- Email: [email protected]
Corresponding principle of critical heat flux in flow boiling
International Journal of Heat and Mass Transfer 136 (2019) 591–596
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Corresponding principle of critical heat flux in flow boiling B.H. Yan a,⇑, C. Wang b, R. Li a a b
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai, Guangdong, China China Ship Development and Design Center, Wuhan, Hubei, China
a r t i c l e
i n f o
Article history: Received 17 October 2018 Received in revised form 3 March 2019 Accepted 6 March 2019
Keywords: Critical heat flux Flow boiling Fluid modeling Corresponding principle
a b s t r a c t An equation of state is a thermodynamic equation relating fluid properties which describe the state of matter under a given set of physical conditions. Considering there are some connections between the properties of different fluids, we believe there are some corresponding relations between the critical heat flux (CHF) of different fluids. A corresponding principle was developed to predict the critical heat flux with the experimental data obtained with modeling fluids. A series of scaling criteria were developed on the basis of fluid similarity and molecular thermodynamics, which is totally different from the conventional scaling methods based on experimental data and empirical correlations. The prediction error of this principle is much lower than conventional prediction methods. The average and maximum error of the corresponding principle is about 10%. Ó 2019 Published by Elsevier Ltd.
1. Introduction Critical heat flux (CHF) is an upper limit of heat flux for nucleate boiling [1–3]. It is important for nuclear power plant designers to be able to predict the CHF and other system parameters at which it occurs. In nuclear engineering, many CHF experiments in water must be performed under high pressure and high temperature conditions that require a very complicated experimental system, very high heating power and very careful operation, resulting in more difficulties and expenditures in experimentation. Along with CHF measurements, CHF prediction methods were also proposed starting in the late 1940s [4]. These physical models depend on the mechanisms controlling the CHF, which changes with flow regime. Then, a more universal CHF prediction methodology, look-up tables for predicting CHF, was derived [5]. The CHF look-up table is basically a normalized CHF databank for watercooled tubes. The primary conditions at which data are scarce or missing are: (i) high flow rates and high vapor qualities, (ii) low flow rates and low vapor qualities, and (iii) pressure range of 0.2–0.4 MPa [5]. Since the existing look-up tables could only be used to predict the CHF of water, it could not be extended to other fluids. If a method which could transfer the CHF data of water to other fluids was created, the CHF databank for all fluids could be enlarged significantly. Meanwhile, the expensive CHF experiments could be substituted for some easier experiments with modeling fluid. ⇑ Corresponding author. E-mail address: [email protected] (B.H. Yan). https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.037 0017-9310/Ó 2019 Published by Elsevier Ltd.
In physics and thermodynamics, an equation of state (EoS) is a thermodynamic equation relating fluid properties which describe the state of matter under a given set of physical conditions [6]. The CHF is correlated with two types of parameters. The first is fluid property, and the other is boundary conditions, such as channel geometry and inlet mass flow rate. There should be a corresponding principle between the CHF values of different fluids. Corresponding principle, which was also called as fluid modeling in nuclear engineering, is used to reduce the high electrical power cost of CHF experiments by replacing water with a modeling fluid having a lower latent heat of vaporization [7–10]. The modeling fluids usually chosen are members of fluorocarbon family because of their very low latent heat, relatively low hazard and well established fluid properties [11]. After the CHF experiments of modeling fluid, these CHF data was used to predict the CHF of water with scaling laws. The main problem with fluid modeling is the interpretation of modeling data. The first systematic analysis to model water CHF behavior with Freon was presented by Barnett [12]. He constructed a model according to the laws of dimensional analysis. Subsequently, many models were proposed by equation-analysis or dimensionless analysis methods [13–19]. As far as the modelling of CHF is concerned, the effects of buoyancy, liquid and vapor thermal conductivities, liquid and vapor Prandtl numbers and secondary properties are negligible [20–23]. Ahmad combined the effect of Reynolds number, Weber-Reynolds number and liquidvapor viscosity ratio in one term, thus creating a compensated distortion technique [11]. This procedure showed limited success but laid the groundwork for the following empirical techniques [24].
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B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
Nomenclature Symbols C CHF Cp D G H Dhfg k L n P Pr q T U x X
Greek letters parameter for boundary conditions a void fraction k thermal conductivity q density r surface tension
P
parameter for molecule characteristics critical heat flux thermal capacity channel diameter mass flow rate coolant enthalpy latent heat fluid conductivity channel length molecule number pressure Prandtl number Heat flux temperature velocity vapor quality Fluid property
Superscripts — dimensionless parameter Subscripts a, b, a, b fluid index e exit i, l, v index for vapor and liquid phases P pressure sat boiling point sub sub-cooling T temperature w wall 1 bulk velocity
Conventional fluid modeling of CHF was based on dimensionless similar relationship, experimental data and completely empirical correlations. These methods are often useful in engineering application, but one must avoid the temptation to use them outside the narrow range of conditions on which they are based [25]. In this work, a universal corresponding principle was established based on the fluid similarity and molecular thermodynamics. The thermal hydraulic parameters in macroscale are determined on the basis of two phase flow and heat transfer and bubble behavior. This principle is proved to be useful in a rather wide range.
CHF ¼ UðPa ; Pb Þ
2. Corresponding principle According to the molecular thermodynamics, the fluid property is correlated with the molecular motion and structure. For a specific fluid, the property parameters like viscosity, specific heat, conductivity, and density etc, could be represented as functions of pressure, temperature, polar factor, and so on. The equations could be expressed as:
X ¼ FðP; T; CÞ
showed that the properties of different fluids could be retransferred, which means that we could find the relation between the CHF of different fluids by the connection of fluid properties. It is well known that the CHF is usually contributed by the pressure, temperature, sub-cooling, viscosity, capacity, channel diameter, length, spacing grid, mixing vane, etc. Here we divide all these factors into two categories, fluid property and boundary condition. Assuming there are two similar CHF phenomena for fluid A and fluid B, as shown in Fig. 1. The burnout phenomena are attributed to the same mechanism. Thus the CHF correlations for these two cases could be expressed as:
ð1Þ
Here X ¼ l; C p ; k; q; Dhfg ; ::: denotes the fluid properties in dimensionless form, P and T are normalized pressure and temperature, respectively. C denotes dimensionless parameter representing the molecule characteristics like polar factor and associated action. As for two fluids A and B, it is easy to have P a ¼ P b and T a ¼ T b . The dimensionless pressure and temperature could also be expressed as the inverse functions of fluid property. Then, we have 1 X a ¼ FðPa ; T a ; C a Þ ¼ FðPb ; T b ; C a Þ ¼ FðF 1 Pb ðX b Þ; F Tb ðX b Þ; C a Þ ¼ GðX b Þ
ð2Þ
ð3Þ
Pa represents the boundary conditions. Pb represents the fluid properties. With the correlations of Pa and Pb for fluids A and B, we can have the CHF correlation between these two fluids. CHF a ¼ UðPa;a ; Pb;a Þ ¼ WðPa;b ; Pb;b Þ ¼ wðCHF b Þ
ð4Þ
Eq. (4) is the theoretical basis of corresponding principle of CHF. It shows that the CHF results of one fluid could be transferred to the CHF data of the other fluid. Here we have two main problems need to be solved. The first one is the similarity of CHF phenomena of two fluids in microscale, as shown in Eq. (2). The other is the correlations between macroscopic parameters of two CHF phenomena similar in microscale, as shown in Eq. (4). The abovementioned analysis reveals that we need to find the exact forms of Eqs. (2) and (4). However, the separation between molecular physics and practical problems in phase behavior is still large. Although the physical properties of every substance depend directly on the nature of the molecules of the substance, a complete understanding of molecular behavior is still unavailable. In other side, we don’t have a commonly used CHF formula, just like that in Eq. (4).
It should be mentioned that the function GðX b Þ is different from the function FðX b Þ since the dimensionless parameter included are
3. Criteria
C a and C b , respectively. Thus, it is reasonable to image that if the working fluid is replaced with the other one with identical fluid properties in CHF experiments, identical CHF results will be obtained. Unfortunately, it is almost impossible to find two fluids with identical properties in engineering application. Eq. (2)
It is difficult to find the connection between two CHF phenomena because the macroscopic phenomenon of CHF is rather complicated. We could not find the corresponding principle of CHF totally based on existing knowledge of molecular thermodynamics, either. However, all macroscopic phenomena are consisted of a large
B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
593
As for a universal fluid-to-fluid scaling law, it is not only applicable for the scaling between fluid A and fluid B, but also for the case between fluid A and fluid A. Since the CHF mechanism is rather complicated, it is nearly impossible to find a upscaling or a downscaling law for the same fluid in two different geometries [26]. Therefore the scaling law for these two geometries should be equal to 1:1. It is easy to find that Eqs. (5) and (6) represent the equal proportional scaling of complex phenomenon. With the similarity of molecular number, we can obtain the length ratio and diameter ratio for prototype fluid and modeling fluid approximately, due to the weak compressibility of liquid phase. Then, the liquid density and vapor density of prototype fluid could be obtained. Since the liquid and vapor density could be expressed as functions of pressure and temperature, the pressure ratio could be obtained as an inverse function of density and temperature, with the already obtained liquid density and vapor density of prototype fluid. Then the saturation properties of prototype fluid could be obtained with the pressure as an input parameter. Considering these two phenomena in similarity, as shown in Fig. 1, it is reasonable to assume that the interface shapes between liquid phase and vapor phase are similar. Thus the void fraction in the channel exit should be identical for prototype fluid and modeling fluid. Then we have
am =ap ¼ 1
Fig. 1. Two DNB phenomena in similarity.
amount of microscopic motions and structures of molecules. It is feasible to find some connections between two CHF phenomena in similarity with the help of molecular thermodynamics. With these connections in microscale and in macroscale, it is possible to find an approximate solution for the corresponding principle. 3.1. Similarity in microscale Since the macroscopic properties could be represented as the sum of microscopic behaviors of all molecules, the identical amount of molecular number is necessary to guarantee the similarity of intermolecular interactions, which is important for the macroscopic phenomenon like fluid property. If the CHF in two cases are similar, the mole numbers of molecules in liquid and vapor phases should be identical, respectively, which could be expressed as:
ni;m =ni;p ¼ 1
ð5Þ
The subscripts ‘‘m” and ‘‘p” stand for modeling fluid and prototype fluid, respectively. n is the molecule number. The subscripts ‘‘i = v” and ‘‘i = l” are vapor phase and liquid phase, respectively. Due to the volume differences between different molecules, the liquid or vapor volumes of two fluids are different. Although the molecules are not absolutely symmetrical in each direction, the volume variation in macroscopic scales should be similar. It is reasonable to believe that the geometry variation is similar in different dimensions after substituting one fluid to another. Thus we have
Li;m =Li;p ¼ Di;m =Di;p
ð6Þ
where L is the channel length, D is the channel diameter. Eq. (6) guarantees the geometry similarity in each direction for both liquid phase and vapor phase. If the test section is not a circular tube, then the geometry similarity in three coordinates should be guaranteed simultaneously for two phases.
