Asme mfc 7m 1987 (2001) scan (american society of mechanical engineers)

Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles ASMEIANSI MFC-7M- 1987 ~ ~ - ~ -~ REAFFIRMED 1992 FOR CURRENT COMMITTEE PERSONNEL PLEASE SEE ASME MANUAL AS-I REAFFIRMED 2001 FOR CURRENT COMMITTEE PERSONNEL PLEASE E-MAIL CS@asme.org S P O N S O R E DA N DP U B L I S H E DB Y T H EA M E R I C A NS O C I E T Y United Engineering Center 345 OF M E C H A N I C A LE N G I N E E R S E a s t 47th Street N e w York, N Y 10017 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled wh A AN M E R I C A N A T I O N A SL T A N D A R D This Standard will be revised when the Society approves the issuance of a new edition There will be no addenda or written interpretations of the requirements of this Standard issued to this edition This code or standard was developed under procedures accredited as meeting the criteria for American National Standards The Consensus Committee that approved thecode or standard was balanced t o assure that individuals from competent and concerned interestshave had an opportunity to participate The proposed code or standard was made available for public review and comment whichprovidesanopportunityforadditionalpublicinputfromindustry, academia, regulatory agencies, and the public-at-large ASME does not "approve," "rate," or "endorse" any item, construction, proprietary device, or activity ASME does not take any position with respect to the validity of any patent rights asserted in connection with any items mentioned in this document, and does not undertake to insure anyone utilizing a standardagainst liability for infringement of any applicable Letters Patent, nor assume any such liability Users of a codeor standard are expressly advised that determinationof the validity of any such patent rights, and the risk of infringement of such rights, is entirely their own responsibility Participation by federal agency representative(s) or person(s) affiliated with industry is not to be interpreted as government or industry endorsement of this code or standard ASME accepts responsibility for only those interpretations issued in accordance with governing ASMEproceduresandpolicieswhichprecludetheissuanceofinterpretations byindividual volunteers No part of this document may be reproduced in any form, in an electronic retrieval system or otherwise, without the prior written permission of thepublisher Copyright 1987 by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS All Rights Reserved Printed in U.S.A Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w Date of Issuance: May 31, 1987 (This Foreword is not part of ASME/ANSI MFC-7M-1987.) This Standard was prepared by Subcommittee 2, Working Group , of the American Society of Mechanical Engineers Committee on Measurement of Fluid Flow in Closed Conduits The Committee is indebted to the many engineers who contributed to this work This Standard is intended to assist the public with the use ofcritical flow nozzles.Critical flow nozzles are especially suited to flow calibration work and precise flowcontrol applications They provide a stableflow of a compressible fluid through aclosed conduit, the rate of which may be determined with a high degree of accuracy The Committee has attempted to blend the best available technical information with common practice to develop this Standard It is as complete a specification as the Committee determined appropriate Some latitude and variation on the application of the Standard tocritical flow venturi nozzles is allowed However, neither these liberties nor this Standard is intended to replace proper judgment in the application of critical flow venturi nozzles This Standard was approved by the American National Standards Institute (ANSI) on February 27, 1987 111 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled wh FOREWORD (The following is the roster of the Committee at the time of approval of this Standard.) OFFICERS R W Miller, Chairman W F Lee, Vice Chairman C J