NOTICE: This is the author's version of a work that was accepted for publication in Chemical Engineering Research and Design. Changes resulting from

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1 NOTICE: This is the author's version of a work that was acceted for ublication in Chemical Engineering Research and Design. Changes resulting from the ublishing rocess, such as eer review, editing, corrections, structural formatting, and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for ublication. A definitive version was subsequently ublished in Chemical Engineering Research and Design,Volume 87, Issue 1, Pages (December 009), DOI: /.cherd

2 Sensitivity of ieline gas flow model to the selection of the equation of state Macie Chaczykowski Warsaw University of Technology, Faculty of Environmental Engineering, Heating and Gas Systems Deartment, Nowowieska 0, Warszawa, Poland, ABSTRACT Real gas effects exert a significant influence on the hydraulics of natural gas transmission ielines. In this article the imlications of the selection of the equation of state for the ieline gas flow model are investigated. A non-isothermal transient gas flow model with AGA-8 and SGERG-88 equations of state was studied. Models with Soave-Redlich-Kwong and Benedict-Webb-Rubin equations of state were solved to illustrate the overall gas flow model inaccuracies. The effect of the selection of different equations of state on the flow arameters is demonstrated and discussed. KEYWORDS Comressibility factor; Equation of state; Natural gas ieline; Transient non-isothermal flow NOMENCLATURE: A - cross-section area of the ie, m, B - second virial coefficient, m 3 /kmol, C - third virial coefficient, m 6 /kmol, c - secific heat at constant ressure, J/(kgK), c v - secific heat at constant volume, J/(kgK), D - ie diameter, mm, f - Fanning friction coefficient, -, g - the net body force er unit mass (the acceleration of gravity) m/s, h - secific enthaly, J/kg, k - heat transfer coefficient, W/(mK), L - ieline length, m, m - heat-transfer element mass, kg, - gas ressure, Pa, q - rate of heat transfer er unit time and unit mass of the gas, W/kg, Q - volumetric flowrate, m 3 /s, R - secific gas constant, J/(kgK), t - time, s, T - gas temerature, K, T soil - soil temerature, K, u - secific internal energy, J/kg, v - secific volume, m 3 /kg, w - flow velocity, m/s, x- satial coordinate, m, z - comressibility factor, -.

3 Greek symbols α - angle between the direction x and the horizontal, ε - ie roughness, mm, λ - thermal conductivity, W/(mK), µ - viscosity of natural gas, N s/m, ρ - density of the gas, kg/m 3, ω - acentric factor, -. Subscrits s - standard conditions, c - critical, m - molar, amb - ambience. Suerscrits k - iteration index Note: Flow rate Q s is shown in the standard conditions of K, 0.1 MPa Substantial derivatives are indicated as d/dt 1. INTRODUCTION Pieline simulations are widely used by gas transmission oerators, that are obliged to ensure that the system is balanced and that deliveries of the gas are maintained. According to EU regulatory framework, the resonsibility for the hysical balance of the system is imosed on the ieline oerator and the balancing should be carried out on a daily and monthly basis. This decision has many ramifications in the field of flow measurement, among other things is an increased imortance of the accuracy of ieline simulation, which is used for the determination of system line-ack (gas network accumulation) on a hourly and daily basis. Pieline leak detection system based on volume balance methods is another examle of where the accuracy of simulation results is an imortant matter. Physical balancing of the system can be considered as a management of system line-ack. The minimum ieline line-ack is the amount of gas in the ieline required to achieve the desired gas flow, and the required delivery ressure. Physical balancing of the system can be achieved by an adequate amount of storage caacities as well as through the variations in the system line-ack. The latter suorts the hourly modulation of gas delivery and suly rates and is determined by means of ieline simulation in order to reclude ressure values above ermissible limits. Extra line-ack is necessary to rovide some flexibility to accommodate variations in gas demand and a safety margin in emergency situations. Seeing that safe and efficient gas transmission requires hysical balancing of the system as a necessary condition to ensure correct technical oeration of the network, ieline oerators, and in articular their disatching centres, control gas transmission arameters such as flowrates and ressures using real time gas network simulators based on transient flow models. Gas flow models with literature review of their solution methods are widely discussed by Thorley and Tiley (1987) and references therein. Many secific contributions were also discussed by Osiadacz (1996). Modelling of ieline flow transients behaviour requires an alication of the equation of state. Modissette (000) rovided a review of equations of state commonly used in the gas industry for comressibility factor calculations. Some equations reviewed included the NX- 3

