Set up the mathematical formulation of the following heat conduction problems. Formulation includes the simplified differential heat equation along with boundary and initial conditions. Do not solve the problems.
a. A slab in 0 ≤×≤ L is initially at a temperature F(x). For times t > 0, the boundary at x = 0 is kept insulated, and the boundary at x = L dissipates heat by convection into a medium at zero temperature. b. A semi-infinite region 0 ≤ × < ∞ is initially at a temperature F(x). For times t > 0, heat is generated in the medium at a constant, uniform rate of g0 (W/m3), while the boundary at x = 0 is kept at zero temperature.
c. A hollow cylinder a ≤ r ≤ b is initially at a temperature F(r). For times t > 0, heat is generated within the medium at a rate of g(r), (W/m3), while both the inner boundary at r = a and outer boundary r = b dissipate heat by convection into mediums at fluid temperature T∞
d. A solid sphere 0 ≤ r ≤ b is initially at temperature F(r). For times t > 0, heat is generated in the medium at a rate of g(r) (W/m3), while the boundary at r = b is kept at a uniform temperature T0.
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