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Fluid Mechanics

Frank M. White

Chapter 5

Dimensional Analysis and Similarity - all with Video Answers

Educators

Chapter Questions

12:44

Problem 1

For axial flow through a circular tube, the Reynolds number for transition to turbulence is approximately 2300 [see Eq. $(6.2)]$, based on the diameter and average velocity. If $d=5 \mathrm{cm}$ and the fluid is kerosene at $20^{\circ} \mathrm{C}$, find the volume flow rate in $\mathrm{m}^{3} / \mathrm{h}$ that causes transition.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
05:43

Problem 2

A prototype automobile is designed for cold weather in Denver, $\mathrm{CO}\left(-10^{\circ} \mathrm{C}, 83 \mathrm{kPa}\right)$. Its drag force is to be tested on a one-seventh-scale model in a wind tunnel at $150 \mathrm{mi} / \mathrm{h}, 20^{\circ} \mathrm{C},$ and 1 atm. If the model and prototype are to satisfy dynamic similarity, what prototype velocity, in $\mathrm{mi} / \mathrm{h},$ needs to be matched? Comment on your result.

Ronald Prasad
Ronald Prasad
Numerade Educator
01:17

Problem 3

The transfer of energy by viscous dissipation is dependent upon viscosity $\mu,$ thermal conductivity $k,$ stream velocity $U$ and stream temperature $T_{0} .$ Group these quantities, if possible, into the dimensionless Brinkman number, which is proportional to $\mu$

Penny Riley
Penny Riley
Numerade Educator
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Problem 4

When tested in water at $20^{\circ} \mathrm{C}$ flowing at $2 \mathrm{m} / \mathrm{s},$ an $8-\mathrm{cm}-$ diameter sphere has a measured drag of 5 N. What will be the velocity and drag force on a 1.5 -m-diameter weather balloon moored in sea-level standard air under dynamically similar conditions?

Victor Salazar
Victor Salazar
Numerade Educator
02:15

Problem 5

An automobile has characteristic length and area of 8 ft and $60 \mathrm{ft}^{2},$ respectively. When tested in sea-level standard air, it has the following measured drag force versus speed:
$$\begin{array}{l|l|l|r}
\mathrm{V}, \mathrm{mi} / \mathrm{h} & 20 & 40 & 60 \\
\hline \text { Drag, lbf } & 31 & 115 & 249
\end{array}$$
The same car travels in Colorado at $65 \mathrm{mi} / \mathrm{h}$ at an altitude of $3500 \mathrm{m}$. Using dimensional analysis, estimate $(a)$ its drag force and
$(b)$ the horsepower required to overcome air drag.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:28

Problem 6

The disk-gap-band parachute in the chapter-opener photo had a drag of 1600 lbf when tested at $15 \mathrm{mi} / \mathrm{h}$ in air at $20^{\circ} \mathrm{C}$ and 1 atm. (a) What was its drag coefficient? $(b)$ If, as stated, the drag on Mars is 65,000 lbf and the velocity is $375 \mathrm{mi} / \mathrm{h}$ in the thin Mars atmosphere, $\rho \approx 0.020 \mathrm{kg} / \mathrm{m}^{3}$
what is the drag coefficient on Mars? $(c)$ Can you explain the difference between $(a)$ and $(b) ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
01:14

Problem 7

A body is dropped on the moon $\left(g=1.62 \mathrm{m} / \mathrm{s}^{2}\right)$ with an initial velocity of $12 \mathrm{m} / \mathrm{s}$. By using option 2 variables, Eq. $(5.11),$ the ground impact occurs at $t^{* *}=0.34$ and $S^{* *}=$ $0.84 .$ Estimate $(a)$ the initial displacement, (b) the final displacement, and ( $c$ ) the time of impact.

Suzanne W.
Suzanne W.
Numerade Educator
04:00

Problem 8

The Archimedes number, Ar, used in the flow of stratified fluids, is a dimensionless combination of gravity $g$, density difference $\Delta \rho,$ fluid width $L,$ and viscosity $\mu .$ Find the form of this number if it is proportional to $g$

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:59

Problem 9

The Richardson number, Ri, which correlates the production of turbulence by buoyancy, is a dimensionless combination of the acceleration of gravity $g,$ the fluid temperature $T_{0}$ the local temperature gradient $\partial T / \partial z,$ and the local velocity gradient $\partial u / \partial z .$ Determine the form of the Richardson number if it is proportional to $g$

Dominador Tan
Dominador Tan
Numerade Educator
03:31

Problem 10

Determine the dimension $\{M L T \Theta\}$ of the following quantities:
(a) $\rho u \frac{\partial u}{\partial x}$
(b) $\int_{1}^{2}\left(p-p_{0}\right) d A$
$(c) \rho c_{p} \frac{\partial^{2} T}{\partial x \partial y}$
$(d) \iiint \rho \frac{\partial u}{\partial t} d x d y d z$
All quantities have their standard meanings; for example, is density.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
06:12

Problem 11

During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to estimate the wave speed of an atomic bomb explosion. He assumed that the blast wave radius $R$ was a function of energy released $E,$ air density $\rho,$ and time $t .$ Use dimensional reasoning to show how wave radius must vary with time.

Ameer Said
Ameer Said
Numerade Educator
02:05

Problem 12

The Stokes number, St, used in particle dynamics studies, is a dimensionless combination of five variables: acceleration of gravity $g,$ viscosity $\mu,$ density $\rho,$ particle velocity $U$ and particle diameter $D$ (a) If $\mathrm{St}$ is proportional to $\mu$ and inversely proportional to $g,$ find its form. $(b)$ Show that $\mathrm{St}$ is actually the quotient of two more traditional dimensionless groups.

James Kiss
James Kiss
Numerade Educator
01:43

Problem 13

The speed of propagation $C$ of a capillary wave in deep water is known to be a function only of density $\rho,$ wavelength $\lambda,$ and surface tension $Y .$ Find the proper functional relationship, completing it with a dimensionless constant. For a given density and wavelength, how does the propagation speed change if the surface tension is doubled?

