Numerical simulation of a one-dimensional shock tube problem at supercritical fluid conditions

Transcription

1 International Journal of Physial Sienes Vol. 3 (1), pp , Deember, 008 Available online at ISSN Aademi Journals Full ength esearh Paper Numerial simulation of a one-dimensional shok tube problem at superritial fluid onditions Hatem Ksibi and Ali Ben Moussa Sfax University, IPEIS P. Box 805 Sfax 3018, Tunisia, CFDTP, ENIS, P. Box W, 3038 Sfax, Tunisia. Aepted 17 July, 007 The numerial omputation of superritial fluid flows is extremely hallenging beause of the omplexity of the physial proesses and the disparity of the spae and time sales involved. Superritial fluids exhibit large density flutuations espeially very lose to the ritial region. In this region, the perfet gas law is no longer valid and has to be replaed by a speifi equation of state (EoS) as, for instane, the Altunin and Gadetskii EoS. In the present work, the problem of hoosing a suitable numerial sheme for dense gas flow omputations in a shok tube is addressed. In partiular, the extension of the lassial oe s sheme to real gas flows is used and its performane is evaluated by omparing with the analytial profile of the dimensionless density obtained by Sod in the shok tube problem. The appliation of this numerial implementation near the ritial region of the fluid gives signifiant differenes ompared to gas dynamis and shows a relevant behaviour of the ompressibility variation and loalises an important gradient of temperature in the shok tube. Key words: Hydrodynamis, shok tube, numerial simulation, real gas flow, ritial region. INTODUCTION A superritial fluid (SCF) may be defined as a substane for whih both temperature and pressure are above the ritial values. When a liquid is heated above its ritial temperature at pressures greater than (or exeeded) the ritial pressure, the transition from liquid to superritial fluid is ontinuous. Close to the ritial density, SCFs display properties that are to some extent intermediate between those of a liquid and a gas. For instane, a SCF may be relatively dense and dissolve ertain solids while being misible with permanent gases and exhibiting high diffusivity and low visosity. In this phase domain, the thermal and mehanial disturbanes are strongly oupled. This speifi behaviour of the superritial fluids is of partiular interest both from the theoretial point of view and for many industrial appliations, suh as the prodution of nanopartiles for medial use and the extration of hemial ompounds. A shok tube onventionally takes the form of a strong smooth wall steel pipe, of either irular or retangular ross-setion, divided into two ompartments initially at different pressure values and separated by a diaphragm, *Corresponding author. hatemksibi@yahoo.fr. see Figure 1. If we suppose that the visous effets are negligible along the tube and we assume that the tube is suffiiently long enough to avoid refletions at the tube ends, the exat solution of the Euler equations an be obtained on the basis of a simple wave analysis. At the bursting of the diaphragm, the disontinuity between the two initial states breaks into leftward and rightward moving waves, separated by uniform solutions: a shok wave (S) followed by a ontat disontinuity (C) moves to the low pressure region and rarefation waves () move to the high pressure side (Sod, 1978). In the experimental studies, shok tubes onven-tionally feature a test-setion, often equipped with obser-vation windows, in whih a shok proess, interation, or other phenomena are studied using speifi diagnosti instrumentation suh as high-speed reording tehni-ques, photography, et. Many shok tubes used in researh studies follow the onventional simple design with lengths of typially a few metres and operating with ompressed air pressures up to several atmospheres. From a numerial point of view, the shok-tube problem is a very interesting test ase beause the exat timedependent solution is analytially known and an be ompared with the omputed solution by applying numerial approximations, (Arina, 004; Guardone and Vigevano,

