The Mathematical Models of Gas Transmission. at Hyper-Pressure

Transcription

1 Alied Mathematical Sciences, ol. 8, 014, no. 14, HIKARI Ltd, htt://dx.doi.org/ /ams he Mathematical Models of Gas ransmission at Hyer-Pressure G. I. Kurbatova Saint-Petersburg State University, Russia N. N. Ermolaeva Saint-Petersburg State University, Russia Coyright 014 G. I. Kurbatova and N. N. Ermolaeva. his is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Abstract Different aroaches to the modeling of thermodynamic rocesses in a gas flow at hyer-ressure are discussed and their equivalence is roved for any form of equation of state. he article resents arguments in favor of the aroach in which temerature and density are chosen as indeendent thermodynamic variables and equation of sate is written in one of its analytical forms, for examle, in the form of Redlich-Kwong equation of state. Exlicit analytical exressions are obtained for the deendences of internal energy and secific heat at constant volume on gas temerature and density. It is roved that simlifications of the model of thermodynamic rocesses, which are based on the assumtion of incomressibility of gas flow, are inadmissible, even in case of negligibly small inertial forces. Keywords: mathematical models, gas-ielines, thermodynamic rocesses, equation of state, Joule-homson effect 1. Introduction Modern gas-ielines oerate at hyer-ressure conditions (~5 MPa) and this comlexifies modeling of thermodynamic rocesses, which is resonsible for adequacy of the mathematical model used.

2 619 G. I. Kurbatova and N. N. Ermolaeva In the 90 s of last century in St.-Petersburg State University (Russia) a quasi-onedimensional model of steady flow of a real multicomonent gas mixture in a sea hyerressure gas-ieline was develoed, with the model taking into consideration radial distribution of gas velocity, relief of the ground, construction of ieline coating, and ossibility of ieline glaciation. his model was described in details in the book Models of sea gas-ielines [1]. he model was successfully used in simulating gas flow in ielines running from Shtokman gas-field in the Barents Sea and in the North Euroean Gas Pieline (Nord Stream) in the Baltic Sea. In [], the validity of reduction of a quasi-one-dimensional model to one-dimensional one was roved for the case of a high value of the Reynolds number (Re>10 6 ). In [3], the model was generalized for the case of unsteady rocesses. he goal of the resent study is to analyze aroaches to the modeling of thermodynamic rocesses in different mathematical models of gas transmission 1, as well as to demonstrate how simlifications of mathematical models can result in rincially wrong inferences about temerature distribution in a gas flow.. General model One of rather general mathematical models of the above-mentioned rocesses is the BK (asil ev-boldyrev-oevodin-kanibolotsky) model ublished in 1978 in [5], and u to now this model is often in use [6,7,8]. Our mathematical model develoed in [1,3] mainly differs from the BK model in the descrition of thermodynamic rocesses, while in both models the resentations of continuity equation (mass conservation), equation of motion (or momentum equation), and energy equation (law of conservation of energy) coincide with an accuracy of u to symbols: continuity equation ( u) 0, t z equation of motion ( u) u u t z 4R energy equation ( u ) g cos ( z), (1) () 1 For examle, a brief review of such models is resented in [4].

3 Mathematical models of gas transmission 6193 t z R W ( e) * u e ( ) ug cos ( z ), and relations between energy, internal energy, and enthaly are (3) e u /, i /. (4) he set of Eqs. (1)-(4) should be sulemented by equation of state (, ) (5) and caloric equation of state either in terms of enthaly i i(, ), (6) or in terms of internal energy (, ). (7) We use the following designations: u, ρ,, and are the flow velocity, the density, the ressure, and the temerature of a gas mixture, corresondingly, which are functions of time t and coordinate z coinciding with the gas-ieline axis; e, ε, and i are the mass densities of energy, internal energy, and enthaly which are also functions of t and z; R is the inner radius of the gas-ieline; W is the exlicit function of the coefficients of inner and external heat exchange, the thicknesses of the layers comrising the coat of the gas-ieline, and the heat conductivities of these layers (examles of function W are resented in [1]); g is the gravity acceleration; α(z) is the angle between the gravity vector and Z-axis; λ is the coefficient of hydraulic resistance, which can be exressed in terms of the Reynolds number Re and the coefficient of relative roughness of inner wall of the ieline k (λ= λ(re,k), for examle, see the emirical equation of Colebrook- White [1]); * is the ambient temerature. heoretically, the model based on Eqs. (1)-(7) is equal to the one which, instead of the energy equation Eq.(3), includes balance equation for internal energy d u u u dt z R W 4R ' * ( ), (8) where d u is the oerator of material derivative. Eq.(8) can be derived from dt t z Eqs. ()-(4) using the well-known rocedure described in [9]. From the view-oint of the simlicity of numerical simulation the model including energy equation, rather than One of these layers could be ice that is tyical for sea gas-ielines.

