VacTran 3 Manual (779748), страница 35
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In otherwords,see also:Annulus calculationsBend calculationsConical pipe calculationsEllipse calculationsRectangle calculationsTriangle calculations© 2011 Professional Engineering Computations336VacTran 318.3.5.2 Bend calculationsFor molecular flowRoth eq 3.130 equivalent lengthL = 2 * P * Bend Radius * Bend Angle /360+1.33 * Bend Angle /180 * DiameterFor viscous flowEntrance K factor = 0 (already a randomizer)Viscous entrance equivalent length = 0{Crane Pipe Bend Equation from page A29}Bend K Factor is a lookup value from handbook tables, based on the bend radius/ inside diameterratio for the bend.Body Equivalent Length = Diameter * Body K FactorCalculate Friction FactorExit K factor = 1 (exit loss)Net K Factor = Entrance K + Body K Factor + Exit KVolume calculationV=* (Diameter)2/4*2*Pi* Bend Radius * Bend Angle/360/1000© 2011 Professional Engineering ComputationsCalculation Formulas18.3.5.3 Rectangle calculationsMolecular flow conductance: rectangleLafferty Rectangle transmission probability equation 2.27L = cma b sides = cmMolecular flow equivalent diameter: rectangleUsing circular pipe transmission probability formula4d3LEquating the two formulas and solving for d4d3L4d3ddd316a3316 abln 43 La2ln 423 16a34 34a32ln 42ln 4baba3a4b3a4bba3a4b3a4b0.7183485a ln 4ba3a4b© 2011 Professional Engineering Computations337338VacTran 3Viscous flow conductance: rectangleLafferty Equation 2.86P = dyne/cm2L = cma b sides = cm= poiseC = cm3/secondViscous flow equivalent diameter: rectangleUsing circular pipe formula4CDPL128Equating the two formulas and solving for DC128128DDD4PL112D4112 La2a 3b 3b 2 0.371aba 3b 31412a2a 3b 3Pb 2 0.371aba2b20.371ab1284 3.3953a 3b 3a2see also:Annulus calculationsBend calculationsConical pipe calculationsElbow and miter calculationsb20.371abEllipse calculationsRectangle calculationsTriangle calculations© 2011 Professional Engineering ComputationsCalculation Formulas33918.3.5.4 Rectangular pipe efficiencyFor long pipes, circular cross sections are the most efficient shape.
In other words, for a given pipe cross sectionarea, a circular cross section has the highest conductance in any flow regime. For the purpose of comparing othercross section shapes, circular pipe is given an efficiency of 100%.For a given cross section area, a circular pipe is more efficient than a rectangular pipe.
At best, a rectangular pipeis about 98% efficient in molecular flow when it has a 1:1 (square) aspect ratio. In other words, it has the sameconductance as a circular pipe that is 2% smaller in area. In viscous flow, this same square pipe is about 94%efficient.The efficiency of the rectangle is further reduced as its aspect ratio decreases from 1:1. For example, at an aspectratio of 1:100, its efficiency is less than 0.1%.The following graph (not generated by VacTran) illustrates the efficiency of a rectangular pipe, assuming no entranceor exit affects. The equations used in the calculation are shown in Rectangle calculations.Rectangular pipe efficiency10.90.8Ratio of sides a/b0.70.6Viscous flow0.5Molecular flow0.40.30.20.100%20%40%60%80%100%Area efficiency relative to round pipe© 2011 Professional Engineering Computations120%340VacTran 318.3.5.5 Annulus calculationsMolecular flow conductance: annulusBerman Equation 6.11am = molecular weightL = cmT = KelvinC = liters/secondK0 from tabled2d100.259 0.500 0.707 0.866 0.966K011.072 1.154 1.254 1.430 1.675Molecular flow equivalent diameter: annulusUsing circular pipe conductance formula(liters/second)Equating the two formulas and solving for dCDD33.81K03K0TmD3L3.81 K 0d2 2 d12d2 d12d2 2 d12d2 d12Tm2d2 2 d12d2 d1 L© 2011 Professional Engineering ComputationsCalculation FormulasViscous flow conductance: annulusLafferty Equation 2.87P = dyne/cm2L = cmd = cm= poiseC = cm3/secondViscous flow equivalent diameter: annulusUsing circular pipe formula4C128DPLEquating the two formulas and solving for DCD1284d2 4D4PLd14128 Ld2 2d12dln 2d1see also:Bend calculationsConical pipe calculationsElbow and miter calculationsEllipse calculationsRectangle calculationsTriangle calculations© 2011 Professional Engineering Computationsd2 42d14d2 2d12dln 2d12P341342VacTran 318.3.5.6 Annular pipe efficiencyFor long pipes, circular cross sections are the most efficient shape.
