Fundamentals of Fluid Mechanics

Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Chapter 12

Turbomachines - all with Video Answers

Educators

Chapter Questions

Problem 1

The rotor shown in Fig. $P 12.1$ rotates clockwise. Assume that the fluid enters in the radial direction and the relative velocity is tangent to the blades and remains constant across the entire rotor. Is the device a pump or a turbine? Explain.

Prabhat Tyagi

Prabhat Tyagi

Numerade Educator

Problem 2

The measured shaft torque on the turbomachine shown in Fig. $P 12.2$ is $-60 \mathrm{N} \cdot \mathrm{m}$ when the absolute velocities are as indicated. Determine the mass flowrate. What is the angular velocity if the magnitude of the shaft power is $1803 \mathrm{N} \cdot \mathrm{m} / \mathrm{s} ?$ Is this machine a pump or a turbine? Explain.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 3

Uniform horizontal sheets of water of 3 -mm thickness issue from the slits on the rotating manifold shown in Fig. P12.3. The velocity relative to the arm is a constant $3 \mathrm{m} / \mathrm{s}$ along each slit. Determine the torque needed to hold the manifold stationary. What would the angular velocity of the manifold be if the resisting tcrque is negligible?

Victor Salazar

Numerade Educator

Problem 4

At a given radial location, a 15 -mph wind against a vindmill (see Video $V 12.1$ ) results in the upstream ( 1 ) and downsream
(2) velocity triangles shown in Fig. P12.4. Sketch an approsriate blade section at that radial location and determine the energy ransferred per unit mass of fluid.

Dominador Tan

Numerade Educator

Problem 5

Sketch how you would arrange four 3 -in.-wide by 12 -in. long thin but rigid strips of shect metal on a hub to create a xindmill like the one shown in Video $V 12.1$. Discuss, with the help of velocity triangles, how you would arrange each blade on the hub and how you would orient your windmill in the wind.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 6

Sketched in Fig. $P 12.6$ are the upstream [section (1) ) and downstream [section (2)] velocity triangles at the arithmetic mean tadius for flow through an axial-flow turbomachine rotor. The axial component of velocity is 50 ft/s at sections (1) and $(2) .$ (a) Label cach velocity vector appropriately. Use $\mathbf{V}$ for absolute velocity, $\mathbf{W}$ for relative velocity, and U for blade velocity, (b) Are you dealng with
¿ turbine or a fan?
(c) Calculate the work per unit mass involved.
(d) Sketch a reasonable blade section. Do you think that the actual blade exit angle will need to be less or greater than $15^{\circ} ?$ Why?

Dominador Tan

Numerade Educator

Problem 7

The radial component of velocity of water leaving the centrifugal pump sketched in Fig. $\mathrm{P} 12.7$ is $45 \mathrm{ft} / \mathrm{s}$. The magritude of the absolute velocity at the pump exit is $90 \mathrm{ft} / \mathrm{s}$, The fluid enters the pump rotor radially. Calculate the shaft work required per unit mass flowing through the pump.

Narayan Hari

Numerade Educator

Problem 8

Water enters a centrifugal pump with an absolute velocity $V_{1}=10 \mathrm{m} / \mathrm{s}$ in the radial direction and leaves with an absolue velocity $V_{2},$ which makes an angle of $\theta_{2}=60^{\circ}$ with the radial direction, as shown in Fig. P12.8. The impeller width (perpendicular to the paper) is $b=0.125 \mathrm{m}, R_{1}=0.125 \mathrm{m},$ and $R_{2}=0.35 \mathrm{m} .$ Find the input
torque $\mathscr{T}$ required to drive the pump if there are no friction losses.

Narayan Hari

Numerade Educator

Problem 9

A centrifugal pump impeller is rotating at 1200 rpm in the direction shown in Fig. P12.9. The flow enters parallel to the axis of rotation and leaves at an angle of $30^{\circ}$ to the radial direction. The absolute exit velocity, $V_{2},$ is $90 \mathrm{ft} / \mathrm{s}$, (a) Draw the velocity triangle for the impeller exit flow.
(b) Estimate the torque necessary to turn the impelier if the fluid is water. What will the impeller rotation speed become i the shaft breaks?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 10

A centrifugal radial water pump has the dimensions shown in Fig. $\mathrm{P} 12.10 .$ The volume rate of flow is $0.25 \mathrm{ft}^{3} / \mathrm{s},$ and the absolute inlet velocity is directed radially outward. The angular veloci-y of the impeller is 960 rpm. The exit velocity as seen from a coordinate system attached to the impeller can be assumed to be tangent to the vane at its trailing edge. The hydraulic efficiency is $82 \%$ and the mechanical efficiency is $96 \% .$ Calculats the power required to drive the pump.

