s and FE X. A. Flow measurement B. properties C. statics D. impulse, and momentum equations E. Pipe and other internal flow 7% of FE Morning Session I

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1 Fundamentals of Engineering (FE) Exam General Section Steven Burian Civil & Environmental Engineering October 26, 2010

2 s and FE X. A. Flow measurement B. properties C. statics D. impulse, and momentum equations E. Pipe and other internal flow 7% of FE Morning Session IV. A. Bernoulli equation and mechanical energy balance B. Hydrostatic pressure C. Dimensionless numbers (e.g., Reynolds Number) D. Laminar and turbulent flow E. Velocity head F. Friction losses (e.g., pipes, p valves, fittings) G. Pipe networks H. Compressible and incompressible flow I. Flow measurement (e.g., orifices, Venturi meters) J. Pumps, turbines, and compressors K. Non-Newtonian Newtonian flow L. Flow through packed beds 10% of FE Afternoon Session

4 FE Statics s! s are substances in either the liquid or gas phase! s cannot support shear, and they deform continuously to minimize applied shear forces

5 FE Statics

Viscosity ("): measure of a fluid s resistance to flow when acted upon by an

6 FE Statics Viscosity! Shear stress (!): force required to slide one unit area layer of a substance over another! Viscosity ("): measure of a fluid s resistance to flow when acted upon by an external force (i.e., ease with which a fluid pours)! As a fluid moves a shear stress is developed in it; magnitude is dependent on viscosity of fluid

7 F/A is the fluid shear stress (!) and the constant of proportionality p is the absolute viscosity ("): FE Statics! # " du dy Newtonian fluids: strains are proportional to the applied shear stress Non-Newtonian Newtonian fluids: fluid shear stress can be computed using the power law The kinematic viscosity is the ratio of the absolute viscosity to mass density: $ % # "

8 FE Statics

9 FE Statics Surface Tension! The skin that seems to form on the free surface of a fluid is caused by intermolecular cohesive forces and is known as surface tension, &! Surface tension can be interpreted as the tensile force between two points a unit distance apart on the surface

10 FE Statics Capillarity! Capillary action is caused by surface tension between the liquid and a vertical solid surface! In water, adhesive forces between liquid molecules and surface are greater than cohesive forces between water molecules; adhesive forces cause water to attach itself to and climb solid vertical surface

11 FE Pressure Statics! Hydrostatic pressure is the pressure a fluid exerts on an immersed object or on container walls! Pressure is equal to the force per unit area of the surface: F P # A

Absolute pressure: measured relative to a

12 FE Statics Pressure! Gage pressure: measured relative to a reference pressure - typically local atmospheric pressure! Absolute pressure: measured relative to a perfect vacuum! Absolute, gage, and atmospheric pressure are related as follows: P abs = P gage + P atm

13 FE Pressure Statics P 1 gage P 1 abs P 2 abs P 2 gage Munson et al. (2002)

14 FE Hydrostatic Pressure Statics 'P P = change in pressure ( = specific weight of fluid 'h h = change in depth in fluid 'P P = ('h ***Incompressible fluid at rest

15 FE Statics Manometry! Manometers measure pressure or pressure differences! Differential manometers: both ends connected to pressure sources! Open manometers: one end open to the atmosphere

16 FE Solving Manometer Problems Statics 1. Select a convenient starting point (usually at one of the endpoints) 2. Using 'P P = (h, write expressions for changes in pressure from starting point to point at opposite end of manometer system. Watch algebraic signs! 3. Equate expression from step 2 to pressure at final point 4. Substitute known values and solve for desired quantity

17 FE Barometers Statics

18 FE Buoyancy Statics! The buoyant force exerted by a fluid is equal to the weight of the fluid displaced and is directed vertically upward (Archimedes Principle): F b = (VV d where F b = the buoyant force ( = specific weight of the fluid V d = displaced volume of the fluid

19 FE Displaced Volume Statics Displaced volume

20 FE Statics Solving Buoyancy Problems! If the object is at rest in the fluid, use the equation of static equilibrium in the vertical direction, )F y = 0! The buoyant force passes vertically through the centroid of the displaced volume of fluid. This point is called the center of buoyancy.

21 FE Forces on Surfaces Statics! Pressure on a horizontal plane surface is uniform over the surface! Resultant t force of pressure distribution ib ti acts through center of pressure of surface and is: R # PA R = resultant vertical force P = pressure on the horizontal surface A = area of submerged horizontal surface

22 FE Forces on Surfaces Statics

23 FE Free Surface Forces on Surfaces O * Statics h C h y y c x R df y R A R y " " I xc y R + Ay # P A # h A c avg ( c y da Centroid, c Center of Pressure, CP I xyc x R # + Ay # c c c x

24 FE Statics Laminar and Turbulent Flow Laminar Flow:! Relatively low velocities! No mixing or a very small degree of mixing! appears to flow in continuous layers with no interaction between the layers Turbulent Flow:! Relatively high velocities! High degree of mixing! motion appears chaotic

25 FE Flow Distribution Statics

26 FE Reynolds Number Statics

27 FE Statics Reynolds Number Circular Pipe Flow Re < 2000 laminar flow 2000 < Re < 4000 transition region Re > 4000 turbulent flow Open Channel Re < 500 laminar flow 500 < Re < 2000 transition region Re > 2000 turbulent flow

28 FE One-Dimensional Flows Statics

29 FE Compressible Flows Statics Continuity: m! # m! 1 m 2 $ A v # $ A v Ideal Gas Law: 2 P # $RT [P is pressure] [R is the gas constant] [T is temperature]

30 FE Statics Incompressible s Continuity Equation: A # v A v

31 FE Bernoulli Equation Statics

32 FE Mechanical Energy Equation Statics

33 FE Friction Loss Statics Valid for laminar and turbulent flow

34 FE Moody Chart Statics

35 FE Statics Friction Loss Hazen-Williams Equation: v # CR S 0.54 v = velocity [m/s] C = roughness coefficient [varies from 140 to 100] R = hydraulic radius = A/WP [m] S = slope of HGL = h L /L [dimensionless] Valid for turbulent t flow of water

36 FE Minor Loss h # h + L f h f,fitting Statics h f,fitting # C 2 v 2g [C = loss coefficient (or resistance coefficient)]

37 FE HGL and EGL Statics

38 FE Statics HGL and EGL Velocity Head (v 2 /2g) Pressure Head (P/() Total Head or Energy Grade Line (EGL) Hydraulic Grade Line HGL Elevation Head (z) Z = 0

39 FE Statics Pump-Turbines Net head added to system by mechanical device 2 1 P v P 1 + z + + h, h # 2 + z + 1 s L 2 ( 2g ( v 2 2 2g

40 FE Open Channel & Pipe Flow Statics

41 FE Impulse- Statics

42 FE Statics Impulse- ) F # Q $ v, Q $ v x 2 2 2x ) F # $, $ y z 1 1 1x Q v Q v 2 2 2y 1 1 1y ) F # Q $ v, Q $ v 2 2 2z 1 1 1z Sum of the external forces Net rate of momentum entering control volume

43 FE Drag Force Statics

44 FE Drag Coefficient Statics

45 FE Pipe Networks Statics

47 Good Luck!!! Steve Burian Department of Civil & Environmental Engineering

Steven Burian Civil & Environmental Engineering September 25, 2013

Fundamentals of Engineering (FE) Exam Mechanics Steven Burian Civil & Environmental Engineering September 25, 2013 s and FE Morning ( Mechanics) A. Flow measurement 7% of FE Morning B. properties Session