# Tanks in series model - Differential equations

1. Jun 13, 2012

### Jacob86

Dear,

I work as a process engineer and havent had to use differential equations or integrated anything for years. I have the following physical problem and please advice me if this is not the correct location on the forum.

Believe me or not but this problem has been researched for over 20 year by many enigneers in a company I can not name but I can guarantee you that solving this problem will potentially save lots of energy and capital.

Situation: 3 reactors ( R1 = 3000, R2 = 3500 and R3 = 9000 m3) in series need to fulfil following requirements
1. Sufficient mixing to avoid excessive dead volume in the reactors
2. At least 2 theoretical tanks per reactor
As the tanks are already there, I would like to do a tracer test for the whole system to determine both requirements by adding a concentration pulse into the feed of the first reactor.
Complication: there are recirculation loops which cannot be disconnected. The recirculation is as follows: Q1 recirculates part of the flow from R2 back to the inlet of R1 and Q2 recirculates from R3 back to the inlet of R2.

Solution: I am familiar with the tanks in series model which could be used when not considering recirculation of tracer loads but this would not help me forward in this case.

My first Question: if I assume ideal mixing in all three tanks and C1, C2 and C3 are the respective concentrations of the tracer in tanks R1, R2 and R3 - can I use the following mass balance equations to model the tracer concentration?
R1: V1*dC1/dt = Q1*C2 - (Qfeed+Q1)*C1
R2: V2*dC2/dt = (Qfeed + Q1)*C1 + Q2*C3 – (Qfeed + Q1 + Q2)*C2
R3: V3*dC3/dt = (Qfeed + Q2)*C2 – (Qfeed +Q2)*C3

My second Question: in order to be able to make assumptions for dead volume and tanks in series I will have to adjust the volumes and add equations between each reactor for each additional theoretical stage. Example bellow is the same as above but with 3 theoretical stages in the first reactor. Is this a correct approach?
R1: (stays the same) (V1/3)*dC1.1/dt = Q1*C2 - (Qfeed+Q1)*C1.1
R1.2: (V1/3)* dC1.2/dt = (Qfeed+Q1)*C1.1 - (Qfeed+Q1)*C1.2
R1.3: (V1/3)* dC1.2/dt = (Qfeed+Q1)*C1.3 - (Qfeed+Q1)*C1.3
R2: V2*dC2/dt = (Qfeed + Q1)*C1.3 + Q2*C3 – (Qfeed + Q1 + Q2)*C2
R3: V3*dC3/dt = (Qfeed + Q2)*C2 – (Qfeed +Q2)*C3

My golden question: if all of the above are correct could a general equation be obtained by integrating over n, m and p tanks in series for all three reactors?