Page 238 - Bird R.B. Transport phenomena

222 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

The term on the left side can be interpreted as the time rate of change of the total ki-
netic and potential energy CK tot + <£> tot) within the "control volume/' whose shape and
volume are changing with time.
We next examine one by one the five terms on the right side:
Term 1 (including the minus sign) contributes only at the entry and exit ports and
gives the rates of influx and efflux of kinetic and potential energy:
Term 1 = {\РМ)$Х + Pi*i<Ui>Si) - (b (4)S 2 + p2^i{v )S ) (7.8-3)
2
2
2
The angular brackets indicate an average over the cross section. To get this result we
have to assume that the fluid density and potential energy per unit mass are constant
over the cross section, and that the fluid is flowing parallel to the tube walls at the entry
and exit ports. The first term in Eq. 7.8-3 is positive, since at plane 1, (—n • v) = (щ •
(гвд)) = v v and the second term is negative, since at plane 2, (—n • v) = (—u • (VL V )) = —v .
2
2 2
2
Term 2 (including the minus sign) gives no contribution on Sf since v is zero there.
On each surface element dS of S there is a force —npdS acting on a surface moving with
m
a velocity v, and the dot product of these quantities gives the rate at which the surround-
ings do work on the fluid through the moving surface element dS. We use the symbol
W^ } to indicate the sum of all these surface terms. Furthermore, the integrals over the
stationary surfaces S and S give the work required to push the fluid into the system at
2
2
plane 1 minus the work required to push the fluid out of the system at plane 2. Therefore
term 2 finally gives
Term 2 = p^S, - p (v )S 2 + Wf (7.8-4)
2
2
Here we have assumed that the pressure does not vary over the cross section at the entry
and exit ports.
Term 3 (including the minus sign) gives no contribution on S^ since v is zero there.
The integral over S can be interpreted as the rate at which the surroundings do work on
m
the fluid by means of the viscous forces, and this integral is designated as W^. At the
entry and exit ports it is conventional to neglect the work terms associated with the vis-
cous forces, since they are generally quite small compared with the pressure contribu-
tions. Therefore we get
Term 3 = W™ (7.8-5)
We now introduce the symbol W m = W^ } + W^ } to represent the total rate at which
the surroundings do work on the fluid within the system through the agency of the mov-
ing surfaces.
Terms 4 and 5 cannot be further simplified, and hence we define

Term 4 = + I p(V • v) dV = -E c (7.8-6)
V(t)

Term 5 = + j (T:VV) dV = -E (7.8-7)
v
V(t)
For Newtonian fluids the viscous loss E v is the rate at which mechanical energy is irre-
versibly degraded into thermal energy because of the viscosity of the fluid and is always
a positive quantity (see Eq. 3.3-3). We have already discussed methods for estimating E v
in §7.5. (For viscoelastic fluids, which we discuss in Chapter 8, E has to be interpreted
v
differently and may even be negative.) The compression term E is the rate at which me-
c
chanical energy is reversibly changed into thermal energy because of the compressiblity
of the fluid; it may be either positive or negative. If the fluid is being regarded as incom-
pressible, then E is zero.
c