Journal of Thermal Science Vol.22, No.4 (2013) 320−326

Received: January 2013 Huanlong Chen: Lecturer

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DOI: 10.1007/s11630-013-0630-1 Article ID: 1003-2169(2013)04-0320-07

Research on Wet Steam Spontaneous Condensing Flows Considering Phase Transition and Slip

Ke CUI1, Huan-long CHEN1, Yan-ping SONG1, Hiroharu OYAMA2

1. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China 2. Mitsubishi Heavy Industries, LTD, Takasago, Hyogo 676-8686 Japan

© Science Press and Institute of Engineering Thermophysics, CAS and Springer-Verlag Berlin Heidelberg 2013

A new dual-fluid model considering phase transition and velocity slip was proposed in this paper and the Cun-ningham correction was used in the droplet resistance calculation. This dual-fluid model was applied to the nu-merical simulations of wet steam flow in a 2D LAVAL nozzle and in the White cascade respectively. The results of two simulations demonstrate that the model is reliable. Meanwhile, the spontaneous condensing flow in White cascade was analyzed and it infers that the irreversible loss caused by condensation accounts for the largest share (about 8.78% of inlet total pressure) in total pressure loss while the loss caused by velocity slip takes the smallest share (nearly 0.42%), and another part of total pressure loss caused by pneumatic factors contributes a less share than condensation, i.e. almost 3.95% of inlet total pressure.

Keywords: dual-fluid model; wet steam; spontaneous condensing flow; LAVAL nozzle; White cascade.

Introduction

Generally speaking, the models for simulation of wet steam spontaneous condensing flow could be divided into three types: single-fluid model, particle trajectory model and dual-fluid model. Single-fluid model ignores the dynamic imbalance brought about by condensation while the particle trajectory model does not consider dif-fusion. A new stochastic particle trajectory model was proposed later. Since more trajectories are required and the Monte-Carlo algorithm [1] must be used, it needs enormous iterations. Dual-fluid model treats continuous phase and discrete phase as continuous medium [2] and takes full account of the influence caused by condensa-tion and turbulence interactions between two phases, so dual-fluid model is a fully coupled two-phase model.

When the multiphase flow is described through using dual-fluid model, the main problem is the numerical

simulation of the discrete phase. Bkhtar and Moham-mandi [3] made a numerical simulation on spontaneous condensing flow in the passages of a 2D cascade and an experiment [4, 5] was conducted as well. Qu and Shen [6] deduced two phase non-equilibrium governing equations by combining droplet nucleation theory and droplet growing theory. Furthermore, they solved these equations by using the time marching method. Zhang [7] developed a quick and accurate simulation method for non –equili-brium wet steam flow, which considered the influence of velocity slip as Zhang’s findings [8]. As for the turbulence of discrete phase, many scholars put forward different solutions. Zhou [9] proposed a turbulent kinetic transport equation theory and deduced a k-ω-k p two phase turbu-lence model for gas-solid turbulent particle movement. Inspired by Zhou's thoughts, Wu et al [10] put forward a k-ε- kp model for simulation of wet steam condensing flow, which was based on the k-ε model for single phase

Ke CUI et al. Research on Wet Steam Spontaneous Condensing Flows Considering Phase Transition and Slip 321

flow simulation. In this paper, a new dual-fluid model considering the

influence of velocity slip was put forward and the Cun-ningham correction was used for droplet resistance cal-culation. Based on the dual-fluid model, numerical simu-lations were conducted on spontaneous condensing flow in a 2D LAVAL nozzle and in the White cascade respec-tively. In addition, the loss which caused by different factors in the spontaneous condensing flow in White cascade was calculated and an analysis on these factors was made finally.

Establishment of the Dual-fluid Model

Reynolds Time-Averaged Equations At first, a Reynolds average was made on the transient

volume-averaged conservation equations. Some fluctuat-ing terms and fluctuating correlation terms were ignored, such as fluctuating terms of mixture density, correlation term between droplet numbers and velocity fluctuating in gas and so on. In order to make the system of equations closed, a gradient simulation [11] was made on the second- order correlation terms and the Reynolds time-averaged equations for wet steam flow were obtained at last.