ð7Þ
where a is void fraction. According to Eq. (7), the ratio of mass quality at the CHF location could be obtained. In the phasic interface, the surface layer is in tension and tends to contract to the smallest area because of the unequal forces acting upon the molecules. The important role of surface tension in phasic interface should be properly evaluated. Since the void fraction and phasic interface are similar for two cases, there should be a similarity between the ratio of bubble velocity and bulk velocity, which plays an important role in classifying the two phase flow pattern.
ðU bubble =U 1 Þm ðU bubble =U 1 Þp
ð8Þ
where U is velocity. The subscripts ‘‘bubble” and ‘‘1” denote bubble and bulk velocity, respectively. The velocity of large bubble of flat spherical shape is a function of surface tension, gravity acceleration, liquid and vapor density [27]. Eq. (8) guarantees the similarity of the relative motion between liquid phase and vapor phase, which is of vital importance for the burnout mechanism. With the similarity of bubble velocity and bulk velocity, the mass flow rate of prototype fluid could be obtained. 3.2. Similarity in macroscale CHF are usually divided to two types, dryout and departure from nucleate boiling (DNB). Here, we divide the CHF calculation of prototype fluid into two types, xe > 0 and xe 6 0. xe is the exit vapor quality. For the case of xe > 0, the wall heat flux could be obtained with heat balance method, which could be expressed as:
qw ¼ Gin DðHout Hin Þ=4L
ð9Þ
where qw is wall heat flux, Gin is inlet mass flow rate, Hout and Hin are the fluid enthalpy in the burnout location and inlet, respectively. For the case of xe 6 0, the temperature field of prototype fluid is similar with that of modeling fluid. Thus the sub-cooling of bulk flow at the onset of nucleate boiling (ONB) should be similar to each other, which could be expressed as
T sub;ONB;m T sub;ONB;p
ð10Þ
The sub-cooling of ONB could be represented by the superheat of ONB DT sup , which is expressed with the correlation proposed by Frost [28], as:
594
DT sup
B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8rT sat qONB Prf ¼ kf qv Dhfg
ð11Þ
where h is the heat transfer coefficient, qONB is the wall heat flux in ONB location, kf is the fluid conductivity, Dhfg is the latent heat, Prf is the fluid Prandtl number.
The Freon fluid is usually used as the modeling fluid to simulate the CHF of water [15]. Since the latent heat of water is much higher than that of Freon fluid due to the effect of hydrogen bond, the heat flux for water (prototype fluid) in vapor phase will be higher than the heat flux in liquid phase, although the uniform heat flux is kept for Freon fluid. The heat flux profiles for modeling fluid and
Fig. 2. Axial flux distribution of modeling fluid and prototype fluid (a) modeling fluid (b) prototype fluid.
Fig. 3. Validation of CHF scaling laws.
B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
prototype fluid are shown in Fig. 2. Therefore, the CHF for channels with non-uniform wall heat flux should be evaluated. The axial flux distribution (AFD) effect on critical power for constant inlet conditions is small but has a strong effect on CHF at high dryout qualities. It is found that the boiling length approach (BLA) gives the best prediction of the AFD effect on CHF, compared with the other methods [7]. The BLA method could be expressed as:
CHF ¼
zCHF
1 zxe¼0
Z
zCHF
00
q ðzÞdz
ð12Þ
zxe¼0
Finally, the scaling criteria for the corresponding principle of CHF and the procedure could be summarized as follows. (a) According to the similarities of molecule number, geometry and void fraction, the density ratio of prototype fluid qv ap =qliq could be obtained with the assumption of phase equilibrium state. Then the pressure of prototype fluid could be calculated. (b) With the pressure and temperature of prototype fluid, the fluid properties could be calculated. Then the channel length and diameter of prototype fluid could be solved according to the similarities of molecule number and geometry. (c) The mass quality of prototype fluid could be obtained easily with the similarity of void fraction. (d) According to the similarity of bulk velocity and bubble velocity, the velocity and mass flow rate of prototype fluid could be solved. (e) Finally, the CHF of prototype fluid could be obtained with Eqs. (9), (10) and (12). It should be mentioned that we could obtain the CHF of prototype fluid without any empirical correlation for the case of xe > 0. However, we have to use the empirical correlation of superheat in Eq. (11) to solve the CHF for the case of xe 6 0, since Eq. (9) is not applicable in high sub-cooling conditions.