Narayan Hari

Numerade Educator

Problem 11

Water is pumped with a centrifugal pump, and measurements made on the pump ndicate that for a flowrate of 240 gpm the required input power is 6 hp. For a pump cfficicncy of $62 \%,$ what is the actual head rise of the water being pumped?

Narayan Hari

Numerade Educator

Problem 12

The performance characteristics of a certain centrifugal pump are deternined from an experimental setup similar to that shown in Fig. $12.10 .$ When the flowrate of a liquid $(S G=0.9)$ through the pump is $120 \mathrm{gpm}$, the pressure gage at ( 1 ) indicates a vacuum of $95 \mathrm{mm}$ of mercury and the pressure gage at (2) indicates a pressure of $80 \mathrm{kPa}$. The diameter of the pipe at the inlet is $110 \mathrm{mm}$ and at the exit it is $55 \mathrm{mm}$. If $z_{2}-z_{1}=0.5 \mathrm{m},$ what is the actual head rise across the pump? Explain how you woud estimate the pump motor power requirement.

Narayan Hari

Numerade Educator

Problem 13

The performance characteristics of a certin centrifugal pump having a 9 -in.-diameter impeller and operatirg at $1750 \mathrm{rpm}$ are determined using an experimental setup similar to that shown in Fig. $12.10 .$ The following data were obtained during a series of tests in which $z_{2}-z_{1}=0, V_{2}=V_{1},$ and the fluid was ware Based on these data, show or plot how the actual head rise, $h_{c}$ and the pump efficiency, $\eta$. vary with the flowrate. What is the design flowrate for this pump?

Dominador Tan

Numerade Educator

Problem 14

Determine algebraic equations for the pump head rise. power, and efficiency as functicns of flow rate based on the data from Problem 12.18 . Analytically determine the Best Efficiency Point and the BEP flow, head rise, and power. How do these values compare with those from Problem $12.18 ?$

Chai Santi

Numerade Educator

Problem 15

In Example $12.3,$ how will the maximum height, $z_{1},$ that the pump can be located above the water surface change if the water temperature is decreased to $40^{\circ} \mathrm{F}$ ?

Narayan Hari

Numerade Educator

Problem 16

A centrifugal pump with a 7 -in.-diameter inpeller has the performance characteristics shown in Fig. $12.12 .$ The pump is used to pump water at $100^{\circ} \mathrm{F}$, and the pump inlet is located $12 \mathrm{ft}$ above the open water surface. When the flowrate is 200 gpm, the head loss between the water surface and the pump inlet is 6 ft of water. Would you expect cavitation in the pump to be a problem? Assume standard atmospheric pressure. Explain how you arrived at your answer.

Narayan Hari

Numerade Educator

Problem 17

Water at $40^{\circ} \mathrm{C}$ is pumped from an open tank through $200 \mathrm{m}$ of $5 \mathrm{C}$ -mm-diameter smooth horizontal pipe as shown in Fig. $P 12.17$ and discharges into the atmosphere with a velocity of $3 \mathrm{m} / \mathrm{s}$. Minor losses are neglaible. (a) If the efficiency of the pump is $70 \%$, how much power is being supplied to the pump?
(b) What is the NPSH at the pump inlet? Neglect losses in the short section of pipe connecting the pump to the tank. Assume standard atmospheric pressure.

Victor Salazar

Numerade Educator

Problem 18

The centrifugal pump shown in Fig. P12.18 is not self. priming. That is, if the water is crained from the punp and pipe as shown in Fig. $P 12.18(d),$ the pump will not draw the water into the pump and start pumping when the pump is turned on. However, if the pump is primed [ie., filled with water as in Fig. P12.18(b)]. the pump does start pumping water when turned on. Explain this behavior.

