TOPICAL PROBLEMS OF FLUID MECHANICS PERFORMANCE OF SIMPLE CONDENSATION MODEL IN HIGH-PRESSURES

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1 TOPICAL PROBLEMS OF FLUID MECHANICS 9 DOI: PERFORMANCE OF SIMPLE CONDENSATION MODEL IN HIGH-PRESSURES V. Hric, J. Halama Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Zikova 93/4, Prague 6, Czech Republic Abstract In this article we deal with the numerical solution of two-phase flow with nonequilibrium condensation in high pressures. We test how simple homogeneous non-equilibrium mixture model (single-fluid model) performs in high-pressure conditions. For this purpose we used our inhouse CFD code with the following limitation: time-dependent compressible Euler equations in D for mixture of dry steam and water droplets, standard method of moments for liquid phase, simple fitting of liquid saturation data, so-called special gas equation provided by Wagner and Pruss (IAPWS-9) used as real equation of state for both superheated and supersaturated dry steam, classical nucleation theory and droplet growth model with appropriate corrections. As a test case we used rapid expansion of high-pressure steam in convergent-divergent nozzle with supersonic outlet in which condensation occurs in the divergent part. Test cases come from work of Bakhtar et al [], []. We used all three nozzle geometries and two distinct cases for each (six test cases in total). Results of our simulations are in a reasonable accordance with the experimental data, several simplifications taking into account. Keywords: nozzle, high pressure, condensation, nucleation, method of moments. Introduction Most works devoted to physical and numerical modelling of non-equilibrium condensation phenomenon limit their attention to conditions within low pressures. In open literature proposed models along with the recommendation of their implementation were validated on many nozzle and turbine geometries under various conditions. In most cases these models employ simple perfect gas equation of state and simplify other parameters of condensation modelling. In low pressures those models perform quite well. However, in higher pressures discrepancies between calculated and measured data become more evident, therefore those classical condensation models are not reliable. The high-pressure condensing flow of steam was studied by e.g. Dykas and Wro blewski [3], Bakhtar and Zidi [] and [], Gyarmathy [4]. Flow model Our CFD code treats both phases as one perfect continuum mixture in which tiny submicron water droplets are homogeneously dispersed in the vapour phase. Creation of liquid phase in the form of small droplets is the result of such conditions in which supercooling (or subcooling in other words) of expanding dry vapour phase hits maximal value thereby so-called nucleation barrier necessary to phase transition is broken. The phase transition releases latent heat, droplets are growing further and flow reverts to equilibrium. The flow of mixture is governed by the Euler system of equation. The four transport equations of so-called liquid moments (moments of droplet number distribution function) are added to the main Euler system.

2 6 Prague, February -, 6. Governing equations Governing equations of our CFD model are as follows W + [F, F ] = S, () t W = [ρ, ρu, ρv, ρe, ρq, ρq, ρq, ρy] T, () F = [ ρu, ρu + p, ρvu, (ρe + p) u, ρq u, ρq u, ρq u, ρyu ] T, (3) F = [ ρv, ρuv, ρv + p, (ρe + p) v, ρq v, ρq v, ρq v, ρyv ] T, (4) [ S =,,,, J, r c J + ṙρq, rcj + ṙρq, 4 3 πρ ( ) ] T L r 3 c J + 3ṙρQ. () Vector W is composed of conservative quantities in which ρ is density of mixture, u and v are average components of velocity vector u, E is total energy of mixture (energy stored in the total surface of droplets is ignored), Q i is the i-th moment of droplet number distribution function and y is liquid mass fraction (wetness). Vectors F, F are flux vectors, p is average static pressure of mixture which is shared by both phases. The source vector S models phasic transiton. Nucleation rate J is the number of new liquid embryos (cores) per unit volume of mixture/vapour and second (volume of droplets is ignored). Liquid cores are able to grow further only if their radii exceed the critical value - critical droplet radius r c. In the opposite case these cores tend to evaporate completely. Growth of droplets ṙ should be applicable both in continuum (small Knudsen number) and free-molecular regime (large Knudsen number). The droplet growth law ṙ = ṙ( r) implements surface-averaged droplet radius according to Hill, Eq. 3.. Nucleation rate Modelling of nucleation rate is a very problematic and controversial matter, cf. Fig. where seven best known models are depicted at low and high pressure and different supercoolings (we took the liberty of not giving any detail of the models). In the engineering approach the model is based on quite old classical isothermal theory and so Becker-Döring nucleation rate model is accepted. This model employs capillary approximation which allows the use of macroscopic description of thermodynamic properties of microscopic nuclei. Along with the basic classical model we used two generally accepted corrections. The first correction is of Courtney and remedies the inconsistency of Gibbs free energy change during nucleation process with the law of mass action. The second correction tries to eliminate discrepancies owing to existing temperature difference between droplets and their vapour surrounds and is proposed by Kantrowicz or alternatively by Feder. We drew a comparison between Kantrowicz and Feder correction in Fig.. Both of the corrections result in slowing down the nucleation rate and therefore shift onset of condensation more downstream. The final form of the nucleation rate is as follows [] J = ξ C ξ K(F ) J classic, (6) ξ C = /S, (7) ξ K = + ρ ( V R RTV ) / ( ), L L α c π RT V RT V (8) ξ F = ( ), (9) + R sv s L c vv + R R σ ρ L RT V r c ( ) / ( ) ( ) σ p Gc J classic = exp, () RT V ρ L k B T V πm 3 G c = 4 3 πσ r c, () r c = σ ρ L (RT V ln S + g r ) (p p s ), () g r = g r (p, T V ) g r (p s, T V ). (3)

