Bernoullis 1738 treatise Hydrodynamica contains many excellent sketches of fl ow

Discuss Newtons second law (the linear momentum relation) in these three forms: a F 5 ma a F 5 d dt (mV) a F 5 d dt a # system V d9b Are they all equally valid? Are they equivalent? Are some forms better for fl uid mechanics as opposed to solid mechanics?

Consider the angular momentum relation in the form a MO 5 d dt c # system (r 3 V) d9d What does r mean in this relation? Is this relation valid in both solid and fl uid mechanics? Is it related to the linear momentum equation (Prob. 3.1)? In what manner?

For steady low-Reynolds-number (laminar) fl ow through a long tube (see Prob. 1.12), the axial velocity distribution is given by u 5 C ( R2 2 r 2 ), where R is the tube radius and r # R . Integrate u ( r ) to fi nd the total volume fl ow Q through the tube

Water at 20 8 C fl ows through a long elliptical duct 30 cm wide and 22 cm high. What average velocity, in m/s, would cause the weight fl ow to be 500 lbf/s?

Water at 20 8 C fl ows through a 5-in-diameter smooth pipe at a high Reynolds number, for which the velocity profi le is approximated by u < Uo ( y/R ) 1/8 , where Uo is the centerline velocity, R is the pipe radius, and y is the distance measured from the wall toward the centerline. If the centerline velocity is 25 ft/s, estimate the volume fl ow rate in gallons per minute.

Water fi lls a cylindrical tank to depth h . The tank has diameter D . The water fl ows out at average velocity Vo from a hole in the bottom of area Ao . Use the Reynolds transport theorem to fi nd an expression for the instantaneous depth change dh/dt .

A spherical tank, of diameter 35 cm, is leaking air through a 5-mm-diameter hole in its side. The air exits the hole at 360 m/s and a density of 2.5 kg/m 3 . Assuming uniform mixing, ( a ) fi nd a formula for the rate of change of average density in the tank and ( b ) calculate a numerical value for ( d / dt ) in the tank for the given data.

Three pipes steadily deliver water at 20 8 C to a large exit pipe in Fig. P3.8. The velocity V2 5 5 m/s, and the exit fl ow rate Q4 5 120 m 3 /h. Find ( a ) V1 , ( b ) V3 , and ( c ) V4 if it is known that increasing Q3 by 20 percent would increase Q4 by 10 percent. P3.8 D2 = 5 cm D3 = 6 cm D1 = 4 cm D4 = 9 cm

A laboratory test tank contains seawater of salinity S and density . Water enters the tank at conditions ( S1 , 1 , A1 , V1 ) and is assumed to mix immediately in the tank. Tank water leaves through an outlet A2 at velocity V2 . If salt is a conservative property (neither created nor destroyed), use the Reynolds transport theorem to fi nd an expression for the rate of change of salt mass Msalt within the tank.

Water fl owing through an 8-cm-diameter pipe enters a porous section, as in Fig. P3.10, which allows a uniform radial velocity vw through the wall surfaces for a distance of 1.2 m. If the entrance average velocity V1 is 12 m/s, fi nd the exit velocity V2 if ( a ) vw 5 15 cm/s out of the pipe walls or ( b ) vw 5 10 cm/s into the pipe. ( c ) What value of vw will make V2 5 9 m/s? 1.2 m vw D = 8 cm V1 V2

Water fl ows from a faucet into a sink at 3 U.S. gallons per minute. The stopper is closed, and the sink has two Problems 191 rectangular overfl ow drains, each 3 /8 in by 1 in. If the sink water level remains constant, estimate the average overfl ow velocity, in ft/s.

The pipe fl ow in Fig. P3.12 fi lls a cylindrical surge tank as shown. At time t 5 0, the water depth in the tank is 30 cm. Estimate the time required to fi ll the remainder of the tank. D = 75 cm 1 m V1 = 2.5 m/s V2 d = 12 cm = 1.9 m/s

The cylindrical container in Fig. P3.13 is 20 cm in diameter and has a conical contraction at the bottom with an exit hole 3 cm in diameter. The tank contains fresh water at standard sea-level conditions. If the water surface is falling at the nearly steady rate dh/dt < 2 0.072 m/s, estimate the average velocity V out of the bottom exit. P3.13 D h(t) V?

The open tank in Fig. P3.14 contains water at 20 8 C and is being fi lled through section 1. Assume incompressible fl ow. First derive an analytic expression for the water-level change dh / dt in terms of arbitrary volume fl ows ( Q1 , Q2 , Q3 ) and tank diameter d . Then, if the water level h is constant, determine the exit velocity V2 for the given data V1 5 3 m/s and Q3 5 0.01 m 3 /s.

Water, assumed incompressible, fl ows steadily through the round pipe in Fig. P3.15. The entrance velocity is constant, u 5 U0 , and the exit velocity approximates turbulent fl ow, u 5 umax (1 2 r / R ) 1/7 . Determine the ratio U0 / umax for this fl ow

An incompressible fl uid fl ows past an impermeable fl at plate, as in Fig. P3.16, with a uniform inlet profi le u 5 U0 and a cubic polynomial exit profi le u < U0 a 3 2 3 2 b where 5 y Compute the volume fl ow Q across the top surface of the control volume. y = 0 CV U0 U0 y = Q? Solid plate, width b into paper Cubic P3.16

Incompressible steady fl ow in the inlet between parallel plates in Fig. P3.17 is uniform, u 5 U0 5 8 cm/s, while downstream the fl ow develops into the parabolic laminar profi le u 5 az ( z0 2 z ), where a is a constant. If z0 5 4 cm and the fl uid is SAE 30 oil at 20 8 C, what is the value of umax in cm/s? 192 Chapter 3 Integral Relations for a Control Volume P3.17 z = z 0 umax z = 0

Gasoline enters section 1 in Fig. P3.18 at 0.5 m 3 /s. It leaves section 2 at an average velocity of 12 m/s. What is the average velocity at section 3? Is it in or out? D2 = 18 cm D3 = 13 cm

Water from a storm drain fl ows over an outfall onto a porous bed that absorbs the water at a uniform vertical velocity of 8 mm/s, as shown in Fig. P3.19. The system is 5 m deep into the paper. Find the length L of the bed that will completely absorb the storm water. 2 m/s Initial depth = 20 cm L? P3.19

Oil (SG 5 0.89) enters at section 1 in Fig. P3.20 at a weight fl ow of 250 N/h to lubricate a thrust bearing. The steady oil fl ow exits radially through the narrow clearance between thrust plates. Compute ( a ) the outlet volume fl ow in mL/s and ( b ) the average outlet velocity in cm/s.

For the two-port tank of Fig. E3.5, assume D1 5 4 cm, V1 5 18 m/s, D2 5 7 cm, and V2 5 8 m/s. If the tank surface is rising at 17 mm/s, estimate the tank diameter.

