- Email: [email protected]
Evaluation of typical interfacial area transport models for bubbly flow
International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Contents lists available at ScienceDirect
International Journal of Advanced Nuclear Reactor Design and Technology journal homepage: http://www.keaipublishing.com/en/journals/ international-journal-of-advanced-nuclear-reactor-design-and-technology/
Evaluation of typical interfacial area transport models for bubbly flow Hang Liu a, *, Jianyong Lai a, Yi Li a, Yulong Zhang a, Xiaoquan Yu a, Minghao Liu a, Liangming Pan b, ** a b
Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu, 610213, China Key Laboratory of Low-grade Energy Utilization Technologies and Systems (Chongqing University), Ministry of Education, Chongqing, 400044, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 March 2019 Received in revised form 22 June 2019 Accepted 11 July 2019 Available online xxx
Interfacial area concentration is one of the most important parameters in two-phase flow. This parameter also reflects the interfacial area of mass, momentum and energy transfer in two-fluid model. The prediction of interfacial area concentration is essential to close the interfacial transfer terms in the two-fluid model. For the modeling of the one-dimensional interfacial area transport equation for adiabatic bubbly flow, existing typical sink and source terms have been reviewed and evaluated by the experimental data of vertical pipes and rod bundles. The interactive mechanisms of bubbles and turbulent eddies have also been analyzed, including bubbles breakup and bubbles coalescence. The evaluation results show the interfacial area transport model of Hibiki and Ishii (Liu and Hibiki) can provide the most accurate prediction of interfacial area concentration for pipes with the relative error of 8.22% for bubbly flow. Their model has also been evaluated by the experiment data of rod bundle with the relative error of 21.89% for bubbly flow. © 2019 Xi'an Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Interfacial area transport Sink terms Source terms Evaluation
1. Introduction Accurate prediction for the behavior of heat exchangers or nuclear reactors is of great important significance for long-term safe operation or accident scenarios. Two-fluid model is the most detailed and accurate macroscopic formulation in two-phase flow models. The behavior prediction of nuclear reactor system under accident conditions is mainly based on the computable code. Reliable two-phase flow models are important to obtain the accurate prediction of the thermal-hydraulic behavior. Two-fluid model is considered as the most accurate and detailed model to simulated two-phase flow [1]. The interfacial transfer terms are very important characteristics of the phase interactions in two-fluid model. These interfacial transfer terms are the essential closure relations. In relation to the modeling of the interfacial transfer terms in the two-fluid model, the concept of the interfacial area transport model
* Corresponding author. ** Corresponding author. E-mail address: [email protected] (H. Liu).
Production and Hosting by Elsevier on behalf of KeAi
has been proposed to develop a constitutive correlation for the interfacial area concentration [2]. provides the state-of-the-art critical review on the modeled source and sink terms of the onegroup interfacial area transport equation for bubbly flow. In the past two decades, numerous studies on interfacial area transport models with different sizes and geometries have been proposed and developed by many researchers./Many researchers have proposed and developed numerous interfacial transport models with different sizes and geometries. Most researchers [3e7] proposed the interfacial area transport model for vertical pipes of medium size. Some researchers provided that models of small size or large size pipes under different gravities [8e11]. [12e14] developed the interfacial area transport models for rectangular geometries with different size, respectively. In additions [15,16], proposed and modified an annular model which was suitable for downward flow. Seldom researchers evaluate the existing models of interfacial area transport models [7]. evluated four interfacial area transport models by two important paraterments (coalesence frequency and efficiency) [17]. conducted an experiment of airwater in a 101.6 mm diameter pipe. The typical interface area transport models were checked by the obtained data. The main purpose presented in this paper is to analyze and evaluate existing typical sink and source terms, and proposes the most accurate model of interfacial area transport for bubbly flow in pipes and rod bundles.
https://doi.org/10.1016/j.jandt.2019.07.001 2468-6050/© 2019 Xi'an Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/).
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
2
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
2. Interfacial area transport equation for bubbly flow
X
Gk ¼ 0
2.1. Two fluid model
(4)
k
X Mik ¼ 0
Two fluid model formulates the conservation equations of mass, momentum and energy for gas phase and liquid phase, respectively. The exchanges of mass, momentum and energy between two phases have also been considered and formulated in the model, which can reflect the internal mechanism and actual process of various physical phenomena. The basic equations of the two-fluid model are as follows: Mass conservation equation:
vak rk þ V,ðak rk vk Þ ¼ Gk vt
(5)
k
X
00
Gk Hik þ ai qki ¼ 0
(6)
k
where Hki means the enthalpy at the interface. The phase interactions are considered through interfacial transfer terms given by the product of an interfacial area concentration and a driving potential. Accurate prediction of interfacial area concentration is critical for the calculation of interfacial transfer terms in two-fluid model.
(1)
Momentum conservation equation:
vak rk vk þ V,ðak rk vk vk Þ ¼ ak Vpk þ V,ak tk þ tTk þ ak rk g vt
2.2. Interfacial area transport equation
(2)
Based on Boltzmann's transport equation [18], formulated the interfacial area transport equation. The general form of the interfacial area transport equation can be expressed as
þvki Gk þ Mik Vak ,tki þ ðpki pk ÞVak Energy conservation equation:
vak rk hk D þ V,ðak rk hk vk Þ ¼ ak Vðpk Þ þ ak k pk vt Dt i h 0 þV, ak qk þ qTk þ ak qik þ fk þhki Gk
vai 1 a 2 2ai va þ V,ai ! þ V, ! v ga vi¼ ½fB fC þ fP þ 3j ai vt 3a vt (3)
¼ FB FC þ FP þ FE (7) where ai , t, ! v i J, a and ! v g indicate the interfacial area concentration, time, interfacial velocity, factor depending on the shape of the bubbles, void fraction and gas average velocity, respectively. FB , FC and FP are respectively the change rates of bubble number density caused by bubble breakup, bubble coalescence and phase change. FB , FC , FP and FE are respectively the change rates of interfacial area concentration due to bubble breakup, bubble coalescence, phase change and bubble volume change. For adiabatic bubbly flow conditions, the form of one-
where ak , rk , vk , Gk , pk , g, tk , tTk , Mik , hk , qk , qTk , qik and fk mean void fraction, density, velocity, mass produced by phase transformation, pressure, acceleration of gravity, average viscous force, turbulent stress, generalized interfacial resistance, specific enthalpy, average heat conduction flux, turbulent heat flux, heat transfer at the interface and viscous dissipation rate of k phase, respectively. In the three conservation equations, the interfacial transport terms characterized phase interactions and satisfy the interfacial transport conditions as follows:
Table 1 Existing typical sink and source terms in interfacial area transport equation for bubbly flow regime. Wu et al.’s model does not include the change term of bubble expansion. [4] neglects the change term of wake entrainment for regular size pipes.
FB
FC
Models FTI [3]
1 G ut 18 TI
! a2i Wecr 1=2 Wecr exp 1 We > Wecr ; GTI ¼ 0:18; a We We
Wecr ¼ 2:0
FRC
"
[6]]
[7]
2
a
GB að1 aÞε1=3 KB s exp 5=3 2=3 ai D11=3 ðamax;B aÞ r D ε f b b KB ¼ 1:59; amax ¼ 0:74
GWE ¼ 0:151
!#
GRC ¼ 0:0565; C1 ¼ 3:0; amax ¼ 0:8
!
GB ¼ 5:02 1010 aRe3 ;
2
a
ai
0
GRC a2 ε1=3 exp@ KC 11=3 Db ðamax;C aÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 5 3 2 6 D r ε b f A GRC ¼ 1:82 3
a
ε1=3 að1 aÞ
1
N\A
Yes
N\A
Yes
N\A
Yes
s
1010 aRe3 ; KC ¼ 1:29; amax ¼ 0:74 2 1=3 2 Wecr 1 rffiffiffiffiffiffiffiffiffiffiffi exp Gb1 12p Gb1 ¼ KC1 12p a ε a rffiffiffiffiffiffiffiffiffiffiffi exp 11=3 11=3 ai We We ai Db We Db 1 þ Gb2 ð1 aÞ a Þ þ K a gð C2 Wecr Wecr 1:6; Gb2 ¼ 0:42; Wecr ¼ 1:24 1=3 1=3 gðaÞ ¼ ðamax a1=3 Þ=amax KC1 ¼ 2:86; KC2 ¼ amax ¼ 0:52 2 1=3 2 1=3 2 a ε að1 aÞ 1 KTI2 1 a ε a 1 rffiffiffiffiffiffiffiffiffiffiffi exp GTI 12p rffiffiffiffiffiffiffiffiffiffiffi exp KC1 12p 11=3 11=3 ai ð1 c3 Þ We We ai Db We Db 1 þ KTI1 ð1 aÞ a Þ þ K a gð C2 Wecr Wecr GTI ¼ 0:16; KTI1 ¼ 0:42; KTI2 ¼ 1:59 1=3 1=3 gðaÞ ¼ ðamax a1=3 Þ=amax KC1 ¼ 2:86; KC2 ¼ amax ¼ 0:52 2
FE
1 N\A GWE ur a2i 3p
1 1 5 GRC ðut a2i Þ4 1=3 1=3 3p amax ðamax a1=3 Þ 1=3 a1=3 max a 1 exp C1 1=3 amax a1=3
[4]
FWE
3
2
KC3
sffiffiffiffiffiffiffiffiffiffiffi ! We Wecr
1:922; KC3 ¼ 1:107;
KC3
sffiffiffiffiffiffiffiffiffiffiffi ! We Wecr
1:922; KC3 ¼ 1:107;
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
dimensional interfacial area transport can be simplified as
v ðhai ihhvi iia Þ ¼ FE þ FB ðFRC þ FWE Þ vz
(8)
where FRC and FWE are respectively bubbles coalescence due to random collision and wake entrainment.
2.3. Evaluation of existing researches Table 1 summaries the existing typical source and sink terms in interfacial area transport equation including bubble breakup, bubble coalescence and volume expansion [3]. proposed the mechanism models of bubble breakup and bubble coalescence and formulated them appropriately. While, the source term of bubble
3
expansion (volume expansion) was been neglected in their model, which is very important for the change of interfacial area concentration, especially for the confined channels. Based on Wu et al.’s model [5], optimized their model by modifying the adjustable parameters and adding the source term of bubble expansion into interfacial area transport equation. Base on the kinetic theory of gases [4], proposed the models of bubble breakup and coalescence, respectively. They assumed that the bubble-bubble and bubbleeddy collision frequencies are similar with the movement of ideal gas molecules. Different from the model of Wu et al. [4], provided the newly coalescence efficiency and breakup efficiency based on the models of [19]. [2] modified the adjustable parameters based on the boundary condition of balance point between bubble breakup and coalescence [6]. proposed the new time scales for bubble coalescence caused by random collision and bubble breakup owing
Fig. 1. Bubble coalescence and break-up models in the open literature [7].
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
4
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
to turbulence impact. In their model, free traveling time and interaction time were considered [7]. considered the turbulent suppression phenomena in interfacial area transport equation. The turbulence suppression was assumed to occur due to the energy exchange between bubbles and turbulent eddies. It should be noted that the models of [6,7] were not be varaficated by the experimental data. Although a large number of typical interfacial area transport models have been proposed and developed by lots of researchers, the evaluation works only be carried out by few researcher. [7] evaluated four interfacial area transport models by two important paraterments including coalescence frequency and efficiency. The coalescence frequencies are divided into two categories: total collision frequency, fc , between two bubbles and total collision frequency, fb , between bubble and turbulent eddy. The efficiencies also can be divided into two categories:coalescence efficiency, hc , and break-up efficiency, hb . Based on the comparison shown in Fig. 1, some assessments of the above models are given below. The differences in the expressions for each model are significant. Weber number must be larger than the critical Weber number for the models of [3,5]. [17] conducted an experiment of air-water in a 101.6 mm diameter pipe. The typical interface area transport models were checked by the obtained data. The results show that the average relative errors of Wu et al.’s model with 34.63%, Hibiki and Ishii's model with 11.37%, Yao and Morel's model with 46.54% and Nguyen et al.’s model with 60.94%, as shown in Fig. 6.3. Through the comparison and discussion of the above results, it is shown that Hibiki and Ishii's model can accurately predict the interfacial
concentration among these eight models (see Fig. 2). In view of these, the result of [17] shows that the model of Hibiki and Ishii could provide better prediction, however, the range of evaluation is narrow and insufficient. To solve this problem, more experimental data are provided in section 3 and section 4 to check these typical interfacial area transport models. 3. Evaluation of existing typical interfacial area transport models by experimental data of pipes [20,21] conducted an series experiments of air-water in a 25.4 mm and 50.8 mm diameter pipes under the atmosphere condition. The detialed experimental date was also provided in their paper inlduding pressure, void fraction, gas velocity and interfacial area concentration along axial direction. Thus, the existing typical interfacial area transport models have been evulated by the experimental data of [4,20,21]. In order to evaluate the accuracy of the four typical models, two statistical parameters are introduced here. The absolute value of mean relative deviation, mrel;ab , and the root mean square error, þ are defined as Eqs. (9) and (10), respectively.