In other words, for a given pipe cross sectionarea, a circular cross section has the highest conductance in any flow regime. For the purpose of comparing othercross section shapes, circular pipe is given an efficiency of 100%.When an annular pipe inner diameter approaches the outer diameter, the efficiency approaches zero. When theinner diameter approaches zero, the efficiency approaches that of a circular pipe.The following graph (not generated by VacTran) illustrates the efficiency of an annular pipe, assuming no entrance orexit affects. The equations used in the calculation are shown in Annulus calculations.Annular pipe efficiency10.90.8Diameter ratio D1/D20.70.6Viscous flow0.5Molecular flow0.40.30.20.100%20%40%60%80%100%Area efficiency relative to round pipe© 2011 Professional Engineering ComputationsCalculation Formulas18.3.5.7 Triangle calculationsMolecular flow conductance: TriangleBerman Equation 6.15aC1 .5Tma3Lm = molecular weightL = cmT = Kelvina = cm (side of equilateral triangle)C = liters/secondMolecular flow equivalent diameter: TriangleFor a long circular cross section pipeC3.81TmD3LEquating the two formulas and solving for DC3.813.81D3D3TmD3L1 .51.5 a 30.3837008a 3© 2011 Professional Engineering ComputationsTma3L343344VacTran 3Viscous flow conductance: TriangleBerman equation 5.23, Page 114P = TorrL = cmA = cm (side of equilateral triangle)C = liters/secondViscous flow equivalent diameter: TriangleBerman equation 5.12c, Page 109C 184D4PLEquating the two formulas and solving for DC 184D4PL184 D 432.2 A 4D432.2A4PL0.175 A 4see also:Annulus calculationsBend calculationsConical pipe calculationsElbow and miter calculationsEllipse calculationsRectangle calculationsTriangle calculations© 2011 Professional Engineering ComputationsCalculation Formulas34518.3.5.8 Triangular pipe efficiencyFor long pipes, circular cross sections are the most efficient shape.
In other words, for a given pipe cross sectionarea, a circular cross section has the highest conductance in any flow regime. For the purpose of comparing othercross section shapes, circular pipe is given an efficiency of 100%.For the case of a triangular pipe with equal length sides, the efficiency is constant with size for viscous andmolecular flow.For viscous flow, triangular pipe efficiency is about 76%.For molecular flow, efficiency is about 96%.© 2011 Professional Engineering Computations346VacTran 318.3.5.9 Ellipse calculationsMolecular flow conductance: ellipseBerman Equation 6.12m = molecular weightL = cmT = Kelvina = semimajor axis cmb = semiminor axis cmC = liters/secondMolecular flow equivalent diameter: ellipseUsing circular pipe formulaCD3LTm3.81Equating the two formulas and solving for DC3.813.81 DDD333D3LTm43.143.13.8111.31243.1LTma 2b 2a2b2a 2b 2a2 b2a 2b 2a2 b2a 2b2a2 b2© 2011 Professional Engineering ComputationsCalculation FormulasViscous flow conductance: ellipseBerman equation 5.21P = dyne/cm2L = cma = semimajor axis cmb = semiminor axis cm= poiseC = cm3/secondViscous flow equivalent diameter: ellipseUsing circular pipe formulaC128D4PLEquating the two formulas and solving for DCDD128D4PLPa 3b 34 L a2 b2128 a 3b 34 a2 b244a 3b 332 2a b2see also:Annulus calculationsBend calculationsConical pipe calculationsElbow and miter calculationsEllipse calculationsRectangle calculationsTriangle calculations© 2011 Professional Engineering Computations347348VacTran 318.3.5.10 Elliptical pipe efficiencyFor long pipes, circular cross sections are the most efficient shape.
In other words, for a given pipe cross sectionarea, a circular cross section has the highest conductance in any flow regime. For the purpose of comparing othercross section shapes, circular pipe is given an efficiency of 100%.When an elliptical pipe minor diameter approaches the major diameter, the efficiency approaches 100%. When theminor diameter approaches zero, the efficiency approaches zero.The following graph (not generated by VacTran) illustrates the efficiency of elliptical pipe ratios, assuming noentrance or exit affects. The equations used in the calculation are shown in Ellipse calculations.Elliptical pipe efficiency10.90.8Diameter ratio a/b0.70.6Viscous flow0.5Molecular flow0.40.30.20.100%20%40%60%80%100%Area efficiency relative to round pipe© 2011 Professional Engineering ComputationsCalculation Formulas18.3.5.11 Comparison of long pipe equationsLong pipe equations for different cross sections are very similar to each other.Molecular flowGeneralformC ge ne r alViscous flowTAM3.64C K ge om e tr yPL:transmission probabilityA: cross section areaCircularpipeC128D4PLcm3/secondliters/second4d3Lliters/secondRectanglecm3/secondAnnulusliters/secondcm3/secondTriangleC1 .5Tma3Lliters/secondliters/secondEllipseliters/second© 2011 Professional Engineering Computationscm3/second349350VacTran 318.3.5.12 Conical pipe calculationsAlternate terms:Conical pipeTube conicalreducerPipe reducerConePipe contractionPipe expansionCaseentrance diameter > exit diameterentrance diameter >> exit diameterentrance diameter < exit diameterentrance diameter << exit diameterFor viscous flowCrane's flow of fluids page A-26d1 = smaller diameterd2 = larger diameterResultreducerequivalent to a pipe entrance (sharp edge), K= 0.5expanderequivalent to a pipe exit, K = 1K factors are used to calculate the equivalent length of pipe in viscous flow.
As with other conductance elements,total equivalent length in viscous flow is a combination of entrance, exit, and body equivalent lengths. However,cones are somewhat special. When the cone angle approaches zero, the cone behaves like a straight pipe, so thebody length dominates the flow loss. When the cone angle approaches 180 degrees, the cone behaves as either apipe entrance or a pipe exit depending on whether it is contracting or expanding relative to the gas flow direction.Therefore, the total equivalent length of pipe is comprised of the cone length (using the smaller diameter) and theexit or entrance equivalent length as described in the next sections.A cone element is intended to be modeled in series with other conductance elements having matching entrance andexit diameters. Be careful to avoid redundant entrance and exit loss settings on adjacent conductance elements.For example:If a cone that is a pipe expansion is downstream from a pipe in series, don't add an exit loss for the upstreampipe.
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