Continuity Equations ( )g g j

j

um

t x

ρ ρ∂ ∂+ = −

∂ ∂ (1)

( )( ) m pjm

j

YuYm

t x

ρρ ∂∂+ =

∂ ∂ (2)

Droplet Number Governing Equation ( )( ) m jm

g

j

NuNI

t x

ρρρ

∂∂+ =

∂ ∂ (3)

Momentum Equations ( )( )

2( ) 3

( )

gg i j effig i

j i

ji lt ij

j j i l

mpi i irp

pu uu gt x x

u uux x x x

Y u u u mG

ρρ ρ

μ μ δ

ρτ

⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞⎪ ⎪⎢ ⎥⎪ ⎪⎜ ⎟ ⎜ ⎟⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎪ ⎪⎩ ⎭

∂∂∂+ =− +Δ

∂ ∂ ∂

∂ ∂∂∂+ + + −∂ ∂ ∂ ∂

+ − −

(4)

( ) ( )

( )

( )

m mpi pi pjm i

j

pi pj mp i pirpj j i

i pi

Yu Yu uYgt x

u u Y u ux x x G

u u m

ρ ρρ

ρμ τ

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∂ ∂+ = +

∂ ∂

∂ ∂∂ + + − +∂ ∂ ∂

−

(5)

Equations (4) and (5) are the momentum equations for gas and liquid respectively.

The Cunningham correction factor G was calculated by using the formula shown as (6):

( )1.741 2.492 0.84 expG Kn

Kn= + + −⎡ ⎤

⎢ ⎥⎣ ⎦ (6)

Where Kn represents the Knudsen number, which is the ratio of the molecular mean free path length to the diameter of particles in the fluid. The Knudsen number is the criterion for determining whether statistical mechan-ics or the continuum mechanics formulation of fluid dy-namics should be used.

Energy equation ( ) ( )( )

( )( )

gg j jeff

j j j j

kji eff kit fg

j

u e pue Tt x x x x

um h hx

ρρλ

τ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

⎡ ⎤⎢ ⎥⎣ ⎦

∂ ∂∂ ∂ ∂+ = −∂ ∂ ∂ ∂ ∂

∂+ − −∂

(7)

Where the λeff and the (τkji)eff were defined in literature [2].

Equations (1) and (4) are both governing equations for gas phase while equations (2) and (5) are governing equations for liquid phase.

Turbulent Model and Turbulent Viscosity Coefficient for Gas Phase

Based on governing equations for gas phase, the tur-bulence kinetic equation and the ε equation were ob-tained.

Turbulence kinetic equation ( )( ) gg j t

j j jk

p g gM

u kk k Gt x x x

G G Y

ρρ μ μσ

ρ ε

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

∂∂ +∂ ∂+ = +∂ ∂ ∂ ∂

+ + + −

(8)

ε equation

1

22 1 3 1

( ) g jg t gkj j j

g pb

uC St x x x

C C C G C Gk v k k

ε

ε ε ε

ρ ερ ε μ μ ε ρσ

ε ε ερ ε

⎛ ⎞ ⎡ ⎤⎜ ⎟ ⎛ ⎞⎜ ⎟ ⎢ ⎥⎜ ⎟⎝ ⎠ ⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

∂∂ +∂ ∂+ = +∂ ∂ ∂ ∂

− + ++

(9)

Likewise, the meaning of the terms in equation (8) and (9) are the same as the terms in literature [2].

The formula (10) is used for the calculation of turbu-lence viscosity coefficient for gas phase.

2

t gkcμμ ρε

= (10)

About the determination of turbulence viscosity coef-ficient μp for liquid phase, Zhou [9] once solved it through the establishment of a two-phase turbulence model which was named as k-ω-kp model for simulation on gas-solid two-phase flow considering vortex and sudden enlarge-ment. Combined with the characteristics of wet steam,

322 J. Therm. Sci., Vol.22, No.4, 2013

the k-ε-kp model was simplified in this paper and the gas turbulent viscosity coefficient was assumed to be equal to the liquid turbulent viscosity coefficient, namely μp=μt. This assumption is proved to be reasonable by the nu-merical results in this paper.

Numerical Validation and Analysis

Spontaneous Condensing Flow in LAVAL Nozzle In order to validate the correctness of the model pro-

posed in this paper, a numerical simulation on spontane-ous flow in a LAVAL nozzle was conducted at first. The geometric parameters and flow conditions were the same as the experiment implemented by Moses and Stein [12]. According to the experimental results, a calculation scheme was designed and given in Table 1. Table1 Calculation scheme

Group Case[12] Inlet total pressure P0(kPa)

Inlet Total temperature T0(°C)

193 43.027 93

424 41.907 103 1

411 42.280 112

2 252 40.050 101

The distributions of static pressure along the centerline

of nozzle for group 1 are shown in Fig.1. Both the ex-perimental and numerical results show that the so called “condensing shock” [13] occurs due to the fact that the condensation releases heat, which makes the vapor heated and leads to a pressure jump. However, it does not mean that the source term, i.e. the ( )fgm e h− − in the

energy equation caused by condensation is positive. It is negative instead because of the double influence of en-ergy taken way by mass transfer and energy added through absorbing the latent heat released by condensa-tion during the whole condensation process.

