Suppression of Condensation Shock in Wet Steam Flow by...

Transaction B: Mechanical EngineeringVol. 17, No. 5, pp. 337{347c Sharif University of Technology, October 2010

Suppression of Condensation Shock in WetSteam Flow by Injection of Water Droplets

in Di�erent Regions of a Laval Nozzle

M.R. Mahpeykar1;� and E. Amirirad1

Abstract. In this paper, a compressible, one-dimensional and two-phase ow is analytically modeledby using a homogeneous classical nucleation rate equation in a Laval nozzle. For droplet growthcalculations, the heat transfer between the droplets and the surrounding steam is modeled by a freemolecular ow model and a semi-empirical two-layer model which is deemed to be valid over a widerange of Knudsen numbers. In order to reduce the condensation shock strength, the water droplets areinjected at the inlet, and just after the throat of the Laval nozzle. According to the results, the nucleationrate is considerably suppressed due to the small droplet injection, and, therefore, the condensation shocknearly disappears, particularly for the inlet injection of water droplets.

Keywords: Two-phases; Nucleation; Condensation shock; Injection; Laval nozzle.

INTRODUCTION

In low-pressure steam turbines used for electricity gen-eration, the presence of condensation leads to problemsof blade erosion and to a loss of turbine e�ciency.

The ow is initially single phase, but after reach-ing the Wilson point, water droplets are developed anda non-equilibrium two-phase ow is generated. Thegrowing water droplets release their latent heat to the ow and this heat addition to the ow causes a pressurerise, called condensation shock.

Because of the irreversible heat transfer in thisregion, the entropy will increase tremendously andcause thermodynamic loss. The drag between largedroplets and vapour generates a friction loss. Also,these droplets are not able to follow the vapor steampath and generate a braking loss, which also leads toerosion.

Reducing the wetness fraction of wet steam ow inthe last stages of steam turbines signi�cantly promotesstage e�ciency and prevents erosion of rotors.

Regarding the literature review of this paper,

1. Department of Mechanical Engineering, Ferdowsi Universityof Mashhad, Mashhad, P.O. Box 91779-48944, Iran.

*. Corresponding author. E-mail: [email protected]

Received 10 November 2009; received in revised form 15 March2010; accepted 3 July 2010

please refer to [1-9]. In these references, the homo-geneous steam two-phase ow is generally modeledthrough one- and two-dimensional analytic and numer-ical (CFD) methods, and the measurement of pressurehas been stated, particularly for laboratory cases.

In this paper, an innovative analytic method forthe one-dimensional homogeneous ow is proposed thatuses a two-layer heat transfer through a four sentenceVirial state equation along with the injection of identi-cal droplets with di�erent sizes in the convergent anddivergent sections.

It can be assumed that the speci�c volume ofdroplets is much smaller in comparison with that ofthe vapour. Hence, the volume of the liquid phase canbe neglected [2].

Considering this assumption, the mixture is takenas the summation of droplets of a speci�ed size, whichare uniformly distributed in the bulk vapour, andthe space between them is �lled by the vapour ofgiven pressure and temperature. The bulk vapourcan be condensed by the growth of existing dropletsor the formation of new ones. Formation of newdroplets can be calculated using various nucleationmodels. The model used in this paper is a classicalhomogeneous nucleation model with Courtney andKantrowitz corrections [3]. This nucleation modelassumes that condensation occurs in the absence ofany impurities. The condensation releases the latent

338 M.R. Mahpeykar and E. Amirirad

heat to the surrounding vapour and, for predicting the ow behavior, it is necessary to derive the equations ofheat and mass exchange between droplets and the bulkvapour.

On the other hand, the mixture, as a whole,must satisfy the conservation laws of mass, momentumand energy. Two-phase e�ects enter into these equa-tions through density and enthalpy. These are non-equilibrium mixture quantities and, therefore, dependon the mass fraction of the liquid and on the vapoursubcooling. Finally, a suitable equation of state isneeded for the surrounding vapour phase treatment.The ow can be described completely using this set ofequations.

