Page 214 - Bird R.B. Transport phenomena

198 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

unsteady-state systems, these are ordinary differential equations, and for steady-state
systems, they are algebraic equations. The macroscopic balances contain terms that ac-
count for the interactions of the fluid with the solid surfaces. The fluid can exert forces
and torques on the surfaces of the system, and the surroundings can do work W m on the
fluid by means of moving surfaces.
The macroscopic balances can be obtained from the equations of change by integrat-
1 2
ing the latter over the entire volume of the flow system: '

(eq. of continuity) dV = macroscopic mass balance

I (eq. of motion) dV = macroscopic momentum balance
J V(t)
(eq. of angular momentum) dV = macroscopic angular momentum balance
•>V(t)
(eq. of mechanical energy) dV = macroscopic mechanical energy balance
JV(t)
The first three of these macroscopic balances can be obtained either by writing the con-
servation laws directly for the macroscopic system or by doing the indicated integra-
tions. However, to get the macroscopic mechanical energy balance, the corresponding
equation of change must be integrated over the macroscopic system.
In §§7.1 to 7.3 we set up the macroscopic mass, momentum, and angular momentum
balances by writing the conservation laws. In §7.4 we present the macroscopic mechani-
cal energy balance, postponing the detailed derivation until §7.8. In the macroscopic me-
chanical energy balance, there is a term called the "friction loss," and we devote §7.5 to
estimation methods for this quantity. Then in §7.6 and §7.7 we show how the set of
macroscopic balances can be used to solve flow problems.
The macroscopic balances have been widely used in many branches of engineering.
They provide global descriptions of large systems without much regard for the details of
the fluid dynamics inside the systems. Often they are useful for making an initial ap-
praisal of an engineering problem and for making order-of-magnitude estimates of vari-
ous quantities. Sometimes they are used to derive approximate relations, which can then
be modified with the help of experimental data to compensate for terms that have been
omitted or about which there is insufficient information.
In using the macroscopic balances one often has to decide which terms can be omit-
ted, or one has to estimate some of the terms. This requires (i) intuition, based on experi-
ence with similar systems, (ii) some experimental data on the system, (iii) flow
visualization studies, or (iv) order-of-magnitude estimates. This will be clear when we
come to specific examples.
The macroscopic balances make use of nearly all the topics covered thus far; there-
fore Chapter 7 provides a good opportunity for reviewing the preceding chapters.

§7.1 THE MACROSCOPIC MASS BALANCE

In the system shown in Fig. 7.0-1 the fluid enters the system at plane 1 with cross section
Si and leaves at plane 2 with cross section S 2. The average velocity is (vi) at the entry
plane and (v 2) at the exit plane. In this and the following sections, we introduce two as-
sumptions that are not very restrictive: (i) at the planes 1 and 2 the time-smoothed veloc-

1
R. B. Bird, Chem. Eng. Sci., 6,123-131 (1957); Chem. Eng. Educ, 27(2), 102-109 (Spring 1993).
2
J. C. Slattery and R. A. Gaggioli, Chem. Eng. Sci., 17', 893-895 (1962).