Evaluation analysis of correlations for predicting void fraction of condensation hydrocarbon refrigerant upward flow in a spiral pipe

Accepted Manuscript Evaluation analysis of correlations for predicting void fraction of condensation hydrocarbon refrigerant upward flow in a spiral pipe Shulei Li, Weihua Cai, Jie Chen, Haochun Zhang, Yiqiang Jiang PII: DOI: Reference:

S1359-4311(17)36249-X https://doi.org/10.1016/j.applthermaleng.2018.05.089 ATE 12228

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

27 September 2017 9 March 2018 23 May 2018

Please cite this article as: S. Li, W. Cai, J. Chen, H. Zhang, Y. Jiang, Evaluation analysis of correlations for predicting void fraction of condensation hydrocarbon refrigerant upward flow in a spiral pipe, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/j.applthermaleng.2018.05.089

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Evaluation analysis of correlations for predicting void fraction of condensation hydrocarbon refrigerant upward flow in a spiral pipe Shulei Li a,d, Weihua Cai a,d*, Jie Chen c, Haochun Zhang a, Yiqiang Jiangb* a. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China b. Department of Building Thermal Energy Engineering, Harbin Institute of Technology, Harbin, China c. CNOOC Gas and Power Group, Beijing, China d. Key Laboratory of Efficient Utilization of Low and Medium Grade Energy (Tianjin University), Ministry of Education of China, Tianjin, China

Abstract: Accurate prediction of void fraction for condensation hydrocarbon refrigerant upward flow in a spiral pipe is very important for the tube-side design of spiral wound heat exchange (SWHE) using in liquid natural gas (LNG) plants. Although scores of void fraction correlations have been developed until now, most of them are focus on two-phase flow in straight tubes and their applicability to spiral pipes needs to be evaluated. In this paper, the condensation void fraction characteristics were numerically investigated based on the verified model. 455 numerical data points of six refrigerants were obtained, under different operating and structural parameters. Based on these data, 96 void fraction correlations were evaluated, which covered the types of homogeneous, slip-ratio, Kαv,H, Lockhart-Martinelli parameter based, drift-flux, general and implicit, and finally the best five correlations were recommended which can well predict condensation void fraction in a spiral pipe within ±5% error. This study will provide some constructive instructions to predict void fraction for condensation hydrocarbon refrigerant upward flow in a spiral pipe, which is helpful in designing more effective SWHE used in large-scale LNG plants.

Keywords: Spiral pipe; Condensation; Void fraction; Hydrocarbon refrigerant; Correlations 1 Introduction

Void fraction, defined as the ratio of area which is occupied

With the extensive applications of spiral wound heat

by the vapor to the area of cross-plane, plays an important

exchanger (SWHE) using in energy and chemical industry

role to predict the pressure drop and heat transfer in

fields, including petroleum, metallurgy, liquefied natural

condensation two-phase flow since it decides the

gas (LNG) and rectisol, lots of attentions have been paid to

condensation flow patterns, which affects the condensation

the study on condensation two-phase flow in a spiral pipe.

flow and heat transfer mechanisms; at low void fractions, the gravity condensation is usually dominated, but with the

* Corresponding authors: [email protected] (W. H. Cai), +86-451-86403254; [email protected] (Y. Q. Jiang), +86-451-86282123

increasing void fraction, shear condensation becomes more Page 1 / 47

obvious. Meanwhile, hydrocarbon refrigerants are usually

al. [8] experimentally investigated condensation void

used as working mediums in LNG field. Therefore, a better

fraction for propane flow in smooth horizontal tubes.

understanding

condensation

Based on experimental results, a new multi-regime drift

hydrocarbon refrigerant flow in a spiral pipe can be

flux model was carried out which could well predict the

contributed to the design and optimization of SWHE.

trends with tube diameter, mass flux and pressure. Dalkilic

of

void

fraction

for

Nowadays, a large number of investigations have been

et al. [9] put forward a void fraction model by an

carried out on void fraction in tubes. Kopke et al. [1],

experimental study on R134a condensation downward

Wilson et al. [2] and Yashar et al. [3, 4] performed a series

annular flow in a vertical smooth tube, which was

of evaporation and condensation experiments on void

associated with film thickness proposed by Soliman [10]

fraction for refrigerant flow in smooth and enhanced

and well coincided with some well-known void fraction

horizontal tubes. The results showed that in smooth tubes,

correlations. They also discussed the effects of void

condensation void fraction tended to increase with the rise

fraction models on prediction for film thickness, two-phase

in vapor quality and mass flux while evaporation void

friction factor, pressure drop and heat transfer coefficient of

fraction was not dependent on mass flux; in enhanced

R134a downward condensation flow in a vertical smooth

horizontal tubes, condensation void fractions were lower

tube [11-14]. Lips et al. [15] and Olivier et al. [16]

than those in smooth tubes while evaporation results

researched the influence of inclination angles on void

showed similar trends found in smooth tubes. Harms et al.

fraction during condensation inside a smooth tube. It was

[5] developed a theoretical void fraction model for annular

found that at low mass flux and vapor quality, void

flow in horizontal tubes where the influence of momentum

fractions increased with the increase of downward

eddy diffusivity damping at vapor-liquid interface was

inclinations but did not change with upward inclinations;

taken into account. Cioncolini and Thome [6] also

however, at high low mass flux and vapor quality, the

established a void fraction correlation for annular flow,

effect of all inclinations became slight. Winkler et al. [17,

which was simplified with most existing correlations and

18] measured void fraction for condensation R134a in

only a function of vapor quality and vapor-liquid density

wavy flow inside small channels. The results indicated that

ratio. Jassim et al. [7] proposed a correlation to predict void

void fraction showed a clear trend with vapor quality, but

fraction for refrigerants flow in smooth horizontal tubes

mass flux and hydraulic diameter had no significant effect

using a probabilistic two-phase flow regime map; in the

on it; they also developed a drift flux model and a slip-ratio

correlation, time fractions were used to provide a weight of

correlation to predict void fraction for wavy flow in small

void fraction models for different flow regimes. Milkie et

channels. A flow pattern independent drift flux void

Page 2 / 47

fraction correlation for different refrigerants flow in

was further verified by Mathure [25]. Xue et al. [26]

straight tubes was proposed by Bhagwat and Ghajar [19]; it

collected 39 void fraction correlations and researched the

had good performance for the whole range of void fraction

accuracy of them for gas-liquid two-phase flows in vertical

when tube diameters and tube orientations were 0.5～

downward pipes based on their own experimental data. It

305mm and -90～90° respectively, while liquid viscosity,

was shown that most correlations predicted low void

system pressure and two phase Reynolds number were

fraction poorly while the accuracy of drift flux ones

0.0001 ～ 0.6Pa·s, 0.1 ～ 18.1MPa and 10 ～ 5 × 106,

showed a significant decline as void fraction was more than 0.83. Xu and Fang [27] investigated the correlations

respectively. Meanwhile, many literatures have been published on the evaluation of void fraction correlations for different two-phase flow in tubes. Diener et al. [20] analyzed the effect of physicochemical properties on void fraction in tubes; the results demonstrated that vapor quality and density ratio had the greatest influence on it, followed by mass flux and viscosity ratio; the least influenced by surface tension. In addition, several void fraction correlations were evaluated for horizontal and vertical upward flows, most of which could not predict mean mixture density well at high void fraction. The comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes was carried out by Woldesemayat and Ghajar [21]. It stated that most of

of void fraction for two-phase refrigerant flow in macro and mini tubes. In their study, 41 correlations were reviewed and evaluated based on 1574 experimental data points collected from 15 published papers. The result showed that the good correlations had better performance for macro tubes than for mini tubes. They also developed a new correlation and its prediction accuracy for mini tubes increased remarkably. Parrales et al. [28] discussed the performance of void fraction correlations to describe the behavior of two-phase flow in helical double pipe evaporators and found that most of good correlations belonged to drift flux models, which indicated that void fraction in evaporators was mainly determined by the velocity slip between vapor phase and liquid phase. In summary, most investigations on void fraction and

existing ones did not have universality for different conditions while drift flux models showed the best performance among them, which was also found in the studies of Godbole et al. [22] and Yan et al. [23]; besides, an improved correlation based on Dix’s model [24] was proposed, which could well predict the whole database and

evaluations of correlations were devoted to two-phase flow in straight tubes and few researches have paid attention to void fraction in coiled tubes expect the study of Parrales et al. [28]. However, their studies focused on an evaporation flow in a helical tube using water as working fluid and the helical diameter is much smaller than 1m, which was far

Page 3 / 47

from the condensation hydrocarbon refrigerant flow inside

2.1. Governing equations

a spiral pipe of SWHE using in LNG field. Moreover,

In this paper, the inhomogeneous two-fluid model

compared with experimental study, numeral simulations

coupling with thermal phase change model was selected to

have great superiority.

calculate condensation multiphase flow in a spiral pipe.

As a consequence, a numeral model verified by

The

Reynolds-averaged

governing

equations

for

experimental data from the existing literature was

conservation of mass, momentum, energy, and turbulent

introduced

quantities can be expressed as follows.

to study the

condensation

hydrocarbon

refrigerant flow inside a spiral pipe. The characteristics of

Mass Conservation

 l l     l l ul    lv (1) t

void fraction were discussed for hydrocarbon refrigerants during condensation upward flow in a spiral pipe under

  v v      v vuv    vl (2) t

different conditions. Then, 96 void fraction correlations were reviewed and evaluated.

where  ,



and u represent the volume fraction,

2. Calculation Methods In this paper, considering the calculation accuracy and computational efficiency, a computational model simplified from SWHE tube side was established to simulate the void

density and velocity, respectively. Subscript l and

v

denote liquid phase and vapor phase, respectively.  lv and  vl are the mass flow rate in per unit volume from vapor

fraction characteristics for condensation hydrocarbon

phase to liquid phase and from liquid phase to vapor phase,

refrigerant upward flow, as shown in Fig. 1. The spiral pipe

respectively;

has a length of 1m which contains 0.6m long section used

 lv  0 represents the positive mass flow rate in per unit

to develop the flow pattern, 0.2m long section used to test

volume from vapor phase to liquid phase. It is important to

void fraction and 0.2m long section used to ensure the

keep track of the direction of mass transfer process.  lv

stability of measurement. The components of refrigerants

can be expressed as follows:

 lv   vl   lv   vl

.

