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Fluid Mechanics EIT Review

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**Shear Stress Tangential force per unit area rate of shear**

change in velocity with respect to distance rate of shear

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**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. ? 1 g2 P1 = 0 h1 3 g1 h2 2 P1 + h1g2 - h2g1 =P3 For small h1 use fluid with high density. Mercury!

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**Differential Manometers**

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

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**Forces on Plane Areas: Inclined Surfaces**

Free surface O q x A’ centroid B’ O center of pressure y The origin of the y axis is on the free surface

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**Statics Fundamental Equations Sum of the forces = 0**

Sum of the moments = 0 pc is the pressure at the __________________ centroid of the area Line of action is below the centroid

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

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

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**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

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**Example using Moments Solution Scheme**

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 hinge water F 8 m 4 m Location of the resultant force Find F using moments about hinge

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**Magnitude of the Force h = _____ 10 m Depth to the centroid pc = ___**

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

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**Location of Resultant Force**

hinge water F 8 m 4 m Fr Slant distance to surface 12.5 m b = 2 m a = 2.5 m cp 0.125 m

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**Force Required to Open Gate**

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

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**Example: Forces on Curved Surfaces**

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

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**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 = m (measured from A) with magnitude of 89.7 kN

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**Example: Forces on Curved Surfaces**

The location of the line of action of the horizontal component is given by A water W1 b h 3 m 2 m W2 2 m (1 m)(2 m)3/12 = m4 4 m x y

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**Example: Forces on Curved Surfaces**

78.5 kN horizontal 0.948 m 4.083 m 89.7 kN vertical 119.2 kN resultant

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**Cylindrical Surface Force Check**

0.948 m 89.7kN 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) = ___

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**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

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**Dimensionless parameters**

Reynolds Number Froude Number Weber Number Mach Number Pressure Coefficient (the dependent variable that we measure experimentally)

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**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

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**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

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**Control Volume Equations**

Mass Linear Momentum Moment of Momentum Energy

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**Conservation of Mass If mass in cv is constant**

2 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]

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**Conservation of Momentum**

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Energy Equation laminar turbulent Moody Diagram

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**Example HGL and EGL z velocity head pressure head elevation pump z = 0**

energy grade line hydraulic grade line z elevation pump z = 0 datum

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**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)

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**Moody Diagram 0.10 0.08 0.06 0.05 0.04 friction factor 0.03 0.02 0.01**

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 0.01 smooth 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 R

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**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)

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**Power and Efficiencies**

Electrical power Shaft power Impeller power Fluid power Motor losses IE bearing losses Tw pump losses Tw gQHp

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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

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**Drag Coefficient on a Sphere**

1000 100 Stokes Law Drag Coefficient 10 1 0.1 0.1 1 10 102 103 104 105 106 107 Reynolds Number

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