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Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 7

DIMENSIONAL ANALYSIS AND MODELING - all with Video Answers

Educators

Chapter Questions

02:40

Problem 1

$ \mathrm{C}$ What is the difference between a dimension and a unit? Give three examples of each.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:57

Problem 2

When performing a dimensional analysis, one of the first steps is to list the primary dimensions of each relevant parameter. It is handy to have a table of parameters and their primary dimensions. We have started such a table for you (Table P7-2C), in which we have included some of the basic parameters commonly encountered in fluid mechanics. As you work through homework problems in this chapter, add to this table. You should be able to build up a table with dozens of parameters.
(TABLE CAN'T COPY)

Farhana Sharmin
Farhana Sharmin
Numerade Educator

Problem 3

List the seven primary dimensions. What is significant about these seven?

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03:59

Problem 4

Write the primary dimensions of the universal ideal gas constant $R_u$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
06:04

Problem 5

On a periodic chart of the elements, molar mass ( $M$ ), also called atomic weight, is often listed as though it were a dimensionless quantity (Fig. P7-5). In reality, atomic weight is the mass of 1 mol of the element. For example, the atomic weight of nitrogen $M_{\text {nitrogen }}=14.0067$. We interpret this as $14.0067 \mathrm{~g} / \mathrm{mol}$ of elemental nitrogen, or in the English system, $14.0067 \mathrm{lbm} / \mathrm{lbmol}$ of elemental nitrogen. What are the primary dimensions of atomic weight?
(FIGURE CAN'T COPY)

AP
Alex Plastow
Numerade Educator
01:17

Problem 6

Some authors prefer to use force as a primary dimension in place of mass. In a typical fluid mechanics problem, then, the four represented primary dimensions $\mathrm{m}, \mathrm{L}, \mathrm{t}$, and T are replaced by F, L, t, and T. The primary dimension of force in this system is \{force $\}=\{\mathrm{F}\}$. Using the results of Prob. 7-4, rewrite the primary dimensions of the universal gas constant in this alternate system of primary dimensions.

Arpit Gupta
Arpit Gupta
Numerade Educator

Problem 7

We define the specific ideal gas constant $R_{\mathrm{gas}}$ for a particular gas as the ratio of the universal gas constant and the molar mass (also called molecular weight) of the gas, $R_{\mathrm{gas}}=$ $R_{\mathrm{u}} / M$. For a particular gas, then, the ideal gas law can be written as follows:
$$
P V=m R_{g a} T \quad \text { or } \quad P=\rho R_{g a} T
$$
where $P$ is pressure, $V$ is volume, $m$ is mass, $T$ is absolute temperature, and $\rho$ is the density of the particular gas. What are the primary dimensions of $R_{\mathrm{gus}}$ ? For air, $R_{\mathrm{uir}}=287.0 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ in standard SI units. Verify that these units agree with your result.

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10:03

Problem 8

The moment of force $(\vec{M})$ is formed by the cross product of a moment arm $(\vec{r})$ and an applied force $(\vec{F})$, as sketched in Fig. P7-8. What are the primary dimensions of moment of force? List its units in primary SI units and in primary English units.
(FIGURE CAN'T COPY)

Anthony Ramos
Anthony Ramos
Numerade Educator
01:14

Problem 9

Write the primary dimensions of each of the following variables from the field of thermodynamics, showing all your work: (a) energy $E$; (b) specific energy $e=E / m ;$ (c) power $W$.

Anand Jangid
Anand Jangid
Numerade Educator
03:23

Problem 10

What are the primary dimensions of electric voltage $(E)$ ?

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:29

Problem 11

You are probably familiar with Ohm's law for electric circuits (Fig. P7-11), where $\Delta E$ is the voltage difference or potential across the resistor, $I$ is the electric current passing through the resistor, and $R$ is the electrical resistance. What are the primary dimensions of electrical resistance? Answer: $\left.\left\{m^1 L^2 t^{-3}\right\}^{-2}\right\}$
(FIGURE CAN'T COPY)

Satpal Satpal
Satpal Satpal
Numerade Educator
01:44

Problem 12

Write the primary dimensions of each of the following variables, showing all your work: (a) acceleration $a$; (b) angular velocity $\omega ;(c)$ angular acceleration $\alpha$.

Donald Albin
Donald Albin
Numerade Educator
06:48

Problem 13

Angular momentum, also called moment of momentum $(\vec{H})$, is formed by the cross product of a moment arm $(\vec{r})$ and the linear momentum $(m \vec{V})$ of a fluid particle, as sketched in Fig. P7-13. What are the primary dimensions of angular momentum? List the units of angular momentum in primary SI units and in primary English units.
(FIGURE CAN'T COPY)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
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Problem 14

Write the primary dimensions of each of the following variables, showing all your work: (a) specific heat at constant pressure $c_p ;(b)$ specific weight $\rho g ;(c)$ specific enthalpy $h$.

Victor Salazar
Victor Salazar
Numerade Educator
01:31

Problem 15

Thermal conductivity $k$ is a measure of the ability of a material to conduct heat (Fig. P7-15). For conduction heat transfer in the $x$-direction through a surface normal to the $x$ direction, Fourier's law of heat conduction is expressed as
$$
\dot{Q}_{\text {condaction }}=-k A \frac{d T}{d x}
$$
where $\dot{Q}_{\text {conduction }}$ is the rate of heat transfer and $A$ is the area normal to the direction of heat transfer. Determine the primary dimensions of thermal conductivity ( $k$ ). Look up a value of $k$ in the appendices and verify that its SI units are consistent with your result. In particular, write the primary SI units of $k$.
(FIGURE CAN'T COPY)

Joseph Liao
Joseph Liao
Numerade Educator
09:52

Problem 16

Write the primary dimensions of each of the following variables from the study of convection heat transfer (Fig. P7-16), showing all your work: (a) heat generation rate $\dot{g}$

Mohammad Mehran
Mohammad Mehran
Numerade Educator
02:28

Problem 17

Thumb through the appendices of your thermodynamics book, and find three properties or constants not mentioned in Probs. 7-1 to 7-16. List the name of each property or constant and its SI units. Then write out the primary dimensions of each property or constant.

Christopher Nilsen
Christopher Nilsen
Numerade Educator
02:28

Problem 18

Thumb through the appendices of this book and/or your thermodynamics book, and find three properties or constants not mentioned in Probs. 7-1 to 7-17. List the name of each property or constant and its English units. Then write out the primary dimensions of each property or constant.