The

term

 lv  mlv Alv (3)

and the conditions of simulation are shown in Tab. 1 and Tab. 2, respectively.

where Alv is the interfacial area density between liquid phase and vapor phase.

mlv

represents the mass flow rate

in per unit interfacial area from vapor phase to liquid phase. It can be calculated based on the thermal phase change Fig.1 Computational model and mesh of a spiral pipe

model. Besides, the constraint can be also given as Page 4 / 47

mlv  mvl .

viscosity and turbulent viscosity, respectively. Subscript

In this paper, vapor phase and liquid phase are both

ref means the reference value, herein, ref  v . g

Flv and Fvl

assumed to be continuous phase. The interfacial area

represents the acceleration of gravity while

density between vapor phase and liquid phase, Alv , can be

denote the interfacial forces acting on liquid phase due to

written as

the presence of vapor phase and on vapor phase due to the

 v l dlv

Alv 

presence of liquid phase per unit volume, respectively; (4)

Flv   Fvl , which can be calculated as follows:

where d lv represents the mean interfacial lengths scale

Flv  FD,lv  F ,lv    lvuv   vlul  (8)

between liquid phase and vapor phase. d lv has been where

deduced in our previous study [29], as shown:

dlv 

1    Nulv tp  v  l Ld lv

where

 , Nulv , tp ,

condensation

surface force acting on vapor phase due to the presence of

8 q Cp L  2m(1  x)Cp d    l

heat

ratio,

l

 lv

and

Nusselt

Cpl

number

FD,lv and F ,lv mean the drag force and the

(5) liquid phase per unit volume, respectively;

lv

represent between



u   vlul 

 lv v

means the momentum transfer associated with interphase mass transfer The drag force

vapor-phase side and liquid-phase side at phase interface, mixture thermal conductivity, latent heat of phase change

FD,lv can be modeled as: FD,lv  CD tp Alv uv  ul  uv  ul  (9)

and specific heat of liquid phase, respectively while m, x, q,

tp

d and L denote mass flux, vapor quality and heat flux,

where CD represents the drag coefficient, CD =0.44;

hydraulic diameter and length of test section, respectively.

represents the vapor-liquid mixture density, which can be

Momentum Conservation

written as follows:

  l l ul      l l ul ul    l ( l  ref )g t





T    l ( l  tl ) ul   ul     l p  Flv  

tp  v v  1  v  l (10) (6)

The surface tension model used in this paper is based on the Continuum Surface Force (CSF) model proposed by

  v vuv      v vuvuv    v ( v  ref )g t





T    v ( v  tv ) uv   uv     vp  Fvl  

where

p

,



and

t

Brackbill et al. [30]. It models the surface tension force as (7) a volume force concentrated at the interface, rather than a surface force. And then

represent pressure, dynamic

Page 5 / 47

F ,lv can be expressed as follows:

F ,lv  f ,lv lv (11)

     v v kv      v vuv kv      v  v  tv t  k  

   k v    (17)

     v v v      v vuv v      v  v  tv t   

    v    (18)

 v  Pkv  v v     vl kl  lv kv 

where:

 lv  l

(12)

f ,lv  lv nlv  s (13) where  lv is the interface delta function; surface tension coefficient;

nlv



 v

kv

 C 1Pkv  C 2 v v    vl  l  lv  v 

is the

      l l kl     l l ul kl      l  l  tl  kl  t  k   (19)  

is the interface normal

 l  Pkl  l  l    lv kv   vl kl 

vector pointing from liquid phase to the vapor phase

       l l  l      l l ul l      l  l  tl   l  t      (20)

(calculated from the gradient of a smoothed volume fraction);  s is the gradient operator on the interface and

 lv

v

 l

is the surface curvature defined by:

klv   nlv (14) Our previous study [29, 31] indicated that the simulation results based on standard k-ε turbulent model have the best agreement with experimental results.

l kl

 C 1Pkl  C 2 l  l    lv  v  vl  l 

where C 1 , C 2 ,

k

and



are constant with the

values of 1.44, 1.92, 1.3 and 1.0, respectively. Pk is the turbulence production due to viscous and buoyancy forces, modeled as follows:

2 Pki  t ui   ui  uiT    ui  3ti ui  i ki   Pkbi (21) 3

Therefore, in this paper, the standard k-ε turbulent model is selected to use in vapor phase and liquid phase. The

where Pkbi is the buoyancy production term and can be turbulent viscosity for vapor phase and liquid phase are written for the full buoyancy model as both modeled as:

tv  C v (

kv2

v

Pkbi  

) (15) where subscript

tl  C l ( where C is a constant given as 0.09. k

kl2

l

) (16)

and



i means any phase, herein, i  v or l .

Energy Conservation

  v v v      v vuv v      v vTv   Qv (23) t

represent turbulent kinetic energy and turbulent dissipation

 l l l     l l ul l     l l Tl   Ql (24) t

rate respectively. The transport equations for k and  can be written as follows:

ti g i (22) i

where Page 6 / 47

 ,

and

T

represent the enthalpy, thermal

conductivity and temperature, respectively. Qv and Ql

be expressed as, respectively.

ql  hl Alv TS  Tl  (30)

denote total interphase heat transfer to vapor phase across the interface with liquid phase and to liquid phase across

qv  hv Alv TS  Tv  (31)

the interface with vapor phase, respectively, Qv  Ql . Based on the thermal phase change model, they can be

Qv  qv  vl  vS (25) Ql  ql  lv lS (26)

qv

and

ql

hv

and

hl

respectively represent the heat

transfer coefficient of liquid phase and vapor phase on one

written as follows:

where

where

denote the sensible interphase heat

transfer to vapor phase across the interface with liquid

side of phase interface between vapor phase and liquid phase. TS is the interfacial temperature, which can be determined

from

considerations

of

equilibrium. Ignoring the effects of surface tension on pressure, the interfacial temperature can be written as

TS  Tsat (32)

phase and to liquid phase across the interface with vapor phase, respectively.

 vS

and

 lS

represent interfacial

thermodynamic

where Tsat represents the saturation temperature.

values of enthalpy carried into vapor phase and liquid

For the vapor phase on one side of the phase interface,

phase due to phase change, respectively. vl vS and lv lS

a zero resistance condition is adopted in this paper. This is

represent heat transfer induced by interphase mass transfer

equivalent to an infinite heat transfer coefficient in vapor

into vapor phase from liquid phase and into liquid phase

phase side of the phase interface, hv   . Its effect is to

from vapor phase, respectively. Then, their relation can be

force the interfacial temperature to be the same as the

given as

vapor phase temperature, as shown here,

qv  ql    vl  vS  lv lS  (27) Substituting Eq. (3) into Eq. (27),

mlv

TS  Tv (33) It is reasonable to force the vapor phase temperature

is given as

qv  ql mlv  (28) Alv   vS   lS 

to be the same as the saturation temperature because the degree of super-cooling is very slight for vapor phase in the condensation flow.

Meanwhile

mlv  0   lS   lsat , vS   v mlv  0   lS   l , vS   vsat Based on the two resistance model,

qv

and

ql

Further, according to the mixture model and the two (29)

resistance model, the following equations can be gotten.

can Page 7 / 47

hlv 

tp Nulv dlv

(34)

1 1 1   (35) hlv hl hv

model together with standard k-ε turbulence model and

where hlv represents the heat transfer coefficient between

scalable wall function was selected to calculate the flow

vapor phase side and liquid phase side.

near the wall. The mass flow, vapor volume fraction and

Then, considering hv   , the heat transfer coefficient

thermal phase change model was adopted while the

temperature were adopted at inlet, while the static pressure was used at outlet and the constant heat flux was

hl can be written as

considered in the wall of spiral pipe. All physical properties

hl 

tp Nulv (36) dlv

of hydrocarbon refrigerant were computed from REFPROP [32].

2.2 Numerical method 2.3 Model Verification The characteristics of void fraction for condensation In the published literatures, there was only the hydrocarbon refrigerant upward flow in a spiral pipe were investigation in Ref. [33] on condensation of hydrocarbon simulated by ANSYS CFX 12.1. Meanwhile, the fluid refrigerant in upward flow inside a spiral tube which is domain was meshed with hexahedral grid, with fine similar to SWHE tube side. Therefore, in order to validate meshes near the wall, as shown in Fig. 1. Three different the numerical model used, the simulations on condensation meshes were performed in simulation: 406,000, 601,600 propane and ethane/propane mixture upward flow in a and 854,000 elements. 601,600 elements were used in this spiral pipe were carried out to compare with experimental study,

a

decision

grid-independent

made

and

in

consideration

computation

of

efficiency.

the

results in Ref. [33] when m=200～350 kg/(m2·s), P=1.2～

The 3.8MPa, x=0.1~0.9, as shown in Fig. 2. It indicates that the

Reynolds-averaged

Navier-Stokes

equations

were deviations between simulation values and experimental

integrated over each control volume, so that the relevant ones are both within ±15% under different operating quantities (such as energy, momentum and mass, etc.) were parameters. This sufficiently proves the feasibility of the conserved in each control volume. Discrete conservation used model. equations in a form of linear set of equations were obtained 3 Results and Discussion by applying Finite Volume Method based on Finite 3.1 Effect of different parameters on void fraction Element Method. The central deferential scheme was Firstly, it discusses the influence of operating applied to treat the diffusion terms. The convection terms parameters (mass flux, vapor quality, heat flux and were treated applying the high resolution scheme, which saturation pressure) and structural parameters (hydraulic was second-order accurate. The inhomogeneous two-fluid diameter, curvature diameter and inclination angle) on void Page 8 / 47

fraction for different hydrocarbon refrigerants condensation

saturation pressure; with the decrease of saturation pressure,

upward flow in a spiral pipe.

