Page 470 - Modelling in Transport Phenomena A Conceptual Approach
P. 470
450 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.
10.2.3 Heating of a Spherical Particle
A spherical particle of radius R is initially at a uniform temperature of To. At t = 0
It
it is exposed to a fluid of temperature T, (T, 7 To). is required to determine
the amount of heat transferred to the spherical particle.
Figure 10.3 Heating of a spherical particle.
Since the heat transfer taka place in the r-direction, Table C.6 in Appendix
C indicates that the only non-zero energy flux component is e,. and it is given by
aT
e, = 4,. = - k - (lO.ZS8)
dr
For a spherical differential volume of thickness Ar, as shown in Figure 10.3, &.
(10.21) is expressed as
a
-
qr(,. 47rr2 - qr(,+*,. 4~(r + AT)^ = - k~~~Arp&p(T Trej)] (10.289)
dt
Dividing Eq. (10.2-89) by 47rAr and letting AT -+ 0 gives
(10.2-90)
e aT
1 a(r2qr)
pep-=--- (10.2-91)
at + &
Substitution of Eq. (10.2-88) into Eq. (10.2-91) gives the governing differential
equation for temperature as
(10.2-92)