Fig.1 Distributions of static pressure along the centerline of nozzle for group 1

When the total pressure is fixed, as the inlet total tem-perature increases, the degree of supercooling decreases, hence the growth of droplet postpones in large scale ac-cording to Hill's droplet growing model [14]. Both the ex-perimental and numerical results in Fig.1 also show that as the inlet total temperature increases, the condensation shock moves downstream.

Fig.2 demonstrates the distribution of humidity along the centerline of nozzle for group 2, which includes two regions in the rapid growth area of humidity. In Li's opinion [13], the region with a large slope corresponds to the period after the nucleation occurs and the other small slope region represents the physical process of droplet growing.

Fig.2 Distribution of humidity along centerline of nozzle for group 2

However, in this paper it suggests that the large slope

region results from the droplet growth under the unsatu-rated steam state. The degree of supercooling for wet steam increases under the unsaturated steam state, hence the growth rate of droplet radius enhances [14], so the hu-midity grows rapidly. Whereas, the latent heat released due to droplets growing makes the wet steam return to the saturate state again, thus the degree of supercooling declines and the process of humidity increase are slowed down.

Generally speaking, the numerical and experimental results match well with each other. It infers that the dual-fluid model proposed in this paper can be used to simulate the wet steam flow in different conditions.

Wet Steam Flow in White Cascade The simulation in this part is only for the L1 case [15].

Table 2 lists the geometrical and aerodynamic parameters of the cascade, and the blade profile is given in Fig.3.

In order to discuss the influence of three factors, i.e. the pneumatic factors, the condensation and the velocity slip on the loss, two other calculation models were pro-

Ke CUI et al. Research on Wet Steam Spontaneous Condensing Flows Considering Phase Transition and Slip 323

posed in this part based on the dual-fluid model. One model was obtained by ignoring the source term, i.e. the ρmY/(trpG)(upi−ui) in the equation (4) and it was called no slip model. Another calculation model named no con-dense model was also acquired with ignoring source terms referred as the piu m− and the ρmY/(trpG)(upi−ui)

in equation (4), along with the ( )t fgm h h− − in equa-

Table 2 Parameters of L1 case [15]

Outlet Mach Number 1.2

Outlet Flow Angle 71°

Chord 137.51mm

Pitch 87.59mm

Stagger Angle 45.3°

Inlet Total Pressure 0.409MPa

Inlet Total Temperature 354K

Back Pressure 0.194MPa

Fig.3 Profile of White cascade

tion (7). In the meantime, the momentum equation (5) for liquid phase was not taken into account in the no con-dense model.

The relationships between these three models are shown in Fig.4. The number “1” represents the inlet total pressure and number “0” represents the local total pres-sure and its value is zero. The expression -“No Con-dense” represents the local total pressure calculated with no condense model. Similarly, the expressions referred as “No Slip” and “Dual Fluid Model” represents the local total pressure calculated with no slip model and dual- fluid model respectively. The difference between two adjacent “local total pressure”s represents the effect of the single factor on total pressure loss, for example the difference between the “No Slip” and the “Dual Fluid Model” represents the loss caused by velocity slip.

Fig.4 Relationships between three calculation models Fig.5 demonstrates the contour of static pressure dis-

tribution obtained with three calculation models respec-tively. Although all of the three simulations capture the dovetail shock nearby the trailing edge, some obvious difference still exists. When no condense model is used, the dovetail shock doesn’t spread to the suction side and the suction side has a lower sudden pressure increase, which is different from the other two models. It implies that the dovetail shock results from the joint action of

Fig.5 Contour of static pressure

324 J. Therm. Sci., Vol.22, No.4, 2013

Fig.6 Static pressure distribution on the blade surface

condensing shock and pneumatic shock, which is consis-tent with White and Young’s experiment [15].

The numerical and experimental result of static pres-sure distributions on the blade surface are demonstrated in Fig.6. The simulation conducted with dual-fluid model or no slip model both capture the spontaneous condens-ing point and a sharp pressure jump on the suction side nearby the trailing edge, though the condensing point appears more closely to the trailing edge in comparison with the experimental results. Moreover, there appears no pressure jump in the simulation with taking no account of condensation. Generally speaking, the two numerical results considering condensation fit better with the ex-periment and the velocity slip has a smaller influence on the wet steam flow than the condensation.

Fig.7 demonstrates the distribution of nucleation rate calculated with dual-fluid model. The nucleation concen-trates in the relatively narrow area, i.e. the throat and the level of nucleation rate reaches to 1026 per kilogram, which becomes the main reason for the pressure jump in Fig.6 with the rapid growth of droplets.

Fig.7 Distribution of nucleation rate Fig.8 shows the distribution of humidity obtained with

dual-fluid model. The humidity near the wall and that near the trailing edge both drop rapidly to zero. The same phenomenon also could be found in Fig.7. Both of them

might be the result of the flow condition, namely, the high temperature in local area. It not only makes the local wet steam overheated but also stops the nucleation and growth of local droplets. Meanwhile, in the flow with shock wave in it, the humidity also declines sharply cross the shock wave because of the temperature increase.

