Steven Burian Civil & Environmental Engineering September 25, 2013

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1 Fundamentals of Engineering (FE) Exam Mechanics Steven Burian Civil & Environmental Engineering September 25, 2013

2 s and FE Morning ( Mechanics) A. Flow measurement 7% of FE Morning B. properties Session C. statics D. impulse, and momentum equations E. Pipe and other internal flow Afternoon (Depends on Discipline) 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, 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 flow L. Flow through packed beds Up to 15% of FE Afternoon Session

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5 FE s Statics s s - substances in liquid or gas phase s cannot support shear; they deform continuously to minimize applied shear forces

6 FE s Statics

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8 FE s Statics Viscosity Shear stress (τ): force required to slide one unit area layer of a substance over another Viscosity (µ): 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

9 FE s Statics F/A is the fluid shear stress (τ) and the constant of proportionality is the absolute viscosity (µ): du τ = µ dy Newtonian fluids: strains are proportional to the applied shear stress Non-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: ν = µ ρ

10 FE s Statics

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12 FE s Statics Surface Tension skin that seems to form on free surface of a fluid; caused by intermolecular cohesive forces and is known as surface tension, σ Surface tension - tensile force between two points a unit distance apart on the surface

13 FE s Statics Capillarity Capillary action: caused by surface tension between liquid and a vertical solid surface Adhesive forces between liquid molecules and surface > cohesive forces between liquid molecules; in water, adhesive forces cause fluid to attach itself to and climb solid vertical surface

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15 FE s Pressure Statics Hydrostatic pressure: pressure of fluid on immersed object or container walls Pressure = force per unit area of surface: P = F A

16 FE s 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

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

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19 FE s Hydrostatic Pressure Statics ΔP = change in pressure γ = specific weight of fluid Δh = change in depth in fluid ΔP = γδh ***Incompressible fluid at rest

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21 FE s Statics Manometry Measure pressure or pressure differences Differential manometers: both ends connected to pressure sources Open manometers: one end open to the atmosphere

22 FE s Barometers Statics

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24 FE s Buoyancy Statics Buoyant force = weight of fluid displaced and is directed vertically upward (Archimedes Principle): F b = γv d where F b = buoyant force γ = specific weight of fluid V d = displaced volume of fluid

25 FE s Displaced Volume Statics Displaced volume

26 FE s Statics Solving Buoyancy Problems If object at rest in fluid, then use equation of static equilibrium in vertical direction, ΣF y = 0 Buoyant force passes vertically through centroid of displaced volume; called the center of buoyancy.

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28 FE s Forces on Surfaces Statics Pressure on horizontal plane is uniform over surface Resultant force of pressure distribution acts through center of pressure of surface and is: R = PA R = resultant vertical force P = pressure on horizontal surface A = area of submerged horizontal surface

29 FE s Forces on Surfaces Statics

30 FE s Statics Forces on Surfaces Free Surface O θ h C y h y c x R df y R A R = P A = γh A R c avg c y I da Centroid, c Center of Pressure, CP xc xyc y = + y x R = + xc Ayc Ayc I

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32 FE s 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

33 FE s Flow Distribution Statics

34 FE s Reynolds Number Statics

35 FE s 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

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38 FE s One-Dimensional Flows Statics

39 FE s Bernoulli Equation Statics

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43 FE s Mechanical Energy Equation Statics

44 FE s Friction Loss Statics Valid for laminar and turbulent flow

45 FE s Moody Chart Statics

46 FE s Minor Loss Statics

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49 FE s HGL and EGL Statics

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

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52 FE s Statics Pump-Turbines Net head added to system by mechanical device 2 P v P 1 + z h h = 2 + z + 1 s L 2 γ 2g γ v 2 2 2g

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54 FE s Open Channel & Pipe Flow Statics

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56 FE s Impulse- Statics

57 FE s Impulse- Statics Σ F x = Q 2 ρ 2 v 2x Q 1 ρ 1 v 1x Σ F y = Q 2 ρ 2 v 2 y Q 1 ρ 1 v 1y Σ F z = Q 2 ρ 2 v 2z Q 1 ρ 1 v 1z Sum of the external forces Net rate of momentum entering control volume

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59 FE s Pipe Networks Statics

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

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