Convection diffusion equation 1D exact solution


by phunnyish
Tags: 1 dimensional, convection-diffusion, heat equation, heat transfer, numerical method
phunnyish
phunnyish is offline
#1
Nov10-12, 11:59 PM
P: 2
I'm looking for the analytical solution for the 1D convection diffusion equation with a constant heat flux.

Boundary conditions:
The domain I'm looking at is x from 0 meters to 1 meter. The temperature at x=0 is T=0 degrees Celsius. At x=1, T=100 C.

I'm given the equation:

d/dx(rho*Cp*u*T)=d/dx(k dT/dx) + q'''

rho=density
Cp=specific heat
u=velocity in x-dir
T=temperature
k=thermal conductivity
q=generated heat (constant)
alpha=k/(rho*Cp)
P=(rho*u*L)/k (Peclet number)
To=0 C=273 K
T_L=100 C=373 K

I get it in the form:

u/alpha (dT/dx) - d^2T/dx^2 = q'''/k (equation 2)

I need to solve for T as a function of x (space).

I found the no generation solution to be

T(x)=(T_L-To)*(exp(P*x/L)-1)/(exp(P)-1)+To;

I tried plugging my equation 2 into a DE solver and I got

T(x)=[a*c1*e^(bx/a)]/(b) + cx/b + c2

c1 and c2 are constants
a=-1
b=u/alpha
c=q'''/k

I tried plugging in my boundary conditions to solve for c1 and c2 multiple times, but the solutions are not coming close to my numerical solutions at all.

I have also searched numerous sites trying to find this solution.

If anyone knows what T(x) is for this type of problem with constant heat flux, please respond to this post. Thank you.
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Vargo
Vargo is offline
#2
Nov12-12, 11:26 AM
P: 350
To me it looks like the DE solver solution is incorrect. Just delete the a=-1 to get the right answer which should be:

Aexp(Px/L) + B + (c/b)x

The boundary conditions give
A+B=T0
Aexp(P)+B+c/b =TL.
Subtracting the two equations gives
A = (TL-T0-c/b) / (exp(P)-1).
B = T0 - A.

T = (TL-T0-c/b)*(exp(Px/L) -1) / (exp(P)-1) + T0 +(c/b)x


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