595
±21.9% and ±11.22%. Since the Stevens & Kirby method was developed for Freon-12 and water CHF data, the prediction error for Freon-12 is much less than the prediction error of Freon-113 and 114. The Stevens & Kirby method is almost impossible to predict the Freon-114 results. 5. Conclusions CHF is tightly correlated with fluid properties. Since the fluid properties could be correlated with an equation of state, the CHF for different fluids could be connected by a similar corresponding principle. The corresponding principle proposed in this work is developed on the basis of fluid similarity and molecular thermodynamics. It is assumed that there is a similarity between the prototype fluid and the modeling fluid which is substituted with identical molecule number. The average error of the corresponding principle is about 10%, which is much less than the existing methods. Since almost no empirical coefficient is used, the corresponding principle is able to be extended in wilder ranges. Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. Acknowledgments The authors are grateful to the financial support of the National Natural Science Foundation of China (Grant No. 51609227). References
4. Validation of corresponding principle Compared with the conventional CHF scaling methods based on experimental data and empirical correlations, the corresponding principle of CHF are developed only on fluid similarity and molecular thermodynamics. In order to validate the reliability of corresponding principle, the CHF data of three kinds of Freon fluids which was also employed in Kiameh’s prediction of CHF for nonaqueous fluid [13], were used and transferred to predict the CHF results of water. The CHF table of Groeneveld (2006) which is the most reliable in predicting CHF of water [29], was used as the CHF data bank to test the accuracy of CHF data obtained by corresponding principle. The validation results are shown in Fig. 3. The validation results were compared with the Ahmad scaling method which was proved to be the most typical and reliable scaling method [13,24] and the Stevens & Kirby method developed based on Freon-12 data [16]. The satisfactory agreement shows that the test results obtained in Freon fluids in a wide range could be well transferred to water conditions by using the corresponding principle. The pressure for water ranges from 5.7 to 15.0 MPa. Most of the prediction error of Freon-113 is less than ±15%, which is much lower than that of Ahmad prediction with an error of ±30%. Most of these CHF results could be predicted with an error of no more than ±20%, which is more superior than conventional methods. Because of the empirical coefficients used in compensated distortion technique, the extension of Ahmad prediction to wider ranges is not reliable. The average errors of CHF prediction for Freon-113, 114 and 12 are ±7.86%, ±10.19% and ±10.39%, which are lower than the Ahmad prediction with mean errors of ±16.06%,
[1] V.S. Nikolayev, D. Chatain, Y. Garrabos, D. Beysens, Experimental evidence of the vapor recoil mechanism in the boiling crisis, Phys. Rev. Let. 97 (2006) 184503. [2] F. He, J.H. Wang, Numerical investigation on critical heat flux and coolant volume required for transpiration cooling with phase change, Energy Conv. Man. 80 (2014) 591–597. [3] N.S. Dhillon, J. Buongiorno, K.K. Varanasi, Critical heat flux maxima during boiling crisis on textured surfaces, Nat. Commun. 6 (2015) 8247. [4] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, 3rd ed., Claredon Press, Oxford, England, 1996. [5] D.C. Groeneveld, A. Ireland, J. Kaizer, A. Vasic, An overview of measurements, data compilations and prediction methods for the critical heat flux in water cooled tubes, Nucl. Eng. Des. 331 (2018) 211–221. [6] B.E. Poling, J.M. Prausnitz, J.P. O’Connellm, The Properties of Gases and Liquids, fifth ed., McGraw Hill, 2004. [7] J. Yang, Effect of Non-uniform Axial Heat Flux Distribution on Critical Heat Flux Master Thesis, University of Ottawa, 2003. [8] Z.J. Zhang, Experimental Study on Fluid to Fluid Modeling of Critical Heat Flux in Flow Boiling Master Thesis, Tsinghua University, Beijing, China, 1984. [9] Z.Q. Lu, D.Q. Shi, Z.J. Zhang, Modeling studies on critical heat flux of convective boiling, in: Two-Phase Flowand Heat Transfer, USA Hemisphere Pub. Corp, USA, 1988, pp. 207–214. [10] J. Chen, J.R. Liao, B. Kuang, H. Zhao, Y.H. Yang, Fluid-to-fluid modeling of critical heat flux in 44 rod bundles, Nucl. Eng. Des. 232 (2004) 47–55. [11] S.Y. Ahmad, Fluid-to-fluid modeling of critical heat flux: A compensated distortion model, Int. J. Heat Mass Transf. 16 (1973) 641–662. [12] P.G. Barnett, The scaling of forced convection boiling heat transfer, United Kingdom Atomic Energy Authority, Report No. AEEW-R134, 1963. [13] B.P. Kiameh, Prediction of Critical Heat Flux for Non-aqueous Fluids in Forced Convective Boiling Master Thesis, University of Ottawa, 1986. [14] C.N. Chen, J.T. Han, T.C. Jen, L. Shao, W.W. Chen, Fluid-to fluid modeling of two phase flow critical heat flux in horizontal helically coiled tubes, Nucl. Eng. Des. 241 (2011) 1430–1437. [15] Y. Katto, A prediction model of subcooled water flow boiling CHF for pressure in the range 0.1–20MPa, Int. J. Heat Mass Transf. 35 (1992) 1115–1123. [16] G.F. Stevens, G.J. Kirby A quantitative comparison between burnout data for water at 1000 psia and Freon-12 at 155 psia. Uniformly heated round tubes,
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B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596 vertical upflow, United Kingdom Atomic Energy Authority, Report No. AEEWR32 7, 1964. R.D. Coffield, A subcooled DNB investigation of Freon 113 and its similarity to subcooled water DNB data, Westinghouse Electric Corporation, Report No. WCAP-7284; PhD thesis, University of Pittsburgh, 1969. D.C. Groeneveld, Similarity of water and Freon dryout data for uniformly heated tubes, Paper 70-HT-27, ASME Fluids and Heat Transfer Meeting, Detroit, Michigan. 1970. G.E. Dix, Freon-water modeling of CHF in round tubes. Paper 70-HT-26, ASME Fluids and Heat Transfer Meeting, Detroit, Michigan, 1970. D.H. Lee, J.D. Obertelli, An experimental investigation of forced convection burnout in high pressure water, Part I - Round tubes with uniform flux distribution, United Kingdom Atomic Energy Authority, Report No. AEEWR213, 1963. J.K. Jones, H.W. Hoffman, Critical heat flux for boiling water in a round bundle as a prelude to boiling potassium, Paper 70-HT-22, ASME Fluids and Heat Transfer Meeting, Detroit, Michigan, 1970. I.T. Aladyev, I.G. Gorlor, L.D. Dodonov, O.S. Fedynskiy, Heat transfer to boiling potassium in uniformly heated tubes, Heat Transf. - Soviet Res. 1 (1969) 14– 26.