Christopher Nilsen
Christopher Nilsen
Numerade Educator
03:09

Problem 19

Explain the law of dimensional homogeneity in simple terms.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
07:25

Problem 20

In Chap. 4 we defined the material acceleration, which is the acceleration following a fluid particle (Fig. P7-20),
$$
\vec{a}(x, y, z, t)=\frac{\partial \vec{V}}{\partial t}+(\vec{V} \cdot \vec{\nabla}) \vec{V}
$$
(a) What are the primary dimensions of the gradient operator $\vec{\nabla}$ ? (b) Verify that each additive term in the equation has the same dimensions.
(FIGURE CAN'T COPY)

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:29

Problem 21

Newton's second law is the foundation for the differential equation of conservation of linear momentum (to be discussed in Chap. 9). In terms of the material acceleration following a fluid particle (Fig. P7-20), we write Newton's second law as follows:
$$
\vec{F}=m \vec{a}=m\left[\frac{\partial \vec{V}}{\partial t}+(\vec{V} \cdot \vec{\nabla}) \vec{V}\right]
$$
Or, dividing both sides by the mass $m$ of the fluid particle,
$$
\frac{\vec{F}}{m}=\frac{\partial \vec{V}}{\partial t}+(\vec{V} \cdot \vec{\nabla}) \vec{V}
$$
Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.

Hava Schwartz
Hava Schwartz
Numerade Educator
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Problem 22

In Chap. 4 we defined volumetric strain rate as the rate of increase of volume of a fluid element per unit volume (Fig. P7-22). In Cartesian coordinates we write the volumetric strain rate as
$$
\frac{1}{V} \frac{D V}{D t}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}
$$
Write the primary dimensions of each additive term, and verify that the equation is dimensionally homogeneous. Show all your work.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator

Problem 23

In Chap. 9 we discuss the differential equation for conservation of mass, the continuity equation. In cylindrical coordinates, and for steady flow,
$$
\frac{1}{r} \frac{\partial\left(r u_r\right)}{\partial r}+\frac{1}{r} \frac{\partial u_\theta}{\partial \theta}+\frac{\partial u_z}{\partial z}=0
$$
Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.

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02:07

Problem 24

Cold water enters a pipe, where it is heated by an external heat source (Fig. P7-24). The inlet and outlet water temperatures are $T_{\text {in }}$ and $T_{\text {out }}$ respectively. The total rate of heat transfer $\dot{Q}$ from the surroundings into the water in the pipe is where $\dot{m}$ is the mass flow rate of water through the pipe, and $c_p$ is the specific heat of the water. Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.
(FIGURE CAN'T COPY)

Naman Kumar
Naman Kumar
Numerade Educator
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Problem 25

The Reynolds transport theorem (RTT) is discussed in Chap. 4. For the general case of a moving and/or deforming control volume, we write the RTT as follows:
$$
\frac{d B_{\mathrm{sys}}}{d t}=\frac{d}{d t} \int_{\mathrm{CV}} \rho b d V+\int_{\mathrm{CS}} \rho b \vec{V}_r \cdot \vec{n} d A
$$
where $\vec{V}_r$ is the relative velocity, i.e., the velocity of the fluid relative to the control surface. Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.

Victor Salazar
Victor Salazar
Numerade Educator
03:37

Problem 26

An important application of fluid mechanics is the study of room ventilation. In particular, suppose there is a source $S$ (mass per unit time) of air pollution in a room of volume V (Fig. P7-26). Examples include carbon monoxide from cigarette smoke or an unvented kerosene heater, gases like ammonia from household cleaning products, and vapors given off by evaporation of volatile organic compounds (VOCs) from an open container. We let $c$ represent the mass concentration (mass of contaminant per unit volume of air). $\dot{V}$ is the volume flow rate of fresh air entering the room. If the room air is well mixed so that the mass concentration $c$ is uniform throughout the room, but varies with time, the differential equation for mass concentration in the room as a function of time is
$$
V \frac{d c}{d t}=S-\dot{V} c-c A_s k_w
$$
where $k_w$ is an adsorption coefficient and $A_s$ is the surface area of walls, floors, furniture, etc., that adsorb some of the contaminant. Write the primary dimensions of the first three additive terms in the equation, and verify that those terms are dimensionally homogeneous. Then determine the dimensions of $k_{w^{-}}$. Show all your work.
(FIGURE CAN'T COPY)

Lottie Adams
Lottie Adams
Numerade Educator

Problem 27

What is the primary reason for nondimensionalizing an equation?

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Problem 28

Consider ventilation of a well-mixed room as in Fig. P7-26. The differential equation for mass concentration in the room as a function of time is given in Prob. 7-26 and is repeated here for convenience,
$$
V \frac{d c}{d t}=S-\dot{V} c-c A_s k_w
$$
There are three characteristic parameters in such a situation: $L$, a characteristic length scale of the room (assume $L=V^{1 / 3}$ ); $\dot{V}$, the volume flow rate of fresh air into the room, and $c_{\text {limit }}$, the maximum mass concentration that is not harmful. (a) Using these three characteristic parameters, define dimensionless forms of all the variables in the equation. (b) Rewrite the equation in dimensionless form, and identify any established dimensionless groups that may appear.

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Problem 29

Recall from Chap. 4 that the volumetric strain rate is zero for a steady incompressible flow. In Cartesian coordinates we express this as
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0
$$
Suppose the characteristic speed and characteristic length for a given flow field are $V$ and $L$, respectively (Fig. P7-29). Define the following dimensionless variables,
$$
\begin{gathered}
x^*=\frac{x}{L}, \quad y^*=\frac{y}{L}, \quad z^*=\frac{z}{L}, \\
u^*=\frac{u}{V} \quad v^*=\frac{v}{V}, \quad \text { and } \quad w^*=\frac{w}{V}
\end{gathered}
$$
Nondimensionalize the equation, and identify any established (named) dimensionless parameters that may appear. Discuss.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 30

In an oscillating compressible flow field the volumetric strain rate is not zero, but varies with time following a fluid particle. In Cartesian coordinates we express this as
$$
\frac{1}{V} \frac{D V}{D t}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}
$$
Suppose the characteristic speed and characteristic length for a given flow field are $V$ and $L$, respectively. Also suppose that $f$ is a characteristic frequency of the oscillation (Fig. P7-30). Define the following dimensionless variables,
$$
\begin{gathered}
t^*=f t, \quad V^*=\frac{V}{L^3}, \quad x^*=\frac{x}{L}, \quad y^*=\frac{y}{L^{\prime}} \\
z^*=\frac{z}{L}, \quad u^*=\frac{u}{V} \quad v^*=\frac{v}{V} \quad \text { and } \quad w^*=\frac{w}{V}
\end{gathered}
$$
Nondimensionalize the equation and identify any established (named) dimensionless parameters that may appear.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
02:33

Problem 31

In Chap. 9 we define the stream function $\psi$ for twodimensional incompressible flow in the $x y$-plane,
$$
u=\frac{\partial \psi}{\partial y} \quad v=-\frac{\partial \psi}{\partial x}
$$
where $u$ and $v$ are the velocity components in the $x$ - and $y$ directions, respectively. (a) What are the primary dimensions of $\psi$ ? (b) Suppose a certain two-dimensional flow has a characteristic length scale $L$ and a characteristic time scale $t$. Define dimensionless forms of variables $x, y, u, v$, and $\psi$. (c) Rewrite the equations in nondimensional form, and identify any established dimensionless parameters that may appear.

