Entrance effect on the interfacial heat transfer and the thermal dispersion in laminar flows through porous media

International Journal of Thermal Sciences 104 (2016) 172e185

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Entrance effect on the interfacial heat transfer and the thermal dispersion in laminar flows through porous media Federico E. Teruel a, b, c, * mico Bariloche, CNEA, Bariloche 8400, Río Negro, Argentina Centro Ato CONICET, Argentina c Instituto Balseiro, Universidad Nacional de Cuyo, Argentina a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 September 2015 Received in revised form 18 December 2015 Accepted 11 January 2016 Available online xxx

Macroscopic coefficients that are needed to complete porous media models, such as the thermal dispersion and the interfacial heat transfer, are in general calculated under thermally and hydrodynamically fully developed conditions. In this study, a laminar flow that thermally develops in a porous structure is simulated to analyze an aspect that has not been addressed in the literature, the entrance effect on the calculation of macroscopic coefficients. Specifically, the simulation of a microscopic steady clet laminar flow in a porous medium formed by staggered square cylinders with ReD ¼ 1, 10 and 75, Pe numbers in the 10-104 range, and porosities between 55 and 95% is presented. The domain simulated has clet numbers. First, nubeen chosen large enough to allow the flow to thermally develop for large Pe merical solutions are space averaged to show that the application of the cellular average is preferred over the generally used volume average. Employing the cellular average, the interfacial heat transfer and the streamwise thermal dispersion are computed in the entire computational domain, from the inlet, where the flow is thermally developing, to the outlet, where fully developed conditions are achieved. Numerical computations for the interfacial heat transfer show a peak at the entrance that gradually decreases to a fully developed value. The value of the peak and the length of the developing region increases with the clet number. Therefore, it is concluded that for laminar flows and large Pe clet numbers porosity and Pe (>500), the assumption that the interfacial heat transfer is a constant defined by its fully developed value implies large errors in the calculation of the energy transferred between phases of the porous medium. The computation of the streamwise thermal dispersion shows the same developing region than that calculated for the interfacial heat transfer. In this region, this coefficient rises monotonically from zero to a fully developed value. Results show that the macroscopic modeling of laminar flows in ordered media clet numbers. cannot neglect the entrance effect for high porosities and large Pe © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Porous media Macroscopic energy equation Volume average Cellular average

1. Introduction The analysis of heat transfer in a heterogeneous system formed by a solid matrix filled with a fluid is required in a vast number of applications of different fields (e.g. bed reactors, pebble nuclear reactors, heat exchangers, vapor generators, oil production, etc.). Generally, this complex solid matrix is difficult to simulate with precise geometrical details, or the computational effort required to do so is unreachable. These reasons have motivated the development of continuum or porous media models that represent a region of the matrix formed by several pores as a macroscopic homogeneous system with uniform properties [1]. One of the techniques to mico Bariloche, CNEA, Bariloche 8400, Río Negro, Argentina. * Centro Ato E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijthermalsci.2016.01.005 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.

rigorously derive continuum models for multiphase systems is the volume-averaging [2]. This technique spatially smooths equations that are valid in one phase to produce equations that are valid everywhere. With this smoothing, the complexity of the geometry is avoided, but the complex physics that take place at the pore-scale still need to be represented. Therefore, partial differential equations that define porous media models, incorporate terms that represent the pore scale physics at a macroscopic level. These additional terms involve the use of macroscopic coefficients that can be calculated from theoretical analyses (e.g. Ref. [3]), experimental results (e.g. Ref. [4]) and/or numerical experiments (e.g. Ref. [5]). The use of the volume-averaging technique to develop macroscopic equations is well documented [2]. When transport equations are volume averaged, several length-scale constrains need to be imposed to avoid a non-local problem. Quintard and Whitaker [6,7]

F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

Nomenclature asf hsf kf kD-xx r0 Asf Cp CA D H P NuD NuD-FD NuD-peak Pe PeD Re ReD TB Ti Tw 〈T〉f

interfacial area per unit volume interfacial or macroscopic heat transfer coefficient fluid thermal conductivity dispersion coefficient in the streamwise direction radius of averaging volume interfacial area fluid specific heat cellular average square-edge length REV’s dimension (REV volume ¼ 2H x H) pore length scale macroscopic Nusselt number ( hsf D/ kf ) fully developed Nusselt number peak Nusselt number clet number Pe clet number based on the Darcy velocity and D Pe Reynolds number Reynolds number based on the Darcy velocity and D bulk temperature inlet fluid temperature wall temperature Vf -normalized space averaged temperature

derived such constrains for ordered and disordered porous media. Additionally, they showed that the weighting function defined as cellular average (CA) is a superior tool than the volume average (VA) to perform the space averaging in ordered media. While VA variables show pore-scale fluctuations, CA variables do not. For instance, the VA value of the interfacial heat transfer coefficient is dependent on the location of the representative elementary volume (REV) in the porous structure [8]. The recommendation given by Quintard and Whitaker to employ the CA over the VA in ordered porous media has not been followed by the research community. As it is shown in the following paragraphs, there is a vast literature that employs the VA as the space average tool to derive macroscopic equations or to calculate macroscopic coefficients. In this study, the superiority of the CA is shown by comparing the VA and the CA tools in different numerical experiments, emphasizing the importance of its use. The study of heat transfer in a saturated fluid flowing through obstacles has been frequently employed by the research community to investigate the performance of porous media models (e.g. Refs. [9,10]). When a space-averaging tool, such as VA or CA, is applied to the fluid energy equation, a macroscopic energy equation is obtained. The latter is characterized by terms that account for the interaction between phases at a pore-scale. In the general case, these terms require the definitions of three additional macroscopic coefficients to complete the macroscopic description [1]: the tortuosity, the thermal dispersion tensor and the interfacial heat transfer. These macroscopic coefficients are needed to complete macroscopic models that allow simulating the energy transfer process without modeling the pore-length-scale [11e13]. In particular, the coupling between the solid and fluid-phase temperatures is partially described with the interfacial heat transfer coefficient. Therefore, extensive literature is dedicated to the numerical computation of the interfacial heat transfer employing the VA tool. Kuwahara et al. [14] carried out numerical simulations in a periodic REV formed by two dimensional staggered squares to compute the interfacial heat transfer coefficient for a variety of flow conditions. These data allowed deriving a correlation for this coefficient dependent on the Reynolds number, Prandtl number and porosity (this correlation was later analyzed in Refs. [15,16]). The methodology employed by

U UD 〈U〉f V Vf Vs VA

173

cross-section averaged streamwise velocity V-normalized space averaged streamwise velocity, Darcy velocity Vf -normalized space averaged streamwise velocity volume of the REV fluid volume inside the REV Solid volume inside the REV volume average

Greek symbols porosity distribution function kinematic fluid viscosity macroscopic non dimensional temperature fluid density local coordinate

f gf n q r x

Additional notations 〈j〉 volume average of j 〈j〉f fluid volume average of j ij space fluctuation of j j