the vapor density decreases rapidly while the liquid density

Fig. 3 and Fig. 4 show the void fraction versus vapor

changes little, resulting in the significant rise in

quality for three hydrocarbon refrigerants under different

liquid-vapor density ratio. In addition, Fig. 4(a) expounds

operating and structural parameters, respectively. The

that with the increasing hydraulic diameter, there is a small

results state that void fraction continuously increases with

increase in vapor fraction owing to the reduction of liquid

vapor quality but the increase rate becomes slow under

contained in the upper film, which was also observed in

different operating and structural parameters. The reason is

Milkie et al.’s study [8]; Fig. 4(b) illustrates that the

that the vapor has occupied most space of the pipe when

curvature diameter affects void fraction to a small extent,

vapor quality becomes large enough, so to further increase

this is because the curvature diameters (D) are always

vapor quality, the void fraction slowly changes. Meanwhile,

much

at the same condition, the void fraction increases when

(d=0.00417-0.00625D), as a result, the curvature effect is

ethane and propane are successively added to methane due

little and can be ignored. From Fig. 4(c), it is found that

to the maximum liquid-vapor density ratio for propane,

when the inclination angle changes from 6 to 14°, the

following by ethane, the minimum for methane, that is to

decrease of vapor fraction is insignificant because that this

say, with the decrease of methane content, the liquid-vapor

change in inclination angle is very small and cannot lead to

density ratio of hydrocarbon refrigerant increases, which

the transformation of flow patterns.

will lead to more tube section occupied by the vapor.

more

Pf,sim (kPa/m)

than the real vapor velocity, resulting in the decrease of

+15% -15%

2 P=1.23.8MPa m=200350kg/(m2s) x=0.10.9

1

vapor-liquid velocity ratio; Fig. 3(b) shows that heat flux has little influence on the void fraction since with the

0 0

1

2

Pf,exp (kPa/m)

increasing heat flux, the local average vapor quality is (a) Frictional pressure drop

unchanged and then the change of vapor-liquid distribution is not evident; Fig. 3(c) explains that as saturation pressure decreases, the void fraction obviously increases due to the fact that the vapor density is usually determined by Page 9 / 47

diameters

C3 C2C3

3

increasing mass flux, the real liquid increases more quickly

hydraulic

4

Besides, Fig. 3(a) indicates that the void fraction slightly increases as the mass flux increases, just because with the

than

3

4

1.0

6 C3 C2C3

5

d=10mm, D=2m, =10 P=3MPa

0.9 0.8

4

-15%

v

hsim(kW/(m2K))

+15%

3

0.7

C1C2C3

m=200kg/(m2s), q=10kW/m2 m=200kg/(m2s), q=20kW/m2 m=600kg/(m2s), q=10kW/m2 m=600kg/(m2s), q=20kW/m2

0.6

2

C1C2

P=1.23.8MPa m=200350kg/(m2s) x=0.10.9

1

0.5 C1

0.4 0.0

0 0

1

2 3 4 hexp(kW/(m2K))

5

0.2

0.4

6

1.0

1.0

Fig.2 Comparison between simulation values and experimental

0.9

results under different operating parameters

0.8 0.7

v

In a word, the void fraction for hydrocarbon

d=10mm, D=2m, =10 q=10kW/m2, m=200kg/(m2s)

0.6

refrigerants condensation upward flow in a spiral pipe is

0.5

mainly influenced by refrigerant components and operating

0.4 C1 C1C2 C1C2C3

0.3

parameters, however, structural parameters have little

0.2 0.0

0.2

P=2MPa P=2MPa P=2MPa

0.4

effect on it. (c)

1.0

P=3MPa P=3MPa P=3MPa

0.6 x(kg/kg)

P=4MPa P=4MPa P=4MPa

0.8

1.0

Under different saturation pressure

Fig.3 Void fraction vs. vapor quality under different operating

0.9

parameters

0.8

m=200kg/(m2s) m=400kg/(m2s) m=600kg/(m2s) m=800kg/(m2s)

C1C2C3

0.6

1.0 0.9

C1C2

0.5 C1

0.4 0.0

0.8

d=10mm, D=2m, =10 q=10kW/m2, P=3MPa

0.2

0.4

0.6 x(kg/kg)

0.8

v

v

0.8

(b) Under different heat fluxes

(b) Heat transfer coefficient

0.7

0.6 x(kg/kg)

0.7

d=6mm d=10mm d=14mm

C1C2C3

1.0 0.6 C1C2

0.5

(a) Under different mass fluxes

D=2m, =10 m=600kg/(m2s), q=10kW/m2, P=3MPa

C1

0.4 0.0

0.2

0.4

0.6 x(kg/kg)

0.8

(a) Under different hydraulic diameters

Page 10 / 47

1.0

v

1.0

and then the vapor velocity is equal to the liquid velocity,

0.9

resulting in the same void fraction to vapor volume fraction.

0.8

It can be written as

0.7

1

C1C2C3

0.6 C1C2

0.5

In order to analyze the capacity of void fraction

d=10mm, =10 m=600kg/(m2s), q=10kW/m2, P=3MPa

C1

0.4 0.0

 v , H  1  1/ x  1 v / l  (37)

D=1.6m D=2.0m D=2.4m

0.2

0.4

0.6 x(kg/kg)

0.8

correlations to predict void fraction for condensation 1.0

hydrocarbon refrigerant upward flow in a spiral pipe, the

(b) Under different curvature diameters

percentages of data points predicted within±5%, ±10%

1.0

and ±15% were given out while the root mean squared

0.9

error (RMSE), the coefficient of determination (R2), the

v

0.8 0.7

=6 =10 =14

C1C2C3

mean absolute relative deviation (MARD) and the mean relative deviation (MRD) were chosen to evaluate the

0.6 C1C2

0.5 0.4 0.0

correlations. They are defined as follows:

D=2m, d=10mm m=600kg/(m2s), q=10kW/m2, P=3MPa

C1

0.2

0.4

0.6 x(kg/kg)

0.8

RMSE 

1.0

2 1 n  v,pre  i  - v,sim  i   (38)  n  1 i 1

(c) Under different inclination angles

n

Fig.4 Void fraction vs. vapor quality under different structural

R2  1 

parameters

   i  - i 1 n

  i 1

3.2 Evaluation of void fraction correlations The void fraction correlations used in this study can

v,sim

v,sim

 i 

 i    v,sim 

2

(39)

2

MARD 

1 n  v,pre  i  - v,sim  i     i  100% (40) n i 1 v,sim

MRD 

1 n   v,pre  i  - v,sim  i       i   100% (41) n i 1  v,sim 

be divided into seven types: homogeneous correlation, slip-ratio correlation, Kαv,H correlation, Lockhart-Martinelli

v,pre

parameter based correlation, drift-flux correlation, general The comparison between predicted void fraction by correlation and implicit correlation [17, 21, 27]. The homogeneous correlation and simulation results is shown description and evaluation of them are discussed as in Tab. 3 and Fig. 5. It can be clearly seen that the follows. homogeneous correlation slightly over-predicts

void

3.2.1 Homogeneous correlation fraction and the deviations are almost all within±5%. In In

the

homogeneous

model,

the

vapor-liquid

two-phase flow is assumed as homogeneous mixture flow;

addition, RMSE, R2 and MARD of homogeneous

Page 11 / 47

correlation are 0.01942, 0.98258 and 2.238%, respectively. 1.0

points within ± 10% error. Then, from Tab. 5, it is illustrated that only the void fraction correlations by

Homogeneous correlation

0.9

+5%

v,pre

0.8

Ahrens [47], Osmachkin-Borisov [42], Wilson et al. [2] -I, -5%

Premoli et al. [43] and Xu-Fang [27] have good

0.7 0.6

performances to predict void fraction for condensation

0.5

hydrocarbon refrigerant upward flow in a spiral pipe; their

0.4

R2 are 0.96477, 0.95110, 0.95722, 0.93779 and 0.92421,

0.3 0.3

0.4

0.5

0.6

v,sim

0.7

0.8

0.9

1.0

respectively, while their MARD are 2.598%, 2.965%,

Fig.5 Void fraction: comparison between homogeneous correlation

3.638%, 3.892% and 4.156%, respectively. Moreover, the

and simulation results

detailed comparison of void fraction between predicted

3.2.2 Slip-ratio correlation

results by existing slip-ratio correlations and simulation

In the separate model, it is thought that the vapor

results is described in Fig. 6. The results indicate that the

velocity is different with liquid velocity in the vapor-liquid

deviations between predicted results based on all slip-ratio

two-phase flow; a slip ratio which defined as the ratio of

correlations and simulation results all decrease with the

vapor velocity to liquid velocity is introduced to calculate

increase of void fraction; when void fraction is more than

the void fraction. It can be given by

0.70, the deviations are all within ±15%. Meanwhile, all 1

 v  1  S 1/ x  1 v / l  (42)

fraction while some correlations with bad performance

where S is the slip ratio and can usually be expressed as

S  A 1/ x  1

a 1

 v / l   l / v  b 1

c

the correlations with good performance predict lower void

(43)

over-predict void fraction and the others are on the contrary.

Substituting Eq. (43) into Eq. (42), the following 3.2.3 Kαv,H correlation equation can be obtained. Since Armand [55] modeled the void fraction as a 1

a b c  v  1  A 1/ x  1  v / l   l / v   (44)  

function of αv,H, many void fraction correlations have been proposed as the following form:

where A, a, b and c are coefficients.

v  Kv, H (45)

In this paper, 24 slip-ratio correlations have been presented in Tab. 4, the evaluation of which is given in Tab.

where K is a coefficient.

5. Herein, the correlation with good performance is defined as the correlation which can predict more than 90% of data

Their descriptions are given in Tab. 6 and their evaluations are shown in Tab. 7.

