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1 Fluid Mechanics FE Review Carrie (CJ) McClelland, P.E. Fluid Mechanics FE Review These slides contain some notes, thoughts about what to study, and some practice problems. The answers to the problems are given in the last slide. In the review session, we will be working some of these problems. Feel free to come to the session, or work the problems on your own. I am happy to answer your questions or those that you bring to the session. Good luck! 1

2 Fluid Mechanics FE Review MAJOR TOPICS Fluid Properties Fluid Statics Fluid Dynamics Fluid Measurements Dimensional Anaysis Fluid Mechanics FE Review It will be very helpful to memorize the following concepts and equations: Specific weight, density, and specific gravity Hydrostatics pressure equation / manometry Force magnitude and location due to hydrostatic pressure for horizontal and vertical plane walls Conservation of mass / continuity Conservation of energy / Bernoulli and Energy Eqn Darcy Eqn Relative roughness equation Drag equation How to use the Moody Diagram Most equations and problems taken from Review Course Book 2

3 Fluid Mechanics Properties Fluids Substances in either the liquid or gas phase Cannot support shear Make sure you use the ideal gas law when calculating properties for gasses!! Fluid Mechanics Properties Property Density Symbol & Equation = Definition Etc. Specific Weight = density x gravity Specific Gravity Viscosity = = Kinematic viscosity = Make sure you know the relationship between density, specific weight, and specific gravity! 3

4 Mass & Weight L of an incompressible liquid exert a force of 20 N at the earth s surface. What force would 2.3 L of this liquid exert on the surface of the moon? The gravitational acceleration on the surface of the moon is 1.67 m/s 2 (A) 0.39 N (B) 0.78 N (C) 3.4 N (D) 4.6 N Fluid Mechanics Fluid Properties Definitions Property Symbol & Equation Definition Vapor Pressure Pressure at which liquid and vapor are in equilibrium Bulk Modulus = Etc. Used to predict cavitation (local pressure < vapor pressure Speed of Sound, =, = Velocity of propegation of a small wave k = Bulk modulus =1.4 4

5 Mach Number 2. A jet aircraft is flying at a speed o f 1700 km/h. The air temperature is 20 C. The molecular weight of air is 29 g/mol. What is the Mach number of the aircraft? (A) (B) 1.38 (C) 1.92 (D) 5.28 Fluid Mechanics Fluid Properties Definitions Property Surface Tension Symbol & Equation = Definition Etc. Capillary Rise 4 h= Distance a liquid will rise (or fall) in a tube d 5

6 Surface Tension 3. A 2 mm (inside diameter) glass tube is placed in a container of mercury. An angle of 40 is measured as illustrated. The density and surface tension of mercury are kg/m 3 and 37.5 x 10-2 N/m, respectively. How high will the mercury rise or be depressed in the tube as a result of capillary action? (A) -4.3 mm (depression) (B) -1.6 mm (depression) (C) 4.2 mm (rise) (D) 6.4 mm (rise) Fluid Mechanics Stresses and Viscosity 9-1b Shear Stress Normal Component: Tangential Component - For a Newtonian fluid: = (v = velocity) - For a pseudoplastic or dilatant fluid: Absolute Viscosity = µ = Ratio of shear stress to rate of shear deformation 6

7 Viscosity 4. A sliding-plate viscometer is used to measure the viscosity of a Newtonian fluid. A force of 25 N is required to keep the top plate moving at a constant velocity of 5 m/s. What is the viscosity of the fluid? (A) N-s/m 2 (B) 0.04 N-s/m 2 (C) 0.2 N-s/m 2 (D) 5.0 N-s/m 2 Fluid Statics 7

8 Manometry 5. An open water manometer is used to measure the pressure in a tank. The tank ifs half-filled with 50,000 kg of a liquid chemical that is not miscible in water. The manometer tube is filled with liquid chemical. What is the pressure in the tank relative to the atmospheric pressure? (A) 1.4 kpa (B) 1.9 kpa (C) 2.4 kpa (D) 3.4 kpa Fluid Statics 9-2c Barometer Atmospheric Pressure 8