4 19, AGA-8, Benedict-Webb-Rubin (BWR) and Soave-Redlich-Kwong (SRK) equation of state. There are two different alications which affect the choice of the equation of state when modelling flow of natural gas in ielines. These are custody transfer measurements and ieline simulation. Custody transfer has its financial asects and legal regulations that imose restrictions on the selection of the equation of state. Natural gas comosition and the range of ressure and temerature values occurring in gas transmission ielines enable the use of equations of state of moderate comlexity in gas industry. Currently AGA-8 and SGERG-88 equations of state are widely used by American and Euroean ieline oerators. In engineering ractice, the gross characterization method of gas comosition is widely used in ieline simulations since a comlete comositional analysis of natural gas often is not available. The method does not consider the detailed gas comosition, but considers hydrocarbons collectively as an equivalent hydrocarbon gas. The molar heating value of the gas mixture with the equivalent hydrocarbon gas is equal to the heating value of the natural gas. The calculation of molar mass, molar heating value and mole fraction of the equivalent hydrocarbon gas is erformed iteratively. The convergence criterion is the absolute difference between the calculated density of the gas mixture with the equivalent hydrocarbon gas and the density of the natural gas at standard conditions. Non-isothermal gas flow models are already widely used in ieline simulations. In regard to comressibility factor, it is treated either as a constant arameter or as a function of temerature and ressure in the flow model. However, we are not aware of any ublished work in the field of single hase natural gas ieline simulation, with more comlex and accurate equations of state than two-arameter correlations in the flow model. The obective in this aer is to focus on a transient gas flow modelling with the alication of various equations of state including those frequently used in gas and etroleum industry. A sensitivity analysis of transient ieline gas flow model to the selection of the equation of state has been carried out. Non-isothermal gas flow model comrises a) AGA-8, b) SGERG- 88, c) BWR and d) SRK equations of state, and their influence on flow arameters, esecially on the gas temerature and ieline line-ack is resented.. TRANSIENT GAS FLOW MODEL The unsteady one-dimensional comressible flow within a gas ieline is described by a set of artial differential equations exressing mass, momentum and energy conservation laws as follows ρ ( ρw + ) = 0 (1) t x ( ρw) ( + ρ ) w f ρw w + = ρ g sinα () t x D w w u + ρ + h + ρw = ρq ρwg sinα (3) t x Eqs. (1) (3) may be rewritten in terms of ressure and the volumetric flow rate under standard conditions instead of density and velocity, resectively. This is a matter of convenience, since these quantities are commonly measured and used in the gas industry. As a result, equation of state which would exress the density in terms of ressure and temerature is needed to close the system of the above equations ρ = (4) zrt 4

