Final answer:
To determine the temperature profile, T(r), of a cylinder with heat generation, one must solve the heat conduction equation including a source term for internal heat generation and apply the boundary condition that T(r) = Ts at the cylinder's surface (r = ro).
Step-by-step explanation:
To find the temperature profile, T(r), within a solid cylinder generating heat (q' > 0), we solve the steady-state heat conduction equation under the assumption of radial symmetry with no dependence on the angular or axial coordinates. The differential form of the heat equation for a cylinder in steady state, without internal heat generation is given by:
kA(T₂ - T₁) / d = Q / t,
where:
k is the thermal conductivity of the material,A is the surface area,d is the thickness,(T₂ - T₁) is the temperature difference across the material,
- Q/t is the rate of heat transfer in watts or kilocalories per second.
However, in our case, there is internal heat generation, so the heat equation needs to include a source term, q', to account for this. The boundary conditions are that at r = r₀, T(r) = Tₛ, where r₀ is the radius of the cylinder and Tₛ is the surface temperature of the cylinder. To solve for T(r), you would integrate the heat equation from 0 to r₀, applying the boundary condition at r = r₀.