Ajay Singhal
Ajay Singhal
Numerade Educator
02:12

Problem 14

Flow in a pipe is often measured with an orifice plate, as in Fig. P5.14. The volume flow $Q$ is a function of the pressure drop $\Delta p$ across the plate, the fluid density $\rho,$ the pipe diameter $D,$ and the orifice diameter $d .$ Rewrite this functional relationship in dimensionless form.

James Kiss
James Kiss
Numerade Educator
01:04

Problem 15

The wall shear stress $\tau_{w}$ in a boundary layer is assumed to be a function of stream velocity $U$, boundary layer thickness $\delta,$ local turbulence velocity $u^{\prime},$ density $\rho,$ and local pressure gradient $d p / d x .$ Using $(\rho, U, \delta)$ as repeating variables, rewrite this relationship as a dimensionless function.

Dominador Tan
Dominador Tan
Numerade Educator
01:17

Problem 16

Convection heat transfer data are often reported as a heat transfer coefficient $h,$ defined by
$$
\begin{aligned}
\dot{Q}=h A \Delta T \\
\text { where } \dot{Q}=\text { heat flow, } \mathrm{J} / \mathrm{s} \\
A &=\text { surface area, } \mathrm{m}^{2} \\
\Delta T &=\text { temperature difference }, \mathrm{K}
\end{aligned}
$$
The dimensionless form of $h$, called the Stanton number, is a combination of $h,$ fluid density $\rho,$ specific heat $c_{p},$ and flow velocity $V$. Derive the Stanton number if it is proportional to $h .$ What are the units of $h ?$

Penny Riley
Penny Riley
Numerade Educator
02:07

Problem 17

If you disturb a tank of length $L$ and water depth $h,$ the surface will oscillate back and forth at frequency $\Omega$ assumed here to depend also upon water density $\rho$ and the acceleration of gravity $g .(a)$ Rewrite this as a dimensionless function. $(b)$ If a tank of water sloshes at $2.0 \mathrm{Hz}$ on earth, how fast would it oscillate on Mars $\left(g \approx 3.7 \mathrm{m} / \mathrm{s}^{2}\right) ?$

Narayan Hari
Narayan Hari
Numerade Educator
02:51

Problem 18

Under laminar conditions, the volume flow $Q$ through a small triangular-section pore of side length $b$ and length $L$ is a function of viscosity $\mu,$ pressure drop per unit length $\Delta p / L,$ and $b .$ Using the pi theorem, rewrite this relation in dimensionless form. How does the volume flow change if the pore size $b$ is doubled?

Narayan Hari
Narayan Hari
Numerade Educator
06:35

Problem 19

The period of oscillation $T$ of a water surface wave is assumed to be a function of density $\rho,$ wavelength $l$, depth $h$ gravity $g$, and surface tension $Y$. Rewrite this relationship in dimensionless form. What results if $Y$ is negligible?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 20

A fixed cylinder of diameter $D$ and length $L,$ immersed in a stream flowing normal to its axis at velocity $U,$ will experience zero average lift. However, if the cylinder is rotating at angular velocity $\Omega,$ a lift force $F$ will arise. The fluid density $\rho$ is important, but viscosity is secondary and can be neglected. Formulate this lift behavior as a dimensionless function.

Victor Salazar
Victor Salazar
Numerade Educator
05:13

Problem 21

In Example 5.1 we used the pi theorem to develop Eq. ( 5.2 ) from Eq. (5.1). Instead of merely listing the primary dimensions of each variable, some workers list the powers of each primary dimension for each variable in an array:
$$
\left.\begin{array}{c|rrrrr}
& F & L & U & \rho & \mu \\
M & 1 & 0 & 0 & 1 & 1 \\
L & 1 & 1 & 1 & -3 & -1 \\
T & -2 & 0 & -1 & 0 & -1
\end{array}\right]
$$
This array of exponents is called the dimensional matrix for the given function. Show that the rank of this matrix (the size of the largest nonzero determinant) is equal to $j=$ $n-k,$ the desired reduction between original variables and the pi groups. This is a general property of dimensional matrices, as noted by Buckingham [1]

Chris Trentman
Chris Trentman
Numerade Educator
01:57

Problem 22

As will be discussed in Chap. 11 , the power $P$ developed by a wind turbine is a function of diameter $D$, air density $\rho$ wind speed $V,$ and rotation rate $\omega .$ Viscosity effects are negligible. Rewrite this relationship in dimensionless form.

James Kiss
James Kiss
Numerade Educator
02:20

Problem 23

The period $T$ of vibration of a beam is a function of its length $L,$ area moment of inertia $I,$ modulus of elasticity $E$ density $\rho,$ and Poisson's ratio $\sigma .$ Rewrite this relation in dimensionless form. What further reduction can we make if $E$ and $I$ can occur only in the product form $E I ?$ Hint: Take $L, \rho,$ and $E$ as repeating variables.

James Kiss
James Kiss
Numerade Educator
03:35

Problem 24

The lift force $F$ on a missile is a function of its length $L$ velocity $V$, diameter $D$, angle of attack $\alpha$, density $\rho$, viscosity $\mu,$ and speed of sound $a$ of the air. Write out the dimensional matrix of this function and determine its rank. (See Prob. P5.21 for an explanation of this concept.) Rewrite the function in terms of pi groups.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:19

Problem 25

The thrust $F$ of a propeller is generally thought to be a function of its diameter $D$ and angular velocity $\Omega,$ the for ward speed $V$, and the density $\rho$ and viscosity $\mu$ of the fluid. Rewrite this relationship as a dimensionless function.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:55

Problem 26

A pendulum has an oscillation period $T$ which is assumed to depend on its length $L,$ bob mass $m,$ angle of swing $\theta,$ and the acceleration of gravity. A pendulum $1 \mathrm{m}$ long, with a bob mass of $200 \mathrm{g}$, is tested on earth and found to have a period of 2.04 s when swinging at $20^{\circ} .(a)$ What is its period when it swings at $45^{\circ} ?$ A similarly constructed pendulum, with $L=30 \mathrm{cm}$ and $m=100 \mathrm{g},$ is to swing on the moon $(g=$ $\left.1.62 \mathrm{m} / \mathrm{s}^{2}\right)$ at $\theta=20^{\circ} .(b)$ What will be its period?

CM
Carlos Marin
Numerade Educator
01:43

Problem 27

In studying sand transport by ocean waves, A. Shields in 1936 postulated that the threshold wave-induced bottom shear stress $\tau$ required to move particles depends on gravity $g$, particle size $d$ and density $\rho_{p},$ and water density $\rho$ and viscosity $\mu .$ Find suitable dimensionless groups of this problem, which resulted in 1936 in the celebrated Shields sand transport diagram.

Ajay Singhal
Ajay Singhal
Numerade Educator
06:07

Problem 28

A simply supported beam of diameter $D$, length $L$, and modulus of elasticity $E$ is subjected to a fluid crossflow of velocity $V,$ density $\rho,$ and viscosity $\mu .$ Its center deflection $\delta$ is assumed to be a function of all these variables. (a) Rewrite this proposed function in dimensionless form. $(b)$ Suppose it is known that $\delta$ is independent of $\mu,$ inversely proportional to $E,$ and dependent only on $\rho V^{2},$ not $\rho$ and $V$ separately. Simplify the dimensionless function accordingly. Hint. Take $L, \rho,$ and $V$ as repeating variables.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:57

Problem 29

When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow and then undergoes transition to turbulence at a time $t_{\mathrm{tr}}$ that depends on the pipe diameter $D,$ fluid acceleration $a,$ density $\rho,$ and viscosity $\mu$ Arrange this into a dimensionless relation between $t_{\mathrm{tr}}$ and $D$

Narayan Hari
Narayan Hari
Numerade Educator
05:58

Problem 30

When a large tank of high-pressure gas discharges through a nozzle, the exit mass flow $\dot{m}$ is a function of tank pressure $p_{0}$ and temperature $T_{0},$ gas constant $R,$ specific heat $c_{p},$ and nozzle diameter $D$. Rewrite this as a dimensionless function. Check to see if you can use $\left(p_{0}, T_{0}, R, D\right)$ as repeating variables.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:12

Problem 31

The pressure drop per unit length in horizontal pipe flow, $\Delta p / L,$ depends on the fluid density $\rho,$ viscosity $\mu,$ diameter $D,$ and volume flow rate $Q .$ Rewrite this function in terms of pi groups.

James Kiss
James Kiss
Numerade Educator
05:21

Problem 32

A weir is an obstruction in a channel flow that can be calibrated to measure the flow rate, as in Fig. P5.32. The volume flow $Q$ varies with gravity $g,$ weir width $b$ into the paper, and upstream water height $H$ above the weir crest. If it is known that $Q$ is proportional to $b$, use the pi theorem to find a unique functional relationship $Q(g, b, H)$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:22

Problem 33

A spar buoy (see Prob. P2.113) has a period $T$ of vertical (heave) oscillation that depends on the waterline crosssectional area $A,$ buoy mass $m,$ and fluid specific weight $\gamma$ How does the period change due to doubling of $(a)$ the mass and ( $b$ ) the area? Instrument buoys should have long periods to avoid wave resonance. Sketch a possible longperiod buoy design.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:28

Problem 34

To good approximation, the thermal conductivity $k$ of a gas (see Ref. 21 of Chap. 1) depends only on the density $\rho$ mean free path $l$, gas constant $R$, and absolute temperature
$T .$ For air at $20^{\circ} \mathrm{C}$ and 1 atm, $k \approx 0.026 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K})$ and $l \approx$ $6.5 \mathrm{E}-8 \mathrm{m} .$ Use this information to determine $k$ for hydrogen at $20^{\circ} \mathrm{C}$ and 1 atm if $l \approx 1.2 \mathrm{E}-7 \mathrm{m}$

Penny Riley
Penny Riley
Numerade Educator
02:47

Problem 35

The torque $M$ required to turn the cone-plate viscometer in Fig. $\mathrm{P} 5.35$ depends on the radius $R,$ rotation rate $\Omega,$ fluid viscosity $\mu,$ and cone angle $\theta .$ Rewrite this relation in dimensionless form. How does the relation simplify it if it is known that $M$ is proportional to $\theta ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
05:57

Problem 36

The rate of heat loss $\dot{Q}_{\text {loss }}$ through a window or wall is a function of the temperature difference between inside and outside $\Delta T$, the window surface area $A$, and the $R$ value of the window, which has units of $\left(\mathrm{ft}^{2} \cdot \mathrm{h} \cdot^{\circ} \mathrm{F}\right) / \mathrm{Btu}$ (a) Using the Buckingham Pi Theorem, find an expression for rate of heat loss as a function of the other three parameters in the problem. (b) If the temperature difference $\Delta T$ doubles, by what factor does the rate of heat loss increase?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:12

Problem 37

The volume flow $Q$ through an orifice plate is a function of pipe diameter $D,$ pressure drop $\Delta p$ across the orifice, fluid density $\rho$ and viscosity $\mu,$ and orifice diameter $d .$ Using $D$ $\rho,$ and $\Delta p$ as repeating variables, express this relationship in dimensionless form.

James Kiss
James Kiss
Numerade Educator
01:32

Problem 38

The size $d$ of droplets produced by a liquid spray nozzle is thought to depend on the nozzle diameter $D$, jet velocity $U$ and the properties of the liquid $\rho, \mu,$ and $Y .$ Rewrite this relation in dimensionless form. Hint: Take $D, \rho,$ and $U$ as repeating variables.

James Kiss
James Kiss
Numerade Educator
05:44

Problem 39

The volume flow $Q$ over a certain dam is a function of dam width $b,$ gravity $g,$ and the upstream water depth $H$ above the dam crest. It is known that $Q$ is proportional to $b$. If $b=$ $120 \mathrm{ft}$ and $H=15$ in., the flow rate is $600 \mathrm{ft}^{3} / \mathrm{s}$. What will be the flow rate if $H=3 \mathrm{ft} ?$

Mahnoor Amin
Mahnoor Amin
Numerade Educator
06:33

Problem 40

The time $t_{d}$ to drain a liquid from a hole in the bottom of a tank is a function of the hole diameter $d,$ the initial fluid volume $v_{0},$ the initial liquid depth $h_{0},$ and the density $\rho$ and viscosity $\mu$ of the fluid. Rewrite this relation as a dimensionless function, using Ipsen's method.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:06

Problem 41

A certain axial flow turbine has an output torque $M$ that is proportional to the volume flow rate $Q$ and also depends on the density $\rho,$ rotor diameter $D,$ and rotation rate
$\Omega .$ How does the torque change due to a doubling of $(a) D$ and $(b) \Omega ?$

James Kiss
James Kiss
Numerade Educator
01:52

Problem 42

When disturbed, a floating buoy will bob up and down at frequency $f$. Assume that this frequency varies with buoy mass $m,$ waterline diameter $d,$ and the specific weight $\gamma$ of the liquid. ( $a$ ) Express this as a dimensionless function. ( $b$ ) If $d$ and $\gamma$ are constant and the buoy mass is halved, how will the frequency change?

Alick Cushing
Alick Cushing
Numerade Educator
07:38

Problem 43

Nondimensionalize the energy equation (4.75) and its boundary conditions $(4.62),(4.63),$ and (4.70) by defining $T^{*}=T / T_{0},$ where $T_{0}$ is the inlet temperature, assumed constant. Use other dimensionless variables as needed from Eqs. $(5.23) .$ Isolate all dimensionless parameters you find, and relate them to the list given in Table 5.2

Kevin Zaborsky
Kevin Zaborsky
Numerade Educator
02:07

Problem 44

The differential energy equation for incompressible twodimensional flow through a "Darcy-type" porous medium is approximately
$$
\rho c_{p} \frac{\sigma}{\mu} \frac{\partial p}{\partial x} \frac{\partial T}{\partial x}+\rho c_{p} \frac{\sigma}{\mu} \frac{\partial p}{\partial y} \frac{\partial T}{\partial y}+k \frac{\partial^{2} T}{\partial y^{2}}=0
$$
where $\sigma$ is the permeability of the porous medium. All other symbols have their usual meanings. (a) What are the appropriate dimensions for $\sigma ?$ (b) Non dimensionalize this equation, using $\left(L, U, \rho, T_{0}\right)$ as scaling constants, and discuss any dimensionless parameters that arise.

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
04:57

Problem 45

A model differential equation, for chemical reaction dynamics in a plug reactor, is as follows:
$$
u \frac{\partial C}{\partial x}=D \frac{\partial^{2} C}{\partial x^{2}}-k C-\frac{\partial C}{\partial t}
$$
where $u$ is the velocity, $D$ is a diffusion coefficient, $k$ is a reaction rate, $x$ is distance along the reactor, and $C$ is the (dimensionless) concentration of a given chemical in the reactor. ( $a$ ) Determine the appropriate dimensions of $D$ and $k$ (b) Using a characteristic length scale $L$ and average velocity $V$ as parameters, rewrite this equation in dimensionless form and comment on any pi groups appearing.

Nicole Smina
Nicole Smina
Numerade Educator
22:53

Problem 46

If a vertical wall at temperature $T_{\mathrm{w}}$ is surrounded by a fluid at temperature $T_{0},$ a natural convection boundary layer flow will form. For laminar flow, the momentum equation is
$$
\rho(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y})=\rho \beta(T-T_{0}) g+\mu \frac{\partial^{2} u}{\partial y^{2}}
$$
to be solved, along with continuity and energy, for $(u, v, T)$ with appropriate boundary conditions. The quantity $\beta$ is the thermal expansion coefficient of the fluid. Use $\rho, g, L$ and $\left(T_{\mathrm{w}}-T_{0}\right)$ to non dimensionalize this equation. Note that there is no "stream" velocity in this type of flow.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:33

Problem 47

The differential equation for small-amplitude vibrations $y(x, t)$ of a simple beam is given by
$$
\rho A \frac{\partial^{2} y}{\partial t^{2}}+E I \frac{\partial^{4} y}{\partial x^{4}}=0
$$
where $\rho=$ beam material density $A=$ cross-sectional area $I=$ area moment of inertia $E=$ Young's modulus Use only the quantities $\rho, E,$ and $A$ to non dimensionalize $y, x$ and $t,$ and rewrite the differential equation in dimensionless form. Do any parameters remain? Could they be removed by further manipulation of the variables?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:35

Problem 48

A smooth steel $(\mathrm{SG}=7.86)$ sphere is immersed in a stream of ethanol at $20^{\circ} \mathrm{C}$ moving at $1.5 \mathrm{m} / \mathrm{s}$. Estimate its drag in N from Fig. $5.3 a .$ What stream velocity would quadruple its drag? Take $D=2.5 \mathrm{cm}$

Narayan Hari
Narayan Hari
Numerade Educator
05:24

Problem 49

The sphere in Prob. P5.48 is dropped in gasoline at $20^{\circ} \mathrm{C}$ Ignoring its acceleration phase, what will its terminal (constant fall velocity be, from Fig. $5.3 a ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
03:26

Problem 50

The parachute in the chapter-opener photo is, of course, meant to decelerate the payload on Mars. The wind tunnel test gave a drag coefficient of about $1.1,$ based upon the projected area of the parachute. Suppose it was falling on earth and, at an altitude of $1000 \mathrm{m},$ showed a steady descent rate of about $18 \mathrm{mi} / \mathrm{h}$. Estimate the weight of the payload.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:29

Problem 51

A ship is towing a sonar array that approximates a submerged cylinder $1 \mathrm{ft}$ in diameter and $30 \mathrm{ft}$ long with its axis normal to the direction of tow. If the tow speed is $12 \mathrm{kn}(1 \mathrm{kn}=1.69 \mathrm{ft} / \mathrm{s}),$ estimate the horsepower required to tow this cylinder. What will be the frequency of vortices shed from the cylinder? Use Figs. 5.2 and 5.3

Sachin Rao
Sachin Rao
Numerade Educator
04:57

Problem 52

When fluid in a long pipe starts up from rest at a uniform acceleration $a$, the initial flow is laminar. The flow undergoes transition to turbulence at a time $t^{*}$ which depends, to first approximation, only upon $a, \rho,$ and $\mu .$ Experiments by P. J. Lefebvre, on water at $20^{\circ} \mathrm{C}$ starting from rest with $1-\mathrm{g}$ acceleration in a 3 -cm-diameter pipe, showed transition at $t^{*}=1.02 \mathrm{s} .$ Use this data to estimate $(a)$ the transition time and $(b)$ the transition Reynolds number $\operatorname{Re}_{D}$ for water flow accelerating at $35 \mathrm{m} / \mathrm{s}^{2}$ in a 5 -cm-diameter pipe.

Narayan Hari
Narayan Hari
Numerade Educator
03:08

Problem 53

Vortex shedding can be used to design a vortex flowmeter (Fig. 6.34 ). A blunt rod stretched across the pipe sheds vortices whose frequency is read by the sensor downstream. Suppose the pipe diameter is $5 \mathrm{cm}$ and the rod is a cylinder of diameter $8 \mathrm{mm}$. If the sensor reads 5400 counts per minute, estimate the volume flow rate of water in $\mathrm{m}^{3} / \mathrm{h}$ How might the meter react to other liquids?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
03:16

Problem 54

A fishnet is made of 1 -mm-diameter strings knotted into $2 \times 2 \mathrm{cm}$ squares. Estimate the horsepower required to tow $300 \mathrm{ft}^{2}$ of this netting at $3 \mathrm{kn}$ in seawater at $20^{\circ} \mathrm{C}$. The net plane is normal to the flow direction.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:41

Problem 55

The radio antenna on a car begins to vibrate wildly at $8 \mathrm{Hz}$ when the car is driven at $45 \mathrm{mi} / \mathrm{h}$ over a rutted road that approximates a sine wave of amplitude $2 \mathrm{cm}$ and wavelength $\lambda=2.5 \mathrm{m}$. The antenna diameter is $4 \mathrm{mm}$. Is the vibration due to the road or to vortex shedding?

James Kiss
James Kiss
Numerade Educator
View

Problem 56

Flow past a long cylinder of square cross-section results in more drag than the comparable round cylinder. Here are data taken in a water tunnel for a square cylinder of side length $b=2 \mathrm{cm}$
$$\begin{array}{c|c|c|c|c}
V, \mathrm{m} / \mathrm{s} & 1.0 & 2.0 & 3.0 & 4.0 \\
\hline \text { Drag, } \mathrm{N} /(\mathrm{m} \text { of depth }) & 21 & 85 & 191 & 335
\end{array}$$
(a) Use these data to predict the drag force per unit depth of wind blowing at $6 \mathrm{m} / \mathrm{s},$ in air at $20^{\circ} \mathrm{C},$ over a tall square chimney of side length $b=55 \mathrm{cm} .(b)$ Is there any uncertainty in your estimate?

Victor Salazar
Victor Salazar
Numerade Educator
03:53

Problem 57

The simply supported 1040 carbon-steel rod of Fig. $\mathrm{P} 5.57$ is subjected to a crossflow stream of air at $20^{\circ} \mathrm{C}$ and 1 atm. For what stream velocity $U$ will the rod center deflection be approximately $1 \mathrm{cm} ?$

Dading Chen
Dading Chen
Numerade Educator
01:49

Problem 58

For the steel rod of Prob. P5.57, at what airstream velocity $U$ will the rod begin to vibrate laterally in resonance in its first mode (a half sine wave)? Hint: Consult a vibration text [34,35] under "lateral beam vibration."

Jonathan Ibarra
Jonathan Ibarra
Numerade Educator
01:25

Problem 59

A long, slender, smooth 3 -cm-diameter flagpole bends alarmingly in $20 \mathrm{mi} / \mathrm{h}$ sea-level winds, causing patriotic citizens to gasp. An engineer claims that the pole will bend less if its surface is deliberately roughened. Is she correct, at least qualitatively?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
06:19

Problem 60

The thrust $F$ of a free propeller, either aircraft or marine, depends upon density $\rho,$ the rotation rate $n$ in $\mathrm{r} / \mathrm{s},$ the diameter $D,$ and the forward velocity $V .$ Viscous effects are slight and neglected here. Tests of a 25 -cm-diameter model aircraft propeller, in a sea-level wind tunnel, yield the following thrust data at a velocity of $20 \mathrm{m} / \mathrm{s}$ :
$$\begin{array}{|c|c|c|c|}
\text { Rotation rate, r/min } & 4800 & 6000 & 8000 \\
\hline \text { Measured thrust, } \mathrm{N} & 6.1 & 19 & 47
\end{array}$$
(a) Use this data to make a crude but effective dimensionless plot.
(b) Use the dimensionless data to predict the thrust, in newtons, of a similar 1.6 -m-diameter prototype propeller when rotating at $3800 \mathrm{r} / \mathrm{min}$ and flying at $225 \mathrm{mi} / \mathrm{h}$ at 4000 -m standard altitude.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:45

Problem 61

If viscosity is neglected, typical pump flow results from Example 5.3 are shown in Fig. P5.61 for a model pump tested in water. The pressure rise decreases and the power required increases with the dimensionless flow coefficient. Curve-fit expressions are given for the data. Suppose a similar pump of $12-\mathrm{cm}$ diameter is built to move gasoline at $20^{\circ} \mathrm{C}$ and a flow rate of $25 \mathrm{m}^{3} / \mathrm{h}$. If the pump rotation speed is $30 \mathrm{r} / \mathrm{s}$, find $(a)$ the pressure rise and ( $b$ ) the power required.

James Kiss
James Kiss
Numerade Educator
05:59

Problem 62

For the system of Prob. P5.22, assume that a small model wind turbine of diameter $90 \mathrm{cm},$ rotating at $1200 \mathrm{r} / \mathrm{min}$ delivers 280 watts when subjected to a wind of $12 \mathrm{m} / \mathrm{s}$. The data is to be used for a prototype of diameter $50 \mathrm{m}$ and winds of $8 \mathrm{m} / \mathrm{s}$. For dynamic similarity, estimate $(a)$ the rotation rate, and $(b)$ the power delivered by the prototype. Assume sea-level air density.

