C2.4 A student sneaks a glass of cola onto a roller coaster ride.
The glass is cylindrical, twice as tall as it is wide, and filled to the brim. He wants to know what percent of the cola he should drink before the ride begins, so that none of it spills during the big drop, in which the roller coaster achieves 0.55-g acceleration at a 45 angle below the horizontal.
Make the calculation for him, neglecting sloshing and assuming that the glass is vertical at all times.
C2.5 Dry adiabatic lapse rate (DALR) is defined as the nega- tive value of atmospheric temperature gradient, dT/dz, when temperature and pressure vary in an isentropic fash- ion. Assuming air is an ideal gas, DALR dT/dz when T T0(p/p0)a, where exponent a (k 1)/k, k cp/cv
is the ratio of specific heats, and T0 and p0are the tem- perature and pressure at sea level, respectively. (a) Assum- ing that hydrostatic conditions exist in the atmosphere, show that the dry adiabatic lapse rate is constant and is given by DALR g(k 1)/(kR), where R is the ideal gas constant for air. (b) Calculate the numerical value of DALR for air in units of C/km.
C2.6 In “soft” liquids (low bulk modulus ), it may be neces- sary to account for liquid compressibility in hydrostatic calculations. An approximate density relation would be
where a is the speed of sound and (p0, 0) are the condi- tions at the liquid surface z 0. Use this approximation to show that the density variation with depth in a soft liq- uid is where g is the acceleration of gravity and z is positive upward. Then consider a vertical wall of width b, extending from the surface (z 0) down to depth z h. Find an analytic expression for the hydrostatic force F on this wall, and compare it with the incom- pressible result F 0gh2b/2. Would the center of pressure be below the incompressible position z 2h/3?
C2.7 Venice, Italy, is slowly sinking, so now, especially in winter, plazas and walkways are flooded during storms.
The proposed solution is the floating levee of Fig. C2.7.
0egz/a2 dp
da2 d or pp0a2(0)
When filled with air, it rises to block off the sea. The levee is 30 m high, 5 m wide, and 20 m deep. Assume a uniform density of 300 kg/m3when floating. For the 1-m sea–lagoon difference shown, estimate the angle at which the levee floats.
C2.7
C2.8 What is the uncertainty in using pressure measurement as an altimeter? A gage on the side of an airplane measures a local pressure of 54 kPa, with an uncertainty of 3 kPa.
The estimated lapse rate that day is 0.007 K/m, with an uncertainty of 0.001 K/m. Effective sea-level temperature is 10C, with an uncertainty of 4C. Effective sea-level pressure is 100 kPa, with an uncertainty of 3 kPa. Estimate the airplane’s altitude and its uncertainty.
C2.9 The ALVIN submersible vehicle in the chapter-opener photo has a passenger compartment which is a titanium sphere of inside diameter 78.08 in and thickness 1.93 in. If the vehi- cle is submerged to a depth of 3850 m in the ocean, estimate (a) the water pressure outside the sphere, (b) the maximum elastic stress in the sphere, in lbf/in2, and (c) the factor of safety of the titanium alloy (6% aluminum, 4% vanadium).
Hinge Storm levee filled
with air to float
Adriatic Sea —25 m deep in a st orm Venice lagoon—24 m deep
Filled with water—no storm
References
1. U.S. Standard Atmosphere, 1976, Government Printing Office, Washington, DC, 1976.
2. J. A. Knauss, Introduction to Physical Oceanography, 2nd ed., Waveland Press, Long Grove, IL, 2005.
3. E. C. Tupper, Introduction to Naval Architecture, 4th ed., Elsevier, New York, 2004.
4. D. T. Greenwood, Advanced Dynamics, Cambridge Univer- sity Press, New York, 2006.
5. R. I. Fletcher, “The Apparent Field of Gravity in a Rotating Fluid System,” Am. J. Phys., vol. 40, July 1972, pp. 959–965.
6. National Committee for Fluid Mechanics Films, Illustrated Experiments in Fluid Mechanics,M.I.T. Press, Cambridge, MA, 1972.
