Experimental study on natural circulation two-phase flow resistance in a vertical 3 × 3 rod bundle channel

International Journal of Heat and Mass Transfer 129 (2019) 920–935

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Experimental study on natural circulation two-phase flow resistance in a vertical 3
3 rod bundle channel Zhiqiang Zhu a,1, Chunping Tian b, Shengzhi Yu b, Changqi Yan a, Jianjun Wang a,⇑,1 a b

Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin 150001, China Wuhan Second Ship Design and Research Institute, Wuhan, Hubei 430205, China

a r t i c l e

i n f o

Article history: Received 26 May 2018 Received in revised form 17 August 2018 Accepted 6 October 2018

Keywords: Rod bundle Two-phase flow Natural circulation Frictional resistance Flow boiling

a b s t r a c t Natural circulation boiling flow resistance characteristics in a 3
3 rod bundle channel are experimentally studied in this paper. In order to achieve a stable two-phase natural circulation flow, the boiling flow phases are identified and measures such as decreasing the inlet fluid subcooling degree and increasing the inlet local resistance are adopted. The ONB during the experiment is determined by the wall temperature drop. The variation of pressure drop components, including the gravity pressure drop, the frictional pressure drop and the acceleration pressure drop, is given. For high mass quality conditions, the mass flow rate is observed declining with the increasing heat flux when the frictional pressure drop and the acceleration pressure drop grow faster than the gravity pressure drop reduction. A few homogeneous flow models and separated flow models for the two-phase frictional pressure drop are compared against experimental results. The homogeneous flow models can well predict the frictional pressure drops for low mass quality conditions and the Duckler model predictions are the most accurate with a mean relative error of 15.7%. In terms of the separated flow models, the Mishima and Sun models predictions are in good agreement with experimental results respectively for low and high mass quality conditions. Based on the Chisholm correlation, a new correlation is fitted by experimental results. The friction factors obtained in single-phase experiments are used to calculate the liquid phase frictional pressure drops. The comparison against experimental results and data from other literatures shows the new correlation can well predict the two-phase frictional pressure drops under given flow conditions. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Rod bundles are widely used as the coolant passage in reactor cores. Flow boiling would occur in Boiling Water Reactors (BWR) and Pressurized Water Reactors (PWR) under accident conditions, significantly affecting the coolant flow and heat transfer characteristics. Because of the complex geometric structures, the two-phase flow characteristics in rod bundle channels are more complex than that in regular channels. Although many researchers investigated the flow characteristics, such as the flow patterns, the void fraction and the pressure drops in rod bundle channels for adiabatic air-water two-phase flow and boiling flow, the results of natural circulation two-phase flow characteristics are scarce in the public literature. Therefore, it is valuable to do research on the natural circulation two-phase flow in a rod bundle channel.

⇑ Corresponding author. E-mail address: [email protected] (J. Wang). Permanent address: 3A Laboratory Building, 145 Nantong Street, Harbin, Heilongjiang 150001, China. 1

https://doi.org/10.1016/j.ijheatmasstransfer.2018.10.021 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

Despite much work on the two-phase flow resistance characteristics, no uniform method for the two-phase pressure drop calculation has been obtained. Lockhart and Martinelli [1] proposed the separated flow model and calculated correlations for two-phase flow resistance. Chisholm [2] concluded the two-phase multiplier / as a functional relation to the Martinelli parameter X. Scholars revised the C value in the Chisholm correlation based on experimental data for various channels. Mishima [3] conducted air-water two-phase flow experiments in small tubes and revised the C value as a function of the tube diameter. Hwang [4] investigated the pressure drop of the two-phase flow in small tubes and found the existing correlations disagreed with the experimental results. The liquid-only Reynolds number, the Martinelli parameter and the confinement number were integrated to obtain the accurate C value. Sun [5] collected a mass of data for the two-phase flow resistance in mini-channels, proposing a new formation of the Chisholm correlation. They found the C value closely related to the ratio of liquid-only and gas-only Reynolds number. Additionally, the exponentiation of the Martinelli parameter was considered a new value rather than one. Except for separated flow

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Nomenclature

General symbols D rod outer diameter (m) De hydraulic equivalent diameter (m) Dth thermal equivalent diameter (m) G mass flow rate (kgm2s1) g gravity acceleration (ms2) h enthalpy (kJkg1) hfg latent heat of vaporization (kJkg1) j superficial velocity (ms1) L length (m) P pressure (MPa) Pi rod pitch (m) q heat flux (kWm2) Re Reynolds number T temperature (K) ugj gas drift velocity (ms1) x mass quality DPacc acceleration pressure drop (Pa) DPf frictional pressure drop (Pa) DPg gravity pressure drop (Pa) DPt total pressure drop (Pa) models, homogeneous flow models were also widely used in the calculation of the two-phase flow pressure drop. Some researchers [6–8] proposed different methods for the two-phase equivalent viscosity calculation to obtain the two-phase frictional factor, furthermore the two-phase multiplier /. A few other researchers [9] directly revised the multiplier based on experimental results. For two-phase flow resistance in rod bundle channels, the experimental research is limited and the data are deficient. Grant [10] investigated the two-phase pressure drop in a rod bundle arranged in a triangular array with a Pi/D (ratio of the rod pitch and the rod outer diameter) of 1.25 and proposed corresponding calculation models. Schrage [11] performed an experimental investigation on the two-phase pressure drop in a horizontal tube bundle. They found the two-phase friction multiplier is related to the mass velocity and the flow patterns for a certain Martinelli parameter. Two-phase flow patterns and frictional pressure drops were experimentally studied by Narrow [12]. The frictional pressure drop was found closely dependent on the flow patterns and the pressure drop cannot be accurately predicted by the homogeneous flow models except for the conditions where the liquid superficial velocities were low enough. Tamai [13] conducted two-phase flow experiments in tight-lattice rod bundle channels with different gap widths and found larger frictional loss in the rod bundles with smaller gap widths. The measured pressure drops were compared with the Martinelli-Nelson’s correlation [14] and the results showed that the two-phase frictional pressure drops accounted for a major component. Yan [15] performed air–water two-phase flow experiments in a 3
3 rod bundle under static and rolling motion conditions. They found that the gas flow regime significantly affected the distribution parameter and divided the experimental data into different flow regimes by the gas Reynolds number. A new calculation method for the two-phase frictional pressure drops was proposed based on the Chisholm correlation. Natural circulation is extensively applied in industrial fields and studied by many researchers for its passive safety features. Ishii [16] conducted a dimensional analysis for a two-phase natural circulation flow and developed a group of similarity parameters which were significant to the frictional and gravity pressure drops calculation. Kamil [17] performed two-phase natural circulation experiments in a closed-loop thermosiphon system and proposed

Greek letters void fraction b volume void fraction l dynamic viscosity (Pas) q density (kgm3) r surface tension (Nm1) k friction factor

a

Superscript and subscripts B boiling cal calculation eq equilibrium exp experimental in inlet l liquid sat saturation sp single-phase tp two-phase tr true v vapor

an empirical correlation to predict the circulation rates. The Martinelli parameter was covered to show the effect of vapor quality. Hsu [18] studied the two-phase natural circulation under a small break loss of coolant accident condition. They found that the natural circulation rate strongly depended on the loop frictional resistance which had little effect on the flow termination yet. This was different from the results of Chen [19] who experimentally studied the flow and heat transfer characteristics in a narrow rectangular channel under natural circulation conditions. They found the saturated boiling natural circulation rate was sensitive to the thermodynamic quality. The circulation rate started to decrease when the thermodynamic quality is large enough because of the increasing frictional resistance. Flow fluctuations are quite likely to occur in two-phase natural circulation loops. Hence, many investigations on two-phase natural circulation fluctuations were conducted for the flow instabilities and rolling motion effects. Su [20] performed theoretical and experimental studies on the density wave oscillation in a two-phase natural circulation, analyzing the effects of experimental parameters such as flow rate and heat flux. Kyung [21] observed three basic flow modes in two-phase natural circulation experiments. In the first periodic circulation mode, the void fraction in the riser increased with the increasing heat flux. Churn or wispy-annular flow patterns occurred in the continuous circulation mode which was a stable mode. With the heat flux increasing, the void fraction started fluctuating and the flow rate decreased instead in the second periodic circulation mode. Wang [22] theoretically and experimentally studied the high volumetric quality natural circulation flow under rolling motion conditions, analyzing pressure drop components and the frictional coefficient. They found that the increase of driving head and the frictional coefficient made the circulation rate tend to be stable. Plenty of studies on two-phase flow resistance in various channels were performed and the flow rate variation characteristics were discussed by many researchers about the two-phase natural circulation. However, few researches on the two-phase frictional resistance in rod bundle channels were found in the open literature, let alone results for the two-phase natural circulation. In order to explore the two-phase flow resistance characteristics under natural circulation conditions, a series of experiments were

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performed in a 3
3 rod bundle channel. The pressure drop results under boiling natural circulation conditions were obtained and the effects of experimental parameters on the frictional resistance were analyzed. Based on the newly obtained experimental results, existing two-phase frictional pressure drop prediction models were evaluated and a new correlation was fitted for natural circulation two-phase flow in the 3
3 rod bundle channel. 2. Experimental apparatus 2.1. Experimental loop The experimental setup diagram is shown in Fig. 1. In the primary circulation loop, deionized water is heated in the rod bundle test section and then flows up through the riser into the condenser. The steam-water mixture is condensed into single-phase water in the condenser and then flows down through the downcomer into the preheater. The fluid is heated by the preheater, which has a maximum heating power of 45 kW, to a desired temperature and flows into the rod bundle test section. Thus, a complete circulation is finished. In order to achieve the natural circulation flow, the water is heated under forced circulation conditions till the temperature difference between the riser and the downcomer is large enough. Then turn off the pump and open another parallel branch. Thus the natural circulation is established. The operational pressure is maintained by the pressurizer, meanwhile, an electromagnetic flowmeter arranged before the test section entrance is used to measure the circulation flow rate. The secondary loop mainly includes a water tank and an air cooling tower, which both serve as heat sinks for the condenser. 2.2. Test section The test section details are shown in Fig. 2. The flow channel is composed of nine heating rods and the ceramic partition board. The rods are arranged in a 3
3 square array with a Pi/D of 1.38.

Two grid spacers are respectively fixed at the upper and lower positions of the heating rods. The heating rods are connected with two copper blocks at both ends and powered by a low-voltage DC supply with a maximum power of 45 kW. The cylindrical stainless steel shell serves as the pressure bearing and sealing part. Each heating rod is divide into three sections: a nickel rod on the top, a stainless rod in the middle and a copper rod at the bottom. Heat mainly generates in the stainless rod because of the higher electrical resistivity compared with the nickel rod and the copper rod. The stainless rod is placed in the middle of the flow channel with an 800-mm-long heating zone. In order to measure the wall temperature of the heating rod, two N-type sheathed thermocouples are brazed on the inner wall in different axial positions inside each rod. Thus, in total eighteen thermocouples are arranged inside nine rods in different six axial positions. With respect to the fluid temperature measurement, two T-type sheathed thermocouples are respectively arranged at the inlet and outlet of the test section. Additionally, another T-type sheathed thermocouple is arranged at the flow channel exit to obtain the exit fluid temperature more accurately. Two pressure drop measurement sections are arranged along the flow channel. One contains a grid spacer and the other doesn’t, respectively measuring the local pressure drop by a Yokogawa EJA110E differential pressure transmitter and the frictional pressure drop by a Rosemount 3051CD differential pressure transmitter. 2.3. Data processing For pressure drop measurement section B, the total pressure drop DPt includes the gravity pressure drop DPg, the frictional pressure drop DPf and the acceleration pressure drop DPacc.

DPt ¼ DPg þ DPf þ DPacc

ð1Þ

For a single-phase section, the gravity pressure drop is expressed as below and the acceleration pressure drop is considered ignored.

Fig. 1. Schematic diagram of the experimental loop. 1 – test section; 2 – differential pressure transmitter; 3 – thermocouple; 4 – condenser; 5 – pump; 6 – valve; 7 – pressurizer; 8 – pressure transmitter; 9 – preheater; 10 – filter; 11 – electromagnetic flowmeter; 12 – water tank; 13 – cooling tower.

Z. Zhu et al. / International Journal of Heat and Mass Transfer 129 (2019) 920–935

(a) side view of the test section

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(b) heating rod structure

(c) cross sectional view of the flow channel Fig. 2. Test section details.

DPg; sp ¼ ql gLsp

ð2Þ

where Lsp is the single-phase section length and ql denotes the liquid density. For a two-phase section, the gravity pressure drop and the acceleration pressure drop are calculated by the separated flow models.

Z

Ltp

DPg;tp ¼ 0

Z

DPacc;tp ¼ 0

½ql ð1  aÞ þ qv agdl

Ltp

( " #) d ð1  xÞ2 x2 G2 dl þ dz ql ð1  aÞ qv a

ð3Þ

ð4Þ

where Ltp is the two-phase section length and qv denotes the vapor density. In order to reduce the heat loss, the whole barrel of the test section is covered with two-tier insulating layers and the flow channel is separated with the barrel through an isolation chamber full of stagnant water. The thermal efficiency g for the heating section is calculated by the ratio of the enthalpy increment to the electric power. It is close to unity and never lower than 95% for the stable single-phase flow in all recorded cases. For the two-phase flow, the enthalpy increment is hard to obtain directly. Therefore, the thermal efficiency for the two-phase flow is replaced by the corresponding single-phase thermal efficiency before ONB.

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g¼

Z. Zhu et al. / International Journal of Heat and Mass Transfer 129 (2019) 920–935

Mðhout  hin Þ UI

ð5Þ

Mass quality is a significant parameter for two-phase pressure drop calculation. Many methods were proposed [23,24], while giving a relatively poor prediction of the mass quality for a low mass flow rate subcooled boiling flow. Sun [25] proposed a model based on thermal equilibrium equations for the Onset of Nucleate Boiling (ONB) and Onset of Saturated Boiling (OSB). Xi [26] found that the model was not completely applicable to the two-phase natural circulation and modified a calculation coefficient based on the natural circulation experimental data. The mass quality in this paper is calculated by the Xi model. The mass quality is calculated as

8 0; xeq < xeq;B ; > > h i > > < 1 ðx  x Þ p2 h2k  p4 h4k þ p6 h6k  ::: eq eq;B ð2kþ1Þ2! 2 ð4kþ1Þ4! ð6kþ1Þ6! xtr ¼ ; > > xeq;B 6 xeq 6 xeq;OSB ; > > : xeq ; xeq;OSB < xeq < 1

ð6Þ

xeq;B ¼ maxðxeq;in ; xeq;ONB Þ

ð7Þ

h ¼ ðxeq  xeq;B Þ=ðxeq;OSB  xeq;B Þ

ð8Þ

k ¼ 0:032ðql ugj Dth =ll Þ h

xeq;OSB

0:33

  1:26 xeq;in min 1; xeq;ONB

i  ð2kþ1Þ2! þ ð4kþ1Þ4!  ð6kþ1Þ6! þ ::: i xeq;B ¼h p2 p4 p6 þ ð4kþ1Þ4!  ð6kþ1Þ6! þ ::: 2  ð2kþ1Þ2! p2

p4

DT sat;ONB ¼ 25ðq=106 Þ xeq;ONB ¼

ð9Þ

ð10Þ

0:25

ð11Þ

expðP=6:2Þ

hONB  hin hfg

ð12Þ

With regard to the void fraction calculation, Chexal [29] developed a method based on the drift flow model for a wide range of two-phase flow conditions and fuel assembly geometries. The void fraction in this paper is calculated by the Chexal–Lellouche correlations, which is used to predict the void fraction in rod bundle channels in the RELAP5/MOD3.2.2 code.

C0 ¼

p6

L K 0 þ ð1  K 0 Þar

ð13Þ

 0:25 ðql  qv Þrg C2C3C4C9 ugj ¼ 1:41 2

ð14Þ

ql

a¼

jg C 0 j þ ugj

ð15Þ

where the parameters L, K0, r and different C values are other parameters in the Chexal–Lellouche correlations. The thermodynamic parameter ranges in the experiments are listed in Table 1. 2.4. Uncertainty analysis

where xeq is the thermodynamic quality and Dth is the thermal equivalent diameter. In the original Xi model, the ONB thermodynamic quality xeq, ONB, which is a significant parameter in this model, is calculated by the Miropolskii [27] correlations. However, the predicted locations of ONB are quite different from the experimental values. Therefore, the Jens and Lottes [28] correlation is used to calculate the xeq, ONB. The calculated mass quality and corresponding thermodynamic quality results along the flow direction are shown in Fig. 3 for different working conditions. The calculated mass quality clearly differs from the other one before saturated boiling. With the mass quality increasing, the two kinds of quality are gradually approaching the same, especially after the OSB point. In this paper, the fully developed saturated boiling in the whole heating section

(a) P = 0.2 MPa q = 128 kW/m2

is hard to occur under most working conditions. Therefore, the thermodynamic quality and the xeq, ONB are not suitable for the pressure drop calculation and the mass quality is necessary to be obtained.

Tsub, in = 10 K

The N-type and T-type sheathed thermocouples respectively have accuracies of ±1 K and ±0.5 K. The uncertainty for the differential pressure transmitter is ±0.055% and the flow rate has an accuracy of ±0.3%. The heating power is measured with an

Table 1 Experimental parameters. Parameters

Range

Operating pressure (MPa) Heat flux in test section (kW/m2) Inlet fluid subcooling (°C) Mass flow rate (kg m2 s1)

0.1–1.0 5–330 10–50 100–600

(b) P = 0.2 MPa q = 184 kW/m2

Fig. 3. Calculated mass quality results.

Tsub, in = 30 K

Z. Zhu et al. / International Journal of Heat and Mass Transfer 129 (2019) 920–935

uncertainty of ±0.5%. Kline & McClintock [30] methodology is used to calculate and evaluate the uncertainties of computed results.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n  2 uX @F U F ¼ t upi @pi i¼1

ð16Þ

where F denotes a function of variables p1, p2, . . . pn; UF means the uncertainty for F and upi is the uncertainty for the variable pi. The relative uncertainty is expressed as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n  2 uX up UF i ¼ t F pi i¼1

ð17Þ

For the experimental parameters in this paper, the calculated relative uncertainty ranges for mass quality and void fraction are respectively 6.2–10.4% and 7.1–11.3%. Moreover, the relative uncertainty range for the two-phase frictional pressure drop is 8.5–14.0%. 3. Experimental results and discussions 3.1. Determination of two-phase section Different from air-water two-phase flow, the boiling flow resistance is much more complex. Since the boiling flow is developing along the flow channel, the determination of the two-phase section is important. Additionally, the flow instability, which is undesirable in the experiments, may occur with the increasing heat flux. In order to accurately investigate the boiling flow resistance, a stable two-phase flow needs to be obtained. Fig. 4 shows the flow rate and pressure drop changes with the increasing heat flux in single-phase, two-phase and instability regions. For the subcooled boiling flow, the flow instability occurs when the heat flux increases at the tb moment. For the saturated boiling flow, methods such as decreasing inlet fluid subcooling degree and increasing local resistance at the inlet are adopted to restrain the flow instability. Thus obtaining the stable saturated boiling flow at the tc moment with the increasing heat flux. More details about the ta and tb moments are shown in Fig. 5. The fluctuation amplitude of measured pressure drops changes obviously when the boiling begins in the pressure drop measurement section. The flow rate and the pressure drop both oscillate intensively when the flow instability occurs. In this paper, the flow resistance in the twophase regions are focused.

925

Apart from the flow phases, the two-phase section length is significant as well. As the demarcation point between the singlephase and the two-phase regions, ONB locations significantly affect the two-phase frictional pressure drops calculation. The ONB is usually identified by visualized methods, the abrupt decrease of wall temperature and the abrupt rise of the heat transfer coefficient. As shown in Fig. 6a, the ONB occurrence in this paper is determined by the wall temperature drop. Limited by the measuring points distribution and based on experimental ONB locations, the Jens and Lottes correlations are used to calculate the xeq, ONB. Fig. 6b shows that for a certain heat flux, the ONB location is determined by the wall temperature variation in different positions and the downstream section between two wall temperature thermocouples is recognized as the ONB region. In order to verify the accuracy of the calculated ONB locations, Fig. 7 shows the calculated ONB location results and corresponding experimental ONB regions with different heat fluxes. It is clear that the variation tendency of the calculated values is reasonable and consistent with the experimental regions. The majority of the calculated locations are within the experimental ONB regions. Fig. 8 shows the comparison of the calculated ONB locations and the experimental ONB regions for all the working conditions. More than 80% calculated values are within the experimental ONB regions. The Jens and Lottes correlation is considered applicable for the ONB location prediction in the experiments, especially when the ONB location is near the outlet of the heating section, namely for relatively lower heat flux. 3.2. Two-phase pressure drop variation The two-phase pressure drops are composed of the gravity pressure drop, the frictional pressure drop and the acceleration pressure drop. Fig. 9 shows the pressure drops variation with the heat flux in pressure drop measurement section B for different inlet fluid subcooling degrees. With the heat flux increasing, the gravity pressure drop decreases but the frictional pressure drop and the acceleration pressure drop increases. For the lower fluid subcooling condition (a), the saturated boiling is easier to achieve. The average mass quality in the pressure drop measurement section can reach a relatively larger value, as shown in Fig. 10. The pressure drops change rapidly first and then the variation tends to be gentle because the void fraction varies fast in the low mass quality conditions. When the saturated boiling is full of the whole pressure drop measurement section, the pressure drop becomes insensitive to the mass quality as well as the heat flux. Addition-

(a) before flow instability Fig. 4. Different phases in flow boiling.

(b) after flow instability

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(a) onset of boiling

(b) onset of flow instability Fig. 5. Flow phases change details.

P = 0.2 MPa Tsub, in = 30 K (a) ONB for a location

Tsub, in = 40 K P = 1.0 MPa q = 128 kW/m2 (b) ONB for a heat flux Fig. 6. Determination of ONB.

P = 0.2 MPa q = 51–169 kW/m2

Tsub, in = 10 K

Fig. 7. ONB location variation with heat flux.

Fig. 8. Comparison of calculated ONB locations and experimental ONB regions.

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(a) P = 0.2MPa, Tsub, in =10 K

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(b) P = 0.2MPa, Tsub, in =30 K

Fig. 9. Pressure drops variation with heat flux.

(a) P = 0.2MPa, Tsub, in =10 K

(b) P = 0.2MPa, Tsub, in =30 K

Fig. 10. Pressure drops variation with mass quality.

ally, the acceleration pressure drop increases faster and is larger than the frictional pressure drop in the low mass quality conditions. However, in the high mass quality conditions, the frictional pressure drop increases a little faster than the acceleration pressure drop. For the higher fluid subcooling condition (b), the subcooled boiling and the saturated boiling both exist in the pressure drop measurement section. The mass quality is limited and the pressure drops vary in the same tendency with the increasing heat flux. Similarly, the acceleration pressure drop is always larger than the frictional pressure drop. Fig. 11 shows the ratio of each kind of pressure drop to the total value. The gravity pressure drop dominates in the low mass quality conditions, however, the acceleration pressure drop and the frictional pressure drop approach the gravity pressure drop with the increasing heat flux. It is known that the driving force, provided by the density difference between the hot leg and the cold leg, increases with the decreasing gravity pressure drop in the test section. The resistance including frictional pressure drop and the acceleration pressure drop increases rapidly as well in the high mass quality conditions, which may lead to the net driving force reduction. Fig. 12 shows the mass flow rate variation with the heat flux. The mass flow rate first increases in the low mass quality con-

ditions and then stay approximately unchanged. When the heat flux continues increasing, the mass flow rate decreases with the heat flux, which means that the resistance grows faster than the gravity pressure drop reduction. Moreover, the flow rate reduction does not occur in the higher fluid subcooling condition for the limited mass quality. 3.3. Comparison with two-phase pressure drop correlations The two-phase pressure drop calculation is complex and the frictional pressure drop is the most important factor. By simplifying the true flow conditions, many methods are proposed by researchers for different flow channels, among which the homogeneous flow models and the separated flow models are evaluated by comparing with the experimental data. The thermodynamic parameters have been given in Table 1 and the mass quality is ranging from 0 to 1.3%. 3.3.1. Homogeneous flow models As typical two-phase frictional pressure drop calculation methods, the calculation methods for single-phase pressure drop are

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(a) P = 0.2MPa, Tsub, in =10 K

(b) P = 0.2MPa, Tsub, in =30 K

Fig. 11. Ratio of each component to total pressure drop.

(a) P = 0.2MPa, Tsub, in =10 K

(b) P = 0.2MPa, Tsub, in =30 K

Fig. 12. Mass flow rate variation with heat flux.

Table 2 Homogeneous flow models. Reference

Correlation

McAdams

1=ltp ¼ x=lv þ ð1  xÞ=ll ltp ¼ xlv þ ð1  xÞll ltp ¼ blv þ ð1  bÞll ltp ¼ ll =½ð1  xÞ þ xðql =qv Þ0:5  ltp ¼ ll ltp ¼ ll lv =½lv þ x1:4 ðll  lv Þ

Cicchitti Duckler Akers [31] Owens [32] Lin [33] Tran

n o /2lo ¼ 1 þ ð4:3C2  1Þ N conf ½xð1  xÞ0:875 þ x1:75

C2 ¼

ðdp=dlÞf;vo ðdp=dlÞf;lo

l qv Þ ; N conf ¼ ½r=gðqDe

0:5

where /2lo denotes a multiplier for the whole two-phase mixture flows as liquid. /2lo is expressed by transforming the frictional pressure drop gradient as Fig. 13. Single-phase friction factors.

used in homogeneous flow models. The two-phase resistance is expressed as

  dPf;tp dPf ¼ /2lo dl dl lo

ð18Þ

/2lo ¼

   ktp q 1þx l 1 klo qv

ð19Þ

The single-phase friction factors in the above correlations are generally calculated by the Blasius correlation for conventional flow channels. In view of the complex structure of the rod bundle

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(a) comparison with McAdams model

(b) comparison with Cicchitti model

(c) comparison with Duckler model

(d) comparison with Akers model

(e) comparison with Owens model

(f) comparison with Lin model

Fig. 14. Comparison of experimental data and homogeneous flow models.

channel, the Blasius correlation is replaced by the fitted correlations obtained from the single-phase natural circulation experiments, as shown in Fig. 13.

ksp ¼ 1:48Re0:44

ð20Þ

Thus, /2lo can be expressed as

/2lo

ll ¼ ltp

!0:44 

 1þx

ql 1 qv

 ð21Þ

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P = 0.2 MPa q = 60–200 kW/m2

(g) comparison with Tran model

Tsub, in = 10 K

Fig. 16. Two-phase equivalent viscosity for homogeneous flow models.

Fig. 14 (continued) Table 4 Separated flow models. Reference

Correlation

Chisholm Mishima Hwang

C ¼ 5; 10; 12; 20 C ¼ 21½1  expð0:319DeÞ 0:82 C ¼ 0:227Re0:452 X 0:32 N conf lo  h  i Rel 0:513 C ¼ 26 1 þ 1000 1  exp 0:27N ; Rel < 2000 conf þ0:8 qffiffiffiffiffiffiffi 0:4 2 1x C 1 v C ¼ 1:79ðRe Þ ¼ 1 þ þ ; Rel > 2000 ; / 1:19 2 l Rel x X X

Sun

C ¼ 21½1  expð0:358=N conf Þ

Zhang [34]

Fig. 15. Void fraction and superficial velocity results.

Many researchers proposed calculation methods for /2lo based on the two-phase equivalent viscosity or the experimental corrections. Several common homogeneous flow correlations are listed in Table 2. As shown in Fig. 14, the calculated frictional pressure drop results by the above models are compared against the experimental data over a large range of working conditions. The homogeneous flow models predictions are in good agreement with experimental data when the frictional pressure drops are relatively small, namely the low mass quality conditions (x < 0.3%). For high mass quality conditions (x > 0.3%), the calculated results are all obviously larger than the experimental data. The one reason is that the energy transfer between the two phases is fast enough for subcooled boiling. The thermal and flow parameters barely change

along the flow channel and the thermodynamic unequilibrium effect can be ignored. However, with the increasing heat flux, the homogeneous flow models have poor predictions for the nonuniform distribution of the two phases. Additionally, as shown in Fig. 15, the void fraction reaches 30% when the mass quality is 0.3%. It is a relatively larger value compared with the void fractions under most experimental conditions. The superficial velocity ratio jg/jl is roughly larger than 2 when the mass quality is over 0.3%, which means the slip between two phases cannot be ignored. It is not consistent with the homogeneous models assumptions. Thus, the bad prediction by homogeneous models for high mass quality conditions is explained. However, it is different from the results of Yan [15], who concluded that the homogeneous models were suitable for air–water two-phase pressure drop predictions in a vertical 3
3 rod bundle channel. The adiabatic flow conditions and the smaller Pi/D (more compact rod bundle structure) that improves the flow velocity are both beneficial to the homogeneous flow models predictions. Table 3 shows the mean relative error (MRE) results for the models predictions. Obviously the mean relative errors for x < 0.3% conditions are much smaller than overall results since the relative errors for x > 0.3% conditions are all larger than 30%.

Table 3 Mean relative errors for homogeneous flow models. Reference

Overall MRE (%)

Within error band (%)

x < 0.3% MRE (%)

Within error band (%)

McAdams Cicchitti Duckler Akers Owens Lin Tran

68.7 69.9 45.1 68.0 70.0 69.8 117.0

32.4 32.4 42.2 32.4 32.4 32.4 30.4

27.2 27.4 15.7 27.2 27.4 27.0 40.7

49.3 49.3 61.2 49.3 49.3 49.6 46.3

Z. Zhu et al. / International Journal of Heat and Mass Transfer 129 (2019) 920–935

(a) comparison with Chisholm model

(b) comparison with Mishima model

(c) comparison with Hwang model

(d) comparison with Sun model

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(e) comparison with Zhang model Fig. 17. Comparison of experimental data and separated flow models.

It is not suitable to discuss this part of experimental results with homogeneous flow models. The Duckler model has the best predictions and the Tran model does the worst, with the mean relative errors of respectively 15.7% and 40.7%. The relative errors of rest models such as McAdams, Cicchitti, Akers, Owens and Lin are all very close, nearly 27%. In the Duckler model, the volume void frac-

tion instead of the mass quality is used to represent the contribution share of the viscosity for each phase. When the mass quality is extremely small, the volume void fraction is small as well. The differences of the viscosity contribution share for different models are limited. However, when the mass quality is a bit higher, even though in the low mass quality conditions (x < 0.3%), the volume

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Table 5 Mean relative errors for separated flow models. Reference

Overall MRE (%)

Within error band (%)

Chisholm Mishima Hwang Sun Zhang

54.2 27.5 67.8 36.4 60.0

30.6 59.7 32.3 41.9 27.4

void fraction contribution share is obviously different from the mass quality. Fig. 16 shows the two-phase equivalent viscosity results for different models. It is evident that the two-phase equivalent viscosity for the volume void fraction based Duckler model is much smaller than other models, since the gas phase viscosity contribution share is larger than others. The two-phase equivalent viscosity for other mass quality based models is close because the mass quality varies little, which accounts for the approximately same mean relative errors for these models. 3.3.2. Separated flow models Different from homogeneous flow models, pressure drop for each phase is calculated respectively and the total result is obtained by a two-phase multiplier. According to Lockhart [1], the two-phase frictional pressure drops are calculated as

    dPf;tp dPf dP f ¼ /2v ¼ /2l dl dl l dl v

ð22Þ

where /2l and /2v denote multipliers respectively for the liquid phase and the gas phase. Chisholm [2] proposed a correlation for the twophase multiplier based on the Martinelli parameter X. The parameter C is significant and many researchers revised it based on different experimental data. Several typical separated flow correlations are listed in Table 4. Likewise, with regard to the single-phase friction factor, the Blasius correlation is replaced by Eq. (20) as well.

C 1 /2l ¼ 1 þ þ 2 X X

ð23Þ

As shown in Fig. 17, the calculated frictional pressure drop results by the above separated flow models are compared against the experimental data. For low pressure drops conditions, the separated flow models also have good predictions. With the pressure drops increasing, the Chisholm, Hwang, and Zhang models are generally larger than the experimental results. The Mishima and Sun

(a) new correlation for the liquid phase multiplier

models have better predictions when the mass quality is relatively larger. Especially the former can predict the pressure drops best except the high mass quality conditions. When the mass quality continues increasing, the Sun model gives best predictions for the frictional pressure drops. Table 5 shows the mean relative error results for the separated flow models predictions. The mean relative errors for Chisholm, Hwang, and Zhang models are all more than 50%, which are considered unsuitable for the experimental frictional pressure drops prediction. Mishima model has the best prediction accuracy as 27.5%, while the relative errors are large for high mass quality conditions. Instead, Sun model can well predict frictional pressure drops under high mass quality conditions and the mean relative error is within 10%. In the Hwang model, the tube diameter, the surface tension and the Reynolds number are considered to revise the C parameter. However, the correlation is obtained by the pressure drops in micro circular tubes with hydraulic equivalent diameters less than 1 mm. Mishima and Zhang models are both based on the results from air-water two-phase flows in circular capillary tubes. Mishima revised the C parameter by the tube diameter and Zhang considered the surface tension and the two phases densities additionally, which explains the remarkable MRE differences between the two models. 3.4. New correlation for two-phase frictional pressure drop Even though the Sun and Mishima models have relatively good predictions, the applicable ranges are different and limited separately. In order to predict the two-phase frictional pressure drops for the rod bundle more accurately, a new correlation is necessary. Based on the evaluation for the two-phase pressure drop models, a revision for the Eq. (23) is reasonable. Chisholm [2] proposed the following equations for the two-phase pressure drop calculation.

/2l ¼

 1n   Ag Ag 1 þ þ 1 Al a1þn Al Z 2 l 1

Al X 1 ¼ ;C ¼ Z þ Z Ag Z

ð24Þ

ð25Þ

For the Eq. (24), the n value is related to the flow conditions. The Eq. (23) is a recommended correlation for engineering design and the original n value is treated as 0. In view of the poor prediction

(b) comparison to experimental data

Fig. 18. Comparison of new correlation results and experimental data.

Z. Zhu et al. / International Journal of Heat and Mass Transfer 129 (2019) 920–935

(a) comparison with Chisholm data

(b) comparison with Mishima data

(c) comparison with Hwang data

(d) comparison with Sun data

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Fig. 19. Comparison of fitted correlation and other data.

/2l ¼ 1 þ

Table 6 Comparison of new correlation to other data. Reference

Data points

Overall MRE (%)

Within error band (%)

Chisholm Mishima Hwang Sun

803 79 78 100

30.4 30.9 81.1 9.1

56.1 51.2 0 95.7

of the Eq. (23), the n value is considered not to be 0. Thus a variable n is added to the Martinelli parameter X as an exponential term in the new correlation. According to the analysis of Sun [5], the parameter C in the Chisholm correlation is not a constant but related to the Laplace number Nconf and the liquid Reynolds number Rel. The C/X depends on the ratio of the liquid and the gas Reynolds number, especially when either one is larger than 2000. Therefore, the parameter C is a function of the Nconf and the Rel. In view of the poor prediction accuracy by the Nconf based Zhang model, the parameter C is considered related to the equivalent diameter De rather than the Nconf. The equivalent diameter for the rod bundle channel in the experiments is constant, thus the parameter C is defined as a constant in the fitted correlation. The fitted correlation is in the form of

C 1 þ Xn X2

ð26Þ

The fitted correlation is shown in Fig. 18a and the parameter C and the power n are respectively 26.8 and 1.5. Fig. 18b shows the comparison of experimental data and predicted two-phase frictional pressure drops by the new correlation. The relative errors of more than 90% predicted values are within ±20% and the overall mean relative error is 12.5%. Fig. 19 shows the comparison of the new correlation in this paper and the experimental data from other literatures. Additionally, the errors of the comparison results are shown in Table 6. The correlation is roughly consistent with the Chisholm and Mishima data while the divergence is relative large. The mean relative errors are approximately 30%. The calculated results are much larger than Hwang data because the range of X in this paper is totally different from Hwang data. The Sun data are in best agreement with the correlation because of the similar experimental conditions. The mean relative error for Sun data is within 10%. However, the predictions are relative poor for low X regions which is probably due to the difference in test section. For the low X conditions, the effect of surface tension and gap size in the mini channels are obvious [5]. According to the comparison with the data from other literatures, the relative errors are within acceptable limits. Thus the calculation results of the fitted correlation are reasonable and valid.

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The new correlation is considered applicable for the two-phase frictional pressure drops prediction in the rod bundle channel under natural circulation conditions. The applicative flow condition ranges included the system pressure 0.1–1 MPa, the mass flow rate 100–600 kg m2 s1 and the mass quality 0–0.13. 4. Conclusions Two-phase flow resistance characteristics under natural circulation conditions in a 3
3 rod bundle channel are experimentally studied in this paper. The flow boiling phases are identified and boiling pressure drops components variation is studied. A few present two-phase pressure drops correlations are evaluated and a new correlation is fitted based on the experimental results. (1) With the increasing heat flux, the flow instability occurs easily under natural circulation subcooled boiling conditions. The flow phases recognition for the stable flow is necessary. The wall temperature drop method for the ONB determination is applicable for the experiments. Compared with the ONB regions recognized by the wall temperature results, the Jens and Lottes correlation is considered reasonable for the ONB location prediction in the experiments. (2) The gravity pressure drops decrease while the frictional pressure drops and the acceleration pressure drops increase with the increasing heat flux. The acceleration pressure drops increase faster than the frictional pressure drops under the low mass quality conditions. For high mass quality conditions, if the resistance including frictional pressure drop and the acceleration pressure drop grows even faster than the gravity pressure drop reduction, the mass flow rate decreases with the increasing heat flux. (3) A few present two-phase pressure drops correlations, including homogeneous flow models and separated flow models, are compared with experimental data over a large range of flow conditions. The homogeneous flow models predictions are in good agreement with experimental data for low mass quality conditions (x < 0.3%) and the Duckler model has the best predictions with a mean relative error of 15.7%. For the separated flow models, the Mishima and Sun models have better predictions respectively for lower and higher mass quality conditions, with mean relative errors of 27.5% and 36.4%. (4) In the form of the Chisholm correlation, a new correlation for the two-phase frictional pressure drop is fitted by the experimental results. The friction factor results for the singlephase natural circulation experiments are adopted in place of the Blasius correlation when calculating the liquid phase frictional pressure drops. The new correlation is tested with data from other literatures for the reasonability and validity. Under the given flow conditions, the new correlation can well predict the two-phase frictional pressure drops in the experiments with a mean relative error of 12.5%. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgements The authors greatly appreciate support from the Natural Science Foundation of China (Grant No. 11675045) and support from the Open Fund Program of the Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education of China (ARES-2017-02). Additionally, the authors are thankful for support

from the Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, China. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.10.021. References [1] R.W. Lockhart, Proposed correlation of data for isothermal two-phase, twocomponent flow in pipes, Chem. Eng. Prog. (1949) 4539–4548. [2] D. Chisholm, A theoretical basis for the Lockhart-Martinelli correlation for twophase flow, Int. J. Heat Mass Transf. 10 (12) (1967) 1767–1778. [3] K. Mishima, T. Hibiki, Some characteristics of air-water two-phase flow in small diameter vertical tubes, Int. J. Multiphase Flow 22 (4) (1996) 703–712. [4] Y.W. Hwang, M.S. Kim, The pressure drop in microtubes and the correlation development, Int. J. Heat Mass Transf. 49 (11–12) (2006) 1804–1812. [5] L. Sun, K. Mishima, Evaluation analysis of prediction methods for two-phase flow pressure drop in mini-channels, Int. J. Multiphase Flow 35 (1) (2009) 47– 54. [6] W.H. Mcadams, Vaporization inside horizontal tubes-II, Benzene oil mixtures, Trans. ASME (1942) 64. [7] A. Cicchitti, C. Lombardi, M. Silvestri, et al., Two-phase cooling experiments: pressure drop, heat transfer and burnout measurements, Energia Nucleare (1960). [8] A.E. Dukler, M. Wicks, R.G. Cleaveland, Pressure drop and hold up in two-phase flow, 1964. [9] T.N. Tran, M.C. Chyu, M.W. Wambsganss, et al., Two-phase pressure drop of refrigerants during flow boiling in small channels: an experimental investigation and correlation development, Int. J. Multiphase Flow 26 (11) (2000) 1739–1754. [10] I.D.R. Grant, I. Murray, Pressure drop on the shell-side of a segmentally baffled shell-and-tube heat exchanger with vertical two-phase flow, Nat. Eng. Lab. (1972). [11] D.S. Schrage, J.T. Hsu, M.K. Jensen, Two-phase pressure drop in vertical crossflow across a horizontal tube bundle, Aiche J. 34 (1) (2010) 107–115. [12] T.L. Narrow, S.M. Ghiaasiaan, S.I. Abdel-Khalik, et al., Gas–liquid two-phase flow patterns and pressure drop in a horizontal micro-rod bundle, Int. J. Multiphase Flow 26 (8) (2000) 1281–1294. [13] H. Tamai, M. Kureta, A. Ohnuki, et al., Pressure drop experiments using tightlattice 37-rod bundles, J. Nucl. Sci. Technol. 43 (6) (2006) 699–706. [14] R.C. Martinelli, D.B. Nelson, Prediction of pressure drop during forcedcirculation boiling of water, Trans. ASME 70 (1948) 695–702. [15] C. Yan, J. Shen, C. Yan, et al., Resistance characteristics of air–water two-phase flow in a rolling 3
3 rod bundle, Exp. Therm. Fluid Sci. 64 (2015) 175–185. [16] M. Ishii, I. Kataoka, Scaling laws for thermal-hydraulic system under single phase and two-phase natural circulation, Nucl. Eng. Design 81 (3) (1984) 411– 425. [17] M. Kamil, S.S. Alam, H. Ali, Prediction of circulation rates in vertical tube thermosiphon reboiler, Int. J. Heat Mass Transf. 38 (4) (1995) 745–748. [18] J.T. Hsu, M. Ishii, T. Hibiki, Experimental study on two-phase natural circulation and flow termination in a loop, Nucl. Eng. Design 186 (3) (1998) 395–409. [19] C. Chen, Studies of flow boiling characteristics in a rectangular narrow channel in non-inertial frame, Harbin Eng. Univ. (2016). [20] S. Guanghui, J. Dounan, K. Fukuda, et al., Theoretical and experimental study on density wave oscillation of two-phase natural circulation of low equilibrium quality, Nucl. Eng. Design 215 (3) (2002) 187–198. [21] I.S. Kyung, S.Y. Lee, Experimental observations on flow characteristics in an open two-phase natural circulation loop, Nucl. Eng. Design (1994). [22] Z. Wang, S. Tan, W. Zhang, et al., Characteristics of high volumetric quality flow of natural circulation under rolling motion, Atom. Energy Sci. Technol. 47 (3) (2013) 371–375. [23] S. Levy, Forced convection subcooled boiling—prediction of vapor volumetric fraction, Int. J. Heat Mass Transf. 10 (7) (1967) 951–965. [24] P. Saha, N. Zuber, Point of net vapor generation and vapor void fraction in subcooled boiling, Analyst 112 (3) (1974) 259–261. [25] Q. Sun, R. Yang, S. Tan, et al., Prediction of true mass quality in subcooled steam-water flow boiling, J. Tsinghua Univ. 44 (11) (2004) 1580–1584. [26] Z. Xi, Q. Sun, J. Chen, et al., Predictive and experimental study of cross-section average void fraction in sub-cooled boiling with natural circulation, Nucl. Power Eng. 04 (2005) 312–316. [27] Z.L. Miropolskii, P.E. Snelobe, A.E. Kalamesebe, Void fraction of water-steam mixture flow with or without heat transfer, Teploenergetika 5 (1971) 374– 379. [28] W.H. Jens, P.A. Lottes, Analysis of heat transfer, burnout, pressure drop and density data for high pressure water, Boiling 2 (9) (1951) 51–59. [29] B. Chexal, G. Lellouche, J. Horowitz, et al., A void fraction correlation for generalized applications, Prog. Nucl. Energy 27 (1992) 255–295. [30] S.J. Kline, F. McClintock, Describing uncertainties in single-sample experiments, Mech. Eng. 75 (1) (1953) 3–8.

Z. Zhu et al. / International Journal of Heat and Mass Transfer 129 (2019) 920–935 [31] W.W. Akers, H.A. Deans, O.K. Crosser, Condensing heat transfer within horizontal tubes, Chem. Eng. Progr. (1957). [32] W.L. Owens, Two-phase pressure gradient, Int. Develop. Heat Transf. (1961) 363–368.

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[33] S. Lin, C.C.K. Kwok, R.Y. Li, et al., Local frictional pressure drop during vaporization of R-12 through capillary tubes, Int. J. Multiphase Flow 17 (1) (1991) 95–102. [34] W. Zhang, Study on Constitutive Equations for Flow Boiling in Mini-Channels, Kyoto University, 2005.