Question

2. (25 points) A spherical metal ball of radius ris heated in an oven to a temperature of T; throughout and is then taken out of the oven and dropped into a large body of water at To where it is cooled by convection with an average convection heat transfer coefficient of h. It is given that there is no internal heat generation, that the thermal conductivity is constant, and that there are no temperature variations in the transient one-dimensional heat transfer. Starting with the general conduction heat equation given below, develop the applicable governing differential equation. Also, provide the appropriate boundary and initial conditions. Do Not Solve the Resulting Equations! rzo (kr 2 ) trèsinoom () + rzsine (ksino A) + égen = pc at at k 12 T. h T;

Answer 1

ASSUMPTIONS :

1. Heat transfer process is given as one dimensional.

2. Heat transfer is given to be transient.

3. Thermal conductivity is constant.

4. Heat generation is absent.

5. At r=ro i.e., at outer surface of sphere when kept in water it undergoes convection process.

The heat conduction equation and boundary conditions are represented in below images;

sol: - Given, Rodius Heated Temperature Ti Water Temperature - Too Heat Transfer Coefficient = h Thermal Conductivity = k K Given Heat Conduction Equation is, . (xv> ;) 4 OT do + IK 8² sinto so 2 or en esimodi 8²sino + &gen = Pc QT dt. K Too hi 토 T: Because of 1-Dimensional Equation and that there is heat generation, The heat Conduction Eqn Conduction Eqn becomes, no internal حم (vara K8 2 JT or + 0 + 0 + 0 = PC &T 82 or ot 2 dT K. ( v «Рc от ot re dr

@ 2 JT dr SC IT K ot :. We know that lz K pc. where, a Thermal diffusivity diffusivity (not) 8 → Density (kg m3 Specific heat J kg The heat Conduction Equation becomes, 182 or Y 8 - Y OT a et Boundary Conditions :- 1 T (8,0) = Ti 2 K OT(YO, !) Y h [T(%o) - Too h 3. T(0) ØY 0 THE ENO