Introduction to Fluid Mechanics

Philip J. Pritchard, Robert W. Fox

Chapter 10

Fluid Machinery - all with Video Answers

Educators

Chapter Questions

Problem 1

Dimensions of a centrifugal pump impeller are
$$\begin{array}{lcc}
\text { Parameter } & \text { Inlet, Section }(\mathrm{l}) & \text { Outlet, Section }(2) \\
\text { Radius, } r(\mathrm{mm}) & 175 & 500 \\
\text { Blade width, } b(\mathrm{mm}) & 50 & 30 \\
\text { Blade angle, } \beta(\mathrm{deg}) & 65 & 70
\end{array}$$
The pump handles water and is driven at 750 rpm. Calculate the theoretical head and mechanical power input if the flow rate is $0.75 \mathrm{m}^{3} / \mathrm{s}$

Rashmi Sinha

Rashmi Sinha

Numerade Educator

Problem 2

The geometry of a centrifugal water pump is $r_{1}=10 \mathrm{cm}$ $r_{2}=20 \mathrm{cm}, b_{1}=b_{2}=4 \mathrm{cm}, \beta_{1}=30^{\circ}, \beta_{2}=15^{\circ},$ and it runs at
speed 1600 rpm. Estimate the discharge required for axial entry, the power generated in the water in watts, and the head produced.

Chai Santi

Numerade Educator

Problem 3

A centrifugal pump running at 3000 rpm pumps water at a rate of $0.6 \mathrm{m}^{3} / \mathrm{min} .$ The water enters axially and leaves the impeller at $5.4 \mathrm{m} / \mathrm{s}$ relative to the blades, which are radial at the exit. If the pump requires $5 \mathrm{kW}$ and is 72 percent efficient, estimate the basic dimensions (impeller exit diameter and width), using the Euler turbomachine equation.

Chai Santi

Numerade Educator

Problem 4

Consider the centrifugal pump impeller dimensions given in Example $10.1 .$ Estimate the ideal head rise and mechanical power input if the outlet blade angle is changed to $60^{\circ}, 70^{\circ}, 80^{\circ},$ or $85^{\circ}$

Chai Santi

Numerade Educator

Problem 5

Dimensions of a centrifugal pump impeller are
$$\begin{array}{lcc}
\text { Parameter } & \text { Inlet, Section }(1) & \text { Outlet, Section }(2) \\
\text { Radius, } r \text { (in.) } & 15 & 45 \\
\text { Blade width, } b \text { (in.) } & 4.75 & 3.25 \\
\text { Blade angle, } \beta(\operatorname{deg}) & 40 & 60
\end{array}$$
The pump is driven at 575 rpm and the fluid is water. Calculate the theoretical head and mechanical power if the flow rate is 80,000 gpm.

Chai Santi

Numerade Educator

Problem 6

Dimensions of a centrifugal pump impeller are
$$\begin{array}{lcc}
\text { Parameter } & \text { Inlet, Section }(1) & \text { Outlet, Section }(2) \\
\text { Radius, } r \text { (in.) } & 3 & 9.75 \\
\text { Blade width, } b \text { (in.) } & 1.5 & 1.125 \\
\text { Blade angle, } \beta(\operatorname{deg}) & 60 & 70
\end{array}$$
The pump is driven at 1250 rpm while pumping water. Calculate the theoretical head and mechanical power input if the flow rate is 1500 gpm.

Chai Santi

Numerade Educator

Problem 7

For the impeller of Problem $10.6,$ determine the rotational speed for which the tangential component of the inlet velocity is zero if the volume flow rate is 4000 gpm. Calculate the theoretical head and mechanical power input.

Chai Santi

Numerade Educator

Problem 8

A centrifugal water pump, with 15 cm diameter impeller and axial inlet flow, is driven at 1750 rpm. The impeller vanes are backward-curved $\left(\beta_{2}=65^{\circ}\right)$ and have axial width $b_{2}=2 \mathrm{cm} .$ For a volume flow rate of $225 \mathrm{m}^{3} / \mathrm{hr}$ determine the theoretical head rise and power input to the pump.

Chai Santi

Numerade Educator

Problem 9

For the impeller of Problem 10.1 , operating at $750 \mathrm{rpm}$ determine the volume flow rate for which the tangential component of the inlet velocity is zero. Calculate the theoretical head and mechanical power input.

Chai Santi

Numerade Educator

Problem 10

Consider the geometry of the idealized centrifugal pump described in Problem $10.11 .$ Draw inlet and outlet velocity diagrams assuming $b=$ constant. Calculate the inlet blade angles required for "shockless" entry flow at the design flow rate. Evaluate the theoretical power input to the pump at the design flow rate.

Chai Santi

Numerade Educator

Problem 11

Consider a centrifugal water pump whose geometry and flow conditions are as follows:
Assume ideal pump behavior with 100 percent efficiency. Find the shutoff head. Calculate the absolute and relative discharge velocities, the total head, and the theoretical power required at the design flow rate.

Chai Santi

Numerade Educator

Problem 12

Consider the centrifugal pump impeller dimensions given in Example $10.1 .$ Construct the velocity diagram for shockless flow at the impeller inlet, if $b=$ constant. Calculate the effective flow angle with respect to the radial impeller blades for the case of no inlet swirl. Investigate the effects on flow angle of (a) variations in impeller width and (b) inlet swirl velocities.

Chai Santi

Numerade Educator

Problem 13

For the impeller of Problem $10.5,$ determine the inlet blade angle for which the tangential component of the inlet velocity is zero if the volume flow rate is 125,000 gpm. Calculate the theoretical head and mechanical power input.

Chai Santi

Numerade Educator

Problem 14

A centrifugal water pump designed to operate at 1300 rpm has dimensions
$$\begin{array}{lcc}
\text { Parameter } & \text { Inlet } & \text { Outlet } \\
\text { Radius, } r(\mathrm{mm}) & 100 & 175 \\
\text { Blade width, } b(\mathrm{mm}) & 10 & 7.5 \\
\text { Blade angle, } \beta(\mathrm{deg}) & & 40
\end{array}$$
Draw the inlet velocity diagram for a volume flow rate of 35 L/s. Determine the inlet blade angle for which the entering velocity has no tangential component. Draw the outlet velocity diagram. Determine the outlet absolute flow angle (measured relative to the normal direction). Evaluate the hydraulic power delivered by the pump, if its efficiency is 75 percent. Determine the head developed by the pump.

Chai Santi

Numerade Educator

Problem 15

A centrifugal pump runs at 1750 rpm while pumping water at a rate of $50 \mathrm{L} / \mathrm{s}$. The water enters axially, and leaves tangential to the impeller blades. The impeller exit diameter and width are $300 \mathrm{mm}$ and $10 \mathrm{mm}$, respectively. If the pump requires $45 \mathrm{kW}$, and is 75 percent efficient, estimate the exit angle of the impeller blades.

Chai Santi

Numerade Educator

Problem 16

A centrifugal water pump designed to operate at 1200 rpm has dimensions
$$\begin{array}{lcc}
\text { Parameter } & \text { Inlet } & \text { Outlet } \\
\text { Radius, } r(\mathrm{mm}) & 90 & 150 \\
\text { Blade width, } b(\mathrm{mm}) & 10 & 7.5 \\
\text { Blade angle, } \beta(\mathrm{deg}) & 25 & 45
\end{array}$$
Determine the flow rate at which the entering velocity has no tangential component. Draw the outlet velocity diagram, and determine the outlet absolute flow angle (measured relative to the normal direction) at this flow rate. Evaluate the hydraulic power delivered by the pump if its efficiency is 70 percent. Determine the head developed by the pump.

