¡Descarga Fluid Mechanics: Control Volume Analysis Exercises y más Esquemas y mapas conceptuales en PDF de Mecánica solo en Docsity!
Problems 253
Go to Appendix G ( WileyPLUS or the book’s website, www. wiley.com/college/munson) for a set of review problems with an- swers. Detailed solutions can be found in the Student Solution
Manual and Study Guide for Fundamentals of Fluid Mechanics, by Munson et al. (© 2013 John Wiley and Sons, Inc.).
Review Problems
Conceptual Questions
5.1C A fluid flows steadily through a pipe with a uniform cross- sectional area. The density of the fluid decreases to half its initial value as it flows through the pipe. The correct statement about the average velocity V is
5.3C The nozzle on a fire hose is connected to the hose via a cou- pling. When the fire hose is in use with water flowing through it and the hose is stationary, the coupling is: a) in equilibrium, so there is no force on the coupling. b) in tension. c) in compression. 5.4C Two fluid jets are pointed at surfaces as shown in the fol- lowing figures. The fluids are incompressible, and the effects of gravity can be neglected. The mass flowrates and the velocities of the jets are identical. The cross-sectional areas of the jets do not change significantly as the fluid flows. The correct statement regarding the horizontal forces is a) F 1 equals 2 F 2. b) F 1 is greater than 0 and F 2 equals 0. c) F 1 equals F 2 /2. d) F 1 equals 0 and F 2 is greater than 0. e) F 1 equals F 2.
Problems
Note: Unless specific values of required fluid properties are given
in the problem statement, use the values found in the tables on the inside of the front cover. Answers to the even-numbered problems are listed at the end of the book. The Lab Problems as well as the videos that accompany problems can be accessed in WileyPLUS or the book’s website, www.wiley.com/college/munson.
Section 5.1.1 Derivation of the Continuity Equation
5.1 Use the Reynolds transport theorem (Eq. 4.19) with B vol- ume and, therefore, b volume/mass 1/density to obtain the
continuity equation for steady or unsteady incompressible flow through a fixed control volume:.
5.2 An incompressible fluid flows horizontally in the x – y plane with a velocity given by
where y and h are in meters and h is a constant. Determine the av- erage velocity for the portion of the flow between y 0 and y h.
u 301 y h 21 ^2 ms, v 0
cv
V nˆ^ dA 0
r 1 r 2 = r 1 /
V 1 V 2 =?
Flow
a) V 2 equals 2 V 1. b) V 2 equals V 1 /2. c) V 2 equals V 1.
d) V 2 equals V 1 /4. e) V 2 equals 4 V 1.
5.2C Water flows steadily into and out of the system with four pipes shown below. The mass flowrate through three of the pipes in kg/s is indicated.
System
4 kg/s
3 kg/s
6 kg/s m •
Nozzle
F 1
F 2
Nozzle
Fluid stream
Fluid stream
Surface
Surface
The mass flowrate in the fourth pipe is a) 13 kg/s. b) 9 kg/s. c) 6 kg/s. d) 5 kg/s. e) 4kg/s.
m
Additional conceptual questions are available in WileyPLUS at the instructor’s discretion.
Section 5.1.2 Fixed, Nondeforming Control Volume—
Uniform Velocity Profile or Average Velocity
5.3 Water flows steadily through the horizontal piping system shown in Fig. P5.3. The velocity is uniform at section (1), the mass flowrate is 146 kg/s at section (2), and the velocity is nonuniform
at section (3). (a) Determine the value of the quantity ,
where the system is the water contained in the pipe bounded by sections (1), (2), and (3). (b) Determine the mean velocity at sec- tion (2). (c) Determine, if possible, the value of the integral
over section (3). If it is not possible, explain what
additional information is needed to do so.
132 rV^ ^ n ˆ^ dA
D
Dt sys
r dV
254 Chapter 5 ■ Finite Control Volume Analysis
146 kg/s Area = 0.03 m 2
(3) Area = 0.07 m^2
4.6 m/s
(1) Area = 0.07 m 2
(2)
■ Figure P5.
0.02 m diameter
0.03 m
Inlet
Blades
0.18 m 60 °
V = 3.0 m/s
■ Figure P5.
5.4 Water flows out through a set of thin, closely spaced blades as shown in Fig. P5.4 with a speed of around the entire circumference of the outlet. Determine the mass flowrate through the inlet pipe.
V 3.0 ms
† 5.5 Estimate the rate (in L/h) that your car uses gasoline when it is being driven on an interstate highway. Determine how long it would take to empty a 360 ml soft-drink container at this flowrate. List all assumptions and show calculations.
5.6 The pump shown in Fig. P5.6 produces a steady flow of 40 L/min through the nozzle. Determine the nozzle exit diameter,
D 2 ,if the exit velocity is to be V 2 30 ms.
■ Figure P5.
Section (1)
Section (2)
D 2
Pump^ V 2
■ Figure P5.
Three 1.0 cm diameter overflow holes (^) Q = 8 L/min
Drain
5.7 Water flows into a sink as shown in Video V5.1 and Fig. P5.7 at a rate of 8 L/min. Determine the average velocity through each of the three 1.0 cm diameter overflow holes if the drain is closed and the water level in the sink remains constant.
5.8 The wind blows through a garage door opening with a speed of 1.5 ms as shown in Fig. P5.8. Determine the aver- age speed, V , of the air through the two openings in the windows.
1 m 1.2 m
2 m 3 m
5 m 3 m
7 m
1 m 1 m
V V
1.5 m/s
20 °
■ Figure P5.
