I have received from my professor the following (summarized) problem:
A first-order exothermic reaction $\ce{A->B}$ took place in a typical tubular plug flow reactor (PFR). The tube was cooled to a constant wall temperature $T_\mathrm{w}$ and the total amount of heat removed from the reactor $Q$ was given in Watts. All needed constants and geometrical parameters and $T_\text{inlet}$ and $T_\text{outlet}$ were provided.
The goal was to find the (constant) value of the wall temperature $T_\mathrm{w}$ that ensures heat $Q$ is being removed from the reactor.
My question is not about the final value but rather the logical aspects. In my professor's solution, they simply used the following formula from a heat exchanger:
$$ Q = U \times A \times LMTD $$
where $LMTD$ is the logarithmic mean temperature difference based on $T_\text{inlet}$, $T_\text{outlet}$ and $T_w$ (to be determined). $Q$, $U$ and $A$ are given.
This approach makes sense for a tubular heat exchanger but as soon as a reaction is happening inside the inner duct, the energy balance doesn't result in the $LMTD$ anymore. The PFR heat balance in this scenario is:
$$\frac{\mathrm{d}T}{\mathrm{d}z} = \frac{k_{W}}{u\rho c_P}\frac{4}{D}\left(T_\mathrm{w}-T\right)+\frac{r(-\Delta_R H)}{u\rho c_P}$$
If the reaction were non-existing ($r=0$), one could derive the $LMTD$ approach to solving this from the balance above. But that's not the case.
What do you think? Given $Q$ (along the entire reactor), $T_\text{inlet}$, $T_\text{outlet}$ and all other constants/parameter, can you calculate $T_\mathrm{w}$? Is my professor's approach correct? If so, how can one derive this formula from the energy balance as one does for a simple heat exchanger?