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1 UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, DECEMBER 10, :00 PM 2.5 HOURS CHE 211F FLUID MECHANICS EXAMINER: PROFESSOR D.G. ALLEN ANSWER ALL SEVEN (7) QUESTIONS IN THE EXAM BOOKLET IF YOU WISH, YOU CAN RETURN THE EXAM PAPER WITH YOUR NAME & USE THE FIGURES ON THE PAPER. Total of 7 pages This exam is marked out of 100 Clearly state any simplifying assumptions. If a particular question requires the solution from a previous part of the same question that you could not solve, show your solution by assuming the answer to the previous part. If you are short of time, setup your equations and describe the solution for part marks. Additional Data for all Questions (some is also with the questions themselves): Fluid properties: Water 16 C : Density = 1000kg/m 3 = 62.4 lb m /ft 3 =1.94 slugs/ft 3 ; Spec wt = 62.4 lb/ft 3 ; Viscosity = Pa s = 2.3 x 105 lb s/ ft 2 Vapour Pressure = 0.25 psi = 1770 Pa; Surface tension=0.073 N/m Mercury 20 C: Density=13,600 kg/m 3 =847 lb m /ft 3 =26.3 slugs/ft 3 ; Spec weight =847 lb/ft 3 ; Viscosity= Pa s Air at 15 C: Density =1.2 kg/m 3 = slugs/ft 3 ; Molecular mass = 29 g/mol Viscosity = 1.79 x 105 Pa s = 3.74 x 107 lb s/ ft 2 SAE Oil 16 C: Density=912 kg/m 3 =57 lb m /ft 3 =1.77 slugs/ft 3 ; Spec weight =57 lb/ft 3 ; Viscosity= 0.38Pa s Constants and Conversions and Gas Law: g = 9.8 m/s 2 = 32.2 ft/s 2 ; 1 ft = m = 12 in; 1 lb (mass) = kg 1 Pa = 1.45 x 104 p.s.i. 1 atm = x 10 5 Pa = 14.7 p.s.i.=760mmhg R= J/mol K Ideal Gas Law PV=nRT ; T(K) = T( 0 C) g c = 32.2 ft lb m /(lb s 2 ); 1 hp = 746 W = 550 ft lb/s; 1 ft lb/s = 1.36 W Other Information:  British gravitational system: 1 lb= (1 slug) (1 ft/s 2 )  Moody Chart & Colebrook formula (see attached) f= (DΔP/L)/(ρv 2 /2)  For laminar flow in pipes, f=64/re, where Re<2100 = Dvρ/µ  Hydraulic diameter relation for rectangle a x b in cross section, D h =2ab/(a+b), use same Moody chart for round pipes  Blausius Equ n f = 0.316(Re ) turbulent flow in smooth pipes and Re < Drag coefficient for a particular shape C D = (Drag force)/(0.5 ρ v 2 A); A is projected area in direction of flow with relative velocity, v. Reynolds number for particle of a particular shape is D p vρ/µ where D p is characteristic particle diameter  Volume of a sphere of radius, r: V= (4/3)П r 3 ; Volume of a cylinder= П r 2 h  Roots of a quadratic are ( b+/ sqrt[b 24ac])/(2a) Page 1 of 7
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3 1) Even though the ideal gas law requires the absolute pressure, in most fluid mechanics problems in class and in life we provide gauge pressures. Why is this the case? Provide at least two reasons. (4 marks) 2) Give 2 reasons why the power that you have to provide to run a pump (e.g. electricity at the cottage) is more than the power actually delivered to the fluid. (4 marks) 3) A solid sphere that is 5 cm in diameter is immersed in a rectangular tank as shown below. The tank has two immiscible fluids, SAE 30 oil and water. The sphere floats (i.e. doesn t move) at the interface with exactly half of its volume in the oil phase and the other half in the water phase. (Total: 16 marks) a) Determine the density of the sphere. (12 marks) b) What is the height, h, of the mercury manometer (4 marks) Open to atmosphere 20 cm Oil Sphere Open to atmosphere L 30 cm Water 10 cm 100 cm Q.3 Page 3 of 7
4 4) In the Figure below, all pipes are 8 cm diameter and the fluid is water. The flowrate through pipe A is measured as m 3 /s. Assume that the friction factor is the same in all three pipes and is equal to The valve on pipe C is a ball valve that is 2/3rds closed in which case it s K l is 210 based on the average velocity in pipe C. (Total: 20 marks) a) Determine the elevation of the surface of tank 1 (z 1 ) relative to z 2. Ignore any other minor losses. (14 marks) b) Estimate the pipe roughness in mm. (6 marks) 1 Z=? 2 Z=0m Water L=100m L=50m B 10 m A C 30 m L=70m VALVE Q.4 Page 4 of 7
