Calculating the Surface Temperature of a Cable with known heat flow

Question

I have a cable which dissipates $\dot Q_{total}$ and I need to determine the surface temperature of the insulation $T_{ins}$ and the surface temperature of the conductor $T_{cond}$.

There is an insulating layer with the thermal resistance $R_{cond}$ and the convective thermal resistance $R_{conv}$. The radiation component is neglected. From the geometry of the insulating layer I can obtain $R_{cond}$, but for $R_{conv}$ I need to calculate the convection coefficient via the Nusselt-number. I have an empirical equation for the Nusselt-number, but the Rayleigh-number calculation requires the value of $T_{ins} - T_{ambient,air}$.

So I have the following equation:

$\dot Q_{total} = (1/R_{total}) \cdot (T_{cond} - T_{ambient,air})$

with $R_{total} = R_{cond} + R_{conv} = R_{cond} + Nu \cdot \lambda/L$.

Where $\lambda$ is the thermal conductivity and $L$ the length of the cable. There are two unknown values in this equation $T_{cond}$ and $T_{ins}$, because $Nu$ depends on $T_{ins}$ and $\dot Q_{total}$ depends on $T_{cond}$. Is there a way to calculate the temperatures or do I have to define a temperature?

Furthermore I cannot determine the film temperature because $T_{ins}$ is unknown. At which temperature should I determine the thermodynamic properties for the Rayleigh- and Prandtl-number calculation?

Summary: Known: heat flow $\dot Q_{total}$, $R_{conv}$ (because geometry is known), $T_{ambient,air}$ Unknown: $T_{ins}$, $T_{cond}$, film temperature $T_{f}$ and therefore also the exact values of Rayleigh, Prandtl and Nusselt-number.

I want the thermal resistance of convection to be as accurate as possible, that is why I wanted to calculate it with these dimensionless quantities

Answer 1

Problem Statement

You have three unknowns: the temperature at the conductor $T_c$, the temperature at the insulation $T_i$, and the air heat transfer coefficient $h_a$. You know the air temperature $T_a$, heat flow per length $\dot{q}/L \equiv q_L$, the geometry of the cable + insulation (radii $r_c$ and $r_i$), and the insulation material (thermal conductivity $k$).

You have two energy balance equations.

$$ q_L = \frac{2\pi\ k}{\ln(r_i/r_c)}\ (T_c - T_i)$$

$$ q_L = 2\pi\ r_i\ h_a\ (T_i - T_a) $$

In the set of unknowns, you also know that $h_a = f(T_i, T_a)$. The relation is solved through a correlation equation, chart, or table using the Nusselt number $Nu$ (and Reynolds number and Prandtl number no doubt). The flow is (ostensibly) laminar (because you mention film temperature, meaning boundary layer theory, meaning laminar flow).

Proposed Approach

When we know $h_a$, we can solve for both unknown temperatures.

$\Rightarrow$ Make a plot of $T_c$ and $T_i$ versus $h_a$ over a range of values for $h_a$ that are expected to be relevant for the physical system.

We do not know $h_a$ until we know $T_i$.

$\Rightarrow$ Calculate $h_a$ as a function of $T_i$. Superimpose the values correspondingly on the above graph.

The answer for the three unknown values will be at the point where the line for $T_i$ versus $h_a$ made using the energy balance equations intersects with the line for $h_a$ versus $T_i$ made using the correlation expressions.

Here is an example of the plot that I believe you might have at the end.

The circles show how to obtain the three unknown values for your system.

You could also iterate through the equations. I prefer a graphical approach.