Simulation of flows at supercritical pressures with a two-fluid code

Transcription

1 Simulation of flos at supercritical pressures ith a to-fluid code Astract Maru Hänninen, Joona Kuri VTT Technical Research Centre of Finland Tietotie 3, Espoo, P.O. Box 1000, FI-0044 VTT, Finland maru.hanninen@vtt.fi, joona.uri@vtt.fi Fossil-fueled poer plants have een operated ith ater at supercritical pressures for decades due to the hih thermal efficiency achievale y increasin the system pressure aove the critical point. Durin the recent years, there has een a reneed interest to develop also ater-cooled nuclear reactors hich or under the supercritical pressure conditions. When such reactor concepts are studied, it is necessary that also the simulation codes used for desin and safety-demonstrations of such reactors are ale to simulate ater flos aove the critical pressure. In APROS process simulation softare, the to-phase flo can e simulated ith a variale level of sophistication: ith a homoeneous model, ith a 5-equation drift-flux model and ith a 6-equation to-fluid model. At supercritical pressures the distinction eteen the liquid and as phases disappears: oilin and condensation are not oserved, ut instead the properties of the fluid vary smoothly from those of a liquid-lie fluid to those of a as-lie fluid, and from the macroscopic point of vie the supercritical-pressure fluid can alays e considered a sinle-phase fluid. Because of this, the homoeneous model ould e ideal for the thermal hydraulic simulation at and aove the critical pressure. Hoever, in the nuclear poer plant applications the homoeneous model is seldom sufficient for the calculation of to-phase flo elo the critical pressure, and thus the six-equation model has to e used in the eneral case. When the six-equation model is applied to supercritical-pressure calculation, the prolems ho the model ehaves near and aove the critical pressure, and ho the phase transition throuh the supercritical-pressure reion is handled, are inevitaly encountered. Aove the critical pressure the heat of evaporation disappears and the hole concept of phase chane is no loner meaninful. In the present paper the use of the six-equation thermal-hydraulic model for supercritical-pressure calculation is descried. The chanes made in the constitutive equations are discussed. The applicaility of the numeric model is demonstrated y simulatin to asic test cases. Keyords Thermal Hydraulics, Supercritical Pressures, System-Code Development (1/14)

2 1. INTRODUCTION The set of constitutive equations needed in the six-equation solution includes friction and heat transfer correlations hich are developed separately for oth phases. The capaility of constitutive equations and the ay ho they are used has to e carefully examined. One possiility to maintain separate liquid and as flos in the numeric model is to use a small evaporation heat and apply the concept of the pseudo-critical line. The pseudo-critical line is an extension of the saturation curve to the supercritical pressure reion: it starts from the point here the saturation curve ends (the critical point), and it can e thouht to approximately divide the supercritical pressure reion to su-reions of pseudo-liquid and pseudo-as, for at the same point here heat capacity c p is maximized, the variation of density as function of temperature is also steepest. The thermo physical properties of ater/steam undero rapid chanes near the pseudocritical line and therefore the quality and accuracy of the steam tales is essential in calculation of flos under supercritical conditions. The steam tales used in APROS are ased on the IAPWS-IF97 recommendation, hich defines the properties of ater ith a sufficient accuracy over ide rane of parameters, includin supercritical pressures. The asic or done ith testin of ne steam and material properties and ne correlations implemented in APROS as made earlier [1]. In the present paper the use of the six-equation thermal-hydraulic model for supercritical-pressure calculation is descried. The chanes made in the constitutive equations are discussed. The applicaility of the numeric model is demonstrated y simulatin an instant emptyin of a horizontal pipe. Several cases ith different initial conditions are calculated and discussed. The calculation of the all-to-fluid heat transfer is examined ith a model of supercritical ater floin throuh an electrically heated pipe.. SIMULATION OF FLOWS WITH THE TWO-FLUID MODEL.1. Principles of the six-equation solution The six-equation solution of the APROS code is ased on the one-dimensional conservation las of mass, momentum and enery. When these las are applied for oth the liquid and as phases, all toether six partial differential equations are otained. These equations can e ritten in the form ( ) ( u ) t z ( u ) ( u ) ui F Fi t z ( h ) ( uh ) p hi qi q Fi u t z t i (1) () (3) In the equations the suscript is either l for the liquid phase or for the as phase, the suscript i refers to the interface eteen the phases, and the suscript to the all of the flo channel. = - l is the mass evaporation rate, F the friction force and q the heat transfer rate. In the enery equation (3) the term h is the total solved enthalpy, i.e. static enthalpy plus the inetic enery per unit mass, h in = ½v here v is the flo (/14)