Victor Salazar

Numerade Educator

Problem 19

A centrifugal pump having a head-capacity relaticanship given by the equation $h_{a}=180-6.10 \times 10^{-4} Q^{2}$, with $h_{a}$ in feet when $Q$ is in $\mathrm{gpm},$ is to be used with a system similar to that shown in Fig. $12.14 .$ For $z_{2}-z_{1}=50 \mathrm{ft},$ what is the expected flowtate if the total length of constant diameter pipe is $600 \mathrm{ft}$ and the fluid is water? Assume the pipe diameter to be 4 in. and the friction actor to be equal to 0.02 . Neglect all minor losses.

Narayan Hari

Numerade Educator

Problem 20

A centrifugal pump having a 6 -in.- diameter impeller and the characteristics shown in Fig. 12.7 is to be used to pump gasoline through $4000 \mathrm{ft}$ of commercial steel 3 -in.-diameter pipe. The pipe connects two reservoirs having open surfaces at the same elevation. Determine the flowrate. Do you think this pump is a good choice? Explain.

Victor Salazar

Numerade Educator

Problem 21

A centrifugal pump having the characteristics shown in Example 12.4 is used to pump water between two large opentanks through 100 ft of 8 -in.-diameter pipe. The pipeline contains four regular flanged $90^{\circ}$ elbows, a check valve, and a fully open globe valve. Other minor losses are negligible. Assume the friction actor $f=0.02$ for the 100 -ft section of pipe. If the static head (difference in height of fluid surfaces in the two tanks) is $30 \mathrm{ft}$, what is the expected flowrate? Do you think this pump is a good choice? Explain.

Victor Salazar

Numerade Educator

Problem 22

Both the suction and discharge piping for the pump shown in Fig. $P 12.22$ consist of 4 -in. I.D. 40 -ft-long plastic pipe. Find the volume flow rate of $60^{\circ} \mathrm{F}$ water through the pump. The connections are glued (equivalent to soldered connections).

Narayan Hari

Numerade Educator

Problem 23

In a chemical processing plant a liquid is pumped from an open tank, through a 0.1 -m-diameter vertical pipe and into another open tark as shown in Fig. $P 12.23(a) .$ A valve is located in the pipe, and the minor loss coefficient for the valve as a function of the valve setting is shown in Fig. $\mathrm{P} 12.23(b)$ The pump head-capacity relationship is givan by the equation $h_{d}=52.0-1.01 \times 10^{3} Q^{2}$ with $h_{a}$ in meters when $Q$ is in $\mathrm{m}^{3} / \mathrm{s}$. Assume the friction factor $f=0.02$ for the pipe, and all minor losses, except for the valve, are negigible. Tre fluid levels in the two tanks can be assumed to remain constant.
(a) Determine the flowrate with the valve widc open.
(b) Determine the required valve setting (percent open) to reduce the flowrate by $50 \%$

Narayan Hari

Numerade Educator

Problem 24

Two of the pumps in Fig. $P 12.24$ are operated in series to supply water through the piping system. Determine the flow rate through the piping system for $10^{\circ} \mathrm{C}$ water and screwed connections. Then find the total power input to the two pumps.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 25

Water is pumped between the two tanks cescribed in Example 12.4 once a day, 365 diys a year, with each pumping period lasting two hours. The water levels in the two tanks remain essentially constant. Estmate the annual cost of the electrical power needed to operate the pump if it were located in your city. You will have to make a reasonable estimate for the efficiency of the motor used to drive the pump. Due to aging, it can be expecied that the overall resistance of the system will increase with time. If the operating point shown in Fig. $\mathrm{E} 12.4 \mathrm{c}$ changes to a point where the flowrate has been reduced to 1000 gpm, what wil be the new annua cost of operating the pump? Assume that the cast of electrical power remains the same.

Victor Salazar

Numerade Educator

Problem 26

A centrifugal pump having an impeller diameter of $1 \mathrm{m}$ is to be constructed so that it will supply a head rise of 200 mat a flowrate of $4.1 \mathrm{m}^{3} / \mathrm{s}$ of water when operating at a speed of $1200 \mathrm{rpm} .$ To study the characteristizs of this pump, a $1 / 5$ scale, geometrizally similar model operated at the same speed is to be tested in the laboratory. Determine the required model discharge and head rise. Assume that both model and prototype operate with the same effciency (and therefore the same flow coefficient)

Narayan Hari

Numerade Educator

Problem 27

Do the head-flowrate data shown in Fig. 12.7 appear to follow the similarity laws as expressed by Eqs. 12.39 and $12.40 ?$ Explain.

