Consider a long, thin, uniform rod of constant cross-section whose temperature distribution is θ(x,t), surrounded by an atmosphere of constant uniform temperature θ0. The sides of the rod are not insulated so that heat is lost from the longitudinal surface at a rate h(θ(x,t)- θ0) per unit length, for some constant h. Derive the equation which describes the temperature distribution in the rod and show that it can be written
∂φ/∂t=k/ρc (∂^2 φ)/(∂x^2 ) - (h/ρAc) φ
where φ(x,t) = (θ(x,t)- θ0), k is the thermal conductivity, ρ the density , c the thermal capacity and A the cross-sectional a

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