid lowing is replaced by an imaginary continuum medium. The derivation o VAM-governing equation requires the Particle size Representative elementary volume FIGURE 16.1 The illustration o a porous medium and considered representative elementary volume (REV).

410 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications deinition o volume average. For an REV shown in Figure 16.1, two types o volume average as total volume average o quantity φ and the intrinsic volume average value o φ can be deined. j j = 1 V x ò V j dv = 1 (16.) j dv V ò x Vx Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 where V is the total volume o REV V x is the volume o considered phase in REV (s or stands or solid or luid phases, respectively). The variable o φ can be velocity, pressure, or temperature in the present study. The ollowing relation exists between the values o real and volume averaged o quantity o φ: j = j- j (16.3) where φ reers to the deviation o volume-averaged value rom the real value o φ or any location in REV. Taking volume integral rom the continuity, momentum, and energy equations over the REV and neglecting thermal tortuosity eect in REV yield the macroscopic motion and energy equations: Ñ V = 0 1 V 1 1 v m + ( V Ñ) V =- Ñ p + Ñ V - t K V e e r e r C - K V V 1 / s T s ( rc k T h A p) s1- e = s ( 1-e)Ñ + s T - T t V ( rc ) p ( ) T e t + r c V Ñ T = k eñ T p ( ) s s s s + h A s T - T kdis T V ( ) + Ñ (16.4) where V is the local average velocity p and T are the intrinsic average pressure and temperature o the luid phase while T s is the average temperature in the solid phase in REV I these equations are compared with the pore-level governing equations, it can be seen that there are new terms in the VAM transport equation involving new parameters called as VAM transport parameters. The new unknown VAM transport parameters are K, h s, C, and k dis named as

Heat and Fluid Flow o Gases in Porous Media with Micropores 411 permeability, interacial heat transer coeicient, Forchheimer coeicient, and thermal dispersion coeicient, respectively. As mentioned previously, these macroscopic transport parameters should be known in order to obtain volume-averaged temperature, velocity, and pressure ields. Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 16.3 HEAT AND FLUID FLOW IN MICROSCALE CHANNELS The eect o pore size along with gas pressure is especially important since most o the porous media contains sub-micron-size pores in addition to existing low-pressure vacuum absorption applications. As the pore size becomes comparable to the mean ree path o the gas molecules, nonequilibrium gas behavior occurs known as the rareaction eects leading to the breakdown o the continuum assumptions and constitutive laws. Nonequilibrium gas lows are classiied by the Knudsen number (Kn = λ/h), which is the ratio o the local gas mean ree path (λ) to the characteristic length scale (H), such as the channel height or pore size. Depending on Kn, transport is considered in the continuum (Kn 0.01), slip (0.01 Kn 0.1), transition (0.1 Kn 10), and ree-molecular (Kn > 10) low regimes [16]. Within the context o continuum luid dynamics, constitutive laws used in the deinition o stress tensor and heat lux vector and the no-slip boundary condition break down with increased Knudsen number. For example in a 1 μm conduit, gas at standard pressure and temperature develops transport in the slip low regime. For such cases, the continuum no-slip boundary condition becomes invalid due to gas rareaction, but the Navier Stokes equations can still be employed with an appropriate boundary conditions deined on the surace. The luid particles adjacent to the boundary surace are not in thermodynamic equilibrium with the wall that there would be slip velocity and temperature jump at the channel wall (For a more detailed discussion on these, readers are reerred to the textbook by Gad-el-Hak [17]). For the slip low regime (0.01 < Kn < 0.1), slip velocity and temperature jump boundary conditions or a microtube can be deined as ollows [18]: T u n m u =- - s l (16.5) s n m wall - st g l T - Tw = - s g + 1 Pr y where s m is the tangential momentum accommodation coeicient s t is the thermal accommodation coeicient g is the speciic heat ratio t wall (16.6) These slip low models are successully employed to consider the eect o rareaction on microscale low [19] while good agreements are obtained with experimental measurements [0]. 16.4 THE CONSIDERED TWO-DIMENSIONAL POROUS MEDIA The schematic o the considered porous medium is shown in Figure 16.. The porous medium is an ininite media consisting o square rods in inline arrangement. Considering the periodicity o the porous structure, an REV with the dimensions H H is employed to investigate the eect o Knudsen number on permeability and interacial convective heat transer coeicient. The dimension o the REV and the square blocks are constant or all studied cases. The height o REV (i.e., H) is 670 μm while the size o square block (i.e., D) is 335 μm. The porosity o porous media is unchanged and the constant is ε = 0.75.

41 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications Representative elementary volume D Flow direction D H Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 y x FIGURE 16. The studied D porous media. TABLE 16.1 Cases Studied or the Determination o Permeability (T = 300 K) Cases Studied Pressure (Pa) Density (kg/m 3 ) Mean Free Path (µm) Knudsen Number Case 1 10135 1.17659 6.70006E 08 0.0001 Case 1013.5 0.11766 6.70006E 07 0.001 Case 3 1013.5 0.011766 6.70006E 06 0.01 Case 4 506.65 0.005883 1.34001E 05 0.0 Case 5 0.65 0.00353 3.35003E 05 0.05 Case 6 101.35 0.001177 1.177000E 03 0.10 The luid lowing through the porous media is air. For determination o permeability or the considered porous media, the dynamic viscosity o air is taken as 18.1 10 6 kg/ms. The air pressure is changed rom 10135 to 101.35 Pa while its temperature is constant at 300 K. Based on the assumed temperature and pressure, the air density changes and dierent mean ree path and consequently dierent Knudsen number are achieved. Table 16.1 shows the studied cases with corresponding density, mean ree path, and Knudsen number. As shown in Table 16.1, in order to investigate permeability change with Knudsen number, six cases are studied. It should be mentioned that the permeability values are obtained in very low Re number as 0.001 to remove the inertia eect in the low ield. In order to calculate the interacial convective heat transer coeicient, the solid particle is assumed at constant temperature o 310 K. A study is perormed or two Reynolds numbers o 10 and 100 and six cases are considered or each Re number. The density o the air and mean ree path are calculated according to the mean temperature which is the arithmetic average o periodic inlet and outlet temperatures and surace temperature. H 16.5 GOVERNING EQUATIONS, BOUNDARY CONDITIONS, AND SOLUTION METHOD The low in the voids between the particles is assumed incompressible and steady. The luid (i.e., air) is Newtonian luid with constant thermophysical properties. The steady orm o the continuity and momentum equations (Equation 16.1) is solved to determine the velocity and pressure ields or the luid low in the voids between the blocks. Ater obtaining the velocity ield in the porous media, the energy equation or luid phase is solved to obtain the temperature distributions in

Heat and Fluid Flow o Gases in Porous Media with Micropores 413 Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 the REV. Considering REV in Figure 16., the boundary conditions or the microscopic equations are chosen as symmetry or the top and the bottom o the REV. The periodic velocity and temperature proiles are used or the inlet and outlet boundaries. The velocity and temperature gradients at the outlet boundary are assumed zero, hence no diusion transport exists. The discretization interval is chosen as 0.0 mm which makes the number o grids as 500 500 or the entire domain. The Fluent commercial code based on the inite volume method is used to solve the governing equations. SIMPLE method is used or handling the pressure velocity coupling. The power law scheme is employed or the discretization o the convection terms in the momentum and energy equations. The approximate errors are set to 10 9 or low equations and 10 1 or temperature ield. 16.5.1 Calculation o Permeability As it is well known, permeability is a tensor quantity and based on Darcy s law or a D low in Cartesian coordinate, and it can be deined as æ ç ç è u v ö K = 1 æ ç m ç K ø è xx xy K K yx yy æ p ö öç x ç ç ø p ç è y ø (16.7) where K xy, K yx, K xy, and K xy are components o the permeability tensor. In the present problem, the permeability is calculated only or x direction since the structure o porous media is symmetrical not only in x and y directions but also with respect to xy and yx diagonals. The permeability is calculated based on velocity and pressure ields obtained rom the continuity and momentum equations. Considering the REV shown in Figure 16., the ollowing boundary conditions are used with σ m =1 assumption: On solid walls: u n u =- l ; n wall Forinlet boundary: u( 0, y) = ( y), v(, 0 y) = 0 For outlet boundaries: u t = 0 uhy (, ) v( H, y) = = 0 x x (16.8) where u n and u t are normal and tangential components o the velocity vector at the solid wall and they can be u or v components according to the position o wall. The mean ree path is shown by λ in the earlier equation. The proile velocity o (y) is obtained by an iterative procedure. A uniorm velocity distribution is assigned or the inlet velocity, and the outlet velocity is obtained by solving the continuity and momentum equations according to the aorementioned boundary conditions. Then, the obtained outlet velocity is used as inlet velocity proile and the same equations are solved again. The procedure continues until the inlet and outlet velocity proiles become identical (i.e., (y)). The Darcy velocity and pressure gradient in x direction or low through the REV are calculated by the ollowing relation: u H H 1 = udxdy H òò 0 0 (16.9)