Benjamin Angeles

Benjamin Angeles

Numerade Educator

Problem 28

A centrifugal fan operating in a duct has the dimensionless parameters
$$C_{Q}=\frac{Q}{\omega D^{3}} \quad \text { and } \quad C_{H}=\frac{\Delta p}{\rho \omega^{2} D^{2}}$$
where $C_{Q}$ is a flow coefficient, $C_{N}$ is a head coefficient, $Q$ is the volume flow rate, $\omega$ is the fan speed, $D$ is the fan diameter, $\rho$ is the fluid density, ard $\Delta p$ is the fan pressure rise. Figure $\mathrm{Pl} 2.28$ shows this fan's performance curve in dimensional form for a fan speed of $\omega_{15}=1500 \mathrm{rpm} .$ Find the fan operating points $(Q \text { and } \Delta p)$ for $\omega_{30}=3000 \mathrm{rpm}$ and corresponding to points $1,3,$ and 5 at $\omega_{15}=$ $1500 \mathrm{rpm} .$ Assume similarity between $1500 \mathrm{rpm}$ and $3000 \mathrm{rpm}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

A centritugal pump has the performance charcteristics of the pump with the 6 -in.-diameter impeller described n Fig. 12.7 Note that the pump in this figure is operating at 3500 rpm. What is the expected head gained if the speed of this pump is reduced to $2800 \mathrm{rpm}$ while operating at peak efficiency?

Narayan Hari

Numerade Educator

Problem 30

A prototype fan has a $20-f$ diameter, an inlet pressure of 14.40 psia, an inlet temperature of $70^{\circ} \mathrm{F},$ and a speed of 90 rpm. $\mathrm{A}$ $\frac{1}{10}-\operatorname{scal} \mathrm{e}$ model of the fan has the same inlet pressure and emperature. an inlet power of $1.24 \mathrm{hp}$, a flow rate of $220 \mathrm{ft}^{3} / \mathrm{min}$, and a speed of $1800 \mathrm{rpm} .$ Find the corresponding input power and flow rate of the prototype fan. Neglect Reynolds number effects.

Narayan Hari

Numerade Educator

Problem 32

In a certain application, a pump is required to deliver 5000 gpm against a 300 -ft head when operating at 1200 rpm. What type of pump would you recommend?

Narayan Hari

Numerade Educator

Problem 33

A centrifugal pump operates at 300 rpm to deliver $20^{\circ} \mathrm{C}$ lubricating oil. A $\frac{1}{s}$ -size, geometrically similar pump deliverng $15^{\circ} \mathrm{C}$ water is used to model the larger pump. How fast should the smaller pump run? Discuss the accuracy of the result.

Narayan Hari

Numerade Educator

Problem 34

A certain axial-flow pump has a specific speed of $N_{x}=5.0$ If the pump is expected to deliver 3000 gpm when operating against a 15 -ft head, at what speed (rpm) should the pump be run? Draw a sketch of the p.ump impeller (front and side views).

Narayan Hari

Numerade Educator

Problem 35

A certain pump is known to have a capacity of $3 \mathrm{m}^{3} / \mathrm{s}$ when operating at a speed of 60 rad/s against a head of $20 \mathrm{m}$. Based on the information in Fig. 12.18 , would you recommend a radialflow, mixed-flow, or axial-flow pump? Draw a sketch of the pump impeller (front and side views).

Narayan Hari

Numerade Educator

Problem 36

The system resistance for a pipeline is given by $\Delta p_{\mathrm{sys}}=2.0 Q^{2}$ where $\Delta p_{\text {ys }}$ is the pressure rise required of a pump to deliver the flow rate $Q$ through the piping system. A pump has the pressure-rise-flow characteristic given by $\Delta p_{p}=30.0-3.0 Q^{2} .$ In both curves, $\Delta p$ is in $\mathrm{kPa}$ and $Q$ is in $\mathrm{m}^{3} / \mathrm{s}$, Find the pump input power if this punp is placed in this piping system and the pump overall eff ciency is $90 \%$

Narayan Hari

Numerade Educator

Problem 37

The axial-flow pump shown in Fig. 12.19 is designad to move 5000 gal/min of water over a head rise of 5 ft of water. Estimate the motor power requirement and the $U, V_{p 2}$ needad to achieve this flowrate on a continuous basis. Comment on any cautions associated with where the pump is placed verticilly in the pipe.

