Consider the problem of anisotropic thermal conduction in a

Consider the problem of anisotropic thermal conduction in a 2-D square domain, with a thermal conductivity tensor given by: k = [k_xx 0 0 k_yy this type of anisotropy is often used to describe heat conduction in composites where, depending on the geometric alignment of matrix, very different thermal conductivities may be engineered in different directions. The energy equation thus becomes: partial differential/partial differential x(k_xx partial differential T/partial differential x) + partial differential/partial differential y(k_yy partial differential T/partial differential y) + q = 0 Discretize the energy equation using the finite volume method.

Solution

As stated we need to discretion of energy equation using The finite volume method.

So,FVM tells us that,

First step is the integration of the equation as stated over the control volume for quantity Fi.

Integration of d/dx(kxxdt/dx)Fi dV + integration of d/dy kyydT/dy Fi dV + q =0

Next we need to apply Gauss divergence theorem.

Integration d Fi / dx dV = integration of Q dV

Via Gauss div theorem we use to convert volume part into surface element dA.