The pitchwise distribution of the static pressure at the position of 1/4 axial chord length away from the trailing edge is demonstrated in Fig.9.

Fig.8 Distribution of humidity

Fig. 9 Pitchwise distribution of static pressure The longitudinal coordinates represent the ratios of the

local static pressure to inlet total pressure. The two nu-merical results obtained with considering the influence of condensation fit better with the experiment than the one takes no account of condensation. The sudden increase of the static pressure (at about 43.3% pitchwise position) is caused by the shock wave and the static pressure in-creases obviously cross the shock wave.

Fig.10 shows the pitchwise humidity distribution. The numerical result obtained with dual-fluid model is almost the same as the result obtained with no slip model and both of them still fit well with the experiment. In the meantime, the humidity decrease (at about 43.3% pitch-wise position) is also the result of shock wave effect, which is consistent with the analysis based on Fig.8.

Ke CUI et al. Research on Wet Steam Spontaneous Condensing Flows Considering Phase Transition and Slip 325

Fig. 10 Pitchwise humidity distribution Fig.11 demonstrates the total pressure distribution at

the same position as the ones in Fig.9 and Fig.10. The longitudinal coordinates are ratios of the local total pres-sure to the inlet total pressure. The no slip model and the dual-fluid model are still useful for the good consistency with the experiment of their results. The total pressure increase (at about 43.3% pitchwise position) in both the calculation and the experiment might be the result of shock wave effect, which seems to be different from the regular pattern in the single phase numerical simulation. However, this phenomenon is reasonable in wet steam flow. As mentioned above, the state of wet steam changes from supercooled to overheated cross the shock wave, thus the momentum of gas increases because of the mo-mentum transfer from liquid phase to gas phase as the droplet evaporates according to the governing equation (4) and (5). In the meantime, the source term, i.e. the

( )t fgm h h− − in the governing equation (7) turns into a

positive term under the influence of condensation. There-fore, the total pressure increases as a consequence of the above two effects.

Fig. 11 Distribution of total pressure The ratios of local total pressure to inlet total pressure

obtained with three calculation models are shown in Ta-

ble 3 respectively. According to relationship depicted in Fig.4, it infers that the loss caused by the pneumatic fac-tors is about 3.95% of the inlet total pressure, another loss caused by condensation occupies a greater percent-age (about 8.78%) than the pneumatic factors and the total pressure loss caused by velocity slip takes the smallest share (about 0.42%).

Table 3 loss caused by different factors

Calculation Model (P02/P01) proportion (%)

No Condense 0.9605 3.95

No Slip 0.8727 12.73

Dual-fluid 0.8685 13.15

The analysis above shows that the velocity slip be-

tween two phases has a smaller effect than condensation on the loss, which is brought by the big difference be-tween the specific volume of gas and that of liquid. The specific volume of liquid increases sharply as the drop-lets evaporates. So the loss caused by the transfer of mass and momentum will play a more important role in the total pressure loss than the effect of velocity slip. How-ever, in other calculation cases, the diameter of droplets might be larger than that discussed in this paper. In that case, the velocity slip will bring bigger resistance and it is worth paying more attention on the effect of velocity slip.

Conclusion

Based on the classical nucleation theory, a dual-fluid model considering the influence of velocity slip was proposed in this paper and this dual- fluid model was used for simulating the spontaneous condensing flow in a 2D LAVAL nozzle and in the White cascade. The main conclusions are as follows.

1. This dual-fluid model considers the features of wet steam and the Cunningham correction is taken for the droplets resistance calculation. Moreover, the result of simulations conducted with this dual-fluid model fit well with the experiment, thus the applicability and correct-ness of this dual-fluid model are verified.

2. The nucleation and condensation has a close rele-vance with the inlet total temperature. As the inlet total temperature increases, the condensing shock moves downstream. In addition, the humidity at the centerline of nozzle increases rapidly under the unsaturated steam state until the latent heat released by droplets growing makes the state of steam return to saturate, which slows down the humidity increase.

3. Condensation has a more important effect on the wet steam flow than slip and it takes the major responsi-bility for the total pressure loss relative to the effect of

326 J. Therm. Sci., Vol.22, No.4, 2013

slip. In this paper, the loss caused by condensation occu-pies about 8.78% of the inlet total pressure. The pneu-matic loss takes a smaller share (about 3.95%) than con-densation while the velocity slip has a minimum effect on the total pressure loss (nearly 0.42%).

Acknowledgement

The authors are grateful for the financial support for this work by the fundamental research funds for the Cen-tral Universities (Grant No. HIT. NSRIF. 201173).

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