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Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 14

TURBOMACHINERY - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the more common term for an energyproducing turbomachine? How about an energy-absorbing turbomachine? Explain this terminology. In particular, from which frame of reference are these terms defined-that of the fluid or that of the surroundings?

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Problem 2

What are the primary differences between fans, blowers, and compressors? Discuss in terms of pressure rise and volume flow rate.

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Problem 3

List at least two common examples of fans, of blowers, and of compressors.

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Problem 4

Discuss the primary difference between a positivedisplacement turbomachine and a dynamic turbomachine. Give an example of each for both pumps and turbines.

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Problem 5

For a pump, discuss the difference between brake horsepower and water horsepower, and also define pump efficiency in terms of these quantities.

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Problem 6

For a turbine, discuss the difference between brake horsepower and water horsepower, and also define turbine efficiency in terms of these quantities.

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Problem 7

Explain why there is an "extra" term in the Bernoulli equation in a rotating reference frame.

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Problem 8

A water pump increases the pressure of the water passing through it (Fig. P14-8). The water is assumed to be incompressible. For each of the three cases listed below, how does average water speed change across the pump? In particular, is $V_{\text {out }}$ less than, equal to, or greater than $V_{\text {in }}$ ? Show your equations, and explain.
(a) Outlet diameter is less than inlet diameter ( $D_{\text {out }}<D_{\text {in }}$ )
(b) Outlet and inlet diameters are equal ( $D_{\text {oat }}=D_{\text {in }}$ )
(c) Outlet diameter is greater than inlet diameter ( $D_{\text {out }}>D_{\text {in }}$ )

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Problem 9

An air compressor increases the pressure ( $P_{\text {out }}>P_{\text {in }}$ ) and the density ( $\rho_{\text {out }}>\rho_{\text {in }}$ ) of the air passing through it (Fig. P14-9). For the case in which the outlet and inlet diameters are equal ( $D_{\text {oat }}=D_{\text {in }}$ ), how does average air speed change across the compressor? In particular, is $V_{\text {oat }}$ less than, equal to, or greater than $V_{\text {in }}$ ? Explain. Answer: less than

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Problem 10

There are three main categories of dynamic pumps. List and define them.

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Problem 11

For each statement about centrifugal pumps, choose whether the statement is true or false, and discuss your answer briefly.
(a) A centrifugal pump with radial blades has higher efficiency than the same pump with backward-inclined blades.
(b) A centrifugal pump with radial blades produces a larger pressure rise than the same pump with backward- or forwardinclined blades over a wide range of $\dot{V}$.
(c) A centrifugal pump with forward-inclined blades is a good choice when one needs to provide a large pressure rise over a wide range of volume flow rates.
(d) A centrifugal pump with forward-inclined blades would most likely have less blades than a pump of the same size with backward-inclined or radial blades.

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Problem 12

Figure P14-12C shows two possible locations for a water pump in a piping system that pumps water from the lower tank to the upper tank. Which location is better? Why?

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Problem 13

Define net positive suction head and required net positive suction head, and explain how these two quantities are used to ensure that cavitation does not occur in a pump.

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Problem 14

Consider flow through a water pump. For each statement, choose whether the statement is true or false, and discuss your answer briefly.
(a) The faster the flow through the pump, the more likely that cavitation will occur.
(b) As water temperature increases, $\mathrm{NPSH}_{\text {required }}$ also increases.
(c) As water temperature increases, the available NPSH also increases.
(d) As water temperature increases, cavitation is less likely to occur.

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Problem 15

Explain why it is usually not wise to arrange two (or more) dissimilar pumps in series or in parallel.

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Problem 16

Consider a typical centrifugal liquid pump. For each statement, choose whether the statement is true or false, and discuss your answer briefly.
(a) $\dot{V}$ at the pump's free delivery is greater than $\dot{V}$ at its best efficiency point.
(b) At the pump's shutoff head, the pump efficiency is zero.
(c) At the pump's best efficiency point, its net head is at its maximum value.
(d) At the pump's free delivery, the pump efficiency is zero.

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Problem 17

Write the equation that defines actual (available) net positive suction head NPSH. From this definition, discuss at least five ways you can decrease the likelihood of cavitation in the pump, for the same liquid, temperature, and volume flow rate.

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Problem 18

Consider steady, incompressible flow through two identical pumps (pumps 1 and 2), either in series or in parallel. For each statement, choose whether the statement is true or false, and discuss your answer briefly.
(a) The volume flow rate through the two pumps in series is equal to $\dot{V}_1+\dot{V}_2$.
(b) The overall net head across the two pumps in series is equal to $H_1+H_2$.
(c) The volume flow rate through the two pumps in parallel is equal to $\dot{V}_1+\dot{V}_2$.
(d) The overall net head across the two pumps in parallel is equal to $H_1+H_2$.

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Problem 19

In Fig. P14-19C is shown a plot of the pump net head as a function of the pump volume flow rate, or capacity. On the figure, label the shutoff head, the free delivery, the pump performance curve, the system curve, and the operating point.

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Problem 20

Suppose the pump of Fig. P14-19C is situated between two water tanks with their free surfaces open to the atmosphere. Which free surface is at a higher elevation-the one corresponding to the tank supplying water to the pump inlet, or the one corresponding to the tank connected to the pump outlet? Justify your answer through use of the energy equation between the two free surfaces.

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Problem 21

Suppose the pump of Fig. P14-19C is situated between two large water tanks with their free surfaces open to the atmosphere. Explain qualitatively what would happen to the pump performance curve if the free surface of the outlet tank were raised in elevation, all else being equal. Repeat for the system curve. What would happen to the operating point-would the volume flow rate at the operating point decrease, increase, or remain the same? Indicate the change on a qualitative plot of $H$ versus $\dot{V}$, and discuss. (Hint: Use the energy equation between the free surface of the tank upstream of the pump and the free surface of the tank downstream of the pump.)

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Problem 22

Suppose the pump of Fig. P14-19C is situated between two large water tanks with their free surfaces open to the atmosphere. Explain qualitatively what would happen to the pump performance curve if a valve in the piping system were changed from 100 percent open to 50 percent open, all else being equal. Repeat for the system curve. What would happen to the operating point-would the volume flow rate at the operating point decrease, increase, or remain the same? Indicate the change on a qualitative plot of $H$ versus $\dot{V}$, and discuss. (Hint: Use the energy equation between the free surface of the upstream tank and the free surface of the downstream tank.)

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Problem 23

Consider the flow system sketched in Fig. P14-23. The fluid is water, and the pump is a centrifugal pump. Generate a qualitative plot of the pump net head as a function of the pump capacity. On the figure, label the shutoff head, the free delivery, the pump performance curve, the system curve, and the operating point. (Hint: Carefully consider the required net head at conditions of zero flow rate.)

