Derive finite difference equations for all the interior

Question

1. Derive finite difference equations for all the interior nodes.

2. Develop a MATLAB code, and determine temperatures at all interior nodes.

The generalized heat equation for laser melting is given by

ρc(T)dT/dt = ddx[kx(T)dT/dx] + d/dy[ky(T)dT/dy] + d/dZ[kz(T)dT/dz] + ρc(T)(VxdT/dx + VydT/dY + V_zdT/dz) + Q

where

T = temperature of the weldment (K)

k(T) = thermal conductivity of the material (J mm ) as a function of temperature

ρ(T)= density of the matenal (J/mms^-1k^-1) as a fun cton of temp.

C(T) = specific heat of the material (J/g^-1 K^-1), as a function of temperature

V_x, V_y and V_z = components of velocity

Q = rate of any internal heat generation. (W/mm³)

Reduce the above equation for steady-state thermal analysis of a system having a stationary heat flux and no-heat generation. Then solve the following problem:

Consider a case where the surface of an aluminum plate is irradiated by a stationary laser so that the steady-state temperature at the laser-irradiated region is defined by Gaussian distribution. The remaining boundaries are maintained room temperature (23°C).

Consider k_x = k_y = 50 W/m²-K, and grid sizes Δx = Δy = 5, mm.