N 〈jðjÞ〉 cal: 〈jðjÞ〉exp: 1 X mrel;ab ¼ 100% N i¼1 〈jðjÞ〉exp: N 〈jðjÞ〉 〈 j ðjÞ〉 X cal: exp: 1 100% mrel;ab ¼ N i¼1 〈jðjÞ〉exp:
(9)
(10)
where N is the number of the sample, 〈jðjÞ〉exp: and 〈jðjÞ〉cal: are the measured and calculated flow parameters, respectively.
Fig. 2. Evaluation of models against with the experimental data of IAC [17].
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Figs. 3e6 show the performance of the four updated models by comparing with experiential data [20,21]. The model of Liu and Hibiki can provide the most accurate prediction among these models. The absolute value of mean relative deviation and the root mean square error are 8.22% and 32.9 m-1, respectively. As shown in Table 2, the prediction of Ishii and Kim's model also performs well with 13.7% mean relative deviation. Correspondingly, the models of Yao and Morel and Nguyen et al. cannot provide accurate predictions. Figs. 7 and 8 are the comparative analysis of sink and source terms of different models under adiabatic steady flow conditions with low and high void fractions, respectively. jg ¼ 1.32 m/s is the gas velocity corresponding to the equilibrium state of bubble coalescence and breakup for 50.0 mm pipe size (detailed in the paper of [2]. Fig. 7 shows that, for the models of Wu et al. and Ishii and Kim, the bubble breakup term caused by turbulence can be neglected at low liquid velocity because their models require Wecrit> 6. In addition, the bubble collision caused by the wake entrainment plays an important role in the change of interfacial area concentration. As can be seen from Figs. 7 and 8, for the model proposed by Ref. [7], when jf approaches 10 m/s, the bubble coalescence phenomenon almost disappears, which is inconsistent with the actual situation. Fig. 9 shows the dependence of the dominant interfacial area transport mechanisms on various flow parameters such as
5
Fig. 5. Evaluation of interfacial area transport model [7].
Fig. 6. Evaluation of interfacial area transport model [2].
Table 2 Predictive performance of updated typical interfacial area transport equation.
Fig. 3. Evaluation of interfacial area transport model [5].
Fig. 4. Evaluation of interfacial area transport model [6].
Comparison of models with experimental data
mrel,ab [%]
RMSE [m1]
Ishii-Kim model (2001) Yao-Morel model (2004) Nguyen et al. model (2013) Liu-Hibiki model (2018)
13.7 110 125 8.22
146 1509 1537 32.9
superficial gas and liquid velocities for existing interfacial area transport models. The ordinate and abscissa in each figure represent the ratio of the interfacial area concentration change rates due to bubble breakup and coalescence, FB ¼ FC , and the superficial liquid velocity, jf, respectively. Figs. 9 and 10 show the contrast effects between sink and source terms in the interfacial area transport model of [2]. Liu and Hibiki's model takes into account the expansion effect of bubbles. Under the condition of low gas velocity (or low void fraction), the source term, FC , (bubbles breakup) decreases sharply, which leads to the interfacial area concentration also decreases sharply-which leads to a sharp decrease in interfacial concentration. This is because the energy dissipation rate per unit mass is composed of bubble expansion and wall friction. Under the condition of low gas velocity, the bubble size is quite small, and the bubble expansion plays the dominate role. However, with the increase of gas velocity, wall
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
6
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Fig. 7. Contrastive analysis of sink and source terms of different models under low void fraction.
Fig. 8. Contrastive analysis of sink and source terms of different models under high void fraction conditions.
4. Evaluation of interfacial area transport model of Hibiki and Ishii (Liu and Hibiki model)by experimental data of rod bundles
Fig. 9. Analysis of dominant items in interface area transport model.
friction plays a leading role. In Fig. 9, when the gas velocity keeps constant, with the increase of liquid velocity, the sink term (bubble coalescence), decreases gradually, while the source term (bubble breakup), increases gradually. This is due to the increase of the liquid velocity enhances the turbulence intensity, resulting in the increase of bubble breakup. In addition, the decrease of void fraction (bubble density also decreases), which also leads to the decrease of bubble coalescence.
Based above discussions and comparison about the existing models, Hibiki and Ishii's model can provide better prediction. While, the evaluation work is still persuasive. Tian et al.’s evaluation only proves that the work of Hibiki and Ishii is better at predicting large pipelines. In section 3, the results only indicate Hibiki's model can predict Hibiki's experiment better than other models. In this section, the experimental data of rod bundles was used to evaluate interfacial area transport model of Hibiki and Ishii (Liu and Hibiki model) for bubbly flow. In relation to develop the interfacial area transport model, accurate axial dataset is necessary to obtained. One-dimensional area averaged interfacial area concentration is plotted against axial direction in Fig. 11 For almost all flow conditions, the interfacial area concentration was obviously increasing at the downstream of spacer grids due to bubble breakup by the shearing effect of spacer grids. In Fig. 11, the interfacial area concentration decrease at the upstream of second spacer grids compared with that at downstream of first spacer grids. The reason is that bubble coalescence makes a dominant contribution to the total interfacial area concentration. When superficial liquid velocity, is constant, the interfacial area enhances with the increasing of superficial gas velocity at low void fraction. It should be noted that this trend is different from the case of high void fraction. The proposed interfacial area transport model has been evaluated with one-dimensional experimental area average data.
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Fig. 10. Comparison of sink and source terms in the interfacial area transport model of [2].
Fig. 11. Axial development of one-dimensional interfacial area concentration.
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
7
8
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Foundation of China (Grant Nos.: 51376201, 51676020) and Natural Science Foundation Project of CQ CSTC (Grant No. cstc2015jcyjB0588). Nomenclature
Fig. 12. Comparison between measured and predicted interfacial area concentrations.
