What is the exact usage of internal heat generation term used in a general heat conduction equation?

Question

∂/∂x (λ ∂T/∂x)+ ∂/∂y (λ ∂T/∂y)+ ∂/∂z (λ ∂T/∂z)+ q_v= ρC_p ∂T/∂t------1

T - Temperature (Temperature of the material) ρ - Density of the material C_p - Specific heat capacity of the material t - time q_v - rate of internal heat generation (W/m3)

The above equation is a 3D Transient heat condcution equation with q_v is the internal heat generation term. I am pretty much confused with term q_v.

I work with laser applications with laser processing of materials. My question is that when I use laser as my source for heating the material. How should I substitute the laser source term into the above heat conduction equation ?. Since q_v is a heat generation term I cannot use laser source into q_v. Is my thing right ?

Another possible way is to incorporate the laser source term as heat flux with a boundary conditon as shown below.

λ ∂T/∂x=q ------- 2

q- laser heat flux (w/m2) λ - thermal conductivity

Which is the right way to include the laser source term ? Either with equation 2 or 1?

Any help will be appreciated?

Answer 1

The second method is correct only if all the energy from the laser is absorbed at the surface. The first method is the way to go. But, you need to determine qv. To do that, you need to use Beer's law to determine the attenuation of the laser energy in the material, and then translate that into the rate of heat "generated" per unit volume.