Numerical Analysis of Rapid Gas Decompression in Pure Nitrogen using 1D and 3D Transient Mathematical Models of Gas Flow in Pipes

Transcription

1 Internatonal Journal of Chemcal and Bologcal Engneerng 6 01 Numercal Analyss of Rapd as Decompresson n Pure Ntrogen usng 1D and 3D Transent Mathematcal Models of as Flow n Ppes Evgeny Burlutsky Abstract The paper presents a numercal nvestgaton on the rapd gas decompresson n pure ntrogen whch s made by usng the one-dmensonal (1D) and three-dmensonal (3D) mathematcal models of transent compressble non-sothermal flud flow n ppes. A 1D transent mathematcal model of compressble thermal multcomponent flud mxture flow n ppes s presented. The set of the mass, momentum and enthalpy conservaton equatons for gas phase s solved n the model. Thermo-physcal propertes of multcomponent gas mxture are calculated by solvng the Equaton of State (EOS) model. The Soave-Redlch-Kwong (SRK-EOS) model s chosen. Ths model s successfully valdated on the expermental data [1] and shows a good agreement wth measurements. A 3D transent mathematcal model of compressble thermal sngle-component gas flow n ppes, whch s bult by usng the CFD Fluent code (ANSYS), s presented n the paper. The set of unsteady Reynolds-averaged conservaton equatons for gas phase s solved. Thermo-physcal propertes of sngle-component gas are calculated by solvng the Real as Equaton of State (EOS) model. The smplest case of gas decompresson n pure ntrogen s smulated usng both 1D and 3D models. The ablty of both models to smulate the process of rapd decompresson wth a hgh order of agreement wth each other s tested. Both, 1D and 3D numercal results show a good agreement between each other. The numercal nvestgaton shows that 3D CFD model s very helpful n order to valdate 1D smulaton results f the expermental data s absent or lmted. T Keywords Mathematcal model, Rapd as Decompresson I. INTRODCTION HE fracture propagaton control s usually made on the bass of the Battelle two-curve method, whch was developed by the Battelle Columbus Laboratores [,3]). Ths method helps to determne the fracture arrest toughness when a ppelne s ruptured. The fracture propagaton speed n a ppelne wall and the decompresson wave speed n gas mxtures are requred to be used n the Battelle analyss. The fracture propagaton s arrested, when the decompresson wave speed n gas mxtures s qucker than the fracture propagaton velocty n a ppelne wall materal. Therefore, the accuracy n the calculaton of the decompresson wave speed n gas mxtures s very mportant n fracture propagaton control analyss. Dr. Evgeny Burlutsky was wth PETROSOFT D&C Sngapore. He s now wth the A*STAR Insttute of Hgh Performance Computng (IHPC), 1 Fusonopols Way, 13863, Sngapore (phone: ; e- mal: burlutskye@hpc.a-star.edu.sg and e.burlutsky@petrosoft-dc.com). The decompresson process n natural gas mxtures s very rapd non-sothermal process. A transent mathematcal model of compressble thermal mult-component flud mxture flow n ppes helps to study the flow behavor n a wde range of the operatng parameters. Extensve expermental measurements of the decompresson wave speed n rch and base natural gas mxtures were made last years [1, 4-7]. The nfluence of shock tube nner dameter, gas mxture composton, pressure, and temperature was carefully examned n detals expermentally. Most of measurements were made on a small-dameter shock tube, where the frcton force nfluences on the flow behavor much stronger compare to large-dameter ppes. However, the nformaton on the decompresson wave speed measurements s extremely lmted n the open source lterature. Papers on the expermental measurements of the gas decompresson n carbon doxde mxtures are absent n the open source lterature too. The program ASDECOM [8], whch s based on the analytcal soluton of the decompresson wave speed determnaton, s used by ol and gas engneers n order to calculate the decompresson wave speed values [1,5] as well. The program predcts the decompresson wave speed values wth a reasonably good level of accuracy. However, the frcton force s not accounted for n the analyss here. The comparson between measured data and ASDECOM calculatons s very poor, when the gas decompresson wave speed s determned from pressure transducers, whch are located far away from the rupture end of the ppe, and where the frcton nfluences on the flow behavor sgnfcantly. The paper presents a numercal nvestgaton on the rapd gas decompresson n pure ntrogen whch s made by usng 1D and 3D mathematcal models of transent compressble thermal flud flow n ppes. A 1D mathematcal model of compressble thermal mult-component flud mxture flow n a ppe s presented. The model s valdated [9,10] on the expermental data [1] and shows a good agreement wth those measurements. A 3D mathematcal model of compressble thermal sngle-component gas flow n ppes, whch s bult by usng the CFD Fluent code (ANSYS), s presented n the paper. The smulaton of rapd gas decompresson n pure ntrogen s made by usng the 3D model and results of calculaton are compared wth 1D predcton. The ablty of both models to smulate the decompresson process wth a 16

2 Internatonal Journal of Chemcal and Bologcal Engneerng 6 01 hgh order of agreement between each other s tested on the smple gas flow case. Both, 1D and 3D numercal results show a very good agreement between each other. II. ONE-DIMENSIONAL MATHEMATICAL MODEL OF TRANSIENT SINLE-PHASE FLOW The set of the mass, momentum and enthalpy conservaton equatons for the gas phase s solved n the mathematcal model. Ths set of equatons for the sngle phase gas mxture n general form s wrtten as [11]: ρ ρ + = 0 (1) t ρ t ρ h t ρ + ρ h + = α P R P = α + t Wall P Here, α s the volume fracton of the gas mxture; ρ s the densty of the gas mxture; s the velocty of the gas mxture; P s the total pressure; R Wall s the frcton term, h s the enthalpy of the flud, t s the tme, z s the axal coordnate. The frcton term s wrtten as [1]: Π ξ Wall ρ R Wall = τ Wall, τ Wall =, S 8 (4) ξ Wall = 64 / Re, Re < ξ Wall = / Re, Re > 1600 Re = ρ D µ (5) ppe / Here, Π s the permeter of the ppe; S s the crosssectonal area of the ppe; τ s the frcton term (.e. as- Wall nteracton); Wall () (3) ξ Wall s the frcton coeffcent; D ppe s the dameter of the ppe; µ s the vscosty of the flud. III. THERMO-PHYSICAL PROPERTIES OF AS MIXTRE IN ONE-DIMENSIONAL MATHEMATICAL MODEL Thermo-physcal flud propertes are modeled by solvng of the Equaton of State (EOS) n the form of the Soave-Redlch- Kwong model [13]. The set of equatons and correlatons (SRK-EOS) may be wrtten as [13]: RT a P = (6) V b V V + b ( ) ( ) N N N z z aa ( 1 k ),b = a = z b (7) a = 1 = 1 C C = 1 R T C T = m 1, PC T C (8) m RT b = P = ω ω (9) Here, V s the volume of the gas mxture; N s the number of components n the gas mxture; T s the temperature of the gas mxture; R s the unversal gas constant; ω s the acentrc factor of the component ; C C P, T are crtcal values of the pressure and temperature, correspondently; z s the mole fracton of the component. The compressblty factor (Z) of the gas mxture s calculated from the followng equaton [13]: Z 3 ( A B B ) Z AB = 0 Z + (10) a P b P A =,B = (11) R T RT The vscosty of gas mxture s calculated by usng of the Lee-onzales-Eakn (LE) correlaton [14]. The algorthm of solvng of the set of One- Dmensonal transent governng equatons of the flud mxture flow n a ppe s based on the Tr-Dagonal