Quantitative Phase Field Modeling of Boiling Phenomena

Transcription

1 Quantitative Phase Fied Modeing of Boiing Phenomena Arnodo Badio Pau Scherrer Institute 5232 Viigen PSI, Schweiz ABSTRACT A quantitative Phase Fied (PF) mode is deveoped for simuating the dynamic of the iquid-vapor interface during boiing of pure substances. In this mode, the interface is described by the variation of a phase indicator fied (phase fied), which is constant in the buk phases and varies smoothy across the transition region (diffuse interface). In contrast to sharp interface modes, where the veocity of the interface due to phase change depends on the oca temperature gradient, the evoution of the phase fied foows a reaxation process that minimizes the free energy of the system. The couping of the phase fied equation to the mass, momentum and energy equations aows for quantitative simuations of the interface dynamics. By eiminating the curvature from the phase fied equation, the convergence of the mode towards the sharp interface imit is no onger controed by the therma capiary ength, but by the thickness of the therma boundary ayer. With water as the test fuid, simuations of a sphericay symmetric bubbe growing in a superheated iquid are performed and compared to known anaytica soutions and experimenta resuts. Good agreement is found for severa superheating eves and saturation pressures. 1 INTRODUCTION After many years of research, much has been gained on the understanding of boiing, but despite of the efforts put into this enterprise, we are sti missing a comprehensive mode, abe to predict or to incorporate a the reevant physica phenomena invoved in this kind of processes. Many questions arise when we think about the fundamenta phenomena governing this compex phase change, for exampe, what is the infuence of a force fied (ike gravity or magnetic fied), what are the ength and times scaes reevant to the dynamic of the interface, what is the reationship between superheating and interface veocity or what are the effects of surface properties. Nowadays, the most advanced modes incude microscopic phenomena at the contact ine and impicit tracking of the iquid-vapor interface during bubbe growth [1]. Nonetheess, these modes athough promising, have not been fuy vaidated and their appicabiity to poar fuids such as water appears to be a ong term goa. Many modes deveoped for immiscibe two phase fows such as Leve-Set [2], VOF [3] CIP [4] and sharp interface tracking techniques [5], have shown great accuracy when, for instance, simuating the raising of air bubbes in water [6]. When these modes are couped to the energy equation and a suitabe evaporation mode is set, it is possibe to simuate boiing from a mesoscopic point of view. Under aminar fow conditions, this kind of simuations aims at resoving the smaest time and ength scaes in the continuum imit, i.e., the interface veocity and therma boundary ayer at the iquid-vapor interface. Ergo these simuations fa into the category of Direct Numerica Simuations (DNS), athough this name has commony 818.1

