• Home
  • Textbooks
  • Fluid Mechanics for Engineers in SI Units
  • Turbomachines

Fluid Mechanics for Engineers in SI Units

David A Chin

Chapter 8

Turbomachines - all with Video Answers

Educators

Chapter Questions

01:05

Problem 1

Water at $20^{\circ} \mathrm{C}$ flows through the pump impeller shown in Figure $8.33,$ where the inner and outer diameters of the impeller are $200 \mathrm{~mm}$ and $500 \mathrm{~mm}$, respectively, and the width of the pump blades is $60 \mathrm{~mm}$. The impeller rotates at 4000 rpm. Under a particular operating condition, water enters in a direction normal to the inflow surface, the component of the outflow velocity normal to the outflow surface, $V_{r},$ is equal to $25 \mathrm{~m} / \mathrm{s}$, and the magnitude of the absolute velocity, $\mathbf{V}$, at the outflow surface is $40 \mathrm{~m} / \mathrm{s}$. Estimate the shaft power required to drive the pump at this operating condition.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:31

Problem 2

A centrifugal blower has a 1000 -mm-diameter impeller with a width of $50 \mathrm{~mm}$ that rotates at a rate of $1800 \mathrm{rpm}$. When the flow rate through the impeller is $0.25 \mathrm{~m}^{3} / \mathrm{s}$, the inflow velocity is normal to the inflow surface and the outflow velocity makes an angle of $40^{\circ}$ with the radial direction as shown in Figure 8.34 . Assuming that the blower is $100 \%$ efficient, estimate the energy per unit mass added to the air as it passes through the blower.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:10

Problem 3

A water pump impeller has radial blades as shown in Figure $8.35 .$ The inner and outer diameters of the impeller are $60 \mathrm{~mm}$ and $150 \mathrm{~mm}$, respectively. When the rotational speed of the impeller is $3250 \mathrm{rpm}$, the pump delivers $12.5 \mathrm{~L} / \mathrm{s}$ and the speed of the water relative to the blade at the exit is equal to $4 \mathrm{~m} / \mathrm{s}$. Determine (a) the width of the impeller at the outflow surface and (b) the power required to drive the pump. Assume water at $20^{\circ} \mathrm{C}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
07:53

Problem 4

A pump impeller has an outer diameter of $400 \mathrm{~mm}$ and a blade width of $50 \mathrm{~mm},$ and it rotates at $800 \mathrm{rpm} .$ At a particular operating condition, the inflow is normal to the inflow surface and the (absolute) outflow velocity has a magnitude of $40 \mathrm{~m} / \mathrm{s}$ and makes an angle of $40^{\circ}$ with the normal to the outflow surface, as shown in Figure 8.36. Estimate the shaft power required to drive the pump under this condition. Assume water at $20^{\circ} \mathrm{C}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
07:53

Problem 5

The pump impeller shown in Figure 8.37 has a diameter of $325 \mathrm{~mm}$, a blade width of $30 \mathrm{~mm}$, and an exit blade angle of $42^{\circ}$; it rotates at 800 rpm. Water at $20^{\circ} \mathrm{C}$ flows through the pump at a flow rate of $75 \mathrm{~L} / \mathrm{s}$, and this flow enters the impeller in a direction normal to the inflow surface. Estimate the required shaft power corresponding to the design flow rate.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:07

Problem 6

Consider the general case of flow through a pump impeller as illustrated in Figure 8.38. (a) Develop a general expression for the power input to the pump in terms of the inner and outer impeller radii, $r_{1}$ and $r_{2},$ respectively; the inner and outer blade widths, $b_{1}$ and $b_{2},$ respectively; the inner and outer blade angles, $\beta_{1}$ and $\beta_{2}$, respectively; rotational speed, $\omega$; volume flow rate, $Q$; and density of the fluid, $\rho$.
(b) Estimate the power input and head added to the pump for $r_{1}=200 \mathrm{~mm}, r_{2}=$ $600 \mathrm{~mm}, b_{1}=60 \mathrm{~mm}, b_{2}=40 \mathrm{~mm}, \beta_{1}=70^{\circ}, \beta_{2}=75^{\circ}, \omega=700 \mathrm{rpm},$ and
$Q=0.90 \mathrm{~m}^{3} / \mathrm{s}$. Assume that the liquid is water at $20^{\circ} \mathrm{C}$

Chai Santi
Chai Santi
Numerade Educator
02:07

Problem 7

A pump impeller similar to the one shown in Figure 8.38 has inner and outer radii of $200 \mathrm{~mm}$ and $400 \mathrm{~mm}$, respectively; inner and outer blade widths of $40 \mathrm{~mm}$ and $35 \mathrm{~mm}$, respectively; and inner and outer blade angles of $25^{\circ}$ and $10^{\circ}$, respectively. The rate of angular rotation is $600 \mathrm{rpm}$. The liquid being pumped is water at $20^{\circ} \mathrm{C}$.
(a) Estimate the volume flow rate required for the entrance velocity to be normal to the inflow surface. (b) Estimate the power added by the pump for the flow rate calculated in part (a).

Chai Santi
Chai Santi
Numerade Educator
02:55

Problem 8

A pump impeller has inner and outer radii of $120 \mathrm{~mm}$ and $280 \mathrm{~mm}$, respectively; inner and outer blade widths of $30 \mathrm{~mm}$ and $25 \mathrm{~mm}$, respectively; and inner and outer blade angles of $50^{\circ}$ and $60^{\circ}$, respectively. The flow rate is $250 \mathrm{~L} / \mathrm{s}$. The liquid being pumped is water at $20^{\circ} \mathrm{C}$. (a) Estimate the angular speed required for the entrance velocity to be normal to the inflow surface. (b) Estimate the power added by the pump for the angular speed calculated in part (a).

Chai Santi
Chai Santi
Numerade Educator
02:55

Problem 9

A pump impeller has inner and outer radii of $200 \mathrm{~mm}$ and $350 \mathrm{~mm}$, respectively, and inner and outer blade widths of $60 \mathrm{~mm}$ and $50 \mathrm{~mm}$, respectively; the outer blade angle is $70^{\circ} .$ The flow rate is $500 \mathrm{~L} / \mathrm{s}$, and the rotational speed is $600 \mathrm{rpm}$. The liquid being pumped is water at $20^{\circ} \mathrm{C}$. (a) Estimate the inner blade angle required for the entrance velocity to be normal to the inflow surface. (b) Estimate the power added by the pump for the angular speed calculated in part (a).

Chai Santi
Chai Santi
Numerade Educator
01:22

Problem 10

A pump impeller is to be designed to have an inner and outer radius of $150 \mathrm{~mm}$ and $300 \mathrm{~mm}$, respectively, and an inner and outer blade width of $90 \mathrm{~mm}$ and $70 \mathrm{~mm}$, respectively. The impeller is to have a rotational speed of $1725 \mathrm{rpm}$ and is to deliver a volume flow rate of $0.3 \mathrm{~m}^{3} / \mathrm{s}$ against a head of $13.8 \mathrm{~m}$. The inflow velocity should be normal to the inflow surface. (a) Design the blade angles at the inflow and outflow surfaces. (b) If the liquid to be pumped is water at $20^{\circ} \mathrm{C}$, what is the power requirement of the pump?

Chai Santi
Chai Santi
Numerade Educator
03:33

Problem 11

A pump delivers $125 \mathrm{~L} / \mathrm{s}$ of water when the rotational speed of the impeller is $650 \mathrm{rpm}$. The inflow velocity is normal to the inflow surface, the magnitude of the outflow velocity relative to the rotating blade is $6 \mathrm{~m} / \mathrm{s}$, and the blade angle at the outflow surface is $90^{\circ}$. The motor consumes $10 \mathrm{~kW}$ of power, and it is estimated that the pump motor has an efficiency of $80 \%$. Estimate the radius and width of the outflow surface. Assume water at $20^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:04

Problem 12

A centrifugal pump uses $9 \mathrm{~kW}$ of power and has an efficiency of $70 \%$. The pump delivers gasoline at a rate of $5 \mathrm{~L} / \mathrm{s}$. Estimate the maximum change in pressure between the inlet and outlet of the pump. How would the estimated pressure change be different if a liquid with higher density was used?

