Air at $100^{\circ} \mathrm{C}$ enters a 125 -mm-diameter duct. Find the volume flow rate at which the flow becomes turbulent. At this flow rate, estimate the entrance length required to establish fully developed flow.

Satpal S.

Numerade Educator

Consider incompressible flow in a circular channel. Derive general expressions for Reynolds number in terms of (a) volume flow rate and tube diameter and (b) mass flow rate and tube diameter. The Reynolds number is 1800 in a section where the tube diameter is $10 \mathrm{mm}$. Find the Reynolds number for the same flow rate in a section where the tube diameter is $6 \mathrm{mm}$

Satpal S.

Numerade Educator

Air at $40^{\circ} \mathrm{C}$ flows in a pipe system in which diameter is decreased in two stages from $25 \mathrm{mm}$ to $15 \mathrm{mm}$ to $10 \mathrm{mm}$ Each section is $2 \mathrm{m}$ long. As the flow rate is increased, which section will become turbulent first? Determine the flow rates at which one, two, and then all three sections first become turbulent. At each of these flow rates, determine which sections, if any, attain fully developed flow.

Satpal S.

Numerade Educator

For flow in circular tubes, transition to turbulence usually Occurs around $R e \approx 2300 .$ Investigate the circumstances under which the flows of (a) standard air and (b) water at $15^{\circ} \mathrm{C}$ become turbulent. On log-log graphs, plot: the average velocity, the volume flow rate, and the mass flow rate, at which turbulence first occurs, as functions of tube diameter.

Suman Saurav T.

Numerade Educator

For the laminar flow in the section of pipe shown in Fig. 8.1 sketch the expected wall shear stress, pressure, and centerline velocity as functions of distance along the pipe. Fxplain significant features of the plots, comparing them with fully developed flow. Can the Bernoulli equation be applied anywhere in the llow field? If so, where? Explain brielly.

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$ \mathrm{An}$ incompressible fluid flows between two infinite stationary parallel plates. The velocity profile is given by $u=$ $u_{\max }\left(A y^{2}+B y+C\right),$ where $A, B,$ and $C$ are constants and $y$ is measured upward from the lower plate. The total gap width is $h$ units. Use appropriate boundary conditions to express the magnitude and units of the constants in terms of $h .$ Develop an expression for volume flow rate per unit depth and evaluate the ratio $\bar{V} / u_{\max }$

Satpal S.

Numerade Educator

The velocity profile for fully developed flow between stationary parallel plates is given by $u=a\left(h^{2} / 4-y^{2}\right),$ where $a$ is a constant, $h$ is the total gap width between plates, and $y$ is the distance measured from the center of the gap. Determine the ratio $V / u_{\max }$

Satpal S.

Numerade Educator

A fluid flows steadily between two parallel plates. The flow is fully developed and laminar. The distance between the plates is $h$

(a) Derive an equation for the shear stress as a function of $y$ Sketch this function.

(b) For $\mu=2.4 \times 10^{-5} \mathrm{lbf} \cdot \mathrm{s} / \mathrm{ft}^{2}, \quad \partial p / \partial x=-4.0 \mathrm{lbf} / \mathrm{ft}^{2} / \mathrm{n}$ and $h=0.05$ in.., calculate the maximum shear stress, in $\mathrm{lbf} / \mathrm{ft}^{2}$

Satpal S.

Numerade Educator

Oil is confined in a 4 -in.- -diameter cylinder by a piston having a radial clearance of 0.001 in. and a length of 2 in. $A$ steady force of 4500 lbf is applied to the piston. Assume the properties of SAE 30 oil at $120^{\circ} \mathrm{F}$. Estimate the rate at which oil leaks past the piston.

Suman Saurav T.

Numerade Educator

$ \mathrm{A}$ viscous oil flows steadily between stationary parallel plates. The flow is laminar and fully developed. The total gap width between the plates is $h=5 \mathrm{mm}$. The oil viscosity is $0.5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$ and the pressure gradient is $-1000 \mathrm{N} / \mathrm{m}^{2} / \mathrm{m} .$ Find the magnitude and direction of the shear stress on the upper plate and the volume flow rate through the channel, per meter of width.

Satpal S.

Numerade Educator

Viscous oil flows steadily between parallel plates. The flow is fully developed and laminar. The pressure gradient is $1.25 \mathrm{kPa} / \mathrm{m}$ and the channel half-width is $h=1.5 \mathrm{mm}$. Cal. culate the magnitude and direction of the wall shear stress at the upper plate surface. Find the volume flow rate through the channel $\left(\mu=0.50 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$

Suman Saurav T.

Numerade Educator

A large mass is supported by a piston of diameter $D=4$ in. and length $L=4$ in. The piston sits in a cylinder closed at the bottom, and the gap $a=0.001$ in. between the cylinder wall and piston is filled with SAE 10 oil at $68^{\circ} \mathrm{F}$. The piston slowly sinks due to the mass, and oil is forced out at a rate of 0.1 gpm. What is the mass (slugs)?

Satpal S.

Numerade Educator

A high pressure in a system is created by a small pistoncylinder assembly. The piston diameter is $6 \mathrm{mm}$ and it extends $50 \mathrm{mm}$ into the cylinder. The radial clearance between the piston and cylinder is $0.002 \mathrm{mm}$. Neglect elastic deformations of the piston and cylinder caused by pressure. Assume the fluid properties are those of SAE $10 \mathrm{W}$ oil at $35^{\circ} \mathrm{C} .$ When the pressure in the cylinder is $600 \mathrm{MPa}$, estimate the leakage rate.

Satpal S.

Numerade Educator

A hydraulic jack supports a load of 9000 kg. The following data are given: Diameter of piston $100 \mathrm{mm}$ Radial clearance between piston and cylinder $\quad 0.05 \mathrm{mm}$ Length of piston $120 \mathrm{mm}$

Estimate the rate of leakage of hydraulic fluid past the piston, assuming the fluid is SAE 30 oil at $30^{\circ} \mathrm{C}$.

Satpal S.

Numerade Educator

A hydrostatic bearing is to support a load of 1000 lbI/ft of length perpendicular to the diagram. The bearing is supplied with $\mathrm{SAE} 10 \mathrm{W} \cdot 30$ oil at $212^{\circ} \mathrm{F}$ and 35 psig through the central slit. since the oil is viscous and the gap is small, the flow may be considered fully developed. Calculate (a) the required width of the bearing pad, (b) the resulting pressure gradient, $d p / d x,$ and

(c) the gap height, if the flow rate is $Q=2.5$ galhrift.

Suman Saurav T.

Numerade Educator

The basic component of a pressure gage tester consists of a piston-cylinder apparatus as shown. The piston, $6 \mathrm{mm}$ in diameter, is loaded to develop a pressure of known magnitude. (The piston length is $25 \mathrm{mm}$.) Calculate the mass, $M,$ required to produce 1.5 MPa (gage) in the cylinder. Determine the leakage flow rate as a function of radial clearance, $a$, for this load if the liquid is SAE 30 oil at $20^{\circ} \mathrm{C}$. Specify the maximum allowable radial clearance so the vertical movement of the piston due to leakage will be less than $1 \mathrm{mm} / \mathrm{min}$

Satpal S.

Numerade Educator

In Section 8.2 we derived the velocity profile between parallel plates (Eq. 8.5 ) by using a differential control volume. Instead, following the procedure we used in Example $5.9,$ derive Eq. 8.5 by starting with the Navier $-$ Stokes equations (Eqs. 5.27 ). Be sure to state all assumptions.

Suman Saurav T.

Numerade Educator

Consider the simple power-law model for a nonNewtonian fluid given by Eq. 2.16. Extend the analysis of Section 8.2 to show that the velocity profile for fully developed laminar flow of a power-law fluid between stationary parallel plates separated by distance $2 h$ may be written $$u=\left(\frac{h}{k} \frac{\Delta p}{L}\right)^{1 / n} \frac{n h}{n+1}\left[1-\left(\frac{y}{h}\right)^{(n+1) / n}\right]$$ where $y$ is the coordinate measured from the channel centerline. Plot the profiles $u / U_{\max }$ versus $y / h$ for $n=0.7,1.0,$ and 1.3

Suman Saurav T.

Numerade Educator

Viscous liquid, at volume flow rate $Q,$ is pumped through the central opening into the narrow gap between the parallel disks shown. The flow rate is low, so the flow is laminar, and the pressure gradient due to convective acceleration in the gap is negligible compared with the gradient caused by viscous forces (this is termed creeping flow). Obtain a general expression for the variation of average velocity in the gap between the disks. For creeping flow, the velocity profile at any cross section in the gap is the same as for fully developed flow between stationary parallel plates. Evaluate the pressure gradient. $d p / d r,$ as a function of radius, Obtain an expression for $p(r)$. Show that the net force required to hold the upper plate in the position shown is

Suman Saurav T.

Numerade Educator

A sealed journal bearing is formed from concentric cylinders. The inner and outer radii are 25 and $26 \mathrm{mm}$, the journal length is $100 \mathrm{mm},$ and it turns at $2800 \mathrm{rpm}$. The gap is filled with oil in laminar motion. The velocity profile is linear across the gap. The torque needed to turn the journal is 0.2 $\mathrm{N} \cdot \mathrm{m} .$ Calculate the viscosity of the oil. Will the torque increase or decrease with time? Why?

Satpal S.

Numerade Educator

Using the profile of Problem $8.18,$ show that the flow rate for fully developed laminar flow of a power-law fluid between stationary parallel plates may be written as $$Q=\left(\frac{h}{k} \frac{\Delta p}{L}\right)^{1 / n} \frac{2 n w h^{2}}{2 n+1}$$ Here $w$ is the plate width. In such an experimental setup the following data on applied pressure difference $\Delta p$ and flow rate $Q$ were obtained:

$$\begin{aligned}

&\Delta p(\mathbf{k P a}) \quad 10 \quad 20 \quad 30 \quad 40 \quad 50 \quad 60 \quad 70 \quad 80 \quad 90 \quad 100\\

&\begin{array}{llllllll}

Q(\mathrm{I} / \mathrm{min}) & 0.451 & 0.759 & 1.01 & 1.15 & 1.41 & 1.57 & 1.66 & 1.85 & 2.05 & 2.25

\end{array}

\end{aligned}$$

Determine if the fluid is pseudoplastic or dilatant, and obtain an experimental value for $n$

Suman Saurav T.

Numerade Educator

Consider fully developed laminar flow between infinite parallel plates separated by gap width $d=0.2$ in. The upper plate moves to the right with speed $U_{2}=5 \mathrm{ft} / \mathrm{s}$; the lower plate moves to the left with speed $U_{1}=2 \mathrm{ft}$ /s. The pressure gradient in the direction of flow is zero. Develop an expression for the velocity distribution in the gap. Find the volume flow rate per unit depth (gpm/ft) passing a given cross section.

Suman Saurav T.

Numerade Educator

Water at $60^{\circ} \mathrm{C}$ flows between two large flat plates. The lower plate moves to the left at a speed of $0.3 \mathrm{m} / \mathrm{s}$; the upper plate is stationary. The plate spacing is $3 \mathrm{mm},$ and the flow is laminar. Determine the pressure gradient required to produce zero net flow at a cross section.

Suman Saurav T.

Numerade Educator

Two immiscible fluids are contained between infinite parallel plates. The plates are separated by distance $2 h$, and the two fluid layers are of equal thickness $h=5 \mathrm{mm}$. The dynamic viscosity of the upper fluid is four times that of the lower fluid, which is $\mu_{10 \text { wer }}=0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$. If the plates are stationary and the applied pressure gradient is $-50 \mathrm{kPa} / \mathrm{m}$ find the velocity at the interface. What is the maximum velocity of the flow? Plot the velocity distribution.

