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Fluid Mechanics

Fluid Mechanics. EIT Review. Shear Stress. Tangential force per unit area. change in velocity with respect to distance. rate of shear. 3. 1. 2. ?. Manometers for High Pressures.

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Fluid Mechanics

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  1. Fluid Mechanics EIT Review

  2. Shear Stress Tangential force per unit area change in velocity with respect to distance rate of shear

  3. 3 1 2 ? Manometers for High Pressures Find the gage pressure in the center of the sphere. The sphere contains fluid with g1 and the manometer contains fluid with g2. What do you know? _____ Use statics to find other pressures. g2 h1 P1 = 0 g1 h2 P1 + h1g2 - h2g1 =P3 Mercury! For small h1 use fluid with high density.

  4. Differential Manometers Water p2 p1 h3 orifice h1 h2 Mercury p1 + h1gw - h2gHg - h3gw = p2 Find the drop in pressure between point 1 and point 2. p1 - p2 = (h3-h1)gw + h2gHg p1 - p2 = h2(gHg - gw)

  5. x centroid center of pressure y Forces on Plane Areas: Inclined Surfaces Free surface O q A’ B’ O The origin of the y axis is on the free surface

  6. Statics • Fundamental Equations • Sum of the forces = 0 • Sum of the moments = 0 centroid of the area pc is the pressure at the __________________ Line of action is below the centroid

  7. b a Ixc yc R Ixc yc Properties of Areas a Ixc yc b d

  8. Ixc yc R b a Ixc yc yc R Properties of Areas

  9. Inclined Surface Summary • The horizontal center of pressure and the horizontal centroid ________ when the surface has either a horizontal or vertical axis of symmetry • The center of pressure is always _______ the centroid • The vertical distance between the centroid and the center of pressure _________ as the surface is lowered deeper into the liquid • What do you do if there isn’t a free surface? coincide below decreases

  10. hinge 8 m water F 4 m Example using Moments An elliptical gate covers the end of a pipe 4 m in diameter. If the gate is hinged at the top, what normal force F applied at the bottom of the gate is required to open the gate when water is 8 m deep above the top of the pipe and the pipe is open to the atmosphere on the other side? Neglect the weight of the gate. Solution Scheme Magnitude of the force applied by the water Location of the resultant force Find F using moments about hinge

  11. hinge 8 m water Fr F 4 m a = 2.5 m b = 2 m Magnitude of the Force h = _____ 10 m Depth to the centroid pc = ___ Fr= ________ 1.54 MN

  12. hinge 8 m water Fr F 4 m a = 2.5 m cp b = 2 m Location of Resultant Force Slant distance to surface 12.5 m 0.125 m

  13. hinge 8 m water Fr F 4 m lcp=2.625 m cp Force Required to Open Gate How do we find the required force? Moments about the hinge =Fltot - Frlcp 2.5 m ltot b = 2 m F = ______ 809 kN

  14. Example: Forces on Curved Surfaces Find the resultant force (magnitude and location) on a 1 m wide section of the circular arc. water W1 + W2 FV = W1 3 m = (3 m)(2 m)(1 m)g + p/4(2 m)2(1 m)g 2 m = 58.9 kN + 30.8 kN W2 = 89.7 kN 2 m FH = x =g(4 m)(2 m)(1 m) = 78.5 kN y

  15. Example: Forces on Curved Surfaces The vertical component line of action goes through the centroid of the volume of water above the surface. A water Take moments about a vertical axis through A. W1 3 m 2 m W2 2 m = 0.948 m (measured from A) with magnitude of 89.7 kN

  16. b h Example: Forces on Curved Surfaces The location of the line of action of the horizontal component is given by A water W1 3 m 2 m W2 2 m (1 m)(2 m)3/12 = 0.667 m4 4 m x y

  17. Example: Forces on Curved Surfaces 78.5 kN horizontal 0.948 m 4.083 m 89.7 kN vertical 119.2 kN resultant

  18. Cylindrical Surface Force Check 89.7kN 0.948 m • All pressure forces pass through point C. • The pressure force applies no moment about point C. • The resultant must pass through point C. C 1.083 m 78.5kN (78.5kN)(1.083m) - (89.7kN)(0.948m) = ___ 0

  19. Curved Surface Trick • Find force F required to open the gate. • The pressure forces and force F pass through O. Thus the hinge force must pass through O! • All the horizontal force is carried by the hinge • Hinge carries only horizontal forces! (F = ________) A water W1 3 m 2 m O F W2 W1 + W2 11.23

  20. Dimensionless parameters • Reynolds Number • Froude Number • Weber Number • Mach Number • Pressure Coefficient • (the dependent variable that we measure experimentally)

  21. Model Studies and Similitude:Scaling Requirements • dynamic similitude • geometric similitude • all linear dimensions must be scaled identically • roughness must scale • kinematic similitude • constant ratio of dynamic pressures at corresponding points • streamlines must be geometrically similar • _______, __________, _________, and _________ numbers must be the same Mach Reynolds Froude Weber

  22. Froude similarity • Froude number the same in model and prototype • ________________________ • define length ratio (usually larger than 1) • velocity ratio • time ratio • discharge ratio • force ratio difficult to change g 11.33

  23. Control Volume Equations • Mass • Linear Momentum • Moment of Momentum • Energy

  24. Conservation of Mass 2 If mass in cv is constant 1 v1 A1 Area vector is normal to surface and pointed out of cv V = spatial average of v [M/t] If density is constant [L3/t]

  25. Conservation of Momentum

  26. Energy Equation laminar turbulent Moody Diagram

  27. Example HGL and EGL velocity head pressure head energy grade line hydraulic grade line elevation z pump z = 0 datum

  28. Smooth, Transition, Rough Turbulent Flow • Hydraulically smooth pipe law (von Karman, 1930) • Rough pipe law (von Karman, 1930) • Transition function for both smooth and rough pipe laws (Colebrook) (used to draw the Moody diagram)

  29. Moody Diagram 0.10 0.08 0.05 0.04 0.06 0.03 0.05 0.02 0.015 0.04 0.01 0.008 friction factor 0.006 0.03 0.004 laminar 0.002 0.02 0.001 0.0008 0.0004 0.0002 0.0001 0.00005 0.01 smooth 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 R

  30. Solution Techniques • find head loss given (D, type of pipe, Q) • find flow rate given (head, D, L, type of pipe) • find pipe size given (head, type of pipe,L, Q)

  31. Power and Efficiencies • Electrical power • Shaft power • Impeller power • Fluid power Motor losses IE bearing losses Tw pump losses Tw gQHp

  32. Manning Formula The Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way... Hydraulic radius for wide channels

  33. Drag Coefficient on a Sphere 1000 Stokes Law 100 Drag Coefficient 10 1 0.1 0.1 1 10 102 103 104 105 106 107 Reynolds Number

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