PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Partial differential equations and boundary-value problems
Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator
Problem
An aluminium cold plate (mass m=1.2kg, specific heat capacity cp=900Jkg−1K−1) is a rectangular slab used to cool electronics mounted on its top face. The mounted chips dissipate a steady total power P=50W inside the plate, distributed however the manufacturer likes. Four side faces and the top face are thermally insulated; the bottom face, of area A=0.030m2, is cooled by a coolant loop that, once running has settled, removes a measured uniform outward heat flux q=1400Wm−2 over that face.
((a)) Steady conduction inside the plate obeys ∇2u=−f/k for internal heat-source density f and conductivity k, with outward heat-flux vector q=−k∇u. Using Green's first identity with ϕ=1, ψ=u, show that a steady state can exist only if ∮∂Vq⋅n^dS=P, i.e. the total heat leaving through the whole boundary must equal the total power generated inside, whatever the (unknown) distribution of f or of the boundary flux.
((b)) Decide whether the cold plate is in a steady state with the measured coolant flux, and if it is not, find the rate of change of the plate's mean temperature, treating the plate as a lumped mass of uniform temperature.