Modeling thermal diffusion using Laplace transform

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Modeling temperature diffusion through a solid body (or fluid diffusion through a porous medium) requires solving a time and space differential equation that is a combination of the Fourier's law (or Darcy's law for the case of fluid diffusion) and the energy consevation equation (mass conservation for the case of fluid diffusion) (Coussy, 1995; Rohsenowa and Cho, 1998). The Laplace's transform allows converting such problem to a simpler one in which the time derivative terms disappear. Analytical solution can be obtained for some particular geometries of the body, otherwise the classical finite element method can be considered to solve the geometric problem in Laplace's transformed space. Solutions obtained in Laplace's space can be transformed to time space by the inverse Laplace's transformation. The later is usually realized by numerical techniques for the general situation (Stehfest, 1970).

Let consider an example of a cylinder hole (circular hole on the transversal plane) located in an infinite medium that has a homogeneous farfield temperature (or fluid pressure for the case of fluid diffusion). A local temperature is applied on the inner surface of the hole. Temperature in the area arround the hole will change during time due to the diffusion process. Below, a tool is provided to simulate that problem.

Rohsenow, W. M., & Cho, Y. I. (1998). Handbook of heat transfer (Vol. 3). J. P. Hartnett (Ed.). New York: McGraw-Hill.
Coussy, O. (1995). Mechanics of porous continua. Wiley.
Stehfest, H. (1970). Algorithm 368: Numerical inversion of Laplace transforms [D5]. Communications of the ACM, 13(1), 47-49.

Input the parameters then click Compute button to run the simulation.

Hole radius (m)

Borehole temperature (°C)

Farfield temperature (°C)

Diffusivity (mm2/s)

Time step (days)
(Use a small or large time step to observe the short or long term diffusion, respectively)

Number of time steps (move this slider to jump between the time steps)