[23] Z. Rotem, E.G. Hauptmann, Hauptmann Dryout in uniformly heated round tubes using CO2 as working fluid, with reference to dryout in water, Progress Report October-December, Dept. of Mech. Eng., The University of British Columbia, Canada, 1970. [24] A. Katsaounis, Zur Übertragbarkeit von Modelluntersuchungen der kritischen Heizflächenbelastung in Heizstabbündeln. GKSS 86/E/57, Geesthacht, 1986. [25] R.M. Tain, D.C. Groeneveld, S.C. Cheng, Limitations of the fluid-to fluid scaling technique for critical heat flux in flow boiling, Int. J. Heat Mass Transf. 38 (1995) 2195–2208. [26] B.H. Yan, Q.L. Wen, Scaling analysis for the ocean motions in single phase natural circulation, Ann. Nucl. Energy 75 (2015) 521–526. [27] Y. Taitel, D. Bornea, A.E. Dukler, Modeling flow pattern transitions for steady upward gas liquid flow in vertical tubes, AIChE J. 26 (1980) 345–354. [28] W. Frost, G.S. Dzakowic, An extension of the method of predicting incipient boiling on commercially finished surfaces. ASME-AIChE Heat Trans. Conf. Paper 67-HT-61. Seattle, 1967. [29] D.C. Groeneveld, J.Q. Shan, A.Z. Vasic, The 2006 CHF look up table, Nucl. Eng. Des. 237 (2006) (2006) 1909–1922.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Corresponding principle of critical heat flux in flow boiling B.H. Yan a,⇑, C. Wang b, R. Li a a b
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai, Guangdong, China China Ship Development and Design Center, Wuhan, Hubei, China
a r t i c l e
i n f o
Article history: Received 17 October 2018 Received in revised form 3 March 2019 Accepted 6 March 2019
Keywords: Critical heat flux Flow boiling Fluid modeling Corresponding principle
a b s t r a c t An equation of state is a thermodynamic equation relating fluid properties which describe the state of matter under a given set of physical conditions. Considering there are some connections between the properties of different fluids, we believe there are some corresponding relations between the critical heat flux (CHF) of different fluids. A corresponding principle was developed to predict the critical heat flux with the experimental data obtained with modeling fluids. A series of scaling criteria were developed on the basis of fluid similarity and molecular thermodynamics, which is totally different from the conventional scaling methods based on experimental data and empirical correlations. The prediction error of this principle is much lower than conventional prediction methods. The average and maximum error of the corresponding principle is about 10%. Ó 2019 Published by Elsevier Ltd.
1. Introduction Critical heat flux (CHF) is an upper limit of heat flux for nucleate boiling [1–3]. It is important for nuclear power plant designers to be able to predict the CHF and other system parameters at which it occurs. In nuclear engineering, many CHF experiments in water must be performed under high pressure and high temperature conditions that require a very complicated experimental system, very high heating power and very careful operation, resulting in more difficulties and expenditures in experimentation. Along with CHF measurements, CHF prediction methods were also proposed starting in the late 1940s [4]. These physical models depend on the mechanisms controlling the CHF, which changes with flow regime. Then, a more universal CHF prediction methodology, look-up tables for predicting CHF, was derived [5]. The CHF look-up table is basically a normalized CHF databank for watercooled tubes. The primary conditions at which data are scarce or missing are: (i) high flow rates and high vapor qualities, (ii) low flow rates and low vapor qualities, and (iii) pressure range of 0.2–0.4 MPa [5]. Since the existing look-up tables could only be used to predict the CHF of water, it could not be extended to other fluids. If a method which could transfer the CHF data of water to other fluids was created, the CHF databank for all fluids could be enlarged significantly. Meanwhile, the expensive CHF experiments could be substituted for some easier experiments with modeling fluid. ⇑ Corresponding author. E-mail address: [email protected] (B.H. Yan). https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.037 0017-9310/Ó 2019 Published by Elsevier Ltd.