3 velocity. The potential enery due to ravity is considered small and has een omitted in the enery equation. The phenomena hich depend on the transverse radients, lie friction and heat transfer eteen the as and liquid phases and eteen the all and oth the phases, have een descried ith empirical correlations. The correlations are stronly dependent on the flo reime and thus usually different correlations have to e used for different flo reimes. The flo reimes hich are treated in the code are uly, annular, droplet and stratified flos. In the calculations, the real prevailin flo mode usually consists of more than one individual flo reime. When the correlations are applied, the different flo reimes are taen into account y eihtin factors. In the supercritical pressure conditions the phenomena related to to phase concept disappear. Therefore the correlations used for to-phase heat transfer and friction must e considered hen the conditions chane from sucritical pressure to-phase area to supercritical pressure area. In the folloin the interfacial heat transfer and all heat transfer as ell as the interfacial friction and all friction are discussed in detail... Calculation of interfacial heat and mass transfer In to-phase flo the heat and mass transfer eteen liquid and as has to e calculated. The calculation of interfacial mass transfer is ased on the requirement that the enery alance over the interface is zero. When it is assumed that the interface is at saturated state the folloin relationship is otained i qi1, i qi, i qi (4) h h, sat 1, sat here q i is the heat floin from the all directly to the interface. The interfacial heat transfer rates are calculated separately for liquid and as side. From liquid side the heat transfer is q i1, i K i1, i h1, stat h1, sat From as side the interface heat transfer is q i, i (5) i, i, stat, sat K h h (6) Dependin on the heat flos the mass transfer is either positive (evaporation) or neative (condensation). The interfacial heat transfer coefficients depend stronly on the void fraction and on the phase flo velocities. When the critical pressure is approached from loer pressure side the evaporation heat is radually disappearin. Actually the decreasin rate of the evaporation heat is acceleratin near the critical point as it can e seen in Fiure 1. Because the present to-fluid model treats the supercritical fluid as a to-component (3/14)

4 mixture, the formalism of interfacial heat and mass transfer has to e extended to the supercritical pressure reime. The ojective of this extension is to treat the fluid as liquid hen its enthalpy is elo the pseudo-critical enthalpy and as as hen its enthalpy is aove the pseudo-critical enthalpy. Additionally it is desirale that the pseudo phase transition occurs as fast as possile so as to minimize the time the pseudo void fraction is not strictly zero or unity, ut ithout introducin any numerical instailities to the solution. In the present model the latent heat of vaporization is set to a constant value L pe at the supercritical pressure reion, and the pseudo-saturation enthalpies are then set to to h h l,sat,sat ( p) h ( p) h Lpe ( p) Lpe ( p) (7) With these definitions, the interfacial heat and mass transfer can e calculated identically to the approach used at sucritical pressures. Because the interfacial heat transfer coefficients, hich at the to-phase reion are normally calculated usin suitale correlations, have no physical relevance, they can e chosen aritrarily to yield the desired effect. In APROS, the interfacial heat transfer coefficients at supercritical pressures are calculated startin from a shape function C1 ( ) (1 ) C 1 1 (8) that defines the mutual relationship for the interfacial heat transfer coefficients of the pseudo-liquid and the pseudo-as for each value of the void fraction (C 1 > 1.0 is a constant that defines the deformation of the shape function). The shape function is then multiplied y a lare value C to et the interfacial heat transfer coefficients K K i il C ( ') C (1 ') (9) Finally, to prevent the phase chane rate from dyin-out too early as the void fraction approaches 0.0 or 1.0, the used in equation (9) is calculated from the pseudo void fraction as 3 ( 0.5) ' 0.5 (10) 0.5 As stated aove, the choice of K i and K il is rather aritrary, and the formulas presented here ere derived throuh a process of trial and error these definitions for the coefficients cause the void fraction to chane from zero to unity (and vice versa) almost instantly in most situations here the ul enthalpy enters the pseudo-to-phase reion eteen h l,sat and h,sat. (4/14)

