How should the control volume method be implemented at an interface between two materials with different thermal conductivities? Illustrate with a steady, one-dimensional example. Neglect contact resistance.
GIVEN
Interface between two different materials with different thermal conductivities
FIND
(a) Difference equation at the interface
ASSUMPTIONS
No heat generation
SKETCH
As shown in the sketch, the node at the interface is i = I. The thermal conductivity to the left of the
interface is kleft and on the right side of the interface it is kright. Since there is no contact resistance or
heat generation, an energy balance for the control volume that straddles the interface is
Simplifying and writing this in the tridiagonal form
The above coefficients would be used to write the Ith row of the tridiagonal matrix. The remaining
rows for internal nodes would be written as before and those for the boundaries would depend on
specified boundary conditions.
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