2 Ksibi and Moussa 315 a b Burster diaphragm introduing the jaobian matrix of the Euler flux A(U) f (U) = U, the system of equations an be written, in semi-onservative form, following: U U + A(U) = 0 t x () S C 0 0,5 1 Figure1 - Shok tube at initial state (a: t= 0. s; b: t >0 s). 004). The initial onfiguration of the shok-tube problem is omposed by two uniform states at the same temperature but at different pressure values, separated by a disontinuity whih is usually loated at the middle of the tube. This partiular initial value problem is known as the iemann problem. In this work we propose a numerial approah, based on the oe averaged tehnique (oe, 1981) written for real gas using the full EoS of Alunin and Gadestkii (1971) whih is available also in the ritial region. This approah, mainly following the Harten et al. (1986) and Montagné et al. (1987) algorithms, allows omputing the distribution along the tube of the thermodynamis quantities at subritial, ritial or superritial initial onditions. Here the high order of the numerial resolution is assoiated to the non linearity of the EoS where thermodynami variables are alulated through a preditionorretion tehnique. Numerial validation is firstly reahed by omparing our results with the Sod s shok tube analytial solution (1978) of a perfet gas dynami due to a pressure falling from 1 to 0.1 bar (Sod, 1978; Toro, 005). The numerial proedure is based on an approximate iemann solver, following the oe tehnique that needs to know an averaged state between the states U and U of the iemann problem at the grid interfaes: f ( U ) f ( U ) = AU (, U ) ( U U ) ˆ (3) To ensure onservativity and onsisteny, the oe matrix (Â) must be evaluated by using the oe average quantities [4]: The average density: (4) ρ = The average veloity u = ρ ρ ρ u ρ + + ρ u (5) ρ H + ρ H And the average total enthalpy: H =. ρ + ρ (6) These averaged quantities are valid both for a perfet gas and a real fluid. The differene between these two ases lies in the alulation of the average of the speed of sound. In the ase of a real fluid it is neessary to reognize speial formulations of the pressure derivatives (Montagné et al., 1987). ρ Governing equations Conservation law approximations Considering a non visous flow in a tube having a onstant setion and assuming a one dimensional problem, the system of equations an be written as: ( U) U f + = 0 t x Where T = ( ρ, ρu, ρ E is the vetor of the U ) f = ( ρ u, ρu + P, ρue + up) onservative variables, the Euler flux vetor. t and x stand for the time and spae oordinates. ρis the density, u the veloity, E the total energy per unit of mass and P is the stati pressure. By T Boundary onditions The tube is supposed to be losed at both ends. The right and left boundaries of the omputational domain are thus ideal solid walls. Sine the integration time is relatively short, the shok and rarefation waves never reah the end walls, and so, onditions at these boundaries are well known. Initially the fluid is at rest and its temperature is presribed at T 0 =305 K (T 0 > T ) in order to over the ritial region of the fluid. Pressures from both sides of the diaphragm are set at different levels. The initial values for the different test-ases an be seen in tables 1, and 3. Numerial methods To obtain a numerial approximation of the Euler equa-

3 316 Int. J. Phys. Si. Table 1. Initial onditions of the perfet gas shok tube. egion Density (kg.m -3 ) Pressure (bar) Temperature (K) EoS (8) ideal gas eft ight Table. Initial onditions of the superritial/subritial shok tube egion Pressure (bar) eft ight Temperature (K) From EoS (14) Table 3. Initial onditions of the near ritial region shok tube. Subritial Critial Superritial egion Pressure (bar) eft 55 ight 50 eft 75 ight 70 eft 105 ight 100 Temperature (K) EoS (14) 305 equation (1), we use a numerial sheme based on oe iemann solver (Montagné et al., 19878]. The sheme used is based on a Finite-Volume approximation. It is of high resolution, that is, it reovers a seond order auray away from disontinuities. It however satisfies the Harten s Total Variation Diminishing (TVD) onstraints to avoid the spurious osillation in the viinity of the disontinuities. A stability ondition is required and, assuming a onstant grid size ( x), the time step ( t) is alulated by using: N t = max CF x ( λi ), (7) Where max ( λ i ) is the maximum wave speed and N CF the required Courant (CF) number. equilibrium properties is an important field of researh. The Altunin and Gadetskii EoS was given to desribe the thermodynami properties of the pure arbon dioxide. Following this speifi EoS, the behaviour of superritial, liquid or gas states of arbon dioxide are aurately fitted. As all thermodynami properties divergene in the viinity of the ritial point (mainly the ompressibility fator), the Altunin and Gadetskii EoS needs other orretive terms to more aurately fit the singular behaviour and to inrease the robustness in the alulation of the thermodynami funtions as, for instane, the speifi heat at onstant volume, the pressure and the speed of sound. The original equation proposed by Altunin and Gadetskii desribes with a great auray the arbon dioxide behaviour in the regions far from the ritial region. However, to supply the Altunin and Gadetskii EoS in the viinity of the ritial point, one must used a speial EoS obtained by digitalizing orretive abaus given in the literature (Angus et al., 1976). This work is followed by a numerial estimation of different thermodynami funtions as C v, in the viinity of the ritial point, using series expansion tehniques (Ksibi and Moussa, 005). The analyti Altunin and Gadestkii EoS is written as: 9 PA Z = = 1 + ρ ρt (8) 6 r i= 0 j= 0 b ij j i ( τ 1) ( ρ 1) Where ρ r =ρ/ρ and τ=t /T. Z is the ompressibility fator and is the perfet gas onstant. The subsript indiates values at the ritial point. The b ij oeffiients an be found in the IUPAC tables, (Angus et al., 1976). A separate EoS was needed for the ritial region within about ± 5 K of the ritial temperature. This equation is expressed in terms of the polar oordinates, r and θ, in the phase plane, entred on the ritial point, Shofield et al (1969). The distanes to the ritial value for the density and the temperature are expressed following: ( T T ) ( 1 b θ ) T = = r (9) ρ = ρ T ρ ρ = r β gθ r (10) The pressure in the viinity of the ritial point (P S ) is dedued from the previous equations by the following parametri equation: Equation of state The development of equations of state (EoS) and their appliations to the orrelation and the predition of phase P = P S P P = r β( δ+ 1) q( θ) + T + ar Where q is a funtion of θ, given by: βδ θ ( 1 θ ) (11)

4 Ksibi and Moussa 317 Figure - Numerial validation of the shok tube problem for sub-ritial onditions, at a dimensionless time t = 0. s q( 4 θ ) = θ θ (1) The omplete EoS is obtained through the ombination between equations (8) and (11) by using a swith funtion f(r), depending on the distane to the ritial point: A [ 1 f (r)]. P S P = f (r).p + (13) Where f(r) is expressed by: n1 n ( 0,01/ r) ( 0,05/ r) f (r) = 1 1 e. 1 e (14) And P A is the analyti pressure dedued from equation (8). Values of different onstants needed to evaluate the pressure at the ritial region P S are: n 1 =3/ ; n =3; a=0.065; β=0.347; δ=4.576; g=1.491, b²= The value of the parameter depends on the relative temperature (T/T); i.e. for T>T, = and for TT, = Numerial results and disussions: Numerial validation The validation of the numerial ode is performed by omparing the numerial solution with the analytial one proposed by Sod and established for a perfet gas (air) flow in a shok tube. The initial onditions are summarized in table 1 where the temperature is obtained by using both the Altunin and Gadestkii and the ideal gas EoS. At the initial state, a diaphragm, loated in the middle of the shok tube, separates the arbon dioxide onsidered as a perfet gas at two different states from both sides of the diaphragm. The density and pressure jumps are hosen so as to reover the three types of wave (shok wave S, ontat disontinuity C, and rarefation waves ) in the tube. The pressure on the left side of the diaphragm is 10 times higher than the one on the right side. At time t = 0 when the diaphragm is broken, the disontinuity between the two initial states leads to leftward and rightward moving waves, separated by uniform states. A shok wave moves to the low pressure region (toward the right side) followed by