Page 67 - Modelling in Transport Phenomena A Conceptual Approach

48 CHAPTER 3. EVTER.F'PHASE TRANSPORT

The entire resistance to heat transfer is assumed to be due to a stagnant film
in the fluid next to the wall. The thickness of the film, &, is such that it provides
the same resistance to heat transfer as the resistance that exists for the actual
convection process. The temperature gradient in the film is constant and is equal
to
(3.2- 10)

Substitution of Eq. (3.2-10) into Eq. (3.2-9) gives

(h=$( (3.2- 1 1)

Equation (3.211) indicates that the thickness of the film, at, determines the value
of h. For this reason the term h is frequently referred to as the film heat transfer
coefficient.

Example 3.2 Energy generation rate per unit volume as a result of fission within
a spherical reactor of radius R is given as a finction of position as
?I? [1- (:,"I
R0
=
where r is the radial distance measured from the center of the sphere. Cooling fluid
at a temperature of Tw Bows ouer the reactor. If the average heat transfer coefi-
cient (h) at the surface of the reactor is known, determine the surface tempemture
of the reactor ut steady-state.

Solution
System: Reactor
Analysis

The inventory rate equation for energy becomes
Rate of energy out = Rate of energy generation (1)
The rate at which energy leaves the sphere by convection is given by Newton's law
of cooling as
Rate of energy out = (47rR2)(h) (T, - T,) (2)
where T,,, is the surface temperature of the sphere.
The rate of energy generation can be determined by integrating W over the vol-
ume of the sphere. The result is
'1
Rate of energy generation = Jd2T Jdr Jd" Ro [1- (G) r2 sin 8 drd8d4

8lr
= (3)