5 Fi. 1: Saturation enthalpies of as and liquid. At supercritical pressures the saturation values are set to follo the pseudo-critical line..3. Wall heat transfer Aove the critical pressure the oilin and condensation phenomena disappear and only one-phase convection occurs. Due to the possile pseudo to-phase conditions convection has to e calculated to oth liquid and as phases. Hoever, to avoid the numerical prolems durin the transition over the critical pressure it is necessary to examine the heat transfer models of sucritical to-phase flo. In to-phase reion elo supercritical pressure there are three asic separate heat transfer zones: etted all (zone 1), dry all (zone 3) and a transition zone eteen etted and dry all (zone ). The selection eteen different heat transfer zones is made on asis of all and fluid temperatures, critical heat flux and minimum film oilin temperatures. In the et all area the to-phase heat transfer can e oilin or condensin. If the all heat flux exceeds the critical heat flux the transition from the et zone to dry zone occurs. If the all temperature exceeds the minimum film oilin temperature the dry all heat transfer is calculated. In the folloin it is discussed ho the all heat transfer correlations ehave near the critical pressure. Also it is descried ho the to-phase heat transfer mechanism must e modified at the supercritical pressure flo. Total heat transfer of the nucleate oilin is calculated as a sum of the forced convection heat transfer to liquid and the nucleate oilin heat transfer. Because the evaporation heat does not exist at the supercritical pressure the oilin part of the heat transfer has to e fainted out. In to-phase area the nucleate oilin heat transfer is calculated ith the Thom correlation [7] h n p / e T Tsat (11) (5/14)

6 In approachin the supercritical pressure the nucleate oilin term is multiplied ith the ratio of evaporation heat to the enthalpy difference of smoothin area. Where h hp h n hcr r hn (1) p hp h cr r min,1. 0 h (13) cr p sm and p sm is the loer limit for the pressure of smoothin area. In the to-phase area the heat flo from the nucleate oilin is divided to the heat flo to the liquid and to the heat flo to the interface q i, hich is used in Equation (4). When the oilin heat transfer is fainted out the one-phase convection is a remainin mode. In case of condensation it is assumed that steam condenses to liquid film on the all (film condensation) and the heat transfer from liquid to all is readily calculated ith one-phase convection correlation. Therefore no special treatment in all heat transfer due to condensation heat transfer is needed durin the transition from sucritical to supercritical pressure. In case of the dry all it is assumed that the steam is in contact ith the all. This means that the one-phase heat transfer to steam is calculated already at sucritical pressure area. In that case the transition is easy if the same forced convection correlation can e used elo and aove the critical point. When the pressure approaches the critical pressure the critical heat flux disappears. The disappearance of the critical heat flux rins aout that the dry all heat transfer is prevailin ith positive heat flux (from all to fluid). The et zone heat transfer is calculated hen the real heat flux is neative. Near the critical pressure the minimum film oilin temperature approaches the saturation temperature as can e seen in Equation (14), here the minimum film oilin temperature is calculated ith the Groeneveld - Steart correlation [7]. T MFB 6 6 r p 6 T 9 10 Pa T 9 10 Pa T, if p 9 10 Pa MFB sat p cr sat (14) The different heat transfer zones have een defined in the folloin ay. The et all zone is used hen The dry all zone is used hen TW T sat Tall T mfs (6/14)

