Validation of local thermal equilibrium (LTE) in porous media for variation in flow rate and permeability: transient analysis

The usage of porous media is one of the effective ways to augment the heat transfer in micro-to-mini scale channels due to its enormously high surface area, enhanced thermophysical properties, and dispersion effect to transfer the thermal energy between the heated channel wall and fluid at a faster rate. However, there is ambiguity regarding the degree of thermal equilibrium within the fluid and solid phases, and this also impacts the selection of an appropriate heat transfer model. In the current study, the numerical analysis of thermo-hydrodynamics for the porous media filled in the channel for incompressible laminar forced convective flow with varying Reynolds number and Darcy number is delt. The validity of the local thermal equilibrium (LTE) condition is performed for transient case to study the effects of the parameters that governs the flow and thermal effects, i.e., Reynolds number (Re) in the range of 10 < Re < 1000 and Darcy number (Da) in the range of 10–6 < Da < 100. The results of the present study show that for higher range of flow rate (Re > 100), the LTE assumption is valid for all ranges of Da, however, for the lower range of flow rate (Re < 100), the LTE assumption deviates for higher Da.


Introduction
The speed of communication and the accessibility of information is astounding in today's fast-paced environment. The requirement for shrinking of electronic devices with highspeed processing applications (such as smartphones, health monitoring devices, etc.) developed enormously because of the improvements in high-speed devices with higher performance. Due to the reduction of the size and improved performance, a lot of thermal energy are being produced, which needs to be dissipated to maintain performance or the devices would break thermally. Thermal management is required because conventional cooling methods, such as air-cooling, fall short of offering suitable solutions for the routine functioning of microscopic electronic equipment. Early 1980s saw a rise in popularity for microdevices as a result of rapid technological advancements in manufacturing, particularly in the electronics sector [1][2][3]. As a result, the use of passive cooling methods such heat pipes (at microscales) is made possible by the miniaturized cooling of these devices with strong heat flux dissipation of the order of 10 4 to 10 6 W/m 2 . The necessity for fundamental research in porous media heat transfer originates from the need for a better understanding of thermal engineering applications that uses porous materials such as the evaporator portion of the heat pipe where heat needs to be dissipated [4,5]. Porous media are distinguished from non-porous media by two fundamental characteristics: a large specific surface area and the tortuous morphology of capillaries. When any fluid passes through the pores, the particles are more exposed or in contact with the interior surfaces of the solid matrix due to the porous structure's incredibly high surface area density, resulting in an exceptionally high heat transmission rate between the solid material and fluid. Additionally, the tortuous geometry of interconnecting pores subjects fluid particles to a series of stagnation, separation, recirculation, and reattachment processes, all of which contribute to dispersion effects thus leading to much better mixing of any fluid resulting in effectively transferring heat within the system [6]. The flow behavior in the channels filled with the porous medium is significantly different than the normal (empty) channel, since a relatively thinner hydrodynamic and thermal boundary layers are observed in the porous media infilled channels [7]. The flow in porous media is mathematically modelled using the Darcy equation, which is a linear relationship between the pressure drop and the mean velocity in the channel. However, this linear relationship is only true for smaller flow velocities and generally identified by the smaller porescale Reynolds number (Re p < 1) [8]. In case of higher flow rates, the numerical analysis of fluid through porous media is performed by extended Darcy equations considering viscous resistance, inertial resistance, and wall effects [9]. The experimental investigation on non-Darcian flows shows the enhancement of heat transfer due the mechanism of thermal dispersion illustrating the intra-pore mixing by flow over the particle [10]. The traditional theory of porous media postulates the existence of a local thermal equilibrium (LTE) between fluid phase and solid phase. According to this theory, the fluid and solid's temperatures quickly approach their equilibrium points. The fluid and solid might not be able to reach a local thermal equilibrium (LTE) rapidly enough in the real world because of their different thermal diffusion capacities [8,9]. In case of heat transfer in porous media, single-phase energy equation, also known as homogeneous energy equation, assumes the solid and fluid phase at thermal equilibrium is a popular choice for many researchers due to its simplicity and computational economy, which is true for limited and ideal applications. The more accurate model is the usage of two-phase energy equation, also known as heterogeneous energy equation, which considers the solid phase and the fluid phase at thermal non-equilibrium, and which is close to the actual situations, however, it is typically complex and computationally expensive to solve. Some researchers have also compared the extent of thermal equilibrium between the solid and fluid phases for variation in parameters, such as flow rate, porosity, permeability [13][14][15]. The numerical simulation of forced convection incompressible flow through porous media using two-phase equation model for local thermal non-equilibrium (LTNE) is used for the energy equation to solve the temperature fields [11,12]. Moreover, the transient version of the conservation equations to capture the time variation of parameters is critical during the initial stage of flow development and the numerical analysis of unsteady forced convection flow of gas (Freon) through a packed bed of spherical particles (of three different materials) is studied and detailed analysis of heat exchange between solid and fluid phases and their effects on flow fields, pressure, and density variations is presented [16].
The degree of thermal equilibrium within the solid and fluid phases is essential to the mathematical modeling of the heat transport process in a porous medium. Therefore, in this study, a transient analysis is performed for the flow rate (Reynolds number, Re) and the porous media morphology (Darcy number, Da) to map the range of regimes in which the local thermal equilibrium (LTE) condition is valid, limiting the use of complicated heterogenous local thermal nonequilibrium (LTNE) equations, which incur high computational costs, when they are not essential.

