On Nonlinear Conductive Heat Transfer Equation

[CSAPL]

I am looking for an analytical solution to the 1D nonlinear heat transfer equation

d^2T/d^2X = c(T) dT/dX

as you can see nonlinearity appears as c is function of temperature; i.e c(T). I want analytical solution . Can you guid me to a sourec where I find the analtical solution of this 1D difuusion equation.
I already posted this in Heat Transfer Forum.

 

[prost]

Isn't the 1D diffusion equation d^2T/d^2x=c*dT/dt ?

your only hope might be a numerical solution.

 

[GregLocock]

CSAPL that will only be analytically solvable for some parametric cases of c(T). so, what is c(T)?

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[zekeman]

Let
p=dT/dx and dp/dx=d^2T/dx^2
Then
d^2T/dx^2=dp/dT*dT/dx=pdp/dT=c(T)*p
Dividing by p
dp/dT=C(T)
And
dp=C(T)dT Integrating both sides
p=F(T)+K where F(T) is the integral of RHS and K is a constant
dT/dx= F(T)+K
dT/[F(T)+K]=dx
integrate again to get x as a function of T

 

[zekeman]

By the way. I don't think you have the diffusion equation there. I think you should have in the steady state
d/dx{K(T)dT/dx}=0
which becomes
K(T)d^2T/dx^2+ dK/dT*(dT/dx)(dT/dX)=0
and if you call -1/K*dK/dT=C(T) you get
d^2T/dx^2=C(T)[dT/dx]^2
Note the square term on the RHS, not what you got. The method of solution is similarto whatI showed. You now get
dp/p=C(T)dT for openers.










]

 

[CSAPL]

For Now Here is
C(T) =(dS/dT) * T + S + A(dS/dT)

Where A is constant and S is an emperical function of T ( based on the Lab data, the best relation suggested in literature is S= B(C/T)^D, where B,C,D are fitting parameters)
However if makes life easy, I can assume this emperical function ,say Polynomial

Thanks

 

[CSAPL]

I am very sorry guys for the typing mistake
Of Course as Prost says it is difussion equation

d^2T/d^2X = c(T) dT/dt

 

[zekeman]

It's a partial differential equartion.

 

[CSAPL]

Yes Zekeman it is a PDE
If one of you is aware of a possible existance of an analytical solution (even if approxinate). please refer me to the source. PLease note that its BCs is also nonlinear .
It is a challenge .. is not it.?

 

[mechj]

Could you use a Galerkin expansion?

Regards,

jdm

"Education is what remains after one has forgotten everything he learned in school." Albert Einstein

 

[CSAPL]

JdM
is not Galerkin expansion based on Finite Difference (numerical approach) ?

 

[mechj]

You can formulate your finite element equations using Galerkin expansion (some finite element formulations are based on this). A Galerkin expansion is basically just minimizing the residual wrt to a weight function. The weight function is basically a shape function, and it can be anything that satisfies the boundary conditions. Your solution will depend on how well this is picked, and also how far the expansion goes. This is still an approximate solution and will probably require some numerical integration, but it will be an analytical solution.

jdm


"Education is what remains after one has forgotten everything he learned in school." Albert Einstein

 

[zekeman]

quote"This is still an approximate solution and will probably require some numerical integration, but it will be an analytical solution."

jdm
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Excuse me, but under my definition of analytical, this will NOT be an analytical solution. He may as well do it piecewise numerical and be done with it.

 

[corus]

I have to agree with zekeman. Given the time and effort to solve such a problem, and the chances of there being an error in the approximate 'analytical' solution, you'd be much better solving the problem numerically. At the end of the day the numerical solution would be much more exact given the assumptions you're bound to make to approximate the non-linear boundary conditions, and non-linear material properties that you effectively have. You should be able to find a free program on the net that can do this without having to develop your own code. You can always test the software if you're not sure about it with known problems that have a true analytical solution.

corus

 

[mechj]

quote: "If one of you is aware of a possible existance of an analytical solution (even if approxinate)."

From MathWorld:
A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Taylor series in a neighborhood of every point.

By this definition if the weight functions were chosen as say trig functions he would have an analytic solution.
But, maybe I am misunderstanding what he means by analytic solution. For his problem there is not an analytic solution, by which I mean you just cannot solve it, but you can generate a closed form approximate solution.

He could also use a perturbation (expansion) techinque such as a Poincare map, or multiple time scales analysis to gain some insight of the system.

And I do agree that a numerical solution would probably be the most effective way of solving the problem, but he was not asking about numerical techniques.

jdm

"Education is what remains after one has forgotten everything he learned in school." Albert Einstein

 

[zekeman]

quote"
From MathWorld:
A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Taylor series in a neighborhood of every point.

By this definition if the weight functions were chosen as say trig functions he would have an analytic solution.
But, maybe I am misunderstanding what he means by analytic solution. For his problem there is not an analytic solution, by which I mean you just cannot solve it, but you can generate a closed form approximate solution."
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It may interest you to know that a differential equation can have analytic coefficients and still be nonlinear and generally, not solvable analytically.It depends on whether or not the coefficients are functions of the independent variable or not.

 

[prost]

an 'analytic function' is not the same as the 'analytical solution,' you are talking about two different things.

 

[mechj]

I see the difference b/w analytic solution and analytic function. Wasn't trying to ruffle any feathers, just help out.

jdm

"Education is what remains after one has forgotten everything he learned in school." Albert Einstein

 

[CSAPL]

Have you guys heard about Integral Methods [Goodman, T. R., “Application of Integral Methods to Transient Nonlinear Heat Transfer” in Advances in Heat Transfer 1, edited by Academic press, 1964, pp. 51-122].?

It is simple and analytical (requires no descritization) If you have experience using it what do you think of it.

 

[zekeman]

Quote"Have you guys heard about Integral Methods [Goodman, T. R., “Application of Integral Methods to Transient Nonlinear Heat Transfer” in Advances in Heat Transfer 1, edited by Academic press, 1964, pp. 51-122].?

It is simple and analytical (requires no descritization) If you have experience using it what do you think of it.'
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I haven't seen it, but will venture to say it will NOT solve analytically any general class of nonlinear PDE of the type we are discussing, or hardly any of any class.
You say it is simple, so let's have an example since access to a 1960s book is almost impossible.