Page 119 - Bird R.B. Transport phenomena

104 Chapter 3 The Equations of Change for Isothermal Systems

QUESTIONS FOR DISCUSSION
1. What is the physical meaning of the term Ax &y(pv )\ in Eq. 3.1-2? What is the physical mean-
z
z
ing of (V • v)? of (V • pv)?
2. By making a mass balance over a volume element (Ar)(rA0)(Az) derive the equation of conti-
nuity in cylindrical coordinates.
3. What is the physical meaning of the term Ax hy(pv v )\ z in Eq. 3.2-2? What is the physical
z x
meaning of [V • pvv]?
4. What happens when / is set equal to unity in Eq. 3.5-4?
5. Equation В in Table 3.5-1 is not restricted to fluids with constant density, even though p is to
the left of the substantial derivative. Explain.
6. In the tangential annular flow problem in Example 3.5-3, would you expect the velocity pro-
files relative to the inner cylinder to be the same in the following two situations: (i) the inner
cylinder is fixed and the outer cylinder rotates with an angular velocity П; (ii) the outer cylin-
der is fixed and the inner cylinder rotates with an angular velocity -П? Both flows are pre-
sumed to be laminar and stable.
7. Suppose that, in Example 3.6-4, there were two immiscible liquids in the rotating beaker.
What would be the shape of the interface between the two liquid regions?
8. Would the system discussed in Example 3.6-5 be useful as a viscometer?
9. In Eq. 3.6-55, explain by means of a carefully drawn sketch the choice of limits in the integra-
tion and the meaning of each factor in the first integrand.
10. What factors would need to be taken into account in designing a mixing tank for use on the
moon by using data from a similar tank on earth?

PROBLEMS

3A.1 Torque required to turn a friction bearing (Fig. and rests in a series of sleeve bearings that give a 0.005 in.
ЗАЛ). Calculate the required torque in Щ • ft and power clearance. The shaft rotates at 50 rpm, the lubricant has a
consumption in horsepower to turn the shaft in the friction viscosity of 5000 cp, and there are 20 bearings, each 1 ft in
bearing shown in the figure. The length of the bearing sur- length. Estimate the fraction of engine power expended in
face on the shaft is 2 in, and the shaft is rotating at 200 rotating the shafts in their bearings. Neglect the effect of
rpm. The viscosity of the lubricant is 200 cp, and its den- the eccentricity.
3
sity is 50 lb /ft . Neglect the effect of eccentricity. Answer: 0.115
w
Answers: 0.32 lb • ft; 0.012 hp = 0.009 kW
;
3A.3 Effect of altitude on air pressure. When standing
at the mouth of the Ontonagon River on the south shore of
Lake Superior (602 ft above mean sea level), your portable
barometer indicates a pressure of 750 mm Hg. Use the
equation of motion to estimate the barometric pressure at
the top of Government Peak (2023 ft above mean sea level)
in the nearby Porcupine Mountains. Assume that the tem-
perature at lake level is 70°F and that the temperature de-
creases with increasing altitude at a steady rate of 3°F per
1000 feet. The gravitational acceleration at the south shore
Fig. ЗАЛ. Friction of Lake Superior is about 32.19 ft/s, and its variation with
bearing.
altitude may be neglected in this problem.
4
Answer: 712 mm Hg = 9.49 X 10 N/nr
3A.2 Friction loss in bearings. 1 Each of two screws on a
large motor-ship is driven by a 4000-hp engine. The shaft 3A.4 Viscosity determination with a rotating-cylinder
that connects the motor and the screw is 16 in. in diameter viscometer. It is desired to measure the viscosities of su-
crose solutions of about 60% concentration by weight at
about 20°C with a rotating-cylinder viscometer such as
1 that shown in Fig. 3.5-1. This instrument has an inner
This problem was contributed by Prof. E. J. Crosby,
University of Wisconsin. cylinder 4.000 cm in diameter surrounded by a rotating