9 Fluid Statics Hydrostatics 6. A 6m x 6m x 6m vented cubical tank is half-filled with water; the remaining space is filled with oil (SG=0.8). What is the total force on one side of the tank? (A) 690 kn (B) 900 kn (C) 950 kn (D) 1.0 MN 9

10 Fluid Statics Examples 9-2f1 Example 1 (FEIM): The tank shown is filled with water. At what depth does the resultant force act? The surface under pressure is a rectangle 1 m at the base and 2.31 m tall. A = bh h I yc = b3 12 Z c = 4 m sin60 = m Fluid Statics Examples 9-2f2 Using the moment of inertia for a rectangle given in the NCEES Handbook, z* = I y c b 3 h = = b2 AZ c 12bhZ c 12Z c = (2.31 m) 2 (12)(4.618 m) = m R depth = (Z c + z*) sin 60 = (4.618 m m) sin 60 = 4.08 m 10

11 Fluid Statics Examples 9-2g Example 2 (FEIM): The rectangular gate shown is 3 m high and has a frictionless hinge at the bottom. The fluid has a density of 1600 kg/m 3. The magnitude of the force F per meter of width to keep the gate closed is most nearly (A) 0 kn/m (B) 24 kn/m (C) 71 kn/m (D) 370 kn/m R is one-third from the bottom (centroid of a triangle from the NCEES Handbook). Taking the moments about R, 2F = F h F w = 1 3 R 70,667 N w = m = 23.6 kn/m 3 p ave = ρgz ave (1600 kg m 3)(9.81m s 2)( 1 )(3 m) 2 = Pa R w = p aveh = (23544 Pa)(3 m) = N/m F + F h = R Therefore, (B) is correct. Hydrostatics 6. A gravity dam has the cross section shown. What is the magnitude of the resultant water force (per meter of width) acting on the face of the dam? (A) 7.85 MN/m (B) 12.3 MN/m (C) 14.6 MN/m (D) 20.2 MN/m 11

12 Buoyancy Archimedes Principle and Buoyancy The buoyant force on a submerged or floating object is equal to the weight of the displaced fluid. A body floating at the interface between two fluids will have buoyant force equal to the weights of both fluids displaced. F buoyant = γ water V displaced Buoyancy 8. A 35 cm diameter solid sphere (ρ = 4500 kg/m 3 ) is suspended by a cable as shown. Half of the sphere is in one fluid (ρ = 1200 kg/m 3 ) and the other half of the sphere is in another (ρ = 1500 kg/m 3 ). What is the tension in the cable? (A) 297 N (B) 593 N (C) 694 N (D) 826 N ρ=1200 kg/m 3 ρ=1500 kg/m 3 12

13 Fluid Dynamics (might be called the field eqn.) Continuity Equation Example 9-3c Example (FEIM): The speed of an incompressible fluid is 4 m/s entering the 260 mm pipe. The speed in the 130 mm pipe is most nearly (A) 1 m/s (B) 2 m/s (C) 4 m/s (D) 16 m/s A 1 v 1 = A 2 v 2 A 1 = 4A 2 so v 2 = 4v 1 = ( 4 ) 4 m = 16 m/s s Therefore, (D) is correct. 13

14 PPI Bernoulli Equation 9. The diameter of a water pipe gradually changes from 5 cm at point A to 15 cm at point B. Point A is 5 m lower than point B. The pressure is 700 kpa at point A and 664 kpa at point B. Friction between the water and the pipe walls is negligible. What is the rate of discharge at point B? (A) m3/s (B) m3/s (C) m3/s (D) m3/s 15 cm 5m Bernoulli Flow from a Jet 10. A liquid with a specific gravity of 0.9 is stored in a pressurized, closed storage tank. The tank is cylindrical with a 10 m diameter. The absolute pressure in the tank above the liquid is 200 kpa. What is the initial velocity of a fluid jet when a 5 cm diameter orifice is opened at point A? (A) 11.3 m/s (B) 18.0 m/s (C) 18.6 m/s (D) 23.9 m/s 14

