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].

Some more information:

  • 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.


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?

  • $\begingroup$ -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. $\endgroup$ – sammy gerbil Sep 13 '17 at 17:12

In my judgment, you are looking at this backwards. You should be assuming (at least as a first approximation) that the temperature in the batch is constant (or at least changing very slowly) rather than the fluid passing through the coil. This will allow you to calculate the fluid temperature through the coil, the exit temperature, and, most importantly, the heat load. So you can determine analytically the current heat load as a function of the current batch temperature. You can then use this to do the heat balance on the batch, and determine the batch temperature as a function of time.

  • $\begingroup$ This is a good problem to simulate. Heating fluid residence time will affect temperature drop, which will affect cold water heating rate. In addition, the cold water should be well stirred. Now, for an alternate question: why couldn't live steam be injected into the cold water? This would eliminate the need for a heating coil, and heat transfer into the cold water would be instantaneous (i.e., no inner and outer heat transfer coefficients, no tube scale, no conduction through metal walls). $\endgroup$ – David White Oct 31 '19 at 5:40

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