Help proving differential equation

Question

I', really stuck on this one HW question:

 

I have to show how

dC/dt = (D * (d^2C/dx^2)) + P
at steady state (so when cD/dt =0)

equals: (d^2C/dz^2) + (Pa^2/D C(a)) = 0

using C= C(x)/C(a) and z=x/a to remove variables

 

thank you so much!

Answer 1

Hello this looks like a classic chemical engineering equation...FUN!

The method they're asking you to employ is called non-dimensionalizing (i wouldn't say we are removing variables) so that your system can be applied to any unit. 

 

at steady state, the equation becomes:

 0 = D(d²C(x)/dx²) + P

 

you told me:

x = az      so equation becomes:

Keep track of the brackets and write it out as you read it.

 

0 = D(d²C(x)/d(az)²) + P = (D/a²)(d²C(x)/dz²) + P 

0 = (d²C(x)/dz²) + Pa²/D

 

You also told me C(x) = C*C(a)

 

0 = d²[C*C(a)]/dz² + Pa²/D

 

0 = d²C/dz² + Pa²/(DC(a))

 

I hope this helps you understand the process Carly. Let me know if you have any questions.