Fluid Mechanics Pp

HydraulicsBy: Engr. Yuri G. Melliza

FLUID MECHANICSProperties of Fluids1. Density ()2.Specific Volume ()3.Specific Weight ()

4. Specific Gravity or Relative DensityFor Liquids: Its specific gravity (relative density) is equal to the ratio of its density to that of water at standard temperature and pressure.For Gases:Its specific gravity (relative density) is equal to the ratio of its density to that of either air or hydrogen at some specified temperature and pressure.where: At standard conditionW = 1000 kg/m3W = 9.81 KN/m3

5. Temperature6. PressureIf a force dF acts on an infinitesimal area dA, the intensity of pressure is,where: F - normal force, KN A - area, m2

PASCAL’S LAW: At any point in a homogeneous fluid at rest the pressures are the same in all directions.yP3 A3AP1 A1xBCzP2 A2Eq. 3 to Eq. 1P1 = P3Eq. 4 to Eq. 2P2 = P3Therefore:P1 = P2 = P3Fx = 0 and Fy = 0P1A1 – P3A3 sin  = 0  1P2A2 – P3A3cos = 0  2From Figure:A1 = A3sin   3A2 = A3cos   4

Atmospheric pressure:The pressure exerted by the atmosphere.At sea level condition:Pa = 101.325 KPa = .101325 Mpa = 1.01325Bar = 760 mm Hg = 10.33 m H2O = 1.133 kg/cm2 = 14.7 psi = 29.921 in Hg = 33.878 ft H2OAbsolute and Gage PressureAbsolute Pressure:is the pressure measured referred to absolute zero and using absolute zero as the base.Gage Pressure:is the pressure measured referred to atmospheric pressure, and using atmospheric pressure as the base

PgageAtmospheric pressurePvacuumPabsPabsAbsolute ZeroPabs = Pa+ PgagePabs = Pa - Pvacuum

7. Viscosity: A property that determines the amount of its resistance to shearing stress.moving platevv+dvdxxvFixed plateS dv/dxS = (dv/dx)S = (v/x) = S/(v/x)where: - absolute or dynamic viscosity in Pa-secS - shearing stress in Pascalv - velocity in m/secx -distance in meters

8. Kinematic Viscosity: It is the ratio of the absolute or dynamic viscosity to mass density. = / m2/sec9. Elasticity: If the pressure is applied to a fluid, it contracts,ifthe pressure is released it expands, the elasticity of a fluid is related to the amount of deformat-ion (contraction or expansion) for a given pressure change. Quantitatively, the degree of elasticity is equal to;Ev = - dP/(dV/V)Where negative sign is used because dV/V is negative for a positive dP.Ev = dP/(d/) because -dV/V = d/ where:Ev - bulk modulus of elasticity, KPadV - is the incremental volume change V - is the original volumedP - is the incremental pressure change

rh10. Surface Tension: CapillarityWhere: - surface tension, N/m - specific weight of liquid, N/m3r – radius, mh – capillary rise, mSurface Tension of Water

FREE SURFACEh11•h2h2•Variation of Pressure with ElevationdP = - dhNote:Negative sign is used because pressure decreases as elevation increases and pressure increases as elevation decreases.

Pressure Head: where: p - pressure in KPa  - specific weight of a fluid, KN/m3h - pressure head in meters of fluidMANOMETERSManometer is an instrument used in measuring gage pressure in length of some liquid column.Open Type Manometer : It has an atmospheric surface and is capable in measuring gage pressure.Differential Type Manometer : It has no atmospheric surface and is capable in measuring differences of pressure.

Fluid AFluid AOpenFluid BManometer FluidManometer FluidOpen Type Manometer Differential Type Manometer

OpenOpenFluid AxyFluid BDetermination of S using a U - TubeSAx = SBy

SSSFree SurfaceMMhpF•C.G.•C.G.•C.P.•C.P.ypeNNForces Acting on Plane SurfacesF - total hydrostatic force exerted by the fluid on any plane surface MNC.G. - center of gravityC.P. - center of pressure

where:Ig- moment of inertia of any plane surface MN with respect to the axis at its centroidsSs - statical moment of inertia of any plane surface MN with respect to the axis SS not lying on its planee - perpendicular distance between CG and CP

Forces Acting on Curved SurfacesFVFree SurfaceDEVertical Projection of ABFC’LCCAC.G.FhC.P.BBB’

A = BC x LA - area of the vertical projection of AB, m2L - length of AB perpendicular to the screen, m V = AABCDEA x L, m3

Dh1 mDP = hTTFF1 mTtTHoop TensionF = 02T = FT = F/2  1S = T/AA = 1t  2

S = F/2(1t)  3From figure, on the vertical projection the pressure P;P = F/AA = 1DF = P(1D)  4substituting eq, 4 to eq. 3S = P(1D)/2(1t)where:S - Bursting Stress KPaP - pressure, KPaD -inside diameter, mt - thickness, m

Laws of BuoyancyAny body partly or wholly submerged in a liquid is subjected to a buoyant or upward force which is equal to the weight of the liquid displaced. 1.Wwhere:W - weight of body, kg, KNBF - buoyant force, kg, KN - specific weight, KN/m3 - density, kg/m3V - volume, m3Subscript: B - refers to the body L - refers to the liquid s - submerged portionVsBFW = BFW = BVBBF = LVsW = BFW = BVB KNBF = LVs KN

2.BFTVsWwhere:W - weight of body, kg, KNBF - buoyant force, kg, KNT - external force T, kg, KN - specific weight, KN/m3 - density, kg/m3V - volume, m3Subscript: B - refers to the body L - refers to the liquid s - submerged portionW = BF - TW = BVB KNBF = LVs KNW = BF - TW = BVBBF = LVs

TWBFVs3.where:W - weight of body, kg, KNBF - buoyant force, kg, KNT - external force T, kg, KN - specific weight, KN/m3 - density, kg/m3V - volume, m3Subscript: B - refers to the body L - refers to the liquid s - submerged portionW = BF + TW = BVB gBF = LVs gW = BF + TW = BVB gBF = LVs g

WTVsBF4.VB = VsW = BF + TW = BVB gBF = LVs gW = BF + TW = BVBBF = LVs

WVsBFT5.VB = VsW = BF - TW = BVB gBF = LVs gW = BF - TW = BVBBF = LVs

Energy and HeadBernoullis Energy equation:2HL = U - QZ21z1Reference Datum (Datum Line)

1. Without Energy head added or given up by the fluid (No work done by the system or on the system:2. With Energy head added to the Fluid: (Work done on the system)3. With Energy head added given up by the Fluid: (Work done by the system)Where: P – pressure, KPa - specific weight, KN/m3 v – velocity in m/sec g – gravitational acceleration Z – elevation, meters m/sec2 + if above datum H – head loss, meters - if below datum

APPLICATION OF THE BERNOULLI'S ENERGY THEOREMNozzleBaseTipQJetwhere: Cv - velocity coefficient

Venturi MeterB. Considering Head lossinlet1throatexit2Meter CoefficientManometerA. Without considering Head loss

Orifice: An orifice is an any opening with a closed perimeterWithout considering Head Lossand from figure: Z1 - Z2 = h, therefore1 ahVena ContractaBy applying Bernoulli's Energy theorem:Let v2 = vt2 awhere:vt - theoretical velocity, m/sech - head producing the flow, metersg - gravitational acceleration, m/sec2But P1 = P2 = Pa and v1is negligible, then

COEFFICIENT OF DISCHARGE(Cd)COEFFICIENT OF VELOCITY (Cv)COEFFICIENT OF CONTRACTION (Cc)where: v' - actual velocityvt - theoretical velocitya - area of jet at vena contractaA - area of orificeQ' - actual flowQ - theoretical flowCv - coefficient of velocityCc - coefficient of contractionCd - coefficient of discharge

Jet Trajctory2dv sinv13v cosR = v cos (2t)If the jet is flowing from a vertical orifice and the jet is initially horizontal where vx = v.v = vxyx

Upper ReservoirSuction GaugeDischarge GaugeLowerReservoirGate ValveGate ValvePUMPS: It is a steady-state, steady-flow machine in which mechanical work is added to the fluid in order to transport the liquid from one point to another point of higher pressure.

1. TOTAL DYNAMIC HEAD4. BRAKE or SHAFT POWERFUNDAMENTAL EQUATIONS2. DISCHARGE or CAPACITY Q = Asvs = Advdm3/sec3. WATER POWER or FLUID POWER WP = QHtKW

5. PUMP EFFICIENCY6. MOTOR EFFICIENCY7. COMBINED PUMP-MOTOR EFFICIENCY

8. MOTOR POWERFor Single Phase Motor

For 3 Phase Motorwhere: P - pressure in KPa T - brake torque, N-m v - velocity, m/sec N - no. of RPM - specific weight of liquid, KN/m3 WP - fluid power, KW Z - elevation, meters BP - brake power, KW g - gravitational acceleration, m/sec2 MP - power input to HL - total head loss, meters motor, KW E - energy, Volts I - current, amperes (cos) - power factor

PenstockHeadraceturbineY – Gross HeadTailraceHYDRO ELECTRIC POWER PLANTA. Impulse Type turbine (Pelton Type)12

1HeadraceGeneratorPenstockY – Gross HeadBZBDraft Tube2B – turbine inletTailraceB. Reaction Type turbine (Francis Type)

Fundamental EquationsWhere:PB – Pressure at turbine inlet, KPavB – velocity at inlet, m/secZB – turbine setting, m - specific weight of water, KN/m31. Net Effective HeadImpulse Type h = Y – HL Y = Z1 – Z2 Y – Gross Head, metersWhere: Z1 – head water elevation, m Z2 – tail water elevation, mB. Reaction Type h = Y – HL Y = Z1 –Z2

2. Water Power (Fluid Power) FP = Qh KWWhere: Q – discharge, m3/sec3. Brake or Shaft PowerWhere: T – Brake torque, N-m N – number of RPMWhere:eh – hydraulic efficiencyev – volumetric efficiencyem – mechanical efficiency4. Turbine Efficiency

5. Generator Efficency6. Generator SpeedWhere:N – speed, RPMf – frequency in cps or Hertzn – no. of generator poles (usually divisible by four)

Turbine-PumpPump-Storage Hydroelectric power plant: During power generation the turbine-pump acts as a turbine and during off-peak period it acts as a pump, pumping water from the lower pool (tailrace) back to the upper pool (headrace).

A 300 mm pipe is connected by a reducer to a 100 mm pipe. Points 1 and 2 are at the same elevation. The pressure at point 1 is 200 KPa. Q = 30 L/sec flowing from 1 to 2, and the energy lost between 1 and 2 is equivalent to 20 KPa. Compute the pressure at 2 if the liquid is oil with S = 0.80. (174.2 KPa)300 mm100 mm12

A venturi meter having a diameter of 150 mm at the throat is installed in a 300 mm water main. In a differential gage partly filled with mercury (the remainder of the tube being filled with water) and connected with the meter at the inlet and at the throat, what would be the difference in level of the mercury columns if the discharge is 150 L/sec? Neglect loss of head. (h=273 mm)

Fluid Mechanics Pp

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