The results show that the interfacial area transport model can provide well prediction and the model agrees well with the experimental data. The mean relative error is 21.89% shown in Fig. 4. It should be noted that there are two main reasons for the error: (1) The effect of spacer grids was not considered in the interfacial area transport equation. The spacer grids will cut big bubbles into small bubbles and produce more turbulence eddied to increases more bubbles coalescence at downstream of spacer grids. (2) The area-averaged parameters of two phase flow are calculated by linear average method which also causes the error when calculating interfacial area transport equation. One-dimensional area-averaged interfacial area concentration is plotted against axial direction in Fig. 12. The blue regions in each figure represent the locations of spacer grids with mixing vanes. The proposed interfacial area transport model considering the effect of spacer grids was evaluated by one-dimensional experimental area-average data. Fig. 11 shows the proposed interfacial area transport model can provide well prediction for interfacial area concentration. The comparison between measured and predicted interfacial area concentrations is plotted in Fig. 12 with the absolute value of mean relative error of 21.89%. 5. Conclusions The existing typical interfacial area transport models have been evaluated by the experimental data of pipes and rod bundles. The results of evaluation has also been analyzed based on the interactive mechanism of bubbles and turbulent eddies. The following conclusions have been made based on the above researches. C The updated typical source and sink terms in interfacial area transport equation have been reviewed and evaluated. The results show Hibiki and Ishii's model (Liu and Hibiki's model) can provide the most accurate prediction for interfacial area concentration for pipes with the error of 8.22%. C The Hibiki and Ishii model's was evaluated by onedimensional experimental area-average data od rod bundles. A fairly good agreement with minor discrepancies has been obtained between the developed interfacial area transport model and measured interfacial area concentration data with the error of 21.89%. Acknowledgements The authors are grateful for the support of the Natural Science
ai D g h Hki i j jf jg k mrel;ab M n P qik RSME
ng ni We Wecr z
Interfacial area concentration Diameter Gravitational acceleration Enthalpy Enthalpy at the interface Interface Mixture volumetric flux Superficial liquid velocity Superficial gas velocity Phase Mean absolute relative deviation Interfacial resistance Exponent Pressure Heat transfer at the interface Root mean square error Velocity of gas phase Interfacial velocity Weber number Critical Weber number Axial direction
Greek symbols Void fraction Mass generation state of gas phase due to evaporation Mass generation state of liquid phase Density difference between phases Liquid density Gas density Surface tension Rate of change of IAC due to bubble breakup Rate of change of IAC due to bubble coalescence Rate of change of IAC due to bubble coalescence caused by random collision FTI Rate of change of IAC due to bubble break caused by turbulent impact FWE Rate of change of IAC due to wake entrainment fB Rate of change of bubble number density due to bubble breakup fC Rate of change of bubble number density due to bubble coalescence fP Rate of change of bubble number density due to phase change j Factor depending on bubble shape
a Gg Gf Dr rf rg s FB FC FRC
Mathematical symbols 〈〉 Area-averaged quantity 〈〈〉〉 Void fraction weighted mean quantity References [1] M. Ishii, T. Hibiki, Thermo-fluid Dynamics of Two-phase Flow, Springer Science & Business Media, 2010. [2] H. Liu, T. Hibiki, Bubble breakup and coalescence models for bubbly flow simulation using interfacial area transport equation, Int. J. Heat Mass Transf 126 (2018) 128e146. [3] Q. Wu, S. Kim, M. Ishii, S. Beus, One-group interfacial area transport in vertical bubbly flow, Int. J. Heat Mass Transf 41 (1998) 1103e1112. [4] T. Hibiki, M. Ishii, One-group interfacial area transport of bubbly flows in vertical round tubes, Int. J. Heat Mass Transf 43 (2000) 2711e2726. [5] M. Ishii, S. Kim, Micro four-sensor probe measurement of interfacial area
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx transport for bubbly flow in round pipes, Nucl. Eng. Des 205 (2001) 123e131. [6] W. Yao, C. Morel, Volumetric interfacial area prediction in upward bubbly two-phase flow, Int. J. Heat Mass Transf 47 (2004) 307e328. [7] V.T. Nguyen, C.H. Song, B.U. Bae, D.J. Euh, Modeling of bubble coalescence and break-up considering turbulent suppression phenomena in bubbly two-phase flow, Int. J. Multiph. Flow 54 (2013) 31e42. [8] T. Hibiki, T. Takamasa, M. Ishii, Interfacial area transport of bubbly flow in a small diameter pipe, J. Nucl. Sci. Technol 38 (2001) 614e620. [9] X. Sun, T.R. Smith, S. Kim, M. Ishii, J. Uhle, Interfacial area of bubbly flow in a relatively large diameter pipe, Exp. Therm. Fluid Sci 27 (2002) 97e109. [10] T. Hibiki, T. Hazuku, T. Takamasa, M. Ishii, Interfacial-area transport equation at reduced-gravity conditions, AIAA J 47 (2009) 1123e1131. [11] T. Smith, J.P. Schlegel, T. Hibiki, M. Ishii, Mechanistic modeling of interfacial area transport in large diameter pipes, Int. J. Multiph. Flow 47 (2012) 1e16. [12] S. Kim, X. Sun, M. Ishii, S.G. Beus, F. Lincoln, Interfacial area transport and evaluation of source and sink terms for confined airewater bubbly flow, Nucl. Eng. Des 219 (2003) 61e75. [13] X. Sun, S. Kim, M. Ishii, S.G. Beus, Model evaluation of two-group interfacial area transport equation for confined upward flow, Nucl. Eng. Des 230 (2004) 27e47. [14] X. Shen, T. Hibiki, T. Ono, K. Sato, K. Mishima, One-dimensional interfacial area
[15]
[16]
[17]
[18] [19] [20]
[21]
9
transport of vertical upward bubbly flow in narrow rectangular channel, Int. J. Heat Mass Transf 36 (2012) 72e82. T. Hibiki, Y. Mi, R. Situ, M. Ishii, Interfacial area transport of vertical upward bubbly two-phase flow in an annulus, Int. J. Heat Mass Transf 46 (2003) 4949e4962. T. Hibiki, H. Goda, S. Kim, M. Ishii, J. Uhle, Axial development of interfacial structure of vertical downward bubbly flow, Int. J. Heat Mass Transf 48 (2005) 749e764. D. Tian, C. Yan, L. Sun, Evaluation of interfacial area transport equation in vertical bubbly two-phase flow in large diameter pipes, Ann. Nucl. Energy 75 (2015) 199e209. G. Kocamustafaogullari, M. Ishii, Foundation of the interfacial area transport equation and its closure relations, Int. J. Heat Mass Transf 38 (1995) 481e493. C. Coulaloglou, L. Tavlarides, Description of interaction processes in agitated liquid-liquid dispersions, Chem. Eng. Sci 32 (1977) 1289e1297. T. Hibiki, S. Hogsett, M. Ishii, Local measurement of interfacial area, interfacial velocity and liquid turbulence in two-phase flow, Nucl. Eng. Des 184 (1998) 287e304. T. Hibiki, M. Ishii, Experimental study on interfacial area transport in bubbly two-phase flows, Int. J. Heat Mass Transf 42 (1999) 3019e3035.