2 818.2 been associated with cacuations of turbuent singe phase fows, where the smaest ength and time scaes are estabished by the dynamics of the smaest eddies. The resoution of the therma boundary ayer at the iquid-vapor interface is one the most difficut task for any mode aiming at mesoscopic simuations (singe bubbes). Experiments of bubbe growth performed in superheated water at 1 bar, have shown that the thickness of the therma boundary ayer is in the order of tens of microns [7]. This poses a great computationa chaenge when simuating system sizes in the order of few miimeters. The situation is even more compex when the bubbes are attached to a surface, where microscopic phenomena at the contact ine might have an important contribution to the overa evaporation rate [8,9]. Therefore any new mode must first be abe to predict accuratey the growth of unbounded bubbes, where the dominant ength scae for the interface dynamics is the thickness of the therma boundary ayer, and then incude the effect of a surface. In this work, I present a newy derived Phase Fied mode for simuating bubbe growth in the diffusioncontroed regime. 2 THE PHASE FIELD MODEL Diffuse interface modes are quite popuar for two-phase fows simuations and the main reason ies in their simpicity for deaing with compex phenomena such as capiarity and topoogic changes of the interface [10]. In these modes, the interface is described by a steep but smooth variation of an order parameter or phase indicator fied, which keeps a constant vaue in the buk phases. This indicator fied is then used to mode the variation of a the physica properties across the interface. The width of the interface is an arbitrary parameter, often reated to the grid spacing, with no physica meaning. By couping the evoution of the indicator fied to conservation aws for mass, momentum and energy, it is possibe to sove a unique set of equation to describe the evoution of the buk as we as the interfacia region, without the need for using any specia agorithm for tracking the interface or satisfying sharp interface baances at the interface. Hence, the most compicated part is to find the appropriate source terms in the mass, momentum and energy equations to account for the phase change. For instance, in sharp interface formuations, these source terms do not exist expicity in the equations, but they are repaced by the so-caed jump conditions, which represent conservation aws satisfied at mathematica surfaces. Phase fied modes aso fa into the smooth interface category, but the path to determine the source terms, which repace the jump conditions, is commony based on thermodynamic arguments. Simpe modes of evaporation, ike the temperature recovery method [11], aows for cacuating the amount of iquid being evaporated based on the oca temperature within the diffuse interface, but the time scae on which this process takes pace, is somewhat arbitrary. This detai at first gance may appear insignificant, but when performing quantitative simuations, an accurate time scae is absoutey crucia, since the interface dynamic wi be primariy dominated by the evaporation rate. Besides, the absorption rate of atent heat by the interface aso depends on the time scae we use to evaporate the iquid. In the temperature recovery method, the evaporation rate depends on the amount of superheating at the interface, that is, the amount by which the interface temperature is above its saturation vaue. When using a source term in the energy equation to account for evaporation, the oca temperature inside the interface must decrease due to the cooing effect of evaporation, but at the same time, heat is fowing towards the interface by diffusion. When the evaporation rate equas the heat fux to the interface, we are basicay satisfying the Stefan condition, where the interface veocity is proportiona to the heat fux at the iquid-vapor interface. Athough this mode captures the phenomenoogy of the boiing process, cannot be used to study boiing in the presence of superheated iquid. In a PF approach, the evaporation rate aso

3 818.3 depends on the oca temperature within the diffuse interface, but the buk iquid can be superheated and the time scae is determined by an asymptotic anaysis of the equations [12]. Furthermore, the evoution of the phase fied is not simpy based on the amount of superheating at the interface, but aso must satisfy the minimization of a functiona that represents the free energy of the entire system. The detais behind PF modes can be found esewhere [12, 13], therefore in the foowing I wi ony present the main ideas. First, we have to buid the free energy functiona that represents the free energy of a system in equiibrium, therefore the temperature must be constant everywhere (isotherma system). Cahn and Hiiard [14] proposed a simpe (but effective) way to do that; they proposed that the free energy of a point within the system f φ (which satisfy cassic coud be expressed as the contribution of a homogeneous part ( ) thermodynamic) and a inhomogeneous part λ φ 2 accounting for oca variations of an order parameter φ, thus the free energy of the system is given by 2 0 F = V f ( φ ) + λ φ + L dv (1) 2 In the present derivation, the inhomogeneous parameter λ was considered constant. If the system is out of equiibrium, this free energy shoud be regarded as a snapshot at a given time. In order to move from a non-equiibrium situation to an equiibrium one, the order parameter must change in the direction that decreases the free energy of the system, i.e., the evoution of the order parameter must satisfy the second aw of thermodynamics. For a nonconserved order parameter, the evoution of the phase fied foows a reaxation process to reach an equiibrium state, which can be expressed mathematicay as φ 1 δf μ = = t τ δφ τ ( φ) (2) Where τ is the reaxation time, δ is the functiona derivative and μ is the transformation potentia, which measures the energy difference between equiibrium and non-equiibrium states. Thus, under equiibrium, the transformation potentia is zero and no phase change is possibe. In order to perform quantitative simuations, equation (2) must be couped to the mass, momentum and energy equations. To derive the mode, we start from a genera conservation aw Φ t = J + S (3) Where Φ can be a scaar or vector quantity. The fux J = uφ + J is composed by convective u Φ and diffusive J d parts and S is a source term. Foowing Sun and Beckermann [15], we consider that the iquid and vapor coexist within the transition region and that each phase has a distinctive mass, veocity and enthapy (see figure 1). Since the phase fied varies continuousy from 0 in the vapor to 1 in the iquid, we can make use of this information to find appropriate interpoation functions for a the physica properties and fied variabes. To obtain the fina form of the mass and energy equations, I have used the foowing definitions Density: ρ = ρ φ + ρ ( 1 φ) v (4) d