Matrx Algorthm (TDMA), also known as the Thomas algorthm [15]. It s a smplfed form of aussan elmnaton that can be used to solve tr-dagonal systems of equatons. The set of unsteady governng equatons s transformed nto the standard form of the dscrete analog of the tr-dagonal system [15] by usng the fully mplct numercal scheme. In ths case the equaton s reduces to the steady state dscretzaton equaton f the tme step goes to nfnty. IV. AS DECOMPRESSION PRORAM A one-dmensonal transent mathematcal model of compressble thermal mult-component flud mxture flow n a ppe was developed under the research proect Multcomponent as mxture flows n ppelnes and wells n PETROSOFT D&C. Ths mathematcal model was mplemented nto the FORTRAN computer code and was named the as Decompresson Program (DP code). More nformaton about the DP code s avalable on V. THREE-DIMENSIONAL MATHEMATICAL MODEL OF TRANSIENT SINLE-PHASE FLOW SIN THE FLENT CODE (ANSYS) Transent smulatons of rapd gas decompresson n pure ntrogen are performed by usng the CFD Fluent code (ANSYS). The unsteady Reynolds-averaged conservaton equatons are solved n order to smulate the Euleran flud flow feld. For unsteady flow case, the set of governng equatons s wrtten as: ρ + ( ρ u ) = 0 (1) t x ( u ) + ( ρ u u τ ) p ρ = (13) t x x Where, x s the Cartesan coordnate (= 1,, 3), u s the absolute flud velocty component n drecton ( x ), p s the pressure, ρ s the flud densty, and τ s the stress tensor component. For the turbulent flow case, the stress tensor s 17

3 Internatonal Journal of Chemcal and Bologcal Engneerng 6 01 wrtten as: u τ = ρ Here, k µ s µ δ u u 3 xk S s the rate of stran tensor, dynamc vscosty of the flud, δ s the Kronecker delta functon (whch s unty when = and zero otherwse), and u are the fluctuatons of the flud velocty as a result of the u Reynolds averagng procedure. These fluctuatons represent the addtonal Reynolds stress due to the turbulent moton. In order to model the turbulent structure of the flud phase, the u component s determned usng the Boussnesq u hypothess. The conservaton equatons are closed usng the Realzable k-ε turbulence model, whch s used together wth the enhanced wall functon treatment for modelng boundary condtons at the wall. VI. EXPERIMENTAL VALIDATION OF 1D DP CODE ON RAPID DECOMPRESSION IN NATRAL AS MIXTRES The presented mathematcal model was successfully valdated [9, 10] on the expermental data on the rapd gas decompresson n base natural gas mxtures [1]. Those expermental measurements were conducted by TCPL (Trans Canada Ppe Lnes) at TCPL as Dynamc Test Faclty n Ddsbury, Alberta, Canada [1]. The man test secton of the faclty s the shock tube, whch s 30 meters long. The nner ppe dameter s mm. The nternal surface of the tube has a roughness, whch s better than 40 mcro-nches. A rupture dsc s placed at one end of the ppe, whch s upon rupturng. A decompresson wave propagates up nto the pressurzed test secton. A few hgh frequency responses Pressure Transducers (PT) are mounted nto the tube n order to capture the tme hstory of the expanson fan [1]. Decompresson wave speed values were determned from the tme between sgnals from PT-P1 and PT-P8 as well as from PT-P5 and PT-P6 (fg. 1). Fg. 1 shows the schematc of the expermental decompresson tube. The computatonal decompresson ppe has a length of 50 meters long. The nner ppe dameter s mm. One end of the shock tube s selected to be the closed end. The rupture dsc s ntroduced nto other end of the ppe. Dstances between the rupture dsc and pressure transducers are shown on fg. 1. Fg. 1 Schematc of the expermental decompresson tube (14) µ s the molecular Three dfferent expermental cases havng a dfferent gas composton, ntal