CHAPTER 1
INTRODUCTION
1.1 Introduction and Background
Porous media has a wide variety of industrial applications such as electronic cooling, geothermal systems and many others. Modeling of non-Darcian transport through porous media has been the subject of interest in recent times, due to its practical applications, which includes electronic air-cooling in microelectronics. The demand for high execution speed and memory capacity for modern computers has led to a high circuit density per unit chip resulting in higher fluxes at the chip level to the order of 10 - 25 W/cm2 for DIPs and PGA respectively (Mahalingam and Berg, 1984; Mahalingam, 1985). For reliable operation these units need to be maintained at a safe level of temperature (1400 C). Natural and forced convection remove only a small amount of heat around 0.001 W/cm2 0 C by natural convection, 0.01W/cm2 0C in forced air and 0.1 W/ cm2 0C in forced liquid (Simons, 1984).
While, the primary mode of removal of heat from the package surface is by convection, the thermal resistance between the coolant and the surface plays a key factor and therefore a large heat transfer coefficient or large contact surface for heat extraction is pursued for package cooling.
Studies (Koh and Colony, 1974; Koh and Stevens, 1975) in forced convection in a porous channel by using a Darcy flow model have shown that under a constant heat flux boundary condition at the boundary the temperature at the wall and the temperature difference between wall and coolant can be drastically reduced by the introduction of porous media in the channel. Also, investigators (Kuo and Tien, 1988; Hunt and Tien, 1988) have utilized a foam material to enhance liquid forced convection cooling and their results show an increase of two to four times in heat transfer as compared with that of laminar slug-flow in a clear duct. Many other researchers have investigated the flow and heat transfer through constant porosity (Poulikakos and Kazmierczak, 1998) and variable porosity categories (Renken and Poulikakos, 1988; Hsai, Cheng and Chen, 1992).
Thermal dispersion caused by the presence of solid matrix plays a key role in heat transfer augmentation. A number of investigations have been carried out (Hunt and Tien, 1988, Hsiao et al., 1992, Hsu and Cheng, 1990) to study the effect of thermal dispersion in porous medium. Later investigations (Cheng and Vortmeyer, 1988; Amiri and Vafai, 1998) have revealed the effect of transverse thermal dispersion in forced convection and their studies show that effect of transverse dispersion is much more important than that of longitudinal dispersion.
The use of local thermal equilibrium (LTE) is used widely to analyze the transport process through the porous media. In such a packed bed operated under steady state conditions, a difference in the local temperatures between the fluid and the particle may exist, but the overall solid and fluid temperature profiles are considered to be identical to each other. In estimating the overall steady-state temperature profile, the heterogeneous packed bed may be considered to be a homogenous single phase. The temperature profiles in the bed are then predicted in terms of the effective thermal conductivities and wall heat transfer coefficients. Several studies have been conducted using the assumption of local thermal equilibrium.
The above assumption is not valid when there is a substantial temperature difference between the solid and the fluid phases or the ratio of the thermal conductivities is large. Dixon and Criswell (1979) investigated the problem of LTNE between the two phases and were the first to obtain a fluid to solid heat transfer correlation. Other researchers (Vafai and Amiri, 1998, Hwang et al., 1994) have obtained their own correlations. Several studies (Amiri and Vafai, 1998, Amiri et al, 1995, Hwang and Chao, 1994, Kuznestov, 1997) employed the two-equation model to study forced convection in porous medium.
1.2 Objectives
The main objective of this work is to numerically analyze using FLUENT 5.5 of a study of non-Darcian forced convection in an asymmetric heated, constant porosity sintered porous channel (Hwang and Chao, 1994; Hwang, Wu and Chao, 1995). Also,
- to develop the model and to analyze the wall temperature distribution; to extend the model to a higher thermal conductivity ratio material namely Copper and analyze the wall temperature distribution;
- analyze the model (for both the materials) for two heat transfer correlations (Hwang et al., 1994, Amiri and Vafai, 1998) considering local thermal non-equilibrium (LTNE) between the two phases and their respective fluid to solid specific area correlations;
- to investigate the wall temperature dependence on variants like particle Reynolds Number (Rep), Heat transfer correlations (HTC), particle diameter (Dp) and solid to fluid thermal conductivity ratio in the constant porosity category.
CHAPTER 2
INTRODUCTION TO POROUS MEDIA AND NAVIER STOKES EQUATIONS
2.1 Definition
Porous medium can be defined as a material consisting of a solid matrix with an interconnected void. The solid matrix is either rigid (non-consolidated) or it undergoes slight deformation (consolidated). The transport in the porous media is achieved by the
flow of fluid in the voids that are interconnected.
In single-phase flow, a single fluid saturates the voids and in two-phase flow a liquid and a gas share the void space. The phases are chemically homogenous, separated by a physical boundary. In nature there are many materials that can be defined as porous medium and some examples of these are: wood, the human lung, Soils, etc.
2.2 Mechanics of Fluid Flow in Porous Media
The flow parameters in the porous medium will clearly be irregular at the pore scale (microscopic) level and therefore are investigated at the macroscopic level. The quantities of interest are measured over areas that cross many pores and such quantities at the macroscopic level are known to change in a regular fashion with respect to space and time. Treating the medium at macroscopic level and obtaining the governing equations as a continuum model achieve the laws governing the flow in porous media.
The phases in disjoint domains are replaced by a continuum, which fills the entire domain for each phase. For each point in the flow domain, representative elementary volumes (REV) are chosen and phase variables are averaged over the REV’s. These averaged values are called the macroscopic values. The length scale of the REV is much larger than the pore scale, but considerably smaller than the length scale of the flow domain.
2.3 Porosity ()
The porosity () of a porous medium is defined as the fraction of the total volume that is occupied by the void spaces i.e.; the total void volume divided by the total volume occupied by the solid matrix and void volumes. (1-) represents the fraction that is occupied by the solid. In defining the porosity we are assuming that all the void spaces are interconnected but in reality some of them may be connected to only one other pore (dead end) or not connected to any other pore (isolated) and in this case we need to define effective porosity which is the volume fraction of the inter connected pores. In non-consolidated media like, particles that are loosely packed the effective porosity and the porosity are equal. For natural media the porosity does not exceed 0.6 .For packed bed of spheres, it varies from 0.245 to 0.476.
2.4 Solid Matrix Structures
Based on the simplicity of the matrix structures we can classify it as follows:
- Very long straight cylinders (circular and other cross sections).
- Spheres (and other three dimensional particles).
- Short and long fibers (circular and other cross sections)
- Rocks, silicon gels, coal, fabrics, etc., have complex topology and in general very difficult to analyze.
2.5 Stokes Flow and Darcy Equation
The first experiments of calculating the bulk resistance to flow of an incompressible fluid through a solid matrix was measured by Darcy (1856). He conducted his experiment on non-consolidated (loosely packed particles), uniform, rigid and isotropic solid matrix. The macroscopic flow was steady, one-dimensional, and driven by gravity. The mass flow rate was measured and the filtration velocity or seepage velocity was determined by dividing the mass flow rate by the product of the fluid density and the cross-section area A of the channel i.e.;
[2.1]
By applying a volumetric force balance to this flow, he discovered that the viscosity of the fluid and the permeability of the solid matrix characterize the bulk resistance to the fluid flow such that
- [2.2]
The permeability of the solid phase k is independent of the nature of the fluid but it depends on the geometry of the medium. It has the dimensions (length) 2 and is called the specific permeability or intrinsic permeability of the medium. In case of single-phase flow it is termed as permeability. The unit for permeability is Darcy. One unit of Darcy is equivalent 9.87×10-13 m2. The Darcy model has been examined extensively and it is not followed for liquid flows at high velocities and for gas at very low and very high velocities.