A

space average of j (volume average or cellular average)

Kuwahara et al., which employs a single cell with periodic boundary conditions for ordered media, is attractive due to its low computational cost. This methodology has been employed frequently for different porous structures, for laminar and turbulent flows and for different flow orientations respect to the porous structure. For instance, Saito and de Lemos [17] developed a correlation for laminar and turbulent flow for the porous structure employed in Ref. [14]. Gamrat et al. [18] also investigated the heat transfer process in a bank of aligned and staggered square rods but including the effect of a volumetric heat source. The effect of the flow direction in the porous structure was considered by Alshare et al. [19]. Based on single cell numerical experiments for an aligned bank of squared rods, the interfacial heat transfer was calculated for aligned and non-aligned flows. A stronger dependence on the Reynolds numbers for nonaligned flows than for aligned flows was obtained for the coefficient. Some studies simulated several REVs of ordered media to study aspects of the heat transfer process that cannot be properly analyzed with single REV simulations. Takemoto et al. [20] simulated the flow transition between steady laminar and oscillatory laminar flow in a bank of aligned cylinders. This study revealed that the solution depends on the initial condition or on the path through which the current state was obtained. Pathak and Ghiaasiaan [21] simulated square rods in a row to study the effect of pulsating laminar flow in the computation of macroscopic parameters. Imani et al. [22] studied the effect of porosity, solid-thermal conductivity ratio and Reynolds number on the heat transferred between phases when no explicit boundary condition, such as constant temperature or constant heat flux, is imposed on the solidefluid interface. As exemplified, calculation of the interfacial heat transfer has been analyzed abundantly for the last two decades. However, to the knowledge of the author, there are no numerical studies that quantify the entrance effect on the calculation of the interfacial heat transfer. This effect needs to be considered to understand its impact on the proper modeling of the energy transfer process in laminar flows in porous media. Other parameter that is relevant in the study of heat transfer in porous media is the thermal dispersion tensor. For ordered media, Koch et al. [3] calculated, in the limit of high porosities and for Stokes flows, a dependence on Pe2 for the streamwise component

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F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

(i.e. parallel or longitudinal) of the dispersion tensor. Numerical clet numbers in two simulations for laminar flow and large Pe dimensional porous structures showed some agreement with the analytical result of [3]. Souto and Moyne [23], calculated the parallel dispersion component for in line squared cylinder obtaining a dependence on Pe1.88 and Pe1.99 for a porosity of 0.36 and 0.84 respectively. For staggered cylinders, the calculated dependence was Pe1.75 and Pe1.86 for a porosity of 0.64 and 0.84 respectively. Different pore-scale simulations show that the dependence on the clet number for the streamwise dispersion can be well described Pe by Pem, where m is dependent on the structure, porosity, fluid properties, Reynolds number and the boundary conditions between phases [1]. There is a vast literature dedicated to the numerical computation of the dispersion tensor employing singleREV simulations. However, as in the case of the interfacial heat transfer, to the knowledge of the author the study of the entrance effect on this quantity has not been carried out. An exception to that is the experimental study of Han et al. [24]. In the latter work [24], the longitudinal and lateral dispersion coefficients were measured clet number range at various axial positions in a packed bed in a Pe from 102 to 104. Measurements showed two main facts: the first is that the longitudinal dispersion was strongly dependent on the axial position; the second one, is the existence of a characteristics clet number, from which length, that is larger when larger the Pe the longitudinal dispersion becomes constant. In the present study, the dependence of the streamwise thermal dispersion on the streamwise coordinate will be quantified for different porosities, clet numbers in the laminar regime Reynolds numbers and Pe allowing to quantify the entrance effect on the calculation of the streamwise thermal dispersion. This effect also needs to be considered to properly model the energy transfer process in laminar flows in porous media and has not been addressed in the literature before. This work is organized as follows. First, the macroscopic energy equation for saturated flows in porous media is presented, and the two coefficients analyzed herein are defined: the interfacial heat transfer and the streamwise thermal dispersion. The computational domain is then described together with the range of parameters simulated. Numerical results are then space-averaged employing the VA and the CA tools. A discussion regarding the benefits of each averaging tool is given to justify the use of the CA tool for further analyses. Lastly, the space evolution of the macroscopic coefficients is presented for all cases simulated. The results clearly show the existence of a macroscopic thermally developing region where the macroscopic coefficients are functions of the streamwise coordinate. It is therefore concluded that the entrance effect cannot be neglected in the macroscopic simulation of heat transfer in laminar flows through porous media. In general, macroscopic models to describe heat transfer in laminar flows in porous media assume thermally fully developed conditions and constant values for macroscopic coefficients. This study shows that this assumption is in general not valid. 2. Macroscopic energy equation and averaging tools

of each averaging volume and x is the position in a local coordinate system specific to each averaging volume. Take this volume as a constant (i.e. no space dependence) equals to the sum of the fluid and solid volumes inside the REV (V ¼ Vf(x) þ Vs(x)). In this treatment, the ideas of Hassanizadeh and Gray [26] and Gray et al. [25] are followed. Defining a distribution function gf(r), as one in the fluid phase and zero in the solid phase,


gf ðrÞ ¼

(1)

the volume-average 〈j〉, and the intrinsic phase cell-average 〈j〉f of any quantity j(r,t) associated with the fluid (scalar, vector or second order tensor) can be defined as:

〈j〉ðx; tÞ ¼

1 V

Z

    j x þ z; t gf x þ z dVz ;

(2)

V

〈j〉f ðx; tÞ ¼

V 〈j〉ðx; tÞ ; 〈j〉ðx; tÞ ¼ Vf ðxÞ fðxÞ

(3)

where the porosity f(x), is defined as:

fðxÞ ¼

1 V

Z

  V ðxÞ : gf x þ z dVz ¼ f V

(4)

V

Moreover, the space-decomposition of Hassanizadeh and Gray [26] can be used to decompose a quantity j(r,t) associated with the fluid in an intrinsic phase cell-average value 〈j〉f(x,t) plus a local fluctuation in space ij(x,t):

jðr; tÞ ¼ 〈j〉f ðx; tÞ þ i jðr; x; tÞ:

(5)

Applying equations (1)e(5) and the volume averaging theorem [27,28] to the microscopic equations (i.e. energy and momentum conservation in the fluid), a macroscopic set of equations can be obtained. For an isothermal fluid that enters to a constant porosity porous medium with constant wall temperature, the macroscopic momentum equations simple reduce to a constant volume average velocity in the streamwise direction. However, the macroscopic energy equation must be capable to accurately describe the behavior of the fluid-average temperature when the fluid flows and transfers heat in the porous medium. Under considerations of steady, incompressible, one-dimensional flow (x-direction) in a constant porosity medium with constant wall temperature, the solid phase is uncoupled from the fluid phase [1] and the transport equation for the macroscopic fluid temperature resumes [9]:

" #   d〈T〉f   d f kf þ kDxx ¼ þ hsf asf Tw  〈T〉f : dx dx dx

f f d〈T〉

rCp f〈U〉 Macroscopic equations are commonly obtained by spatial averaging of the microscopic ones over a REV of the porous medium. A REV should be the smallest differential volume that results in meaningful local average properties [2]. In addition, the macroscopic description of a system is useful when the system admits a separation of length scales between the pore-scale, the REV-scale and the scale at which properties are studied [1,25]. Consider a porous medium and an averaging volume (REV) with centroid in the position x and radius r0. For averaging purposes, an auxiliary coordinate system r ¼ x þ x is defined, so that x describes the origin

if r2Vf ; if r2Vs ;

1 0

(6) Which is a convectionediffusion equation with the intrinsic average temperature as dependent variable. The equation is also characterized by two macroscopic coefficients, the interfacial heat transfer (hsf) and the streamwise thermal dispersion (kD-xx). These two macroscopic parameters have been defined from modeling assumptions and conservation criteria. The interfacial heat transfer is defined to assure the conservation of energy [9]:

F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

Z 1 V

hsf ¼

.

kf VT$d A Asf

 : asf Tw  〈T〉f

(7)

And the streamwise thermal dispersion is defined employing a diffusion hypothesis [1] following the ideas of Taylor [29] and Aris [30]:

Z i

1 Vf

kDxx ¼ rCp

ux i T dV

V

Vx 〈T〉f

:

(8)

Equations (7) and (8) allow computing hsf and kD-xx when the velocity and temperature fields are known for a given porous structure. With this knowledge, the macroscopic energy equation (6) is, in principle, completed. Equations (7) and (8), based on the VA definition, have been employed frequently in the literature. Nevertheless, Quintard and Whitaker [6,7] showed that for ordered media volume average quantities present pore-scale fluctuations. This means that the VA value of a macroscopic quantity depends on the location of the REV in the porous structure. Instead of the VA, Quintard and Whitaker [7] recommended the use of the CA for ordered media. This averaging makes use of a specific weighting function that satisfies the mathematical requirements to obtain meaningful macroscopic values without pore-scale fluctuations. In this study, a periodic structure (i.e. an ordered porous medium) is employed, thus the CA is the preferred tool. However, the calculation of the interfacial heat transfer and streamwise thermal dispersion is carried out with both, the VA and the CA, to show the superiority of the CA. Following the formalism of Refs. [6,7], the VA can be defined employing the weighting function mV defined as:


mV ¼

1=V 0

if jx  rj  r o ; if jx  rj > r o :

(9)

And a volume averaged quantity is then calculated employing the convolution product * as:

Z 〈j〉ðxÞ ¼ mV *j ¼

mV ðx  rÞjðrÞdV:

(10)

R3

The CA must be defined by a weighting function mC so that the quantities mC *gf , mC *ðgf xÞ, mC *ðgf xxÞ, etc., are constants. This ensures that the average of a linear function does not show pore-scale fluctuations. A weighting function defined as mC ¼ mV * mV satisfies these requirements [7]. Mathematically, mC is the double application of the volume average weighting function and can be explicitly computed to show that the cellular average is equivalent to apply a triangular-shape function. For the one-dimensional case and for a REV of size 2r0, the VA and the CA are defined respectively as:




1=r0 if jx  rj  r0 mV ¼ ; 0 if jx  rj > r0 8 > x þ 2r0 if  2r0  x  0 1 < mC ¼ 2 2r0  x if 0  x  2r0 4r0 > : 0 if jx  rj > 2r0 :

175

or a signal without pore-scale fluctuations. This aspect is explained in detail in Refs. [6,7]. To complete a correct description of the weighting function, it is needed to impose that space-averaged variables are infinitely differentiable [6,7]. Therefore, either for the VA or for the CA, a new weighting function mg is introduced to satisfy this condition. Thus, for example, mC is redefined as mC ¼ mg * mV * mV where mg 2C ∞ . Note that mg can be any function that removes the discontinuity of the volume average weighting function at the boundary without modifying the value of the macroscopic variable respect to the double application of the VA. For practical purposes, this function does not need to be specified, but it is a tool that assures the differentiability of macroscopic variables. As this study deals with numerical results, macroscopic coefficients are calculated averaging in the REV-space microscopic data with the appropriate weighting function. For the CA, the weighting function has an explicit representation that consists in a triangular shape function [7]. Nevertheless, for discrete data, the CA is easily computed carrying out a double volume average in the REV. Additionally, if the CA is employed to average microscopic equations, the macroscopic energy equation model (eq. (6)) and definitions given in equations (7) and (8) are still valid, but macroscopic quantities have to be redefined. In this case, averaged quantities and space-fluctuations have to be based on definitions given for the CA. In Section 3 and for the sake of clarity, volume and cellular averages are defined for the specific porous structure simulated in this study. 3. Numerical method and domain of study A schematic diagram of the domain selected for the simulation is shown in Fig. 1. The fluid flows from left to right, entering the porous medium after flowing a distance equals to H as a clear flow. The porous medium extends in the streamwise direction from x ¼ 0 to a location between x ¼ 48H and x ¼ 220H, depending on flow conditions. Therefore, the porous region has between 24 and 110 REVs in a row (the REV is chosen to be a cell of 2H  H in the streamwise and spanwise directions respectively). As it will be understood later, this large domain is needed to allow the flow to thermally develop clet numbers. The fluidesolid interface is set to a lower for large Pe temperature than that in the fluid at the entrance. This is done at the location x ¼ 6H to achieve a smooth transition in the volume average temperature at the entrance region and to allow the hydrodynamic development of the flow (Fig. 1). To save computational time, only the bottom half of the REV (H/2) is simulated. This simplification is valid considering that simulations of a single REV with periodic BCs evolve to steady solutions at the Reynolds numbers simulated in this study [8,31]. Note that the domain shown in Fig. 1 corresponds to a cross flow heat exchanger in a staggered configuration where the constant wall temperature BC can be associated with a phase change in the tube side. The governing equations for the fluid phase (mass, momentum and energy respectively) are given as follows: .

V$ u ¼ 0;

(12)

v u  . . 1 . þ V u $ u ¼ Vp þ V2 u ; Re vt

(13)

.  vT 1 þ V$ u T ¼ V2 T: vt Pe

(14)

.

(11)

A relatively easy way to understand how mC operates is to consider that mV applied to a linear function yields a constant plus a periodic function; and that mV applied to a periodic function yields a constant. Therefore, mC applied to a linear function yields a constant

Boundary conditions are standard for all the boundaries of the domain, except at the outlet, where periodic BCs are applied. On the solid walls BCs resume:

176

F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

Fig. 1. Geometry of the domain simulated (smallest domain 49H  H/2, largest domain 221H  H/2). Free stream entering to the porous medium.




.

.

u ¼ 0 ;

T¼

Ti Tw

 x < 6H : x > 6H

(15)

On the inlet of the domain (uniform field): .

u ¼ ðP=H; 0Þ;

T ¼ Ti :

(16)

On the bottom and top horizontal lines of the domain (symmetry): .

.

Vn u ¼ 0 ;

Vn T ¼ 0:

(17)

And on the outlet of the domain (periodicity): .

.

4. Results and discussion

u ðxo ; yÞ ¼ u ðxo  2H; yÞ;

(18)

Tðxo ; yÞ ¼ Tw þ tðTðxo  2H; yÞ  Tw Þ;

(19)

where xo indicates the x-coordinate of the outlet; and t is defined as:

t¼

TB ðxÞ  Tw ; TB ðx  2HÞ  Tw

and 180  45 for 55, 75 and 95% porosity respectively. The porosity is defined as f ¼ 1  D2 =H 2 according to Fig. 1. The Reynolds number based on the Darcy velocity and size of the obstacles, ReD, was varied from 1 to 75. As it was mentioned, three different porosities were simulated 55, 75 and 95%. The PeD number, defined as ReDPr, was varied from 50 to 104 for the cases of 55 and 75% porosity, and between 10 and 5000 for 95% porosity. Two particular cases, f ¼ 0.55, ReD ¼ 1, PeD ¼ 104 and f ¼ 0.75, ReD ¼ 75, PeD ¼ 104, are not presented due to poor numerical convergence. It is important to note that the Re numberp based ffiffiffiffiffiffiffiffiffiffiffiffion the flow rate ðRe ¼ Up=vÞ can be calculated as Re ¼ ReD = 1  f.

(20)

where TB is the bulk temperature of the fluid. Equation (13), the periodicity for temperature, has been discussed in detail in Refs. [8,14]. To solve the set of equations (12)e(14) under BCs given in eqs. (15)e(19) the SIMPLER algorithm developed by Patankar [32] was employed. To model the diffusion and the convective terms, the central difference and the QUICK scheme were employed, respectively [33,34]. To evolve the initial condition to the steady state a backward Euler scheme was used. The solver has been fully tested and validated for different geometries, including those presented in this study [35,36]. Periodic variables were solved in an iterative manner, and profiles at the outlet were obtained from previous time steps according to eqs. (18) and (19). Simulations were considered to reach convergence when normalized residuals were lower than 106. It has been carefully checked that numerical solutions conserve energy in a global sense (domain) and in local sense (REV); additional details are available in Refs. [35,36]. The domain was discretized using a uniform and structured grid of squares, and a systematic grid refinement study was carried out. Macroscopic quantities reported in this study were found to be independent of any further grid refinement for the calculation of macroscopic variables. The grid resolution employed for each REV was 180  45 (2H  H/2, streamwise  vertical direction), 128  32

4.1. The use of different averaging tools to compute macroscopic coefficients In this section it is shown that the CA is preferred over the VA as it yields macroscopic values without pore-scale fluctuations. Consider a REV with centroid at position x in the streamwise direction (Fig. 1), the two dimensional velocity, the pressure and the temperature of the fluid are known in this volume after solving the microscopic set of equations. Therefore, the VA temperature can be calculated as:

T

VA

ðxÞ ¼

1 fH 2

ZH

ZH=2 Tðx þ x; yÞdy;

dx H

(21)

0

where T(x,y) is the microscopic numerical solution. The spacefluctuation of the temperature respect to the VA temperature is: i VA

T

ðx; yÞ ¼ Tðx; yÞ  T

VA

ðxÞ:

(22)

For the particular porous medium under consideration, the CA can be computed as:

T

CA

1 ðxÞ ¼ 2H

ZH T

VA

ðx þ xÞdx:

(23)

H

And the space-fluctuation respect to CA temperature is: i CA

T

ðx; yÞ ¼ Tðx; yÞ  T

CA

ðxÞ:

(24)

Averages and space-fluctuations for the streamwise velocity can be computed as it is shown in equations (21)e(24) for the temperature.

F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

177

Fig. 2. NuD calculated with VA and CA operations as a function of the REV position in the porous structure. Cases for ReD ¼ 10, and PeD ¼ 50, 100, 500, f ¼ 0.55 (a) and f ¼ 0.95 (b). The maximum difference between the VA and CA quantities is pointed out together with the difference at x/2H ¼ 0. The abscissa, x/2H, corresponds to the centroid of the REV.

Therefore, definitions given in eqs. (7) and (8) allow the calculation of the interfacial heat transfer coefficient and the streamwise thermal dispersion employing both, the VA and the CA [8]. First, fully developed (FD) values of the macroscopic coefficients are analyzed. For that, a REV located far enough from the inlet in Fig. 1 is chosen to avoid the thermally developing region. Moreover, the centroid of the REV is moved a 2H-distance in the porous structure to obtain the dependence of the macroscopic parameter on the REV-location. This dependence has been discussed with some detail in Refs. [8,37]. Fig. 2 shows the non-dimensional interfacial heat transfer (NuD ¼ hsfD/kf) computed employing both, the VA and the CA tools, as a function of the location of the REV-centroid in the porous structure. Qualitatively, it can be said that VA signals present pore-scale fluctuations and CA signals are practically constant, independent on the REV location in the porous structure. For both porosities presented in Fig. 2 (55 and 95%), the clet number the higher the fluctuations. This fact is lower the Pe explained considering that for large PeD, the VA temperature is

practically constant in the scale of the REV (2H). Additionally, and for a fixed PeD, the lower the porosity, the higher the fluctuations. It is important to note that calculations presented in the literature [5,14,15,17e19] correspond to a point in Fig. 2. In general, this point is chosen to have a symmetric REV respect to its centroid. The location of the centroid of this REV is x/2H ¼ 0 in Fig. 2 and this particular REV has been drawn in Fig. 1. Although in the present study the CA is computed as indicated in eq. (23) (i.e. moving the REV in the structure), it is shown in Ref. [8] that under FD conditions the CA can be computed employing data from a single REV. The streamwise thermal dispersion computed with the VA and with the CA tools is shown in Fig. 3. Comparing for the same flow conditions, this quantity presents smaller fluctuations respect to the CA than those computed for the interfacial heat transfer. For f ¼ 0.55 and f ¼ 0.95, both averaging coincide for PeD > 500 and PeD > 100 respectively. As single REV simulations are generally found in the literature, it is of interest to quantify the maximum difference between a VA

Fig. 3. kD-xx calculated with VA and CA operations as a function of the REV position in the porous structure. Cases for ReD ¼ 10, and PeD ¼ 50, 100, 500, f ¼ 0.55 (a) and f ¼ 0.95 (b). The maximum difference between the VA and CA quantities is pointed out together with the difference at x/2H ¼ 0. The abscissa, x/2H, corresponds to the centroid of the REV.

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F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

Table 1 Percentage difference between the VA and CA values. MD stands for maximum difference and SR stands for difference respect to a symmetric REV. PeD

f ¼ 0.55

f ¼ 0.75

ReD ¼ 1 MD a) Interfacial heat 10 50 22.6 100 11.9 200 6.3 500 2.9

ReD ¼ 10 SR

MD

ReD ¼ 75

f ¼ 0.95

ReD ¼ 1

ReD ¼ 10

ReD ¼ 75

ReD ¼ 1

ReD ¼ 10

ReD ¼ 75

SR

MD

SR

MD

SR

MD

SR

MD

SR

MD

SR

MD

SR

MD

SR

2.4 1.4 0.8 0.4

29.1 15.9 8.6 4.0

3.2 1.8 1.0 0.5

9.6 5.2 2.9 1.3

1.3 0.8 0.5 0.2

10.8 5.9 3.2 1.5

1.9 1.1 0.7 0.3

12.6 7.2 4.1 2.0

2.2 1.3 0.8 0.4

9.4 2.4 1.4 0.8 0.4

1.7 0.5 0.3 0.2 0.1

11.0 2.9 1.6 0.9 0.4

2.7 0.9 0.6 0.3 0.2

11.5 3.1 1.8 1.0 0.5

3.0 1.1 0.6 0.4 0.2

6.2 2.1 0.7 0.2

8.8 3.1 1.1 0.3

6.0 2.1 0.7 0.2

1.9 0.6 0.2 0.1

1.7 0.6 0.2 0.1

2.0 0.7 0.3 0.1

1.6 0.6 0.2 0.1

2.5 0.9 0.4 0.1

1.5 0.6 0.2 0.1

2.9 0.3 0.1 0.1 0.0

2.6 0.3 0.1 0.0 0.0

3.5 0.3 0.1 0.0 0.0

2.9 0.3 0.1 0.0 0.0

4.0 0.3 0.1 0.0 0.0

3.2 0.3 0.1 0.0 0.0

transfer coefficient 2.0 1.1 0.6 0.3

24.4 13.0 6.9 3.1

b) Streamwise thermal dispersion 10 50 7.4 6.8 7.2 100 2.4 2.3 2.4 200 0.8 0.7 0.8 500 0.2 0.2 0.2

quantity and a CA quantity. And also, the difference between the CA and the single-REV value obtained for the symmetric-REV with centroid located at x/2H ¼ 0 in Figs. 2 and 3. Table 1 shows these differences as a percentage of the CA value for the interfacial heat transfer (a) and for the streamwise thermal dispersion (b), for all cases simulated in this study with PeD  500 (for larger PeD numbers both averaging do not show differences). For the interfacial heat transfer coefficient, the symmetric REV presents results with differences below 5% for all cases simulated. This suggests that this location in the porous structure is a good choice to approximate the CA with the VA in the calculation of NuD. The maximum difference calculated is approximately 30% (f ¼ 0.55 and ReD ¼ 75), and to assure maximum differences below 5% for all cases simulated the PeD has to be approximately larger than 500. Part b) of Table 1 shows that the streamwise thermal dispersion, presents smaller fluctuation respect to the CA than that calculated for the interfacial heat transfer coefficient. However, the symmetric REV is not the best location to obtain a VA value close to the CA value (Fig. 3). For PeD > 100, maximum differences are below 5% for all cases simulated. Figs. 2 and 3, and Table 1 exemplified the drawbacks of employing the VA as an averaging tool. Volume average FD values may vary significatively with the location of the REV in the porous structure. Therefore, these values cannot be univocally determined. Additionally, it is shown that the REV location generally found in the literature, yields a VA value for NuD close to that calculated with the CA but it does not for the calculation of kD-xx.

Pore-scale fluctuations that characterized VA signals in ordered media appear also in regions of the flow where fully developed conditions have not been reached. This aspect is shown averaging microscopic results for the case of 75% porosity, ReD ¼ 10 and PeD between 50 and 500 (Fig. 4). The REV is moved in the numerical grid from the inlet to the outlet to yield a 1D macroscopic temperature and to yield macroscopic coefficients for each macroscopic point in the porous structure. It is important to note that in Fig. 4 and subsequent figures, a non-dimensional streamwise coordinate is used, x* ¼ x/2H, where x* corresponds to the centroid of the REV. The origin of this coordinate system (x* ¼ 0) is indicated in Fig. 1 and corresponds to x ¼ 5H in the microscopic coordinate system. Fig. 4a shows the non-dimensional temperature, defined as qðx* Þ ¼ ðT A ðx* Þ  Tw Þ=ðTi  Tw Þ, where TA(x*) represents a spaceaverage temperature, VA or CA. The non-dimensional macroscopic coefficients NuD and kD-xx/kfPe are also shown in Fig. 4b and c respectively, as a function of the streamwise coordinate. The physical behavior of these macroscopic quantities is discussed in the following sections; the focus here is on the difference between the two averaging tools employed. Again and even in the developing region, variables computed with the VA tool show pore-scale fluctuations or a poor macroscopic representation, being more significant this effect for NuD. The CA tool yields clear and well resolved signals. This result emphasizes the importance in using the CA tool to represent macroscopic variables in ordered media. This tool yields well defined macroscopic quantities in regions

Fig. 4. Space dependence of q (a), NuD (b) and kD-xx (c) for f ¼ 75%, ReD ¼ 10 and PeD ¼ 50, 200 and 500.

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179

Fig. 5. Calculated NuD for 55% porosity. Space dependence. a) ReD ¼ 1, b) ReD ¼ 10, c) ReD ¼ 75.

where coefficients are constant or even space dependent. Note that for the rest of the study only CA values will be presented. 4.2. Developing values for macroscopic coefficients Porous media models for problems such as the one under study, generally assume that macroscopic coefficients are constant (i.e. they are not space dependent). One example of this is that the developing region is neglected [38,39]. For the particular case under study, it is known that the microscopic flow thermally develops in the porous structure. However, it is fair to ask how this developing is shown at a macroscopic scale. For example, making an analogy with a clear channel flow with isothermal walls, it may be expected that the interfacial heat transfer shows a peak at the entrance that evolves to a fully developed value [40]. If this aspect is shown in macroscopic data obtained from microscopic numerical results, it should be captured by the macroscopic energy equation model presented in eq. (6). The macroscopic coefficients are now presented analyzing the macroscopic signal obtained from the CA of a large set of numerical results. First, the space evolution of the interfacial heat transfer coefficient is considered (NuD). Figs. 5e7 show this quantity for f ¼ 0.55, f ¼ 0.75 and f ¼ 0.95 respectively, with the PeD and ReD as parameters. Three different regions may be identified in these three figures. First, there is a non-heated region where the NuD starts at zero value. Second, there is a region of approximately 1-REV in length, where it value rises from zero to its maximum (peak). This is the first REV with boundary condition Tw < Ti (actually the peak

