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Numerical investigation for the supersonic steamjetting flow in the thermal vapor compressorYong Yang a , Shengqiang Shen a , Shihe Zhou a , Rui Liu a & Huan Yu aa Key Laboratry of Liaoning Province for Desalination, School of Power and Engineering,Dalian University of Technology, Dalian, 116024, China Phone: Tel. +86 411 84709822 Fax:Tel. +86 411 84709822Version of record first published: 11 Feb 2013.

To cite this article: Yong Yang , Shengqiang Shen , Shihe Zhou , Rui Liu & Huan Yu (2013): Numerical investigationfor the supersonic steam jetting flow in the thermal vapor compressor, Desalination and Water Treatment,DOI:10.1080/19443994.2012.750585

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Numerical investigation for the supersonic steam jetting flow in thethermal vapor compressor

Yong Yang, Shengqiang Shen*, Shihe Zhou, Rui Liu, Huan Yu

Key Laboratry of Liaoning Province for Desalination, School of Power and Engineering, Dalian University ofTechnology, Dalian 116024, ChinaTel. +86 411 84709822; Fax: +86 411 84707963; email: zzbshen@dlut.edu.cn

Received 27 September 2012; Accepted 8 November 2012

ABSTRACT

The thermal vapor compressor (TVC) is an essential part that governs the overall process in themultiple-effect distillation––TVC system. The supersonic steam jetting flow always occurs inthe TVC equipment, and the flow in the TVC is very complex due to the strong compressibility,non-equilibrium phase change, and supersonic turbulent flow of the stream. So, to improve theperformance of an ejector system, an investigation of the characteristics for the high speed flowinside the ejector is often required. In the present study, the supersonic steam jetting flow withnon-equilibrium phase change and condensation shock was numerically studied based on thecomputational fluid dynamics method. A conservative compressible numerical model couplingwith non-isothermal classical nucleation model and droplet growth model, in which the ther-modynamic state and properties of steam was calculated by the Virial Equations, was devel-oped and used to predict the supersonic non-equilibrium jetting flow. The effects of variousoperating pressures on the nozzle performance were investigated. The research explored theeffect of steam pressure, temperature, supercooling level, and super-saturation ratio on theonset of the nucleation and the intensity of condensation shock in the supersonic steam flow.

Keywords: Thermal vapor compressor; Supersonic jetting flow; Shock; CFD

1. Introduction

Nucleation and condensation of steam alwaysoccurs in several natural and industrial processesincluding cloud formation, flight of aircraft in humidcondition, power generation equipment, turbomechan-ical flow, and so on. With the expansion of the desali-nation industry [1,2], multiple-effect distillation (MED)with thermal vapor compressor (TVC) systembecomes more attractive than other thermal systemsdue to effectiveness, easier operation and mainte-nance, and good economic characteristics, so MED–

TVC desalination technology is taking a considerableportion in the desalination field, and the market sizeis growing rapidly especially in china [3–5]. The MEDdesalination plant is normally built in a dual-purposepower plant with electricity and fresh water produc-tion to increase the economy and thermal efficiency.In this case, the heating steam for the MED desalina-tion plant is extracted from steam turbine, while thesteam is normally with much higher pressure andtemperature than the steam inducted into the desali-nation plant. Thus, to increase the energy utilizationefficiency, the TVC device is usually used to make thevapor recycle as the pressure reducer. Since theperformance of a TVC is affected obviously by the*Corresponding author.

Desalination and Water Treatmentwww.deswater.com

doi: 10.1080/19443994.2012.750585

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pressures of entrained vapor and motive steam, thusthe thermodynamic state of the heating steam and theposition of TVC in a MED desalination plant willaffect its performance significantly. Therefore, theheating steam parameters, and the TVC suction posi-tion and its performance, are concerned by both aca-demic researchers and engineers.

A schematic diagram of the MED–TVC desalina-tion system is shown in Fig. 1 [6]. It includes mainlyevaporators, a TVC device, and an end condenser.Other auxiliary equipments include distillate flashingboxes, brine flashing boxes, a venting system, seawater feeding, and brine and distillate expelling facili-ties. The TVC is an essential part that governs theoverall process in the MED system. Accurate predic-tion of the TVC performance promotes the reliabilityof the process, increases the TVC entraining efficiency,and thus improves the performance of MED signifi-cantly. The flow and heat transfer in the TVC is verycomplex due to the strong compressibility, non-equi-librium phase change, and supersonic turbulent flowof the stream. Thus, studying the shock structure ofthe supersonic jets, visualizing the turbulent structuresof the flow in the ejectors, and understanding thenon-equilibrium phase change in the transonic steamflow are very important for predicting, and possiblycontrolling, when the resulting flow phenomena occurin the TVC system [7,8]. The current research focuseson the investigation of the spontaneous condensationand the condensation shocks in the rapid expansionsin the TVC system.