Using a two-phase Eulerian-Lagrangian model,conservation equations for the two-phase mixture aresolved in a Eulerian frame of reference, but nucle-ation and droplet heat and mass exchange equationsare calculated following uid particles in steady one-dimensional ows. Therefore, four unknowns, namelytemperature, pressure, velocity and density of vapour,can be calculated [4].

TWO-PHASE FLOW EQUATIONSNUCLEATION RATE MODEL

For non-equilibrium ow calculations, there must be acriterion for the onset of nucleation and the appearanceof the second phase (liquid). The present model con-siders only homogenous nucleation in a pure substanceand relies heavily on the classical nucleation theory.The relevant nucleation theory is brie y presented here.

The essence of the model is that, in order fornucleation to occur in a owing stream undergoingexpansion, molecular clusters (the minute embryo ofa liquid droplet) must overcome a free-energy barrierassociated with increase in the surface-free-energy ofthe droplets for the onset of phase change. As asubstance is increasingly supercooled, the chances ofa molecular cluster moving over the free energy barrierincrease.

The change of Gibbs free energy for a mass,mr, of a supercooled vapour at constant pressure andtemperature, which forms a liquid droplet at the samepressure and temperature, can be calculated in theabsence of magnetic and electrical �elds [5]:

�G = �mrRTG[�G1] + 4�r2�r;

�G1 = ln�

��S(TG)

�+ 2B1[�� S(TG)]

+32B2[�2 � �2

S(TG)] +43B3[�3 � �3

S(TG)]; (1)

where �G is the change of Gibbs free energy, mr is thedroplet mass, R is the gas constant for vapour, T is the

temperature, r is the droplet radius, � is the surfacetension, � is the density, and B parameters are thevirial coe�cients given in the Appendix. Furthermore,the subscripts r, G and s denote the radius, vapourand saturation phases, respectively.

Based on the thermodynamic equilibrium, there isa minimum radius, r� (the critical radius), that mustbe attained in order to form a stable nucleus fromsupersaturated vapour. For this condition, the Gibb'sfree energy of formation of a critical cluster is denotedby �G�. By di�erentiating Equation 1 with respect tor, �G� is obtained [6]:

�G� =16��3

r3�2L(�G1)2 ; (2)

where subscript L denotes the liquid phase.The critical radius is equal to [6]:

r� =2�r

3�2L(�G1)2 : (3)

r� becomes increasingly smaller as the level of super-cooling increases, since P=Ps(TG) increases with higherlevels of supercooling. In Equation 3, liquid surfacetension, �r for a at surface is used.

In the vapour phase, there exists a statisticallysteady state population of molecular clusters of varioussizes. As supercooling increases and r� becomessmaller, there will eventually be enough clusters atthe size of r� (or greater) to start the condensationprocess and the reversion back to equilibrium. Usinga Boltzmann distribution function to represent thecluster population, combined with the change in freeenergy at r�, and subjecting the result to re�nementsby Courtney and Kantrowitz leads to the followingequation as given in [3]:

JKa =

(1 +

qc�G�r

�RTG2�

�0:5� l2

RT 2G� l

2TG

�)�1

�r

2N3

�qP 2p�r�L(RTG)2

exp

264 �16�N�3r

3�2L(RTG)3

nln�

PPs(TG)

�o2

375 : (4)

JKa gives the steady-state number of droplets nucle-ated per unit volume per unit time. In Equation 4,qc is the condensation coe�cient, K is Boltzmann'sconstant, l is the equilibrium latent heat, P is thepressure and N is the number of molecules per unitmass.

It should be noted that no universally applicablenucleation model is yet available for condensing ows,

Suppression of Nucleating Shock by Injecting Droplets 339

however, the general form described here has beensuccessfully used for steam at low pressures by anumber of researchers [7,8].

