# Governing Equations

The basic equations of Darcy flow coupled to thermal transport are: \nabla \cdot \vec{u} = 0 \\ \vec{u} = -\frac{\mathbf{K}}{\mu} (\nabla p - \rho \vec{g}) \\ C\left( \frac{\partial T}{\partial t} + \epsilon \vec{u}\cdot\nabla T \right) - \nabla\cdot k \nabla T = 0 where $$\vec{u}$$$is the fluid velocity, $$\epsilon$$$ is porosity, $$\mathbf{K}$$$is the permeability tensor, $$\mu$$$ is fluid viscosity, $$p$$$is the pressure, $$\vec{g}$$$ is the gravity vector, and $$T$$$is the temperature. The parameters $$\rho$$$, $$C$$$, and $$k$$$ are the porosity-dependent density, heat capacity, and thermal conductivity of the combined fluid/solid medium, defined by: \rho \equiv \epsilon \rho_f + (1-\epsilon) \rho_s \\ C \equiv \epsilon \rho_f {c_p}_f + (1-\epsilon) \rho_s {c_p}_s \\ k \equiv \epsilon k_f + (1-\epsilon) k_s where $$\epsilon$$$is the porosity, $$c_p$$$ is the specific heat, and the subscripts $$f$$$and $$s$$$ refer to fluid and solid, respectively.

We shall henceforth assume that $$\vec{g} = \vec{0}$$$. Taking the divergence of the second equation, and imposing the divergence-free condition leads to the following system of two equations in the unknowns $$p$$$ and $$T$$$: -\nabla \cdot \frac{\mathbf{K}}{\mu} \nabla p = 0 \\ C\left( \frac{\partial T}{\partial t} + \epsilon \vec{u}\cdot\nabla T \right) - \nabla \cdot k \nabla T = 0 where $$\vec{u} = -\frac{\mathbf{K}}{\mu} \nabla p$$$. That is, the pressure satisfies Laplace's equation (with possibly non-constant coefficients) and the temperature satisfies the linear convection-diffusion equation.

# Values for Spherical Media

We next compute some representative values for the mean fluid velocity induced by a given pressure head for the flow of water through a packed spherical medium, using the experimental apparatus described by Pamuk and Ozdemir (2012). Based on Fig. 2, we estimate the following values for the non-dimensionalized pressure drop P' \equiv \frac{\Delta P}{L} \frac{D^2}{\mu u_m} and Re_D \equiv \frac{\rho u_m D}{\mu}

The Reynolds number based on the test chamber diameter:

Name $$P'$$$$$Re_D$$$
First Medium $$4 \times 10^6$$$700 Second Medium $$0.5 \times 10^6$$$ 950

Given these values and following geometric specifications and fluid properties of water at 30C:

Property Value
Viscosity, $$\mu$$$$$7.98 \times 10^{-4}$$$ Pa-s
Density, $$\rho$$$$$995.7$$$ kg/m$$^3$$$Test chamber length, $$L$$$ $$304$$$mm Test chamber diameter, $$D$$$ $$51.4$$$mm We obtain the following dimensional pressure drop and mean velocity values: Name $$\Delta P$$$ $$u_m$$$First Medium $$4008.8$$$ Pa $$1.09 \times 10^{-2}$$$m/s Second Medium $$680.07$$$ Pa $$1.48 \times 10^{-2}$$$m/s # The Cell Peclet Restriction The continuous Galerkin finite element method is known to exhibit spurious numerical oscillations unless the cell Peclet number, $$Pe$$$, satisfies: Pe \equiv \frac{|\vec{u}| h}{2 \kappa} < 1 where $$h$$$is a representative finite element grid cell size, and $$\kappa\equiv\frac{k}{\rho c}$$$ is the thermal diffusivity. All of the parameters in the definition of $$Pe$$$depend on the fluid/solid medium except the velocity magnitude $$|\vec{u}|$$$, and the grid size $$h$$$. This formula can therefore be used to determine a maximum-allowable cell size $$h$$$ depending on the flow velocity magnitude.

The following thermal properties for water and steel were utilized.

Water Value
Thermal conductivity, $$k$$$$$0.6$$$ W/m-K
Specific heat capacity, $$c_p$$$$$4181.3$$$ J/kg-K
Density, $$\rho$$$$$995.7$$$ kg/m$$^3$$$Steel Value Thermal conductivity, $$k$$$ $$18$$$W/m-K Specific heat capacity, $$c_p$$$ $$466$$$J/kg-K Density, $$\rho$$$ $$8000$$$kg/m$$^3$$$

Using these properties and a porosity of $$\epsilon = .25952$$$yields the combined fluid/solid properties: Combined Value Thermal conductivity, $$k$$$ $$13.48$$$W/m-K Specific heat capacity, $$c_p$$$ $$1430.19$$$J/kg-K Density, $$\rho$$$ $$6182.24$$$kg/m$$^3$$$

and a combined thermal diffusivity of \kappa \approx 1.525 \times 10^{-6} \; \mathrm{m}^2/\mathrm{s}

Assuming a velocity of approximately 1 cm/s, the Peclet condition gives us the restriction h < \frac{2 \kappa}{|\vec{u}|} \approx 3 \times 10^{-4} \; \mathrm{m}

The cell Peclet number is only a necessary (but not sufficient) condition for numerical oscillations to be present in the solution, and is valid for the steady convection-diffusion equation. Numerical timestepping schemes may add artificial diffusion of $$O(\Delta t)$$\$, inhibiting numerical oscillations, but producing overly-diffusive solutions.