5.9 The human circulatory system consists of a complex branch- ing pipe network ranging in diameter from the aorta (largest) to the capillaries (smallest). The average radii and the number of these vessels are shown in the table. Does the average blood velocity in- crease, decrease, or remain constant as it travels from the aorta to the capillaries?
256 Chapter 5 ■ Finite Control Volume Analysis
*5.19 To measure the mass flowrate of air through a 15 cm inside-diameter pipe, local velocity data are collected at different radii from the pipe axis (see Table). Determine the mass flowrate corresponding to the data listed in the following table.
5.21 Various types of attachments can be used with the shop vac shown in Video V5.2. Two such attachments are shown in Fig. P5.21—a nozzle and a brush. The flowrate is (a) Deter- mine the average velocity through the nozzle entrance, (b) Assume the air enters the brush attachment in a radial direction all around the brush with a velocity profile that varies linearly from 0 to along the length of the bristles as shown in the figure. Determine the value of Vb.
Vb
Vn.
0.03 m (^3) s.
5.22 An appropriate turbulent pipe flow velocity profile is
where centerline velocity, local radius, pipe radius, and vector along pipe centerline. Determine the ratio of av- erage velocity, to centerline velocity, for (a) (b) (c) (d) Compare the different velocity profiles.
5.23 As shown in Fig. P5.23, at the entrance to a 1 m wide channel the velocity distribution is uniform with a velocity V. Further downstream the velocity profile is given by where u is in m/s and y is in m. Determine the value of V.
u 4 y 2 y^2 ,
GO
n 8, n 10.
u , uc , n 4, n 6,
iˆ^ unit
uc r R
V uc a
R r R
b
(^1) n iˆ
5.20 Two rivers merge to form a larger river as shown in Fig. P5.20. At a location downstream from the junction 1 before the two streams completely merge^2 , the nonuniform veloc- ity profile is as shown and the depth is 1.8 m. Determine the value of V.
5.24 An incompressible flow velocity field (water) is given as
V
r
eˆ r^
r
eˆ u^ ms
5.17 Fresh water flows steadily into an open 200 L drum ini- tially filled with seawater. The fresh water mixes thoroughly with the seawater, and the mixture overflows out of the drum. If the fresh water flowrate is 40 L/min, estimate the time in seconds required to decrease the difference between the density of the mixture and the density of fresh water by 50%.
Section 5.1.2 Fixed, Nondeforming Control Volume—
Nonuniform Velocity Profile
5.18 A water jet pump 1 see Fig. P5.18 2 involves a jet cross- sectional area of and a jet velocity of 30 m/s. The jet is surrounded by entrained water. The total cross-sectional area asso- ciated with the jet and entrained streams is These two fluid streams leave the pump thoroughly mixed with an average velocity of 6 m/s through a cross-sectional area of Deter- mine the pumping rate 1 i.e., the entrained fluid flowrate 2 involved in liters/s.
0.075 m 2.
0.075 m 2.
0.01 m 2 ,
■ Figure P5.
1.2 m/s
0.6 m/s
Depth = 0.6 m
Depth = 1.5 m
24 m
15 m
0.8 V
V
21 m
9 m
■ Figure P5.
Entrained water
Entrained water
30 m/s jet
6 m/s
r (cm) Axial Velocity (m/s) 0 9 0.5 9. 1.0 8. 1.5 8. 2.0 8. 2.5 8. 3.0 8. 3.5 8. 4.0 8. 4.5 8. 5.0 7. 5.5 7. 6.0 7. 6.5 6. 7.0 6. 7.4 5. 7.5 5. 7.6 4. 7.6 0
■ Figure P5.
Q = 0.03 m^3 /s
Q = 0.03 m 3 /s
Vn
5 cm dia.
4 cm
Vb 8 cm dia.
■ Figure P5.
u = 4 y – 2 y^2
x
0.3 m 0.2 m y
V
Problems 257
where r is in meters. (a) Calculate the mass flowrate through the cylindrical surface at m from to m as shown in Fig. P5.24 a. (b) Show that mass is conserved in the annular control volume from m to m and to m as shown in Fig. P5.24 b.
r 1 r 2 z 0 z 1
r 1 z 0 z 1
5.25 Flow of a viscous fluid over a flat plate surface results in the development of a region of reduced velocity adjacent to the wetted surface as depicted in Fig. P5.25. This region of reduced flow is called a boundary layer. At the leading edge of the plate, the veloc- ity profile may be considered uniformly distributed with a value U. All along the outer edge of the boundary layer, the fluid velocity component parallel to the plate surface is also U. If the x -direction velocity profile at section 122 is
develop an expression for the volume flowrate through the edge of the boundary layer from the leading edge to a location downstream at x where the boundary layer thickness is d.
u U
a
y d
b
(^1) 7
Section 5.1.2 Fixed, Nondeforming Control Volume—
Unsteady Flow
5.26 Air at standard conditions enters the compressor shown in Fig. P5.26 at a rate of It leaves the tank through a 3 cm diameter pipe with a density of 1.8 kg/m^3 and a uniform speed of 210 m/s. (a) Determine the rate (kg/s) at which the mass of air in the tank is increasing or decreasing. (b) Determine the average time rate of change of air density within the tank.
0.3 m^3 s.
■ Figure P5.
z
( a )
1 m
1 m
z
( b )
1 m
2 m
1 m
■ Figure P5.
U U
x
δ
Section (1)
Section (2) Outer edge of boundary layer
■ Figure P5.