Narayan Hari
Narayan Hari
Numerade Educator
05:04

Problem 63

The Keystone Pipeline in the Chapter 6 opener photo has $D=36$ in. and an oil flow rate $Q=590,000$ barrels per day barrel $=42$ U.S. gallons . Its pressure drop per unit length, $\Delta p / L,$ depends on the fluid density $\rho,$ viscosity $\mu$ diameter $D,$ and flow rate $Q .$ A water-flow model test, at $20^{\circ} \mathrm{C},$ uses a 5 -cm-diameter pipe and yields $\Delta p / L \approx 4000$ $\mathrm{Pa} / \mathrm{m} .$ For dynamic similarity, estimate $\Delta p / L$ of the pipeline. For the oil take $\rho=860 \mathrm{kg} / \mathrm{m}^{3}$ and $\mu=0.005 \mathrm{kg} / \mathrm{m} \cdot \mathrm{s}$

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
01:39

Problem 64

The natural frequency $\omega$ of vibration of a mass $M$ attached to a rod, as in Fig. P5.64, depends only on $M$ and the stiffness $E I$ and length $L$ of the rod. Tests with a 2-kg mass attached to a 1040 carbon steel rod of diameter $12 \mathrm{mm}$ and length $40 \mathrm{cm}$ reveal a natural frequency of $0.9 \mathrm{Hz}$ Use these data to predict the natural frequency of a $1-\mathrm{kg}$ mass attached to a 2024 aluminum alloy rod of the same size.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
10:25

Problem 65

In turbulent flow near a flat wall, the local velocity $u$ varies only with distance $y$ from the wall, wall shear stress $\tau_{w},$ and fluid properties $\rho$ and $\mu .$ The following data were taken in the University of Rhode Island wind tunnel for airflow, $\rho=0.0023$ slug/ft $^{3}, \mu=3.81 \mathrm{E}-7$ $\operatorname{slug} /(\mathrm{ft} \cdot \mathrm{s}),$ and $\tau_{w}=0.029 \mathrm{lbf} / \mathrm{ft}^{2}$
$$\begin{array}{l|l|l|l|l|l|l}
y, \text { in } & 0.021 & 0.035 & 0.055 & 0.080 & 0.12 & 0.16 \\
\hline u, \mathrm{ft} / \mathrm{s} & 50.6 & 54.2 & 57.6 & 59.7 & 63.5 & 65.9
\end{array}$$
(a) Plot these data in the form of dimensionless $u$ versus dimensionless $y,$ and suggest a suitable power-law curve fit. $(b)$ Suppose that the tunnel speed is increased until $u=$ $90 \mathrm{ft} / \mathrm{s}$ at $y=0.11$ in. Estimate the new wall shear stress, in $\mathrm{Ibf} / \mathrm{ft}^{2}$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:33

Problem 66

A torpedo $8 \mathrm{m}$ below the surface in $20^{\circ} \mathrm{C}$ seawater cavitates at a speed of $21 \mathrm{m} / \mathrm{s}$ when atmospheric pressure is $101 \mathrm{kPa}$ If Reynolds number and Froude number effects are negligible, at what speed will it cavitate when running at a depth of $20 \mathrm{m} ?$ At what depth should it be to avoid cavitation at $30 \mathrm{m} / \mathrm{s} ?$

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
07:25

Problem 67

A student needs to measure the drag on a prototype of characteristic dimension $d_{p}$ moving at velocity $U_{p}$ in air at standard atmospheric conditions. He constructs a model of characteristic dimension $d_{m},$ such that the ratio $d_{p} / d_{m}$ is some factor $f$. He then measures the drag on the model at dynamically similar conditions (also with air at standard atmospheric conditions). The student claims that the drag force on the prototype will be identical to that measured on the model. Is this claim correct? Explain.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:25

Problem 68

A student needs to measure the drag on a prototype of characteristic dimension $d_{p}$ moving at velocity $U_{p}$ in air at standard atmospheric conditions. He constructs a model of characteristic dimension $d_{m},$ such that the ratio $d_{p} / d_{m}$ is some factor $f$. He then measures the drag on the model at dynamically similar conditions (also with air at standard atmospheric conditions). The student claims that the drag force on the prototype will be identical to that measured on the model. Is this claim correct? Explain.
$$\begin{array}{|l|l|c|l|l|l|l|}
\Omega, \text { rev/min } & 0 & 3000 & 6000 & 9000 & 12000 & 15000 \\
\hline F, \mathrm{N} & 0 & 850 & 2260 & 2900 & 3120 & 3300
\end{array}$$
(a) Reduce this data to the two dimensionless groups and make a plot. ( $b$ ) Use this plot to predict the lift of a cylinder with $D=5 \mathrm{cm}, L=80 \mathrm{cm},$ rotating at $3800 \mathrm{rev} / \mathrm{min}$ in water at $U=4 \mathrm{m} / \mathrm{s}$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 69

A simple flow measurement device for streams and channels is a notch, of angle $\alpha,$ cut into the side of a dam, as shown in Fig. P5.69. The volume flow $Q$ depends only on $\alpha,$ the acceleration of gravity $g,$ and the height $\delta$ of the upstream water surface above the notch vertex. Tests of a model notch, of angle $\alpha=55^{\circ},$ yield the following flow rate data:
$$\begin{array}{l|c|c|c|r}
\delta, \mathrm{cm} & 10 & 20 & 30 & 40 \\
\hline Q, \mathrm{m}^{3} / \mathrm{h} & 8 & 47 & 126 & 263
\end{array}$$
(a) Find a dimensionless correlation for the data. $(b)$ Use the model data to predict the flow rate of a prototype notch, also of angle $\alpha=55^{\circ},$ when the upstream height $\delta$ is $3.2 \mathrm{m}$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 70

A diamond-shaped body, of characteristic length 9 in, has the following measured drag forces when placed in a wind tunnel at sea-level standard conditions:
$$\begin{array}{|c|c|c|c|c|c|}
V, \mathrm{ft} / \mathrm{s} & 30 & 38 & 48 & 56 & 61 \\
\hline F, \text { 1bf } & 1.25 & 1.95 & 3.02 & 4.05 & 4.81
\end{array}$$
Use these data to predict the drag force of a similar 15 -in diamond placed at similar orientation in $20^{\circ} \mathrm{C}$ water flowing at $2.2 \mathrm{m} / \mathrm{s}$

Victor Salazar
Victor Salazar
Numerade Educator
04:42

Problem 71

The pressure drop in a venturi meter (Fig. P3.128) varies only with the fluid density, pipe approach velocity, and diameter ratio of the meter. A model venturi meter tested in water at $20^{\circ} \mathrm{C}$ shows a 5 -kPa drop when the approach velocity is $4 \mathrm{m} / \mathrm{s}$ A geometrically similar prototype meter is used to measure gasoline at $20^{\circ} \mathrm{C}$ and a flow rate of $9 \mathrm{m}^{3} / \mathrm{min}$. If the prototype pressure gage is most accurate at $15 \mathrm{kPa}$, what should the upstream pipe diameter be?

