Chapter 5 Flow Of An Incompressible Ideal Fluid
Chapter 51/85Flow of an Incompressible Ideal Fluid
Chapter 5 Flow of an Incompressible Ideal FluidContents5.1 Euler’s Equation5.2 Bernoulli’s Equation5.3 Bernoulli Equation for the One-Dimensional flow5.4 Application of Bernoulli’s Equation5.5 The Work-Energy Equation5.6 Euler’s Equation for Two-Dimensional Flow5.7 Bernoulli’s Equation for Two-Dimensional Flow Stream Function andVelocity Potential2/85
Chapter 5 Flow of an Incompressible Ideal FluidObjectives-Apply Newton’s 2nd law to derive equation of motion, Euler’s equation-Introduce the Bernoulli and work-energy equations, which permit us topredict pressures and velocities in a flow-field-Derive Bernoulli equation and more general work-energy equation basedon a control volume analysis3/85
Chapter 5 Flow of an Incompressible Ideal Fluid What is ideal fluid?-An ideal fluid is a fluid assumed to be inviscid.-In such a fluid there are no frictional effects between moving fluidlayers or between these layers and boundary walls.-There is no cause for eddy formation or energy dissipation due tofriction.-Thus, this motion is analogous to the motion of a solid body on africtionless plane.[Cf] real fluid – viscous fluid4/85
Chapter 5 Flow of an Incompressible Ideal Fluid Why we first deal with the flow of ideal fluid instead of real fluid?-Under the assumption of frictionless motion, equations areconsiderably simplified and more easily assimilated by thebeginner in the field.-These simplified equations allow solution of engineering problemsto accuracy entirely adequate for practical use in many cases.-The frictionless assumption gives good results in real situationswhere the actual effects of friction are small.[Ex] the lift on a wing5/85
Chapter 5 Flow of an Incompressible Ideal Fluid Incompressible fluid; ρ 0 (t , x, y, z ) constant density negligibly small changes of pressure and temperature thermodynamic effects are disregarded6/85
7/855.1 Euler's EquationgdW sin θ
8/855.1 Euler's EquationEuler (1750) first applied Newton's 2nd law to the motion of fluid particles.Consider a streamline and select a small cylindrical fluid system F maPressure forceGravitational force(i) dF pdA ( p dp )dA dW sin θ dp dA ρ gdAdsdzds dp dA ρ g dAdz(ii) dm ρ dAds (density volume)sin θ dzds
9/855.1 Euler's Equationa(iii) dV dV dsdV Vdtds dtdsdV dpdA ρ gdAdz ( ρ dsdA)VdsDividing by ρ dA gives the one-dimensional Euler's equationdpρ VdV gdz 0Divide by gdp 1 VdV dz 0γ g V2 dp0 d dz γ 2g d (V 2 ) 2V dV
5.1 Euler's EquationFor incompressible fluid flow, p V2 d z 0 γ 2g 1-D Euler's equation (Eq. of motion)10/85
11/855.2 Bernoulli's EquationFor incompressible fluid flow, integrating 1-D Euler's equation yieldsBernoulli equationV2z const. Hγ 2gp(5.1)where H total headBetween two points on the streamline, (5.1) givesV12p2 V22 z1 z2γ 2gγ 2gp pressure headγp1zkg m/s 2m2 potential head (elevation head), mV2 velocity head2g( m s) 2 mmskg m/s 2 mm3
12/855.2 Bernoulli's EquationHenri de Pitot(1695 1771)Pitot tubemanometer
13/855.2 Bernoulli's EquationBernoulli Family:JacobJohann - NikolausDaniel
5.3 Bernoulli Equation for the One-Dimensional flow14/85Bernoulli Eq. is valid for a single streamline or infinitesimal streamtube acrosswhich variation of p, V andz is negligible.This equation can also be applied to large stream tubes such as pipes, canals.Consider a cross section of large flow through which all streamlines areprecisely straight and parallel.