DROPLETS GROWTH EQUATIONS INSUPERSATURATED VAPOUR

The droplets grow by capturing molecules from bulkvapour. But, this mass transfer is a�ected also bythe transfer rate of latent heat of phase change fromdroplets to the surrounding vapour. Bakhtar and Zidipresented a semi-empirical relation for droplet growth,as explained in [9]:

dmr

dt= 4�r2�

drdt

=qcp

8�R1 + ar

h�GpTG � �s(TL; r)pTLi : (5)

Denoting the energy released per unit mass of vapour,the rate of heat loss from the droplet to the vapourmay be written as:

ldmr

dt= 4�r2�r(TL � TG) +

d(CLmr)dt

: (6)

In this expression, �r is the coe�cient of heat transferfrom a droplet with radius r to the surrounding vapour,and l is latent heat. In the right hand side of theabove expression, the second term can be neglected,with respect to the �rst term, and therefore, the heattransfer equation can be written as [10]:

ldmr

dt= 4�r2�r(TL � TG): (7)

The Knudsen number, Kn, plays a key role in thecoe�cient of heat transfer due to the vast range ofdroplet radii. The Knudsen number is the ratio ofthe mean free path, �l, of vapour molecules to dropletdiameter.

Kn =�l

2r: (8)

If the Knudsen number is much greater than unity,intermolecular collisions are not important. This isknown as the free molecular ow and an expression forthe coe�cient of heat transfer can be obtained fromthe kinetic theory [9]:

�r =PGp

2�RTG G + 1 G

CPG; (9)

where is the ratio of speci�c heat and CPG is thevapour speci�c heat at constant pressure.

At the other extreme, for a Knudsen numbermuch less than unity, the vapour phase behaves as acontinuum [11].

�r =�Gr; (10)

where � is the coe�cient of thermal conductivity.Figure 1 illustrates Knudsen numbers versus dropletdiameter with temperature as a parameter. The criticalradius of ow in a steam turbine is of the order between10�3 and 10�4 �m, whilst the mean diameter of �nedroplets after the last stage is of the order between 0.1and 1 �m. Therefore, the Knudsen number has a widerange from free molecular ow to so-called slip ow.Unfortunately, it is di�cult to model the coe�cientof heat transfer in the intermediate Knudsen numberrange.

By hypothesizing that the surrounding space of adroplet consists of two separate layers, it is assumedthat the exchange regime in the internal layer is freemolecular and, in the external layer, a continuumregime is dominant. A two-layer model, as shown inFigure 2, provides an expression for the heat transfercoe�cient applicable to a wide range of Knudsennumbers.

The droplet is assumed enclosed in a volumegoverned by free molecular ow. Surrounding this,the outer region is governed by continuum ow. Thedivision between the two regions occurs at � times themean free path.

Considering the conservation of energy at themiddle plate:

ldmr

dt= 4�r2 Pip

2�RTi

� + 1

2

�Cp(TL � Ti); (11)

ldmr

dt= 4�(r + �l)2 �

r + �l(Ti � TG): (12)

Figure 1. Knudsen number versus droplet diameter [12].


Figure 2. Temperature distribution around a droplet inthe Langmuir model [12].

Eliminating Ti between the two above expressions andrearranging the result, it is concluded that [12]:

ldmr

dt= 4�r2 �

rh

1(1+2�Kn) + 3:78KnPr

i (TL � TG):(13)

Hence, the heat transfer coe�cient in this model canbe calculated as:

�r =�

rh

1(1+2�Kn) + 3:78Kn

Pr

i ;� = 0:75; Pr =

CP�G�

; (14)

where �G is the kinematic viscosity of vapour.

TWO-PHASE GAS DYNAMICSEQUATIONS

By considering the steady one-dimensional ow overan incremental distance dx along the channel, thefundamental equation of ow can be written as:

d�G�G

+dAA

+duu

+dWL

W �WL= 0; (15)

where, A is the area, W is the overall mass ow rateand u is the velocity.

State Equation [10]:

P�GRTG

= 1 +B1�G +B2�2G +B3�3

G; (16)

where B parameters are virial coe�cients; these arefunctions of temperature as given in the Appendix.

dPP�Xd�G

�G� Y dTG

TG= 0;

where:

X =�GP

�@P@�G

�TG

=1 + 2B1�2

G + 3B2�2G + 4B3�3

G1 +B1�1

G +B2�2G +B3�3

G;

Y =TGP

�@P@�G

��G

= 1 +�GTG

�dB1dTG + �G dB2

dTG + �2GdB3dTG

�1 +B1�1

G +B2�2G +B3�3

G: (17)

MACH NUMBER

The speed of sound, a, in a single phase gas is:

a =

s P�G

: (18)

Thus:

Z = Ma2 =u2� P�G

� : (19)

Di�erentiating Equation 19 leads to:

dZZ

=2dMaMa

= 2duu

+d�G�G� dP

P: (20)

MOMENTUM EQUATION

Considering the ow as a whole, the change in momen-tum of the system with the friction factor of f acrossdx can be written as:

AdP +Af�Gu2

G2De

dx = �d[WGuG +WLuL]: (21)

Expanding this equation, denoting dWG = �dWL anddividing it by A � P , regarding (uG�uL)

AP dWL as beingsmall in comparison with other terms, and re-arrangingyields [4]:

dPP

=� f�Gu2G

2PdxDe� (W �WL)uG

APduGuG

� WLuLAP

duLuL

: (22)

ENERGY EQUATION

Considering the ow of energy into and out of thecontrol volume and assuming the heat loss through thesides of the duct to be small, the energy equation canbe written as:


d�(W �WL)

�hG +

u2

2

�+WL

�hL +

u2

2

��= 0:

(23)

Since the phase change occurs within the controlvolume, the term dWL(hG � hL) will appear whileexpanding this equation. It represents the heat releasedto, or absorbed from, the ow by condensation onto,or evaporating from, the droplets. Having consideredthe released energy associated with the phase change,the bulk of the vapour may then be regarded as aperfect gas for which the term dhG can be replaced byCpdTG. Substituting this into Equation 23, denoting(hG�hL) by l, regarding the term u2

G�u2L

2 as being smallcompared to l, neglecting WLdl, dividing throughoutby WCPTG, and rearranging results in [13], yields:

dTGTG

=��

1� WL

W

�u2G

CpTGduGuG

+l

CpTGdWL

W

� WL

Wu2L

CpTGduLuL

: (24)

Equations 17, 20, 22 and 24 can be solved for the fourunknowns dTG

TG , dPP , du

u and d�G�G , providing that duL

uL ,dAA and dWL

WLare known as explained in [2] or [6].

CALCULATING THE INTERACTIONBETWEEN THE INJECTED AND NEWDROPLETS

There are two groups of droplets inside each element of�x. Group (a) includes the droplets formed and grownwithin the element in the case of a considerable rateof nucleation. Group (b) includes the droplets formedin the upstream of the element entrance, which growwithin the element of �x. In the element of �x, thetime of �t is divided by n to give a suitable time scale(�t):

�t =�xu;

and:

�t =�tn: (25)

The new droplets grow with a radius of r� and the massof the liquid phase related to the droplets of group (a)at the time of nucleation is calculated as:

WL;a =�Jst�t(AU)

43��

��Lr�32

+�L1r31 +�L2r3

2 +� � �+�Ln r3n2

�: (26)

Taking the number of droplets in group (b) asNi, if ri isthe mean input radius and re is the mean output radius

of the element, the net mass of the grown dropletswithin the element of �x is equal to:

WL;b = Ni43�(�ler

3e � �lir3

i ): (27)

It should be noted that in this research, the droplets ofgroup (b) are initially injected and, then, new dropletsare produced in the nucleation zone.

The total mass of the injected and formed dropletsin the element of �x is:

WL =43�Nir3

i �Li +WL(a) +WL(b);

N = Ni + Jst�t(A�x): (28)

In order to decrease the calculation complexity, themean diameters and temperatures of the droplets atthe end of element �x are used as inputs for the nextelement.

Using the method of mean square root, thedroplet radius at the end of the element is determinedas:

�r =�SA

4�N

� 12

;

where:

SA = 4�Nir2e

+ J�t(A�x)4��r�22

+ r21 + � � �+ r2

n2

�: (29)

Calculations are continued for other elements using thismethod.

ENTROPY CALCULATION

The explored ow entropy at each point is a combi-nation of three parameters, namely; vapour entropy,liquid entropy and the entropy related to the formationof the droplet surface, each of which can be calculatedfrom the following equations:

S = SG + SL + Ss: (30)

Using the equation of state, along with thermodynamicequations, the following equation is derived for thespecial entropy of gas or vapour [7]:

SG = R�(�B1�) +

��dB1

dT

��+ S0; (31)

where S0 is calculated from:

S0 = 0:30773� 0:46153 ln(�) + 1:1095 ln(TG)

+ 7:11756�10�4TG � 3495T 2G: (32)


According to classic thermodynamics, the special en-tropy of water droplets at temperature TL can becalculated with acceptable accuracy using:

SL = CL ln�TLTD

�: (33)

In this equation, CL is the special temperature of waterand TD is the datum temperature, which is equal to273.15 Kelvin.

According to the thermodynamics of droplet sur-face formation, the entropy per unit surface is calcu-lated from the following equation [14]:

Ss = �AD�@�@T

�P: (34)

Considering all the above points, the total entropywould be:

S = WGSG +WLSL + Ss: (35)

Dividing the above equation by the total discharge, thespecial entropy of the total ow is calculated from:

s = (1� w)SG + wSL +SsW: (36)

Considering the fact that the ow is adiabatic and thewetness at the inlet is constant, the outlet entropy canbe a good basis for estimating the generated entropy.

When the radius of the droplet at the inlet issmall (e.g. R = 0:2 micron), after a rising trend,the supersaturation ratio starts to decrease and, at thechannel end (X=L = 2:0), reaches a low value. Consid-ering the signi�cant decrease in the outlet entropy inthis state, presuming more condensation for the outlet ow will not have much e�ect on the increase of outletentropy.

INVESTIGATION OF THE HEATTRANSFER COEFFICIENT

The two-phase compressible ow �eld in a one di-mensional Laval nozzle is solved, using both modelsfor the heat transfer coe�cient, which was describedpreviously. Figure 3 shows the pressure ratio distribu-tion along the nozzle computed by both models. Bycomparing the results to experimental results reportedby Young [15], it is clearly observed that both non-equilibrium condensation ow calculations predict thepressure increase due to sudden heat release fromgrowing droplets, but agreement with the experimentis better for the two-layer model.

Hence, in this research, the two layer heat transfermodel is used to solve the droplet growth along thenozzle.

Figure 3. Comparison of computed pressure ratio alongthe nozzle with the experiment (P0 = 150 kPa , T0 = 377K).

INJECTING WATER DROPLETS AT THEINLET OF THE LAVAL NOZZLE

In order to calculate the ow variables in theconvergent-divergent nozzle, they are studied in thetwo-phase mixture, and the saturation line is crossedby the supersonic wet steam non-equilibrium ow inthe convergent part.

The rate of injecting the liquid droplets at theinlet of the Laval nozzle is calculated by giving thewetness fraction and droplet diameter. It is assumedthat the mass of vapour phase is decreased by the sameamount as the mass of injected droplets.

The number of droplets can be varied by changingthe wetness fraction for a given droplet diameter or bychanging the diameter for a given wetness fraction.

RESULTS AND DISCUSSION

Droplets are injected at the inlet and just after thethroat of the nozzle with a geometry that is shown inFigure 4.

Stagnation pressure and temperature at the inletof the nozzle are considered as 70 kPa and 366 K.

Injected droplets are performed with 0.5% of wet-ness fraction and radiuses between 2 and 0.2 microns.Injected droplets provide surfaces for the condensationof vapour.

Figure 4. Laval nozzle geometry.


Since Gibbs free energy of condensation on thesurface is less than that of the nucleation, it is expectedthat the nucleation rate will be decreased.

On the other hand, in a certain wetness fraction,the smaller injecting droplet diameter gives a highernumber of droplets and, therefore, greater surfaces areprovided for the condensation of vapour. So, it iscon�rmed that droplets with a smaller radius cause agreater reduction in nucleation rate.

Figure 5 represents the variation of nucleationrate along the nozzle length and proves this fact.

Figure 6 represents the variation of pressure ratioalong the nozzle length. It is observed that the injecteddroplets reduce the condensation shock strength. Sincethe nucleation is reduced by injection of droplets, lesslatent heat is given to the vapour ow and, therefore,condensation shock strength is reduced.

As the nucleation rate reduction is greater for thesmaller injected droplets, it is forecast that the strengthof the condensation shock has a greater decrease forthe smaller injected droplets. The thermodynamiclosses in two-phase ow may be estimated by theentropy generation between the inlet and outlet of thenozzle.

Table 1 shows the outlet entropy for various diam-eters of injected droplets at the inlet of the Laval nozzle.

It is clearly observed that the outlet entropy isreduced signi�cantly as the result of droplet injection.On the other hand, it is seen that with a constant

Figure 5. E�ect of droplets injection on nucleation (0.5%inlet wetness).

wetness fraction, smaller injected droplets cause muchreduction in the outlet entropy.

As expected, the more droplets injected, the lessheat is given to the ow. Therefore, the increasedtemperature is smaller for vapour, as shown in Figure 7.

In steam turbines, the purpose is to transformthe ow enthalpy to kinetic energy. Fortunately,the injection droplets into wet steam ow preventa decrease in uid Mach number as shown in Fig-ure 8.

Figure 6. E�ect of droplets injection on pressure changes(0.5% inlet wetness).

Figure 7. E�ect of droplets injection on vapourtemperature changes (0.5% inlet wetness).

Table 1. E�ect of injected droplets on outlet entropy.

Wetness at Inlet = 0:5% Wetness at Inlet = 0%

Radius (�m) 0.2 0.3 0.6 1 2 0

Outlet Entropy�

kJkg.K

�5.65 6.62 6.81 6.85 6.91 7.51


Figure 8. E�ect of droplets injection on Mach number(0.5% inlet wetness).

Figure 9. E�ect of droplets injection on supersaturatingratio (0.5% inlet wetness).

It is observed that the injection of droplets re-duces the degree of supercooling or the supersaturatingratio to a level much lower than the case withoutinjection as shown in Figure 9.

Variation of the droplets mean radius is repre-sented in Figure 10. Because of initial vaporization,mean radius is decreased at the inlet. On accountof the nucleation process after the throat, there aremany very small droplets in that region and thesevery small droplets reduce the mean diameter of thedroplets.

LOCATION EFFECT OF INJECTIONDROPLETS

The injection of droplets is performed with di�erentinlet radius and wetness fractions at the inlet andat the throat of the Laval nozzle. In Figure 11,

Figure 10. Variation of mean droplet radius along nozzle(0.5% inlet wetness, injected droplet radius = 0.2 micron).

Figure 11. Variation of pressure ratio along nozzle length(0.5% inlet wetness, injected droplet radius = 0.2 micron).

variation of the pressure ratio along the nozzle lengthfor these cases, are shown. It is observed that theinjection of droplets causes a tremendous reduction inthe condensation shock strength. It is also observedthat the injection of droplets at the inlet has bettere�ects on the ow characteristics. Variations of wetnessfractions along the nozzle length, for these cases, areshown in Figure 12. It is observed that the inlet dropletinjection leads to a lower amount of wetness along thenozzle length.

The reason for the injection of droplets at the inletof the converging channel being more e�ective thaninjection at the throat is as follows:

i) Pressure distribution along the channel length issteadier.

ii) When injecting small droplets at the inlet, nucle-


Figure 12. Variation of wetness fraction along nozzle.

ation does not occur while the injection at thethroat is accompanied by nucleation.

iii) The amount of wetness in the diverging channelsection when the droplets are injected at the inletis less than that of the state where the injection isat the throat as shown in Figure 10.

iv) Due to the formation of less wetness when inject-ing droplets at the channel inlet, the generatedentropy is less than when the droplets are injectedat the throat.

In fact, the injected droplets at the inlet acceleratethe ow in the convergent section and reduce therate of nucleation in the divergent part and, there-fore, the condensation shock strength is considerablydecreased.

SUMMARY OF CONCLUSION

1. Using a two-layer heat transfer model, we cansuccessfully predict the ow behavior.

2. Injecting the droplets at the inlet of the nozzleaccelerates the ow in the convergent section,causes severe reduction of the nucleation rate in thediverging part and results in the disappearance ofcondensation shock.

3. Injection at the inlet of the nozzle is better thaninjection droplets after the nozzle throat in thesupersonic region.

Hence, it is concluded that the injection of dropletswith suitable droplet radius or wetness fraction at theinlet of actual steam turbines, causes a reduction inthe nucleation rate and results in the disappearance ofcondensation shock, consequently, improving turbinee�ciency.

NOMENCLATURE

A areaa speed of soundB1; B2; B3 viral coe�cientsCp speci�c heat at constant pressureCL special temperaturef friction factor�G change in Gibbs free energyJ rate of formation of critical droplets

per unit volume and timeKn Knudsen numberl latent heat�l mean free path of vapour moleculesMa mach numbermr mass of dropletP vapour pressurePs(TG) saturation pressure at TGqc condensation coe�cientR gas constant for vapour phaser radius of dropletr� critical radiusT temperatureTs(P ) saturation temperature at P�T degree of supercooling [Ts(P )� TG]U velocityv speci�c volumeW total mass ow ratex distance along duct axisX;Y functions of temperature and density

in equation of state�r coe�cient of heat transfer ratio of speci�c heats�G kinematic viscosity of vapour� dryness fraction� density of mixture� coe�cient of thermal conductivity� surface tension�s(TL; r) density corresponding to saturation

pressure at temperature TL over asurface of curvature r

Subscripts

G vapour phaseL liquid phase0 stagnation conditions saturation

Superscripts

� critical droplet


REFERENCES

1. Bakhtar, F., Henson, R.J.K. and Mashmoushi, H. \Onthe performance of a cascade of a turbine rotor tipsection in wet steam, part 5: Theoretical treatment",IMechE, 220(4), Part C, pp. 457-472 (2006).

2. White, A.J. and Handslow, M.J. \Modelling dropletsize distribution in polydispersed wet-steam ows",International Journal of Heat and Mass Transfer, 43,pp. 1844-1873 (2000)

3. Bakhtar, F., Young, J.B., White, A.J. and Simpson,D.A. \Classical nucleation theory and its applicationto condensing steam ow calculations", Proc. Instn.Mech. Engnrs. Part C: J. Mechanical EngineeringScience, 219(C12), pp. 1315-1333 (2005).

4. Mahpeykar, M.R., Amirirad, E. and Lakzian, E. \Thee�ects of water injections in wet steam ow in di�erentregions of a mini laval nozzle", ASME ICNMM2008,Darmstadt, Germany (June 23-25, 2007).

5. Bakhtar, F. and Mahpeykar, M.R. \On the perfor-mance of a cascade of turbine rotor tip section bladingin nucleating steam Part 3: Theoretical treatment",Proc. Instn. Mech. Engrs., 211(3), Part C, pp. 195-210 (1997).

6. Mahpeykar, M.R. and Taymourtash, A.R. \The e�ectsof friction factor and inlet stagnation conditions onthe self condensation of steam in a supersonic nozzle",Scientia Iranica, 11(4), pp. 269-282 (Oct. 2004)

7. Mashmoushy, F., Mahpeykar, M.R. and Bakhtar, F.\Studies of nucleating and wet steam ows in two-dimensional cascades", Journal of Mechanical Engi-neering Science IMechE, 218(8), Part C, pp. 843-858(2004).

8. Gerber, A.G. and Kermani, M.J. \A pressurebased eulerian-eulerian multi-phase model for non-equilibrium condensation in transonic steam ow",International Journal of Heat and Mass Transfer, 47,pp. 2217-2231 (2004).

9. Bakhtar, F. and Zidi, K. \On the self di�usion of watervapour", Proc. Instn. Mech. Engrs., 199(C2), pp. 159-164 (1985).

10. Bakhtar, F., White, A.J. and Mashmoushi, H. \Theo-retical treatments of 2-D, 2-phase ows of steam andcomparison with cascade measurements", I. Mech. E.,219, Part C, pp. 1335-1355 (2005).

11. Young, J.B. \The condensation and evaporation ofliquid droplet in a pure vapour at arbitrary Knudsennumber", International Journal of Heat and MassTransfer, 34(7), pp. 1649-1661 (1991).

12. Senoo, S. and White, A.J. \Numerical simulationof unsteady wet steam ows with non-equilibriumcondensation in the nozzle and the steam turbine",ASME Joint US-European Fluid Engineering SummerMeeting (2006)

13. Mahpeykar, M.R., Lakzian, E. and Amirirad, E.\Reduction of thermodynamic losses in a supersonicnucleating steam nozzle by spraying water droplets",

Scientia Iranica, Trans. B, Mechanical Engineering,16(3), pp. 253-262 (2009).

14. Hiwase, S.D., Datta, A. and Som, S.K. \Entropybalance and exergy analysis of the process of dropletcombustion", Journal of Physics D: Applied Physics,31(13), pp. 1601-1610 (1998).

15. Young, J.B. and Bakhtar, F. \A comparison betweentheoretical calculations and experimental measure-ments of droplet sizes in nucleating steam ows",Trans. Inst. Fluid Flow Machinery, pp. 259-276(1976).

APPENDIX

The virial coe�cients of the Vukalovich state equation:

B1 = � eGTG

� �1 + b;

B2 = �b�1 + 4�21�2;

B3 = 32b�21�2;

where:

�1 =CGTW1G

; �2 = 1� KTW2G

;

and:e = 63:2; b = 0:00085

C = 0:3900� 106; G = 47:053;

K = 22:7; n = 0:355� 10�7;

m1 = 1:968; m2 = 2:957;

W1 = (3 + 2m1)=2; W2 = (3m2 � 4m1)=2:

The thermodynamic properties:

h = pv �RT 2G

1vdB1

dTG+

12v2

dB2

dTG+

13v3

dB3

dTG

!+ 1:111177TG + 3:55878� 10�4T�2

G

� 6991:96TG

+ 2070:54;

s = R+ 0:30773 + 1:111177 lnTG + 7:11756

� 10�4TG � 3495:98T 2G

;

Cv =@@TG

(h� pv)v;

Cp = Cv �TG�@p@TG

�2

v�@p@v

�T

;


R = 0:46151 kJ/kg.K:

BIOGRAPHIES

Mohammad Reza Mahpeykar received his B.S.degree in Mechanical Engineering from Ferdowsi Uni-versity of Mashhad in 1981, his M.S. degree inThermodynamics and Fluid Mechanics from Birming-ham University, UK, in 1987, and a Ph.D. degreein Energy Conversion from the same university in1992.

He has published more than 7 books on Mechanicsand Industrial Engineering and about 50 papers inrespected journals and international conference pro-ceedings. Professor Mahpeykar is Chief Editor of theJournal of Applied Sciences and Computational inMechanics (published by Ferdowsi University of Mash-had). He has also been involved in about 15 applied

funded projects for various industries. He has wonacademic awards for supervising some distinguishedM.S. and Ph.D. theses and also received an ExcellentEngineer award from Khorasan province. He is nowAssociate Professor at Ferdowsi University of Mashhad(since 1992).

Ehsan Amirirad received a B.S. degree in MechanicalEngineering from the Iran University of Science andTechnology in 2005, and a M.S. degree in EnergyConversion from Ferdowsi University of Mashhad in2007. He is now a Ph.D. student and has been workingon his doctoral project in `Two-Phase Flow Theory inSteam Turbines' since 2007. He has published morethan 5 papers in journals and international conferenceproceedings. He was a �rst class student during hisM.S. course, when taking his Ph.D. entrance examsand other comprehensive tests.

Suppression of Condensation Shock in Wet Steam Flow by...

Documents

Transcript of Suppression of Condensation Shock in Wet Steam Flow by...