James Kiss
James Kiss
Numerade Educator
06:56

Problem 32

In an oscillating incompressible flow field the force per unit mass acting on a fluid particle is obtained from Newton's second law in intensive form (see Prob. 7-21),
$$
\frac{\vec{F}}{m}=\frac{\partial \vec{V}}{\partial t}+(\vec{V} \cdot \vec{\nabla}) \vec{V}
$$
Suppose the characteristic speed and characteristic length for a given flow field are $V_{\infty}$ and $L$, respectively. Also suppose that $\omega$ is a characteristic angular frequency ( $\mathrm{rad} / \mathrm{s}$ ) of the oscillation (Fig. P7-32). Define the following nondimensionalized variables,
$$
t^*=\omega t, \quad \vec{x}^*=\frac{\vec{x}}{L}, \quad \vec{\nabla}^*=L \vec{\nabla}, \quad \text { and } \quad \vec{V}^*=\frac{\vec{V}}{V_x}
$$
Since there is no given characteristic scale for the force per unit mass acting on a fluid particle, we assign one, noting that $\{\vec{F} / m\}=\left\{\mathrm{L} / \mathrm{t}^2\right\}$. Namely, we let

$$
(\vec{F} / m)^*=\frac{1}{\omega^2 L} \vec{F} / m
$$
Nondimensionalize the equation of motion and identify any established (named) dimensionless parameters that may appear.
(FIGURE CAN'T COPY)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:00

Problem 33

A wind tunnel is used to measure the pressure distribution in the airflow over an airplane model (Fig. P7-33). The air speed in the wind tunnel is low enough that compressible effects are negligible. As discussed in Chap. 5, the Bernoulli equation approximation is valid in such a flow situation everywhere except very close to the body surface or wind tunnel wall surfaces and in the wake region behind the model. Far away from the model, the air flows at speed $V_{\infty}$ and pressure $P_\alpha$, and the air density $\rho$ is approximately constant. Gravitational effects are generally negligible in airflows, so we write the Bernoulli equation as
$$
P+\frac{1}{2} \rho V^2=P_{\infty}+\frac{1}{2} \rho V_{\infty}^2
$$
Nondimensionalize the equation, and generate an expression for the pressure coefficient $C_p$ at any point in the flow where the Bernoulli equation is valid. $C_p$ is defined as
$$
C_p=\frac{P-P_x}{\frac{1}{2} \rho V_x^2}
$$
(FIGURE CAN'T COPY)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:51

Problem 34

List the three primary purposes of dimensional analysis.

Sohini Lahiri
Sohini Lahiri
Numerade Educator
00:41

Problem 35

List and describe the three necessary conditions for complete similarity between a model and a prototype.

Katelyn Chen
Katelyn Chen
Numerade Educator
04:28

Problem 36

A student team is to design a human-powered submarine for a design competition. The overall length of the prototype submarine is 2.24 m , and its student designers hope that it can travel fully submerged through water at $0.560 \mathrm{~m} / \mathrm{s}$. The water is freshwater (a lake) at $T=15^{\circ} \mathrm{C}$. The design team builds a one-eighth scale model to test in their university's wind tunnel (Fig. P7-36). A shield surrounds the drag balance strut so that the aerodynamic drag of the strut itself does not influence the measured drag. The air in the wind tunnel is at $25^{\circ} \mathrm{C}$ and at one standard atmosphere pressure. At what air speed do they need to run the wind tunnel in order to achieve similarity? Answer: $61.4 \mathrm{~m} / \mathrm{s}$
(FIGURE CAN'T COPY)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:20

Problem 37

Repeat Prob. 7-36 with all the same conditions except that the only facility available to the students is a much smaller wind tunnel. Their model submarine is a one-twentyfourth scale model instead of a one-eighth scale model. At what air speed do they need to run the wind tunnel in order to achieve similarity? Do you notice anything disturbing or suspicious about your result? Discuss.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:12

Problem 38

A lightweight parachute is being designed for military use (Fig. P7-38E). Its diameter $D$ is 24 ft and the total weight $W$ of the falling payload, parachute, and equipment is 230 lbf. The design terminal settling speed $V_t$ of the parachute at this weight is $20 \mathrm{ft} / \mathrm{s}$. A one-twelfth scale model of the parachute is tested in a wind tunnel. The wind tunnel temperature and pressure are the same as those of the prototype, namely $60^{\circ} \mathrm{F}$ and standard atmospheric pressure. (a) Calculate the drag coefficient of the prototype.
(FIGURE CAN'T COPY)

Narayan Hari
Narayan Hari
Numerade Educator
08:10

Problem 39

Some wind tunnels are pressurized. Discuss why a research facility would go through all the extra trouble and expense to pressurize a wind tunnel. If the air pressure in the tunnel increases by a factor of 1.5 , all else being equal (same wind speed, same model, etc.), by what factor will the Reynolds number increase?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 40

This is a follow-up to Prob. 7-36. The students measure the aerodynamic drag on their model submarine in the wind tunnel (Fig. P7-36). They are careful to run the wind tunnel at conditions that ensure similarity with the prototype submarine. Their measured drag force is 2.3 N . Estimate the drag force on the prototype submarine at the conditions given in Prob. 7-36.

Victor Salazar
Victor Salazar
Numerade Educator
07:25

Problem 41

The aerodynamic drag of a new sports car is to be predicted at a speed of $60.0 \mathrm{mi} / \mathrm{h}$ at an air temperature of $25^{\circ} \mathrm{C}$. Automotive engineers build a one-fourth scale model of the car (Fig. P7-41E) to test in a wind tunnel. The temperature of the wind tunnel air is also $25^{\circ} \mathrm{C}$. The drag force is measured with a drag balance, and the moving belt is used to simulate the moving ground (from the car's frame of reference). Determine how fast the engineers should run the wind tunnel to achieve similarity between the model and the prototype.
(FIGURE CAN'T COPY)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 42

This is a follow-up to Prob. 7-41E. The aerodynamic drag on the model in the wind tunnel (Fig. P7-41E) is measured to be 36.5 lbf when the wind tunnel is operated at the speed that ensures similarity with the prototype car. Estimate the drag force (in lbf) on the prototype car at the conditions given in Prob. 7-41E.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 43

Consider the common situation in which a researcher is trying to match the Reynolds number of a large prototype vehicle with that of a small-scale model in a wind tunnel. Is it better for the air in the wind tunnel to be cold or hot? Why? Support your argument by comparing wind tunnel air at $10^{\circ} \mathrm{C}$ and at $50^{\circ} \mathrm{C}$, all else being equal.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 44

Some students want to visualize flow over a spinning baseball. Their fluids laboratory has a nice water tunnel into which they can inject multicolored dye streaklines, so they decide to test a spinning baseball in the water tunnel (Fig. P7-44). Similarity requires that they match both the Reynolds number and the Strouhal number between their model test and the actual baseball that moves through the air at $80 \mathrm{mi} / \mathrm{h}$ and spins at 300 rpm . Both the air and the water are at $20^{\circ} \mathrm{C}$. At what speed should they run the water in the water tunnel, and at what rpm should they spin their baseball?
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
02:54

Problem 45

Using primary dimensions, verify that the Archimedes number (Table 7-5) is indeed dimensionless.

Keshav Singh
Keshav Singh
Numerade Educator
01:58

Problem 46

Using primary dimensions, verify that the Grashof number (Table 7-5) is indeed dimensionless.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 47

Using primary dimensions, verify that the Rayleigh number (Table 7-5) is indeed dimensionless. What other established nondimensional parameter is formed by the ratio of Ra and Gr? Answer: the Prandtl number

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03:54

Problem 48

Consider a liquid in a cylindrical container in which both the container and the liquid are rotating as a rigid body (solid-body rotation). The elevation difference $h$ between the center of the liquid surface and the rim of the liquid surface is a function of angular velocity $\omega$, fluid density $\rho$, gravitational acceleration $g$, and radius $R$ (Fig. P7-48). Use the method of repeating variables to find a dimensionless relationship between the parameters. Show all your work. Answer: $h / R=f(\mathrm{Fr})$
(FIGURE CAN'T COPY)

Jacob Schulze
Jacob Schulze
Numerade Educator
06:43

Problem 49

Consider the case in which the container and liquid of Prob. 7-48 are initially at rest. At $t=0$ the container begins to rotate. It takes some time for the liquid to rotate as a rigid body, and we expect that the liquid's viscosity is an additional relevant parameter in the unsteady problem. Repeat Prob. 7-48, but with two additional independent parameters included, namely, fluid viscosity $\mu$ and time $t$. (We are interested in the development of height $h$ as a function of time and the other parameters.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:33

Problem 50

A periodic Karman vortex street is formed when a uniform stream flows over a circular cylinder (Fig. P7-50). Use the method of repeating variables to generate a dimensionless relationship for Kármán vortex shedding frequency $f_k$ as a function of free-stream speed $V$, fluid density $\rho$, fluid viscosity $\mu$, and cylinder diameter $D$. Show all your work.
(FIGURE CAN'T COPY)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:33

Problem 51

Repeat Prob. 7-50, but with an additional independent parameter included, namely, the speed of sound $c$ in the fluid. Use the method of repeating variables to generate a dimensionless relationship for Kármán vortex shedding frequency $f_k$ as a function of free-stream speed $V$, fluid density $\rho$, fluid viscosity $\mu$, cylinder diameter $D$, and speed of sound $c$. Show all your work.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:21

Problem 52

A stirrer is used to mix chemicals in a large tank (Fig. P7-52). The shaft power $\dot{W}$ supplied to the stirrer blades is a function of stirrer diameter $D$, liquid density $\rho$, liquid viscosity $\mu$, and the angular velocity $\omega$ of the spinning blades. Use the method of repeating variables to generate a dimensionless relationship between these parameters. Show all your work and be sure to identify your $\Pi$ groups, modifying them as necessary.
(FIGURE CAN'T COPY)

Narayan Hari
Narayan Hari
Numerade Educator
01:13

Problem 53

Repeat Prob. 7-52 except do not assume that the tank is large. Instead, let tank diameter $D_{\text {tank }}$ and average liquid depth $h_{\text {tank }}$ be additional relevant parameters.

James Macpherson
James Macpherson
Numerade Educator
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Problem 54

A boundary layer is a thin region (usually along a wall) in which viscous forces are significant and within which the flow is rotational. Consider a boundary layer growing along a thin flat plate (Fig. P7-54). The flow is steady. The boundary layer thickness $\delta$ at any downstream distance $x$ is a function of $x$, free-stream velocity $V_x$, and fluid properties $\rho$ (density) and $\mu$ (viscosity). Use the method of repeating variables to generate a dimensionless relationship for $\delta$ as a function of the other parameters. Show all your work.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator

Problem 55

Miguel is working on a problem that has a characteristic length scale $L$, a characteristic velocity $V$, a characteristic density difference $\Delta \rho$, a characteristic (average) density $\rho$, and of course the gravitational constant $g$, which is always available. He wants to define a Richardson number, but does not have a characteristic volume flow rate. Help Miguel define a characteristic volume flow rate based on the parameters available to him, and then define an appropriate Richardson number in terms of the given parameters.

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01:24

Problem 56

Consider fully developed Couette flow-flow between two infinite parallel plates separated by distance $h$, with the top plate moving and the bottom plate stationary as illustrated in Fig. P7-56. The flow is steady, incompressible, and two-dimensional in the $x y$-plane. Use the method of repeating variables to generate a dimensionless relationship for the $x$ component of fluid velocity $u$ as a function of fluid viscosity $\mu$, top plate speed $V$, distance $h$, fluid density $\rho$, and distance y. Show all your work. Answer: $u / V=f(\operatorname{Re}, y(h)$
(FIGURE CAN'T COPY)

Penny Riley
Penny Riley
Numerade Educator

Problem 57

Consider developing Couette flow-the same flow as Prob. 7-56 except that the flow is not yet steady-state, but is developing with time. In other words, time $t$ is an additional parameter in the problem. Generate a dimensionless relationship between all the variables.

