THE DETERMINATION OF CRITICAL FLOW FACTORS FOR NATURAL GAS MIXTURES. Part 3: The Calculation of C* for Natural Gas Mixtures

Transcription

1 A REPORT ON THE DETERMINATION OF CRITICAL FLOW FACTORS FOR NATURAL GAS MIXTURES Par 3: The Calculain f C* fr Naural Gas Mixures FOR NMSPU Deparmen f Trade and Indusry 151 Buckingham Palace Rad Lndn SW1W 9SS Prjec N: DSDC51 14 April 1997 Repr N: 162/96

2 Nainal Engineering Labrary Densiy Flw Cenre Repr n THE DETERMINATION OF CRITICAL FLOW FACTORS FOR NATURAL GAS MIXTURES Par 3: The Calculain f C* fr Naural Gas Mixures FOR NMSPU Deparmen f Trade and Indusry 151 Buckingham Palace Rad Lndn SW1W 9SS S U M M A R Y This is he hird f hree reprs describing he wrk n he calculain f C* fr naural gas mixures. The ms accurae and wide ranging equain f sae fr use wih such mixures can be used nly deermine he mechanical prperies. A knwledge f he calric prperies is als required in he calculain f he criical flw facr. This repr describes he sluin f he hermdynamic equains fr flw in a snic nzzle and he applicain f he mehdlgy calculae he criical flw facrs fr several naural gas mixures. Prepared by: Mr J T R Wasn... Apprved by: Mr J T R Wasn... Dae: 14 April 1997 fr W Pan Direcr and General Manager Prjec N: DSDC51 Repr N: 162/96 Page 1 f 16

3 Nainal Engineering Labrary CONTENTS 1 INTRODUCTION... 3 Page 2 THE CALCULATION OF CRITICAL FLOW THROUGH SONIC NOZZLES FROM THERMODYNAMIC EQUATIONS OF STATE METHOD OF SOLUTION OF THE THERMODYNAMIC AND FLOW EQUATIONS SOFTWARE IMPLEMENTATION AND VALIDATION C* TABLES FOR VARIOUS NATURAL GAS MIXTURES REFINEMENT OF C* CALCULATION FOR A SPECIFIC PIPELINE GEOMETRY... 9 REFERENCES LIST OF TABLES LIST OF FIGURES Prjec N: DSDC51 Repr N: 162/96 Page 2 f 16

4 Nainal Engineering Labrary 1 INTRODUCTION The bjecive f his prjec is derive reliable values fr he criical flw facr, C *, f naural gas mixures. This is he hird repr n his pic. The w previus reprs [1,2] cvered he exensin f he American Gas Assciain s equain f sae, AGA8, include he calculain f calric prpery daa. This repr ulines he mehdlgy fr he calculain f he criical flw facr and her flw parameers frm he AGA8 equain f sae. The link beween equains f sae and snic nzzle echnlgy is given in he paper by R. C. Jhnsn [3] n real gas effec n snic nzzles. Hydrdynamic cnsiderains require ha a he hra f he nzzle he enhalpy differs frm ha a he upsream sagnain cndiins by an amun prprinal he square f he velciy a he hra and fr criical flw his relaes he speed f sund f he gas a he hra. Given his relainship i is hen pssible deermine he pin n he isenrpe, passing hrugh he upsream sagnain pin, which crrespnds he cndiins a he hra f he nzzle. Tw separae mehds fr deermining he same have been develped, deails f he secnd and faser mehd are given in his dcumen. When esed bh mehds yielded exacly he same values fr he criical flw facr and her parameers fr mehane, argn and nirgen. The inpu requiremens are: (1) he pressure and emperaure a upsream sagnain cndiins; and (2) he rai f he nzzle diameer ha f he pipeline a he upsream meering pin. The parameers which can be calculaed are: (i) (ii) (iii) (iv) (v) he criical flw facr, C * ; he mass flw per uni area hrugh he nzzle hra; he velciy f he gas a he hra; he criical pressure, emperaure and densiy rais; and all f he hermphysical prperies f engineering ineres fr he gas bh a sagnain cndiins and a he hra f he nzzle. Prjec N: DSDC51 Repr N: 162/96 Page 3 f 16

