International Communications in Heat and Mass Transfer 36 (2009) 902–907

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International Communications in Heat and Mass Transfer

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Numerical simulation on non-equilibrium spontaneous condensationin supersonic steam flow☆

Yong Yang, Shengqiang Shen 1,⁎

Key Lab. of Ocean Energy Utilization and Energy Conservation of Ministry of Edu., Daian University of Technology, Dalian 116024, PR China

☆ Communicated by P. Cheng and W.Q. Tao.⁎ Corresponding author. School of Energy and Power

of Technology, Dalian, Liaoning 116024, PR China.E-mail addresses: zzbshen@dlut.edu.cn (S. Shen), no

(Y. Yang).1 Postal address: School of Energy and Power Eng

Technology, Dalian, Liaoning 116024, PR China.

0735-1933/$ – see front matter © 2009 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2009.06.001

a b s t r a c t

a r t i c l e i n f o

Available online 28 June 2009

Keywords:SupersonicSpontaneous condensationNon-equilibriumSteam

It is well known that the release of the latent heat by the homogenously nucleating steam flow can have astrong effect on the flow dynamics when the flow is transonic or supersonic with shock waves present. Thispaper describes a conservative two-dimensional compressible numerical model for supersonic non-equilibrium spontaneous condensing steam flow based on the classical nucleation theory and the Virial styleequation of state. Quantitative validation of the numerical model was accomplished and the results showed agood agreement between numerical simulation and the experimental data. On the basis of the comparison ofthe proposed simulation and the dry isentropic expansion solution, the physical description of thecondensation shock in Laval nozzles is concluded. And the effect of the condensation shock on the nucleationprocess itself and the resulting droplet size distribution was revealed.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Dropwise condensation and droplet growth in supersonic expan-sions occur in a number of natural and technological processes, suchas the formation of aerosols, wet steam flows in steam turbines, theflight of aircraft in humid conditions, droplet-spray combustionprocesses or in phase separation devices. Thus, understandingcondensation in supersonic flows is important for predicting, andpossibly controlling the condensation phenomenon, when the phasetransitionwill occur under the high super saturation achieved in theserapid expansions.

Laval nozzles have been used extensively to examine condensationof both mixtures of condensable vapor in carrier gas and pure vapors,especially steam [1–3]. In Laval nozzles, the cross-sectional flow areafirst decreases to the throat and then increases monotonically in thesupersonic region. As the steam flows through the nozzle it expandsand the temperature falls rapidly. Because the equilibrium vaporpressure decreases exponentially with temperature, very high supersaturation can be achieved. Eventually the steam acquires sufficientsupercooling for spontaneous nucleation of droplets in the divergingsection of the nozzle, where the flow is supersonic. The subsequentrelease of latent heat thus results in a deceleration of the flow and arise in pressure, known traditionally as the “condensation shock”.

Engineering, Dalian University

tebookdlut@gmail.com

ineering, Dalian University of

l rights reserved.

Experimental investigations have been undertaken to quantify thethermodynamic losses prevalent in transonic flowswith phase changein steam systems [4,5]. Measurements of nozzle pressure distributionsand the use of light scattering data to infer droplet sizes have been usedextensively to test the theories of nucleation and droplet growth [6].

In the paper, a conservative two-dimensional compressiblenumerical model is developed under the Euler coordinate system.The numerical simulation for the prediction of the non-equilibriumsteam flow with spontaneous condensation in the conventional Lavalnozzle is completed. The method described here, is based on a time-accurate Euler solver and suitable for the accurate investigation ofunsteady phenomena. The adaptive mesh was employed, therefore, tobe able to focus on regions of rapid flow change. The success of themethod was measured by the ability to predict measured pressuredistributions, droplet sizes and the mixture conditions for thesubsequent single-phase inviscid flow calculations. The contributionof the present paper is that the comparison between the non-equilibrium simulation and the dry isentropic expansion solution hasbeen accomplished for the first time, and the physical description ofthe condensation shock in Laval nozzles is concluded.

2. Mathematical modeling

2.1. Governing conservation equations for mass, momentum and energy

Different calculation methods for condensing steam flow havebeen presented many times in the literature (e.g. [3,7]). In the presentpaper, the mathematical model describing the condensing steamconsists of the continuous vapor phase at temperature T and pressureP, interspersed with a large number of spherical liquid droplets. It is

Nomenclature

B The second Virial coefficient (m3/kg)C The third Virial coefficient (m6/kg2)E Total specific internal energy (J/kg)h Static enthalpy (J/kg)I Nucleation rate (1/m3 s)Kb Boltzmann constant (=1.3807×10−23 J/K)m Mass (kg)P Pressure (Pa)r Radius of droplets (m)r ̅ Mean droplet radius of droplets (m)r⁎