Chai Santi

Numerade Educator

Problem 17

Repeat the analysis for determining the optimum speed for an impulse turbine of Example $10.13,$ using the Euler turbomachine equation.

Chai Santi

Numerade Educator

Problem 18

Kerosene is pumped by a centrifugal pump. When the flow rate is 350 gpm, the pump requires 18 hp input, and its efficiency is 82 percent. Calculate the pressure rise produced by the pump. Express this result as (a) feet of water and (b) feet of kerosene.

Chai Santi

Numerade Educator

Problem 18

A compressor has been designed for entrance conditions of 14.7 psia and $70^{\circ} \mathrm{F}$. To economize on the power required, it is being tested with a throttle in the entry duct to reduce the entry pressure. The characteristic curve for its normal design speed of 3200 rpm is being obtained on a day when the ambient temperature is $58^{\circ} \mathrm{F}$. At what speed should the compressor be run? At the point on the characteristic curve at which the mass flow would normally be $125 \mathrm{lbm} / \mathrm{s}$ the entry pressure is 8.0 psia. Calculate the actual mass flow rate during the test.

Narayan Hari

Numerade Educator

Problem 19

A centrifugal pump designed to deliver water at $70 \mathrm{cfm}$ has dimensions
$$\begin{array}{lcc}
\text { Parameter } & \text { Inlet } & \text { Outlet } \\
\text { Radius, } r \text { (in.) } & 14 & 7 \\
\text { Blade width, } b \text { (in.) } & 0.4 & 0.3 \\
\text { Blade angle, } \beta\left(^{\circ}\right) & 20 & 45
\end{array}$$
Draw the inlet velocity diagram. Determine the design speed if the entering velocity has no tangential component. Draw the outlet velocity diagram. Determine the outlet absolute flow angle (measured relative to the normal direction). Evaluate the theoretical head developed by the pump. Estimate the minimum mechanical power delivered to the
pump.

Chai Santi

Numerade Educator

Problem 19

The turbine for a new jet engine was designed for entrance conditions of 160 psia and $1700^{\circ} \mathrm{F}$, ingesting $500 \mathrm{lbm} / \mathrm{s}$ at a speed of $500 \mathrm{rpm},$ and exit conditions of 80 psia and $1350^{\circ} \mathrm{F}$. If the altitude and fueling for the engine were changed such that the entrance conditions were now 140 psia and $1600^{\circ} \mathrm{F}$, calculate the new operating speed, mass flow rate, and exit conditions for similar operation, i.e. equal efficiency.

Chai Santi

Numerade Educator

Problem 20

In the water pump of Problem $10.8,$ the pump casing acts as a diffuser, which converts 60 percent of the absolute velocity head at the impeller outlet to static pressure rise. The head loss through the pump suction and discharge channels is 0.75 times the radial component of velocity head leaving the impeller. Estimate the volume flow rate, head rise, power input, and pump efficiency at the maximum efficiency point. Assume the torque to overcome bearing, seal, and spin losses is 10 percent of the ideal torque at $Q=0.065 \mathrm{m}^{3} / \mathrm{s}.$

Chai Santi

Numerade Educator

Problem 20

We have seen many examples in Chapter 7 of replacing working fluids in order to more easily achieve similitude between models and prototypes. Describe the effects of testing an air compressor using helium as the working fluid on the dimensionless and dimensional parameters we have discussed for compressible flow machines.

Chai Santi

Numerade Educator

Problem 21

The theoretical head delivered by a centrifugal pump at shutoff depends on the discharge radius and angular speed of the impeller. For preliminary design, it is useful to have a plot showing the theoretical shutoff characteristics and approximating the actual performance. Prepare a log-log plot of impeller radius versus theoretical head rise at shutoff with standard motor speeds as parameters. Assume the fluid is water and the actual head at the design flow rate is 70 percent of the theoretical shutoff head. (Show these as dashed lines on the plot.) Explain how this plot might be used for preliminary design.

Chai Santi

Numerade Educator

Problem 22

Use data from Appendix $D$ to choose points from the performance curves for a Peerless horizontal split case Type $16 \mathrm{A} 18 \mathrm{B}$ pump at 705 and 880 nominal rpm. Obtain and plot curve-fits of total head versus delivery for this pump, with an 18.0 -in.-diameter impeller.

Chai Santi

Numerade Educator

Problem 23

Use data from Appendix D to choose points from the performance curves for a Peerless horizontal split case Type 4AE12 pump at 1750 and 3550 nominal rpm. Obtain and plot curve-fits for total head versus delivery at each speed for this pump, with a 12 -in.-diameter impeller.

Chai Santi

Numerade Educator

Problem 24

Data from tests of a water suction pump operated at 2000 rpm with a 12 -in. diameter impeller are $$\begin{array}{lrrrrr}
\text { Flow rate, } Q(\mathrm{cfm}) & 36 & 50 & 74 & 88 & 125 \\
\text { Total head, } H(\mathrm{ft}) & 190 & 195 & 176 & 162 & 120 \\
\text { Power input, } \vartheta(\mathrm{hp}) & 25 & 30 & 35 & 40 & 46
\end{array}$$
Plot the performance curves for this pump; include a curve of efficiency versus volume flow rate. Locate the best efficiency point and specify the pump rating at this point.

Chai Santi

Numerade Educator

Problem 25

A 9 -in.-diameter centrifugal pump, running at 900 rpm with water at $68^{\circ} \mathrm{F}$ generates the following performance data:
$$\begin{array}{lcrrrrr}
\text { Flow rate, } Q(\mathrm{cfm}) & 0 & 200 & 400 & 600 & 800 & 1000 \\
\text { Total head, } H(\mathrm{ft}) & 23.0 & 22.3 & 21.0 & 19.5 & 17.0 & 12.5 \\
\text { Power input, } \mathscr{P}(\mathrm{hp}) & 3.13 & 3.50 & 4.06 & 4.47 & 4.88 & 5.09
\end{array}$$
Plot the performance curves for this pump; include a curve of efficiency versus volume flow rate. Locate the best efficiency point. What is the specific speed for this pump?

Chai Santi

Numerade Educator

Problem 26

An axial-flow fan operates in seal-level air at 1350 rpm and has a blade tip diameter of 3 ft and a root diameter of $2.5 \mathrm{ft} .$ The inlet angles are $\alpha_{1}=55^{\circ}, \beta_{1}=30^{\circ},$ and at the exit $\beta_{2}=60^{\circ} .$ Estimate the flow volumetric flow rate, horsepower, and the outlet angle, $\alpha_{2}.$

Chai Santi

Numerade Educator

Problem 27

Write the turbine specific speed in terms of the flow coefficient and the head coefficient.

Chai Santi

Numerade Educator

Problem 28

Data measured during tests of a centrifugal pump driven at 3000 rpm are $$\begin{array}{lcc}
& \text { Inlet, Section } & \text { Outlet, Section } \\
\text { Parameter } & \text { (1) } & \text { (2) } \\
\text { Gage pressure, } p \text { (psi) } & 12.5 & \\
\text { Elevation above datum, } z \text { (ft) } & 6.5 & 32.5 \\
\text { Average speed of flow, } \bar{V} \text { (ft/s) } & 6.5 & 15
\end{array}$$
The flow rate is 65 gpm and the torque applied to the pump shaft is 4.75 lbf . ft. The pump efficiency is 75 percent, and the electric motor efficiency is 85 percent. Find the electric power required, and the gage pressure at section (2).

Chai Santi

Numerade Educator

Problem 29

The kilogram force (kgf), defined as the force exerted by a kilogram mass in standard gravity, is commonly used in European practice. The metric horsepower (hpm) is defined as $1 \mathrm{hpm} \equiv 75 \mathrm{m} \cdot \mathrm{kgf} / \mathrm{s}$. Develop a conversion relating metric horsepower to U.S. horsepower. Relate the specific speed for a hydraulic turbine-calculated in units of rpm, metric horsepower, and meters-to the specific speed calculated in U.S. customary units.

Narayan Hari

Numerade Educator

Problem 30

Write the pump specific speed in terms of the flow coefficient and the head coefficient.

Narayan Hari

Numerade Educator

Problem 31

A small centrifugal pump, when tested at $N=2875 \mathrm{rpm}$ with water, delivered $Q=0.016 \mathrm{m}^{3} / \mathrm{s}$ and $H=40 \mathrm{m}$ at its best efficiency point $(\eta=0.70) .$ Determine the specific speed of the pump at this test condition. Sketch the impeller shape you expect. Compute the required power input to the pump.

Narayan Hari

Numerade Educator

Problem 32

Typical performance curves for a centrifugal pump, tested with three different impeller diameters in a single casing, are shown. Specify the flow rate and head produced by the pump at its best efficiency point with a 12 -in. diameter impeller. Scale these data to predict the performance of this pump when tested with 11 in. and 13 in. impellers. Comment on the accuracy of the scaling procedure.

Chai Santi

Numerade Educator

Problem 33

A pump with $D=500 \mathrm{mm}$ delivers $Q=0.725 \mathrm{m}^{3} / \mathrm{s}$ of
water at $H=10 \mathrm{m}$ at its best efficiency point. If the specific speed of the pump is $1.74,$ and the required input power is 90 $\mathrm{kW},$ determine the shutoff head, $H_{0},$ and best efficiency, $\eta$ What type of pump is this? If the pump is now run at 900 rpm, by scaling the performance curve, estimate the new flow rate, head, shutoff head, and required power.

Chai Santi

Numerade Educator

Problem 34

At its best efficiency point $(\eta=0.87),$ a mixed-flow pump, with $D=16$ in., delivers $Q=2500 \mathrm{cfm}$ of water at $H=140 \mathrm{ft}$ when operating at $N=1350 \mathrm{rpm} .$ Calculate the specific speed of this pump. Estimate the required power input. Determine the curve-fit parameters of the pump performance curve based on the shutoff point and the best efficiency point. Scale the performance curve to estimate the flow, head, efficiency, and power input required to run the same pump at 820 rpm.

Chai Santi

Numerade Educator

Problem 35

A pumping system must be specified for a lift station at a wastewater treatment facility. The average flow rate is 110 million liters per day and the required lift is $10 \mathrm{m} .$ Nonclogging impellers must be used; about 65 percent efficiency is expected. For convenient installation, electric motors of $37.5 \mathrm{kW}$ or less are desired. Determine the number of motor/ pump units needed and recommend an appropriate operating speed.

Chai Santi

Numerade Educator

Problem 36

A centrifugal water pump operates at $1750 \mathrm{rpm} ;$ the impeller has backward-curved vanes with $\beta_{2}=60^{\circ}$ and $b_{2}=1.25 \mathrm{cm} .$ At a flow rate of $0.025 \mathrm{m}^{3} / \mathrm{s},$ the radial outlet velocity is $V_{n_{2}}=3.5 \mathrm{m} / \mathrm{s}$. Estimate the head this pump could deliver at $1150 \mathrm{rpm}.$

Narayan Hari

Numerade Educator

Problem 37

A set of eight $30-\mathrm{kW}$ motor-pump units is used to deliver water through an elevation of $30 \mathrm{m}$. The efficiency of the pumps is specified to be 65 percent. Estimate the delivery (liters per day) and select an appropriate operating speed.

Narayan Hari

Numerade Educator

Problem 38

Appendix $D$ contains area bound curves for pump model selection and performance curves for individual pump models. Use these data to verify the similarity rules for a Peerless Type $4 \mathrm{AE} 12$ pump, with impeller diameter $D=11.0$ in., operated at 1750 and 3550 nominal rpm.

Chai Santi

Numerade Educator

Problem 39

Appendix D contains area bound curves for pump model selection and performance curves for individual pump models. Use these data and the similarity rules to predict and plot the curves of head $H(\mathrm{ft})$ versus $Q(\mathrm{gpm})$ of a Peerless Type 10 AE12 pump, with impeller diameter $D=12$ in., for nominal speeds of $1000,1200,1400,$ and 1600 rpm.

Chai Santi

Numerade Educator

Problem 40

Consider the Peerless Type $16 \mathrm{A} 18 \mathrm{B}$ horizontal split case centrifugal pump (Appendix D). Use these performance data to verify the similarity rules for (a) impeller diameter change and (b) operating speeds of 705 and 880 rpm (note the scale change between speeds).

Chai Santi

Numerade Educator

Problem 41

Use data from Appendix D to verify the similarity rules for the effect of changing the impeller diameter of a Peerless Tvoe $4 \mathrm{AE} 12$ pump operated at 1750 and 3550 nominal rpm.

Chai Santi

Numerade Educator

Problem 42

Performance curves for Peerless horizontal split case pumps are presented in Appendix D. Develop and plot a curve-fit for a Type $10 \mathrm{AE} 12$ pump driven at 1150 nominal rpm using the procedure described in Example 10.6 .

Chai Santi

Numerade Educator

Problem 43

Performance curves for Peerless horizontal split case pumps are presented in Appendix
D. Develop and plot curve-fits for a Type $16 \mathrm{A} 18 \mathrm{B}$ pump, with impeller diameter $D=18.0$ in., driven at 705 and 880 nominal rpm. Verify the effects of pump speed on scaling pump curves using the procedure described in Example 10.6 .

Chai Santi

Numerade Educator

Problem 44

Catalog data for a centrifugal water pump at design conditions are $Q=250$ gpm and $\Delta p=18.6$ psi at 1750 rpm. A laboratory flume requires 200 gpm at 32 ft of head. The only motor available develops 3 hp at 1750 rpm. Is this motor suitable for the laboratory flume? How might the pump/ motor match be improved?

Chai Santi

Numerade Educator

Problem 45

Problem 10.21 suggests that pump head at best efficiency is typically about 70 percent of shutoff head. Use pump data from Appendix D to evaluate this suggestion. A further suggestion in Section 10.4 is that the appropriate scaling for tests of a pump casing with different impeller diameters is $Q \propto D^{2} .$ Use pump data to evaluate this suggestion.