Narayan Hari

Numerade Educator

Problem 38

A propeller-driven airplane is traveling with a velocity $V_{1}=200 \mathrm{moh}(294 \mathrm{ft} / \mathrm{s}) .$ The propeller diameter is $10 \mathrm{ft}$ and it
rotates at 3000 rpm. Figure $\mathrm{P} 12.38$ shows a popeller crosssectional profile and the velocity diagrams for a stort section of the propeller at a radius of $3.0 \mathrm{ft}$. The outlet relative velocity $W_{2}$ is assumed tangent to the propeller at the outlet, so $\beta_{1}^{\prime}=\beta_{2}=50^{\circ}$ The air density is constant as it flows over the propeller. Assume that the flow area for the mass flow rate interacting with this short section of the propeller is the same upstreem and downstream (inlet and outlet) from the propeller (ie.. $A_{1}=A_{2}$ ). Produce the valocity diagram dowastream from the propeller by finding $U, W_{2}, V_{2},$ and $a_{2}$

Victor Salazar

Numerade Educator

Problem 39

For the fan of both Examples 5.19 and 5.28 discuss what fluid flow properties you would need to measure to estimate fan efficency.

Dominador Tan

Numerade Educator

Problem 40

A lossless motor drives the fan shown in Fig. $P 12,40$ at
$40 \mathrm{Hz}$. The power input to the motor is 40 amps at 440 volts. For the geometry shown, what is the discharge flow rate of air through the fan? Assume that the tangential component of the velocity leaving the impeller is equal to that of the impeller at that point. The exit air temperature $T_{2}=15^{\circ} \mathrm{C}$

Narayan Hari

Numerade Educator

Problem 41

A centrifugal fan has a power input of $25 \mathrm{kW}$, an inner radius of $R_{1}=0.5 \mathrm{m},$ an outer radius of $R_{2}=1.0 \mathrm{m},$ and delivers $100 \mathrm{kg} / \mathrm{s}$ of air. There are no friction losses, the air inlet absolute velocity has no tangential component, and the outlet absolute velocity has a tangential component equal to the blade velocity at the outer radius $R_{2}$. The rotor depth (i.e., blade height) is $1.0 \mathrm{m}$ What is the required rotational speed of the rotor?

Narayan Hari

Numerade Educator

Problem 42

An axial fan operating at 1000 rp. has the characteristics shown in Fig. $\mathrm{P} 12.42 .$ It delivers $15^{\circ} \mathrm{C}$ atmospheric air throagh a $50-\mathrm{cm}$ I.D., galvanized, sheet-metal, horizontal duct having a length of $175 \mathrm{m}$ and seven $90^{\circ}$ long-radius elbows. For constant air density what is the flow rate if the duct discharges to the atmosphere $^{\text {th }}$

Narayan Hari

Numerade Educator

Problem 43

A model fan with wheel diameter 32 in. is tested at a speed of $1750 \mathrm{rpm}$. The test fluid is air with density $0.075 \mathrm{lbm} / \mathrm{ft}$. At its $\mathrm{BEP}$, the fan produces $8000 \mathrm{ft}^{3} / \mathrm{min}$ at total pressure rise of 8 in. $\mathrm{H}_{2} \mathrm{O}$. A geometrically similar fan is to handle $200,000 \mathrm{ft}^{3} / \mathrm{min}$ of flue gas with density $0.050 \mathrm{lbm} / \mathrm{ft}^{3}$ and 30 in. $\mathrm{H}_{2} \mathrm{O}$ total pressure rise. Determine the required size and speed of the flue gas fan. Note any assumptions and/or limitations.

Narayan Hari

Numerade Educator

Problem 44

A fan is to produce a total pressure rise of 6 in. $\mathrm{H}_{2} \mathrm{O}$ and a flow of $4000 \mathrm{ft}^{3} / \mathrm{min}$. Two mators are available, $3550 \mathrm{rpn}$ and $1160 \mathrm{rpm} .$ For each motor, specify the best (most efficient) type of fan to use and sketch the impeller.

Narayan Hari

Numerade Educator

Problem 45

An inward-flow radial turbine (see Fig. $P 12.48$ ) involves a nozzle angle, $a_{1},$ of $60^{\circ}$ and an inlet rotor tip speed, $U_{1},$ cf $3 \mathrm{m} / \mathrm{s}$. The ratio of rotor inlet to outlet diameters is $2.0 .$ The absolute velocity leaving the rotor at section (2) is radial with a magnitude of $6 \mathrm{m} / \mathrm{s}$, Determine the energy transfer per unit mass of fluid flowing through this turbine if the fluid is
(a) air.
(b) water.