Mayukh Banik
Mayukh Banik
Numerade Educator
06:38

Problem 13

A pump is operated at a rotational speed of $2500 \mathrm{rpm}$, and has an efficiency of $80 \%$ at its operating condition. The pump is driven by a motor that has an efficiency of $90 \%$ and delivers a shaft torque of $10 \mathrm{~N} \cdot \mathrm{m}$. The discharge side of the pump is at an elevation $3 \mathrm{~m}$ higher than the suction side of the pump. Measurements taken when the pump is delivering $7.5 \mathrm{~L} / \mathrm{s}$ show pressure on the suction side and velocity of $92 \mathrm{kPa}$ (gauge) and $1.8 \mathrm{~m} / \mathrm{s}$, respectively, and the velocity on the discharge side is $4.8 \mathrm{~m} / \mathrm{s}$. Estimate the power delivered to the motor and the pressure on the discharge side of the pump. Assume water at $20^{\circ} \mathrm{C}$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 14

The blade of a turbine is illustrated in Figure $8.39,$ where the relative inflow velocity makes an angle of $70^{\circ}$ with the inflow surface and the relative outflow velocity makes an angle of $60^{\circ}$ with the outflow surface. Relative velocities are measured relative to the tip velocities of the blades mounted on the rotor. The inflow surface has a radius of $1.75 \mathrm{~m}$, the outflow surface has a radius of $1.05 \mathrm{~m}$, and the width of the rotor is $0.5 \mathrm{~m}$. Under design conditions, the flow rate through the turbine is $35 \mathrm{~m}^{3} / \mathrm{s}$ and the rotational speed is $150 \mathrm{rpm}$. Estimate the shaft power extracted by the turbine. Assume water at $20^{\circ} \mathrm{C}$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 15

Water at $20^{\circ} \mathrm{C}$ flows through a turbine at a rate of $400 \mathrm{~L} / \mathrm{s}$, and the velocities on the inflow and outflow surfaces of the rotor are shown in Figure $8.40 .$ The absolute velocity and the velocity relative to the rotating blade are represented by $\mathbf{V}$ and $\mathbf{W}$, respectively, and the subscripts 1 and 2 indicate conditions at the inflow and outflow surfaces, respectively. On the inflow surface, the incoming flow makes an angle of $25^{\circ}$ with the tangent to the surface, and the velocity relative to the rotor should make an angle of $55^{\circ}$ with the inflow surface to match the rotor angle. On the outflow surface, it is assumed that the velocity relative to the moving blade will make an angle of $50^{\circ}$ with the outflow surface so that it matches the inner blade angle. The inflow and outflow surfaces have radii of $800 \mathrm{~mm}$ and $400 \mathrm{~mm}$, respectively, and the rotor rotates at $130 \mathrm{rpm}$. (a) Determine the required width of the rotor. (b) Estimate the shaft power generated by the turbine.

Victor Salazar
Victor Salazar
Numerade Educator
02:55

Problem 16

A proposed pump design has an impeller with inner and outer radii of $125 \mathrm{~mm}$ and $225 \mathrm{~mm}$, respectively; widths of inflow and outflow areas of $50 \mathrm{~mm}$ and $45 \mathrm{~mm}$ respectively; and blade angles at the inflow and outflow surfaces of $35^{\circ}$ and $15^{\circ}$, respectively. The impeller rotates at $1140 \mathrm{rpm}$, and the estimated efficiency of the pump is $85 \%$. Estimate (a) the optimal flow rate through the pump, (b) the head added by the pump at the optimal flow rate, and (c) the estimated power requirement to run the pump. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
01:19

Problem 17

Pumps in a homologous series of centrifugal pumps are all driven by 1200 -rpm motors. For a 500 -mm size pump in the homologous series, the best efficiency of $81 \%$ occurs when the flow rate is $250 \mathrm{~L} / \mathrm{s}$ and the total dynamic head is $63.7 \mathrm{~m}$. What is the best efficiency operating point for a 250 -mm size pump in the homologous series? Estimate the efficiency of the smaller pump.

Chai Santi
Chai Santi
Numerade Educator
01:32

Problem 18

Pump manufacturers typically present the performance characteristics of their pumps in dimensional form as shown in Figure 8.41 , where the flow rate is in $\mathrm{L} / \mathrm{min}$ and the head added by the pump is in $\mathrm{m}$. The efficiencies in percent at various operational states are represented by dashed lines on the performance curve. The performance curves shown in Figure 8.41 are for homologous pumps with a motor speed of $3500 \mathrm{rpm}$ and impeller sizes of $178 \mathrm{~mm}, 165 \mathrm{~mm}, 152 \mathrm{~mm}, 140 \mathrm{~mm},$ and $127 \mathrm{~mm} .$
(a) Plot on a single graph the nondimensional performance curves for each of the pumps. (b) Compare the nondimensional performance curves and assess whether these pumps are from a homologous series. (c) Identify the maximum efficiency and specific speed for each impeller size.

Dominador Tan
Dominador Tan
Numerade Educator
View

Problem 19

Water at $15^{\circ} \mathrm{C}$ is pumped at a rate of $20 \mathrm{~L} / \mathrm{s}$ using a $5 \mathrm{~kW}$ pump. If the efficiency of the pump is $80 \%,$ what is the head added to the water as it passes through the pump?

Ankur S
Ankur S
Numerade Educator
03:20

Problem 20

A Model $X$ pump with an impeller diameter of $300 \mathrm{~mm}$ and rotational speed of 1800 rpm has its maximum efficiency at a flow rate of $300 \mathrm{~L} / \mathrm{s},$ at which point the head added by the pump is $65 \mathrm{~m}$ and the brake horsepower is $240 \mathrm{~kW}$. A geometrically similar Model $\mathrm{X} 1$ pump with an impeller size of $250 \mathrm{~mm}$ is driven by a $1400-$ rpm motor. The working fluid is water at $20^{\circ} \mathrm{C}$. (a) Determine the flow coefficient, head coefficient, and power coefficient of the Model $\mathrm{X}$ pump when it is operating at its most efficient point. (b) What is the flow rate, head added, and brake horsepower of the Model $\mathrm{X} 1$ pump when it is operated at its most efficient point?

Chai Santi
Chai Santi
Numerade Educator
02:55

Problem 21

A pump with a rotary speed of 1725 rpm delivers $25 \mathrm{~L} / \mathrm{s}$ at its most efficient operating point. Under this condition, the inflow velocity is normal to the inflow surface of the impeller, the component of the velocity normal to the outflow surface of the impeller is $4 \mathrm{~m} / \mathrm{s}$, and the efficiency of the pump is $80 \%$. The width of the impeller at the outflow surface is $15 \mathrm{~mm}$, and the blade angle at the outflow surface is $50^{\circ}$.
(a) Estimate the head added by the pump. (b) Use the affinity laws to estimate the head added and the flow rate delivered by the pump when the rotational speed is changed to $1140 \mathrm{rpm}$.

Chai Santi
Chai Santi
Numerade Educator
02:56

Problem 22

A pump manufacturer makes three homologous series of pumps with three different specific speeds. The manufacturer documents the performance of each homologous series by nondimensional functional relationships. An engineer identifies the desired homologous series for a particular project as the one that has a best efficiency point with a flow coefficient of $0.035,$ a head coefficient of $0.14,$ and a power coefficient of $0.006 .$ For a particular pump with an impeller diameter of $600 \mathrm{~mm}$ and a rated rotational speed of $1140 \mathrm{rpm}$, determine the rated discharge, total dynamic head, brake horsepower, and efficiency of the pump. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
02:56

Problem 23

A pump is to be chosen from a homologous series that has a best efficiency point with a flow coefficient of 0.04 , a head coefficient of $0.15,$ and a power coefficient of $0.0065 .$ The desired pump is to have a best efficiency flow rate of $250 \mathrm{~L} / \mathrm{s}$ and a motor with a speed of $900 \mathrm{rpm}$. What is the impeller size of the required pump? Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
03:20

Problem 24

An existing pump has a 500 -mm-diameter impeller, and under conditions of maximum efficiency, the efficiency of the pump is $80 \%,$ the flow rate is $250 \mathrm{~L} / \mathrm{s}$, the head added is $25 \mathrm{~m}$, and the power consumed by the pump is $55 \mathrm{~kW}$. The pump is to be refurbished by replacing the existing impeller with a geometrically similar impeller that has a diameter of $400 \mathrm{~mm}$. (a) Neglecting scale effects, estimate the flow rate, head added, and power consumption of the pump with the new impeller when the pump is operating under conditions of maximum efficiency. (b) Quantify the scale effect on the efficiency of the pump and assess whether the scale effect is expected to be significant.

Chai Santi
Chai Santi
Numerade Educator
02:56

Problem 25

A homologous series of centrifugal pumps are driven by 2500 -rpm motors. For a 500 -mm size within this series, the manufacturer claims that the best efficiency of $80 \%$ occurs when the flow rate is $600 \mathrm{~L} / \mathrm{s}$ and the head added by the pump is $90.5 \mathrm{~m} .$ What would be the best efficiency operating point for a 400 -mm size within this homologous series? Estimate the corresponding efficiency.

