Page 232 - Bird R.B. Transport phenomena

216 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

will be assumed to be flat. In Fig. 7.6-5(b) we show an approximate velocity profile at plane 2,
which we use in the application of the macroscopic balances. The standard orifice meter equa-
tion is obtained by applying the macroscopic mass and mechanical energy balances.
SOLUTION (a) Mass balance. For a fluid of constant density with a system for which S } = S 2 = S, the
mass balance in Eq. 7.1-1 gives
<0i> = (v ) (7.6-41)
2
With the assumed velocity profiles this becomes

v, = jv o (7.6-42)

and the volume rate of flow is w = pv S.
}
(b) Mechanical energy balance. For a constant-density fluid in a flow system with no eleva-
tion change and no moving parts, Eq. 7.4-5 gives

2 ( ) 2 ( ) P
V
The viscous loss E v is neglected, even though it is certainly not equal to zero. With the as-
sumed velocity profiles, Eq. 7.6-43 then becomes

When Eqs. 7.6-42 and 44 are combined to eliminate v , we can solve for v ] to get
0
T

We can now multiply by pS to get the volume rate of flow. Then to account for the errors in-
troduced by neglecting E and by the assumptions regarding the velocity profiles we include
v
a discharge coefficient, C , and obtain
d

Experimental discharge coefficients have been correlated as a function of S /S and the
o
2
4
Reynolds number. For Reynolds numbers greater than 10 , C approaches about 0.61 for all
d
practical values of S /S.
o
This example has illustrated the use of the macroscopic balances to get the general form of
the result, which is then modified by introducing a multiplicative function of dimensionless
groups to correct for errors introduced by unwarranted assumptions. This combination of
macroscopic balances and dimensional considerations is often used and can be quite useful.
§7.7 USE OF THE MACROSCOPIC BALANCES
FOR UNSTEADY-STATE PROBLEMS

In the preceding section we have illustrated the use of the macroscopic balances for solv-
ing steady-state problems. In this section we turn our attention to unsteady-state prob-
lems. We give two examples to illustrate the use of the time-dependent macroscopic
balance equations.

2
G. L. Tuve and R. E. Sprenkle, Instruments, 6, 202-205, 225, 232-234 (1935); see also R. H. Perry and
С. Н. Chilton, Chemical Engineers' Handbook, McGraw-Hill, New York, 5th edition (1973), Fig. 5-18; Fluid
Meters: Their Theory and Applications, 6th edition, American Society of Mechanical Engineers, New York
(1971), pp. 58-65; Measurement of Fluid Flow Using Small Bore Precision Orifice Meters, American
Society of Mechanical Engineers, MFC-14-M, New York (1995).