# 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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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.