4 6194 G. I. Kurbatova and N. N. Ermolaeva balance equation for internal energy, aears to be referable because it tolerates the usage of conservative difference schemes. 3. Modeling of thermodynamic rocesses here are two aroaches to the modeling of thermodynamic rocesses in gas flow. In the first one [5], and are chosen as indeendent thermodynamic variables; the equation of state Eq.(5) and the caloric equation in terms of enthaly Eq.(6) take the forms Z(, ) R g, (9) i(, ) cd d 0 0, (10) where Z(,) is the comressibility factor, is the secific volume (=1/ρ), Rg is the constant equal to Ro/M (Ro is the gas constant and M is the molecular weight of the gas mixture). c() is the temerature deendence of secific heat at constant ressure, which is considered unknown. Numerous works based on this aroach differ just in the deendence Z(,) used [5,10,11]. In the first aroach the balance equation for internal energy Eq.(8) is rewritten as d d * u u c ( ). dt dt R W 4R (11) he derivation of Eq.(11) from Eq.(8) is based on the following relations: d d i d i d d i, (1) dt dt dt dt dt i i, c, (13) d 1 1 u. dt z (14) Eqs. (13) are the well-known thermodynamic equalities [1], Eq. (14) ensues from the continuity equation Eq. (1) and definition of material derivative. Eqs. (1), (), (11), and (9) form a closed system of equations for unknown functions ρ, u,, and, which, being sulemented by initial and boundary conditions, enables one to calculate all characteristics of gas flow.

5 Mathematical models of gas transmission 6195 In the second aroach [1], the indeendent thermodynamic variables are ρ and, while one of the analytical forms of the generalized an der Waals equation is used as equation of state. By now, a vast number of such analytical forms has been roosed, but the most known ones are the Redlich-Kwong, Soave, Peng-Robinson, Benedict- Webb-Rubin equations of state [7, 11, 13]. In our works [1,3] as well as in [4,7] the two-arameter Redlich-Kwong equation of state is used, for this equation is considered one of the best which works in a wide range of variations of, ρ and, including hyerressure range. For a gas mixture it can be written as [13]: R c 0, (15) where 1/ 1 M 1 c R M and br 0 c Mc, Ωa and Ωb are the known constants, a 0 5/ c c c and c are the critical ressure and temerature of a gas mixture of given chemical comosition. he values c and c can be determined using the tables resented in [13]. In the second aroach, caloric equation is written in terms of internal energy ε(,), and the balance equation of internal energy Eq. (8) is transformed into d u * u u c ( ). dt z R W 4R (16) he derivation of Eq.(16) from Eq.(8) is based on the following relations: d d d dt dt dt,, c, (17) d dt 1 u z, where c is the secific heat at constant volume, which for a real (unideal) gas mixture is a function of ρ and. he derivation of the first relation of the set of equations Eq.(17) can be found in [1]. In accordance with the Redlich-Kwong equation of state the derivative ( / ) is determined as h 1 c 1 (1 ) 3/. (18) Eqs. (1), (), (16), and (15) form a closed set of equations for unknown functions ρ, u,, and.

