Chapter 2 Heat Conduction Equation2-29 We consider a thin ring shaped volume element of width Δzand thickness Δrin a cylinder. The density of the cylinder is ρand the specific heat is C. In general, an energy balanceon this ring element during a small time interval Δt can be expressed as tEQQQQzzzrrrΔΔ=−+−Δ+Δ+element)()(&&&&But the change in the energy content of the element can be expressed as ΔΔΔΔΔΔEmCTTCrrzTTttt t=EEttelement=−−=−+++()()ρπ2Substituting, (&&)()( )QQCrrzTTtrrrzzz−+−=−+ΔΔρπ2Δzr+Δr r Dividing the equation above by 2rr zΔΔgives −−−−=−1212ππ+r zrr rzCtrzΔΔr2Noting that the heat transfer surface areas of the element for heat conduction in the r and z directions are Arz==2Δand ,Δrespectively, and taking the limit as ΔΔΔt, and →0 yields tTCzTkzTkrrTkrrr∂∂ρ=⎟⎠⎞⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛
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