Tank volume = 0.6 m^3 3 cm 210 m/s 1.8 kg/m 3
0.3 m^3 /s
Compressor
1.2 kg/m 3
5.27 Estimate the time required to fill with water a cone- shaped container (see Fig. P5.27) 1.5 m high and 1.5 m across at the top if the filling rate is 76 L/min.
■ Figure P5.
1.5 m
1.5 m
5.28 How long would it take to fill a cylindrical-shaped swimming pool having a diameter of 8 m to a depth of 1.5 m with water from a garden hose if the flowrate is 1.0 liter/s?
Section 5.1.3 Moving, Nondeforming Control Volume
† 5.29 For an automobile moving along a highway, describe the
control volume you would use to estimate the flowrate of air across the radiator. Explain how you would estimate the velocity of that air.
Section 5.1.4 Deforming Control Volume
5.30 A hypodermic syringe (see Fig. P5.30) is used to apply a vaccine. If the plunger is moved forward at the steady rate of 20 mm/s and if vaccine leaks past the plunger at 0.1 of the volume flowrate out the needle opening, calculate the average velocity of the needle exit flow. The inside diameters of the syringe and the needle are 20 mm and 0.7 mm.
■ Figure P5.
Q leak (^) Q out
5.31 The Hoover Dam (see Video V2.4 ) backs up the Colorado River and creates Lake Mead, which is approximately 185 km long and has a surface area of approximately 580 square kilo- meters. If during flood conditions the Colorado River flows into the lake at a rate of 1300 m^3 /s and the outflow from the dam is 227 m^3 /s, how many meters per 24-hour day will the lake level rise? 5.32 Storm sewer backup causes your basement to flood at the steady rate of 2.5 cm of depth per hour. The basement floor area is 139 m^2. What capacity 1 L/min 2 pump would you rent to (a) keep the water accumulated in your basement at a constant level until the storm sewer is blocked off, and (b) reduce the water accumulation in your basement at a rate of 7.5 cm/h even while the backup problem exists? 5.33 (See Fluids in the News article “New 6 LPF standards,” Section 5.1.2.) When a toilet is flushed, the water depth, h , in the tank as a function of time, t , is as given in the table. The size of the rec- tangular tank is 50 cm by 20 cm. (a) Determine the volume of water used per flush, liter per flush. (b) Plot the flowrate for 0 t 6 s.
t (s) h (cm) 0 14. 0.5 13. 1.0 12. 2.0 8. 3.0 6. 4.0 3. 5.0 1. 6.0 0
to hold the wedge stationary are FH and F (^) V , respectively. Gravity is negligible, and the fluid speed remains constant. Determine the force ratio, FH / FV.
Problems 259
5.42 Water enters the horizontal, circular cross-sectional, sudden contraction nozzle sketched in Fig. P5.42 at section (1) with a uniformly distributed velocity of 7.5 m/s and a pressure of 500 kPa. The water exits from the nozzle into the atmosphere at section (2) where the uniformly distributed velocity is 30 m/s. Determine the axial component of the anchoring force required to hold the contraction in place.
†5.43 A truck carrying chickens is too heavy for a bridge that it needs to cross. The empty truck is within the weight limits; with the chickens it is overweight. It is suggested that if one could get the chickens to fly around the truck (i.e., by banging on the truck’s side) it would be safe to cross the bridge. Do you agree? Explain.
5.44 Exhaust (assumed to have the properties of stan- dard air) leaves the 1 m diameter chimney shown in Video V5.4 and Fig. P5.44 with a speed of 2 m/s. Because of the wind, after a few diameters downstream the exhaust flows in a horizontal direction with the speed of the wind, 5 m/s. Determine the horizontal com- ponent of the force that the blowing wind exerts on the exhaust gases.
5.45 Air flows steadily between two cross sections in a long, straight section on 30 cm inside-diameter pipe. The static temper- ature and pressure at each section are indicated in Fig P5.45. If the average air velocity at section (2) is 320 m/s, determine the aver- age air velocity at section (1). Determine the frictional force ex- erted by the pipe wall on the air flowing between sections (1) and (2). Assume uniform velocity distributions at each section.
5.46 Water flows steadily from a tank mounted on a cart as shown in Fig. 5.46. After the water jet leaves the nozzle of the tank, it falls and strikes a vane attached to another cart. The cart’s wheels are frictionless, and the fluid is inviscid. (a) Determine the speed of the water leaving the tank, V 1 , and the water speed leaving the cart, V 2. (b) Determine the tension in rope A. (c) Determine the tension in rope B.
5.47 Determine the magnitude and direction of the anchor- ing force needed to hold the horizontal elbow and nozzle combi- nation shown in Fig. P5.47 in place. Atmospheric pressure is 100 kPa(abs). The gage pressure at section (1) is 100 kPa. At section (2), the water exits to the atmosphere.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
FH FV
V V
V
Free jet
q = 30°
D 1 = 7.5 cm p 1 = 500 kPa V 1 = 7.5 m/s
p 2 = 0 kPa
V 2 = 30 m/s
Section (2)
Section (1)
5 m/s 5 m/s
2 m/s 1 m
D = 30 cm
Section (1) Section (2) p 1 = 690 kPa (abs) T 1 = 300 K
p 2 = 127 kPa (abs) T 2 = 252 K V 2 = 320 m/s
V 2
V 1
2 m
4 m
Nozzle area = 0.01 m^2 Horizontal free jets
Rope A Rope B ■ Figure P5.
■ Figure P5.
■ Figure P5.