6 6196 G. I. Kurbatova and N. N. Ermolaeva It is arguable that Eq. (11) and Eq. (16) are equivalent for any equation of sate. o make sure that it is true let us show that d d d u c c. dt dt dt z (19) Using the known thermodynamic relations [1] c c,, (0) Eq. (0) and Eq. (14), and introducing the following designations a, b, (1) let us transform the right side of Eq. (19) as follows: d u d d c a ( c ab) a dt z dt dt d d a d c ba. dt dt b dt For any equation of state =(,) the equality () d d d, dt dt dt is valid. Given Eq. (0) and Eq. (1), this equality can be rewritten as d b d d b dt a dt dt that allows writing the right side of Eq. () as follows d d a d d d c b a c b. dt dt b dt dt dt hus, Eq. (19) is valid and, therefore, the equivalence of Eq.(11) and Eq.(16) is roved. his evidences the theoretical arity of the two aroaches to the modeling of thermodynamic rocesses in gas flow. he choice of either one aroach or another is related to the choice of equation of state. If the equation of state in the form of Eq. (9) is used, secific volume can be resented as an exlicit function of and and, therefore, the derivative ( / ) can be easily calculated, that makes the first aroach exedient. If an analytical form of equation of state is used, ressure is an exlicit

7 Mathematical models of gas transmission 6197 function of and and, therefore, the derivative ( / ) can be calculated. In this case the second aroach is more exedient. If both equations of state are caable of giving similar and adequate interrelations between values, ρ, and, both aroaches can be used. However, the roblem to find an exression which could adequately describe the comressibility factor Z(,) in a wide range of and aears to be rather difficult. hat is why the use of suitable analytical forms of equation of state and, therefore, the second aroach is more referable, esecially in the case of hyer-ressure. 4. Deendences ε(ρ,) and c(ρ,) Within the frames of the second aroach let us find the deendences of internal energy and secific heat on density and temerature. Internal energy ε being a function of and, that is ε=ε(,), its total differential dε can be written as d c d d. Above we have written Eq. (17) from which one can obtain the well-known Helmholtz s equation. (3) After integration of Eq. (3) we have (, ) (, 0 ) d. 0 (4) Let 0, then any gas mixture can be considered an ideal gas and, therefore, ε(,0) ε0(), where ε0() is the internal energy of an ideal gas [14]. Now let us calculate the integral in the right side of Eq. (4). For the Redlich-Kwong equation of state the following exression 3 c ( ) is valid, and the integral can be calculated as 3 c d 3 c d 1/ 1/ ( ) 1/, ln(1 ),

8 6198 G. I. Kurbatova and N. N. Ermolaeva so that the internal energy ε(ρ,) of a gas mixture obeying the Redlich-Kwong equation of state becomes equal to 3 c (, ) 0( ) ln(1 ). 1/ his exression for ε(ρ,) allows finding c as an exlicit function of ρ and : c d 0 3 c (, ) 3/ d 4 ln(1 ). Internal energy of an ideal gas ε0 is known to be a function of temerature only: ε0()= ĉ, where ĉ is the secific heat of an ideal gas (including ideal gas mixtures). hus, the secific heat and the internal energy of a real gas mixture obeying the Redlich- Kwong equation of state take the forms c 3 c, ) cˆ ln(1 ), (5) 3/ 4 ( 3 c (, ) cˆ ln(1 ). (6) 1/ Numerical simulations of the model based on Eqs. (1), (), (16), and (15) show that allowance for the deendence c(ρ,) (Eq. (5)) in solving Eq. (16) mostly influences the resultant temerature field and it roves to be rather imortant in the comuter modeling and engineering of sea gas-ielines in the North. Above it is mentioned that for numerical simulations the model based on the energy equation Eq. (3) is more referable. In this case the functions to be found are ρ, u, ε, and. Eq. (6) enables one to exress in the right side of Eq. (3) via functions ρ and ε. Eq. (6) is a cubic equation with resect to 1/, with only one root having hysical meaning. his root can be easily found by means of standard techniques. In the first aroach to the modeling of thermodynamic rocesses the functions to be found are ρ, u, i, and. he deendence i(,) can be found from Eq. (10) and Eq. (9) and this makes ossible to resent (in the right side of Eq. (3)) as a function of enthaly i and temerature. 5. Simlification of mathematical models In solving certain roblems, simlified variants of the model Eqs. (1)-(7) are often used. he simlifications are based on the following assumtions; all the rocesses are steady, isothermal, and adiabatic, as well as the inertial and gravity forces are ignored.

