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This study aims to numerically examine the fluid flow and heat transfer in a porous microchannel saturated with power-law fluid. The governing momentum and energy equations are solved by using the finite difference technique. The present study focuses on the slip flow regime, and the flow in porous media is modeled using the modified Darcy-Brinkman-Forchheimer model for power-law fluids. Parametric studies are conducted to examine the effects of Knudsen number, Darcy number, power law index, and inertia parameter. Results are given in terms of skin friction and Nusselt number. It is found that when the Knudsen number and the power law index decrease, the skin friction on the walls decreases. This effect is reduced slowly while the Darcy number decreases until it reaches the Darcy regime. Consequently, with a very low permeability the effect of power law index vanishes. The numerical results indicated also that when the power law index decreases the fully-developed Nusselt number increases considerably especially, in the limit of high permeability, that is, nonDarcy regime. As far as Darcy regime is concerned the effects of the Knudsen number and the power law index of the fully-developed Nusselt number is very little.

Fluid flow and heat transfer in porous media has been a subject of continuous interest during past decades because of the wide range of engineering applications. In addition to conventional applications including solar receivers, building thermal insulation materials, packed bed heat exchangers, and energy storage units, new applications in the emerging field of microscale heat transfer have existed. However, microchannels are now used in several industries and equipment such as cooling of electronic package, microchannel heat sinks, microchannel heat exchanger, microchannel fabrication, and cooling, and heating of different devices [

One of the major difficulties in trying to predict the gaseous transport in micron sized devices can be attributed to the fact that the continuum flow assumption implemented in the Navier-Stokes equations breaks down when the mean free path of the molecules (

However, there is a certain limit of the channel size with which one can still apply Navier-Stokes equations with some modifications on the boundary conditions [

The continuum model is valid for very small Knudsen number flows (

The appropriate flow and heat transfer models for a given gas flow problem depend on the range of Knudsen number. A classification of the different gas flow regimes is given as follows:

Convection heat transfer in circular and noncircular microchannels has been solved over the years [

Laminar forced convection of Newtonian fluid flow in microchannels filled with a porous medium has been solved by numerical and analytical means over the years [

A theoretical and numerical analysis of the fully-developed forced convection in a porous channel saturated with a power-law fluid in porous channel has been investigated recently [

However, little of information on the related literature regarding the flow and heat transfer of power law fluids through porous microchannels. That said, in this study, the forced convection of heat and fluid flow of power law fluids through parallel plate microchannels filled with porous media were considered. The aim of the present study is to investigate the effects of Knudsen number, Darcy number and the inertia parameter on the hydrodynamic and thermal behavior of a power law fluid flow between infinitely long parallel-plates microchannels filled with porous media.

The analysis is carried out for unsteady state, incompressible and laminar forced convection flow between parallel-plates microchannel filled with porous medium and heated with uniform wall temperature at the walls. The flow is assumed to be hydrodynamically fully-developed. The porous medium is saturated with a single phase nonNewtonian fluid described by the power law model and assumed to be in local thermodynamic equilibrium with the fluid.

As a result of the continuity equation, the flow is a unidirectional and is expressed in terms of the axial velocity

In the momentum equation, a modified Darcy’s law for power-law fluids was used where

In the energy equation, the thermal dispersion conductivity of the porous media is assumed to be constant and is incorporated into the effective thermal conductivity. The axial heat conduction effects are usually negligible for nearly parallel flows. Momentum and energy transfer between the liquid molecules and the surface requires specification of interactions between the impinging molecules and the surface. From the macroscopic point of view, it is sufficient to know some average parameters in terms of the so-called tangential momentum (

Under the above assumptions and by using the nondimensional variables listed in the nomenclature, the equations of motion and energy equation are reduced to the following form:

The quantities of primary interest in this study are the friction factor and Nusselt number. These are defined as follows:

The governing equations are solved numerically using the finite difference technique. The governing momentum and energy equations are not coupled, consequently the numerical solution proceeds by first solving the velocity distribution from the momentum equation, and then solving the energy equation for the temperature distribution.

The momentum equation (

In a similar manner, the energy equation is discretized using the same numerical scheme. In contrast to the momentum equation, it should be mentioned that the steady state form of the energy equation is discretized directly and therefore a time dependent solution was not obtained. When the momentum equation is discretized and applied at every point in the finite difference grid, a system of nonlinear finite difference (algebraic) equations was obtained. To overcome the difficulties in solving such a system, the nonlinear term (last term) in (

Based on the above approach, the resulting systems of algebraic equations obtained by discretizing the momentum and energy equations are tri-diagonal, which are best solved by using Thomas algorithm. The adequacy of the grid is verified by comparing the results of different grid sizes. A mesh refinement study was carried out in order to ensure grid independent solutions. It was found that the obtained numerical solution for the momentum equation is invariant beyond a grid size of 75 points in the y-direction. Therefore, all velocity profiles are obtained using this grid size. Similar refinement study was carried out for the energy equation. It was found that a grid size of

In order to verify the validity and accuracy of the numerical model, the present numerical results were compared with corresponding integral solution results for the case of fully developed forced convection in porous macrochannel saturated with a power-law fluid [

Comparison of results for fully developed Nusselt number with analytical results of Chen and Hadim.

The effect of the Knudsen number and the power law index on the axial fully-developed velocity profiles is shown in Figure

Effect of Nudsen number and power law indexon the axial velocity profile for

The effect of the Darcy number and the power law index on the axial fully developed velocity profiles is shown in Figure

Effect of Darcy number and power law index on the axial velocity profile for

In contrast, Figure

Effects of inertia parameter and power law index on the axial velocity profile for

The combined effects of the Knudsen number, power law index and Darcy number on the skin friction, are clearly presented in Figure

Effects of Knudsen number and power law index on the skin friction with Darcy number for

The combined effects of the Kn number, power law index, and the inertia parameter on the skin friction are clearly presented in Figure

Effect of Knudsen number and power law index on the skin friction with inertia parameter for

The effect of the Kn number and power law index on the variation of the fully developed Nusselt number with inertia parameter is shown in Figure

Effects of Knudsen number and power law index on the variation of the fully developed Nusselt number with inertia parameter.

Figure

Effects of Knudsen number and power law index on the variation of the fully developed Nusselt number with Darcy number (a)

The present numerical solutions are conducted for steady laminar forced convection flow between parallel-plate microchannels filled with porous medium and saturated with a power-law fluid. In this study, the slip flow regime (

Specific heat, J/kg

Inertia coefficient

Modified inertia coefficient

Darcy number,

Hydraulic diameter,

Channel height, m

Local heat transfer coefficient

Intrinsic permeability of the porous medium,

Modified permeability of the porous medium,

Knudsen number, (

Thermal conductivity of the fluid, W/m-K

Effective thermal conductivity of the fluid saturated porous medium, W/m-K

Power law index, m

Local Nusselt number, (

Fully developed Nusselt number

Pressure, Pa

Dimensionless pressure,

Prandtl number,

Reynolds number,

Temperature, K

Time, s

Reference time, (

Nondimensional axial velocity, (

Axial velocity component, m/s

Dimensionless axial coordinate, (

Axial coordinate, m

Dimensionless transverse coordinate, (

Transverse coordinate, m.

Thermal diffusivity (

Mean free path of the gas molecules

Tangential momentum accommodation coefficient

Thermal accommodation coefficient

Specific heat ratio, (

Microinertia parameter, (

Porosity of the porous media

Dynamic viscosity

Consistency index of the power law fluid

Fluid density

Nondimensional temperature,

Dimensionless mean temperature,

Dimensionless time,

Shear stress at the wall (

Effective (i.e., fluid-saturated porous medium)

Fluid

Mean

Pressure

Solid

Volume

Wall.