Chai Santi
Chai Santi
Numerade Educator
04:10

Problem 26

A 1: 4 scale model of a water pump is operated at a speed of $4500 \mathrm{rpm}$. At its best efficiency point, the efficiency of the model pump is $84 \%,$ and the model delivers a flow rate of $0.7 \mathrm{~m}^{3} / \mathrm{s}$ with an added head of $4.9 \mathrm{~m}$. If the full-scale pump has a rotational speed of $120 \mathrm{rpm}$, what is the flow rate and head delivered by the fullscale pump operating at its most efficient point? What is the power requirement of the full-scale pump at its best efficiency point? Assume water at $20^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:50

Problem 27

Affinity laws are typically used to identify homologous values of flow rate, head, and power between geometrically similar pumps. Develop an affinity law for relating homologous values of torque.

Chai Santi
Chai Santi
Numerade Educator
View

Problem 28

A manufacturer tests a 1: 10 scale model of a pump in the laboratory. The model pump has an impeller diameter of $200 \mathrm{~mm}$ and a rotational speed of $3450 \mathrm{rpm},$ and when the head across the pump is $40 \mathrm{~m}$, the pump delivers a flow rate of $10 \mathrm{~L} / \mathrm{s}$ at an efficiency of $84 \%$. The prototype pump is to develop the same head as the scale model; however, because of its increased size, the prototype pump is expected to have an efficiency of $90 \%$. (a) What is the power supplied to the model pump? (b) What is the rotational speed, flow rate, and power supplied to the prototype pump under homologous conditions? Assume water at $20^{\circ} \mathrm{C}$.

Victor Salazar
Victor Salazar
Numerade Educator
03:15

Problem 29

A pump has an impeller diameter of $300 \mathrm{~mm}$ and a rotational speed of $1500 \mathrm{rpm}$. At the best efficiency operating point, the pump adds a head of $9 \mathrm{~m}$ at a flow rate of $25 \mathrm{~L} / \mathrm{s} .$ What is the specific speed of the pump? What type of pump is this likely to be?

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
04:42

Problem 30

Express the specific speed of a pump in terms of the head coefficient and flow coefficient at the best efficiency point.

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 31

A pump is required to deliver $600 \mathrm{~L} / \mathrm{s}$ from a lake to a storage reservoir. Application of the energy equation to the pipeline system shows that the head that must be added by the pump is $84 \mathrm{~m}$. If the pump motor is to have a rotational speed of $2000 \mathrm{rpm}$, what is the specific speed of the required pump? What type of pump is required?

Victor Salazar
Victor Salazar
Numerade Educator
03:15

Problem 32

The head and flow rate to be delivered by a particular pump are to be changed. The existing pump is to be retained with a new motor installed. The existing pump has a specific speed of $4.5 .$ Under the new operating conditions, the pump is required to deliver $400 \mathrm{~L} / \mathrm{s}$ with an added head of $8 \mathrm{~m}$. What rotational speed should the new motor have? What type of pump is this?

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
04:58

Problem 33

What is the highest synchronous speed for a motor driving a pump?

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
02:28

Problem 34

A pump is to be selected such that it operates at or near its most efficient state when delivering $100 \mathrm{~L} / \mathrm{s}$ with an added head of $50 \mathrm{~m}$. A manufacturer has five models of homologous pumps, with specific speeds of $0.95,0.81,0.61,0.36,$ and $0.25 .$ Motors with any of the standard-rated speeds can be provided with these pumps. Identify the best choice of specific speed and the rotational speed of the motor that should be used to drive the pump.

Chai Santi
Chai Santi
Numerade Educator
03:20

Problem 35

A pump has an impeller diameter of $450 \mathrm{~mm}$, and at its most efficient operating point, it delivers water at a flow rate of $650 \mathrm{~L} / \mathrm{s}$ with an added head of $9.5 \mathrm{~m}$. The specific speed of the pump is $1.5,$ and the shaft power delivered by the motor is $80 \mathrm{~kW}$. (a) Estimate the shutoff head of the pump. (b) Estimate the efficiency of the pump at its best operating point. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
04:53

Problem 36

A pump with an impeller size of $250 \mathrm{~mm}$ is operated at a rotational speed of $1200 \mathrm{rpm}$. A review of the pump performance data shows that the pump has a shutoff head of $8.2 \mathrm{~m}$, and at the best efficiency operating point, the pump adds a head of $6.5 \mathrm{~m}$ at a flow rate of $20 \mathrm{~L} / \mathrm{s}$. (a) Use the performance data to fit a parabolic performance curve of the form $h_{\mathrm{p}}=a-b Q^{2},$ where $h_{\mathrm{p}}$ is the head added by the pump in $\mathrm{m}, Q$ is the flow rate in $\mathrm{L} / \mathrm{s},$ and $a$ and $b$ are constants. (b) If the rotational speed of the pump impeller is increased to $1800 \mathrm{rpm}$, estimate the parabolic performance curve at the adjusted speed.

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 37

A prototype water pump has a specific speed of $1.2,$ and when operating at its most efficient state, it delivers $5 \mathrm{~L} / \mathrm{s}$ with an added head of $10 \mathrm{~m}$. Operation of the pump is to be tested using a $\frac{1}{5}-$ scale model with various test fluids that have dynamic viscosities in the range of $5-10$ times that of water. Viscous effects should be accurately accounted for in the model. What range of rotational speeds, flow rates, and heads will be required in model testing? Assess whether accurately accounting for viscous effects is realistic.

Victor Salazar
Victor Salazar
Numerade Educator
03:07

Problem 38

Pressure measurements are taken in the suction and discharge pipes of a pump as shown in Figure 8.42 . When the flow rate through the pump is $10 \mathrm{~L} / \mathrm{s}$, the measured (gauge) pressure on the suction side is $-20 \mathrm{kPa}$ and the measured (gauge) pressure on the discharge side is $100 \mathrm{kPa}$. The measurement section on the discharge side is $0.83 \mathrm{~m}$ above the measurement section on the suction side. The diameters of the suction and discharge pipes are $150 \mathrm{~mm}$ and $75 \mathrm{~mm}$, respectively. The pumped fluid is water at $10^{\circ} \mathrm{C},$ and the head loss between the suction and discharge sides of the pump can be assumed to be negligible. Estimate the head added by the pump.

James Kiss
James Kiss
Numerade Educator
03:03

Problem 39

A centrifugal pump is to be used to pump water at a rate of $50 \mathrm{~L} / \mathrm{s}$ from a sump to a storage reservoir, where the water surface in the storage reservoir is $25 \mathrm{~m}$ higher than the water surface in the sump. The suction and discharge line from the pump have a total length of $48 \mathrm{~m}$, a diameter of $125 \mathrm{~mm}$, and an estimated roughness height of $0.5 \mathrm{~mm}$. The sum of the local loss coefficients in the pipeline is equal to $4,$ the specific speed of the pump is 0.8 , and the manometric efficiency of the pump is $80 \%$. In the homologous series from which the pump is selected, the impeller blades are forward-curved and make an angle of $55^{\circ}$ with the outflow surface, the width of the impeller is equal to $12.5 \%$ of the diameter, and the blades cover $7 \%$ of the outer area of the impeller. Inflow to the impeller is normal to the inflow surface. What size impeller should be selected for this application? Assume water at $20^{\circ} \mathrm{C}$.

Prashant Bana
Prashant Bana
Numerade Educator
01:32

Problem 40

The performance characteristics of a pump are determined using the setup shown in Figure $8.43,$ where simultaneous measurements are taken of the flow rate, $Q,$ the pressures, $p_{1}$ and $p_{2},$ at the suction and discharge pipes sections, respectively, and the power consumption, $P$, of the pump. These measurements are as follows:
$$
\begin{array}{l|l|l|l|l|l|l|l}
Q \text { (L/s) } & 1.26 & 2.52 & 3.79 & 5.05 & 6.31 & 7.57 & 8.83 \\
\hline p_{2}-p_{1}(\mathrm{kPa}) & 671 & 665 & 653 & 629 & 593 & 557 & 498 \\
\hline P(\mathrm{~kW}) & 3.54 & 4.22 & 4.97 & 5.60 & 6.27 & 7.20 & 7.77
\end{array}
$$
The discharge pipe section has an elevation $0.38 \mathrm{~m}$ higher than the suction pipe section, and the temperature of the water used in the test is $20^{\circ} \mathrm{C}$. (a) Plot the performance curve of the pump, showing $h_{\mathrm{p}}$ versus $Q$ and $\eta$ versus $Q,$ where $h_{\mathrm{p}}$ is the head added by the pump and $\eta$ is the efficiency. Both of these relationships should be shown on the same graph. (b) What is the flow rate through the pump when it is operating at maximum efficiency? What is the maximum efficiency of the pump?