160 mm
300 mm
Section (2)
Section (1)
y
x
Water
V 2
V 1
p 1 = 100 kPa V 1 = 2 m/s
5.48 Water is added to the tank shown in Fig. P5.48 through a vertical pipe to maintain a constant (water) level. The tank is placed
Jet area F = 625 mm^2
Jet area = 1250 mm^2
Frictionless surface
Constant water level
1 m
1 m
on a horizontal plane which has a frictionless surface. Determine the horizontal force, F , required to hold the tank stationary. Neglect all losses. 5.49 Water flows as two free jets from the tee attached to the pipe shown in Fig. P5.49. The exit speed is 15 m/s. If viscous effects and gravity are negligible, determine the x and y compo- nents of the force that the pipe exerts on the tee.
GO
260 Chapter 5 ■ Finite Control Volume Analysis
5.50 A nozzle is attached to a vertical pipe and discharges water into the atmosphere as shown in Fig. P5.50. When the discharge is 0.1 m^3 /s, the gage pressure at the flange is 40 kPa. Determine the vertical component of the anchoring force required to hold the nozzle in place. The nozzle has a weight of 200 N, and the vol- ume of water in the nozzle is 0.012 m 3. Is the anchoring force di- rected upward or downward?
5.51 The hydraulic dredge shown in Fig. P5.51 is used to dredge sand from a river bottom. Estimate the thrust needed from the propeller to hold the boat stationary. Assume the specific grav- ity of the sand/water mixture is SG 1.4.
5.52 A static thrust stand is to be designed for testing a specific jet engine, knowing the following conditions for a typi- cal test.
estimate a nominal thrust to design for. 5.53 A vertical jet of water leaves a nozzle at a speed of 10 m/s and a diameter of 20 mm. It suspends a plate having a mass of 1.5 kg as indicated in Fig. P5.53. What is the vertical distance h?
exhaust gas pressure 0 kPa
intake static temperature 267 K
intake static pressure 79 kPa
intake cross section area 1 m 2
exhaust gas velocity 500 ms
intake air velocity 210 ms
5.54 A horizontal, circular cross-sectional jet of air having a diameter of 15 cm strikes a conical deflector as shown in Fig. P5.54. A horizontal anchoring force of 22 N is required to hold the cone in place. Estimate the nozzle flowrate in m^3 /s. The magnitude of the velocity of the air remains constant.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
y x
V = 15 m/s
V = 15 m/s
Area = 0.3 m^2
Area = 0.5 m^2
Area = 1 m^2
Pipe Tee
Area = 0.02 m^2
Area = 0.01 m^2
g
30 °
p = 40 kPa
0.10 m 3 /s
Nozzle
0.6 m diameter 30 °
10 m/s
3 m 2 m^ Prop
h
15 cm
60 ° FA = 22 N
5.55 A vertical, circular cross-sectional jet of air strikes a conical deflector as indicated in Fig. P5.55. A vertical anchoring force of 0.1 N is required to hold the deflector in place. Determine the mass (kg) of the deflector. The magnitude of velocity of the air remains constant.
0.1 m
V = 30 m/s
FA = 0.1 N
60 °
262 Chapter 5 ■ Finite Control Volume Analysis
5.63 Water flows steadily into and out of a tank that sits on fric- tionless wheels as shown in Fig. P5.63. Determine the diameter D so that the tank remains motionless if F 0.
5.64 The rocket shown in Fig. P5.64, is held stationary by the horizontal force, Fx , and the vertical force, Fz. The velocity and pressure of the exhaust gas are 1500 m/s and 138 kPa at the nozzle exit, which has a cross section area of 390 cm^2. The exhaust mass flowrate is constant at 10 kg/s. Determine the value of the restrain- ing force Fx. Assume the exhaust flow is essentially horizontal.
5.65 A horizontal circular jet of air strikes a stationary flat plate as indicated in Fig. P5.65. The jet velocity is 40 m/s and the jet diameter is 30 mm. If the air velocity magnitude remains con- stant as the air flows over the plate surface in the directions shown, determine: (a) the magnitude of FA , the anchoring force required to
GO
hold the plate stationary; (b) the fraction of mass flow along the plate surface in each of the two directions shown; (c) the magnitude of FA , the anchoring force required to allow the plate to move to the right at a constant speed of 10 m/s. 5.66 Air discharges from a 5 cm diameter nozzle and strikes a curved vane, which is in a vertical plane as shown in Fig. P5.66. A stagnation tube connected to a water U-tube manometer is lo- cated in the free air jet. Determine the horizontal component of the force that the air jet exerts on the vane. Neglect the weight of the air and all friction.
5.67 Water is sprayed radially outward over 180 as indi- cated in Fig. P5.67. The jet sheet is in the horizontal plane. If the jet velocity at the nozzle exit is 6 m/s, determine the direction and magnitude of the resultant horizontal anchoring force required to hold the nozzle in place.
5.68 A sheet of water of uniform thickness ( h 0.01 m) flows from the device shown in Fig. P5.68. The water enters verti- cally through the inlet pipe and exits horizontally with a speed that varies linearly from 0 to 10 m/s along the 0.2 m length of the slit. Determine the y component of anchoring force necessary to hold this device stationary.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
3.0 m
1.2 m/s
0.45 m
F
d
D (^) d
Fz
Fx
Vj = 40 m/s
Dj = 30 mm
V 3
V 2
30 °
90 °
FA
18 cm
Fixed vane
5 cm dia.