As shown in Fig. 2, a typical TVC constructioncomprises four distinct parts: a convergent divergent

nozzle, a suction chamber, a mixing chamber attachedto a constant area duct, and a diffuser. The typicalprocesses inside an ejector begin with high tempera-ture and high pressure stream from the generator,entering the ejector through the convergent divergentnozzle. This stream is accelerated and expanded tosupersonic speed at the nozzle exit where it creates anaerodynamic duct to entrain the low pressure and lowtemperature secondary stream into the mixing cham-ber. The secondary stream is accelerated to sonicvelocity and mixes with the primary stream in theconstant area duct. The region of supersonic flow isterminated by a normal shock wave further down theduct or in the diffuser [9]. In practice, two chokingphenomena exist in the ejector performance [10], onein the primary flow through the nozzle and the otherin the entrained flow. In addition to the choking inthe nozzle, the second choking of an ejector results inthe acceleration of the entrained flow from a stagnantstate at the suction port to a supersonic flow in theconstant-area section.

The entrainment ratio (x, the ratio of mass flowrate of secondary fluid to that of primary fluid) isaffected by many factors. The physical phenomenainvolve supersonic flow, shock interactions, and tur-bulent mixing of two streams inside the ejector enclo-sure. It is so complicated that the design of an ejectorto date still heavily relies on trials and errors methodsalthough a number of gas-dynamic theories for ejectoranalysis were developed by several researchers. Fig. 3gives the typical performance curve for a fixed struc-ture TVC [11]. The flow in the TVC device is usuallyin three modes: critical mode (with choking flow in

Fig. 1. Schematic diagram of MED–TVC system [6].

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the mixing chamber), subcritical mode (without chok-ing flow in the mixing chamber), and malfunctionmode (with reversed flow in the mixing chamber)depended on different back pressure (PC, dischargepressure of TVC). When PC is below the critical value(PC

⁄), the entrainment ratio remains constant. WhenPC is increased higher than PC

⁄, the entrainment ratiobegins to fall down rapidly. A TVC should be oper-ated at the critical mode to keep steady pumping per-formances, so it is very important to obtain thepumping performance curve, especially the criticalpoint accurately in theoretical analysis and engineer-ing applications.

In the TVC system, the working characteristic ofthe TVC has great impact on the system performance,and a slight change of the inlet or back pressure willcause great change for the TVC entraining efficiency.For example, a minor change of the suction pressurecan lead to a drastic change on the TVC operatingperformance and the increase of the motive steampressure can slightly improve the performance of theTVC under a certain range [12,13]. And in the caseof steam expanding rapidly in the primary nozzle,

condensation will occur in the form of homogenousnucleation after a high supersaturated state isattained. In such situation, the steam flow will beaffected by the latent heat released from the condensa-tion of vapor and the “condensation shocks” willoccur in the type of pressure raise. In some earlyresearch, it is indicated that the pressure has greateffect on the condensation in a supersonic nozzle [14].So, exploring the effect of steam pressure on the onsetof the nucleation and the intensity of condensationshock is an urgent research task.

To improve the performance of TVC and toinvestigate the characteristics for the heat transferand high speed flow-in inside the nozzle in the TVCsystem, The present study reveals the pressure char-acteristics of the supersonic steam flow with homoge-nous nucleation and the condensation shocks in thenozzle. The supersonic steam flow with non-equilib-rium phase change and condensation shock wasnumerically studied based on the computational fluiddynamics (CFD) method. And the special phenomenain the supersonic steam flow in the nozzle wereinvestigated. The effects of various operating pres-sures and temperatures on the nozzle performancewere investigated.

2. Mathematical model

2.1. Governing equations for the two-phase mixture

In this paper, a conservative two-dimensional com-pressible numerical model is developed under Eule-rain coordinate system. The numerical simulation forthe prediction of the non-equilibrium steam flow withspontaneous condensation in the conventional Lavalnozzle is completed. To make the simulation availabil-ity and efficient, the condensing steam flow isassumed to be a mixture of steam and monodisperseddroplets. That is, all droplets are of the same size atone point in the flow and the interactions betweendroplets are neglected. Since droplet sizes are

Fig. 2. Schematic diagram of the TVC construction.

