# 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? Mar 28, 2019 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. Mar 28, 2019 at 12:42
• Okay, thank you for teaching me on how to use this forum :D Mar 29, 2019 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.