Page 12 / 47

Schrage et al. [66], Massina [56] and Xiong-Chung [69]

1.0 Ahrens [47] Osmachkin-Borisov [42] Wilson [2]-I Premoli et al. [43] Xu-Fang [27]

0.9

v,pre

0.8

are less than 10%. The specific comparison of void fraction

+10%

between predicted results by these correlations and

-10%

0.7 0.6

simulation results is plotted in Fig. 7. It demonstrates that

0.5

the correlations with good performances predict slightly

0.4

lower for low void fraction while predict almost just right

0.3 0.3

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

for high void fraction. The correlations with poor

1.0

performances predict lower void fraction and their (a) Correlations with good performance

v,pre

deviations decrease with the increase of void fraction; 1.0 +15% Lockhart-Martinelli [34] Spedding-Spence [52] 0.9 Hamersma-Hart [50] El-Boher et al. [51] 0.8 Chen [49] -15% 0.7 0.6 0.5 Hart et al. [53] Smith [41] 0.4 Hibiki et al. [54] Zivi [37]-II 0.3 Thom [39] Chisholm [45] v=0.7 0.2 Zivi [37]-I Rigot [44] 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

when void fraction reaches more than 0.7, which will be within ±15% as void fraction reaches more than 0.7. 3.2.4 Lockhart-Martinelli parameter based correlation In Lockhart-Martinelli parameter based correlations, the void fraction is calculated by a function of Xtt, Ft or Re, as presented in Tab. 8, which gives out eleven

v,sim

Lockhart-Martinelli parameter based correlations. Their

(b) Correlations with poor performance

evaluations were discussed in Tab. 9 and Fig. 8. The results

Fig.6 Void fraction: comparison between existing slip-ratio

demonstrate that MARD of all correlations are less than correlations and simulation results (without considering the

10%, but only the Harms et al.’s correlation [5] has good

correlation whose MARD is more than 10%)

It can be seen from Tab. 7 that the void fraction correlations by Loscher-Reinhardt [61], Moussali [63] and Greskovich-Cooper [62] have good performances; and they can predict more than 90% of data points within±5% error and almost 100% of data points within ± 10% error. Simultaneously, their R2 are 0.99048, 0.99037 and 0.98486, respectively while their MARD are 1.490%, 1.530% and

performances with R2 and MARD of 0.93339 and 4.032%, respectively. Also, it can be seen that the deviations of Harms et al.’s correlation [5] is almost negative with MAD of -2.959%; but other correlations almost have positive deviations at low void fraction while the deviations become within±15% and distribute disorganized as void fraction is more than 0.70.

1.681%, respectively. Besides, except the correlations with good performances, only MARD of the correlations by Page 13 / 47

1.0 0.9

1.0 Loscher-Reinhardt [61] Moussali [63] Greskovich-Cooper [62]

+15%

0.9 +10%

0.8 -15%

-10%

0.7

v,pre

v,pre

0.8

0.6

0.7

0.5

0.5

0.4

0.4

0.3 0.3

0.4

0.5

0.6

0.7

0.8

v,sim

0.9

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

1.0

(b) Correlations with poor performance

(a) Correlations with good performance

Fig.8 Void fraction: comparison between existing

1.0 0.9

v=0.7

0.3 0.3

1.0

Yashar et al. [4] Wilson et al. [76] Tandon et al. [73] Wallis [70] Domanski-Didion [72] Butterworth [71] Ali et al. [74] Wilson et al. [2]-II Graham et al. [75] Kopke et al. [1]

0.6

+15%

Schrage et al. [66] Massina [56] Xiong-Chung [69]

Lockhart-Martinelli parameter based correlations and simulation results (without considering the correlation whose MARD is more

0.8

v,pre

-15%

0.7

than 10%)

0.6

3.2.5 Drift flux correlation

0.5

Zuber and Findlay [79] put forward a drift flux model

0.4 v=0.7

0.3 0.3

0.4

0.5

0.6

to predict void fraction. In their model, the distribution

0.7

0.8

v,sim

0.9

1.0

parameters (C0) and drift velocity (Uvm) were introduced to

(b) Correlations with poor performance

consider the effect of variable density and velocity slip in

Fig.7 Void fraction: comparison between existing K αv,H correlations

the vapor-liquid two-phase flow, respectively. The general

and simulation results (without considering the correlation whose

form of drift flux correlation is shown as follows:

MARD is more than 10%)

v 

1.0 Harms et al. [5]

0.9

+10%

where Uv and Um represent superficial vapor velocity and

0.8

v,pre

Uv U mx ,U v  ,U m  v (46) C0U m  U vm v  v,H

-10%

mixture velocity, respectively.

0.7 0.6

There are 23 drift flux correlations descripted and

0.5

evaluated in Tab. 10 and Tab. 11, respectively. The result 0.4 0.3 0.3

states that the correlations proposed by Milkie et al. [8], 0.4

0.5

0.6

0.7

v,sim

0.8

(a) Correlations with good performance

0.9

1.0

Bestion [92] and Woldesemayat-Ghajar [21] have good performances; their R2 are 0.98197, 0.97129 and 0.93338, respectively, while their MARD are 2.062%, 2.190% and Page 14 / 47

4.256%, respectively. Furthermore, MARD of correlations

1.0 Rouhani-Axelsson [82] Filimonov et al. [77] Dix [24] Qazi et al. [93] Sun et al. [86]

0.9

by Rouhani-Axelsson [82], Filimonov et al. [77], Dix [24],

0.8

v,pre

Qazi et al. [93] and Sun et al. [86] are less than 10% while MARD of other 15 ones are all more than 10%; this may

+15%

-15%

0.7 0.6 0.5

be because that they were carried out based on different

0.4

flow patterns (bubble flow, slug flow) with those in our

0.3 0.3

study (as discussed in our previous study [29, 31], flow

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

1.0

patterns are mainly stratified flow and annular flow). Fig. 9 (b) Correlations with poor performance

shows the comparison of void fraction between predicted

Fig.9 Void fraction: comparison between existing drift flux

results by the above eight drift flux correlations and

correlations and simulation results (without considering the

simulation results. It reveals that all drift flux correlations

correlation whose MARD is more than 10%)

predict lower void fraction and their deviations almost do

3.2.6 Generational correlation

not change with the increase of void fraction, which is

The generational correlation is the one which does not

different from the correlations of other types mentioned

belong to the above five types. Tab. 12 presents ten

above; this is due to the fact that the effects of different

generational correlations and their evaluations were shown

flow patterns (void fraction) had been taken into account in

in Tab. 13. It expounds that the correlations developed by

the development of these correlations, especially those

El Hajal et al. [103], Cioncolini-Thome [6], Jassim et al. [7]

ones with good performances.

and Steiner [101] have good performances with R 2 of 0.99229, 0.98589, 0.93577 and 0.92523 and MARD of

1.0 0.9

Milkie et al. [8] Bestion [92] Woldesemayat-Ghajar [21]

1.313%, 1.792%, 4.305% and 4.654%, respectively. +10%

v,pre

0.8

Simultaneously, the correlations by Huq-Loth [100] and

-10%

0.7

Minami-Brill [99] also have MARD less than 10%. The

0.6 0.5

distribution of the deviations for the above six generational

0.4

correlations are drawn in Fig. 10. It explicates that the

0.3 0.3

0.4

0.5

0.6

0.7

v,sim

0.8

(a) Correlations with good performance

0.9

1.0

deviations all decrease with the increasing void fraction while most of them are negative except at low void fraction.

Page 15 / 47

Bhagwat-Ghajar’s correlation [19] almost do not change

1.0 El hajal et al. [103] Cioncolin-Thome [6] Jassim et al. [7] Steiner [101]

0.9

v,pre

0.8

while those of Ozaki et al.’s correlation [114] decrease.

+10%

Moreover, MARD of correlations by Levy [105] and

-10%

0.7 0.6

Hughmark [106] are less than 10% and their deviations

0.5

decrease as the void fraction increases; when void fraction

0.4

is more than 0.7, their deviations are within±15%.

0.3 0.3

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

1.0

1.0

1.0 v,pre

0.8

-10%

0.7 0.6

-15%

v,pre

+10%

0.8

+15%

Huq-Loth [100] Minami-Brill [99]

0.9

Bhagwat-Ghajar [19] Ozaki et al. [114]

0.9

(a) Correlations with good performance

0.7

0.5

0.6

0.4

0.5

0.3 0.3

0.4 0.3 0.3

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

1.0

0.4

0.5

0.6

0.7

0.8

v,sim

0.9

1.0

(a) Correlations with good performance

1.0

(b) Correlations with poor performance

0.9 Fig.10 Void fraction: comparison between existing generational

+15%

Levy [105] Hughmark [106]

0.8

v,pre

correlations and simulation results (without considering the correlation whose MARD is more than 10%)

-15%

0.7 0.6 0.5

3.2.7 Implicit correlation

0.4

The above correlations which were discussed in this

v=0.7

0.3

study are all explicit correlations. However, some implicit

0.3

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

1.0

correlations also have been proposed until now, as shown (b) Correlations with poor performance

in Tab. 14. Their evaluations were given in Tab. 15 and Fig.

Fig.11 Void fraction: comparison between existing implicit

11. It explains that the correlations by Bhagwat-Ghajar [19]

correlations and simulation results (without considering the

and Ozaki et al. [114] have good performances, R2 of

correlation whose MARD is more than 10%)

which are 0.97782 and 0.88827, respectively, while MARD

3.2.8 The correlations with good performances

of which are 2.119% and 5.564%, respectively. Meanwhile,

The correlations with good performances which were

they both predict a little lower void fraction; with the

mentioned above are summarized in Tab. 16. It indicates

increase

that

of

void

fraction,

the

deviations

of

Page 16 / 47

the

correlations

by

El

hajal

et

al.

[103],

Loscher-Reinhardt [61], Moussali [63], Greskovich-Cooper

structural

parameters

(hydraulic

diameter,

curvature

[62] and Cioncolini-Thome [6] are the best five void

diameter and inclination angle) have little effect on it.

fraction correlations whose MARD is less than 2%. At the

(2) Through the evaluation and analysis of existing

same time, they all can predict void fraction for

correlations, 19 void fraction correlations with good

condensation hydrocarbon refrigerant upward flow in a

performance are identified, which can predict more than 90%

spiral pipe just right within±5% error, as shown in Fig. 12.

of data points within ±10% error, the best five ones of which in ranking order are proposed by El hajal et al. [103],

1.0 0.9 0.8

v,pre

Loscher-Reinhardt [61], Moussali [63], Greskovich-Cooper

El hajal et al. [103] Loscher-Reinhardt [61] Moussali [63] Greskovich-Cooper [62] Cioncolin-Thome [6]

+5%

[62] and Cioncolini-Thome [6], whose MARD are less

-5%

0.7

than 2% and the deviations are all almost within ±5%.