414 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications d p 1 é =- ê dx H( H - D) ê ë H-D/ ò D/ H-D/ ù p dy- p dyú x= 0 ò x= H ú D/ û (16.10) Hence the permeability or x direction can be obtained rom the ollowing Darcy equation: Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 u kxx d p =- (16.11) m dx Considering Figure 16., the permeability tensor or the present problem takes the ollowing orm: æ ç ç è u v ö K xx = 1 æ 0 m ç K ø è0 yy æ p ö öç x ç ç ø p ç è y ø (16.1) For the studied cases, the values o K xx and K yy are equal to each other due to symmetrical geometry o REV. 16.5. Calculation o Interacial Heat Transer Coeicient For a thermal nonequilibrium condition, the heat transer between the solid phase surace and the luid lowing in the voids can be calculated by using the interacial convective heat transer concept. Mathematically, the convective heat transer between the solid and luid can be calculated by the ollowing relation: k hs Ass( Ts - T ) = n ÑTdA V ò (16.13) where h s is the interacial convective heat transer coeicient A s is the solid luid interace area A ss is the speciic solid luid interace area (i.e., A ss = A s /V). Correspondingly, the interacial Nusselt number can be deined as ollows: A s Nu s hs H = (16.14) k where H represents the length and height o the REV. Although dierent characteristic lengths such as hydraulic diameter and the block height may be employed to deine Nusselt number or the studied porous structure, the dimension o the REV is selected in this study. The same characteristic

Heat and Fluid Flow o Gases in Porous Media with Micropores 415 length is also selected in many reported studies [4,5,7]. Considering the REV in Figure 16., the thermal boundary conditions or the PSM equations are as ollows: On the solid walls T T - Tw = -kl y wall Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 On the top and bottom For inlet boundary For outlet boundary T = 0 y T(, 0 y) = gy ( ) (16.15) T = 0 x where κ is a parameter that represents the degree o temperature jump, deined rom the temperature jump boundary condition, κ = (( σ t )/σ t )(γ/(γ + 1))(1/Pr), and κ = 0 corresponds to no temperature jump at the wall, while κ = 1.667 is a typical value or air, which is the working luid in many engineering applications and is taken so in this study. As seen in Equation 16.15, a temperature jump is assumed at the interace between the solid and luid phases. The unction o g(y) is the temperature proile that provides thermal periodicity or the inlet and outlet boundaries o the REV. In order to determine the unction o g(y), it is assumed that a thermally ully developed convection heat transer is valid which means that no change o the dimensionless temperature should be observed in sequential REVs in the low direction. A uniorm temperature proile which is dierent rom the solid temperature is deined or the luid inlet boundary and then the temperature ield or the entire domain and consequently or the outlet boundary is obtained. The temperature at the inlet is determined rom the dimensionless temperature proile at the outlet boundary. The iterative process continues until no change in the dimensionless temperature distribution at the inlet and outlet is observed. The dimensionless temperature equality or achieving thermally periodic low is T - T T - T b w w inlet T - T = T - T b w w outlet (16.16) where T b is the bulk temperature and it can be obtained by the ollowing relation or the considered REV: T ò A b = ò A utda da (16.17) Furthermore, ollowing dimensionless temperature distribution is deined to provide temperature distribution o dierent REV in the same range: q * T - T = T - T max min min (16.18)

416 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications where T min and T max are the minimum and maximum temperatures in the considered temperature ield. 16.6 RESULTS AND DISCUSSION In this section, the results or permeability and interacial heat transer coeicient are presented separately. Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 16.6.1 Permeability As discussed previously, a commercial code is used to solve the microscopic governing equations or the considered porous media. In order to deine slipping boundary conditions or solid walls, a user deined unction (UDF) is written and adapted to program. The validation o written UDF is checked with ully developed velocity proiles in a channel obtained analytically. Figure 16.3 shows the dimensionless velocity proile obtained by the employed code and the corresponding analytical results or a straight channel or three dierent Knudsen numbers. As shown, there is a good agreement between the analytical solution and the obtained computational results illustrating the correctness o the written UDF code. Figure 16.4 shows the distribution o longitudinal dimensionless velocity component or two Knudsen numbers o 0.0001 and 0.1. The dimensionless velocity is deined as the ratio o velocity component to the mean inlet velocity value. A high-velocity gradient at the solid horizontal surace or Kn = 0.0001 can be observed. By increasing the Knudsen number rom 0.0001 to 0.1, the gradient o velocity at the solid surace decreases considerably and this causes a lower pressure drop through the REV. Hence, it may be expected that the increase in Knudsen number increases the permeability o luid through the porous medium. The values o permeability or the cases given in Table 16.1 are obtained and nondimensionalized with H. The results are presented in Figure 16.5. Dimensionless ully developed velocity 1.6 1. 0.8 0.4 Numerical solution Analytical solution Kn=0.001 Kn = 0.0 Kn=0.1 0 0 0.5 0.5 Dimensionless length 0.75 1 FIGURE 16.3 Comparison o computational results with the analytical solution.

Heat and Fluid Flow o Gases in Porous Media with Micropores 417 0.0 0. 0.4 0.5 0.7 0.9 1.1 Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 (a) FIGURE 16.4 Distribution o dimensionless longitudinal velocity component or low in REV: (a) Kn = 0.0001 and (b) Kn = 0.1. K/H 0.0 0.018 0.016 0.014 0.01 0.0001 (b) 0.001 0.01 Kn FIGURE 16.5 The change o dimensionless permeability with Kn number. The permeability value increases considerably with Knudsen number indicating high permeability at lower Knudsen number. Although many equations were suggested in the literature or the determination o permeability under rareied eect, it seems that the relationship presented by Kuwahara and Nakayama [14] is the most suitable equation or the studied porous media in this work. 0.1 K K ns = 1+ C Kn (16.19)

418 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications 1.6 1.5 Present study Kuwahara and Nakayama [14] 1.4 Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 K/K ns 1.3 1. 1.1 1 0.0001 0.001 Kn 0.01 0.1 FIGURE 16.6 The comparison between the obtained numerical results with the correlation suggested by Kuwahara and Nakayama [14]. where K ns is the permeability o the porous media or no-slip boundary condition. The coeicient o C is deined as - C = 9 1 1 ( e) / e (16.0) Figure 16.6 shows the comparison o the obtained results with correlation suggested by Kuwahara and Nakayama. As shown, good agreement between the correlation and the numerical results obtained in this study is observed. 16.6. Interacial Convective Heat Transer Similar to the calculation o permeability, UDF codes are written to provide temperature jump or horizontal and vertical interaces between solid and luid boundaries. The written code and program are checked by comparing numerical results with Res. [1,] or a channel and it is shown in Figure 16.7. A good agreement between the computational results o present and reported studies can be observed. Figure 16.8 shows the dimensionless temperature distribution or two porous media with two dierent Reynolds and Knudsen numbers. Figure 16.8a shows the temperature contours in porous media or Re = 10. It should be mentioned that the dimensionless temperature at the solid luid is 1 (i.e., θ* = 1). For Kn = 0.0001, the temperature o luid at the region close to the solid surace is around 1 (i.e., θ* = 1) due to negligible temperature jump at the solid surace. By increasing Kn number rom 0.0001 to 0.1, the eect o the slipping boundary condition can be observed clearly. The interace temperature or luid region is considerably reduced and this causes luid lowing in

