In Chap. 9, we discuss the differential equation for

Problem 1P List the seven primary dimensions. What is significant about these seven?

Problem 2P 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 P7-3), 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 P7–3 Parameter Name Parameter Symbol Primary Dimensions Acceleration a L1t-2 Angle ?, ?, etc. 1 (none) Density ? m1L-3 Force F m1L1t-2 Frequency f t-1 Pressure P m1L-1t-2 Surface tension ?s m1L-2 Velocity V L1t-1 Viscosity ? m1L-1t-2 Volume flow rate L3t-1

Problem 3P Consider the table of Prob. 7-3 where the primary dimensions of several variables are listed in the mass- length-time system. Some engineers prefer the force-length- time system (force replaces mass as one of the primary dimensions). Write the primary dimensions of three of these (density, surface tension, and viscosity) in the force- length-time system.

Problem 4P Write the primary dimensions of the universal ideal gas constant Ru.(Hint: Use the ideal gas law, PV = nRuT where P is pressure, V is volume, T is absolute temperature, and n is the number of moles of the gas.)

Problem 5P 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. P7-6). In reality, atomic weight is the mass of 1 mol of the element. For example, the atomic weight of nitrogen Mnitrogen = 14.0067. We interpret this as 14.0067 g/mol of elemental nitrogen, or in the English system. 14.0067 lbm/lbmol of elemental nitrogen. What are the primary dimensions of atomic weight?

Problem 6P 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 m, L, t, and T are replaced by F, L, t, and T. The primary dimension of force in this system is {force} = {F}. Using the results of Prob. 7-5, rewrite the primary dimensions of the universal gas constant in this alternate system of primary dimensions.

Problem 7P We define the specific ideal gas constant Rgas for a particular gas as the ratio of the universal gas constant and the molar mass (also called molecular weight) of the gas, Rgas = Ru/M. For a particular gas, then, the ideal gas law is written as follows: where P is pressure, V is volume, m is mass, T is absolute temperature, and ? is the density of the particular gas. What are the primary dimensions of Rgas? For air, Rair = 287.0 J/kg ? K in standard SI units. Verify that these units agree with your result

Problem 8P The moment of force is formed by_the cross product of a moment arm and an applied force as sketched in Fig. P7-9. What are the primary dimensions of moment of force? List its units in primary SI units and in primary English units.

Problem 9P 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.

Problem 10P What are the primary dimensions of electric voltage (E)?(Hint: Make use of the fact that electric power is equal to voltage times current.)

Problem 11P You are probably familiar with Ohm's law for electric circuits (Fig. P7-12), where ?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?

Problem 13P Write the primary dimensions of each of the following variables, showing all your work: (a) acceleration a; (b) angular velocity ?; (c) angular acceleration ?.

Problem 12P Write the primary dimensions of each of the following variables, showing all your work: (a) acceleration a; (b) angular velocity ?; (c) angular acceleration ?.

Problem 14P Write the primary dimensions of each of the following variables, showing all your work: (a) specific heat at constant pressure cp; (b) specific weight ?g; (c) specific enthalpy h.

Problem 15P Thermal conductivity k is a measure of the ability of a material to conduct heat (Fig. P7-16). 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 where Q?conduction is the rate of heat transfer and A is the area S normal to the direction of heat transfer. Determine the primary dimensions of thermal conductivity (k). Look up a value of kin the appendices and verify that its SI units are consistent I; with your result. In particular, write the primary SI units of k.

Problem 16P Write the primary dimensions of each of the following variables from the study of convection heat transfer (Fig. P7-17), showing all your work: (a) heat generation rate ?(Hint: rate of conversion of thermal energy per unit volume); (b) heat flux q?(Hint: rate of heat transfer per unit area); (c) heat transfer coefficient h(Hint: heat flux per unit temperature difference).

Problem 17P Thumb through the appendices of 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 SI units. Then write out the primary dimensions of each property or constant.

Problem 18P Explain the law of dimensional homogeneity in simple terms.

