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Fluid Mechanics

Frank M. White

Chapter 3

Integral Relations for a Control Volume - all with Video Answers

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Chapter Questions

01:25

Problem 1

Discuss Newton's second law (the linear momentum relation) in these three forms:
$$\begin{array}{cc}
\sum \mathbf{F}=m \mathbf{a} & \sum \mathbf{F}=\frac{d}{d t}(m \mathbf{V}) \\
\sum \mathbf{F}=\frac{d}{d t}\left(\int_{\mathrm{system}} \mathbf{V} \rho d^{\gamma}\right)
\end{array}$$
Are they all equally valid? Are they equivalent? Are some forms better for fluid mechanics as opposed to solid mechanics?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:41

Problem 2

Consider the angular momentum relation in the form
$$\sum \mathbf{M}_{o}=\frac{d}{d t}\left[\int_{\mathrm{syscm}}(\mathbf{r} \times \mathbf{V}) \rho d^{v}\right]$$
What does $\mathbf{r}$ mean in this relation? Is this relation valid in both solid and fluid mechanics? Is it related to the linear momentum equation (Prob. 3.1)? In what manner?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:40

Problem 3

For steady low-Reynolds-number (laminar) flow through a long tube (see Prob. 1.12 ), the axial velocity distribution is given by $u=C\left(R^{2}-r^{2}\right),$ where $R$ is the tube radius and $r \leq R .$ Integrate $u(r)$ to find the total volume flow $Q$ through the tube.

Amany Waheeb
Amany Waheeb
Numerade Educator
01:45

Problem 4

Water at $20^{\circ} \mathrm{C}$ flows through a long elliptical duct $30 \mathrm{cm}$ wide and $22 \mathrm{cm}$ high. What average velocity, in $\mathrm{m} / \mathrm{s}$, would cause the weight flow to be 500 lbf/s?

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
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Problem 5

Water at $20^{\circ} \mathrm{C}$ flows through a 5 -in-diameter smooth pipe at a high Reynolds number, for which the velocity profile is approximated by $u \approx U_{\mathrm{o}}(y / R)^{1 / 8},$ where $U_{\mathrm{o}}$ is the centerline velocity, $R$ is the pipe radius, and $y$ is the distance measured from the wall toward the centerline. If the centerline velocity is $25 \mathrm{ft} / \mathrm{s}$, estimate the volume flow rate in gallons per minute.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 6

Water fills a cylindrical tank to depth $h$. The tank has diameter $D$. The water flows out at average velocity $V_{\mathrm{o}}$ from a hole in the bottom of area $A_{\mathrm{o}} .$ Use the Reynolds transport theorem to find an expression for the instantaneous depth change $d h / d t$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:12

Problem 7

A spherical tank, of diameter $35 \mathrm{cm},$ is leaking air through
a 5 -mm-diameter hole in its side. The air exits the hole at $360 \mathrm{m} / \mathrm{s}$ and a density of $2.5 \mathrm{kg} / \mathrm{m}^{3} .$ Assuming uniform mixing, $(a)$ find a formula for the rate of change of average density in the tank and ( $b$ ) calculate a numerical value for $(d \rho / d t)$ in the tank for the given data.

Penny Riley
Penny Riley
Numerade Educator
04:58

Problem 8

Three pipes steadily deliver water at $20^{\circ} \mathrm{C}$ to a large exit pipe in Fig. $\mathrm{P} 3.8 .$ The velocity $V_{2}=5 \mathrm{m} / \mathrm{s},$ and the exit flow rate $Q_{4}=120 \mathrm{m}^{3} / \mathrm{h} .$ Find $(a) V_{1},(b) V_{3},$ and $(c) V_{4}$ if it is known that increasing $Q_{3}$ by 20 percent would increase $Q_{4}$ by 10 percent.

RZ
Rubeena Zulfiqar
Numerade Educator
02:43

Problem 9

A laboratory test tank contains seawater of salinity $S$ and density $\rho .$ Water enters the tank at conditions $\left(S_{1}, \rho_{1}, A_{1},\right.$ $V_{1}$ ) and is assumed to mix immediately in the tank. Tank water leaves through an outlet $A_{2}$ at velocity $V_{2} .$ If salt is a "conservative" property (neither created nor destroyed), use the Reynolds transport theorem to find an expression for the rate of change of salt mass $M_{\text {salt }}$ within the tank.

Supratim Pal
Supratim Pal
Numerade Educator
03:39

Problem 10

Water flowing through an 8 -cm-diameter pipe enters a porous section, as in Fig. P3.10, which allows a uniform radial velocity $v_{w}$ through the wall surfaces for a distance of $1.2 \mathrm{m}$. If the entrance average velocity $V_{1}$ is $12 \mathrm{m} / \mathrm{s}$, find the exit velocity $V_{2}$ if $(a) v_{w}=15 \mathrm{cm} / \mathrm{s}$ out of the pipe walls or $(b) v_{w}=10 \mathrm{cm} / \mathrm{s}$
into the pipe.
(c) What value of $v_{w}$ will make $V_{2}=9 \mathrm{m} / \mathrm{s} ?$

Amany Waheeb
Amany Waheeb
Numerade Educator
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Problem 11

Water flows from a faucet into a sink at 3 U.S. gallons per minute. The stopper is closed, and the sink has two rectangular overflow drains, each $^{3} / \mathrm{g}$ in by $1^{1 / 4}$ in. If the $\operatorname{sink}$ water level remains constant, estimate the average overflow velocity, in ft/s.

Victor Salazar
Victor Salazar
Numerade Educator
01:11

Problem 12

The pipe flow in Fig. $\mathrm{P} 3.12$ fills a cylindrical surge tank as shown. At time $t=0,$ the water depth in the tank is $30 \mathrm{cm}$ Estimate the time required to fill the remainder of the tank.

Anand Jangid
Anand Jangid
Numerade Educator
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Problem 13

The cylindrical container in Fig. $\mathrm{P} 3.13$ is $20 \mathrm{cm}$ in diameter and has a conical contraction at the bottom with an exit
hole $3 \mathrm{cm}$ in diameter. The tank contains fresh water at standard sea-level conditions. If the water surface is falling at the nearly steady rate $d h / d t \approx-0.072 \mathrm{m} / \mathrm{s}$, estimate the average velocity $V$ out of the bottom exit.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 14

The open tank in Fig. $P 3.14$ contains water at $20^{\circ} \mathrm{C}$ and is being filled through section 1. Assume incompressible flow. First derive an analytic expression for the water-level change $d h / d t$ in terms of arbitrary volume flows $\left(Q_{1}, Q_{2}\right.$, $Q_{3}$ and tank diameter $d$. Then, if the water level $h$ is constant, determine the exit velocity $V_{2}$ for the given data $V_{1}=$ $3 \mathrm{m} / \mathrm{s}$ and $Q_{3}=0.01 \mathrm{m}^{3} / \mathrm{s}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
05:56

Problem 15

Water, assumed incompressible, flows steadily through the round pipe in Fig. $P 3.15 .$ The entrance velocity is constant, $u=U_{0},$ and the exit velocity approximates turbulent flow, $u=u_{\max }(1-r / R)^{1 / 7} .$ Determine the ratio $U_{0} / u_{\max }$ for this flow.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:15

Problem 16

An incompressible fluid flows past an impermeable flat plate, as in Fig. $\mathrm{P} 3.16,$ with a uniform inlet profile $u=U_{0}$ and a cubic polynomial exit profile
\[u \approx U_{0}\left(\frac{3 \eta-\eta^{3}}{2}\right) \text { where } \eta=\frac{y}{\delta}\]
Compute the volume flow $Q$ across the top surface of the control volume.

Amany Waheeb
Amany Waheeb
Numerade Educator
06:12

Problem 17

Incompressible steady flow in the inlet between parallel plates in Fig. $\mathrm{P} 3.17$ is uniform, $u=U_{0}=8 \mathrm{cm} / \mathrm{s},$ while downstream the flow develops into the parabolic laminar profile $u=a z\left(z_{0}-z\right),$ where $a$ is a constant. If $z_{0}=4 \mathrm{cm}$ and the fluid is SAE 30 oil at $20^{\circ} \mathrm{C}$, what is the value of $u_{\max }$ in $\mathrm{cm} / \mathrm{s}$ ?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:05

Problem 18

Gasoline enters section 1 in Fig. $\mathrm{P} 3.18$ at $0.5 \mathrm{m}^{3} / \mathrm{s}$. It leaves section 2 at an average velocity of $12 \mathrm{m} / \mathrm{s}$. What is the average velocity at section $3 ?$ Is it in or out?

Hubert Agamasu
Hubert Agamasu
Numerade Educator
01:23

Problem 19

Water from a storm drain flows over an outfall onto a
porous bed that absorbs the water at a uniform vertical velocity of $8 \mathrm{mm} / \mathrm{s},$ as shown in Fig. $\mathrm{P} 3.19 .$ The system is $5 \mathrm{m}$ deep into the paper. Find the length $L$ of the bed that will completely absorb the storm water.

Naman Kumar
Naman Kumar
Numerade Educator
04:27

Problem 20

Oil (SG $=0.89$ ) enters at section 1 in Fig. P3.20 at a weight flow of $250 \mathrm{N} / \mathrm{h}$ to lubricate a thrust bearing. The steady oil flow exits radially through the narrow clearance between thrust plates. Compute $(a)$ the outlet volume flow in $\mathrm{mL} / \mathrm{s}$ and $(b)$ the average outlet velocity in $\mathrm{cm} / \mathrm{s}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
01:30

Problem 21

For the two-port tank of Fig. E3.5, assume $D_{1}=4 \mathrm{cm},$ $V_{1}=18 \mathrm{m} / \mathrm{s}, D_{2}=7 \mathrm{cm},$ and $V_{2}=8 \mathrm{m} / \mathrm{s}$. If the tank surface is rising at $17 \mathrm{mm} / \mathrm{s}$, estimate the tank diameter.