5.3 Bernoulli Equation for the One-Dimensional flowW15/85
5.3 Bernoulli Equation for the One-Dimensional flowi) Forces, normal to the streamlines, on the element of fluid are in equilibrium acceleration toward the boundary is zero. F 0( p1 p2 )ds γ h ds cos α 0 ( p1 p2 )ds γ ( z2 z1 )dsp1p z1 2 z2γγ the same result as that in Ch. 2α ( z2 z1 ) / hcos (2.6) p z quantity is constant over the flow cross section normal to theγ streamlines when they are straight and parallel. This is often called a hydrostatic pressure distribution( z pγ const. for fluid at rest).16/85
5.3 Bernoulli Equation for the One-Dimensional flow17/85ii) In ideal fluid flows, distribution of velocity over a cross section of a flowcontaining straight and parallel streamlines is uniform because of theabsence of friction. All fluid particles pass a given cross section at the same velocity, V(average velocity)V1 V2Combine (i) and (ii)V12p2 V22 z1 z2γ 2gγ 2gp1
5.3 Bernoulli Equation for the One-Dimensional flow18/85 Bernoulli equation can be extended from infinitesimal to the finitestreamtube. Total head H is the same for every streamline in the streamtube. Bernoulli equation of single streamline may be extended to apply to 2- and3-dimensional flows.[IP 5.1] p. 129Water is flowing through a section of cylindrical pipe.γ 9.8 103 N/m3pC 35 kPa,
5.3 Bernoulli Equation for the One-Dimensional flow19/85
5.3 Bernoulli Equation for the One-Dimensional flow[Sol]pAγ zA pBγ zB pCγ zC 1.2 p A pC γ ( zC z A ) 35 10 (9.8 10 ) 29.9 kPacos30 2 33 1.2 40.1 kPapB pC γ ( zC z B ) 35 103 (9.8 103 ) cos30 2 pC 35 103 3.57 m above point C . The hydraulic grade line is3γ 9.8 1020/85
5.4 Applications of Bernoulli's Equation Bernoulli's equationV2 z H const.γ 2gp where velocity is high, pressure is low. Torricelli's theorem (1643) special case of the Bernoulli equation.21/85
5.4 Applications of Bernoulli's Equation22/85
23/855.4 Applications of Bernoulli's EquationApply Bernoulli equation to points 1 and 2V12p2 V22 z1 z2γ 2gγ 2gp1p1 p 0V1 0 (for very large reservoir); atmV22 p2 z1 z2 2g γV22z1 z2 h γ 2gp2(a)
24/855.4 Applications of Bernoulli's EquationApply Newton's 2nd law in the vertical direction at section 2 F madF ( p dp )dA pdA γ dAdz dpdA γ dAdzdm ρ dAdza g dAdp γ dAdz ( ρ dAdz ) g dp γ dz γ dz dp 0 no pressure gradient across the jet at section 2. p p p p2ABC p A patm 0 (gage)(b)
5.4 Applications of Bernoulli's EquationThus, combining (a) and (b) givesV22h 2g V2 2 gh equal to solid body falling from rest through a height h .[IP 5.2] p.131 Flow in the pipeline for water intake25/85
5.4 Applications of Bernoulli's Equation26/85
5.4 Applications of Bernoulli's EquationFind: p1 , p2 , p3 , p4 and eleveation at point 6[Sol](i) Bernoulli's Eq. between & ⑤V02p5 V52 z0 z5γ 2gγ 2gp0p p patm 0, V 0050V52 90 60 2gV5 24.3 m/sCalculate Q using Eq. (4.4)πQ AV 24.3 (0.125) 2 0.3 m3 /s427/85
28/855.4 Applications of Bernoulli's Equation(ii) Apply Continuity equation, Eq. (4.5)2 125 AV Q AV V 1 15 51 V5 300 44V12 125 V52 125 30 ) 0.9 m ( 2 g 300 2 g 300 V1 0.9(2 9.8) 4.2 m/s V3 V444V22 125 V52 125 ( 30 ) 4.58 m,2 g 200 2 g 200 V 24.58(2 9.8) 9.5 m/s(4.5)Continuityequation
5.4 Applications of Bernoulli's Equation(iii) B. E. & ①V12of H2O head90 72γ 2gp 1 18 0.9 17.1 mof H2O headγwp13p 17.1(9.8 10 ) 167.5 kPa1(iv) B. E. & ②pp90 2 87 4.58 2 1.58 mγγ 15.48 103p2 1.58(9.8 10 ) 15.48 kPa 116 mmHg vacuum133.3 15.48 kPa below patm329/85
