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Combined Residence Time Distributions

rtd residence time distribution kinetics. reaction engineering

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#1 bobbingthroner

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Posted 01 December 2022 - 01:57 PM

Hello,

 

I am having a problem with this question regarding residence time distributions and was wondering if anyone could help.

 

I have been given experimental data for two reactors in series and been asked to calculate the combined residence time. The full question is in the excel sheet and the answer is given but I have not been able to calculate it on my own.

 

I have managed to work out the residence time and residence time distributions of the reactors separately but not together. This is where I'm stuck and need help.

 

If anyone could give tips or advice it would be greatly appreciated.

 

Thank you

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#2 Pilesar

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Posted 02 December 2022 - 07:16 AM

Post an answer when you figure it out. I am not uninterested, but am not expert at such questions. As for a tip, I see that reactor 2 will not see any pulse at all until time 48 when it gets a '1' pulse. Then you would use the R2 distribution (shifted with its zero to time 48) with the '1' pulse to see its output as if '1' were the only pulse it would see. But then at time 54, reactor 2 will see a '5' pulse. You would use the R2 distribution beginning at time 54 with the '5' pulse to see the reactor output. The '1' and '5' distributions would be additive for reactor 2 for the subsequent time periods. Each subsequent time period would have a different pulse to reactor 2 because of the change caused by reactor 1.

  That is where I get stuck. If it helps, I think reactor 1 is plug flow and reactor 2 is CSTR based on the response time observed. The missing piece is the size of the pulse in the experimental data. I don't know how a pulse of '1' or a pulse of '5' would compare to the initial pulse to reactor 2 in the experiment. Since the answer is expected without giving the experimental pulse value, then that means that either the pulse value can be inferred from the experimental results or that it is not really needed and I am following a wrong path. Maybe the experimental pulse value must be assumed to answer the question???

  I remember being taught in my remote past about how to combine distributions but it has left my mind through disuse. If I were forced to give answer, I would likely use spreadsheet to calculate every time step and then form the distribution from the results. It would be very inefficient and only be an approximation of a smooth curve, but it would be close and I would not have to relearn distribution math. You should probably get your results in a better way because it is expected of you. Good luck.



#3 breizh

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Posted 02 December 2022 - 10:19 PM

Hi,

Consider the paper attached to support your work.

review the calculation of sigma^2 accordingly.

 

residence time is not tau 1+tau 2, but tau +tau 2 + Teta, Teta being the time delay = 138 (pb statement) -126 (individual times) = 12 minutes.

 

Perform your calculation as you did, or I did (2nd time) by shifting c1 (t) to c1(t-12), similarly E1(t) to E1(t-12), no change for C2(t) and E2(t) and you will find 

tau 1 = 114.72 '

tau 2 = 23.44 '

 

Similar result with C2(t) shifted to C2(t-12) and no change on C1(t); E2(t) to E2(t-12), E1(t) with no change.

tau 1 = 102.744 '

tau 2 = 35.444 '

 

Hope I put you on the right direction, unfortunately I'm too rusty to calculate the convolution product to get RTD as indicated in the paper, I let it to you!

 

 

Good luck

Breizh 



#4 breizh

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Posted 06 December 2022 - 02:14 AM

Hi,

Attached my calculations

Breizh .



#5 Pilesar

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Posted 06 December 2022 - 08:31 AM

Thanks for the calcs, Breizh! So the concentrations are not for a reactant or a product of the reaction but for an inert tracer. I had not understood that before. And this means the size of the pulse can be calculated by summing the outlet quantity. Since the distributions of the two reactors are normalized, they can be put on the same basis even though their pulses were different quantities. 

  The article by Toson that you attached looks to be a great resource if I ever want to analyze such a system. I started following your detailed calcs in the spreadsheet but did not make it through them all. I think I will be content to get a 'D' grade from this class!

  When I was designing ammonia plants 35 years ago or so, I struggled with characterizing the radial flow reactors in the synthesis loop because of their back-mixing. An approach like this might have helped.






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