15 Flow in Pipes Fluid Dynamics and Friction Steady, Incompressible Flow Reynolds Number 9-3g For a Newtonian fluid: D = hydraulic diameter = 4R H ν = kinematic viscosity µ = dynamic viscosity For a pseudoplastic or dilatant fluid: 15

16 Fluid Dynamics and Friction 11. A steel pipe with an inside diameter of 25 mm is 20 m long and carries water at a rate of 4.5 m 3 /h. Assuming the specific roughness of the pipe is m, the water has an absolute viscosity of 1.00 x 10-3 Pa-s and a density of 1000 kg/m 3, what is the friction factor? (A) (B) (C) (D) Non-Circular Conduits, Open Channel Flow, and Partially Full Pipes 16

17 PPI Hydraulic Radius 12. What is the hydraulic radius of the trapezoidal irrigation canal shown? (A) 1.63 m (B) 2.00 m (C) 2.13 m (D) 4.00 m Hydraulic Grade Line and Energy Grade Line Pump Power 17

18 Multi-path Pipelines Multi-path pipelines 13. The Darcy friction factor for both of the pipes shown is The total flow rate is 300 m 3 /h. What is the flow rate through the 250 mm pipe? (A) 0.04 m 3 /s (B) 0.05 m 3 /s (C) 0.06 m 3 /s (D) 0.07 m 3 /s 18

19 Implulse-Momentum Be sure to familiarize yourself with how this equation works for bends, enlargements, contractions, jet propulsion, fixed blades, moving blades, and impulse turbines. Impulse-Momentum 9-6a Pipe Bends, Enlargements, and Contractions 19

20 Impulse-Momentum 14. Water is flowing at 50 m/s through a 15 cm diameter pipe. The pipe makes a 90 degree bend, as shown. What is the reaction on the water in the z-direction at the bend? (A) -44 kn (B) -33 kn (C) 14 kn (D) 44 kn Flow and Pressure Measurement 20

21 PPI Fluid Mechanics 9-10a Fluid Measurements Pitot Tube measures flow velocity The static pressure of the fluid at the depth of the pitot tube (p0) must be known. For incompressible fluids and compressible fluids with M 0.3, Flow and Pressure Measurement 15. The density of air flowing in a duct is 1.15 kg/m3. A pitot tube is placed in the duct as shown. The static pressure in the duct is measured with a wall tap and pressure gage. Use the gage readings to determine the velocity of the air. (A) 42 m/s (B) 102 m/s (C) 110 m/s (D) 150 m/s 21

22 Fluid Mechanics Fluid Measurements 9-10c Venturi Meters measures the flow rate in a pipe system The changes in pressure and elevation determine the flow rate. In this diagram, z 1 = z 2, so there is no change in height. Fluid Mechanics Fluid Measurements 9-10e Orifices 22

23 Orifice Flow 16. Water flows out of a tank at 12.5 m/s from an orifice located 9m below the surface. The crosssectional area of the orifice is m 2, and the coefficient of discharge is What is the diameter D, at the vena contracta? (A) 4.2 cm (B) 4.5 cm (C) 4.7 cm (D) 4.8 cm Similitude In other words Dimensionless parameters must be equal between the model and prototype (Each dimensionless parameter is a ratio of different types of forces being exerted on the fluid) For example: For completely submerged models/prototypes and pipe flow, the Reynolds numbers must be equal For Weirs, dams, ships, and open channels, the Froude numbers must be equal 23

24 Similitude Dimensionless Parameters Similitude 17. A nuclear submarine is capable of a top underwater speed of 65 km/h. How fast would a 1/20 scale model of the submarine have to be moved through a testing pool filled with seawater for the forces on the submarine and model to be dimensionally similar? (A) 0.90 m/s (B) 18 m/s (C) 180 m/s (D) 360 m/s 24

25 Drag Typically these problems will be plug and chug Drag Coefficients for Spheres and Circular Flat Disks ANSWERS!! 1. B 2. B 3. A 4. B 5. A 6. B 7. C 8. C 9. C 10.C 11.C 12.C 13.C 14.D 15.A 16.D 17.D 25

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