Free air jet
Stagnation tube Air
Water
Open
30 °
20 cm (^) 1.0 cm V = 6 m/s
0.2 m
h = 0.01 m
x
y 0 m/s
10 m/s
Q
5.69 The results of a wind tunnel test to determine the drag on a body (see Fig. P5.69) are summarized below. The upstream [sec- tion (1)] velocity is uniform at 30 m/s. The static pressures are given by p 1 p 2 101 kPa. The downstream velocity distribution, which is symmetrical about the centerline, is given by
where u is the velocity in m/s and y is the distance on either side of the centerline in feet (see Fig. P5.69). Assume that the body shape does not change in the direction normal to the paper. Calculate the drag force (reaction force in x direction) exerted on the air by the body per unit length normal to the plane of the sketch.
u 30 y 7 1 m
u 30 19 3 y 211 y 2 y 1 m
Problems 263
5.70 A variable mesh screen produces a linear and axisym- metric velocity profile as indicated in Fig. P5.70 in the airflow through a 0.6 m diameter circular cross-sectional duct. The static pressures upstream and downstream of the screen are 1.4 kPa and 1.0 kPa and are uniformly distributed over the flow cross-sectional area. Neglecting the force exerted by the duct wall on the flowing air, calculate the screen drag force.
5.71 Consider unsteady flow in the constant diameter, horizontal pipe shown in Fig. P5.71. The velocity is uniform throughout the entire pipe, but it is a function of time:. Use the x com- ponent of the unsteady momentum equation to determine the pres- sure difference p 1 p 2. Discuss how this result is related to Fx max.
V u 1 t 2 i ˆ
5.72 In a laminar pipe flow that is fully developed, the axial ve- locity profile is parabolic. That is,
as is illustrated in Fig. P5.72. Compare the axial direction momen- tum flowrate calculated with the average velocity, with the axial direction momentum flowrate calculated with the nonuniform velocity distribution taken into account.
u ,
u uc c 1 a
r R
b
2 d
†5.73 Water from a garden hose is sprayed against your car to rinse dirt from it. Estimate the force that the water exerts on the car. List all assumptions and show calculations. 5.74 A Pelton wheel vane directs a horizontal, circular cross-sectional jet of water symmetrically as indicated in Fig. P5. and Video V5.6. The jet leaves the nozzle with a velocity of 30 m/s. Determine the x -direction component of anchoring force required to (a) hold the vane stationary, (b) confine the speed of the vane to a value of 30 m/s to the right. The fluid speed magnitude re- mains constant along the vane surface.
5.75 The thrust developed to propel the jet ski shown in Video V9.18 and Fig. P5.75 is a result of water pumped through the vehicle and exiting as a high-speed water jet. For the conditions shown in the figure, what flowrate is needed to produce a 1.3 kN thrust? Assume the inlet and outlet jets of water are free jets.
5.76 Thrust vector control is a technique that can be used to greatly improve the maneuverability of military fighter aircraft. It consists of using a set of vanes in the exit of a jet engine to deflect the exhaust gases as shown in Fig. P5.76. (a) Determine the pitch- ing moment (the moment tending to rotate the nose of the aircraft
u Body (^) 1 m
1 m
V 2 = 30 m/s
V 1 = 30 m/s
Section (2)
Section (1)
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
Variable mesh screen
Section (1) Section (2) p 1 = 1.4 kPa V 1 = 30 m/s
D = 0.6 m
p 2 = 1.0 kPa
u ( t )
D x
(1) (2)
ρ= density
uc
u R r
45 °
45 °
D = 2.5 cm
30 m/s
( a )
45 °
45 °
D = 2.5 cm
30 m/s 30 m/s
( b )
y
x
9.0 cm diameter outlet jet
30 ° 160 cm^2 inlet area
5.83 A snowplow mounted on a truck clears a path 3.6 m through heavy wet snow, as shown in Figure P5.83. The snow is 20 cm deep and its density is 160 kg/m 3. The truck travels at 48 km/h. The snow is discharged from the plow at an angle of 45 from the direction of travel and 45 above the horizontal, as shown in Figure P5.83. Estimate the force required to push the plow.
Problems 265
Section 5.2.3 Derivation of the Moment-of-Momentum
Equation
5.84 Describe a few examples (include photographs/images) of turbines where the force/torque of a flowing fluid leads to rotation of a shaft.
5.85 Describe a few examples (include photographs/images) of pumps where a fluid is forced to move by “blades” mounted on a rotating shaft.
5.86 An incompressible fluid flows outward through a blower as indicated in Fig. P5.86. The shaft torque involved, T shaft, is esti- mated with the following relationship:
where mass flowrate through the blower, r 2 outer radius of blower, and V u 2 tangential component of absolute fluid velocity leaving the blower. State the flow conditions that make this formula valid.
m
T shaft m
r 2 V u 2
Section 5.2.4 Application of the Moment-of-Momentum
Equation
5.87 Water enters a rotating lawn sprinkler through its base at the steady rate of 60 L/min as shown in Fig. P5.87. The exit cross-sectional area of each of the two nozzles is 0.26 cm^2 , and the flow leaving each nozzle is tangential. The radius from the axis of rotation to the centerline of each nozzle is 20 cm. (a) Determine the resisting torque required to hold the sprinkler head stationary. (b) Determine the resisting torque associated with the sprinkler ro- tating with a constant speed of 500 rev/min. (c) Determine the an- gular velocity of the sprinkler if no resisting torque is applied.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
5.89 (See Fluids in the News article titled “Tailless Helicopters,” Section 5.2.4.) Shown in Fig. P5.89 is a toy “helicopter” powered
5.88 Five liters/s of water enter the rotor shown in Video V5.10 and Fig. P5.88 along the axis of rotation. The cross-sectional area of each of the three nozzle exits normal to the relative velocity is 18 mm^2. How large is the resisting torque required to hold the rotor stationary? How fast will the rotor spin steadily if the resisting torque is reduced to zero and (a) u 0 , (b) u 30 , (c) u 60 ?
GO
= 45° (in plane of blade)
d = 20 cm
θ
U = 48 km/h
V 2 V (^) θ 2
ω
r (^2) r 1
Q = 60 L/min
r = 20 cm Nozzle exit area = 0.26 cm^2
Q = 5 liters/s
r = 0.5m Nozzle exit area normal to relative velocity = 18 mm^2
θ
Balloon
ω
by air escaping from a balloon. The air from the balloon flows ra- dially through each of the three propeller blades and out through small nozzles at the tips of the blades. Explain physically how this flow can cause the rotation necessary to rotate the blades to produce the needed lifting force. 5.90 A water turbine wheel rotates at the rate of 50 rpm in the di- rection shown in Fig. P5.90. The inner radius, r 2 , of the blade row is 0.6 m, and the outer radius, r 1 , is 1.2 m. The absolute velocity vector at the turbine rotor entrance makes an angle of 20 with the tangential direction. The inlet blade angle is 60 relative to the tan- gential direction. The blade outlet angle is 120. The flowrate is 0.6 m 3 /s. For the flow tangent to the rotor blade surface at inlet and outlet, determine an appropriate constant blade height, b , and the corresponding power available at the rotor shaft.
266 Chapter 5 ■ Finite Control Volume Analysis
5.91 A water turbine with radial flow has the dimensions shown in Fig. P5.91. The absolute entering velocity is 15 m/s, and it makes an angle of 30 with the tangent to the rotor. The absolute exit velocity is directed radially inward. The angular speed of the rotor is 120 rpm. Find the power delivered to the shaft of the tur- bine.
5.92 A fan (see Fig. P5.92) has a bladed rotor of 30 cm outside di- ameter and 13 cm inside diameter and runs at 1725 rpm. The width of each rotor blade is 2.5 cm from blade inlet to outlet. The volume flowrate is steady at 6.51 m^3 /min, and the absolute velocity of the air at blade inlet, V 1 , is purely radial. The blade discharge angle is 30
measured with respect to the tangential direction at the outside di- ameter of the rotor. (a) What would be a reasonable blade inlet an- gle (measured with respect to the tangential direction at the inside diameter of the rotor)? (b) Find the power required to run the fan. 5.93 The radial component of velocity of water leaving the cen- trifugal pump sketched in Fig. P5.93 is 9 m/s. The magnitude of the absolute velocity at the pump exit is 18 m/s. The fluid enters the pump rotor radially. Calculate the shaft work required per unit mass flowing through the pump.
5.94 An axial flow turbomachine rotor involves the upstream (1) and downstream (2) velocity triangles shown in Fig. P5.94. Is this turbomachine a turbine or a fan? Sketch an appropriate blade section and determine energy transferred per unit mass of fluid.
5.95 An inward flow radial turbine (see Fig. P5.95) involves a nozzle angle, 1 , of 60 and an inlet rotor tip speed, U 1 , of 6 m/s. The ratio of rotor inlet to outlet diameters is 1.8. The absolute ve- locity leaving the rotor at section (2) is radial with a magnitude of
W 1
W 2
V 1
50 rpm
r 2 = 0.6 m
r 1 = 1.2 m
60 °
120 °
20 °
Section (1) Section (2) Q = 0.6 m^3 /s
b
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
■ Figure P5.
0.3 m
Section (1) Section^ (2)
r 1 = 0.6 m
120 rpm
r 2 = 0.3 m
V 2
V 1 = 15 m/s 30 °
V 1
1725 rpm
Q = 6.5 m^3 /min
2.5 cm
13 cm
30 cm
30 °
V 1
VR 2 = 9 m/s
V 2 = 18 m/s
0.15 m
m 2000 rpm
U 1 = 9 m/s
V 1 = 6 m/s
=
60 °
W 1 U 2 = 9 m/s
W 1 W 2
1
W 2
5.104 A siphon is used to draw water at 70 F from a large con- tainer as indicated in Fig. P5.104. The inside diameter of the siphon line is 2.5 cm and the pipe centerline rises 0.9 m above the essentially constant water level in the tank. Show that by varying the length of the siphon below the water level, h , the rate of flow through the siphon can be changed. Assuming frictionless flow, determine the maximum flowrate possible through the siphon. The limiting condi- tion is the occurrence of cavitation in the siphon. Will the actual maximum flow be more or less than the frictionless value? Explain.
268 Chapter 5 ■ Finite Control Volume Analysis
5.105 A water siphon having a constant inside diameter of 8 cm is arranged as shown in Fig. P5.105. If the friction loss be- tween A and B is 0.8 V^2 /2, where V is the velocity of flow in the siphon, determine the flowrate involved.
5.106 Water flows through a valve (see Fig. P5.106) at the rate of 450 kg/s. The pressure just upstream of the valve is 620 kPa and the pressure drop across the valve is 345 kPa. The inside diameters of the valve inlet and exit pipes are 30 and 60 cm. If the flow through the valve occurs in a horizontal plane, determine the loss in available energy across the valve.
5.107 A gas expands through a nozzle from a pressure of 2068 kPa to a pressure of 34 kPa. The enthalpy change involved, hˇ^ 1 hˇ^ 2 , is 350 kJ/kg. If the expansion is adiabatic but with frictional effects and the inlet gas speed is negligibly small, determine the exit gas velocity. 5.108 For the 180 elbow and nozzle flow shown in Fig. P5.108, determine the loss in available energy from section (1) to section (2). How much additional available energy is lost from sec- tion (2) to where the water comes to rest?