The convergingdiverging nozzle shown in Fig. P3.22 expands and accelerates dry air to supersonic speeds at the exit, where p2 5 8 kPa and T2 5 240 K. At the throat, p1 5 284 kPa, T1 5 665 K, and V1 5 517 m/s. For steady compressible fl ow of an ideal gas, estimate ( a ) the mass fl ow in kg/h, ( b ) the velocity V2 , and ( c ) the Mach number Ma 2 . P3.20 h = 2 mm D = 10 cm 2 1 2 D1 = 3 mm 1 2 Air D1 = 1 cm D2 = 2.5 cm

Water enters the bottom of the cone in Fig. P3.24 at a uniformly increasing average velocity V 5 Kt . If d is very small, derive an analytic formula for the water surface rise h ( t ) for the condition h 5 0 at t 5 0. Assume incompressible fl ow. P3.24 h(t) V = Kt Diameter d Cone

Water enters the bottom of the cone in Fig. P3.24 at a uniformly increasing average velocity V 5 Kt . If d is very small, derive an analytic formula for the water surface rise h ( t ) for the condition h 5 0 at t 5 0. Assume incompressible fl ow. P3.24 h(t) V = Kt Diameter d Cone

As will be discussed in Chaps. 7 and 8, the fl ow of a stream U0 past a blunt fl at plate creates a broad low-velocity wake behind the plate. A simple model is given in Fig. P3.25, with only half of the fl ow shown due to symmetry. The velocity profi le behind the plate is idealized as dead air (near-zero velocity) behind the plate, plus a higher velocity, decaying vertically above the wake according to the variation u < U0 1 DU e 2 z/L , where L is the plate height and z 5 0 is the top of the wake. Find DU as a function of stream speed U0 . U0 U0 Width u b into paper Dead air (negligible velocity) Exponential curve

A thin layer of liquid, draining from an inclined plane, as in Fig. P3.26, will have a laminar velocity profi le u < U0 (2 y / h 2 y 2 / h2 ), where U0 is the surface velocity. If the plane has width b into the paper, determine the volume rate of fl ow in the fi lm. Suppose that h 5 0.5 in and the fl ow rate per foot of channel width is 1.25 gal/min. Estimate U0 in ft/s.

Consider a highly pressurized air tank at conditions ( p0 , 0 , T0 ) and volume 0 . In Chap. 9 we will learn that, if the tank is allowed to exhaust to the atmosphere through a well-designed converging nozzle of exit area A , the outgoing mass fl ow rate will be m # 5 p0 A 1RT0 where < 0.685 for air This rate persists as long as p0 is at least twice as large as the atmospheric pressure. Assuming constant T0 and an ideal gas, ( a ) derive a formula for the change of density 0 ( t ) within the tank. ( b ) Analyze the time Dt required for the density to decrease by 25 percent.

Air, assumed to be a perfect gas from Table A.4, fl ows through a long, 2-cm-diameter insulated tube. At section 1, the pressure is 1.1 MPa and the temperature is 345 K. At section 2, 67 meters further downstream, the density is 1.34 kg/m 3 , the temperature 298 K, and the Mach number is 0.90. For one-dimensional fl ow, calculate ( a ) the mass fl ow; ( b ) p2 ; (c) V2 ; and ( d ) the change in entropy between 1 and 2. ( e ) How do you explain the entropy change?

In elementary compressible fl ow theory (Chap. 9), compressed air will exhaust from a small hole in a tank at the mass fl ow rate m # < C , where is the air density in the tank and C is a constant. If 0 is the initial density in a tank of volume 9 , derive a formula for the density change ( t ) after the hole is opened. Apply your formula to the following case: a spherical tank of diameter 50 cm, with initial pressure 300 kPa and temperature 100 8 C, and a hole whose initial exhaust rate is 0.01 kg/s. Find the time required for the tank density to drop by 50 percent.

For the nozzle of Fig. P3.22, consider the following data for air, k 5 1.4. At the throat, p1 5 1000 kPa, V1 5 491 m/s, and T1 5 600 K. At the exit, p2 5 28.14 kPa. Assuming isentropic steady fl ow, compute ( a ) the Mach number Ma 1 ; ( b ) T2 ; ( c ) the mass fl ow; and ( d ) V2 .

A bellows may be modeled as a deforming wedge-shaped volume as in Fig. P3.31. The check valve on the left (pleated) end is closed during the stroke. If b is the bellows width into the paper, derive an expression for outlet mass fl ow m # 0 as a function of stroke ( t ). h h Stroke L d  h m0 (t) (t)

Water at 20 8 C fl ows steadily through the piping junction in Fig. P3.32, entering section 1 at 20 gal/min. The average velocity at section 2 is 2.5 m/s. A portion of the fl ow is diverted through the showerhead, which contains 100 holes of 1-mm diameter. Assuming uniform shower fl ow, estimate the exit velocity from the showerhead jets. P3.32 (3) (2) (1) d = 4 cm d = 1.5 cm d = 2 cm

In some wind tunnels the test section is perforated to suck out fl uid and provide a thin viscous boundary layer. The test section wall in Fig. P3.33 contains 1200 holes of 5-mm diameter each per square meter of wall area. The suction velocity through each hole is Vs 5 8 m/s, and the testsection entrance velocity is V1 5 35 m/s. Assuming incompressible steady fl ow of air at 20 8 C, compute ( a ) V0 , ( b ) V2 , and ( c ) Vf , in m/s. Df = 2.2 m D0 = 2.5 m Vf V2 V1 V0 L = 4 m

A rocket motor is operating steadily, as shown in Fig. P3.34. The products of combustion fl owing out the exhaust nozzle approximate a perfect gas with a molecular weight of 28. For the given conditions calculate V2 in ft/s. P3.34 1 3 2 Liquid oxygen: 0.5 slug/s Liquid fuel: 0.1 slug/s 1100 F 15 lbf/in2 D2 = 5.5 in

In contrast to the liquid rocket in Fig. P3.34, the solid- propellant rocket in Fig. P3.35 is self-contained and has no entrance ducts. Using a control volume analysis for the conditions shown in Fig. P3.35, compute the rate of mass loss of the propellant, assuming that the exit gas has a molecular weight of 28. Propellant Propellant Combustion: 1500 K, 950 kPa

The jet pump in Fig. P3.36 injects water at U1 5 40 m/s through a 3-in pipe and entrains a secondary fl ow of water U2 5 3 m/s in the annular region around the small pipe. The two fl ows become fully mixed downstream, where U3 is approximately constant. For steady incompressible fl ow, compute U3 in m/s. Inlet Mixing region Fully mixed U1 D1 = 3 in U2 U3 D2 = 10 in

If the rectangular tank full of water in Fig. P3.37 has its right-hand wall lowered by an amount , as shown, water will fl ow out as it would over a weir or dam. In Prob. P1.14 we deduced that the outfl ow Q would be given by Q 5 Cbg1/2 3/2 where b is the tank width into the paper, g is the acceleration of gravity, and C is a dimensionless constant. Assume that the water surface is horizontal, not slightly curved as in the fi gure. Let the initial excess water level be o . Derive a formula for the time required to reduce the excess water level to ( a ) o /10 and ( b ) zero. P3.37 Q 3/2 L

An incompressible fl uid in Fig. P3.38 is being squeezed outward between two large circular disks by the uniform downward motion V0 of the upper disk. Assuming onedimensional radial outfl ow, use the control volume shown to derive an expression for V ( r ). h(t) r V CV CV V(r)? V0 Fixed circular disk

A wedge splits a sheet of 20 8 C water, as shown in Fig. P3.39. Both wedge and sheet are very long into the paper. If the force required to hold the wedge stationary is F 5 124 N per meter of depth into the paper, what is the angle of the wedge? F 4 cm 6 m/s 6 m/s 6 m/

The water jet in Fig. P3.40 strikes normal to a fi xed plate. Neglect gravity and friction, and compute the force F in newtons required to hold the plate fi xed. P3.40 F Plate Dj = 10 cm Vj = 8 m/s

In Fig. P3.41 the vane turns the water jet completely around. Find an expression for the maximum jet velocity V0 if the maximum possible support force is F0 . F0 0, V0, D0

Very small pressure differences pA 2 pB can be measured accurately by the two-fl uid differential manometer in Fig. P2.42. Density 2 is only slightly larger than that of the upper fl uid 1 . Derive an expression for the proportionality between h and pA 2 pB if the reservoirs are very large. 110 Chapter 2 Pressure Distribution in a Fluid P2.42 h1 pA 1 pB 1 h 2

The traditional method of measuring blood pressure uses a sphygmomanometer , fi rst recording the highest ( systolic ) and then the lowest ( diastolic ) pressure from which fl owing Korotkoff sounds can be heard. Patients with dangerous hypertension can exhibit systolic pressures as high as 5 lbf/in 2 . Normal levels, however, are 2.7 and 1.7 lbf/in 2 , respectively, for systolic and diastolic pressures. The manometer uses mercury and air as fl uids. ( a ) How high in cm should the manometer tube be? ( b ) Express normal systolic and diastolic blood pressure in millimeters of mercury.