Gomez, Secrerary COMMITTEE PERSONNEL R B Abernethy, Pratt & Whitney Aircraft, West Palm Beach, Florida N A Alston, Diederich Standard Corp., Boulder, Colorado H P Bean, El Paso, Texas S R Beitler, The Ohio State University, Columbus, Ohio M Bradner, Foxboro, Massachusetts E E Buxton, St Albans, West Virginia J S Castorina, Naval Ship System Engineering Station, Philadelphia, Pennsylvania G.P Corpron, Rosemount Inc., Eden Prairie, Minnesota C F Cusick, Philadelphia, Pennsylvania D G Darby, D o w Chemical Co., Lake Jackson, Texas R B Dowdell, University of Rhode Island, Kingston, Rhode Island A G Ferron, Alden Research Lab, Holden, Massachusetts R L Galley, Antioch, California D Halmi, Primary Flow Signal Inc., Pawtucket, Rhode Island B T Jeffries, Ponca City, Oklahoma E H Jones, Jr., Chevron Oil Field Res Co., La Habra, California L J Kemp, Palos Verdes Estate, California C A Kemper, Kaye Instruments Inc., Bedford, Massachusetts D R Keyser, NADC, Aero-Mechanical Branch, Warminster, Pennsylvania C P Kittredge, Princeton, New Jersey C G Langford, E I DuPont de Nemours and Co., Wilmington, Delaware E D Mannherz, Fischer & Porter Co., Warminster, Pennsylvania G E Mattingly, National Bureau of Standards, Gaithersburg, Maryland R V Moorse, Union Carbide Corp., Tonawanda, New York M H November, Hacienda Heights, California B D Powell, Pratt & Whitney Aircraft Group, West Palm Beach, Florida W M Reese, Jr., Arlington, Texas P G Scott, The Foxboro Co., Foxboro, Massachusetts H E Snider, A W W A Standards Committee, Kansas City, Missouri D A Sullivan, Southeastern Massachusetts University, Sandwich, Massachusetts R G Teyssandier, Daniel Industries Inc., Houston, Texas V Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w ASME STANDARDS COMMITTEE MFFCC Measurement of Fluid Flow in Closed Conduits W M Reese, Jr., Chairman, Arlington, Texas H P.Bean, El Paso, Texas S R Beitler, The Ohio State University, Columbus, Ohio M Bradner, The Foxboro Co., Foxboro, Massachusetts E E Buxton, St Albans, West Virginia C F Cusick, Philadelphia, Pennsylvania P J Disimile, University of Cincinnati, Cincinnati, Ohio R B Dowdell, University of Rhode Island, Kingston, Rhode Island W A Fling, Jr., Cities Service Oil and Gas Corp., Tulsa, Oklahoma G B Golden, Houston Lighting and Power, Bacliff, Texas D Halmi, Primary Flow Signal Inc., Pawtucket, Rhode Island G E Mattingly, National Bureau of Standards, Gaithersburg, Maryland R M Reimer, General Electric Co., Cincinnati, Ohio R G Teyssandier, Daniel industries, Inc., Houston, Texas ' Subcommittee 2, Working Group - Critical Flow Measurement E H Jones, Chairman, Chevron Oil Field Services Company, La Habra, California D Halmi, Primah Flow Signal Inc., Pawtucket, Rhode Island R M Reimer, General Electric Co., Cincinnati, Ohio R E Smith, J?: Sverdrup Tech Inc., Arnold Air Force Station, Tennessee R G Teyssandier, Daniel Industries, Inc., Houston, Texas C R Varner, Vernon, Connecticut vi Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w SUBCOMMITTEE - PRESSURE DIFFERENTIAL DEVICES Foreword Standards Committee Roster 2.1 2.2 Symbols Definitions 1 3.1 3.2 3.3 State Equation Flow Rate in Ideal Conditions Flow Rate in Real Conditions Scope and Field Symbols and BasicEquations of Application Definitions Applications For Which the Method is Suitable Standard Critical Flow Venturi Nozzles 5.1 5.2 General Upstream Pipeline Large Upstream Space Downstream Requirements Pressure Measurement Drain Holes Temperature Measurement Density Measurement 8 8 8 10 10 10 Method of MassFlow Rate Computation Discharge Coefficient Computation of Real Gas Critical Flow Function Conversion of Measured Pressure and Temperature to Stagnation Conditions Maximum Permissible Downstream Pressure Uncertainties in the Measurement of Flow Rate Figures 6 Calculation Methods 7.1 7.2 7.3 7.4 5 General Requirements Standard Venturi Nozzles Installation Requirements 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 7.5 v 111 Toroidal Throat Venturi Nozzle Cylindrical Throat Venturi Nozzle Installation Requirements foran Upstream Pipework Configuration vii 10 10 10 11 11 11 11 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w CONTENTS Detail of Pressure Tap Maximum Permissible Back Ratio for Critical Flow Venturi Nozzles Table Symbols 12 Venturi Nozzle Discharge Coefficients References from Which Standard Critical Flow Venturi Nozzle Discharge Coefficients Were Obtained C Example Flow Calculation D Critical Flow Functions EThe Critical Flow Coefficient 13 Appendices A B Figure C1 Sectional View of the Nozzle andPipe Tables A1 A2 A3 El E2 E3 E4 E5 E6 ToroidalThroat Venturi Nozzle Discharge Coefficient Cylindrical Throat Venturi Nozzle Discharge Coefficient Comparison of Theoretical and Experimental Discharge Coefficients fortheToroidalThroat Nozzle Table of Fluids for Various Equations of State Critical Flow Coefficient for Nitrogen Critical Flow Coefficient for Oxygen Critical Flow Coefficient forArgon Critical Flow Coefficient for Methane Critical Flow Coefficient forCarbon Dioxide viii 15 17 25 29 17 13 13 13 32 33 33 33 34 34 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled whe MEASUREMENT OF GAS FLOWBY MEANS OF CRITICAL FLOW VENTURI NOZZLES SCOPE AND FIELD OF APPLICATION 2.2 Definitions This Standard applies only to the steady flow of single-phase gasesand deals withdevices for which direct calibration experiments have been made, sufficient in number and quantity to enable inherent systems of applicationsto be basedon their resultsand coefficients to be given with certain predictable limits of uncertainty The critical flow venturi nozzles dealt with can only beused within limitsthat arespecified, for example nozzle throat to inlet diameter ratio and Reynolds number This Standardspecifies the geometry and method of use (installation and operating conditions) of critical flow venturi nozzles inserted in a system to determine systhe mass flow rate of the gas flowing through the tem It also gives necessary information for calculating the flow rate and its associated uncertainty This Standard applies only to venturi nozzles in which the flow is critical Critical flow exists whenthe mass flow rate through the venturi nozzle is the maximum possible for theexisting upstream conditions At criticalflow or choked conditions, the average gas velocity at the nozzle throat closely approximates the local sonic velocity Information is given in this Standard for cases in which: (a) the pipeline upstream of the venturi nozzle is of circular cross section; or (b) it can be assumed that there is a large space upstream of the venturi nozzle The venturi nozzles specified in this Standard are called primary devices Other instruments for the measurement are known as secondary devices This Standard covers primary devices; secondary devices will be mentioned only occasionally 2.2.1 Pressure Measurement wall pressure tap - hole drilled in the wall of a conduit, the inside edge of which is flush with the inside surface of the conduit static pressure of a gas - the actual pressure of the flowing gas, which can be measured by connecting a pressure gauge to a wall pressure tap Only the value of the absolute static pressure is used in this Standard stagnation pressure of a gas - pressure that would exist in the gasif the flowing gas stream were brought to rest by an isentropic process Only the value of the absolute stagnation pressure is used in this Standard 2.2.2 Temperature Measurement static temperature of a gas - actual temperature of the flowing gas Only the value of the absolute static temperature is used in this Standard stagnation temperature of a gas - temperature that would exist in the gas if the flowing gas stream were brought torest byan adiabaticprocess Onlythe value of the absolute stagnation temperature is used in this Standard 2.2.3 Critical Flow Nozzles venturi nozzle - a convergent divergent restriction inserted in a system intended for the measurement of flow rate throat - the minimum diametersection of the venturi nozzle critical venturi nozzle - a venturi nozzle for which the nozzle geometrical configuration and conditions of use are such that the flow rateis critical 2.2.4 Flow massflow rate - the mass of gasper unit time passing through the venturi nozzle In this Standard, flow rate is always the steady-state or equilibrium mass flow rate throat Reynolds number - In this Standard thenozzle throat Reynolds numberis calculated from thegas velocity, density at thenozzle throat, andgas viscosity SYMBOLS AND DEFINITIONS 2.1 Symbols Thesymbols used in thisStandardare Table listedin Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled wh ASME/ANSI MFC-7M-1987 TABLE Description SYMBOLS Dimensions [Note (111 Symbol SI (Metric) Unit US (Customary) Unit A* Area of venturi nozzle throat L2 m2 in.’ A2 Area of venturi nozzle exit L2 m2 in.’ B Bias c Coefficient of discharge Dimensionless CRi Critical flow function for onedimensional isentropic flow of a real gas Dimensionless C *; (Critical flow function foronedimensional isentropic flow of a perfect gas Dimensionless Real gas critical flow coefficient for one-dimensional real gas flow Dimensionless D Diameter of upstream conduit L m in d Diameter of venturi nozzle throat L m in e Relative uncertainty Dimensionless h Specific enthalpy of thegas L2 T-’ J/kg BTU/lbm M Molecular mass M kg/kgmole Ibm/lbm-mole Ma Mach number Dimensionless Pl Absolute static pressure of the gas at the nozzle inlet ML-’T-’ Pa Ibf/in.* p2 Absolute static pressure of the gas at nozzle exit ML-’T-~ Pa Ibf/in.’ PO Absolute stagnation pressure of the gas at nozzle inlet ML- T - ~ Pa Ibf/in.’ P* Absolute static pressure of the gas at nozzle throat ML-’ T - ~ Pa Ibf/in.’ P*; Absolute static pressure of the gas at nozzle throat for onedimensional isentropic flow of a perfect gas ML-’T-= Pa Ibf/in.* Ratio of nozzle exit static pressure t o stagnation pressure for onedimensional isentropic flow of a perfect gas Dimensionless Qm Mass flow rate MT- Ibm/sec 9m; Mass flow rate for one-dimensional isentropic flow MT-’ Ibm/sec Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled wh MEASUREMENT OF GAS FLOW BY MEANS OF CRITICAL FLOW VENTURI NOZZLES ASME/ANSI MFC-7M-1987 TABLE Description ASMElANSl MFC-7M-1987 SYMBOLS(CONT’D) Dimensions [Note Unit (111 Symbol Red Nozzle throat Reynolds number Dimensionless ‘C Radius curvature ofof L r* Critical pressure ratio P*/Po S Specific entropy theof nozzle inlet gas temperature Absolute static at nozzle throat T* Two-tailed Student‘s t s5 t uRSS, Uncertainty the at level 95% confidence u99 Uncertainty the at level 99Y0confidence TO Absolute stagnation temperature of the gas V* Throat sonic velocity flow u95 UADD L2T-2e-’ OR e K OR Z Compressibility factor Dimensionless ZO Compressibility factor at CY Temperature probe constant Dimensionless d/D Dimensionless heatsspecific Y of The ratio in BTUllbm-OR ft/sec LT To and Po BTU Ibm-mole-OR e V velocity fluidAverage of m US (Customary) Dimensionless -’ LT -’ P J L2T-2e-’ kg-mole-K Universal gas constant R SI (Metric) Unit ft/sec Dimensionless Ratio Dimensionless x exponent Isentropic Dimensionless P* Dynamic viscosity nozzle throat PO ML-’ T - ’ Pass Ibm/ft-sec Dynamic viscosity theof gas at stagnation conditions M L - ‘ 7-’ Pa-s Ibm/ft-sec eo Gas stagnation density condiat tions at nozzle inlet ML- kg~m-~ Ibm/ft3 e* Gas at density ML- kg.m-3 Ibm/ft3 deviation Standard of the gas at nozzle throat Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w MEASUREMENT OF GAS FLOW BY MEANS OF CRITICAL FLOW VENTURI NOZZLES The equation for the discharge coefficient Cis given in para.7.2.2 Using this expression for Cand that for Reynolds the number at the nozzle throat theflow rate may beexpressed as follows: - 1.525 ()-'/'I CR -4 PO This equation maybe solved by first assuming that the Reynolds number is infinite and then iterating the solution using the calculated flow For an infinite Reynolds number 4, = ad * - (0.9935) - - (C,) (0.9935) (C,) ~ ( ) ~(0.9935) PO m (145) & 4(1.98586) (459.67 + 77) (778.2)/16.043 (C,) x 106 4(8314.41)(273.15 (100)2 + 25)/16.043 where the factor g , is included for consistency of units.' Using the value of CRobtained from the tables in Appendix E (CR = 0.6754): q, = 1.9064 lbm/sec , = 0.8649 kg/s 'The numerical value of g, is 32.174 lbm-ft/lbf-sec2 18 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w From para 7.1 the flow equation is 10- lbf-sec ft2(1.005 X [using 2.1 X lo-’ Pa-s) for the viscosity] Red = - (4) (1.9064) (12) (32.174) (T)(1) (2.1 X lo-’) (4) (0.8649) (100) (T)(2.54) (1.005 X = 4.31 X lo6 = 4.31 X lo6 q, = 1.9188[0.9935 - 1.525 (4.31 X lo6) - ‘ / ] = 1.9049 lbm/sec q, = (0.8706) = [0.9935 - 1.525 (4.31 X lo6) - ‘ I = 0.8643 kg/s Another iteration would little to change the answer C2 THE EFFECT OF PERFECT GAS CONSUMPTION It is interesting to find the difference in the calculated flow if it were assumed that the flow was that of a perfect gas The perfect gas critical flow function is where y = 1.321 C*i = 0.6710 19 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w Substituting t h i s value into the expression fortheReynoldsnumber (Us) q, = 1.8925 lbm/sec q, = 0.8587 kg/s The percentage difference is (Us) C3 EFFECT OF TEMPERATURE AND PRESSURE CORRECTIONS ON THE MEASURED PRESSURE AND TEMPERATURE The equation inpara 7.4 can be used to correct the measured pressure and temperature to stagnation conditions Equations from para 7.4 Po=P1 To=T1 The flow in the conduit is Solving for the velocity ( + - x - Ma,2> ( + -x Ma:), - 1) with a! = I Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w Using Ceiin place of C, in the flow rate equation (Us) e= PI = (145) (144) (16.043) ( R / M ) TI (1.98586) (536.67) (778.2) = 0.40398 lbm/ft3 e = PI = (1 x lo6) (16.043) (R/M) T, (8314.41) (273.15 + 25) = 6.472 kg/m3 The minimum area of the plenum At = Ird,where D must be at least 4 X d The maximum velocity in the plenum (Us) v, = (1.9049) (0.4039) (0.08727) v, = (0.8643) (6.472) (0.008107) 21 = = 54.04 ft/sec 16.47 m/s Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled wh For the calculation of the plenum Mach number, assume that the gas is a perfect gas The density at the plenum v;F VT = = 1482.3 ft/sec H =gc R TI = J (1.321) (273.15 = (1.321) (96.33) (32.174) (536.67) + 25) (8314.41) (16.043) = 45 1.79 m/s The Mach number at the plenum (Us) Ma, = 54.04/1482.3 = 0.03646 Mal = 16.47/451.79 = 0.03646 The corrected pressure Po [ +1 1.321 ~ 0*321 (0.03646)2] = P, = P , (1.000878) ~~~~ o.321 Po = (145) (1.000878) = 145.127 Ibf/im2 Po = (1 x lo6) (1.000878) = 1.000878 x lo6 MPa 22 Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w The speed of sound at plenum conditions [ + y(0.03646)2] ; with a = (Us) To = Ti (1.0002134) = 536.784 O R (SI) To = Ti (1.0002134) = 298.22 K Substituting these values into the equation for the mass flow 4, = ?r (0.6745) [0.9935 - 1.525 (4.31 x lo6) -'I21 (145.127) 4 ,/(96.33) (536.784) s = 1.9064 lbm/sec 4, = A (0.6754) [0.9935 - 1.525 (4.31 x 106)-'/2] (1.000878 x lo6) (2.54)2 (298.22) (8314.41)/(16.043) (loo)* = 0.8650 kg/s which gives a percentage difference of %Error = 1.9064 - 1.9046 x 1.9064 = o.lqo %Error 0.8650 - 0.8641 x 0.8650 = = 23 o.lqo Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled w The corrected temperature To = T, CRITICAL FLOW FUNCTIONS (This Appendix contains supplementary information for the convenience of the reader It is not a partof ASME/ANSI MFC-7M-1987.) In practice it is difficult to measure the state of the fluid at the throat of the nozzle To avoid doing this the critical flow function is used This function is ob- D l GENERAL FLOW EQUATIONS Critical flow functions are derived by reducing the mass flow equation for a nozzle The mass flow through a conduit is qm = tained by assuming that the flow is isentropic, onedimensional, and that the fluid is a perfect gas None of these assumptions are true for areal fluid Isentropic, one-dimensional flow requires the en- Aev