5.109 An automobile engine will work best when the back pres- sure at the interface of the exhaust manifold and the engine block is minimized. Show how reduction of losses in the exhaust manifold, piping, and muffler will also reduce the back pressure. How could losses in the exhaust system be reduced? What primarily limits the minimization of exhaust system losses? 5.110 (See Fluids in the News article titled “Smart Shocks,” Section 5.3.3.) A 890 N force applied to the end of the piston of the shock absorber shown in Fig. P5.110 causes the two ends of the shock absorber to move toward each other with a speed of 1.5 m/s. Determine the head loss associated with the flow of the oil through the channel. Neglect gravity and any friction force between the pis- ton and cylinder walls.
■ Figure P5.
0.75 m
1.0 m
1.5 m 6 m
3 m
■ Figure P5.
2.5 cm
0.9 m
h
■ Figure P5.
A
1.2 m
1.2 m
3.6 m
8 cm
B
■ Figure P5.
60 cm
30 cm
■ Figure P5.
15 cm
30 cm
Section (2)
Section (1)
y
x
p 1 = 103 kPa V 1 = 1.5 m/s
■ Figure P5.
Piston
Oil
Channel 2.5 cm diameter
p = 0
890 N
Gas
Problems 269
Section 5.3.2 Application of the Energy Equation—
With Shaft Work
†5.111 Based on flowrate and pressure rise information, estimate the power output of a human heart.
5.112 What is the maximum possible power output of the hy- droelectric turbine shown in Fig. P5.112?
5.113 Oil ( SG 0.88) flows in an inclined pipe at a rate of 0.14 m 3 /s as shown in Fig. P5.113. If the differential reading in the mercury manometer is 0.9 m, calculate the power that the pump supplies to the oil if head losses are negligible.
GO
5.114 The pumper truck shown in Fig. P5.114 is to deliver 0.042 m 3 /s to a maximum elevation of 18 m above the hydrant. The
pressure at the 10 cm diameter outlet of the hydrant is 69 kPa. If head losses are negligibly small, determine the power that the pump must add to the water. 5.115 The hydroelectric turbine shown in Fig. P5.115 passes 500 kL/s across a head of 180 m. What is the maximum amount of power output possible? Why will the actual amount be less?
5.116 A pump is to move water from a lake into a large, pressurized tank as shown in Fig. P5.116 at a rate of 4000 liters in 10 min or less. Will a pump that adds 2.2 kW to the water work for this purpose? Support your answer with appropriate calculations. Repeat the problem if the tank were pressurized to 300, rather than 200, kPa.
5.117 Water is supplied at 4.5 m^3 /s and 415 kPa to a hy- draulic turbine through a 1 m inside-diameter inlet pipe as indicated in Fig. P5.117. The turbine discharge pipe has a 1.2 m inside diam- eter. The static pressure at section (2), 3 m below the turbine inlet, is 25 cm Hg vacuum. If the turbine develops 1.9 MW, determine the power lost between sections (1) and (2).
■ Figure P5.
50 m
6 m/s
1 m
Turbine
■ Figure P5.
30 cm
Oil
0.9 m
H
h
15 cm
P
■ Figure P5.
Hydrant
18 m
69 kPa 10 cm diameter
■ Figure P5.
180 m
Turbine
■ Figure P5.
6 m
Air
Pump
p = 200 kPa
■ Figure P5.
Turbine
3 m
Section (1)
p 1 Q D 1
= 415 kPa = 4.5 m 3 /s = 1 m
p 2
D 2
= 25 cm Hg vacuum = 1.2 m
Section (2)
Problems 271
is 30 mm, the pressure is 400 kPa. If the loss in energy across the pump due to fluid friction effects is 170 N m/kg, determine the hydraulic efficiency of the pump. 5.126 Water is to be pumped from the large tank shown in Fig. P5.126 with an exit velocity of 6 m/s. It was determined that the original pump (pump 1) that supplies 1 kW of power to the wa- ter did not produce the desired velocity. Hence, it is proposed that an additional pump (pump 2) be installed as indicated to increase the flowrate to the desired value. How much power must pump 2 add to the water? The head loss for this flow is hL 250 Q^2 , where hL is in m when Q is in m^3 /s.
5.127 (See Fluids in the News article titled “Curtain of Air,” Section 5.3.3.) The fan shown in Fig. P5.127 produces an air curtain to separate a loading dock from a cold storage room. The air curtain is a jet of air 3 m wide, 0.15 m thick moving with speed V 9 m/s. The loss associated with this flow is loss K (^) LV^2 /2, where KL 5. How much power must the fan supply to the air to produce this flow?
Section 5.3.3 Application of the Energy Equation—
Combined with Linear Momentum
5.128 If a 0.5 W motor is required by a ventilating fan to produce a 60 cm stream of air having a velocity of 12 m/s as shown in Fig. P5.128, estimate (a) the efficiency of the fan and (b) the thrust of the supporting member on the conduit enclosing the fan.
5.129 Air flows past an object in a pipe of 2 m diameter and exits as a free jet as shown in Fig. P5.129. The velocity and pressure upstream are uniform at 10 m/s and 50 N/m^2 , respectively. At the pipe exit the velocity is nonuniform as indicated. The shear stress along the pipe wall is negligible. (a) Determine the head loss asso- ciated with a particle as it flows from the uniform velocity upstream
■ Figure P5.
V = 6 m/s Pump
Pipe area = 0.02 m^2
Nozzle area = 0.01 m^2
2 m
Pump
■ Figure P5.