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Problem 24

Suppose the pump of Fig. P14-23 is operating at free delivery conditions. The pipe, both upstream and downstream of the pump, has an inner diameter of 2.0 cm and nearly zero roughness. The minor loss coefficient associated with the sharp inlet is 0.50 , each valve has a minor loss coefficient of 2.4 , and each of the three elbows has a minor loss coefficient of 0.90 . The contraction at the exit reduces the diameter by a factor of 0.60 ( $60 \%$ of the pipe diameter), and the minor loss coefficient of the contraction is 0.15 . Note that this minor loss coefficient is based on the average exit velocity, not the average velocity through the pipe itself. The total length of pipe is 6.7 m , and the elevation difference is $\left(z_1-z_2\right)$ $=4.6 \mathrm{~m}$. Estimate the volume flow rate through this piping system.

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Problem 25

Repeat Prob. 14-24, but with a rough pipe-pipe roughness $\varepsilon=0.50 \mathrm{~mm}$. Assume that a modified pump is used, such that the new pump operates at its free delivery conditions, just as in Prob. 14-24. Assume all other dimensions and parameters are the same as in that problem. Do your results agree with intuition? Explain

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Problem 26

(Gs) Consider the piping system of Fig. P14-23, with all the dimensions, parameters, minor loss coefficients, etc., of Prob. 14-24. The pump's performance follows a parabolic curve fit, $H_{\text {available }}=H_0-a \dot{V}^2$, where $H_0=17.6 \mathrm{~m}$ is the pump's shutoff head, and $a$ $=0.00426 \mathrm{~m} /(\mathrm{Lpm})^2$ is a coefficient of the curve fit. Estimate the operating volume flow rate $\dot{V}$ in Lpm (liters per minute), and compare with that of Prob. 14-24. Discuss.

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Problem 27

The performance data for a centrifugal water pump are shown in Table P14-27E for water at $77^{\circ} \mathrm{F}$ (gpm $=$ gallons per minute). (a) For each row of data, calculate the pump efficiency (percent). Show all units and unit conversions for full credit. (b) Estimate the volume flow rate (gpm) and net head $(\mathrm{ft})$ at the BEP of the pump.
$$
\begin{array}{rcl}
\dot{\text { V }}, \mathrm{gpm} & H, \mathrm{ft} & \text { bhp, hp } \\
\hline 0.0 & 19.0 & 0.06 \\
4.0 & 18.5 & 0.064 \\
8.0 & 17.0 & 0.069 \\
12.0 & 14.5 & 0.074 \\
16.0 & 10.5 & 0.079 \\
20.0 & 6.0 & 0.08 \\
24.0 & 0.0 & 0.078 \\
\hline
\end{array}
$$

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Problem 28

Transform each column of the pump performance data of Prob. 14-27E to metric units: $\dot{V}$ into Lpm (liters per minute), $H$ into m , and bhp into W. Calculate the pump efficiency (percent) using these metric values, and compare to those of Prob. 14-27E.

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Problem 29

For the centrifugal water pump of Prob. (ft), (f), bhp (hp), and $\eta_{\text {pump }}$ (percent) as functions of $\dot{V}$ (gpm), using symbols only (no lines). Perform linear least-squares polynomial curve fits for all three parameters, and plot the fitted curves as lines (no symbols) on the same plot. For consistency, use a first-order curve fit for $H$ as a function of $\dot{V}^2$, use a second-order curve fit for bhp as a function of both $\dot{V}$ and $\dot{V}^2$, and use a third-order curve fit for $\eta_{\text {pump }}$ as a function of $\dot{V}, \dot{V}^2$, and $\dot{V}^3$. List all curve-fitted equations and coefficients (with units) for full credit. Calculate the BEP of the pump based on the curve-fitted expressions.

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Problem 30

Suppose the pump of Probs. 14-27E and 14-29E is used in a piping system that has the system requirement $H_{\text {required }}=\left(z_2-z_1\right)+b \dot{\cup}^2$, where elevation difference $z_2-z_1=15.5 \mathrm{ft}$, and coefficient $b=0.00986 \mathrm{ft} /(\mathrm{gpm})^2$. Estimate the operating point of the system, namely, $\ddot{V}_{\text {operating }}$ (gpm) and $H_{\text {operating }}(\mathrm{ft})$. Answers: $9.14 \mathrm{gpm}, 16.3 \mathrm{ft}$

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Problem 31

The performance data for a centrifugal water pump are shown in Table P14-31 for water at $20^{\circ} \mathrm{C}$ ( Lpm $=$ liters per minute). (a) For each row of data, calculate the pump efficiency (percent). Show all units and unit conversions for full credit. (b) Estimate the volume flow rate (Lpm) and net head ( m ) at the BEP of the pump.
$$
\begin{array}{rrr}
\dot{\text { V}}, \mathrm{Lpm} & H, \mathrm{~m} & \text { bhp, W } \\
\hline 0.0 & 47.5 & 133 \\
6.0 & 46.2 & 142 \\
12.0 & 42.5 & 153 \\
18.0 & 36.2 & 164 \\
24.0 & 26.2 & 172 \\
30.0 & 15.0 & 174 \\
36.0 & 0.0 & 174 \\
\hline
\end{array}
$$

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Problem 32

For the centrifugal water pump of Prob. 14-31, plot the pump's performance data: $H(\mathrm{~m})$, bhp (W), and $\eta_{\text {pump }}$ (percent) as functions of $\dot{V}$ (Lpm), using symbols only (no lines). Perform linear least-squares polynomial curve fits for all three parameters, and plot the fitted curves as lines (no symbols) on the same plot. For consistency, use a first-order curve fit for $H$ as a function of $\dot{V}^2$, use a secondorder curve fit for bhp as a function of both $\dot{V}$ and $\dot{V}^2$, and use a third-order curve fit for $\eta_{\text {pump }}$ as a function of $\dot{V}, \dot{V}^2$, and $\dot{V}^3$. List all curve-fitted equations and coefficients (with units) for full credit. Calculate the BEP of the pump based on the curve-fitted expressions.

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Problem 33

Suppose the pump of Probs. 14-31 and 14-32 is used in a piping system that has the system requirement $H_{\text {required }}=\left(z_2-z_1\right)+b \dot{V}^2$, where the elevation difference $z_2-z_1=10.0 \mathrm{~m}$, and coefficient $b=0.0185 \mathrm{~m} /(\mathrm{Lpm})^2$. Estimate the operating point of the system, namely, $\ddot{V}_{\text {operating }}$ (Lpm) and $H_{\text {operating }}(\mathrm{m}$ ).