9 Mathematical models of gas transmission 6199 Let us dwell uon the assumtion ertaining to the ignoring of inertial forces, which is used in many similar studies. Main conditions of the exloitation of different modern gas-ielines are about the same; gas density in the hyer-ressure range is equal to about 160 kg/m 3 and, in case of the gas rate Q in the ie and the ie diameter D being equal to about 450 kg/s and 1 m, corresondingly, such density conditions rather low gas flow velocity u (u 3.5 m/s). Under these conditions Mach s number is 1 that enables the gas comressibility not to be taken into consideration. However, comlete ignoration of gas comressibility in the modeling of gas transmission rocesses is inadmissible, because, as it is shown below, it leads to absurd temerature distributions in gas-ielines. Let us consider a steady gas flow in a horizontal gas-ieline thermally isolated from outside ambient. Now we will use the first aroach to the modeling of thermodynamic rocesses. he above-mentioned assumtions allow reducing the model Eqs. (1), (), (11), and (9) to the following set of equations: us Q const, (7) d uc u u ( u ), (8) D d d u u u, (9) D R Z(, ), (30) g where Eq. (7) is the integral of the continuity equation Eq. (1), S is the area of the ieline cross-section. Let us ignore the inertial forces in the equation of motion Eq. (8) and write u. (31) hen the reduced equation of motion takes the form d u u. (3) D Using Eq. (3) we rewrite Eq. (9) as d 1 d d, c (33) where μ is the Joule-homson coefficient. For the equation of state Eq. (30) this coefficient is known [1] to be defined as Rg Z c,

10 600 G. I. Kurbatova and N. N. Ermolaeva that allows rewriting Eq. (33) as follows: d Rg Z d. (34) c he set of equations Eqs. (7), (3), (34), and (30) resents a simlified model of the relevant rocesses, which results from the assumtion Eq. (31). In the second aroach to the modeling of thermodynamic rocesses we have to use Eq. (16), instead of Eq. (11) used in the first aroach. aking into account the reduced equation of motion Eq. (3), Eq. (16) can be transformed into the form d du d uc u. (35) It is above mentioned that in the second aroach one of known analytical forms of equation of state should be chosen as actual equation of state to be used in the modeling, for examle, the Redlich-Kwong equation of state Eq. (15). Note, Eq. (33) and Eq. (35) are equivalent for any equation of state. he same way as in general case, this fact ensues from the thermodynamic identities Eq. (0), the equality du u d, and obvious relations d 1 d and d d d. It is essential that in both aroaches, that is, in Eq. (34) and Eq. (35), the gas comressibility is taken into consideration. he model based on Eqs. (7), (3), (34), and (30) is used in many similar studies, for examle [16]. Simulations based on the general model as well as the simlified models are resented in our revious work [15] where the following inference has been reasoned. If the equations of state Eq. (15) and Eq. (30) give the same deendence =(ρ,), the results of simulations based on the model Eqs. (7), (3), (34), (30) and the model Eqs. (7), (3), (35), and (15) coincide and they rovide qualitatively correct evolution of temerature field, that is, temerature decreases with a dro in ressure. We show what would haen if inertial forces were comletely ignored. Simlification of the equation of motion Eq. (8) resulting in the equation of motion Eq. (3) formally validates the equality

