Case Study: Tumour growth
Heat flux and thermal resistance: Suppose we have two different materials with widths d1 and d2, conductivities k1 and k2, and which are joined together at x = 0. The equilibrium temperatures U1(x) and U2(x) both satisfy the basic equilibrium heat equations
d2U1/dx2 = 0, d2U2/dx2 = 0.
The temperature on the inside (x = -d1) is held (by a thermostat) at a temperature ui and the temperature on the outside (x = d2) is held (by a thermostat) at a temperature uo.
1) Assuming the temperature and heat flux are continuous where the materials join at x = 0, what are the boundary conditions?
2) Show that the heat flux is given by
J = ui - u0 /(d1/k1 + d2k2)
3) Interpret this in terms of thermal resistances
Textbook: MATHEMATICAL MODELLING WITH CASE STUDIES - Using Maple and MATLAB - Third Edition by B. Barnes and G. R. Fulford
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