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Problem 34

Suppose you are looking into purchasing a water pump with the performance data shown in Table P14-34. Your supervisor asks for some more information about the pump. (a) Estimate the shutoff head $H_0$ and the free delivery $\dot{V}_{\max }$ of the pump. [Hint: Perform a leastsquares curve fit (regression analysis) of $H_{\text {available }}$ versus $\dot{V}^2$, and calculate the best-fit values of coefficients $H_0$ and $a$ that translate the tabulated data of Table P14-34 into the parabolic expression, $H_{\text {avilable }}=H_0-a \dot{V}^2$. From these coefficients, estimate the free delivery of the pump.] (b) The application requires 57.0 Lpm of flow at a pressure rise across the pump of 5.8 psi . Is this pump capable of meeting the requirements? Explain.
$$
\begin{array}{cc}
\dot{\dot{y}}, \mathrm{Lpm} & \mathrm{H}, \mathrm{~m} \\
\hline 20 & 21 \\
30 & 18.4 \\
40 & 14 \\
50 & 7.6 \\
\hline
\end{array}
$$

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Problem 35

A manufacturer of small water pumps lists the performance data for a family of its pumps as a parabolic curve fit, $H_{\text {available }}=H_0-a \dot{V}^2$, where $H_0$ is the pump's shutoff head and $a$ is a coefficient. Both $H_0$ and $a$ are listed in a table for the pump family, along with the pump's free delivery. The pump head is given in units of feet of water column, and capacity is given in units of gallons per minute. (a) What are the units of coefficient $a$ ? (b) Generate an expression for the pump's free delivery $\dot{V}_{\text {max }}$ in terms of $H_0$ and $a$. (c) Suppose one of the manufacturer's pumps is used to pump water from one large reservoir to another at a higher elevation. The free surfaces of both reservoirs are exposed to atmospheric pressure. The system curve simplifies to $H_{\text {required }}=\left(z_2-z_1\right)+$ $b \dot{V}^2$. Calculate the operating point of the pump ( $\dot{V}_{\text {opentaing }}$ and $\left.H_{\text {operting }}\right)$ in terms of $H_0, a, b$, and elevation difference $z_2-z_1$.

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Problem 36

The performance data of a water pump follow the curve fit $H_{\text {available }}=H_0-a \dot{V}^2$, where the pump's shutoff head $H_0=5.30 \mathrm{~m}$, coefficient $a=0.0453 \mathrm{~m} /(\mathrm{Lpm})^2$, the units of pump head $H$ are meters, and the units of $\dot{V}$ are liters per minute ( Lpm ). The pump is used to pump water from one large reservoir to another large reservoir at a higher elevation. The free surfaces of both reservoirs are exposed to atmospheric pressure. The system curve simplifies to $H_{\text {required }}=\left(z_2\right.$ $\left.-z_1\right)+b \dot{V}^2$, where elevation difference $z_2-z_1=3.52 \mathrm{~m}$, and coefficient $b=0.0261 \mathrm{~m}(\mathrm{Lpm})^2$. Calculate the operating point of the pump ( $\dot{V}_{\text {operating }}$ and $H_{\text {operating }}$ ) in appropriate units (Lpm and meters, respectively).

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Problem 37

A water pump is used to pump water from one large reservoir to another large reservoir that is at a higher elevation. The free surfaces of both reservoirs are exposed to atmospheric pressure, as sketched in Fig. P14-37E. The dimensions and minor loss coefficients are provided in the figure. The pump's performance is approximated by the expression $H_{\text {available }}=H_0-a \dot{V}^2$, where the shutoff head $H_0=125 \mathrm{ft}$ of water column, coefficient $a=2.50 \mathrm{ft} / \mathrm{gpm}^2$, available pump head $H_{\text {availbble }}$ is in units of feet of water column, and capacity $\dot{V}$ is in units of gallons per minute (gpm). Estimate the capacity delivered by the pump.

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Problem 38

For the pump and piping system of Prob. 14-37E, plot the required pump head $H_{\text {required }}$ ( ft of water column) as a function of volume flow rate $\dot{V}(\mathrm{gpm})$. On the same plot, compare the available pump head $H_{\text {awailble }}$ versus $\dot{V}$, and mark the operating point. Discuss.

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Problem 39

Suppose that the two reservoirs in Prob. 14-37E are 1000 ft further apart horizontally, but at the same elevations. All the constants and parameters are identical to those of Prob. 14-37E except that the total pipe length is 1124 ft instead of 124 ft . Calculate the volume flow rate for this case and compare with the result of Prob. 14-37E. Discuss.

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Problem 40

Paul realizes that the pump being used in Prob. 14-37E is not well-matched for this application, since its shutoff head ( 125 ft ) is much larger than its required net head (less than 30 ft ), and its capacity is fairly low. In other words, this pump is designed for highhead, low-capacity applications, whereas the application at hand is fairly low-head, and a higher capacity is desired. Paul tries to convince his supervisor that a less expensive pump, with lower shutoff head but higher free delivery, would result in a significantly increased flow rate between the two reservoirs. Paul looks through some online brochures, and finds a pump with the performance data shown in Table P14-40E. His supervisor asks him to predict the volume flow rate between the two reservoirs if the existing pump were replaced with the new pump. (a) Perform a least-squares curve fit (regression analysis) of $H_{\text {available }}$ versus $\dot{V}^2$, and calculate the best-fit values of coefficients $H_0$ and $a$ that translate the tabulated data of Table P14-40E into the parabolic expression $H_{\text {aveilable }}=H_0-a \dot{V}^2$. Plot the data points as symbols and the curve fit as a line for comparison. (b) Estimate the operating volume flow rate of the new pump if it were to replace the existing pump, all else being equal. Compare to the result of Prob. $14-37 \mathrm{E}$ and discuss. Is Paul correct? (c) Generate a plot of required net head and available net head as functions of volume flow rate and indicate the operating point on the plot.
$$
\begin{array}{cc}
\dot{\boldsymbol{v}} \text {, gpm } & H, \mathrm{ft} \\
\hline 0 & 38 \\
4 & 37 \\
8 & 34 \\
12 & 29 \\
16 & 21 \\
20 & 12 \\
24 & 0 \\
\hline
\end{array}
$$

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Problem 41

A water pump is used to pump water from one large reservoir to another large reservoir that is at a higher elevation. The free surfaces of both reservoirs are exposed to atmospheric pressure, as sketched in Fig. P14-41. The dimensions and minor loss coefficients are provided in the
$$
\begin{aligned}
z_2-z_1 & =7.85 \mathrm{~m} \text { (elevation difference) } \\
D & =2.03 \mathrm{~cm} \text { (pipe diameter) } \\
K_{L, \text { entraine }} & =0.50 \text { (pipe entrance) } \\
K_{L, \text { valve }} & =17.5 \text { (valve) } \\
K_{L, \text { elbow }} & =0.92 \text { (each elbow-there are } 5 \text { ) } \\
K_{L, \text { exit }} & =1.05 \text { (pipe exit) } \\
L & =176.5 \mathrm{~m} \text { (total pipe length) } \\
\varepsilon & =0.25 \mathrm{~mm} \text { (pipe roughness) }
\end{aligned}
$$
Figure Can't Copy
figure. The pump's performance is approximated by the expression $H_{\text {available }}=H_0-a \dot{V}^2$, where shutoff head $H_0=$ 24.4 m of water column, coefficient $a=0.0678 \mathrm{~m} / \mathrm{Lpm}^2$, available pump head $H_{\text {available }}$ is in units of meters of water column, and capacity $\ddot{V}$ is in units of liters per minute (Lpm). Estimate the capacity delivered by the pump.