Kelly Hughes
Kelly Hughes
Numerade Educator
04:26

Problem 22

The converging-diverging nozzle shown in Fig. $P 3.22$ expands and accelerates dry air to supersonic speeds at the exit, where $p_{2}=8 \mathrm{kPa}$ and $T_{2}=240 \mathrm{K} .$ At the throat, $p_{1}=$ $284 \mathrm{kPa}, T_{1}=665 \mathrm{K},$ and $V_{1}=517 \mathrm{m} / \mathrm{s} .$ For steady
compressible flow of an ideal gas, estimate ( $a$ ) the mass flow in $\mathrm{kg} / \mathrm{h}$
(b) the velocity $V_{2},$ and
$(c)$ the Mach number $\mathrm{Ma}_{2}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:23

Problem 23

The hypodermic needle in Fig. P3.23 contains a liquid serum $(\mathrm{SG}=1.05) .$ If the serum is to be injected steadily at $6 \mathrm{cm}^{3} / \mathrm{s}$, how fast in in/s should the plunger be advanced
(a) if leakage in the plunger clearance is neglected and
(b) if leakage is 10 percent of the needle flow?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:34

Problem 24

Water enters the bottom of the cone in Fig. $\mathrm{P} 3.24$ at a uniformly increasing average velocity $V=K t .$ If $d$ is very small, derive an analytic formula for the water surface rise $h(t)$ for the condition $h=0$ at $t=0 .$ Assume incompressible flow.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:04

Problem 25

As will be discussed in Chaps. 7 and 8 , the flow of a stream $U_{0}$ past a blunt flat plate creates a broad low-velocity wake behind the plate. A simple model is given in Fig. P3.25, with only half of the flow shown due to symmetry. The velocity profile behind the plate is idealized as "dead air" (near-zero velocity) behind the plate, plus a higher velocity, decaying vertically above the wake according to the variation $u \approx$ $U_{0}+\Delta U e 2^{I / L},$ where $L$ is the plate height and $z=0$ is the top of the wake. Find $\Delta U$ as a function of stream speed $U_{0}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:22

Problem 26

A thin layer of liquid, draining from an inclined plane, as in Fig. P3.26, will have a laminar velocity profile $u \approx$ $U_{0}\left(2 y / h-y^{2} / h^{2}\right),$ where $U_{0}$ is the surface velocity. If the plane has width $b$ into the paper, determine the volume rate of flow in the film. Suppose that $h=0.5$ in and the flow rate per foot of channel width is 1.25 gal/min. Estimate $U_{0}$ in ft/s.

Amany Waheeb
Amany Waheeb
Numerade Educator
03:34

Problem 27

Consider a highly pressurized air tank at conditions $\left(p_{0}, \rho_{0}\right.$
$T_{0}$ ) and volume $v_{0} .$ In Chap. 9 we will learn that, if the $\tan \mathrm{k}$ is allowed to exhaust to the atmosphere through a well-designed converging nozzle of exit area $A,$ the outgoing mass flow rate will be
$$\dot{m}=\frac{\alpha p_{0} A}{\sqrt{R T_{0}}} \text { where } \alpha \approx 0.685 \text { for air }$$
This rate persists as long as $p_{0}$ is at least twice as large as the atmospheric pressure. Assuming constant $T_{0}$ and an ideal gas, $(a)$ derive a formula for the change of density $\rho_{0}(t)$ within the tank.
(b) Analyze the time $\Delta t$ required for the density to decrease by 25 percent.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:03

Problem 28

Air, assumed to be a perfect gas from Table $\mathrm{A} .4,$ flows through a long, 2 -cm-diameter insulated tube. At section 1, the pressure is $1.1 \mathrm{MPa}$ and the temperature is $345 \mathrm{K}$. At section 2,67 meters further downstream, the density is $1.34 \mathrm{kg} / \mathrm{m}^{3},$ the temperature $298 \mathrm{K},$ and the Mach number is $0.90 .$ For one-dimensional flow, calculate $(a)$ the mass flow;
$(b) p_{2} ;(c) V_{2} ;$ and $(d)$ the change in entropy between 1 and 2. (e) How do you explain the entropy change?

Chai Santi
Chai Santi
Numerade Educator
03:34

Problem 29

In elementary compressible flow theory (Chap. 9), compressed air will exhaust from a small hole in a tank at the mass flow rate $\dot{m} \approx C \rho,$ where $\rho$ is the air density in the $\tan \mathrm{k}$ and $C$ is a constant. If $\rho_{0}$ is the initial density in a $\tan \mathrm{k}$ of volume $\gamma,$ derive a formula for the density change $\rho(t)$ after the hole is opened. Apply your formula to the following case: a spherical tank of diameter $50 \mathrm{cm},$ with initial pressure $300 \mathrm{kPa}$ and temperature $100^{\circ} \mathrm{C}$, and a hole whose initial exhaust rate is $0.01 \mathrm{kg} / \mathrm{s}$. Find the time required for the tank density to drop by 50 percent.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:26

Problem 30

For the nozzle of Fig. $\mathrm{P} 3.22,$ consider the following data for air, $k=1.4 .$ At the throat, $p_{1}=1000 \mathrm{kPa}, V_{1}=491 \mathrm{m} / \mathrm{s}$,and $T_{1}=600 \mathrm{K} .$ At the exit, $p_{2}=28.14 \mathrm{kPa} .$ Assuming isentropic steady flow, compute ( $a$ ) the Mach number $Ma_{1}$.
(b) $T_{2} ;(c)$ the mass flow; and $(d) V_{2}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:01

Problem 31

A bellows may be modeled as a deforming wedge-shaped volume as in Fig. P3.31. The check valve on the left (pleated) end is closed during the stroke. If $b$ is the bellows width into the paper, derive an expression for outlet mass flow $\dot{m}_{0}$ as a function of stroke $\theta(t)$.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:51

Problem 32

Water at $20^{\circ} \mathrm{C}$ flows steadily through the piping junction in Fig. $P 3.32,$ entering section 1 at 20 gal/min. The average velocity at section 2 is $2.5 \mathrm{m} / \mathrm{s}$. A portion of the flow is diverted through the showerhead, which contains 100 holes of $1-\mathrm{mm}$ diameter. Assuming uniform shower flow, estimate the exit velocity from the showerhead jets.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:22

Problem 33

In some wind tunnels the test section is perforated to suck out fluid and provide a thin viscous boundary layer. The test section wall in Fig. $\mathrm{P} 3.33$ contains 1200 holes of $5-\mathrm{mm}$ diameter each per square meter of wall area. The suction velocity through each hole is $V_{s}=8 \mathrm{m} / \mathrm{s},$ and the testsection entrance velocity is $V_{1}=35 \mathrm{m} / \mathrm{s}$. Assuming incompressible steady flow of air at $20^{\circ} \mathrm{C}$, compute
$(a) V_{0},(b) V_{2}$,
and $(c) V_{f}$, in $\mathrm{m} / \mathrm{s}$.

James Kiss
James Kiss
Numerade Educator
06:30

Problem 34

A rocket motor is operating steadily, as shown in Fig. P3.34. The products of combustion flowing out the exhaust nozzle approximate a perfect gas with a molecular weight of 28. For the given conditions calculate $V_{2}$ in $\mathrm{ft} / \mathrm{s}$.

Chai Santi
Chai Santi
Numerade Educator
04:05

Problem 35

In contrast to the liquid rocket in Fig. $\mathrm{P} 3.34$, the solid-propellant rocket in Fig. $\mathrm{P} 3.35$ is self-contained and has no entrance ducts. Using a control volume analysis for the conditions shown in Fig. $P 3.35,$ compute the rate of mass loss of the propellant, assuming that the exit gas has a molecular weight of 28.

Chai Santi
Chai Santi
Numerade Educator
02:57

Problem 36

The jet pump in Fig. $P 3.36$ injects water at $U_{1}=40 \mathrm{m} / \mathrm{s}$ through a 3 -in pipe and entrains a secondary flow of water $U_{2}=3 \mathrm{m} / \mathrm{s}$ in the annular region around the small pipe. The two flows become fully mixed downstream, where $U_{3}$ is approximately constant. For steady incompressible flow, compute $U_{3}$ in $\mathrm{m} / \mathrm{s}$.

Averell Hause
Averell Hause
Carnegie Mellon University
05:17

Problem 37

If the rectangular tank full of water in Fig. $P 3.37$ has its right-hand wall lowered by an amount $\delta,$ as shown, water will flow out as it would over a weir or dam. In Prob. P1.14 we deduced that the outflow $Q$ would be given by
\[Q=C b g^{1 / 2} \delta^{3 / 2}\]
where $b$ is the tank width into the paper, $g$ is the acceleration of gravity, and $C$ is a dimensionless constant. Assume that the water surface is horizontal, not slightly curved as in the figure. Let the initial excess water level be $\delta_{0} .$ Derive a formula for the time required to reduce the excess water level to $(a) \delta_{\mathrm{o}} / 10$ and $(b)$ zero.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 38

An incompressible fluid in Fig. $\mathrm{P} 3.38$ is being squeezed outward between two large circular disks by the uniform downward motion $V_{0}$ of the upper disk. Assuming onedimensional radial outflow, use the control volume shown to derive an expression for $V(r)$.

Victor Salazar
Victor Salazar
Numerade Educator
05:19

Problem 39

A wedge splits a sheet of $20^{\circ} \mathrm{C}$ water, as shown in Fig. P3.39. Both wedge and sheet are very long into the paper. If the force required to hold the wedge stationary is $F=$ $124 \mathrm{N}$ per meter of depth into the paper, what is the angle $\theta$ of the wedge?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:39

Problem 40

The water jet in Fig. $\mathrm{P} 3.40$ strikes normal to a fixed plate. Neglect gravity and friction, and compute the force $F$ in newtons required to hold the plate fixed.

Paul Gabriel
Paul Gabriel
Numerade Educator
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Problem 41

In Fig. $\mathrm{P} 3.41$ the vane turns the water jet completely around. Find an expression for the maximum jet velocity $V_{0}$ if the maximum possible support force is $F_{0}$.

Victor Salazar
Victor Salazar
Numerade Educator
02:42

Problem 42

A liquid of density $\rho$ flows through the sudden contraction in Fig. $P 3.42$ and exits to the atmosphere. Assume uniform conditions $\left(p_{1}, V_{1}, D_{1}\right)$ at section 1 and $\left(p_{2}, V_{2}, D_{2}\right)$ at section $2 .$ Find an expression for the force $F$ exerted by the fluid on the contraction.

Narayan Hari
Narayan Hari
Numerade Educator
07:47

Problem 43

Water at $20^{\circ} \mathrm{C}$ flows through a 5 -cm-diameter pipe that has
a $180^{\circ}$ vertical bend, as in Fig. $\mathrm{P} 3.43 .$ The total length of pipe between flanges 1 and 2 is $75 \mathrm{cm}$. When the weight flow rate is $230 \mathrm{N} / \mathrm{s}, p_{1}=165 \mathrm{kPa}$ and $p_{2}=134 \mathrm{kPa}$. Neglecting pipe weight, determine the total force that the flanges must withstand for this flow.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:07

Problem 44

When a uniform stream flows past an immersed thick cylinder, a broad low-velocity wake is created downstream, idealized as a $V$ shape in Fig. $P 3.44 .$ Pressures $p_{1}$ and $p_{2}$ are approximately equal. If the flow is two-dimensional and incompressible, with width $b$ into the paper, derive a formula for the drag force $F$ on the cylinder. Rewrite your result in the form of a dimensionless drag coefficient based on body length $C_{D}=F /\left(\rho \mathrm{U}^{2} b L\right)$.