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Two immiscible fluids are contained between infinite parallel plates. The plates are separated by distance $2 h,$ and the two fluid layers are of equal thickness $h$; the dynamic viscosity of the upper fluid is three times that of the lower fluid. If the lower plate is stationary and the upper plate moves at constant speed $U=20 \mathrm{ft} / \mathrm{s}$, what is the velocity at the interface? Assume laminar flows, and that the pressure gradient in the direction of flow is zero.

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The record-read head for a computer disk-drive memory storage system rides above the spinning disk on a very thin film of air (the film thickness is $0.25 \mu \mathrm{m}$ ). The head location is $25 \mathrm{mm}$ from the disk centerline; the disk spins at $8500 \mathrm{rpm} .$ The record-read head is $5 \mathrm{mm}$ square. For standard air in the gap between the head and disk, determine

(a) the Reynolds number of the flow,

(b) the viscous shear stress, and (c) the power required to overcome viscous shear.

Satpal S.

Numerade Educator

The dimensionless velocity profile for fully developed laminar flow between infinite parallel plates with the upper plate moving at constant speed $U$ is shown in Fig. $8.6 .$ Find the pressure gradient $\partial p / \partial x$ at which (a) the upper plate and (b) the lower plate expericnce zero shear stress, in terms of $U, a,$ and $\mu .$ Plot the dimensionless velocity profiles for these cases

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Consider steady, fully developed laminar flow of a viscous liquid down an inclined surface. The liquid layer is of constant thickness, $h$, Use a suitably chosen differential control volume to obtain the velocity profile. Develop an expression for the volume flow rate.

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Consider steady, incompressible, and fully developed laminar flow of a viscous liquid down an incline with no pressure gradient. The velocity profile was derived in Frample $5.9 .$ Plot the velocity profile. Calculate the kinematic viscosity of the liquid if the film thickness on a $30^{\circ}$ slope is $0.8 \mathrm{mm}$ and the maximum velocity is $15.7 \mathrm{mm} / \mathrm{s}$

Satpal S.

Numerade Educator

Two immiscible fluids of equal density are flowing down a surface inclined at a $60^{\circ}$ angle. The two fluid layers are of equal thickness $h=10 \mathrm{mm}$; the kinematic viscosity of the upper fluid is $1 / 5$ th that of the lower fluid, which is $\nu_{\text {lower }}=$ $0.01 \mathrm{m}^{2} / \mathrm{s} .$ Find the velocity at the interface and the velocity at the free surface. Plot the velocity distribution.

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The velocity distribution for flow of a thin viscous film down an inclined plane surface was developed in Example 5.9. Consider a film $7 \mathrm{mm}$ thick, of liquid with $\mathrm{SG}=1.2$ and dynamic viscosity of $1.60 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$. Derive an expression for the shear stress distribution within the film. Calculate the maximum shear stress within the film and indicate its direction. Evaluate the volume flow rate in the film, in $\mathrm{mm}^{3} / \mathrm{s}$ per millimeter of surface width. Calculate the film Reynolds number based on average velocity.

Satpal S.

Numerade Educator

Consider fully developed flow between parallel plates with the upper plate moving at $U=5$ ft/s; the spacing between the plates is $a=0.1$ in. Determine the flow rate per unit depth for the case of zero pressure gradicnt. If the fluid is air, evaluate the shear stress on the lower plate and plot the shear stress distribution across the channel for the zero pressure gradient case. Will the flow rate increase or decrease if the pressure gradient is adverse? Determine the pressure gradient that will give zero shear stress at $y=0.25 a$ Plot the shear stress distribution across the channel for the latter case.

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Glycerin at $59^{\circ} \mathrm{F}$ flows between parallel plates with gap width $b=0.1$ in. The upper plate moves with speed $U=2 \mathrm{fl} / \mathrm{s}$ in the positive $x$ direction. The pressure gradient is $\partial p / \partial x=$ -50 psi/ft. Locate the point of maximum velocity and determine its magnitude (let $y=0$ at the bottom plate). Determine the volume of flow (gal/ft) that passes a given cross section $(x=\text { constant })$ in 10 s. Plot the velocity and shear stress distributions.

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The velocity profile for fully developed flow of castor oil at $20^{\circ} \mathrm{C}$ between parallel plates with the upper plate moving is given by Eq. 8.8 . Assume $U=1.5 \mathrm{m} / \mathrm{s}$ and $a=5$ $\mathrm{mm} .$ Find the pressure gradient for which there is no net flow in the $x$ direction. Plot the expected velocity distribution and the expected shear stress distribution across the channel for this flow, For the case where $u=1 / 3 \mathrm{U}$ at $y / a=0.5,$ plot the expected velocity distribution and shear stress distribution across the channel. Comment on features of the plots.

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The velocity profile for fully developed flow of carbon tetrachloride at $68^{\circ} \mathrm{F}$ between parallel plates (gap $a=$ $0.05 \text { in. }),$ with the upper plate moving, is given by Eq. 8.8 Assuming a volume flow rate per unit depth is 1.5 gpm/ft for zero pressure gradient, find $U$. Evaluate the shear stress on the lower plate. Would the volume flow rate increase or decrease with a mild adverse pressure gradient? Calculate the pressure gradient that will give zero shear stress at $y / a=0.25 .$ Plot the velocity distribution and the shear stress distribution for this case.

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Free-surface waves begin to form on a laminar liquid film flowing down an inclined surface whenever the Reynolds number, based on mass flow per unit width of film, is larger than about 33 . Estimate the maximum thickness of a laminar film of water that remains free from waves while flowing down a vertical surface.

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Microchips are supported on a thin air film on a smooth horizontal surface during one stage of the manufacturing process. The chips are $11.7 \mathrm{mm}$ long and $9.35 \mathrm{mm}$ wide and have a mass of 0.325 g. The air film is 0.125 mm thick. The initial speed of a chip is $V_{0}=1.75 \mathrm{mm} / \mathrm{s} ;$ the chip slows as the result of viscous shear in the air film. Analyze the chip motion during deceleration to develop a differential equation for chip speed $V$ versus time $t .$ Calculate the time required for a chip to lose 5 percent of its initial speed. Plot the variation of chip speed versus time during deceleration. Explain why it looks as you have plotted it.

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A viscous-shear pump is made from a stationary housing with a close-fitting rotating drum inside. The clearance is small compared with the diameter of the drum, so flow in the annular space may be treated as flow between parallel plates. Fluid is dragged around the annulus by viscous forces. Fvaluate the performance characteristics of the shear pump (pressure differential, input power, and efficiency) as functions of volume flow rate. Assume that the depth normal to the diagram is $b$

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The clamping force to hold a part in a metal-turning operation is provided by high-pressure oil supplied by a pump. Oil leaks axially through an annular gap with diameter $D$ Icngth $L,$ and radial clearance $a$. The inner member of the annulus rotates at angular speed $\omega$, Power is required both to pump the oil and to overcome viscous dissipation in the annular gap. Develop expressions in terms of the specified geometry for the pump power, $9_{p^{\prime}}$ and the viscous dissipation power, $9_{v}$ Show that the total power requirement is minimized when the radial clearance, $a$, is chosen such that $\mathscr{P}_{\mathrm{L}}=3 \mathscr{P}_{p}$

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The efficiency of the viscous-shear pump of Fig. $P 8.39$ is given by $$\eta=6 q \frac{(1-2 q)}{(4-6 q)}$$ where $q=Q / a b R \omega$ is a dimensionless flow rate $(Q$ is the Aow rate at pressure differential $\Delta p,$ and $b$ is the depth normal to the diagram). Plot the efficiency versus dimensionless flow rate, and find the flow rate for maximum efficiency. Explain why the efficiency peaks, and why it is zero at certain values of $q$

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Automotive design is tending toward all-wheel drive to improve vehicle performance and safety when traction is poor. An all-wheel drive vehicle must have an interaxle differential to allow operation on dry roads. Numerous vehicles are being built using multiplate viscous drives for interaxle differentials. Perform the analysis and design needed to define the torque transmitted by the differential for a given speed difference, in terms of the design parameters. Identify suitable dimensions for a viscous differential to transmit a torque of $150 \mathrm{N} \cdot \mathrm{m}$ at a speed loss of $125 \mathrm{rpm}$ using lubricant with the properties of SAE 30 oil. Discuss how to find the minimum material cost for the viscous differential, if the plate cost per square meter is constant.

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An inventor proposes to make a "viscous timer" by placing a weighted cylinder inside a slightly larger cylinder containing viscous liquid, creating a narrow annular gap close to the wall. Analyze the flow field created when the apparatus is inverted and the mass begins to fall under gravity. Would this system make a satisfactory timer? If so, for what range of time intervals? What would be the effect of a temperature change on measured time?

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A journal bearing consists of a shaft of diameter $D=35$ $\mathrm{mm}$ and length $I_{0}=50 \mathrm{mm}$ (moment of inertia $I=0.125$ $\left.\mathrm{kg} \cdot \mathrm{m}^{2}\right)$ installed symmetrically in a stationary housing such that the annular gap is $\delta=1 \mathrm{mm}$. The fluid in the gap has viscosity $\mu=0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$. If the shaft is given an initial angular velocity of $\omega=500 \mathrm{rpm},$ determine the time for the shaft to slow to $100 \mathrm{rpm} .$ On another day, an unknown fluid is tested in the same way, but takes 10 minutes to slow from 500 to $100 \mathrm{rpm} .$ What is its viscosity?

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In Example 8.3 we derived the velocity profile for laminar flow on a vertical wall by using a differential control volume. Instead, following the procedure we used in Example $5.9,$ derive the velocity profile by starting with the Navier-Stokes equations (Fqs. 5.27 ). Be sure to state all assumptions.

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A continuous belt, passing upward through a chemical bath at speed $U_{0},$ picks up a liquid film of thickness $h,$ density $\rho,$ and viscosity $\mu .$ Gravity tends to make the liquid drain down, but the movement of the belt keeps the liquid from running off completely. Assume that the flow is fully developed and laminar with zero pressure gradicnt, and that the atmosphere produces no shear stress at the outer surface of the film. State clearly the boundary conditions to be satisfied by the velocity at $y=0$ and $y=h$. Obtain an expression for the velocity profile.

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A wet paint film of uniform thickness, $\delta$, is painted on a vertical wall. The wet paint can be approximated as a Bingham fluid with a yield stress, $\tau_{y},$ and density, $\rho .$ Derive an expression for the maximum value of $\delta$ that can be sustained without having the paint flow down the wall. Calculate the maximum thickness for lithographic ink whose yield stress $\tau_{y}=40$ Pa and density is approximately $1000 \mathrm{kg} / \mathrm{m}^{3}$

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When dealing with the lubrication of bearings, the governing equation describing pressure is the Reynolds equation, generally written in 10 as $$\frac{d}{d x}\left(\frac{h^{3}}{\mu} \frac{d p}{d x}\right)+6 U \frac{d h}{d x}=0$$ where $h$ is the step height and $U$ is the velocity of the lower surface. Step bearings have a relatively simple design and are used with low-viscosity fluids such as water, gasoline, and solvents. The minimum film thickness in these applications is quite small. The step height must be small enough for good load capacity, yet large enough for the bearing to accommodate some wear without losing its load capacity by becoming smooth and flat. Beginning with the 1 D equation for fluid motion in the $x$ direction, show that the pressure distribution in the step bearing is as shown, where $$p_{s}=\frac{6_{p}\left(h_{2}-h_{1}\right)}{\frac{h_{1}^{3}}{L_{1}}+\frac{h_{2}^{3}}{L_{2}}}$$

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Consider first water and then SAF $10 \mathrm{W}$ lubricating oil flowing at $40^{\circ} \mathrm{C}$ in a $6-\mathrm{mm}$ -diameter tube. Determine the maximum flow rate (and the corresponding pressure gradient, \partialp/\partialx) for each fluid at which laminar flow would be expected.

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For fully developed laminar flow in a pipe, determine the radial distance from the pipe axis at which the velocity equals the average velocity.