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Problem 42

For the pump and piping system of Prob. 14-41, plot required pump head $H_{\text {required }}$ ( m of water column) as a function of volume flow rate $V$ (Lpm). On the same plot, compare available pump head $H_{\text {available }}$ versus $\dot{V}$, and mark the operating point. Discuss.

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Problem 43

Suppose that the free surface of the inlet reservoir in Prob. $14-41$ is 5.0 m higher in elevation, such that $z_2-z_1$ $=2.85 \mathrm{~m}$. All the constants and parameters are identical to those of Prob. 14-41 except for the elevation difference. Calculate the volume flow rate for this case and compare with the result of Prob. 14-41. Discuss.

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Problem 44

April's supervisor asks her to find a replacement pump that will increase the flow rate through the piping system of Prob. 14-41 by a factor of 2 or greater. April looks through some online brochures, and finds a pump with the performance data shown in Table P14-44. All dimensions and parameters remain the same as in Prob. 14-41-only the pump is changed. (a) Perform a leastsquares curve fit (regression analysis) of $H_{\text {available }}$ versus $\dot{V}^2$, and calculate the best-fit values of coefficients $H_0$ and $a$ that translate the tabulated data of Table P14-44 into the parabolic expression $H_{\text {available }}=H_0-a \dot{V}^2$. Plot the data points as symbols and the curve fit as a line for comparison. (b) Use the expression obtained in part (a) to estimate the operating volume flow rate of the new pump if it were to replace the existing pump, all else being equal. Compare to the result of Prob. $14-41$ and discuss. Has April achieved her goal? (c) Generate a plot of required net head and available net head as functions of volume flow rate, and indicate the operating point on the plot.
$$
\begin{array}{cc}
\dot{V}, \mathrm{Lpm} & \mathrm{H}, \mathrm{~m} \\
\hline 0 & 46.5 \\
5 & 46 \\
10 & 42 \\
15 & 37 \\
20 & 29 \\
25 & 16.5 \\
30 & 0 \\
\hline
\end{array}
$$

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Problem 45

Calculate the volume flow rate between the reservoirs of Prob. 14-41 for the case in which the pipe diameter is doubled, all else remaining the same. Discuss.

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Problem 46

Comparing the results of Probs. $14-41$ and $14-45$, the volume flow rate increases as expected when one doubles the inner diameter of the pipe. One might expect that the Reynolds number increases as well. Does it? Explain.

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Problem 47

Repeat Prob. 14-41, but neglect all minor losses. Compare the volume flow rate with that of Prob. 14-41. Are minor losses important in this problem? Discuss.

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Problem 48

Consider the pump and piping system of Prob. 14-41. Suppose that the lower reservoir is huge, and its surface does not change elevation, but the upper reservoir is not so big, and its surface rises slowly as the reservoir fills. Generate a curve of volume flow rate $\dot{V}$ (Lpm) as a function of $z_2-z_1$ in the range 0 to the value of $z_2-z_1$ at which the pump ceases to pump any more water. At what value of $z_2-z_1$ does this occur? Is the curve linear? Why or why not? What would happen if $z_2-z_1$ were greater than this value? Explain.

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Problem 49

A local ventilation system (a hood and duct system) is used to remove air and contaminants produced by a welding operation (Fig. P14-49E). The inner diameter (ID) of the duct is $D=9.06 \mathrm{in}$, its average roughness is 0.0059 in , and its total length is $L=34.0 \mathrm{ft}$. There are three elbows along the duct, each with a minor loss coefficient of 0.21 . Literature from the hood manufacturer lists the hood entry loss coefficient as 4.6 based on duct velocity. When the damper is fully open, its loss coefficient is 1.8 . A squirrel

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Problem 50

For the duct system and fan of Prob. 14-49E, partially closing the damper would decrease the flow rate. All else being unchanged, estimate the minor loss coefficient of the damper required to decrease the volume flow rate by a factor of 2 .

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Problem 51

Repeat Prob. 14-49E, ignoring all minor losses. How important are the minor losses in this problem? Discuss.

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Problem 52

A local ventilation system (a hood and duct system) is used to remove air and contaminants from a pharmaceutical lab (Fig. P14-52). The inner diameter (ID) of the duct is $D=150 \mathrm{~mm}$, its average roughness is 0.15 mm , and its total length is $L=24.5 \mathrm{~m}$. There are three elbows along the duct, each with a minor loss coefficient of 0.21 . Literature from the hood manufacturer lists the hood entry loss coefficient as 3.3 based on duct velocity. When the damper is fully open, its loss coefficient is 1.8 . The minor loss coefficient through the $90^{\circ}$ tee is 0.36 . Finally, a one-way valve is installed to prevent contaminants from a second hood from flowing "backward" into the room. The minor loss coefficient of the (open) one-way valve is 6.6 . The performance data of the fan fit a parabolic curve of the form $H_{\text {avilable }}=H_0-a \dot{V}^2$, where shutoff head $H_0=60.0 \mathrm{~mm}$ of water column, coefficient $a$ $=2.50 \times 10^{-7} \mathrm{~mm}$ of water column per $(\mathrm{Lpm})^2$, available head $H_{\text {available }}$ is in units of mm of water column, and capacity $\dot{V}$ is in units of Lpm of air. Estimate the volume flow rate in Lpm through this ventilation system.

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Problem 53

For the duct system of Prob. 14-52, plot required fan head $H_{\text {required }}$ ( mm of water column) as a function of volume flow rate $\dot{V}(\mathrm{Lpm})$. On the same plot, compare available fan head $H_{\text {amillble }}$ versus $\dot{V}$, and mark the operating point. Discuss.

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Problem 54

Repeat Prob. 14-52, ignoring all minor losses. How important are the minor losses in this problem? Discuss.

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Problem 55

Suppose the one-way valve of Fig. P14-52 malfunctions due to corrosion and is stuck in its fully closed position (no air can get through). The fan is on, and all other conditions are identical to those of Prob. 14-52. Calculate the gage pressure (in pascals and in mm of water column) at a point just downstream of the fan. Repeat for a point just upstream of the one-way valve.

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Problem 56

A centrifugal pump is used to pump water at $77^{\circ} \mathrm{F}$ from a reservoir whose surface is 20.0 ft above the centerline of the pump inlet (Fig. P14-56E). The piping system consists of 67.5 ft of PVC pipe with an ID of 1.2 in and negligible average inner roughness height. The length of pipe from the bottom of the lower reservoir to the pump inlet is 12.0 ft . There are several minor losses in the piping system: a sharpedged inlet ( $K_L=0.5$ ), two flanged smooth $90^{\circ}$ regular elbows ( $K_L=0.3$ each), two fully open flanged globe valves ( $K_L=6.0$ each), and an exit loss into the upper reservoir ( $K_L$ $=1.05$ ). The pump's required net positive suction head is provided by the manufacturer as a curve fit: $\mathrm{NPSH}_{\text {required }}$ $=1.0 \mathrm{ft}+\left(0.0054 \mathrm{ft} / \mathrm{gpm}^2\right) \dot{V}^2$, where volume flow rate is in gpm. Estimate the maximum volume flow rate (in units of gpm ) that can be pumped without cavitation.