5 Nainal Engineering Labrary 2 THE CALCULATION OF CRITICAL FLOW THROUGH SONIC NOZZLES FROM THERMODYNAMIC EQUATIONS OF STATE The mass flw per uni area f real gases hrugh criical flw nzzles is given by Jhnsn (3) as: * G = C. p / R. T, (2.1) where p and T R M C * are he pressure and emperaure f he gas a upsream sagnain cndiins, in pascal and kelvin respecively, is he specific gas cnsan = /M J/kg.K, is he mlar mass, in kg/ml, and is he criical flw facr, dimensinless. The assumpins invlved in he derivain f eqn (2.1) are: (1) he flw is isenrpic; and (2) he flw is ne dimensinal. Jhnsn (3) shwed ha he energy equain requires ha he change in enhalpy, dh, beween upsream sagnain and nzzle hra cndiins, is relaed he flw velciy, V, by dh = V. dv. Inegraing he laer expressin beween upsream sagnain cndiins (subscrip ) and he hra f he nzzle (subscrip ) yields: 2 2 h h = V / 2 + V / 2, (2.2) where nrmally he upsream flw velciy, V 0, is aken be zer. V 0 is here reained in he wrking equain allw fr finie flw a he upsream meering pin. Rearranging and dividing bh sides by he speed f sund a he hra, w, we have: ( h h ) / w + ( V / w ) = ( V / w ) = Μ, where Μ is he Mach number f he gas a he hra f he nzzle. Fr criical flw a he nzzle he Mach number is uniy. Thus frm he abve, he enhalpy and enrpy cndiins be saisfied a he hra are: and 2 2 h = h ( w V ) / 2 ; (2.3) s = s (2.4) Prjec N: DSDC51 Repr N: 162/96 Page 4 f 16

6 Nainal Engineering Labrary Given a reliable hermdynamic frmulain fr he es r calibrain gas i is herefre pssible slve he hermdynamic equain f sae fr he sae pin wih enhalpy h and enrpy s. This sae pin crrespnds he cndiins a he hra f he nzzle. Once his has been deermined he hermdynamic prperies f he gas a he hra f he nzzle are knwn and all f he required flw parameers can be calculaed. 3 METHOD OF SOLUTION OF THE THERMODYNAMIC AND FLOW EQUATIONS Given he pressure, p, and emperaure, T, a upsream sagnain cndiins he hermdynamic equain f sae can be slved yield he crrespnding values f enhalpy, h, and enrpy, s, f he fluid. A he nzzle hra he enrpy f he fluid is aken be unchanged bu he enhalpy is reduced by he increased kineic energy f he flw. The cndiins a he hra are n knwn and have be esablished by ierain. Since he equain f sae fr he fluid is a naural funcin f bh densiy and emperaure i is mre efficien wrk in erms f hese variables. We begin by making an iniial guess f he densiy and emperaure, ρ (1) and T (1) a he hra cndiins. Fr snic nzzles, reliable iniial esimaes are: ρ (1) 0.5.ρ, and T (1) /K = T /K 50. The values f ρ (1) and T (1) can hen be refined using he fllwing prcedure. Since bh h and s can be cnsidered be funcins f densiy and emperaure, we have n differeniain ha: h = ( h / ρ ). ρ + ( h / T ) ρ. T, T and (3.1) s = ( s / ρ ). ρ + ( s / T ) ρ. T. T Subsiuing frm hermdynamics fr he fur parial derivaives in erms f readily calculable prpery values we bain: 2 h = v. k.( 1 T. α ). ρ + ( c v + v. α. k ). T, and (3.2) 2 s = v. α. k. ρ + ( c / T ). T, where α is he cefficien f cubical expansin, α = ( 1 / v ).( v / T ) p, k is he ishermal bulk mdulus, k = v.( p / v ) T, and v is he mlar vlume, v = 1/ ρ. v Prjec N: DSDC51 Repr N: 162/96 Page 5 f 16