In physics and thermodynamics, an equation of state (EoS) is a thermodynamic equation relating fluid properties which describe the state of matter under a given set of physical conditions [6]. The CHF is correlated with two types of parameters. The first is fluid property, and the other is boundary conditions, such as channel geometry and inlet mass flow rate. There should be a corresponding principle between the CHF values of different fluids. Corresponding principle, which was also called as fluid modeling in nuclear engineering, is used to reduce the high electrical power cost of CHF experiments by replacing water with a modeling fluid having a lower latent heat of vaporization [7–10]. The modeling fluids usually chosen are members of fluorocarbon family because of their very low latent heat, relatively low hazard and well established fluid properties [11]. After the CHF experiments of modeling fluid, these CHF data was used to predict the CHF of water with scaling laws. The main problem with fluid modeling is the interpretation of modeling data. The first systematic analysis to model water CHF behavior with Freon was presented by Barnett [12]. He constructed a model according to the laws of dimensional analysis. Subsequently, many models were proposed by equation-analysis or dimensionless analysis methods [13–19]. As far as the modelling of CHF is concerned, the effects of buoyancy, liquid and vapor thermal conductivities, liquid and vapor Prandtl numbers and secondary properties are negligible [20–23]. Ahmad combined the effect of Reynolds number, Weber-Reynolds number and liquidvapor viscosity ratio in one term, thus creating a compensated distortion technique [11]. This procedure showed limited success but laid the groundwork for the following empirical techniques [24].
592
B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
Nomenclature Symbols C CHF Cp D G H Dhfg k L n P Pr q T U x X
Greek letters parameter for boundary conditions a void fraction k thermal conductivity q density r surface tension
P
parameter for molecule characteristics critical heat flux thermal capacity channel diameter mass flow rate coolant enthalpy latent heat fluid conductivity channel length molecule number pressure Prandtl number Heat flux temperature velocity vapor quality Fluid property
Superscripts — dimensionless parameter Subscripts a, b, a, b fluid index e exit i, l, v index for vapor and liquid phases P pressure sat boiling point sub sub-cooling T temperature w wall 1 bulk velocity
Conventional fluid modeling of CHF was based on dimensionless similar relationship, experimental data and completely empirical correlations. These methods are often useful in engineering application, but one must avoid the temptation to use them outside the narrow range of conditions on which they are based [25]. In this work, a universal corresponding principle was established based on the fluid similarity and molecular thermodynamics. The thermal hydraulic parameters in macroscale are determined on the basis of two phase flow and heat transfer and bubble behavior. This principle is proved to be useful in a rather wide range.
CHF ¼ UðPa ; Pb Þ
2. Corresponding principle According to the molecular thermodynamics, the fluid property is correlated with the molecular motion and structure. For a specific fluid, the property parameters like viscosity, specific heat, conductivity, and density etc, could be represented as functions of pressure, temperature, polar factor, and so on. The equations could be expressed as:
X ¼ FðP; T; CÞ
showed that the properties of different fluids could be retransferred, which means that we could find the relation between the CHF of different fluids by the connection of fluid properties. It is well known that the CHF is usually contributed by the pressure, temperature, sub-cooling, viscosity, capacity, channel diameter, length, spacing grid, mixing vane, etc. Here we divide all these factors into two categories, fluid property and boundary condition. Assuming there are two similar CHF phenomena for fluid A and fluid B, as shown in Fig. 1. The burnout phenomena are attributed to the same mechanism. Thus the CHF correlations for these two cases could be expressed as:
ð1Þ
Here X ¼ l; C p ; k; q; Dhfg ; ::: denotes the fluid properties in dimensionless form, P and T are normalized pressure and temperature, respectively. C denotes dimensionless parameter representing the molecule characteristics like polar factor and associated action. As for two fluids A and B, it is easy to have P a ¼ P b and T a ¼ T b . The dimensionless pressure and temperature could also be expressed as the inverse functions of fluid property. Then, we have 1 X a ¼ FðPa ; T a ; C a Þ ¼ FðPb ; T b ; C a Þ ¼ FðF 1 Pb ðX b Þ; F Tb ðX b Þ; C a Þ ¼ GðX b Þ
ð2Þ
ð3Þ
Pa represents the boundary conditions. Pb represents the fluid properties. With the correlations of Pa and Pb for fluids A and B, we can have the CHF correlation between these two fluids. CHF a ¼ UðPa;a ; Pb;a Þ ¼ WðPa;b ; Pb;b Þ ¼ wðCHF b Þ
ð4Þ
Eq. (4) is the theoretical basis of corresponding principle of CHF. It shows that the CHF results of one fluid could be transferred to the CHF data of the other fluid. Here we have two main problems need to be solved. The first one is the similarity of CHF phenomena of two fluids in microscale, as shown in Eq. (2). The other is the correlations between macroscopic parameters of two CHF phenomena similar in microscale, as shown in Eq. (4). The abovementioned analysis reveals that we need to find the exact forms of Eqs. (2) and (4). However, the separation between molecular physics and practical problems in phase behavior is still large. Although the physical properties of every substance depend directly on the nature of the molecules of the substance, a complete understanding of molecular behavior is still unavailable. In other side, we don’t have a commonly used CHF formula, just like that in Eq. (4).