7 At the critical pressure point as indicated aove Tmfs T sat resultin in that the zone ceases to exist at the supercritical pressure area, i.e. only the heat transfer zones 1 or 3 are used. At the supercritical pressure condition it can e assumed that the heat transfer correlations are the same for pseudo liquid phase (zone 1) and in pseudo as phase (zone 3). The zone 1 heat transfer is applied hen the averae enthalpy is less than pseudo saturation enthalpy, and the zone 3 heat transfer is calculated hen the averae enthalpy is aove the pseudo saturation enthalpy. Aove the critical pressure point the heat transfer from all to oth pseudo liquid and pseudo as is calculated ith the correlation of Jacson and Hall [5]. The exponent n depends on the ratios of ul, all and pseudo critical temperatures. The correlation ives ood values for the supercritical pressure heat transfer, ut it does not predict the deterioration of heat transfer. Nu Re 0.4, T n T T T 0.8 Pr 0.5 1, T T 0.3 c c p p, 1, n if T if T if T T T T T T 1.T or 1.T and T T T T (15).4. Interfacial friction Because aove the critical pressure the distinction eteen as and liquid does not exist, a ood assumption to velocities of pseudo as and liquid is that they flo ith nearly the same velocity. In the to-phase flo at sucritical pressure the interfacial friction is stronly dependent on the flo reime that is prevailin in the flo and the velocities of phases may larely differ from each other. The larest velocity difference is related to stratified flo. Different interfacial friction correlations are used for the different flo reimes. The modeled flo reimes are stratified flo and non-stratified flo consistin of uly, annular and droplet flo. The final value for the interfacial friction is then otained as a eihted averae of the different correlations. Void fraction, rate of stratification and rate of entrainment are used as eihtin coefficients. The final interfacial friction is calculated usin interfacial friction of different flo reimes and the eihin factors for stratification and entrainment (Equation 16). F RF i is R(1 E) 1 1 F F EF (16) i ia id (7/14)

8 In the definition of the rate of stratification the eneral form of the Kelvin-Helmholtz staility criterion is used (see Equation 17). Because the liquid and steam densities approach each other at the supercritical pressure the stratification disappears, i.e. the stratification rate R approaches zero. u ul D 1 l l (17) Because the surface tension disappears at the critical pressure the entrainment eihin factor approaches to 1 providin that the pseudo void fraction is not zero E 1 u l f ( ) (18) Because near the critical pressure the stratification disappears and droplet flo is prevailin flo reime the interfacial friction inherently ecomes lare in approachin the critical pressure. At the critical pressure it is required that the pseudo liquid and as flos nearly ith the same velocity. Therefore for the interfacial friction a lare numer is used. In order to avoid instailities the linear interpolation eteen the sucritical interface coefficient and the lare supercritical coefficient is performed over the pressure interval around the critical pressure..5. Wall Friction The friction eteen one phase (liquid or as) and the all of the flo channel is calculated ith the formula F 1 f u D H u (19) The quantity f is the friction pressure loss coefficient or friction factor for phase. The used friction factors of phases consist of the sinle phase friction factor and the to phase friction multiplier: f f, c (0) sp The purpose of the to phase friction multiplier is to expand the pressure drop calculation of sinle phase flo and to estimate the phase distriution on the all of the flo channel. The multiplier is defined separately for the folloin flo reimes: stratified flo, non-stratified flo ithout droplet entrainment and non-stratified flo ith droplet entrainment. The rate of stratification and rate of entrainment are used as eihtin coefficients hen the multiplier is calculated. (8/14)

9 R c, st 1 Rc ns R Ec c, (1) c l R cl, st 1 1 l, ne E cl, en () In case of the supercritical pressure the stratification ecomes zero and the entrainment approaches one and therefore in the calculations of the multipliers the non-stratified part is used for the as and the entrainment part is used for the liquid. 1.5 c, ns c (3) 1 c l c l, en (4) 1 l l In Equations 3 and 4 it can e seen that the all friction is mainly calculated for liquid only hen void fraction approaches 0 and hen the void fraction is near 1 the all friction multiplier to as ecomes predominant. At the sucritical area the sinle phase friction factors are calculated either ith Blasius equation (smooth pipes) or ith the Coleroo equation, hich taes into account the rouhness of pipes. In the supercritical pressure reion the correlation of Kirillov et al. [3] is recommended: 1 f sp, 1.8lo10(Re ) 1.64 (5) The friction factor f is calculated for oth pseudo liquid and pseudo as. The phase friction multipliers from Equations 3 and 4 are then used to calculate the phase friction factors that are finally used for the all friction calculation. 0.4 (9/14)