Amit Srivastava
Amit Srivastava
Numerade Educator
03:21

Problem 45

Water enters and leaves the 6 -cm-diameter pipe bend in Fig. $\mathrm{P} 3.45$ at an average velocity of $8.5 \mathrm{m} / \mathrm{s} .$ The horizontal force to support the bend against momentum change is $300 \mathrm{N} .$ Find $(a)$ the angle $\phi ;$ and $(b)$ the vertical force on the bend.

Narayan Hari
Narayan Hari
Numerade Educator
02:14

Problem 46

When a jet strikes an inclined fixed plate, as in Fig. $\mathrm{P} 3.46$, it breaks into two jets at 2 and 3 of equal velocity $V=V_{\text {jet }}$ but unequal flows $\alpha Q$ at 2 and $(1-\alpha) Q$ at section $3, \alpha$ being a fraction. The reason is that for frictionless flow the fluid can exert no tangential force $F_{t}$ on the plate. The condition $F_{t}=0$ enables us to solve for $\alpha .$ Perform this analysis, and find $\alpha$ as a function of the plate angle
$\theta .$ Why doesn't the answer depend on the properties of the jet?

Chai Santi
Chai Santi
Numerade Educator
07:11

Problem 47

A liquid jet of velocity $V_{j}$ and diameter $D_{j}$ strikes a fixed hollow cone, as in Fig. $\mathrm{P} 3.47$, and deflects back as a conical sheet at the same velocity. Find the cone angle $\theta$ for which the restraining force $F=\frac{3}{2} \rho A_{j} V_{j}^{2}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:53

Problem 48

The small boat in Fig. $\mathrm{P} 3.48$ is driven at a steady speed $V_{0}$ by a jet of compressed air issuing from a 3 -cm-diameter hole at $V_{e}=343 \mathrm{m} / \mathrm{s}$. Jet exit conditions are $p_{e}=1$ atm and $T_{e}=30^{\circ} \mathrm{C} .$ Air drag is negligible, and the hull drag is $k V_{0}^{2}$, where $k \approx 19 \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2} .$ Estimate the boat speed $V_{0}$ in $\mathrm{m} / \mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
06:22

Problem 49

The horizontal nozzle in Fig. $P 3.49$ has $D_{1}=12$ in and $D_{2}=6$ in, with inlet pressure $p_{1}=38$ lbf/in $^{2}$ absolute and $V_{2}=56 \mathrm{ft} / \mathrm{s} .$ For water at $20^{\circ} \mathrm{C},$ compute the horizontal force provided by the flange bolts to hold the nozzle fixed.

Keshav Singh
Keshav Singh
Numerade Educator
01:25

Problem 50

The jet engine on a test stand in Fig. $\mathrm{P} 3.50$ admits air at $20^{\circ} \mathrm{C}$ and 1 atm at section $1,$ where $A_{1}=0.5 \mathrm{m}^{2}$ and $V_{1}=$ $250 \mathrm{m} / \mathrm{s}$. The fuel-to-air ratio is $1: 30 .$ The air leaves section 2 at atmospheric pressure and higher temperature, where $V_{2}=900 \mathrm{m} / \mathrm{s}$ and $A_{2}=0.4 \mathrm{m}^{2} .$ Compute the horizontal test stand reaction $R_{x}$ needed to hold this engine fixed.

Penny Riley
Penny Riley
Numerade Educator
07:15

Problem 51

A liquid jet of velocity $V_{j}$ and area $A_{j}$ strikes a single $180^{\circ}$ bucket on a turbine wheel rotating at angular velocity $\Omega,$ as in Fig. P3.51. Derive an expression for the power $P$ delivered to this wheel at this instant as a function of the system parameters. At what angular velocity is the maximum power delivered? How would your analysis differ if there were many, many buckets on the wheel, so that the jet was continually striking at least one bucket?

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:00

Problem 52

A large commercial power washer delivers 21 gal/min of water through a nozzle of exit diameter one-third of an inch. Estimate the force of the water jet on a wall normal to
the jet.

Chai Santi
Chai Santi
Numerade Educator
03:13

Problem 53

Consider incompressible flow in the entrance of a circular tube, as in Fig. P3.53. The inlet flow is uniform, $u_{1}=U_{0}$. The flow at section 2 is developed pipe flow. Find the wall drag force $F$ as a function of $\left(p_{1}, p_{2}, \rho, U_{0}, R\right)$ if the flow at section 2 is
(a) Laminar: $u_{2}=u_{\max }\left(1-\frac{r^{2}}{R^{2}}\right)$
(b) Turbulent: $u_{2} \approx u_{\max }\left(1-\frac{r}{R}\right)^{1 / 7}$

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
01:35

Problem 54

For the pipe-flow-reducing section of Fig. P3.54, $D_{1}=$ $8 \mathrm{cm}, D_{2}=5 \mathrm{cm},$ and $p_{2}=1$ atm. All fluids are at $20^{\circ} \mathrm{C}$. If $V_{1}=5 \mathrm{m} / \mathrm{s}$ and the manometer reading is $h=58 \mathrm{cm},$ estimate the total force resisted by the flange bolts.

Dominador Tan
Dominador Tan
Numerade Educator
02:35

Problem 55

In Fig. $P 3.55$ the jet strikes a vane that moves to the right at constant velocity $V_{c}$ on a frictionless cart. Compute $(a)$ the force $F_{x}$ required to restrain the cart and $(b)$ the power $P$ delivered to the cart. Also find the cart velocity for which
$(c)$ the force $F_{x}$ is a maximum and
$(d)$ the power $P$ is a maximum.

Alexander Allen
Alexander Allen
Numerade Educator
03:42

Problem 56

Water at $20^{\circ} \mathrm{C}$ flows steadily through the box in Fig. $\mathrm{P} 3.56$, entering station (1) at 2 $\mathrm{m} / \mathrm{s}$. Calculate the
$(a)$ horizontal and $(b)$ vertical forces required to hold the box stationary against the flow momentum.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:09

Problem 57

Water flows through the duct in Fig. P3.57, which is 50 $\mathrm{cm}$ wide and $1 \mathrm{m}$ deep into the paper. Gate $B C$ completely closes the duct when $\beta=90^{\circ} .$ Assuming one-dimensional flow, for what angle $\beta$ will the force of the exit jet on the plate be $3 \mathrm{kN} ?$

Chai Santi
Chai Santi
Numerade Educator
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Problem 58

The water tank in Fig. $\mathrm{P} 3.58$ stands on a frictionless cart and feeds a jet of diameter $4 \mathrm{cm}$ and velocity $8 \mathrm{m} / \mathrm{s}$, which is deflected $60^{\circ}$ by a vane. Compute the tension in the supporting cable.

Victor Salazar
Victor Salazar
Numerade Educator
07:19

Problem 59

When a pipe flow suddenly expands from $A_{1}$ to $A_{2},$ as in Fig. $P 3.59,$ low-speed, low-friction eddies appear in the corners and the flow gradually expands to $A_{2}$ downstream. Using the suggested control volume for incompressible steady flow and assuming that $p \approx p_{1}$ on the corner annular ring as shown, show that the downstream pressure is given by
$$p_{2}=p_{1}+\rho V_{1}^{2} \frac{A_{1}}{A_{2}}\left(1-\frac{A_{1}}{A_{2}}\right)$$
Neglect wall friction.

Ronald Prasad
Ronald Prasad
Numerade Educator
01:35

Problem 60

Water at $20^{\circ} \mathrm{C}$ flows through the elbow in Fig. $\mathrm{P} 3.60$ and exits to the atmosphere. The pipe diameter is $D_{1}=10 \mathrm{cm},$ while $D_{2}=3 \mathrm{cm} .$ At a weight flow rate of $150 \mathrm{N} / \mathrm{s}$, the pressure $p_{1}=2.3$ atm (gage). Neglecting the weight of water and elbow, estimate the force on the flange bolts at section 1.

Dominador Tan
Dominador Tan
Numerade Educator
04:36

Problem 61

A $20^{\circ} \mathrm{C}$ water jet strikes a vane mounted on a tank with frictionless wheels, as in Fig. P3.61. The jet turns and falls into the tank without spilling out. If $\theta=30^{\circ},$ evaluate the horizontal force $F$ required to hold the tank stationary.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:35

Problem 62

Water at $20^{\circ} \mathrm{C}$ exits to the standard sea-level atmosphere through the split nozzle in Fig. P3.62. Duct areas are $A_{1}=$ $0.02 \mathrm{m}^{2}$ and $A_{2}=A_{3}=0.008 \mathrm{m}^{2} .$ If $p_{1}=135 \mathrm{kPa}$ (absolute) and the flow rate is $Q_{2}=Q_{3}=275 \mathrm{m}^{3} / \mathrm{h}$, compute the force on the flange bolts at section 1.

Dominador Tan
Dominador Tan
Numerade Educator
View

Problem 63

Water flows steadily through the box in Fig. P3.63. Average velocity at all ports is $7 \mathrm{m} / \mathrm{s}$. The vertical momentum force on the box is $36 \mathrm{N}$. What is the inlet mass flow?

Victor Salazar
Victor Salazar
Numerade Educator
02:00

Problem 64

The 6 -cm-diameter $20^{\circ} \mathrm{C}$ water jet in Fig. $\mathrm{P} 3.64$ strikes a plate containing a hole of $4-\mathrm{cm}$ diameter. Part of the jet passes through the hole, and part is deflected. Determine the horizontal force required to hold the plate.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
04:53

Problem 65

The box in Fig. $\mathrm{P} 3.65$ has three 0.5 -in holes on the right side. The volume flows of $20^{\circ} \mathrm{C}$ water shown are steady, but the details of the interior are not known. Compute the force, if any, that this water flow causes on the box.

