7-Problems.pdf

Problems

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Problems

Energy and Power

7.1 From the list below, select one topic that is interesting to you. Then, use references such as the

Internet to research your topic and prepare one page of documentation that you could use to present your

topic to your peers.

a. Explain how hydroelectric power is produced.

b. Explain how a Kaplan turbine works, how a Francis turbine works, and the differences between these

two types of turbines.

c. Explain how a horizontalaxis wind turbine is used to produce electrical power.

d. Explain how a steam turbine is used to produce electrical power.

7.2 Skim this chapter and identify three realworld applications that are motivating to you. For each

application, write a paragraph that describes what you already know about the application and why this

application has appeal to you.

7.3 Using Section 7.1 and other resources, answer the following questions. Strive for depth, clarity, and

accuracy. Also, strive for effective use of sketches, words, and equations.

a. What are the common forms of energy? Which of these forms are relevant to fluid mechanics?

b. What is work? Describe three example of work that are relevant to fluid mechanics.

c. What are the most common units of power?

d. List three significant differences between power and energy.

7.4 Apply the grid method to each situation.

a. Calculate the energy in joules used by a 1 hp pump that is operating for 6 hours. Also, calculate the

cost of electricity for this time period. Assume that electricity costs $0.15 per kWhr.

b. A motor is being to used to turn the shaft of a centrifugal pump. Apply Eq. 7.2b to calculate the

power in watts corresponding to a torque of 100 lbfin and a rotation speed of 850 rpm.

c. A turbine produces a power of 7500 ftlbf/s. Calculate the power in hp and in watts.

Answer:

a. cost = $0.67,

b. P = 1010 W,

c. P = 13.6 hp

7.5 Estimate the power required to spray water out of the spray bottle that is pictured in Fig. 7.1. Hint:

Make appropriate assumptions about the number of sprays per unit time and the force exerted by the finger.

7.6 The sketch shows a common consumer product called the Water Pik. This device uses a motor to drive a

piston pump that produces a jet of water ( d = 3 mm, T = 10°C) with a speed of 25 m/s. Estimate the

minimum electrical power in watts that is required by the device. Hints: (a) Assume that the power is used

only to produce the kinetic energy of the water in the jet; and (b) in a time interval t , the amount of mass

that flows out the nozzle is m and the corresponding amount of kinetic energy is ( mV 2 /2).

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Problems

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PROBLEM 7.6

Answer:

P = 55.2 W

7.7 An engineer is considering the development of a small wind turbine ( D = 1.25 m) for home applications.

The design wind speed is 15 mph at T = 10°C and p = 0.9 bar. The efficiency of the turbine is η = 20%,

meaning that 20% of the kinetic energy in the wind can be extracted. Estimate the power in watts that can be

produced by the turbine. Hint: In a time interval t , the amount of mass that flows through the rotor is

and the corresponding amount of kinetic energy in this flow is ( m V 2 /2).

PROBLEM 7.7

Kinetic Energy Correction Factor (α)

7.8 Using Section 7.3 and other resources, answer the questions below. Strive for depth, clarity, and

accuracy while also combining sketches, words, and equations in ways that enhance the effectiveness of

your communication.

a. What is the kineticenergy correction factor? Why do engineers use this term?

b. What is the meaning of each variable (α, A , V ) that appears in Eq. 7.21?

c. What values of α are commonly used?

7.9 For this hypothetical velocity distribution in a wide rectangular channel, evaluate the kineticenergy

correction factor α

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Problems

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PROBLEM 7.9

7.10 For these velocity distributions in a round pipe, indicate whether the kineticenergy correction factor α is

greater than, equal to, or less than unity.

Answer:

a. α = 1.0,

b. α > 1.0,

c. α > 1.0,

d. α > 1.0

7.11 Calculate α for case ( c ) in Prob. 7.10.

7.12 Calculate α for case ( d ) in Prob. 7.10.

PROBLEMS 7.10, 7.11 and 7.12

Answer:

α = 27/20

7.13 An approximate equation for the velocity distribution in a pipe with turbulent flow is

where V max is the centerline velocity, y is the distance from the wall of the pipe, r 0 is the radius of the pipe,

and n is an exponent that depends on the Reynolds number and varies between 1/6 and 1/8 for most

applications. Derive a formula for α as a function of n . What is α if n = 1/7?

