On Nonlinear Conductive Heat Transfer Equation

[CSAPL]

I looking looking for an analytical solution to 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

 

[btrueblood]

unless c(T) is described by an equation, there is no analytical solution beyond the equation you've written. Depending on the form of c(T), there may not be a closed-form (classical, analytical) solution, and the problem would need to be solved numerically.

 

[imok2]

This may be of some help:

http://farside.ph.utexas.edu/teaching/329/lectures/node82.html#fdiff3

 

[sailoday28]

Is the derivation from the steady state one dimensional Fourier equation d/dx(-kdt/dx) ?

Regards

 

[insult2injury]

There are a number of conduction heat transfer books that will cover this type of problem. More information is definitely needed to determine if an analytical solution exists.

I2I

 

[CSAPL]

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
Hopfully this answers sailoday28 question . BTW site posted by imok2 is based on FD method(numerical)

 

[corus]

Why an analytical solution when a numerical solution will be musch easier, and much quicker to arrive at? Your data is only arrived at from lab data with the associated errors of curve fitting, so what purpose does an analytical solution serve?

corus

 

[sailoday28]

corus (Mechanical)I assume when you state analytical, you mean a closed form solution. A closed form solution will give an idication of how accurate a numerial solution is.

Clearly, not all problems can be solved with a closed form solution.
For example, varying surface temperature shock on the inside surface of a long insulated hollow cylinder with varying thermal conductivity. In that case, a numerical solution can be performed with a step shock of the cylinder with a fixed thermal conductivity. Closed form solutions are available. If the numerical solution does not give reasonable results, then other things would have to be considered. Such as node length, time increment, or other means.

Regards

 

[CSAPL]

I am very sorry guys for the typing mistake
Of Course it is the diffusion equation

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

I am trying to avoid writing a special FD code for this equation. I believe that analytical solutions (even if approximate) is more acceptable in industry than the numerical ones. Engineers are not required to be programming practitioner (as numerical solution requires reasonable programing knowledge). Analytical solution is much faster to get and more effective and controllable than numerical one for problems involving Simple Geometry (1D) and single medium and simple BCs. In fact it is good practice that Engineers should go to numerical solution when the analytical one (even in approximate manner) is impossible.

 

[corus]

The solution to the problem is trivial of course, and is
T = constant. This approximate analytical solution satisfes both sides of the PDE. QED.

corus

 

[CSAPL]

corus
Thank you for this astonishing constant solution

 

[sailoday28]

corus (Mechanical)Is you solution based on boundary conditions and initial conditions of T= the same constant?

I don't believe that is what CSAPL (Geotechnical)had in mind. Even for steady state, we would normally have boundary conditions different then T= the same constant at both boundaries.

Regards

 

[corus]

I haven't seen any boundary conditions so far, but as an approximation, T=constant satisfies them all. It all depends on how approximate is approximate.
This approximation satisfies the PDE, which was the original question, and satisfies the later requests that an approximate analytical solution would do. There's no fee.

corus

 

[sailoday28]

corus (Mechanical)Is T=ax+b where a and b are constants a better solution when no boundary conditions or initial conditions are given.
Lets give real free good advice.

Regards