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Problem 58

The speed of sound $c$ in an ideal gas is known to be a function of the ratio of specific heats $k$, absolute temperature $T$, and specific ideal gas constant $R_{\text {gas }}$ (Fig. P7-58). Showing all your work, use dimensional analysis to find the functional relationship between these parameters.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
03:05

Problem 59

Repeat Prob. 7-58, except let the speed of sound $c$ in an ideal gas be a function of absolute temperature $T$, universal ideal gas constant $R_u$, molar mass (molecular weight) $M$ of the gas, and ratio of specific heats $k$. Showing all your work, use dimensional analysis to find the functional relationship between these parameters.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:05

Problem 60

Repeat Prob. 7-58, except let the speed of sound $c$ in an ideal gas be a function only of absolute temperature $T$ and specific ideal gas constant $R_{\text {gas }}$. Showing all your work, use dimensional analysis to find the functional relationship between these parameters. Answer: $c / \sqrt{R_{\mathrm{gs}} T}=$ constant

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
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Problem 61

Repeat Prob. 7-58, except let speed of sound $c$ in an ideal gas be a function only of pressure $P$ and gas density $\rho$. Showing all your work, use dimensional analysis to find the functional relationship between these parameters. Verify that your results are consistent with the equation for speed of sound in an ideal gas, $c=\sqrt{k R_{\mathrm{gas}} T}$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 62

When small aerosol particles or microorganisms move through air or water, the Reynolds number is very small ( $\mathrm{Re} \ll 1$ ). Such flows are called creeping flows. The aerodynamic drag on an object in creeping flow is a function only of its speed $V$, some characteristic length scale $L$ of the object, and fluid viscosity $\mu$ (Fig. P7-62). Use dimensional analysis to generate a relationship for $F_D$ as a function of the independent variables.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator

Problem 63

A tiny aerosol particle of density $\rho_p$ and characteristic diameter $D_p$ falls in air of density $\rho$ and viscosity $\mu$ (Fig. P7-63). If the particle is small enough, the creeping flow approximation is valid, and the terminal settling speed of the particle $V$ depends only on $D_p, \mu$, gravitational constant $g$, and the density difference ( $\rho_p-\rho$ ). Use dimensional analysis to generate a relationship for $V$ as a function of the independent variables. Name any established dimensionless parameters that appear in your analysis.
(FIGURE CAN'T COPY)

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Problem 64

Combine the results of Probs. 7-62 and 7-63 to generate an equation for the settling speed $V$ of an aerosol particle falling in air (Fig. P7-63). Verify that your result is consistent with the functional relationship obtained in Prob. 7-63. For consistency, use the notation of Prob. 7-63.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 65

You will need the results of Prob. 7-64 to do this problem. A tiny aerosol particle falls at steady settling speed $V$. The Reynolds number is small enough that the creeping flow approximation is valid. If the particle size is doubled, all else being equal, by what factor will the settling speed go up? If the density difference $\left(\rho_p-\rho\right.$ ) is doubled, all else being equal, by what factor will the settling speed go up?

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Problem 66

An incompressible fluid of density $\rho$ and viscosity $\mu$ flows at average speed $V$ through a long, horizontal section of round pipe of length $L$, inner diameter $D$, and inner wall roughness height $\varepsilon$ (Fig. P7-66). The pipe is long enough that the flow is fully developed, meaning that the velocity profile does not change down the pipe. Pressure decreases (linearly) down the pipe in order to "push" the fluid through the pipe to overcome friction. Using the method of repeating variables, develop a nondimensional relationship between pressure drop $\Delta P=P_1-P_2$ and the other parameters in the problem. Be sure to modify your $\Pi$ groups as necessary to achieve established nondimensional parameters, and name them.
(FIGURE CAN'T COPY)

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03:40

Problem 67

Consider laminar flow through a long section of pipe, as in Fig. P7-66. For laminar flow it turns out that wall roughness is not a relevant parameter unless $\varepsilon$ is very large. The volume flow rate $\dot{V}$ through the pipe is in fact a function of pipe diameter $D$, fluid viscosity $\mu$, and axial pressure gradient $d P / d x$. If pipe diameter is doubled, all else being equal, by what factor will volume flow rate increase? Use dimensional analysis.

Guilherme Barros
Guilherme Barros
Numerade Educator
04:41

Problem 68

The rate of heat transfer to water flowing in a pipe was analyzed in Prob. 7-24. Let us approach that same problem, but now with dimensional analysis. Cold water enters a pipe, where it is heated by an external heat source (Fig. P7-68). The inlet and outlet water temperatures are $T_{\text {in }}$ and $T_{\text {out }}$, respectively. The total rate of heat transfer $\dot{Q}$ from the surroundings into the water in the pipe is known to be a function of mass flow rate $\dot{m}$, the specific heat $c_p$ of the water, and the temperature difference between the incoming and outgoing water. Showing all your work, use dimensional analysis to find the functional relationship between these parameters, and compare to the analytical equation given in Prob. 7-243
(FIGURE CAN'T COPY)

Salamat Ali
Salamat Ali
Numerade Educator
03:34

Problem 69

Define wind tunnel blockage. What is the rule of thumb about the maximum acceptable blockage for a wind tunnel test? Explain why there would be measurement errors if the blockage were significantly higher than this value.

Chai Santi
Chai Santi
Numerade Educator
03:34

Problem 70

What is the rule of thumb about the Mach number limit in order that the incompressible flow approximation is reasonable? Explain why wind tunnel results would be incorrect if this rule of thumb were violated.

Chai Santi
Chai Santi
Numerade Educator
00:26

Problem 71

Although we usually think of a model as being smaller than the prototype, describe at least three situations in which it is better for the model to be larger than the prototype.

David Collins
David Collins
Numerade Educator
02:33

Problem 72

Discuss the purpose of a moving ground belt in wind tunnel tests of flow over model automobiles. Can you think of an alternative if a moving ground belt is unavailable?

Kajal Gautam
Kajal Gautam
Numerade Educator
10:41

Problem 73

Use dimensional analysis to show that in a problem involving shallow water waves (Fig. P7-73), both the Froude number and the Reynolds number are relevant dimensionless parameters. The wave speed $c$ of waves on the surface of a liquid is a function of depth $h$, gravitational acceleration $g$, fluid density $\rho$, and fluid viscosity $\mu$. Manipulate your $\Pi$ 's to get the parameters into the following form:
$$
\mathrm{Fr}=\frac{c}{\sqrt{g h}}=f(\operatorname{Re}) \quad \text { where } \operatorname{Re}=\frac{\rho c h}{\mu}
$$
(FIGURE CAN'T COPY)

Abid Hussain
Abid Hussain
Numerade Educator
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Problem 74

Water at $20^{\circ} \mathrm{C}$ flows through a long, straight pipe. The pressure drop is measured along a section of the pipe of length $L=1.3 \mathrm{~m}$ as a function of average velocity $V$ through the pipe (Table P7-74). The inner diameter of the pipe is $D$ $=10.4 \mathrm{~cm}$. (a) Nondimensionalize the data and plot the Euler number as a function of the Reynolds number. Has the experiment been run at high enough speeds to achieve Reynolds number independence? (b) Extrapolate the experimental data to predict the pressure drop at an average speed of $80 \mathrm{~m} / \mathrm{s}$. Answer: $1,940,000 \mathrm{~N} / \mathrm{m}^2$
(TABLE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
12:16

Problem 75

In the model truck example discussed in Section 7-5, the wind tunnel test section is 2.6 m long, 1.0 m tall, and 1.2 m wide. The one-sixteenth scale model truck is 0.991 m long, 0.257 m tall, and 0.159 m wide. What is the wind tunnel blockage of this model truck? Is it within acceptable limits according to the standard rule of thumb?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:50

Problem 76

Consider again the model truck example discussed in Section 7-5, except that the maximum speed of the wind tunnel is only $50 \mathrm{~m} / \mathrm{s}$. Aerodynamic force data are taken for wind tunnel speeds between $V=20$ and $50 \mathrm{~m} / \mathrm{s}$-assume the same data for these speeds as those listed in Table 7-7. Based on these data alone, can the researchers be confident that they have reached Reynolds number independence?

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
12:16

Problem 77

A small wind tunnel in a university's undergraduate fluid flow laboratory has a test section that is 20 by 20 in in cross section and is 4.0 ft long. Its maximum speed is 160 $\mathrm{ft} / \mathrm{s}$. Some students wish to build a model 18 -wheeler to study how aerodynamic drag is affected by rounding off the back of the trailer. A full-size (prototype) tractor-trailer rig is 52 ft long, 8.33 ft wide, and 12 ft high. Both the air in the wind tunnel and the air flowing over the prototype are at $80^{\circ} \mathrm{F}$ and atmospheric pressure. (a) What is the largest scale model they can build to stay within the rule-of-thumb guidelines for blockage? What are the dimensions of the model truck in inches? (b) What is the maximum model truck Reynolds number achievable by the students? (c) Are the students able to achieve Reynolds number independence? Discuss.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:34

Problem 78

A one-sixteenth scale model of a new sports car is tested in a wind tunnel. The prototype car is 4.37 m long, 1.30 m tall, and 1.69 m wide. During the tests, the moving ground belt speed is adjusted so as to always match the speed of the air moving through the test section. Aerodynamic drag force $F_D$ is measured as a function of wind tunnel speed; the experimental results are listed in Table P7-78. Plot drag coefficient $C_D$ as a function of the Reynolds number Re , where the area used for calculation of $C_D$ is the frontal area of the model car (assume $A=$ width $\times$ height), and the length scale used for calculation of Re is car width $W$. Have we achieved dynamic similarity? Have we achieved Reynolds number independence in our wind tunnel test? Estimate the aerodynamic drag force on the prototype car traveling on the highway at $29 \mathrm{~m} / \mathrm{s}(65 \mathrm{mi} / \mathrm{h})$. Assume that both the wind tunnel air and the air flowing over the prototype car are at $25^{\circ} \mathrm{C}$ and atmospheric pressure.
(TABLE CAN'T COPY)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:24

Problem 79

For each statement, choose whether the statement is true or false and discuss your answer briefly.
(a) Kinematic similarity is a necessary and sufficient condition for dynamic similarity.
(b) Geometric similarity is a necessary condition for dynamic similarity.
(c) Geometric similarity is a necessary condition for kinematic similarity.
(d) Dynamic similarity is a necessary condition for kinematic similarity.

Holly Miner
Holly Miner
Numerade Educator
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Problem 80

Think about and describe a prototype flow and a corresponding model flow that have geometric similarity, but not kinematic similarity, even though the Reynolds numbers match. Explain.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 81

There are many established nondimensional parameters besides those listed in Table 7-5. Do a literature search or an Internet search and find at least three established, named nondimensional parameters that are not listed in Table 7-5. For each one, provide its definition and its ratio of significance, following the format of Table 7-5. If your equation contains any variables not identified in Table 7-5, be sure to identify those variables.

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04:24

Problem 82

Write the primary dimensions of each of the following variables from the field of solid mechanics, showing all your work: (a) moment of inertia $I$; (b) modulus of elasticity $E$, also called Young's modulus; (c) strain $\varepsilon$; (d) stress $\sigma$.
(e) Finally, show that the relationship between stress and strain (Hooke's law) is a dimensionally homogeneous equation.

Amit Srivastava
Amit Srivastava
Numerade Educator
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Problem 83

Force $F$ is applied at the tip of a cantilever beam of length $L$ and moment of inertia $I$ (Fig. P7-83). The modulus of elasticity of the beam material is $E$. When the force is applied, the tip deflection of the beam is $z_d$. Use dimensional analysis to generate a relationship for $z_d$ as a function of the independent variables. Name any established dimensionless parameters that appear in your analysis.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
01:56

Problem 84

An explosion occurs in the atmosphere when an antiaircraft missile meets its target (Fig. P7-84). A shock wave (also called a blast wave) spreads out radially from the explosion. The pressure difference across the blast wave $\Delta P$ and its radial distance $r$ from the center are functions of time $t$, speed of sound $c$, and the total amount of energy $E$ released by the explosion. (a) Generate dimensionless relationships between $\Delta P$ and the other parameters and between $r$ and the other parameters. (b) For a given explosion, if the time $t$ since the explosion doubles, all else being equal, by what factor will $\Delta P$ decrease?
(FIGURE CAN'T COPY)

Chai Santi
Chai Santi
Numerade Educator
01:58

Problem 85

The Archimedes number listed in Table 7-5 is appropriate for buoyant particles in a fluid. Do a literature search or an Internet search and find an alternative definition of the Archimedes number that is appropriate for buoyant fluids (e.g., buoyant jets and buoyant plumes, heating and air-conditioning applications). Provide its definition and its ratio of significance, following the format of Table 7-5. If your equation contains any variables not identified in Table 7-5, be sure to identify those variables. Finally, look through the established dimensionless parameters listed in Table 7-5 and find one that is similar to this alternate form of the Archimedes number.

