# Computation of equivalent thermal resistance and thermal capacitance from FE model

I need to compute the equivalent thermal resistance and thermal capacitance of a structure used for heat transfer. For illustration purpose let’s say it’s the 2D problem of the following figure. In the top surface (nodes 1 to 4) I have a constant heat generation boundary condition (BC) and in the bottom surface (nodes 15 to 20) I have a convection BC. I am interested in the equivalent resistance and capacitance between the top and bottom surfaces given the Finite element matrices.

I found in this answer the hint that I need to use static condensation to compute the reduced matrix $$K_{cc} - K_{ci}K_{ii}^{-1}K_{ic}$$. My first question is: how to transform that matrix into a scalar quantity (resistance and capacitance)?

My second question is regarding the following approach: I would intuitively just impose at $$t_0$$ the constant heat generation $$Q_V$$ in the top surface and make a transient simulation. For the equivalent resistance I would take $$R_{th}$$ = ( average temp top surface - average temp bottom surface) / Average power flowing from top surface downwards. For the equivalent capacitance I guess I would fit an exponential function to the transient temperature, get the time constant $$\tau$$ and compte $$C_{th}$$ as $$\tau/R_{th}$$. My question is: how does this intuitive solution relate to the formal method using static condensation?

• I think that you want to compute those properties over your FEM solution. – nicoguaro Mar 5 '20 at 14:46
• That would be the second approach that I mentioned above. My question is: is there a relationship between this approach and the more formal approach using static condensation?? – Ken Grimes Mar 5 '20 at 18:24
• No, I don't think those two processes are related. – nicoguaro Mar 5 '20 at 18:39
• Can you define the "equivalent thermal resistance" and "thermal capacitance" in mathematical terms? – Wolfgang Bangerth Mar 5 '20 at 20:37
• I will use the term impedance for both capacitance and resistance. Intuitively, I would say an impedance Z1 is equivalent to an impedance Z2 if the ratio temperature difference/heat power is the same for both. Two examples: The equivalent impedance of the series connection of Za and Zb is one single impedance of value Za+Zb and in a parallel connection of Za and Zb the equivalent impedance is Za*Zb/(Za+Zb) – Ken Grimes Mar 6 '20 at 9:23