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On Nonlinear Conductive Heat Transfer Equation

On Nonlinear Conductive Heat Transfer Equation

On Nonlinear Conductive Heat Transfer Equation

(OP)
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.

RE: On Nonlinear Conductive Heat Transfer Equation

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

your only hope might be a numerical solution.

RE: On Nonlinear Conductive Heat Transfer Equation

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.

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

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.










]

RE: On Nonlinear Conductive Heat Transfer Equation

(OP)
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

RE: On Nonlinear Conductive Heat Transfer Equation

(OP)
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

RE: On Nonlinear Conductive Heat Transfer Equation

It's a partial differential equartion.

RE: On Nonlinear Conductive Heat Transfer Equation

(OP)
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.?

RE: On Nonlinear Conductive Heat Transfer Equation

Could you use a Galerkin expansion?

Regards,

jdm

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

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

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

jdm
-----------------------------------------------------------
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.

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

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."
--------------------------------------------------------
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.  

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

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

RE: On Nonlinear Conductive Heat Transfer Equation

(OP)
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.
 

RE: On Nonlinear Conductive Heat Transfer Equation

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.'
-----------------------------------------------------------
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.

RE: On Nonlinear Conductive Heat Transfer Equation

(OP)

let me tell what I am doing regading this issue so that we can share knowledge. As you know that the nonlinear equation is d^2T/d^2x=C*dT/dt
and the nasty  C(T) =(dS/dT) * T + S + A(dS/dT). Now if the relation between S  and T is emperical (fitting curve),why not find S=f(T) for which
(dS/dT) * T + S + A(dS/dT)= constant (C) So I am just solving this equation so that I will get families of S =f(t)for which the nonlinear equation above becomes linear
afterwards I will optimize the constants of this simple differentail equations to find the best curve that fits my emperical data S and T.  Makes sense to you.Your comment is appreciated
P.S  zekeman: you can find info about Integral Methods  for NLHT problem in any HT book

Thanks

 

RE: On Nonlinear Conductive Heat Transfer Equation

What you are saying, is that you will enter a universe of constants, each equalling C, to get a universe of f(x)s and pick the one that comes closest to the empirical f(x). If you are extraordiarily lucky, you may have an infinitesimal shot, although coming close is no guarantee of a proper solution.
Why are you going through all this labor? What is wrong with a numerical solution?
If you can share this with us, what is the application? Maybe one of us could shed some light on it.
And by the way, I am aware of integral methods, but they will not solve general nonlinear PDEs as I mentioned. Want to win a Nobel Prize in mathematics, just find a general solution to your problem. Oh, they don't give the prize in mathematics, so make it a problem in economics and then you may.
Good luck.

RE: On Nonlinear Conductive Heat Transfer Equation

(OP)
zekeman,
Well the idea of trying to get T given in an analytical formula is because I am going to use it in another close formula. Plus you  talk about developing numerical solution as if it were piece of cake and requires few minutes to finish without any troubles.
P.S. I have no intention now to develop Zekeman's (or soory Goodman's) Integral Method in heat transfer But if i decide and win Nobel price I will give you some bucks

RE: On Nonlinear Conductive Heat Transfer Equation

To CSAPL- This reply is a bit late,however, I became interested in your problem and determined that you may already have the answer if you had DE in college. Esbach,3rd Edition has a series of solutions for second order DE on pages 326 thru 327.
You stated that C(T)=(dS/dT)*T + S +A(dS/dT).  I am quiet certain that dS and dX are the same, therefore, plugging c(T) in your equation d^2T/dX^2 would yield
     d^2T/dX^2 = T + A + X(T)dT/dX
which could be solved adding the results of Esbach's case 1 and case 3 the latter being shown by Zekman in your reply.

RE: On Nonlinear Conductive Heat Transfer Equation

Further digging into your question.  Your DE and C(T) equations that you presented simplifies to a non-linear DE:

     T" = x(T)T'+T+A;  A is a constant per your definition

     let T+A=I(P*dX);  I is the symbol for integral

     differentiate T+A becoming dT=PdX then P'= T"

     make substitution to above DE: P'=x(T)P+I(pdX)
                              then: P'/P=x(T)+(I(PdX))/P
                                        =x(T)+I(dX)
                              then: dP/P=x(T)dX+(I(dX))dX
                               I(dP/P)=I(x(T)dX)+I(I(dX))dX
                        since x(T)=F(T) then dx=f(T)dT     
                             I(x(T)dX)=I(F(T)dT)    
       Final differential equation:
                        I(dP/P)=I(F(T)dT)+I(I(dX))dX    
Make sure imits of integrations are appropriately changed.

RE: On Nonlinear Conductive Heat Transfer Equation

chicopee
I do not think CSAPL meant in his equation that dS and dX are the same" Why you said "I am quiet certain dS and dX are not the same. According to CSAPL explaination S is a function of T while  X is the coordinate

CSAPL can correct me if i am wrong
d^2T/d^2X = c(T) dT/dt
C(T)=(dS/dT)*T + S +A(dS/dT)

RE: On Nonlinear Conductive Heat Transfer Equation

helppoorpeople- what is the significance of S?

RE: On Nonlinear Conductive Heat Transfer Equation

helppoorpeople- Yep, somehow I missed CSALP equation for S.

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