Narayan Hari
Narayan Hari
Numerade Educator
01:09

Problem 86

Consider steady, laminar, fully developed, twodimensional Poiseuille flow-flow between two infinite parallel plates separated by distance $h$, with both the top plate and bottom plate stationary, and a forced pressure gradient $d P / d x$ driving the flow as illustrated in Fig. P7-86. ( $d P / d x$ is constant and negative.) The flow is steady, incompressible, and two-dimensional in the xy-plane. The flow is also fully developed, meaning that the velocity profile does not change with downstream distance $x$. Because of the fully developed nature of the flow, there are no inertial effects and density does not enter the problem. It turns out that $u$, the velocity component in the $x$-direction, is a function of distance $h$, pressure gradient $d P / d x$, fluid viscosity $\mu$, and vertical coordinate $y$. Perform a dimensional analysis (showing all your work), and generate a dimensionless relationship between the given variables.
(FIGURE CAN'T COPY)

AP
Andreas Papavassiliou
Numerade Educator
01:15

Problem 87

Consider the steady, laminar, fully developed, twodimensional Poiseuille flow of Prob. 7-86. The maximum velocity $u_{\text {max }}$ occurs at the center of the channel. (a) Generate a dimensionless relationship for $u_{\max }$ as a function of distance between plates $h$, pressure gradient $d P / d x$, and fluid viscosity $\mu$. (b) If the plate separation distance $h$ is doubled, all else being equal, by what factor will $u_{\max }$ change? (c) If the pressure gradient $d P / d x$ is doubled, all else being equal, by what factor will $u_{\max }$ change? (d) How many experiments are required to describe the complete relationship between $u_{\max }$ and the other parameters in the problem?

Dominador Tan
Dominador Tan
Numerade Educator

Problem 88

The pressure drop $\Delta P=P_1-P_2$ through a
long section of round pipe can be written in
terms of the shear stress $\tau_w$ along the wall. Shown in Fig. P7-88 is the shear stress acting by the wall on the fluid. The shaded blue region is a control volume composed of the fluid in the pipe between axial locations 1 and 2 . There are two dimensionless parameters related to the pressure drop: the Euler number Eu and the Darcy friction factor $f$. (a) Using the control volume sketched in Fig. P7-88, generate a relationship for $f$ in terms of Eu (and any other properties or parameters in the problem as needed). (b) Using the experimental data and conditions of Prob. 7-74 (Table P7-74), plot the Darcy friction factor as a function of Re. Does $f$ show Reynolds number independence at large values of Re ? If so, what is the value of $f$ at very high $\operatorname{Re}$ ?
(FIGURE CAN'T COPY)

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Problem 89

Oftentimes it is desirable to work with an established dimensionless parameter, but the characteristic scales available do not match those used to define the parameter. In such cases, we create the needed characteristic scales based on dimensional reasoning (usually by inspection). Suppose for example that we have a characteristic velocity scale $V$, characteristic area $A$, fluid density $\rho$, and fluid viscosity $\mu$, and we wish to define a Reynolds number. We create a length scale $L=\sqrt{A}$, and define
$$
\operatorname{Re}=\frac{\rho V \sqrt{A}}{\mu}
$$
In similar fashion, define the desired established dimensionless parameter for each case: (a) Define a Froude number, given $\dot{V}^{\prime}=$ volume flow rate per unit depth, length scale $L$, and gravitational constant $g$. (b) Define a Reynolds number, given $\dot{V}^{\prime}=$ volume flow rate per unit depth and kinematic viscosity $\nu$. (c) Define a Richardson number, given $\dot{V}^{\prime}=$ volume flow rate per unit depth, length scale $L$, characteristic density difference $\Delta \rho$, characteristic density $\rho$, and gravitational constant $g$.

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Problem 90

A liquid of density $\rho$ and viscosity $\mu$ flows by gravity through a hole of diameter $d$ in the bottom of a tank of diameter $D$ (Fig. P7-90). At the start of the experiment, the liquid surface is at height $h$ above the bottom of the tank, as sketched. The liquid exits the tank as a jet with average velocity $V$ straight down as also sketched. Using dimensional analysis, generate a dimensionless relationship for $V$ as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
01:43

Problem 91

Repeat Prob. 7-90 except for a different dependent parameter, namely, the time required to empty the tank $t_{\text {enppty }}$. Generate a dimensionless relationship for $t_{\text {empty }}$ as a function of the following independent parameters: hole diameter $d$, tank diameter $D$, density $\rho$, viscosity $\mu$, initial liquid surface height $h$, and gravitational acceleration $g$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
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Problem 92

A liquid delivery system is being designed such that ethylene glycol flows out of a hole in the bottom of a large tank, as in Fig. P7-90. The designers need to predict how long it will take for the ethylene glycol to completely drain. Since it would be very expensive to run tests with a full-scale prototype using ethylene glycol, they decide to build a onequarter scale model for experimental testing, and they plan to use water as their test liquid. The model is geometrically similar to the prototype (Fig. P7-92). (a) The temperature of the ethylene glycol in the prototype tank is $60^{\circ} \mathrm{C}$, at which $\nu$ $=4.75 \times 10^{-6} \mathrm{~m}^2 / \mathrm{s}$. At what temperature should the water in the model experiment be set in order to ensure complete similarity between model and prototype? (b) The experiment is run with water at the proper temperature as calculated in part (a). It takes 4.53 min to drain the model tank. Predict how long it will take to drain the ethylene glycol from the prototype tank.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
03:34

Problem 93

Liquid flows out of a hole in the bottom of a tank as in Fig. P7-90. Consider the case in which the hole is very small compared to the tank ( $d \ll D$ ). Experiments reveal that average jet velocity $V$ is nearly independent of $d, D, \rho$, or $\mu$. In fact, for a wide range of these parameters, it turns out that $V$ depends only on liquid surface height $h$ and gravitational acceleration $g$. If the liquid surface height is doubled, all else being equal, by what factor will the average jet velocity increase? Answer: $\sqrt{2}$

Jacob Shpiece
Jacob Shpiece
Numerade Educator
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Problem 94

An aerosol particle of characteristic size $D_p$ moves in an airflow of characteristic length $L$ and characteristic velocity $V$. The characteristic time required for the particle to adjust to a sudden change in air speed is called the particle relaxation time $\tau_p$,
$$
\tau_p=\frac{\rho_p D_p^2}{18 \mu}
$$
Verify that the primary dimensions of $\tau_p$ are time. Then create a dimensionless form of $\tau_p$, based on some characteristic velocity $V$ and some characteristic length $L$ of the airflow (Fig. P7-94). What established dimensionless parameter do you create?
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
06:48

Problem 95

Compare the primary dimensions of each of the following properties in the mass-based primary dimension system (m, L, t, T, I, C, N) to those in the force-based primary dimension system (F, L, t, T, I, C, N): (a) pressure or stress; (b) moment or torque; (c) work or energy. Based on your results, explain when and why some authors prefer to use force as a primary dimension in place of mass.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:39

Problem 96

In Example 7-7, the mass-based system of primary dimensions was used to establish a relationship for the pressure difference $\Delta P=P_{\text {inside }}-P_{\text {outside }}$ between the inside and outside of a soap bubble as a function of soap bubble radius $R$ and surface tension $\sigma_s$ of the soap film (Fig. P7-96). Repeat the dimensional analysis using the method of repeating variables, but use the force-based system of primary dimensions instead. Show all your work. Do you get the same result?
(FIGURE CAN'T COPY)

Averell Hause
Averell Hause
Carnegie Mellon University

Problem 97

Many of the established nondimensional parameters listed in Table 7-5 can be formed by the product or ratio of two other established nondimensional parameters. For each pair of nondimensional parameters listed, find a third established nondimensional parameter that is formed by some manipulation of the two given parameters: (a) Reynolds number and Prandtl number; (b) Schmidt number and Prandtl number; (c) Reynolds number and Schmidt number.

