Hello Researchers,
I am trying to develop a transient heat transfer model for a solid-solid device with internal heat generation using the basic Fourier's equation. I have attached an image that can well describe my problem. I am encountering a challenge in my model , i.e., When trying to obtain an exact solution using inverse Laplace transform approach, the equation involves computing residues for many terms , which makes the model too complicated.
Can anyone suggest me a model that avoids complexity by providing a compact infinite series solution? Thanks
If possible please send the current picture of the mesh and the various parameters which you have used.
The picture shows a homework for a numerical solution of a parabolic equation, you are talking about the derivation of the exact solution.
So, what is the key of your question?
Is this linked paper any help?
Otherwise see if you library has Stephen Whitaker (2013) Fundamental Principles of Heat Transfer.
http://www.sciencedirect.com/science/article/pii/S0017931015001568
Dear Filippo,
Thanks for taking time to read mu question. My key question is , how can we obtain a simple analytical solution to the transient heat conduction problem which involves two segments connected with zero contact resistance? In general, higher the number of segments, higher is the complexity involved. I am looking for a method that could solve the transient heat problem that accounts free convection on the surface and the heat conduction in the element with internal heat generation(Joule heating). Let me know if you want to have a look at the governing equations. Thank you
Dear Ian,
Thanks very much. This paper is a close one to my problem.
Dear Filippo and Sharba,
I have attached the governing equations for the two segmented device. The objective is to get theta_1 and theta_2 in terms of position(x) and time(t) for the corresponding thermal conditions of my problem. Any papers in literature or any suitable textbooks or anyother materials will be surely helpful inaddition to your valuable suggestions. Thank you !
Analytical solutions for the heat equation is a classical issue, you can find a lot of material also searching in Google.
Just one example:https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/heateqni.pdf
Hi
Please check with the following book, hope this will help you:
Carnahan B., Luther H.A. and Wilkes J.O., 1969 “Applied Numerical Methods”, John Wiley and sons, New York.
Hello,
For transient problems in multilayer systems, especially in the case of 1D transfer, the quadrupole method is particularly efficient. A first step is to perform a Laplace transform. Then the [temperature, flux] vectors at two different positions are related through a 2x2 matrix involving the effusivity and the diffusivity of the material in between. Quadrupole matrices related to different materials are then multiplied together and one finally makes use of the boundary conditions for closing the problem. The temperature can be then expressed at any depth. For the inverse Laplace transformation, there are very efficient numerical methods (e.g. De Hoog method). Localised or distributed heat sources can also be present.
See, e.g. D. Maillet, S. André, J.-C. Batsale, A. Degiovanni, C. MoyneThermal Quadrupoles: Solving the Heat Equation Through Integral Transforms
John Wiley and Sons, New-York (2000)
and https://doi.org/10.1016/j.ijthermalsci.2014.02.007
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