5 Eq. (1) can be rearranged in the form 1 dρ w + = 0 (5) ρ dt x Rewriting Eq. (4) in logarithmic form and differentiating with resect to time yields 1 dρ 1 d 1 dt 1 dz = (6) ρ dt dt T dt z dt For a given gas comosition, the comressibility factor can be exressed as a function of z = z, T thus ressure and temerature ( ) z z dz = d + dt (7) T T Substituting Eq. (7) into Eq. (6) we obtain 1 dρ 1 1 z d 1 1 z dt = + (8) ρ dt z dt T z T dt T The velocity in terms of ressure, temerature and volumetric flowrate at standard conditions ρsqs zrt w = (9) A Differentiation of the logarithmic form of the Eq. (9) with resect to satial coordinate yields 1 w 1 Qs 1 1 T 1 z = + + (10) w x Qs x x T x z x The last term on the right-hand side of the Eq. (10) can be substituted using the Eq. (7). Therefore 1 w 1 Qs 1 1 T 1 z 1 z T = (11) w x Qs x x T x z x z T T x Substituting Eqs. (8) and (11) into the Eq. (5) gives the following form of continuity equation exressed by quantities which are directly measured 1 1 z 1 1 z T ρszrt Qs + + = 0 (1) z t T z T T t A x Eq. () can be rewritten in the form dw 1 fw w g sinα = 0 (13) dt ρ x D Differentiating the logarithmic form of Eq. (9) with resect to time we obtain 1 dw 1 dqs 1 dρ = (14) w dt Qs dt ρ dt Substituting Eq. (8) into Eq. (14) and combining this with Eq. (13) results in Qs ρsqs zrt Q s 1 1 z ρsqs zrt + Qs + t A x z t a x T (15) 1 1 z T ρ fzrt sqs zrt T A ρsqs Qs + Qs = 0 T z T t A x ρs x DA Some terms in Eq. (3) can be canceled out since the mass and the momentum are also conserved. Eq. (3) can be converted using Eq. (1) to the following form dh dw ρ ρw = ρq ρwg sinα (16) dt dt t Combining this with Eq. () gives the following derivation 5

6 3 dh d f ρw ρ = ρq (17) dt dt D By using the thermodynamic identities dh du d = + ρ, d d u = c d v T + T v T v the following form of energy equation is obtained T ρsqs zrt T RT ρsqs zrt 1 1 z + + zt + t A x cv A T z T ρ 1 Q s 1 1 z 1 1 z T + + Qs x z x T z T T x 3 f zrt ρs Qs q = 0 cvd A cv The first and the second term of Eq. (18) reresent the time rate of change of the temerature of the gas as it flows along the ieline. The third term reresents the real gas effects resulting from the enthaly deendence on ressure and temerature, and in the range of ressure and temerature values reresentative for natural gas transmission ielines it has a ositive value, causing the exanding gas to cool. The fourth term reresents heating of the gas from friction. The last term in Eq. (18) reresents the heat transfer from the gas to the ieline surroundings and has a significant effect on the gas arameters obtained from the solution of the above model. Heat transfer in gas ielines was studied by Gersten et al. (001). Based on a steady-state non-isothermal gas flow model in both onshore and offshore ielines, they showed that considering heat transfer reduces uncertainties in lanned transort caacities and ressure losses. Paer (Osiadacz and Chaczykowski, 001a) resents comarison of isothermal and non-isothermal ieline gas flow models. Non-isothermal model contained simlified form of energy equation with enthaly and internal energy calculated from ideal gas equations, and the steady-state heat transfer term for calculation of the heat transfer from the gas to the surrounding soil. Nevertheless, it has been shown that there exists a significant difference in the ressure rofile along the ieline between isothermal and non-isothermal flow rocesses. The use of an isothermal model may lead to significant errors in calculation of the energy consumtion of the drivers of the comressors. The work of Modisette (00) concluded that the accuracy of the heat transfer model affects both line-ack and ressure loss in gas ielines. Recently, non-isothermal transient flow of natural gas in a ieline was studied by Abbasour and Chaman (008). They showed that the effect of cooling of the gas due to exansion is significant on the temerature distribution, and the effect of treating the gas in a non-isothermal manner is very necessary for ieline flow calculation accuracies, esecially for raid transient rocess. Various methods are used for estimation of the heat transfer term in the energy equation, most of which assume modification of steady-state heat flow exression. Generally, models describing the heat transfer to the surroundings in gas ielines are one-dimensional, due to the lack of accurate data describing ground roerties. (18) 3. HEAT-TRANSFER MODEL In the energy equation, the heat transfer term q reresents the amount of heat exchanged between unit mass of gas and the surroundings er unit time. Alication of Fourier s law to 6