Water fl ows downward in a pipe at 45 8 , as shown in Fig. P2.44. The pressure drop p1 2 p2 is partly due to gravity and partly due to friction. The mercury manometer reads a 6-in height difference. What is the total pressure drop p1 2 p2 in lbf/in 2 ? What is the pressure drop due to friction only between 1 and 2 in lbf/in 2 ? Does the manometer reading correspond only to friction drop? Why? P2.44 5 ft Flow 1 2 45 6 in Mercury

In Fig. P2.45, determine the gage pressure at point A in Pa. Is it higher or lower than atmospheric? P2.45 45 cm 30 cm 15 cm 40 cm patm Air Oil, SG = 0.85 Water Mercury

In Fig. P2.46 both ends of the manometer are open to the atmosphere. Estimate the specifi c gravity of fl uid X . P2.46 7 cm 4 cm 6 cm 9 cm 5 cm 12 cm SAE 30 oil Water Fluid X 10 cm P2.47 The cylindrical tank in Fig. P2.47 is being fi lled

The cylindrical tank in Fig. P2.47 is being fi lled with water at 20 8 C by a pump developing an exit pressure of 175 kPa. At the instant shown, the air pressure is 110 kPa and H 5 35 cm. The pump stops when it can no longer raise the water pressure. For isothermal air compression, estimate H at that time. Problems 111 P2.47 75 cm H 50 cm Air 20 C Water Pump

The system in Fig. P2.48 is open to 1 atm on the right side. ( a ) If L 5 120 cm, what is the air pressure in container A ? ( b ) Conversely, if pA 5 135 kPa, what is the length L ? 15 cm 32 cm Air 18 cm 35 Mercury Water L

Conduct the following experiment to illustrate air pressure. Find a thin wooden ruler (approximately 1 ft in length) or a thin wooden paint stirrer. Place it on the edge of a desk or table with a little less than half of it hanging over the edge lengthwise. Get two full-size sheets of newspaper; open them up and place them on top of the ruler, covering only the portion of the ruler resting on the desk as illustrated in Fig. P2.49. ( a ) Estimate the total force on top of the newspaper due to air pressure in the room. ( b ) Careful! To avoid potential injury, make sure nobody is standing directly in front of the desk. Perform a karate chop on the portion of the ruler sticking out over the edge of the desk. Record your results. ( c ) Explain your results. Newspaper Ruler Desk

A small submarine, with a hatch door 30 in in diameter, is submerged in seawater. ( a ) If the water hydrostatic force on the hatch is 69,000 lbf, how deep is the sub? ( b ) If the sub is 350 ft deep, what is the hydrostatic force on the hatch?

Gate AB in Fig. P2.51 is 1.2 m long and 0.8 m into the paper. Neglecting atmospheric pressure, compute the force F on the gate and its center-of-pressure position X . 8 m 6 m 1.2 m F 40

Example 2.5 calculated the force on plate AB and its line of action, using the moment-of-inertia approach. Some teachers say it is more instructive to calculate these by direct integration of the pressure forces. Using Figs. P2.52 and E2.5 a , ( a ) fi nd an expression for the pressure variation p ( ) along the plate; ( b ) integrate this expression to fi nd the total force F ; ( c ) integrate the moments about point A to fi nd the position of the center of pressure. 112 Chapter 2 Pressure Distribution in a Fluid P2.52 8 ft B A 6 ft

The Hoover Dam, in Arizona, encloses Lake Mead, which contains 10 trillion gallons of water. The dam is 1200 ft wide and the lake is 500 ft deep. ( a ) Estimate the hydrostatic force on the dam, in MN. ( b ) Explain how you might analyze the stress in the dam due to this hydrostatic force.

In Fig. P2.54, the hydrostatic force F is the same on the bottom of all three containers, even though the weights of liquid above are quite different. The three bottom shapes and the fl uids are the same. This is called the hydrostatic paradox . Explain why it is true and sketch a free body of each of the liquid columns. P2.54 (a) F (b) (c) F F

Gate AB in Fig. P2.55 is 5 ft wide into the paper, hinged at A , and restrained by a stop at B . The water is at 20 8 C. Compute ( a ) the force on stop B and ( b ) the reactions at A if the water depth h 5 9.5 ft. P2.55 pa Water pa 4 ft

In Fig. P2.55, gate AB is 5 ft wide into the paper, and stop B will break if the water force on it equals 9200 lbf. For what water depth h is this condition reached?

The square vertical panel ABCD in Fig. P2.57 is submerged in water at 20 8 C. Side AB is at least 1.7 m below the surface. Determine the difference between the hydrostatic forces on subpanels ABD and BCD . P2.57 A D B C 60 cm

In Fig. P2.58, the cover gate AB closes a circular opening 80 cm in diameter. The gate is held closed by a 200 - kg mass as shown. Assume standard gravity at 20 8 C. At what water level h will the gate be dislodged? Neglect the weight of the gate. P2.58 Water 30 cm 3 m 200 kg

Gate AB has length L and width b into the paper, is hinged at B , and has negligible weight. The liquid level h remains at the top of the gate for any angle . Find an analytic expression for the force P , perpendicular to AB , required to keep the gate in equilibrium in Fig. P2.59. P2.59 B h Hinge A P L

In Fig. P2.60, vertical, unsymmetrical trapezoidal panel ABCD is submerged in fresh water with side AB 12 ft below the surface. Since trapezoid formulas are complicated, ( a ) estimate, reasonably, the water force on the panel, in lbf, neglecting atmospheric pressure. For extra credit, ( b ) look up the formula and compute the exact force on the panel. Problems 113 P2.60 C A B D 8 ft 6 ft 9 ft

Gate AB in Fig. P2.61 is a homogeneous mass of 180 kg, 1.2 m wide into the paper, hinged at A , and resting on a smooth bottom at B . All fl uids are at 20 8 C. For what water depth h will the force at point B be zero? P2.61 Water B A Glycerin 1 m h 2 m

Gate AB in Fig. P2.62 is 15 ft long and 8 ft wide into the paper and is hinged at B with a stop at A . The water is at 20 8 C. The gate is 1-in-thick steel, SG 5 7.85. Compute the water level h for which the gate will start to fall. P2.62 Water

The tank in Fig. P2.63 has a 4-cm-diameter plug at the bottom on the right. All fl uids are at 20 8 C. The plug will pop out if the hydrostatic force on it is 25 N. For this condition, what will be the reading h on the mercury manometer on the left side? h 50 2 cm H Water Plug, D = 4 cm Mercury

Gate ABC in Fig. P2.64 has a fi xed hinge line at B and is 2 m wide into the paper. The gate will open at A to release water if the water depth is high enough. Compute the depth h for which the gate will begin to open. Water at 20C A 20 cm B C 1 m h

Gate AB in Fig. P2.65 is semicircular, hinged at B , and held by a horizontal force P at A . What force P is required for equilibrium? P2.65 P A B 5 m Water 3 m Gate: Side view

Dam ABC in Fig. P2.66 is 30 m wide into the paper and made of concrete (SG 5 2.4). Find the hydrostatic force on surface AB and its moment about C . Assuming no seepage of water under the dam, could this force tip the dam over? How does your argument change if there is seepage under the dam? P2.66 60 m C A B Water 20C 80 m Dam

Generalize Prob. P2.66 as follows. Denote length AB as H , length BC as L , and angle ABC as . Let the dam material have specifi c gravity SG. The width of the dam is b . Assume no seepage of water under the dam. Find an analytic relation between SG and the critical angle c for which the dam will just tip over to the right. Use your relation to compute c for the special case SG 5 2.4 (concrete).