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Problem 57

Repeat Prob. 14-56E, but at a water temperature of $150^{\circ}$ F. Discuss.

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Problem 58

A self-priming centrifugal pump is used to pump water at $25^{\circ} \mathrm{C}$ from a reservoir whose surface is 2.2 m above the centerline of the pump inlet (Fig. P14-58). The pipe is PVC pipe with an ID of 24.0 mm and negligible average inner roughness height. The pipe length from the submerged pipe inlet to the pump inlet is 2.8 m . There are only two minor losses in the piping system from the pipe inlet to the pump inlet: a sharp-edged reentrant inlet ( $K_L=0.85$ ), and a flanged smooth $90^{\circ}$ regular elbow ( $K_L=0.3$ ). The pump's required net positive suction head is provided by the manufacturer as a curve fit: $\mathrm{NPSH}_{\text {required }}=2.2 \mathrm{~m}$ $+\left(0.0013 \mathrm{~m} / \mathrm{Lpm}^2\right) \dot{V}^2$, where volume flow rate is in Lpm. Estimate the maximum volume flow rate (in units of Lpm) that can be pumped without cavitation.

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Problem 59

Repeat Prob. 14-58, but at a water temperature of $80^{\circ} \mathrm{C}$. Repeat for $90^{\circ} \mathrm{C}$. Discuss.

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Problem 60

Repeat Prob. 14-58, but with the pipe diameter increased by a factor of 2 (all else being equal). Does the volume flow rate at which cavitation occurs in the pump increase or decrease with the larger pipe? Discuss.

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Problem 61

Two water pumps are arranged in series. The performance data for both pumps follow the parabolic curve fit $H_{\text {availhble }}=H_0-a \dot{V}^2$. For pump $1, H_0=5.30 \mathrm{~m}$ and coefficient $a=0.0438 \mathrm{~m} / \mathrm{Lpm}^2$; for pump 2, $H_0=7.80 \mathrm{~m}$ and coefficient $a=0.0347 \mathrm{~m} / \mathrm{Lpm}^2$. In either case, the units of net pump head $H$ are m , and the units of capacity $\dot{V}$ are Lpm. Calculate the combined shutoff head and free delivery of the two pumps working together in series. At what volume flow rate should pump 1 be shut off and bypassed? Explain.

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Problem 62

The same two water pumps of Prob. 14-61 are arranged in parallel. Calculate the shutoff head and free delivery of the two pumps working together in parallel. At what combined net head should pump 1 be shut off and bypassed? Explain.

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Problem 63

The two-lobe rotary pump of Fig. P14-63E moves 0.145 gal of a coal slurry in each lobe volume $\dot{V}_{\text {lobe }}$. Calculate the volume flow rate of the slurry (in gpm) for the case where $\tilde{n}=300 \mathrm{rpm} . \quad$

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Problem 64

Repeat Prob. 14-63E for the case in which the pump has three lobes on each rotor instead of two, and $\dot{V}_{\text {lobe }}$ $=0.087 \mathrm{gal}$.

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Problem 65

A two-lobe rotary positive-displacement pump, similar to that of Fig. $14-30$, moves $3.64 \mathrm{~cm}^3$ of tomato paste in each lobe volume $\dot{V}_{\text {lobe }}$. Calculate the volume flow rate of tomato paste for the case where $\dot{n}=400 \mathrm{rpm}$.

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Problem 66

Consider the gear pump of Fig. 14-26c. Suppose the volume of fluid confined between two gear teeth is $0.350 \mathrm{~cm}^3$. How much fluid volume is pumped per rotation?

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Problem 67

A centrifugal pump rotates at $\dot{n}=750 \mathrm{rpm}$. Water enters the impeller normal to the blades ( $\alpha_1=0^{\circ}$ ) and exits at an angle of $35^{\circ}$ from radial ( $\alpha_2=35^{\circ}$ ). The inlet radius is $r_1=12.0 \mathrm{~cm}$, at which the blade width $b_1=18.0 \mathrm{~cm}$. The outlet radius is $r_2=24.0 \mathrm{~cm}$, at which the blade width $b_2$ $=14.0 \mathrm{~cm}$. The volume flow rate is $0.573 \mathrm{~m}^3 / \mathrm{s}$. Assuming 100 percent efficiency, calculate the net head produced by this pump in cm of water column height. Also calculate the required brake horsepower in W .

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Problem 68

A vane-axial flow fan is being designed with the stator blades upstream of the rotor blades (Fig. P14-68). To reduce expenses, both the stator and rotor blades are to be constructed of sheet metal. The stator blade is a simple circular are with its leading edge aligned axially and its trailing edge at angle $\beta_{\text {st }}=26.6^{\circ}$ from the axis as shown in the sketch. (The subscript notation indicates stator trailing edge.) There are 18 stator blades. At design conditions, the axialflow speed through the blades is $31.4 \mathrm{~m} / \mathrm{s}$, and the impeller rotates at 1800 rpm . At a radius of 0.50 m , calculate the leading and trailing edge angles of the rotor blade, and sketch the shape of the blade. How many rotor blades should there be?

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Problem 69

Give at least two reasons why turbines often have greater efficiencies than do pumps.

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Problem 70

Briefly discuss the main difference in the way that dynamic pumps and reaction turbines are classified as centrifugal (radial), mixed flow, or axial.

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Problem 71

What is a draft tube, and what is its purpose? Describe what would happen if turbomachinery designers did not pay attention to the design of the draft tube.

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Problem 72

Name and briefly describe the differences between the two basic types of dynamic turbine.

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Problem 73

Discuss the meaning of reverse swirl in reaction hydroturbines, and explain why some reverse swirl is desirable. Use an equation to support your answer. Why is it not wise to have too much reverse swirl?

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Problem 74

Prove that for a given jet speed, volume flow rate, turning angle, and wheel radius, the maximum shaft power produced by a Pelton wheel occurs when the turbine bucket moves at half the jet speed.

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Problem 75

A Pelton wheel is used to produce hydroelectric power. The average radius of the wheel is 1.83 m , and the jet velocity is $102 \mathrm{~m} / \mathrm{s}$ from a nozzle of exit diameter equal to 10.0 cm . The turning angle of the buckets is $\beta=165^{\circ}$. (a) Calculate the volume flow rate through the turbine in $\mathrm{m}^3 / \mathrm{s}$. (b) What is the optimum rotation rate (in rpm ) of the wheel (for maximum power)? (c) Calculate the output shaft power in MW if the efficiency of the turbine is 82 percent.

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Problem 76

Some engineers are evaluating potential sites for a small hydroelectric dam. At one such site, the gross head is 650 m , and they estimate that the volume flow rate of water through each turbine would be $1.5 \mathrm{~m}^3 / \mathrm{s}$. Estimate the ideal power production per turbine in MW.

