- #1

Makadamij

- 7

- 2

- Homework Statement
- Try to understand the solving method of the three heat equations given in the following article.

- Relevant Equations
- 1-dimensional heat equations - see the pictures below, please:

Hello everyone!

I'm analysing a scientific paper regarding the theory of the photoacoustic effect. The subject of investigation is a so called photoacoustic cell, in which a higly light-absorbing solid material is placed on a backing material and the rest of the cell is filled with gas (see picture below). Only the solid material is capable of absorbing light and converting it into heat. The cell is illuminated by sinosoidal chopped monochromatic light flux. The goal is to find the settled temperature distribution inside the cell. Therefore, three heat equations have been set up in picture pg65_2, two homogenous ones for the backing material and the gas and a non homogeneous heat equation for the solid, because heat is being created inside of it because of light absorption. The general solutions to these three equations are in picture pg66_1, not taking into account the boundary conditions of heat flux continuity.

I wanted to understand how they came up with these solution without using the boundary conditions, therefore I tried to solve the two homogenous ones via variable separation. Although no initial conditions are given, I assumed they are zero, becuase the backing material and gas do not absorb any light and therefore their temperature at t=0 is equal to the ambient temperature. However, then I didn't know how to calculate the b_n variables, since they would all became zero, hence the solution would be zero.

I would be very thankful I you could give me some advice on how to solve those equations or at least get some insight of the methods the researchers have used.

I'm analysing a scientific paper regarding the theory of the photoacoustic effect. The subject of investigation is a so called photoacoustic cell, in which a higly light-absorbing solid material is placed on a backing material and the rest of the cell is filled with gas (see picture below). Only the solid material is capable of absorbing light and converting it into heat. The cell is illuminated by sinosoidal chopped monochromatic light flux. The goal is to find the settled temperature distribution inside the cell. Therefore, three heat equations have been set up in picture pg65_2, two homogenous ones for the backing material and the gas and a non homogeneous heat equation for the solid, because heat is being created inside of it because of light absorption. The general solutions to these three equations are in picture pg66_1, not taking into account the boundary conditions of heat flux continuity.

I wanted to understand how they came up with these solution without using the boundary conditions, therefore I tried to solve the two homogenous ones via variable separation. Although no initial conditions are given, I assumed they are zero, becuase the backing material and gas do not absorb any light and therefore their temperature at t=0 is equal to the ambient temperature. However, then I didn't know how to calculate the b_n variables, since they would all became zero, hence the solution would be zero.

I would be very thankful I you could give me some advice on how to solve those equations or at least get some insight of the methods the researchers have used.