Page 473 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 453

The rate of energy entering into the sphere, Q, is given by
%,>
Q = 4=R2 k-

= -4rRk(T, -To) - (10.2-113)

Substitution of Eq. (10.2112) into Eq. (10.2-113) results in

(10.2-114)

The amount of heat transferred can be calculated from
t
&= J Qdt=- L2 lr QdT (10.2-115)
0 CY
Substitution of Eq. (10.2114) into Eq. (10.2-115) yields

00
6 (sin A, - A, cos A,)2
(A, - sin A, cos A,) [I - exp (-A:.)] (10.2-116)

where Q,, is the amount of heat transferred to the sphere when the driving force is
constant and equal to its greatest (or, initial) value, Le.,

4
Qo = ,rp P~P(T~ (10.2-117)
-To)
Example 10.4 Due to an unexpected cold spell, air temperature drops down to
-3°C accompanied by a wind blowing at a velocity of 3m/s in Florida. Farmers
have to toke precautions in order to avoid post in their orange orchards. If frost
formation starts when the surface temperature of the orange reaches O"C, use your
engineering judgement to estimate the time the farmers have to take precautions.
Assume the oranges are spherical in shape with a diameter of lOcm and at an
initial uniform temperature of 10°C. The thermal conductivity and the thermal
digusivity of an orange are 0.51 W/ m. K and 1.25 x m2/ s, respectively.
Solution

Physical properties

Initially the film temperature is (-3 + 10)/2 = 3.5 "C.
Y = 13.61 x 10-6m2/s
For air at 3.5 "C (276.5 K) : k = 24.37 x W/ m. K
Pr = 0.716