Isosceles triangle gate AB in Fig. P2.68 is hinged at A and weighs 1500 N. What horizontal force P is required at point B for equilibrium? A P 3 m Gate 50 B Oil, SG = 0.83 1 m 2 m

Consider the slanted plate AB of length L in Fig. P2.69. ( a ) Is the hydrostatic force F on the plate equal to the weight of the missing water above the plate? If not, correct this hypothesis. Neglect the atmosphere. ( b ) Can a missing water theory be generalized to curved surfaces of this type? P2.69

The swing-check valve in Fig. P2.70 covers a 22.86-cm diameter opening in the slanted wall. The hinge is 15 cm from the centerline, as shown. The valve will open when the hinge moment is 50 N m. Find the value of h for the water to cause this condition. P2.70 Water at 20C 15 cm Hinge Air

In Fig. P2.71 gate AB is 3 m wide into the paper and is connected by a rod and pulley to a concrete sphere (SG 5 2.40). What diameter of the sphere is just suffi cient to keep the gate closed? P2.71 A 4 m B Concrete sphere, SG = 2.4 8 m 6 m Wate

In Fig. P2.72, gate AB is circular. Find the moment of the hydrostatic force on this gate about axis A . P2.72 A B Water 3 m 2 m

Gate AB is 5 ft wide into the paper and opens to let fresh water out when the ocean tide is dropping. The hinge at A is 2 ft above the freshwater level. At what ocean level h will the gate fi rst open? Neglect the gate weight. P2.73 A B h Stop 10 ft Tide range Seawater, SG = 1.025

Find the height H in Fig. P2.74 for which the hydrostatic force on the rectangular panel is the same as the force on the semicircular panel below. P2.74 2R H

The cap at point B on the 5-cm-diameter tube in Fig. P2.75 will be dislodged when the hydrostatic force on its base reaches 22 lbf. For what water depth h does this occur? P2.75 1 m Water Oil, SG = 0.8 B 2 m h

Panel BC in Fig. P2.76 is circular. Compute ( a ) the hydrostatic force of the water on the panel, ( b ) its center of pressure, and ( c ) the moment of this force about point B

The circular gate ABC in Fig. P2.77 has a 1-m radius and is hinged at B . Compute the force P just suffi cient to keep the gate from opening when h 5 8 m. Neglect atmospheric pressure.

Panels AB and CD in Fig. P2.78 are each 120 cm wide into the paper. ( a ) Can you deduce, by inspection, which panel has the larger water force? ( b ) Even if your deduction is brilliant, calculate the panel forces anyway.

The Saturn V rocket in the chapter opener photo was powered by fi ve F-1 engines, each of which burned 3945 lbm/s of liquid oxygen and 1738 lbm of kerosene per second. The exit velocity of burned gases was approximately 8500 ft/s. In the spirit of Prob. P3.34, neglecting external pressure forces, estimate the total thrust of the rocket, in lbf. P3.78 1 2 u V1 1 D1 Blades V2 2 Air jet

A river of width b and depth h1 passes over a submerged obstacle, or drowned weir, in Fig. P3.80, emerging at a new fl ow condition ( V2 , h2 ). Neglect atmospheric pressure, and assume that the water pressure is hydrostatic at both sections 1 and 2. Derive an expression for the force exerted by the river on the obstacle in terms of V1 , h1 , h2 , b , , and g . Neglect water friction on the river bottom. V2, h2 V1, h1 Width b into paper

Torricellis idealization of effl ux from a hole in the side of a tank is V 5 12 gh, as shown in Fig. P3.81. The cylindrical tank weighs 150 N when empty and contains water at 20 8 C. The tank bottom is on very smooth ice (static friction coeffi cient < 0.01). The hole diameter is 9 cm. For what water depth h will the tank just begin to move to the right? P3.81 30 cm h Water 1 m Static friction

The model car in Fig. P3.82 weighs 17 N and is to be accelerated from rest by a 1-cm-diameter water jet moving 202 Chapter 3 Integral Relations for a Control Volume at 75 m/s. Neglecting air drag and wheel friction, estimate the velocity of the car after it has moved forward 1 m. x V Vj P3.82

Gasoline at 20 8 C is fl owing at V1 5 12 m/s in a 5-cmdiameter pipe when it encounters a 1-m length of uniform radial wall suction. At the end of this suction region, the average fl uid velocity has dropped to V2 5 10 m/s. If p1 5 120 kPa, estimate p2 if the wall friction losses are neglected.

Air at 20 8 C and 1 atm fl ows in a 25-cm-diameter duct at 15 m/s, as in Fig. P3.84. The exit is choked by a 90 8 cone, as shown. Estimate the force of the airfl ow on the cone. P3.84 25 cm 90 40 cm 1 cm

The thin-plate orifi ce in Fig. P3.85 causes a large pressure drop. For 20 8 C water fl ow at 500 gal/min, with pipe D 5 10 cm and orifi ce d 5 6 cm, p1 2 p2 < 145 kPa. If the wall friction is negligible, estimate the force of the water on the orifi ce plate. P3.85 1 2 P3.86 For the wat

For the water jet pump of Prob. P3.36, add the following data: p1 5 p2 5 25 lbf/in 2 , and the distance between sections 1 and 3 is 80 in. If the average wall shear stress between sections 1 and 3 is 7 lbf/ft 2 , estimate the pressure p3 . Why is it higher than p1 ?

A vane turns a water jet through an angle , as shown in Fig. P3.87. Neglect friction on the vane walls. ( a ) What is the angle for the support force to be in pure compression? ( b ) Calculate this compression force if the water velocity is 22 ft/s and the jet cross section is 4 in 2 . P3.87 V V F 25

The boat in Fig. P3.88 is jet-propelled by a pump that develops a volume fl ow rate Q and ejects water out the stern at velocity Vj . If the boat drag force is F 5 kV2 , where k is a constant, develop a formula for the steady forward speed V of the boat. V Pump Vj Q

Consider Fig. P3.36 as a general problem for analysis of a mixing ejector pump. If all conditions ( p , , V ) are known at sections 1 and 2 and if the wall friction is negligible, derive formulas for estimating ( a ) V3 and ( b ) p3 .

As shown in Fig. P3.90, a liquid column of height h is confi ned in a vertical tube of cross-sectional area A by a stopper. At t 5 0 the stopper is suddenly removed, exposing the bottom of the liquid to atmospheric pressure. Using a control volume analysis of mass and vertical momentum, derive the differential equation for the downward motion V ( t ) of the liquid. Assume one-dimensional, incompressible, frictionless fl ow.