7 calculate the overall heat-transfer between the gas and the ground, for a discretisation section of a ieline, yields qρ Adx = k ( T Tamb ) dx where k is an overall heat transfer coefficient and T amb is the ambient temerature at the sane horizontal level as ie axis, but at a sufficient lateral distance from the ie. Therefore k qρ = ( T Tamb ) (19) A There exists an analytical steady-state solution for k for a cylinder near a half-lane, which corresonds to the geometry of a buried ieline. Nevertheless, it is a common ractice to calculate k as for a concentric cylindrical layer, with the distance between the outer boundary and the ie equal to the burial deth of the ie. The ambient temerature is fixed and equal to the ground temerature at the same horizontal level as the ie axis, and at a sufficient lateral distance from the ie. Fig. 1. Heat transfer area discretization scheme In this work, however, the rocess of heat transfer from the gas to the surrounding environment is described using unsteady heat transfer model, so that the descrition of heat flux could take into consideration the effect of heat caacity of the surroundings of a ieline. Using the element method, one-dimensional axial-symmetric heat exchange rocess can be exressed by the following set of equations, reresenting thermal balances of the elements coaxial cylindrical layers (Fig. 1). k0 qρ = ( T T1 ) A m1c 1 T1 = k0 ( T T1 ) k1 ( T1 T ) dx t mc T = k1 ( T1 T ) k ( T T3 ) (0) dx t mnc n Tn = kn 1 ( Tn 1 Tn ) kn ( Tn Tamb ) dx t where n is the number of discretization sections of heat-transfer area (equal to number of elements), m i is element mass (i = 1,... n), c i is the is the secific heat of element i, m i c i is the element heat caacity, dx is the discretization section of a ieline, T i is the element temerature and k i is the heat transfer coefficient between elements (i 1) and i (k 0 denotes heat transfer coefficient between the gas and the first element). In case of one-dimensional aroach, the rocess of heat transfer may be modelled by a minimum two cylindrical layers 7

8 as heat caacitors. Assuming substantially different heat caacity of the layers, so that their time constants were different, the near and the remote surrounding of the ieline would therefore resond to temerature changes quickly and slowly, resectively. It has been assumed for the urose of heat-transfer area discretization in this study, that every element has the same thermal resistivity. Thus, the temerature difference between consecutive ground sections (element surfaces) are equal in steady state, and the initial condition can be accurately modelled. The technique for heat transfer modelling resented above and its alicability to calculate flow arameters in the gas ieline has been evaluated in the case study resented in this work. 4. EQUATIONS OF STATE According to AGA8/199 (Comressibility Factor of Natural Gas and Related Hydrocarbon Gases, AGA Reort No. 8, American Gas Association, Arlington, VA.) and ISO 113-3:1997 (Natural gas calculation of comression factor Part 3: Calculation using hysical roerties), the equation of state for the calculation of comressibility factor of natural gas is in the form of the virial exansion Z = 1 + Bρm + Cρm (1) where ρ m molar density of the gas, kmol/m 3. It is convenient to rewrite Eq. (1) as a series in owers of ressure instead of molar density, which would be somewhat better form considering the deendent variables of the system of Eqs. (1), (15) and (18). An equivalent form used for calculation of the derivatives of comressibility factor is Z = 1 + B + C () Virial coefficients in Eq. () are calculated from the original virial coefficients by equating (1) and () and solving the original virial exansion for. The new virial coefficients in terms of B, C are B B = RT C B C = ( RT ) Therefore the equations for the first derivative of the comressibility factor with resect to temerature and ressure are z db B dc db = + B + 3 ( B C ) T RT dt T RT dt dt R T ( ) 8 ( C B ) z B = + RT T ( RT ) For given gas comosition the virial coefficients are functions of temerature only. The AGA8/199 and ISO 113-3:1997 standards give constants, gas arameters and mixing rules for the calculation of the virial coefficients in Eq. (1). SRK equation of state and BWR equation of state were taken for comarison of the flow models. The following form of SRK equation, allowing for convenient comressibility factor calculations was used in this work where A = aα ( RT ) ( ) 3 Z Z Z A B B AB + = 0 (3)

