How would you include contact resistance between the two materials Derive the appropriate difference equations.

Question

GIVEN

Interface between two materials with different thermal conductivities and contact resistance at the

interface

FIND

(a) The appropriate difference equations

SKETCH

Answer 1

Let the contact resistance be Rc. The interface is located at node i = I. Represent temperatures to the

left of the interface with TL, i and to the right of the interface with TR, i. Thermal conductivity to the left

of the interface is kleft and to the right of the interface is kright. We have drawn two half control

volumes, one just to the left of the interface and one just to the right of the interface.

An energy balance on the left control volume is



Writing these equations in the tridiagonal form we have



From these equations, the coefficients for the tridiagonal matrix can be defined



The vector of nodal temperatures in would be modified to look like



The coefficients with subscripts L, I would appear in the row corresponding to TL, I and those with

subscripts R, I would appear in the row corresponding to TR, I. Remaining coefficients would be

determined as for any other one-dimensional steady problems including those determined by the

boundary conditions.