Dominador Tan
Dominador Tan
Numerade Educator
08:18

Problem 41

A pump is required to deliver $600 \mathrm{~L} / \mathrm{s}(\pm 10 \%)$ through a 300 -mm-diameter $\mathrm{PVC}$ pipe from a well to a reservoir. The water level in the well is $1.5 \mathrm{~m}$ below the ground surface, and the water surface in the reservoir is $2 \mathrm{~m}$ above the ground surface. The delivery pipe is $300 \mathrm{~m}$ long, and local losses can be neglected. A pump manufacturer suggests using a pump with a performance curve given by
$$
h_{\mathrm{p}}=6-6.67 \times 10^{-5} Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added in meters and $Q$ is the flow rate in liters per second. Is this pump adequate? Explain.

Ronald Prasad
Ronald Prasad
Numerade Educator
03:20

Problem 42

A pump with the performance curve shown in Figure 8.44 is being considered to pump water from a lower reservoir to a higher reservoir where the difference in water surface elevations is $9.0 \mathrm{~m}$. The conduit is $100 \mathrm{~m}$ long, is $100 \mathrm{~mm}$ in diameter, and has negligible roughness. Estimate the maximum flow rate that can be achieved if this pump is used. At what efficiency would the pump operate? Assess the desirability of using this pump.

Chai Santi
Chai Santi
Numerade Educator
View

Problem 43

Water is pumped from a lower reservoir to a higher reservoir. The water surface in the higher reservoir is $10 \mathrm{~m}$ above the water surface in the lower reservoir. The piping system consists of 200 -mm-diameter ductile iron pipe with a length of $2 \mathrm{~km}$ and minor losses equal to 6.2 times the velocity head. The pump characteristics are shown in Table 8.3. (a) Determine the expected flow rate through the system, assuming fully turbulent flow. (b) What is the power of the motor required to drive the pump?
$$
\begin{array}{l|c|c|c|c|c|c}
\text { Discharge (L/s) } & 0 & 10 & 20 & 30 & 40 & 50 \\
\hline \text { Total head (m) } & 25 & 23.2 & 20.8 & 17.0 & 12.4 & 7.3 \\
\hline \text { Efficiency (\%) } & -1 & 45 & 65 & 71 & 65 & 45
\end{array}
$$

Victor Salazar
Victor Salazar
Numerade Educator
06:58

Problem 44

The pumped-storage system illustrated in Figure 8.45 is designed to exchange $2 \mathrm{~m}^{3} / \mathrm{s}$ through a 1220 -mm-diameter, 3.2 -km-long ductile iron pipeline lined with bitumen. The elevation difference between the water surfaces in the upper and lower reservoirs is $61 \mathrm{~m},$ and the pump/turbine is to operate 8 hours during the day as a turbine (to generate electricity) and 8 hours during the night as a pump (to return the water to the upper reservoir). The pump efficiency is $85 \%,$ the turbine efficiency is $90 \%$, the cost of pumping is $\$ 0.06 / \mathrm{kWh}$, and the revenue from turbine operations (selling electricity) is $\$ 0.12 / \mathrm{kWh}$. Determine the annual profit from this operation. Neglect the effect of storage on reservoir elevations. If the pump performance curve is given by $h_{\mathrm{p}}=80-3.5 Q^{2}$ where $h_{\mathrm{p}}$ is in $\mathrm{m}$ and $Q$ is in $\mathrm{m}^{3} / \mathrm{s}$, estimate the change in profit when the elevation difference is $65 \mathrm{~m}$. For an elevation difference of $65 \mathrm{~m}$, assume that the flow is fully turbulent.

Ronald Prasad
Ronald Prasad
Numerade Educator
03:20

Problem 45

A pump is to be selected to deliver water from a well to a treatment plant through a 300 -m-long pipeline. The temperature of the water is $20^{\circ} \mathrm{C}$, the average elevation of the water surface in the well is $5 \mathrm{~m}$ below the ground surface, the pump is $50 \mathrm{~cm}$ above the ground surface, and the water surface in the receiving reservoir at the water treatment plant is $4 \mathrm{~m}$ above the ground surface. The delivery pipe is made of ductile iron $\left(k_{\mathrm{s}}=0.26 \mathrm{~mm}\right)$ with a diameter of $800 \mathrm{~mm}$. If the selected pump has a performance curve of $h_{\mathrm{p}}=12-0.1 Q^{2},$ where $Q$ is in $\mathrm{m}^{3} / \mathrm{s}$ and $h_{\mathrm{p}}$ is in $\mathrm{m}$, what is the flow rate through the system? Calculate the specific speed of the required pump and state the type of pump required when the speed of the pump motor is $1200 \mathrm{rpm}$. Neglect local losses.

Chai Santi
Chai Santi
Numerade Educator
01:32

Problem 46

A pump is to be used to withdraw water from a reservoir at a rate of $1500 \mathrm{~L} / \mathrm{s}$. When operating at this flow rate, the head loss in the suction pipe is estimated to be $2.3 \mathrm{~m}$, and the pump specifications give the required net positive suction head as $2.9 \mathrm{~m}$. Standard sea-level atmospheric conditions are expected at the site, and under worstcase conditions, the temperature of the water in the reservoir is $25^{\circ} \mathrm{C}$. What is the maximum allowable elevation of the suction side of the pump above the reservoir water surface?

Penny Riley
Penny Riley
Numerade Educator
00:53

Problem 47

Under design conditions, a water pump is expected to deliver $500 \mathrm{~L} / \mathrm{s}$ with an added head of $75 \mathrm{~m}$. The diameter of the suction pipe is $550 \mathrm{~mm}$. Under worst-case conditions, the temperature of the water is expected to be $80^{\circ} \mathrm{C}$ and atmospheric pressure is $96 \mathrm{kPa}$. The manufacturer states that the cavitation parameter of the pump is equal to 0.12 . A pressure gauge is installed on the suction side of the pump to detect conditions when cavitation is likely to occur. At what pressure reading is cavitation likely to occur?

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
01:32

Problem 48

Water at $20^{\circ} \mathrm{C}$ is to be pumped out of a reservoir at a rate of $20 \mathrm{~L} / \mathrm{s}$ through a vertical 150 -mm-diameter ductile iron pipeline with an estimated roughness height of $0.3 \mathrm{~mm}$. Appurtenances installed in the intake pipe are expected to contribute to a total local head loss coefficient of 12 . The pump being considered for installation has a required net positive suction head of $5.5 \mathrm{~m}$. (a) What is the maximum elevation of the pump relative to the water surface in the reservoir? (b) If the total head loss coefficient can be reduced to 1.2 , how much higher can the pump be placed? Assume standard atmospheric conditions.

Penny Riley
Penny Riley
Numerade Educator
01:38

Problem 49

Tests on a pump under standard atmospheric conditions show that when water at $20^{\circ} \mathrm{C}$ is pumped at $60 \mathrm{~L} / \mathrm{s}$ and the head added by the pump is $40 \mathrm{~m},$ cavitation occurs when the pressure head plus velocity head on the suction side of the pump is $3.9 \mathrm{~m}$.
(a) Determine the required net positive suction head and the cavitation number of the pump. (b) If this same pump is operated on a mountain under the same flow rate and added head condition but the temperature of the water is $5^{\circ} \mathrm{C}$ and the atmospheric pressure is $90 \mathrm{kPa}$, by how much must the elevation of the pump above the sump reservoir be reduced compared with the test condition? Assume that the friction loss in the suction pipe remains approximately the same and that the sump reservoir is open to the atmosphere in both cases.

Chai Santi
Chai Santi
Numerade Educator
04:53

Problem 50

A pump with an impeller diameter of $225 \mathrm{~mm}$ and a rotational speed of $1725 \mathrm{rpm}$ pumps water at a temperature of $80^{\circ} \mathrm{C}$. Cavitation begins on the suction side of the pump when the volume flow rate is $50 \mathrm{~L} / \mathrm{s},$ the pressure on the suction side of the pump is $80 \mathrm{kPa}$, and the corresponding velocity in the suction pipe is $5 \mathrm{~m} / \mathrm{s}$. (a) What is the required net positive suction head of the pump? (b) If a geometrically similar pump has a rotational speed of $1140 \mathrm{rpm}$ and an impeller diameter of $675 \mathrm{~mm}$, what required net positive suction head is expected in the larger pump?