9 b B = RT R Tc a = RTc b = c c T α = 1 + m 1 T c m = 0, ,574ω 0,176ω For iterative comressibility factor calculations Newton s method was used. BWR equation of state is in the following form 3 C cρ γρ = RT ρ + B0RT A0 ρ + ( brt a) ρ + aαρ + ( 1+ γρ ) e T T The values of eight coefficients: A 0, B 0, C 0, a, b, c, α and γ deend on gas comosition only. Alication of Newton s method enables iterative density calculation. Once the density is known, comressibility factor is calculated from the Eq. (4). 5. SOLUTION METHOD Eqs. (11), (14), (17) comrise the set of hyerbolic artial differential equations with ressure, flow and temerature as a function of time and location. It is solved by the method of lines with a five-oint biased uwind aroximation scheme for satial derivatives T ρnqn zrt RT ρnqn zrt 1 1 z = X ( T ) zt + t A cv A T z T ρ z 1 1 z X ( Qn ) X ( ) + + X ( T ) Qn z T z T T 3 f zrt ρn Q n q + + = 0, = 0, 1,..., N cvd A c v (4) z 1 1 z T ρnzrt 1 1 z = + X ( Qn ), = 0,1,..., N t T z T z t T a z T Q ρnqn zrt n 1 1 z ρnqn zrt = X ( Qn ) + Qn + X ( ) t A z t a T 1 1 z T ρ nqn zrt T A fzrt ρnqn Qn Qn + + X ( ), = 0,1,..., N T z T t A x ρn DA T where is the satial coordinate discretisation section index and X ( T ) =, X ( ) =, x x Qn X ( Qn ) =. The five-oint differentiation formula for satial derivative of ressure is x given below as an examle 9

10 d ( x0 ) dx d ( x1 ) dx ( x0 ) d ( x) ( x 1) dx ( x ) 1 X ( ) 4 = = 1 x + O( x ) d ( xn ) ( xn ) dx ( xn 1) d ( xn 1) ( xn ) dx d ( xn ) dx This aroximation is fourth-order correct, i.e. the truncation error is roortional to x 4. The derivative (x) aroximated at oint x is based on function values at grid oints x, x 1, x, x +1, x +. For a detailed descrition of the solution algorithm see aer (Osiadacz and Chaczykowski, 001b). 6. SIMULATION RESULTS AND DISCUSSION In the case study, different equations of state were incororated into the gas flow model in order to determine the influence on the flow arameters such as ressure, temerature, flowrate and line-ack. Line-ack calculations enable the ieline oerator to ustify for how long the sulies of the gas could be continued in case of roduction stos, and otimize its oeration strategy. The Yamal-West Euroe ieline on Polish territory was selected as a test network, in articularly the 363 km ie section between Kondratki and Wloclawek comressor stations (Fig. ). This is a tyical onshore gas transmission system with a maximum oerating ressure 8,4 MPa. Fig.. Section of the gas transortation system In the numerical calculations the following data were used: Gas: The gas is a 9-comonent mixture with a molar comosition x of CH , C H , C 3 H , i-c 4 H , n-c 4 H , i-c 5 H , n-c 5 H , N and CO The density ρ n = kg/m 3, dynamic viscosity µ = 1.59 µpa s, thermal conductivity λ = W/m K. Pie: The distance between the comressor stations is L = 363 km and the ie diameter d o = 14 mm. The average roughness of the ie ε = mm. The roerties of the ie wall are listed in Table 3. 10