a ontat disontinuity moving at the loal speed of the flow. Expansion waves move to the high pressure side (toward the left side). Sine the omputation stops before the waves reah the end-walls of the shok tube, the boundary onditions are trivial at both ends of the shok tube. The simulations are performed by using a CF number equal to 0.. In these alulations, we have used a regular grid with 100 grid-nodes. From Figure, we show that the present ode has aptured with auray the jumps aross disontinuities. The veloities of disontinuity propagations have also been aurately omputed. When the simulation is performed on the sub-ritial onditions (T >T and ρ <ρ ), differenes with the exat ideal gas solution are notieable on the onstant states around the ontat disontinuity. We must also notie that the veloity of the ontat disontinuity is slightly more important than in the ase of an ideal gas. These differenes are only due to the thermodynami behaviour of the fluid within the sub-ritial region (EoS (8)) whih indues a higher flow veloity than the ideal gas onfiguration between the last expansion wave and the shok wave. Shok tube at high pressure initial onditions Numerial simulations are also performed by onsidering the arbon dioxide at superritial state on the left side, whereas the arbon dioxide is set at subritial onditions, on the right side (see table ). Numerial results greatly improve the auray of the apture of rarefation, ontat disontinuity, and shok waves. In Figure 3, the dimensionless density of the arbon dioxide is depited along the tube axis at the several times. We show that at every time, S, C and waves are well aptured along the tube. During time, shok wave (S) reahes the left side tube while keeping a onstant density behind the shok. The ontat disontinuity is aptured at the half of the tube for the different ases. At

5 318 Int. J. Phys. Si. 7,6x10 6 Pressure (Pa) 7,4x10 6 7,x10 6 7,0x10 6 6,8x10 6 t=0.08 s t=0.18 s t=0.47 s t=0.66 s 6,6x10 6 6,4x10 6 Figure 3- Dimensionless density profiles along the shok tube for different time steps. 0,0 0, 0,4 0,6 0,8 1,0 Shok tube axis Figure 6. Pressure profiles along the shok tube for different time steps. 310 Temperature (K) t=0.08 s t=0.18 s t=0.47 s t=0.66 s 300 0,0 0, 0,4 0,6 0,8 1,0 Shok tube axis Figure 4- Temperature profiles along the shok tube for different time steps. Veloity (m/s) 10 t=0.08 s t=0.18 s t=0.47 s 8 t=0.66 s ,0 0, 0,4 0,6 0,8 1,0 Shok tube axis Figure 5. Veloity profiles along the shok tube for different time steps. the other side, the rarefation wave reahes the right side while keeping a onstant density at the right end of the tube. The temperature evolutions are shown on figure 4. egarding the temperature amplitudes in the shok tube, it is soliited to an important thermal gradient (about six degrees). In the left ompartment, the bursting of the diaphragm involves a loal expansion upstream of the wave C that generates a fall of temperature. Whereas, the ontribution of the matter implies a loal ompression downstream of the wave C whih generates an elevation of the temperature. This behaviour is supported by the hyper ompressibility of the arbon dioxide in the viinity of its ritial point. Conerning the speed profile along the tube (figure 5) the initially stagnant fluid, moves abruptly in the both sides of the tube when the diaphragm is burst. At t = 0,66s, the fluid has pratially a onstant speed along the tube (about 9 m/s) exept at its end walls where no motion onditions are imposed, the fluid veloity exeeds Similarly to the density profile, figure 6 shows the pressure behaviour in the tube for the different shoks waves S, C and at different time steps. We note that the depressurisation goes proportionally to time in the left side whih involves a ompression of the fluid inside the right ompartment. Shok tube behaviour at ritial initial onditions In order to show the importane of the Sod problem modelling in the viinity of the ritial region, numerial implementations are seleted so as to over different regions of the phase diagram of the arbon dioxide, mainly: the subritial zone (T>T C and P<P C ); the ritial zone and the superritial zone (T>T C and P>P C ).