In three dimensions, Eq. [2.2] is generalized as
[2.3]
where K is now a second order tensor. For an isotropic medium the pressure gradient and the velocity vector v are parallel and permeability is a scalar .In this case Eq. [2.3] reduces to
[2.4]
Typical values of permeability are 10-9 to 10-22 for clean sand 1.9´10-12 to 2.4´10-16 for spherical packing (well shaken).
2.6 Permeability
The permeability K of the porous medium is a measure of the flow conductance of the matrix. Various models are used for the determining the relationship between permeability and the matrix property parameters (porosity and other structure variables). The most widely used among them are the Capillary models and the Hydraulic radius model. In case of packed bed of spheres the Hydraulic radius model is most widely used for determining the relationship between the porosity and the permeability of the medium. . The effective particle diameter (Dp) is determined and using the hydraulic radius theory of Carmen-Kozenzy which leads to the relationship:
[2.5]
where the constant 180 is obtained by seeking the best fit with experimental data.
The Carmen-Kozenzy equation gives satisfactory results for particles that are of spherical shape and whose diameters fall within a narrow range and is widely used.
2.7 Equation of Continuity
The conservation of mass applied to the porous media as a continuum model; based on the R.E.V concept is defined as follows:
[2.6]
The equation is obtained by considering the mass flux into a representative elementary volume to the increase of the fluid within the volume.
2.8 Momentum equation:
For low filtration velocities the Darcy equation (Eq. [2.4]) can be used for each R.E.V in the domain.
Darcy’s equation is linear to the filtration velocity and is valid when the filtration velocity is small or the Rep (Reynolds number based on the particle diameter) is of the order of unity or smaller. As increases the form drag due to the solid is considerable as compared to the surface drag due to friction and transition to non-linear drag is quite smooth in the range of Rep 2 to 20.Thus the Darcy equation does not hold true and the equation is replaced by an modification of the Darcy equation to take into account form drag.
The modified equation also known as the Darcy Forchheimer equation is as shown below:
[2.7]
where is the dimensionless form drag constant or the Ergun coefficient
and the term is also called the Forchheimer term and is proportional to the square of the filtration velocity and hence the name quadratic drag. The value of varies with the nature of the porous medium and is not a constant. The deviation of the Darcy law begins when ReK0.5 (Reynolds Number based on the square root of permeability) is greater than or equal to 0.2.
2.9 Energy Equation for Porous Medium
The energy equation of an isotropic porous medium neglecting viscous dissipation, radiative effects and work done by pressure changes can be defined for an elemental volume of the medium as follows:
for solid phase,
, [2.8]
and for the fluid phase
[2.9]
where the subscripts s and f refer to the solid and fluid phases respectively. c is the specific heat of the solid and is the specific heat at constant pressure of the fluid; k is the thermal conductivity, is the porosity of the medium and [W/m3 ] is the heat production per unit volume.
Considering local thermal equilibrium (LTE) between the solid and the fluid phases i.e.; where the solid and the fluid phase temperature are equal (Ts = Tf = T) and setting the same in equation 2 and 2 we get:
[2.10]
where
is the overall heat capacity per unit volume, overall thermal conductivity, and overall heat production per unit volume of the medium.
2.10 Overall Thermal conductivity of the porous medium:
When the heat conduction between the two phases is in parallel, then the overall thermal conductivity is the weighted arithmetic mean of the conductivities of the two phases i.e.;
[2.11]
If the heat conduction takes place in series, with all the heat flux passing through both the solid and the fluid, then the overall thermal conductivity is the Harmonic mean of the two conductivities.
[2.12]
where ke = effective thermal conductivity of the porous medium.
The assumption of Local thermal equilibrium is not valid when there is a substantial temperature difference between the two phases or when the thermal conductivities of the two phases are very different. In this case of local thermal non-equilibrium (LTNE) between the two phases, the energy equation is replaced by the following two equations: