## 13.70. SOLID70 - 3-D Thermal Solid

Matrix or VectorShape Functions Integration Points
Conductivity Matrix and Heat Generation Load Vector Equation 11–224 2 x 2 x 2
Specific Heat Matrix Equation 11–224. Matrix is diagonalized as described in Lumped Matrices Same as conductivity matrix
Convection Surface Matrix and Load Vector Equation 11–224 specialized to the face2 x 2

### 13.70.1. Other Applicable Sections

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.

### 13.70.2. Fluid Flow in a Porous Medium

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:

 f = mass flux (output as MASS FLUX)

The fluid velocity components are:

 (13–140)

where:

 Sx = fluid velocity in the x-direction (output as FLUID VELOCITY (X))

and the maximum fluid velocity is:

 (13–141)

where:

 S = total fluid velocity (output as FLUID VELOCITY (TOTAL))