occurs at x* z 1.25 for all simulations carried out). And there is a third region where NuD gradually decreases to its FD value in a length that depends on flow conditions, fluid properties and porosity. The physical behavior of this macroscopic coefficient is completely analog to that found for the convective heat transfer coefficient in a clear channel flow with constant wall temperature. Analyzing the dependence of this coefficient on simulation conditions, it is found that for a fixed porosity, the larger the ReD and PeD, the larger the NuD. For fixed ReD and PeD, the higher the porosity, the smaller the NuD. For low PeD values (PeD < 100e200), the peak in NuD is not present due to the relatively large molecular diffusion and the short developing region (a couple of REVs). In particular, for large PeD and high porosities, the developing region is relatively large (e.g. more than 15 REVs for the 95% porosity case, Fig. 7), suggesting that the energy transferred will be largely underestimated by a model that assumes a constant FD interfacial heat transfer coefficient. This region is, from the macroscopic point of view, a thermally developing region. Fig. 8 shows a comparison of NuD for PeD ¼ 1000 and different ReD and f. The objective of this figure is to show that the peak location (x* z 1.25) is approximately independent of the parameters simulated. Additionally, it shows qualitatively that for PeD ¼ 1000, the length of the thermally developing region is strongly dependent on the porosity. Figs. 6e9 show that for the simulated cases and especially, for large PeD numbers, the heat transfer coefficient in the developing region is significantly higher than that calculated in the FD region. This implies that the energy transferred from the liquid to the

Fig. 6. Calculated NuD for 75% porosity. Space dependence. a) ReD ¼ 1, b) ReD ¼ 10, c) ReD ¼ 75.

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Fig. 7. Calculated NuD for 95% porosity. Space dependence. a) ReD ¼ 1, b) ReD ¼ 10, c) ReD ¼ 75.

solid can be largely underestimated if this effect is not considered by the macroscopic model. Fig. 9 helps to quantify this fact showing the peak to fully developed ratio of NuD as a function of the PeD. For PeD ¼ 1000, the ratio is between 10% (f ¼ 0.55) and 45% (f ¼ 0.95). Table 2 also suggests that the use of a constant interfacial heat transfer has to be well justified for laminar flows in ordered media. The streamwise location ðx*FD Þ where the difference between the local NuD and its FD value becomes lower than 5% is shown in Table 2 for PeD  500 (this value may be used to define a thermal entry length making an analogy with the heated channel flow). This value is strongly dependent on the porosity. For PeD ¼ 1000, the region extends from 2 to 15 REVs for f ¼ 0.55 and f ¼ 0.95 respectively, and for PeD ¼ 5000 for 6e65 REVs. Table 2 together with Fig. 9 allow concluding that for high porosities and/or large PeD there is a relatively large region (larger

than four REVs) where the macroscopic coefficient differs significatively from its FD value. Now, the space evolution of the streamwise dispersion coefficient is analyzed. Figs. 10e12 show cases for 55, 75 and 95% porosity respectively (note that the ordinate is in log-scale). This macroscopic diffusion coefficient rises from zero (no-heated zone) to a FD value. A relatively large developing length for large PeD numbers and high porosities is found (e.g. more than 15 REVs for 95% porosity and PeD > 500). Similar to that found for the interfacial heat transfer coefficient, the larger the ReD and PeD, the larger the FD developed value of kD-xx/kf. However, the higher the porosity, the higher the values of kD-xx/kf. For a fixed PeD, the macroscopic dispersion coefficient is also presented in Fig. 13. The developing distance is strongly dependent on the porosity being larger when higher the porosity.

Fig. 8. Calculated NuD for PeD ¼ 1000, ReD and porosity as parameters.

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181

Fig. 9. NuD-peak/NuD-FD ratio as a function of PeD.

Results obtained for the macroscopic thermal dispersion coefficient suggests, as those discussed for NuD, that considering this parameter as a constant is an approximation that has to be carefully tested for laminar flow with PeD > 500. This coefficient scales the diffusion process in the streamwise direction. Therefore, the assumption of a FD value overestimates the diffusion of the temperature in the streamwise direction. Although the literature in the clet subject is scarce, Hang et al. [24] measured, for different Pe numbers and different particle distributions of a sphere packing, the longitudinal dispersion in the developing region at five different probe locations in the streamwise direction. Results are qualitatively similar to those showed in Figs. 10e12 for large PeD clet number or numbers. Hang et al. obtained that the larger the Pe the larger the size range of particles distributions, the larger the developing length. Based on present numerical results for both macroscopic coefficients, it is concluded that the developing region is not well described by the model given in eq. (6) or, alternatively, that the model may be valid considering that macroscopic coefficients are space dependent in the developing region. 4.3. Fully developed macroscopic values Extensive literature has been dedicated to develop correlations to compute macroscopic coefficients as a function of the fluid, flow

and porous medium characteristics [4,5,14,15,17,18]. However, these results are calculated employing the VA tool in single-REV simulations. It is then of interest to compute FD values employing the CA tool and to compare with available data. Fig. 14 shows NuD. Results for ReD ¼ 1 are well compared with the correlation presented by Nakayama [16] for the same porous structure than that analyzed here:

NuD ¼

2 þ 12

1f 1=3 þ ð1 þ fÞ1=2 Re0:6 : D Pr f

(25)

The large differences are found for low PeD values and low porosities as it is expected according to Fig. 2. For fix porosity and Reynolds number, equation (25) proposes a dependence of Pr1/3. Present results agree better with an exponent between 0.2 and 0.3, depending on the porosity and Reynolds number. Fig. 15 shows fully developed CA values for kD-xx/kf as a function of PeD, and with the porosity and ReD as parameters. Present results scale well with Pe2D (this functionality is added to Fig. 15). For fixed porosity and ReD, a least-square fitting with Pem D yields values for the exponent m that vary between 1.84 and 2.01 for all cases simulated. Taylor [29] and Aris [30] predicted a dependence of Pe2D for the streamwise dispersion for laminar flows. This dependence was also theoretically derived by Koch et al. [3] for Stokes flows in ordered media in the limit of high porosities. This theoretical

Table 2 Space location ðx*FD Þ from which NuD differs in less than 5% respect to its fully developed value. PeD

500 1000 5000 10,000

f ¼ 0.55

f ¼ 0.75

ReD ¼ 1

ReD ¼ 10

1.5 2.3 8.8 *

1.5 2.2 8.4 14.7

f ¼ 0.95

ReD ¼ 75

ReD ¼ 1

ReD ¼ 10

ReD ¼ 75

ReD ¼ 1

ReD ¼ 10

ReD ¼ 75

1.9 5.8 8.7

2.4 4.2 18 30.3

2.3 4.1 16 25.6

1.5 2.4 6.9 *

7.2 15.3 83 *

7.5 12.4 58.2 *

6 12 65 *

182

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Fig. 10. Calculated kD-xx/kf number for 55% porosity. Space dependence. a) ReD ¼ 1, b) ReD ¼ 10, c) ReD ¼ 75.