Heat and Fluid Flow o Gases in Porous Media with Micropores 419 1. Cetin et al. [1,] Present study Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 Dimensionless ully developed temperature 0.8 0.4 0 0 0.5 0.5 Dimensionless length Kn=0.1 Kn=0.0 Kn=0.001 0.75 1 FIGURE 16.7 Comparison o obtained computational and reported temperature proiles or a straight channel. the voids which cannot be heated. Figure 16.8b shows the same porous media; however, the Re number is increased to 100. The same behavior or temperature changes observed or Re = 10 can also be seen or luid lowing in the voids with Re = 100. The Kn number has an important eect on the temperature ield o luid and it reduces the heat transer rate in the voids. Hence, a reduction in interacial heat transer coeicient is expected. The change in interacial Nusselt number with Knudsen or two Reynolds number o Re = 10 and 100 is presented in Figure 16.9. As expected, the Nu number considerably decreases with Kn number or both Re = 10 and 100 lows. The interesting point o Figure 16.9 is that the dierence between Nu number o lows with Re = 10 and 100 depends on the Kn number. By increasing Kn number, the dierence between Nu numbers is reduced. Our study shows that it is possible to express a relation or interacial heat transer coeicient based on no-slip Nusselt number. A relation or the determination o Nu s or porous media with ε = 0.75 is ound as ollows: Nu Nu s ns, s = 0. 9915-11. 757Kn + 57. 83Kn (16.1) where Nu ns,s is the interacial Nusselt number or porous media with no-slip boundary condition. Figure 16.10 shows the comparison o the suggested relation with the obtained numerical results.

40 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications 0.0 Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 Kn = 0.0001 0. 0.4 0.6 0.8 1.0 Kn = 0.1 (a) Kn = 0.0001 Kn = 0.1 (b) FIGURE 16.8 16.7 Dimensionless temperature distribution in the porous media: (a) Re = 10 and (b) Re = 100. CONCLUSION A numerical study on heat and luid low in a D porous media with microchannel is perormed. The related PSM-governing equations and boundary conditions are solved and both permeability and interacial heat transer coeicient are calculated by using volume-averaged method. It is observed that or porous media with microchannel particularly at low gas pressure, the Kn number has a signiicant eect on both permeability and interacial heat transer coeicient. Hence, the traditional corrections reported in the literature or permeability and interacial heat transer coeicient cannot be used. Two-dimensional porous media studies showed that an increase in Kn number rom 0.0001 to 0.1 increased the permeability by 51%. The reduction o interacial heat transer coeicient with Kn number depends on Re number. The interacial heat transer coeicients become closer to each other with an increase in the Kn number.

Heat and Fluid Flow o Gases in Porous Media with Micropores 41 10 9 8 Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 Nu 7 6 5 4 3 0.0001 Re =10 Re = 100 0.001 0.01 FIGURE 16.9 The change in Nu number with Kn or two Re numbers o 10 and 100. Nu s / Nu ns,s 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Re =10 Re = 100 Equation 16.1 Kn 0.1 0. 0.0001 0.001 Kn 0.01 0.1 FIGURE 16.10 Comparison o the suggested correlation with obtained numerical results.