Narayan Hari
Narayan Hari
Numerade Educator
03:11

Problem 51

Water is to be pumped out of a well and stored in an above-ground reservoir. The water surface in the well is $3 \mathrm{~m}$ below the ground surface, water is to be pumped through a 100 -m-long, 50 -mm-diameter galvanized iron line and exit $19.3 \mathrm{~m}$ above the ground, and water is to be delivered to the upper reservoir at a rate of at least $370 \mathrm{~L} / \mathrm{min}$ when the upper reservoir is empty. The sum of the local loss coefficients in the system is $1.8 .$ A local pump salesperson suggests that you choose a pump from a set of pumps with performance curves shown in Figure $8.46 .$ Determine whether this set of pumps is worthy of consideration and, if so, what particular pump size is required. What is the maximum height the pump can be placed above ground?

James Kiss
James Kiss
Numerade Educator
03:17

Problem 52

Consider the $165-\mathrm{mm}$ pump with the performance curve shown in Figure $8.46 .$ The motor on this pump is changed from one with a rotational speed of 3500 rpm to one with a rotational speed of 2500 rpm. (a) Plot the performance curve of the modified pump, showing both the head added by the pump and the efficiency of the pump as a function of the flow rate. Use the following units in your plot: flow rate in $\mathrm{L} / \mathrm{s}$, head in $\mathrm{m}$, and efficiency in $\%$. (b) What are the flow rate, head added, and efficiency of the modified pump at its best efficiency point? (c) Calculate and compare the specific speeds of the existing and modified pumps and use these results to infer the type of pump.

Narayan Hari
Narayan Hari
Numerade Educator
03:11

Problem 53

A pump is located $2 \mathrm{~m}$ below the water surface in a reservoir as shown in Figure 8.47 . The piping between the pump and the reservoir consists of $3 \mathrm{~m}$ of 150 -mm-diameter ductile iron pipe (DIP).It is estimated that the DIP has an equivalent sand roughness of $0.25 \mathrm{~mm}$. The valves and bends between the source reservoir and the pump are estimated to have a total local loss coefficient of $50 .$ Under design conditions, the water in the reservoir has a temperature of $25^{\circ} \mathrm{C}$ and the atmospheric pressure is $101.3 \mathrm{kPa}$. The pump specifications require that the pump have a minimum net positive suction head of $4 \mathrm{~m}$. What is the maximum allowable flow rate through the system so as to avoid cavitation?

James Kiss
James Kiss
Numerade Educator
01:32

Problem 54

A pump lifts water through a 100 -mm-diameter ductile iron pipe from a lower to upper reservoir (Figure 8.48 ). If the difference in elevation between the water surfaces in the reservoirs is $10 \mathrm{~m}$ and the performance curve of the 2400 -rpm pump is given by
$$
h_{\mathrm{p}}=15-0.1 Q^{2}
$$
where $h_{\mathrm{p}}$ is in $\mathrm{m}$ and $Q$ is in $\mathrm{L} / \mathrm{s}$, estimate the flow rate through the system. If the pump manufacturer gives the required net positive suction head under these operating conditions as $1.5 \mathrm{~m}$, what is the maximum height above the lower reservoir that the pump can be placed and maintain the same operating conditions?

Penny Riley
Penny Riley
Numerade Educator
02:39

Problem 55

Water is being pumped from Point $\mathrm{A}$ in a lower reservoir to Point $\mathrm{F}$ in an upper reservoir through a 30 -m-long PVC pipe of diameter $150 \mathrm{~mm}$ (see Figure 8.49 ). An open gate valve is located at $\mathrm{C} ; 90^{\circ}$ bends (threaded) are located at $\mathrm{B}, \mathrm{D},$ and $\mathrm{E} ;$ and the pump performance curve is given by
$$
h_{\mathrm{p}}=20-4713 Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added by the pump in $\mathrm{m}$ and $Q$ is the flow rate in $\mathrm{m}^{3} / \mathrm{s}$. The dimensionless specific speed of the pump is $1.10 .$ Assume that the flow is turbulent (in the smooth, rough, or transition range) and the temperature of the water is $20^{\circ} \mathrm{C}$.
(a) Write the energy equation between the upper and lower reservoirs, accounting for entrance, exit, and local losses between $\mathrm{A}$ and $\mathrm{F}$. (b) Calculate the flow rate and velocity in the pipe. (c) If the required net positive suction head at the pump operating point is $3.0 \mathrm{~m}$, assess the potential for cavitation in the pump. (For this analysis, assume that the head loss in the suction pipe is negligible.) (d) Use the affinity laws to estimate the pump performance curve when the rotational speed is changed from $800 \mathrm{rpm}$ to $1600 \mathrm{rpm} .$

Chai Santi
Chai Santi
Numerade Educator
03:11

Problem 56

The performance curve of a Goulds Model 3656 irrigation pump with several impeller sizes is shown in Figure $8.50 .$ The pump is to be used to deliver water at $20^{\circ} \mathrm{C}$ from a pond to the center of a large field located $100 \mathrm{~m}$ from the pond. The pump is expected to deliver a maximum flow rate of $380 \mathrm{~L} / \mathrm{min}$ through $107 \mathrm{~m}$ of 600 -mm-diameter steel pipe having an equivalent sand roughness of $0.01 \mathrm{~mm}$. The water surface elevation in the pond during the irrigation season is $10.00 \mathrm{~m},$ and the ground elevation of the field is $15.00 \mathrm{~m}$. (a) Which of the impeller sizes shown in Figure 8.50 would you select for the job? (b) Using the efficiency of the pump under the operating conditions, calculate the size of the motor in kilowatts that must be used to drive the pump.
(c) What is the maximum height above the pond that the pump could be located?

James Kiss
James Kiss
Numerade Educator
02:56

Problem 57

If the performance curve of a certain pump model is given by
$$
h_{\mathrm{p}}=30-0.05 Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added in $\mathrm{m}$ and $Q$ is the flow rate in $\mathrm{L} / \mathrm{s}$, what is the performance curve of a pump system containing $n$ of these pumps in series? What is the performance curve of a pump system containing $n$ of these pumps in parallel?

Chai Santi
Chai Santi
Numerade Educator
01:26

Problem 58

A pump is placed in a pipe system in which the energy equation (i.e., the system curve) is given by
$$
h_{\mathrm{p}}=15+0.03 Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added by the pump in $\mathrm{m}$ and $Q$ is the flow rate in the system in $\mathrm{L} / \mathrm{s}$. The performance curve of the pump is
$$
h_{\mathrm{p}}=20-0.08 Q^{2}
$$
What is the flow rate in the system? If the pump was replaced by two identical pumps in parallel, what would be the flow rate in the system? If the pump was replaced by two identical pumps in series, what would be the flow rate in the system?

Narayan Hari
Narayan Hari
Numerade Educator
03:20

Problem 59

A wastewater pump station is required to handle a design flow rate of $1000 \mathrm{~L} / \mathrm{s}$, and the pump station must provide an added head of $9 \mathrm{~m}$. Pump units are available, each having a power demand of $35 \mathrm{~kW}$ and an overall efficiency of under optimal conditions $62 \%$ under optimal conditions. The specific speed of each unit is around 1.5. (a) If the pump units are to be placed in parallel to accommodate the design flow, how many pump units are required? (b) What motor speed would be required for optimal performance of each unit? Assume that the properties of the wastewater are approximately equal to those of water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
View

Problem 60

Nine pump units are placed in parallel at a pump station. Each unit has a power demand of $40 \mathrm{~kW}$ and adds $35 \mathrm{~m}$ of head under optimal conditions. The best efficiency of each unit is $60 \%$. The liquid being pumped is water at $20^{\circ} \mathrm{C}$. When all units are operating under optimal conditions, what is the flow rate delivered by the pump station?

Ankur S
Ankur S
Numerade Educator
03:11

Problem 61

Water is to be pumped from a river to an elevated reservoir through a 250 -mmdiameter pipe that has an equivalent sand roughness of $0.1 \mathrm{~mm}$. The water surface in the reservoir is $70 \mathrm{~m}$ above the water surface in the river, and the length of pipeline is $5 \mathrm{~km}$. Initially, a flow rate between the river and the reservoir of $35 \mathrm{~L} / \mathrm{s}$ will be required. However, it is expected that the required flow rate will double to $70 \mathrm{~L} / \mathrm{s}$ at some time in the future. The chief engineer on the project has proposed that a pump from a homologous series with performance characteristics shown in Figure 8.51 be used to deliver the initial flow rate of $35 \mathrm{~L} / \mathrm{s}$, and that the pump station be designed such that additional pumps can be added in parallel to handle the increased flow when it becomes necessary. (a) Determine the required pump size needed to deliver $35 \mathrm{~L} / \mathrm{s}$. (b) What would be the efficiency of the pump when it is delivering $35 \mathrm{~L} / \mathrm{s} ?$ (c) What is the maximum height that this pump can be placed above the water surface of the river? (d) How many pumps would be needed to deliver $70 \mathrm{~L} / \mathrm{s}$. Clearly explain your calculations. Assume that the flow in the pipeline will generally be fully turbulent. For part (c) assume that the entrance loss and friction head loss in the suction pipe is negligible.

James Kiss
James Kiss
Numerade Educator
06:26

Problem 62

A 20 -km-long, 1120 -mm-diameter steel pipe with an estimated roughness height of $0.05 \mathrm{~mm}$ is to deliver water from a water supply reservoir through a system of parallel pumps as shown in Figure $8.52 .$ Both the intake at $\mathrm{A}$ and the exit from the pump system at $\mathrm{B}$ are at an elevation of $5 \mathrm{~m} ;$ the water level in the supply reservoir is $2 \mathrm{~m}$ above the intake at $\mathrm{A} ;$ and the elevation at the delivery point, $\mathrm{C}$, is $15 \mathrm{~m}$. There is negligible head loss due to friction between $\mathrm{A}$ and $\mathrm{B},$ and the performance curve for each pump in the system is given by
$$
h_{\mathrm{p}}=65-7.6 \times 10^{-8} Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added by the pump in $\mathrm{m}$ and $Q$ is the flow through the pump in $\mathrm{m}^{3} / \mathrm{d}$. When the system is delivering $48000 \mathrm{~m}^{3} / \mathrm{d}$ at $\mathrm{C},$ the required pressure at C is 448 kPa. (a) Determine how many pumps are required. (b) Assuming that the flow is fully turbulent, what is the actual flow rate at $\mathrm{C}$ when the number of pumps determined in part (a) is used? The temperature of the water is $20^{\circ} \mathrm{C}$.

Satpal Satpal
Satpal Satpal
Numerade Educator
03:11

Problem 63

The water supply system shown in Figure 8.53 is to be constructed such that water is delivered from a reservoir at $\mathrm{A}$ to two communities located at $\mathrm{C}$ and $\mathrm{D}$. The pipe lengths, diameters, and demand flow rates are as follows:
$$
\begin{array}{cccc}
\hline \text { Line } & \begin{array}{c}
\text { Length } \\
(\mathrm{km})
\end{array} & \begin{array}{c}
\text { Diameter } \\
(\mathrm{mm})
\end{array} & \begin{array}{c}
\text { Flow } \\
(\mathrm{L} / \mathrm{s})
\end{array} \\
\hline \text { AB } & 1.05 & 200 & 27 \\
\text { BC } & 2.80 & 150 & 12 \\
\text { BD } & 2.50 & 150 & 15 \\
\hline
\end{array}
$$
The water surface elevation in the supply reservoir is $3.00 \mathrm{~m},$ and the elevations of the delivery pipes at $\mathrm{C}$ and $\mathrm{D}$ are $2.00 \mathrm{~m}$ and $5.00 \mathrm{~m}$, respectively. Under the given demand conditions, it is desired to have water pressures of at least $350 \mathrm{kPa}$ at $\mathrm{C}$ and
D. All pipes are to be made of ductile iron.
(a) Determine the minimum power that must be delivered by the pump. (b) A pump manufacturer will be able to match the minimum power operating condition by providing several "micro-pumps" in series where each micro-pump has a performance given by
$$
h_{\mathrm{p}}=0.455 D-4000 Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added in $\mathrm{m}, D$ is the size of the micro-pump in $\mathrm{cm},$ and $Q$ is the flow rate in $\mathrm{m}^{3} / \mathrm{s}$. The manufacturer can deliver any size, $D,$ in the range of $40-50 \mathrm{~cm} .$ How many and what size micro-pumps will be needed?

James Kiss
James Kiss
Numerade Educator
02:56

Problem 64

The performance curve for a variable-speed pump operating at $600 \mathrm{rpm}$ is given by
$$
h_{\mathrm{p}}=6-0.05 Q^{2}
$$
where $h_{\mathrm{p}}$ is the head added by the pump in $\mathrm{m}$ and $Q$ is the flow rate in $\mathrm{m}^{3} / \mathrm{min}$. This pump is installed in a system where energy considerations require a system curve given by
$$
h_{\mathrm{p}}=3+0.042 Q^{2}
$$
Find the flow rate in the system when the pump is operating at 600 rpm and compare this with the flow rate when the pump is operating at $1200 \mathrm{rpm}$.

Chai Santi
Chai Santi
Numerade Educator
01:52

Problem 65

A large centrifugal fan generates a flow of $7 \mathrm{~m}^{3} / \mathrm{s}$ with a motor speed of $1140 \mathrm{rpm}$. A smaller geometrically similar fan has a motor speed of $1725 \mathrm{rpm}$, operates at the same efficiency as the larger fan, and generates the same pressure increase. What flow rate is generated by the smaller fan?

Narayan Hari
Narayan Hari
Numerade Educator
02:18

Problem 66

A single-jet Pelton wheel hydropower plant is to be operated such that the shaft power developed by the Pelton wheel is equal to $15 \mathrm{MW}$ when the head just upstream of the nozzle is equal to $1600 \mathrm{~m}$. The Pelton wheel is constructed such that is has a diameter of $4 \mathrm{~m}$, has a deflection angle of $170^{\circ},$ and rotates at a controlled speed of 600 rpm. Determine the appropriate nozzle diameter to be used in the project. Assume water at $20^{\circ} \mathrm{C}$ and a nozzle loss coefficient of 0.03 .

Chai Santi
Chai Santi
Numerade Educator
07:15

Problem 67

An existing Pelton wheel has a diameter of $3 \mathrm{~m}$ and is controlled to rotate at $150 \mathrm{rpm}$. The buckets on the Pelton wheel have a deflection angle of $170^{\circ} .$ The power output from the Pelton wheel is controlled by adjusting the jet nozzle diameter. If the desired power is $850 \mathrm{~kW},$ what is the required nozzle diameter for the most efficient operation of the Pelton wheel? Assume water at $20^{\circ} \mathrm{C}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
00:58

Problem 68

A Pelton wheel has a diameter of $2.7 \mathrm{~m}$ and buckets with a deflection angle of $167^{\circ}$. The nozzle and water supply characteristics are fixed such that under design conditions, the incident jet has a diameter of $150 \mathrm{~mm}$ and a velocity of $12 \mathrm{~m} / \mathrm{s}$. What is the maximum power that can be extracted by this Pelton wheel? What is the rotational speed of the wheel under the maximum power condition? Assume water at $20^{\circ} \mathrm{C}$.

James Kiss
James Kiss
Numerade Educator
02:48

Problem 69

A 4 -m-diameter Pelton wheel is driven by a jet from a single 240 -mm-diameter nozzle, and the pressure and velocity in the pipeline just upstream of the nozzle is $4 \mathrm{MPa}$ and $6 \mathrm{~m} / \mathrm{s}$, respectively. The nozzle loss coefficient is estimated as 0.03 . The buckets on the Pelton wheel have a deflection angle of $165^{\circ} .$ Estimate the following: (a) the maximum power that can be generated by the Pelton wheel, (b) the rotational speed at the maximum power state, (c) the runaway rotational speed, (d) the torque on the wheel shaft at the maximum-power state, and (e) the torque on the wheel shaft when the wheel is held stationary. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
00:55

Problem 70

A manufacturer cites the wheel efficiency of an impulse turbine as $90 \%$ when the turbine is operating at its maximum efficiency state. Estimate the blade angle of the buckets mounted on the turbine.

Chai Santi
Chai Santi
Numerade Educator
00:58

Problem 71

A Pelton wheel has an average radius of $2 \mathrm{~m},$ and its buckets have a deflection angle of $165^{\circ}$. According to the manufacturer, the Pelton wheel has a wheel efficiency of $80 \%$ under optimal operating conditions. In a particular installation, the turbine is driven by a jet with a diameter of $160 \mathrm{~mm}$ and a velocity of $100 \mathrm{~m} / \mathrm{s}$.
(a) Estimate the optimal rotation rate of the turbine. (b) Estimate the maximum power that can be extracted by the turbine. Assume water at $20^{\circ} \mathrm{C}$.

James Kiss
James Kiss
Numerade Educator
01:36

Problem 72

Show that if the effective head on a Pelton wheel is $h_{\mathrm{e}}$, the velocity coefficient of the nozzle is $C_{\mathrm{v}}$, and the bucket speed of the wheel is $U$, then the theoretical maximum efficiency is attained by the Pelton when
$$
U=\frac{1}{2} C_{\mathrm{v}} \sqrt{2 g h_{\mathrm{e}}}
$$

Dominador Tan
Dominador Tan
Numerade Educator
12:53

Problem 73

The peripheral velocity factor of a Pelton wheel, $\phi$, is defined as $\phi=U / \sqrt{2 g h_{\mathrm{e}}}$ where $U$ is the bucket speed, and $h_{\mathrm{e}}$ is the effective head on the turbine. Problem 8.72 shows that the efficiency of a Pelton wheel is maximized when $\phi=\frac{1}{2} C_{\mathrm{v}}$, where $C_{\mathrm{v}}$ is the velocity coefficient of the nozzle. Show that the maximum efficiency of a Pelton wheel turbine, $\eta_{\mathrm{t}},$ under these conditions is given by
$$
\eta_{\mathrm{t}}=\frac{1}{2} C_{\mathrm{v}}^{2}(1-\cos \beta)
$$
where $\beta$ is the bucket angle.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
05:06

Problem 74

The head loss in the nozzle of a Pelton wheel is sometimes represented by a local head loss coefficient, $k_{\mathrm{j}}$, and is sometimes represented by a nozzle velocity coefficient, $C_{\mathrm{v}}$. Show that the relationship between $C_{\mathrm{v}}$ and $k_{\mathrm{j}}$ is given by
$$
C_{\mathrm{v}}=\sqrt{\frac{1}{1+k_{\mathrm{j}}}}
$$

Chai Santi
Chai Santi
Numerade Educator
03:20

Problem 75

A Pelton wheel has a diameter of $5 \mathrm{~m}$ and a bucket angle of $165^{\circ} .$ The effective head at the nozzle is $550 \mathrm{~m}$, and the nozzle is set such that the velocity coefficient is 0.94 and the diameter of the jet is $130 \mathrm{~mm}$. Estimate the optimal flow rate, rotational speed of the wheel, and power output. Assume water at $20^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 76

A Pelton wheel is to be designed to harness the available hydropower from a site where the effective head on the turbine will be $160 \mathrm{~m}$ and the reliable flow rate through the turbine will be $6 \mathrm{~m}^{3} / \mathrm{s}$. The wheel is to have a rotational speed of $500 \mathrm{rpm}$, the bucket angle will be $165^{\circ}$, and the nozzle is expected to have a velocity coefficient of 0.92 . (a) What diameter wheel would maximize the efficiency of the turbine? (b) What power can be expected from the turbine? (c) Assess whether a different type of turbine should be considered for this site. Assume water at $20^{\circ} \mathrm{C}$.

Victor Salazar
Victor Salazar
Numerade Educator
06:58

Problem 77

The Pelton wheel shown in Figure 8.54 is driven by two jets, where each jet has a diameter of $40 \mathrm{~mm}$ and a velocity of $50 \mathrm{~m} / \mathrm{s}$. Each bucket mounted on the Pelton wheel deflects its incident jet by $160^{\circ} .$ The diametric distance between the centers of the buckets is $2 \mathrm{~m}$, and it can be assumed that the reaction forces on the each of the buckets act at the center of the bucket. (a) What torque is required to hold the Pelton wheel stationary? (b) If the Pelton wheel is allowed to "freewheel" such that the shaft torque is negligible, at what speed will the Pelton wheel rotate? (c) If a governor controls the rotational speed at $15.7 \mathrm{rad} / \mathrm{s}$, what power can be generated by the Pelton wheel? Assume water at $20^{\circ} \mathrm{C}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
03:21

Problem 78

The impulse turbine system shown in Figure 8.55 uses water at $20^{\circ} \mathrm{C}$ from a reservoir with a water surface elevation that is $80 \mathrm{~m}$ above the elevation of the nozzle. The length of the 300 -mm-diameter delivery pipe is $600 \mathrm{~m}$, and the delivery pipe has an estimated roughness height of $0.5 \mathrm{~mm} .$ The nozzle at the end of the pipe is estimated to have a nozzle loss coefficient of $0.05,$ and the buckets on the impulse turbine have a deflection angle of $160^{\circ}$. (a) Identify the optimum nozzle diameter and state the corresponding power that can be extracted by the turbine. (b) Determine the optimum nozzle diameter and corresponding power derived by assuming fully turbulent flow in the supply pipe and compare your result with that obtained in part (a). Comment on the appropriateness of assuming fully turbulent flow.

Chai Santi
Chai Santi
Numerade Educator
03:21

Problem 79

At a Pelton wheel installation, the water surface elevation in the supply reservoir is $85 \mathrm{~m}$ above the nozzle; the delivery line has a diameter of $600 \mathrm{~mm}$, a length of $300 \mathrm{~m}$, and a roughness height of $8 \mathrm{~mm}$. The discharge nozzle has a diameter of $50 \mathrm{~mm}$ and a head loss coefficient of $0.8 .$ The bucket friction loss coefficient is $0.5,$ the velocity of water relative to the bucket at the exit from the bucket is $2 \mathrm{~m} / \mathrm{s},$ and the absolute velocity of the water leaving the bucket is $6 \mathrm{~m} / \mathrm{s}$. Determine the power that could be derived from the system and the hydraulic efficiency of the turbine.

Chai Santi
Chai Santi
Numerade Educator
View

Problem 80

A Francis turbine unit rotating at 900 rpm is to be designed to deliver a head of $180 \mathrm{~m}$ when operating under design conditions. Based on experience with similar units, the overall efficiency of the turbine unit is expected to be $82 \%$ and the hydraulic efficiency is expected to be $90 \%$. The guide vanes are oriented to direct the inflow at an angle of $30^{\circ}$ to the inflow surface, and the blades in the runner are to be at an angle of $60^{\circ}$ to the inflow surface as shown in Figure $8.56 .$ The width of the runner is $20 \%$ of the diameter of the inflow surface, and the outflow velocity is normal to the outflow surface of the runner. What outer diameter of the turbine runner is required?

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 81

A Francis turbine has an overall efficiency of $92 \%$ and a hydraulic efficiency of $95 \%$. The turbine runner has an outer diameter of $2.5 \mathrm{~m}$, has a width of $0.35 \mathrm{~m}$, and rotates at $300 \mathrm{rpm} .$ The turbine is located $4 \mathrm{~m}$ above the discharge reservoir surface, and the head loss in the draft tube can be assumed negligible. Under design conditions, the flow rate through the turbine is $20 \mathrm{~m}^{3} / \mathrm{s}$ and the velocity and pressure at the entrance to the scroll case are $10 \mathrm{~m} / \mathrm{s}$ and $2447 \mathrm{kPa}$, respectively. The flow exits the runner in a direction normal to the outflow surface. (a) Estimate the power output of the turbine unit. (b) What is the specific speed of the turbine? (c) Estimate the guide vane and runner blade angles. Assume water at $20^{\circ} \mathrm{C}$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 82

The inflow surface of a runner in a reaction turbine has a diameter and width of $500 \mathrm{~mm}$ and $60 \mathrm{~mm}$, respectively, and the outflow surface has a diameter and width of $350 \mathrm{~mm}$ and $80 \mathrm{~mm}$, respectively. On both the inflow and outflow surfaces, the blades occupy $6 \%$ of the flow area. Under design conditions, the inflow and outflow velocity components are shown in Figure $8.57,$ where $\mathbf{V}, \mathbf{U},$ and $\mathbf{W}$ are the absolute velocity, tip velocity, and velocity relative to the moving runner, respectively. The guide vane directs the inflow velocity at an angle of $24^{\circ}$ to the inflow surface, the runner blade is at an angle of $80^{\circ}$ to the inflow surface, and the flow in the runner should be aligned with the runner blade. The runner blade makes an angle of $30^{\circ}$ with the outflow surface. It is estimated that the head available to the turbine is $65 \mathrm{~m}$, of which $10 \%$ is lost due to hydraulic friction within the turbine, and $5 \%$ is lost due to mechanical friction in the turbine system. (a) At what speed should the turbine be operated so that shock losses are minimized, which means that the flow in the runner is aligned with the runner blade on inflow? (b) What is the power output of the turbine? Assume water at $20^{\circ} \mathrm{C}$.