0.6

Acknowledgment

0.5 0.4

The authors are grateful for the support of the research

0.3 0.3

0.4

0.5

0.6

0.7

v,sim

0.8

0.9

funds from 863 program on National high technology

1.0

research and development program (2013AA09A216), the

Fig.12 Void fraction: comparison between the best five correlations

program of the Ministry of Industry and Information

and simulation results

Technology in China ([2013]418) and Key Laboratory of 4 Conclusions Efficient Utilization of Low and Medium Grade Energy In this paper, a numerical model was established to (Tianjin University), Ministry of Education of China investigate the condensation void fraction characteristics (201604-501). for hydrocarbon refrigerant upward flow in a spiral pipe. The influences of refrigerant compositions, operating parameters and structural parameters on void fraction were

Nomenclature Alv= interfacial area density between liquid phase and -1

vapor phase, m discussed.

Meanwhile,

96

existing

void

fraction

correlations were evaluated, the best five ones of which were elected to predict void fraction of condensation hydrocarbon refrigerant upward flow in a spiral pipe. Some

C0 = distribution parameters CD = drag coefficient, 0.44 C , C 1 , C 2 = constant with the values of 0.09,1.44 and

1.92, respectively

main conclusions are drawn as follows: (1) The void fraction increases with the increase of vapor quality and mass flux but decreases with the

Cp = specific heat at constant pressure, J/(kg·K) d = hydraulic diameter, m

increasing saturation pressure; the heat flux and all Page 17 / 47





d* = non-dimensional diameter, d  /  g  l  v  

0.5

hlv = heat transfer coefficient between vapor phase side and 2

liquid phase side, W/(m ·K) dlv = mean interfacial length scale between the liquid phase

2

2

k = turbulent kinetic energy, m /s and vapor phase, m

K = the ratio between real and homogeneous void fraction D = curvature diameter, m L = length of test section, m f = friction factor m = mass flux, kg/(m2·s)

f ,lv = surface force acting on vapor phase due to the

mlv = mass flow rate in per unit interfacial area from vapor

2

presence of the liquid phase per unit area, N/m

phase to liquid phase, kg/(m2·s)

Flv = interfacial forces acting on liquid phase due to the

MARD = mean absolute relative deviation, Eq. (40)

presence of the vapor phase, N/m3

MRD = mean relative deviation, Eq. (41)

FD,lv = drag force acting on vapor phase due to the presence

nlv = interface normal vector pointing from liquid phase to

3

of the liquid phase per unit volume, N/m

the vapor phase

F ,lv = surface force acting on vapor phase due to the

Nulv = Nusselt number between vapor phase side and liquid

presence of the liquid phase per unit volume, N/m3

phase side, hlv dlv / lv

Frl = liquid Froude number, m2 (1  x)2 / ( l2 gd )

p =pressure, Pa

Frl0 = full-liquid Froude number, m2 / ( l2 gd )

P = saturation pressure, MPa Pcr = cirtical pressure, MPa

Frm= mixture Froude number, m x / (   v, H gd )

Pk = turbulence production, Pa /s

Frso = Soliman's modified Froude number, in Ref. [10]

Pkbi = buoyancy turbulence production, Pa /s

Frv0 = full-vapor Froude number, m2 / ( v2 gd )

Prl = liquid Prandtl number, l Cpl / l

2 2





Ft = Froude rate, x m /  v gd (1  x)  3

2

2

2 v

2

2

0.5

q= heat flux, W/m

qv, ql = sensible interphase heat transfer to the vapor phase g = gravity acceleration, m/s

2

across the interface with the liquid phase and to the liquid Gal = liquid Galileo number,  gd 2 l

3



2 l

phase across the interface with the vapor phase per unit

h= heat transfer coefficient, W/(m2·K)

volume, respectively, W/m

hv, hl = heat transfer coefficient of liquid phase and vapor

Qv, Ql = total interphase heat transfer to the vapor phase

phase on one side of the phase interface, respectively,

across the interface with the liquid phase and to the liquid

W/(m2·K)

phase across the interface with the vapor phase per unit

3

Page 18 / 47

 = thermal conductivity, W/(m·K)

volume, respectively, W/m3 R = coefficient of determination, Eq. (39)

 = enthalpy, J/kg

Rel = liquid Reynolds number, m(1  x)d / l

 lv = latent heat of phase change, J/kg

2

Rel0 = full-liquid Reynolds number, md / l

 vS ,  lS = interfacial values

Rem = mixture Reynolds number, lU m d / l

of enthalpy carried into vapor

phase and liquid phase due to phase change, respectively,

RMSE= root mean squared error, Eq. (38)

J/kg

S = slip ratio

 = surface tension coefficient, N/ m

T = temperature, K

k , 

= constant with the values of 1.3 and 1.0,

u = velocity, m/s respectively Ul = superficial liquid velocity, m/s

 lv = interface delta function, m-1

Um = mixture velocity, m/s Uvm =drift velocity, m/s

Pf = Frictional pressure drop, Pa/m

Uv = superficial vapor velocity, m/s

 lv = surface curvature, m-1

Wel = liquid Weber number, m2 (1  x)2 d / (l )



Wel0 = full-liquid Weber number, m2 d / (l )

 = condensation heat ratio

Wev0 = full-vapor Weber number, m2 d / (v )

 lv = mass flow rate in per unit volume from vapor phase

x = vapor quality Xtt

x

3

to liquid phase, kg/(m ·s)

= 1

 1

0.9

Lockhart-Martinelli

parameter,

      v

= turbulent dissipation rate, m2/s3

1 0.5 l

l

1 0.1 v

 lv = positive mass flow rate in per unit volume from vapor phase, kg/(m3·s)

Greek Symbols

Subscripts

 = volume fraction or void fraction

A = annular

 = inclination angle

ann = annular

 = density, kg/m3

cr = cirtical

 = dynamic viscosity, Pa·s

exp =experimental value

l* = liquid viscous number,

t = turbulent viscosity, Pa·s

H = homogeneous

l  1.5 l /  g  l  v  

0.5

i = any int = intermittent

Page 19 / 47

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l = liquid phase

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ref = reference

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sat = saturation

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sim = simulated value

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Page 25 / 47

Tab.1 Components of the refrigerants in this study Tab.2 Conditions of simulation in this study Tab.3 Comparison between predicted void fraction by homogeneous correlation and simulation results Tab.4 Slip-ratio correlations Tab.5 Comparison between predicted void fractions by existing slip-ratio correlations and simulation results Tab.6 K αv,H correlations Tab.7 Comparison between predicted void fractions by existing K αv,H correlations and simulation results Tab.8 Lockhart-Martinelli parameter based correlations Tab.9 Comparison between predicted void fractions by existing Lockhart-Martinelli parameter based correlations and simulation results Tab.10 Drift flux correlations Tab.11 Comparison between predicted void fractions by existing drift flux correlations and simulation results Tab.12 Generational correlations Tab.13 Comparison between predicted void fractions by existing generational correlations and simulation results Tab.14 Implicit correlations Tab.15 Comparison between predicted void fractions by existing implicit correlations and simulation results Tab.16 Comparison between predicted void fractions by existing correlations with good performance and simulation results

Page 26 / 47

Tab.1 Components of the refrigerants in this study

Logogram

Component of the refrigerant (Molar ratio) Methane

Ethane

Propane

Nitrogen

C3

-

-

100%

-

C2C3

-

50%

50%

-

C1

100%

-

-

-

C1C2

90%

10%

-

-

C1C2C3

65%

25%

10%

-

C1C2C3N2

65%

25%

5%

5%

Page 27 / 47

Tab.2 Conditions of simulation in this study

Refrigerant

m

q

P

x

D

d



Data points

kg/(m ·s)

kW/m

MPa

kg/kg

m

mm



C3

150～350

5～20

1.2～2.0

0.15～0.95

2

14

10

63

C2C3

200～300

5～20

2.0～3.8

0.10～0.90

2

14

10

56

C1

200～800

10～20

2.0～4.0

0.15～0.90

1.6～2.4

6～14

6～14

84

2

2

C1C2

200～800

10～20

2.0～4.0

0.15～0.90

1.6～2.4

6～14

6～14

84

C1C2C3

200～800

10～20

2.0～4.0

0.15～0.90

1.6～2.4

6～14

6～14

84

C1C2C3N2

200～800

10～20

2.0～4.0

0.15～0.90

1.6～2.4

6～14

6～14

84

Page 28 / 47

Tab.3 Comparison between predicted void fraction by homogeneous correlation and simulation results

Correlations Homogeneous

Percentage of data points predicted within ±5%

±10%

±15%

97.582%

100.000%

100.000%

Page 29 / 47

RMSE

R2

0.01942

0.98258

MARD

MRD

(%)

(%)

2.238

2.238

Tab.4 Slip-ratio correlations No. 1

Author/source Lockhart-Martinelli [34]

2

Fauske [35]

3

Baroczy [36]

4

Zivi [37] -I

Void fraction correlation

A  0.28, a  0.64, b  0.36, c  0.07 A  1, a  1, b  0.5, c  0 A  1, a  0.74, b  0.65, c  0.13 A  1, a  1, b  2 / 3, c  0 1

Zivi [37] -II 5 Petalas-Aziz [38]

6

Thom [39]

7

Turner-Wallis [40]

1  E  v / l 1/ x  1  3 2 A  , a  1, b  , c  0 1  E 1/ x  1 3   0.074 0.2 E  1  1/ 0.735  v l 2 l 2 m 2 x 2   v,1H  1   

1

A  1, a  1, b  0.89, c  0.18 A  1, a  0.72, b  0.4, c  0.08

  /   0.4 1/ x  1  A  0.4  0.6  l v  , a  1, b  1, c  0  1  0.4 1/ x  1  A  1   0.6  1.5 v2, H  Frl 00.25 1  P / Pcr  0.5

