# Heat conduction in a hollow cylinder with low conductvity coefficient - Why do I get negative temperatures?

I have a hollow cylinder with a heat source inside. The generated heat flow is $$\dot{Q}_g = 14\ 962 \ \text{W}$$ , the cylinder has the radius $$r = 20 \ \text{mm}$$, the length $$l = 50 \ \text{mm}$$ and the thickness $$t = 5 \ \text{mm}$$. The inside temperature is $$T_1 = 1300 \ \text{K}$$. If the material is for example stainless steel with a conductivity of $$\lambda = 30 \ \text{W/(m * K)}$$ I get a reasonable outside wall temperature $$T_2$$:

$$T_2 = T_1 - \dot{Q}_g \cdot \frac{\text{ln}(\frac{r + t}{r})}{2 \pi \cdot \lambda \cdot l} = 946 \ \text{K}$$

But if I have a material with a low conductivity like $$\lambda = 1 \ \text{W/(m * K)}$$, the same equations gets me a negative outside wall temperature:

$$T_2 = -9328 \ \text{K}$$

That doesn't make sense to me. I think I made some stupid mistake, so is there another equation for this case or did I made a false assumption? Or something else? I already checked the units, and it looks all right. The unit for the conductivity is watt per meter kelvin, not milli kelvin. This looked wrong in the first version, sorry.

I would be really happy, if somebody could help me with this. Thanks!

• Can you show your calculations? I get $T_2$ of 1289 K for the low conductivity. Also, look out that the units relate properly, you sometimes have K, sometimes mK, also sometimes mm. Figuring out the units is a good way to verify results. Aug 12 '20 at 17:19
• Ah, right, thermodynamics is not my field, but W/m*K makes more sense indeed :) Aug 13 '20 at 11:20
• 15 kW is a very high energy output for such a small volume. Not quite nuclear reactor levels, but still high. I think the calculated temperature is "correct", at least mathematically. In other words, the ~10,000$K$ temperature difference is what would be required to dissipate this much power over such a small area. The fact that a negative temperature is non-physical is not part of the heat conduction equation. Aug 18 '20 at 19:02