Matthew Baker
Matthew Baker
Numerade Educator
02:13

Problem 66

The tank in Fig. $\mathrm{P} 3.66$ weighs $500 \mathrm{N}$ empty and contains $600 \mathrm{L}$ of water at $20^{\circ} \mathrm{C} .$ Pipes 1 and 2 have equal diameters of $6 \mathrm{cm}$ and equal steady volume flows of $300 \mathrm{m}^{3} / \mathrm{h}$. What should the scale reading $W$ be in $\mathrm{N} ?$

Janielle Madlansacay
Janielle Madlansacay
Numerade Educator
05:46

Problem 67

For the boundary layer of Fig. $3.10,$ for air, $\rho=1.2 \mathrm{kg} / \mathrm{m}^{3}$, let $h=7 \mathrm{cm}, U_{\mathrm{o}}=12 \mathrm{m} / \mathrm{s}, b=2 \mathrm{m},$ and $L=1 \mathrm{m} .$ Let the
velocity at the exit, $x=L,$ approximate a turbulent flow:
$u / U_{o} \approx(y / \delta)^{1 / 7} .$ Calculate
$(a) \delta ;$ and
(b) the friction $\operatorname{drag} D$.

Satpal Satpal
Satpal Satpal
Numerade Educator
02:58

Problem 68

The rocket in Fig. $\mathrm{P} 3.68$ has a supersonic exhaust, and the exit pressure $p_{e}$ is not necessarily equal to $p_{a} .$ Show that the force $F$ required to hold this rocket on the test stand is $F=\rho_{e} A_{e} V_{e}^{2}+A_{e}\left(p_{e}-p_{a}\right) .$ Is this force $F$ what we term the thrust of the rocket?

James Kiss
James Kiss
Numerade Educator
02:00

Problem 69

A uniform rectangular plate, $40 \mathrm{cm}$ long and $30 \mathrm{cm}$ deep into the paper, hangs in air from a hinge at its top (the $30-\mathrm{cm} \text { side }) .$ It is struck in its center by a horizontal $3-\mathrm{cm}-$ diameter jet of water moving at $8 \mathrm{m} / \mathrm{s}$. If the gate has a mass of $16 \mathrm{kg}$, estimate the angle at which the plate will hang from the vertical.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
02:13

Problem 70

The dredger in Fig. $P 3.70$ is loading sand $(S G=2.6)$ onto a barge. The sand leaves the dredger pipe at $4 \mathrm{ft} / \mathrm{s}$ with a weight flow of 850 lbf/s. Estimate the tension on the mooring line caused by this loading process.

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
01:30

Problem 71

Suppose that a deflector is deployed at the exit of the jet engine of Prob. P3.50, as shown in Fig. P3.71. What will the reaction $R_{x}$ on the test stand be now? Is this reaction sufficient to serve as a braking force during airplane landing?

Penny Riley
Penny Riley
Numerade Educator
02:22

Problem 72

When immersed in a uniform stream, a thick elliptical cylinder creates a broad downstream wake, as idealized in Fig. $P 3.72 .$ The pressure at the upstream and downstream sections are approximately equal, and the fluid is water at $20^{\circ} \mathrm{C} .$ If $U_{0}=4 \mathrm{m} / \mathrm{s}$ and $L=80 \mathrm{cm},$ estimate the drag
force on the cylinder per unit width into the paper. Also compute the dimensionless drag coefficient $C_{D}=$ $2 F /\left(\rho U_{0}^{2} b L\right)$.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:36

Problem 73

A pump in a tank of water at $20^{\circ} \mathrm{C}$ directs a jet at $45 \mathrm{ft} / \mathrm{s}$ and 200 gal/min against a vane, as shown in Fig. P3.73. Compute the force $F$ to hold the cart stationary if the jet follows
$(a)$ path $A$ or $(b)$ path $B .$ The tank holds 550 gal of water at this instant.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
15:37

Problem 74

Water at $20^{\circ} \mathrm{C}$ flows down through a vertical, $6-\mathrm{cm}-$ diameter tube at 300 gal/min, as in Fig. P3.74. The flow then turns horizontally and exits through a $90^{\circ}$ radial duct segment $1 \mathrm{cm}$ thick, as shown. If the radial outflow is uniform and steady, estimate the forces $\left(F_{x}, F_{y}, F_{z}\right)$ required to support this system against fluid momentum changes.

David Morabito
David Morabito
Numerade Educator
View

Problem 75

A jet of liquid of density $\rho$ and area $A$ strikes a block and splits into two jets, as in Fig. P3.75. Assume the same velocity $V$ for all three jets. The upper jet exits at an angle $\theta$ and area $\alpha A .$ The lower jet is turned $90^{\circ}$ downward. Neglecting fluid weight, $(a)$ derive a formula for the forces $\left(F_{x} F_{y}\right)$ required to support the block against fluid momentum changes.
(b) Show that $F_{y}=0$ only if $\alpha \geq 0.5$
$(c)$ Find the values of $\alpha$ and $\theta$ for which both $F_{x}$ and $F_{y}$ are zero.

Victor Salazar
Victor Salazar
Numerade Educator
01:35

Problem 76

A two-dimensional sheet of water, $10 \mathrm{cm}$ thick and moving at $7 \mathrm{m} / \mathrm{s}$, strikes a fixed wall inclined at $20^{\circ}$ with respect to the jet direction. Assuming frictionless flow, find $(a)$ the normal force on the wall per meter of depth, and find the widths of the sheet deflected ( $b$ ) upstream and
$(c)$ downstream along the wall.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:35

Problem 77

Water at $20^{\circ} \mathrm{C}$ flows steadily through a reducing pipe bend, as in Fig. P3.77. Known conditions are $p_{1}=350 \mathrm{kPa}, D_{1}=$ $25 \mathrm{cm}, V_{1}=2.2 \mathrm{m} / \mathrm{s}, p_{2}=120 \mathrm{kPa},$ and $D_{2}=8 \mathrm{cm}$
Neglecting bend and water weight, estimate the total force that must be resisted by the flange bolts.

Dominador Tan
Dominador Tan
Numerade Educator
00:56

Problem 78

A fluid jet of diameter $D_{1}$ enters a cascade of moving blades at absolute velocity $V_{1}$ and angle $\beta_{1},$ and it leaves at absolute velocity $V_{2}$ and angle $\beta_{2},$ as in Fig. P3.78. The blades move at velocity $u$. Derive a formula for the power $P$ delivered to the blades as a function of these parameters.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
05:19

Problem 79

The Saturn V rocket in the chapter opener photo was powered by five $F-1$ engines, each of which burned $3945 \mathrm{lbm} / \mathrm{s}$ of liquid oxygen and 1738 lbm of kerosene per second. The exit velocity of burned gases was approximately $8500 \mathrm{ft} / \mathrm{s}$. In the spirit of Prob. P3.34, neglecting external pressure forces, estimate the total thrust of the rocket, in lbf.

Prashant Bana
Prashant Bana
Numerade Educator
01:29

Problem 80

A river of width $b$ and depth $h_{1}$ passes over a submerged obstacle, or "drowned weir," in Fig. P3.80, emerging at a new flow condition $\left(V_{2}, h_{2}\right) .$ Neglect atmospheric pressure, and assume that the water pressure is hydrostatic at both sections 1 and $2 .$ Derive an expression for the force exerted by the river on the obstacle in terms of $V_{1}, h_{1}, h_{2}, b, \rho,$ and
$g .$ Neglect water friction on the river bottom.

Penny Riley
Penny Riley
Numerade Educator
07:29

Problem 81

Torricelli's idealization of efflux from a hole in the side of
a tank is $V=\sqrt{2 g h}$, as shown in Fig. P3.81. The cylindrical tank weighs $150 \mathrm{N}$ when empty and contains water at $20^{\circ} \mathrm{C} .$ The tank bottom is on very smooth ice (static friction coefficient $\zeta \approx 0.01$ ). The hole diameter is $9 \mathrm{cm} .$ For what water depth $h$ will the tank just begin to move to the right?

Sirat Shah
Sirat Shah
Numerade Educator
02:30

Problem 82

The model car in Fig. P3.82 weighs $17 \mathrm{N}$ and is to be accelerated from rest by a 1 -cm-diameter water jet moving at $75 \mathrm{m} / \mathrm{s}$. Neglecting air drag and wheel friction, estimate the velocity of the car after it has moved forward $1 \mathrm{m}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
17:07

Problem 83

Gasoline at $20^{\circ} \mathrm{C}$ is flowing at $V_{1}=12 \mathrm{m} / \mathrm{s}$ in a $5-\mathrm{cm}-$ diameter pipe when it encounters a 1 -m length of uniform radial wall suction. At the end of this suction region, the average fluid velocity has dropped to $V_{2}=10 \mathrm{m} / \mathrm{s}$. If $p_{1}=$ $120 \mathrm{kPa},$ estimate $p_{2}$ if the wall friction losses are neglected.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:06

Problem 84

Air at $20^{\circ} \mathrm{C}$ and 1 atm flows in a 25 -cm-diameter duct at $15 \mathrm{m} / \mathrm{s},$ as in Fig. $\mathrm{P} 3.84 .$ The exit is choked by a $90^{\circ}$ cone, as shown. Estimate the force of the airflow on the cone.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:43

Problem 85

The thin-plate orifice in Fig. $\mathrm{P} 3.85$ causes a large pressure drop. For $20^{\circ} \mathrm{C}$ water flow at $500 \mathrm{gal} / \mathrm{min},$ with pipe $D=$ $10 \mathrm{cm}$ and orifice $d=6 \mathrm{cm}, p_{1}-p_{2} \approx 145 \mathrm{kPa} .$ If the wall
friction is negligible, estimate the force of the water on the orifice plate.

Chai Santi
Chai Santi
Numerade Educator
03:02

Problem 86

For the water jet pump of Prob. $\mathrm{P} 3.36,$ add the following data: $p_{1}=p_{2}=25 \mathrm{lbf} / \mathrm{in}^{2},$ and the distance between sections 1 and 3 is 80 in. If the average wall shear stress between sections 1 and 3 is 7 lbf/ft $^{2}$, estimate the pressure $p_{3} .$ Why is it higher than $p_{1} ?$

Chai Santi
Chai Santi
Numerade Educator
03:08

Problem 87

A vane turns a water jet through an angle $\alpha,$ as shown in Fig. $P 3.87 .$ Neglect friction on the vane walls.
(a) What is the angle $\alpha$ for the support force to be in pure compression?
(b) Calculate this compression force if the water velocity is
$22 \mathrm{ft} / \mathrm{s}$ and the jet cross section is $4 \mathrm{in}^{2}$.

Chai Santi
Chai Santi
Numerade Educator
17:18

Problem 88

The boat in Fig. $\mathrm{P} 3.88$ is jet-propelled by a pump that develops a volume flow rate $Q$ and ejects water out the stern at velocity $V_{j} .$ If the boat drag force is $F=k V^{2},$ where $k$ is a constant, develop a formula for the steady forward speed $V$ of the boat.