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Problem 77

A hydroelectric power plant is being designed. The gross head from the reservoir to the tailrace is 1065 ft , and the volume flow rate of water through each turbine is $203,000 \mathrm{gpm}$ at $70^{\circ} \mathrm{F}$. There are 12 identical parallel turbines, each with an efficiency of 95.2 percent, and all other mechanical energy losses (through the penstock, etc.) are estimated to reduce the output by 3.5 percent. The generator
itself has an efficiency of 94.5 percent. Estimate the electric power production from the plant in MW.

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Problem 78

A Francis radial-flow hydroturbine is being designed with the following dimensions: $r_2=2.00 \mathrm{~m}, r_1=1.42 \mathrm{~m}, b_2$ $=0.731 \mathrm{~m}$, and $b_1=2.20 \mathrm{~m}$. The runner rotates at $\dot{n}$ $=180 \mathrm{rpm}$. The wicket gates turn the flow by angle $\alpha_2=30^{\circ}$ from radial at the runner inlet, and the flow at the runner outlet is at angle $\alpha_1=10^{\circ}$ from radial (Fig. P14-78). The volume flow rate at design conditions is $340 \mathrm{~m}^3 / \mathrm{s}$, and the gross head provided by the dam is $H_{\text {gross }}=90.0 \mathrm{~m}$. For the preliminary design, irreversible losses are neglected. Calculate the inlet and outlet runner blade angles $\beta_2$ and $\beta_1$, respectively, and predict the power output (MW) and required net head (m). Is the design feasible?

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Problem 79

Reconsider Prob. 14-78. Using EES (or other) software, investigate the effect of the runner outlet angle $\alpha_1$ on the required net head and the output power. Let the outlet angle vary from $-20^{\circ}$ to $20^{\circ}$ in increments of $1^{\circ}$, and plot your results. Determine the minimum possible value of $\alpha_1$ such that the flow does not violate the laws of thermodynamics.

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Problem 80

A Francis radial-flow hydroturbine has the following dimensions, where location 2 is the inlet and location 1 is the outlet: $r_2=2.00 \mathrm{~m}, r_1=1.30 \mathrm{~m}, b_2=0.85 \mathrm{~m}$, and $b_1$ $=2.10 \mathrm{~m}$. The runner blade angles are $\beta_2=66^{\circ}$ and $\beta_1$ $=18.5^{\circ}$ at the turbine inlet and outlet, respectively. The runner rotates at $\dot{n}=100 \mathrm{rpm}$. The volume flow rate at design conditions is $80.0 \mathrm{~m}^3 / \mathrm{s}$. Irreversible losses are neglected in this preliminary analysis. Calculate the angle $\alpha_2$ through which the wicket gates should turn the flow, where $\alpha_2$ is measured from the radial direction at the runner inlet (Fig. P14-78). Calculate the swirl angle $\alpha_1$, where $\alpha_1$ is measured from the radial direction at the runner outlet (Fig. P14-78). Does this turbine have forward or reverse swirl? Predict the power output (MW) and required net head (m).

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Problem 81

A Francis radial-flow hydroturbine has the following dimensions, where location 2 is the inlet and location 1 is the outlet: $r_2=6.60 \mathrm{ft}, r_1=4.40 \mathrm{ft}, b_2=2.60 \mathrm{ft}$, and $b_1$ $=7.20 \mathrm{ft}$. The runner blade angles are $\beta_2=82^{\circ}$ and $\beta_1=46^{\circ}$ at the turbine inlet and outlet, respectively. The runner rotates at $\dot{n}=120 \mathrm{rpm}$. The volume flow rate at design conditions is $4.70 \times 10^6 \mathrm{gpm}$. Irreversible losses are neglected in this preliminary analysis. Calculate the angle $\alpha_2$ through which the wicket gates should turn the flow, where $\alpha_2$ is measured from the radial direction at the runner inlet (Fig. P14-78). Calculate the swirl angle $\alpha_1$, where $\alpha_1$ is measured from the radial direction at the runner outlet (Fig. P14-78). Does this turbine have forward or reverse swirl? Predict the power output (hp) and required net head ( ft ).

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Problem 82

Using EES or other software, adjust the runner blade trailing edge angle $\beta_1$ of Prob. $14-81 \mathrm{E}$, keeping all other parameters the same, such that there is no swirl at the turbine outlet. Report $\beta_1$ and the corresponding shaft power.

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Problem 83

Look up the word affinity in a dictionary. Why do you suppose some engineers refer to the turbomachinery scaling laws as affinity laws?

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Problem 84

For each statement, choose whether the statement is true or false, and discuss your answer briefly.
(a) If the rpm of a pump is doubled, all else staying the same, the capacity of the pump goes up by a factor of about 2 .
(b) If the rpm of a pump is doubled, all else staying the same, the net head of the pump goes up by a factor of about 2 .
(c) If the rpm of a pump is doubled, all else staying the same, the required shaft power goes up by a factor of about 4 .
(d) If the rpm of a turbine is doubled, all else staying the same, the output shaft power of the turbine goes up by a factor of about 8 .

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Problem 85

Discuss which dimensionless pump performance parameter is typically used as the independent parameter. Repeat for turbines instead of pumps. Explain.

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Problem 86

Consider the pump of Prob. 14-41. The pump diameter is 1.80 cm , and it operates at $n=4200 \mathrm{rpm}$. Nondimensionalize the pump performance curve, i.e., plot $C_H$ versus $C_Q$. Show sample calculations of $C_H$ and $C_Q$ at $V=14.0 \mathrm{Lpm}$.

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Problem 87

Calculate the pump specific speed of the pump of Prob. 14-86 at the best efficiency point for the case in which the BEP occurs at 14.0 Lpm . Provide answers in both dimensionless form and in customary U.S. units. What kind of pump is it? Answers: $0.199,545$, centrifugal

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Problem 88

Consider the fan of Prob. 14-52. The fan diameter is 30.0 cm , and it operates at $\dot{n}=600 \mathrm{rpm}$. Nondimensionalize the pump performance curve, i.e., plot $C_H$ versus $C_Q$. Show sample calculations of $C_H$ and $C_Q$ at $\dot{V}=13,600 \mathrm{Lpm}$.

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Problem 89

Calculate the pump specific speed of the fan of Prob. 14-88 at the best efficiency point for the case in which the BEP occurs at $13,600 \mathrm{Lpm}$. Provide answers in both dimensionless form and in customary U.S. units. What kind of fan is it?

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Problem 90

Calculate the pump specific speed of the pump of Example 14-11 at its best efficiency point. Provide answers in both dimensionless form and in customary U.S. units. What kind of pump is it?

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Problem 91

Len is asked to design a small water pump for an aquarium. The pump should deliver 18.0 Lpm of water at a net head of 1.6 m at its best efficiency point. A motor that spins at 1200 rpm is available. What kind of pump (centrifugal, mixed, or axial) should Len design? Show all your calculations and justify your choice. Estimate the maximum pump efficiency Len can hope for with this pump.

