I have received from my professor the following problem:
A first-order exothermic reaction A -> B takes place in a tubular reactor (typical PFR). The tube is cooled to a constant wall temperature $T_w$ and the total amount of heat removed from the reactor is given ($Q$, in Watts). All needed constants and geometrical parameters were given, as well as $T_{inlet}$ and $T_{oulet}$.
The goal is to find the (constant) value of the wall temperature $T_w$ that ensures $Q$ is being removed from the reactor.
My question is not so much about the final number, but rather the logic to get there. In my professor's official solution, they simply used the following formula from a heat exchanger:
$$ Q = UA\Delta T_{lm}$$ $$\Delta T_{lm} = f(T_{inlet},T_{outlet},T_{w})$$
where $\Delta T_{lm}$ is the logarithmic mean temperature difference. $Q$, $U$, and $A$ are given in the question. One would then isolate $T_w$ and solve for it, as all other values are given.
This formula is easily derived from a simple heat exchanger, but as soon as a chemical reaction is happening inside the inner duct, the energy balance doesn't result in this aforementioned equation anymore as there is now a term for the enthalpy of reaction in the balance.
The reactor heat balance in this scenario would be:
$$u\rho c_{p}\frac{dT}{dz}=\frac{4U}{D}\left ( T- T_{w} \right) + r\left (-\Delta_{R}H\right )$$
where $\Delta_{R}H$ is the reaction enthalpy, $r$ is the reaction rate, $D$ is the reactor diameter and $z$ is the main axis-coordinate of the reactor, assuming there's no temperature profile radially. $U$, $u$, $\rho$, and $c_p$ are heat transfer coefficient, flow velocity, specific mass and heat capacity, respectively.
If the reaction were non-existing ($r = 0$), one could derive the mean-log approach to solving this from the balance above. But that's not the case in a chemical reactor.
Do you believe this problem is solvable? I would try to numerically solve the energy balance above but I don't have $T_w$ to begin with so I don't know how I could tackle this simulation. If my professor's approach is correct, why can't I derive that formula from the energy balance?
Thanks a lot for any input!
Best regards.