Mahnoor Amin
Mahnoor Amin
Numerade Educator
03:42

Problem 72

A one-twelfth-scale model of a large commercial aircraft is tested in a wind tunnel at $20^{\circ} \mathrm{C}$ and 1 atm. The model chord length is $27 \mathrm{cm},$ and its wing area is $0.63 \mathrm{m}^{2}$. Test results for the drag of the model are as follows:
$$\begin{array}{l|l|l|l|l}
\mathrm{V}, \mathrm{mi} / \mathrm{h} & 50 & 75 & 100 & 125 \\
\hline \text { Drag, } \mathrm{N} & 15 & 32 & 53 & 80
\end{array}$$
In the spirit of Fig. $5.8,$ use this data to estimate the drag of the full-scale aircraft when flying at $550 \mathrm{mi} / \mathrm{h}$, for the same angle of attack, at $32,800 \mathrm{ft}$ standard altitude.

Chai Santi
Chai Santi
Numerade Educator
09:06

Problem 73

The power $P$ generated by a certain windmill design depends on its diameter $D$, the air density $\rho$, the wind velocity $V$, the rotation rate $\Omega,$ and the number of blades $n .(a)$ Write this relationship in dimensionless form. A model windmill, of diameter $50 \mathrm{cm},$ develops $2.7 \mathrm{kW}$ at sea level when $V=40 \mathrm{m} / \mathrm{s}$ and when rotating at $4800 \mathrm{r} / \mathrm{min}$
(b) What power will be developed by a geometrically and dynamically similar prototype, of diameter $5 \mathrm{m}$, in winds of $12 \mathrm{m} / \mathrm{s}$ at $2000 \mathrm{m}$ standard altitude? $(c)$ What is the appropriate rotation rate of the prototype?

Calin Lupas
Calin Lupas
Numerade Educator
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Problem 74

A one-tenth-scale model of a supersonic wing tested at $700 \mathrm{m} / \mathrm{s}$ in air at $20^{\circ} \mathrm{C}$ and 1 atm shows a pitching moment of 0.25 $\mathrm{kN} \cdot \mathrm{m} .$ If Reynolds number effects are negligible, what will the pitching moment of the prototype wing be if it is flying at the same Mach number at 8 -km standard altitude?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 75

According to the web site $U S G S$ Daily Water Data for the Nation, the mean flow rate in the New River near Hinton, $\mathrm{WV},$ is $10,100 \mathrm{ft}^{3} / \mathrm{s}$. If the hydraulic model in Fig. 5.9 is to match this condition with Froude number scaling, what is the proper model flow rate?

Victor Salazar
Victor Salazar
Numerade Educator
13:04

Problem 76

A 2 -ft-long model of a ship is tested in a freshwater tow tank. The measured drag may be split into "friction" drag (Reynolds scaling) and "wave" drag (Froude scaling). The model data are as follows:
$$\begin{array}{|l|l|l|l|l|l|l|}
\text { Tow speed, ft/s } & 0.8 & 1.6 & 2.4 & 3.2 & 4.0 & 4.8 \\
\hline \text { Friction drag, lbf } & 0.016 & 0.057 & 0.122 & 0.208 & 0.315 & 0.441 \\
\hline \text { Wave drag, lbf } & 0.002 & 0.021 & 0.083 & 0.253 & 0.509 & 0.697
\end{array}$$
The prototype ship is $150 \mathrm{ft}$ long. Estimate its total drag when cruising at $15 \mathrm{kn}$ in seawater at $20^{\circ} \mathrm{C}$

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:08

Problem 77

A dam $75 \mathrm{ft}$ wide, with a nominal flow rate of $260 \mathrm{ft}^{3}$, is to be studied with a scale model $3 \mathrm{ft}$ wide, using Froude scaling. (a) What is the expected flow rate for the model? (b) What is the danger of only using Froude scaling for this test? $(c)$ Derive a formula for a force on the model as compared to a force on the prototype.

Ronald Prasad
Ronald Prasad
Numerade Educator
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Problem 78

A prototype spillway has a characteristic velocity of $3 \mathrm{m} / \mathrm{s}$ and a characteristic length of $10 \mathrm{m} .$ A small model is constructed by using Froude scaling. What is the minimum scale ratio of the model that will ensure that its minimum Weber number is $100 ?$ Both flows use water at $20^{\circ} \mathrm{C}$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 79

An East Coast estuary has a tidal period of $12.42 \mathrm{h}$ (the semidiurnal lunar tide) and tidal currents of approximately $80 \mathrm{cm} / \mathrm{s} .$ If a one-five-hundredth-scale model is constructed with tides driven by a pump and storage apparatus, what should the period of the model tides be and what model current speeds are expected?

Danielle Fairburn
Danielle Fairburn
Numerade Educator
02:21

Problem 80

A prototype ship is $35 \mathrm{m}$ long and designed to cruise at $11 \mathrm{m} / \mathrm{s}(\text { about } 21 \mathrm{kn}) .$ Its drag is to be simulated by a $1-\mathrm{m}-$ long model pulled in a tow tank. For Froude scaling find (a) the tow speed, (b) the ratio of prototype to model $\operatorname{drag},$ and $(c)$ the ratio of prototype to model power.