4 Linear momentum: ρu ρ φu + ρ v ( φ ) u v Enthapy: = φ + ( 1 φ) = 1 (5) H H H v (6) Hu = H φu + H 1 φ u (7) Enthapy advective fux v ( ) v Soenoida fieds u = u = 0 (8) v φ 1 0 vapor u v u iquid Figure 1. Schematic of the phase distribution within the diffuse interface. The figure aso depicts the numerica setup for the sucking probem. The detais of the derivation of the present mode wi be presented in a forthcoming pubication and ony the fina set of equations wi be written. For the sake of simpicity, I wi consider the computationa setup depicted in figure 1. In this case, known as the sucking probem [16], the veocity of the vapor phase is zero, hence mass conservation states 1 1 u & (9) ρ ρv Mass: ( φ) = Γ Where Γ & is the mass evaporation rate per unit voume. Since the evaporating iquid suffers a arge expansion, a certain amount of mass must eave the system to aow the free movement of the interface. In the case under study, iquid is exiting the system at the right boundary. By negecting viscous dissipation and considering a quasi-isobaric process, the variation of the interna energy of a differentia fuid eement is given by the variation of its enthapy, thus T t ρ ( T + j ) + ( L ( c c ) ΔT )Γ& (10) Energy: c + c u φ T = k( φ) P P The ast term in the R.H.S. represents a inearization of the atent heat absorbed during evaporation (or reeased during condensation). The mixture heat capacity c P, therma conductivity k ( φ) and the phenomenoogica current j at are given by P = ρ cp φ + ρv cpv ( 1 φ) ( φ) = k φ + k ( 1 φ) c (11) k v (12) j = a( ) W ( L ( c c ) ΔT ) Γ& at φ sat P Pv n (13) at sat P Pv

5 818.5 The unusua energy fux j at is non-zero ony within the diffuse interface, and vanishes in the case of equa therma conductivities and in the sharp interface imit ( W 0). This current is somehow simiar to those proposed by McFadden and co-workers [13] in their version of a phase fied mode, based on an energy/entropy approach. In their artice, they expicity expressed some concerns about the physica vaidity of this current, but justified its use as a mean for faciitating computations and modeing. In our mode, this current is competey unphysica and appears as a necessity to recover a oca thermodynamic equiibrium for artificiay enarged interface thicknesses. As seen in moecuar dynamics studies of boiing [17], the rea thickness of a iquid-vapor interface is in the range of few angstroms at atmospheric pressure. This utra-thin interface sets a difficut constraint for any mode, due to the fact that there must be severa grid points inside the diffuse interface to resove the derivatives of the phase fied. To be abe to perform simuations at much arger scaes, we must increase artificiay the interface thickness to vaues computationay tractabe. In this interface thickening process, some spurious effects are introduced in the energy equation, which have to be baanced by the phenomenoogica current. Karma [12] proposed a simiar expression when deriving a mode for dendritic growth, which he caed anti-trapping current, thus in resembance to his idea, I used the subindex at. The functions entering this current are ( ) ( q( φ) p( φ) φ) φ = 2φ ( 1 φ ) ( φ ) ( k v k )φ a (14) p = (15) ( φ) = ( k k 1 ) φ + 1 q (16) v In order to cose the equations, we have to provide an expression for the transformation potentia μ, evaporation rate and reaxation time. 2 2 ( φ) = W ( φ κ φ ) + φ( 1 φ )( 2φ 1) 2a φ( φ)( T ) μ T sat (17) Γ & = ρ v μ φ (18) ( ) τ 2 sat τ = 2W ρv cpv kv (19) The constant a 2 is defined in the foowing way I = , J = , G = ( kv k ), K = ( kv k ) (20) sat a0 = I J, a1 = ( JG K ) J, a2 = cpv a0 2 ( Lsata1 ) These constants were determined by an asymptotic anaysis of the current mode. Detais on this kind of anaysis can be found in [13]. 3 RESULTS Numerica simuations of singe bubbe growth in superheated water were carried out at 1 and 0.39 bar. The bubbe is assumed to be stationary and therefore the veocity fied, due to the iquid expansion, is obtained by direct integration of the mass conservation equation (9). Figure 2, shows the schematic for a stationary bubbe with spherica symmetry. A uniform 1D spherica grid with 160 equidistant points was used for both saturation pressures. The system size was 1.5 mm and 9 mm for simuations at 1 and 0.39 bar respectivey. The interface thickness was set equa to W = 1. 5Δx for a the simuations and the physica properties of the fuids were taken from the ASME steam tabes. The initia temperature distributions were obtained from Scriven s anaytica soution [18].