5 5) Professor Allen wants to provide water from the lake to a Bunkie (sleeping cabin) at his cottage. The basic diagram showing the separate pump and piping system for the Bunkie is found below. He wants to provide water to the Bunkie tank which will be at a pressure of 40 psig and he wants a flowrate of 60 L/min. All the piping is 1.5 inch diameter plastic pipe (smooth) and the total length of pipe is 48 m. He plans to use a centrifugal pump located 30 m along the pipe from the lake that is at the same elevation as the cottage tank that will operate at 85% efficiency. The Bunkie tank bottom is 3 metres above the cottage tank; when full the Bunkie tank water level is 0.5m. The cottage tank bottom is 5 metres above the lake level. The lake is 2 m deep. Assume a temperature of 16 0 C and neglect minor losses. Specify the pump requirements both for horsepower pump (W) and Net Positive Suction Head (Total: 16 marks) Cottage Tank Bunkie Tank 40 PSIG 0.5 m 18 m 3 m 5 m 30 m 2 m Lake Q.5 Page 5 of 7
6 1) As seen in the Mythbusters video in class, to study drag around a pickup truck, they simulated air flow around a pickup truck by putting a model truck into a tank filled with water that was pumped at a certain velocity past the truck. For this problem, assume that the characteristic diameter (i.e. D p in the Reynold s number) is simply given by the width of the truck facing into the flow and the projected area into the flow is a rectangle. Also assume that the real truck is 2 m wide (i.e. D p,truck = 2m) and the model truck is 10 cm wide (D p,model = 0.10m). The height of the rectangle projected into the flow for each truck is half its width (i.e. 1 m tall for the real truck, so rectangle for cross sectional area is 2m x 1m; 5 cm tall for the model, so rectangle for cross sectional area is 10cm x 5 cm). (Total: 16 marks) a) In the Mythbusters show they tested the real truck moving at two speeds: 40 and 90 km/h. In order to keep the same flow regime (i.e. degree of turbulence), estimate the two water velocities they should use in the model system with water to simulate the flow of air around the real truck. Assume the air would be at 15 0 C and the water would be at 16 0 C. (8 marks) b) Estimate the ratio of the Drag Force (in N) on the moving real truck at 90 km/h in air and on the model truck in water at the same flow regime. (8 marks) v Air or water Q.6 Page 6 of 7
7 2) As you know, when a viscous fluid flows past a stationary object (e.g. flat plate, pipe wall, etc), the velocity of the fluid is assumed to be zero at the surface and then there is a boundary layer that develops where the fluid velocity increases from zero to the free stream velocity. This is illustrated below for a fluid flowing across a flat plate. We propose that the wall shear stress (τ w ) is a function of the distance from the edge of the flat plate (x), the free stream velocity (v), fluid density (ρ) and viscosity (µ). Using the wall shear stress and viscosity as nonrepeating variables, determine the dimensionless groups that describe this flow situation. (12 marks) Free stream velocity, V Boundary layer thickness δ (x) Boundary layer y x Q.7 Flat Plate 3) Determine the average velocity for a fluid flowing in a rectangular channel (below) that is 1 ft deep if the velocity profile from the channel bottom (y=0) to the channel surface (y=1 ft) is u=4y2y 2, where u is in ft/s and y is in ft. Assume the channel is wide enough so velocity at any point y is constant (i.e. u only varies in the y direction). (12 marks) 1 ft U=4y2y 2 Q.8 Page 7 of 7
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