tropy to be equal at the nozzle throat and plenum These assumptions also allow one to set the change in enthalpy equal to one-half of the average fluid velocity squared These conditions are expressed mathematically below where V = average velocity e = average density A = cross-sectional area For a critical flow venturi nozzle the average velocity is the speed of sound of the fluid at local conditions Thus the mass flow may be written as shown in Eq (D2) so - s * = V** - = (h, - h*) 2 PERFECT GAS CRITICAL FLOW FUNCTION In addition, for a perfect gas the compressibility factor is 1, and the specific heats and isentropic exponent are all constants.a Thus for perfect gas one can write the following equations: The sonic velocity of a fluid is defined for a perfect gas by Eq (D3) v* =4x")T* 03) This velocity is the speed at which a pressure wave will move through the fluid In the general case, the speed of sound is function a of the frequency of the pressure wave At very high frequencies this speed is reduced because of the ability of the molecules to transfer energy However, at low frequencies the speed is the same as the compression rate of the fluid In order to calculate the flow fora critical flowventuri nozzle, Eqs (D2) and (D3) are combined to yield Eq (D4) p* Po = ( L )- 1) X/(X xfl - 1) =(z) I/(X eo x+l - - e* T* To ~ + qm = A * @ * x ( R I M ) T* 25 P* = P o X(+ ll ) l = T* eo(-+) = TO(&) X/(X - 1) ] / ( x - 1) Copyrighted material licensed to Stanford University by Thomson Scientific (www.techstreet.com), downloaded on Oct-05-2010 by Stanford University User No further reproduction or distribution is permitted Uncontrolled wh APPENDIX D qm = A * eo [x(&)( and by replacing eo with (~9) PO Z ( R/ M ) TO This leads to the definition of the isentropic real gas critical flow function shown below 1’” Equation (D5) is usable on a perfect gas withisentropic, one-dimensional flow It should be noted that isentropic expansion relations used to translate equations the from the nozzle throat conditionsto the upstreamplenum assume that the gas is a perfect gas However, both the equation of state and the expression for thesonic velocity have assumed that the gas is a real gas Despite this inconsistency, Eq (D9) may be used in some cases with acceptable error D3 REAL GAS CRITICAL FLOW FUNCTION It is often desired to write the flow equations in a form that allows for real gas effects In a real gas the ratio of the specific heats is not constant and the isentropic exponent x is defined by Eq (D6) x = - (p e 1/2 + l)/(x - 1) c*i= [( L)( x+l x x + I)/(x - ) D4 REAL GAS CRITICAL FLOW COEFFICIENT aa&? p s To extend the range of application and to improve the accuracy of the computed flow the critical flow function presented thus farcan be replaced with a factor called the critical flow coefficient This coefficient It is still assumed that thesonic velocity may be described by an adiabatic compression of the fluid which is assumed to be isentropic Thus theacoustic velocity is defined by Eq (D7) may be thought of as a factor in the flow rate equation which accounts for the real gas effects of fluid The coefficient still assumes that theflow is isentropic and one-dimensional The isentropic critical flow coefficient is defined by Eq (D10) Combining Eqs (D6) and (D7) leads to Eq (D8), the definition of the sonic velocity in terms of ( x , P, e) The value of this factor is obtained by integrating thermodynamic functionsfortheentropyand enthalpy of the fluid from the plenum to the nozzle throat conditions along constant temperature and constant density paths These integrations are performed until the entropy at both points is equal and the change in enthalpy is equal to one-half the sonic velocity at the throat squared A further description of this procedure, along with suggested references, is presented in Appendix E The nonisentropic multidimensional effects of the flow field are accounted for by the discharge coefficient By utilizing the equation of state P = Z