6 818.6 The interface position predicted by PF simuations was in good agreement with the anaytica soution proposed by Scriven for both saturation pressures and superheatings. Nonetheess, it is evident that for sma initia bubbe sizes, the therma boundary ayer is too thin and ceary under resoved by the mesh. Figure 3a presents the resuts for two different initia bubbe sizes 0.1 mm A and 0.45 mm B. For case A, the correct trend for the interface position is ony achieved when the therma boundary ayer is we resoved (at about 4 miiseconds), whereas by setting an initia size of 0.45 mm, the boundary ayer is much more thicker than in the previous case, and consequenty we captured by the mesh from the beginning of the simuation. This resut shows the imperative need for resoving the therma boundary ayer when performing DNS of bubbe growth. For water at 0.39 bar, the initia bubbe size was 2 mm and the agreement with Scriven s soution is aso good. Figure 2. Schematic of the numerica setup used to simuate bubbe growth in an unbounded superheated iquid. The figure dispays a temperature jump at the interface to iustrate out-of-equiibrium effects such as the interface therma resistance, which are not considered in the present mode. (a) (b) Figure 3. Comparison between predicted and measured bubbe size. Red ines represent PF simuations and cose symbos correspond to experimenta measurements.

7 818.7 Detaied experiments on bubbe growth in unbounded systems are extremey scarce and the ony experimenta data known to me, were obtained by Dergarabedian [7] in 1952 and Lien in 1969 [19]. A comparison between these experimenta data and PF simuations shows a fair agreement (see figure 3). For instance, the sope of curve R i (t) for the two cases under study has the right trend, but the actua bubbe size is over estimated. Since the experimenta measurements were performed such a ong time ago, it was not possibe to gather enough information to estimate the eve of uncertainty in the bubbe size. Nonetheess, since the sope of the curve represents the change of the bubbe size (rather than the size itsef), its vaue is assumed to have a ower degree of uncertainty. The mode was aso soved in 3D and cacuations of bubbe growth in water at 10 bar were performed to compare the resuts with the anaytica soution. To speedup the cacuations, I took advantage of the spherica symmetry and simuate ony 1/8 of the bubbe, as seen in figure 4(b,c). The system size was 256 μm 3 and three eves of mesh refinement were used 16 3, 32 3 and Bubbe size as a function of time is presented in figure 4(a) and the resuts show a convergent behavior towards the anaytica soution. (a) (b) (c) Figure 4: Three dimensiona Phase Fied simuations of bubbe growth. (a) Time evoution of bubbe radius, (b) initia condition, (c) fina stage of the simuation. Vector fied indicates the fow pattern at the iquid-vapor interface generated by the iquid expansion. Coor map represents temperature distribution. Figure 4(b) shows the initia condition for the Phase Fied simuations at the finest grid. It can be seen the sma extent of the therma boundary ayer compared to the bubbe radius, which sets a difficut numerica constraint. In fact, it is the thickness of the therma boundary ayer (not the bubbe size) that sets up the scae of the system that is feasibe to be simuated. 4 CONCLUSIONS A newy derived quantitative Phase Fied mode for simuating iquid-vapor phase transitions has been presented. Comparison with known anaytica soutions and experimenta data on bubbe growth in the diffusion controed regime, reveaed a good performance of the mode for different eves of superheating and saturation conditions. The fu mode, not presented here to compy with space requirements, incudes surface tension and couping with the momentum equations. New numerica resuts and the fu derivation of the mode wi be presented in forthcoming pubications.