3 TOPICAL PROBLEMS OF FLUID MECHANICS 6 Factor ξ C is Courtney correction, ξ K is Kantrowicz correction, ξ F is Feder correction, S = p/p s is supersaturation ratio, α c is surface heat transfer coefficient between droplet with critical radius and surrounding vapour. Subscripts ( ) V, ( ) L, ( ) s, ( ) c pertain to vapour quantity, liquid quantity, saturated value, critical value, respectively. Surface tension σ acts on the flat surface of liquid water, G c is Gibbs free energy of formation of cluster with critical radius, L is equilibrium latent heat, k B is Boltzmann constant, m is mass of one molucule of water. The term g r is the difference in residual specific Gibbs free energy between actual and saturated state. If perfect gas equation of state were used, this term vanishes. nucleation number Jn / #.m-3.s- J-class J-class-court J-class-kantr J-class-court-kantr J-wolkstrey J-halle J-class-court-kantr-icct 3 nucleation number Jn / #.m-3.s- 3 3 J-class J-class-court J-class-kantr J-class-court-kantr J-wolkstrey J-halle J-class-court-kantr-icct 3 supercooling dt / K supercooling dt / K Figure : Models of nucleation rate for water at. MPa (left) and MPa (right) nonisothermal correction / Kantrowicz Feder nonisothermal correction / Kantrowicz Feder supercooling dt / K supercooling dt / K Figure : Nonisothermal corrections of nucl. rate at. MPa (left) and MPa (right).3 Droplet growth The evolution of droplet radius is modelled by the droplet growth model provided its radius exceeds the critical value r c. This growth is controlled by Knudsen number which is defined as the ratio of collision mean free path of vapour molecules to droplet diameter Kn = L cmfp, (4) r L cmfp = µ ( ) / V π. () ρ V RT V At the beginning of the condensation process droplet radii are very small compared to mean free path of vapour molecules (Kn ) and flow is in free molecular regime. Growth of larger droplets is in continuum regime (Kn ). Hertz-Knudsen droplet growth model which should be applicable only in free molecular regime is as follows ṙ HK = ρ L [ p (πrt V ) / p s,r = p s,r (T L, r) = p s (T L ) exp p s,r (πrt L ) ] ( σ ) ρ L RT L r, (6) (7)

4 6 Prague, February -, 6 For continuum droplet growth law we used ṙ C = λ V r T s T V lρ L, (8) where l = h V h L is nonequilibrium local latent heat and λ V is thermal conductivity of vapour phase (calculated by the relevant IAPWS release). For calculation of liquid temperature we adopted Gyarmathy s approach which is proposed for submicron droplets ( r < e 6 m) T L = T s (T s T V ) r c r The final droplet growth is a combination of Hertz-Knudsen and continuum model (9) ṙ = ζṙ C + ( ζ)ṙ HK, () ζ = exp(.693 Kn), () where ζ is Knudsen number dependent blending function. This combination results in the fact that both droplet growth models equally contribute to the final growth at Kn =..4 Equation of state and other properties As equation of state (EOS) for vapor phase we chose so-called special gas equation developed by Wagner and Pruss. This EOS represents the experimental data to within their uncertainties provided densities do not exceed kg/m 3 and temperatures are in the range from 73 to 73 K. Moreover, this EOS is recommended by IAPWS organization for calculating thermodynamic properties in supersaturated (metastable) region. Therefore, we find this EOS perfect candidate in case of high-pressures. This EOS is in the form of equation for nondimensional Helmholtz free energy, so all important thermodynamic properties can be calculated without integration. The equation is comprised of nine ideal terms and seven residual terms. One drawback is that its independent variables are density and temperature, which means we do not have explicit relation for pressure, since in every time level we know new values of density and internal energy. For complete formulation and more details see [6]. Mixture properties are calculated by means of liquid mass fraction (wetness) y = m L m () ρ = + y ρ L ρ V (3) e = ye L + ( y)e V (4) h = yh L + ( y)h V () w = ( y) / w V (6) The last equation is for speed of sound in mixture w. We accept the fact that the volume occupied by droplets is small compared to total volume of mixture. That is why we ignore the first term in Eq. 3. All liquid water properties are roughly approximated by polynomial fitting of liquid saturation curve. 3 Numerical solution 3. Solution procedure We employed cell-centered finite volume method on unstructured body-fitted grid. We are interested in stationary solution. In every time step we solve our nonhomogeneous governing system, Eq., in the three stages by implementing symmetrical time operator splitting W n+ = S ( t/) C ( t) S ( t/) W n, (7)

5 TOPICAL PROBLEMS OF FLUID MECHANICS 63 where S and C are solution operators which solve following subsystems of equations S : W t C : W t = S(W ), (8) + [F, F ] = (9) Both nonhomogeneous system, Eq. 8, and homogeneous system, Eq. 9, are solved by nd order -stage explicit Runge-Kutta method. Sources are calculated according the scheme in Tab.. Hill s average droplet radius is as follows r = r = { ( ) / Q Q... W 7 = ρy > e... otherwise. (3) Table : Algorithm for evaluation of condensation sources S > ( + e ) S < ( e ) r > r c nucleation and growth J, ṙ r r c only nucleation J, ṙ = r > evaporation J =, ṙ r = nothing J =, ṙ = otherwise nothing J =, ṙ = For spatial derivatives we used basic AUSM+ scheme without reconstruction of flow quantities on faces. In every time step we know updated value of mixture density and mixture internal energy, vapour density and vapour internal energy are extracted by the following procedure ρ n+ V = ( y n+) ρ n+, (3) V = en+ y n+ e n+ L (TV n) y n+. (3) e n+ Calculation of liquid internal energy e n+ L uses vapour temperature form previous time level. This simplification can be justifiable in case of stationary solution. Further, for static pressure p = p V = p L equation of state for vapour phase is applied and in case of IAPWS-9 special gas equation, calculation consists of two steps Step : e V = e V (ρ V, T V ), (33) Step : p V = p V (ρ V, T V ). (34) Eq. 33 is nonlinear algebraic equation for vapour temperature T V, then Eq. 34 is explicit for pressure. 3. Boundary conditions Following description of prescribed boundary conditions concerns all test cases. At the inlet of the nozzle we consider dry steam is in the subsonic regime, so we have to prescribe three quantities and the last one is extrapolated from the interior of the nozzle. The prescribed quantities are flow angle β, stagnation entropy s and stagnation enthalpy h which are calculated from given stagnation temperature T and stagnation pressure p (calculation is performed by the IAPWS-9 special gas equation), static pressure p ex is extrapolated from interior. In order to calculate density and temperature we solve the system of two nonlinear algebraic equations at the inlet s (ρ, T ) s =, (3) p (ρ, T ) p ex =. (36) We consider the flow angle β varies linearly from rad at the center line location to its maximal value at the wall. At the wall location prescribed flow vector and normal vector of the wall form

6 64 Prague, February -, 6 Table : Geometrical data of Bakhtar s experimental nozzles parameter nozzle S nozzle M nozzle L L/mm L/mm R/mm m/mm a/mm b/ b orig / c/mm c orig /mm d/ Table 3: Test cases conditions test number parameter nozzle S nozzle M nozzle L p /MPa T /K p /MPa T /K a right angle. Outflow from the nozzle is in supersonic regime, therefore all four quantities are extrapolated and none is prescribed. At the wall boundary we prescribed slip condition u n = (n is unity wall normal vector) and pressure is extrapolated from interior. 4 Results 4. Test cases We used work by Bakhtar and Zidi who experimentally investigated pressure distribution and droplet sizes in three convergent-divergent nozzles, namely nozzle S (small), nozzle M (medium), nozzle L (large). Fig. 3 shows sketch of all parameters defining the nozzle geometry. These parameters are summarizes in Tab.. Original values of parameters b and c are slightly corrected due to the fact that functions for the width of the nozzles w = w(x) were not continuous in the position x = m, cf. Fig. 3. Nozzle S was designed for expansion rate ṗ = (/p) dp/dt = 3 s, nozzle M for ṗ = s and nozzle L for ṗ = s. Six test cases in total were chosen. For all the test cases static pressure distribution along the center lines of the nozzles is available. Droplet radius measurement data are available only for test cases S-, M-, M-. Figure 3: Sketch of Bakhtar s nozzles geometry

