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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU February 2006 Qualifying Exam This problem has to do with diffusion of heat in a slab of material. For the purposes of this problem we will ignore thermal expansion. As such, we may define the internal energy density in terms of the heat capacity per unit mass such that the total energy of the sample is given by: p c ( ) ( ) ∫ = t r T t r rc d E p , , 3 v v ρ where ρ is the mass density and T is the temperature. The “heat current density” is defined by ( ) t r T k j th , v v v ∇ − = where is the “thermal conductivity”. th k (a) Assuming that there are no sources or sinks and that the density is constant derive the heat diffusion equation T t T 2 ∇ = ∂ ∂ κ expressing the diffusivity constant κ in terms of ρ , , th p k c . What are the units of κ ? The heart of each diffusion equation comes from the assumption that there are no sources or sinks of energy: namely, that q t E v v ⋅ ∇ − = ∂ ∂...
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