Droplet critical radius (m)S Super saturation ratioT Temperature (K)Tsubc Supercooling level (K)u X-axial velocity (m/s)v Y-axial velocity (m/s)

Greek symbolsβ Mass fraction of the condensed phaseγ The ratio of specific heat capacitiesΓ Mass generation rate (kg/m3 s)η Number density for droplets (1/m3)θ Nonisothermal correction factorρ Density (kg/m3)σ Liquid surface tension (N/m)

Subcripts0 Stagnation state conditiong Gas (vapor)l Liquids Saturated state condition

Superscripts()→

Vector() ̅ Average of a variable

903Y. Yang, S. Shen / International Communications in Heat and Mass Transfer 36 (2009) 902–907

assumed that the liquid is monodispersed; that is, that all droplets areof the same size at one point in the flow and the interactions betweendroplets are neglected.

To make the simulation available and efficient, the condensingsteam flow is assumed to be adiabatic and inviscid. Since droplet sizesare sufficiently small, it is assumed that the volume of the condensedliquid droplets is negligible and the velocity slip between the dropletsand gaseous-phase is zero. We have adopted the Eulerian–Eulerianapproach formodeling the condensing steam flow. And the two-phaseflow is modeled using the conservation-type two-dimensionalcompressible Navier-Stokes equations, with the transport equationsfor the liquid-phase mass-fraction (β) and the number of liquid-droplets per unit volume (η).

Under the foregoing assumptions, the Euler Equations may bewritten in integral form as follows:

AUAt

+AFAx

+AGAy

= 0 ð1Þ

where x, y and t are the space and time coordinates respectively. U, Fand G are defined as:

U =

ρρuρvρE

2664

3775; F =

ρuρu2 + P

ρuvρE + Pð Þu

2664

3775 ; G =

ρvρuv

ρv2 + PρE + Pð Þv

2664

3775

And the mixture density, the internal energy and specific enthalpyare determined as follows:

ρ = ρg = 1 − βð Þ ð2Þ

E = h0 − P = ρ ð3Þ

h0 = h + u2 + v2� �

= 2 ð4Þ

h = 1− βð Þhg + hl ð5Þ

To model wet steam, two additional transport equations areneeded [8]. The first transport equation governs the mass fraction ofthe condensed liquid phase (β) and the other transport equation (η)determines the number of droplets per unit volume. The twoequations are combined to the model in the following expression:

AρAt

β + j ρ v→β� �

= C ð6Þ

AρAt

η + j ρ v→η� �

= ρI ð7Þ

And the number of droplets per unit volume (η) is calculated asfollows:

η =β

1− βð ÞVd ρl = ρg

� � ð8Þ

Vd =43πr3 ð9Þ

where ρl,Vd, r ̅ denote the liquid density, the average droplet volumeand the average droplet radius.

2.2. Nucleation model and droplet growth model for nucleating particles

Experimental and theoretical studies of expansion of steamthrough supersonic nozzles have been conducted by various investi-gators. While several theories have been proposed for obtaining thenucleation rate during spontaneous condensation, the two maintheories of interest are the classical theory and the theory developedby Deich et al.[9]. The most widely adopted classical nucleation theoryis discussed by many authors. In the present study, the non-equilibrium flow calculation considers only homogenous nucleationin pure substance and relies on the classical nucleation theorycorrected for non-isothermal effects by Young [10].

In the model, the classical homogeneous nucleation theorydescribes the formation of a liquid-phase in the form of dropletsfrom a supersaturated phase in the absence of impurities or foreignparticles, and the nucleation rate is given by:

I =qc

1 + θρ2v

ρl

! ffiffiffiffiffiffiffiffiffiffiffiffi2πRTM3

mπ

sexp −4πr2T σ

3KbT

!ð10Þ

where qc is the condensation coefficient, Kb is the Boltzmann constant,Mm is the mass of onemolecule, σ is the liquid surface tension, and θ isa non-isothermal correction factor which is given by the followingrelation:

θ =2 γ − 1ð Þγ + 1ð Þ

hlvRT

� �hlvRT

− 0:5� �

ð11Þ

where hlv is the latent heat of evaporation at pressure P.Based on the nucleation model just described the quantity of

droplets at a location in the continuous gas phase is known, and therate at which these droplets grow can be derived on the basis of heat

Fig. 1. Geometry of the converging–diverging nozzle (Nozzle B used by Moore et al.,1973).

Fig. 2. Pressure distribution along the nozzle centerline comparedwith the Experimentsof Moore et al.(1973).