10 3. TESTING OF THE MODEL 3.1. Testin of the transition from supercritical to sucritical pressures Aility of the numerical model to calculate transition from the supercritical pressure reion to the sucritical pressure reion as tested y calculatin an imainary experiment, in hich a closed pipe initially filled ith ater at a supercritical pressure as nearly instantaneously emptied. This simulated experiment is identical to the Edards-O Brien lodon test [6], except for the initial pressure hich as aout 7 MPa in the oriinal real-life experiment. Because this test is purely imainary, experimental data is not availale, and thus no definitive conclusions can e dran on the validity of the model ased on the simulation results. Thus, the simulation serves only to demonstrate that the numerical model is ale to simulate the transient. In the simulated experiment a straiht horizontal pipe of m lenth and 73 mm inner diameter is initially closed and filled ith ater at state p 0, T 0. Then an orifice ith cross-sectional area of 87 % from the pipe s cross-sectional area is opened at one end of the pipe in an interval of 10 ms. The openin of the orifice causes a depressurization ave to travel throuh the pipe, hich in turn causes the liquid in pipe to flash. Expansion of the fluid in the pipe maes it flo out of the pipe ith a very lare velocity, hich is, hoever, limited y the critical velocity at the rea orifice. This experiment as simulated several times ith different initial temperatures T 0, to ensure that the numerical model can cope ith passin throuh the critical point (.064 MPa, C) in hich the thermophysical property chanes are the steepest. The initial temperatures are listed in tale 1, the initial pressure as 5.0 MPa for all tests. Tale 1: Initial temperatures of the lodon simulations Test 01 Test 0 Test 03 Test 04 Test 05 Test 06 Test 07 T 0 [ C] Time-ehavior of pressure and void fraction durin three of these simulations is presented in fiure. In test 01 pressure drops in the hole pipe very quicly due to the hih velocity of sound at the initial state (c m/s). The local minimum oserved at the closed end at aout s is caused y reflection of the depressurization ave, and it s occurrence in time is in excellent areement ith the time the depressurization ave should travel the pipe (4.096 m / 1039 m/s s). Later rides in the plot are caused y the pressure ave travelin ac and forth in the pipe ith its velocity altered y loerin enthalpy of the fluid, and evaporation. (10/14)

11 Fi. : Behavior of pressure and void fraction durin the lodon tests 01, 03 and 06. Location is the distance from the rea orifice. Notice also the different vie anles in the pressure and void fraction raphs. Test 03 is similar to test 01, ut the sonic velocity in the initial state is very lo due to the initial temperature hich is near the pseudo-critical temperature (c m/s). This manifests itself as sloer depressurization of the closed end. Finally in test 06, the pipe contains initially (pseudo-)as and thus the depressurization is still slo (c m/s), ut the flashin doesn t happen. Hoever, in this case enthalpy of fluid drops elo the steam saturation enthalpy and this causes some of the fluid to temporarily condensate. (11/14)

12 As a conclusion of these simulations, it can e stated that APROS is capale of calculatin this ind of a fast transition from supercritical to sucritical pressures, and the results seem at least quantitatively correct. Hoever, real-life test data is still needed to validate the code. 3.. Testin of the pseudo-phase transition The transition from pseudo-liquid state to pseudo-as phase and ac as tested y simulatin flo throuh a thin heated pipe in hich the pseudo-critical state as passed. The diameter of the vertical pipe (4 mm) corresponded to a typical hydraulic diameter of the flo channels in a planned supercritical ater reactor (SCWR). In the simulation a constant pseudo-liquid mass flo of 0.04 / at 310 C as let flo in the loer end of the pipe, and the upper end of the pipe as ept at constant pressure of 5.0 MPa. The pipe as heated ith 50 W of poer eteen times 1.0 s and 10.0 s. This poer as enouh to raise the outlet temperature aove the pseudo-critical temperature (384 C at 5.0 MPa), up to 405 C. The pipe as divided to 0 internal nodes, and the all-to-fluid heat transfer as calculated ith the correlation of Jacson and Hall. Development of temperature and void fraction distriutions in the pipe are presented in fiure 3. As can e readily seen, the void fraction chanes eteen zero and unity almost instantly hen the pseudo-critical temperature is passed. In the first node in hich the pseudo-critical temperature is passed, the void fraction chanes only to aout 90 %, ut in the next node the void fraction is already at aout 99 %. This happens ecause the superheatin of the pseudo-liquid phase is not sufficient to evaporate the hole mass contents of the node in this case. Susequent simulations ith varied heat flos and node volumes revealed similar development for the void fraction distriution. Fi. 3: Development of temperature and void fraction durin the pseudo-phase transition test. Althouh the pseudo-phase transition is not fully optimal (i.e. the void fraction does not chane eteen zero and unity in a sinle time step), it can e still considered very satisfactory, as the (pseudo-)void fraction is insinificant as lon as the pressure stays aove the critical point. The pseudo-void fraction distriution can have effect on any real (measurale) variales only hen simulatin situations here the pressure is alloed to drop elo the critical point. Because such transitions happen typically very (1/14)

13 fast, and ecause many other phenomena are involved in the transition, it is the present authors elief that the effect of the void fraction distriution is insinificant even in these inds of simulations. Hoever, experimental data is needed to verify this arument. 4. CONCLUSION The most important chanes implemented into APROS thermal hydraulic system code, in order to mae it or at the supercritical pressures reion, have een presented. The ehavior of constitutive equations in to-phase flo at the transition from sucritical pressure to supercritical pressure has een descried. The aility of the ne models has een demonstrated y calculatin to asic transients. The simulations ith various initial states and oundary conditions prove that the transition from sucritical to supercritical conditions taes place in a sound manner ithout numerical prolems. Also at supercritical pressure the transition from pseudo liquid to pseudo as due to heat transfer taes place fast and ithout any numerical prolems. In order to closer examine the accuracy of model experimental data is needed. ACKNOWLEDGMENTS The authors appreciate to Fortum Nuclear Services Ltd for fundin this or toether ith VTT Technical Research Centre of Finland. Nomenclature C constant c specific heat capacity, speed of sound E rate of entrainment F force ravitational acceleration h total specific enthalpy (i.e. includin inetic enery) K heat transfer coefficient L evaporation heat Nu Nusselt numer p pressure Pr Prandtl numer q heat transfer rate R rate of stratification Re Reynolds numer t time T temperature u velocity z location void fraction (volume fraction of the as phase) mass transfer rate inclination dynamic (asolute) viscosity inematic viscosity density surface tension friction Suscripts ul as h hydraulic (13/14)

14 i interface phase (l or ) l liquid pseudo-critical pe pseudo-evaporation sat saturation stat static (ithout inetic enery) all REFERENCES 1. J. Kuri, Simulation of thermal hydraulics at supercritical pressures ith APROS, Thesis for the deree of Master of Science in Technoloy, Helsini University of Technoloy, Department of Enineerin Physics and Mathematics, T. Siionen, Numerical Method for One-Dimensional To-Phase Flo, Numerical Heat Transfer, 1, 1 18, I. Pioro, R. Duffey and T. Dumouchel, Hydraulic resistance of fluid floin in channels at supercritical pressures (survey), Nuclear Enineerin and Desin, 31, , I. Pioro, H. Khartail and R. Duffey, Heat transfer to supercritical fluids floin in channels empirical correlations (survey), Nuclear Enineerin and Desin, 30, 69 91, J. D. Jacson and W. B. Hall, Forced convection Heat Transfer to Fluids at Supercritical Pressure, in Turulent Forced Convection in Channels and Bundles,, edited y S. Kaaç and D. B. Spaldin, Hemisphere, A. R. Edards and T. P. O Brien, Studies of phenomena connected ith the depressurization of ater reactors, Journal of British Nuclear Enery Society, 9, , D. C. Groeneveld and C. W. Snoe, A Comprehensive Examination of Heat Transfer Correlations Suitale for Reactor Safety Analysis, Multiphase Science and Technoloy,, , (14/14)