Page 246 - Bird R.B. Transport phenomena

230 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

7C.1 End corrections in tube viscometers (Fig. 7C.1). 3 In analyzing tube-flow viscometric data to
determine viscosity, one compares pressure drop versus flow rate data with the theoretical
expression (the Hagen-Poiseuille equation of Eq. 2.3-21). The latter assumes that the flow is
fully developed in the region between the two planes at which the pressure is measured. In
an apparatus such as that shown in the figure, the pressure is known at the tube exit (2) and
also above the fluid in the reservoir (1). However, in the entrance region of the tube, the
velocity profiles are not yet fully developed. Hence the theoretical expression relating the
pressure drop to the flow rate is not valid.
There is, however, a method in which the Hagen-Poiseuille equation can be used, by
making flow measurements in two tubes of different lengths, L and L ; the shorter of the two
B
A
tubes must be long enough so that the velocity profiles are fully developed at the exit. Then
the end section of the long tube, of length L - L , will be a region of fully developed flow. If
B
A
(
>
we knew the value of ^? - 3 for this region, then we could apply the Hagen-Poiseuille
A
0
equation.
Show that proper combination of the mechanical energy balances, written for the sys-
tems 1-2, 3-4, and 0-4 gives the following expression for 9> - 9> when each viscometer has
4
0
the same flow rate.
(7C.1-1)
L
where 2P = p 0 + pgz . Explain carefully how you would use Eq. 7C.1-1 to analyze experimen-
0
0
tal measurements. Is Eq. 7C.1-1 valid for ducts with noncircular, uniform cross section?
Run Л Run В

Fig. 7C.1. Two tube viscometers with
the same flow rate and the same exit
pressure. The pressures p A and p are
B
'- Plane 4 maintained by an inert gas.

7D.1 Derivation of the macroscopic balances from the equations of change. Derive the macro-
scopic mass and momentum balances by integrating the equations of continuity and motion
over the flow system of Fig. 7.0-1. Follow the procedure given in §7.8 for the macroscopic me-
chanical energy balance, using the Gauss divergence theorem and the Leibniz formula.

3 A. G. Fredrickson, PhD Thesis, University of Wisconsin (1959); Principles and Applications of
Rheology, Prentice-Hall, Englewood Cliffs, N.J. (1964), §9.2.