6 Ksibi and Moussa 319 Figure 9. Density falls under the ation of the various shok waves (S, C, ). Figure 7. Temperature profiles along the shok tube for different expansion ratios (effet of the ritial region behaviour). Figure 8. Dimensionless density profiles along the shok tube for different expansion ratios (effet of the ritial region behaviour) In these three test ases, the pressure jump (P -P ) between the left and the right sides of the tube is kept onstant and equal to 5 bars. The initial temperature is onsidered onstant along the tube and set at 310 K. On the Figure 7, we illustrate the profile of the temperature along the shok tube at t = 0.8 s. The temperature is maintained onstant at the end-walls. We initially distinguish an anti symmetry of the temperature variation ompared to the position of the diaphragm (x = 0.5). The subsoni ase (expansion from 55 to 50 bars) provides a small variation in the temperature of 0.5 K. This amplitude limbs to 4 K in the superritial ase (from 105 to 100 bars) and generates mainly a strong gradient making it possible to indue in a relevant way a thermal transfer in the shok tube. Conerning the variation of the arbon dioxide density along the shok tube, it varies by marking the various waves C, and S in different ways relating to the pressure level. Indeed, the variation of in the superritial ase (P =105 and P =100 bars) implies a fall of this one on three various levels similarly with the ase of shok tube of a perfet gas, (Figure 8). Nevertheless, the state being dense like a liquid, the variation of remains tiny and lower than.5%. In the subritial ase (P =55 and P =50 bars), the density follows the same profile and loates the waves C, and S orretly. Moreover, we notie that the dimensionless density variation beomes more amplified. Applying a pressure level around the ritial one (P = 73.8 bars), the variation of the density appears exatly as the last two ases but a signifiant derease of the density is reahed (about 0%). Here, the high ompressibility of the state implies a non linearity of the density falls with the pressure level differently to the temperature osillations as desribed before. Indeed, figure 9 shows this singular behaviour of the dimensionless density values aross different waves in the shok tube. For the same differene in pressure (P - P ), the variation of the density is very signifiant while passing by the ritial region, ompared with the subritial and superritial ases. Conlusion This paper disusses the propagation of shok waves through a shok tube ontaining initially superritial fluid. It was found that the numerial alulations give similar results presented by several authors for a perfet gas. Appliations of the superritial state by using a speifi

7 30 Int. J. Phys. Si. EoS, valid near and far from the ritial region, in a shok tube problem give similar profiles of density, pressure and veloity to that of perfet gas. Nevertheless, omputing the shok tube problem at high pressure levels shows a partiular behaviour of the fluid density near the ritial region and an important thermal gradient in the shok tube. EFEENCES Sod GA (1978). A survey of several finite differene methods for systems of nonlinear hyperboli onservation laws. J. Computational Physis, (7): Arina (004). Numerial simulation of near-ritial fluids. Applied Numerial Mathematis (51): Guardone A, Vigevano (004): Assessment of thermodynami models for dense gas dynamis. Physis of Fluids 16, (11): oe P (1981). Approximate iemann solvers, parameter vetors, and differene shemes. J. Computational Physis, Altunin W, Gadetskii OG (1971). Equation of State and Thermodynami Properties of iquid and Gaseous Carbon Dioxide. Teploenergetika, 18, (3): Yee HC, Harten A (1986). Impliit TVD Shemes for Hyperboli Conservation aws in Curvilinear CoordinatesAIAA J. 5: Toro EF (005). iemann Solvers and Numerial Methods for Fluid Dynamis - A Pratial Introdution, XIX Edition, Springer Published, Montagné J, Yee HC, Harten A (1987). in: Dr Hill library (Eds), Proeedings of 7th GAMM Conferene on Numerial Methods on Fluid Mehanis, ouvain la Neuve. Angus S, Armstrong B, de euk KM (1976) International Thermodynami Tables of the Fluid State Carbon Dioxide 3. IUPAC Pergamon Press, Oxford. Ksibi H, Moussa AB (005). Numerial implementation of the Altinun and Gadetskii EoS at the ritial region og the pure arbon dioxide. Proeedings of 10 th European Meeting on Superritial Fluids, eations, Materials and Natural Produts Proessing, Strasbourg. Shofield PJD (1969). itster and J. T. Ho, Correlation between Critial Coeffiients and Critial Exponents. Phys. ev. etters, 3: