Page 360 - Bird R.B. Transport phenomena

342 Chapter 11 The Equations of Change for Nonisothermal Systems

To illustrate the solution of problems in which the energy equation plays a signifi-
cant role, we solve a series of (idealized) problems. We restrict ourselves here to steady-
state flow problems and consider unsteady-state problems in Chapter 12. In each
problem we start by listing the postulates that lead us to simplified versions of the equa-
tions of change.

EXAMPLE 11.4-1 Show how to set up the equations for the problem considered in §10.8—namely, that of find-
ing the fluid temperature profiles for the fully developed laminar flow in a tube.
Steady-State Forced-
Convection Heat SOLUTION
Transfer in Laminar
Flow in a Circular Tube ^ e a s s u m e constant physical properties, and we postulate a solution of the following form:
v = b v (r), & = ^(2), and T = T(r, 2). Then the equations of change, as given in Appendix B,
z z
may be simplified to
Continuity: 0 - 0 (11.4-1)
—f
Motion: (11.4-2)

Energy: (11.4-3)

The equation of continuity is automatically satisfied as a result of the postulates. The equation
of motion, when solved as in Example 3.6-1, gives the velocity distribution (the parabolic ve-
locity profile). This expression is then substituted into the convective heat transport term on
the left side of Eq. 11.4-3 and into the viscous dissipation heating term on the right side.
Next, as in §10.8, we make two assumptions: (i) in the 2 direction, heat conduction is
2
much smaller than heat convection, so that the term d T/dz 2 can be neglected, and (ii) the
flow is not sufficiently fast that viscous heating is significant, and hence the term \x(dvjdr) 1
can be omitted. When these assumptions are made, Eq. 11.4-3 becomes the same as Eq. 10.8-
12. From that point on, the asymptotic solution, valid for large 2 only, proceeds as in §10.8.
Note that we have gone through three types of restrictive processes: (i) postulates, in which
a tentative guess is made as to the form of the solution; (ii) assumptions, in which we elimi-
nate some physical phenomena or effects by discarding terms or assuming physical proper-
ties to be constant; and (iii) an asymptotic solution, in which we obtain only a portion of the
entire mathematical solution. It is important to distinguish among these various kinds of
restrictions.

EXAMPLE 11.4-2 Determine the temperature distribution in an incompressible liquid confined between two
coaxial cylinders, the outer one of which is rotating at a steady angular velocity fl 0 (see §10.4
Tangential Flow in an and Example 3.6-3). Use the nomenclature of Example 3.6-3, and consider the radius ratio к to
Annulus with Viscous be fairly small so that the curvature of the fluid streamlines must be taken into account.
Heat Generation The temperatures of the inner and outer surfaces of the annular region are maintained at
T and T u respectively, with Т ФТ . Assume steady laminar flow, and neglect the tempera-
K
К
Х
ture dependence of the physical properties.
This is an example of a forced convection problem: The equations of continuity and mo-
tion are solved to get the velocity distribution, and then the energy equation is solved to get
the temperature distribution. This problem is of interest in connection with heat effects in
coaxial cylinder viscometers 1 and in lubrication systems.

J. R. Van Wazer, J. W. Lyons, K. Y. Kim, and R. E. Colwell, Viscosity and Flow Measurement, Wiley,
1
New York (1963), pp. 82-85.