# 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 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 at 18:24
• No, I don't think those two processes are related. – nicoguaro Mar 5 at 18:39
• Can you define the "equivalent thermal resistance" and "thermal capacitance" in mathematical terms? – Wolfgang Bangerth Mar 5 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 at 9:23

Thermal mass is much more complicated than thermal resistance since unsteady conduction involves propagation. This means that spatial discretization can't be avoided. A reduced-order system can be obtained though by modal analysis and truncation; I.e. projecting down to the slowest few eigenmodes, which is effectively.replacing the finite element discretization with a global spectral one. The mechanics of that, once the generalized algebraic eigenvalue problem on the conduction and mass matrices has been solved (use ARPACK or similar to get just the first few modes) is very like the statical condensation that you've already hit on.

For your first question, I'd say that actually thermal resistance R is only a scalar by shorthand, really its reciprocal, thermal conductance, is a 2×2 matrix of a special form: [[1, -1], [-1, 1]]/ R. This matrix acts on the 2-vector of plate temperatures and gives the 2-vector of inward heat transfer rates. This matrix in this form can be derived by statical condensation using two 'supernodes', one for each plate. This way of thinking about resistance becomes essential if there are more than two plates. The coefficients of the multiplate conductance matrix can be computed from solutions of the steady heat equation either from an integral of the heat flux over each plate or from a volume integral (which is kinetic energy in the hydrodynamic analogue or dissipated power in the electrical). I wrote a worked example in Python, see https://kinnala.github.io/scikit-fem-docs/examples/ex13.html, and a longer explanation in 'Three ways to compute multiport inertance' (2019) The ANZIAM Journal, 60, C140–C155, https://doi.org/10.21914/anziamj.v60i0.14058 .

Going back to the transient problem and its reduced-basis approximation, if you do want a scalar representation, just use a single eigenmode. If you include more than one mode, each mode has its own thermal capacitance and thermal resistance, the product being the timescale of the mode. As a circuit, a modal model has a parallel branch between the plates for each mode and each modal branch has a resistance and a shunt capacitance (capacitance to ground). That the modal branches are in parallel corresponds to the simultaneous diagonalization of the mass and conduction matrices in the generalized algebraic eigenproblem.

• Thanks! Wow, very insightful answer. It will take me some time to go through it – Ken Grimes Apr 20 at 22:37
• I missed the bit about the boundary conditions (Neumann above, Robin below), sorry. The plates or supernodes strictly only apply to uniform Dirichlet conditions; modal truncation is still valid though, just solve the eigenproblem with the appropriate homogeneous boundary conditions. – G. D. McBain Apr 20 at 23:29
• There's nothing wrong with linking to your own work but you should mention that you're one of the authors of both scikit-fem and the paper you referenced. – Daniel Shapero Apr 20 at 23:58
• Ah, O. K., sorry. Actually I only suppressed the list of authors for concision and because I couldn't think of an idiom that wasn't clumsy. I can only claim a tiny fraction of the credit for scikit-fem but I did contribute ex13. – G. D. McBain Apr 21 at 0:36