Page 220 - Bird R.B. Transport phenomena

204 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

Here K tot — f\pv dV and O tot = /рФ dV are the total kinetic and potential energies within
2
the system. According to Eq. 7.4-1, the total mechanical energy (i.e., kinetic plus poten-
tial) changes because of a difference in the rates of addition and removal of mechanical
energy, because of work done on the fluid by the surroundings, and because of com-
pressibility effects and viscous dissipation. Note that, at the system entrance (plane 1),
the force p^S multiplied by the velocity (у ) gives the rate at which the surroundings do
г
x
work on the fluid. Furthermore, W m is the work done by the surroundings on the fluid
by means of moving surfaces.
The macroscopic mechanical energy balance may now be written more compactly as

+ O ) = -A i V f + Ф + n \v> + W w - E - E v (7.4-2)
tot
c
in which the terms E and E are defined as follows:
c
v
E =- f p(V- v) dV and E v = - j (T:VV) dV (7.4-3,4)
c
V(t) V(t)
The compression term E is positive in compression and negative in expansion; it is zero
c
when the fluid is assumed to be incompressible. The term E is the viscous dissipation (or
v
friction loss) term, which is always positive for Newtonian liquids, as can be seen from Eq.
3.3-3. (For polymeric fluids, which are viscoelastic, E v is not necessarily positive; these
fluids are discussed in the next chapter.)
If the total kinetic plus potential energy in the system is not changing with time, we get

(7.4-5)

which is the steady-state macroscopic mechanical energy balance. Here h is the height above
some arbitrarily chosen datum plane.
Next, if we assume that it is possible to draw a representative streamline through
the system, we may combine the A(p/p) and E terms to get the following approximate re-
c
lation (see §7.8)
\ dp (7.4-6)

Then, after dividing Eq. 7.4-5 by w = w = w, we get
x 2
(7.4-7)

Here ру„ = W /w and E v = E /w. Equation 7.4-7 is the version of the steady-state me-
m
v
chanical energy balance that is most often used. For isothermal systems, the integral
term can be calculated as long as an expression for density as a function of pressure is
available.
Equation 7.4-7 should now be compared with Eq. 3.5-12, which is the "classical"
Bernoulli equation for an inviscid fluid. If, to the right side of Eq. 3.5-12, we just add the
work W done by the surroundings and subtract the viscous dissipation term E and rein-
m v/
terpret the velocities as appropriate averages over the cross sections, then we get Eq. 7.4-7.
This provides a "plausibility argument" for Eq. 7.4-7 and still preserves the fundamental
idea that the macroscopic mechanical energy balance is derived from the equation of mo-
tion (that is, from the law of conservation of momentum). The full derivation of the macro-
scopic mechanical energy balance is given in §7.8 for those who are interested.
3
3
Notes for turbulent flow: (i) For turbulent flows we replace (v ) by (Г; ), and ignore the
contribution from the turbulent fluctuations, (ii) It is common practice to replace the