11 Soil: The thermal conductivity λ = 1.0 W/m K, density ρ = 1640 kg/m 3, secific heat at constant ressure c = 1530 (J/kg K) and the ie deth z = 1.5 m. The soil temerature is 3.1 C. The boundary conditions are 0, t = 8.4 MPa T ( ) ( t) = (, ) = ( ) n 0, K Q L t f t where f(t) is deicted in Figure 3 with a time interval, t [0, 40]. The eriod function is arbitrarily selected. In case of AGA, BWR and SRK equations of state, values of the derivatives of comressibility factor were calculated based on olynomial regression with degree two. Van der Waals (classical quadratic) mixing rules were used to calculate the coefficients of SRK equation of state for gas mixture. Tests have shown that the convergence in the solutions of SRK and BWR equations is fast, and the results are obtained within three iterations in this articular case study. Fanning friction coefficient was calculated using Colebrook-White equation. Fig. 3. Change of flowrate at x = L (boundary condition) Figures 4-6 resent a grahical interretation of the calculated flow arameters at the inlet and outlet nodes of the ieline. Figures 4 and 5 clearly show a minor effect of the tye of the equation of state in gas flow model on ressure and flowrate values in the ieline. Changes of the results obtained from different equations of state are insignificant. It can be seen from Figure 6 that the value of the gas temerature at the outlet node dros below the surrounding soil temerature (3.1 C), which is a demonstration of the effect of cooling of the gas due to exansion. The value of the gas temerature decreases when increasing the flowrate, because larger ressure dro in the ieline and larger exansion of the gas downstream of comressor station is observed. The temerature tends to be lower in the case of SGERG and AGA equations of state. From this we can assume that comressor station ower, calculated based on the simulation results obtained from these equations, will be a little smaller than in the case of the SRK and BWR equations of state (reduced cost and/or higher throughut). 11

12 Figure 7 shows the influence of different equations of state on the ieline line-ack. Differences between the equations exist but are relatively moderate. The largest difference between line-ack value with AGA and BWR equations amounts to m 3 which corresonds to 5,9% of the actual flowrate ( m 3 /h) at the outlet node of the ieline, and is equal to 0,3% of the average value of the line-ack obtained from the solutions of the all analysed equations of state. Fig. 4. Change of suction ressure at Wloclawek (x = L) Fig. 5. Change of flowrate at Kondratki (x = 0) 1

13 Fig. 6. Change of temerature at x = L Fig. 7. Change of ieline line-ack 7. CONCLUSION This aer has considered the ossible imrovements in the simulation accuracy by studying the influence of the equation of state on the non-isothermal flow model, which is used in the evaluation of control and oerating strategies of the tyical onshore gas ieline. The results of comarison of flow arameters and ieline line-ack values show relatively small influence of the tye of the equation of state on the simulation results. The character of the results cannot be easily generalized, but closer observation suggests that the form of the 13

14 equation of state might be regarded as a contributory factor in leak detection systems based on volume balance methods. There are many arameters associated with a mathematical descrition of fluid flow transients within the ieline, for which it is difficult to determine aroriate values: friction factor, heat transfer coefficient, seed of sound, ambient temerature, actual ieline geometry. By using measured resonse data for the ieline under a range of oeration conditions and regimes, comarison with the simulation results can be made. It is then ossible to validate the model with estimated values of these arameters, and rovide a means of establishing an imroved efficiency of the overall oeration of the ieline. REFERENCES Abbasour, M. and Chaman, K.S., 008, Nonisothermal transient flow in natural gas ieline, J. Al. Mech., Trans. ASME, 75 (3): Gersten, K., Paenfuss, H.D., Kurschat, T., Genillon, F., Fernandez, P., Ravell, N., 001, Heat Transfer in Gas Pielines, Oil Gas Eur. Mag., 7 (1): Modisette, J.L., 000, Equations of state tutorial, Proceedings of the PSIG The 3th Annual Meeting, Savannach. Modisette, J., 00, Pieline thermal models, Proceedings of the PSIG The 34th Annual Meeting, Portland. Osiadacz, A.J., 1996, Different transient models - Limitations, advantages and disadvantages, Proceedings of the PSIG - The 8th Annual Meeting, San Francisco. Osiadacz, A.J. and Chaczykowski, M., 001a, Comarison of isothermal and non-isothermal ieline gas flow models, Chem. Eng. J., 81 (1-3): Osiadacz, A. and Chaczykowski, M., 001b, The thermodynamics of ieline gas flow, Arch. Thermodyn. (1-): Thorley, A.R.D. and Tiley, C.H., Unsteady and transient flow of comressible fluids in ielines - a review of theoretical and some exerimental studies, 1987, Int. J. Heat Fluid Flow, 8 (1):