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07:45

Problem 98

The Stanton number is listed as a named, established nondimensional parameter in Table 7-5. However, careful analysis reveals that it can actually be formed by a combination of the Reynolds number, Nusselt number, and Prandtl number. Find the relationship between these four dimensionless groups, showing all your work. Can you also form the Stanton number by some combination of only two other established dimensionless parameters?

Bhumika Jayee
Bhumika Jayee
Numerade Educator
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Problem 99

Consider a variation of the fully developed Couette flow problem of Prob. 7-56-flow between two infinite parallel plates separated by distance $h$, with the top plate moving at speed $V_{\text {top }}$ and the bottom plate moving at speed $V_{\text {bottom }}$ as illustrated in Fig. P7-99. The flow is steady, incompressible, and two-dimensional in the $x y$-plane. Generate a dimensionless relationship for the $x$-component of fluid velocity $u$ as a function of fluid viscosity $\mu$, plate speeds $V_{\text {top }}$ and $V_{\text {bottom }}$, distance $h$, fluid density $\rho$, and distance $y$.
(FIGURE CAN'T COPY)

Victor Salazar
Victor Salazar
Numerade Educator
01:20

Problem 100

What are the primary dimensions of electric charge $q$, the units of which are coulombs (C)?

cm
Charles Magnusen
Numerade Educator
06:31

Problem 101

What are the primary dimensions of electrical capacitance $C$, the units of which are farads?

Sikandar Baig
Sikandar Baig
Numerade Educator
02:01

Problem 102

In many electronic circuits in which some kind of time scale is involved, such as filters and time-delay circuits (Fig. P7-102-a low-pass filter), you often see a resistor ( $R$ ) and a capacitor $(C)$ in series. In fact, the product of $R$ and $C$ is called the electrical time constant, $R C$. Showing all your work, what are the primary dimensions of $R C$ ? Using dimensional reasoning alone, explain why a resistor and capacitor are often found together in timing circuits.
(FIGURE CAN'T COPY)

Manish Haldankar
Manish Haldankar
Numerade Educator
03:03

Problem 103

From fundamental electronics, the current flowing through a capacitor at any instant of time is equal to the capacitance times the rate of change of voltage across the capacitor,
$$
I=C \frac{d E}{d t}
$$
Write the primary dimensions of both sides of this equation, and verify that the equation is dimensionally homogeneous. Show all your work.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
02:21

Problem 104

A common device used in various applications to clean particle-laden air is the reverse-flow cyclone (Fig. P7-104). Dusty air (volume flow rate $\dot{V}$ and density $\rho$ ) enters tangentially through an opening in the side of the cyclone and swirls around in the tank. Dust particles are flung outward and fall out the bottom, while clean air is drawn out the top. The reverse-flow cyclones being studied are all geometrically similar; hence, diameter $D$ represents the only length scale required to fully specify the entire cyclone geometry. Engineers are concerned about the pressure drop $\delta P$ through the cyclone. (a) Generate a dimensionless relationship between the pressure drop through the cyclone and the given parameters. Show all your work. (b) If the cyclone size is doubled, all else being equal, by what factor will the pressure drop change? (c) If the volume flow rate is doubled, all else being equal, by what factor will the pressure drop change?

Narayan Hari
Narayan Hari
Numerade Educator
06:02

Problem 105

An electrostatic precipitator (ESP) is a device used in various applications to clean particle-laden air. First, the dusty air passes through the charging stage of the ESP, where dust particles are given a positive charge $q_p$ (coulombs) by charged ionizer wires (Fig. P7-105). The dusty air then enters the collector stage of the device, where it flows between two oppositely charged plates. The applied electric field strength between the plates is $E_f$ (voltage difference per unit distance). Shown in Fig. P7-105 is a charged dust particle of diameter $D_p$. It is attracted to the negatively charged plate and moves toward that plate at a speed called the drift velocity $w$. If the plates are long enough, the dust particle impacts the negatively charged plate and adheres to it. Clean air exits the device. It turns out that for very small particles the drift velocity depends only on $q_p, E_f, D_p$, and air viscosity $\mu$. (a) Generate a dimensionless relationship between the drift velocity through the collector stage of the ESP and the given parameters. Show all your work. (b) If the electric field strength is doubled, all else being equal, by what factor will the drift velocity change? (c) For a given ESP, if the particle diameter is doubled, all else being equal, by what factor will the drift velocity change?
(FIGURE CAN'T COPY)

Keshav Singh
Keshav Singh
Numerade Educator
02:09

Problem 106

When a capillary tube of small diameter $D$ is inserted into a container of liquid, the liquid rises to height $h$ inside the tube (Fig. P7-106). $h$ is a function of liquid density $\rho$, tube diameter $D$, gravitational constant $g$, contact angle $\phi$, and the surface tension $\sigma_s$ of the liquid. (a) Generate a dimensionless relationship for $h$ as a function of the given parameters. (b) Compare your result to the exact analytical equation for $h$ given in Chap. 2. Are your dimensional analysis results consistent with the exact equation? Discuss.
(FIGURE CAN'T COPY)

James Kiss
James Kiss
Numerade Educator
03:19

Problem 107

Repeat part (a) of Prob. 7-106, except instead of height $h$, find a functional relationship for the time scale $t_{\text {rise }}$ needed for the liquid to climb up to its final height in the capillary tube.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:54

Problem 108

Sound intensity $I$ is defined as the acoustic power per unit area emanating from a sound source. We know that $I$ is a function of sound pressure level $P$ (dimensions of pressure) and fluid properties $\rho$ (density) and speed of sound $c$.
(a) Use the method of repeating variables in mass-based primary dimensions to generate a dimensionless relationship for $I$ as a function of the other parameters. Show all your work. What happens if you choose three repeating variables? Discuss. (b) Repeat part (a), but use the force-based primary dimension system. Discuss.

Dominador Tan
Dominador Tan
Numerade Educator
03:45

Problem 109

Repeat Prob. 7-108, but with the distance $r$ from the sound source as an additional independent parameter.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator