The Autocatalytic Reaction running in a CSTR

The material balance for the CSTR reads:

dcA/dt = (CA,in/tau) - (CA,out/tau) - k* CA*CB

with:
CB = CB,0 +(CA,0 - CA)

we get:

dcA/dt = (CA,in/tau) - (CA,out/tau) - k* CA(CB,0 +(CA,0 - CA))

and for the logistic equation:

CA,j+1 = CA,j + (CA,0 *dt/tau) - (CA,j *dt/tau) -
-----------------------(k* CA,j*dt ( CB,0+CA,0-CA,j))

I hope that you are able to rewrite that for unisim with: CA,0 = X0 ; CB,0 = Y0 ; k = m and tau = n !!! see snapshot

The given data was:

* here we have some strange situation: the reaction will not start without an initial B, - but here we could, for instance with a 'dosing slider', add some B for a short time. I preferred for my example to add a very small amount (0.001) of B continuously

How to calculate tau (50%) ??:

At 50 % conversion we have (starting from 1 mol/l) CA = 0.5 and CB = 0.5 mol/l

r(50%) = 1*(0.5*0.5) = 0.25 (mol/l sec)

for the space time we write (as we know!!):

tau = tR = CA,0(U/r(U))

that gives:

tau = 1( 0.5/0.25) = 2 sec !!!!

If you run a simulation with unisim, you should get a result like presented in the snapshot, with a stationary value of CA = 0.5 mol/l. If you run several simulations with mounting values for tau and look at the conversion of A, you should get something like the following snapshot (last image, scroll down). You see that the space time gets longer and longer for a little 'gain' in conversion, after having reached 50% conversion.

For the realization of an optimal reactor set, you should take a serial combination of a CSTR, running with 50% conversion, and a TFR for the 'rest' of conversion. A real-time simulation trying to find an optimum in space time for such a stationary running reactor combination is not very informative. The normal calculation and comparison of the space time (or reactor volumina) is sufficient. This can be found e.g. in the book 'Chemische Reaktionstechnik' of Jens Hagen or in the archive material on our exercises to the lecture 'Reaktionstechnik und Katalyse'. You remember the formulas ? or not ? Can you show the advantage of the choice of such a serial combination in a 1/r(U)-plot ?

Very comprehensive are some values of an example, calculated in our exercises: for a given reaction you need for 99% conversion in a CSTR a reactor volume of 661 m3, for a series of CSTR (50%) and TFR (rest) a volume of 43.6 m3 (CSTR = 30.4 m3, TFR = 13.3 m3) and if you take an initial B of 0.01 for a TFR alone: 61 m3; - for a loop reactor with an optimal R of 0.188 you get a volume of 49.5 m3

What about taking a loop rector for reaching an adapted back-mixing pattern ? Let us have a look at it !

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