4 Microscale and Nanoscale Heat Transer: Analysis, Design, and Applications ACKNOWLEDGMENT The authors want to thank M.Sc. student Saa Sabet or his great eort in obtaining computational results and it would have been diicult to inish this study on time without his help. Downloaded by [Izmir Institute o Technology], [MURAT BARISIK] at 07:7 04 February 016 REFERENCES 1. C. Yang, A. Nakayama, A synthesis o tortuosity and dispersion in eective thermal conductivity o porous media, Int. J. Heat Mass Transer, 53 (15), 3 330 (010).. S. Sharma, D.A. Siginer, Permeability measurement methods in porous media o iber reinorced composites, Appl. Mech. Rev., 63 (), 0080 0080-19 (010). 3. T. Ozgumus, M. Mobedi, U. Ozkol, A. Nakayama, Thermal dispersion in porous media; a review on the experimental studies or packed beds, Appl. Mech. Rev., 65 (3), 031001 031001-19 (013). 4. F. Kuwahara, A. Nakayama, Numerical determination o thermal dispersion coeicients using a periodic porous structure, J. Heat Transer, 11 (1), 160 163 (1999). 5. M.B. Saito, M.J.S. de Lemos, A correlation or interacial heat transer coeicient or turbulent low over an array o square rods, J. Heat Transer, 18, 444 45 (006). 6. A. Nakayma, F. Kuwahara, Numerical modeling o convective heat transer in porous media using microscopic structure, Hand Book o Porous Media, Edited by Vaai K., Marcel Dekker Inc., New York, pp. 441 487 (000). 7. T. Ozgumus, M. Mobedi, Eect o pore to throat size ratio on interacial heat transer coeicient o porous media, J. Heat Transer, 137 (1), 0160 (014). 8. A. Nakayama, F. Kuwahara, T. Umemoto, T. Hayashi, Heat and luid low within an anisotropic porous medium, J. Heat Transer, 14, 746 753 (00). 9. M. Peszynska, A. Trykozko, Pore-to-core simulations o low with large velocities using continuum models and imaging data, Comput. Geosci., 17, 63 645, (013). 10. A. Akolkar, J. Petrasch, Tomography-based characterization and optimization o luid low through porous media, Transp. Porous Med., 95, 535 550 (01). 11. E. Ucar, M. Mobedi, G. Altıntas, E. Glatt, Eect o voxel size in low direction on permeability and Forchheimer coeicients determined by using micro-tomography images o a porous medium, Progress in Computational Fluid Dynamics (in press). 1. N. Jeoong, Advanced Study about the permeability or micro-porous structures using the Lattice Boltzmann method, Transp. Porous Med., 83, 71 88 (010). 13. T.L. Vu, G. Lauriat, O. Manca, Forced convection o air through networks o square rods or cylinders embedded in microchannels, Microluid Nanoluid, 16, 87 304 (014). 14. F. Kuwahara, A. Nakayama, Numerical simulation o rareied gas low through a porous media, Trans. Jpn. Soc. Mech. Eng., 69, 189 195 (1999). 15. R.N. Xu, Y.L. Huang, P.X. Jiyang, B.X. Wang, Internal heat transer coeicients in microporous media with rareaction eects, Sci. China, Technol. Sci., 55 (10), 869 876 (01). 16. A. Beskok, G.E. Karniadakis, Simulation o heat and momentum transer in complex micro-geometries. J. Thermophys. Heat Transer, 8, 355 370 (1994). 17. M. Gad-el-Hak, The MEMS Handbook, CRC Press, New York (001). 18. G.E. Karniadakis, A. Beskok, N. Aluru, Micro Flows and Nano Flows: Fundamentals and Simulation, Springer-Verlag, New York, 005. 19. H.P. Kavehpour, M. Faghri, Y. Asako, Eects o compressibility and rareaction on gaseous lows in microchannels. Numer. Heat Transer, Part A, 3, 677 696 (1997). 0. E.B. Arkilic, K.S. Breuer, M.A. Schmidt, Gaseous low in microchannels. Application o microabrication to luid mechanics. ASME FED, 197, 57 66 (1994). 1. B. Cetin, Analysis o single phase convective heat transer in microtubes and microchannels, Master thesis, Middle East Technical University, 005.. B. Cetin, H. Yuncu, S. Kakac, Gaseous low in microchannels with viscous dissipation. Int. J. Transp. Phenom., 8, 97 315 (006).