8 818.8 ACKNOWLEDGMENTS The author gratefuy acknowedge the financia support from Swissnucear for the project Muti-Scae Modeing Anaysis of convective boiing (MSMA). REFERENCES [1] V. Dhir, Numerica Simuations of Poo-Boiing Heat Transfer, AIChE Journa, 47, 2001, pp [2] W. Osson. G. Kreiss and S. Zahedi, A Conservative Leve Set Method for two Phase Fow II, J. Comp. Phys., 225, 2007, pp [3] D. Gueyffier, J. Li, A. Nadim, R. Scardovei and S. Zaeski, Voume-of-Fuid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensiona Fows, J. Comp. Phys., 152, 1999, pp [4] T. Nakamura, R. Tanaka, T. Yabe and K. Takizawa, Exacty Conservative Semi- Lagrangian Scheme for Muti-dimensiona Hyperboic Equations with Directiona Spitting Technique, J. Comp. Phys, 174, 2001, pp [5] H. Liu, S. Krishnan, S. Marea and H.S. Udaykumar, Sharp Interface Cartesian Grid Method II: A Technique for Simuating Dropet Interactions with Surfaces of Arbitrary Shape, J. Comp. Phys., 210, 2005, pp [6] Y. Sato, N. Bojan, A Conservative Loca Interface Sharpening Scheme for the Constrained Interpoation Profie Method, Int. J. Numer. Meth. F. (submitted 2011). [7] P. Dergarabedian, The Rate of Growth of Vapor Bubbes in Superheated Water, Ph.D. thesis CALTECH, [8] J. Kern and P. Stephan, Theoretica Mode for Nuceate Boiing Heat and Mass Transfer of Binary Mixtures, J. Heat Trans., 125, 2003, pp [9] J. Lay and V. Dhir, Shape of a Vapor Stem During Nuceate Boiing of Saturated Liquids, J. Heat. Trans., 117, 1995, pp [10] Y. Sun and C. Beckermann, Sharp Interface Tracking Using the Phase-Fied Equation, J. Comp. Phys., 220, 2007, pp [11] K. Takase, Y. Ose and T. Kunugi, Numerica Study on Direct-Contact Condensation of Vapor in Cod Water, Fusion Eng. Des., 63-64, 2002, pp [12] A. Karma, Phase-Fied Formuation for Quantitative Modeing of Aoy Soidification, Phys. Rev. Lett. 87, 2001, [13] G. McFadden, A. Wheeer and D. Anderson, Thin Interface Asymptotics for and Energy/Entropy Approach to Phase-Fied Modes with Unequa Conductivities, Phys. D., 144, 2000, pp [14] J. Cahn and J. Hiiard, Free Energy of a Nonuniform System. I. Interfacia Free Energy, J. Chem. Phys. 28, 1958, pp

9 818.9 [15] Y. Sun and C. Beckermann, Diffuse Interface Modeing of Two-Phase Fows Based on Averaging Mass and Momentum Equations, Phys. D, 198, 2004, pp [16] S. Wech and J. Wison, A Voume of Fuid Based Method for Fuid Fows with Phase Change, J. Comp. Phys., 160, 2000, pp [17] B. Shi and V. Dhir Moecuar dynamics simuation of the density and surface tension of water by partice-partice partice-mesh method, J. Chem. Phys., 124, 2006, [18] L. E. Scriven, On the Dynamics of Phase Growth, Chem. Engng. Sci., 10, 1959, pp [19] Y. Lien, Bubbe Growth Rates at Reduced Pressure, Ph.D. thesis MIT, 1969.