Fig. 3. Entrainment ratio as a function of dischargepressure Pc [11].

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sufficiently small, it is assumed that the volume of thecondensed liquid droplets is negligible and the veloc-ity slip between the droplets and gaseous-phase iszero. And the density of the mixture is approximatedas the following equation by neglecting the volumeoccupied by the liquid droplets,

q ¼ qg=ð1� bÞ ð1Þ

where b is the overall mass fraction of the liquidphase.

Under the foregoing assumptions, the Euler equa-tions may be written in integral form as follows:

oUot

þ oFox

þ oGoy

¼ 0 ð2Þ

where x, y, and t are the space and time coordinatesand U, F, and G respectively defined as:

U ¼qquqvqE

264

375F ¼

ququ2 þ Pquv

ðqEþ PÞu

264

375G ¼

qvquv

qv2 þ PðqEþ PÞv

264

375

where P, u, v, E denote pressure, x-wise velocity,y-wise velocity, and internal energy, respectively. Andthe internal energy and specific enthalpy are deter-mined as follows:

E ¼ h0 � P=q ð3Þ

h0 ¼ hþ ðu2 þ v2Þ=2 ð4Þ

h ¼ ð1� bÞhg þ bhl ð5Þ

where h0, hg, and hl are the enthalpy for the stagnationstate condition, the gas phase, and the condensedliquid phase, respectively.

To model wet steam, two additional transportequations are needed [15]. The first transport equationgoverns the mass fraction of the condensed liquidphase (b) and the other transport equation (g) deter-mines the number of droplets per unit volume. Thetwo equations are combined to the model in the fol-lowing expression:

oqot

bþrðq~vbÞ ¼ C ð6Þ

oqot

gþrðq~vgÞ ¼ PI ð7Þ

where C is the mass generation rate due to condensa-tion and evaporation and I is the nucleation rate. Andthe number of droplets per unit volume (g) is calcu-lated as follows:

g ¼ bð1� bÞVdðql=qgÞ

ð8Þ

Vd ¼ 4

3p�r 3 ð9Þ

where ql, Vd, and �r denote the liquid density, the aver-age droplet volume, and the average droplet radius,respectively.

2.2. Nucleation model and droplet growth model fornucleating particles

In the model, the classical homogeneous nucleationtheory describes the formation of a liquid-phase in theform of droplets from a supersaturated phase inthe absence of impurities or foreign particles, and thenucleation rate is given by:

I ¼ qc1þ h

q2v

ql

� � ffiffiffiffiffiffiffiffiffiffiffiffi2pRTM3

mp

sexp �4pr2�r

3KbT

� �ð10Þ

where qc is the evaporation coefficient, Kb is the Boltz-mann constant, Mm the is mass of one molecule, r isthe liquid surface tension, and h is a non-isothermalcorrection factor which is given by the following rela-tion:

h ¼ 2ðc� 1Þðcþ 1Þ

hlvRT

� �hlvRT

� 0:5

� �ð11Þ

where hlv is the specific enthalpy of evaporation atpressure P and c is the ratio of specific heat capacities.

Based on the nucleation model just described thequantity of droplets at a location in the continuousgas phase is known, and the rate at which these drop-lets grow can be derived on the basis of heat transferconditions surrounding the droplet [16]. This energytransfer relation can be written as:

o�rot

¼ P

hlvql

ffiffiffiffiffiffiffiffiffiffiffiffi2pRT

p cþ 1

2cCpðTl � TÞ ð12Þ

where Tl is the liquid droplet temperature.Based on the above theory, the mass generation

rate C is given by the sum of mass increase due to

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nucleation and also due to growth/demise of thesedroplets and therefore, C is written as:

C ¼ 4

3pqlIr

3� þ 4pqlgI�r

2o�rot

ð13Þ

where r⁄ is the Kelvin–Helmholtz critical dropletradius, above which the droplet will grow and belowwhich the droplet will evaporate. And r⁄ is given asfollows:

r� ¼ 2rqlRTLnS

ð14Þ

where S is the supersaturation ratio defined as theratio of vapor pressure to the equilibrium saturationpressure:

S ¼ P

PsatðTÞ ð15Þ

After considering the nucleation model and thedroplet growth model, we can get the source termsfor the mass, momentum, and energy exchangebetween the gas and liquid phases. As in the simu-lation, it is assumed that the velocity slip betweenthe droplets and gaseous-phase is zero, so themomentum source between the two phases can beneglected.