Air curtain (0.15 m thickness)
Open door 3 m
V = 9 m/s
Fan
■ Figure P5.
60 cm 12 m/s
■ Figure P5.
2 m dia. 1 m dia.^ 4 m/s
12 m/s
Exit
Wake
Air
p = 50 N/m^2 V = 10 m/s
of the object to a location in the wake at the exit plane of the pipe. (b) Determine the force that the air exerts on the object.
5.130 Near the downstream end of a river spillway, a hy- draulic jump often forms, as illustrated in Fig. P5.130 and Video V10.11. The velocity of the channel flow is reduced abruptly across the jump. Using the conservation of mass and linear momentum principles, derive the following expression for h 2 ,
The loss of available energy across the jump can also be determined if energy conservation is considered. Derive the loss expression
jump loss
g 1 h 2 h 123 4 h 1 h 2
h 2
h 1 2
B
a
h 1 2
b
2
2 V^21 h 1 g
5.131 Water flows steadily down the inclined pipe as indi- cated in Fig P5.131. Determine the following: (a) the difference in pressure p 1 p 2 , (b) the loss between sections (1) and (2), (c) the net axial force exerted by the pipe wall on the flowing water be- tween sections (1) and (2).
■ Figure P5.
V 1
h 1 h 2
■ Figure P5.
1.5 m
15 cm
30 °
Mercury
Section (2)
Section (1)
Flow
15 cm
5.132 Water flows steadily in a pipe and exits as a free jet through an end cap that contains a filter as shown in Fig. P5.132. The flow is in a horizontal plane. The axial component, Ry , of the anchoring force needed to keep the end cap stationary is 270 N. Determine the head loss for the flow through the end cap.
272 Chapter 5 ■ Finite Control Volume Analysis
Section 5.3.4 Application of the Energy Equation to
Nonuniform Flows
5.135 Water flows vertically upward in a circular cross-sectional pipe. At section (1), the velocity profile over the cross-sectional ■^ Figure P5.
U 1 = 9 m/s
V r1 = 6 m/s
60 °
2 1
r 1 r 2
5.134 Two water jets collide and form one homogeneous jet as shown in Fig. P5.134. (a) Determine the speed, V , and direction, u, of the combined jet. (b) Determine the loss for a fluid particle flowing from (1) to (3), from (2) to (3). Gravity is negligible.
5.133 When fluid flows through an abrupt expansion as indi- cated in Fig. P5.133, the loss in available energy across the expan- sion, lossex, is often expressed as
where A 1 cross-sectional area upstream of expansion, A 2 cross-sectional area downstream of expansion, and V 1 velocity of flow upstream of expansion. Derive this relationship.
lossex a 1
A 1
A 2
b
2 V^2
1 2
■ Figure P5.
Area = 93 cm^2
Area = 111 cm^2
Ry = 270 N
V = 3 m/s
Rx
Pipe
Filter 30 °
■ Figure P5.
Section (1) Section (2)
■ Figure P5.
V 2 = 6 m/s
V
V 1 = 4 m/s
θ
0.12 m
0.10 m (1)
(2)
(3)
90 °
area is uniform. At section (2), the velocity profile is
where V local velocity vector, wc centerline velocity in the ax- ial direction, R pipe inside radius, and r radius from pipe axis. Develop an expression for the loss in available energy between sec- tions (1) and (2). 5.136 A small fan moves air at a mass flowrate of 0.002 kg/s. Upstream of the fan, the pipe diameter is 6.4 cm, the flow is laminar, the velocity distribution is parabolic, and the kinetic energy coeffi- cient, a 1 , is equal to 2.0. Downstream of the fan, the pipe diameter is 2.5 cm, the flow is turbulent, the velocity profile is quite flat, and the kinetic energy coefficient, a 2 , is equal to 1.08. If the rise in static pressure across the fan is 103 Pa and the fan shaft draws 0.18 W, compare the value of loss calculated: (a) assuming uniform velocity distributions, (b) considering actual velocity distributions.
Section 5.3.5 Combination of the Energy Equation
and the Moment-of-Momentum Equation
5.137 Air enters a radial blower with zero angular momen- tum. It leaves with an absolute tangential velocity, V u, of 60 m/s. The rotor blade speed at rotor exit is 50 m/s. If the stagnation pres- sure rise across the rotor is 3 kPa, calculate the loss of available en- ergy across the rotor and the rotor efficiency. 5.138 Water enters a pump impeller radially. It leaves the impeller with a tangential component of absolute velocity of 10 m/s. The im- peller exit diameter is 60 mm, and the impeller speed is 1800 rpm. If the stagnation pressure rise across the impeller is 45 kPa, deter- mine the loss of available energy across the impeller and the hy- draulic efficiency of the pump. 5.139 Water enters an axial-flow turbine rotor with an ab- solute velocity tangential component, V u, of 4.5 m. The corre- sponding blade velocity, U , is 15 m. The water leaves the rotor blade row with no angular momentum. If the stagnation pressure drop across the turbine is 83 kPa, determine the hydraulic effi- ciency of the turbine. 5.140 An inward flow radial turbine (see Fig. P5.140) involves a nozzle angle, a 1 , of 60 and an inlet rotor tip speed, U 1 , of 9 m/s. The ratio of rotor inlet to outlet diameters is 2.0. The radial compo- nent of velocity remains constant at 6 m/s through the rotor, and the flow leaving the rotor at section (2) is without angular momentum. If the flowing fluid is water and the stagnation pressure drop across the rotor is 110 kPa, determine the loss of available energy across the rotor and the hydraulic efficiency involved.
V wc a
R r R
b
(^1) 7 k ˆ