Fig. 11. Calculated kD-xx/kf number for 75% porosity. Space dependence. a) ReD ¼ 1, b) ReD ¼ 10, c) ReD ¼ 75.

development agrees well with the experimental results of Gunn and Pryce for a cubic array of spheres with f ¼ 0.8 [41]. Nakayama et al. [42] also derived a quadratic dependence for laminar flow and a 7/8 dependence for turbulent flows. Numerical results, however, have calculated an exponent that depends on the structure of the ordered media. Souto et al. [23] discussed the accuracy of the prediction by Koch, showing that numerical computations for aligned squared cylinder yield an exponent that varies between 1.88 and 1.99 for porosities between 0.36 and 0.84. Saada et al. [43]

also obtained a quadratic dependence for aligned squared cylinders. Kuwahara and Nakayama obtained a quadratic dependence for PeD < 10 and a linear dependence for PeD > 10 for numerical experiments in aligned squared cylinders where the ReD was varied between 102 and 103 [5]. However, the linear dependence obtained in Ref. [5] can be explained by the change in the flow pattern inside the REV. For porous media flows as the one under consideration herein, the flow becomes oscillatory for ReD larger than approximately 200 [44]. All the literature reviewed allows to

Fig. 12. Calculated kD-xx/kf number for 95% porosity. Space dependence. a) ReD ¼ 1, b) ReD ¼ 10, c) ReD ¼ 75.

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183

Fig. 13. Calculated kD-xx/kf for PeD ¼ 1000. ReD and porosity as parameters.

Fig. 14. Fully developed values for NuD as a function of the PeD number and with the ReD number and porosity as parameters. Correlation given in Ref. [16] is added for ReD ¼ 1.

184

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Fig. 15. Fully developed values for kD-xx/kf as a function of PeD and with ReD and porosity as parameters. A straight line with slope equals to 2 is shown for comparison purposes.

conclude that the dependence on PeD found in this study for the streamwise thermal dispersion is correct. Data presented in Figs. 14 and 15 can be employed to develop correlations as a function of ReD, PeD and f (e.g. kDxx =kf ¼ f ðReD ; PeD ; fÞ). Correlations of this type for ordered porous media can be found in Refs. [4,5,14]. Moreover and recently, Ref. [38] proposes a relationship to determine kD-xx as a function of hsf (i.e. kDxx ¼ f ðhsf ; PeD ; fÞ). 5. Concluding remarks In this study, a large set of numerical data simulating the thermal developing of a laminar flow in an ordered porous medium was analyzed. First, it was shown that the CA is a superior tool than the VA to space average microscopic data. This observation is relevant because the suboptimal tool VA is the most commonly used until now. While the CA yields macroscopic values without pore-scale fluctuations in both, developing and fully developed regions, the clet number VA does not. It was also shown that the smaller the Pe and the smaller the porosity, the larger the fluctuations of the VA quantities. Based on the range of parameters simulated, it can be concluded that VA quantities are a good approximation of the CA quantities for PeD numbers larger than 100. Fully developed values of the interfacial heat transfer and the streamwise thermal dispersion were calculated employing the CA. These values were well compared with VA values reported in the literature. The interfacial heat transfer shows good agreement with the correlation presented in Ref. [16]. The streamwise thermal dispersion was found to scale approximately with Pe2D for fixed ReD and f. This dependence has been found in theoretical, numerical and experimental studies in the laminar regime. Generally, macroscopic energy equation models employed in the literature assume constant, fully developed values, for the macroscopic coefficients. This assumption, that neglects the entrance effect, was tested in this study computing the coefficients

in a laminar flow that thermally develops in an ordered porous structure. Results show that, from the macroscopic point of view, the flow is characterized by a developing region that can be several REVs in length. In this region, both, the interfacial heat transfer and the streamwise thermal dispersion are dependent on the streamwise coordinate. Therefore, the macroscopic energy equation model may show large differences with an experiment if the entrance effect is not considered. In the developing region, the macroscopic heat transfer coefficient shows a similar behavior to that found for the convective heat transfer coefficient in a clear channel flow. A peak is found at the entrance followed by a decrease to a fully developed value. The length of this developing region depends on ReD, PeD and f. For ReD ¼ 10 and PeD ¼ 1000, the developing length is 2, 4, 12 REVs for 55, 75 and 95% porosity respectively (5% difference respect to fully developed values); and the NuD-peak/NuD-FD ratio is 1.4, 1.2 and 1.1 respectively. Therefore, the assumption of a constant fully developed heat transfer coefficient underestimates the energy transferred between the solid and fluid-phases. A developing region is also found for the streamwise thermal dispersion. This coefficient evolves from zero to the fully developed value in a length that increases with ReD, PeD and f. The developing region for this quantity can be of several REVs in length. For instance, for PeD ¼ 1000 and f ¼ 0.95, this length is larger than 15 REVs. Therefore, the assumption of a constant fully developed streamwise thermal dispersion overestimate the diffusion of the temperature in the streamwise direction. This study gave new insight on the impact of the entrance effect on the energy transfer process in laminar flows through porous media, an aspect heretofore unconsidered. For ordered media, large PeD and high porosities, the entrance effect is shown to be significant at a macroscopic scale. This suggests, again, that the macroscopic temperature calculated with a macroscopic model that does not account for the entrance effect would be largely over/under

F.E. Teruel / International Journal of Thermal Sciences 104 (2016) 172e185

estimated respect to an experiment. Therefore, further research is needed to improve the macroscopic energy equation model to correctly capture the physical process that occurs in the thermally developing region. New research is under way to quantify this effect on the behavior of the macroscopic temperature.

Acknowledgments This work was supported by grants from Universidad Nacional de Cuyo (PB 2013-2015), from ANPCyT-FONCyT (PICT 2012-2575) and from CONICET (PIP 112 201301-00829 CO).