5.4 Applications of Bernoulli's Equation[Re]1 bar 1000 mb(millibar) 100 kPa 100 kN m 2 105 N m 2mmHg 101.325 kPa(105 pascal) 1013mb 29.92 in. Hgpatm 760 1 mmHg 133.3 Pa 133.3 N/m 2(v) B. E. & ③90 p3p3γ 0.9 78 12 0.9 11.1mγp3 108.8 kPa30/85
31/855.4 Applications of Bernoulli's Equation(vi) B. E. & ④p4γ 31 0.9 30.1mp4 295.0 kPa(vii) Velocity at the top of the trajectory V6 24.3 cos 30 21.0 m/s(5.2)Apply B. E. & ⑥21.02 El. 90 67.5m2g(5.3)
32/855.4 Applications of Bernoulli's EquationPoint 0Point 1Point 2Point 3Point 4Pressure, kPa0167.5-15.8108.7294.9Velocity, m/s04.224.614.224.22Elevation, m9072877859
5.4 Applications of Bernoulli's Equation33/85 CavitationAs velocity or potential head increase, the pressure within a flowing fluid drops. Pressure does not drop below the absolute zero of pressure.( patm 103 millibar 100 kPa pabs 0 pgage 100 kPa) Actually, in liquids the absolute pressure can drop only to the vapor pressureof the liquid.For water,pv
5.4 Applications of Bernoulli's EquationTemperaturepv10 1.23kPa15 1.70kPa20 2.34kPa34/85
5.4 Applications of Bernoulli's Equation35/85[IP 5.3] p.134 Cavitation at the throat of pipe constrictionpB 96.5 kPa barometric pressure.What diameter of constriction can be expected to produce incipientcavitation at the throat of the constriction?Water at 40 γ 9.73 kN/m3 ; pv 7.38 kPapvγ7.38 103 N/m 2 0.76 m339.73 10 N/mpB patm 96.5 103 N/m 2 9.92 m339.73 10 N/mγγ
5.4 Applications of Bernoulli's Equation36/85
5.4 Applications of Bernoulli's Equation(i) Bernoulli Eq. between ① and pc Vc2V12z1 zc γ 2gγ 2gp1V1 0, p1 pB , pc pvVc2 11 9.92 0 3 0.76 2gVc2 17.16 m Vc 18.35 m s2gIncipient cavitation37/85
5.4 Applications of Bernoulli's Equation(ii) Bernoulli Eq. between ① and ②V12p2 V22 z2 z1 γ 2gγ 2gp1V1 0, p1 p2 pBV2211 9.92 0 0 9.92 2gV2 14.69 m s38/85
5.4 Applications of Bernoulli's Equation(iii) Continuity between ② and Q A AcVc2V2π4(0.15) (14.69) 2π4d c2 (18.35) dc 0.134 m 134 mm[Cp] For incipient cavitation,critical gage pressure at point C ispcγ) gage patm pv (9.92 0.76) 9.16 mγ γ39/85
5.4 Applications of Bernoulli's Equation Bernoulli Equation in terms of pressure11p1 ρ V12 γ z1 p2ρ V22 γ z222p1 static pressure1ρ V12 dynamic pressure2γ z potential pressure Stagnation pressure, pSApply Bernoulli equation between 0 and S112p0 ρ V0 γ z0 pSρ VS2 γ zS2240/85
5.4 Applications of Bernoulli's Equation41/85
5.4 Applications of Bernoulli's Equation z0 zS ; VS 01p0 ρV02 pS 02V0 2( pS p0 )ρ[IP 5.4] p.136 Pitot-static tubeWhat is the velocity of the airstream, V0?ρ air 1.23kg m3 γ W 9810 N m3 2 V0 ( pS p0 ) ρa 1242/85
5.4 Applications of Bernoulli's Equation43/85
5.4 Applications of Bernoulli's EquationBy the way,p1 p2p1 pS 0.15ρ air g ; p2 p0 0.15γ w pS p0 0.15(γ w ρ air g ) 0.15(9,810 1.23 9.81) 1, 469.7 paV02 (1, 469.7) 48.9 m/s1.23[Cf] Ifγ air γ γwThen,pS p0 γh V0 2 gh44/85
5.4 Applications of Bernoulli's Equation45/85 Bernoulli principle for open flow-Flow over the spillway weir: a moving fluid surface in contact with theatmosphere and dominated by gravitational action-At the upstream of the weir, the streamlines are straight and paralleland velocity distribution is uniform.-At the chute way, Section 2, the streamlines are assumed straight andparallel, the pressures and velocities can be computed from the onedimensional assumption.
5.4 Applications of Bernoulli's Equation46/85[IP 5.6] p.139 Flow over a spillwayAt section 2, the water surface is at elevation 30.5 m and the 60 spillwayface is at elevation 30.0 m. The velocity at the water surface at section 2 is6.11 m/s.[Sol] 1mThickness of sheet flow (30.5 30) / cos60 Apply 1-D assumption across the streamline at section ②pw.s.p zw.s. b zbγγ pb γ ( zw. s. zb ) 9.8 103 (0.5) 4.9 kPa26.1Elevation of energy line H 30.5 32.4 m2g
5.4 Applications of Bernoulli's Equation47/85
5.4 Applications of Bernoulli's EquationApply B.E. between ② and p2γ 222bVp V z2 b zb2gγ 2gVelocity is the same at boththe surface and the bottom4.9 Vb2 30.0 Vb 6.11m s32.4 9.8 2 gq h2V2 1 6.11 6.11 m 2 s per meter of spillway length48/85
5.4 Applications of Bernoulli's EquationApply Bernoulli equation between ① and ②21 6.11 y1 29.0 32.4 2 g y1 y1 3.22 mV 1q 6.11 1.9 m sh1 3.22h1 y149/85
5.5 The Work-Energy Equation50/85For pipelines containing pumps and turbines, the mechanical work-energyequation can be derived via a control volume analysis. pump add energy to the fluid systemturbine extract energy from the fluid system Bernoulli equation mechanical work-energy equation for ideal fluid flow
5.5 The Work-Energy Equation51/85
52/855.5 The Work-Energy EquationApply mechanical work-energy principle to fluid flow work done on a fluid system is exactly balanced by the change in the sumof the kinetic energy ( KE ) and potential energy ( PE ) of the system.dW dE(1)where dW the increment of work done; dE resulting incremental changein energy Heat transfer and internal energy are neglected.[Cf] The first law of Thermodynamics Heat transfer and internal energy are included.
53/855.5 The Work-Energy EquationDividing (1) by dt yieldsdW dE dtdt(2)(i)Apply the Reynolds Transport Theorem to evaluate the rate of change ofan extensive property, in this case energy steady state form of the Reynolds Transport TheoremdE dt c . s .out i ρ v dA c . s .in i ρ v dA(3)where i energy per unit massPotential energyV2 i gz 2Kinetic energy(4)
54/855.5 The Work-Energy EquationSubstituting (4) into (3) givesdE dtwheredEdt V 2 V 2 c.s.out gz 2 ρ v dA c.s.in gz 2 ρ v dA(5) the rate of energy increase for the fluid system Even in steady flow, the fluid system energy can change with timebecause the system moves through the control volume where bothvelocity and elevation can change.Since the velocity vector is normal to the cross sectional area and thevelocity is uniform over the two cross sections, integration of RHS of (5)yields
55/855.5 The Work-Energy Equation dEV2 2 V12 ρ gz2 V2 A2 ρ gz1 V1 A1dt2 2 V2 V ρ g z2 V2 A2 ρ g z1 V1 A12g 2g 221(6)Continuity equation is Q V V1 A12 A2(7)Substituting the Continuity equation into (6) gives dEV2 2 V12 Qγ z2 z1 dt2g2g (5.4)
56/855.5 The Work-Energy Equation(ii)Now, evaluate the work done by the fluid system ( dW )1) Flow work done via fluid entering or leaving the control volume Pressure work p Area Distance2) Shaft work done by pump and turbine3) Shear work done by shearing forces action across the boundary of thesystem Wshear 0 for inviscid fluid Pressure work consider only pressure forces at the control surface, p1A1 and p2A2 Net pressure work rate pressure forceⅹdistance / time pressureforceⅹvelocity p1 AV1 1 p2 A2V2(8)
57/855.5 The Work-Energy Equation Shaft workWT 0 (energy is extracted from the system)W p 0 (energy is put in) Net shaft work rate Qγ EP Qγ ET(9)where EP ( ET ) work done per unit weight of fluid flowingCombining the two net-work-rate equations, Eqs. (8) and (9), yields p1 p2 Qγ E ENet work rate PT γγ (5.5) p1 p2 V2 2 V12 γ Qγ z2 zQEEPT 1 γγ2g 2 g (5.6)Equating Eqs. (5.4) and (5.5), we get
58/855.5 The Work-Energy EquationCollecting terms with like subscripts givesV12p2 V2 2 E P z2 ETz1 γ 2gγ 2gp1Head, m(5.7) Work-energy equation used in real fluid flow situations Work-energy W/O E p and ET is identical to the Bernoulli equation for idealfluid. Addition of mechanical energy ( E p ) or extraction ( ET ) cause abrupt risesof falls of energy line.
59/855.5 The Work-Energy Equation Power of machinesW work Force distance m g E ρ vol. g E vol. γ Power ttttimetime tKilowatts (kW) of machine γ QEP or ET1000Horsepower (hp) of machine γ Q 1 hp 0.746 kWEP or ET550 E γQ E (5.8a)(5.8b)
5.5 The Work-Energy Equation60/85[IP 5.7] p.145 Work done by pumpThe pump delivers a flowrate of 0.15 m3/s of water. How much power mustthe pump supply to the water to maintain gage readings of 250 mm ofmercury vacuum on the suction side of the pump and 275 kPa of pressureon the discharge side?[Sol]p1 250 mm of Hg 760 mmHg 250 133.3 N/m 2 33,325 N/m 2p1 33,325 3.39 mγ9800 p2 275 kPa 100 kPap2 275 103 28.1 mγ9800
5.5 The Work-Energy Equation61/85
5.5 The Work-Energy EquationApply Continuity Equation Q AV A2V21 10.15 4.8 m sπ2( 0.2 )4V124.82 1.16 m2 g 2 9.8V1 0.15 8.5 m sπ2( 0.15)42V28.52 3.68 m2 g 2 9.8V2 62/85
63/855.5 The Work-Energy EquationApply Work-Energy equation between ① & ②V12p2 V22 z1 E p z2 ETγ 2gγ 2gp1(5.7) 3.39 1.16 0 E 28.1 3.68 3p Ep 37.0 mQγ ( E p ) 0.15(9800)(37.0)Pump power 54.4 kW10001000(5.8b) The local velocity in the pump passage may be considerably largerthan the average velocity in the pipes. There is no assurance that the pump will run cavitation-free.
5.6 Euler's Equations for Two-Dimensional Flow64/85 Two-Dimensional Flow The solution of flowfield problems is much more complex than thesolution of 1D flow. Partial differential equations for the motion for real fluid are usuallysolved by computer-based numerical methods. present an introduction to certain essentials and practical problems Euler’s equations for a vertical two-dimensional flowfield may bederived by applying Newton's 2nd law of motion to differentialsystem dxdz. F ma
5.6 Euler's Equations for Two-Dimensional Flow65/85
5.6 Euler's Equations for Two-Dimensional FlowForce: pdFx dxdz x pdFz dxdz ρ gdxdz zAcceleration for steady flow: u u w x z w w az u w x z ax ux - direction: z - direction: ufor unsteady flow t p u udxdz ρ dxdz u w x z x p w w ρ dxdz u w dxdz ρ gdxdz z z x 66/85
5.6 Euler's Equations for Two-Dimensional FlowEuler's equation for 2-D flow 1 p u u u wρ x x z(5.9a) w w1 p u w gρ z x z(5.9b) Equation of Continuity for 2-D flow of ideal fluid u w 0 x zUnknowns:p, u , wEquations:3 simultaneous solution for non-linear PDE(4.11)67/85
5.7 Bernoulli's Equation for Two-Dimensional FlowBernoulli’s equation can be derived by integrating the Euler's equationsfor a uniform density flow. 1 p u u dx uw dx z ρ x x(a) 1dz ρ(b)(a) (b): p w w uwg dz z x z 1 p p u u w wdx dz udx wdx udz wdz gdz ρ x z x z x z68/85
5.7 Bernoulli's Equation for Two-Dimensional Flow u w w u u dx u dz wdx wdz x z x z w dwu du u w u u wdz u dz w dx wdx gdz x z z x w u (udz wdx )ξ(udz wdx ) x z 69/85
5.7 Bernoulli's Equation for Two-Dimensional FlowBy the way, p pdpdx dz x z u u dudx dz x z dw ξ w wdx dz x z w u x zd (u 2 ) 2u du u u u dx u dz22 x z70/85
5.7 Bernoulli's Equation for Two-Dimensional FlowIncorporating these terms and dividing by g givesdp γ11d (u 2 w2 ) (udz wdx )ξ dz2gg(c)Integrating (c) yieldspγ 1 21(u w2 ) z H ξ (udz wdx)2ggwhere H constant of integrationSubstituting resultant velocity, V2V u 2 w2(d)71/85
5.7 Bernoulli's Equation for Two-Dimensional FlowV21 z H ξ (udz wdx)γ 2ggp(5.10)(i) For irrotational (potential) flow ξ 0V2 z Hγ 2gp(5.11) Constant H is the same to all streamlines of the 2-D flowfield.(ii) For rotational flow ( ξ 0 ) : ξ (udz wdx) 0(5.12)However, along a streamline for steady flow,w dz udz wdx 0u dx(e)72/85
5.7 Bernoulli's Equation for Two-Dimensional Flow73/85Substituting (e) into (5.10) givesp V2 z Hγ 2g(5.13) H is different for each streamline.[Re]For ideal incompressible fluid, for larger flow through which all streamlinesare straight and parallel (irrotational flow) Bernoulli equation can be applied to any streamline.
5.8 Stream Function and Velocity PotentialThe concepts of the stream function and the velocity potential can beused for developing of differential equations for two-dimensional flow.5.8.1 Stream functionDefinition of the stream function is based on the continuity principle andthe concept of the streamline. provides a mathematical means of solving for two-dimensionalsteady flowfields.74/85
5.8 S
Chapter 5 Flow of an Incompressible Ideal Fluid Contents 5.1 Euler’s Equation. 5.2 Bernoulli’s Equation. 5.3 Bernoulli Equation for the One- Dimensional flow. 5.4 Application of Bernoulli’s Equation. 5.5 The Work-Energy Equation. 5.6 Euler’s Equation for Two- Dimensional Flow. 5.7 Bernoulli’s Equation for Two- Dimensional Flow Stream .
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .
About the husband’s secret. Dedication Epigraph Pandora Monday Chapter One Chapter Two Chapter Three Chapter Four Chapter Five Tuesday Chapter Six Chapter Seven. Chapter Eight Chapter Nine Chapter Ten Chapter Eleven Chapter Twelve Chapter Thirteen Chapter Fourteen Chapter Fifteen Chapter Sixteen Chapter Seventeen Chapter Eighteen
18.4 35 18.5 35 I Solutions to Applying the Concepts Questions II Answers to End-of-chapter Conceptual Questions Chapter 1 37 Chapter 2 38 Chapter 3 39 Chapter 4 40 Chapter 5 43 Chapter 6 45 Chapter 7 46 Chapter 8 47 Chapter 9 50 Chapter 10 52 Chapter 11 55 Chapter 12 56 Chapter 13 57 Chapter 14 61 Chapter 15 62 Chapter 16 63 Chapter 17 65 .
HUNTER. Special thanks to Kate Cary. Contents Cover Title Page Prologue Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter
Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 . Within was a room as familiar to her as her home back in Oparium. A large desk was situated i
This product provides a local flow indication and automatically signals the operator or PLC if flow is too high or too low. Uses of the Flow-Alert flow meter include: bearing lubrication, case drain verification, gun drill cooling, pump flow confirmation, etc. FLOW-ALERT FLOW SWITCH 6000 PSI / 414 BARS MAX GPM LPM OIL FIGURE 1- Flow-AlERt Flow .