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Problem 92

A large water pump is being designed for a nuclear reactor. The pump should deliver 2500 gpm of water at a net head of 45 ft at its best efficiency point. A motor that spins at 300 rpm is available. What kind of pump (centrifugal, mixed, or axial) should be designed? Show all your calculations and justify your choice. Estimate the maximum pump efficiency that can be hoped for with this pump. Estimate the power (brake horsepower) required to run the pump.

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Problem 93

Consider the pump of Prob. 14-91. Suppose the pump is modified by attaching a different motor, for which the rpm is half that of the original pump. If the pumps operate at homologous points (namely, at the BEP) for both cases, predict the volume flow rate and net head of the modified pump. Calculate the pump specific speed of the modified pump, and compare to that of the original pump. Discuss.

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Problem 94

Verify that turbine specific speed and pump specific speed are related as follows: $N_{\mathrm{St}}=N_{\mathrm{Sp}} \sqrt{\eta_{\text {turbine }}}$.

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Problem 95

Consider a pump-turbine that operates both as a pump and as a turbine. Under conditions in which the rotational speed $\omega$ and the volume flow rate $\dot{V}$ are the same for the pump and the turbine, verify that turbine specific speed and pump specific speed are related as

$$
\begin{aligned}
& N_{\mathrm{St}}=N_{\mathrm{Sp}} \sqrt{\eta_{\text {turtine }}}\left(\frac{H_{\text {pump }}}{H_{\text {turbine }}}\right)^{3 / 4} \\
& =N_{\text {Sp }}\left(\eta_{\text {lurtine }}\right)^{5 / 4}\left(\eta_{\text {pump }}\right)^{3 / 4}\left(\frac{\text { bhp }_{\text {pump }}}{\text { bhp }_{\text {urutine }}}\right)^{3 / 4}
\end{aligned}
$$

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Problem 96

Apply the necessary conversion factors to prove the relationship between dimensionless turbine specific speed and conventional U.S. turbine specific speed, $N_{\mathrm{St}}=43.46 N_{\mathrm{St}, \mathrm{us}}$.
Note that we assume water as the fluid and standard earth gravity.

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Problem 97

Calculate the turbine specific speed of the Round Butte hydroturbine of Fig 14-89. Does it fall within the range of $N_{\mathrm{St}}$ appropriate for that type of turbine?

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Problem 98

Calculate the turbine specific speed of the Smith Mountain hydroturbine of Fig 14-90. Does it fall within the range of $N_{\mathrm{St}}$ appropriate for that type of turbine?

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Problem 99

Calculate the turbine specific speed of the Warwick hydroturbine of Fig 14-91. Does it fall within the range of $N_{\mathrm{St}}$ appropriate for that type of turbine?

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Problem 100

Calculate the turbine specific speed of the turbine of Example $14-12$ for the case where $\alpha_1=10^{\circ}$. Provide answers in both dimensionless form and in customary U.S. units. Is it in the normal range for a Francis turbine? If not, what type of turbine would be more appropriate?

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Problem 101

Calculate the turbine specific speed of the turbine in Prob. 14-80. Provide answers in both dimensionless form and in customary U.S. units. Is it in the normal range for a Francis turbine? If not, what type of turbine would be more appropriate?

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Problem 102

Calculate the turbine specific speed of the turbine in Prob. 14-81E using customary U.S. units. Is it in the normal range for a Francis turbine? If not, what type of turbine would be more appropriate?

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Problem 103

Calculate the turbine specific speed of the turbine in Prob. 14-78. Provide answers in both dimensionless form and in customary U.S. units. Is it in the normal range for a Francis turbine? If not, what type of turbine would be more appropriate?

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Problem 104

A one-fifth scale model of a water turbine is tested in a laboratory at $T=20^{\circ} \mathrm{C}$. The diameter of the model is 8.0 cm , its volume flow rate is $17.0 \mathrm{~m}^3 / \mathrm{h}$, it spins at 1500 rpm , and it operates with a net head of 15.0 m . At its best efficiency point, it delivers 450 W of shaft power. Calculate the efficiency of the model turbine. What is the most likely kind of turbine being tested? Answers: $64.9 \%$, impulse

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Problem 105

The prototype turbine corresponding to the onefifth scale model turbine discussed in Prob. 14-104 is to operate across a net head of 50 m . Determine the appropriate rpm and volume flow rate for best efficiency. Predict the brake horsepower output of the prototype turbine, assuming exact geometric similarity.

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Problem 106

Prove that the model turbine (Prob. 14-104) and the prototype turbine (Prob. 14-105) operate at homologous points by comparing turbine efficiency and turbine specific speed for both cases.

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Problem 107

In Prob. 14-106, we scaled up the model turbine test results to the full-scale prototype assuming exact dynamic similarity. However, as discussed in the text, a large
prototype typically yields higher efficiency than does the model. Estimate the actual efficiency of the prototype turbine. Briefly explain the higher efficiency.

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Problem 108

A group of engineers is designing a new hydroturbine by scaling up an existing one. The existing turbine (turbine A) has diameter $D_{\mathrm{A}}=1.50 \mathrm{~m}$, and spins at $\dot{n}_{\mathrm{A}}$ $=150 \mathrm{rpm}$. At its best efficiency point, $\dot{V}_{\mathrm{A}}=162 \mathrm{~m}^3 / \mathrm{s}, H_{\mathrm{A}}$ $=90.0 \mathrm{~m}$ of water, and bhp ${ }_{\mathrm{A}}=132 \mathrm{MW}$. The new turbine (turbine B) will spin at 120 rpm , and its net head will be $H_{\mathrm{B}}$ $=110 \mathrm{~m}$. Calculate the diameter of the new turbine such that it operates most efficiently, and calculate $\dot{V}_{\mathrm{B}}$ and bhp $\mathrm{p}_{\mathrm{B}}$.

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Problem 109

Calculate and compare the efficiency of the two turbines of Prob. 14-108. They should be the same since we are assuming dynamic similarity. However, the larger turbine will actually be slightly more efficient than the smaller turbine. Use the Moody efficiency correction equation to predict the actual expected efficiency of the new turbine. Discuss.

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Problem 110

Calculate and compare the turbine specific speed for both the small (A) and large (B) turbines of Prob. 14-108. What kind of turbine are these most likely to be?

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Problem 111

For each statement, choose whether the statement is true or false, and discuss your answer briefly.
(a) A gear pump is a type of positive-displacement pump.
(b) A rotary pump is a type of positive-displacement pump.
(c) The pump performance curve (net head versus capacity) of a positive-displacement pump is nearly vertical throughout its recommended operating range at a given rotational speed.
(d) At a given rotational speed, the net head of a positivedisplacement pump decreases with fluid viscosity.

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Problem 112

The common water meter found in most homes can be thought of as a type of turbine, since it extracts energy from the flowing water to rotate the shaft connected to the volume-counting mechanism (Fig. P14-112C). From the point of view of a piping system, however (Chap. 8), what kind of device is a water meter? Explain.

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Problem 113

Pump specific speed and turbine specific speed are "extra" parameters that are not necessary in the scaling laws for pumps and turbines. Explain, then, their purpose.

