Simulating Heat Diffusion from Laser Beam on Sample

[datsyuk]

Homework Statement

Hello, I am trying to simulate the heat diffusion from a laser beam striking a sample. My model/concept is very simple in that it assumes the temperature distribution will be a radially symetric sphere (i.e T=T(r,t)).

I would like to plot a temperature profile evolving through time as a function of position from the initial contact of the laser beam. I am not making a 2D/3D map of the temperature distribution but simply a position vs T plot evolve through time.

My question is that can I use the solution for the heat diffusion equation in 1D cartesian coordinates to simulate the temperature distribution from the initial point of contact or would I have to use the spherical heat diffusion equation. The reason is that the solution for 1d cartesian is readily available while in spherical coordinates obtaining the solution is a bit more difficult.

Homework Equations

Heat Diffusion Equations
Cartesian

dT/dt=a(d^2/dx^2), where a represents thermal diffusivity

Spherical

1/r^2(d(r^2dT/dr)dr)=1/a(dT/dt), since no dependence on theta or phi

The Attempt at a Solution

I am currently simulating the heat diffusion using the solution from the 1d cartesian coordinate heat diffusion equation but I am not sure if this is entirely correct. Also, I would imagine that my temperature profiles would be the same as long as a set my position axis to be centered at the origin (point of contact of laser beam).

 

[LawrenceC]

Is the sphere completely immersed by the laser beam or is the laser beam relatively small with respect to the sphere so that only a point on the sphere is being heated? Furthermore, does the beam pass through the surface of the sphere?

 

[datsyuk]

Thanks for your response. I've attached a simple diagram to show what it looks like.

I'm not irradiated a sphere. I will be irradiated a flat horizontal target (e.g. microscope slide). The sphere represents the heat diffusing from the target. Also the beam is a short pulse and therefore once I have irradiated the sample resulting in the initial temperature increase, all there will be left will be the heat diffusing from where the beam initially made contact with the sample.

In other words, I am introducing a temperature increase on the target with the laser beam (with negligible diameter). After the initial temperature increase, the beam is not present but what we have is heat diffusing in all directions, including a direction parallel to the flat target, and diffusion representing radial symetry. Therefore if we made a 3D map of the temperature at a certain time point, we would notice a sphere and one in which the temperature was only dependent on the radial position from the origin (which in our case would be the point of contact of the beam onto the target).

I only care about the temperature profile as a function of horizontal position from the point of contact of the beam.

 

[LawrenceC]

Since it is a slide with only a point being heated, I would consider solving it using cylindrical coordinates in one dimension - Fourier equation. The axis would be perpendicular to the slide. The complication is that the slide will lose heat from its surface by both convection and radiation. The latter will cause the PDE to be nonlinear. If you omit radiation loss from the surface, the equation can be solved by omitting the non-homogeneous convective loss term (leaving the Fourier equation), then accounting for it by the method called variation of parameters. The solution would be a combination of separation of variables for the homogeneous equation, the variation of parameters for the non-homogeneous heat loss term.

Obvously the best way to solve it is to use a finite element program such as ABAQUS. If you make too many assumptions, you are assuming the problem away.

 

[DeeNos]

I understand your concern about using the 1D cartesian coordinate heat diffusion equation for a simulation that is radially symmetrical. While the solution may be readily available, it may not accurately represent the behavior of heat diffusion in a spherical system. It would be more appropriate to use the spherical heat diffusion equation, as it takes into account the radial dependence of the temperature distribution.

However, if you are only interested in the general behavior of the temperature distribution and not the exact values, using the 1D cartesian solution may be sufficient. In this case, setting the position axis to be centered at the origin would ensure that the temperature profiles are the same as long as the initial point of contact is the same.

In summary, it is recommended that you use the spherical heat diffusion equation for a more accurate simulation, but the 1D cartesian solution may also be used depending on your specific goals for the simulation.