7.14 An approximate equation for the velocity distribution in a rectangular channel with turbulent flow is

where u max is the velocity at the surface, y is the distance from the floor of the channel, d is the depth of

flow, and n is an exponent that varies from about 1/6 to 1/8 depending on the Reynolds number. Derive a

formula for α as a function of n . What is the value of α for n = 1/7?

Answer:

α = ( n + 1) 3 /(3 n + 1), α = 1.05

7.15 The following data were taken for turbulent flow in a circular pipe with a radius of 3.5 cm. Evaluate the

kinetic energy correction factor. The velocity at the pipe wall is zero.

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Problems

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R (cm) V (m/s) R (cm) V (m/s)

0.0

32.5 2.8

22.03

0.5

32.44 2.9

21.24

1.0

32.27 3.0

20.49

1.5

31.22 3.1

19.6

2.0

28.21 3.2

18.69

2.25 26.51 3.25 18.16

2.5

24.38 3.3

17.54

2.6

23.7 3.35 17.02

2.7

22.88 3.4

16.14

Energy Equation

7.16 Using Section 7.3 and other resources, answer the questions below. Strive for depth, clarity, and

accuracy. Also, strive for effective use of sketches, words, and equations.

a. What is conceptual meaning of the first law of thermodynamics for a system?

b. What is flow work? How is the equation for flow work (Eq. 7.15) derived?

c. What is shaft work? How is shaft work different than flow work?

7.17 Using Section 7.3 and other resources, answer the questions below. Strive for depth, clarity, and

accuracy. Also, strive for effective use of sketches, words, and equations.

a. What is head? How is head related to energy? To power?

b. What is head of a turbine?

c. How is head of a pump related to power? To energy?

d. What is head loss?

7.18 Using Sections 7.3 and 7.6 and using other resources, answer the following questions. Strive for

depth, clarity, and accuracy. Also, strive for effective use of sketches, words and equations.

a. What are the five main terms in the energy equation 7.29? What does each term mean?

b. How are terms in the energy equation related to energy? To power?

c. What assumptions are required for using the energy equation 7.29?

d. How is the energy equation 7.29 similar to the Bernoulli equation? How is it different? Give three

important similarities and three important differences.

7.19 Using the energy equation 7.29, prove that fluid in a pipe will flow from a location with high piezometric

head to a location with low piezometric head. Assume there are no pumps or turbines and that the pipe has

a constant diameter.

7.20 Water flows at a steady rate in this vertical pipe. The pressure at A is 10 kPa, and at B it is 98.1 kPa. Then

the flow in the pipe is (a) upward, (b) downward, or (c) no flow. ( Hint: see problem 7.19).

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PROBLEM 7.20

Answer:

(b)

7.21 Determine the discharge in the pipe and the pressure at point B . Neglect head losses. Assume α = 1.0 at all

locations.

PROBLEM 7.21

7.22 A pipe drains a tank as shown. If x = 10 ft, y = 4 ft, and head losses are neglected, what is the pressure at

point A and what is the velocity at the exit? Assume α = 1.0 at all locations.

PROBLEMS 7.22 and 7.23

Answer:

p A = 250 psf, V 2 = 30.0 ft/s

7.23 A pipe drains a tank as shown. If x = 10 m, y = 1.5 m, and head losses are neglected, what is the pressure

at point A and what is the velocity at the exit? Assume α = 1.0 at all locations.

7.24 For this system, the discharge of water is 0.1 m 3 /s, x = 1.0 m, y = 2.0 m, z = 7.0 m, and the pipe diameter is

30 cm. Neglecting head losses, what is the pressure head at point 2 if the jet from the nozzle is 10 cm in

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