# How would I go about solving this transient convection problem if the mean fluid temperature is constantly changing?

Let's say I have a ceramic slab on a conveyor belt that is initially at $450\,^{\circ}\mathrm{C}$ and there is air being blown over it at a speed of $35 \frac{m}{s}$ with an ambient temperature of $18\,^{\circ}\mathrm{C}$ until the slab reaches a temperature of $35\,^{\circ}\mathrm{C}$.

I understand the overall procedure of the problem. I have to find the Reynolds number, the Nusselt number, etc. But here is where I am confused.

The properties that can be found in the back of the book rely on the mean fluid temperature which is $T_{f}=\frac{T_{s}+T_{\infty}}{2}$ where $T_{s}$ is the surface temperature and $T_{\infty}$ is the free stream temperature. Once that temperature is found, the properties can be determined from the tables in the back.

But this problem involves a surface temperature that is constantly changing which means that the mean fluid temperature is constantly changing. This causes the Reynolds and Nusselt numbers to constantly change. I could easily do this problem if it weren't for transient convection, so is there a way to solve the problem that I am having here?

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## 2 Answers

If your ceramic slab has a high enough heat capacity then it should be a reasonable approximation if you just consider $T_s$ to be a function of time and let $T_f(t)=(T_s(t)+T_\infty)/2$. In theory you could plug this into the formulas you'd use for the static case you'd get the heat flux as a function of $T_s$, and you can use that together with the heat capacity to get $\frac{dT_s}{dt}$ in terms of $T_s$, which is a differential equation that you can then solve (numerically if necessary) to get $T_s$ as a function of time. But since the calculation involves looking up numbers in a table this is likely to be tricky - the only obvious way I can see is to write a computer program to interpolate between the table values at each time step.

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Have you tried to solve the problem using Newton's law of cooling (convection) and Fourier law of conductance (conduction)? They are both linear in temperature gradient, so you should get simple solveable differential equation. I think the coefficients for these two laws are not temperature dependant.

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Newton's law of cooling uses the convection coefficient h and since it's not given in the problem, the average convection coefficient across the slab must be calculated from the Nusselt and Reynold's numbers. The problem here is that the film temperature is constantly changing which causes the properties in the back of the book that are use to calculate both the Reynold's and Nusselt numbers to constantly change and the convection coefficient is constantly changing. So yes, it does depend on the surface temperature –  Greg Harrington Apr 19 '12 at 20:52
OK, I guess you know all I know, and beyond. I don't think you'll find answer here. Most of the audience here are theoretical physicist and I know all this stuff only because I happen to be connected to civil engineering... –  Pygmalion Apr 19 '12 at 21:04
I'm a mechanical engineering student taking heat transfer. It was worth a try at least since there is not a mechanical engineering page :/ –  Greg Harrington Apr 19 '12 at 21:09
OK, good luck then! –  Pygmalion Apr 19 '12 at 21:16