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
Contents lists available at ScienceDirect
International Journal of Advanced Nuclear Reactor Design and Technology journal homepage: http://www.keaipublishing.com/en/journals/ international-journal-of-advanced-nuclear-reactor-design-and-technology/
Evaluation of typical interfacial area transport models for bubbly flow Hang Liu a, *, Jianyong Lai a, Yi Li a, Yulong Zhang a, Xiaoquan Yu a, Minghao Liu a, Liangming Pan b, ** a b
Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu, 610213, China Key Laboratory of Low-grade Energy Utilization Technologies and Systems (Chongqing University), Ministry of Education, Chongqing, 400044, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 March 2019 Received in revised form 22 June 2019 Accepted 11 July 2019 Available online xxx
Interfacial area concentration is one of the most important parameters in two-phase flow. This parameter also reflects the interfacial area of mass, momentum and energy transfer in two-fluid model. The prediction of interfacial area concentration is essential to close the interfacial transfer terms in the two-fluid model. For the modeling of the one-dimensional interfacial area transport equation for adiabatic bubbly flow, existing typical sink and source terms have been reviewed and evaluated by the experimental data of vertical pipes and rod bundles. The interactive mechanisms of bubbles and turbulent eddies have also been analyzed, including bubbles breakup and bubbles coalescence. The evaluation results show the interfacial area transport model of Hibiki and Ishii (Liu and Hibiki) can provide the most accurate prediction of interfacial area concentration for pipes with the relative error of 8.22% for bubbly flow. Their model has also been evaluated by the experiment data of rod bundle with the relative error of 21.89% for bubbly flow. © 2019 Xi'an Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Interfacial area transport Sink terms Source terms Evaluation
1. Introduction Accurate prediction for the behavior of heat exchangers or nuclear reactors is of great important significance for long-term safe operation or accident scenarios. Two-fluid model is the most detailed and accurate macroscopic formulation in two-phase flow models. The behavior prediction of nuclear reactor system under accident conditions is mainly based on the computable code. Reliable two-phase flow models are important to obtain the accurate prediction of the thermal-hydraulic behavior. Two-fluid model is considered as the most accurate and detailed model to simulated two-phase flow [1]. The interfacial transfer terms are very important characteristics of the phase interactions in two-fluid model. These interfacial transfer terms are the essential closure relations. In relation to the modeling of the interfacial transfer terms in the two-fluid model, the concept of the interfacial area transport model
* Corresponding author. ** Corresponding author. E-mail address: [email protected] (H. Liu).
Production and Hosting by Elsevier on behalf of KeAi
has been proposed to develop a constitutive correlation for the interfacial area concentration [2]. provides the state-of-the-art critical review on the modeled source and sink terms of the onegroup interfacial area transport equation for bubbly flow. In the past two decades, numerous studies on interfacial area transport models with different sizes and geometries have been proposed and developed by many researchers./Many researchers have proposed and developed numerous interfacial transport models with different sizes and geometries. Most researchers [3e7] proposed the interfacial area transport model for vertical pipes of medium size. Some researchers provided that models of small size or large size pipes under different gravities [8e11]. [12e14] developed the interfacial area transport models for rectangular geometries with different size, respectively. In additions [15,16], proposed and modified an annular model which was suitable for downward flow. Seldom researchers evaluate the existing models of interfacial area transport models [7]. evluated four interfacial area transport models by two important paraterments (coalesence frequency and efficiency) [17]. conducted an experiment of airwater in a 101.6 mm diameter pipe. The typical interface area transport models were checked by the obtained data. The main purpose presented in this paper is to analyze and evaluate existing typical sink and source terms, and proposes the most accurate model of interfacial area transport for bubbly flow in pipes and rod bundles.
https://doi.org/10.1016/j.jandt.2019.07.001 2468-6050/© 2019 Xi'an Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/).
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
2
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
2. Interfacial area transport equation for bubbly flow
X
Gk ¼ 0
2.1. Two fluid model
(4)
k
X Mik ¼ 0
Two fluid model formulates the conservation equations of mass, momentum and energy for gas phase and liquid phase, respectively. The exchanges of mass, momentum and energy between two phases have also been considered and formulated in the model, which can reflect the internal mechanism and actual process of various physical phenomena. The basic equations of the two-fluid model are as follows: Mass conservation equation:
vak rk þ V,ðak rk vk Þ ¼ Gk vt
(5)
k
X
00
Gk Hik þ ai qki ¼ 0
(6)
k
where Hki means the enthalpy at the interface. The phase interactions are considered through interfacial transfer terms given by the product of an interfacial area concentration and a driving potential. Accurate prediction of interfacial area concentration is critical for the calculation of interfacial transfer terms in two-fluid model.
(1)
Momentum conservation equation:
vak rk vk þ V,ðak rk vk vk Þ ¼ ak Vpk þ V,ak tk þ tTk þ ak rk g vt
2.2. Interfacial area transport equation
(2)
Based on Boltzmann's transport equation [18], formulated the interfacial area transport equation. The general form of the interfacial area transport equation can be expressed as
þvki Gk þ Mik Vak ,tki þ ðpki pk ÞVak Energy conservation equation:
vak rk hk D þ V,ðak rk hk vk Þ ¼ ak Vðpk Þ þ ak k pk vt Dt i h 0 þV, ak qk þ qTk þ ak qik þ fk þhki Gk
vai 1 a 2 2ai va þ V,ai ! þ V, ! v ga vi¼ ½fB fC þ fP þ 3j ai vt 3a vt (3)
¼ FB FC þ FP þ FE (7) where ai , t, ! v i J, a and ! v g indicate the interfacial area concentration, time, interfacial velocity, factor depending on the shape of the bubbles, void fraction and gas average velocity, respectively. FB , FC and FP are respectively the change rates of bubble number density caused by bubble breakup, bubble coalescence and phase change. FB , FC , FP and FE are respectively the change rates of interfacial area concentration due to bubble breakup, bubble coalescence, phase change and bubble volume change. For adiabatic bubbly flow conditions, the form of one-
where ak , rk , vk , Gk , pk , g, tk , tTk , Mik , hk , qk , qTk , qik and fk mean void fraction, density, velocity, mass produced by phase transformation, pressure, acceleration of gravity, average viscous force, turbulent stress, generalized interfacial resistance, specific enthalpy, average heat conduction flux, turbulent heat flux, heat transfer at the interface and viscous dissipation rate of k phase, respectively. In the three conservation equations, the interfacial transport terms characterized phase interactions and satisfy the interfacial transport conditions as follows:
Table 1 Existing typical sink and source terms in interfacial area transport equation for bubbly flow regime. Wu et al.’s model does not include the change term of bubble expansion. [4] neglects the change term of wake entrainment for regular size pipes.