Chai Santi

Numerade Educator

Problem 46

White [53] suggests modeling the efficiency for a centrifugal pump using the curve-fit, $\eta=a Q-b Q^{3},$ where and $b$ are constants. Describe a procedure to evaluate $a$ and $b$ from experimental data. Evaluate $a$ and $b$ using data for the Peerless Type 10 AE 12 pump, with impeller diameter $D=12.0$ in., at 1760 rpm (Appendix D). Plot and illustrate the accuracy of the curve-fit by comparing measured and predicted efficiencies for this pump.

Chai Santi

Numerade Educator

Problem 47

A fan operates at $Q=6.3 \mathrm{m}^{3} / \mathrm{s}, H=0.15 \mathrm{m},$ and
$N=1440$ rpm. A smaller, geometrically similar fan is planned in a facility that will deliver the same head at the same efficiency as the larger fan, but at a speed of 1800 rpm. Determine the volumetric flow rate of the smaller fan.

Narayan Hari

Numerade Educator

Problem 48

A 1 / 3 scale model of a centrifugal water pump, when running at $N_{m}=5100 \mathrm{rpm},$ produces a flow rate of $Q_{m}=1 \mathrm{m}^{3} / \mathrm{s}$ with a head of $H_{m}=5.4 \mathrm{m} .$ Assuming the model and prototype efficiencies are comparable, estimate the flow rate, head, and power requirement if the design speed is 125 rpm.

Narayan Hari

Numerade Educator

Problem 49

Sometimes the variation of water viscosity with temperature can be used to achieve dynamic similarity. A model pump delivers $0.10 \mathrm{m}^{3} / \mathrm{s}$ of water at $15^{\circ} \mathrm{C}$ against a head of $27 \mathrm{m},$ when operating at $3600 \mathrm{rpm} .$ Determine the water temperature that must be used to obtain dynamically similar operation at 1800 rpm. Estimate the volume flow rate and head produced by the pump at the lower-speed test condition. Comment on the $N P S H$ requirements for the two tests.

Chai Santi

Numerade Educator

Problem 50

A large deep fryer at a snack-food plant contains hot oil that is circulated through a heat exchanger by pumps. Solid particles and water droplets coming from the food product are observed in the flowing oil. What special factors must be considered in specifying the operating conditions for the pumps?

Chai Santi

Numerade Educator

Problem 51

Data from tests of a pump operated at $1450 \mathrm{rpm}$, with 12.3 -in. diameter impeller, are Develop and plot a curve-fit equation for $N P S H R$ versus volume flow rate in the form $N P S H R=a+b Q^{2}$, where $a$ and $b$ are constants. If the $N P S H A=20 \mathrm{ft}$, estimate the maximum allowable flow rate of this pump.

Chai Santi

Numerade Educator

Problem 52

A four-stage boiler feed pump has suction and discharge lines of $10 \mathrm{cm}$ and $7.5 \mathrm{cm}$ inside diameter. At $3500 \mathrm{rpm}$, the pump is rated at $0.025 \mathrm{m}^{3} / \mathrm{s}$ against a head of $125 \mathrm{m}$ while handling water at $115^{\circ} \mathrm{C}$. The inlet pressure gage, located $50 \mathrm{cm}$ below the impeller centerline, reads $150 \mathrm{kPa}$. The pump is to be factory certified by tests at the same flow rate, head rise, and speed, but using water at $27^{\circ} \mathrm{C}$. Calculate the $\mathrm{NPSHA}$ at the pump inlet in the field installation. Evaluate the suction head that must be used in the factory test to duplicate field suction conditions.

Chai Santi

Numerade Educator

Problem 53

The net positive suction head required $(N P S H R)$ by a pump may be expressed approximately as a parabolic function of volume flow rate. The $N P S H R$ for a particular pump operating at $1800 \mathrm{rpm}$ is given as $H_{r}=H_{0}=A Q^{2},$ where $H_{0}=10 \mathrm{ft}$ of water and $A=7.9 \mathrm{ft} / \mathrm{cfs}^{2} .$ Assume the pipe system supplying the pump suction consists of a reservoir, whose surface is $22 \mathrm{ft}$ above the pump centerline, a square entrance, $20 \mathrm{ft}$ of 6 -in. (nominal) cast-iron pipe, and a $90^{\circ}$elbow. Calculate the maximum volume flow rate at $70^{\circ} \mathrm{F}$ for which the suction head is sufficient to operate this pump without cavitation.

Chai Santi

Numerade Educator

Problem 54

A centrifugal pump, operating at $N=2265$ rpm, lifts water between two reservoirs connected by $300 \mathrm{ft}$ of 6 in. and $100 \mathrm{ft}$ of 3 in. cast-iron pipe in series. The gravity lift is $25 \mathrm{ft}$ Estimate the head requirement, power needed, and hourly cost of electrical energy to pump water at 200 gpm to the higher reservoir. Assume that electricity costs $12 \mathrm{e} / \mathrm{kWhr}$ and that the electric motor efficiency is 85 percent.

Chai Santi

Numerade Educator

Problem 55

For the pump and flow system of Problem 10.53 calculate the maximum flow rate for hot water at various temperatures and plot versus water temperature. (Be sure to consider the density variation as water temperature is varied.

Chai Santi

Numerade Educator

Problem 56

A centrifugal pump is installed in a piping system with $L=300 \mathrm{m}$ of $D=40 \mathrm{cm}$ cast-iron pipe. The downstream reservoir surface is $15 \mathrm{m}$ lower than the upstream reservoir. Determine and plot the system head curve. Find the volume flow rate (magnitude and direction) through the system when the pump is not operating. Estimate the friction loss, power requirement, and hourly energy cost to pump water at $1 \mathrm{m}^{3} / \mathrm{s}$ through this system.

Chai Santi

Numerade Educator

Problem 57

Part of the water supply for the South Rim of Grand Canyon National Park is taken from the Colorado River $[54] .$ A flow rate of 600 gpm, taken from the river at elevation $3734 \mathrm{ft}$, is pumped to a storage tank atop the South Rim at $7022 \mathrm{ft}$ elevation. Part of the pipeline is above ground and part is in a hole directionally drilled at angles up to $70^{\circ}$ from the vertical; the total pipe length is approximately $13,200 \mathrm{ft}$. Under steady flow operating conditions, the frictional head loss is 290 ft of water in addition to the static lift. Estimate the diameter of the commercial steel pipe in the system. Compute the pumping power requirement if the pump efficiency is 61 percent.

Chai Santi

Numerade Educator

Problem 58

A Peerless horizontal split-case type $4 \mathrm{AE} 12$ pump with 11.0 -in.-diameter impeller, operating at $1750 \mathrm{rpm}$, lifts water between two reservoirs connected by $200 \mathrm{ft}$ of 4 in. and $200 \mathrm{ft}$ of 3 in. cast-iron pipe in series. The gravity lift is $10 \mathrm{ft}$ Plot the system head curve and determine the pump operating point.

Chai Santi

Numerade Educator

Problem 59

A pump transfers water from one reservoir to another through two cast-iron pipes in series. The first is 3000 ft of 9 in. pipe and the second is $1000 \mathrm{ft}$ of 6 in. pipe. A constant flow rate of 75 gpm is tapped off at the junction between the two pipes. Obtain and plot the system head versus flow rate curve. Find the delivery if the system is supplied by the pump of Example $10.6,$ operating at $1750 \mathrm{rpm}$

Chai Santi

Numerade Educator

Problem 60

Performance data for a pump are
$$\begin{array}{lrrrrrrr}
\boldsymbol{H}(\mathbf{f t}) & 90 & 87 & 81 & 70 & 59 & 43 & 22 \\
\boldsymbol{Q}(\mathbf{c f m}) & 0 & 50 & 100 & 150 & 200 & 250 & 300
\end{array}$$
The pump is to be used to move water between two open reservoirs with an elevation increase of $24 \mathrm{ft}$. The connecting pipe system consists of 1750 ft of commercial steel pipe containing two $90^{\circ}$ elbows and an open gate valve. Find the (b) 10 -in., and flow rate if we use (a) 8 -in., (c) $12-$ in. (nominal) pipe.

Chai Santi

Numerade Educator

Problem 61

Performance data for a pump are
$$\begin{array}{lrrrrrrr}
\boldsymbol{H}(\mathrm{ft}) & 179 & 176 & 165 & 145 & 119 & 84 & 43 \\
\boldsymbol{Q}(\mathrm{gpm}) & 0 & 500 & 1000 & 1500 & 2000 & 2500 & 3000
\end{array}$$
Estimate the delivery when the pump is used to move water between two open reservoirs, through 1200 ft of 12 in. commercial steel pipe containing two $90^{\circ}$ elbows and an open gate valve, if the elevation increase is 50 ft. Determine the gate valve loss coefficient needed to reduce the volume flow rate by half.

Chai Santi

Numerade Educator

Problem 62

Consider again the pump and piping system of Problem $10.61 .$ Determine the volume flow rate and gate valve loss coefficient for the case of two identical pumps installed in series.

Chai Santi

Numerade Educator

Problem 63

The resistance of a given pipe increases with age as deposits form, increasing the roughness and reducing the pipe diameter (see Fig. 8.14 ). Typical multipliers to be applied to the friction factor are given in [15]
$$\begin{array}{ccc}
\text { Pipe Age } & \text { Small Pipes, } & \text { Large Pipes, } \\
\text { (years) } & \mathbf{4 - 1 0} \text { in. } & \mathbf{1 2 - 6 0} \text { in. } \\
\text { New } & 1.00 & 1.00 \\
10 & 2.20 & 1.60 \\
20 & 5.00 & 2.00 \\
30 & 7.25 & 2.20 \\
40 & 8.75 & 2.40 \\
50 & 9.60 & 2.86 \\
60 & 10.0 & 3.70 \\
70 & 10.1 & 4.70
\end{array}$$
Consider again the pump and piping system of Problem 10.61. Estimate the percentage reductions in volume flow rate that occur after (a) 20 years and (b) 40 years of use, if the pump characteristics remain constant. Repeat the calculation if the pump head is reduced 10 percent after 20 years of use and 25 percent after 40 years.

Chai Santi

Numerade Educator

Problem 64

Consider again the pump and piping system of Problem $10.61 .$ Determine the volume flow rate and gate valve loss coefficient for the case of two identical pumps installed in parallel.

Chai Santi

Numerade Educator

Problem 65

Consider again the pump and piping system of Problem $10.64 .$ Estimate the percentage reductions in volume flow rate that occur after (a) 20 years and (b) 40 years of use, if the pump characteristics remain constant. Repeat the calculation if the pump head is reduced 10 percent after 20 years of use and 25 percent after 40 years. (Use the data of Problem 10.63 for increase in pipe friction factor with age.)

Chai Santi

Numerade Educator

Problem 66

Consider again the pump and piping system of Problem $10.62 .$ Estimate the percentage reductions in volume flow rate that occur after (a) 20 years and (b) 40 years of use, if the pump characteristics remain constant. Repeat the calculation if the pump head is reduced 10 percent after 20 years of use and 25 percent after 40 years. (Use the data of Problem 10.63 for increase in pipe friction factor with age.)

Chai Santi

Numerade Educator

Problem 67

The city of Englewood, Colorado, diverts water for municipal use from the South Platte River at elevation $1610 \mathrm{m}$ $[54] .$ The water is pumped to storage reservoirs at $1620-\mathrm{m}$ elevation. The inside diameter of the steel water line is 68.5 $\mathrm{cm} ;$ its length is $1770 \mathrm{m} .$ The facility is designed for an initial capacity (flow rate) of $3200 \mathrm{m}^{3} / \mathrm{hr},$ with an ultimate capacity of $3900 \mathrm{m}^{3} / \mathrm{hr} .$ Calculate and plot the system resistance curve. Ignore entrance losses. Specify an appropriate pumping system. Estimate the pumping power required for steady-state operation, at both the initial and ultimate flow rates.

Chai Santi

Numerade Educator

Problem 68

A pump in the system shown draws water from a sump and delivers it to an open tank through $400 \mathrm{m}$ of new, $10-\mathrm{cm}-$ diameter steel pipe. The vertical suction pipe is $2 \mathrm{m}$ long and includes a foot valve with hinged disk and a $90^{\circ}$ standard elbow. The discharge line includes two $90^{\circ}$ standard elbows, an angle lift check valve, and a fully open gate valve. The design flow rate is 800 L/min. Find the head losses in the suction and discharge lines. Calculate the $N P S H A .$ Select a pump suitable for this application.

Anand Jangid

Numerade Educator

Problem 69

Consider the flow system described in Problem 8.175 Select a pump appropriate for this application. Check the NPSHR versus the NPSHA for this system.

Chai Santi

Numerade Educator

Problem 70

Consider the flow system and data of Problem 10.68 and the data for pipe aging given in Problem $10.63 .$ Select pump(s) that will maintain the system flow at the desired rate for $(\text { a }) 10$ years and (b) 20 years. Compare the delivery produced by these pumps with the delivery by the pump sized for new pipes only.

Chai Santi

Numerade Educator

Problem 71

Consider the flow system shown in Problem 8.176 Select an appropriate pump for this application. Check the pump efficiency and power requirement compared with those in the problem statement.

Chai Santi

Numerade Educator

Problem 72

Consider the flow system shown in Problem 8.124 Assume the minimum $N P S H R$ at the pump inlet is $15 \mathrm{ft}$ of water. Select a pump appropriate for this application. Use the data for increase in friction factor with pipe age given in Problem 10.65 to determine and compare the system flow rate after 10 years of operation.

Chai Santi

Numerade Educator

Problem 73

Consider the pipe network of Problem $8.189 .$ Select a pump suitable to deliver a total flow rate of 300 gpm through the pipe network.

Chai Santi

Numerade Educator

Problem 74

A fire nozzle is supplied through 300 ft of 3 -in. diameter canvas hose (with $e=0.001 \mathrm{ft}$ ). Water from hydrant is supplied at 50 psig to a booster pump on board the pumper truck. At design operating conditions, the pressure at the nozzle inlet is 100 psig, and the pressure drop along the hose is 33 psi per 100 ft of length. Calculate the design flow rate and the maximum nozzle exit speed. Select a pump appropriate for this application, determine its efficiency at this operating condition, and calculate the power required to drive the pump.

Chai Santi

Numerade Educator

Problem 75

A pumping system with two different static lifts is shown. Each reservoir is supplied by a line consisting of $1000 \mathrm{ft}$ of 6 -in. cast-iron pipe. Evaluate and plot the system head versus flow curve. Explain what happens when the pump head is less than the height of the upper reservoir. Calculate the flow rate delivered at a pump head of 85 ft.

Chai Santi

Numerade Educator

Problem 76

Consider the flow system shown in Problem 8.90 Evaluate the $N P S H A$ at the pump inlet. Select a pump appropriate for this application. Use the data on pipe aging from Problem 10.63 to estimate the reduction in flow rate after 10 years of operation.

Chai Santi

Numerade Educator

Problem 77

Consider the gasoline pipeline flow of Problem 8.142 Select pumps that, combined in parallel, supply the total flow requirement. Calculate the power required for 4 pumps in parallel. Also calculate the volume flow rates and power required when only $1,2,$ or 3 of these pumps operates.

Chai Santi

Numerade Educator

Problem 78

Consider the chilled water circulation system of Problem $8.178 .$ Select pumps that may be combined in parallel to supply the total flow requirement. Calculate the power required for 3 pumps in parallel. Also calculate the volume flow rates and power required when only 1 or 2 of these pumps operates.

Chai Santi

Numerade Educator

Problem 79

Water for the sprinkler system at a lakeside summer home is to be drawn from the adjacent lake. The home is located on a bluff $33 \mathrm{m}$ above the lake surface. The pump is located on level ground $3 \mathrm{m}$ above the lake surface. The sprinkler system requires $40 \mathrm{L} / \mathrm{min}$ at $300 \mathrm{kPa}$ (gage). The piping system is to be 2 -cm-diameter galvanized iron. The inlet section (between the lake and pump inlet) includes a reentrant inlet, one standard $45^{\circ}$ elbow, one standard $90^{\circ}$ elbow, and $20 \mathrm{m}$ of pipe. The discharge section (between the pump outlet and the sprinkler connection) includes two standard $45^{\circ}$ elbows and $45 \mathrm{m}$ of pipe. Evaluate the head loss on the suction side of the pump. Calculate the gage pressure at the pump inlet. Determine the hydraulic power requirement of the pump. If the pipe diameter were increased to $4 \mathrm{cm} .$ would the power requirement of the pump increase, decrease, or stay the same? What difference would it make if the pump were located halfway up the hill?

Chai Santi

Numerade Educator

Problem 80

Consider the fire hose and nozzle of Problem 8.179 Specify an appropriate pump to supply four such hoses simultaneously. Calculate the power input to the pump.

Chai Santi

Numerade Educator

Problem 81

Manufacturer's data for a submersible utility pump are $$\begin{array}{lrlrrrrr}
\text { Discharge height (ft) } & 0.3 & 0.7 & 1.5 & 3.0 & 4.5 & 6.0 & 8.0 \\
\text { Water flow rate (L/min) } & 77.2 & 75 & 71 & 61 & 51 & 26 & 0
\end{array}$$
The owner's manual also states, "Note: These ratings are based on discharge into 25 -mm pipe with friction loss neglected. Using 20 -mm garden hose adaptor, performance will be reduced approximately 15 percent." Plot a performance curve for the pump. Develop a curve-fit equation for the performance curve; show the curve-fit on the plot. Calculate and plot the pump delivery versus discharge height through a 15 -m length of smooth 20 -mm garden hose. Compare with the curve for delivery into 25 -mm pipe.

Chai Santi

Numerade Educator

Problem 82

Consider the swimming pool filtration system of Problem $8.190 .$ Assume the pipe used is 20 -mm PVC (smooth plastic). Specify the speed and impeller diameter and estimate the efficiency of a suitable pump.

Chai Santi

Numerade Educator

Problem 83

Water is pumped from a lake (at $z=0$ ) to a large storage tank located on a bluff above the lake. The pipe is 3-in.-diameter galvanized iron. The inlet section (between the lake and the pump) includes one rounded inlet, one standard $90^{\circ}$ elbow, and $50 \mathrm{ft}$ of pipe. The discharge section (between the pump outlet and the discharge to the open tank $)$ includes two standard $90^{\circ}$ elbows, one gate valve, and $150 \mathrm{ft}$ of pipe. The pipe discharge (into the side of the tank) is at $z=70 \mathrm{ft}$. Calculate the system flow curve. Estimate the system operating point. Determine the power input to the pump if its efficiency at the operating point is 80 percent. Sketch the system curve when the water level in the upper tank reaches $z=90 \mathrm{ft}$. If the water level in the upper tank is at $z=75 \mathrm{ft}$ and the valve is partially closed to reduce the flow rate to $0.1 \mathrm{ft}^{3} / \mathrm{s}$, sketch the system curve for this operating condition. Would you expect the pump efficiency to be higher for the first or second operating condition? Why?

Chai Santi

Numerade Educator

Problem 84

10.84 Performance data for a centrifugal fan of 3 -ft diameter, tested at $750 \mathrm{rpm},$ are Plot the performance data versus volume flow rate. Calculate static efficiency, and show the curve on the plot. Find the best efficiency point, and specify the fan rating at this point.

Chai Santi

Numerade Educator

Problem 85

Using the fan of Problem 10.84 , determine the minimum size square sheet-metal duct that will carry a flow of $200 \mathrm{ft}^{3} / \mathrm{s}$ over a distance of $50 \mathrm{ft}$. Estimate the increase in delivery if the fan speed is increased to 1000 rpm.

Chai Santi

Numerade Educator

Problem 86

The performance data of Problem 10.84 are for a 36-in.-diameter fan wheel. The fan also is manufactured with $42-, 48-, 54-,$ and 60 -in. diameter wheels. Pick a standard fan to deliver $600 \mathrm{ft}^{3} / \mathrm{s}$ against a $1-\mathrm{in.} \mathrm{H}_{2} \mathrm{O}$ static pressure rise. Determine the required fan speed and input power required.

Chai Santi

Numerade Educator

Problem 87

Consider the fan and performance data of Problem 10.84. At $Q=200 \mathrm{ft}^{3} / \mathrm{s},$ the dynamic pressure is equal to 0.25 in. of water. Evaluate the fan outlet area. Plot total pressure rise and input horsepower for this fan versus volume flow rate. Calculate the fan total efficiency, and show the curve on the plot. Find the best efficiency point, and specify the fan rating at this point.

Chai Santi

Numerade Educator

Problem 88

Performance characteristics of a Howden Buffalo axial flow fan are presented below. The fan is used to power a wind tunnel with 1 -ft square test section. The tunnel consists of a smooth inlet contraction, two screens (each with loss coefficient $K=0.12$ ), the test section, and a diffuser where the cross section is expanded to 24 in. diameter at the fan inlet. Flow from the fan is discharged back to the room. Calculate and plot the system characteristic curve of pressure loss versus volume flow rate. Estimate the maximum air flow speed available in this wind tunnel test section.

