Page 88 - Bird R.B. Transport phenomena

Problems 73

(b) By making a mass balance over the gauge, develop an approximate relation between
gauge pressure /?, - p 0 and rate of climb v z for a long continued constant-rate climb. Neglect
change of air viscosity, and assume changes in air density to be small.
(c) Develop an approximate expression for the "relaxation time" f rcl of the indicator—that is,
the time required for the gauge pressure to drop to 1 /e of its initial value when the external
pressure is suddenly changed from zero (relative to the interior of the gauge) to some differ-
ent constant value, and maintained indefinitely at this new value.
(d) Discuss the usefulness of this type of indicator for small aircraft.
(e) Justify the plus and minus signs in the figure.
Answers: (a) dp/dz = -pg = -(pM/RT)g
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(b) pi - p 0 ~ v z(8/xL/7rR )(MgV/R^T), where R ? is the gas constant and M is the mole-
cular weight.
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= (m/7r)(/jiVL/7TD p), where p = \(p, + p 0)
(c) t 0
2D.1 Rolling-ball viscometer. An approximate analysis of the rolling-ball experiment has been
8
given, in which the results of Problem 2B.3 are used. Read the original paper and verify the
results.
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2D.2 Drainage of liquids (see Fig. 2D.2). How much liquid clings to the inside surface of a large
vessel when it is drained? As shown in the figure there is a thin film of liquid left behind on
the wall as the liquid level in the vessel falls. The local film thickness is a function of both z
(the distance down from the initial liquid level) and t (the elapsed time).

Initial level of liquid

8(z, t) = thickness of film

I Liquid level moving
I downward with
* speed s
Fig. 2D.2 Clinging of a viscous fluid to wall of
Wall of containing vessel
vessel during draining.

s
H. W. Lewis, Anal. Chem., 25, 507 (1953); R. B. Bird and R. M. Turian, bid. Eng. Chem. Fundamentals,
3, 87 (1964); J. Sestak and F. Ambros, Rheoi Ada, 12, 70-76 (1973).
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J. J. van Rossum, Appl. Sci. Research, A7,121-144 (1958); see also V. G. Levich, Physicochemical
Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ. (1962), Chapter 12.