It should be mentioned that the function GðX b Þ is different from the function FðX b Þ since the dimensionless parameter included are
3. Criteria
C a and C b , respectively. Thus, it is reasonable to image that if the working fluid is replaced with the other one with identical fluid properties in CHF experiments, identical CHF results will be obtained. Unfortunately, it is almost impossible to find two fluids with identical properties in engineering application. Eq. (2)
It is difficult to find the connection between two CHF phenomena because the macroscopic phenomenon of CHF is rather complicated. We could not find the corresponding principle of CHF totally based on existing knowledge of molecular thermodynamics, either. However, all macroscopic phenomena are consisted of a large
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593
As for a universal fluid-to-fluid scaling law, it is not only applicable for the scaling between fluid A and fluid B, but also for the case between fluid A and fluid A. Since the CHF mechanism is rather complicated, it is nearly impossible to find a upscaling or a downscaling law for the same fluid in two different geometries [26]. Therefore the scaling law for these two geometries should be equal to 1:1. It is easy to find that Eqs. (5) and (6) represent the equal proportional scaling of complex phenomenon. With the similarity of molecular number, we can obtain the length ratio and diameter ratio for prototype fluid and modeling fluid approximately, due to the weak compressibility of liquid phase. Then, the liquid density and vapor density of prototype fluid could be obtained. Since the liquid and vapor density could be expressed as functions of pressure and temperature, the pressure ratio could be obtained as an inverse function of density and temperature, with the already obtained liquid density and vapor density of prototype fluid. Then the saturation properties of prototype fluid could be obtained with the pressure as an input parameter. Considering these two phenomena in similarity, as shown in Fig. 1, it is reasonable to assume that the interface shapes between liquid phase and vapor phase are similar. Thus the void fraction in the channel exit should be identical for prototype fluid and modeling fluid. Then we have
am =ap ¼ 1
Fig. 1. Two DNB phenomena in similarity.
amount of microscopic motions and structures of molecules. It is feasible to find some connections between two CHF phenomena in similarity with the help of molecular thermodynamics. With these connections in microscale and in macroscale, it is possible to find an approximate solution for the corresponding principle. 3.1. Similarity in microscale Since the macroscopic properties could be represented as the sum of microscopic behaviors of all molecules, the identical amount of molecular number is necessary to guarantee the similarity of intermolecular interactions, which is important for the macroscopic phenomenon like fluid property. If the CHF in two cases are similar, the mole numbers of molecules in liquid and vapor phases should be identical, respectively, which could be expressed as:
ni;m =ni;p ¼ 1
ð5Þ
The subscripts ‘‘m” and ‘‘p” stand for modeling fluid and prototype fluid, respectively. n is the molecule number. The subscripts ‘‘i = v” and ‘‘i = l” are vapor phase and liquid phase, respectively. Due to the volume differences between different molecules, the liquid or vapor volumes of two fluids are different. Although the molecules are not absolutely symmetrical in each direction, the volume variation in macroscopic scales should be similar. It is reasonable to believe that the geometry variation is similar in different dimensions after substituting one fluid to another. Thus we have
Li;m =Li;p ¼ Di;m =Di;p
ð6Þ
where L is the channel length, D is the channel diameter. Eq. (6) guarantees the geometry similarity in each direction for both liquid phase and vapor phase. If the test section is not a circular tube, then the geometry similarity in three coordinates should be guaranteed simultaneously for two phases.
ð7Þ
where a is void fraction. According to Eq. (7), the ratio of mass quality at the CHF location could be obtained. In the phasic interface, the surface layer is in tension and tends to contract to the smallest area because of the unequal forces acting upon the molecules. The important role of surface tension in phasic interface should be properly evaluated. Since the void fraction and phasic interface are similar for two cases, there should be a similarity between the ratio of bubble velocity and bulk velocity, which plays an important role in classifying the two phase flow pattern.
ðU bubble =U 1 Þm ðU bubble =U 1 Þp
ð8Þ
where U is velocity. The subscripts ‘‘bubble” and ‘‘1” denote bubble and bulk velocity, respectively. The velocity of large bubble of flat spherical shape is a function of surface tension, gravity acceleration, liquid and vapor density [27]. Eq. (8) guarantees the similarity of the relative motion between liquid phase and vapor phase, which is of vital importance for the burnout mechanism. With the similarity of bubble velocity and bulk velocity, the mass flow rate of prototype fluid could be obtained. 3.2. Similarity in macroscale CHF are usually divided to two types, dryout and departure from nucleate boiling (DNB). Here, we divide the CHF calculation of prototype fluid into two types, xe > 0 and xe 6 0. xe is the exit vapor quality. For the case of xe > 0, the wall heat flux could be obtained with heat balance method, which could be expressed as:
qw ¼ Gin DðHout Hin Þ=4L
ð9Þ
where qw is wall heat flux, Gin is inlet mass flow rate, Hout and Hin are the fluid enthalpy in the burnout location and inlet, respectively. For the case of xe 6 0, the temperature field of prototype fluid is similar with that of modeling fluid. Thus the sub-cooling of bulk flow at the onset of nucleate boiling (ONB) should be similar to each other, which could be expressed as
T sub;ONB;m T sub;ONB;p
ð10Þ
The sub-cooling of ONB could be represented by the superheat of ONB DT sup , which is expressed with the correlation proposed by Frost [28], as:
594
DT sup
B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8rT sat qONB Prf ¼ kf qv Dhfg
ð11Þ
where h is the heat transfer coefficient, qONB is the wall heat flux in ONB location, kf is the fluid conductivity, Dhfg is the latent heat, Prf is the fluid Prandtl number.