Problem 19P In Chap. 4, we defined the material acceleration. which is the acceleration following a fluid particle (Fig.P7-21), (a) What are the primary dimensions of the gradient operator Verify that each additive term in the equation has the same dimensions.

Problem 20P 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. P7-21), we write Newton's second law as follows: Or. dividing both sides by the mass m of the fluid particle, Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.

Problem 21P In Chap. 4 we defined volumetric strain rate as the rate of increase of volume of a fluid element per unit volume (Fig. P7-23). In Cartesian coordinates we write the volumetric strain rate as Write the primary dimensions of each additive term, and verify that the equation is dimensionally homogeneous. Show all your work.

Problem 22P In Chap. 9, we discuss the differential equation for conservation of mass, the continuity equation. In cylindrical coordinates, and for steady flow, Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.

Problem 23P Cold water enters a pipe, where it is heated by an external heat source (Fig. P7-25). The inlet and outlet water temperatures are Tin and Tout respectively. The total rate of heat transfer Q?? from the surroundings into the water in the pipe is where m? is the mass flow rate of water through the pipe, and cp 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.

Problem 24P 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: where 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, (Hint: Since B can be any property of the flow—scalar, vector, or even tensor—it can have a variety of dimensions. S0, just let the dimensions of B be those of B itself, {B}, Also,b. is defined as B per unit mass.)

Problem 25P 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.P7-21). 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).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 where kw is an adsorption coefficient and As 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 kw. Show all your work.

Problem 26P What is the primary reason for nondimensionalizing an equation?

Problem 27P Consider ventilation of a well-mixed room as in Fig. P7-27. The differential equation for mass concentration in the room as a function of time is given in Prob. 7-27 and is repeated here for convenience, There are three characteristic parameters in such a situation:L, a characteristic length scale of the room (assume L = V1/3);V?, the volume flow rate of fresh air into the room, and climit the maximum mass concentration that is not harmful. (a) Using these three characteristic parameters, define dimensionless forms of all the variables in'the equation. (Hint: For,example, define c* = c/climit.) (b) Rewrite the equation in dimensionless form, and identify any established dimension: less groups that may appear.

Problem 28P Recall from Chap. 4 that the volumetric strain rate is zero for a steady incompressible flow. In Cartesian coordinates we express this as Suppose the characteristic speed and characteristic length for a given flow field are V and L, respectively (Fig. P7-30). Define the following dimensionless variables, Nondimensionalize the equation and identify any established (named) dimensionless parameters that may appear. Discuss.

Problem 30P In Chap. 9, we define the stream function ? for two- dimensional incompressible flow in the xy-plane. where u and ? are the velocity components in the.x- and y- directions, respectively, (a) What are the primary dimensions of ?? (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, ?. and ? (c) Rewrite the equations in nondimensional form, and identify any established dimensionless parameters that may appear.

Problem 29P 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 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. P7-31 ). Define the following dimensionless variables, Nondimensionalize the equation and identify any established (named) dimensionless parameters that may appear.

Problem 31P 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. 7-22), Suppose the characteristic speed and characteristic length for a given flow field are V? and L, respectively. Also suppose that ? is a characteristic angular frequency (rad/s) of the oscillation (Fig. P7-33). Define the following nondimensionalized variables Since there is no given characteristic scale for the force per unit mass acting on a fluid particle, we assign one, noting that . Namely, we let Nondimensionalize the equation of motion and identify any established (named) dimensionless parameters that may appear.

Problem 32P A wind tunnel is used to measure the pressure distribution in the airflow over an airplane model (Fig. P7-34). 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? and pressure P? , and the air density ? is approximately constant. Gravitational effects are generally negligible in airflows. so we write the Bernoulli equation as Nondimensionalize the equation, and generate an expression for the pressure coefficient Cp, at any point in the flow where the Bernoulli equation is valid. Cp is defined as

Problem 33P List the three primary purposes of dimensional analysis.

Problem 34P List and describe the three necessary conditions for complete similarity between a model and a prototype.