7 TOPICAL PROBLEMS OF FLUID MECHANICS 6 3. Pressure p / MPa 3.. dry- wet- exp- dry- wet- exp Nucleation number Jn / #.m-3.s- Supercooling dt / K wet- wet- exp- 3 3 Sauter droplet radius r3 / nm Supersaturation S / wet- wet Wetness y / Figure 4: Nozzle S center line distribution 4. Center line results Fig. 4 shows distribution of important wet steam quantites along the center line of nozzle S. Nucleation onset is slightly shifted in downstream direction comparing to pressure experimental data. That concerns both test case and test case. Our results gave smaller expansion rates, which can be approximately visualized by pressure derivative in x-direction. Droplet size measurement was provided only for test case and that is substantially bigger (338 nm at location x =.7 m) comparing to our CFD result (69 nm at the same location). The wet-steam flow is characterized by quite smaller maximal supersaturations (approx..7) and supercoolings (approx. 4 K). Dashed lines present CFD simulations in which condensation is completely switched off. Maximum wetness ranges between. and 6 %. Results for nozzle M are presented in Fig.. The pressure distributions are in quite good accordance with the experiments. Droplet radii at selected location are under-predicted again. Outlet wetness is about 6. %. The final results are for nozzle L, cf. Fig. 6. In front of the nozzle throat the calculated expansion rates are smaller (pressure falls slowly). Droplet size measurement is not available, calculated sizes are close to nm at the location x =.7 m. Wetness at exit location is in the range 7. 8 %. Comparison of Knudsen number is in Fig. 7. In all cases its value is below., essential part is below., therefore flow is in continuum regime and this is the consequence of high pressure.

8 66 Prague, February -, 6 3. Pressure p / MPa 3.. dry- wet- exp- dry- wet- exp Nucleation number Jn / #.m-3.s- Supercooling dt / K wet- exp- wet- exp- 3 Sauter droplet radius r3 / nm Supersaturation S / wet- wet Wetness y / Figure : Nozzle M center line distribution

9 TOPICAL PROBLEMS OF FLUID MECHANICS Pressure p / MPa 3.. dry- wet- exp-. dry- wet- exp Nucleation number Jn / #.m-3.s- Supercooling dt / K wet- wet Sauter droplet radius r3 / nm Supersaturation S / wet- wet Wetness y / Figure 6: Nozzle L center line distribution Knudsen number Kn / S- M- L- S- M- L Figure 7: Knudsen number center line distribution

10 68 Prague, February -, 6 Conclusion We presented results of numerical simulation of nonequilibrium condensation of high-pressure steam during its expansion in convergent-divergent nozzles using our in-house CFD code with real equation of state provided by IAPWS release. Despite the many simplifications which we adopted in the CFD code (single-fluid mixture model, ignoring the volume of liquid phase, only inviscid and spatially st order Euler equations in D), we can conclude that obtained results are qualitatively acceptable. The biggest deviations are in estimation of droplet radii and we should subject droplet growth modelling to more detailed analysis in the future. Acknowledgment The financial support for the present project was partly provided by the Grant OHK-6/6. References [] Bakhtar F. & Zidi K.: Nucleation Phenomena in Flowing High-Pressure Steam Part : Experimental Results. Journal of Power and Energy, vol. 3, no. 3: (989) pp. 9. [] Bakhtar F. & Zidi K.: Nucleation Phenomena in Flowing High-Pressure Steam Part : Theoretical Analysis. Journal of Power and Energy, vol. 4, no. 4: (99) pp [3] Dykas, S. & Wróblewski W.: Numerical Modelling of Steam Condensing Flow in Low and Highpressure Nozzles. International Journal of Heat and Mass Transfer, vol., no. : () pp [4] Gyarmathy, G.: Nucleation of Steam in High-pressure Nozzle Experiments. Journal of Power and Energy, vol. 9, no. 6: () pp.. [] Bakhtar F. et al.: Classical Nucleation Theory and Its Application to Condensing Steam Flow Calculations. Journal of Mechanical Engineering Science, vol. 9, no. : () pp [6] Wagner, W & Pruss A.: The IAPWS Formulation 99 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. Journal of Physical and Chemical Reference Data, vol. 3, no. : () pp