Frictional pressure drop during steam stratified condensation flow in vacuum horizontal tube

International Journal of Heat and Mass Transfer 115 (2017) 979–990

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Frictional pressure drop during steam stratified condensation flow in vacuum horizontal tube Yaoxuan Wang, Shengqiang Shen ⇑, Dayuan Yuan Key Laboratory of Liaoning Province for Desalination, School of Energy and Power Engineering, Dalian University of Technology, Dalian, China

a r t i c l e

i n f o

Article history: Received 10 June 2017 Received in revised form 19 August 2017 Accepted 26 August 2017 Available online 4 September 2017 Keywords: Condensation Horizontal tube Frictional pressure drop Multi-effect evaporation Desalination

a b s t r a c t The frictional pressure drop of steam condensation flow in vacuum horizontal tube was studied experimentally. The steam saturation temperature changes from 50 to 70 °C, the steam mass flux varies from 2 to 10 kg/(m2
s), vapor quality range are from 0 to 1 and the temperature difference between steam and cooling water are 3, 5 and 8 °C respectively. 205 experimental data were obtained in the experiment and compared with 25 existing frictional pressure drop models in three different kinds. All the experimental conditions are stratified flow and the flow states are turbulent and laminar flow in steam and liquid phase respectively. The frictional pressure drop increases with mass flux and vapor quality. It decreases with saturation temperature and has less relationship with temperature difference. Five models with the highest prediction accuracy are Quibén’s model, Chisholm’s model, Zhang’s model, Sun’s model, Lee’s model. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Steam condensation flow in horizontal tube is widely used in various industrial fields, such as chemical industry, air conditioning and desalination. A review of refrigerants condensation flow inside and outside tube was proposed by Cavallini et al. [1]. Later, Miyara [2], and Dalkilic [3] also gave comprehensive reviews of condensation flow, including the heat transfer, flow pattern, void fraction and pressure drop. Wang et al. [4] investigated the heat transfer characteristics of steam condensation flow in vacuum horizontal tube with inner diameter of 18 mm experimentally. Condensation heat transfer of vapor and noncondensable gas mixtures in horizontal tube with inner diameter of 27.5 mm when the inlet pressure is 0.2 MPa is shown in Wu and Vierow’s study [5]. Similar experiment with inner diameter of 16 mm at the pressure 0.1 MPa was conducted by Ren et al. [6]. Thome [7] proposed a new condensation model for horizontal tube including two types of heat transfer mechanisms in the tube: film condensation at the top of tube and convective condensation at the bottom. Two-phase flow is a very complex thermodynamic process. The frictional pressure drop of two-phase flow plays an important role in design and optimization of heat exchanger, especially for multieffect evaporation desalination plant. The steam condenses in vacuum horizontal tube in multi-effect evaporation desalination plant, and the pressure drop of this process has a great influence on the ⇑ Corresponding author. E-mail address: [email protected] (S. Shen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.08.088 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

thermal performance of desalination system [8]. Kouhikamali et al. [9] investigated the pressure drop in the heat exchangers of multi-effect evaporation with thermal vapor compression system and showed that condensation pressure drop in the tube has most influence on the system performance among all different kinds of pressure losses. Considering condensation pressure drop in tube increases the specific heat transfer surface area by about 7% than neglecting it. The frictional pressure drop of refrigerant condensation flow at high or atmospheric pressure has been studied extensively, but there are few studies about the frictional pressure drop of steam condensation flow under vacuum conditions. The main factors influencing the frictional pressure drop are steam mass flux, vapor quality and saturation temperature. Many experiments [10–13] showed that the frictional pressure drop increased with the mass flux and vapor quality. Col et al. [14] investigated propane condensation flow in minichannel and found that the pressure drop increases with vapor quality and the increment increases with mass flux. Zhuang et al. [15] measured R170 condensation flow in a 4 mm diameter horizontal tube. The frictional pressure drop increases with mass flux and the effect of mass flux weakens as the saturation temperature increases. Quibén and Thome [16] measured the pressure drop of R134a, R22 and R410A experimentally. They found that the frictional pressure drop increased to the maximum firstly and then decreased with the increasing vapor quality. The mechanism of this phenomenon is the transitions of flow patterns. Charnay et al. [17] got the same experimental results with Quibén and Thome [16]. Zhang et al.

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Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990

Nomenclature A B C Co cp D f Fr G g j⁄ L m n p Q Re r T We X x Y

area coefficient coefficient confinement number specific heat at constant pressure, kJ/(kg
°C) diameter, m friction factor Froude number mass flux, kg/(m2
s) gravity, m/s2 dimensionless velocity length, m mass flow rate, kg/s number pressure, Pa heat transfer rate, kW Reynolds number latent heat, kJ/kg temperature, °C Webb number Lockhart-Martinelli parameter vapor quality Chisholm parameter

Greek symbols D difference a void fraction d film thickness, m

[18] investigated pressure drop of R22, R410A and R407C in horizontal tube with two different diameters. The experimental results showed that vapor quality had less relationship with frictional pressure drop when the vapor quality was higher than 0.8. A large number of experimental studies [10,12,13,17–20] showed that the frictional pressure drop decreased with the saturation temperature. The increase of saturation temperature leads to the increase of vapor density. So the vapor velocity decreases at the same mass flux, and the friction between vapor and condensate and tube wall is weakened. Steam condensation flow pressure drop at very low mass flux was reported by Guo et al. [21], the flow pattern observed in the experiment is stratified flow. The effects of tube length and inclination were studied experimentally, but the effects of mass flux, vapor quality and saturation temperature were not mentioned. Charmay et al. [17] compared their experimental data with different existing models. They approved that the homogeneous models proposed by Cicchitti et al. [22], Awad and Muzychka [23], the separated models proposed by Müller-Steinhagen and Heck [24], Zhang and Webb [25] and Friedel [26] are more accurate. Xu et al. [27] collected 3480 data of in-tube two-phase flow from existing literatures, in which the tube diameter varied from 0.0695 mm to 14 mm and the mass flux changed from 8 to 6000 kg/(m2
s), and then assessed and analyzed 28 models. The results showed that the homogeneous models proposed by Beattie and Whalley [28] and McAdams et al. [29], the separated models proposed by Müller-Steinhagen and Heck [24] and Sun and Mishima [30] made the most accurate prediction. Ould Didi et al. [31] measured the frictional pressure drop of 5 different refrigerants in horizontal tube and obtained 788 data, which matched well with models proposed by Müller-Steinhagen and Heck [24] and Grönnerud [32]. The model proposed by Müller-Steinhagen and

l h

q r

/2

dynamic viscosity, Pa
s angle, ° density, kg/m3 surface tension, N/m two phase friction multiplier

Subscripts annular annular c cooling water d dimensionless exp experimental frict frictional i ordinal number in inlet l liquid lo liquid only mom momentum out outlet pred predicted s steam strat stratified static static tp two-phase tt turbulent-turbulent v vapor vo vapor only

Heck [24] is the most suitable for annular flow, while the model proposed by Grönnerud [32] is the most suitable for intermittent flow and stratified wavy flow. The experiment of Wang et al. [20] showed that Müller-Steinhagen and Heck [24] model gave the most accurate prediction of frictional pressure drop. Col et al.’s experiments [14] showed that the pressure drop calculated by Friedel [26] model is in good agreement with the measured values. Guo et al.’s data [21] showed that the existing correlations were not valid for condensation at very low mass flux. 2. Two-phase frictional pressure drop prediction models The empirical models are widely used to calculate the twophase frictional pressure drop, due to the complexity of the twophase flow. The models have the advantages of easy calculation and high accuracy. But they can only be applied to certain experimental conditions, on which the models based. These models usually can be divided into two categories, homogeneous models and separated models. In recent years, the phenomenological model based on flow pattern has been proposed. 2.1. Homogeneous model Homogeneous model is the simplest model to analyze multiphase flow. The vapor and liquid flow are assumed to be at the same velocity, so in this case the two-phase flow is considered as idealized single-phase fluid flow. The frictional pressure drop of two-phase flow can be calculated as a single-phase flow by using the mixture fluid properties, shown in Eq. (1):


 Dp G2 ¼ f DL frict 2Dqtp tp

ð1Þ

Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990

where qtp is the two-phase density which can be calculated by Eq. (2) and ftp is the friction coefficient of two-phase flow, which is a piecewise function of two-phase Reynolds number Retp, shown in the Eqs. (3) and (4):

1

qtp

¼

1x

ql

þ

f tp ¼ 64=Retp

x

ð2Þ

qv for Retp < 2100

f tp ¼ 0:316=Re0:25 tp

ð3Þ

for Retp > 2100

ð4Þ

where two-phase Reynolds number Retp is defined as:

Retp ¼

GD

ð5Þ

ltp

where G is the mass flux, ltp is the two-phase dynamic viscosity. The differences between different homogeneous models are how to calculate ltp, and many scholars proposed the expressions of ltp. These expressions are the functions of vapor quality x, and satisfy the condition that the ltp is ll and lv respectively when the vapor quality is 0 and 1. McAdams et al. [29] first gave a two-phase dynamic viscosity correlation with the same expression of two-phase density. Cicchitti et al. [22] proposed a simple correlation based on mass average value of viscosity. Dukler et al. [33] introduced the density into the correlation. Beattie and Whalley [28] proposed a theoretical correlation including the homogeneous void fraction. Awad and Muzychka [23] used an analogy between thermal conductivity of porous media and viscosity in two-phase flow and defined new correlations. Six well-known homogeneous models are summarized and listed in Table 1. 2.2. Separated model The vapor phase and liquid phase flow at different velocities in the separated models. This kind of model is the same as the real situation, but less complicated than the real situation. The separated model also could be divided into the /2l , /2v based model and /2lo , /2vo based model. 2.2.1. The /2l , /2v based separated model The two-phase frictional pressure drop is the product of the frictional pressure drop of single-phase flow across the tube alone and the two phase multiplier. Taking the liquid phase as example, the calculation process is shown as following equations:



 Dp Dp ¼ /2 DL frict DL l l

ð6Þ


 Dp ½Gð1  xÞ2 ¼ fl DL l 2Dql

ð7Þ

Table 1 Six homogeneous frictional pressure drop models. Time

Author

Correlation

1942

McAdams et al. [29]

1 x 1x ltp ¼ lv þ ll

1960

Cicchitti et al. [22]

ltp ¼ xlv þ ð1  xÞll

1964

Dukler et al. [33]

1982

Beattie and Whalley [28]

1991

Lin et al. [34]

ll ltp ¼ qtp xqlvv þ ð1xÞ ql ltp ¼ ll ð1  aÞð1 þ 2:5aÞ þ lv a a ¼ ½1 þ ðð1  xÞ=xÞðqv =ql Þ1 ll lv ltp ¼ lv þx1:4 ðll lv Þ

2008

Awad and Muzychka [23] Definition 3

h

i

ll lv Þx ltp ¼ ll 22lll lþþllvv2ð þðll lv Þx

Gð1  xÞD

Rel ¼

ll

981

ð8Þ

where fl is calculated by Eqs. (3) or (4) with the liquid physical properties and mass flux. Lockhart and Martinelli [35] studied benzene, kerosene and water flowing with air in tube with the diameter from 1.5 to 25.8 mm experimentally and proposed that two phase multiplier /2l and /2v are the function of Lockhart-Martinelli parameter X, which was defined as Eq. (9). According to the flow state, gasliquid two phase flow can be divided into four cases: laminarlaminar flow, turbulent-laminar flow, laminar-turbulent flow and turbulent-turbulent flow. Lockhart and Martinelli drew the curves of /2l and X for four different cases. The graphs are not convenient in calculation, so Chisholm [36] creatively put forward Eq. (10) to calculate /2l , according to the graphs.

 X¼

ðDp=DLÞl ðDp=DLÞv

/2l ¼ 1 þ

0:5

C 1 þ X X2

ð9Þ

ð10Þ

where C is a coefficient which values are 5, 10, 12 and 20 respectively for four different cases. Chisholm model is a significant progress in two-phase frictional pressure drop. Since then, the researchers focus on the correlation of C which is more applicable for certain conditions, and the coefficient C is affected by many factors. Mishima and Hibiki [37] measured the frictional pressure drop of air-water flow in tubes with the inner diameters varying from 1 to 4 mm, and proposed that the coefficient C is a function of inner diameter. Then Zhang et al. [38] added non-dimensional confinement number Co to Mishima and Hibiki’s correlation [37] to predict the adiabatic gas-liquid and vapor-liquid frictional pressure drop in mini-channels. Co is a main parameter in mini channels and first defined by Kew and Cornwell [39] as shown in Eq. (11). This parameter represents the ratio of surface tension force to buoyancy force, and takes into account the impact of surface tension, density and diameter at the same time.

Co ¼

½r=ðgðql  qv ÞÞ0:5 D

ð11Þ

Two-phase frictional characteristics of R22, R134a and R407C inside a 6.5 mm tube were reported by Wang et al. [40] and a new correlation of C was proposed in their research. The new correlation was a function of Relo, X and physical properties. Lee and Mudawar [41] introduced Weber number We into the correlations in order to consider the effect of surface tension. We is defined as the ratio of inertia forces and surface tension forces, shown in Eq. (12). Based on the pressure drop experimental data of five kinds of refrigerants in stainless steel horizontal tube with the diameter changing from 0.5 to 3 mm, Pamitran et al. [42] proposed a correlation similar to Lee and Mudawar [41]. Pamitran calculated the Retp by using the two phase dynamic viscosity of Beattie and Whalley [28]. Hwang and Kim [43] measured pressure drop of R134a in micro tubes and summarized a new correlation, which is a function of Relo, X and Co.

We ¼

G2 D

rq

ð12Þ

Sun and Mishima [30] changed the coefficient of X in Chisholm model and proposed the new expression of C. Yu et al. [44] proposed a new expression of two phase multiplier different from the Chisholm model. Nine /2l , /2v based separated models are summarized and listed in Table 2.

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Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990

Table 2 Nine /2l , /2v based separated frictional pressure drop models. Time

Author

Correlation

Working fluid and diameter

1967 1996

C = 5, 10, 12 and 20 respectively C ¼ 21½1  expð0:319DÞ

1997

Chisholm [36] Mishima and Hibiki [37] Wang et al. [40]

2002

Yu et al. [44]

0:5 /2l ¼ ½18:65ðqv =ql Þ0:5 ðð1  xÞ=xÞðRe0:1 v =Rel Þ

2005

Lee and Mudawar [41]

C ¼ 2:16Re0:047 We0:60 laminar-laminar lo lo

benzene, kerosene and water flow with air D = 1.5–25.8 mm air-water D = 1–4 mm R22, R134a, R407C D = 6.5 mm Water D = 2.98 mm R134a 231 lm wide  713 lm deep

2006

Hwang and Kim [43]

X 0:32 Co0:82 C ¼ 0:227Re0:452 lo

2009

Sun and Mishima [30]

C ¼ 4:566  106 X 0:128 Re0:938 ðqv =ql Þ2:15 ðll =lv Þ5:1 lo 1:9

0:23 laminar-turbulent C ¼ 1:45Re0:25 lo Welo

C /2l ¼ 1 þ X 1:19 þ X12

C ¼ 1:79ðRev =Rel Þ0:4 ðð1  xÞ=xÞ0:5 2010

Zhang et al. [38]

C ¼ 21½1  expð0:674=CoÞ liquid-gas C ¼ 21½1  expð0:142=CoÞ liquid-vapor

2010

Pamitran et al. [42]

0:433 C ¼ 0:003Re1:23 tp Wetp

R22, R134a, R410A, R290 and R744 D = 0.5, 1.5, 3.0 mm

2.2.2. The /2lo , /2vo based separated model Two phase frictional pressure drop is the product of the frictional pressure drop of single-phase flow across the tube alone with the total mass flux of two-phase and the two phase multiplier. Choosing the liquid phase as example, calculation process is shown as following equations:



 Dp Dp ¼ /2 DL frict DL lo lo
 Dp G2 ¼ f DL lo 2Dql lo

R134a D = 0.244, 0.430, 0.792 mm R123, R134a, R22, R236ea, R245fa, R404a, R407C, R410a, R507, CO2, water and air. D = 0.506–12 mm a variety of data sets collected from the literature

ð13Þ

ð14Þ

a model similar to the Friedel’s model [26], based on the experimental data of seven refrigerant flow in a tube with diameter of 8 mm. The models proposed by Grönnerud [32] and Müller-Steinhagen and Heck [24] are also widely used, and have high accuracy. Grönnerud [32] developed the model by using about 1000 data of R12 and R717 in horizontal tube with a diameter of 26.2 mm. Müller-Steinhagen and Heck [24] collected a data bank including 9300 data of pressure drop for a variety of fluid including airwater, argon and R12. Nine /2lo , /2vo based separated models are summarized and listed in Table 3. 2.3. Other model

Relo ¼

GD

ð15Þ

ll

Based on Baroczy’s study [45], Chisholm [46] presented a correlation of two phase multiplier /2lo , which contains Chisholm parameter Y and coefficient B. Y is defined as Eq. (16) and B is a piecewise function of mass flux and Y.

 Y¼

ðDp=DLÞvo ðDp=DLÞlo

0:5 ð16Þ

Tran et al. [47] measured pressure drop of three refrigerants boiling flow in small channels and used confinement number Co as a substitute for the coefficient B in Chisholm’s model [46]. Yoon et al. [48] investigated pressure drop of carbon dioxide evaporation flow and introduced Weber number We to Chisholm’s model [46]. Xu and Fang [49] summarized large number of previous experimental data, and the model suitable for condensation pressure drop was obtained. The new model is similar to these models and contains the two-phase Froude number Frtp and Weber number Wetp. These two parameters are defined as Eqs. (12) and (17). 2

G Fr ¼ gDq2

ð17Þ

The model presented by Friedel [26] can be applied to horizontal flow and vertical up flow when the ratio of liquid and vapor viscosity is less than 1000 and mass flux is less than 2000 kg/(m2
s). This correlation is accurate and has been widely cited. Chen et al. [50] added the piecewise function of We, Re and Bo to Friedel’s model [26], to make the calculation results matching with their experimental value better. Zhang and Webb [25] simplified Friedel’s model [26] by using reduced pressure instead of the ratio of two phase physical properties. Cavallini et al. [51] also put forward

El Hajal et al. [53] developed a new version for two-phase flow pattern map of condensation inside horizontal tubes. A new twophase frictional pressure drop model has been developed by Quibén and Thome [16,54] based on the new flow pattern map and experimental data of three refrigerants. This approach used the two-phase friction factor to calculate the frictional pressure drop which was similar to the homogeneous models. There are different correlations for different flow patterns. For stratified flow, the friction factor is calculated by Eq. (18).

f tp ¼ hstrat f v þ ð1  hstrat Þf annular

ð18Þ

⁄

here h strat is the nondimensionalized stratified angle for stratified flow regime and the fv and fannular are the friction factors for singlephase vapor flow and annular flow regime respectively. hstrat is calculated from a simple explicit approximation for a given void fraction which was proposed by Biberg [55], shown in Eq. (20). The correlation of friction factor for annular flow regime is shown in Eq. (21).

hstrat ¼

hstrat 2p

ð19Þ

i9 8 3p1=3 h > 1  2ð1  aÞ þ ð1  aÞ1=3  a1=3 > < pð1  aÞ þ 2 = hstrat ¼ 2p  2 h i > :  1 ð1  aÞa½1  2ð1  aÞ 1 þ 4ðð1  aÞ2 þ a2 Þ > ; 200 ð20Þ #0:4    1:2 " d ðql  qv Þgd2 lv 0:08 0:034 f annular ¼ 0:67 Wel D r ll

ð21Þ

983

Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990 Table 3 Nine /2lo , /2vo based separated frictional pressure drop models. Time

Author

Correlation

1973

Chisholm [46]

/2lo

Working fluid and diameter

1979

Friedel [26]

ð1xÞ /2lo ¼ ð1  xÞ2 þ x2 qql ffvo þ 3:24x Fr 0:045 We0:035

2

¼ 1 þ ðY  1Þ½Bx

0:875

0:875

ð1  xÞ

v lo

H ¼ ðql =qv Þ

0:91

tp

0:19

ðlv =ll Þ

þx

0:224

0:78

1:75

Steam, air-water, and nitrogen-mercury



25,000 data points of air-water and refrigerants D > 4 mm

H

tp

0:7

ð1  lv =ll Þ   0:25

 1

1979

Grönnerud [32]

1986

Müller-Steinhagen and Heck [24]

1992

Souza et al. [52]

/2lo /2lo

2000

Tran et al. [47]

/2lo ¼ 1 þ ð4:3Y 2  1Þ½Cox0:875 ð1  xÞ0:875 þ x1:75 

2002

Cavallini et al. [51]

/2lo ¼ ð1  xÞ2 þ x2 ql fvo þ 1:262x We0:1458

/2lo

¼ 1 þ f Fr ½x þ 4ðx

1:8



ql qv

0:5 x10 f Fr Þ

lv ll

R12 and R717 D = 26.2 mm

2

f Fr ¼ Fr 0:3 lo þ 0:0055½lnð1=Fr lo Þ Fr lo < 1 f Fr ¼ 1 Fr lo > 1 2 3

¼ Y x þ ð1  xÞ

1=3

9300 data points of air-water, argon, R12

½1 þ 2xðY 2  1Þ

0:169Fr lo 1:773 ¼ ½1:376 þ ð4:172 þ 5:480Fr lo  1:564Fr2lo ÞX tt ð1  xÞ1:75

qf

v lo

0:6978

H

vo

1:181

ðlv =ll Þ ð1  lv =ll Þ3:477 h i B 0:875 ¼ 1 þ 4:2ðY  1Þ Wevo x ð1  xÞ0:875 þ x1:75

H ¼ ðql =qv Þ

0:3278

2004

Yoon et al. [48]

/2lo

2013

Xu and Fang [49]

/2lo ¼ Y 2 x3 þ ð1  x2:59 Þ h i 1 þ 2x1:17 ðY 2  1Þ þ 0:00775x0:475 Fr0:535 We0:188 tp tp

2

0:632

where d is the film thickness, which is calculated by the following equation for annular flow:

d¼

Dð1  aÞ 4

ð22Þ

3. Experimental apparatus Fig. 1 is the schematic diagram of experimental apparatus. The experimental section is made up of five double-pipe heat exchangers. The water is heated into steam in the evaporator and it flows into the inner tube of the experimental section. The steam is condensed completely or partly when flowing through the experimental sections. The uncondensed steam and condensate flow together into the steam-liquid separator at the end of test section, where steam and condensate are separated by gravity. Finally, uncondensed steam flows into the end condenser and condenses into water completely. A vacuum pump is connected to the condenser keeping the system at vacuum pressure. A cooling water tank and a water pump provide cooling water in parallel to annular space of five experimental sections. The inner tube is made of aluminium brass. The length of each test tube is 2000 mm and inner diameter is 18 mm. The cooling water temperatures at the inlet and outlet of each experimental section and the tube wall temperatures are measured by T-type thermocouples, which were cali-

R12, R134a D = 10.9 mm R12, R113, R134a D = 2.46, 2.92 mm and rectangular 4.06  1.7 mm R22, R134a, R125, R32, R236er, R407c, R410a D = 8 mm CO2 D = 7.53 mm 525 data points of 9 refrigerants D = 0.1–10.07 mm

brated by thermostatic water bath with the accuracy of 0.1 °C. Pressure drops of steam condensation flow in each experimental section are measured by differential pressure transducers with the accuracy of ±1 Pa. The absolute pressures at evaporator, inlet and outlet of the experimental sections are measured by pressure transducers which precision is ±28 Pa. The data of pressure and pressure drop are recorded in the interval of 3 s. Each parameter is recorded 200 times and the average of these values are calculated by a computer equipped with a National Instruments SCXI data acquisition system. The experiment was carried out at the saturation temperature ranging from 50 to 70 °C, which corresponds to an absolute pressure from 12.3 to 31.2 kPa. The steam mass flux varies between 2 and 10 kg/(m2
s) at the inlet of test section. The temperature differences between steam and cooling water are set up at 3, 5 and 7 °C at the end of experimental section respectively. 4. Data processing and error analysis 4.1. Data processing The heat transfer rate Qi in test section i is calculated by following equation:

Q i ¼ mc;i cp ðT c;out;i  T c;in;i Þ

Fig. 1. Schematic diagram of experimental apparatus.

ð23Þ

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Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990

where mc,i is the mass flux of cooling water in the experimental section i and cp is the specific heat of water at constant pressure. Tc,in,i and Tc,out,i are the cooling water temperatures at the inlet and outlet of experimental section i respectively. The vapor quality at the outlet of evaporator is 1. The vapor quality in inlet and outlet of section i can be calculated by the following equations:

Pi1

1 Qi mr

xi;in ¼ 1 

ð24Þ

Pi xi;out ¼ 1 

1Q i mr

ð25Þ

where m is the mass flux of steam, r is the latent heat of vaporization. So the average vapor quality xi in the section i is:

xi ¼

1 ðxi;in þ xi;out Þ 2

ð26Þ

The unit pressure drop (Dp/DL) is the sum of static pressure drop (Dp/DL)static, momentum pressure drop (Dp/DL)mom and frictional pressure drop (Dp/DL)frict, as shown in the following equation:





 Dp Dp Dp Dp þ þ ¼ DL DL static DL mom DL frict

ð27Þ

The static pressure drop (Dp/DL)static is zero when steam flows in horizontal tube and the momentum pressure drop (Dp/DL)mom is calculated by the following equation:

("

Dpmom ¼ G2

# ð1  xÞ2 x2 þ ql ð1  aÞ qv a

"

 out

# ) ð1  xÞ2 x2 þ ql ð1  aÞ qv a

ð28Þ

in

where a is the void fraction. It can be calculated by Rouhani and Axelsson’s drift flux model [56], shown in following equation:

a¼

x

qs

(


  0:25 )1 x 1x 1:18 g rðql  qs Þ þ ½1 þ 0:2ð1  xÞ þ G qs ql q2l ð29Þ

The pressure drop in each experimental section was measured by differential pressure transducer. Hence, the frictional pressure drop equals to the pressure drop minus the momentum pressure drop, shown in the following equation:




 Dp Dp Dp  ¼ DL frict DL DL mom

ð30Þ

In order to verify the accuracy of prediction of each model, two parameters (MRE and MAE) are introduced. MRE is the mean relative error, and MAE is the mean absolute relative error. The calculation formulas are shown below:

" # n ðDp=DLÞfrict;pred;i  ðDp=DLÞfrict;exp;i 1X MRE ¼  100% ðDp=DLÞfrict;exp;i n i¼1 MAE ¼

" # n jðDp=DLÞfrict;pred;i  ðDp=DLÞfrict;exp;i j 1X  100% n i¼1 ðDp=DLÞfrict;exp;i

ð31Þ

ð32Þ

where the subscripts pred and exp represent predicted value and experimental value respectively. n is the total number of experimental data and 205 data were used in this article. 4.2. Error analysis The mass flow rate of cooling water was controlled accurately to ensure that the increment of cooling water temperature is larger than 1 °C. So the uncertainty is less than 0.14 °C. All uncertainties are list in Table 4.

Table 4 The uncertainties of experimental parameters. Parameters

Uncertainty

Mass flow rate of cooling water mc Increment of cooling water temperature Tc,out  Tc,in Heat transfer rate Q Mass flux of steam G Vapor quality x Heat transfer length L Void friction a Steam temperature Ts Momentum pressure drop (Dp/DL)mom Pressure drop (Dp/DL) Frictional pressure drop (Dp/DL)frict

±1.3% ±0.14 °C ±14% ±2.5% ±14% ±2 mm ±4.5% ±0.05 °C ±7.82% ±0.5% ±1.83%

5. Experimental results and analysis 5.1. Flow pattern Flow pattern is one of the main factors that influence the pressure drop of steam-liquid two-phase flow. There are kinds of flow patterns in horizontal tube flow. The flow pattern can be determined by using the flow pattern map. Breber [57] used LockhartMartinelli parameter Xtt and dimensionless steam velocity j⁄ g as the horizontal and vertical coordinates, where j⁄ g is calculated by Eq. (33). This flow pattern map is widely used in condensation heat transfer process to determine the flow pattern. The flow patterns are divided into four zones in Breber’s map, so the wavy flow and stratified flow could not be distinguished clearly in the map. The flow pattern map proposed by El Hajal [53] is more accurate and able to distinguish each flow pattern. The horizontal and vertical coordinates of this map are vapor quality and mass flux respectively. The discriminant of stratified flow changing to the wavy flow is shown in Eq. (34).

Gx  jv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dg qv ðql  qv Þ

Gstrat

" #1=3 ð226:3Þ2 Ald A2vd qv ðql  qv Þll g ¼ þ 20x x2 ð1  xÞp3

ð33Þ

ð34Þ

where Ald and Avd are the vapor and liquid flow cross-sectional area respectively, the calculation methods are shown below.

Ald ¼

Avd ¼

Að1  aÞ D2 Aa D2

ð35Þ

ð36Þ

The steam mass flux is very low in the experiment, the gravity force makes the condensate gathered at the bottom of tube. The flow pattern observed in the range of experimental conditions is only stratified flow. The experimental observations could be compared with the flow pattern maps. The position of each experimental condition in the two flow pattern maps is shown in Fig. 2. In Breber’s map, in most cases, they are in the wavy and stratified flow region and in a few cases, they are in the transition region between wavy flow and annular flow. The flow pattern tends to change from stratified flow to annular flow with the increase of mass flux. In El Hajal’s map, all the positions are in the stratified flow region rather than wavy flow region, which is more consistent with the experimental observations.

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(1)Breber flow pattern map [57]

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(2)El Hajal flow pattern map [53]

Fig. 2. Positions of experimental conditions in the flow pattern maps.

5.2. Flow state The flow state of steam-liquid two-phase flow also affects the frictional pressure drop, and the flow state can be divided into four types depending on whether the steam or liquid flow is laminar or turbulent. Re = 2100 is the boundary between laminar and turbulent flow. The steam and liquid phase Re numbers are shown in Fig. 3. The steam mass flux varies between 2 and 10 kg/(m2
s) at the inlet of test section, and the steam saturation temperature changes from 50 to 70 °C. The steam mass flux is low in the experiment, but the steam viscosity is very low in the vacuum state, so the steam flow state is turbulent. When the condensate gathered in the bottom of the tube, the condensate flowed forward by the shear force between the steam and liquid. The velocity is low and the liquid viscosity is high so the liquid flow state is laminar. 5.3. Influence factors of frictional pressure drop The frictional pressure drop when mass flux changes from 6.3 to 9.8 kg/(m2
s) and vapor quality varies from 0.2 to 1 at the saturation temperature 60 °C is presented in Fig. 4. The frictional pressure drop increases with vapor quality, while the increment rises

Fig. 3. Flow state of steam-liquid two-phase flow.

Fig. 4. Effect of mass flux and vapor quality on frictional pressure drop.

significantly with the mass flux. The increment rises from 242 Pa/m to 541 Pa/m when the mass flux changes from 6.3 to 9.8 kg/(m2
s). The steam velocity increases with vapor quality when the inlet mass flux is constant. So the pressure drop increases with vapor quality. The pressure drop increases with mass flux obviously. The pressure drop rises sharply as the vapor quality is high. The steam velocity is the main factor of pressure drop. The shear force between steam and liquid film rises with mass flux, so the pressure drop increases. The effect of saturation temperature on frictional pressure drop is shown in Fig. 5. The average frictional pressure drop declines with saturation temperature at different mass fluxs. The effect of saturation temperature on pressure drop is mainly reflected in the physical properties. The steam velocity and water dynamic viscosity both declines with the increasing saturation temperature. So the friction between steam and condensate is weakened, and the pressure drop decreases with saturation temperature. The effect of temperature difference on frictional pressure drop when the mass flux is 8 kg/(m2
s) and saturation temperature is 60 °C are shown in Fig. 6. The temperature differences between steam and cooling water are 3, 5 and 7 °C respectively. The experimental data at different temperature differences are almost in the

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two most accurate models to calculate the two-phase viscosity. The great difference between the vapor and liquid densities under the experimental conditions were also considered in these two models. Beattie and Whalley [28] also introduced the homogeneous void friction into the model. In addition, Dukler’s model [33] is simpler than the Beattie’s model [28]. 6.2. Comparison with /2l , /2v based separated models The comparisons of frictional pressure drop between experi-

Fig. 5. Effect of saturation temperature on frictional pressure drop.

mental value and predicted value calculated by 9 /2l , /2v based separated models are shown in Fig. 8. The flow states are laminar and turbulent flow respectively in liquid and steam phase, so the coefficient C is 12 in Chisholm’s model [36], which can predict the experimental value very well. The MRE and MAE are only 1.2% and 15.5% respectively, and the percentage of data predicted within ±30% error band accounts for 92.7%. The Chisholm’s model [36] is the simplest model and also the most accurate /2l , /2v based separated model for predicting the experimental value. Later, the Mishima’s model [37] and Zhang’s model [38] which modified the coefficient C of Chisholm’s model [36] also predicted the experiment accurately and the MREs are 21.8% and 3.7% respectively. Sun’s model [30] was based on a large number of experimental data and predicted the experimental data very well. The MRE of Sun’s model [30] is slightly larger than the MREs of Chisholm’s model [36] and Zhang’s model [38] and the proportion of data within 30% error is slightly smaller than the proportions of these two models. The Lee’s model [41] and Hwang’s model [43] are based on frictional pressure drop in micro channel, and the predicted values don’t match with the experimental value, with the MREs 21.0% and 71.3% respectively. Wang’s model [40] and Yu’s model [44] also have large error in predicting the experimental value. Pamitran’s model [42] has the largest error among this kind of models. 6.3. Comparison with /2lo , /2vo based separated models

Fig. 6. Effect of temperature difference on frictional pressure drop.

same curve, which means temperature difference has less effect on the frictional pressure drop.

6. Comparison with existing models 6.1. Comparison with homogeneous models The comparisons between experimental value and predicted value of frictional pressure drop calculated by 6 homogeneous models are shown in Fig. 7. The predicted results of Cicchitti’s model [22] and Awad’s model [33] are obviously higher than the experimental value. The MREs are 255.4% and 198.2% respectively. They have a poor coherence with experimental results. While the predicted results of other four homogeneous models are relatively good. The prediction accuracies of McAdams’ model [29] and Lin’s model [34] are low when the frictional pressure drop is low but more accurate at other conditions. The MREs of these two models are 27.8% and 35.5% respectively. The Dukler’s model [33] and Beattie’s model [28] are the most accurate homogeneous models for predicting experimental values. The MREs are only 10.6% and 11.0%, and the MAEs are 26.3% and 26.6% respectively. The percentages of data predicted within ±30% error band account for more than 70%. The vapor and liquid densities were added to these

The comparisons of frictional pressure drop between experimental value and predicted value calculated by 9 /2lo , /2vo based separated models are shown in Fig. 9. The predicted values calculated by this kind of models are quite higher than the experimental values. The model proposed by Cavallini et al. [51] and Xu and Fang [49] predicted the most accurate results among this kind of models, with the MREs 85.5% and 96.7% respectively. The data points which prediction errors are within ±30% account for only 7.3% and 8.8% of the total data points in these two models. The predicted values calculated by other models in this kind are 100% higher than the experimental values. 6.4. Comparison with other model The flow pattern is the stratified flow under the experimental conditions. The comparison of frictional pressure drop between experimental value and predicted value calculated by Quibén’s phenomenological model [54] is shown in Fig. 10. The model predicted the experimental data pretty well, and the MRE and MAE are only 4.9% and 15.1%. The percentage of data predicted within ±30% error band accounts for 93.2%. But the calculation of this model is quite complicated. The MREs, MAEs and percentages of data predicted within ±30% error band of the 25 models are summarized in Table 5. The prediction accuracy of /2lo , /2vo based separated model is worst, some models in homogeneous models and the /2l , /2v based separated models could predict the experimental results well. Five models with the highest prediction accuracy are Quibén’s model [54],

Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990

Fig. 7. Comparisons between experimental value and predicted value based on homogeneous models of frictional pressure drop.

Fig. 8. Comparisons between experimental value and predicted value based on /2l , /2v based separated models of frictional pressure drop.

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Fig. 9. Comparisons between experimental value and predicted value based on /2lo , /2vo based separated models of frictional pressure drop.

7. Conclusions Three kinds of frictional pressure drop models are reviewed in this article. The frictional pressure drop of steam condensation flow in vacuum horizontal tube is studied experimentally. The steam mass flux is no more than 10 kg/(m2
s), and the saturation temperature changes from 50 to 70 °C. Some conclusions are summarized as following.

Fig. 10. Comparison between experimental value and predicted value based on Quibén and Thome’s model of frictional pressure drop.

Chisholm’s model [36], Zhang’s model [38], Sun’s model [30] and Lee’s model [41] respectively. Quibén’s model [54] is a phenomenological model, and the next four models belong to the /2l , /2v based separated models. Besides, the Dukler’s model [33] and Beattie’s model [28] also predict the experimental value well.

1. The flow pattern is stratified flow under the experimental conditions, which is in accordance with the prediction of El Hajal flow pattern map [53]. The flow states are turbulent and laminar flow in steam and liquid phase respectively. 2. The frictional pressure drop increases with vapor quality and mass flux. It decreases with saturation temperature and has less relationship with temperature difference. 3. The homogeneous models proposed by Dukler [33] and Beattie [28], the /2l , /2v based separated models proposed by Chisholm [36], Zhang [38] and Sun [30], and Quibén’s phenomenological model [54] predict the experimental values accurately. The predicted values calculated by /2lo , /2vo based separated models are quite higher than the experimental values. Quibén’s model [54] is the most accurate model in these kinds of models, but the calculation process of this model is quite complicated. The prediction accuracy of Chisholm’s model [36] is slightly lower than the Quibén’s model [54], but the calculation process is the simplest.

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Y. Wang et al. / International Journal of Heat and Mass Transfer 115 (2017) 979–990 Table 5 Prediction results of the 25 models. Model

MRE

MAE

Error within 30%

McAdams et al. [29] Cicchitti et al. [22] Dukler et al. [33] Beattie and Whalley [28] Lin et al. [34] Awad and Muzychka [23]

27.8% 255.4% 10.6% 11.0% 35.5% 198.2%

40.4% 257.0% 26.3% 26.6% 46.9% 198.9%

57.1% 21.0% 71.7% 70.7% 49.8% 28.3%

The /2l , /2v based separated models 7 Chisholm [36] 8 Mishima and Hibiki [37] 9 Wang et al. [40] 10 Yu et al. [44] 11 Lee and Mudawar [41] 12 Hwang and Kim [43] 13 Sun and Mishima [30] 14 Zhang et al. [38] 15 Pamitran et al. [42]

1.2% 21.8% 26.4% 53.2% 21.0% 71.3% 7.7% 3.7% 104.7%

15.5% 28.4% 26.7% 53.2% 21.4% 71.4% 18.7% 16.7% 104.8%

92.7% 63.9% 62.9% 0.5% 82.9% 28.8% 84.9% 89.8% 35.1%

The /2lo , /2vo based separated models 16 Chisholm [46] 17 Friedel [26] 18 Grönnerud [32] 19 Müller-Steinhagen and Heck [24] 20 Souza et al. [52] 21 Tran et al. [47] 22 Cavallini et al. [51] 23 Yoon et al. [48] 24 Xu and Fang [49]

260.0% 276.7% 405.6% 101.7% 292.4% 250.4% 85.5% 220.3% 96.7%

260.0% 276.7% 405.6% 101.7% 292.4% 250.4% 85.5% 220.3% 96.7%

7.3% 0% 0% 2.9% 0% 0% 7.3% 0% 8.8%

Other model 25

4.9%

15.1%

93.2%

Homogeneous models 1 2 3 4 5 6

Quibén and Thome [54]

Acknowledgement The research is supported by the Key Project of National Science Foundation of China (No. 51336001) and National Science and Technology Support Program (2014BAB09B00). Conflict of interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. References [1] A. Cavallini, G. Censi, D. Del Col, L. Doretti, G.A. Longo, L. Rossetto, C. Zilio, Condensation inside and outside smooth and enhanced tubes – a review of recent research, Int. J. Refrig. 26 (4) (2003) 373–392. [2] A. Miyara, Condensation of hydrocarbons – a review, Int. J. Refrig. 31 (4) (2008) 621–632. [3] A. Dalkilic, S. Wongwises, Intensive literature review of condensation inside smooth and enhanced tubes, Int. J. Heat Mass Transf. 52 (15) (2009) 3409–3426. [4] Y. Wang, X. Mu, S. Shen, W. Zhang, Heat transfer characteristics of steam condensation flow in vacuum horizontal tube, Int. J. Heat Mass Transf. 108 (2017) 128–135. [5] T. Wu, K. Vierow, Local heat transfer measurements of steam/air mixtures in horizontal condenser tubes, Int. J. Heat Mass Transf. 49 (15) (2006) 2491–2501. [6] B. Ren, L. Zhang, H. Xu, J. Cao, Z. Tao, Experimental study on condensation of steam/air mixture in a horizontal tube, Exp. Therm. Fluid Sci. 58 (2014) 145– 155. [7] J.R. Thome, J. El Hajal, A. Cavallini, Condensation in horizontal tubes, part 2: new heat transfer model based on flow regimes, Int. J. Heat Mass Transf. 46 (18) (2003) 3365–3387. [8] S. Zhou, Y. Guo, X. Mu, S. Shen, Effect of design parameters on thermodynamic losses of the heat transfer process in LT-MEE desalination plant, Desalination 375 (2015) 40–47. [9] R. Kouhikamali, A.S. Kojidi, M. Asgari, F. Alamolhoda, The effect of condensation and evaporation pressure drop on specific heat transfer surface area and energy consumption in MED–TVC plants, Desalin. Water Treat. 46 (1– 3) (2012) 68–74.

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