Page 378 - Modelling in Transport Phenomena A Conceptual Approach

358 CHAPTER 9. STEADY MICROSCOPIC BACANCES WITH GEN.

9.2.3.2 Solid sphere

Consider a solid sphere of radius R with a constant surface temperature of TR.
The solution obtained for a hollow sphere, Eq. (9.2-55) is also valid for this case.
However, since the temperature must have a finite value at the center, i.e., r = 0,
then 15'1 must be zero and the temperature distribution becomes
Jd' k(T) dT = - Jd' $ [Jd' %(u) u2 du] dr + cz (9.2-67)

The use of the boundary condition
at r=R T=TR (9.2-68)
gives the solution in the form
6 $ (9.2-69)
k(T) dT = lR [Jd' R(u) u2 du] dr

W Case (i) k= constant
Simplification of Eq. (9.2-69) gives

(9.2-70)

Case (ii) k = constant; R = constant
In this case Eq. (9.2-69) simplifies to
T=TR+- R R2 [1-(;)']

6k (9.2-71)
which implies that the variation of temperature with respect to the radial position
is parabolic with the maximum temperature at the center of the sphere.

Macroscopic equation
The integration of the governing equation, &. (9.2-53) over the volume of the
system gives the macroscopic energy balance as
Jd"
- 1'" /" 1" $ -$ (r2k g) r2 sinBdrdOd4 = 1'" Jd" R r2 sin 8 drd8d4
(9.2-72)
Integration of Eq. (9.2-72) yields
(-k%) 4nR2=4x1 R Rr'dr
(9.2-73)
A-
Rate of energy out Rate of energy
generation