Problem 37P 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 0.520 m/s. The water is freshwater (a lake) at T = 15°C. The design team builds a one-eighth scale model to test in their university's wind tunnel (Fig. P7-39). 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°C and at one standard atmosphere pressure. At what air; speed do they need to run the wind tunnel in order to achieve similarity?

Problem 38P Repeat Prob. 7-39 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-twenty-fourth 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 your results.

Problem 39P This is a follow-up to Prob. 7-39. The students measure the aerodynamic drag on their model submarine in the wind tunnel (Fig. P7-39). They are careful to run the wind, tunnel at conditions that ensure similarity with the prototype submarine. Their measured drag force is 2.45 N. Estimate the drag force on the prototype submarine at the conditions given; in Prob. 7-39.

Problem 41P 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 1.8, all else being equal (same wind speed, same model, etc.), by what factor will the Reynolds number increase?

Problem 44P 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°C and at 50°C, all else being equal.

Problem 45P Using primary dimensions, verify that the Archimedes number (Table 7-5) is indeed dimensionless.

Problem 46P Using primary dimensions, verify that the Grashof number (Table 7-5) is indeed dimensionless.

Problem 47P 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?

Problem 48P 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 ?. fluid density ?, gravitational acceleration g. and radius R (Fig. P7-51). Use the method of repeating variables to find a dimensionless relationship between the parameters. Show all your work.

Problem 49P Consider the case in which the container and liquid of Prob. 7-51 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. 7-51, but with two additional independent parameters included, namely, fluid viscosity ? and time t. (We are interested in the development of height h as a function of time and the other parameters.)

Problem 50P A periodic Kármán vortex street is formed when a uniform stream flows over a circular cylinder (Fig. P7-53). Use the method of repeating variables to generate a dimensionless relationship for Kármán vortex shedding frequency fk as a function of free-stream speed V, fluid density ?, fluid viscosity ?, and cylinder diameter D. Show all your work.

Problem 51P Repeat Prob. 7-53, 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 fk as a function of free-stream speed V, fluid density ?, fluid viscosity ?, cylinder diameter D, and speed of sound c. Show all your work.

Problem 53P Repeat Prob. 7-55 except do not assume that the tank is large. Instead, let tank diameter Dtank and average liquid; depth htank be additional relevant parameters.

Problem 52P A stirrer is used to mix chemicals in a large tank (Fig. P7-55). The shaft power W? supplied to the stirrer blades is a function of stirrer diameter D, liquid density ?, liquid viscosity ?, and the angular velocity ? 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 ? groups, modifying them as necessary.

Problem 54P 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. P7-57). The flow is steady. The boundary layer thickness ? at any downstream distance x is a function of x, free-stream velocity V? and fluid properties ? (density) and ? (viscosity). Use the method of repeating variables to generate a dimensionless relationship for ? as a (function of the other parameters. Show all your work.

Problem 55P The Richardson number is defined as Miguel is working on a problem that has a characteristic length scale L, a characteristic velocity V, a characteristic density difference ??, a characteristic (average) density ?, 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.

Problem 56P Consider fully developed Couette flow—flow between two infinite parallel plates separated by distance h, with the) plate moving and the bottom plate stationary as illusstrated in Fig.P7-59. The flow is steady, incompressible, and two-dimensional in the xy-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; ?,, top plate speed V, distance h, fluid density p, and distance y. Show all your work.

Problem 57P Consider developingCouette flow–the same flow as Prob, 7?59 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.

Problem 58P 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 gas (Fig. P7-6l). Showing all your work, use dimensional analysis to find the functional relationship between these parameters.

Problem 59P Repeat Prob. 7-61, except let the speed of sound c in an ideal gas be a function of absolute temperature T, universal ideal gas constant Ru, 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.

Problem 60P Repeat Prob. 7-61, except let the speed of sound c in an ideal gas be a function only of absolute temperature T and specific ideal gas constant Rgas. Showing all your work, use dimensional analysis to find the functional relationship between these parameters.

Problem 61P Repeat Prob. 7-61, except let speed of sound c in an ideal gas be a function only of pressure P and gas density ?. 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,

Problem 63P A tiny aerosol particle of density ?p and characteristic diameter Dp falls in air of density ? and viscosity ? (Fig. P7-66). If the particle is small enough, the creeping flow approximation is valid, and the terminal settling speed of the particle V depends only on Dp, ?, gravitational constant g, and the density difference (?p – ?). 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.

Problem 64P Combine the results of Probs. 7-65 and 7-66 to generate an equation for the settling speed V of an aerosol particle falling in air (Fig. P7-66). Verify that your result is consistent with the functional relationship obtained in Prob. 7-66. For consistency, use the notation of Prob. 7-66. (Hint: For a particle falling at constant settling speed, the particle's net weight must equal its aerodynamic drag. Your final result should be an equation for V that is valid to within some unknown constant.)

Problem 65P You will need the results of Prob. 7-67 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 (?p – ?) is doubled, all else being equal, by what factor will the settling speed go up?

Problem 66P An incompressible fluid of density ? and viscosity ?. flows at average speed V through a long, horizontal section of round pipe of length L, inner diameter D, and inner wall roughness height ? (Fig. P7-69). 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 ?P = P1, ? P2 and the other parameters in the problem. Be sure to modify your ? groups as necessary to achieve established nondimensional parameters, and name them (Hint: For consistency, choose D rather than L or ? as one of your repeating parameters.)

Problem 67P Consider laminar flow through a long section of pipe; as in Fig. P7-69. For laminar flow it turns out that will; roughness is not a relevant parameter unless ? is very large. The volume flow rate V? through the pipe is a function of pipe: diameter D, fluid viscosity ?, and axial pressure gradient dP/dx. If pipe diameter is doubled, all else being equal, by what factor will volume flow rate increase? Use dimensional analysis.

Problem 68P One of the first things you learn in physics class is the law of universal gravitation, where F is the attractive force between two bodies, m1, and m2 are masses of the two bodies, r is the distance between the two bodies, and G is the universal gravitational constant equal to (6.67428 ± 0.00067) × 10 ?11 [the units of G are not given here], (a) Calculate the SI units of G. For consistency, give your answer in terms of kg, m, and s. (b) Suppose you don’t remember the law of universal gravitation, but you are clever enough to know that F is a function of G, m1,m2, and r. Use dimensional analysis and the method of repeating variables (show all your work) to generate a nondimensional expression for F = F(G, m1, m2, r). Give your answer as ?1 = function of (?2, ? 3,…). (c) Dimensional analysis cannot yield the exact form of the function. However, compare your result to the law of universal gravitation to find the form of the function (e.g., ?1, = ? 22 or some other functional form).

When small aerosol particles or microorganisms move through air or water, the Reynolds number is very small (Re << 1). Such flows are called creeping flows. The aerodynamics drag on an object in creeping flow is a function only of its V, some characteristic length scale L of the object, and fluid viscosity ????, (Fig. P7-65). Use dimensional analysis to generate a relationship for FD as a function of the independent variables.

Problem 69P Jen is working on a spring-mass-damper system, as shown in Fig. P7-72. She remembers from her dynamic systems class that the damping ratio ? is a nondimensional property of such systems and that ? is a function of spring constantk, mass m, and damping coefficient c. Unfortunately, she does not recall the exact form of the equation for ? However, she is taking a fluid mechanics class and decides to use her newly acquired knowledge about dimensional analysis to recall the form of the equation. Help Jen develop the equation for ? using the method of repeating variables, showing all of your work. (Hint: Typical units for k are N/m and those for c are N ? s/m.)

Problem 70P Bill is working on an electrical circuit problem. He remembers from his electrical engineering class that voltage drop ?E is a function of electrical current I and electrical resistance R. Unfortunately, he does not recall the exact form of the equation for ?E. However, he is taking a fluid mechanics class and decides to use his newly acquired knowledge about dimensional analysis to recall the form of the equation. Help Bill develop the equation for ?E using the method of repeating variables, showing all of your work. Compare this to Ohm’s law—does dimensional analysis give you the correct form of the equation?

