Page 239 - Bird R.B. Transport phenomena

§7.8 Derivation of the Macroscopic Mechanical Energy Balance 223

When all the contributions are inserted into Eq. 7.8-2 we finally obtain the macro-
scopic mechanical energy balance:

| (X tot + O tot) = (b

+ (v 2)S 2) + W m E c - (7.8-8)
-
& ) S 2 p 2 E v
If, now, we introduce the symbols w x = p^{v^)S A and zv 2 = p 2(v 2)S 2 for the mass rates of
flow in and out, then Eq. 7.8-8 can be rewritten in the form of Eq. 7.4-2. Several assump-
tions have been made in this development, but normally they are not serious. If the situ-
ation warrants, one can go back and include the neglected effects.
It should be noted that the above derivation of the mechanical energy balance does
not require that the system be isothermal. Therefore the results in Eqs. 7.4-2 and 7.8-8 are
valid for nonisothermal systems.
To get the mechanical energy balance in the form of Eq. 7.4-7 we have to develop an
approximate expression for E c. We imagine that there is a representative streamline run-
ning through the system, and we introduce a coordinate s along the streamline. We as-
sume that pressure, density, and velocity do not vary over the cross section. We further
imagine that at each position along the streamline, there is a cross section S(s) perpendic-
ular to the s-coordinate, so that we can write dV = S(s)ds. If there are moving parts in the
system and if the system geometry is complex, it may not be possible to do this.
We start by using the fact that (V • pv) = 0 at steady state so that

V
Ec = ~j p(V • v) dV = + j - (v • Vp) dV (7.8-9)
V V
Then we use the assumption that the pressure and density are constant over the cross
section to write approximately

, )S(s)ds (7.8-10)
as/
Even though p, v, and S are functions of the streamline coordinate s, their product, w = pvS,
is a constant for steady-state operation and hence may be taken outside the integral. This
gives

2
\
i p ds J i ds P/
Then an integration by parts can be performed:
2
2 f2 -i dp d s I = w A /w\ JrW f i (7 8 12)
v
i " Ji ^ J " W )M *"
When this result is put into Eq. 7.4-5, the approximate relation in Eq. 7.4-7 is obtained. Be-
cause of the questionable nature of the assumptions made (the existence of a representative
streamline and the constancy of p and p over a cross section), it seems preferable to use Eq.
7.4-5 rather than Eq. 7.4-7. Also, Eq. 7.4-5 is easily generalized to systems with multiple inlet
and outlet ports, whereas Eq. 7.4-7 is not; the generalization is given in Eq. (D) of Table 7.6-1.

QUESTIONS FOR DISCUSSION
1. Discuss the origin, meaning, and use of the macroscopic balances, and explain what assump-
tions have been made in deriving them.
2. How does one decide which macroscopic balances to use for a given problem? What auxiliary
information might one need in order to solve problems with the macroscopic balances?
3. Are friction factors and friction loss factors related? If so, how?