The steam equation of state used in the study,which relates the pressure to the vapor density andthe temperature, is given by

p ¼ qvRTð1þ Bqv þ Cq2vÞ ð16Þ

where B and C are the second and the third Virialcoefficients.

3. Effect of inlet pressure on condensation shock

3.1. Numerical validation: comparison vs. experimentalvalues

The following example describes an application ofthe present model to the converging–diverging nozzlefor the non-equilibrium condensing steam flow. Thegeometries of the converging–diverging nozzle (Fig. 4)were taken to be the same as the Nozzle B used in theexperiment of Moore et al. [17]. To retain the calcula-tion speed advantage coming with the use of regularblock-structure element, the multi-block techniquewas applied to make the concentration of grid densityfocused on the areas where significant phenomenawas expected.

The validation of the numerical results was accom-plished with the experiment conducted by Moore [17]who obtained centerline pressure distributions for aseries of converging–diverging nozzle configurationsand the results showed a good agreement betweennumerical simulation and the experiment data. InFig. 5 the pressure distribution along the centerline ofthe nozzle is compared with the experimental valuesfor Nozzle B used in the experiment of Moore. Asshow in the Fig. 5, the profiles of the pressuredistribution for the dry isentropic expansion and thenon-equilibrium overlap before the location of thecondensation shock where the pressure profile risesabruptly calculated by the non-equilibrium model.The simulation results are in good agreement withexperiments for the converging–diverging nozzlebefore the location of the condensation shock, and theappearance of condensation shock is predicted accu-rately where a great amount of latent heat of conden-sate droplets is released causing an abrupt rise ofpressure and temperature.

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5X (m)

Y (m

)

Position

X

Y

1 2 3 4

-0.25 -0.2 0 0.5

0.05635 0.05635 0.05 0.072

Position 1

2

3

4

Throat

Fig. 4. Geometry of the nozzle.

-0.25 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Distance from Throat X / m

Pres

sure

Rat

io /

(P/P

oin)

The Proposed ModelThe Isentropic ModelExperiment moore et al. (1973)

Fig. 5. Pressure distribution along the nozzle centerlinecompared with the experiment.

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To validate the capability of the non-equilibriummodel capture the condensation shock, another com-parison has been made for the structure of the conden-sation shock. Fig. 6 shows a schlieren picture of a“Condensation shock” with Iso-Mach lines beginningat the sonic line [18]. The bright parabolic area corre-sponds to the continuous supersonic compression fromthe heat addition which creates the characteristic obli-que shocks. Fig. 7 gives the structure of the calculatedcondensation shock with the Iso-Mach lines. The simu-lation result shows good agreement in the structure ofthe condensation shock with the schlieren picture.

3.2. Effect of inlet pressure on condensation shock

To investigate the effect of inlet parameters on theshock in the nozzle, the research explored the effect of

steam pressure, temperature, supercooling level, andsuper-saturation ratio on the onset of the nucleationand the intensity of condensation shock in the super-sonic steam flow.

The simulation results clearly show the process ofthe forming and weakening of condensation shockin the nozzle and the influence of the inlet pressureon the condensation shock. Fig. 8 gives the pressureprofiles along the nozzle centerline for various inletpressure values. Figs. 9–11, respectively, show the dis-tribution of the temperature (T), the super saturationratio (S), and the nucleation rate (I) along the center-line of the nozzle under the conditions with differentinlet pressure values. And the numerical simulation

Fig. 6. Schlieren picture of condensation shock withIso-Mach lines.

Fig. 7. Simulation result for the condensation chock withIso-Mach Lines.

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Pres

sure

Rat

io (

P/P oi

n)

Distance from throat X(m)

Isentropic model

Pin=20 kPa

Pin=25 kPa

Pin=30 kPa

Pin=35 kPa

Pin=40 kPa

Pin=45 kPa

Fig. 8. Pressure profiles along the nozzle centerline forvarious inlet pressure.

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5220

240

260

280

300

320

340

360

380

400

420

Tem

pera

ture

T(K

)

Distance from throat X(m)

Isentropic model

Pin=20kPa

Pin=25kPa

Pin=30kPa

Pin=35kPa

Pin=40kPa

Pin=45kPa

Fig. 9. Temperature profiles along the nozzle centerline forvarious inlet pressure.

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that used the proposed model has been comparedwith the dry isentropic expansion solution.