References [1] M. Kaviany, Principles of Heat Transfer in Porous Media, Springer-Verlag, 1991. [2] S. Whitaker, The Method of Volume Averaging, Kluwer, 1999. [3] D. Koch, R. Cox, H. Brenner, J.F. Brady, The effect of order on dispersion in porous media, J. Fluid Mech. 200 (1989) 173e188. [4] N. Wakao, S. Kaguei, Heat and Mass Transfer in Packed Beds, Gorden and Breach, New York, 1982, pp. 243e295. [5] F. Kuwahara, A. Nakayama, Numerical determination of thermal dispersion coefficients using a periodic porous structure, J. Heat Transfer 121 (1999) 160e163. [6] M. Quintard, S. Whitaker, Transport in ordered and disordered porous media I: the cellular average and the use of weighting functions, Transp. Porous Media 14 (1994) 163e177. [7] M. Quintard, S. Whitaker, Transport in ordered and disordered porous media II: generalized volume averaging, Transp. Porous Media 14 (1994) 179e206. [8] F. Teruel, L. Díaz, Calculation of the interfacial heat transfer coefficient in porous media employing numerical simulations, Int. J. Heat Mass Transfer 60 (2013) 406e412. [9] A. Nakayama, F. Kuwahara, T. Hayashi, Numerical modelling for threedimensional heat and fluid flow through a bank of cylinders in yaw, J. Fluid Mech. 498 (2004) 139e159. [10] A.A. Alshare, T.W. Simon, P.J. Strykowski, Simulations of flow and heat transfer in a serpentine heat exchanger having dispersed resistance with porouscontinuum and continuum models, Int. J. Heat Mass Transfer 53 (2010) 1088e1099. [11] B. Buonomo, O. Manca, G. Lauriat, Forced convection in micro-channels filled with porous media in local thermal non-equilibrium conditions, Int. J. Therm. Sci. 77 (2014) 206e222. [12] L. Cheng, A.V. Kuznetsov, Heat transfer in a laminar flow in a helical pipe filled with a fluid saturated porous medium, Int. J. Therm. Sci. 44 (2005) 787e798. [13] M.E. Nimvari, M. Maerefat, M.K. El-Hossaini, Numerical simulation of turbulent flow and heat transfer in a channel partially filled with a porous media, Int. J. Therm. Sci. 60 (2012) 131e141. [14] F. Kuwahara, M. Shirota, A. Nakayama, A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media, Int. J. Heat Mass Transfer 44 (2001) 1153e1159. [15] J. Pallares, F.X. Grau, A modification of a Nusselt number correlation for forced convection in porous media, Int. Commun. Heat Mass Transfer 37 (2010) 1187e1190. [16] A. Nakayama, A note on the confusion associated with the interfacial heat transfer coefficient for forced convection in porous media, Int. J. Heat Mass Transfer 79 (2014) 1e2. [17] M.B. Saito, M.S.J. de Lemos, A correlation for interfacial heat transfer coefficient for turbulent flow over and array of square rods, J. Heat Transfer 128 (2006) 444e452. [18] G. Gamrat, M. Favre-Marinet, S. Le Person, Numerical study of heat transfer over banks of rods in small Reynolds number cross-flow, Int. J. Heat Mass Transfer 51 (2008) 853e864.

185

[19] A.A. Alshare, P.J. Strykowski, T.W. Simon, Modeling of unsteady and steady fluid flow, heat transfer and dispersion in porous media using unit cell scale, Int. J. Heat Mass Transfer 53 (2010) 2294e2310. [20] Y. Takemoto, K. Kawanishi, J. Mizushima, Heat transfer in the flow through a bundle of tubes and transitions of the flow, Int. J. Heat Mass Transfer 53 (2010) 5411e5419. [21] M.G. Pathak, S.M. Ghiaasiaan, Convective heat transfer and thermal dispersion during laminar pulsating flow in porous media, Int. J. Therm. Sci. 50 (2011) 440e448. [22] G.R. Imani, M. Maerefat, K. Hooman, Estimation of heat flux bifurcation at the heated boundary of a porous medium using a pore-scale numerical simulation, Int. J. Therm. Sci. 54 (2012) 109e118. [23] H.P. Amaral Souto, C. Moyne, Dispersion in two-dimensional periodic porous media. Part II. Dispersion tensor, Phys. Fluids 9 (1997) 2253. [24] N. Han, J. Bhakta, R.G. Carbonell, Longitudinal and lateral dispersion in packed beds: effect of column length and particle size distribution, AIChE J. 31 (1985) 277e288. [25] W.G. Gray, A. Leijnse, R.L. Kolar, C.A. Blain, Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems, CRC Press, 1993. [26] M. Hassanizadeh, W.G. Gray, General conservation equations for multi-phase systems: 1. Averaging procedure, Adv. Water Resour. 2 (1979) 131e144. [27] J.C. Slattery, Single phase flow through porous media, Am. Inst. Chem. Eng. J. 15 (1969) 866e872. [28] S. Whitaker, Advances in theory of fluid motion in porous media, Ind. Eng. Chem. 61 (1969) 14e28. [29] G.I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. A 219 (1953) 186e203. [30] R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. A 235 (1956) 67e77. [31] F.E. Teruel, Rizwan-uddin, Characterization of a porous medium employing numerical tools: permeability and pressure-drop from Darcy to turbulence, Int. J. Heat Mass Transfer 52 (2009) 5878e5888. [32] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, 1980. [33] B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Eng. 19 (1979) 59e98. [34] T. Hayase, J.A.C. Humphrey, R. Greif, A consistently formulated QUICK scheme for fast and stable convergence using finite-volume iterative calculation procedures, J. Comput. Phys. 98 (1982) 108e118. rica para el ca lculo de para metros [35] L. Díaz, Desarrollo de una herramienta nume  picos en medios porosos bidimensionales (Thesis for the Degree of macrosco Mechanical Engineer), Instituto Balseiro, Universidad Nacional de Cuyo, Argentina, 2012. [36] F.E. Teruel, Macroscopic Turbulence Modeling and Simulation for Flow through Porous Media (Ph.D. thesis), University of Illinois, Urbana-Champaign, IL, 2007. [37] F. Teruel, Calculation of the interfacial heat transfer and streamwise dispersion coefficients employing double volume average, Proc. FEDSM 1C (2013) art. 16119. [38] Y. Sano, F. Kuwahara, M. Mobedi, A. Nakayama, Effects of thermal dispersion on heat transfer in cross-flow tubular heat exchangers, Heat Mass Transfer 48 (2011) 183e189. [39] C. Yang, A. Nakayama, A synthesis of tortuosity and dispersion in effective thermal conductivity of porous media, Int. J. Heat Mass Transfer 53 (2010) 3222e3230. [40] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, Springer-Verlag, 1991. [41] D.J. Gunn, C. Pryce, Dispersion in packed beds, Trans. Inst. Chem. Engrs. 47 (1969) T341eT350. [42] A. Nakayama, F. Kuwahara, Y. Kodama, An equation for thermal dispersion flux transport and its mathematical modelling for heat and fluid flow in a porous medium, J. Fluid Mech. 563 (2006) 81e96. [43] M.A. Saada, S. Chikh, A. Campo, Analysis of hydrodynamics and thermal dispersion in porous media by means of a local approach, Heat Mass Transfer 42 (2006) 995e1006. [44] A. Dybbs, R.V.A. Edwards, A new look at porous media fluid mechanics e Darcy to turbulent, in: Fundamentals of Transport Phenomena in Porous Media, Martinus Nijhoff Publishers, 1984, pp. 199e256.