On Nonlinear Conductive Heat Transfer Equation
On Nonlinear Conductive Heat Transfer Equation
(OP)
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
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





RE: On Nonlinear Conductive Heat Transfer Equation
RE: On Nonlinear Conductive Heat Transfer Equation
RE: On Nonlinear Conductive Heat Transfer Equation
Regards
RE: On Nonlinear Conductive Heat Transfer Equation
I2I
RE: On Nonlinear Conductive Heat Transfer Equation
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)
RE: On Nonlinear Conductive Heat Transfer Equation
corus
RE: On Nonlinear Conductive Heat Transfer Equation
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
RE: On Nonlinear Conductive Heat Transfer Equation
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.
RE: On Nonlinear Conductive Heat Transfer Equation
T = constant. This approximate analytical solution satisfes both sides of the PDE. QED.
corus
RE: On Nonlinear Conductive Heat Transfer Equation
Thank you for this astonishing constant solution
RE: On Nonlinear Conductive Heat Transfer Equation
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
RE: On Nonlinear Conductive Heat Transfer Equation
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
RE: On Nonlinear Conductive Heat Transfer Equation
Lets give real free good advice.
Regards