Page 380 - Modelling in Transport Phenomena A Conceptual Approach

360 CHAPTER 9. STEADY MICROSCOPIC BALANCES WTH GEN.

Equation (4) gives the surface temperature as
2 RoR
TR = T, + - - (5)
‘
15 (h)
Another way of calculating the surface temperature is to equate Newton’s law of
cooling and Fourier’s law of heat conduction at the surface of the sphere, i-e.,

(h) (TR -T,) = -k -
Ir=R
Prom Eq. (3)

Substituting Eq. (7) into Eq. (6) and solving for TR results in Eq. (5).
Therefore, the tempemture distribution within the reactor in tern of the known
quantities is given by

2 R,R 7 %,R2 RoR2 1 r 2 1 r 4
T=T,+-- 15 (h) +---- 2k [?(d -E(d] (8)
60 k
The rate of heat loss can be calculated from Eq. (9.2-73) as
Qloss = 47r R, 1 [I - ( i)2]
R
dr
r2
8n
R,
= - R3 (9)
15
Note that the calculation of the rate of heat loss does not require the temperature
distribution to be known.

9.3 HEAT TRANSFER WITH CONVECTION

9.3.1 Laminar Flow Forced Convection in a Pipe

Consider the laminar flow of an incompressible Newtonian fluid in a circular pipe
under the action of a pressure gradient as shown in Figure 9.9. The velocity
distribution is given by Eqs. (9.1-79) and (9.1-84) as

(9.3-1)

Suppose that the fluid, which is at a uniform temperature of To for z < 0, is
started to be heated for z > 0 and we want to develop the governing equation for
temperature.