Chai Santi
Chai Santi
Numerade Educator
07:18

Problem 81

An airplane, of overall length $55 \mathrm{ft}$, is designed to fly at $680 \mathrm{m} / \mathrm{s}$ at $8000-\mathrm{m}$ standard altitude. A one-thirtieth-scale model is to be tested in a pressurized helium wind tunnel at $20^{\circ} \mathrm{C}$. What is the appropriate tunnel pressure in atm? Even at this (high) pressure, exact dynamic similarity is not achieved. Why?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:37

Problem 82

A one-fiftieth-scale model of a military airplane is tested at $1020 \mathrm{m} / \mathrm{s}$ in a wind tunnel at sea-level conditions. The model wing area is $180 \mathrm{cm}^{2} .$ The angle of attack is $3^{\circ} .$ If the measured model lift is $860 \mathrm{N},$ what is the prototype lift, using Mach number scaling, when it flies at $10,000 \mathrm{m}$ standard altitude under dynamically similar conditions? Note: Be careful with the area scaling.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
07:42

Problem 83

A one-fortieth-scale model of a ship's propeller is tested in a tow tank at $1200 \mathrm{r} / \mathrm{min}$ and exhibits a power output of 1.4 $\mathrm{ft} \cdot \mathrm{lbf} / \mathrm{s} .$ According to Froude scaling laws, what should the revolutions per minute and horsepower output of the prototype propeller be under dynamically similar conditions?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:55

Problem 84

A prototype ocean platform piling is expected to encounter currents of $150 \mathrm{cm} / \mathrm{s}$ and waves of 12 -s period and $3-\mathrm{m}$ height. If a one-fifteenth-scale model is tested in a wave channel, what current speed, wave period, and wave height should be encountered by the model?

Supratim Pal
Supratim Pal
Numerade Educator
01:32

Problem 85

As shown in Example $5.3,$ pump performance data can be nondimensionalized. Problem P5.61 gave typical dimensionless data for centrifugal pump "head," $H=$ $\Delta p / \rho g,$ as follows:
$$
\frac{g H}{n^{2} D^{2}} \approx 6.0-120\left(\frac{Q}{n D^{3}}\right)^{2}
$$
where $Q$ is the volume flow rate, $n$ the rotation rate in $\mathrm{r} / \mathrm{s}$ and $D$ the impeller diameter. This type of correlation allows one to compute $H$ when $(\rho, Q, D)$ are known. $(a)$ Show how to rearrange these pi groups so that one can size the pump, that is, compute $D$ directly when $(Q, H, n)$ are known.
(b) Make a crude but effective plot of your new function. $(c)$ Apply part $(b)$ to the following example: Find $D$ when $H=37 \mathrm{m}, Q=0.14 \mathrm{m}^{3} / \mathrm{s},$ and $n=35 \mathrm{r} / \mathrm{s} .$ Find the pump diameter for this condition.

Dominador Tan
Dominador Tan
Numerade Educator
01:55

Problem 86

Solve Prob. P5.49 for glycerin at $20^{\circ} \mathrm{C}$, using the modified sphere-drag plot of Fig. 5.11

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:01

Problem 87

In Prob. P5.61 it would be difficult to solve for $\Omega$ because it appears in all three of the dimensionless pump coefficients. Suppose that, in Prob. $5.61, \Omega$ is unknown but $D=12 \mathrm{cm}$ and $Q=25 \mathrm{m}^{3} / \mathrm{h} .$ The fluid is gasoline at $20^{\circ} \mathrm{C}$. Rescale the coefficients, using the data of Prob. P5.61, to make a plot of dimensionless power versus dimensionless rotation speed. Enter this plot to find the maximum rotation speed $\Omega$ for which the power will not exceed $300 \mathrm{W}$

Narayan Hari
Narayan Hari
Numerade Educator
01:15

Problem 88

Modify Prob. P5.61 as follows: Let $\Omega=32 \mathrm{r} / \mathrm{s}$ and $Q=24$ $\mathrm{m}^{3} / \mathrm{h}$ for a geometrically similar pump. What is the maximum diameter if the power is not to exceed $340 \mathrm{W}$ ? Solve this problem by rescaling the data of Fig. P5.61 to make a plot of dimensionless power versus dimensionless diameter. Enter this plot directly to find the desired diameter.

Chai Santi
Chai Santi
Numerade Educator
10:25

Problem 89

Wall friction $\tau_{\mathrm{w}},$ for turbulent flow at velocity $U$ in a pipe of diameter $D,$ was correlated, in $1911,$ with a dimensionless correlation by Ludwig Prandtl's student
H. Blasius:
$$
\frac{\tau_{w}}{\rho U^{2}} \approx \frac{0.632}{(\rho U D / \mu)^{1 / 4}}
$$
Suppose that $\left(\rho, U, \mu, \tau_{\mathrm{w}}\right)$ were all known and it was desired to find the unknown velocity $U$. Rearrange and rewrite the formula so that $U$ can be immediately calculated.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:56

Problem 90

Knowing that $\Delta p$ is proportional to $L,$ rescale the data of Example 5.10 to plot dimensionless $\Delta p$ versus dimensionless viscosity. Use this plot to find the viscosity required in the first row of data in Example 5.10 if the pressure drop is increased to 10 kPa for the same flow rate, length, and density.

Supratim Pal
Supratim Pal
Numerade Educator
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Problem 91

The traditional "Moody-type" pipe friction correlation in Chap. 6 is of the form
$$
f=\frac{2 \Delta p D}{\rho V^{2} L}=\operatorname{fcn}\left(\frac{\rho V D}{\mu}, \frac{\varepsilon}{D}\right)
$$
where $D$ is the pipe diameter, $L$ the pipe length, and $\varepsilon$ the wall roughness. Note that pipe average velocity $V$ is used on both sides. This form is meant to find $\Delta p$ when $V$ is known. (a) Suppose that $\Delta p$ is known, and we wish to find $V$. Rearrange the above function so that $V$ is isolated on the left-hand side. Use the following data, for $\varepsilon / D=0.005,$ to make a plot of your new function, with your velocity parameter as the ordinate of the plot.
$$\begin{array}{c|c|c|c|c|c}
f & 0.0356 & 0.0316 & 0.0308 & 0.0305 & 0.0304 \\
\hline p V D / \mu & 15,000 & 75,000 & 250,000 & 900,000 & 3,330,000
\end{array}$$
(b) Use your plot to determine $V$, in $\mathrm{m} / \mathrm{s}$, for the following pipe flow: $D=5 \mathrm{cm}, \varepsilon=0.025 \mathrm{cm}, L=10 \mathrm{m},$ for water flow at $20^{\circ} \mathrm{C}$ and 1 atm. The pressure drop $\Delta p$ is $110 \mathrm{kPa}$

Victor Salazar
Victor Salazar
Numerade Educator