Extend Prob. P3.90 to include a linear (laminar) average wall shear stress resistance of the form < cV , where c is a constant. Find the differential equation for dV / dt and then solve for V ( t ), assuming for simplicity that the wall area remains constant. Pro

A more involved version of Prob. P3.90 is the elbowshaped tube in Fig. P3.92, with constant cross-sectional area A and diameter D ! h , L . Assume incompressible fl ow, neglect friction, and derive a differential equation for dV / dt when the stopper is opened. Hint: Combine two control volumes, one for each leg of the tube. P3.92 V2 V1 pa h L

According to Torricellis theorem, the velocity of a fl uid draining from a hole in a tank is V < (2 gh ) 1/2 , where h is the depth of water above the hole, as in Fig. P3.93. Let the hole have area Ao and the cylindrical tank have cross-section area Ab Ao . Derive a formula for the time to drain the tank completely from an initial depth ho . P3.93 Water h V

A water jet 3 in in diameter strikes a concrete (SG 5 2.3) slab which rests freely on a level fl oor. If the slab is 1 ft wide into the paper, calculate the jet velocity which will just begin to tip the slab over. P3.94 36 in 20 in 3 in 8 in

A tall water tank discharges through a well-rounded orifi ce, as in Fig. P3.95. Use the Torricelli formula of Prob. P3.81 to estimate the exit velocity. ( a ) If, at this instant, the force F required to hold the plate is 40 N, what is the depth h ? ( b ) If the tank surface is dropping at the rate of 2.5 cm/s, what is the tank diameter D ? P3.95 h D d = 4 cm

Extend Prob. P3.90 to the case of the liquid motion in a frictionless U-tube whose liquid column is displaced a distance Z upward and then released, as in Fig. P3.96. h1 h3 z z Equilibrium position h2 0 Liquid column length L = h1 + h2 + h3

Extend Prob. P3.96 to include a linear (laminar) average wall shear stress resistance of the form < 8 V/D , where is the fl uid viscosity. Find the differential equation for dV / dt and then solve for V ( t ), assuming an initial displacement z 5 z0 , V 5 0 at t 5 0. The result should be a damped oscillation tending toward z 5 0.

As an extension of Example 3.9, let the plate and its cart (see Fig. 3.9 a ) be unrestrained horizontally, with frictionless wheels. Derive ( a ) the equation of motion for cart velocity Vc ( t ) and ( b ) a formula for the time required for the cart to accelerate from rest to 90 percent of the jet velocity (assuming the jet continues to strike the plate horizontally). ( c ) Compute numerical values for part ( b ) using the conditions of Example 3.9 and a cart mass of 2 kg.

Let the rocket of Fig. E3.12 start at z 5 0, with constant exit velocity and exit mass fl ow, and rise vertically with zero drag. ( a ) Show that, as long as fuel burning continues, the vertical height S ( t ) reached is given by S 5 Ve Mo m # 3ln 2 1 14 , where 5 1 2 m # t Mo ( b ) Apply this to the case Ve 5 1500 m/s and Mo 5 1000 kg to fi nd the height reached after a burn of 30 seconds, when the fi nal rocket mass is 400 kg.

Suppose that the solid-propellant rocket of Prob. P3.35 is built into a missile of diameter 70 cm and length 4 m. The system weighs 1800 N, which includes 700 N of propellant. Neglect air drag. If the missile is fi red vertically from rest at sea level, estimate ( a ) its velocity and height at fuel burnout and ( b ) the maximum height it will attain.

Water at 20 8 C fl ows steadily through the tank in Fig. P3.101. Known conditions are D1 5 8 cm, V1 5 6 m/s, and D2 5 4 cm. A rightward force F 5 70 N is required to keep the tank fi xed. ( a ) What is the velocity leaving section 2? ( b ) If the tank cross section is 1.2 m 2 , how fast is the water surface h ( t ) rising or falling? P3.101 2 1 h (t)

As can often be seen in a kitchen sink when the faucet is running, a high-speed channel fl ow ( V1 , h1 ) may jump to a low-speed, low-energy condition ( V2 , h2 ) as in Fig. P3.102. The pressure at sections 1 and 2 is approximately hydrostatic, and wall friction is negligible. Use the continuity and momentum relations to fi nd h2 and V2 in terms of ( h1 , V1 ).

Suppose that the solid-propellant rocket of Prob. P3.35 is mounted on a 1000-kg car to propel it up a long slope of 15 8 . The rocket motor weighs 900 N, which includes 500 N of propellant. If the car starts from rest when the rocket is fi red, and if air drag and wheel friction are neglected, estimate the maximum distance that the car will travel up the hill

A rocket is attached to a rigid horizontal rod hinged at the origin as in Fig. P3.104. Its initial mass is M0 , and its exit properties are m # and Ve relative to the rocket. Set up the differential equation for rocket motion, and solve for the angular velocity ( t ) of the rod. Neglect gravity, air drag, and the rod mass. P3.104 y x R m, Ve, pe = pa

Extend Prob. P3.104 to the case where the rocket has a linear air drag force F 5 cV , where c is a constant. Assuming no burnout, solve for ( t ) and fi nd the terminal angular velocitythat is, the fi nal motion when the angular acceleration is zero. Apply to the case M0 5 6 kg, R 5 3 m, m # 5 0.05 kg/s, Ve 5 1100 m/s, and c 5 0.075 N ? s/m to fi nd the angular velocity after 12 s of burning.

Actual airfl ow past a parachute creates a variable distribution of velocities and directions. Let us model this as a circular air jet, of diameter half the parachute diameter, which is turned completely around by the parachute, as in Fig. P3.106. ( a ) Find the force F required to support Problems 205 the chute. ( b ) Express this force as a dimensionless drag coeffi cient , CD = F /[() V2 ( / 4) D2 ] and compare with Table 7.3. P3.106 D D/2 , V

The cart in Fig. P3.107 moves at constant velocity V0 5 12 m/s and takes on water with a scoop 80 cm wide that dips h 5 2.5 cm into a pond. Neglect air drag and wheel friction. Estimate the force required to keep the cart moving. Water V0 h P3.107

A rocket sled of mass M is to be decelerated by a scoop, as in Fig. P3.108, which has width b into the paper and dips into the water a depth h , creating an upward jet at 60 8 . The rocket thrust is T to the left. Let the initial velocity be V0 , and neglect air drag and wheel friction. Find an expression for V ( t ) of the sled for ( a ) T 5 0 and ( b ) fi nite T ? 0. P3.108 M V 60 h Water

For the boundary layer fl ow in Fig. 3.10, let the exit velocity profi le, at x 5 L , simulate turbulent fl ow, u < U0(y/) 1/7 . ( a ) Find a relation between h and . ( b ) Find an expression for the drag force F on the plate between 0 and L

Repeat Prob. P3.49 by assuming that p1 is unknown and using Bernoullis equation with no losses. Compute the new bolt force for this assumption. What is the head loss between 1 and 2 for the data of Prob. P3.49?

As a simpler approach to Prob. P3.96, apply the unsteady Bernoulli equation between 1 and 2 to derive a differential equation for the motion z ( t ). Neglect friction and compressibility.

A jet of alcohol strikes the vertical plate in Fig. P3.112. A force F < 425 N is required to hold the plate stationary. Assuming there are no losses in the nozzle, estimate ( a ) the mass fl ow rate of alcohol and ( b ) the absolute pressure at section 1. Alcohol, SG = 0.79 V1 V2 F D1 = 5 cm D2 = 2 cm pa = 101 kPa

An airplane is fl ying at 300 mi/h at 4000 m standard altitude. As is typical, the air velocity relative to the upper surface of the wing, near its maximum thickness, is 26 percent higher than the planes velocity. Using Bernoullis equation, calculate the absolute pressure at this point on the wing. Neglect elevation changes and compressibility.