Problem description
Forced flow through a 2D microchannel, filled with homogenous and isotropic porous media is studied in the present work. Transient numerical investigation on 2D microchannel of dimensions (L × H) of 20 mm × 0.267 mm as shown in Fig. 1 are performed. The flow is assumed to be laminar, unsteady, and incompressible neglecting the gravity effects considering the size of the porous media in which the fluid flows. Whereas the properties of porous media are assumed to be homogenous and isotropic, as provided in the experimental study [17]. The thermophysical properties and parameters of porous media are described in Tables 1 and 2. The validity of local thermal equilibrium is verified for different flow rates (10 < Re < 1000) and porous media conductance (10 -6 < Da < 10 0 ).
There are two stages at which the transient results are presented and discussed: initial stage and final stage. Initial stage refers to the time while the simulation has just begun and the flow variables are developing at a faster rate, whereas the final stage refers to the time at which the flow variables have achieved the developed state and no further development can occur. In each parametric study, transient simulations are performed from the initial  condition and ran till the steady state condition is reached. Darcy-Forchheimer-Brinkman equation is used to solve momentum equation which accounts for the additional pressure drop due to viscous and inertial resistances, along with the wall effect. whilst the thermal equilibrium amongst the solid and fluid phases in porous media gives rise to two unique formulations for the energy equation.
As described earlier, in case of thermal non-equilibrium (LTNE), two energy equations are solved separately for each phase, whereas, in case of thermal equilibrium (LTE), a single energy equation is solved.

Solution methodology
The numerical simulation of forced flow through a 2D microchannel, containing porous media, is investigated for the validity of local thermal equilibrium (LTE) conditions. The flow is laminar, unsteady, and incompressible, where the gravity effects are neglected. The homogenous and isotropic porous media used in this study has acquired its properties from sintered metal structures [17]. As the size of the grid is quite small (approx. 1.3 μm) for the geometry used in the present study, therefore for transient simulations, it requires a very small time-step size (approx. 10 -6 s) for converged solutions, by considering the Courant-Friedrichs-Lewy stability condition [18]. In this study, a mini-channel is chosen as a computational domain as shown in the Fig. 1. The dimensions of the microchannels (L × H) are of 20 mm × 0.267 mm in x and y directions, respectively. The physical properties of the materials and the parameters used for the porous matrix are displayed in Tables 1 and 2, respectively. Numerical simulations in the present study are performed on the commercial CFD solver ANSYS Fluent, which use the Finite Volume Method (FVM) to solve the governing equations [19]. SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm is used for pressure-velocity coupling [20]. Spatial discretization of pressure is done using second order scheme and second order upwind is used for the momentum and energy equation. For the transient simulations, the time is discretized and solve using second order implicit scheme. Furthermore, during the simulation, the convergence criteria for continuity, momentum and energy equations are kept at 10 -6 , 10 -6 , and 10 -12 , respectively. Forchheimer-Brinkman extended Darcy model is used for the momentum equation, whereas, the LTNE and LTE versions of energy equations are solved separately for each case to establish the deviation in their corresponding results. The equations are written as follows:

LTNE (T s ≠ T f ):
for fluid phase for solid phase The relations for pore-scale heat transfer coefficient, h sf and interfacial area density, a Sf are given as follows [16]:

LTE (T s = T f = T):
where the effective material properties of the porous medium, such as specific heat capacity and thermal conductivity, are calculated using the volume fraction weighted average of solid and fluid material properties, as shown in Eq. 9. For the transient simulations, the initial condition for the pressure and temperature are set at 0 Pa and 298 K, respectively. The initial velocity for each location is set as the inlet velocity, where the boundary conditions at the c P e = c P f + (1 − ) c P s ; k e = k f + (1 − )k s inlet are set at uniform velocity (calculated according to the Re) in the streamwise direction and uniform temperature at 298 K. Outflow conditions are used at the outlet, where zero gradients persist in the normal direction to the outlet surface for the velocity and temperature. The top and bottom walls are set as no-slip condition and provided with the uniform heat flux of the value 100 kW/m 2 . Pressure variation within the channel and resulting velocity profile are important quantities in the hydrodynamic study which determines the extent of the convective and dispersive effects. Additionally, most important quantities in the present study are temperatures obtained using different forms of the energy equations. The LTNE equation provides solid and fluid temperatures separately, whereas, the LTE equation provide the equilibrium or a common temperature. Determination of relative difference between solid and fluid temperatures, and their deviation in comparison to the equilibrium temperature is the major aim of this study. The aforementioned quantities are shown and discussed in the next section using the nondimensional variables of lengths, velocity, pressure, and temperature, as shown below: The streamwise and transverse directions are nondimensionalized by using the channel length and width, respectively. The velocity and pressure are nondimensionalized by using the inlet velocity and dynamic pressure created by the inlet velocity, respectively. Moreover, the temperature is nondimensionalized by the inlet temperature, constant heat flux provided at the walls, channel width, and the thermal conductivity according to the model used. In case of LTNE, two temperature field are obtained for the solid and fluid, therefore, while converting them to the nondimensional temperature, thermal conductivity of solid and fluid as shown in Table 1 are used, whereas, in case of LTE, a single value of temperature is obtained common to both solid and fluid, therefore, effective thermal conductivity is used which is calculated by the relation given in Eq. 9. Throughout the results, these dimensionless quantities are shown and used in the discussion accordingly.
Hq wall 4 Results and discussion

Grid independence study
Grid independent study is performed by choosing five different grid sizes and structured mesh of non-uniform density with very fine mesh at the walls to coarser mesh towards the core to obtain the accurate resolution of the gradients at walls. Total number of nodes used for this are 450, 1800, 4050, 7200, 11,250. Figure 2 shows the calculation of pressure drop in the channel and average temperature at the outlet for different of grids used. Relative change is smaller for the grids 4050, 7200 and 11,250, therefore, after comparing the results for the tested values, it is found that 4050 grids are optimal for executing the computations with the required accuracy and judicious use of computational resources.