As shown in the figures, in the nozzle, the cross-sectional flow area first decreases to the throat andthen increases monotonically in the supersonic region.As the steam flows through the nozzle, it expandsand the temperature falls rapidly. Because the equilib-rium vapor pressure decreases exponentially withtemperature, very high supersaturation can beachieved. Eventually, the steam acquires sufficientsupercooling for spontaneous nucleation of droplets inthe diverging section of the nozzle, where the flow issupersonic. The subsequent release of latent heat thusresults in a deceleration of the flow and a rise in pres-sure, known traditionally as the “condensationshock”.

At the location of the condensation shock, a signifi-cant amount of liquid droplets generate, approxi-mately 1021 droplets per second per unit volume(Fig. 11). And a sharp rise in wetness fraction

achieves, reflecting the rapid growth of the dropletsimmediately following the peak nucleation. Referringagain to Fig. 10, it is shown that after peak nucleation,the super saturation ratio rapidly drops to near equi-librium conditions (s� 1), where the supercoolinglevel of the steam is lower than 5K. This near equilib-rium condition prevails in the remaining length of thenozzle for the supersonic outflow case.

As shown in Table 1 and Fig. 12, based on the com-parison between the different conditions with severalinlet pressure values (Pin = 20, 25, 30, 35, 40, and45 kPa), the location of the condensation shock can berevealed clearly moving to the outlet of the nozzlefrom the throat (from 0.028 to 0.112m) with the pres-sure becoming lower. And as shown in Fig. 12, whenthe inlet pressure decreases, the strength of the con-densation shock is also weakened. Because at the sameinlet temperature, the higher the inlet pressure is, thehigher super saturation ratio the steam will achieves atthe inlet of the nozzle. As shown in Fig. 9, when thesteam is expanded in the converging–diverging nozzle,

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

0

2

4

6

8

10

12

14

Supe

r Sa

tura

tion

Rat

io S

Distance from throat X(m)

Pin=20 kPa

Pin=25 kPa

Pin=30 kPa

Pin=35 kPa

Pin=40 kPa

Pin=45 kPa

Fig. 10. Super saturation ratio profiles along the nozzlecenterline for various inlet pressure.

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.51E7

1E11

1E15

1E19

1E23

Distance from throat X(m)

Nuc

lear

Rat

e J(

1/m

3 s)

Pin=20 kPa

Pin=25 kPa

Pin=30 kPa

Pin=35 kPa

Pin=40 kPa

Pin=45 kPa

Fig. 11. Nucleation rate profiles along the nozzle centerlinefor various inlet pressure.

Table 1The main characteristics of the condensation chocks for different inlet pressures

Inletpressure P(kPa)

Onset point of thecondensation shock X (m)

Super saturation ratioof onset point (S)

Nucleation rate I of onsetpoint I (1/m�3 s�1)

Mach number of theonset point (Ma)

20 0.112 12.13 3.99� 1021 1.48

25 0.087 10.53 3.88� 1021 1.41

30 0.053 8.57 3.05� 1021 1.34

35 0.042 7.95 2.96� 1021 1.30

40 0.033 7.18 2.81� 1021 1.25

45 0.028 6.46 1.83� 1021 1.20

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the temperature decreases rapidly to make the isentro-pic expansion vapor crosses the saturation line quicklyand stays in a supersaturated state until the spontane-ous condensation occurs suddenly where the ultimatesupercooling achieves. As shown in Fig. 12, when theinlet pressure is higher, the ultimate supercoolingachieves much easier after the throat and the dropletradius grows much larger and more rapidly, althoughthe nucleation rate becomes a little smaller.

4. Effect of back pressure on condensation shock

To examine the effect of the back pressure on thespontaneous condensation, for the same inlet bound-ary conditions at inflow (Pin = 25 kPa and Toin =358.6K), a series of back pressures (Pb = 7, 12, 15, 16.5,20, and 22 kPa) were obtained at the exit plane.

As is apparent in Figs. 13–15, when the out flow issupersonic (Pb = 7 kPa), the spontaneous homogenousnucleation and the condensation shock will appear asforecast. While in the case with a subsonic out flow(Pb = 12, 15, and 16.5 kPa), aerodynamic shock appearsin the supersonic flow. The flow conditions upstreamof the aerodynamic shock are the same as those forsupersonic outflow case (Pb = 7 kPa). Through the aero-dynamic shock, liquid droplets rapidly evaporate inresponse to the rapid pressure and temperature riseacross the shock (Figs. 13 and 15). Downstream of theaerodynamic shock, the temperature falls abruptly due

Fig. 12. Main characteristics of onset for condensation under different inlet pressures.