Water fl ows through a circular nozzle, exits into the air as a jet, and strikes a plate, as shown in Fig. P3.114. The force required to hold the plate steady is 70 N. Assuming steady, frictionless, one-dimensional fl ow, estimate ( a ) the velocities at sections (1) and (2) and ( b ) the mercury manometer reading h . P3.114 F Water at 20C Hg D1 = 10 cm D2 = 3 cm Air h

A free liquid jet, as in Fig. P3.115, has constant ambient pressure and small losses; hence from Bernoullis equation z 1 V2 /(2 g ) is constant along the jet. For the fi re nozzle in the fi gure, what are ( a ) the minimum and ( b ) the maximum values of for which the water jet will clear the corner of the building? For which case will the jet velocity be higher when it strikes the roof of the building? P3.115 40 ft V1 = 100 ft/s X 50 ft

For the container of Fig. P3.116 use Bernoullis equation to derive a formula for the distance X where the free jet leaving horizontally will strike the fl oor, as a function of h and H . For what ratio h / H will X be maximum? Sketch the three trajectories for h / H 5 0.25, 0.5, and 0.75. P3.116 h Free H jet X

Water at 20 8 C, in the pressurized tank of Fig. P3.117, fl ows out and creates a vertical jet as shown. Assuming steady frictionless fl ow, determine the height H to which the jet rises. Air 75 kPa (gage) Water H? 85 c

Bernoullis 1738 treatise Hydrodynamica contains many excellent sketches of fl ow patterns related to his frictionless relation. One, however, redrawn here as Fig. P3.118, seems physically misleading. Can you explain what might be wrong with the fi gure? Jet Jet

A long fi xed tube with a rounded nose, aligned with an oncoming fl ow, can be used to measure velocity. Measurements are made of the pressure at (1) the front nose and (2) a hole in the side of the tube further along, where the pressure nearly equals stream pressure. ( a ) Make a sketch of this device and show how the velocity is calculated. ( b ) For a particular sea-level airfl ow, the difference between nose pressure and side pressure is 1.5 lbf/in 2 . What is the air velocity, in mi/h?

The manometer fl uid in Fig. P3.120 is mercury. Estimate the volume fl ow in the tube if the fl owing fl uid is ( a ) gasoline and ( b ) nitrogen, at 20 8 C and 1 atm. P3.120 1 in 3 in

In Fig. P3.121 the fl owing fl uid is CO 2 at 20 8 C. Neglect losses. If p1 5 170 kPa and the manometer fl uid is Meriam red oil (SG 5 0.827), estimate ( a ) p2 and ( b ) the gas fl ow rate in m 3 /h.

The cylindrical water tank in Fig. P3.122 is being fi lled at a volume fl ow Q1 5 1.0 gal/min, while the water also drains from a bottom hole of diameter d 5 6 mm. At time t 5 0, h 5 0. Find and plot the variation h ( t ) and the eventual maximum water depth hmax . Assume that Bernoullis steady-fl ow equation is valid.

The air-cushion vehicle in Fig. P3.123 brings in sea-level standard air through a fan and discharges it at high velocity through an annular skirt of 3-cm clearance. If the vehicle weighs 50 kN, estimate ( a ) the required airfl ow rate and ( b ) the fan power in kW. h = 3 cm W = 50 kN V D = 6 m

A necked-down section in a pipe fl ow, called a venturi, develops a low throat pressure that can aspirate fl uid upward from a reservoir, as in Fig. P3.124. Using Bernoullis equation with no losses, derive an expression for the velocity V1 that is just suffi cient to bring reservoir fl uid into the throat. Water V1 V2 , p2 = pa h pa Water D2 D1

Suppose you are designing an air hockey table. The table is 3.0 3 6.0 ft in area, with 1 16 -in-diameter holes spaced every inch in a rectangular grid pattern (2592 holes total). The required jet speed from each hole is estimated to be 50 ft/s. Your job is to select an appropriate blower that will meet the requirements. Estimate the volumetric fl ow rate (in ft 3 /min) and pressure rise (in lb/in 2 ) required of the blower. Hint: Assume that the air is stagnant in the large volume of the manifold under the table surface, and neglect any frictional losses.

The liquid in Fig. P3.126 is kerosene at 20 8 C. Estimate the fl ow rate from the tank for ( a ) no losses and ( b ) pipe losses hf < 4.5 V2 /(2 g ). P3.126 5 ft D = 1 in V pa = 14.7 lbf/in2 abs p = 20 lbf/in2 abs Air:

In Fig. P3.127 the open jet of water at 20 8 C exits a nozzle into sea-level air and strikes a stagnation tube as shown. Sea-level air 12 cm (1) Open jet 4 cm Water H

A venturi meter , shown in Fig. P3.128, is a carefully designed constriction whose pressure difference is a measure of the fl ow rate in a pipe. Using Bernoullis equation for steady incompressible fl ow with no losses, show that the fl ow rate Q is related to the manometer reading h by Q 5 A2 11 2 (D2/D1)4 B 2gh(M 2 ) where M is the density of the manometer fl uid. h 1 2

A water stream fl ows past a small circular cylinder at 23 ft/s, approaching the cylinder at 3000 lbf/ft 2 . Measurements at low (laminar fl ow) Reynolds numbers indicate a maximum surface velocity 60 percent higher than the stream velocity at point B on the cylinder. Estimate the pressure at B .

In Fig. P3.130 the fl uid is gasoline at 20 8 C at a weight fl ow of 120 N/s. Assuming no losses, estimate the gage pressure at section 1. p1 8 cm Open jet 2 5 cm 12 m

In Fig. P3.131 both fl uids are at 20 8 C. If V1 5 1.7 ft/s and losses are neglected, what should the manometer reading h ft be? P3.131 3 in 1 1 in 2 10 ft 2 ft Mercury h Water

Extend the siphon analysis of Example 3.14 to account for friction in the tube, as follows. Let the friction head loss in the tube be correlated as 5.4( Vtube ) 2 /(2 g ), which approximates turbulent fl ow in a 2-m-long tube. Calculate the exit velocity in m/s and the volume fl ow rate in cm 3 /s, and compare to Example 3.14.

If losses are neglected in Fig. P3.133, for what water level h will the fl ow begin to form vapor cavities at the throat of the nozzle? P3.133 2 1 D2 = 8 cm D1 = 5 cm pa = 100 kPa Water at 30C h Open jet

For the 40 8 C water fl ow in Fig. P3.134, estimate the volume fl ow through the pipe, assuming no losses; then explain what is wrong with this seemingly innocent question. If the actual fl ow rate is Q 5 40 m 3 /h, compute ( a ) the head loss in ft and ( b ) the constriction diameter D that causes cavitation, assuming that the throat divides the head loss equally and that changing the constriction causes no additional losses. Problems 209 P3.134 25 m 10 m 5 cm D

The 35 8 C water fl ow of Fig. P3.135 discharges to sea-level standard atmosphere. Neglecting losses, for what nozzle diameter D will cavitation begin to occur? To avoid cavitation, should you increase or decrease D from this critical value? P3.135 1 2 3 3 in1 in D

Air, assumed frictionless, fl ows through a tube, exiting to sea-level atmosphere. Diameters at 1 and 3 are 5 cm, while D2 5 3 cm. What mass fl ow of air is required to suck water up 10 cm into section 2 of Fig. P3.136? P3.136 1 2 3 Water Air 10 cm

In Fig. P3.137 the piston drives water at 20 8 C. Neglecting losses, estimate the exit velocity V2 ft/s. If D2 is further constricted, what is the limiting possible value of V2 ? Water D1 = 8 in D2 = 4 in V2 pa

For the sluice gate fl ow of Example 3.10, use Bernoullis equation, along the surface, to estimate the fl ow rate Q as a function of the two water depths. Assume constant width b .