Chai Santi

Numerade Educator

Problem 89

Consider again the axial-flow fan and wind tunnel of Problem $10.88 .$ Scale the performance of the fan as it varies with operating speed. Develop and plot a "calibration curve" showing test section flow speed (in $\mathrm{m} / \mathrm{s}$ ) versus fan speed (in $\mathrm{rpm})$

Chai Santi

Numerade Educator

Problem 90

Experimental test data for an aircraft engine fuel pump are presented below. This gear pump is required to supply jet fuel at 450 pounds per hour and 150 psig to the engine fuel controller. Tests were conducted at $10,96,$ and 100 percent of the rated pump speed of 4536 rpm. At each constant speed, the back pressure on the pump was set, and the flow rate measured. On one graph, plot curves of pressure versus delivery at the three constant speeds. Estimate the pump displacement volume per revolution. Calculate the volumetric efficiency at each test point and sketch contours of constant $\eta_{v} .$ Evaluate the energy loss caused by valve throttling at 100 percent speed and full delivery to the engine.

Chai Santi

Numerade Educator

Problem 91

A hydraulic turbine is designed to produce 36,000 hp at 95 rpm under 50 ft of head. Laboratory facilities are available to provide $15 \mathrm{ft}$ of head and to absorb $50 \mathrm{hp}$ from the model turbine. Determine (a) the appropriate model test speed and scale ratio and (b) volume flow rate, assuming a model efficiency of 86 percent.

Chai Santi

Numerade Educator

Problem 92

Preliminary calculations for a hydroelectric power generation site show a net head of $2350 \mathrm{ft}$ is available at a water flow rate of $75 \mathrm{ft}^{3} / \mathrm{s}$. Compare the geometry and efficiency of Pelton wheels designed to run at (a) $450 \mathrm{rpm}$ and (b) 600 rpm.

Chai Santi

Numerade Educator

Problem 93

Conditions at the inlet to the nozzle of a Pelton wheel are $p=700$ psig and $V=15$ mph. The jet diameter is $d=7.5$ in. and the nozzle loss coefficient is $K_{\text {nozzle }}=0.04 .$ The wheel diameter is $D=8$ ft. At this operating condition, $\eta=0.86 .$ Calculate
(a) the power output, (b) the normal operating speed,
(c) the approximate runaway speed,
(d) the torque at normal operating speed, and (e) the approximate torque at zero speed.

Chai Santi

Numerade Educator

Problem 94

The reaction turbines at Niagara Falls are of the Francis type. The impeller outside diameter is $4.5 \mathrm{m}$. Each turbine produces $54 \mathrm{MW}$ at $107 \mathrm{rpm},$ with 93.8 percent efficiency under $65 \mathrm{m}$ of net head. Calculate the specific speed of these units. Evaluate the volume flow rate to each turbine. Estimate the penstock size if it is $400 \mathrm{m}$ long and the net head is 83 percent of the gross head.

Chai Santi

Numerade Educator

Problem 95

Francis turbine Units $19,20,$ and $21,$ installed at the Grand Coulee Dam on the Columbia River, are very large $[55] .$ Each runner is $32.6 \mathrm{ft}$ in diameter and contains 550 tons of cast steel. At rated conditions, each turbine develops 820,000 hp at 72 rpm under 285 ft of head. Efficiency is nearly 95 percent at rated conditions. The turbines operate at heads from 220 to 355 ft. Calculate the specific speed at rated operating conditions. Estimate the maximum water flow rate through each turbine.

Chai Santi

Numerade Educator

Problem 96

Measured data for performance of the reaction turbines at Shasta Dam near Redding, California, are shown in Fig. $10.39 .$ Each turbine is rated at 103,000 hp when operating at $138.6 \mathrm{rpm}$ under a net head of $380 \mathrm{ft}$. Evaluate the specific speed and compute the shaft torque developed by each turbine at rated operating conditions. Calculate and plot the water flow rate per turbine required to produce rated output power as a function of head.

Chai Santi

Numerade Educator

Problem 97

Figure 10.37 contains data for the efficiency of a large Pelton waterwheel installed in the Tiger Creek Power House of Pacific Gas \& Electric Company near Jackson, California.
This unit is rated at $26.8 \mathrm{MW}$ when operated at $225 \mathrm{rpm}$ under a net head of $360 \mathrm{m}$ of water. Assume reasonable flow angles and nozzle loss coefficient, and water at $15^{\circ} \mathrm{C}$ Determine the rotor radius, and estimate the jet diameter and the mass flow rate of water.

Chai Santi

Numerade Educator

Problem 98

An impulse turbine is to develop $15 \mathrm{MW}$ from a single wheel at a location where the net head is $350 \mathrm{m}$. Determine the appropriate speed, wheel diameter, and jet diameter for singleand multiple-jet operation. Compare with a double-overhung wheel installation. Estimate the required water consumption.

Chai Santi

Numerade Educator

Problem 99

An impulse turbine under a net head of 33 ft was tested at a variety of speeds. The flow rate and the brake force needed to set the impeller speed were recorded:
$$\begin{array}{ccc}
\text { Wheel Speed } & \text { Flow rate } & \text { Brake Force (lbf) } \\
(\mathrm{rpm}) & (\mathrm{cfm}) & (R=0.5 \mathrm{ft}) \\
0 & 7.74 & 2.63 \\
1000 & 7.74 & 2.40 \\
1500 & 7.74 & 2.22 \\
1900 & 7.44 & 1.91 \\
2200 & 7.02 & 1.45 \\
2350 & 5.64 & 0.87 \\
2600 & 4.62 & 0.34 \\
2700 & 4.08 & 0.09
\end{array}$$
Calculate and plot the machine power output and efficiency as a function of water turbine speed.

Chai Santi

Numerade Educator

Problem 100

In U.S. customary units, the common definition of specific speed for a hydraulic turbine is given by Eq. $10.13 \mathrm{b}$ Develop a conversion between this definition and a truly dimensionless one in SI units. Evaluate the specific speed of an impulse turbine, operating at 400 rpm under a net head of $1190 \mathrm{ft}$ with 86 percent efficiency, when supplied by a single 6-in.-diameter jet. Use both U.S. customary and SI units. Estimate the wheel diameter.

Chai Santi

Numerade Educator

Problem 101

According to a spokesperson for Pacific Gas \& Electric Company, the Tiger Creek plant, located east of Jackson, California, is one of 71 PG\&E hydroelectric powerplants. The plant has $373 \mathrm{m}$ of gross head, consumes $21 \mathrm{m}^{3} / \mathrm{s}$ of water, is rated at $60 \mathrm{MW}$, and operates at $58 \mathrm{MW}$. The plant is claimed to produce $0.785 \mathrm{kW} \cdot \mathrm{hr} /\left(\mathrm{m}^{2} \cdot \mathrm{m}\right)$ of water and $336.4 \times 10^{6} \mathrm{kW} \cdot \mathrm{hr} / \mathrm{yr}$ of operation. Estimate the net head at the site, the turbine specific speed, and its efficiency. Comment on the internal consistency of these data.

Chai Santi

Numerade Educator

Problem 102

Design the piping system to supply a water turbine from a mountain reservoir. The reservoir surface is $320 \mathrm{m}$ above the turbine site. The turbine efficiency is 83 percent, and it must produce $30 \mathrm{kW}$ of mechanical power. Define the minimum standard-size pipe required to supply water to the turbine and the required volume flow rate of water. Discuss the effects of turbine efficiency, pipe roughness, and installing a diffuser at the turbine exit on the performance of the installation.

Chai Santi

Numerade Educator

Problem 103

A small hydraulic impulse turbine is supplied with water through a penstock with diameter $D$ and length $L ;$ the jet diameter is $d .$ The elevation difference between the reservoir surface and nozzle centerline is $Z$. The nozzle head loss coefficient is $K_{\text {nozyle }}$ and the loss coefficient from the reservoir to the penstock entrance is $K_{\text {entrance }} .$ Determine the water jet speed, the volume flow rate, and the hydraulic power of the jet, for the case where $Z=300 \mathrm{ft}, L=1000 \mathrm{ft}, D=6 \mathrm{in.}$
$K_{\text {entrance }}=0.5, K_{\text {nozzle }}=0.04,$ and $d=2$ in., if the pipe is made from commercial steel. Plot the jet power as a function of jet diameter to determine the optimum jet diameter and the resulting hydraulic power of the jet. Comment on the effects of varying the loss coefficients and pipe roughness.

Chai Santi

Numerade Educator

Problem 104

The propeller on a fanboat used in the Florida Everglades moves air at the rate of $50 \mathrm{kg} / \mathrm{s}$. When at rest, the speed of the slipstream behind the propeller is $45 \mathrm{m} / \mathrm{s}$ at a location where the pressure is atmospheric. Calculate (a) the propeller diameter, (b) the thrust produced at rest, and (c) the thrust produced when the fanboat is moving ahead at $15 \mathrm{m} / \mathrm{s}$ if the mass flow rate through the propeller remains constant.

Chai Santi

Numerade Educator

Problem 105

A fanboat in the Florida Everglades is powered by a propeller, with $D=1.5 \mathrm{m},$ driven at maximum speed, $N=1800 \mathrm{rpm},$ by a $125 \mathrm{kW}$ engine. Estimate the maximum thrust produced by the propeller at (a) standstill and (b) $V=12.5 \mathrm{m} / \mathrm{s}$

Chai Santi

Numerade Educator

Problem 106

A jet-propelled aircraft traveling at $225 \mathrm{m} / \mathrm{s}$ takes in $50 \mathrm{kg} / \mathrm{s}$ of air. If the propulsive efficiency (defined as the ratio of the useful work output to the mechanical energy input to the fluid) of the aircraft is 45 percent, determine the speed at which the exhaust is discharged relative to the aircraft.

Chai Santi

Numerade Educator

Problem 107

Analyze the air flow relative to a blade element of a Darrieus wind turbine rotating about its troposkien axis. Develop a numerical model for the blade element. Calculate the power coefficient developed by the blade element as a function of tip-speed ratio. Compare your result with the general trend of power output for Darrieus rotors shown in Fig. 10.50.

Chai Santi

Numerade Educator

Problem 108

The propulsive efficiency, $\eta,$ of a propeller is defined as the ratio of the useful work produced to the mechanical energy input to the fluid. Determine the propulsive efficiency of the moving fanboat of Problem $10.104 .$ What would be the efficiency if the boat were not moving?

Chai Santi

Numerade Educator

Problem 109

The propeller for the Gossamer Condor humanpowered aircraft has $D=12 \mathrm{ft}$ and rotates at $N=107 \mathrm{rpm}$ Additional details on the aircraft are given in Problem 9.174 Estimate the dimensionless performance characteristics and efficiency of this propeller at cruise conditions. Assume the pilot expends 70 percent of maximum power at cruise. (See Reference [56] for more information on human-powered flight.)

Chai Santi

Numerade Educator

Problem 110

Equations for the thrust, power, and efficiency of propulsion devices were derived in Section $10.6 .$ Show that these equations may be combined for the condition of constant thrust to obtain
\[
\eta=\frac{2}{1+\left(1+\frac{F_{T}}{\frac{\rho V^{2}}{2} \frac{\pi D^{2}}{4}}\right)^{1 / 2}}
\]
Interpret this result physically.

Chai Santi

Numerade Educator

Problem 111

The National Aeronautics \& Space Administration (NASA) and the U.S. Department of Energy (DOE) cosponsor a large demonstration wind turbine generator at Plum Brook, near Sandusky, Ohio [47]. The turbine has two blades, with a radius of $63 \mathrm{ft}$, and delivers maximum power when the wind speed is above $V=16$ knots. It is designed to produce 135 hp with powertrain efficiency of 74 percent. The rotor is designed to operate at a constant speed of 45 rpm in winds over 5 knots by controlling system load and adjusting blade angles. For the maximum power condition, estimate the rotor tip speed and power coefficient.

Chai Santi

Numerade Educator

Problem 112

A typical American multiblade farm windmill has $D=7 \mathrm{ft}$ and is designed to produce maximum power in winds with $V=15$ mph. Estimate the rate of water delivery, as a function of the height to which the water is pumped, for this windmill.

Chai Santi

Numerade Educator

Problem 113

A model of an American multiblade farm windmill is to be built for display. The model, with $D=1 \mathrm{m},$ is to develop full power at $V=10 \mathrm{m} / \mathrm{s}$ wind speed. Calculate the angular speed of the model for optimum power generation. Estimate the power output.

Chai Santi

Numerade Educator

Problem 114

A large Darrieus vertical axis wind turbine was built by the U.S. Department of Energy near Sandia, New Mexico $[48] .$ This machine is $18 \mathrm{m}$ tall and has a $5-\mathrm{m}$ radius; the area swept by the rotor is over $110 \mathrm{m}^{2}$. If the rotor is constrained to rotate at $70 \mathrm{rpm},$ plot the power this wind turbine can produce in kilowatts for wind speeds between 5 and 50 knots.

Chai Santi

Numerade Educator

Problem 115

Lift and drag data for the NACA 23015 airfoil section are presented in Fig. $9.17 .$ Consider a two-blade horizontalaxis propeller wind turbine with NACA 23015 blade section. Analyze the air flow relative to a blade element of the rotating wind turbine. Develop a numerical model for the blade element. Calculate the power coefficient developed by the blade element as a function of tip-speed ratio. Compare your result with the general trend of power output for high-speed two-bladed turbine rotors shown in Fig. 10.50 .

Chai Santi

Numerade Educator

Problem 116

Aluminum extrusions, patterned after NACA symmetric airfoil sections, frequently are used to form Darrieus wind turbine "blades." Below are section lift and drag coefficient data [57] for a NACA 0012 section, tested at $R e=6 \times 10^{6}$ with standard roughness (the section stalled for $\left.\alpha>12^{\circ}\right)$
Analyze the air flow relative to a blade element of a Darrieus wind turbine rotating about its troposkien axis. Develop a numerical model for the blade element. Calculate the power coefficient developed by the blade element as a function of tip-speed ratio. Compare your result with the general trend of power output for Darrieus rotors shown in Fig. 10.50

Chai Santi

Numerade Educator

Problem 117

A prototype air compressor with a compression ratio of 7 is designed to take $8.9 \mathrm{kg} / \mathrm{s}$ air at 1 atmosphere and $20^{\circ} \mathrm{C}$ The design point speed, power requirement, and efficiency are $600 \mathrm{rpm}, 5.6 \mathrm{MW},$ and 80 percent, respectively. A 1: 5 scale model of the prototype is built to help determine operability for the prototype. If the model takes in air at identical conditions to the prototype design point, what will the mass flow and power requirement be for operation at 80 percent efficiency?

Chai Santi

Numerade Educator