Page 478 - Bird R.B. Transport phenomena

458 Chapter 15 Macroscopic Balances for Nonisothermal Systems

balance, which is a statement of the law of conservation of energy. Therefore, in general,
both balances are needed for problem solving. The mechanical energy balance is not "an
alternative form" of the energy balance.
In fact, if we subtract the mechanical energy balance in Eq. 15.2-1 from the total en-
ergy balance in Eq. 15.1-2 we get the macroscopic balance for the internal energy

— = -Ш-w + Q + E + E (15.2-6)
dt c v
This states that the total internal energy in the system changes because of the difference
in the amount of internal energy entering and leaving the system by fluid flow, because
of the heat entering (or leaving) the system through walls of the system, because of
the heat produced (or consumed) within the fluid by compression (or expansion), and
because of the heat produced in the system because of viscous dissipation heating.
Equation 15.2-6 cannot be written a priori, since there is no conservation law for inter-
nal energy. It can, however, be obtained by integrating Eq. 11.2-1 over the entire flow
system.

§15.3 USE OF THE MACROSCOPIC BALANCES
TO SOLVE STEADY-STATE PROBLEMS
WITH FLAT VELOCITY PROFILES
The most important applications of the macroscopic balances are to steady-state prob-
lems. Furthermore, it is usually assumed that the flow is turbulent so that the variation
of the velocity over the cross section can be safely neglected (see "Notes" after Eqs. 7.2-3
and 7.4-7). The five macroscopic balances, with these additional restrictions, are summa-
rized in Table 15.3-1. They have been generalized to multiple inlet and outlet ports to ac-
commodate a larger set of problems.

Table 15.3-1 Steady-State Macroscopic Balances for Turbulent Flow in Nonisothermal Systems
w
Mass: 2 i ~ 2^2 = 0 (A)

и
n
Momentum: S^i^i + P\S\) \ ~ 2(^2^2 + Р2$г) 2 + w g = F ^ (B)
tot s
Angular momentum: S f c w + ^7 S )[r 1 X u j - ^(v iv 2 + p S )[r 2 x u ] + T exfc = T ^ s (C)
1 1
2
2
2
2
Mechanical energy: ^ ( ^ + gh } + ^ - 2 ( ^ 2 + gh 2 + )w 2 = - W + E + E v (D)
g
m
c
(Total) energy: 2&>i + %K + H > i " 2&>2 + gh 2 + H )ii7 = - W - Q (E)
2
2
ni
Notes:
n
All formulas here imply flat velocity profiles.
** £u>i = w la + w ]b + w [c + • • •, where w ]n = p ]nv hlS bl, and so on.
c
h-i and h 2 are elevations above an arbitrary datum plane.
li
H} and H 2 are enthalpies per unit mass relative to some arbitrarily chosen reference state (see Eq. 9.8-8).
c
All equations are written for compressible flow; for incompressible flow, E c = 0. The quantities E c and
E v are defined in Eqs. 7.3-3 and 4.
• Uj and u 2 are unit vectors in the direction of flow.