In the spillway fl ow of Fig. P3.139, the fl ow is assumed uniform and hydrostatic at sections 1 and 2. If losses are neglected, compute ( a ) V2 and ( b ) the force per unit width of the water on the spillway. 5 m V1 V2 0.7 m

For the water channel fl ow of Fig. P3.140, h1 5 1.5 m, H 5 4 m, and V1 5 3 m/s. Neglecting losses and assuming uniform fl ow at sections 1 and 2, fi nd the downstream depth h2 , and show that two realistic solutions are possible. P3.140 h1 h2 V2

For the water channel fl ow of Fig. P3.141, h1 5 0.45 ft, H 5 2.2 ft, and V1 5 16 ft/s. Neglecting losses and assuming uniform fl ow at sections 1 and 2, fi nd the downstream depth h2 ; show that two realistic solutions are possible. 210 Chapter 3 Integral Relations for a Control Volume P3.141 h2 h1 V1 V2 H

A cylindrical tank of diameter D contains liquid to an initial height h0 . At time t 5 0 a small stopper of diameter d is removed from the bottom. Using Bernoullis equation with no losses, derive ( a ) a differential equation for the freesurface height h ( t ) during draining and ( b ) an expression for the time t0 to drain the entire tank

The large tank of incompressible liquid in Fig. P3.143 is at rest when, at t 5 0, the valve is opened to the atmosphere. Assuming h < constant (negligible velocities and accelerations in the tank), use the unsteady frictionless Bernoulli equation to derive and solve a differential equation for V ( t ) in the pipe. h constant L Valve V (t)

A fi re hose, with a 2-in-diameter nozzle, delivers a water jet straight up against a ceiling 8 ft higher. The force on the ceiling, due to momentum change, is 25 lbf. Use Bernoullis equation to estimate the hose fl ow rate, in gal/min. [ Hint: The water jet area expands upward.]

The incompressible fl ow form of Bernoullis relation, Eq. (3.54), is accurate only for Mach nu m bers less than about 0.3. At higher speeds, variable density must be accounted for. The most common assumption for compressible fl uids is isentropic fl ow of an ideal gas, or p 5 Ck , where k 5 cp/c. Substitute this rel a tion into Eq. (3.52), integrate, and eliminate the constant C . Compare your compressible result with Eq. (3.54) and comment.

The pump in Fig. P3.146 draws gasoline at 20 8 C from a reservoir. Pumps are in big trouble if the liquid vaporizes (cavitates) before it enters the pump. ( a ) Neglecting losses and assuming a fl ow rate of 65 gal/min, fi nd the limitations on ( x , y , z ) for avoiding cavitation. ( b ) If pipe friction losses are included, what additional limitations might be important? Pump y z x D = 3 cm patm = 100 kPa Gasoline, SG = 0.68

The very large water tank in Fig. P3.147 is discharging through a 4-in-diameter pipe. The pump is running, with a performance curve hp < 40 2 4 Q2 , with hp in feet and Q in ft 3 /s. Estimate the discharge fl ow rate in ft 3 /s if the pipe friction loss is 1.5( V 2 /2 g ). P3.147 30 ft Pump Q, V

By neglecting friction, ( a ) use the Bernoulli equation between surfaces 1 and 2 to estimate the volume fl ow through the orifi ce, whose diameter is 3 cm. ( b ) Why is the result to part ( a ) absurd? ( c ) Suggest a way to resolve this paradox and fi nd the true fl ow rate. Problems 211 pa 2.5 m 4 m 1 m 1

The horizontal lawn sprinkler in Fig. P3.149 has a water fl ow rate of 4.0 gal/min introduced vertically through the center. Estimate ( a ) the retarding torque required to keep the arms from rotating and ( b ) the rotation rate (r/min) if there is no retarding torque. P3.149 R = 6 in d = in

In Prob. P3.60 fi nd the torque caused around fl ange 1 if the center point of exit 2 is 1.2 m directly below the fl ange center.

The wye joint in Fig. P3.151 splits the pipe fl ow into equal amounts Q /2, which exit, as shown, a distance R0 from the axis. Neglect gravity and friction. Find an expression for the torque T about the x axis required to keep the system rotating at angular velocity V . Q Q Q x T, 2 2 R0 >> Dpipes R0

Modify Example 3.19 so that the arm starts from rest and spins up to its fi nal rotation speed. The moment of inertia of the arm about O is I0 . Neglecting air drag, fi nd d / dt and integrate to determine the angular velocity ( t ), assuming 5 0 at t 5 0

The three-arm lawn sprinkler of Fig. P3.153 receives 20 8 C water through the center at 2.7 m 3 /h. If collar friction is negligible, what is the steady rotation rate in r/min for ( a ) 5 0 8 and ( b ) 5 40 8 ? P3.153 d = 7 mm R = 15 cm

Water at 20 8 C fl ows at 30 gal/min through the 0.75-indiameter double pipe bend of Fig. P3.154. The pressures are p1 5 30 lbf/in 2 and p2 5 24 lbf/in 2 . Compute the torque T at point B necessary to keep the pipe from rotating. P3.154 50 1 2 3 ft B

The centrifugal pump of Fig. P3.155 has a fl ow rate Q and exits the impeller at an angle 2 relative to the blades, as shown. The fl uid enters axially at section 1. Assuming incompressible fl ow at shaft angular velocity , derive a formula for the power P required to drive the impeller. Vrel, 2 2 Blade R2 R1 Q T, P, b2

A simple turbomachine is constructed from a disk with two internal ducts that exit tangentially through square holes, as in Fig. P3.156. Water at 20 8 C enters normal to the disk at the center, as shown. The disk must drive, at 250 r/min, a small device whose retarding torque is 1.5 N ? m. What is the proper mass fl ow of water, in kg/s? Q 2 cm 2 cm 32 cm

Reverse the fl ow in Fig. P3.155, so that the system operates as a radial-infl ow turbine . Assuming that the outfl ow into section 1 has no tangential velocity, derive an expression for the power P extracted by the turbine

Revisit the turbine cascade system of Prob. P3.78, and derive a formula for the power P delivered, using the angular momentum theorem of Eq. (3.59).

A centrifugal pump impeller delivers 4000 gal/min of water at 20 8 C with a shaft rotation rate of 1750 r/min. Neglect losses. If r1 5 6 in, r2 5 14 in, b1 5 b2 5 1.75 in, Vt1 5 10 ft/s, and Vt2 5 110 ft/s, compute the absolute velocities ( a ) V1 and ( b ) V2 and ( c ) the horsepower required. ( d ) Compare with the ideal horsepower required.

The pipe bend of Fig. P3.160 has D1 5 27 cm and D2 5 13 cm. When water at 20 8 C fl ows through the pipe at 4000 gal/min, p1 5 194 kPa (gage). Compute the torque required at point B to hold the bend stationary. P3.160 1 2 V2, p2 = pa B 50 cm 50 cm V1, p1 C

Extend Prob. P3.46 to the problem of computing the center of pressure L of the normal face Fn , as in Fig. P3.161. (At the center of pressure, no moments are required to hold the plate at rest.) Neglect friction. Express your result in terms of the sheet thickness h1 and the angle between the plate and the oncoming jet 1. P3.161 , V V h1 h2 V Fn L h3

The waterwheel in Fig. P3.162 is being driven at 200 r/min by a 150-ft/s jet of water at 20 8 C. The jet diameter is 2.5 in. Assuming no losses, what is the horsepower developed by the wheel? For what speed V r/min will the horsepower developed be a maximum? Assume that there are many buckets on the waterwheel. 75 4

3 A rotating dishwasher arm delivers at 60 8 C to six nozzles, as in Fig. P3.163. The total fl ow rate is 3.0 gal/min. Each nozzle has a diameter of 3 16 in. If the nozzle fl ows are equal and friction is neglected, estimate the steady rotation rate of the arm, in r/min. Problems 213 5 in 40 5 in 6 in P3.163

A liquid of density fl ows through a 90 8 bend as shown in Fig. P3.164 and issues vertically from a uniformly porous section of length L . Neglecting pipe and liquid weight, derive an expression for the torque M at point 0 required to hold the pipe stationary. Q x y 0 d

There is a steady isothermal fl ow of water at 20 8 C through the device in Fig. P3.165. Heat-transfer, gravity, and temperature effects are negligible. Known data are D1 5 9 cm, Q1 5 220 m 3 /h, p1 5 150 kPa, D2 5 7 cm, Q2 5 100 m 3 /h, p2 5 225 kPa, D3 5 4 cm, and p3 5 265 kPa. Compute the rate of shaft work done for this device and its direction. P3.165 1 2 3

A power plant on a river, as in Fig. P3.166, must eliminate 55 MW of waste heat to the river. The river conditions upstream are Qi 5 2.5 m 3 /s and Ti 5 18 8 C. The river is 45 m wide and 2.7 m deep. If heat losses to the atmosphere and ground are negligible, estimate the downstream river conditions ( Q0 , T0 ). P3.166 Q0, T0 Qi , Ti Power plant Q Q T + T T

For the conditions of Prob. P3.166, if the power plant is to heat the nearby river water by no more than 12 8 C, what should be the minimum fl ow rate Q , in m 3 /s, through the plant heat exchanger? How will the value of Q affect the downstream conditions ( Q0 , T0 )?

