Page 316 - Bird R.B. Transport phenomena

300 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

The two integration constants are determined from the boundary conditions
B.C.1: atx = 0, T = T 0 (10.4-7)
B.C. 2: atx = b, T = T b (10.4-8)
This yields finally, for T Ф T o
b
£rJ-H'-f) + j|
n

1
Here Br = /jLvl/k(T b — T o) is the dimensionless Brinkman number, which is a measure of
the importance of the viscous dissipation term. If T b = T o, then Eq. 10.4-9 can be written
as

and the maximum temperature is at x/b = \.
If the temperature rise is appreciable, the temperature dependence of the viscosity
has to be taken into account. This is discussed in Problem 10C.1.
2
The viscous heating term S v = ix{v b/b) may be understood by the following argu-
ments. For the system in Fig. 10.4-2, the rate at which work is done is the force acting on
the upper plate times the velocity with which it moves, or (-r Y2WL)(^). The rate of en-
ergy addition per unit volume is then obtained by dividing this quantity by WLb, which
2
gives (-T xzv b/b) = fi(v b/b) . This energy all appears as heat and is hence S v.
In most flow problems viscous heating is not important. However if there are large
velocity gradients, then it cannot be neglected. Examples of situations where viscous
heating must be accounted for include: (i) flow of a lubricant between rapidly moving
parts, (ii) flow of molten polymers through dies in high-speed extrusion, (iii) flow of
highly viscous fluids in high-speed viscometers, and (iv) flow of air in the boundary
layer near an earth satellite or rocket during reentry into the earth's atmosphere. The
first two of these are further complicated because many lubricants and molten plastics
are non-Newtonian fluids. Viscous heating for non-Newtonian fluids is illustrated in
Problem 10B.5.

§10.5 HEAT CONDUCTION WITH A CHEMICAL HEAT SOURCE
A chemical reaction is being carried out in a tubular, fixed-bed flow reactor with inner
radius R as shown in Fig. 10.5-1. The reactor extends from z — — o° to z = +°° and is di-
vided into three zones:
Zone I: Entrance zone packed with noncatalytic spheres
Zone II: Reaction zone packed with catalyst spheres, extending from z = 0 to z = L
Zone III: Exit zone packed with noncatalytic spheres
It is assumed that the fluid proceeds through the reactor tube in "plug flow"—that is,
with axial velocity uniform at a superficial value v Q = W/TTR P (see text below Eq. 6.4-1
2
for the definition of "superficial velocity"). The density, mass flow rate, and superficial

1
H. C. Brinkman, Appl Sci. Research, A2,120-124 (1951), solved the viscous dissipation heating
problem for the Poiseuille flow in a circular tube. Other dimensionless groups that may be used for
characterizing viscous heating have been summarized by R. B. Bird, R. C. Armstrong, and O. Hassager,
Dynamics of Polymeric Liquids, Vol. 1, 2nd edition, Wiley, New York (1987), pp. 207-208.