David Morabito
David Morabito
Numerade Educator
01:11

Problem 89

Consider Fig. $\mathrm{P} 3.36$ as a general problem for analysis of a mixing ejector pump. If all conditions $(p, \rho, V)$ are known at sections 1 and 2 and if the wall friction is negligible, derive formulas for estimating
$(a) V_{3}$ and $(b) p_{3}$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
05:53

Problem 90

As shown in Fig. $\mathrm{P} 3.90$, a liquid column of height $h$ is confined in a vertical tube of cross-sectional area $A$ by a stopper. At $t=0$ the stopper is suddenly removed, exposing the bottom of the liquid to atmospheric pressure. Using a control volume analysis of mass and vertical momentum, derive the differential equation for the downward motion $V(t)$ of the liquid. Assume one-dimensional, incompressible, frictionless flow.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:45

Problem 91

Extend Prob. P3.90 to include a linear (laminar) average wall shear stress resistance of the form $\tau \approx c V$, where $c$ is a constant. Find the differential equation for $d V / d t$ and then solve for $V(t),$ assuming for simplicity that the wall area remains constant.

Narayan Hari
Narayan Hari
Numerade Educator
00:56

Problem 92

A more involved version of Prob. P3.90 is the elbowshaped tube in Fig. $\mathrm{P} 3.92,$ with constant cross-sectional area $A$ and diameter $D \ll h, L .$ Assume incompressible flow, neglect friction, and derive a differential equation for $d V / d t$ when the stopper is opened. Hint: Combine two control volumes, one for each leg of the tube.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:10

Problem 93

According to Torricelli's theorem, the velocity of a fluid draining from a hole in a tank is $V \approx(2 g h)^{1 / 2},$ where $h$ is the depth of water above the hole, as in Fig. P3.93. Let the hole have area $A_{o}$ and the cylindrical tank have cross-section $\operatorname{area} A_{b} \gg A_{o} .$ Derive a formula for the time to drain the tank completely from an initial depth $h_{o}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:53

Problem 94

A water jet 3 in in diameter strikes a concrete $(\mathrm{SG}=2.3)$ slab which rests freely on a level floor. If the slab is $1 \mathrm{ft}$ wide into the paper, calculate the jet velocity which will just begin to tip the slab over.

Keshav Singh
Keshav Singh
Numerade Educator
01:14

Problem 95

A tall water tank discharges through a well-rounded orifice, as in Fig. $\mathrm{P} 3.95 .$ Use the Torricelli formula of Prob. P3.81 to estimate the exit velocity. ( $a$ ) If, at this instant, the force
$F$ required to hold the plate is $40 \mathrm{N}$, what is the depth $h ?$
(b) If the tank surface is dropping at the rate of $2.5 \mathrm{cm} / \mathrm{s}$, what is the tank diameter $D ?$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:50

Problem 96

Extend Prob. $\mathrm{P} 3.90$ to the case of the liquid motion in a frictionless U-tube whose liquid column is displaced a distance $Z$ upward and then released, as in Fig. P3.96.

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
02:45

Problem 97

Extend Prob. $\mathrm{P} 3.96$ to include a linear (laminar) average wall shear stress resistance of the form $\tau \approx 8 \mu V / D$, where $\mu$ is the fluid viscosity. Find the differential equation for $d V / d t$ and then solve for $V(t),$ assuming an initial displacement $z=z_{0}, V=0$ at $t=0 .$ The result should be a damped oscillation tending toward $z=0$.

Narayan Hari
Narayan Hari
Numerade Educator
03:10

Problem 98

As an extension of Example 3.9 , let the plate and its cart (see Fig. $3.9 a$ ) be unrestrained horizontally, with frictionless wheels. Derive $(a)$ the equation of motion for cart velocity $V_{c}(t)$ and $(b)$ a formula for the time required for the cart to accelerate from rest to 90 percent of the jet velocity (assuming the jet continues to strike the plate horizontally).
$(c)$ Compute numerical values for part ( $b$ ) using the conditions of Example 3.9 and a cart mass of $2 \mathrm{kg}$.

Chai Santi
Chai Santi
Numerade Educator
11:44

Problem 99

Let the rocket of Fig. E3.12 start at $z=0,$ with constant exit velocity and exit mass flow, and rise vertically with zero drag. (a) Show that, as long as fuel burning continues, the vertical height $S(t)$ reached is given by
$$S=\frac{V_{e} M_{o}}{\dot{m}}[\operatorname{Gin} \zeta-\zeta+1], \text { where } \zeta=1-\frac{\dot{m} t}{M_{o}}$$
(b) Apply this to the case $V_{e}=1500 \mathrm{m} / \mathrm{s}$ and $M_{o}=1000 \mathrm{kg}$ to find the height reached after a burn of 30 seconds, when the final rocket mass is $400 \mathrm{kg}$.

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
08:45

Problem 100

Suppose that the solid-propellant rocket of Prob. P3.35 is built into a missile of diameter $70 \mathrm{cm}$ and length $4 \mathrm{m}$. The system weighs $1800 \mathrm{N},$ which includes $700 \mathrm{N}$ of propellant. Neglect air drag. If the missile is fired vertically from rest at sea level, estimate $(a)$ its velocity and height at fuel burnout and ( $b$ ) the maximum height it will attain.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:39

Problem 101

Water at $20^{\circ} \mathrm{C}$ flows steadily through the tank in Fig. P3.101. Known conditions are $D_{1}=8 \mathrm{cm}, V_{1}=6 \mathrm{m} / \mathrm{s},$ and $D_{2}=4 \mathrm{cm} .$ A rightward force $F=70 \mathrm{N}$ is required to keep the tank fixed. ( $a$ ) What is the velocity leaving section 2?
$(b)$ If the tank cross section is $1.2 \mathrm{m}^{2}$, how fast is the water surface $h(t)$ rising or falling?

Amany Waheeb
Amany Waheeb
Numerade Educator
03:12

Problem 102

As can often be seen in a kitchen sink when the faucet is
running, a high-speed channel flow $\left(V_{1}, h_{1}\right)$ may "jump" to a low-speed, low-energy condition $\left(V_{2}, h_{2}\right)$ as in Fig. $\mathrm{P} 3.102$.
The pressure at sections 1 and 2 is approximately hydrostatic, and wall friction is negligible. Use the continuity and momentum relations to find $h_{2}$ and $V_{2}$ in terms of $\left(h_{1}, V_{1}\right)$.

Chai Santi
Chai Santi
Numerade Educator
03:49

Problem 103

Suppose that the solid-propellant rocket of Prob. P3.35 is mounted on a 1000 -kg car to propel it up a long slope of $15^{\circ} .$ The rocket motor weighs $900 \mathrm{N},$ which includes $500 \mathrm{N}$ of propellant. If the car starts from rest when the rocket is fired, and if air drag and wheel friction are neglected, estimate the maximum distance that the car will travel up the hill.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
02:58

Problem 104

A rocket is attached to a rigid horizontal rod hinged at the origin as in Fig. P3.104. Its initial mass is $M_{0},$ and its exit properties are $\dot{m}$ and $V_{e}$ relative to the rocket. Set up the differential equation for rocket motion, and solve for the angular velocity $\omega(t)$ of the rod. Neglect gravity, air drag, and the rod mass.

James Kiss
James Kiss
Numerade Educator
02:49

Problem 105

Extend Prob. P3.104 to the case where the rocket has a linear air drag force $F=c V,$ where $c$ is a constant. Assuming no burnout, solve for $\omega(t)$ and find the terminal angular velocity-that is, the final motion when the angular acceleration is zero. Apply to the case $M_{0}=6 \mathrm{kg}, R=3 \mathrm{m}, \dot{m}=$ $0.05 \mathrm{kg} / \mathrm{s}, V_{e}=1100 \mathrm{m} / \mathrm{s},$ and $c=0.075 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}$ to find the angular velocity after 12 s of burning.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
00:57

Problem 106

Actual airflow past a parachute creates a variable distribution of velocities and directions. Let us model this as a
circular air jet, of diameter half the parachute diameter, which is turned completely around by the parachute, as in Fig. P3.106.
(a) Find the force $F$ required to support the chute.
(b) Express this force as a dimensionless drag coefficient, $C_{D}=F /\left[(1 / 2) \rho V^{2}(\pi / 4) D^{2}\right]$ and compare with Table 7.3.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
00:55

Problem 107

The cart in Fig. $P 3.107$ moves at constant velocity $V_{0}=$ $12 \mathrm{m} / \mathrm{s}$ and takes on water with a scoop $80 \mathrm{cm}$ wide that dips $h=2.5 \mathrm{cm}$ into a pond. Neglect air drag and wheel friction. Estimate the force required to keep the cart moving.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
03:10

Problem 108

A rocket sled of mass $M$ is to be decelerated by a scoop, as in Fig. P3.108, which has width $b$ into the paper and dips into the water a depth $h,$ creating an upward jet at $60^{\circ} .$ The rocket thrust is $T$ to the left. Let the initial velocity be $V_{0}$, and neglect air drag and wheel friction. Find an expression for $V(t)$ of the sled for $(a) T=0$ and $(b)$ finite $T \neq 0$.

Chai Santi
Chai Santi
Numerade Educator
05:09

Problem 109

For the boundary layer flow in Fig. 3.10 , let the exit velocity profile, at $x=L,$ simulate turbulent flow, $u \approx U_{0}(y / \delta)^{1 / 7}$. (a) Find a relation between $h$ and $\delta$. ( $b$ ) Find an expression for the drag force $F$ on the plate between 0 and $L$.

Satpal Satpal
Satpal Satpal
Numerade Educator
02:24

Problem 110

Repeat Prob. P3.49 by assuming that $p_{1}$ is unknown and using Bernoulli's equation with no losses. Compute the new bolt force for this assumption. What is the head loss between 1 and 2 for the data of Prob. $\mathrm{P} 3.49 ?$

Amany Waheeb
Amany Waheeb
Numerade Educator
02:59

Problem 111

As a simpler approach to Prob. P3.96, apply the unsteady Bernoulli equation between 1 and 2 to derive a differential equation for the motion $z(t) .$ Neglect friction and compressibility.

Dominador Tan
Dominador Tan
Numerade Educator
07:19

Problem 112

A jet of alcohol strikes the vertical plate in Fig. P3.112. A force $F \approx 425 \mathrm{N}$ is required to hold the plate stationary. Assuming there are no losses in the nozzle, estimate ( $a$ ) the mass flow rate of alcohol and ( $b$ ) the absolute pressure at section 1.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:45

Problem 113

An airplane is flying at $300 \mathrm{mi} / \mathrm{h}$ at $4000 \mathrm{m}$ standard altitude. As is typical, the air velocity relative to the upper surface of the wing, near its maximum thickness, is 26 percent higher than the plane's velocity. Using Bernoulli's equation, calculate the absolute pressure at this point on the wing. Neglect elevation changes and compressibility.