-0.25 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.54681

1214161820 2224

Distance From Throat X / m

Pres

sure

Rat

io (

P / P

oin

)

Pb=22KPa

20KPa

7027.5Pa

16.5KPa 15KPa 12KPa

Normal Aerodynamic Shock

Condensation Shock

T in=358.6 K P in=25 KPa

Fig. 13. Pressure distribution along the nozzle centerlinefor various back pressure.

-0.25 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distance From Throat X / m

Froz

en M

ach

Num

ber (

Mf)

T in=358.6 K P in=25 kPa

Pb=22 kPa

20kPa

16.5kPa 15kPa 12kPa

7 kPa

Fig. 14. Mach number distribution along the nozzlecenterline for various back pressure.

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to reacceleration of the flow and, after another peakpoint, the flow will decelerate with a small negativevalue of supercooling level to support heat movementtoward the liquid phase as evaporation continues ondroplet surfaces and wetness fraction reduces.

As shown in the figures, with the back pressureincreases the normal aerodynamic shock movestoward the nozzle throat, and the strength of the aero-dynamic shock decreases. If the normal aerodynamicshock passes through the location of the condensationshock (Pb = 20, 22 kPa), the spontaneous condensationwill be weakened or even not occur at all. As theappearance of the aerodynamic shock will cause signif-icant energy dissipation to make the steam flow decel-erate to subsonic, the supercooling level achieves toonly several degrees or even superheated conditions.

5. Conclusions

In the present study, the supersonic non-equilib-rium steam flow with spontaneous condensation andcondensation shock was numerically studied based onthe CFD method. A conservative compressible numer-ical model coupling with non-isothermal classicalnucleation model and droplet growth model wasdeveloped and used to predict the spontaneous con-densation phenomenon with homogenous nucleationin the supersonic steam flow. The model has been val-idated against the experimental data and the resultsare in excellent agreement with experiments.

Based on the simulation, the effect of inlet pressurechanging and outlet pressure changing on the thermo-dynamic and aerodynamic properties of condensationshock was studied. The research explored the effect ofsteam pressure, temperature, supercooling level, andsuper-saturation ratio on the onset of the nucleationand the intensity of condensation shock in the super-

sonic steam flow. And the main conclusions wereobtained as follows:

(1) Different from the isentropic expansion, in super-sonic steam flow non-equilibrium spontaneouscondensation will occur in the form of “condensa-tion shock” downstream of the nozzle throat,when the supercooling, and the super saturationratio achieve a high level.

(2) The location of the condensation shock can berevealed, clearly moving to the outlet of the noz-zle from the throat with the inlet pressure becom-ing lower. And when the inlet pressuredecreases, the strength of the condensation shockis also weakened.

(3) With the back pressure increases the normal aero-dynamic shock will appears, and the normalaerodynamic shock moves toward the nozzlethroat. The aerodynamic shock has a complexinfluence on the homogeneous nucleation conden-sation. If the normal aerodynamic shock passesthrough the location of the condensation shock,the spontaneous condensation will be weakenedor even not occur at all.

Nomencalture

B — the second Virial coefficient (m3/kg)

C — the third Virial coefficient (m6/kg2)

Cp — specific heat capacity (J/kg K)

E — total specific internal energy (J/kg)

e — specific internal energy (J/kg)

h — static enthalpy (J/kg)

hlv — specific enthalpy of evaporation (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 ratio

T — temperature (K)

Tsubc — supercooling level (K)

u — X-axial velocity (m/s)

v — Y-axial velocity (m/s)

Greek symbols

b — mass fraction of the condensed phase

c — the ratio of specific heat capacities

C — mass generation rate (kg/m3 s)

h — nonisothermal correction factor

q — density (kg/m3)

r — liquid surface tension (N/m)

-0.25 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5240

260

280

300

320

340

360

380

400

Distance From Throat X(m)

Tem

pera

ture

T (K

)

Isentropic Expansion Expansion with Condensation

T in= 358.6 K P in= 25 kPa

———

Pb=22 kPa

20 kPa

16.5 kPa 12 kPa

7 kPa

15 kPa

Fig. 15. Temperature distribution along the nozzlecenterline for various back pressure.

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Subscripts

0 — stagnation state condition

g — gas (vapor)

l — liquid

Superscripts

! — vector

— — average of a variable

Acknowledgments

This research is supported by the National NaturalScience Foundation of China (No. 51206015) and thePostdoctoral Science Foundation of China (2012T50253).

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