Narayan Hari

Numerade Educator

Problem 46

The frictionless converging stationary nozzle of the hydraulic turbine shown in Fig. $\mathrm{P} 12.46$ has an inlet pressure $p_{0}=480 \mathrm{kPa}$
negligible inlet velocity $V_{0 .}$ and an exit pressure $p_{1}=101 \mathrm{kPa}$ The velocity $V_{1}$ is used to drive the axial flow turbine, wich has
a rotational speed of $185.7 \mathrm{rpm}$, an outside radius of $R=1.20 \mathrm{m}$ and a blade height of $h=0.40 \mathrm{m} .$ Determine velocities $V_{1}$ and $V_{2}$ and the power transmitted to the turbine in terms of $a_{1}$ and $\alpha_{2}$. The fluid density is constant and the velocities of the fluid relative to the blade are directed as shown in Fig. $\mathrm{Pl} 2.46$. Assume that the velocities are uniform over the flow inlet and outlet areas and use an average blade velocity at the blade midheight $(r=1.0 \mathrm{cm})$ and
a water temperature of $20^{\circ} \mathrm{C}$

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 47

A water turbine wheel rotates at the rate of 100 rpm in the diection shown in Fig. $P 12.47$. The inner radius, $r_{2},$ of the blade row is $1 \mathrm{ft}$, and the outer radius, $r_{1}$, is $2 \mathrm{ft}$. The absolute velocity vector at the turbine rotor entrance makes an angle of $20^{\circ}$ with the tengential direction. The inlet blade engle is $60^{\circ}$ relative to the tangential direction. The blade outlet angle is $120^{\circ} .$ The flowrate is $10 \mathrm{ft}^{3} / \mathrm{s}$. For the flow tangent to the rosor blade surface at inlet and outlet, determine an appropriate constant blade height, $b,$ and the corresponding power available at the rosor shaft. Is the shaft power greater or less than the power lost by the fluid? Explain.

Victor Salazar

Numerade Educator

Problem 48

A sketch of the arithmetic mean radius blade sections of an axial-flow water turbine stage is shown in Fig. $\mathrm{P} 12.48$. The rotor speed is $1500 \mathrm{rpm}$. (a) Sketch and label velocity triangles for the flow entering and leaving the rotor row. Use $V$ for absolute velocity, $\mathbf{W}$ for relative velocity, and $\mathbf{U}$ for blade velocity. Assume flow enters and leaves each blade row at the blade angles shown.
(b) Calculate the work per unit mass delivered at the shaft.

Narayan Hari

Numerade Educator

Problem 49

Figure $P 12.49$ shows a piping system with frictional losses of $h_{\mathrm{L} 1-2}=4.0 Q^{2},$ with $h_{\mathrm{L} .1-2}$ in $\mathrm{ft}$ and $Q$ in $\mathrm{gal} / \mathrm{min}$. The turbine performance characteristics are given by $h_{r}=20+12.0 Q$ where $h_{t}$ is the turbine tead in $f t$ and $Q$ is in gal/min, Find the flow rate $Q$

Narayan Hari

Numerade Educator

Problem 50

A small Pclton wheel is used to power an oscillating lawn sprinkler as shown in Video $V 12.4$ and Fig. $P 12.50 .$ The arithmetic mean radius of the turbine is 1 in. and the exit angle of the blade is $135^{\circ}$ relative to the blade motion. Water is surplied through a single 0.20 -in.- -diameter nozzle at a speed of $50 \mathrm{ft} / \mathrm{s}$ Determine the flowrate, the maximum torque developed, and the maximum power developed by this turbine.

