Maximum space time yield for a simple consecutive reaction
My proposal for managing the task:
- look at the snapshot of the simulation of the batch
reaction and run a simulation with the given data: when k1 > k2 (here even
10 fold !!), you observe a maximum curve for the concentration of compound B.
If you run a batch reaction at the reaction time corresponding to
Bmax, you will get most B (as possible). The same statement is valid
for the 'not-back-mixed brother' of the batch STR, namely the TFR, you have
only to apply a coordinate transformation from time to space.
- derive the formula for the optimal reaction time
Tauopt by differentiating CB to time in the material
balance equation and set dcB/dt = 0 . For Tauopt you will
get:
Tauopt= (ln(k1/k2))/(k1-k2)
and
therefore:
Bmax = CA,0
(k1/k2)(k2/(k2-k1))
- Input the given data and compare the results to the
values obtained for the real-time simulations, hope
they are identical!! You see that you can't take a maximum conversion, or ? By
the way: realize that you have bad chances for that optimization, when k1
<< k2 !!! try it with a simulation!! Clear?
- write the differential equation for the reaction
running in a CSTR, take the continuous operation mode and apply the same
mathematical procedure as above. Here you will find for
Tauopt:
Tauopt = 1/((k1*k2)0.5)
and for Bmax:
Bmax= cA,0
(1/((k2/k1)0.5 + 1)2)
- look at the snapshot for an example of the same
reaction running in a CSTR, do some simulations (and
see further informations) for different Taus close to
the Tauopt of the batch and finally with the value received by the
formula above. Don't forget that you have to regard the 'constant' stationary
values in this case, not maximums observed in the start-up phase !!. The
results should be equivalent!!
- Compare the optimal values, - the space time for STR
and TFR should be shorter than the value for the CSTR. Can you explain that
phenomenologically ? No ?
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