Victor Salazar
Victor Salazar
Numerade Educator
01:05

Problem 83

Flow enters a Francis turbine at an angle of $20^{\circ}$ to the inflow surface as shown in Figure $8.58 .$ The runner has an outer diameter of $1600 \mathrm{~mm}$ and an inner diameter of $900 \mathrm{~mm}$, and the width of the inflow surface is $80 \mathrm{~mm}$. When the runner rotates at $120 \mathrm{rpm},$ the flow rate through the runner is $2.9 \mathrm{~m}^{3} / \mathrm{s}$ and the outflow velocity is normal to the outflow surface. Estimate the power produced by the turbine. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
01:28

Problem 84

A hydropower plant utilizes eight Francis turbines. The change in head across the power plant is $250 \mathrm{~m}$, and the flow through each turbine unit is $12 \mathrm{~m}^{3} / \mathrm{s}$. The estimated efficiency of each turbine unit is $95 \%$, and the efficiency of the generator and supporting power delivery systems is $91 \%$. Estimate the power-generating capacity of the hydropower facility. Assume water at $20^{\circ} \mathrm{C}$.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:27

Problem 85

A hydropower installation with reaction-type (Francis) turbines is to be located where the downstream water surface elevation is $100 \mathrm{~m}$ below the water surface elevation in the upstream reservoir. The 2.0 -m-diameter concrete-lined penstock is $500 \mathrm{~m}$ long and has an estimated roughness height of $15 \mathrm{~mm}$. When the flow rate through the system is $20 \mathrm{~m}^{3} / \mathrm{s}$, the combined head loss in the turbine and draft tube is $5.0 \mathrm{~m}$, and the average velocity in the tailrace is $0.80 \mathrm{~m} / \mathrm{s}$. Estimate the power that can be extracted from the system.

Penny Riley
Penny Riley
Numerade Educator
01:05

Problem 86

A large hydropower facility uses several Francis turbines in its power plant. Each turbine has a rated power output of $50 \mathrm{MW}$ at a head of $70 \mathrm{~m}$, and has a maximum hydraulic efficiency of $90 \%$. The generator in each unit has a rotational speed of 120 rpm. (a) Estimate the specific speed of each unit and verify that a Francis turbine is the best type of turbine to use. (b) Estimate the optimal flow rate through each turbine unit. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
01:05

Problem 87

A Francis turbine with a runner diameter of $600 \mathrm{~mm}$ has been working well at a particular hydropower installation for many years, and an analysis of available performance data indicate that the efficiency of this turbine is $85 \%$ when it is operating at its best efficiency point. It is planned that a geometrically similar turbine with a runner diameter of $2500 \mathrm{~mm}$ will be used at another site. Estimate the maximum efficiency that can be expected from the larger unit?

Chai Santi
Chai Santi
Numerade Educator
01:27

Problem 88

A hydropower facility that is under design is to accommodate a design flow rate of $44 \mathrm{~m}^{3} / \mathrm{s},$ and under this condition, the available head is $40 \mathrm{~m}$. The turbine under consideration has a shaft rotation rate of $150 \mathrm{rpm}$ and can generate a shaft power $12 \mathrm{MW}$ when operated at maximum efficiency. (a) What is the hydraulic efficiency of the turbine under consideration? (b) What type of turbine is being considered? (c) If the available head during operation is reduced to $18 \mathrm{~m}$ and the turbine is operated at maximum efficiency, what shaft power can be extracted by the turbine? Assume water at $20^{\circ} \mathrm{C}$.

Penny Riley
Penny Riley
Numerade Educator
03:31

Problem 89

A site being considered for hydropower development has an available head of $800 \mathrm{~m}$ and a reliable flow rate of $2.5 \mathrm{~m}^{3} / \mathrm{s}$. A generator operating at a speed of $600 \mathrm{rpm}$ is feasible, and the expected hydraulic efficiency of a turbine unit at this site is $95 \%$. What type of turbine should be considered for use?

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:51

Problem 90

A hydropower facility is reported to have several turbine units, each with a rated flow rate of $8 \mathrm{~m}^{3} / \mathrm{s}$ at a head of $40 \mathrm{~m}$. The rotational speed of the runner in each turbine is $140 \mathrm{rpm}$, and the hydraulic efficiency of each unit is $95 \%$. Estimate the type and size of each unit.

Chai Santi
Chai Santi
Numerade Educator
01:28

Problem 91

A hydroelectric power plant is reported to have three turbine units, with each unit having a rated power output of $520 \mathrm{~kW}$, a runner size of $800 \mathrm{~mm}$, and a rotation speed of $150 \mathrm{rpm}$. Under optimal operating conditions, when the available head is $12 \mathrm{~m}$, the flow through each turbine unit is $6 \mathrm{~m}^{3} / \mathrm{s}$. What type of turbine unit is likely being used at this site? Assume water at $20^{\circ} \mathrm{C}$.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:37

Problem 92

A proposed turbine is being designed to generate a power of $30 \mathrm{MW},$ with a generator rotational speed of $150 \mathrm{rpm}$ and an available head of $22 \mathrm{~m}$. A model of the turbine is to be tested in the laboratory, where the available head is $6 \mathrm{~m}$, the power is $45 \mathrm{~kW},$ and the model turbine is expected to have a hydraulic efficiency of $95 \%$. What length scale, rotational speed, and flow rate should be used in the model tests? Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
01:05

Problem 93

A Francis turbine that is operating successfully at a particular hydropower facility has a runner of diameter $2.5 \mathrm{~m}$, has a rotational speed of $120 \mathrm{rpm}$, and generates $160 \mathrm{MW}$ of power when the available head is $60 \mathrm{~m}$ and the flow rate through the turbine is $350 \mathrm{~m}^{3} / \mathrm{s}$. It is anticipated that these optimal conditions can be scaled up at a new hydropower facility, which will use a turbine with the same rotational speed, but the available head will be $90 \mathrm{~m}$. Estimate the runner size, flow rate, power delivered, and efficiency expected at the upscaled facility. Assume water at $20^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
05:16

Problem 94

The performance of a turbine is being studied using a $\frac{1}{5}$ -scale model. The prototype (full-scale) turbine operates at a design head of $35 \mathrm{~m}$ when the flow rate through the turbine is $64.1 \mathrm{~m}^{3} / \mathrm{s}$ and the angular speed of the runner is $600 \mathrm{rpm}$. The model is to be tested at a head of $12 \mathrm{~m}$. (a) What should be the angular speed and flow rate in the model to achieve similarity with the prototype? (b) If the shaft power generated in the model is measured as $110 \mathrm{~kW}$ and the efficiency in the prototype is assumed to be $5 \%$ better than the efficiency in the model, estimate the power that is generated in the prototype under design conditions. (c) What is the specific speed of the turbine, and what should be its type? Assume water at $20^{\circ} \mathrm{C}$.

WM
William Mead
Numerade Educator
01:27

Problem 95

A particular site in a river valley has an available head for hydropower generation of $35 \mathrm{~m}$. Turbines used at such sites by the local power authority typically have a rotational speed of 130 rpm and optimal efficiencies of approximately $85 \% .$ It is desired to generate $25 \mathrm{MW}$ of power at this site. (a) What type of turbine is required? (b) Approximately what flow rate through the turbine would generate the required power at the most efficient operating condition? Assume water at $20^{\circ} \mathrm{C}$.

Penny Riley
Penny Riley
Numerade Educator
01:05

Problem 96

A Francis turbine is located at an elevation of $2000 \mathrm{~m}$ above sea level. The discharge from the runner is $5 \mathrm{~m}$ above the tailwater pool, the water flowing through the runner is at a temperature of $15^{\circ} \mathrm{C}$, and the critical cavitation parameter of the runner is given by the manufacturer as 0.25 . The head loss in the draft tube is negligible. Estimate the maximum available head that can be accommodated without the occurrence of cavitation.

Chai Santi
Chai Santi
Numerade Educator
01:27

Problem 97

A proposed hydropower plant is to be located at a site with an available head of $9 \mathrm{~m}$, and it is desired to obtain a (shaft) power of $35 \mathrm{MW}$ from this site. The axial-flow turbine units under consideration operate at an angular speed of $150 \mathrm{rpm}$, have a specific speed of $5,$ and have an estimated maximum efficiency of $80 \%$.
(a) How may of these units are required? (b) What total flow rate must be available to generate the desired power? Assume water at $20^{\circ} \mathrm{C}$.

Penny Riley
Penny Riley
Numerade Educator
03:01

Problem 98

(a) What is the typical efficiency associated with a hydropower installation? (b) If a potential site for hydropower generation has an average available head of $350 \mathrm{~m}$ and an average flow rate of $0.5 \mathrm{~m}^{3} / \mathrm{s}$, estimate the amount of hydropower that could reasonably be expected from this site.

Chai Santi
Chai Santi
Numerade Educator
01:52

Problem 99

A hydroelectric project is to be developed where the 85 percentile flow rate is $1540 \mathrm{~m}^{3} / \mathrm{s},$ and upstream storage to a height of $20 \mathrm{~m}$ above the downstream stage is required to meet all the water-supply demands. Estimate the maximum installed capacity that would be appropriate for these conditions.

Mayukh Banik
Mayukh Banik
Numerade Educator