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Using Eq. A.3 in Appendix A for the viscosity of water, find the viscosity at $-20^{\circ} \mathrm{C}$ and $120^{\circ} \mathrm{C}$. Plot the viscosity over this range. Find the maximum laminar flow rate (Lhr) in a 7.5 -mm-diameter tube at these temperatures. Plot the max. imum laminar flow rate over this temperature range.

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$ \mathrm{A}$ hypodermic needle, with inside diameter $d=0.005 \mathrm{in}$ and length $L=1$ in., is used to inject saline solution with viscosity five times that of water. The plunger diameter is $D=0.375$ in.; the maximum force that can be exerted by a thumb on the plunger is $F=7.5$ lbf. Estimate the volume flow rate of saline that can be produced.

Rashmi S.

Numerade Educator

In engineering science, there are often analogies to be made between disparate phenomena. For example, the applied pressure difference, $\Delta p,$ and corresponding volume flow rate, $Q,$ in a tube can be compared to the applied DC voltage, $V,$ across and current, $I,$ through an electrical resistor, respectively. By analogy, find a formula for the "resistance" of laminar flow of fluid of viscosity, $\mu$, in a tube length of $L$ and diameter $D$, corresponding to electrical resistance, $R$. For a tube $250 \mathrm{mm}$ long with inside diameter $7.5 \mathrm{mm},$ find the maximum flow rate and pressure difference for which this analogy will hold for (a) kerosene and (b) castor oil (both at $40^{\circ} \mathrm{C}$ ). When the flow exceeds this maximum, why does the analogy fail?

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Consider fully developed laminar flow in the annulus between two concentric pipes. The outer pipe is stationary, and the inner pipe moves in the $x$ direction with speed $V$ Assume the axial pressure gradient is zero $(\partial p / \partial x=0)$ Obtain a general expression for the shear stress, $\tau,$ as a function of the radius, $r,$ in terms of a constant, $C_{1}$. Obtain a general expression for the velocity profile, $u(r),$ in terms of two constants, $C_{1}$ and $C_{2}$. Obtain expressions for $C_{1}$ and $C_{2}$.

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Consider fully developed laminar flow in a circular pipe. Use a cylindrical control volume as shown. Indicate the forces acting on the control volume. Using the momentum equation, develop an expression for the velocity distribution.

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Consider fully developed laminar flow in the annular space formed by the two concentric cylinders shown in the diagram for Problem $8.53,$ but with pressure gradient, $\partial p / \partial x$ and the inner cylinder stationary. Let $r_{0}=R$ and $r_{i}=k R$ Show that the velocity profile is given by $$u=-\frac{R^{2}}{4 \mu} \frac{\partial p}{\partial x}\left[1-\left(\frac{r}{R}\right)^{2}+\left(\frac{1-k^{2}}{\ln (1 / k)}\right) \ln \frac{r}{R}\right]$$ Obtain an expression for the location of the maximum velocity as a function of $k$. Plot the location of maximum velocity $(\alpha=r / R)$ as a function of radius ratio $k$. Compare the limiting case, $k \rightarrow 0,$ with the corresponding expression for flow in a circular pipe.

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For the flow of Problem 8.55 show that the volume flow rate is given by $$Q=-\frac{\pi R^{4}}{8 \mu} \frac{\partial p}{\partial x}\left[\left(1-k^{4}\right)-\frac{\left(1-k^{2}\right)^{2}}{\ln (1 / k)}\right]$$ Find an expression for the average velocity. Compare the limiting case, $k \rightarrow 0,$ with the corresponding expression for flow in a circular pipe.

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It has been suggested in the design of an agricultural sprinkler that a structural member be held in place by a wire placed along the centerline of a pipe; it is surmised that a relatively small wire would have little effect on the pressure drop for a given flow rate. Using the result of Problem $8.56,$ derive an expression giving the percentage change in pressure drop as a function of the ratio of wire diameter to pipe diameter for laminar flow. Plot the percentage change in pressure drop as a function of radius ratio $k$ for $0.001 \leq$ $k \leq 0.10$

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Consider fully developed pressure-driven flow in a cylindrical tube of radius, $R$, and length, $L=10 \mathrm{mm}$, with flow generated by an applied pressure gradient, $\Delta p$, Tests are performed with room temperature water for various values of $R,$ with a fixed flow rate of $Q=10 \mu \mathrm{L} / \mathrm{min}$. The hydraulic resistance is defined as $R_{\mathrm{hyd}}=\Delta p / Q$ (by analogy with the electrical resistance $R_{\text {elec }}=\Delta V / I,$ where $\Delta V$ is the electrical potential drop and $I$ is the electric current). Calculate the required pressure gradient and hydraulic resistance for the range of tube radii listed in the table. Based on the results, is it appropriate to use a pressure gradient to pump fluids in microchannels, or should some other driving mechanism be used?

$$\begin{array}{|l|l|l|}

\hline R(\mathrm{mm}) & \Delta p(\mathrm{Pa}) & R_{\text {hyd }}\left(\mathrm{Pa} \cdot \mathrm{s} / \mathrm{m}^{3}\right) \\

\hline 1 & & \\

\hline 10^{-1} & & \\

\hline 10^{-2} & & \\

\hline 10^{-3} & & \\

\hline 10^{-4} & & \\

\hline

\end{array}$$

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The figure schematically depicts a conical diffuser, which is designed to increase pressure and reduce kinetic energy. We assume the angle $\alpha$ is small $\left(\alpha<10^{\circ}\right)$ so that $\tan \alpha \approx \alpha$ and $r_{r}=r_{i}+\alpha l,$ where $r_{i}$ is the radius at the diffuser inlet, $r_{e}$ is the radius at the exit, and $l$ is the length of the diffuser. The flow in a diffuser is complex, but here we assume that each layer of fluid in the diffuser flow is laminar, as in a cylindrical tube with constant cross-sectional area. Based on reasoning similar to that in Section $8.3,$ the pressure difference $\Delta p$ between the ends of a cylindrical pipe is

$$\Delta p=\frac{8 \mu}{\pi} Q \int_{0}^{x} \frac{1}{r^{4}} d x$$ where $x$ is the location in the diffuser, $\mu$ is the fluid dynamic viscosity, and $Q$ is the flow rate. The equation above is applicable to flows in a diffuser assuming that the inertial force and exit effects are negligible. Derive the hydraulic resistance, $R_{\mathrm{hyd}}=\Delta p / Q,$ of the diffuser.

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Consider blood flow in an artery. Blood is nonNewtonian; the shear stress versus shear rate is described by the Casson relationship:

$$\left\{\begin{array}{ll}

\sqrt{\tau}=\sqrt{\tau_{c}}+\sqrt{\mu \frac{d u}{d r}} & \text { for } \tau \geq \tau_{c} \\

\tau=0 & \text { for } \tau<\tau_{c}

\end{array}\right.$$

where $\tau_{e}$ is the critical shear stress, and $\mu$ is a constant having the same dimensions as dynamic viscosity. The Casson relationship shows a linear relationship between $\sqrt{\tau}$ and $\sqrt{d u / d r},$ with intercept $\sqrt{\tau_{c}}$ and slope $\sqrt{\mu} .$ The Casson relationship approaches Newtonian behavior at high values of deformation rate. Derive the velocity profile of steady fully developed blood flow in an artery of radius $R$. Determine the flow rate in the blood vessel. Calculate the flow rate due to a pressure gradient $d p / d x=-100 \mathrm{Pa} / \mathrm{m},$ in an artery of radius $R=1 \mathrm{mm},$ using the following blood data: $\mu=3.5 \mathrm{cP}$ $\tau_{c}=0.05$ dynes $/ \mathrm{cm}^{2}$

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Using Eq. $2.16,$ derive the velocity profile, flow rate, and average velocity of a non-Newtonian fluid in a circular tube. For a flow rate of $Q=1 \mu \mathrm{L} / \mathrm{min}$ and $R=1 \mathrm{mm},$ with $k$ having a value of unity in standard SI units, compare the required pressure gradients for $n=0.5,1.0,$ and $1.5 .$ Which fluid requires the smallest pump for the same pipe length?

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The classic Poiseuille flow (Eq. 8.12 ), is for no-slip conditions at the walls. If the fluid is a gas, and when the mean free path, $l$ (the average distance a molecule travels before collision with another molecule), is comparable to the length-scale $L$ of the flow, slip will occur at the walls, and the flow rate and velocity will be increased for a given pressure gradient. In Eq. $8.11, c_{1}$ will still be zero, but $c_{2}$ must satisfy the slip condition $u=l \partial u / \partial r$ at $r=R .$ Derive the velocity profile and flow rate of gas flow in a micro- or nanotube which has such a slip velocity on the wall. Calculate the flow rate when $R=10 \mathrm{m}, \mu=1.84 \times 10^{-5} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$ the mean free path $l=68 \mathrm{nm},$ and $-\partial p / \partial x=1.0 \times 10^{6} \mathrm{Pa} / \mathrm{m}$

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The following solution: $$u=u_{0}\left(1-\frac{y^{2}}{a^{2}}-\frac{z^{2}}{b^{2}}\right)$$ can be used as a model for the velocity profile of fully developed pressure-driven flow in a channel with an elliptic cross section. The center of the ellipse is $(y, z)=(0,0),$ and the major axis of length $a$ and the minor axis of length $b$ are parallel to the $y$ axis and $z$ axes, respectively. The axial pressure gradient, $\partial p / \partial x,$ is constant. Based on the NavierStokes equations, determine the maximum velocity $u_{0}$ in terms of $a, b,$ viscosity $\mu,$ and $\partial p / \partial x,$ Letting $(\rho, \phi)$ be the radial and azimuthal polar coordinates, respectively, of a unit disk $(0 \leq \rho \leq 1 \text { and } 0 \leq \phi \leq 2 \pi),$ the coordinates $(y, z)$ and the velocity $u(y, z)$ can be expressed as functions of $(\rho, \phi)$ $$y(\rho, \phi)=a \rho \cos \phi \quad z(\rho, \phi)=b \rho \sin \phi \quad u(\rho, \phi)=u_{0}\left(1-\rho^{2}\right)$$ The flow rate is $Q=\int u(y, z) d y d z=a b \int_{0}^{2 \pi} \int_{0}^{1} \rho u(\rho, \phi) d \rho d \phi$

Derive the flow rate of fully developed pressure-driven flow in an elliptic pipe. Compare the flow rates in a channel with an elliptic cross section with $a=1.5 R$ and $b=R$ and in a pipe of radius $R$ with the same pressure gradient.

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For pressure-driven, steady, fully developed laminar flow of an incompressible fluid through a straight channel of length $I_{1},$ we can define the hydraulic resistance as $R_{\text {hyd }}=$ $\Delta p / Q,$ where $\Delta p$ is the pressure drop and $Q$ is the flow rate (analogous to the electrical resistance $R_{\mathrm{cloc}}=\Delta V / I,$ where $\Delta V$ is the electrical potential drop and $I$ is the electric current). The following table summarizes the hydraulic resistance of channels with different cross sectional shapes [30] Calculate the hydraulic resistance of a straight channel with the listed cross-sectional shapes using the following parameters:

$\mu=1 \mathrm{mPa} \cdot \mathrm{s}(\text { water }), L=10 \mathrm{mm}, a=100 \mu \mathrm{m}, b=33 \mu \mathrm{m}$ $h=100 \mu \mathrm{m},$ and $w=300 \mathrm{gm} .$ Based on the calculated hydraulic resistance, which shape is the most energy efficient to pump water?

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8.65 In a food industry plant, two immiscible fluids are pumped through a tube such that Iluid $1\left(\mu_{1}=0.5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$ forms an inner core and fluid $2\left(\mu_{2}=5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$ forms an outer annulus. The tube has $D=5 \mathrm{mm}$ diameter and length $L=5 \mathrm{m}$. Derive and plot the velocity distribution if the applied pressure difference, $\Delta p,$ is 5 MPa.

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A horizontal pipe carries fluid in fully developed turbulent flow. The static pressure difference measured between two sections is 750 psi. The distance between the sections is $15 \mathrm{ft}$, and the pipe diameter is 3 in. Calculate the shear stress, $\tau_{w},$ that acts on the walls.

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One end of a horizontal pipe is attached using glue to a pressurized tank containing liquid, and the other has a cap attached. The inside diameter of the pipe is 3 in., and the tank pressure is 30 psig. Find the force the glue must withstand with the cap on, and the force it must withstand when the cap is off and the liquid is discharging to atmosphere.

Satpal S.

Numerade Educator

Kerosene is pumped through a smooth tube with inside diameter $D=30 \mathrm{mm}$ at close to the critical Reynolds number. The flow is unstable and fluctuates hetween laminar and turbulent states, causing the pressure gradient to intermittently change from approximately $-4.5 \mathrm{kPa} / \mathrm{m}$ to $-11 \mathrm{kPa} / \mathrm{m} .$ Which pressure gradient corresponds to laminar. and which to turbulent, flow? For each flow, compute the shear stress at the tube wall, and sketch the shear stress distributions.

Satpal S.

Numerade Educator

The pressure drop between two taps separated in the streamwise direction by $30 \mathrm{ft}$ in a horizontal, fully developed channel flow of water is 1 psi. The cross section of the channel is a 1 in. $\times 9 \frac{1}{2}$ in. rectangle. Calculate the average wall shear stress.

Satpal S.

Numerade Educator

A liquid drug, with the viscosity and density of water, is to be administered through a hypodermic needle. The inside diameter of the needle is $0.25 \mathrm{mm}$ and its length is $50 \mathrm{mm}$. Determine (a) the maximum volume flow rate for which the flow will be laminar, (b) the pressure drop required to deliver the maximum flow rate, and (c) the corresponding wall shear stress.

Satpal S.

Numerade Educator

The "pitch-drop" experiment has been running continuously at the University of Queensland since 1927 (http Www.physics.uq.edu.au/physics_museum/pitchdrop.shtml). In this experiment, a funnel pitch is being used to measure the viscosity of pitch. Flow averages at about one drop $-p e r$ decade! Viscosity is calculated using the volume flow rate equation $$Q=\frac{\Psi}{t}=\frac{\pi D^{4} \rho g}{128 \mu}\left(1+\frac{h}{L}\right)$$ where $D$ is the diameter of the flow from the funnel, $h$ is the depth to the pitch in the main body of the funnel, $L$ is the length of the funnel stem, and $t$ is the elapsed time. Compare this equation with Fq. $8.13 \mathrm{c}$ using hydrostatic force instead of a pressure gradient. After the 6 th drop in $1979,$ they measured that it took 17,708 days for $4.7 \times 10^{-5} \mathrm{m}^{3}$ of pitch to fall. Given the measurements $D=9.4 \mathrm{mm}, h=75 \mathrm{mm}$ $I_{1}=29 \mathrm{mm},$ and $\rho_{\text {pitch }}=1.1 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3},$ what is the viscosity of the pitch?

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Consider the empirical "power-law" profile for turbuIent pipe flow, Eq. $8.22 .$ For $n=7$ determine the value of $r / R$ at which $u$ is equal to the average velocity, $\bar{V}$. Plot the results over the range $6 \leq n \leq 10$ and compare with the case of fully developed laminar pipe flow, Eq. 8.14

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Laufer [5] measured the following data for mean velocity in fully developed turbulent pipe flow at $R c_{U}=50,000$

$$\begin{array}{llllllll}

\bar{u} / U & 0.996 & 0.981 & 0.963 & 0.937 & 0.907 & 0.866 & 0.831 \\

y / r & 0.898 & 0.794 & 0.691 & 0.588 & 0.486 & 0.383 & 0.280 \\

\bar{u} / U & 0.792 & 0.742 & 0.700 & 0.650 & 0.619 & 0.551 & \\

y / R & 0.216 & 0.154 & 0.093 & 0.062 & 0.041 & 0.024 &

\end{array}$$

In addition, Laufer measured the following data for mean velocity in fully developed turbulent pipe flow at $R e_{U}=500,000$

$$\begin{array}{lllllll}

\hline u / U & 0.997 & 0.988 & 0.975 & 0.959 & 0.934 & 0.908 \\

y / R & 0.898 & 0.794 & 0.691 & 0.588 & 0.486 & 0.383 \\

\bar{u} / U & 0.874 & 0.847 & 0.818 & 0.771 & 0.736 & 0.690 \\

y / R & 0.280 & 0.216 & 0.154 & 0.093 & 0.062 & 0.037

\end{array}$$

Using Excel's trendline analysis, fit each set of data to the "power-law" profile for turbulent flow, Eq. 8.22 , and obtain a value of $n$ for each set. Do the data tend to confirm the validity of Eq. $8.22 ?$ Plot the data and their corresponding trendlines on the same graph.

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Equation 8.23 gives the power-law velocity profile exponent, $n,$ as a function of centerline Reynolds number, Rev, for fully developed turbulent flow in smooth pipes. Equation 8.24 relates mean velocity, $V$, to centerline velocity, $U,$ for various values of $n .$ Prepare a plot of $\bar{V} / U$ as a function of Reynolds number, $R e_{\mathbb{V}}$

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A momentum coefficient, $\beta,$ is defined by $$\int_{A} u \rho u d A=\beta \int_{A} V \rho u d A=\beta \dot{m} V$$ Evaluate $\beta$ for a laminar velocity profile, Eq. 8.14 , and for a "power-law" turbulent velocity profile, Eq. $8.22 .$ Plot $\beta$ as a function of $n$ for turbulent power-law profiles over the range $6 \leq n \leq 10$ and compare with the case of fully developed laminar pipe flow.

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Consider fully developed laminar flow of water between stationary parallel plates. The maximum flow speed, plate spacing, and width are $20 \mathrm{ft} / \mathrm{s}, 0.075$ in. and 1.25 in., respectively. Find the kinetic energy coefficient, $\alpha$

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Consider fully developed laminar flow in a circular tube. Evaluate the kinetic energy coefficient for this flow.

Satpal S.

Numerade Educator

Show that the kinetic energy coefficient, $\alpha$, for the "power law" turbulent velocity profile of Eq. 8.22 is given by Eq. 8.27 Plot $a$ as a function of $R e_{\bar{v}},$ for $R e_{\bar{v}}=1 \times 10^{4}$ to $1 \times 10^{7}$ When analyzing pipe flow problems it is common practice to assume $\alpha \approx 1 .$ Plot the error associated with this assumption as a function of $R e_{\bar{Y}},$ for $R e_{\bar{Y}}=1 \times 10^{4}$ to $1 \times 10^{7}$

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Measurements are made for the flow configuration shown in Fig. $8.12 .$ At the inlet, section (1), the pressure is $70 \mathrm{kPa}$ (gage), the average velocity is $1.75 \mathrm{m} / \mathrm{s}$, and the ele. vation is $2.25 \mathrm{m}$. At the outlet, section $(2),$ the pressure, average velocity, and elevation are 45 kPa (gage), $3.5 \mathrm{m} / \mathrm{s}$ and $3 \mathrm{m}$, respectively. Calculate the head loss in meters. Convert to units of energy per unit mass.

Satpal S.

Numerade Educator

Water flows in a horizontal constant-area pipe; the pipe diameter is $75 \mathrm{mm}$ and the average flow speed is $5 \mathrm{m} / \mathrm{s}$. At the pipe inlet, the gage pressure is $275 \mathrm{kPa}$, and the outlet is at atmospheric pressure. Determine the head loss in the pipe. If the pipe is now aligned so that the outlet is $15 \mathrm{m}$ above the inlet, what will the inlet pressure need to be to maintain the same flow rate? If the pipe is now aligned so that the outlet is $15 \mathrm{m}$ below the inlet, what will the inlet pressure need to be to maintain the same flow rate? Finally, how much lower than the inlet must the outlet be so that the same

flow rate is maintained if both ends of the pipe are at atmospheric pressure (i.e., gravity feed)?

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For the flow configuration of Fig. $8.12,$ it is known that the head loss is $1 \mathrm{m}$. The pressure drop from inlet to outlet is $50 \mathrm{kPa},$ the velocity doubles from inlet to outlet, and the elevation increase is $2 \mathrm{m}$. Compute the inlet water velocity.

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For a given volume flow rate and piping system, will the pressure loss be greater for hot water or cold water? Explain.

Satpal S.

Numerade Educator

Consider the pipe flow from the water tower of Example 8.7 . After another 5 years the pipe roughness has increased such that the flow is fully turbulent and $f=0.035$ Find by how much the flow rate is decreased.

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Consider the pipe flow from the water tower of Prob. lem 8.83 . To increase delivery, the pipe length is reduced from 600 ft to 450 ft (the flow is still fully turbulent and $f=0.035$ ). What is the flow rate?

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Water flows from a horizontal tube into a large tank. The tube is located $2.5 \mathrm{m}$ below the free surface of water in the tank. The head loss is $2 \mathrm{J} / \mathrm{kg}$. Compute the average flow speed in the tube.

Satpal S.

Numerade Educator

The average flow speed in a constant-diameter section of the Alaskan pipeline is $2.5 \mathrm{m} / \mathrm{s}$. At the inlet, the pressure is $8.25 \mathrm{MPa}$ (gage) and the elevation is $45 \mathrm{m} ;$ at the outlet, the pressure is $350 \mathrm{kPa}(\text { gage })$ and the elevation is $115 \mathrm{m}$. Calculate the head loss in this section of pipeline.

Satpal S.

Numerade Educator

At the inlet to a constant-diameter section of the Alaskan pipeline, the pressure is $8.5 \mathrm{MPa}$ and the elevation is $45 \mathrm{m} ;$ at the outlet the elevation is $115 \mathrm{m}$. The head loss in this section of pipeline is 6.9 kJ/kg, Calculate the outlet pressure.

Satpal S.

Numerade Educator

Water flows at $10 \mathrm{L} / \mathrm{min}$ through a horizontal $15-\mathrm{mm}$ diameter tube. The pressure drop along a $20-\mathrm{m}$ length of tube is 85 kPa. Calculate the head loss.

Satpal S.

Numerade Educator

Iaufer [5] measured the following data for mean velocity near the wall in fully developed turbulent pipe flow at $R e_{U}=50,000(U=9.8 \mathrm{ft} / \mathrm{s} \text { and } R=4.86 \mathrm{in.})$ in air:

$$\begin{aligned}

&\begin{array}{llllllll}

\bar{u} / U & 0.343 & 0.318 & 0.300 & 0.264 & 0.228 & 0.221 & 0.179 & 0.152 & 0.140

\end{array}\\

&y / R \quad 0.00820 .00750 .00710 .00610 .00550 .00510 .00410 .00340 .0030

\end{aligned}$$

Plot the data and obtain the best-fit slope, $d \bar{u} / d y$. Use this to estimate the wall shear stress from $\tau_{w}=\mu d u / d y,$ Compare this value to that obtained using the friction factor $f$ computed using

(a) the Colebrook formula (Fa 8.37 ), and

(b) the Blasius correlation (Eq. 8.38 ).

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Water is pumped at the rate of $0.075 \mathrm{m}^{3} / \mathrm{s}$ from a reservoir $20 \mathrm{m}$ above a pump to a free discharge $35 \mathrm{m}$ above the pump. The pressure on the intake side of the pump is $150 \mathrm{kPa}$ and the pressure on the discharge side is $450 \mathrm{kPa}$ All pipes are commercial steel of $15 \mathrm{cm}$ diameter. Determine (a) the head supplied by the pump and (b) the total head loss between the pump and point of free discharge.

Satpal S.

Numerade Educator

$ \mathrm{A}$ smooth, 75 -mm-diameter pipe carries water $\left(65^{\circ} \mathrm{C}\right)$ horizontally. When the mass flow rate is $0.075 \mathrm{kg} / \mathrm{s}$, the pressure drop is measured to be 7.5 Pa per $100 \mathrm{m}$ of pipe. Based on these measurements, what is the friction factor? What is the Reynolds number? Does this Reynolds number generally indicate laminar or turbulent flow? Is the flow actually laminar or turbulent?