FB
FC
Models FTI [3]
1 G ut 18 TI
! a2i Wecr 1=2 Wecr exp 1 We > Wecr ; GTI ¼ 0:18; a We We
Wecr ¼ 2:0
FRC
"
[6]]
[7]
2
a
GB að1 aÞε1=3 KB s exp 5=3 2=3 ai D11=3 ðamax;B aÞ r D ε f b b KB ¼ 1:59; amax ¼ 0:74
GWE ¼ 0:151
!#
GRC ¼ 0:0565; C1 ¼ 3:0; amax ¼ 0:8
!
GB ¼ 5:02 1010 aRe3 ;
2
a
ai
0
GRC a2 ε1=3 exp@ KC 11=3 Db ðamax;C aÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 5 3 2 6 D r ε b f A GRC ¼ 1:82 3
a
ε1=3 að1 aÞ
1
N\A
Yes
N\A
Yes
N\A
Yes
s
1010 aRe3 ; KC ¼ 1:29; amax ¼ 0:74 2 1=3 2 Wecr 1 rffiffiffiffiffiffiffiffiffiffiffi exp Gb1 12p Gb1 ¼ KC1 12p a ε a rffiffiffiffiffiffiffiffiffiffiffi exp 11=3 11=3 ai We We ai Db We Db 1 þ Gb2 ð1 aÞ a Þ þ K a gð C2 Wecr Wecr 1:6; Gb2 ¼ 0:42; Wecr ¼ 1:24 1=3 1=3 gðaÞ ¼ ðamax a1=3 Þ=amax KC1 ¼ 2:86; KC2 ¼ amax ¼ 0:52 2 1=3 2 1=3 2 a ε að1 aÞ 1 KTI2 1 a ε a 1 rffiffiffiffiffiffiffiffiffiffiffi exp GTI 12p rffiffiffiffiffiffiffiffiffiffiffi exp KC1 12p 11=3 11=3 ai ð1 c3 Þ We We ai Db We Db 1 þ KTI1 ð1 aÞ a Þ þ K a gð C2 Wecr Wecr GTI ¼ 0:16; KTI1 ¼ 0:42; KTI2 ¼ 1:59 1=3 1=3 gðaÞ ¼ ðamax a1=3 Þ=amax KC1 ¼ 2:86; KC2 ¼ amax ¼ 0:52 2
FE
1 N\A GWE ur a2i 3p
1 1 5 GRC ðut a2i Þ4 1=3 1=3 3p amax ðamax a1=3 Þ 1=3 a1=3 max a 1 exp C1 1=3 amax a1=3
[4]
FWE
3
2
KC3
sffiffiffiffiffiffiffiffiffiffiffi ! We Wecr
1:922; KC3 ¼ 1:107;
KC3
sffiffiffiffiffiffiffiffiffiffiffi ! We Wecr
1:922; KC3 ¼ 1:107;
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
dimensional interfacial area transport can be simplified as
v ðhai ihhvi iia Þ ¼ FE þ FB ðFRC þ FWE Þ vz
(8)
where FRC and FWE are respectively bubbles coalescence due to random collision and wake entrainment.
2.3. Evaluation of existing researches Table 1 summaries the existing typical source and sink terms in interfacial area transport equation including bubble breakup, bubble coalescence and volume expansion [3]. proposed the mechanism models of bubble breakup and bubble coalescence and formulated them appropriately. While, the source term of bubble
3
expansion (volume expansion) was been neglected in their model, which is very important for the change of interfacial area concentration, especially for the confined channels. Based on Wu et al.’s model [5], optimized their model by modifying the adjustable parameters and adding the source term of bubble expansion into interfacial area transport equation. Base on the kinetic theory of gases [4], proposed the models of bubble breakup and coalescence, respectively. They assumed that the bubble-bubble and bubbleeddy collision frequencies are similar with the movement of ideal gas molecules. Different from the model of Wu et al. [4], provided the newly coalescence efficiency and breakup efficiency based on the models of [19]. [2] modified the adjustable parameters based on the boundary condition of balance point between bubble breakup and coalescence [6]. proposed the new time scales for bubble coalescence caused by random collision and bubble breakup owing
Fig. 1. Bubble coalescence and break-up models in the open literature [7].
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
4
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
to turbulence impact. In their model, free traveling time and interaction time were considered [7]. considered the turbulent suppression phenomena in interfacial area transport equation. The turbulence suppression was assumed to occur due to the energy exchange between bubbles and turbulent eddies. It should be noted that the models of [6,7] were not be varaficated by the experimental data. Although a large number of typical interfacial area transport models have been proposed and developed by lots of researchers, the evaluation works only be carried out by few researcher. [7] evaluated four interfacial area transport models by two important paraterments including coalescence frequency and efficiency. The coalescence frequencies are divided into two categories: total collision frequency, fc , between two bubbles and total collision frequency, fb , between bubble and turbulent eddy. The efficiencies also can be divided into two categories:coalescence efficiency, hc , and break-up efficiency, hb . Based on the comparison shown in Fig. 1, some assessments of the above models are given below. The differences in the expressions for each model are significant. Weber number must be larger than the critical Weber number for the models of [3,5]. [17] conducted an experiment of air-water in a 101.6 mm diameter pipe. The typical interface area transport models were checked by the obtained data. The results show that the average relative errors of Wu et al.’s model with 34.63%, Hibiki and Ishii's model with 11.37%, Yao and Morel's model with 46.54% and Nguyen et al.’s model with 60.94%, as shown in Fig. 6.3. Through the comparison and discussion of the above results, it is shown that Hibiki and Ishii's model can accurately predict the interfacial
concentration among these eight models (see Fig. 2). In view of these, the result of [17] shows that the model of Hibiki and Ishii could provide better prediction, however, the range of evaluation is narrow and insufficient. To solve this problem, more experimental data are provided in section 3 and section 4 to check these typical interfacial area transport models. 3. Evaluation of existing typical interfacial area transport models by experimental data of pipes [20,21] conducted an series experiments of air-water in a 25.4 mm and 50.8 mm diameter pipes under the atmosphere condition. The detialed experimental date was also provided in their paper inlduding pressure, void fraction, gas velocity and interfacial area concentration along axial direction. Thus, the existing typical interfacial area transport models have been evulated by the experimental data of [4,20,21]. In order to evaluate the accuracy of the four typical models, two statistical parameters are introduced here. The absolute value of mean relative deviation, mrel;ab , and the root mean square error, þ are defined as Eqs. (9) and (10), respectively.
N 〈jðjÞ〉 cal: 〈jðjÞ〉exp: 1 X mrel;ab ¼ 100% N i¼1 〈jðjÞ〉exp: N 〈jðjÞ〉 〈 j ðjÞ〉 X cal: exp: 1 100% mrel;ab ¼ N i¼1 〈jðjÞ〉exp:
(9)
(10)
where N is the number of the sample, 〈jðjÞ〉exp: and 〈jðjÞ〉cal: are the measured and calculated flow parameters, respectively.
Fig. 2. Evaluation of models against with the experimental data of IAC [17].
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Figs. 3e6 show the performance of the four updated models by comparing with experiential data [20,21]. The model of Liu and Hibiki can provide the most accurate prediction among these models. The absolute value of mean relative deviation and the root mean square error are 8.22% and 32.9 m-1, respectively. As shown in Table 2, the prediction of Ishii and Kim's model also performs well with 13.7% mean relative deviation. Correspondingly, the models of Yao and Morel and Nguyen et al. cannot provide accurate predictions. Figs. 7 and 8 are the comparative analysis of sink and source terms of different models under adiabatic steady flow conditions with low and high void fractions, respectively. jg ¼ 1.32 m/s is the gas velocity corresponding to the equilibrium state of bubble coalescence and breakup for 50.0 mm pipe size (detailed in the paper of [2]. Fig. 7 shows that, for the models of Wu et al. and Ishii and Kim, the bubble breakup term caused by turbulence can be neglected at low liquid velocity because their models require Wecrit> 6. In addition, the bubble collision caused by the wake entrainment plays an important role in the change of interfacial area concentration. As can be seen from Figs. 7 and 8, for the model proposed by Ref. [7], when jf approaches 10 m/s, the bubble coalescence phenomenon almost disappears, which is inconsistent with the actual situation. Fig. 9 shows the dependence of the dominant interfacial area transport mechanisms on various flow parameters such as
5
Fig. 5. Evaluation of interfacial area transport model [7].
Fig. 6. Evaluation of interfacial area transport model [2].
Table 2 Predictive performance of updated typical interfacial area transport equation.
Fig. 3. Evaluation of interfacial area transport model [5].
Fig. 4. Evaluation of interfacial area transport model [6].
Comparison of models with experimental data
mrel,ab [%]
RMSE [m1]
Ishii-Kim model (2001) Yao-Morel model (2004) Nguyen et al. model (2013) Liu-Hibiki model (2018)
13.7 110 125 8.22
146 1509 1537 32.9
superficial gas and liquid velocities for existing interfacial area transport models. The ordinate and abscissa in each figure represent the ratio of the interfacial area concentration change rates due to bubble breakup and coalescence, FB ¼ FC , and the superficial liquid velocity, jf, respectively. Figs. 9 and 10 show the contrast effects between sink and source terms in the interfacial area transport model of [2]. Liu and Hibiki's model takes into account the expansion effect of bubbles. Under the condition of low gas velocity (or low void fraction), the source term, FC , (bubbles breakup) decreases sharply, which leads to the interfacial area concentration also decreases sharply-which leads to a sharp decrease in interfacial concentration. This is because the energy dissipation rate per unit mass is composed of bubble expansion and wall friction. Under the condition of low gas velocity, the bubble size is quite small, and the bubble expansion plays the dominate role. However, with the increase of gas velocity, wall
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
6
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Fig. 7. Contrastive analysis of sink and source terms of different models under low void fraction.
Fig. 8. Contrastive analysis of sink and source terms of different models under high void fraction conditions.
4. Evaluation of interfacial area transport model of Hibiki and Ishii (Liu and Hibiki model)by experimental data of rod bundles
Fig. 9. Analysis of dominant items in interface area transport model.
friction plays a leading role. In Fig. 9, when the gas velocity keeps constant, with the increase of liquid velocity, the sink term (bubble coalescence), decreases gradually, while the source term (bubble breakup), increases gradually. This is due to the increase of the liquid velocity enhances the turbulence intensity, resulting in the increase of bubble breakup. In addition, the decrease of void fraction (bubble density also decreases), which also leads to the decrease of bubble coalescence.
Based above discussions and comparison about the existing models, Hibiki and Ishii's model can provide better prediction. While, the evaluation work is still persuasive. Tian et al.’s evaluation only proves that the work of Hibiki and Ishii is better at predicting large pipelines. In section 3, the results only indicate Hibiki's model can predict Hibiki's experiment better than other models. In this section, the experimental data of rod bundles was used to evaluate interfacial area transport model of Hibiki and Ishii (Liu and Hibiki model) for bubbly flow. In relation to develop the interfacial area transport model, accurate axial dataset is necessary to obtained. One-dimensional area averaged interfacial area concentration is plotted against axial direction in Fig. 11 For almost all flow conditions, the interfacial area concentration was obviously increasing at the downstream of spacer grids due to bubble breakup by the shearing effect of spacer grids. In Fig. 11, the interfacial area concentration decrease at the upstream of second spacer grids compared with that at downstream of first spacer grids. The reason is that bubble coalescence makes a dominant contribution to the total interfacial area concentration. When superficial liquid velocity, is constant, the interfacial area enhances with the increasing of superficial gas velocity at low void fraction. It should be noted that this trend is different from the case of high void fraction. The proposed interfacial area transport model has been evaluated with one-dimensional experimental area average data.
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Fig. 10. Comparison of sink and source terms in the interfacial area transport model of [2].
Fig. 11. Axial development of one-dimensional interfacial area concentration.
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
7
8
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx
Foundation of China (Grant Nos.: 51376201, 51676020) and Natural Science Foundation Project of CQ CSTC (Grant No. cstc2015jcyjB0588). Nomenclature
Fig. 12. Comparison between measured and predicted interfacial area concentrations.
The results show that the interfacial area transport model can provide well prediction and the model agrees well with the experimental data. The mean relative error is 21.89% shown in Fig. 4. It should be noted that there are two main reasons for the error: (1) The effect of spacer grids was not considered in the interfacial area transport equation. The spacer grids will cut big bubbles into small bubbles and produce more turbulence eddied to increases more bubbles coalescence at downstream of spacer grids. (2) The area-averaged parameters of two phase flow are calculated by linear average method which also causes the error when calculating interfacial area transport equation. One-dimensional area-averaged interfacial area concentration is plotted against axial direction in Fig. 12. The blue regions in each figure represent the locations of spacer grids with mixing vanes. The proposed interfacial area transport model considering the effect of spacer grids was evaluated by one-dimensional experimental area-average data. Fig. 11 shows the proposed interfacial area transport model can provide well prediction for interfacial area concentration. The comparison between measured and predicted interfacial area concentrations is plotted in Fig. 12 with the absolute value of mean relative error of 21.89%. 5. Conclusions The existing typical interfacial area transport models have been evaluated by the experimental data of pipes and rod bundles. The results of evaluation has also been analyzed based on the interactive mechanism of bubbles and turbulent eddies. The following conclusions have been made based on the above researches. C The updated typical source and sink terms in interfacial area transport equation have been reviewed and evaluated. The results show Hibiki and Ishii's model (Liu and Hibiki's model) can provide the most accurate prediction for interfacial area concentration for pipes with the error of 8.22%. C The Hibiki and Ishii model's was evaluated by onedimensional experimental area-average data od rod bundles. A fairly good agreement with minor discrepancies has been obtained between the developed interfacial area transport model and measured interfacial area concentration data with the error of 21.89%. Acknowledgements The authors are grateful for the support of the Natural Science
ai D g h Hki i j jf jg k mrel;ab M n P qik RSME
ng ni We Wecr z
Interfacial area concentration Diameter Gravitational acceleration Enthalpy Enthalpy at the interface Interface Mixture volumetric flux Superficial liquid velocity Superficial gas velocity Phase Mean absolute relative deviation Interfacial resistance Exponent Pressure Heat transfer at the interface Root mean square error Velocity of gas phase Interfacial velocity Weber number Critical Weber number Axial direction
Greek symbols Void fraction Mass generation state of gas phase due to evaporation Mass generation state of liquid phase Density difference between phases Liquid density Gas density Surface tension Rate of change of IAC due to bubble breakup Rate of change of IAC due to bubble coalescence Rate of change of IAC due to bubble coalescence caused by random collision FTI Rate of change of IAC due to bubble break caused by turbulent impact FWE Rate of change of IAC due to wake entrainment fB Rate of change of bubble number density due to bubble breakup fC Rate of change of bubble number density due to bubble coalescence fP Rate of change of bubble number density due to phase change j Factor depending on bubble shape
a Gg Gf Dr rf rg s FB FC FRC
Mathematical symbols 〈〉 Area-averaged quantity 〈〈〉〉 Void fraction weighted mean quantity References [1] M. Ishii, T. Hibiki, Thermo-fluid Dynamics of Two-phase Flow, Springer Science & Business Media, 2010. [2] H. Liu, T. Hibiki, Bubble breakup and coalescence models for bubbly flow simulation using interfacial area transport equation, Int. J. Heat Mass Transf 126 (2018) 128e146. [3] Q. Wu, S. Kim, M. Ishii, S. Beus, One-group interfacial area transport in vertical bubbly flow, Int. J. Heat Mass Transf 41 (1998) 1103e1112. [4] T. Hibiki, M. Ishii, One-group interfacial area transport of bubbly flows in vertical round tubes, Int. J. Heat Mass Transf 43 (2000) 2711e2726. [5] M. Ishii, S. Kim, Micro four-sensor probe measurement of interfacial area
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001
H. Liu et al. / International Journal of Advanced Nuclear Reactor Design and Technology xxx (xxxx) xxx transport for bubbly flow in round pipes, Nucl. Eng. Des 205 (2001) 123e131. [6] W. Yao, C. Morel, Volumetric interfacial area prediction in upward bubbly two-phase flow, Int. J. Heat Mass Transf 47 (2004) 307e328. [7] V.T. Nguyen, C.H. Song, B.U. Bae, D.J. Euh, Modeling of bubble coalescence and break-up considering turbulent suppression phenomena in bubbly two-phase flow, Int. J. Multiph. Flow 54 (2013) 31e42. [8] T. Hibiki, T. Takamasa, M. Ishii, Interfacial area transport of bubbly flow in a small diameter pipe, J. Nucl. Sci. Technol 38 (2001) 614e620. [9] X. Sun, T.R. Smith, S. Kim, M. Ishii, J. Uhle, Interfacial area of bubbly flow in a relatively large diameter pipe, Exp. Therm. Fluid Sci 27 (2002) 97e109. [10] T. Hibiki, T. Hazuku, T. Takamasa, M. Ishii, Interfacial-area transport equation at reduced-gravity conditions, AIAA J 47 (2009) 1123e1131. [11] T. Smith, J.P. Schlegel, T. Hibiki, M. Ishii, Mechanistic modeling of interfacial area transport in large diameter pipes, Int. J. Multiph. Flow 47 (2012) 1e16. [12] S. Kim, X. Sun, M. Ishii, S.G. Beus, F. Lincoln, Interfacial area transport and evaluation of source and sink terms for confined airewater bubbly flow, Nucl. Eng. Des 219 (2003) 61e75. [13] X. Sun, S. Kim, M. Ishii, S.G. Beus, Model evaluation of two-group interfacial area transport equation for confined upward flow, Nucl. Eng. Des 230 (2004) 27e47. [14] X. Shen, T. Hibiki, T. Ono, K. Sato, K. Mishima, One-dimensional interfacial area
[15]
[16]
[17]
[18] [19] [20]
[21]
9
transport of vertical upward bubbly flow in narrow rectangular channel, Int. J. Heat Mass Transf 36 (2012) 72e82. T. Hibiki, Y. Mi, R. Situ, M. Ishii, Interfacial area transport of vertical upward bubbly two-phase flow in an annulus, Int. J. Heat Mass Transf 46 (2003) 4949e4962. T. Hibiki, H. Goda, S. Kim, M. Ishii, J. Uhle, Axial development of interfacial structure of vertical downward bubbly flow, Int. J. Heat Mass Transf 48 (2005) 749e764. D. Tian, C. Yan, L. Sun, Evaluation of interfacial area transport equation in vertical bubbly two-phase flow in large diameter pipes, Ann. Nucl. Energy 75 (2015) 199e209. G. Kocamustafaogullari, M. Ishii, Foundation of the interfacial area transport equation and its closure relations, Int. J. Heat Mass Transf 38 (1995) 481e493. C. Coulaloglou, L. Tavlarides, Description of interaction processes in agitated liquid-liquid dispersions, Chem. Eng. Sci 32 (1977) 1289e1297. T. Hibiki, S. Hogsett, M. Ishii, Local measurement of interfacial area, interfacial velocity and liquid turbulence in two-phase flow, Nucl. Eng. Des 184 (1998) 287e304. T. Hibiki, M. Ishii, Experimental study on interfacial area transport in bubbly two-phase flows, Int. J. Heat Mass Transf 42 (1999) 3019e3035.
Please cite this article as: H. Liu et al., Evaluation of typical interfacial area transport models for bubbly flow, International Journal of Advanced Nuclear Reactor Design and Technology, https://doi.org/10.1016/j.jandt.2019.07.001