Linear Steady-State Heat Transfer Analysis

Heat transfer analysis solves for unknown temperatures and fluxes under thermal loading.

Temperature represents the amount of thermal energy available, and fluxes represent the flow of thermal energy.

Conduction deals with thermal energy exchange by molecular motion. Free convection deals with thermal energy exchange between solids and surrounding fluids. Thermal loading is defined as energy flows into and out of the system.

In linear steady-state analysis, material properties such as conductivity and convection coefficient are linear. Temperature and fluxes at the final thermal equilibrium state are of interest. The basic finite element equation is:

(K_{C} + H) T = f

$(K_{C} + H) T = f$

Where,

$K_{C}$ $K_{C}$: Conductivity matrix
$H$ $H$: Boundary convection matrix due to free convection
$T$ $T$: An unknown nodal temperature
$f$ $f$: Thermal loading vector; The system of linear equation is solved to find nodal temperature $T$ $T$ .

Thermal load vector can be expressed as:

f = f_{B} + f_{H} + f_{Q}

$f = f_{B} + f_{H} + f_{Q}$

Where,

$f_{B}$ $f_{B}$: Power due to heat flux at boundary specified by QBDY1 card
$f_{H}$ $f_{H}$: Boundary convection vector due to convection specified by CONV or CONVG entries
$f_{Q}$ $f_{Q}$: Power vector due to internal heat generation specified by QVOL card.

The matrix on the left hand side of Equation 1 is singular unless temperature boundary conditions are specified. The equilibrium equation is solved simultaneously for the unknown temperatures, using a Gauss elimination method that exploits the sparseness and symmetry for computational efficiency. Once the unknown temperatures at the nodal points of the elements are calculated, temperature gradient $\nabla T$ $\nabla T$ can be calculated according to element shape functions. Element fluxes can be calculated by using:

Q_{F l u x} = K_{C} (\nabla T)

$Q_{F l u x} = K_{C} (\nabla T)$

Where,

K_{C}

$K_{C}$ is the conductivity of the material.

Table 1. Heat Transfer and Structural Analysis Analog
	Heat Transfer	Structural
Unknown	Temperature	Displacement
	Temperature gradient	Strain
	Flux	Stress
$K_{C}$ $K_{C}$	Conductivity matrix	Stiffness matrix
$H$ $H$	Boundary convection matrix	Elastic foundation stiffness matrix
$f$ $f$	Thermal loading vector	Load vector
$Q_{V o l}$ $Q_{V o l}$	Element volumetric	Gravity load

The thermal loads and boundary conditions are defined in the Bulk Data section of the input deck. They need to be referenced in the Subcase Information Entry section using an SPC or MPC and LOAD statement in a SUBCASE.

Forced Convection is available for Linear Steady-State Heat Transfer via Darcy Flow Analysis and 1D Forced Convection Analysis via CAFLUID.