pressure, and temperature values are smulated usng the DP code [9, 10]. Smulaton results of one test case (Table I) are shown. The followng gas mxture composton, ntal pressure and ntal temperature were used to perform the smulaton (Table I): TABLE I AS COMPOSITION (MOLE %), INITIAL PRESSRE (MPA) AND TEMPERATRE (K) Case 1 P 0.67 ntal T 64.7 ntal N CO 1.79 C C C C n-c C n-c Rapd gas decompresson n base natural gas mxtures havng the nlet and boundary condtons (Table I), whch are dentcal to the expermental one, s smulated by usng the presented mathematcal model. Predctons are started wth the ntal pressure of 0.67 MPa n each computatonal cell of the ppe. New values of the velocty, temperature, densty and pressure are calculated after each tme step. The upper lmt of the tme step was selected from the pont of vew of the numercal stablty. Pressuree values were collected at PT locatons P1, P8, P5 and P6 after each tme step started from the begnnng. Fg. shows the evoluton of pressure values at P1&P8 (fg. (a)) and P5&P6 (fg. (b)) PT locatons. Pressure [bara] Pressure [bara] PT at P1 experm ent PT at P8 experm ent P T a t P 1 p re d ( D P co d e ) P T a t P 8 p re d ( D P co d e ) T m e [m s] a) PT at P1 experment PT at P8 experment P T at P 1 p re d ( D P c od e) P T at P 8 p re d ( D P c od e) T m e [m s] b) Fg. Pressure tme hstory at PT-P1&P8 (a) and PT-P5&P6 (b) case 1 18

4 Internatonal Journal of Chemcal and Bologcal Engneerng 6 01 Fg. 3 shows the decompresson wave speed, whch was determned from P1&P8 (fg. 3(a)) and P5&P6 (fg. 3(b)) locatons, correspondently. Pressure values are normalzed on the ntal pressure before rupturng. Expermental ponts are shown n all fgures as symbols. Contnues lnes represent predctons, whch are made by usng the proposed DP code. Calculatons of the decompresson wave speed by usng analytcal ASDECOM [8] are made by [1] settng up the same gas composton, ntal pressure and temperate (.e. case1). 1.0 Pressure rato [-] Pressure rato [-] experment (calc. from P1&P8) DP code (calc. from P1&P8) ASDECOM Decompresson Wave Speed [m/s] a) 1.0 experment (calc. from P5&P6) 0.9 DP code (calc. from P5&P6) ASDECOM decompresson process n base natural gases much better than other analytcal and mathematcal models, whch are avalable from the open source lterature. However, expermental data on the rapd gas decompresson process n gases are lmted. Therefore, the paper proposes to use 3D CFD model as a valdaton tool for 1D model calculatons when expermental measurements are absent. The dea to conduct a numercal study of a smple gas flow n a ppe by usng of 1D and 3D models s tested here. VII. NMERICAL ANALYSIS OF RAPID AS DECOMPRESSION IN PRE NITROEN SIN 1D AND 3D MODELS The numercal analyss of rapd gas decompresson n pure ntrogen usng 1D and 3D models s performed. The comparson of both predctedd results wll show the ablty of both approaches (1D and 3D) on successful smulaton of smple flows wth a hgh level of agreement between each other. It wll gve the addtonal confdence n 1D smulaton n cases when expermental measurements are absent or lmted. The same decompresson tube (Table I) s selected for smulatons usng both 1D and 3D codes. The predcton of the decompresson flow starts from no-flow stagnant condtons n the ppe. The ntal pressure s 10 MPa. The ntal temperature s 60 K. The PT locatons (PT-1,, 3, 4) for the case of pure ntrogen smulaton are dffered compare to the prevous case (fg. 4). All PT are located near the rupture end of the ppe. The flow through the rupture dsc begns after when rupturng s started Decompresson Wave Speed [m/s] b) Fg. 3 Decompresson wave speed as a functon of pressure rato Those analytcal data are taken from [1] n order to