11 Mathematical models of gas transmission 601 du u 0, from which for steady roblems it follows that u const and const. (36) If these conditions are acceted as valid ones, we come to a rincially incorrect result, that is, rincially inadmissible behavior of temerature. o make sure of this incorrectness it is easy to use the second aroach to the modeling of thermodynamic rocesses. Indeed, if du/=0, from Eq. (35) it follows that d 1 d, (37) c that is, the derivatives of functions and with resect to z have different signs. It means that a decrease in ressure should lead to an increase in temerature. It is to this conclusion that the authors of several works come, for examle [10]. However, this result contradicts well-known exerimental data and, in addition, is not in agreement with neither the known equations of state nor the results of simulations based on rather general models in which the assumtion +ρu is not used. 6. Conclusions It is roved that even if gas flow velocity is very small, the assumtion that ρ and u are constants is inadmissible. In our oinion there are no reasons to use simlifications leading to a decrease in the accuracy of the calculations of main characteristics of gas flow, for the calculations based on general models which take into consideration inertial forces is not critically comlicated. wo aroaches to the modeling of thermodynamic rocesses in a gas flow have been analyzed, an equivalence of two balance equations for internal energy used in the aroaches has been roved, and it has been shown that the second aroach is more referable in the case of modeling of gas flow taking lace under hyer-ressure conditions. Exlicit deendences of internal energy and secific heat at constant volume on temerature and gas density have been found for the two-arameter Redlich-Kwong equation of state. Acknowledgments he author acknowledges Professor. A. Pavlovsky for long-term collaboration, Academician N. F. Morozov and Professor N.. Egorov for their continuous interest in

12 60 G. I. Kurbatova and N. N. Ermolaeva this work and suort, as well as Professor D. K. agantsev for fruitful discussions and assistance in writing this article. References [1] G. I. Kurbatova, E. A. Poova, B.. Filiov,. B. Filliov, K. B. Filliov, Models of sea gas-ielines, St.-Petersburg State University, St.-Petersburg, 005. [] G. I. Kurbatova, E. A. Poova, Problem of allowance for velocity rofiles in simulation of a turbulent flow in ies, Gazette of St.-Petersburg State University, series 10, () (007) [3] E.. Grunicheva, G. I. Kurbatova, E. A. Poova, Non-stationary non-isothermal flow of gas mix in the sea gas ielines, Matem. Mod., 3 (4) (011) [4]. E. Seleznev, G. S. Klishiv,.. Aleshin, S. N. Pryalov,.. Kiselev, A. L. Boichenko,.. Motlokhov, Numerical analysis and otimization of gasdynamic conditions of natural gas transmission, Moscow, URSS, 003. [5] O. F. asil ev, E. A. Bondarev, A. F. oevodin, M. A. Kanibolotsky, Nonisothermal gas flow in ies, Nauka, Novosibirsk S.O., [6]. I. Zubov,. N. Koterov,. M. Krivtsov, A.. Shiilin, Unsteady gas dynamic henomena in ieline through the Black sea, Matem. Mod., 13 (4) (001) [7]. E. Seleznev,.. Aleshin, S. N. Pryalov, Basics of numerical modeling of long-distance ielines, KomKniga, Moscow, 005. [8] M. Chaczykowski, ransient flow in natural gas ieline he effect of ieline thermal mode l, Al. Math. Model., 34 (010) [9] C. ruesdell, A first course in rational continuum mechanics, Maryland, he Johns Hokins University, Baltimore, 197. [10] M.. Lur e, O.. Pyatakova, Particularities of thermal calculations of longdistance gas-ielines with allowance for the inversion of the Joule-homson effect, Gas Industry, () (010) [11] J. L. Modisette, Equations of State utorial, Proceedings of the PSIG he 3- nd Annual Meeting, Savannach, (8) (000) 1-1. [1] Ch. Kittel, hermal Physics, John Wiley and Sons, Inc., New York, 1969.

13 Mathematical models of gas transmission 603 [13] R. C. Reid, J. M. Prausnitz, h. K. Sherwood, he roerties of gases and liquids, MeGraw Hill Book Comany, New York. St. Louis, San Francisco, [14] D. Kondeudi, I. Prigogine, Modern thermodynamics. From heat engines to dissiative structures, John Wiley & Sons, Chichester, New York, Weinheim, Brisbade, Singaore, oronto, [15] G. I. Kurbatova, E. A. Poova, On difference in mathematical models of gas transmission in ielines, Gazette of St.-Petersburg State University, series 10, (3) (011) [16] A.D.evyashev,.S. Smirnova, he method of the aroached decision of the Cauchy roblem for system of the equations of stationary flow of gas in the ieline, Radioelektronika i informatika, (1) (009) Received: June 15, 014