Heat rate through composite wall with non-constant thermal conductivity

Question

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.

Answer 1

You are after a method to reduce a position or temperature variation in thermal conductivity to one value over the entire object. You want to substitute this one value into series + parallel thermal resistance equations.

The starting theoretical formulation for heat flow $\dot{q}$ (W) through an area $A$ (m$^2$) in a material at a temperature $T$ (K) with a thermal conductivity $k$ (W/m K) is below.

$$ \dot{q} = - \nabla \bullet k\ A\ \nabla T $$

This is expanded in Cartesian, cylindrical, or spherical coordinates as needed. When area is independent of position along the direction of heat flow, the expression becomes

$$ \frac{\dot{q}}{A} = J_q = - \nabla \bullet k\ \nabla T $$

An example is when the rods in your system are cylindrical and oriented along the direction of heat flow.

Analytical analysis can be simple or complicated depending on how $k$ depends on position and temperature.

One approach to reduce a complicated system with $k(position, temperature)$ is to develop the base analytics for a position and temperature averaged value for thermal conductivity $\langle k \rangle$ that represents the entire object. Suppose that $k = k_o + \beta T$ as a baseline. Integrate this over a range $T_o$ to $T_f$ to obtain

$$ \langle k\rangle \Delta T = k_o \Delta T + \frac{\beta}{2}\left(T_f^2 - T_o^2\right) $$

This leads to the expression $\langle k \rangle = k_o + \beta\langle T\rangle$ where $\langle T \rangle$ is the mean temperature over the expanse for the heat flow.

A possible iterative approach at this point that avoids numerical modeling involves these steps:

• neglect the temperature variation and solve the resistance problem using $\langle k \rangle = k_o$
• determine the temperatures across the object
• calculate a first iteration for $\langle k \rangle_1 = k_o + \beta \langle T \rangle_1$
• repeat the analysis of the resistance problem with the new thermal conductivity
• re-calculate the temperatures across the object
• iterate until the change is considered negligible

The next level of effort in analysis would use numerical methods.