Problem 71P Albert Einstein is pondering how to write his (soon- to-be-famous) equation. He knows that energy E is a function of mass m and the speed of light c, but he doesn’t know the functional relationship (E = m2c? E = mc4?). Pretend that Albert knows nothing about dimensional analysis, but since you are taking a fluid mechanics class, you help Albert come up with his equation. Use the step-by-step method of repeating variables to generate a dimensionless relationship between these parameters, showing all of your work. Compare this to Einstein’s famous equation—does dimensional analysis give you the correct form of the equation?

Problem 72P A liquid of density ? and viscosity ? is pumped at volume flow rate V? through a pump of diameter D. The blades of the pump rotate at angular velocity ?. The pump supplies a pressure rise ?P to the liquid. Using dimensional analysis, generate a dimensionless relationship for ?P as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. Hint: For consistency (and whenever possible), it is wise to choose a length, a density, and a velocity (or angular velocity) as repeating variables.

Problem 73P A propeller of diameter D rotates at angular velocity.? in a liquid of density ? and viscosity ?. The required torque T is determined to be a function of D, ?, ?, and ?. Using dimensional analysis, generate a dimensionless relationship. Identify any established nondimensional parameters that appear in your result. Hint: For consistency (and when- ever possible), it is wise to choose a length, a density, and a: velocity (or angular velocity) as repeating variables.

Problem 74P Repeat Prob. 7-76 for the case in which the propeller operates in a compressible gas instead of a liquid.

Problem 75P In the study of turbulent flow, turbulent viscous dissipation rate ? (rate of energy loss per unit mass) is known to be a function of length scale l and velocity scale u? of the large-scale turbulent eddies. Using dimensional analysis (Buckingham pi and the method of repeating variables) and showing all of your work, generate an expression for e as a function of l and u'.

Problem 76P The rate of heat transfer to water flowing in a pipe was analyzed in Prob. 7-25. 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. P7-79). The inlet and outlet water temperatures are Tin and Tout, respectively. The total rate of heat transfer Q? from the surroundings into the water in the pipe is known to be a function of mass flow rate m?, the specific heat cp 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. 7-25. (Note: We are pretending that we do not know the analytical equation.)

Problem 77P 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.

Problem 78P 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.

Problem 79P 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.

Problem 80P Discuss the purpose of a moving ground belt in wind tunnel tests of flow over model automobiles. Think of an alternative if a moving ground belt is unavailable.

Problem 81P Consider again the model truck example discussed in Section 7-5, except that the maximum speed of the wind tunnel is only 50 m/s. Aerodynamic force data are taken for wind tunnel speeds between V = 20 and 50 m/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?

Problem 82P Use dimensional analysis to show that in a problem involving shallow water waves (Fig. P7-85), 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 ?. and fluid viscosity ?. Manipulate your ?’s to get the parameters into the following form:

Problem 83P Water at 20°C flows through a long, straight pipe. The pressure drop is measured along a section of the pipe of length L= 1.3 m as a function of average velocity V through the pipe (Table P7-86). The inner diameter of the pipe is D= 10.4 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 m/s. TABLE P7-86 V, m/s ?P,N/m2 0.5 77.0 1 306 2 1218 4 4865 6 10,920 8 19,440 10 30,340 15 68,330 20 121,400 25 189,800 30 273,200 35 372,100 40 485,300 45 614,900 50 758,700

Problem 84P In the model truck example discussed in Section 7- 5,the wind tunnel test section is 2.6mlong, 1.0 m tall, and 1.2 m wide. The one-sixteenth seale 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?

Problem 86P 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 FD is measured as a function of wind tunnel speed; the experimental results are listed in Table P7-89. Plot drag coefficient CD as a function of the Reynolds number Re, where the area used for calculation of CD is the frontal area of the model car (assume A= width × 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 31.3 m/s (70 mi/h). Assume that both the wind tunnel air and the air flowing over the prototype car are at 25°C and atmospheric pressure. TABLE P7-89 V,m/s FD, N 10 0.29 15 0.64 20 0.96 25 1.41 30 1.55 35 2.10 40 2.65 45 3.28 50 4.07 55 4.91

Problem 87P 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.

Problem 88P 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.

Problem 90P 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 ?; (d) stress ?(e) Finally, show that the relationship between stress and strain (Hooke’s law) is a dimensionally homogeneous equation.

Problem 91P Force F is applied at the tip of a cantilever beam of length L and moment of inertia I (Fig. P7-94). The modulus of elasticity of the beam material is E. When the force is applied, the tip deflection of the beam is zd. Use dimensional analysis to generate a relationship for zd as a function of the independent variables. Name any established dimensionless parameters that appear in your analysis.

Problem 92P An explosion occurs in the atmosphere when an antiaircraft missile meets its target (Fig. P7-95). A shock wave(also called a blast wave) spreads out radially from the explosion. The pressure difference across the blast wave ?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 ?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 ?P decrease?

Problem 95P Consider the steady, laminar, fully developed, two- dimensional Poiseuille flow of Prob. 7-97. The maximum velocity umax occurs at the center of the channel, (a) Generate a dimensionless relationship for umax as a function of distance between plates h, pressure gradient dP/dx, and fluid viscosity ?. (b) If the plate separation distance h is doubled, all else being equal, by what factor will umax change? (c) If the pressure gradient dP/dx is doubled, all else being equal, by what factor will umax change? (d) How many experiments are required to describe the complete relationship between umax and the other parameters in the problem?

Problem 94P Consider steady, laminar, fully developed. two. dimensional 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 dP/dx driving the flow as illustrated in Fig. P7-97. (dP/dx 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 ?, the velocity component in the ?-direction, isa function of distance h , pressure gradient dP/dx, fluid viscosity ?, and vertical coordinate y.Perform a dimensional analysis (showing all your work), and generate a dimensionless relationship between the given variables.

Problem 96P The pressure drop ?P = P1 - P2 through a long section of round pipe can be written in terms of the shear stress ?w. along the wall. Shown in Fig. P7-99 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. P7-99, 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. 7-86 (Table P7-86), 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 Re?

Problem 97P 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 ?, and fluid viscosity ?., and we wish to define a Reynolds number. We create a length scale . and define In similar fashion, define the desired established dimensionless parameter for each case: (a) Define a Froude number, given = volume flow rate per unit depth, length scale L, and gravitational constant g. (b) Define a Reynolds number, given = volume flow rate per unit depth and kinematic viscosity v. (c) Define a Richardson number (see Table 7-5), given = volume flow rate per unit depth, length scale L, characteristic density difference ??, characteristic density ?, and gravitational constant g.

Problem 98P A liquid of density ? and viscosity ? flows by graveity through a hole of diameter d in the bottom of a tank of diameter D (Fig. P7-101). 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 dimensionl analysis, generate a dimensionless relationship for V as t function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. (Hint. There are three length scales in this problem.: For consistency, choose h as your length scale.)

Problem 99P Repeat Prob. 7-101 except for a different dependent parameter, namely, the time required to empty the tank tempty. Generate a dimensionless relationship for tempty as a function of the following independent parameters: hole diameter d, tank diameter D, density ?, viscosity ?,, initial liquid surface height h, and gravitational acceleration g.

Problem 100P 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. P7-101. 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 one- quarter scale model for experimental testing, and they plan to use water as their test liquid. The model is geometrically similar to the prototype (Fig. P7-103). (a) The temperature of the ethylene glycol in the prototype tank is 60°C, at which v = 4.75 × 10-6 m2/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.

Problem 101P Liquid flows out of a hole in the bottom of a tank as in Fig. P7-101. Consider the case in which the hole is very small compared to the tank (d ? D). Experiments reveal that average jet velocity V is nearly independent of d, D, ?, or ?. 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 veloc- I ity increase?

Problem 102P An aerosol particle of characteristic size Dp 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 ??, Verify that the primary dimensions of ?p are time. Then create a dimensionless form of ?p, based on some characteristic velocity V and some characteristic length L of the airflow (Fig. P7-105). What established dimensionless parameter do you create?

Problem 103P 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.

Problem 104P In Example 7-7, the mass-based system of primary dimensions was used to establish a relationship for the pressure difference ?P = Pinside - Poutside between the inside and outside of a soap bubble as a function of soap bubble radius R and surface tension ?s of the soap film (Fig. P7-107). 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?

Problem 105P 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.

Problem 106P 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?

Problem 107P Consider a variation of the fully developed Couette flow problem of Prob. 7-59—flow between two infinite parallel plates separated by distance h, with the top plate moving at speed Vtop and the bottom plate moving at speed Vbottom as illustrated in Fig. P7-110. The flow is steady, incompressible, and two-dimensional in the xy-plane. Generate a dimensionless relationship for the x-component of fluid velocity u as a function of fluid viscosity ?., plate speeds Vtop and Vbottom distance h, fluid density ?, and distance y. (Hint: Think carefully about the list of parameters before rushing into the algebra.)

Problem 108P What are the primary dimensions of electric charge q, the units of which are coulombs (C)? (Hint: Look up the fundamental definition of electric current.)

Problem 109P What are the primary dimensions of electrical capacitance C. the units of which are farads? (Hint: Look up the fundamental definition of electrical capacitance.)

Problem 110P In many electronic circuits in which some kind of time scale is involved, such as filters and time-delay circuits (Fig. P7-113—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, RC. Showing all your work, what are the primary dimensions of RC? Using dimensional reasoning alone, explain why a resistor and capacitor are often found together in timing circuits.

Problem 111P 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, Write the primary dimensions of both sides of this equation, and verify that the equation is dimensionally homogeneous. Show all your work.

Problem 112P A common device used in various applications to clean particle-laden air is the reverse-flow cyclone (Fig. P7-115). Dusty air (volume flow rate and density ?) 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 ?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?

Problem 113P 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 qp (coulombs) by; charged ionizer wires (Fig. P7-116). 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 Ef (voltage difference per unit distance). Shown in Fig. P7-116 is a charged dust particle of diameter Dp. 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 qp, Ef, Dp, and air viscosity ?. (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?

Problem 114P Experiments are being designed to measure the horizontal force F on a fireman’s nozzle, as shown in Fig. P7-117. Force F is a function of velocity V1, pressure drop ?P = P, – P2, density ?, viscosity ?, inlet area A1 outlet area A2, and length L. Perform a dimensional analysis for F = f(V1 ?P, ?, ?, A1, A2, L). For consistency, use V1, A1, and ? as the repeating parameters and generate a dimension- less relationship. Identify any established nondimensional parameters that appear in your result.

Problem 116P 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. P7-119). h is a function of liquid density ?, tube diameter D, gravitational constant g, contact angle ?. and the surface tension ?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.

Problem 117P Repeat part (a) of Prob. 7-119, except instead of height h, find a functional relationship for the time scale trise needed for the liquid to climb up to its final height in the capillary tube. (Hint: Check the list of independent parameters in Prob. 7-119. Are there any additional relevant parameters?)

Problem 119P Repeat Prob. 7-121. but with the distance r from the sound source as an additional independent parameter.

Problem 118P 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 ? (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.

Problem 120P Engineers at MIT have developed a mechanical model of a tuna fish to study its locomotion. The “Robotuna” shown in Fig. P7-123 is 1.0 m long and swims at speeds up to 2.0 m/s. Real bluefin tuna can exceed 3.0 m in length and have been clocked at speeds greater than 13 m/s. How fast would the 1.0-m Robotuna need to swim in order to match the Reynolds number of a real tuna that is 2.5 m long and swims at 12.5 m/s?