Integration of mass balance differential equations

Dear all,

Background:
I am a consultant process engineer and new to this forum, I hope to comply with most rules here. I thought this was the appropriate location to post this problem.

I came accross a client who has been trying to solve the following problem for over 20 years. Any help would be welcome. The context of this problem is to adjust the theoretical "Tanks In Series" model to account for recirculation loops.

I have been working for quite some time on this and boiled the problem down to the following as included attached to this post.

Attachment 24160

This is the overall mass balance for three tanks in series.
V_x is the volume of each reactor (fixed values)
C_x is the concentration in function of time (variable in function of time)
Q_x is the flow rate of the stream on the diagram (fixed values)

This model would be sufficient if I assume ideal mixing, which is not the case. Therefore I broak up the model as follows:

Attachment 24162

Now following picture has the differntial equations representing the mass balance for each theoretical tank.

Attachment 24163

With n,m and p being the theoretical stages in each tank.
Eventually here is my question, can the above equations be solved to give a function of C_3,m(t)?