Page 475 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 455

Example 10.5 A 2-kg spherical rump roast is placed into a 175°C oven. How
long does it take for the center to reach 80°C if the initial temperature is 5OC7
The average heat transfer coeficient in the oven is 15 W/ m2. K and the physical
properties of meat are given as: p = 1076kg/m3, k = 0.514W/m.K, Cp =
3.431 kJ/ kg. K.

Solution
The diameter of the roast is
(,) 6M 'I3
D=
= [-] 113 = 0.153111

The Biot number is

- (15)(0.153/2)
- = 2.23
0.514
From Eq. (10.2-log), the first root is A1 = 2.101. Considering only the first term
of the series in Eq. (10.2-112) gives
sin(A1 E)
e= sinA1 - A1 cosA1 e- - (3)
A1 - sinA1 cos A1 I
Since

the temperature at the center, T,, is given by

Substitution of the values into Eq. (5) gives
175 - 80 sin 120.4 - 2.101 cos 120.41
175 - 5 =2( 2.101 - sin 120.4~0s 120.4 (6)
in which 2.101 rad = 120.4". Solving for r yields

r = 0.226
Therefore, the time is
rR2
t=

- (0.226) (0. 153/2)2 = 9500s E 2.64h
-
0.514/ [(1076)(3431)]