I just started a job working with cryogenics and I am trying to solve an interesting heat conduction problem that I was hoping someone could help me along with.
I have a structure that spans from room temp (call it 300K) to 4K, and I want to calculate the heat rate at the 4K surface. The geometry of the structure is simple, just rods, plates, and tubes. Everything is under vacuum and there are heat shields and multi-layer super insulation so convection and radiation can be ignored, I am only looking at conduction.
Typically for these types of problems I just use the NIST material database (https://trc.nist.gov/cryogenics/materials/materialproperties.htm) to integrate the thermal conductivity over the temperature range, then multiply by the cross sectional area and divide by the length to get the heat rate (aka heat leak) in Watts.
This time I would like to treat the structure as a composite wall with each component contributing its own thermal resistance like the classic heat transfer problems we have seen where one sums up the resistance contributed by each element and uses the total to find the heat rate. https://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Staedy%20Conduction%20Heat%20Transfer.pdf
The complication is that the thermal conductivity of the materials used varies greatly over this temperature range so I must use k(T) and integrate somehow but I am stuck on how to implement it. To start I can ignore the thermal conductance of the mating interfaces, I can bake that in later. Any thoughts?
p.s. Yes I am aware this can be solved with software but I want a clean solution that is more analytical to understand the problem better and inform the design.