# Calculating the Surface Temperature of a Cable with known heat flow

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

• Please clarify again in your statements what the known and unknowns are. I realize the geometry and materials values are known. In particular, is the heat flow $\dot{q}$ known? Is the only known the air temperature? Also, why not just use an estimated value for the external convection coefficient? How accurate do you want the answer to be? – Jeffrey J Weimer Mar 28 '19 at 2:58
• Thank you. Just FYI, the recommended approach is to edit the original question and add the requested clarifications there rather than putting them in the comments. With this change, I can likely make a few suggestions. – Jeffrey J Weimer Mar 28 '19 at 12:42
• Okay, thank you for teaching me on how to use this forum :D – satu Mar 29 '19 at 9:58

# 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.