Narayan Hari

Numerade Educator

Problem 51

A windmill has an approach velocity $V_{1}=24 \mathrm{mph}(35.3 \mathrm{fl} / \mathrm{s})$ and a blade diameter of $100 \mathrm{ft}$. The blade rotates at 15 rpm. Figure $P 12.51$ shows a blade cross-sectional profile and the velocity diagrams for a short section of the blade at a radius of $r=25.0 \mathrm{ft}$ The outlet relative velocity $W_{2}$ is tangent to the blade at the outlet so $\beta_{2}=\beta_{2}^{\prime}=30^{\circ} .$ The air density is constant as it flows over the blade. Assume that the flow area for the mass flow rate that interacts with this short section of the blade is the same upstream and downstream of the blade (i.e., $A_{1}=A_{2}$ ). Find the velocity diagram downstream of the blade by finding $U, W_{2}, V_{2},$ and $x_{2}$

Victor Salazar

Numerade Educator

Problem 52

The single-stage, axial-flow turbomachine shown ir Fig. P12.52 involves water flow at a volumetric flowrate of $9 \mathrm{m}^{3} / \mathrm{s}$ The rotor revolves at 600 rpm. The inner and outer radii of the annular flow path through the stage are 0.46 and $061 \mathrm{m},$ and $\beta_{2}=60^{\circ} .$ The flow entering the rotor row and leaving the stator row is axial when vieved from he stationary casing. Is this device a turbine or a pump? Est mate the amount of power transferred to or from the fluid.

Narayan Hari

Numerade Educator

Problem 53

For an air turbine of a dentist's drill like the one shown in Fig. E12.8 and Video $V 12.5$, calculate the average blade speed associated with a rotational speed of 350,000 rpm. Estimate the air pressure needed to run this turbine.

Narayan Hari

Numerade Educator

Problem 54

A Pelton wheel has a diameter of $2 \mathrm{m}$ and develops $500 \mathrm{kW}$ when rotating 180 rpm. What is the average force of the water against the blades? If the turbiac is operating at maximum efficiency, determine the speed of the water jet from the nozzle and the mass flowrate.

Narayan Hari

Numerade Educator

Problem 55

A Pelion wheel has a diameter of $2 \mathrm{m}$ and develops $500 \mathrm{kW}$ when rotating 180 rpm. What is the average force of the water against the blades? If the turbiale is operating at maximum efficiency, determine the speed of the water jet from the nozzle and the mass flowrate.

Narayan Hari

Numerade Educator

Problem 56

Water to run a Pelton wheel is supplied by a penstcck of length $\ell$ and diameter $D$ with a friction factor $f$. If the only losses associated with the flow in the penstock are due to pipe friction. show that the maximum power output of the turbine occurs when the nozzle diameter, $D_{1},$ is given by $D_{1}=D /(2 f \ell / D)^{1 / 4}$

Prabhat Tyagi

Numerade Educator

Problem 57

A Pelton wheel is supplied with water from a lake at an elevation $H$ above the turbine. The penstock that supplies the water to the wheel is of length $\ell$, diameter $D$, ard friction factor $f$. Minor losses are negligible. Show that the power developed by the turbine is maximum when the velocity head at the nozzle exit is $2 \mathrm{H} / 3$ Noie: The result of Problem 12.56 may be of use.

Dominador Tan

Numerade Educator

Problem 58

Water flows through the Pelton wheel turbine shown in Fig. $12.25 .$ For simplicity we assume that the water is turned $180^{\circ}$ by the blade. Show, based on the energy equation (Eq. 5.84 , that the maximum power output occurs when the absolute velocity of the fluid exiting the turbine is zero.

Vidhi Bhatt

Numerade Educator

Problem 59

A 1 -m-diameter Pelton wheel rotates at 300 rpm Which of the following heads (in meters) would be best suited for this
(c) 40 turbine: (a) 2, (b) 5.
(d) $70,$ or
(e) $140 ?$ Explain.

Narayan Hari

Numerade Educator

Problem 60

Draft tubes as shown in Fig. $\mathrm{P} 12.60$ are often installed at the exit of Kaplan and Francis turbines. Explain why such draft tubes are advantageous.

Narayan Hari

Numerade Educator

Problem 61

Turbines are to be designed to develop 30,000 horsepower while operating under a head of $70 \mathrm{ft}$ and an angular vebcity of $60 \mathrm{rpm} .$ What type of turbine is best suited for this rurpose? Estimate the flowrate needed

Anand Jangid

Numerade Educator

Problem 62

Water at 400 psi is available to operate a turbine at 1750 rpm. What type of turbine would you suggest to use if the turbine should have an output of approximately 200 hp?

