A solid sphere of radius has density

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A solid sphere of radius R=0.10 mR = 0.10\,\mathrm{m} has density ρm=8000 kg m−3\rho_{\mathrm{m}} = 8000\,\mathrm{kg\,m^{-3}}, specific heat capacity cp=450 J kg−1 K−1c_p = 450\,\mathrm{J\,kg^{-1}\,K^{-1}} and thermal conductivity K=15 W m−1 K−1K = 15\,\mathrm{W\,m^{-1}\,K^{-1}}. A tiny heater of steady power Ph=5.0 WP_{\mathrm{h}} = 5.0\,\mathrm{W} sits at the centre r=0\mathbf{r} = \mathbf{0}. The outer surface r=Rr = R is held in a bath at T=0∘CT = 0^\circ\mathrm{C} for all time (a Dirichlet condition). (a) For an arbitrary volume VV inside the sphere, write energy conservation using the divergence theorem and Fourier's law q=−K∇T\mathbf{q} = -K\nabla T, and show that the temperature obeys ∂tT=D∇2T+Phρmcp δ3(r),D=Kρmcp.\partial_t T = D\nabla^2 T + \frac{P_{\mathrm{h}}}{\rho_{\mathrm{m}}c_p}\,\delta^3(\mathbf{r}), \qquad D = \frac{K}{\rho_{\mathrm{m}}c_p}. (b) After the heater has been on a long time the temperature settles to a steady profile T∞(r)T_\infty(r) obeying ∇2T∞=−Ph δ3(r)/K\nabla^2 T_\infty = -P_{\mathrm{h}}\,\delta^3(\mathbf{r})/K inside the sphere, with T∞(R)=0∘CT_\infty(R) = 0^\circ\mathrm{C}. Using ∇2(1/r)=−4πδ3(r)\nabla^2(1/r) = -4\pi\delta^3(\mathbf{r}), verify that T∞(r)=Ph4πK(1r−1R),0<r≤R,T_\infty(r) = \frac{P_{\mathrm{h}}}{4\pi K}\left(\frac1r - \frac1R\right), \qquad 0 \lt r \le R, satisfies both the equation and the boundary condition. (c) Prove that T∞(r)T_\infty(r) is the only function with this singularity at the origin that satisfies ∇2T=−Phδ3(r)/K\nabla^2 T = -P_{\mathrm{h}}\delta^3(\mathbf{r})/K inside the sphere and T(R)=0∘CT(R) = 0^\circ\mathrm{C}, by an energy argument on the difference of two candidate solutions. (d) Evaluate T∞(r)T_\infty(r) at r=1.0 cmr = 1.0\,\mathrm{cm}, giving the steady temperature rise there above the bath temperature.

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