8

Smith [41]

9

Osmachkin-Borisov [42]

10

Premoli et al. [43]

Frl 0  m2 /  l2 gd  , a  1, b  1, c  0

A  1  F1 y / 1  yF2   yF2

0.5

F1  1.578 Rel00.19  l / v 

, Rel 0  md / l

0.22

, a  1, b  1, c  0

F2  0.0273Wel 0 Rel00.51  l / v 

0.08

1

y  1/ x  1 v / l   ,Wel 0  m 2 d / l 

11

Rigot [44]

A  2, a  1, b  1, c  0

12

Chisholm [45]

A  1  x   l / v  x 

13

Madsen [46]

A  1, a 

0.5

, a  1, b  1, c  0

0.5log  l / v   log 1/  0  1 ， 0  0.302 log  l / v   log 1/  0  1

b  0.5, c  0

A  1  0.10902  B 0.972458  1  0.208073  B 0.48578  1

14

Ahrens [47]

15

Spedding-Chen [48]

16

Chen [49]

17

Hamersma-Hart [50]

0.2

    B   l   v  , a  1, b  1, c  0  v   l  A  2.22, a  0.65, b  0.65, c  0 A  0.286, a  0.6, b  0.33, c  0.07 A  0.26, a  0.67, b  0.33, c  0

A  0.27 Frl 18

El-Boher et al. [51]

0.177

 Rel     Wel 

0.067

, a  0.69, b  0.69, c  0.378

m 2 1  x  m 2 1  x  d m 1  x  d Frl  , We  , Rel  l 2 l gd l l 2

Page 30 / 47

2

Tab.4 (continued) No.

Author/source

Void fraction correlation





A  0.45  0.08exp  min(0, 25  100U l2 

1

19

Spedding-Spence [52]

20

Hart et al. [53]

21

Wilson et al. [2] -I

22

Winkler et al. [18] -I

   x  A  1  1.105 Rel00.19  l    , a  1, b  1, c  0  v   1  x  A  1.583, a  0.349, b  0.638, c  0.072

23

Xu-Fang [27]

A  1  2Frl00.2v3.5, H , a  1, b  1, c  0

U l  m 1  x  / l , a  0.65, b  0.65, c  0

A  1  108  l /  g  Rel0.726  , a  1, b  1, c  0 0.5

0.72

0.5

  /   e 1/ x  1  A  e  1  e   l v  , a  1, b  1, c  0 1  e 1/ x  1     0.292 m2 d e  min 0.209 Wev0.50 l*0.2 d *0.5  ,1 , Wev 0     0.5

24

Hibiki et al. [54]

v

d* 

d

 /  g  l  v  

Page 31 / 47

, l* 

l  1.5 l /  g  l  v  

0.5

Tab.5 Comparison between predicted void fractions by existing slip-ratio correlations and simulation results Percentage of data points Correlations

predicted within

RMSE

R2

MARD

MRD

(%)

(%)

±5%

±10%

±15%

Ahrens [47]

79.78%

91.43%

100.00%

0.02762

0.96477

2.784

-2.598

Osmachkin-Borisov [42]

78.24%

94.51%

99.12%

0.03254

0.95110

3.066

-2.965

Wilson et al. [2] -I

79.78%

100.00%

100.00%

0.03043

0.95722

3.638

-3.638

Premoli et al. [43]

67.03%

93.63%

100.00%

0.03670

0.93779

3.979

-3.892

Xu-Fang [27]

62.86%

94.95%

100.00%

0.04051

0.92421

4.199

-4.156

Hart et al. [53]

60.66%

82.20%

91.65%

0.04922

0.88811

5.453

-5.453

Lockhart-Martinelli [34]

73.63%

82.42%

86.37%

0.06270

0.81841

6.319

4.372

Smith [41]

46.59%

75.16%

96.48%

0.05550

0.85775

6.322

-6.322

Hibiki et al. [54]

47.25%

73.63%

93.19%

0.05829

0.84305

6.475

-6.475

Chen [49]

74.95%

83.30%

88.57%

0.06253

0.81943

6.499

3.477

Zivi [37] –II

49.45%

72.53%

88.57%

0.05743

0.84766

6.501

-6.501

Spedding-Spence [52]

66.15%

78.46%

84.40%

0.06551

0.80181

6.538

6.267

Hamersma-Hart [50]

69.23%

77.58%

83.74%

0.07093

0.76763

7.012

6.180

Thom [39]

52.31%

69.45%

83.30%

0.06527

0.80326

7.116

-7.113

Chisholm [45]

32.97%

70.33%

99.12%

0.06499

0.80492

7.554

-7.554

El-Boher et al. [51]

54.95%

77.58%

84.18%

0.08751

0.64633

9.107

5.015

Zivi [37] -I

42.42%

64.84%

73.85%

0.08597

0.65866

9.583

-9.583

Rigot [44]

44.62%

61.76%

75.38%

0.08582

0.65982

9.656

-9.653

Baroczy [36]

15.60%

38.46%

70.99%

0.09943

0.54334

11.790

-11.790

Winkler et al. [18]-I

7.91%

30.33%

72.09%

0.11184

0.42234

11.994

-10.065

Madsen [46]

2.42%

31.87%

66.81%

0.10428

0.49771

12.309

-12.307

Fauske [35]

23.74%

41.54%

51.65%

0.14524

0.02567

16.630

-16.630

Turner-Wallis [40]

1.54%

16.70%

32.97%

0.18082

-0.51012

21.631

-21.631

Spedding-Chen [48]

1.10%

11.87%

24.40%

0.19600

-0.77438

23.890

-23.890

Page 32 / 47

Tab.6 K αv,H correlations No. 1

Author/source Armand [55]

Void fraction correlation

2

Massina [56]

3

Bankoff [57]

K  0.833 K  0.833  0.167 x K  0.71  0.0145P

4

Nishino-Yamazaki [58]

K  1  1  v,H / v,H

5

Kowalczewski [59]

K  1  0.7 v,1H 1   v , H 

6

Guzhov et al. [60]

K  0.81 1  exp  2.2 Frm0.5  , Frm  m2 x 2 / v2 v, H gd

7

Loscher-Reinhardt [61]

 P  K  1    Pcr 

8

Greskovich-Cooper [62]

K  1  0.671 sin   





0.22



0.39 v,H

0.5

Frl 00.045 1  P / Pcr 



1   

0.8

v,H

0.263

Frm0.5  

0.25 l0

Fr

 P  1   Pcr  

1

1  x  v   30.4 y   11 K  1 ,y  60 1  1.6 y 1  3.2 y   x l  0.5 2 K  1   v,1H 1   v , H  Frl 0.2 0 1  P / Pcr 

9

Moussali [63]

10

Kutucuglu [64]

11

Chisholm [65]

12

Schrage et al. [66]

K  1  0.123Frl 00.0955 ln  x 

13

Czop et al. [67]

K  1.097  0.285 / v, H

14

Zhao-Bi [68]

K  0.838

15

Xiong-Chung [69]

K

0.5 K   v , H  1   v , H    

C v,0.5 H

1  1  C  

0.5 v, H

Page 33 / 47

,C 

2

1

1

0.266 1  13.8exp  6880d 

3.4



Tab.7 Comparison between predicted void fractions by existing K αv,H correlations and simulation results Percentage of data points Correlations

predicted within

RMSE

R2

MARD

MRD

(%)

(%)

±5%

±10%

±15%

Loscher-Reinhardt [61]

98.02%

100.00%

100.00%

0.01436

0.99048

1.490

0.859

Moussali [63]

96.92%

100.00%

100.00%

0.01444

0.99037

1.530

-1.373

Greskovich-Cooper [62]

92.97%

99.34%

100.00%

0.01811

0.98486

1.681

-1.120

Schrage et al. [66]

58.02%

79.78%

89.89%

0.05326

0.86899

5.802

-5.801

Massina [56]

46.81%

84.40%

100.00%

0.05000

0.88453

5.909

-5.909

Xiong-Chung [69]

51.43%

70.11%

84.18%

0.06144

0.82567

6.893

-6.885

Kowalczewski [59]

30.33%

60.66%

74.29%

0.09957

0.54209

11.573

-11.573

Chisholm [65]

1.10%

14.29%

51.43%

0.11570

0.38167

13.863

-13.863

Zhao-Bi [68]

0.00%

0.00%

73.63%

0.12348

0.29578

14.324

-14.324

Armand [55]

0.00%

0.00%

50.55%

0.12780

0.24565

14.836

-14.836

Kutucuglu [64]

21.98%

47.03%

62.86%

0.14717

-0.00029

16.002

-15.990

Guzhov et al. [60]

0.00%

0.00%

2.42%

0.14769

-0.00745

17.191

-17.191

Nishino-Yamazaki [58]

1.10%

12.09%

27.69%

0.17871

-0.47511

22.143

-22.143

Bankoff [57]

0.00%

0.00%

0.00%

0.20006

-0.84856

23.283

-23.283

Czop et al. [67]

0.00%

0.00%

0.00%

0.18536

-0.58684

23.409

-23.409

Page 34 / 47

Tab.8 Lockhart-Martinelli parameter based correlations No.