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Problem 114

What is a pump-turbine? Discuss an application where a pump-turbine is useful.

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Problem 115

For two dynamically similar pumps, manipulate the dimensionless pump parameters to show that $D_{\mathrm{B}}$ $=D_A\left(H_A / H_B\right)^{1 / 4}\left(\dot{V}_{\mathrm{B}} / V_A\right)^{1 / 2}$. Does the same relationship apply to two dynamically similar turbines?

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Problem 116

For two dynamically similar turbines, manipulate the dimensionless turbine parameters to show that $D_{\mathrm{B}}$ $=D_{\mathrm{A}}\left(H_{\mathrm{A}} / H_{\mathrm{B}}\right)^{3 / 4}\left(\rho_{\mathrm{A}} / \rho_{\mathrm{B}}\right)^{1 / 2}\left(\mathrm{bh} \mathrm{p}_{\mathrm{B}} / \mathrm{bh} \mathrm{p}_{\mathrm{A}}\right)^{1 / 2}$. Does the same relationship apply to two dynamically similar pumps?

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Problem 117

Develop a general-purpose computer application (using EES or other software) that employs the affinity laws to design a new pump (B) that is dynamically similar to a given pump (A). The inputs for pump A are diameter, net head, capacity, density, rotational speed, and pump efficiency. The inputs for pump B are density ( $\rho_{\mathrm{B}}$ may differ from $\rho_{\mathrm{A}}$ ), desired net head, and desired capacity. The outputs for pump B are diameter, rotational speed, and required shaft power. Test your program using the following inputs: $D_{\mathrm{A}}=5.0 \mathrm{~cm}, H_{\mathrm{A}}=120 \mathrm{~cm}, \dot{V}_{\mathrm{A}}$ $=400 \mathrm{~cm}^3 / \mathrm{s}, \rho_{\mathrm{A}}=998.0 \mathrm{~kg} / \mathrm{m}^3, \dot{n}_{\mathrm{A}}=1725 \mathrm{rpm}, \eta_{\text {pump. } \mathrm{A}}$ $=81$ percent, $\rho_{\mathrm{B}}=1226 \mathrm{~kg} / \mathrm{m}^3, H_{\mathrm{B}}=450 \mathrm{~cm}$, and $\dot{V}_{\mathrm{B}}$ $=2400 \mathrm{~cm}^3 / \mathrm{s}$. Verify your results manually. Answers: $D_{\mathrm{e}}$ $=8.80 \mathrm{~cm}, \dot{n}_{\mathrm{B}}=1898 \mathrm{rpm}$, and bhp $\mathrm{p}_{\mathrm{B}}=160 \mathrm{~W}$

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Problem 118

Experiments on an existing pump (A) yield the following BEP data: $D_{\mathrm{A}}=10.0 \mathrm{~cm}, H_{\mathrm{A}}$ $=210 \mathrm{~cm}, \dot{V}_{\mathrm{A}}=1350 \mathrm{~cm}^3 / \mathrm{s}, \rho_{\mathrm{A}}=998.0 \mathrm{~kg} / \mathrm{m}^3, \dot{n}_{\mathrm{A}}$ $=1500 \mathrm{rpm}, \eta_{\text {pump, } \mathrm{A}}=87$ percent. You are to design a new pump (B) that has the following requirements: $\rho_{\mathrm{B}}$ $=998.0 \mathrm{~kg} / \mathrm{m}^3, H_{\mathrm{B}}=570 \mathrm{~cm}$, and $\dot{V}_{\mathrm{B}}=3670 \mathrm{~cm}^3 / \mathrm{s}$. Apply the computer program you developed in Prob. 14-117 to calculate $D_{\mathrm{B}}(\mathrm{cm}), \dot{n}_{\mathrm{B}}(\mathrm{rpm})$, and bhp $(\mathrm{W})$. Also calculate the pump specific speed. What kind of pump is this (most likely)?

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Problem 119

Develop a general-purpose computer application (using EES or other software) that employs the affinity laws to design a new turbine (B) that is dynamically similar to a given turbine (A). The inputs for turbine A are diameter, net head, capacity, density, rotational speed, and brake horsepower. The inputs for turbine B are density ( $\rho_{\mathrm{B}}$ may differ from $\rho_{\mathrm{A}}$ ), available net head, and rotational speed. The outputs for turbine B are diameter, capacity, and brake horsepower. Test your program using the following inputs: $D_{\mathrm{A}}=1.40 \mathrm{~m}, H_{\mathrm{A}}=80.0 \mathrm{~m}, \dot{V}_{\mathrm{A}}=162 \mathrm{~m}^3 / \mathrm{s}, \rho_{\mathrm{A}}$ $=998.0 \mathrm{~kg} / \mathrm{m}^3, \dot{n}_{\mathrm{A}}=150 \mathrm{rpm}, \mathrm{bhp}_{\mathrm{A}}=118 \mathrm{MW}, \rho_{\mathrm{B}}$ $=998.0 \mathrm{~kg} / \mathrm{m}^3, H_{\mathrm{B}}=95.0 \mathrm{~m}$, and $\dot{n}_{\mathrm{B}}=120 \mathrm{rpm}$. Verify your results manually. Answers: $D_{\mathrm{B}}=1.91 \mathrm{~m}, \dot{V}_{\mathrm{B}}=328 \mathrm{~m}^3 / \mathrm{s}$, and $\mathrm{bhp}_{\mathrm{B}}=283 \mathrm{MW}$

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Problem 120

Experiments on an existing turbine (A) yield the following data: $D_{\mathrm{A}}=86.0 \mathrm{~cm}, H_{\mathrm{A}}$ $=22.0 \mathrm{~m}, \dot{V}_{\mathrm{A}}=69.5 \mathrm{~m}^3 / \mathrm{s}, \rho_{\mathrm{A}}=998.0 \mathrm{~kg} / \mathrm{m}^3, \dot{n}_{\mathrm{A}}=240 \mathrm{rpm}$, $\mathrm{bhp}_{\mathrm{A}}=11.4 \mathrm{MW}$. You are to design a new turbine (B) that has the following requirements: $\rho_{\mathrm{B}}=998.0 \mathrm{~kg} / \mathrm{m}^3, H_{\mathrm{B}}$ $=95.0 \mathrm{~m}$, and $\dot{n}_{\mathrm{B}}=210 \mathrm{rpm}$. Apply the computer program you developed in Prob. 14-119 to calculate $D_{\mathrm{B}}$ (m), $\dot{V}_{\mathrm{B}}$ $\left(\mathrm{m}^3 / \mathrm{s}\right.$ ), and $\mathrm{bhp}_{\mathrm{B}}$ (MW). Also calculate the turbine specific speed. What kind of turbine is this (most likely)?

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Problem 121

Calculate and compare the efficiency of the two turbines of Prob. 14-120. They should be the same since we are assuming dynamic similarity. However, the larger turbine will actually be slightly more efficient than the smaller turbine. Use the Moody efficiency correction equation to predict the actual expected efficiency of the new turbine. Discuss.

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