An 8-cm-diamcler spherical potato that is initially at a uniform temperature of

Question

An 8-cm-diamcler spherical potato that is initially at a uniform temperature of 25 degree C is baked in an oven at 170 degree C until a temperature sensor inserted to the center of the potato indicates a reading of 70 degree C. The potato is then taken out of the oven and wrapped in thick towels so that almost no heat is lost from the baked potato. Assuming the heat transfer coefficient in the oven 1 to be 25 W/m^2. degree C, determine: the time required for baking the potato in the oven? the temperature at the surface, T_s = T(r t). the amount of heat transferred, Q (in J). the final equilibrium temperature of the potato after it is wrapped. Data: The properties of the potato are given to be k = 0.6 W/m. degree C rho = 1100 kg/m^3 C_p = 3.9 kJ/kg. degree C

Explanation / Answer

Assumptions:
1) Heat conduction in the potato is one dimensional in the radial direction
2) thermal properties do not change and therefore heat transfer co-effcient remain constant

Given:
ro = 0.04 m ; Ti = 25o C ; T = 170oC ; h = 25 W/m2oC
find the Biot number
Bi = hro / K = 25 * 0.04 / 0.6 = 1.667
(i) calculate from the expression below,
(To - T)  / (Ti - T) = A e-^2 *
Consider A and from the tables
calculate time taken using the formula = t / ro2  
(ii) At surface r = ro
(T(ro) - T ) /  (Ti - T)  = A e-^2 * * [ sin ( r / ro) / ( r / ro) ]
T(ro) = Ts , therefore find T(ro) which will give you surface temperature
(iv) T eq = Ti + mCp
assume uniform temperature throughout the potato
(iii) Q = mCp ( Ti - Teq )

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