7. J. P. Holman, Experimental Methods for Engineers, 7th ed., McGraw-Hill, New York, 2000.
8. R. P. Benedict, Fundamentals of Temperature, Pressure, and Flow Measurement,3d ed., Wiley, New York, 1984.
9. T. G. Beckwith and R. G. Marangoni, Mechanical Measurements, 5th ed., Addison-Wesley, Reading, MA, 1993.
45° 45°
Hinge
Deep water Shallow water
Draft
D2.3
D2.2
b h
Y L
W
Circular arc block Fluid:
Pivot arm Pivot Counterweight
Side view of block face R
D2.2 A laboratory apparatus used in some universities is shown in Fig. D2.2. The purpose is to measure the hydrostatic force on the flat face of the circular-arc block and compare it with the theoretical value for given depth h. The coun- terweight is arranged so that the pivot arm is horizontal when the block is not submerged, whence the weight W can be correlated with the hydrostatic force when the sub- merged arm is again brought to horizontal. First show that the apparatus concept is valid in principle; then derive a formula for W as a function of h in terms of the system parameters. Finally, suggest some appropriate values of Y, L, and so on for a suitable apparatus and plot theoretical Wversus h for these values.
D2.3 The Leary Engineering Company (see Popular Science, November 2000, p. 14) has proposed a ship hull with hinges that allow it to open into a flatter shape when entering shallow water. A simplified version is shown in Fig. D2.3. In deep water, the hull cross section would be triangular, with large draft. In shallow water, the hinges would open to an angle as high as 45. The dashed line indicates that the bow and stern would be closed.
Make a parametric study of this configuration for various , assuming a reasonable weight and center of gravity loca- tion. Show how the draft, the metacentric height, and the ship’s stability vary as the hinges are opened. Comment on the effectiveness of this concept.
10. J. W. Dally, W. F. Riley, and K. G. McConnell, Instrumentation for Engineering Measurements,2d ed., Wiley, New York, 1993.
11. E. N. Gilbert, “How Things Float,” Am. Math. Monthly, vol.
98, no. 3, 1991, pp. 201–216.
12. R. J. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, 4th ed., Wiley, New York, 2005.
13. R. W. Miller, Flow Measurement Engineering Handbook, 3d ed., McGraw-Hill, New York, 1996.
14. L. D. Clayton, E. P. EerNisse, R. W. Ward, and R. B.
Wiggins, “Miniature Crystalline Quartz Electromechanical
Structures,” Sensors and Actuators, vol. 20, Nov. 15, 1989, pp. 171–177.
15. A. Kitai (ed.), Luminescent Materials and Applications, John Wiley, New York, 2008.
16. B. G. Liptak (ed.), Instrument Engineer’s Handbook: Process Measurement and Analysis,4th ed., vol. 1, CRC Press, Boca Raton, FL, 2003.
17. D. R. Gillum, Industrial Pressure, Level and Density Mea- surement, Insrument Society of America, Research Triangle Park, NC, 1995.
138
Table tennis ball suspended by an air jet. The control volume momentum principle, studied in this chapter, requires a force to change the direction of a flow. The jet flow deflects around the ball, and the force is the ball’s weight. (Courtesy of Paul Silverman/Fundamental Photographs.)
139
Motivation. In analyzing fluid motion, we might take one of two paths: (1) seeking to describe the detailed flow pattern at every point (x, y, z) in the field or (2) work- ing with a finite region, making a balance of flow in versus flow out, and determining gross flow effects such as the force or torque on a body or the total energy exchange.
The second is the “control volume” method and is the subject of this chapter. The first is the “differential” approach and is developed in Chap. 4.
We first develop the concept of the control volume, in nearly the same manner as one does in a thermodynamics course, and we find the rate of change of an arbitrary gross fluid property, a result called the Reynolds transport theorem. We then apply this theorem, in sequence, to mass, linear momentum, angular momentum, and energy, thus deriving the four basic control volume relations of fluid mechanics.
There are many applications, of course. The chapter includes a special case of frictionless, shaft-work-free momentum and energy: the Bernoulli equation. The Bernoulli equation is a wonderful, historic relation, but it is extremely restrictive and should always be viewed with skepticism and care in applying it to a real (viscous) fluid motion.
It is time now to really get serious about flow problems. The fluid statics applications of Chap. 2 were more like fun than work, at least in this writer’s opinion. Statics problems basically require only the density of the fluid and knowledge of the posi- tion of the free surface, but most flow problems require the analysis of an arbitrary state of variable fluid motion defined by the geometry, the boundary conditions, and the laws of mechanics. This chapter and the next two outline the three basic approaches to the analysis of arbitrary flow problems:
1. Control volume, or large-scale, analysis (Chap. 3).
2. Differential, or small-scale, analysis (Chap. 4).
3. Experimental, or dimensional, analysis (Chap. 5).