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October 29th, 2017, 12:49 PM   #1
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This differential equation is driving me crazy...

Hello There, I am david, i am new in the forum and really hope you cold help me.

I am supose to solve this differential equations which is making me going crazy

[1] d(Ca)/d(t) = (-k*Ca*Cb) -(q*Ca/V)
[2] d(Cb)/d(t) = (-k*Ca*Cb) + (q*(Cbo-Cb)/V)
[3] d(Cc)/d{t) = (k*Ca*Cb) - (q*Cc/V)
[4] d(Cd)/d(t) = (k*Ca*Cb) - (q*Cd/V)

This equations are the result of a molar-mass balance in a semibatch reactor.

I am suppose to graphic reaction velocity (r) vs time (t)

There is some extra information

[1] k = 2.2
(2) q = 0.05
(3] Cbo",0.025
[4] Vo=5
(5] Cao = 0.05
[6] r = k*Ca*Cb
[7 J V = Vo + q*t
[8] X = (Cao*Vo-Ca*V)/(Cao·Vo)

In the book i got this from they solve the problem with polymath but my teacher want us to do it manually (a real pain the in ass) If somebody could tell me at least how i am suppose to resolve the first equation i would really apreciate, thanks for taking your time to read this and i apoligize if my grammar is not correct
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October 29th, 2017, 04:59 PM   #2
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Any ideas on how to begin?
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October 29th, 2017, 05:49 PM   #3
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I tried a couple of things (variable separations, 1st order linear differential equation but kinda got stuck because Cb also depens on t) so I dont know if there is other way to solve it, maybe using a system of ecuations?
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October 29th, 2017, 07:59 PM   #4
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Are you sure you have the equations right? I'm getting a seriously ugly expression for Ca involving a number of Gamma and Bessel functions. (I'd show you the solution using Wolfram Alpha, but it doesn't know how to do it. I had to do it on my personal copy of Mathematica.)

Here's a quick rundown of what I did to get the equation for Ca:
1) Subtract equations 1 and 2 to remove the CaCb terms.

2) Set X = Ca - Cb.

3) Solve for X(t)

4) Find Cb = Ca - X

5) Put this into equation 1.

You get the equation $\displaystyle C_a'(t) = -k C_a^2 + k P e^{-qt/V} + k C_{b0} C_a$
Where P is an integration constant.

-Dan

Last edited by topsquark; October 29th, 2017 at 08:05 PM.
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October 29th, 2017, 08:00 PM   #5
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It is already a system of equations with 4 variables. As it is nonlinear and coupled I don't know what you mean by "solve by hand". The best you can hope for is to expand a solution in a power series since the RHS is analytic.
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October 30th, 2017, 10:46 AM   #6
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My teacher said we could make this system of equations using laplace transform, i will give it a shot, Thanks for thanking your time for answering me, i really appreciate it
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