Multnomah Falls in the Columbia River Gorge has a sheer drop of 543 ft. Using the steady fl ow energy equation, estimate the water temperature change in 8 F caused by this drop.

When the pump in Fig. P3.169 draws 220 m 3 /h of water at 20 8 C from the reservoir, the total friction head loss is 5 m. The fl ow discharges through a nozzle to the atmosphere. Estimate the pump power in kW delivered to the water. Water 6 m 2 m Pump Ve De D = 12 cm = 5 cm

A steam turbine operates steadily under the following conditions. At the inlet, p 5 2.5 MPa, T 5 450 8 C, and V 5 40 m/s. At the outlet, p 5 22 kPa, T 5 70 8 C, and V 5 225 m/s. ( a ) If we neglect elevation changes and heat transfer, how much work is delivered to the turbine blades, in kJ/kg? ( b ) If the mass fl ow is 10 kg/s, how much total power is delivered? ( c ) Is the steam wet as it leaves the exit?

Consider a turbine extracting energy from a penstock in a dam, as in Fig. P3.171. For turbulent pipe fl ow (Chap. 6), the friction head loss is approximately hf 5 CQ2 , where the constant C depends on penstock dimensions and the properties of water. Show that, for a given penstock geometry and variable river fl ow Q , the maximum turbine power possible in this case is Pmax 5 2 gHQ /3 and occurs when the fl ow rate is Q 5 1H/(3C). Penstock Q Turbine H

The long pipe in Fig. P3.172 is fi lled with water at 20 8 C. When valve A is closed, p1 2 p2 5 75 kPa. When the valve is open and water fl ows at 500 m 3 /h, p1 2 p2 5 160 kPa. What is the friction head loss between 1 and 2, in m, for the fl owing condition? P3.172 2 1 Constantdiameter pipe A

A 36-in-diameter pipeline carries oil (SG 5 0.89) at 1 million barrels per day (bbl/day) (1 bbl 5 42 U.S. gal). The friction head loss is 13 ft/1000 ft of pipe. It is planned to place pumping stations every 10 mi along the pipe. Estimate the horsepower that must be delivered to the oil by each pump.

The pump-turbine system in Fig. P3.174 draws water from the upper reservoir in the daytime to produce power for a city. At night, it pumps water from lower to upper reservoirs to restore the situation. For a design fl ow rate of 15,000 gal/min in either direction, the friction head loss is 17 ft. Estimate the power in kW ( a ) extracted by the turbine and ( b ) delivered by the pump. Water at 20C Pumpturbine 1 2 Z1 = 150 ft Z 2 = 25 ft

Water at 20 8 C is delivered from one reservoir to another through a long 8-cm-diameter pipe. The lower reservoir has a surface elevation z2 5 80 m. The friction loss in the pipe is correlated by the formula hloss < 17.5( V2 /2 g ), where V is the average velocity in the pipe. If the steady fl ow rate through the pipe is 500 gallons per minute, estimate the surface elevation of the higher reservoir.

A fi reboat draws seawater (SG 5 1.025) from a submerged pipe and discharges it through a nozzle, as in Fig. P3.176. The total head loss is 6.5 ft. If the pump effi ciency is 75 percent, what horsepower motor is required to drive it? P3.176 D = 2 in 120 ft/s D = 6 in Pump 6 ft 10 ft

A device for measuring liquid viscosity is shown in Fig. P3.177. With the parameters ( , L, H, d ) known, the fl ow rate Q is measured and the viscosity calculated, assuming a laminar-fl ow pipe loss from Chap. 6, hf 5 (32 LV )/( gd2 ). Heat transfer and all other losses are negligible. ( a ) Derive a formula for the viscosity of the fl uid. ( b ) Calculate for the case d 5 2 mm, = 800 kg/m 3 , L 5 95 cm, H 5 30 cm, Problems 215 and Q 5 760 cm 3 /h. ( c ) What is your guess of the fl uid in part ( b )? ( d ) Verify that the Reynolds number Re d is less than 2000 (laminar pipe fl ow). P3.177 Water level H L

The horizontal pump in Fig. P3.178 discharges 20 8 C water at 57 m 3 /h. Neglecting losses, what power in kW is delivered to the water by the pump? P3.178 D2 = 3 cm D1 = 9 cm 120 kPa 400 kPa Pump

Steam enters a horizontal turbine at 350 lbf/in 2 absolute, 580 8 C, and 12 ft/s and is discharged at 110 ft/s and 25 8 C saturated conditions. The mass fl ow is 2.5 lbm/s, and the heat losses are 7 Btu/lb of steam. If head losses are negligible, how much horsepower does the turbine develop?

Water at 20 8 C is pumped at 1500 gal/min from the lower to the upper reservoir, as in Fig. P3.180. Pipe friction losses are approximated by hf < 27 V2 /(2 g ), where V is the average velocity in the pipe. If the pump is 75 percent effi cient, what horsepower is needed to drive it? P3.180 z2 = 150 ft z1 = 50 ft D = 6 in Pump

A typical pump has a head that, for a given shaft rotation rate, varies with the fl ow rate, resulting in a pump performance curve as in Fig. P3.181. Suppose that this pump is 75 percent effi cient and is used for the system in Prob. 3.180. Estimate ( a ) the fl ow rate, in gal/min, and ( b ) the horsepower needed to drive the pump. Head, ft Flow rate, ft3/s 300 200 100 0 01234

The insulated tank in Fig. P3.182 is to be fi lled from a highpressure air supply. Initial conditions in the tank are T 5 20 8 C and p 5 200 kPa. When the valve is opened, the initial mass fl ow rate into the tank is 0.013 kg/s. Assuming an ideal gas, estimate the initial rate of temperature rise of the air in the tank. Valve Air supply: Tank : = 200 L T1 = 20C P1 = 1500 kPa

The pump in Fig. P3.183 creates a 20 8 C water jet oriented to travel a maximum horizontal distance. System friction head losses are 6.5 m. The jet may be approximated by the trajectory of frictionless particles. What power must be delivered by the pump? 216 Chapter 3 Integral Relations for a Control Volume 2 m 25 m Jet Pump 15 m D = 10 cm De = 5 cm

The large turbine in Fig. P3.184 diverts the river fl ow under a dam as shown. System friction losses are hf 5 3.5 V2 /(2 g ), where V is the average velocity in the supply pipe. For what river fl ow rate in m 3 /s will the power extracted be 25 MW? Which of the two possible solutions has a better conversion effi ciency? P3.184 Turbine D = 4 m z2 = 10 m z3 = 0 m z1 = 50 m

Kerosine at 20 8 C fl ows through the pump in Fig. P3.185 at 2.3 ft 3 /s. Head losses between 1 and 2 are 8 ft, and the pump delivers 8 hp to the fl ow. What should the mercury manometer reading h ft be? P3.185 Mercury V1 h? V2 5 ft D2 = 6 in D1 = 3 in Pump