904 Y. Yang, S. Shen / International Communications in Heat and Mass Transfer 36 (2009) 902–907

transfer conditions surrounding the droplet [11]. This energy transferrelation can be written as:

ArAt

=P

hlvρl

ffiffiffiffiffiffiffiffiffiffiffiffi2πRT

p γ + 12γ

Cp Tl − Tð Þ ð12Þ

Departures from thermal equilibrium for the vapor are quantifiedby the supercooling level (Tsubc),

Tsubc = Ts − T ð13Þ

where Ts is the saturation temperature at the local pressure P. And thedroplet temperatureTl is assumed tomaintain its quasi-steady value [12].

Tl = Ts −2σTsρlhlvr

ð14Þ

This assumption is equivalent to neglecting droplet temperaturerelaxation and is justified because the time constant associated withthis process is extremely short. The difference between the liquid andsaturation temperatures given in Eq. (14) is due to surface curvatureeffects and is significant only for very small droplets.

Based on the above theory, the mass generation rate Γ during thenonequilibrium condensation process is given by the sum of massincrease due to nucleation and also due to growth/demise of thesedroplets and therefore, Γ is written as:

C =43πρlIr

3T + 4πρlηI r

2 ArAt

ð15Þ

where r⁎is the Kelvin-Helmholtz critical droplet radius. The droplet

will grow as its radius is larger than r⁎, otherwise the droplet will

evaporate. And r⁎is given as follows [10]:

rT =2σ

ρlRTLnSð16Þ

where S is the super saturation ratio defined as the ratio of vaporpressure to the equilibrium saturation pressure:

S =P

Psat Tð Þ ð17Þ

2.3. Equation of state

The study of non-equilibrium thermodynamics (in the context ofhigh-speed compressible flows) involves conditions of supercooledstates, and an equation of state reliable for extrapolation into

supercooled conditions must be utilized. In the present work theequation of state reported in the study of Vukalovich [13] is used,which was tested for extrapolation into supercooled states [14]. Thisequation of state, utilizing a Virial formulation, is reliable for both highand low pressure conditions ranging from 0.01 to 100 bar, and over atemperature range of 273.15–1000 K. The Virial equation of state alongwith relations for vapor pressure, liquid density and specific heat dataprovides the basis for calculating all properties required in thesimulations, for both equilibrium and supercooled states [15].

The steam equation of state used in the study, which relates thepressure to the vapor density and the temperature, is given by:

p = ρvRT 1 + Bρv + Cρ2v

� �ð18Þ

where B and C are the second and the third virial coefficients.

2.4. CFD solution methodology

In the study, the non-equilibrium condensation model has beenimplemented within the commercial CFD code FLUENT, whichprovides the framework for the solution of the hydrodynamic equa-tions. The liquid and gas phase conservation equations are discretizedusing a conservative finite-volume integration over a control volume.Second-order upwind schemes are selected, and in this approach,high-order accuracy is achieved at cell faces through a Taylor seriesexpansion of the cell-centered solution about the cell centroid.

In the simulation, turbulence is modeled via the standard κ-εmodel. The source terms in the κ-ε equations do not consider theinfluence of the liquid droplets on turbulence conditions. However,this effect is indirectly modeled via the velocity field introduced to theκ-ε equations in each iteration. This is assumed adequate for dropletsof submicron size. The mesh and model were created in a twodimension (2-D) domain. The geometries of the converging–diver-ging nozzle (Fig. 1) were taken to be the same as the Nozzle B used inthe experiment of Moore et al. (1973). To retain the calculation speedadvantage coming with the use of a regular block-structure element,the multi-block technique was applied to make the concentration ofgrid density focused on the areas where significant phenomena wereexpected. The boundary conditions and the assumptions made wereas follows:

(1) At the nozzle inflow, subsonic flow was specified using totalpressure (P0in), total temperature (T0in) and flow angle normalto the inlet plane. In addition a turbulent intensity of 0.05, andan eddy length scale of 0.075 of the nozzle inlet diameter wereused.

Fig. 3. Droplet radius distribution (r) along the nozzle centerline compared with theexperimental data with the droplet growth rate (∂ r ̅/∂t) profile.

905Y. Yang, S. Shen / International Communications in Heat and Mass Transfer 36 (2009) 902–907

(2) At the wall a no-slip adiabatic wall condition was used and itwas assumed that the liquid droplets, upon impact with thewall, were reflected back into the flow with a coefficient ofrestitution equal to 0.1.

(3) Assuming that the flow is symmetrical about the nozzlecenterline, symmetry conditions were enforced for all flowvariables along this plane.

(4) At the exit plane, supersonic outflow conditions were applied,and all of the flow parameters were extrapolated from theinterior of the domain to the exit plane.

