Using Mathcad to Plot a system of Polynomials (Chemical Reactor Design)

[FrankButcher]

Hi,

I'm an undergraduate Chemical Engineer and I am doing a problem designing a CSTR for a transesterification reaction. The reaction involves three reversible steps, so in total six reactions with six rate constants. I have kinetically modelled the steady-state material balance equations for the reaction, however because of the order of the polynomials am not aware of a simple hand-calculation method to rearrange the concentrations in to functions of kinetic parameters and residence time only, which would allow me to plot them against each other in mathcad.

The reactions taking place are:

A+B<->C+D r1=k1*C.A*C.B; r.2=k.2*C.C*C.D

C+B<->E+D r3=k3*C.C*C.B; r.4=k.4*C.E*C.D

D+B<->F+D r5=k5*C.D*C.B; r.6=k.2*C.F*C.D

I've typed out the material balances for the reaction in the attached document, I put in boolean equals because I thought that might be the right first step, however any help would be appreaciated..

I'm wondering if there's any way that I can enter the equations as functions of each other, and allow mathcad to do the rearrangement work and simply plot C.A;C.B;C.C;C.D;C.E and C.F against Tau on the time axis.

I had considered using a solve block to find individual values of each parameter at any stated value of tau; however a graphical representation would be more ideal for choosing a value of conversion at any Tau and sizing the reactor from this. In the sheet I haven't actually expressed them as functions of tau i.e. C.A(tau) however this is what I'd like the end result to be like, alternatively I could plot them against a set of tau:0s,0.1s;1000s for example.

Any help would be greatly appreciated.

Regards

Fraser

[HarveyHensley]

Frasher,

The CSTR worksheet here will show you how to get the concentrations as a function of tau (and temperature if you needed). That worksheet is developed for mass fractions, but you should be able to do the same in concentrations.

Also, your mass balances are incorrect. You have left out the outlet flows.

[FrankButcher]

Ahh so I have, so I can add on a C.i for each component at the end of each equation and use that method. Thanks, I will give it a try and get back to you on how I get on.

Fraser

[ViktorKorobov]

Fraser,

Your balance equations are incorrect. The residence time (tau) should be in the denominator.

See in attached file (Mathcad 14).

Viktor

[HarveyHensley]

Viktor,

His equations were correct with respect to tau. You have just divided the entire equation by tau. Note that in Fraser's equations, tau was in the reaction terms, not in the flow terms which you show.

[ViktorKorobov]

Dear Harvey,

You're absolutely right.

For the steady state in the reactor does not matter the location parameter tau (numerator or denominator). But look what happens if tau is in the numerator! (attachment).

Viktor

[HarveyHensley]

Viktor,

Your mass balances are incorrect. Check the flow terms for the correct species in each equation. When you correct those, the curves are smooth with the tau in the reaction terms.