Fundamentals of Momentum, Heat and Mass Transfer, 6th Edition International Student Version - Chapter 11

11.1 Through a series of tests on pipe flow, H. Darcy derived an equation for the friction loss in pipe flow as...in which f is a dimensionless coefficient that depends on (a) the average velocity v of the pipe flow; (b) the pipe diameter D; (c) the fluid density ρ; (d) the fluid viscosity μ; and (e) the average pipe wall uneveness π (length). Using the Buckingham p theorem, find a dimensionless function for the coefficient f.
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11.2 The maximum pitching moment that is developed by the water on a flying boat as it lands is noted as cmax The following are the variables involved in this action:Α = angle made by flight path of plane with horizontalβ = angle defining attitude of planeM = mass of planeL = length of hullρ = density of waterg = acceleration of gravityR = radius of gyration of plane about axis of pitching
a. According to the Buckingham p theorem, how many independent dimensionless groups should there be characterizing this problem?
b. What is the dimensional matrix of this problem? What is its rank?
c. Evaluate the appropriate dimensionless parameters for this problem.

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11.3 The performance of a journal bearing around a rotating shaft is a function of the following variables: Q, the rate of flow lubricating oil to the bearing in volume per unit time; D, the bearing diameter; N, the shaft speed in revolutions per minute; μ, the lubricant viscosity; ρ, the lubricant density; and σ, the surface tension of the lubricating oil. Suggest appropriate parameters to be used in correlating experimental data for such a system.
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11.4 The functional frequency n of a stretched string is a function of the string length L, its diameter D, the mass density ρ, and the applied tensile force T. Suggest a set of dimensionless parameters relating these variables.
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11.5 A large amount of energy E is suddenly released in the air as in a point of explosion. Experimental evidence suggests that the radius r of the high-pressure blast wave depends on time t as well as the energy E and ρ the density of the ambient air.
a. Using the Buckingham method, find the equation for r as a function of t, ρ, and E.
b. Show that the speed of the wave front decreases as r increases.

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11.6 The size d of droplets produced by a liquid spray nozzle is thought to depend upon the nozzle diameter D, jet velocity V, and the properties of the liquid ρ, μ, and σ. Rewrite this relation in dimensionless form. Take D, ρ, and V as repeating variables.
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11.7 In natural-convection problems, the variation of density due to the temperature difference ΔT creates an important buoyancy term in the momentum equation. If a warm gas at TH moves through a gas at temperature T0 and if the density change is only due to temperature changes, the equation of motion becomes...Show that the ratio of gravity (buoyancy) to inertial forces acting on a fluid element is...where L and V0 are reference lengths and velocity, respectively.
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11.8 During the development of a 300-ft ship, it is desired to test a 10% scale model in a towing tank to determine the drag characteristics of the hull. Determine how the model is to be tested if the Froude number is to be duplicated.
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11.9 An estimate is needed on the lift provided by a hydrofoil wing section when it moves through water at 60 mph. Test data are available for this purpose from experiments in a pressurized wind tunnel with an airfoil section model geometrically similar to but twice the size of the hydrofoil. If the lift F1 is a function of the density ρ of the fluid, the velocity u of the flow, the angle of attack θ, the chord length D, and the viscosity μ, what velocity of flow in the wind tunnel would correspond to the hydrofoil velocity for which the estimate is desired? Assume the same angle of attack in both cases, that the density of the air in the pressurized tunnel is 5.0 × 10−3 slugs/ft3, that its kinematic viscosity is 8.0 × 10−5 ft2/s, and that the kinematic viscosity of the water is approximately 1.0 × 10−5 ft2/s. Take the density of water to be 1.94 slugs/ft3.
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11.10 A 40% scale model of an airplane is to be tested in a flow regime where unsteady flow effects are important. If the fullscale vehicle experiences the unsteady effects at a Mach number of 1 at an altitude of 40,000 ft, what pressure must the model be tested at to produce an equal Reynolds number? The model is to be tested in air at 70°F. What will the timescale of the flow about the model be relative to the full-scale vehicle?
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11.11 A coating operation is creating materials for the electronics industry. The coating requires a specific volumetric flow rate Q, solution density ρ, solution viscosity μ, substrate coating velocity v, solution surface tension σ, and the length of the coating channel L. Determine the dimensionless groups formed from the variables involved using the Buckingham method. Choose the groups so that Q, σ, and μ appear in one group only.
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