The Freon fluid is usually used as the modeling fluid to simulate the CHF of water [15]. Since the latent heat of water is much higher than that of Freon fluid due to the effect of hydrogen bond, the heat flux for water (prototype fluid) in vapor phase will be higher than the heat flux in liquid phase, although the uniform heat flux is kept for Freon fluid. The heat flux profiles for modeling fluid and
Fig. 2. Axial flux distribution of modeling fluid and prototype fluid (a) modeling fluid (b) prototype fluid.
Fig. 3. Validation of CHF scaling laws.
B.H. Yan et al. / International Journal of Heat and Mass Transfer 136 (2019) 591–596
prototype fluid are shown in Fig. 2. Therefore, the CHF for channels with non-uniform wall heat flux should be evaluated. The axial flux distribution (AFD) effect on critical power for constant inlet conditions is small but has a strong effect on CHF at high dryout qualities. It is found that the boiling length approach (BLA) gives the best prediction of the AFD effect on CHF, compared with the other methods [7]. The BLA method could be expressed as:
CHF ¼
zCHF
1 zxe¼0
Z
zCHF
00
q ðzÞdz
ð12Þ
zxe¼0
Finally, the scaling criteria for the corresponding principle of CHF and the procedure could be summarized as follows. (a) According to the similarities of molecule number, geometry and void fraction, the density ratio of prototype fluid qv ap =qliq could be obtained with the assumption of phase equilibrium state. Then the pressure of prototype fluid could be calculated. (b) With the pressure and temperature of prototype fluid, the fluid properties could be calculated. Then the channel length and diameter of prototype fluid could be solved according to the similarities of molecule number and geometry. (c) The mass quality of prototype fluid could be obtained easily with the similarity of void fraction. (d) According to the similarity of bulk velocity and bubble velocity, the velocity and mass flow rate of prototype fluid could be solved. (e) Finally, the CHF of prototype fluid could be obtained with Eqs. (9), (10) and (12). It should be mentioned that we could obtain the CHF of prototype fluid without any empirical correlation for the case of xe > 0. However, we have to use the empirical correlation of superheat in Eq. (11) to solve the CHF for the case of xe 6 0, since Eq. (9) is not applicable in high sub-cooling conditions.
595
±21.9% and ±11.22%. Since the Stevens & Kirby method was developed for Freon-12 and water CHF data, the prediction error for Freon-12 is much less than the prediction error of Freon-113 and 114. The Stevens & Kirby method is almost impossible to predict the Freon-114 results. 5. Conclusions CHF is tightly correlated with fluid properties. Since the fluid properties could be correlated with an equation of state, the CHF for different fluids could be connected by a similar corresponding principle. The corresponding principle proposed in this work is developed on the basis of fluid similarity and molecular thermodynamics. It is assumed that there is a similarity between the prototype fluid and the modeling fluid which is substituted with identical molecule number. The average error of the corresponding principle is about 10%, which is much less than the existing methods. Since almost no empirical coefficient is used, the corresponding principle is able to be extended in wilder ranges. Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. Acknowledgments The authors are grateful to the financial support of the National Natural Science Foundation of China (Grant No. 51609227). References
4. Validation of corresponding principle Compared with the conventional CHF scaling methods based on experimental data and empirical correlations, the corresponding principle of CHF are developed only on fluid similarity and molecular thermodynamics. In order to validate the reliability of corresponding principle, the CHF data of three kinds of Freon fluids which was also employed in Kiameh’s prediction of CHF for nonaqueous fluid [13], were used and transferred to predict the CHF results of water. The CHF table of Groeneveld (2006) which is the most reliable in predicting CHF of water [29], was used as the CHF data bank to test the accuracy of CHF data obtained by corresponding principle. The validation results are shown in Fig. 3. The validation results were compared with the Ahmad scaling method which was proved to be the most typical and reliable scaling method [13,24] and the Stevens & Kirby method developed based on Freon-12 data [16]. The satisfactory agreement shows that the test results obtained in Freon fluids in a wide range could be well transferred to water conditions by using the corresponding principle. The pressure for water ranges from 5.7 to 15.0 MPa. Most of the prediction error of Freon-113 is less than ±15%, which is much lower than that of Ahmad prediction with an error of ±30%. Most of these CHF results could be predicted with an error of no more than ±20%, which is more superior than conventional methods. Because of the empirical coefficients used in compensated distortion technique, the extension of Ahmad prediction to wider ranges is not reliable. The average errors of CHF prediction for Freon-113, 114 and 12 are ±7.86%, ±10.19% and ±10.39%, which are lower than the Ahmad prediction with mean errors of ±16.06%,
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