compare those predctons wth the DP calculatons (fg. 3). Broken curves represent ASDECOM numercal results [1]. All predctons on the tme evoluton of pressure values at dfferent PT locatons, whch are performed by usng the DP model, are n good agreement wth the expermental data. The presented 1D model predcts the decompresson wave speed, whch s determned from P5 and P6 PT locatons, much better than the analytcal ASDECOMP. The analytcal gas decompresson model ASDECOM [8] s one of the most commonly used engneerng software n ol and gas feld applcatons. However, the frcton between the gas and the ppe wall s not accounted for n the model. Ths fact s sgnfcantly lmted the applcaton of ASDECOM to the decompresson wave calculaton. The presented 1D mathematcal model of transent compressble thermal multppes predcts component gas mxture flow n the Fg. 4 Schematc of the decompresson tube Basc smulatons of the decompresson process n pure ntrogen by usng CFD Fluent code (ANSYS) are performed on the computatonal doman havng totally 300,000 computatonal cells. The length of the computatonal ppe s 10 meters long. A mesh refnement study was performed. Smulatons, whch are made on the refned computatonal mesh show grd ndependence compare to predctons, whch are performed on the basc computatonal mesh. Predctons of the decompresson n pure ntrogen are performed wth dfferent tme steps too. Tme step refnement study shows the tme-step ndependence of the results startng from the tme step, whch s equal to 10 5 seconds. Most of smulatons are 19

5 Internatonal Journal of Chemcal and Bologcal Engneerng made wth the tme step equal to 10 seconds. The real gas Redlch-Kwong (RK-EOS) model s chosen for predctons. Predctons, whch are made usng 1D DP code, are shown as symbols n fg. 5. Contnues and broken lnes represent calculatons, whch are performed by usng 3D CFD model. Pressure [bara] Pressure rato [-] 100 PT-1 DP code PT- DP code PT-3 DP code 90 PT-4 DP code PT-1 CFD Fluent 80 PT- CFD Fluent PT-3 CFD Fluent 70 PT-4 CFD Fluent Tme [ms] a) 1.0 pred (DP code) 0.9 pred (CFD Fluent) Decompresson Wave Speed [m/s] 350 b) Fg. 5 Predcted pressure tme hstory at dfferent PT locatons (a) and decompresson wave speed as a functon of pressure rato (b) Predcted values of the pressure tme evoluton at four dfferent PT locatons are shown n fg. 5(a). Calculatons, whch are performed by usng the 1D DP code and 3D CFD code, show a good agreement between each other for each PT locaton. The decompresson wave speed s determned from PT-1 and PT- (fg. 4). The dstrbuton of the decompresson wave speed values n pure ntrogen s shown n fg. 5(b). Smulaton results, whch are performed by usng 3D CFD code and 1D DP code, are shown n fgure both. Those numercal results are n a very good agreement wth each other n a wde range of pressure values. VIII. CONCLSION A numercal nvestgaton of the rapd gas decompresson n pure ntrogen, whch s made by usng the 1D and 3D transent mathematcal models of compressble thermal flud flow n ppes s performed. Results of ths numercal study are presented n the paper. A 1D mathematcal model of compressble thermal multcomponent flud mxture flow n ppes s presented. The set of the mass, momentum and enthalpy conservaton equatons for gas phase s solved n the model. Thermo-physcal propertes of mult-component gas mxture are calculated by solvng the Equaton of State (EOS) model. The model valdaton on the expermental data [1] shows a good agreement wth those measurements. Presented 1D mathematcal model predcts the decompresson n natural gas mxtures much better than the analytcal ASDECOM. A commercal 3D transent mathematcal model of compressble thermal snglecomponent gas flow n ppes s presented n the paper too. The set of unsteady Reynolds-averaged conservaton equatons for the gas phase s solved. Thermo-physcal propertes of snglecomponent gas are calculated by solvng the Real as Equaton of State (EOS) model. The smplest case of the gas decompresson n pure ntrogen s smulated usng both 1D and 3D models. The ablty of both models to smulate the process of the rapd decompresson n pure ntrogen wth a hgh order of agreement wth each other s tested. Both 1D and 3D numercal results show a good agreement between each other. Ths numercal nvestgaton shows that 3D CFD model s very helpful n order to valdate 1D smulaton results f the expermental data s absent or lmted. ACKNOWLEDMENT The author would lke to acknowledge the fnancal support of PETROSOFT D&C to develop the presented mathematcal model and to create the computer program (DP code). The author would lke to acknowledge the support of current employer A*STAR Insttute of Hgh Performance Computng (IHPC), Sngapore for gvng access to CFD Fluent code (ANSYS). REFERENCES [1] K.K. Botros, W. Studznsk, J. eerlgs, A. lover, Measurement of decompresson wave speed n rch gas mxtures usng a decompresson tube, Amercan as Assocaton Proceedngs (AA-003), 003. [] R.J. Eber, T.A. Bubenk, W.A. Maxey, Fracture control for natural gas ppelnes, PRCI Report Number L51691, [3] R.J. Eber, L. Carlson, B. Les, Fracture control requrements for gas transmsson ppelnes, Proceedngs of the Fourth Internatonal Conference on Ppelne Technology, p. 437, 004. [4] K.K. Botros, W. Studznsk, J. eerlgs, A. lover, Determnaton of decompresson wave speed n rch gas mxtures, The Canadan Journal of Chemcal Engneerng, vol. 8, pp , 004. [5] K.K. Botros, J. eerlgs, J. Zhou, A. lover, Measurements of flow parameters and decompresson wave speed follow rapture of rch gas ppelnes, and comparson wth ASDECOM, Internatonal Journal of Pressure Vessels and Ppng, vol. 84, pp , 007. [6] K.K. Botros, J. eerlgs, R.J. Eber, Measurement of decompresson wave speed n rch gas mxtures at hgh pressures (370 bars) usng a specalzed rupture tube, Journal of Pressure Vessel Technology, vol. 13, , 010. [7] K.K. Botros, J. eerlgs, B. Rothwell, L. Carlson, L. Fletcher, P. Venton, Transferablty of decompresson wave speed measured by a small-dameter shock tube to full sze ppelnes and mplcatons for determnng requred fracture propagaton resstance, Internatonal Journal of Pressure Vessels and Ppng, vol. 87, pp , 010. [8] ADECOM, Computer code for the calculaton of gas decompresson speed that s ncluded n Fracture Control Technology for Natural as 0

6 Internatonal Journal of Chemcal and Bologcal Engneerng 6 01 Ppelnes, by R.J. Eber, T.A. Bubenk, W.A. Maxey, N-18 report 08, AA Catalog N L51691, [9] E. Burlutsky, Mathematcal model of compressble non-sothermal flow of mult-component natural gas mxture n a ppe, Internatonal Scentfc Conference on Mechancs, S-Petersburg, Russa, January 01, to be publshed [10] E. Burlutsky, Mathematcal modellng of non-sothermal multcomponent flud flow n ppes applyng to rapd gas decompresson n rch and base natural gases, Internatonal Conference on Flud Mechancs, Heat Transfer and Thermodynamcs ICFMHTT-01 (15-17 Jan 01), Zurch, Swtzerland, to be publshed [11].B. Walls, One-dmensonal two-phase flows, Mcraw Hll, New York, [1] P.R.H. Blasus, Das Aehnlchketsgesetz be Rebungsvorgangen n Fluessgketen, Forschungsheft, vol. 131, pp. 1 41, [13]. Soave, Equlbrum constants from a modfed Redlch-Kwong equaton of state, Chemcal Engneerng Scence, vol. 7, pp , [14] A.L. Lee, M.N. onzales, B.E. Eakn, The vscosty of natural gases, Journal of Petroleum Technology, pp , 010. [15] S. Patankar, Numercal heat transfer and flud flow, Hemsphere Publshng, New York,