Satpal S.

Numerade Educator

$ \mathrm{A}$ small-diameter capillary tube made from drawn aluminum is used in place of an expansion valve in a home refrigerator, The inside diameter is $0.5 \mathrm{mm}$. Calculate the corresponding relative roughness. Comment on whether this tube may be considered "smooth" with regard to fluid flow.

Satpal S.

Numerade Educator

The Colebrook equation (Eq. 8.37 ) for computing the turbulent friction factor is implicit in $f$. An explicit expres$\operatorname{sion}[31]$ that gives reasonable accuracy is $$f_{0}=0.25\left[\log \left(\frac{e / D}{3.7}+\frac{5.74}{R e^{0.9}}\right)\right]^{-2}$$ Compare the accuracy of this expression for $f$ with $\mathrm{Eq}, 8.37$ by computing the percentage discrepancy as a function of $R e$ and $e / D,$ for $R e=10^{4}$ to $10^{8},$ and $e^{1} D=0,0.0001,0,001$ $0.01,$ and $0.05 .$ What is the maximum discrepancy for these $R e$ and $e / D$ values? Plot $f$ against $R e$ with $e / D$ as a parameter.

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The Moody diagram gives the Darcy friction factor, $f,$ in terms of Reynolds number and relative roughness. The Fanning friction factor for pipe flow is defined as $$f_{F}=\frac{\tau_{w}}{\frac{1}{2} \rho \nabla^{2}}$$ where $\tau_{w}$ is the wall shear stress in the pipe. Show that the relation between the Darcy and Fanning friction factors for fully developed pipe flow is given by $f=4 f_{F}$

Satpal S.

Numerade Educator

We saw in Section 8.7 that instead of the implicit Colebrook equation (Eq. 8.37 ) for computing the turbulent friction factor $f,$ an explicit expression that gives reasonable accuracy is $$\frac{1}{\sqrt{f}}=-1.8 \log \left[\left(\frac{c / D}{3.7}\right)^{1.11}+\frac{6.9}{R e}\right]$$ Compare the accuracy of this expression for $f$ with $\mathrm{F}$ g. 8.37 by computing the percentage discrepancy as a function of $R e$ and $e / D,$ for $R e=10^{4}$ to $10^{8},$ and $e / D=0,0.0001,0.001$ $0.01,$ and $0.05 .$ What is the maximum discrepancy for these Re and e/D values? Plot $f$ against $R e$ with $e j D$ as parameter.

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8.97 Water flows at $25 \mathrm{L} / \mathrm{s}$ through a gradual contraction, in which the pipe diameter is reduced from $75 \mathrm{mm}$ to $37.5 \mathrm{mm}$ with a $150^{\circ}$ included angle. If the pressure before the contraction is $500 \mathrm{kPa}$, estimate the pressure after the contraction. Recompute the answer if the included angle is changed to $180^{\circ}$ (a sudden contraction).

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Water flows through a 25 -mm-diameter tube that suddenly enlarges to a diameter of $50 \mathrm{mm}$. The flow rate through the enlargement is 1.25 Liter/s. Calculate the pressure rise across the enlargement. Compare with the value for frictionless flow.

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Water flows through a 2 -in.-diameter tube that suddenly contracts to 1 in. diameter. The pressure drop across the contraction is 0.5 psi. Determine the volume flow rate.

Satpal S.

Numerade Educator

Air at standard conditions flows through a sudden expansion in a circular duct. The upstream and downstream duct diameters are $75 \mathrm{mm}$ and $225 \mathrm{mm}$, respectively. The pressure downstream is $5 \mathrm{mm}$ of water higher than that upstream. Determine the average speed of the air approaching the expansion and the volume flow rate.

Satpal S.

Numerade Educator

In an undergraduate laboratory, you have been assigned the task of developing a crude flow meter for measuring the flow in a 45 -mm-diameter water pipe system. You are to install a 22.5 -mm-diameter section of pipe and a water manometer to measure the pressure drop at the sudden contraction. Derive an expression for the theoretical calibration constant $k$ in $Q=k \sqrt{\Delta h},$ where $Q$ is the volume flow rate in $\mathrm{L} / \mathrm{min}$, and $\Delta h$ is the manometer deflection in $\mathrm{mm} .$ Plot the theoretical calibration curve for a flow rate range of 10 to 50 L/min. Would you expect this to be a practical device for measuring flow rate?

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Water flows from a larger pipe, diameter $D_{1}=100 \mathrm{mm}$ into a smaller one, diameter $D_{2}=50 \mathrm{mm},$ by way of a reentrant device. Find the head loss between points (1) and (2).

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Flow through a sudden contraction is shown. The minimum flow area at the vena contracta is given in terms of the area ratio by the contraction coefficient [32] $$C_{c}=\frac{A_{c}}{A_{2}}=0.62+0.38\left(\frac{A_{2}}{A_{1}}\right)^{3}$$ The loss in a sudden contraction is mostly a result of the vena contracta: The fluid accelerates into the contraction, there is flow separation (as shown by the dashed lines), and the vena contracta acts as a miniature sudden expansion with significant secondary flow losses. Use these assumptions to obtain and plot estimates of the minor loss coefficient for a sudden contraction, and compare with the data presented in Fig. 8.15

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Water flows from the tank shown through a very short pipe. Assume the flow is quasi-steady. Estimate the flow rate at the instant shown. How could you improve the flow system if a larger flow rate were desired?

Satpal S.

Numerade Educator

Consider again flow through the elbow analyzed in Example $4.6 .$ Using the given conditions, calculate the minor head loss coefficient for the elbow.

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Air flows out of a clean room test chamber through a 150-mm-diameter duct of length L. The original duct had a square edged entrance, but this has been replaced with a well-rounded one. The pressure in the chamber is $2.5 \mathrm{mm}$ of water above ambient. Losses from friction are negligible compared with the entrance and exit losses. Fstimate the increase in volume flow rate that results from the change in entrance contour.

Satpal S.

Numerade Educator

A water tank (open to the atmosphere) contains water to a depth of $5 \mathrm{m} .$ A 25 -mm-diameter hole is punched in the bottom. Modeling the hole as square-edged, estimate the flow rate (L/s) exiting the tank. If you were to stick a short section of pipe into the hole, by how much would the flow rate change? If instead you were to machine the inside of the hole to give it a rounded edge $(r=5 \mathrm{mm})$, by how much would the flow rate change?

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A conical diffuser is used to expand a pipe flow from a diameter of $100 \mathrm{mm}$ to a diameter of $150 \mathrm{mm}$. Find the minimum length of the diffuser if we want a loss coefficient

(a) $K_{\text {diffuser }} \leq 0.2,$

(b) $K_{\text {diffuser }} \leq 0.35$

Satpal S.

Numerade Educator

A conical diffuser of length 6 in. is used to expand a pipe flow from a diameter of 2 in. to a diameter of 3.5 in. For a water flow rate of 750 gal/min, estimate the static pressure rise. What is the approximate value of the loss coefficient?

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Space has been found for a conical diffuser $0.45 \mathrm{m}$ long in the clean room ventilation system described in Problem $8.106 .$ The best diffuser of this size is to be used. Assume that data Irom Fig. 8.16 may be used. Determine the appropriate diffuser angle and area ratio for this installation and estimate the volume flow rate that will be delivered after it is installed.

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By applying the basic equations to a control volume starting at the expansion and ending downstream, analyze flow through a sudden expansion (assume the inlet pressure $p_{1}$ acts on the area $A_{2}$ at the expansion). Develop an expression for and plot the minor head loss across the expansion as a function of area ratio, and compare with the data of Fig. 8.15

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Water al $45^{\circ} \mathrm{C}$ enters a shower head through a circular tube with $15.8 \mathrm{mm}$ inside diameter. The water leaves in 24 streams, each of $1.05 \mathrm{mm}$ diameter. The volume flow rate is $5.67 \mathrm{L} / \mathrm{min} .$ Estimate the minimum water pressure needed at the inlet to the shower head. Evaluate the force needed to hold the shower head onto the end of the circular tube. Indicate clearly whether this is a compression or a tension force.

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Analyze flow through a sudden expansion to obtain an expression for the upstream average velocity $V_{1}$ in terms of the pressure change $\Delta p=p_{2}-p_{1},$ area ratio $A R,$ fluid density $\rho,$ and loss coefficient $K .$ If the flow were frictionless, would the flow rate indicated by a measured pressure change be higher or lower than a real flow, and why? Conversely, if the flow were frictionless, would a given flow generate a larger or smaller pressure change, and why?

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Water discharges to atmosphere from a large reservoir through a moderately rounded horizontal nozzle of $25 \mathrm{mm}$ diameter. The free surface is $2.5 \mathrm{m}$ above the nozzle exit plane. Calculate the change in thow rate when a short section of 50 -mm-diameter pipe is attached to the end of the nozzle to form a sudden expansion. Determine the location and estimate the magnitude of the minimum pressure with the sudden expansion in place. If the flow were frictionless (with the sudden expansion in place), would the minimum pressure be higher, lower, or the same? Would the flow rate be higher, lower, or the same?

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Water flows steadily from a large tank through a length of smooth plastic tubing, then discharges to atmosphere. The tubing inside diameter is $3.18 \mathrm{mm},$ and its length is $15.3 \mathrm{m}$ Calculate the maximum volume flow rate for which flow in the tubing will remain laminar. Estimate the water level in the tank below which flow will be laminar (for laminar Ilow, $a=2$ and $K_{\mathrm{ent}}=1.4$ ).

Satpal S.

Numerade Educator

You are asked to compare the behavior of fully developed laminar flow and fully developed turbulent flow in a horizontal pipe under different conditions. For the same flow rate, which will have the larger centerline velocity? Why? If the pipe discharges to atmosphere, what would you expect the trajectory of the discharge stream to look like (for the same flow rate)? Sketch your expectations for each case. For the same flow rate, which flow would give the larger wall shear stress? Why? Sketch the shear stress distribution $\pi / \tau_{\mathrm{w}}$ as a function of radius for each flow. For the same Reynolds number, which flow would have the larger pressure drop per unit length? Why? For a given imposed pressure differential, which flow would have the larger flow rate? Why? Most of the remaining problems in this chapter involve determination of the turbulent friction factor $f$ from the Reynolds number $R e$ and dimensionless roughness $e / D$. For approximate calculations, $f$ can be read from Fig. $8.13 ;$ a more accurate approach is to use this value (or some other value, even $f=1$ ) as the first value for iterating in $\mathrm{Eq} .8 .37 .$ The most convenient approach is to use solution of Eq. 8.37 programmed into (or built-into) your calculator, or programmed into an Excel workbook. Hence, most of the remaining problems benefit from use of Excel. To avoid needless duplication, the computer symbol will only be used next to remaining problems in this chapter when it has an additional benefit (e.g., for iterating to a solution, or for graphing

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Estimate the minimum level in the water tank of Problem 8.115 such that the flow will be turbulent.

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A laboratory experiment is set up to measure pressure drop for flow of water through a smooth tube. The tube diameter is $15.9 \mathrm{mm},$ and its length is $3.56 \mathrm{m}$. Flow enters the tube from a reservoir through a square-edged entrance. Calculate the volume flow rate needed to obtain turbulent flow in the tube. Evaluate the reservoir height differential required to obtain turbulent flow in the tube.

Satpal S.

Numerade Educator

A benchtop experiment consists of a reservoir with a $500-m m-$ long horizontal tube of diameter $7.5 \mathrm{mm}$ attached to its base. The tube exits to a sink. A flow of water at $10^{\circ} \mathrm{C}$ is

to be generated such that the Reynolds number is 10,000 What is the flow rate? If the entrance to the tube is squareedged, how deep should the reservoir be? If the entrance to the tube is well-rounded, how deep should the reservoir be?