Fig. 4. Centerline values for (4–1) the frozen mach number Mf, (4–2) the Super satu

3. Results and discussion

3.1. Comparison vs. experimental values in literature

Quantitative validation of the numerical model is accomplishedusing the converging–diverging nozzle based on the experimentconducted by Moore et al. (1973) and the results showed a goodagreement between numerical simulation and the experiment data.The pressure distribution along the centerline is compared with theexperimental values, as shown in Fig. 2. As it is apparent in this figure,the final result is in excellent agreement with the experiment for theconverging–diverging nozzle. Fig. 3 shows the distribution of dropletradius comparedwith the experimental data given at a single location.The agreement of the computed droplet radius with the experiment isreasonable considering the considerable uncertainty in the measure-ment of droplet sizes at this scale.

3.2. Physical description of supersonic condensing steam flow withhomogeneous nucleation

To qualitatively describe the physics of supersonic steam flowwithor without non-equilibrium condensation, a sample of the calculatedresults for the converging–diverging nozzle are shown in Fig. 2 toFig. 6. The boundary conditions for the results were P0in=25 kPa andT0in=354.6 K at the inflow with supersonic outflow condition.

With comparison to the dry isentropic expansion model and theexperimental values, Fig. 2 gives the distribution of pressure along thenozzle centerline, in which, the solid-line corresponds to the pressureprofile of condensing steam flow with homogeneous nucleation, and

ration ratio S, (4–3) the supercooling level Tsubc, and (4–4) the nucleation rate I.

Fig. 5. Comparison between the proposed model and the isentropic model forcenterline temperature (T) and the local saturation temperature (Ts).

Fig. 6. Centerline values for wetness fraction (β) and liquid mass generation rate (Γ).

906 Y. Yang, S. Shen / International Communications in Heat and Mass Transfer 36 (2009) 902–907

the dashdotted-line corresponds to the pressure profile of the dryisentropic expansion. One can see in Fig. 2 that the proposed modelused in this study predicts close distribution of the pressure over thenozzle length, in particular the location of the condensation shock,where the pressure ratio starts to deviate the isentropic pressure ratiodownstream of the throat.

Fig. 3 gives the droplet radius distribution compared with theexperiments and the droplet growth rate (∂r_/∂t) along the nozzlecenterline. Fig. 4 shows the distribution of the frozen mach number(Mf), the super saturation ratio (S), the supercooling level (Tsubc) andthe nucleation rate (I) along the centerline of the nozzle calculated bythe proposed model. Fig. 5 gives the centerline values of thetemperature (T) compared with the local saturation temperature(Ts) and the temperature calculated by the isentropic model. Fig. 6gives the centerline values for wetness fraction (β) and liquid massgeneration rate (Γ).

As is apparent in the figures when the steam is expanded in theconverging–diverging nozzle, the temperature decreases rapidly tomake the isentropic expansion vapor cross the saturation line quicklyand stay in a supersaturated state until the spontaneous condensationoccurs suddenly where the supercooling achieves near 40 K, with theresulting droplets travelling downstream to reach droplets sizes closeto 15 nm. As the critical conditions are approached the supersaturation ratio and the nucleation rate increase dramatically asseen in Fig. 4, with the biggest super saturation ratio of more than 10and the peak nucleation rate approaching the order of 1021 at the samelocation X=0.061(m) downstream of the nozzle throat. The release

of latent heat to the flow, induces a shock at the location causing anincrease in temperature as well as pressure and as a result thenucleation pulse is terminated abruptly.

As shown in Figs. 4–6, after the condensation shock, the super-cooling decreases as the vapor temperature increases. Consequently,the supercooled vapor becomes closer and closer to its equilibriumstate and further expansion would take place almost like anequilibrium isentropic expansion again. But there will still be a verysmall amount of supercooling at the end of spontaneous condensationand the droplets still grow in a relatively low rate (only a few micronsper second) with the resulting liquid mass generation rate still abovezero. Accordingly, because of the latent heat release caused by thedroplets' growth, the calculated temperature difference between theproposed model and the isentropic model ΔT becomes bigger andbigger, as shown in Fig. 5.

4. Conclusions

The numerical simulation for the prediction of the non-equili-brium steam flowwith spontaneous condensation in the conventionalLaval nozzle is completed. The success of the simulationwasmeasuredby the quantitative validation based on the experiment. The differencebetween the dry isentropic expansion and the non-equilibrium flowwith spontaneous condensation has been investigated particularly.Based on the CFD simulation for the supersonic steam flow, we haveobtained the following conclusions:

(1) Different from the isentropic expansion, in supersonic steamflow non-equilibrium spontaneous condensation will occur inthe form of “condensation shock” downstream of the nozzlethroat at a certain value, when the supercooling and the supersaturation ratio achieve a high level.

(2) The qualitative description of the thermodynamic processes ofspontaneous condensation, droplet growth, condensationshock and the other non-equilibrium phenomenon in thesupersonic steam have been revealed.

Acknowledgement

The support of the National Natural Science Foundation of ChinaNo. 50576007 is gratefully acknowledged.

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