Validation of the model
The validation of the governing equations is carried out by comparing the numerical solution given by Vafai and Sozen (1990) [16] with the present results by considering the same initial and boundary conditions (P o = 100 kPa; T w,o = 280 K, T inlet = 300 K) used by them, where the nondimensional temperature which is more relevant to the constant temperature boundary condition is defined as below: As shown in Fig. 3, the plots of nondimensional pressure and nondimensional temperature with respect to transient variations indicate reasonable accuracy with the numerical model used in this study. The time is nondimensionalized

Effect of reynolds number
Reynolds number may be used as the nondimensional representation of the rate at which the fluid is flown in the porous media, since the material properties of the fluid and channel width are constant. For the higher value of Reynolds number, fluids ability to advect the heat from the porous media also gets high, which also leads to better thermal dispersion and mixing. In this section, the effect of Reynolds number is analysed for the transient case through pressure, velocity, and temperature, for the covered range of 10 < Re < 1000. From the Fig. 4a, b, overall dimensionless pressure drops along the channel are found to be approximately 40 and 11 × 10 3 for Re = 10 and Re = 1000, respectively. In both cases, depicted in Fig. 4c, d, the nondimensional velocity profiles are almost constant with a sharp gradient observed near the walls, similar to the plug flow, due to the presence of porous media. However, the thickness of the hydrodynamic boundary layer is higher in Re = 10 case, and extremely thin which makes it almost invisible in case of Re = 1000.
From the Fig. 5a-d, the nondimensional temperature contours for initial stage and final stage are shown for Re = 10 and Re = 1000, respectively. At the initial stage, the flow is developing as evidently seen by nearly uniform temperature at LTE condition for the entire flow domain except at the entry region. On the other hand, at the final stage, flow is developed, reaching almost the steady state, as no further improvement in the solution was observed. The Fig. 5a, b shows the growth of the thermal boundary layer for LTE and LTNE cases at Re = 10. At the initial stage, the thermal boundary layer is developing, and the thermal energy (heat) is not penetrated to the core of the channel. However, at the final stage, the thermal boundary layer is developed with the thicker boundary layer due to the low advection caused by the lower Re. The temperature profiles are similar to the state as if it tends to the steady state condition. The Fig. 5c, d shows the growth of the thermal boundary layer for LTE and LTNE cases at Re = 1000, where the sharp gradients are observed at the walls due to the high advection. The temperature scales are compared with LTE condition which shows greater and lower temperature scale values for Re = 10 and Re = 1000 respectively. It is also observed that the LTE model shows better dispersion of the temperature for the Re = 1000 case ( Fig. 5c) even at the beginning of the time. Figure 6 shows the nondimensional temperatures for LTNE (fluid and solid), and LTE (equilibrium) cases, averaged at the outlet boundary at the beginning and end of the transient simulation. At low Re (Re = 10), dispersion of thermal energy is weaker due to lower advection, resulting in a thermal non-equilibrium, whereas this situation is reversed for higher Re (Re = 1000). In addition to that, at the initial stage, there is a sudden imparting of inertial effects (due to the transient term in the governing

