6.2. Derivation of Heat Flow Matrices

As stated before, the variable T was allowed to vary in both space and time. This dependency is separated as:

(6–22)

where:

T = T(x,y,z,t) = temperature
{N} = {N(x,y,z)} = element shape functions
{Te} = {Te(t)} = nodal temperature vector of element

Thus, the time derivatives of Equation 6–22 may be written as:

(6–23)

δT has the same form as T:

(6–24)

The combination {L}T is written as

(6–25)

where:

[B] = {L}{N}T

Now, the variational statement of Equation 6–11 can be combined with Equation 6–22 thru Equation 6–25 to yield:

(6–26)

Terms are defined in Heat Flow Fundamentals. ρ is assumed to remain constant over the volume of the element. On the other hand, c and may vary over the element. Finally, {Te}, , and {δTe} are nodal quantities and do not vary over the element, so that they also may be removed from the integral. Now, since all quantities are seen to be premultiplied by the arbitrary vector {δTe}, this term may be dropped from the resulting equation. Thus, Equation 6–26 may be reduced to:

(6–27)

Equation 6–27 may be rewritten as:

(6–28)

where:

= element specific heat (thermal damping) matrix
= element mass transport conductivity matrix
= element diffusion conductivity matrix
= element convection surface conductivity matrix
= element mass flux vector
= element convection surface heat flow vector
= element heat generation load

Comments on and modifications of the above definitions:

  1. is not symmetric.

  2. is calculated as defined above, for SOLID90 only. All other elements use a diagonal matrix, with the diagonal terms defined by the vector .

  3. is frequently diagonalized, as described in Lumped Matrices.

  4. If exists and has been diagonalized and also the analysis is a transient (Key = ON on the TIMINT command), has its terms adjusted so that they are proportioned to the main diagonal terms of . , the heat generation rate vector for Joule heating is treated similarly, if present. This adjustment ensures that elements subjected to uniform heating will have a uniform temperature rise. However, this adjustment also changes nonuniform input of heat generation to an average value over the element.

  5. For phase change problems, is evaluated from the enthalpy curve (Tamma and Namnuru([43])) if enthalpy is input (input as ENTH on MP command). This option should be used for phase change problems.