Narayan Hari

Numerade Educator

Problem 63

It is desired to produce 50,000 hp with a head of $50 \mathrm{ft}$ and an angular velocity of $100 \mathrm{rpm} .$ How many turbines would be needed if the specific speed is to be
(a) 50
(b) $100 ?$

Narayan Hari

Numerade Educator

Problem 64

Test data for the small Francis turbine shown in Fig $\mathrm{P} 12.64$ is given in the following table. The test was run at a constant $32.8-\mathrm{ft}$ head just upstream of the turbine. The Prony brake or the turbine output shaft was adjusted to give various angular ve ocities, and the force on the brake arm, $F,$ was recorded. Use the given data to plot curves of torque as a function of angular velocity and turbine efficiency as a function of angular velocity.

Narayan Hari

Numerade Educator

Problem 65

Obtain photographs/images of a variety of turbo compressor rotors and categorize them as axial-flow or radial-flew compressors, Explain bricfly how they are used. Note any unusual features. Repeat for compressible flow turbines.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 66

An axial flow compressor stage shown in Fig. $P / 2.66$ has the inlet and outlet velocity diagrams shown. Calculate the vork per unit mass. Quantities are $U_{1}=U_{2}=U=762 \mathrm{ft} / \mathrm{s}, V_{1}=440 \mathrm{ft} / \mathrm{s}, W_{1}=$
880 ft's, $\alpha_{1}=90^{\circ}, V_{2}=545 \mathrm{ft} / \mathrm{s}, W_{2}=622 \mathrm{ft} / \mathrm{s},$ and $\alpha_{2}=53.8^{\circ}$

Dading Chen

Numerade Educator

Problem 67

The axial flow gas turbine stage shown in Fig. $P 12.67$ has a mean jade radius of $R=5.0$ in. a rotational speed of $15,000 \mathrm{rpm}$ a mass flow rate of $10.0 \mathrm{lbm} / \mathrm{s}, W_{1}=972 \mathrm{ft} / \mathrm{s}, V_{1}=1550 \mathrm{ft} / \mathrm{s}, \alpha_{1}=$
$20^{\circ}, W_{2}=874$ ifs, $V_{2}=483 \mathrm{ft} / \mathrm{s},$ and $\alpha_{2}=99.4^{\circ} .$ Fird the power
transmitted from the gas to the turbine blade

Dading Chen

Numerade Educator

Problem 68

A centrifugal ar compressor has a rotor inrer diameter of $D_{1}=2.0$ in.., a rotor oater diameter of $D_{2}=6.5$ in, a rotor depth of 10 in. and a rotor rotational speed of 3600 rpm. The fluid relative velocities $(W)$ are purely radial. The compressor delivers an air mass flow rate of $1.0 \mathrm{lom} / \mathrm{s}$ with $T_{1}=70^{\circ} \mathrm{F}, p_{1}=4.7 \mathrm{psia}, T_{2}=$
$240^{\circ} \mathrm{F},$ and $p_{2}=33.1$ psia. Find the power transferied by the rotor to the air and the inlet relative velocity $W$

Narayan Hari

Numerade Educator

Problem 69

The axial flow steam turbine rotcr shown in Fig. $\mathrm{P} 12.69$ has a blade outer radius $R_{0}=2.40 \mathrm{ft},$ a blade innar radius $\mathrm{R}_{i}=$ $2.00 \mathrm{ft},$ a steam inlet pressure $p_{1}=200$ psia, a stean inlet density $\rho_{1}=0.296 \mathrm{lbm} / \mathrm{ft}^{3},$ and an inlet absolute velocity $V_{1}=1000 \mathrm{ft} / \mathrm{s}$
making an angle of $70^{\circ}$ with the axial direction. The steam outlet pressure $p_{2}=50$ psia and outlet density is $0.1014 \mathrm{lbm} / \mathrm{ft}^{3} . \beta_{2}=$ $40^{\circ} .$ The rotor rotates at 3600 rom. Using a blade velocity at the blade midheight $(R=2.20 \mathrm{ft}) .$ find the power transerred from the steam to the rotor.

Victor Salazar

Numerade Educator

Problem 70

The figure below shows a nozzle vane and a rotor blade for an axial flow gas rurbine stage. The blade speed is $800 \mathrm{ftls}$ The absolut verity leaving the stage is identical to the absolute velocity entering the stage and both are purely in the axial direction.
(a) Draw and label the velocity diagrams. Show values for all three velocities and the absolute and relative angles on each diagram.
(b) Calculate the work per unit mass for this stage.

Victor Salazar

Numerade Educator