Page 361 - Bird R.B. Transport phenomena

§11.4 Use of the Equations of Change to Solve Steady-State Problems 343

SOLUTION We begin by postulating that v = & v (r), that 9> = 9>(r, z), and that T = T(r). Then the simplifi-
e 0
cation of the equations of change leads to Eqs. 3.6-20, 21, and 22 (the г-, в-, and z-components
of the equation of motion), and the energy equation

+ а г— — (11.4-4)

When the solution to the ^-component of the equation of motion, given in Eq. 3.6-29, is substi-
tuted into the energy equation, we get

(11.4-5)
dr \ dr (1 - к ) г 4
2 2
This is the differential equation for the temperature distribution. It may be rewritten in terms
of dimensionless quantities by putting

«-я 0 = т-т к N = WT, - T ) (1 - к ) (11.4-6,7,8)
2 2
K
The parameter N is closely related to the Brinkman number of §10.4. Equation 11.4-5 now
becomes

(11.4-9)

This is of the form of Eq. C.l-11 and has the solution

c (11.4-10)
2
The integration constants are found from the boundary conditions
B.C1: atf = #c, 0 = 0 (11.4-11)
B.C. 2: at f = 1, 0 = 1 (11.4-12)
Determination of the constants then leads to
In
0 = 1 - (11.4-13)
In к
When N = 0, we obtain the temperature distribution for a motionless cylindrical shell of
thickness R(l - к) with inner and outer temperatures T and 7V If N is large enough, there
K
will be a maximum in the temperature distribution, located at
2 In (1/ic)
(11.4-14)

with the temperature at this point greater than either T or TV
K
Although this example provides an illustration of the use of the tabulated equations of
change in cylindrical coordinates, in most viscometric and lubrication applications the clear-
ance between the cylinders is so small that numerical values computed from Eq. 11.4-13 will
not differ substantially from those computed from Eq. 10.4-9.

EXAMPLE 11.4-3 A liquid is flowing downward in steady laminar flow along an inclined plane surface, as
shown in Figs. 2.2-1 to 3. The free liquid surface is maintained at temperature T , and the solid
Steady Flow in a surface at x = 8 is maintained at T . At these temperatures the liquid viscosity o has values /x
0
8
Nonisothermal Film and /UL^, respectively, and the liquid density and thermal conductivity may be assumed con-
stant. Find the velocity distribution in this nonisothermal flow system, neglecting end effects