Author/source

1

Wallis [70]

 v  1  X tt0.8 

2

Butterworth [71]

 v  1  0.28 X tt0.71 

Domanski-Didion [72]

0.8 0.378  , X tt  10 1  X tt  v    0.823  0.157 ln  X tt  , X tt  10

3

4

Tandon et al. [73]

Void fraction correlation 0.378

, X tt   x 1  1

0.9

      v

1 0.5 l

1 0.1 v

l

1

 Rel00.315 Rel00.63 1  1.928  0.9293 , Rel 0  1125  F ( X tt ) F 2 ( X tt )  v   0.088 0.176 1  0.38 Rel 0  0.0361 Rel 0 , Re  1125 l0  F ( X tt ) F 2 ( X tt ) F ( X tt )  0.15  X tt1  2.85 X tt0.476 

5

Ali et al. [74]

6

Graham et al. [75]

7

Wilson et al. [2] -II

 v  1  1  20 X tt1  X tt2 

1/2





0, Ft  0.01032, Ft  x3m 2 /   2 gd (1  x)  0.5   v  v   2 1  e 10.3ln( Ft )0.0328ln ( Ft ) , Ft  0.01032  v  1.005  0.0229ln( Ft / m)  0.0062ln 2 ( Ft / m)

8

Kopke et al. [1]

 v , H , Ft  0.044  3 v   i  ai ln( Ft ) 1.045  e i0 , 0.044  Ft  454  a0  1, a1  0.342, a2  0.0268, a3  0.00597

9

Yashar et al. [4]

 v  1  1/ Ft  X tt 

Wilson et al. [76]

1  1.84 Ft 1  3.11X 0.21 , X  1/ Ft  2 tt tt  v    0.35 1  0.5Ft 1  1.2 X tt  , X tt  1/ Ft  2 

Harms et al. [5]

 10.06 Re 0.875 1.74  0.104 Re0.5 2  l l   v  1  0.5  1.655   1.376  7.242 X   tt  

10

11

Page 35 / 47

0.321

2

Tab.9 Comparison between predicted void fractions by existing Lockhart-Martinelli parameter based correlations and simulation results Percentage of data points Correlations

predicted within

RMSE

±5%

±10%

±15%

R2

MARD

MRD

(%)

(%)

Harms et al. [5]

71.43%

93.85%

98.90%

0.03797

0.93339

4.032

-2.959

Yashar et al. [4]

74.73%

85.93%

89.23%

0.05438

0.86344

5.154

4.242

Wilson et al. [76]

61.10%

90.99%

92.75%

0.05116

0.87914

5.709

-0.355

Tandon et al. [73]

72.75%

84.84%

89.23%

0.05811

0.84402

5.878

3.371

Wallis [70]

76.26%

83.74%

90.11%

0.06112

0.82747

6.216

3.406

Domanski-Didion [72]

76.26%

83.74%

90.11%

0.06112

0.82747

6.216

3.406

Butterworth [71]

73.85%

82.86%

87.47%

0.06198

0.82259

6.266

4.189

Ali et al. [74]

73.63%

81.32%

84.84%

0.06870

0.78204

7.081

4.408

Wilson et al. [2] -II

69.89%

81.98%

86.81%

0.06845

0.78359

7.246

3.320

Graham et al. [75]

49.89%

79.12%

84.84%

0.07942

0.70871

8.351

3.884

Kopke et al. [1]

59.78%

72.31%

79.34%

0.09879

0.54925

9.994

8.564

Page 36 / 47

Tab.10 Drift flux correlations No.

Author/source

Void fraction correlation

1

Filimonov et al. [77]

C0  1,U vm   0.65  0.0385P  d / 0.063

2

Nicklin et al. [78]

C0  1.2,U vm  0.35 gd

3

Zuber-Findlay [79]

C0  1.2,U vm  1.53U1 ,U1   g  l  v  / l2 

4

Hughmark [80]

5

Gregory-Scott [81]

C0  1.2,U vm  0 C0  1.19,U vm  0

6

Rouhani-Axelsson [82]

C0  1  0.2(1  x) Frl00.25 ,U vm  1.18U1

7

Bonnecaze et al. [83]

C0  1.2,U vm  0.35 gd 1  v / l 

8

Dix [24]

C0   v, H   v , H 1/  v , H  1

9

Mattar-Gregory [84]

C0  1.3,U vm  0.7

10

Ishii [85]

C0  1.2  0.2

11

Sun et al. [86]

C0   0.82  0.18P / Pcr  ,U vm  1.41U1

12

Jowitt [87]

C0  1  0.796 e

13

Anklam-Miller [88]

C0   0.82  0.18P / Pcr  ,U vm  1.53U1

14

Kataoka-Ishii [89]

  v / l  0.157 U vm  0.0019d *0.809  v / l  l*0.562U1

15

Morooka et al. [90]

16

Kokal-Stanislav [91]

C0  1.08,U vm  0.45 C0  1.2,U vm  0.345U 2

17

Bestion [92]

C0  1,U vm  0.188 gd  l / v  1

18

Qazi et al. [93]

C0   0.82  0.18P / Pcr  ,U vm  0.35 gd

19

Mishima-Hibiki [94]

C0  1.2  0.510e691d ,Uvm  0

 v / l 0.1

0.25

0.25

,U vm  2.9U1

v , U  0.35U 2 , U 2  l vm

gd  l  v 

l

1

 0.061

l v1

 ,U

vm

 0.034





l v1  1

1

C0  1.2  0.2

1

C0  0.00257 P  1.0062

20

Maier-Coddington [95]

U vm  m  aP 2  bP  c    0.00563P 2  0.123P  0.8  a  6.73 107 , b  8.81105 , c  0.00105

21

Woldesemayat-Ghajar [21]

22

Winkler et al. [18] -II

C0   v , H   v , H 1/  v , H  1

 v / l 0.1

F  d ,     d 1  cos   

1.22  1.22sin  

C0  1.197,U vm  0.064

Page 37 / 47

0.25

,U vm  2.9U1 F  d ,   0.101325/ P 

Tab.10 (continued) No.

Author/source

Void fraction correlation

C0  1, Frso  10, U vm  U vm , w ; Frso  20,U vm  U vm , A ; 10  Frso  20, U vm 

 20  Frso U vm,w   Frso  10 U vm, A 10

  1  1.09 X tt0.039  0.5 0.025 Re1.59   Gal , Rel  1250 l X tt    Frso   1.5 0.039   1.04  1  1.09 X tt 0.5  Gal , Rel  1250 1.26 Rel  X tt    1.5

23

Milkie et al. [8]

Gal  l2 gd 3 l2 ,U vm , w  1.47 U1U m1  U vm , A  49.11  x 

0.11

Page 38 / 47

U1U m1 

2.028

Um

0.96

Um

Tab.11 Comparison between predicted void fractions by existing drift flux correlations and simulation results Percentage of data points Correlations

predicted within

RMSE

R2

MARD

MRD

(%)

(%)

±5%

±10%

±15%

Milkie et al. [8]

91.65%

98.90%

100.00%

0.01976

0.98197

2.062

-0.637

Bestion [92]

89.45%

98.02%

99.78%

0.02493

0.97129

2.190

-1.858

Woldesemayat-Ghajar [21]

68.57%

98.46%

100.00%

0.03798

0.93338

4.256

-4.121

Rouhani-Axelsson [82]

57.14%

86.59%

96.70%

0.04931

0.88768

5.389

-5.389

Filimonov et al. [77]

57.80%

83.08%

93.19%

0.05458

0.86241

5.587

-5.587

Dix [24]

31.43%

74.73%

93.85%

0.06773

0.78812

7.668

-7.656

Qazi et al. [93]

4.84%

63.96%

98.90%

0.08282

0.68317

9.361

-9.361

Sun et al. [86]

5.27%

54.73%

97.14%

0.08644

0.65492

9.790

-9.790

Anklam-Miller [88]

4.40%

50.77%

96.70%

0.08815

0.64109

10.015

-10.015

Kataoka-Ishii [89]

0.44%

24.84%

99.78%

0.09540

0.57964

10.965

-10.965

Ishii [85]

0.00%

15.38%

96.04%

0.10071

0.53156

11.701

-11.701

Winkler et al. [18] -II

0.22%

4.62%

85.27%

0.11791

0.35790

13.446

-13.446

Morooka et al. [90]

0.00%

26.59%

68.79%

0.11481

0.39115

13.709

-13.709

Gregory-Scott [81]

0.00%

0.00%

81.54%

0.12146

0.31862

14.086

-14.086

Hughmark [80]

0.00%

0.00%

51.43%

0.12751

0.24904

14.801

-14.801

Mishima-Hibiki [94]

0.00%

0.00%

49.45%

0.12796

0.24376

14.855

-14.855

Bonnecaze et al. [83]

0.00%

0.00%

5.71%

0.14013

0.09307

16.516

-16.516

Kokal-Stanislav [91]

0.00%

0.00%

4.62%

0.14079

0.08449

16.609

-16.609

Nicklin et al. [78]

0.00%

0.00%

3.52%

0.14191

0.06991

16.763

-16.763

Zuber-Findlay [79]

0.00%

0.00%

1.76%

0.14626

0.01201

17.327

-17.327

Maier-Coddington [95]

0.00%

2.86%

25.93%

0.19879

-0.82509

23.778

-23.778

Mattar-Gregory [84]

0.00%

0.00%

0.00%

0.24983

-1.88272

30.028

-30.028

Jowitt [87]

0.00%

0.00%

0.00%

0.33107

-4.06226

39.026

-39.026

Page 39 / 47

Tab.12 Generational correlations No.

Author/source

Void fraction correlation

1

Flanigan [96]

 v  1  3.063  mx / v 

Yamazaki-Yamaguchi [97]

3

Beattie-Sugawara [98]

 



v  2

1.006 1

1  A  y 1 

1  A  y 

1 2

 4A

2A

1  x  g  ,y   x l 

1

6  gd  l  v  l2 1.0, B  2 10 A , B  6  l 2  0.57, B  2 10  v , H / C0   v v , H / mx U vm  ,  v , H   v , H ,TR     2/  1 8 C0 1  v    1   v,H  1  0.20  ,  v , H   v , H ,TR   1  v , H ,TR    0.8(C0  lU vm / m)  v , H ,TR  , U vm  0.35U 2 1  0.8( l  v )U vm / m

C0  1  2.6 0.0716 Rel00.237  0.0008

v  e 4

Minami-Brill [99]

  ln Z1  9.21 /8.7115

4.3374

1.84(1  x)0.575 v g 0.0924 0.0451l0.1  P  Z1    0.425 0.7201 0.0277 m l xd  0.101325 

2 1  x 

0.05

2

Huq-Loth [100]

v  1 

6

Steiner [101]

 x 1 x  U1  x   v  1.12  0.12 x      1.18 1  x   v  l  m   v

7

Gomez et al. [102]

8

El hajal et al. [103]

5

1  2 x  1  4 x 1  x  l / v  1 

0.45  /180  2.4810  v  1  e

6

Rem

0.5

 , Re   U d /  m l m l

 v   v, H   v,Steiner  / ln  v, H /  v,Steiner 

 v  Fint liq v ,Graham  Fstrat v ,Yashar  Fann v ,Steiner i s/ x    1 x    1  Fint liq  Fstrat  , i  0.0243 Xi  8.07

Fint liq  1  x  , Fstrat  1  x  i

9

Jassim et al. [7]

Fann

i

 v  1 1 s  , Xs  Frv0.5  0  4.44 0.45 Xs 0.025 Xs  l 

0.65

Xi  Wev0.40  l / v  , Frv 0  m 2 /   v2 gd 

10

Cioncolini-Thome [6]

  hx n v  , h  2.129  3.129  v  n 1   h  1 x  l  n  0.3487  0.6513  v / l 

Page 40 / 47

0.515

0.2186

1

Tab.13 Comparison between predicted void fractions by existing generational correlations and simulation results Percentage of data points Correlations

predicted within

RMSE

±5%

±10%

±15%

R2

MARD

MRD

(%)

(%)

El hajal et al. [103]

96.70%

100.00%

100.00%

0.01292

0.99229

1.313

-1.263

Cioncolini-Thome [6]

93.85%

97.80%

99.56%

0.01748

0.98589

1.792

-0.279

Jassim et al. [7]

63.74%

96.92%

99.12%

0.03729

0.93577

4.305

-3.658

Steiner [101]

62.42%

89.23%

100.00%

0.04024

0.92523

4.654

-4.654

Huq-Loth [100]

52.53%

80.66%

94.73%

0.05051

0.88218

5.411

-5.321

Minami-Brill [99]

15.16%

66.81%

86.59%

0.08320

0.68029

9.316

-8.781

Beattie-Sugawara [98]

5.71%

12.09%

36.26%

0.12809

0.24221

14.956

-14.956

Yamazaki-Yamaguchi [97]

1.10%

12.75%

34.95%

0.13729

0.12943

16.682

-16.682

Gomez et al. [102]

9.01%

18.68%

32.53%

0.25371

-1.97308

26.191

-24.364

Flanigan [96]

0.00%

0.22%

6.37%

0.27795

-2.56833

31.415

-31.415

Page 41 / 47

Tab.14 Implicit correlations No. 1

2

Author/source

Void fraction correlation

Hoogendoorn [104]

 mx  v  1  x v    0.6  1  v  1  v   v  1   v x l  

Levy [105]

x   v 1  2 v    v 

1  2 v 

2

0.85

 A v  / A 

A  2  l / v 1   v    v 1  2 v  2

3

Hughmark [106]

  0.16376  a1Z  a2 Z 2  a3 Z 3   v , H , Z  10   v   2  b  0.003585Z  0.00001436 Z   v , H , Z  10 a1  0.31037, a2  0.03525, a3  0.001366, b  0.75545 1/6

  md Z     v v  (1   v ) l 

1/8

  m2 x 2  2 2 2  gd v  v , H (1   v , H ) 

0.5   v0.25  0.068948   1     0.5  P    1   v   1.2  0.24  0.35 v2, H gd v U m1

4

Fujie [107]

 v x  v 1  x l 1   v

5

Shipley [108]

 v , H  v1

6

Liao et al. [109]

 v , H  v1  1.2  0.2 1  e18

7

Zhao et al. [110]

1   v   v 1/ x  1 v / l  l / v  

8

Gomez et al. [111]

9

Hibiki-Ishii [112]





v

  

1 0.5 l

v

 0.33U1U m1 0.875

 v, H v1  1.15  1.53U1U m1 sin  1  v  1   v  1  8.165U U 1 1   0.5   v , H  v1  1  2 m  v   v  4  v / l  0.5

 v , H  v1  C0   0.0246cos   1.606U1 sin  U m1 10

Choi et al. [113]

C0 

2 1   Rem /1000 

 v , H  v1  1.1  0.1

2



Ozaki et al. [114]

v

1  1000 / Re m 

U vm  1.41e 1.39U vU1 U1 1   v  U vm , P  0.0019d *0.809  v l1 

Page 42 / 47



1.75



/ l 

0.5

2

 v / l   U vmU m1

1

11



1.2  0.2 1  e18v

1



 1  e 1.39U vU1 U vm , P

0.157

l*0.562U1

Tab.14 (continued)

12

No. Author/source Bhagwat-Ghajar [19] 

Void fraction correlation v,H

  C0  U vmU m1 , ftp  0.0716 Rel00.237  0.0008 1 v

0.21 v 

2 1    v / l 2 cos    v    2  1  cos  l       C0   2 2  Re m   1000  1  1     1000   Re m 

   0.5  0.15 C0,1  0.2 1   v    2.6   v , H     l    U vm   0.35sin   0.45cos   U 2

 C0,1

ftp  1  x  

1   v C2

 0.434 / log   / 0.001  0.15 ,   / 0.001  10 l l  C2    1,  l / 0.001  10

Page 43 / 47

1.5

Tab.15 Comparison between predicted void fractions by existing implicit correlations and simulation results Percentage of data points Correlations

predicted within

RMSE

R2

MARD

MRD

(%)

(%)

±5%

±10%

±15%

Bhagwat-Ghajar [19]

85.27%

99.56%

100.00%

0.02192

0.97782

2.119

-1.909

Ozaki et al. [114]

32.75%

99.78%

100.00%

0.04918

0.88827

5.564

-5.564

Levy [105]

43.30%

68.13%

85.49%

0.07199

0.76061

7.928

-7.928

Hughmark [106]

10.55%

55.60%

98.24%

0.07582

0.73451

9.184

-9.184

Liao et al. [109]

0.66%

38.90%

100.00%

0.09026

0.62370

10.273

-10.273

Choi et al. [113]

0.22%

28.79%

100.00%

0.09282

0.60210

10.640

-10.640

Gomez et al. [111]

0.00%

7.47%

100.00%

0.09798

0.55657

11.370

-11.370

Fujie [107]

1.76%

21.54%

72.09%

0.10488

0.49194

12.703

-12.703

Hibiki-Ishii [112]

44.84%

60.22%

67.69%

0.12714

0.25344

12.893

-12.890

Shipley [108]

0.00%

0.00%

0.00%

0.16482

-0.25464

19.603

-19.603

Hoogendoorn [104]

0.00%

0.00%

10.33%

0.23355

-1.51925

27.426

-27.426

Zhao et al. [110]

1.54%

14.29%

17.36%

0.31396

-3.55271

37.388

-37.388

Page 44 / 47

Tab.16 Comparison between predicted void fractions by existing correlations with good performance and simulation results Percentage of data points predicted Correlations

within

RMSE

R2

MARD

MRD

(%)

(%) -1.263

±5%

±10%

±15%

El hajal et al. [103]

96.70%

100.00%

100.00%

0.01292

0.99229

1.313

Loscher-Reinhardt [61]

98.02%

100.00%

100.00%

0.01436

0.99048

1.490

0.859

Moussali [63]

96.92%

100.00%

100.00%

0.01444

0.99037

1.530

-1.373

Greskovich-Cooper [62]

92.97%

99.34%

100.00%

0.01811

0.98486

1.681

-1.120

Cioncolini-Thome [6]

93.85%

97.80%

99.56%

0.01748

0.98589

1.792

-0.279

Milkie et al. [8]

91.65%

98.90%

100.00%

0.01976

0.98197

2.062

-0.637

Bhagwat-Ghajar [19]

85.27%

99.56%

100.00%

0.02192

0.97782

2.119

-1.909

Bestion [92]

89.45%

98.02%

99.78%

0.02493

0.97129

2.190

-1.858

Homogeneous

97.58%

100.00%

100.00%

0.01942

0.98258

2.238

2.238

Ahrens [47]

79.78%

91.43%

100.00%

0.02762

0.96477

2.784

-2.598

Osmachkin-Borisov [42]

78.24%

94.51%

99.12%

0.03254

0.95110

3.066

-2.965

Wilson et al. [2]-I

79.78%

100.00%

100.00%

0.03043

0.95722

3.638

-3.638

Premoli et al. [43]

67.03%

93.63%

100.00%

0.03670

0.93779

3.979

-3.892

Harms et al. [5]

71.43%

93.85%

98.90%

0.03797

0.93339

4.032

-2.959

Xu-Fang [27]

62.86%

94.95%

100.00%

0.04051

0.92421

4.199

-4.156

Woldesemayat-Ghajar [21]

68.57%

98.46%

100.00%

0.03798

0.93338

4.256

-4.121

Jassim et al. [7]

63.74%

96.92%

99.12%

0.03729

0.93577

4.305

-3.658

Steiner [101]

62.42%

89.23%

100.00%

0.04024

0.92523

4.654

-4.654

Ozaki et al. [114]

32.75%

99.78%

100.00%

0.04918

0.88827

5.564

-5.564

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Fig.1 Computational model and mesh of a spiral pipe Fig.2 Comparison between simulation values and experimental results under different operating parameters Fig.3 Void fraction vs. vapor quality under different operating parameters Fig.4 Void fraction vs. vapor quality under different structural parameters Fig.5 Void fraction: comparison between homogeneous correlation and simulation results Fig.6 Void fraction: comparison between existing slip-ratio correlations and simulation results Fig.7 Void fraction: comparison between existing Kαv,H correlations and simulation results Fig.8 Void fraction: comparison between existing Lockhart-Martinelli parameter based correlations and simulation results Fig.9 Void fraction: comparison between existing drift flux correlations and simulation results Fig.10 Void fraction: comparison between existing generational correlations and simulation results Fig.11 Void fraction: comparison between existing implicit correlations and simulation results Fig.12 Void fraction: comparison between the best five correlations and simulation results

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Highlights (1) A model was established to investigate the condensation flow. (2) The characteristics of condensation void fraction in a spiral pipe were analyzed. (3) 96 existing void fraction correlations were reviewed and evaluated. (4) Five correlations which can well predict condensation void fraction were recommended.

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