Derivation of Heat Flow Matrices has a complete derivation of the matrices and load vectors of a general thermal analysis element. Mass transport is discussed in PLANE55 - 2-D Thermal Solid.

An option (KEYOPT(7) = 1) is available to convert SOLID70 to a nonlinear steady-state fluid flow element. Pressure is the variable rather than temperature. From Equation 6–22, the element conductivity matrix is:

(13–134)

[B] is defined by Equation 6–22 and for this option, [D] is defined as:

(13–135)

where:

= absolute permeability of the porous medium in the x direction (input as KXX on MP command)

ρ = mass density of the fluid (input as DENS on MP command)

μ = viscosity of the fluid (input as VISC on MP command)

β = visco-inertial parameter of the fluid (input as C on MP command)

S = seepage velocity (at centroid from previous iteration, defined below)

α = empirical exponent on S (input as MU on MP command)

For this option, no “specific heat” matrix or “heat generation” load vector is computed.

The pressure gradient components are computed by:

(13–136)

where:

= pressure gradient in the x-direction (output as PRESSURE GRADIENT (X))

{Te} = vector of element temperatures (pressures)

The pressure gradient is computed from:

(13–137)

where:

The mass flux components are:

(13–138)

The vector sum of the mass flux components is:

(13–139)

where:

The fluid velocity components are:

(13–140)

where:

and the maximum fluid velocity is:

(13–141)

where: