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Fluid Mechanics 4th

Frank M. White

Chapter 11

Turbomachinery

Educators


Problem 1

Describe the geometry and operation of a peristaltic positive-displacement pump which occurs in nature.

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Problem 2

What would be the technical classification of the following turbomachines: $(a)$ a household fan, $(b)$ a wind$\operatorname{mill},(c)$ an aircraft propeller, $(d)$ a fuel pump in a car $(e)$ an eductor, $(f)$ a fluid-coupling transmission, and $(g)$ a power plant steam turbine?

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Problem 3

A PDP can deliver almost any fluid, but there is always a limiting very high viscosity for which performance will deteriorate. Can you explain the probable reason?

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Problem 4

Figure $11.2 c$ shows a double-screw pump. Sketch a $\sin$ gle-screw pump and explain its operation. How did Archimedes' screw pump operate?

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Problem 5

What type of pump is shown in Fig. P11.5? How does it operate?

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Problem 6

Figure P11.6 shows two points a half-period apart in the operation of a pump. What type of pump is this? How does it work? Sketch your best guess of flow rate versus time for a few cycles.

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Problem 7

A piston PDP has a 5 -in diameter and a 2 -in stroke and operates at $750 \mathrm{r} / \mathrm{min}$ with 92 percent volumetric efficiency. ( $a$ ) What is its delivery, in gal/min? ( $b$ ) If the pump delivers SAE $10 \mathrm{W}$ oil at $20^{\circ} \mathrm{C}$ against a head of $50 \mathrm{ft}$ what horsepower is required when the overall efficiency is 84 percent?

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Problem 8

A centrifugal pump delivers 550 gal/min of water at $20^{\circ} \mathrm{C}$ when the brake horsepower is 22 and the efficiency is 71 percent. ( $a$ ) Estimate the head rise in $\mathrm{ft}$ and the pressure rise in $\mathrm{lbf} / \mathrm{in}^{2} .(b)$ Also estimate the head rise and horsepower if instead the delivery is 550 gal/min of gasoline at $20^{\circ} \mathrm{C}$.

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Problem 9

Figure $\mathrm{P} 11.9$ shows the measured performance of the Vickers model PVQ40 piston pump when delivering SAE $10 \mathrm{W}$ oil at $180^{\circ} \mathrm{F}\left(\rho \approx 910 \mathrm{kg} / \mathrm{m}^{3}\right) .$ Make some general observations about these data vis-à-vis Fig. 11.2 and your intuition about the behavior of piston pumps.

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Problem 10

Suppose that the piston pump of Fig. P11.9 is used to deliver 15 gal/min of water at $20^{\circ} \mathrm{C}$ using 20 brake horsepower. Neglecting Reynolds-number effects, use the figure to estimate $(a)$ the speed in $\mathrm{r} / \mathrm{min}$ and $(b)$ the pressure rise in $\mathrm{lbf} / \mathrm{in}^{2}$.

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Problem 11

A pump delivers $1500 \mathrm{L} / \mathrm{min}$ of water at $20^{\circ} \mathrm{C}$ against a pressure rise of $270 \mathrm{kPa}$. Kinetic- and potential-energy changes are negligible. If the driving motor supplies $9 \mathrm{kW},$ what is the overall efficiency?

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Problem 12

In a test of the centrifugal pump shown in Fig. P11.12, the following data are taken: $p_{1}=100 \mathrm{mmHg}$ (vacuum) and $p_{2}=500 \mathrm{mmHg}$ (gage). The pipe diameters are $D_{1}=12 \mathrm{cm}$ and $D_{2}=5 \mathrm{cm} .$ The flow rate is $180 \mathrm{gal} / \mathrm{min}$ of light oil (SG = 0.91). Estimate ( $a$ ) the head developed, in meters; and ( $b$ ) the input power required at 75 percent efficiency.

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Problem 13

A 20 -hp pump delivers 400 gal/min of gasoline at $20^{\circ} \mathrm{C}$ with 75 percent efficiency. What head and pressure rise result across the pump?

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Problem 14

A pump delivers gasoline at $20^{\circ} \mathrm{C}$ and $12 \mathrm{m}^{3} / \mathrm{h}$. At the inlet $p_{1}=100 \mathrm{kPa}, z_{1}=1 \mathrm{m},$ and $V_{1}=2 \mathrm{m} / \mathrm{s} .$ At the exit $p_{2}=500 \mathrm{kPa}, z_{2}=4 \mathrm{m},$ and $V_{2}=3 \mathrm{m} / \mathrm{s} .$ How much power is required if the motor efficiency is 75 percent?

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Problem 15

A lawn sprinkler can be used as a simple turbine. As shown in Fig. $\mathrm{P} 11.15,$ flow enters normal to the paper in the center and splits evenly into $Q / 2$ and $V_{\mathrm{rel}}$ leaving each nozzle. The arms rotate at angular velocity $\omega$ and do work on a shaft. Draw the velocity diagram for this turbine. Neglecting friction, find an expression for the power delivered to the shaft. Find the rotation rate for which the power is a maximum.

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Problem 16

For the "sprinkler turbine" of Fig. P11.15, let $R=18 \mathrm{cm}$ with total flow rate of $14 \mathrm{m}^{3} / \mathrm{h}$ of water at $20^{\circ} \mathrm{C}$. If the nozzle exit diameter is $8 \mathrm{mm}$, estimate $(a)$ the maximum power delivered in $\mathrm{W}$ and $(b)$ the appropriate rotation rate in $\mathrm{r} / \mathrm{min}$.

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Problem 17

A centrifugal pump has $d_{1}=7$ in, $d_{2}=13$ in, $b_{1}=4$ in, $b_{2}=3$ in, $\beta_{1}=25^{\circ},$ and $\beta_{2}=40^{\circ}$ and rotates at 1160 $\mathrm{r} / \mathrm{min} .$ If the fluid is gasoline at $20^{\circ} \mathrm{C}$ and the flow enters the blades radially, estimate the theoretical ( $a$ ) flow rate in gal/min, ( $b$ ) horsepower, and ( $c$ ) head in ft.

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Problem 18

A jet of velocity $V$ strikes a vane which moves to the right at speed $V_{c},$ as in Fig. P11.18. The vane has a turning angle $\theta$. Derive an expression for the power delivered to the vane by the jet. For what vane speed is the power maximum?

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Problem 19

A centrifugal pump has $r_{2}=9$ in, $b_{2}=2$ in, and $\beta_{2}=$ $35^{\circ}$ and rotates at $1060 \mathrm{r} / \mathrm{min} .$ If it generates a head of $180 \mathrm{ft},$ determine the theoretical $(a)$ flow rate in gal/min and $(b)$ horsepower. Assume near-radial entry flow.

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Problem 20

Suppose that Prob. 11.19 is reversed into a statement of the theoretical power $P_{w} \approx 153$ hp. Can you then compute the theoretical ( $a$ ) flow rate and ( $b$ ) head? Explain and resolve the difficulty which arises.

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Problem 21

The centrifugal pump of Fig. P11.21 develops a flow rate of 4200 gal/min of gasoline at $20^{\circ} \mathrm{C}$ with near-radial absolute inflow. Estimate the theoretical ( $a$ ) horsepower, ( $b$ ) head rise and ( $c$ ) appropriate blade angle at the inner radius.

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Problem 22

A 37 -cm-diameter centrifugal pump, running at 2140 $\mathrm{r} / \mathrm{min}$ with water at $20^{\circ} \mathrm{C},$ produces the following performance data:
$$\begin{array}{l|l|l|l|l|l|c|c}
Q, \mathrm{m}^{3} / \mathrm{s} & 0.0 & 0.05 & 0.10 & 0.15 & 0.20 & 0.25 & 0.30 \\
\hline H, \mathrm{m} & 105 & 104 & 102 & 100 & 95 & 85 & 67 \\
\hline P, \mathrm{kW} & 100 & 115 & 135 & 171 & 202 & 228 & 249
\end{array}$$
(a) Determine the best efficiency point. ( $b$ ) Plot $C_{H}$ ver$\operatorname{sus} C_{Q} .(c)$ If we desire to use this same pump family to deliver 7000 gal/min of kerosine at $20^{\circ} \mathrm{C}$ at an input power of $400 \mathrm{kW}$, what pump speed (in $\mathrm{r} / \mathrm{min}$ ) and impeller size (in $\mathrm{cm}$ ) are needed? What head will be developed?

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Problem 23

If the 38 -in-diameter pump of Fig. $11.7 b$ is used to deliver $20^{\circ} \mathrm{C}$ kerosine at $850 \mathrm{r} / \mathrm{min}$ and $22,000 \mathrm{gal} / \mathrm{min},$ what $(a)$ head and $(b)$ brake horsepower will result?

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Problem 24

Figure $\mathrm{P} 11.24$ shows performance data for the Taco, Inc. model 4013 pump. Compute the ratios of measured shutoff head to the ideal value $U^{2} / g$ for all seven impeller sizes. Determine the average and standard deviation of this ratio and compare it to the average for the six impellers in Fig. 11.7.

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Problem 25

At what speed in $\mathrm{r} / \mathrm{min}$ should the 35 -in-diameter pump of Fig. $11.7 b$ be run to produce a head of $400 \mathrm{ft}$ at a discharge of 20,000 gal/min? What brake horsepower will be required? Hint: Fit $H(Q)$ to a formula.

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Problem 26

Determine if the seven Taco, Inc., pump sizes in Fig. P11.24 can be collapsed into a single dimensionless chart of $C_{H}, C_{P},$ and $\eta$ versus $C_{Q},$ as in Fig. $11.8 .$ Comment on the results.

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Problem 27

The 12 -in pump of Fig. $\mathrm{P} 11.24$ is to be scaled up in size to provide a head of 90 ft and a flow rate of 1000 gal/min at BEP. Determine the correct ( $a$ ) impeller diameter, ( $b$ ) speed in r/min, and ( $c$ ) horsepower required.

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Problem 28

Tests by the Byron Jackson Co. of a 14.62 -in-diameter centrifugal water pump at $2134 \mathrm{r} / \mathrm{min}$ yield the following data:
$$\begin{array}{l|c|c|c|c|c|c}
Q, \mathrm{ft}^{3} / \mathrm{s} & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline H, \mathrm{ft} & 340 & 340 & 340 & 330 & 300 & 220 \\
\hline \mathrm{bhp} & 135 & 160 & 205 & 255 & 330 & 330
\end{array}$$
What is the BEP? What is the specific speed? Estimate the maximum discharge possible.

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Problem 29

If the scaling laws are applied to the pump of Prob. 11.28 for the same impeller diameter, determine $(a)$ the speed for which the shutoff head will be $280 \mathrm{ft},(b)$ the speed for which the BEP flow rate will be $8.0 \mathrm{ft}^{3} / \mathrm{s},$ and $(c)$ the speed for which the BEP conditions will require $80 \mathrm{hp}$.

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Problem 30

A pump from the same family as Prob. 11.28 is built with $D=18$ in and a BEP power of 250 hp for gasoline (not water). Using the scaling laws, estimate the resulting $(a)$ speed in $\mathrm{r} / \mathrm{min},(b)$ flow rate at $\mathrm{BEP}$, and $(c)$ shutoff head.

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Problem 31

Figure $\mathrm{P} 11.31$ shows performance data for the Taco, Inc. model 4010 pump. Compute the ratios of measured shutoff head to the ideal value $U^{2} / g$ for all seven impeller sizes. Determine the average and standard deviation of this ratio, and compare it to the average of $0.58 \pm 0.02$ for the seven impellers in Fig. P11.24. Comment on your results.

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Problem 32

Determine if the seven Taco, Inc., impeller sizes in Fig. P11.31 can be collapsed into a single dimensionless chart of $C_{H}, C_{P},$ and $\eta$ versus $C_{Q},$ as in Fig. $11.8 .$ Comment on the results.

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Problem 33

Clearly the maximum efficiencies of the pumps in Figs. P11.24 and P11.31 decrease with impeller size. Compare $\eta_{\max }$ for these two pump families with both the Moody and the Anderson correlations, Eqs. $(11.29) .$ Use the central impeller size as a comparison point.

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Problem 34

You are asked to consider a pump geometrically similar to the 9 -in-diameter pump of Fig. P11.31 to deliver 1200 gal/min at 1500 r/min. Determine the appropriate $(a)$ impeller diameter, (b) BEP horsepower, (c) shutoff head, and $(d)$ maximum efficiency. The fluid is kerosine, not water.

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Problem 35

An 18 -in-diameter centrifugal pump, running at 880 $\mathrm{r} / \mathrm{min}$ with water at $20^{\circ} \mathrm{C},$ generates the following performance data:
$$\begin{array}{l|c|c|c|c|c|c}
Q, \operatorname{gal} / \min & 0.0 & 2000 & 4000 & 6000 & 8000 & 10,000 \\
\hline H, \mathrm{ft} & 92 & 89 & 84 & 78 & 68 & 50 \\
\hline P, \mathrm{hp} & 100 & 112 & 130 & 143 & 156 & 163
\end{array}$$
Determine (a) the $\mathrm{BEP}$, (b) the maximum efficiency, and
(c) the specific speed. (d) Plot the required input power versus the flow rate.

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Problem 36

Plot the dimensionless performance curves for the pump of Prob. 11.35 and compare with Fig. $11.8 .$ Find the appropriate diameter in inches and speed in r/min for a geometrically similar pump to deliver 400 gal/min against a head of $200 \mathrm{ft}$. What brake horsepower would be required?

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Problem 37

The efficiency of a centrifugal pump can be approximated by the curve fit $\eta \approx a Q-b Q^{3},$ where $a$ and $b$ are constants. For this approximation, $(a)$ what is the ratio of $Q^{*}$ at BEP to $Q_{\max } ?$ If the maximum efficiency is 88 percent, what is the efficiency at ( $b$ ) $\frac{1}{3} Q_{\max }$ and $(c) \frac{4}{3} Q^{*} ?$

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Problem 38

A 6.85 -in pump, running at $3500 \mathrm{r} / \mathrm{min}$, has the following measured performance for water at $20^{\circ} \mathrm{C}$:
$$\begin{array}{l|c|c|c|c|c|c|c|c|c}
Q, \mathrm{gal} / \mathrm{min} & 50 & 100 & 150 & 200 & 250 & 300 & 350 & 400 & 450 \\
\hline H, \mathrm{ft} & 201 & 200 & 198 & 194 & 189 & 181 & 169 & 156 & 139 \\
\hline \eta, \% & 29 & 50 & 64 & 72 & 77 & 80 & 81 & 79 & 74
\end{array}$$
(a) Estimate the horsepower at BEP. If this pump is rescaled in water to provide 20 bhp at 3000 r/min, determine the appropriate ( b ) impeller diameter, (c) flow rate, and (d) efficiency for this new condition.

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Problem 39

The Allis-Chalmers D30LR centrifugal compressor delivers $33,000 \mathrm{ft}^{3} / \mathrm{min}$ of $\mathrm{SO}_{2}$ with a pressure change from 14.0 to $18.0 \mathrm{lbf} / \mathrm{in}^{2}$ absolute using an 800 -hp motor at $3550 \mathrm{r} / \mathrm{min} .$ What is the overall efficiency? What will the flow rate and $\Delta p$ be at $3000 \mathrm{r} / \mathrm{min} ?$ Estimate the diameter of the impeller.

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Problem 40

The specific speed $N_{s},$ as defined by Eqs. $(11.30),$ does not contain the impeller diameter. How then should we size the pump for a given $N_{s} ?$ Logan [7] suggests a parameter called the specific diameter $D_{s},$ which is a dimensionless combination of $Q, g H,$ and $D .(a)$ If $D_{s}$ is proportional to $D$, determine its form. ( $b$ ) What is the relationship, if any, of $D_{s}$ to $C_{Q^{*},} C_{H^{*}},$ and $C_{P^{*}} ?(c)$ Estimate $D_{s}$ for the two pumps of Figs. 11.8 and 11.13.

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Problem 41

It is desired to build a centrifugal pump geometrically similar to that of Prob. 11.28 to deliver 6500 gal/min of gasoline at $20^{\circ} \mathrm{C}$ at $1060 \mathrm{r} /$ min. Estimate the resulting $(a)$ impeller diameter, ( $b$ ) head, ( $c$ ) brake horsepower, and $(d)$ maximum efficiency.

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Problem 42

An 8 -in model pump delivering $180^{\circ} \mathrm{F}$ water at 800 gal/min and $2400 \mathrm{r} / \mathrm{min}$ begins to cavitate when the inlet pressure and velocity are 12 lbf/in $^{2}$ absolute and $20 \mathrm{ft} / \mathrm{s}$ respectively. Find the required NPSH of a prototype which is 4 times larger and runs at $1000 \mathrm{r} / \mathrm{min}$.

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Problem 43

The 28 -in-diameter pump in Fig. $11.7 a$ at $1170 \mathrm{r} / \mathrm{min}$ is used to pump water at $20^{\circ} \mathrm{C}$ through a piping system at 14,000 gal/min. (a) Determine the required brake horsepower. The average friction factor is $0.018$ .(b) If there is 65 ft of 12 -indiameter pipe upstream of the pump, how far below the surface should the pump inlet be placed to avoid cavitation?

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Problem 44

The pump of Prob. 11.28 is scaled up to an 18 -in diameter, operating in water at best efficiency at $1760 \mathrm{r} / \mathrm{min}$. The measured NPSH is $16 \mathrm{ft}$, and the friction loss between the inlet and the pump is $22 \mathrm{ft}$. Will it be sufficient to avoid cavitation if the pump inlet is placed 9 ft below the surface of a sea-level reservoir?

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Problem 45

Determine the specific speeds of the seven Taco, Inc. pump impellers in Fig. P11.24. Are they appropriate for centrifugal designs? Are they approximately equal within experimental uncertainty? If not, why not?

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Problem 46

The answer to Prob. 11.40 is that the dimensionless "specific diameter" takes the form $D_{s}=D\left(g H^{*}\right)^{1 / 4} / Q^{* 1 / 2}$ evaluated at the BEP. Data collected by the author for 30 different pumps indicate, in Fig. P11.46, that $D_{s}$ correlates well with specific speed $N_{s}$. Use this figure to estimate the appropriate impeller diameter for a pump which delivers 20,000 gal/min of water and a head of $400 \mathrm{ft}$ when running at $1200 \mathrm{r} / \mathrm{min}$. Suggest a curve-fit formula to the data. Hint: Use a hyperbolic formula.

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Problem 47

A typical household basement sump pump provides a discharge of 5 gal/min against a head of 15 ft. Estimate
$(a)$ the maximum efficiency and $(b)$ the minimum horsepower required to drive such a pump at $1750 \mathrm{r} / \mathrm{min}$.

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Problem 48

Compute the specific speeds for the pumps in Probs. $11.28,11.35,$ and 11.38 plus the median sizes in Figs. P11.24 and P11.31. Then determine if their maximum efficiencies match the values predicted in Fig. 11.14.

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Problem 49

Data collected by the author for flow coefficient at BEP for 30 different pumps are plotted versus specific speed in Fig. P11.49. Determine if the values of $C_{\mathrm{Q}}^{*}$ for the five pumps in Prob. 11.48 also fit on this correlation. If so, suggest a curve-fitted formula for the data.

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Problem 50

Data collected by the author for power coefficient at BEP for 30 different pumps are plotted versus specific speed in Fig. P11.50. Determine if the values of $C_{P}^{*}$ for the five pumps in Prob. 11.48 also fit on this correlation. If so, suggest a curve-fitted formula for the data.

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Problem 51

An axial-flow pump delivers $40 \mathrm{ft}^{3} / \mathrm{s}$ of air which enters at $20^{\circ} \mathrm{C}$ and 1 atm. The flow passage has a 10 -in outer radius and an 8 -in inner radius. Blade angles are $\alpha_{1}=$ $60^{\circ}$ and $\beta_{2}=70^{\circ},$ and the rotor runs at $1800 \mathrm{r} / \mathrm{min} .$ For the first stage compute ( $a$ ) the head rise and ( $b$ ) the power required.

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Problem 52

An axial-flow fan operates in sea-level air at $1200 \mathrm{r} / \mathrm{min}$ and has a blade-tip diameter of $1 \mathrm{m}$ and a root diameter of $80 \mathrm{cm} .$ The inlet angles are $\alpha_{1}=55^{\circ}$ and $\beta_{1}=30^{\circ}$ while at the outlet $\beta_{2}=60^{\circ} .$ Estimate the theoretical values of the $(a)$ flow rate, $(b)$ horsepower, and $(c)$ outlet angle $\alpha_{2}$.

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Problem 53

If the axial-flow pump of Fig. 11.13 is used to deliver 70,000 gal/min of $20^{\circ} \mathrm{C}$ water at $1170 \mathrm{r} / \mathrm{min},$ estimate $(a)$ the proper impeller diameter, $(b)$ the shutoff head, $(c)$ the shutoff horsepower, and $(d) \Delta p$ at best efficiency.

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Problem 54

The Colorado River Aqueduct uses Worthington Corp. pumps which deliver $200 \mathrm{ft}^{3} / \mathrm{s}$ at $450 \mathrm{r} / \mathrm{min}$ against a head of $440 \mathrm{ft}$. What types of pump are these? Estimate the impeller diameter.

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Problem 55

We want to pump $70^{\circ} \mathrm{C}$ water at $20,000 \mathrm{gal} / \mathrm{min}$ and 1800 r/min. Estimate the type of pump, the horsepower required, and the impeller diameter if the required pressure rise for one stage is $(a) 170 \mathrm{kPa}$ and $(b) 1350 \mathrm{kPa}$.

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Problem 56

A pump is needed to deliver 40,000 gal/min of gasoline at $20^{\circ} \mathrm{C}$ against a head of $90 \mathrm{ft}$. Find the impeller size, speed, and brake horsepower needed to use the pump families of $(a)$ Fig. 11.8 and $(b)$ Fig. $11.13 .$ Which is the better design?

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Problem 57

Performance data for a 21 -in-diameter air blower running at $3550 \mathrm{r} / \mathrm{min}$ are as follows:
$$\begin{array}{l|c|c|c|c|c}
\Delta p, \operatorname{inH}_{2} \mathrm{O} & 29 & 30 & 28 & 21 & 10 \\
\hline Q, \mathrm{ft}^{3} / \mathrm{min} & 500 & 1000 & 2000 & 3000 & 4000 \\
\hline \mathrm{bhp} & 6 & 8 & 12 & 18 & 25
\end{array}$$
Note the fictitious expression of pressure rise in terms of water rather than air. What is the specific speed? How does the performance compare with Fig. $11.8 ?$ What are $C_{Q}^{*}, C_{H}^{*},$ and $C_{P}^{*} ?$

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Problem 58

The Worthington Corp. model A-12251 water pump, operating at maximum efficiency, produces $53 \mathrm{ft}$ of head at $3500 \mathrm{r} / \mathrm{min}, 1.1 \mathrm{bhp}$ at $3200 \mathrm{r} / \mathrm{min},$ and $60 \mathrm{gal} / \mathrm{min}$ at 2940 r/min. What type of pump is this? What is its efficiency, and how does this compare with Fig. $11.14 ?$ Estimate the impeller diameter.

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Problem 59

Suppose it is desired to deliver $700 \mathrm{ft}^{3} / \mathrm{min}$ of propane gas (molecular weight $=44.06$ ) at 1 atm and $20^{\circ} \mathrm{C}$ with a single-stage pressure rise of 8.0 inH $\mathrm{O}$. Determine the appropriate size and speed for using the pump families of $(a)$ Prob. 11.57 and $(b)$ Fig. $11.13 .$ Which is the better design?

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Problem 60

A 45 -hp pump is desired to generate a head of $200 \mathrm{ft}$ when running at BEP with $20^{\circ} \mathrm{C}$ gasoline at $1200 \mathrm{r} / \mathrm{min}$ Using the correlations in Figs. P11.49 and P11.50, determine the appropriate ( $a$ ) specific speed, ( $b$ ) flow rate and $(c)$ impeller diameter.

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Problem 61

A mine ventilation fan, running at $295 \mathrm{r} / \mathrm{min}$, delivers 500 $\mathrm{m}^{3} / \mathrm{s}$ of sea-level air with a pressure rise of $1100 \mathrm{Pa}$. Is this fan axial, centrifugal, or mixed? Estimate its diameter in $\mathrm{ft}$. If the flow rate is increased 50 percent for the same diameter, by what percentage will the pressure rise change?

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Problem 62

The actual mine ventilation fan discussed in Prob. 11.61 had a diameter of $20 \mathrm{ft}[20, \mathrm{p} .339] .$ What would be the proper diameter for the pump family of Fig. 11.14 to provide $500 \mathrm{m}^{3} / \mathrm{s}$ at $295 \mathrm{r} / \mathrm{min}$ and BEP? What would be the resulting pressure rise in $\mathrm{Pa} ?$

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Problem 63

The 36.75 -in pump in Fig. $11.7 a$ at $1170 \mathrm{r} / \mathrm{min}$ is used to pump water at $60^{\circ} \mathrm{F}$ from a reservoir through $1000 \mathrm{ft}$ of 12 -in-ID galvanized-iron pipe to a point $200 \mathrm{ft}$ above the reservoir surface. What flow rate and brake horsepower will result? If there is $40 \mathrm{ft}$ of pipe upstream of the pump, how far below the surface should the pump inlet be placed to avoid cavitation?