7 Nainal Engineering Labrary The flw cndiins be saisfied n h and s fr criical flw in a snic nzzle are: h = h [ h + ( w 2 V 2 ) / 2 ] = 0, and (3.3) s = s s = 0. Cmbining he hermdynamic and flw equains, eqns (3.2) and (3.3), we herefre bain: h [ h + ( w V ) / 2 = v. k.( 1 T. α ). ρ + ( c v + v. α. k ). T, and (3.4) s s = v 2. α. k. ρ + ( c / T ). T, which yield n sluin fr ρ and T: v 2 2 { h [ h + ( w V ) / 2 ]}. c v T.( s s ).( c v + v. α. k ) ρ =, 2 2 v. k.( c v + T. v. α. k ) and (3.5) ( s s ).( 1 T. α ) + α.{ h [ h + ( w 2 V 2 ) / 2 ]} T = T.. 2 c + T. v. α. k v In he laer, h and s are he knwn values f he enhalpy and enrpy a he upsream cndiins crrespnding he measured emperaure, T, and pressure, p. The hermdynamic quaniies a he nzzle hra, h, s, w, c v, v, α, and k, are calculaed a ( 1 ) ( 1 ) and T frm he Helmhlz energy frmulain fr he fluid: The emperaure T in ρ eqn (3.5) is aken as he curren esimae f he hra emperaure, T ( ) The calculaed ρ and T values frm eqn (3.5) hen yield imprved esimaes fr bh he densiy and emperaure a he nzzle hra, namely: ( ρ n + 1 ) ( ) ( ) = ρ n + ϕ. ρ n (, and T n + 1 ) ( T n ) ( T n ) = + ϕ., (3.6) where ϕ a muliplier, has a value clse uniy. ( n +1 ) This ieraive prcess is repeaed unil cnvergence is bained in bh ρ 1. ( n +1 ) and T. In pracice, i has been fund ha fr high sagnain pressures (f several hundred bar r mre) he cnvergence rae can be dubled by seing he parameer ϕ in he abve equains a value f 0.9. Prjec N: DSDC51 Repr N: 162/96 Page 6 f 16

8 Nainal Engineering Labrary Once he densiy and emperaure a he hra are esablished all her hermdynamic prperies and flw parameers can be calculaed fr he gas a his cndiin, namely: Nzzle hra velciy: V = w Mass flw per uni area a he nzzle: G = ρ. w Criical flw facr: * C = ρ. w. R. T / p Rai f he hra upsream pressure: p c = p / p Rai f he hra upsream emperaure: T c = T / T Rai f he hra upsream densiy: ρ c = ρ / ρ 4 SOFTWARE IMPLEMENTATION AND VALIDATION Frran subruines have been develped calculae he criical flw facr and her flw parameers fr naural gas mixures frm he exended AGA8 equain f sae. The mehd is general and can be applied a wide range f gases and gas mixures. The requiremens n he hermdynamic package/s be linked he ruines are: (a) (b) (c) (d) (e) he hermdynamic prperies f he gas can be calculaed given emperaure and pressure as independen variables; he hermdynamic prperies f he gas can be calculaed given emperaure and densiy as independen variables; he hermdynamic prperies reurned by he package include he se f parameers required by eqn (3.2); he lwer emperaure limi f he hermdynamic frmulain is a leas 50 kelvin belw ha f he sagnain emperaure; and he sae f he fluid r fluid mixure bh a sagnain and a he nzzle hra perain he single phase gaseus regin. Cndiins (a) (d) are saisfied by NEL s implemenain f he exended AGA8 equain f sae. In he absence f a rbus phase equilibrium package, hwever, i is n pssible check ha cndiin (e) is me. Difficulies f usage may herefre be encunered wih cerain mixures, namely: (i) (ii) rich naural gas mixures a sagnain emperaures f less han 60 C; and lean naural gas mixures a sagnain emperaures belw 10 C. I shuld be ned ha he AGA8 equain f sae was develped fr use wih lean naural gas mixures in he single phase gaseus regin and is n suied fr use clse he phase bundary r in phase equilibria calculains. Thugh recen measuremen wrk n rich Prjec N: DSDC51 Repr N: 162/96 Page 7 f 16

9 Nainal Engineering Labrary naural gas mixures a NEL have exended he range f applicabiliy f he AGA8 equain he laer limiains remain. The sfware implemenain has been carefully validaed fr a number f gases as fllws: (1) C * values and her flw parameers have been calculaed frm he sfware package fr hree pure fluids (mehane, nirgen, and carbn dixide) ver a range f pressures and emperaures and he values cmpared agains hse frm separae implemenains f he laes hermdynamic frmulains fr hese fluids [4,5,6]. (2) C * values and her flw parameers have been calculaed frm he sfware fr a dry air mixure (78.12% N 2, 20.96% O 2 and 0.92% Ar) and validaed agains values calculaed frm an implemenain f a recen hermdynamic frmulain fr dry CO 2 free air [7,8]. In each es he differences in he criical flw facrs were wihin he uncerainies f he AGA8 equain. 5 C* TABLES FOR VARIOUS NATURAL GAS MIXTURES The mean cmpsiins f naural gas mixures frm cerain fields are well knwn and have been published in he lieraure. Six f such mixures have been seleced illusrae he perain f he sfware package, see Table 1. Tables f C* values have been calculaed fr each f he naural gas mixures given in Table 1 fr a range f sagnain emperaures and pressures, see Tables 2 8. In each f hese es examples he upsream velciy was aken as zer. Table 1 Mean Percenage Mlar Cmpsiin f Varius Naural Gases Cmpnens Amarill Bacn Ekfisk High N 2 High CO 2 S Fergus CH N CO C 2 H C 3 H i C 4 H n C 4 H i C 5 H n C 5 H C Prjec N: DSDC51 Repr N: 162/96 Page 8 f 16

10 Nainal Engineering Labrary 6 REFINEMENT OF C* CALCULATION FOR A SPECIFIC PIPELINE GEOMETRY Any pracical calculain f C* mus ake accun f he finie velciy f he gas a he upsream meering psiin. Tabulaed values f C*, such as hse derived in he previus secin, are nrmally calculaed n he assumpin ha he gas has zer velciy upsream f he nzzle. This, as far as he auhr is aware, applies all published ables f C* and is a limiain f his mehd f prmulgain. The pracical calculain f C* requires a knwledge f he diameers f bh he nzzle hra and he pipeline a he upsream meering pin and a furher level f ierain deermine he upsream velciy frm he mass flw rae a he nzzle. ( 1 ) We cmmence he ierain n he assumpin ha he upsream velciy,v, is zer ( 1 ) hen, using he mehdlgy f Secin 3, bain an esimae fr he densiy, ρ, and ( 1 ) velciy,v, f he gas a he nzzle hra. Frm he gemery f he paricular nzzle and pipeline we can hen bain an imprved esimae fr he upsream ( n +1 ) velciy,v, frm he mass balance equain; namely: V = V.( ρ / ρ ). β, (5.1) ( n + 1 ) ( n ) ( n ) 2 where β is he rai f he diameer f he nzzle hra ha f he pipeline a he upsream meering pin. ( n +1 ) This calculain prcess is repeaed unil cnvergence in V is bained; his is nrmally achieved wihin hree ierains. The value f C * fr he specified meering cndiins and pipeline gemery fllws as a resul. A schemaic f he enire calculain prcess is given in Fig. 1. Fig. 2 shws he dependence f he calculaed values f C* wih he diameer rai, β, fr argn a a specified sagnain emperaure and pressure. This figure shws ha he calculaed values f C * are dependen n β fr values f β > This limi will be fluid and cndiin dependen herefre unless β is very small he refinemen ulined in his secin shuld be applied all calculains f C *. The rue value f C * fr any gas r gas mixure, peraining specific meering cndiins and pipeline gemery, can be bained by: (1) ieraive calculain using cmpuer sfware, r (2) inerplain in a series f ables fr C * fr he specified mixure a eiher seleced values f V r β. Prjec N: DSDC51 Repr N: 162/96 Page 9 f 16

11 Nainal Engineering Labrary The firs f hese mehds, because f bh is speed and is abiliy handle a wide range f mixure cmpsiins and pipeline gemeries, ffers a much mre saisfacry sluin. A sfware package, Gasfl, being develped a NEL incrpraes all f he abve mehdlgy. REFERENCES 1 WATSON, J. T. R. The Deerminain f Criical Flw Facrs fr Naural Gas Mixures, Par 1: The Thermdynamic Prperies f he Cmpnens f Naural Gas and Naural Gas Mixures in he Ideal Gas Sae. Flw Cenre Repr N 72/96 prduced fr he NMSPU. Eas Kilbride, Glasgw: Nainal Engineering Labrary, December WATSON, J. T. R. The Deerminain f Criical Flw Facrs fr Naural Gas Mixures, Par 2: The Real Gas Cnribuin he Thermdynamic Prperies f Naural Gas Mixures. Flw Cenre Repr N 72/96 prduced fr he NMSPU. Eas Kilbride, Glasgw: Nainal Engineering Labrary, February JOHNSON, R. C. Real Gas Effecs in Criical Flw Thrugh Nzzles and Tabulaed Thermdynamic Prperies. NASA Technical Ne TN D 2565, Lewis Research Cener, SETZMANN, U. and WAGNER, W. A new equain f sae and ables f hermdynamic prperies f mehane cvering he range frm he meling line 625 K a pressures up 1000 MPa. J. Phys. Chem. Ref. Daa 1991, 20(6), pp SPAN, R., LEMMON, E., JACOBSEN, R. T. and WAGNER, W. Privae cmmunicain; be published in J. Phys. Chem. Ref. Daa. [Nirgen]. 6 SPAN, R. and WAGNER, W. Privae cmmunicain; be published in J. Phys. Chem. Ref. Daa. (Carbn dixide). 7 JACOBSEN, R. T., PENONCELLO, S. G., BEYERLEIN, S. W., CLARKE, W. P., and LEMMON, E. W. A hermdynamic prpery frmulain fr air. Fluid Phase Equilibria, 1992, 79, WATSON, J. T. R. Subprgram fr he Accurae Deerminain f he Thermphysical Prperies f Dry CO 2 Free Air a Elevaed Pressure. Repr N FPT(DS)SR08, Fluids and Prcess Technlgies Divisin. Eas Kilbride, Glasgw: Nainal Engineering Labrary, April Prjec N: DSDC51 Repr N: 162/96 Page 10 f 16

12 Nainal Engineering Labrary LIST OF FIGURES 1 Calculain f C* frm an Equain f Sae 2 Dependence f C* n β fr Argn a T = 298 K, p = 7 MPa. LIST OF TABLES 1 Mean Percenage Mlar Cmpsiin f Varius Naural Gases. 2 C * Values fr Amarill Field 3 C * Values fr Bacn Field 4 C * Values fr Ekfisk Field 5 C * Values fr High N 2 Gas Fields (USA) 6 C * Values fr High CO 2 Gas Fields (USA) 7 C * Values fr S Fergus Field. Prjec N: DSDC51 Repr N: 162/96 Page 11 f 16

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18 Table 2 C* Values fr Amarill Field p/ Table 3 C* Values fr Bacn Field p/ Ne: In Tables 2 7, is in Celsius n he ITS 90 scale and p is in bar.

19 Table 4 C* Values fr Ekfisk Field p/ Table 5 C* Values fr High N 2 Gas Fields (USA) p/ Ne: In Tables 2 7, is in Celsius n he ITS 90 scale and p is in bar.

20 Table 6 C* Values fr High CO 2 Gas Fields (USA) p/ Table 7 C* Values fr S Fergus Field p/ Ne: In Tables 2 7, is in Celsius n he ITS 90 scale and p is in bar.