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As discussed in Problem $8.52,$ the applied pressure difference, $\Delta p,$ and corresponding volume flow rate, $Q,$ for laminar flow in a tube can be compared to the applied DC voltage $V$ across, and current $I$ through, an electrical resistor, respectively. Investigate whether or not this analogy is valid for turbulent flow by plotting the "resistance" $\Delta p / Q$ as a function of $Q$ for turbulent flow of kerosene (at $40^{\circ} \mathrm{C}$ ) in a tube $250 \mathrm{mm}$ long with inside diameter $7.5 \mathrm{mm}$.

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Plot the required reservoir depth of water to create flow in a smooth tube of diameter $10 \mathrm{mm}$ and length $100 \mathrm{m}$ for a tlow rate range of 1 L $/$ min through 10 L $/$ min.

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Oil with kinematic viscosity $\nu=7.5 \times 10^{-4} \mathrm{ft}^{2} / \mathrm{s}$ flows at 45 gpm in a 100 -fi-long horizontal drawn-tubing pipe of 1 in. diameter. By what percentage ratio will the energy loss increase if the same flow rate is maintained while the pipe diameter is reduced to 0.75 in.?

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A water system is used in a laboratory to study flow in a smooth pipe. The water is at $10^{\circ} \mathrm{C}$. To obtain a reasonable range, the maximum Reynolds number in the pipe must be $100,000 .$ The system is supplied from an overhead constant head tank. The pipe system consists of a square-edged entrance, two $45^{\circ}$ standard elbows, two $90^{\circ}$ standard clbows, and a fully open gate valve. The pipe diameter is $7.5 \mathrm{mm},$ and the total length of pipe is $1 \mathrm{m}$. Calculate the minimum height of the supply tank above the pipe system discharge to reach the desired Reynolds number. If a pressurized chamber is used instead of the reservoir, what will be the required pressure?

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Water from a pump flows through a 9 -in.-diameter commercial steel pipe for a distance of 4 miles from the pump discharge to a reservoir open to the atmosphere. The level of the water in the reservoir is $50 \mathrm{ft}$ above the pump discharge, and the average speed of the water in the pipe is $10 \mathrm{ft} / \mathrm{s},$ Calculate the pressure at the pump discharge.

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Water is to flow by gravity from one reservoir to a lower one through a straight, inclined galvanized iron pipe. The pipe diameter is $50 \mathrm{mm},$ and the total length is $250 \mathrm{m}$ Each reservoir is open to the atmosphere, Plot the required elevation difference $\Delta z$ as a function of flow rate $Q,$ for $Q$ ranging from 0 to $0.01 \mathrm{m}^{3} / \mathrm{s}$. Estimate the fraction of $\Delta z$ due to minor losses.

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A 5 -cm-diameter potable water line is to be run through a maintenance room in a commercial building. Three possible layouts for the water line are proposed, as shown. Which is the best option, based on minimizing losses? Assume galvanized iron, and a flow rate of $350 \mathrm{L} / \mathrm{min}$.

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In an air-conditioning installation, a flow rate of 1750 cfm of air at $50^{\circ} \mathrm{F}$ is required. A smooth sheet metal duct of rectangular section $(0.75 \mathrm{ft} \text { by } 2.5 \mathrm{ft})$ is to be used. Determine the pressure drop (inches of water) for a 1000 -ft horizontal duct section.

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A system for testing variable-output pumps consists of the pump, four standard elbows, and an open gate valve forming a closed circuit as shown. The circuit is to absorb the energy added by the pump. The tubing is 75 -mm-diameter cast iron, and the total length of the circuit is 20 -m. Plot the pressure difference required from the pump for water flow rates $Q$ ranging from $0.01 \mathrm{m}^{3} / \mathrm{s}$ to $0.06 \mathrm{m}^{3} / \mathrm{s}$

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A pipe friction experiment is to be designed, using water, to reach a Reynolds number of 100,000 . The system will use 5 -cm smooth PVC pipe from a constant-head tank to the flow bench and $20 \mathrm{m}$ of smooth 2.5 -cm PVC line mounted horizontally for the test section. The water level in the constant-head tank is $0.5 \mathrm{m}$ above the entrance to the 5-cm PVC line. Determine the required average speed of water in the 2.5 -cm pipe. Estimate the feasibility of using a constant-head tank. Calculate the pressure difference expected between taps $5 \mathrm{m}$ apart in the horizontal test section.

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Two reservoirs are connected by three clean cast-iron pipes in series, $L_{1}=600 \mathrm{m}, \quad D_{1}=0.3 \mathrm{m}, \quad L_{2}=900 \mathrm{m}$

$D_{2}=0.4 \mathrm{m}, L_{3}=1500 \mathrm{m},$ and $D_{3}=0.45 \mathrm{m} .$ When the dis-

charge is $0.11 \mathrm{m}^{3} / \mathrm{s}$ of water at $15^{\circ} \mathrm{C}$, determine the difference in elevation between the reservoirs.

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Consider flow of standard air at $1250 \mathrm{ft}^{3} / \mathrm{min}$. Compare the pressure drop per unit length of a round duct with that for rectangular ducts of aspect ratio $1,2,$ and 3. Assume that all ducts are smooth, with cross-sectional areas of $1 \mathrm{ft}^{2}$

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8.131 Consider flow of standard air at $1250 \mathrm{ft}^{3} / \mathrm{min}$. Compare the pressure drop per unit length of a round duct with that for rectangular ducts of aspect ratio $1,2,$ and

3. Assume that all ducts are smooth, with cross-sectional areas of $1 \mathrm{ft}^{2}$

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Water, at volume flow rate $Q=0.75 \mathrm{ft}^{3} / \mathrm{s},$ is delivered by a fire hose and nozzle assembly. The hose $(L=250 \mathrm{ft}$ $D=3$ in., and $e^{\prime} D=0.004$ ) is made up of four 60 -ft sections joined by couplings. The entrance is square-edged; the minor loss coefficient for each coupling is $K_{c}=0.5,$ based on mean velocity through the hose. The nozzle loss coefficient is $K_{n}=0.02,$ based on velocity in the exit jet, of $D_{2}=1$ in. diameter. Estimate the supply pressure required at this flow rate.

Satpal S.

Numerade Educator

Flow in a tube may alternate between laminar and turbulent states for Reynolds numbers in the transition zone. Design a bench-top experiment consisting of a constant-head cylindrical transparent plastic tank with depth graduations, and a length of plastic tubing (assumed smooth) attached at the base of the tank through which the water flows to a measuring container. Select tank and tubing dimensions so that the system is compact, but will operate in the transition zone range. Design the experiment so that you can easily increase the tank head from a low range (laminar flow) through transition to turbulent flow, and vice versa. (Write instructions for students on recognizing when the flow is laminar or turbulent.) Generate plots (on the same graph) of tank depth against Reynolds number, assuming laminar or turbulent flow.

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A small swimming pool is drained using a garden hose. The hose has $20 \mathrm{mm}$ inside diameter, a roughness height of $0.2 \mathrm{mm},$ and is $30 \mathrm{m}$ long. The free end of the hose is located $3 \mathrm{m}$ below the elevation of the bottom of the pool. The average velocity at the hose discharge is $1.2 \mathrm{m} / \mathrm{s}$. Fstimate the depth of the water in the swimming pool. If the flow were inviscid, what would be the velocity?

Satpal S.

Numerade Educator

When you drink you beverage with a straw, you need to overcome both gravity and friction in the straw. Fstimate the fraction of the total effort you put into quenching your thirst of each factor, making suitable assumptions about the liquid and straw properties, and your drinking rate (for example, how long it would take you to drink a 12 -oz drink if you drank it all in one go (quite a feat with a straw). Is the flow laminar or turbulent? (Ignore minor losses.)

Eric M.

Numerade Educator

The hose in Problem 8.135 is replaced with a larger diameter hose, diameter $25 \mathrm{mm}$ (same length and roughness). Assuming a pool depth of $1.5 \mathrm{m}$, what will be the new average velocity and flow rate?

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What flow rate (gpm) will be produced in a 75 -mmdiameter water pipe for which there is a pressure drop of 425 kPa over a 200 -m length? The pipe roughness is $2.5 \mathrm{mm}$. The water is at $0^{\circ} \mathrm{C}$

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A compressed air drill requires $0.25 \mathrm{kg} / \mathrm{s}$ of air at 650 kPa (gage) at the drill. The hose from the air compressor to the drill is $40 \mathrm{mm}$ inside diameter. The maximum compressor discharge gage pressure is 670 kPa; air leaves the compressor at $40^{\circ} \mathrm{C}$. Neglect changes in density and any effects of hose curvature. Calculate the longest hose that may be used.

Satpal S.

Numerade Educator

You recently bought a house and want to improve the flow rate of water on your top floor. The poor flow rate is due to three reasons: The city water pressure at the water meter is poor $(p-200 \mathrm{kPa} \text { gage }) ;$ the piping has a small diameter $(D=1.27 \mathrm{cm})$ and has been crudded up, increasing its roughness $(e / D=0.05) ;$ and the top floor of the house is $15 \mathrm{m}$ higher than the water meter. You are considering two options to improve the flow rate: Option 1 is replacing all the piping after the water meter with new smooth piping with a diameter of $1.9 \mathrm{cm} ;$ and option 2 is installing a booster pump while keeping the original pipes. The booster pump has an outlet pressure of 300 kPa. Which option would be more effective? Neglect minor losses.

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The students of Phi Gamma Delta are putting a kiddy pool on a porch attached to the second story of their house and plan to fill it with water from a garden hose. The kiddy pool has a diameter of $5 \mathrm{ft}$, and is $2.5 \mathrm{ft}$ deep. The porch is 18 ft above the faucet. The garden hose is very smooth on the inside, has a length of $50 \mathrm{ft}$, and a diameter of $5 / 8$ in. If the water pressure at the faucet is 60 psi, how long will it take to fill the pool? Neglect minor losses.

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Gasoline flows in a long. underground pipeline at a constant temperature of $15^{\circ} \mathrm{C}$. Two pumping stations at the same elevation are located $13 \mathrm{km}$ apart. The pressure drop between the stations is $1.4 \mathrm{MPa}$. The pipeline is made from 0.6-m-diameter pipe. Although the pipe is made from commercial steel, age and corrosion have raised the pipe roughness to approximately that for galvanized iron. Compute the volume flow rate.

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Water flows steadily in a horizontal 125 -mm-diameter cast-iron pipe. The pipe is $150 \mathrm{m}$ long and the pressure drop between sections $(\mathrm{D} \text { and }(2) \text { is } 150 \mathrm{kPa}$. Find the volume flow rate through the pipe.

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Water llows steadily in a 125 -mm-diameter cast-iron pipe $150 \mathrm{m}$ long. The pressure drop between sections (1) and (2) is $150 \mathrm{kPa}$, and section (2) is located $15 \mathrm{m}$ above section

(1). Find the volume flow rate.

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Two open standpipes of equal diameter are connected by a straight tube, as shown. Water flows by gravity from one standpipe to the other. For the instant shown, estimate the rate of change of water level in the left standpipe.

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Two galvanized iron pipes of diameter $D$ are connected to a large water reservoir, as shown. Pipe $A$ has length $L$ and pipe $B$ has length $2 L .$ Both pipes discharge to atmosphere. Which pipe will pass the larger flow rate? Justify (without calculating the flow rate in cach pipe). Compute the flow rates if $H=10 \mathrm{m}, D=50 \mathrm{mm},$ and $L=10 \mathrm{m}$

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Galvanized iron drainpipes of diameter $50 \mathrm{mm}$ are located at the four corners of a building, but three of them become clogged with debris. Find the rate of downpour $(\mathrm{cm} /$ min) at which the single functioning drainpipe can no longer drain the roof. The building roof area is $500 \mathrm{m}^{2},$ and the height is $5 \mathrm{m}$. Assume the drainpipes are the same height as the building, and that both ends are open to atmosphere. Ignore minor losses.

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$\mathrm{A}$ mining engineer plans to do hydraulic mining with a high-speed jet of water. A lake is located $H=300 \mathrm{m}$ above the mine site. Water will be delivered through $L=900 \mathrm{m}$ of fire hose; the hose has inside diameter $D=75 \mathrm{mm}$ and relative roughness $e / D=0.01 .$ Couplings, with equivalent length $L_{c}=20 D,$ are located every $10 \mathrm{m}$ along the hose. The nozzle outlet diameter is $d=25 \mathrm{mm}$. Its minor loss coefficient is $K=0.02$ based on outlet velocity. Estimate the maximum outlet velocity that this system could deliver. Determine the maximum force exerted on a rock face by this water jet.

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Investigate the effect of tube roughness on flow rate by computing the flow generated by a pressure difference $\Delta p=100 \mathrm{kPa}$ applied to a length $L=100 \mathrm{m}$ of tubing, with diameter $D=25 \mathrm{mm}$. Plot the flow rate against tube relative roughness $e / D$ for $e / D$ ranging from 0 to 0.05 (this could be replicated experimentally by progressively roughening the tube surface). Is it possible that this tubing could be roughened so much that the flow could be slowed to a laminar flow rate?

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Investigate the effect of tube length on water flow rate by computing the flow generated by a pressure difference $\Delta p=100 \mathrm{kPa}$ applied to a length $L$ of smooth tubing, of diameter $D=25 \mathrm{mm}$. Plot the flow rate against tube length for flow ranging from low speed laminar to fully turbulent.

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For the pipe flow into a reservoir of Example 8.5 consider the effect of pipe roughness on flow rate, assuming the pressure of the pump is maintained at 153 kPa. Plot the flow rate against pipe roughness ranging from smooth $(e=0)$ to very rough $(e=3.75 \mathrm{mm})$. Also consider the effect of pipe length (again assuming the pump always produces 153 kPa) for smooth pipe. Plot the flow rate against pipe length for

\[

L=100 \mathrm{m} \text { through } L=1000 \mathrm{m}

\]

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Water for a fire protection system is supplied from a water tower through a 150 -mm cast-iron pipe. A pressure gage at a fire hydrant indicates 600 kPa when no water is flowing. The total pipe length between the elevated tank and the hydrant is $200 \mathrm{m}$. Determine the height of the water tower above the hydrant. Calculate the maximum volume flow rate that can be achieved when the system is flushed by opening the hydrant wide (assume minor losses are 10 percent of major losses at this condition). When a fire hose is attached to the hydrant, the volume flow rate is $0.75 \mathrm{m}^{3} / \mathrm{min}$. Determine the reading of the pressure gage at this flow condition.

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The siphon shown is fabricated from 50 -mm-id. drawn aluminum tubing. The liquid is water at $15^{\circ} \mathrm{C}$. Compute the volume flow rate through the siphon. Estimate the minimum pressure inside the tube.

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A large open water tank has a horizontal cast iron drainpipe of diameter $D=1$ in. and length $L=2 \mathrm{ft}$ attached at its base. If the depth of water is $h=3$ ft, find the llow rate $(\mathrm{gpm})$ if the pipe entrance is

(a) reentrant,

(b) square-edged, and

(c) rounded $(r=0.2 \text { in. })$

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A tank containing $30 \mathrm{m}^{3}$ of kerosene is to be emptied by a gravity feed using a drain hose of diameter $15 \mathrm{mm}$ roughness $0.2 \mathrm{mm},$ and length $1 \mathrm{m}$. The top of the tank is open to the atmosphere and the hose exits to an open chamber. If the kerosene level is initially $10 \mathrm{m}$ above the drain exit, estimate (by assuming steady flow) the initial drainage rate. Estimate the flow rate when the kerosene level is down to $5 \mathrm{m},$ and then down to $1 \mathrm{m}$. Based on these three estimates, make a rough estimate of the time it took to drain to the 1 -m level.

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Consider again the Roman water supply discussed in Example $8.10 .$ Assume that the 50 ft length of horizontal constant-diameter pipe required by law has been installed. The relative roughness of the pipe is 0.01 . Estimate the flow rate of water delivered by the pipe under the inlet conditions of the example. What would be the effect of adding the same diffuser to the end of the 50 ft pipe?

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You are watering your lawn with an old hose, Because lime deposits have built up over the years, the 0.75 -in.-i.d. hose now has an average roughness height of 0.022 in. One 50 -ft length of the hose, attached to your spigot, delivers 15 gpm of water $\left(60^{\circ} \mathrm{F}\right) .$ Compute the pressure at the spigot, in psi. Fstimate the delivery if two 50 -ft lengths of the hose are connected. Assume that the pressure at the spigot varies with flow rate and the water main pressure remains constant at 50 psig.

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In Example 8.10 we found that the flow rate from a water main could be increased (by as much as 33 percent) by attaching a diffuser to the outlet of the nozzle installed into the water main. We read that the Roman water commissioner required that the tube attached to the nozzle of each customer's pipe be the same diameter for at least 50 feet from the public water main. Was the commissioner overly conservative? Using the data of the problem, estimate the length of pipe (with $e / D=0.01$ ) at which the system of pipe and diffuser would give a flow rate equal to that with the nozzle alone. Plot the volume flow ratio $Q / Q_{i}$ as a function of $L / D,$ where $L$ is the length of pipe between the nozzle and the diffuser, $Q_{i}$ is the volume flow rate for the nozzle alone, and $Q$ is the actual volume flow rate with the pipe inserted between nozzle and diffuser.

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Your boss, from the "old school," claims that for pipe flow the flow rate, $Q \propto \sqrt{\Delta p}$, where $\Delta p$ is the pressure dif ference driving the flow. You dispute this, so perform some calculations. You take a 1 -in.- -diameter commercial steel pipe and assume an initial flow rate of 1.25 gal/min of water. You then increase the applicd pressure in equal increments and compute the new flow rates so you can plot $Q$ versus $\Delta p,$ as computed by you and your boss. Plot the two curves on the same graph. Was your boss right?

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For Problem $8.146,$ what would the diameter of the pipe of length $2 L$ need to be to generate the same flow as the pipe of length $L ?$

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A hydraulic press is powered by a remote highpressure pump. The gage pressure at the pump outlet is 3000 psi, whereas the pressure required for the press is 2750 psi (gage), at a flow rate of $0.02 \mathrm{ft}^{3} / \mathrm{s}$. The press and pump are connected by 165 ft of smooth, drawn steel tubing. The fluid is SAE $10 \mathrm{W}$ oil at $100^{\circ} \mathrm{F}$. Determine the minimum tubing diameter that may be used.

Satpal S.

Numerade Educator

$\mathrm{A}$ pump is located $4.5 \mathrm{m}$ to one side of, and $3.5 \mathrm{m}$ above a reservoir. The pump is designed for a flow rate of $6 \mathrm{L} / \mathrm{s} .$ For satisfactory operation, the static pressure at the pump inlet must not be lower than $-6 \mathrm{m}$ of water gage. Determine the smallest standard commercial steel pipe that will give the required performance.

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Determine the minimum size smooth rectangular duct with an aspect ratio of 3 that will pass $1 \mathrm{m}^{3} / \mathrm{s}$ of $10^{\circ} \mathrm{C}$ air with a head loss of $25 \mathrm{mm}$ of water per $100 \mathrm{m}$ of duct.

Khoobchandra A.

Numerade Educator

A new industrial plant requires a water flow rate of 5.7 $\mathrm{m}^{3} / \mathrm{min} .$ The gage pressure in the water main, located in the street $50 \mathrm{m}$ from the plant, is $800 \mathrm{kPa}$. The supply line will require installation of 4 elbows in a total length of $65 \mathrm{m}$. The gage pressure required in the plant is 500 kPa. What size galvanized iron line should be installed?

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Air at $40^{\circ} \mathrm{F}$ flows in a horizontal square cross-section duct made from commercial stecl. The duct is 1000 ft long. What size (length of a side) duct is required to convey 1500 cfm of air with a pressure drop of 0.75 in. $\mathrm{H}_{2} \mathrm{O} ?$

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Investigate the effect of tube diameter on water flow rate by computing the flow generated by a pressure difference, $\Delta p=100 \mathrm{kPa},$ applied to a length $L=100 \mathrm{m}$ of smooth tubing. Plot the flow rate against tube diameter for a range that includes laminar and turbulent flow.

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What diameter water pipe is required to handle 0.075 $\mathrm{m}^{3} / \mathrm{s}$ and a $500 \mathrm{kPa}$ pressure drop? The pipe length is $175 \mathrm{m}$ and roughness is $2.5 \mathrm{mm}$

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A large reservoir supplies water for a community. A portion of the water supply system is shown. Water is pumped from the reservoir to a large storage tank before being sent on to the water treatment facility. The system is designed to provide $1310 \mathrm{L} / \mathrm{s}$ of water at $20^{\circ} \mathrm{C}$. From $B$ to $C$ the system consists of a square-edged entrance, $760 \mathrm{m}$ of pipe, three gate valves, four $45^{\circ}$ elbows, and two $90^{\circ}$ elbows. Gage pressure at $C$ is 197 kPa. The system between $F$ and $G$ contains $760 \mathrm{m}$ of pipe, two gate valves, and four $90^{\circ}$ elbows. All pipe is $508 \mathrm{mm}$ diameter, cast iron. Calculate the average velocity of water in the pipe, the gage pressure at section $F,$ the power input to the pump (its efficicncy is 80 percent), and the wall shear stress in section $F G$

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An air-pipe friction experiment consists of a smooth brass tube with $63.5 \mathrm{mm}$ inside diameter, the distance between pressure taps is $1.52 \mathrm{m}$. The pressure drop is indicated by a manometer filled with Meriam red oil. The centerline velocity $U$ is measured with a pitot cylinder. At one flow condition, $U=23.1 \mathrm{m} / \mathrm{s}$ and the pressure drop is $12.3 \mathrm{mm}$ of oil. For this condition, evaluate the Reynolds number based on average flow velocity. Calculate the friction factor and compare with the value obtained from Eq. 8.37 (use $n=7$ in the power-law velocity profile).

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Oil has been flowing from a large tank on a hill to a tanker at the wharf. The compartment in the tanker is nearly full and an operator is in the process of stopping the flow. A valve on the wharf is closed at a rate such that 1 MPa is maintained in the line immediately upstream of the valve. Assume:

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Problem 8.171 describes a situation in which flow in a long pipeline from a hilltop tank is slowed gradually to avoid a large pressure rise. Expand this analysis to predict and plot the closing schedule (valve loss coefficient versus time) needed to maintain the maximum pressure at the valve at or below a given value throughout the process of stopping the flow from the tank.

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A pump draws water at a steady flow rate of $25 \mathrm{lbm} / \mathrm{s}$ through a piping system. The pressure on the suction side of the pump is -2.5 psig. The pump outlet pressure is 50 psig. The inlet pipe diameter is 3 in.; the outlet pipe diameter is 2 in. The pump efficiency is 70 percent. Calculate the power required to drive the pump.

Satpal S.

Numerade Educator

The pressure rise across a water pump is 35 psi when the volume flow rate is 500 gpm. If the pump efficiency is 80 percent, determine the power input to the pump.

Satpal S.

Numerade Educator

A 125 -mm-diameter pipeline conveying water at $10^{\circ} \mathrm{C}$ contains $50 \mathrm{m}$ of straight galvanized pipe, 5 fully open gate valves, 1 fully open angle valve, 7 standard $90^{\circ}$ elbows, 1 square-edged entrance from a reservoir, and 1 free discharge. The entrance conditions are $p_{1}=150 \mathrm{kPa}$ and $z_{1}=15 \mathrm{m},$ and exit conditions are $p_{2}=0 \mathrm{kPa}$ and $z_{2}=30 \mathrm{m} .$ A centrifugal pump is installed in the line to move the water. What pressure rise must the pump deliver so that the volume flow rate will be $Q=50 \mathrm{L} / \mathrm{s}^{\circ}$

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Cooling water is pumped from a reservoir to rock drills on a construction job using the pipe system shown. The flow rate must be 600 mpg and water must leave the spray nozzle at $120 \mathrm{ft} / \mathrm{s}$. Calculate the minimum pressure needed at the pump outlet. Estimate the required power input if the pump efficiency is 70 percent.