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Problem 64

In Prob. 11.63 the operating point is off design at an efficiency of only 77 percent. Is it possible, with the similarity rules, to change the pump rotation speed to deliver the water near BEP? Explain your results.

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Problem 65

The 38 -in pump of Fig. $11.7 a$ is used in series to lift $20^{\circ} \mathrm{C}$ water $3000 \mathrm{ft}$ through $4000 \mathrm{ft}$ of 18 -in-ID cast-iron pipe. For most efficient operation, how many pumps in series are needed if the rotation speed is $(a) 710 \mathrm{r} / \mathrm{min}$ and $(b)$ $1200 \mathrm{r} / \mathrm{min} ?$

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Problem 66

It is proposed to run the pump of Prob. 11.35 at $880 \mathrm{r} / \mathrm{min}$ to pump water at $20^{\circ} \mathrm{C}$ through the system in Fig. P1 1.66 The pipe is 20 -cm-diameter commercial steel. What flow rate in $\mathrm{ft}^{3} / \mathrm{min}$ will result? Is this an efficient application?

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Problem 67

The pump of Prob. $11.35,$ running at $880 \mathrm{r} / \mathrm{min},$ is to pump water at $20^{\circ} \mathrm{C}$ through $75 \mathrm{m}$ of horizontal galvanized-iron pipe. All other system losses are neglected.Determine the flow rate and input power for $(a)$ pipe di ameter $=20 \mathrm{cm}$ and $(b)$ the pipe diameter found to yield maximum pump efficiency.

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Problem 68

A 24 -in pump is homologous to the 32 -in pump in Fig. 11.7a. At 1400 r/min this pump delivers 12,000 gal/min of water from one reservoir through a long pipe to another $50 \mathrm{ft}$ higher. What will the flow rate be if the pump speed is increased to $1750 \mathrm{r} / \mathrm{min} ?$ Assume no change in pipe friction factor or efficiency.

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Problem 69

The pump of Prob. 11.38 , running at $3500 \mathrm{r} / \mathrm{min}$, is used to deliver water at $20^{\circ} \mathrm{C}$ through $600 \mathrm{ft}$ of cast-iron pipe to an elevation $100 \mathrm{ft}$ higher. Determine $(a)$ the proper pipe diameter for BEP operation and ( $b$ ) the flow rate which results if the pipe diameter is 3 in.

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Problem 70

The pump of Prob. 11.28 , operating at $2134 \mathrm{r} / \mathrm{min}$, is used with $20^{\circ} \mathrm{C}$ water in the system of Fig. P11.70. $(a)$ If it is operating at BEP, what is the proper elevation $z_{2} ?(b)$ If $z_{2}=225 \mathrm{ft},$ what is the flow rate if $d=8$ in.?

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Problem 71

The pump of Prob. 11.38 , running at 3500 r/min, delivers water at $20^{\circ} \mathrm{C}$ through 7200 ft of horizontal $5-$ in -diameter commercial-steel pipe. There are a sharp entrance, sharp exit, four $90^{\circ}$ elbows, and a gate valve. Estimate (a) the flow rate if the valve is wide open and (b) the valve closing percentage which causes the pump to operate at BEP. (c) If the latter condition holds continuously for 1 year, estimate the energy cost at $10 \not c \mathrm{kWh}$.

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Problem 72

Performance data for a small commercial pump are as follows:
$$\begin{array}{c|c|c|c|c|c|c|c|c}
Q, \text { gal/min } & 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 \\
\hline H, \mathrm{ft} & 75 & 75 & 74 & 72 & 68 & 62 & 47 & 24
\end{array}$$
This pump supplies $20^{\circ} \mathrm{C}$ water to a horizontal $\frac{5}{8}$ -indiameter garden hose $(\epsilon \approx 0.01 \text { in })$ which is $50 \mathrm{ft}$ long. Estimate $(a)$ the flow rate and $(b)$ the hose diameter which would cause the pump to operate at BEP.

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Problem 73

The piston pump of Fig. P11.9 is run at 1500 r/min to deliver SAE $10 \mathrm{W}$ oil through $100 \mathrm{m}$ of vertical $2-\mathrm{cm}-\mathrm{di}$ ameter wrought-iron pipe. If other system losses are neglected, estimate $(a)$ the flow rate, $(b)$ the pressure rise and $(c)$ the power required.

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Problem 74

The 32 -in pump in Fig. $11.7 a$ is used at $1170 \mathrm{r} / \mathrm{min}$ in a system whose head curve is $H_{s}(\mathrm{ft})=100+1.5 Q^{2},$ with $Q$ in thousands of gallons of water per minute. Find the discharge and brake horsepower required for $(a)$ one pump, ( $b$ ) two pumps in parallel, and (c) two pumps in series. Which configuration is best?

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Problem 75

Two 35 -in pumps from Fig. $11.7 b$ are installed in parallel for the system of Fig. P11.75. Neglect minor losses. For water at $20^{\circ} \mathrm{C}$, estimate the flow rate and power required if $(a)$ both pumps are running and $(b)$ one pump is shut off and isolated.

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Problem 76

Two 32 -in pumps from Fig. $11.7 a$ are combined in parallel to deliver water at $60^{\circ} \mathrm{F}$ through $1500 \mathrm{ft}$ of horizontal pipe. If $f=0.025,$ what pipe diameter will ensure a flow rate of 35,000 gal/min for $n=1170 \mathrm{r} / \mathrm{min} ?$

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Problem 77

Two pumps of the type tested in Prob. 11.22 are to be used at $2140 \mathrm{r} / \mathrm{min}$ to pump water at $20^{\circ} \mathrm{C}$ vertically upward through $100 \mathrm{m}$ of commercial-steel pipe. Should they be in series or in parallel? What is the proper pipe diameter for most efficient operation?

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Problem 78

Suppose that the two pumps in Fig. P11.75 are modified to be in series, still at $710 \mathrm{r} / \mathrm{min}$. What pipe diameter is required for BEP operation?

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Problem 79

Two 32 -in pumps from Fig. $11.7 a$ are to be used in series at $1170 \mathrm{r} / \mathrm{min}$ to lift water through $500 \mathrm{ft}$ of vertical cast-iron pipe. What should the pipe diameter be for most efficient operation? Neglect minor losses.

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Problem 80

It is proposed to use one 32 - and one 28 -in pump from Fig. $11.7 a$ in parallel to deliver water at $60^{\circ} \mathrm{F}$. The system-head curve is $H_{s}=50+0.3 Q^{2},$ with $Q$ in thousands of gallons per minute. What will the head and delivery be if both pumps run at $1170 \mathrm{r} / \mathrm{min} ?$ If the 28 -in pump is reduced below $1170 \mathrm{r} / \mathrm{min},$ at what speed will it cease to deliver?

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Problem 81

Reconsider the system of Fig. P6.68. Use the Byron Jackson pump of Prob. 11.28 running at $2134 \mathrm{r} / \mathrm{min},$ no scaling, to drive the flow. Determine the resulting flow rate between the reservoirs. What is the pump efficiency?

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Problem 82

The S-shaped head-versus-flow curve in Fig. P11.82 occurs in some axial-flow pumps. Explain how a fairly flat system-loss curve might cause instabilities in the operation of the pump. How might we avoid instability?

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Problem 83

The low-shutoff head-versus-flow curve in Fig. P11.83 occurs in some centrifugal pumps. Explain how a fairly flat system-loss curve might cause instabilities in the operation of the pump. What additional vexation occurs when two of these pumps are in parallel? How might we avoid instability?

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Problem 84

Turbines are to be installed where the net head is $400 \mathrm{ft}$ and the flow rate 250,000 gal/min. Discuss the type, number, and size of turbine which might be selected if the generator selected is $(a)$ 48-pole, 60 -cycle $(n=150$ $\mathrm{r} / \mathrm{min})$ and (b) 8 -pole $(n=900 \mathrm{r} / \mathrm{min}) .$ Why are at least two turbines desirable from a planning point of view?

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Problem 85

Turbines at the Conowingo Plant on the Susquehanna River each develop 54,000 bhp at 82 r/min under a head of $89 \mathrm{ft}$. What type of turbines are these? Estimate the flow rate and impeller diameter.

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Problem 86

The Tupperware hydroelectric plant on the Blackstone River has four 36 -in-diameter turbines, each providing $447 \mathrm{kW}$ at $200 \mathrm{r} / \mathrm{min}$ and $205 \mathrm{ft}^{3} / \mathrm{s}$ for a head of $30 \mathrm{ft}$ What type of turbines are these? How does their performance compare with Fig. $11.21 ?$

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Problem 87

An idealized radial turbine is shown in Fig. P11.87. The absolute flow enters at $30^{\circ}$ and leaves radially inward. The flow rate is $3.5 \mathrm{m}^{3} / \mathrm{s}$ of water at $20^{\circ} \mathrm{C}$. The blade thickness is constant at $10 \mathrm{cm} .$ Compute the theoretical power developed at 100 percent efficiency.

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Problem 88

A certain turbine in Switzerland delivers 25,000 bhp at $500 \mathrm{r} / \mathrm{min}$ under a net head of $5330 \mathrm{ft}$. What type of turbine is this? Estimate the approximate discharge and size.

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Problem 89

A Pelton wheel of 12 -ft pitch diameter operates under a net head of $2000 \mathrm{ft}$. Estimate the speed, power output and flow rate for best efficiency if the nozzle exit diameter is 4 in.

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Problem 90

An idealized radial turbine is shown in Fig. P11.90. The absolute flow enters at $25^{\circ}$ with the blade angles as shown. The flow rate is $8 \mathrm{m}^{3} / \mathrm{s}$ of water at $20^{\circ} \mathrm{C}$. The blade thickness is constant at $20 \mathrm{cm} .$ Compute the theoretical power developed at 100 percent efficiency.

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Problem 91

The flow through an axial-flow turbine can be idealized by modifying the stator-rotor diagrams of Fig. 11.12 for energy absorption. Sketch a suitable blade and flow arrangement and the associated velocity vector diagrams. For further details see chap. 8 of Ref. 25.

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Problem 92

At a proposed turbine installation the available head is $800 \mathrm{ft},$ and the water flow rate is 40,000 gal/min. Discuss the size, speed, and number of turbines which might be suitable for this purpose while using $(a)$ a Pelton wheel and (b) a Francis wheel.

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Problem 93

Figure $\mathrm{P} 11.93$ shows a cutaway of a cross-flow or "Banki" turbine [55], which resembles a squirrel cage with slotted curved blades. The flow enters at about 2 o'clock, passes through the center and then again through the blades, leaving at about 8 o'clock. Report to the class on the operation and advantages of this design, including idealized velocity vector diagrams.

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Problem 94

A simple cross-flow turbine, Fig. $\mathrm{P} 11.93,$ was constructed and tested at the University of Rhode Island. The blades were made of PVC pipe cut lengthwise into three $120^{\circ}$ arc pieces. When it was tested in water at a head of 5.3 ft and a flow rate of 630 gal/min, the measured power output was 0.6 hp. Estimate $(a)$ the efficiency and $(b)$ the power specific speed if $n=200$ rev/min.

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Problem 95

One can make a theoretical estimate of the proper diameter for a penstock in an impulse turbine installation, as in Fig. P11.95. Let $L$ and $H$ be known, and let the turbine performance be idealized by Eqs. (11.38) and $(11.39) .$ Account for friction loss $h_{f}$ in the penstock, but neglect minor losses. Show that $(a)$ the maximum power is generated when $h_{f}=H / 3,(b)$ the optimum jet velocity is $(4 g H / 3)^{1 / 2},$ and $(c)$ the best nozzle diameter is $D_{j}=\left[D^{5} /(2 f L)\right]^{1 / 4},$ where $f$ is the pipe-friction factor .

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Problem 96

Apply the results of Prob. 11.95 to determining the opti$\operatorname{mum}(a)$ penstock diameter and $(b)$ nozzle diameter for the data of Prob. 11.92 with a commercial-steel penstock of length $1500 \mathrm{ft}$.

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Problem 97

Consider the following nonoptimum version of Prob. $11.95: H=450 \mathrm{m}, L=5 \mathrm{km}, D=1.2 \mathrm{m}, D_{j}=20 \mathrm{cm}$ The penstock is concrete, $\epsilon=1 \mathrm{mm} .$ The impulse wheel diameter is $3.2 \mathrm{m}$. Estimate $(a)$ the power generated by the wheel at 80 percent efficiency and $(b)$ the best speed of the wheel in $\mathrm{r} / \mathrm{min}$. Neglect minor losses.

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Problem 98

Francis and Kaplan turbines are often provided with draft tubes, which lead the exit flow into the tailwater region,
as in Fig. P11.98. Explain at least two advantages in using a draft tube.

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Problem 99

Turbines can also cavitate when the pressure at point 1 in Fig. $\mathrm{P} 11.98$ drops too low. With NPSH defined by Eq. $(11.20),$ the empirical criterion given by Wislicenus [4] for savitation is $$N_{s s}=\frac{(\mathrm{r} / \mathrm{min})(\mathrm{gal} / \mathrm{min})^{1 / 2}}{[\mathrm{NPSH}(\mathrm{ft})]^{3 / 4}} \geq 11,000$$ Use this criterion to compute how high $z_{1}-z_{2},$ the impeller eye in Fig. $\mathrm{P} 11.98$, can be placed for a Francis turbine with a head of $300 \mathrm{ft}, N_{s p}=40,$ and $p_{a}=14 \mathrm{lbf} / \mathrm{in}^{2}$ absolute before cavitation occurs in $60^{\circ} \mathrm{F}$ water.

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Problem 100

One of the largest wind generators in operation today is the ERDA/NASA two-blade propeller HAWT in Sandusky, Ohio. The blades are $125 \mathrm{ft}$ in diameter and reach maximum power in $19 \mathrm{mi} / \mathrm{h}$ winds. For this condition estimate $(a)$ the power generated in $\mathrm{kW},(b)$ the rotor speed in $\mathrm{r} / \mathrm{min},$ and $(c)$ the velocity $V_{2}$ behind the rotor.

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Problem 101

A Darrieus VAWT in operation in Lumsden, Saskatchewan, that is $32 \mathrm{ft}$ high and $20 \mathrm{ft}$ in diameter sweeps out an area of $432 \mathrm{ft}^{2}$. Estimate $(a)$ the maximum power and $(b)$ the rotor speed if it is operating in $16 \mathrm{mi} / \mathrm{h}$ winds.

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Problem 102

An American 6-ft diameter multiblade HAWT is used to pump water to a height of $10 \mathrm{ft}$ through 3 -in-diameter cast-iron pipe. If the winds are $12 \mathrm{mi} / \mathrm{h}$, estimate the rate of water flow in gal/min.

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Problem 103

A very large Darrieus VAWT was constructed by the U.S. Department of Energy near Sandia, New Mexico. It is $60 \mathrm{ft}$ high and $30 \mathrm{ft}$ in diameter, with a swept area of $1200 \mathrm{ft}^{2} .$ If the turbine is constrained to rotate at $90 \mathrm{r} / \mathrm{min}$ use Fig. 11.31 to plot the predicted power output in $\mathrm{kW}$ versus wind speed in the range $V=5$ to $40 \mathrm{mi} / \mathrm{h}$.

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