# Transient Heat Transfer: Batch Reactor / Heat Exchanger

I am attempting to design a batch reactor with an internal heating coil. The reactor must be heated from 20 to 80°C within one hour [t= 3600]. The aim is to find the required area of the heating coil, $A$, and the mass flow rate of the heating fluid, $µ$ [H2O].

• The heating fluid is available at 95°C.
• The reactor contains 200kg water + 100kg steel.
• Overall efficiency of process: 0.65.
• $U$ = 350 $wm^{-2}K^{-1}$ [forced convection, water/water]
• Pinch temp. is usually kept to a minimum of 10K, but that seems unimportant in this case.

So far I have used the equations from Time in the simple Heat Equation and solved for $A$ using the assumption that the heating fluid does not change temperature.

$$-\frac{uA}{mc_p}t=\ln\Big[\frac{T_{\infty}-T_2}{T_{\infty}-T_1}\Big]$$

And dividing the answer by the efficiency [good enough approximation]. Then $Q_{\min}$ and $Q_{\max}$ could be found which can be then converted into $µ_{\min}$ and $µ_{\max}$ by assuming a $ΔT$ in the heating fluid. This last assumption is however not consistent with the previous assumption.

It feels like I am very close to being able to reiterate this, or perhaps a substitution for $T_{\infty}$, or rederiving the initial equation with $T_{LM}$, or $T_{\infty}$ in terms of $t$.

$µ$ in terms of $t$ would also be lovely! :)

Any clues on how to solve this?

• -1. Not clear what you are asking. Your difficulty seems to be mathematical, not physical. If you are close to a solution, what is stopping you? Asking for help with solving a problem is off topic here. You need to identify a conceptual difficulty about physics - it is not clear what conceptual difficulty you are having. – sammy gerbil Sep 13 '17 at 17:12