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8.177 You are asked to size a pump for installation in the water supply system of the Willis Tower (formerly the Sears Tower) in Chicago. The system requires 100 gpm of water pumped to a reservoir at the top of the tower $340 \mathrm{m}$ above the street. City water pressure at the street-level pump inlet is 400 kPa (gage). Piping is to be commercial steel. Determine the minimum diameter required to keep the average water velocity below $3.5 \mathrm{m} / \mathrm{s}$ in the pipe. Calculate the pressure rise required across the pump. Estimate the minimum power needed to drive the pump.

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Air conditioning on a university campus is provided by chilled water $\left(10^{\circ} \mathrm{C}\right)$ pumped through a main supply pipe. The pipe makes a loop $5 \mathrm{km}$ in length. The pipe diameter is $0.75 \mathrm{m}$ and the material is steel. The maximum design volume flow rate is $0.65 \mathrm{m}^{3} / \mathrm{s}$. The circulating pump is driven by an electric motor. The efficiencies of pump and motor are $\eta_{p}=85$ percent and $\eta_{m}=85$ percent, respectively. Electricity $\cos t$ is $14 \& /(\mathrm{kW} \cdot \mathrm{hr}) .$ Determine (a) the pressure drop, (b) the rate of energy addition to the water, and (c) the daily cost of electrical energy for pumping.

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A fire nozzle is supplied through $100 \mathrm{m}$ of $3.5-\mathrm{cm}$ diameter, smooth, rubber-lined hose. Water from a hydrant is supplied to a booster pump on board the pumper truck at $350 \mathrm{kPa}(\text { gage }) .$ At design conditions, the pressure at the nozzle inlet is 700 kPa (gage), and the pressure drop along the hose is 750 kPa per $100 \mathrm{m}$ of length. Determine (a) the design flow rate, (b) the nozzle exit velocity, assuming no losses in the nozzle, and (c) the power required to drive the booster pump, if its efficiency is 70 percent.

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Heavy crude oil $\left(S G=0.925 \text { and } \nu=1.0 \times 10^{-4} \mathrm{m}^{2} / \mathrm{s}\right)$ is pumped through a pipeline laid on flat ground. The line is made from steel pipe with $600 \mathrm{mm}$ i.d. and has a wall thickness of $12 \mathrm{mm}$. The allowable tensile stress in the pipe wall is limited to $275 \mathrm{MPa}$ by corrosion considerations. It is important to keep the oil under pressure to ensure that gases remain in solution. The minimum recommended pressure is 500 kPa. The pipeline carries a flow of 400,000 barrels (in the petroleum industry, a "barrel" is 42 gal) per day. Determine the maximum spacing between pumping stations. Compute the power added to the oil at each pumping station.

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The volume flow rate through a water fountain on a college campus is $0.075 \mathrm{m}^{3} / \mathrm{s}$. Each water stream can rise to a height of $10 \mathrm{m}$. Estimate the daily cost to operate the fountain. Assume that the pump motor efficiency is 85 percent, the pump efficiency is 85 percent, and the cost of clectricity is $14 \& /(\mathrm{kW} \cdot \mathrm{hr})$

Satpal S.

Numerade Educator

Petroleum products are transported over long distances by pipcline, e.g., the Alaskan pipeline (see Example 8.6). Estimate the energy needed to pump a typical petroleum product, expressed as a fraction of the throughput energy carried by the pipeline. State and critique your assumptions clearly.

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The pump testing system of Problem 8.128 is run with a pump that generates a pressure difference given by $\Delta p=$ $750-15 \times 10^{4} Q^{2}$ where $\Delta p$ is in $k P a,$ and the generated flow rate is $Q \mathrm{m}^{3} / \mathrm{s}$. Find the water flow rate, pressure difference, and power supplied to the pump if it is 70 percent efficient.

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A water pump can generate a pressure difference $\Delta p$ (psi) given by $\Delta p=145-0.1 Q^{2},$ where the flow rate is $Q \mathrm{ft}^{3} / \mathrm{s},$ It supplies a pipe of diameter 20 in., roughness 0.5 in.. and length 2500 ft. Find the flow rate, pressure difference, and the power supplied to the pump if it is 70 percent efficient. If the pipe were replaced with one of roughness 0.25 in. how much would the flow increase, and what would the required power be?

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A square cross-section duct $(0.35 \mathrm{m} \times 0.35 \mathrm{m} \times 175$ $\mathrm{m}$ ) is used to convey air $\left(\rho=1.1 \mathrm{kg} / \mathrm{m}^{3}\right)$ into a clean room in an electronics manufacturing facility. The air is supplied by a fan and passes through a filter installed in the duct. The duct friction factor is $f=0.003,$ the filter has a loss coefficient of $K=3 .$ The fan performance is given by $\Delta p=2250-250 Q-$ $150 Q^{2},$ where $\Delta p(P$ a) is the pressure generated by the fan at flow rate $Q\left(\mathrm{m}^{3} / \mathrm{s}\right) .$ Determine the flow rate delivered to the room.

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The head versus capacity curve for a certain fan may be approximated by the equation $H=30-10^{-7} Q^{2},$ where $H$ is the output static head in inches of water and $Q$ is the air flow rate in $\mathrm{ft}^{3} / \mathrm{min}$. The fan outlet dimensions are $8 \times 16$ in. Determine the air flow rate delivered by the fan into a $200 \mathrm{ft}$ straight length of $8 \times 16$ in. rectangular duct.

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The water pipe system shown is constructed from galvanized iron pipe. Minor losses may be neglected. The inlet is at $400 \mathrm{kPa}$ (gage), and all exits are at atmospheric pressure. Find the flow rates $Q_{0}, Q_{1}, Q_{2}, Q_{3},$ and $Q_{4-}$

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A cast-iron pipe system consists of a 500 -ft section of water pipe, after which the flow branches into two $300-f t$ sections. The two branches then meet in a final 250 -ft section. Minor losses may be neglected. All sections are 1.5 -in. diameter, except one of the two branches, which is 1 -in. diameter. If the applied pressure across the system is 100 psi, find the overall flow rate and the flow rates in each of the two

branches.

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A swimming pool has a partial-flow filtration system. Water at $75^{\circ} \mathrm{F}$ is pumped from the pool through the system shown. The pump delivers 30 gpm. The pipe is nominal $3 / 4$ in. PVC (i.d. -0.824 in.). The pressure loss through the filter is approximately $\Delta p=0.6 Q^{2},$ where $\Delta p$ is in psi and $Q$ is in gpm. Determine the pump pressure and the flow rate through each branch of the system.

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Why does the shower temperature change when a toilet is flushed? Sketch pressure curves for the hot and cold water supply systems to explain what happens.

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A square-edged orifice with corner taps and a water manometer are used to meter compressed air. The following data are given:

Inside diameter of air line

Orifice plate diameter

Upstream pressure Temperature of air

Manometer deflection

Calculate the volume flow rate in the line, expressed in cubic meters per hour.

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Water at $65^{\circ} \mathrm{C}$ flows through a 75 -mm-diameter orifice installed in a 150 -mm-i.d. pipe. The flow rate is $20 \mathrm{L} / \mathrm{s}$. Determine the pressure difference between the corner taps.

Satpal S.

Numerade Educator

$\mathrm{A}$ smooth $200-\mathrm{m}$ pipe, $100 \mathrm{mm}$ diameter connects two reservoirs (the entrance and exit of the pipe are sharpedged). At the midpoint of the pipe is an orifice plate with diameter $40 \mathrm{mm}$. If the water levels in the reservoirs differ by $30 \mathrm{m},$ estimate the pressure differential indicated by the orifice plate and the flow rate.

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A venturi meter with a 3 -in. -diameter throat is placed in a 6 -in.-diameter line carrying water at $75^{\circ} \mathrm{F}$, The pressure drop between the upstream tap and the venturi throat is 12 in. of mercury. Compute the rate of flow.

Satpal S.

Numerade Educator

Consider a horizontal 2 in. $\times 1$ in. venturi with water flow. For a differential pressure of 25 psi, calculate the volume flow rate (gpm).

Satpal S.

Numerade Educator

Gasoline flows through a $2 \times 1$ in. venturi meter. The differential pressure is $380 \mathrm{mm}$ of mercury. Find the volume flow rate.

Satpal S.

Numerade Educator

Air flows through the venturi meter described in Problem $8.195 .$ Assume that the upstream pressure is 60 psi. and that the temperature is everywhere constant at $68^{\circ} \mathrm{F}$ Determine the maximum possible mass flow rate of air for which the assumption of incompressible flow is a valid engineering approximation. Compute the corresponding differential pressure reading on a mercury manometer.

Satpal S.

Numerade Educator

Air flow rate in a test of an internal combustion engine is to be measured using a flow nozzle installed in a plenum. The engine displacement is 1.6 liters, and its max. imum operating speed is 6000 rpm. To avoid loading the engine, the maximum pressure drop across the nozzle should not exceed $0.25 \mathrm{m}$ of water. The manometer can be read to $\pm 0.5 \mathrm{mm}$ of water. Determine the flow nozzle diameter that should be specified. Find the minimum rate of air flow that can be metered to ±2 percent using this setup.

Satpal S.

Numerade Educator

Water at $10^{\circ} \mathrm{C}$ flows steadily through a venturi. The pressure upstream from the throat is $200 \mathrm{kPa}$ (gage). The throat diameter is $50 \mathrm{mm}$; the upstream diameter is $100 \mathrm{mm}$. Fstimate the maximum flow rate this device can handle without cavitation.

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Derive Eq. $8.42,$ the pressure loss coefficient for a diffuser assuming ideal (frictionless) flow.

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Consider a flow nozzle installation in a pipe. Apply the basic equations to the control volume indicated, to show that the permanent head loss across the meter can be expressed, in dimensionless form, as the head loss coefficicnt, $$C_{l}=\frac{p_{1}-p_{3}}{p_{1}-p_{2}}=\frac{1-A_{2} / A_{1}}{1+A_{2} / A_{1}}$$ Plot $C_{l}$ as a function of diameter ratio, $D_{2} / D_{1}$

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Drinking straws are to be used to improve the air flow in a pipe-flow experiment. Packing a section of the air pipe with drinking straws to form a "laminar flow element" might allow the air flow rate to be measured directly, and simultaneously would act as a flow straightener. To evaluate this idea, determine

(a) the Reynolds number for flow in each drinking straw,

(b) the friction factor for llow in each straw, and

(c) the gage pressure at the exit from the drinking straws. (For laminar flow in a tube, the entrance loss coefficient is $\left.K_{\mathrm{ent}}=1.4 \text { and } \alpha=2.0 .\right)$ Comment on the utility of this idea.

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In some western states, water for mining and irrigation was sold by the "miner's inch," the rate at which water flows through an opening in a vertical plank of 1 in. $^{2}$ area, up to 4 in. tall, under a head of 6 to 9 in. Develop an equation to predict the flow rate through such an orifice. Specify clearly the aspect ratio of the opening, thickness of the plank, and datum level for measurement of head (top, bottom, or middle of the opening). Show that the unit of measure varies from 38.4 (in Colorado) to 50 (in Arizona, Idaho, Nevada, and Utah) miner's inches equal to $1 \mathrm{ft}^{3} / \mathrm{s}$

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The volume flow rate in a circular duct may be measured by "pitot traverse," i.e., by measuring the velocity in each of several area segments across the duct, then summing. Comment on the way such a traverse should be set up. Quantify and plot the expected error in measurement of flow rate as a function of the number of radial locations used in the traverse.

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