Review Of Basic Fluid Mechanics Hydromechanics VVR090 Hydrostatics Fluids at rest: • the pressure at any point in a fluid is the same in every direction • in a continuous fluid with constant density the pressure increases linearly with depth and the pressure is the same along horizontal planes dp = −γ = −ρg dz Constant density: p − pzzh11=−γ() − =γ 1 Pressure Definitions p Pressure head: h = γ (pressure may be expressed in height of a fluid column; e.g., mm Hg, m H2O) p Static head (piezometric head): + z γ p Hydrostatic pressure distribution: +=z const. γ Force on a plane area Total force: FhA= γ c Ic Center of pressure: yypc=+ yc A 2 Force on Curved Surfaces Look at horizontal (Fx) and vertical components (Fz) separately. Total force: 22 Ftot=+FF x z Flowing Fluid Basic equations for conservation of: • mass (continuity equation) • energy (involves potential and kinetic energy + work) • momentum (involves momentum fluxes + forces) Analysis through control volumes 3 Classification of flow types • 1-D, 2-D, and 3-D • real and ideal fluid • incompressible - compressible • steady – unsteady • laminar – turbulent • established – unestablished • uniform – non-uniform • subcritical – supercritical • subsonic - supersonic Continuity Equation 1-D, steady, compressible: ρ111AVAV=ρ 2 22 1-D, steady, incompressible: A11VAV= 2 2 stream tube 4 Energy Equation With energy losses (hL) and energy input (hM): ⎛⎞⎛⎞pV11 p 2 V 2 ⎜⎟⎜⎟++zh12 +M = ++ z ++ hL ⎝⎠⎝⎠γγ1222gg No losses: ⎛⎞pV ⎜⎟++z = const. Bernoulli’s equation ⎝⎠γ 2g Definitions in Fluid Flow • Pressure head (p/g) • Elevation head (z) • Static head (piezometric head) (p/g + z) • Velocity head (V2/2g) • Total (energy) head (p/g + z + V2/2g) • Hydraulic grade line Plane and parallell • Energy line streamlines p +=z const. γ 5 Energy and Hydraulic Grade Lines exit loss flow between reservoirs Momentum Equations Newton’s second law (vector relationship) ∑ FQVVx =ρ()21xx − ∑ FQVVyyy=ρ()21 − ∑ FQVVzzz=ρ()21 − 6 Pipe Flow Laminar – turbulent flow Characterized by Reynolds number: DVρ DV Re == μν Critical Re-value for transition to turbulence: 2000 (pipe flow) Head Loss in Pipes Wall shear stress L ULU22 hf== f L Dg242 Rg A R = (hydraulic P radius) Moody’s diagram 7 Minor Losses in Pipelines • expansion (e.g., exit) • contraction (e.g., entrance) • bends • fittings (e.g., valves) V 2 General expression: hK= LL2g Pipe Configurations Pipes in parallell: QQQ=+++123 Q ... hhLL====123 h L h L... Pipes in series: QQ===12 Q ... hhhLL=+++123 L h L... 8 Pumps and Turbines in the System Pumps supply the system with energy Turbines extract energy from the system Pumping Between Two Reservoirs Pump and system curve System curve: 2 HzCQP =Δ + PP 9.