Narayan Hari
Narayan Hari
Numerade Educator
01:54

Problem 114

Water flows through a circular nozzle, exits into the air as a jet, and strikes a plate, as shown in Fig. P3.114. The force required to hold the plate steady is $70 \mathrm{N}$. Assuming steady, frictionless, one-dimensional flow, estimate ( $a$ ) the velocities at sections (1) and (2) and $(b)$ the mercury manometer reading $h$.

Chai Santi
Chai Santi
Numerade Educator
01:34

Problem 115

A free liquid jet, as in Fig. $\mathrm{P} 3.115,$ has constant ambient pressure and small losses; hence from Bernoulli's equation $z+V^{2} /(2 g)$ is constant along the jet. For the fire nozzle in the figure, what are $(a)$ the minimum and $(b)$ the maximum values of $\theta$ for which the water jet will clear the corner of the building? For which case will the jet velocity be higher when it strikes the roof of the building?

Amany Waheeb
Amany Waheeb
Numerade Educator
04:22

Problem 116

For the container of Fig. $\mathrm{P} 3.116$ use Bernoulli's equation to derive a formula for the distance $X$ where the free jet leaving horizontally will strike the floor, as a function of $h$ and
$H .$ For what ratio $h / H$ will $X$ be maximum? Sketch the three trajectories for $h / H=0.25,0.5,$ and 0.75.

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
05:04

Problem 117

Water at $20^{\circ} \mathrm{C},$ in the pressurized tank of Fig. $\mathrm{P} 3.117$, flows out and creates a vertical jet as shown. Assuming steady frictionless flow, determine the height $H$ to which the jet rises.

Prashant Bana
Prashant Bana
Numerade Educator
02:34

Problem 118

Bernoulli's 1738 treatise Hydrodynamica contains many excellent sketches of flow patterns related to his frictionless relation. One, however, redrawn here as Fig. P3.1 18, seems physically misleading. Can you explain what might be wrong with the figure?

Stanley Enemuo
Stanley Enemuo
Numerade Educator
05:25

Problem 119

A long fixed tube with a rounded nose, aligned with an oncoming flow, can be used to measure velocity. Measurements are made of the pressure at (1) the front nose and
(2) a hole in the side of the tube further along, where the pressure nearly equals stream pressure.
(a) Make a sketch of this device and show how the velocity is calculated.
(b) For a particular sea-level airflow, the difference between nose pressure and side pressure is $1.5 \mathrm{lbf} / \mathrm{in}^{2}$. What is the air velocity, in $\mathrm{mi} / \mathrm{h} ?$

Averell Hause
Averell Hause
Carnegie Mellon University
03:42

Problem 120

The manometer fluid in Fig. P3.120 is mercury. Estimate the volume flow in the tube if the flowing fluid is
(a) gasoline and $(b)$ nitrogen, at $20^{\circ} \mathrm{C}$ and 1 atm.

Prashant Bana
Prashant Bana
Numerade Educator
01:03

Problem 121

In Fig. $P 3.121$ the flowing fluid is $\mathrm{CO}_{2}$ at $20^{\circ} \mathrm{C}$. Neglect losses. If $p_{1}=170 \mathrm{kPa}$ and the manometer fluid is Meriam red oil $(\mathrm{SG}=0.827),$ estimate $(a) p_{2}$ and $(b)$ the gas flow rate in $\mathrm{m}^{3} / \mathrm{h}$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:43

Problem 122

The cylindrical water tank in Fig. $\mathrm{P} 3.122$ is being filled at a volume flow $Q_{1}=1.0$ gal/min, while the water also drains from a bottom hole of diameter $d=6 \mathrm{mm} .$ At time $t=0, h=0 .$ Find and plot the variation $h(t)$ and the eventual maximum water depth $h_{\max } .$ Assume that Bernoullis's steady-flow equation is valid.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:57

Problem 123

The air-cushion vehicle in Fig. P3.123 brings in sea-level standard air through a fan and discharges it at high velocity through an annular skirt of $3-\mathrm{cm}$ clearance. If the vehicle weighs $50 \mathrm{kN}$, estimate (a) the required airflow rate and $(b)$ the fan power in $\mathrm{kW}$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
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Problem 124

A necked-down section in a pipe flow, called a venturi, develops a low throat pressure that can aspirate fluid upward from a reservoir, as in Fig. P3.124. Using Bernoulli's equation with no losses, derive an expression for the velocity $V_{1}$ that is just sufficient to bring reservoir fluid into the throat.

Victor Salazar
Victor Salazar
Numerade Educator
03:44

Problem 125

Suppose you are designing an air hockey table. The table is $3.0 \times 6.0 \mathrm{ft}$ in area, with $\frac{1}{16}$ -in-diameter holes spaced every inch in a rectangular grid pattern (2592 holes total). The required jet speed from each hole is estimated to be $50 \mathrm{ft} / \mathrm{s}$. Your job is to select an appropriate blower that will meet the requirements. Estimate the volumetric flow rate (in $\left.\mathrm{ft}^{3} / \mathrm{min}\right)$ and pressure rise (in $\mathrm{lb} / \mathrm{in}^{2}$ ) required of the blower. Hint: Assume that the air is stagnant in the large volume of the manifold under the table surface, and neglect any frictional losses.

Willis James
Willis James
Numerade Educator
02:00

Problem 126

The liquid in Fig. $P 3.126$ is kerosene at $20^{\circ} \mathrm{C}$. Estimate the flow rate from the tank for $(a)$ no losses and $(b)$ pipe losses $h_{f} \approx 4.5 V^{2} /(2 g)$.

Ameer Said
Ameer Said
Numerade Educator
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Problem 127

In Fig. $P 3.127$ the open jet of water at $20^{\circ} \mathrm{C}$ exits a nozzle into sea-level air and strikes a stagnation tube as shown.
If the pressure at the centerline at section 1 is $110 \mathrm{kPa}$, and losses are neglected, estimate $(a)$ the mass flow in $\mathrm{kg} / \mathrm{s}$ and (b) the height $H$ of the fluid in the stagnation tube.

Victor Salazar
Victor Salazar
Numerade Educator
03:50

Problem 128

A venturi meter, shown in Fig. P3.128, is a carefully designed constriction whose pressure difference is a measure of the flow rate in a pipe. Using Bernoulli's equation for steady incompressible flow with no losses, show that the flow rate $Q$ is related to the manometer reading $h$ by
$$Q=\frac{A_{2}}{\sqrt{1-\left(D_{2} / D_{1}\right)^{4}}} \sqrt{\frac{2 g h\left(\rho_{M}-\rho\right)}{\rho}}$$
where $\rho_{M}$ is the density of the manometer fluid.

Adnan Gill
Adnan Gill
Numerade Educator
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Problem 129

A water stream flows past a small circular cylinder at $23 \mathrm{ft} / \mathrm{s},$ approaching the cylinder at $3000 \mathrm{lbf} / \mathrm{ft}^{2}$. Measurements at low (laminar flow) Reynolds numbers indicate a maximum surface velocity 60 percent higher than the stream velocity at point $B$ on the cylinder. Estimate the pressure at $B$.

Victor Salazar
Victor Salazar
Numerade Educator
01:28

Problem 130

In Fig. $\mathrm{P} 3.130$ the fluid is gasoline at $20^{\circ} \mathrm{C}$ at a weight flow of $120 \mathrm{N} / \mathrm{s}$. Assuming no losses, estimate the gage pressure at section 1.

Penny Riley
Penny Riley
Numerade Educator
02:36

Problem 131

In Fig. $\mathrm{P} 3.131$ both fluids are at $20^{\circ} \mathrm{C}$. If $V_{1}=1.7 \mathrm{ft} / \mathrm{s}$ and losses are neglected, what should the manometer reading $h$ ft be?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:54

Problem 132

Extend the siphon analysis of Example 3.14 to account for friction in the tube, as follows. Let the friction head loss in the tube be correlated as $5.4\left(V_{\text {tube }}\right)^{2} /(2 g),$ which approximates turbulent flow in a 2 -m-long tube. Calculate the exit velocity in $\mathrm{m} / \mathrm{s}$ and the volume flow rate in $\mathrm{cm}^{3} / \mathrm{s},$ and $\mathrm{com}-$ pare to Example 3.14.

Anand Jangid
Anand Jangid
Numerade Educator
03:36

Problem 133

If losses are neglected in Fig. P3.133, for what water level $h$ will the flow begin to form vapor cavities at the throat of the nozzle?

Narayan Hari
Narayan Hari
Numerade Educator
01:56

Problem 134

For the $40^{\circ} \mathrm{C}$ water flow in Fig. $\mathrm{P} 3.134$, estimate the volume flow through the pipe, assuming no losses; then explain what is wrong with this seemingly innocent question. If the actual flow rate is $Q=40 \mathrm{m}^{3} / \mathrm{h},$ compute ( $a$ ) the head loss in $\mathrm{ft}$ and $(b)$ the constriction diameter $D$ that causes cavitation, assuming that the throat divides the head loss equally and that changing the constriction causes no additional losses.

Penny Riley
Penny Riley
Numerade Educator
01:40

Problem 135

The $35^{\circ} \mathrm{C}$ water flow of Fig. $\mathrm{P} 3.135$ discharges to sea-level standard atmosphere. Neglecting losses, for what nozzle diameter $D$ will cavitation begin to occur? To avoid cavitation, should you increase or decrease $D$ from this critical value?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:16

Problem 136

Air, assumed frictionless, flows through a tube, exiting to sea-level atmosphere. Diameters at 1 and 3 are $5 \mathrm{cm},$ while $D_{2}=3 \mathrm{cm} .$ What mass flow of air is required to suck water up $10 \mathrm{cm}$ into section 2 of Fig. $\mathrm{P} 3.136 ?$

Anand Jangid
Anand Jangid
Numerade Educator
01:20

Problem 137

In Fig. $\mathrm{P} 3.137$ the piston drives water at $20^{\circ} \mathrm{C}$. Neglecting losses, estimate the exit velocity $V_{2}$ ft/s. If $D_{2}$ is further constricted, what is the limiting possible value of $V_{2} ?$

James Kiss
James Kiss
Numerade Educator
05:21

Problem 138

For the sluice gate flow of Example 3.10 , use Bernoulli's equation, along the surface, to estimate the flow rate $Q$ as a function of the two water depths. Assume constant width $b$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:42

Problem 139

In the spillway flow of Fig. $P 3.139,$ the flow is assumed uniform and hydrostatic at sections 1 and $2 .$ If losses are neglected, compute $(a) V_{2}$ and $(b)$ the force per unit width of the water on the spillway.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
07:07

Problem 140

For the water channel flow of Fig. $\mathrm{P} 3.140, h_{1}=1.5 \mathrm{m}$, $H=4 \mathrm{m},$ and $V_{1}=3 \mathrm{m} / \mathrm{s} .$ Neglecting losses and assuming uniform flow at sections 1 and 2 , find the downstream depth $h_{2},$ and show that $t w o$ realistic solutions are possible.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
07:07

Problem 141

For the water channel flow of Fig. $\mathrm{P} 3.141, h_{1}=0.45 \mathrm{ft}$ $H=2.2 \mathrm{ft},$ and $V_{1}=16 \mathrm{ft} / \mathrm{s} .$ Neglecting losses and assuming uniform flow at sections 1 and $2,$ find the downstream depth $h_{2} ;$ show that $t w o$ realistic solutions are possible.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
05:10

Problem 142

A cylindrical tank of diameter $D$ contains liquid to an initial height $h_{0} .$ At time $t=0$ a small stopper of diameter $d$ is removed from the bottom. Using Bernoulli's equation with no losses, derive ( $a$ ) a differential equation for the freesurface height $h(t)$ during draining and
$(b)$ an expression for the time $t_{0}$ to drain the entire tank.

Amany Waheeb
Amany Waheeb
Numerade Educator
05:04

Problem 143

The large tank of incompressible liquid in Fig. P3.143 is at rest when, at $t=0,$ the valve is opened to the atmosphere. Assuming $h \approx$ constant (negligible velocities and accelerations in the tank), use the unsteady friction-less Bernoulli equation to derive and solve a differential equation for $V(t)$ in the pipe.

Prashant Bana
Prashant Bana
Numerade Educator
01:45

Problem 144

A fire hose, with a 2 -in-diameter nozzle, delivers a water jet straight up against a ceiling $8 \mathrm{ft}$ higher. The force on the ceiling, due to momentum change, is 25 lbf. Use Bernoulli's equation to estimate the hose flow rate, in gal/min. [Hint: The water jet area expands upward.]

Anand Jangid
Anand Jangid
Numerade Educator
02:59

Problem 145

The incompressible flow form of Bernoulli's relation, Eq. $(3.54),$ is accurate only for Mach numbers less than about
$0.3 .$ At higher speeds, variable density must be accounted for. The most common assumption for compressible fluids is isentropic flow of an ideal gas, or $p=C \rho^{k},$ where $k=$ $c_{p} / c_{v}$ Substitute this relation into Eq. $(3.52),$ integrate, and eliminate the constant
$C .$ Compare your compressible result with Eq. (3.54) and comment.

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 146

The pump in Fig. $\mathrm{P} 3.146$ draws gasoline at $20^{\circ} \mathrm{C}$ from a reservoir. Pumps are in big trouble if the liquid vaporizes (cavitates) before it enters the pump. (a) Neglecting losses and assuming a flow rate of 65 gal/min, find the limitations on $(x, y, z)$ for avoiding cavitation.
(b) If pipe friction losses are included, what additional limitations might be important?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 147

The very large water tank in Fig. P3.147 is discharging through a 4 -in-diameter pipe. The pump is running, with a performance curve $h_{p} \approx 40-4 Q^{2},$ with $h_{p}$ in feet and $Q$ in $\mathrm{ft}^{3} / \mathrm{s} .$ Estimate the discharge flow rate in $\mathrm{ft}^{3} / \mathrm{s}$ if the pipe friction loss is $1.5\left(V^{2} / 2 g\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
03:35

Problem 148

By neglecting friction, $(a)$ use the Bernoulli equation between surfaces 1 and 2 to estimate the volume flow through the orifice, whose diameter is $3 \mathrm{cm} .(b)$ Why is the result to part $(a)$ absurd? $(c)$ Suggest a way to resolve this paradox and find the true flow rate.

Chai Santi
Chai Santi
Numerade Educator
02:05

Problem 149

The horizontal lawn sprinkler in Fig. $P 3.149$ has a water flow rate of 4.0 gal/min introduced vertically through the center. Estimate ( $a$ ) the retarding torque required to keep the arms from rotating and $(b)$ the rotation rate (r/min) if there is no retarding torque.

Chai Santi
Chai Santi
Numerade Educator
03:28

Problem 150

In Prob. P3.60 find the torque caused around flange 1 if the center point of exit 2 is $1.2 \mathrm{m}$ directly below the flange center.

Donald Albin
Donald Albin
Numerade Educator
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Problem 151

The wye joint in Fig. $P 3.151$ splits the pipe flow into equal amounts $Q / 2,$ which exit, as shown, a distance $R_{0}$ from the axis. Neglect gravity and friction. Find an expression for the torque $T$ about the $x$ axis required to keep the system rotating at angular velocity $\Omega$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:34

Problem 152

Modify Example 3.19 so that the arm starts from rest and spins up to its final rotation speed. The moment of inertia of the arm about $O$ is $I_{0} .$ Neglecting air drag, find $d \omega / d t$ and integrate to determine the angular velocity $\omega(t),$ assuming $\omega=0 \text { at } t=0$.

Manish Jain
Manish Jain
Numerade Educator
02:46

Problem 153

The three-arm lawn sprinkler of Fig. P3.153 receives $20^{\circ} \mathrm{C}$ water through the center at $2.7 \mathrm{m}^{3} / \mathrm{h}$. If collar friction is negligible, what is the steady rotation rate in $\mathrm{r} / \mathrm{min}$ for $(a) \theta=0^{\circ}$ and $(b) \theta=40^{\circ} ?$

Chai Santi
Chai Santi
Numerade Educator
06:43

Problem 154

Water at $20^{\circ} \mathrm{C}$ flows at 30 gal/min through the 0.75 -in-diameter double pipe bend of Fig. P3.154. The pressures are $p_{1}=30$ lbf/in $^{2}$ and $p_{2}=24$ lbf/in $^{2}$. Compute the torque $T$ at point $B$ necessary to keep the pipe from rotating.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
07:53

Problem 155

The centrifugal pump of Fig. $\mathrm{P} 3.155$ has a flow rate $Q$ and exits the impeller at an angle $\theta_{2}$ relative to the blades, as shown. The fluid enters axially at section 1. Assuming in-compressible flow at shaft angular velocity $\omega$, derive a formula for the power $P$ required to drive the impeller.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:21

Problem 156

A simple turbo-machine is constructed from a disk with two internal ducts that exit tangentially through square holes, as in Fig. P3.156. Water at $20^{\circ} \mathrm{C}$ enters normal to the disk at the center, as shown. The disk must drive, at $250 \mathrm{r} / \mathrm{min}$, a small device whose retarding torque is $1.5 \mathrm{N} \cdot \mathrm{m} .$ What is the proper mass flow of water, in $\mathrm{kg} / \mathrm{s} ?$

Penny Riley
Penny Riley
Numerade Educator
01:54

Problem 157

Reverse the flow in Fig. $P 3.155,$ so that the system operates as a radial-inflow turbine. Assuming that the outflow into section 1 has no tangential velocity, derive an expression for the power $P$ extracted by the turbine.

Narayan Hari
Narayan Hari
Numerade Educator
10:46

Problem 158

Revisit the turbine cascade system of Prob. P3.78, and derive a formula for the power $P$ delivered, using the angular momentum theorem of Eq. (3.59).

Jonathan Ibarra
Jonathan Ibarra
Numerade Educator
04:53

Problem 159

A centrifugal pump impeller delivers 4000 gal/min of water at $20^{\circ} \mathrm{C}$ with a shaft rotation rate of $1750 \mathrm{r} / \mathrm{min}$. Neglect losses. If $r_{1}=6$ in, $r_{2}=14$ in, $b_{1}=b_{2}=1.75$ in,$V_{t 1}=10 \mathrm{ft} / \mathrm{s},$ and $V_{t 2}=110 \mathrm{ft} / \mathrm{s},$ compute the absolute
velocities $(a) V_{1}$ and $(b) V_{2}$ and $(c)$ the horsepower required. $(d)$ Compare with the ideal horsepower required.

Narayan Hari
Narayan Hari
Numerade Educator
01:35

Problem 160

The pipe bend of Fig. $P 3.160$ has $D_{1}=27 \mathrm{cm}$ and $D_{2}=$ $13 \mathrm{cm} .$ When water at $20^{\circ} \mathrm{C}$ flows through the pipe at 4000 $\operatorname{gal/min}, p_{1}=194 \mathrm{kPa}$ (gage). Compute the torque required at point $B$ to hold the bend stationary.

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 161

Extend Prob. $\mathrm{P} 3.46$ to the problem of computing the center of pressure $L$ of the normal face $F_{n},$ as in Fig. P3.161. (At the center of pressure, no moments are required to hold the plate at rest.) Neglect friction. Express your result in terms of the sheet thickness $h_{1}$ and the angle $\theta$ between the plate and the oncoming jet 1.

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
03:58

Problem 162

The waterwheel in Fig. $P 3.162$ is being driven at $200 \mathrm{r} / \mathrm{min}$ by a 150 -ft/s jet of water at $20^{\circ} \mathrm{C}$. The jet diameter is 2.5 in. Assuming no losses, what is the horsepower developed by the wheel? For what speed $\Omega$ r/min will the horsepower developed be a maximum? Assume that there are many buckets on the waterwheel.

Narayan Hari
Narayan Hari
Numerade Educator
07:00

Problem 163

A rotating dishwasher arm delivers at $60^{\circ} \mathrm{C}$ to six nozzles, as in Fig. $\mathrm{P} 3.163 .$ The total flow rate is $3.0 \mathrm{gal} / \mathrm{min}$. Each nozzle has a diameter of $\frac{3}{16}$ in. If the nozzle flows are equal and friction is neglected, estimate the steady rotation rate of the arm, in r/min.

Supratim Pal
Supratim Pal
Numerade Educator
02:24

Problem 164

A liquid of density $\rho$ flows through a $90^{\circ}$ bend as shown in Fig. $\mathrm{P} 3.164$ and issues vertically from a uniformly porous section of length $L .$ Neglecting pipe and liquid weight, derive an expression for the torque $M$ at point 0 required to hold the pipe stationary.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:25

Problem 165

There is a steady isothermal flow of water at $20^{\circ} \mathrm{C}$ through the device in Fig. $\mathrm{P} 3.165 .$ Heat-transfer, gravity, and temperature effects are negligible. Known data are $D_{1}=9 \mathrm{cm}$, $Q_{1}=220 \mathrm{m}^{3} / \mathrm{h}, p_{1}=150 \mathrm{kPa}, D_{2}=7 \mathrm{cm}, Q_{2}=100 \mathrm{m}^{3} / \mathrm{h}$
$p_{2}=225 \mathrm{kPa}, D_{3}=4 \mathrm{cm},$ and $p_{3}=265 \mathrm{kPa}$. Compute the rate of shaft work done for this device and its direction.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:58

Problem 166

A power plant on a river, as in Fig. $\mathrm{P} 3.166,$ must eliminate $55 \mathrm{MW}$ of waste heat to the river. The river conditions upstream are $Q_{i}=2.5 \mathrm{m}^{3} / \mathrm{s}$ and $T_{i}=18^{\circ} \mathrm{C} .$ The river is $45 \mathrm{m}$ wide and $2.7 \mathrm{m}$ deep. If heat losses to the atmosphere and ground are negligible, estimate the downstream river conditions $\left(Q_{0}, T_{0}\right)$.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:31

Problem 167

For the conditions of Prob. P3.166, if the power plant is to heat the nearby river water by no more than $12^{\circ} \mathrm{C}$, what should be the minimum flow rate $Q,$ in $\mathrm{m}^{3} / \mathrm{s},$ through the plant heat exchanger? How will the value of $Q$ affect the downstream conditions $\left(Q_{0}, T_{0}\right) ?$

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
03:58

Problem 168

Multnomah Falls in the Columbia River Gorge has a sheer drop of $543 \mathrm{ft}$. Using the steady flow energy equation, estimate the water temperature change in $^{\circ} \mathrm{F}$ caused by this drop.

Dominique Jan Tan
Dominique Jan Tan
Numerade Educator
01:11

Problem 169

When the pump in Fig. $P 3.169$ draws $220 \mathrm{m}^{3} / \mathrm{h}$ of water at $20^{\circ} \mathrm{C}$ from the reservoir, the total friction head loss is $5 \mathrm{m}$. The flow discharges through a nozzle to the atmosphere. Estimate the pump power in $\mathrm{kW}$ delivered to the water.

Penny Riley
Penny Riley
Numerade Educator
04:08

Problem 170

A steam turbine operates steadily under the following conditions. At the inlet, $p=2.5 \mathrm{MPa}, T=450^{\circ} \mathrm{C},$ and $V=$ $40 \mathrm{m} / \mathrm{s} .$ At the outlet, $p=22 \mathrm{kPa}, T=70^{\circ} \mathrm{C},$ and $V=$ $225 \mathrm{m} / \mathrm{s}$. ( $a$ ) If we neglect elevation changes and heat transfer, how much work is delivered to the turbine blades, in $\mathrm{kJ} / \mathrm{kg} ?(b)$ If the mass flow is $10 \mathrm{kg} / \mathrm{s}$, how much total power is delivered?
$(c)$ Is the steam wet as it leaves the exit?

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
01:36

Problem 171

Consider a turbine extracting energy from a penstock in a $\operatorname{dam}, \text { as in Fig. } P 3.171 . \text { For turbulent pipe flow (Chap. } 6)$, the friction head loss is approximately $h_{f}=C Q^{2},$ where the constant $C$ depends on penstock dimensions and the properties of water. Show that, for a given penstock geometry and variable river flow $Q,$ the maximum turbine power possible in this case is $P_{\max }=2 \rho g H Q / 3$ and occurs when the flow rate is $Q=\sqrt{H /(3 C)}$.

Dominador Tan
Dominador Tan
Numerade Educator
17:03

Problem 172

The long pipe in Fig. $\mathrm{P} 3.172$ is filled with water at $20^{\circ} \mathrm{C}$. When valve $A$ is closed, $p_{1}-p_{2}=75$ kPa. When the valve is open and water flows at $500 \mathrm{m}^{3} / \mathrm{h}, p_{1}-p_{2}=160 \mathrm{kPa}$ What is the friction head loss between 1 and $2,$ in $\mathrm{m}$, for the flowing condition?

Ronald Prasad
Ronald Prasad
Numerade Educator
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Problem 173

A 36 -in-diameter pipeline carries oil $(\mathrm{SG}=0.89)$ at
1 million barrels per day (bbl/day) (1 bbl = 42 U.S. gal). The friction head loss is $13 \mathrm{ft} / 1000 \mathrm{ft}$ of pipe. It is planned to place pumping stations every $10 \mathrm{mi}$ along the pipe. Estimate the horsepower that must be delivered to the oil by each pump.

Victor Salazar
Victor Salazar
Numerade Educator
03:16

Problem 174

The pump-turbine system in Fig. P3.174 draws water from the upper reservoir in the daytime to produce power for a city. At night, it pumps water from lower to upper reservoirs to restore the situation. For a design flow rate of 15,000 gal/min in either direction, the friction head loss is 17 ft. Estimate the power in $\mathrm{kW}$ ( $a$ ) extracted by the turbine and $(b)$ delivered by the pump.

Chai Santi
Chai Santi
Numerade Educator
08:59

Problem 175

Water at $20^{\circ} \mathrm{C}$ is delivered from one reservoir to another
through a long 8 -cm-diameter pipe. The lower reservoir has a surface elevation $z_{2}=80 \mathrm{m} .$ The friction loss in the pipe is correlated by the formula $h_{\text {loss }} \approx 17.5\left(V^{2} / 2 g\right),$ where $V$ is the average velocity in the pipe. If the steady flow rate through the pipe is 500 gallons per minute, estimate the surface elevation of the higher reservoir.

Ronald Prasad
Ronald Prasad
Numerade Educator
04:36

Problem 176

A fireboat draws seawater $(\mathrm{SG}=1.025)$ from a submerged pipe and discharges it through a nozzle, as in Fig. $\mathrm{P} 3.176$. The total head loss is 6.5 ft. If the pump efficiency is 75 percent, what horsepower motor is required to drive it?

Anand Jangid
Anand Jangid
Numerade Educator
02:51

Problem 177

A device for measuring liquid viscosity is shown in Fig. P3.177. With the parameters $(\rho, L, H, d)$ known, the flow rate $Q$ is measured and the viscosity calculated, assuming a laminar-flow pipe loss from Chap. $6, h_{\mathrm{f}}=(32 \mu L V) /\left(\rho g d^{2}\right)$. Heat transfer and all other losses are negligible.
$(a)$ Derive a formula for the viscosity $\mu$ of the fluid.
(b) Calculate $\mu$ for the case $d=2 \mathrm{mm}, \rho=800 \mathrm{kg} / \mathrm{m}^{3}, L=95 \mathrm{cm}, H=30 \mathrm{cm}$, and $Q=760 \mathrm{cm}^{3} / \mathrm{h} .(c)$ What is your guess of the fluid in part $(b) ?(d)$ Verify that the Reynolds number $\operatorname{Re}_{d}$ is less than 2000 (laminar pipe flow).

Amit Srivastava
Amit Srivastava
Numerade Educator
04:48

Problem 178

The horizontal pump in Fig. P3.178 discharges 20^{ C water } at $57 \mathrm{m}^{3} / \mathrm{h}$. Neglecting losses, what power in $\mathrm{kW}$ is delivered to the water by the pump?

Ameer Said
Ameer Said
Numerade Educator
01:08

Problem 179

Steam enters a horizontal turbine at $350 \mathrm{lbf} / \mathrm{in}^{2}$ absolute, $580^{\circ} \mathrm{C},$ and $12 \mathrm{ft} / \mathrm{s}$ and is discharged at $110 \mathrm{ft} / \mathrm{s}$ and $25^{\circ} \mathrm{C}$
saturated conditions. The mass flow is $2.5 \mathrm{lbm} / \mathrm{s}$, and the heat losses are 7 Btu/lb of steam. If head losses are negligible, how much horsepower does the turbine develop?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
06:38

Problem 180

Water at $20^{\circ} \mathrm{C}$ is pumped at $1500 \mathrm{gal} / \mathrm{min}$ from the lower to the upper reservoir, as in Fig. P3.180. Pipe friction losses are approximated by $h_{f} \approx 27 V^{2} /(2 g),$ where $V$ is the average velocity in the pipe. If the pump is 75 percent efficient, what horsepower is needed to drive it?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:55

Problem 181

A typical pump has a head that, for a given shaft rotation rate, varies with the flow rate, resulting in a pump performance curve as in Fig. P3.181. Suppose that this pump is 75 percent efficient and is used for the system in Prob. $3.180 .$ Estimate $(a)$ the flow rate, in gal/min, and $(b)$ the horsepower needed to drive the pump.

Chai Santi
Chai Santi
Numerade Educator
05:02

Problem 182

The insulated tank in Fig. $P 3.182$ is to be filled from a highpressure air supply. Initial conditions in the tank are $T=$ $20^{\circ} \mathrm{C}$ and $p=200 \mathrm{kPa} .$ When the valve is opened, the initial mass flow rate into the tank is $0.013 \mathrm{kg} / \mathrm{s}$. Assuming an ideal gas, estimate the initial rate of temperature rise of the air in the tank.

Keshav Singh
Keshav Singh
Numerade Educator
07:36

Problem 183

The pump in Fig. $P 3.183$ creates a $20^{\circ} \mathrm{C}$ water jet oriented to travel a maximum horizontal distance. System friction head losses are $6.5 \mathrm{m}$. The jet may be approximated by the trajectory of frictionless particles. What power must be delivered by the pump?

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:27

Problem 184

The large turbine in Fig. $\mathrm{P} 3.184$ diverts the river flow under a dam as shown. System friction losses are $h_{f}=3.5 V^{2} /(2 g)$ where $V$ is the average velocity in the supply pipe. For what river flow rate in $\mathrm{m}^{3} / \mathrm{s}$ will the power extracted be $25 \mathrm{MW} ?$ Which of the $t w o$ possible solutions has a better "conversion efficiency"?

Penny Riley
Penny Riley
Numerade Educator
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Problem 185

Kerosine at $20^{\circ} \mathrm{C}$ flows through the pump in Fig. $\mathrm{P} 3.185$ at $2.3 \mathrm{ft}^{3} / \mathrm{s} .$ Head losses between 1 and 2 are $8 \mathrm{ft}$, and the pump delivers 8 hp to the flow. What should the mercury manometer reading $h$ ft be?

Victor Salazar
Victor Salazar
Numerade Educator