Effect of Darcy number
Darcy number (Da) represents the degree of permeability provided to the fluid when it passes through the porous medium. Low Darcy number represents that the porous media is tightly packed and higher visco-inertial resistances are offered to the fluid, therefore, the fluid flowing through it has higher thermal dispersion and vice-versa for a high Darcy number. The Da has been varied in the range of 10 -6 < Da < 10 0 for the two values of the Reynolds numbers as Re = 10 and 1000.
From the Fig. 7a, b, the nondimensional pressure drops along the channel at Re = 10 are 12 × 10 3 and 15 × 10 6 for Da = 10 0 and Da = 10 -6 , respectively, which shows that as Da decreases, the conductance of the flow reduces with huge drop in pressure. The velocity profile from the Fig. 7c has visible gradients at the walls due to high Da (Da = 10 0 ) compared to the Fig. 7d which has sharp gradient at the walls with almost plug like flow as the Da is low (Da = 10 -6 ), due to the high dispersion within the porous media.
At Re = 10, for Da = 10 0 and Da = 10 -6 , there is a low fluid advection due to which the thermal energy in the solid porous matrix is not taken away by the fluid and high temperatures are observed in the solid temperature profiles as seen from the Figs. 8a-d. But the scales of the temperature values in the Fig. 8b-d for the final stage shows the significant difference in their temperature values for the solid and the fluid, proving the thermal equilibrium condition is not established in these regimes.
From the Fig. 9a, b, at Re = 1000, the nondimensional pressure drops along the channel are approximately found to be 10 × 10 3 and 16 × 10 4 for Da = 10 0 and Da = 10 -6 , respectively, which shows that as Da decreases, the conductance of the flow reduces with huge drop in pressure. The velocity profiles from the Fig. 9c, d has the sharp gradient at the walls with almost plug like flow, irrespective of Da values, showing the dominance of inertial effects over the dispersion effects.
At Re = 1000 for Da = 10 0 and Da = 10 -6 , due to the high fluid advection, the thermal energy in the solid porous matrix is taken away by the fluid and temperature scales observed in the solid, fluid and LTE cases are same as seen from the Fig. 10a-d proving that the thermal equilibrium between the two phases (solid and fluid) is reached at high Re (Re = 1000) irrespective of Da values. In this particular situation (Re = 1000), it is observed that the temperature profiles are almost identical for corresponding cases of the initial and final stage, irrespective of the level of permeability. This fact also establishes the dominance of the Reynolds number over the Darcy number for assigning the validity of the LTE condition. In addition to that, it is also observed that the growth of the thermal boundary layer is much faster in the LTE case since it has enhanced effective thermal conductivity. From the Fig. 11, it can be concluded that for a lower value of permeability (Da = 10 -6 ), the dispersion ability of the porous medium is greatly improved, however, at the cost of higher pressure drop across the channel. Dispersion causes better mixing and therefore leads to a thermal equilibrium between the two phases. Similarly, at higher permeability (Da = 10 0 ), the dispersion effects are weaker, which leads to improper mixing and results into a thermal non-equilibrium between the two phases. However, when the convection effects are very high (Re = 1000), the effect of medium permeability is over-shadowed, therefore the thermal equilibrium is found irrespective of the change in Da in this case (Fig. 11b), when the scales of the nondimensional temperature is compared for the low and high advection cases.

Conclusions
In order to determine the proper regimes for the LTE assumption of heat transfer, numerical simulations of incompressible, unsteady, and laminar flow in a 2D microchannel containing the porous material and uniformly heated with heat flux at the walls were carried out in the current work. Effect of Reynolds number and Influence of Darcy number, two parametric studies, have been investigated. In both parametric studies, the flow was developing both hydrodynamically and thermally. Due to similar geometric, flow, and heating conditions, these insights can be applied to thermal management applications and heat pipes. The parametric investigations allow for the following inferences:

Effect of time
In the transient study, two stages are defined, namely (a) initial stage: where the flow variables are just started to develop; and (b) final stage: where the flow variables are developed. At the initial stage, as the flow is developing, the inertial disturbances are high and there is less difference in the temperature between the two phases (solid and fluid). At the final stage, as the flow is developed, and the inertial disturbances caused by the transient term gets lower resulting in poor thermal dispersion and inadequate mixing leading to the poor exchange of the thermal energy. Therefore, a significant difference in temperature between solid and fluid phases is observed.

Effect of Reynolds number (Re)
At low Re (Re = 10), the advection is less, and dispensing of the thermal energy is low, resulting in a thermal non-equilibrium and vice versa for higher Re (Re = 1000). Thus, in the present study, the observed result shows the thermal equilibrium condition is applicable for Re > 100 and the thermal non-equilibrium equations are applicable for Re < 100. Also, the role of Reynolds number is dominating as compared to the Darcy number, since for high Re (Re = 1000), the thermal equilibrium is observed between the two phases irrespective of the variation of the Da.

Effect of Darcy number (Da)
For a low Da, the dispersion ability of the porous medium is improved at the cost of high-pressure losses. The thermal dispersion causes better mixing and therefore leads to a thermal equilibrium between the two phases. On the other hand, at higher Da, the dispersion effects are low which leads to thermal non-equilibrium. Thus, at low Re (Re = 10), the thermal equilibrium condition is applicable at low Da (Da = 10 -6 ), whereas the thermal equilibrium condition is observed in the whole range of Da values at higher Re (Re = 1000).