The 22nd Annual International Conference on Mechanical Engineering-ISME2014 22-24 April, 2014, Mech. Eng. Dept., Faculty of Eng., Shahid Chamran University, Ahvaz, Iran.
ISME2014-XXXX
A New Void Fraction Correlation for Simulation of Two-Phase Flows Using Genetic Algorithm M. Charmiyan1, B. Nezamzadeh 2 , M. Davazdahemami 3 1 2
Department of Mechanical Engineering, Isfahan University of Technology;
[email protected] Department of Mechanical Engineering, Isfahan University of Technology;
[email protected] 3 Department of Mechanical Engineering, Isfahan University of Technology;
[email protected]
parameters of practical interest, such as flow pattern transitions, pressure drop and void fraction. Void fraction ( ), which is defined as the volume of space occupied by the gas, is one of the critical parameters involved in predicting the pressure loss and heat transfer [2] in any gas–liquid system. Due to the complexity and lack of understanding of the basic physics of the problem, the majority of the analyses are based on empirical correlations for void fraction. In order to measure the void fraction, an appreciable number of experimental facilities have been built under different operating conditions. These setups were operated with different fluids under a broad range of temperatures, pressures and pipe inclinations spanning from vertically upwards to vertically downwards. Although a huge number of studies have been carried out and are still ongoing to accurately predict the void fraction for the known operating conditions, design engineers are faced with the difficult task of choosing the right correlation amongst the available correlations. The difficulty originates from the fact that most of the correlations have some form of restrictions attached to them. The idea of combining different correlations to form a unique correlation which supports various flow conditions would involve its own challenges. In order to achieve an appropriate estimation of void fraction, general expression for “slip ratio” correlation family was investigated by Butterworth [1]. Most of the correlations in this category have fixed coefficients in the associated expression. The other slip ratio models are due to Osmachkin and Borisov [3] and Mochizuki and Ishii [4]. Although Butterworth correlation predicted the void fraction well, it was not good enough for a vast type of applications, because the correlation did not include the mass flux (G), which plays an important role in evaluation of void fraction [5 and 6]. This study aims at presenting a unique and efficient void fraction correlation that could satisfactorily match most of the available experimental data, without working with complex expressions that require iterative schemes or multiple solutions. In order to obtain such a correlation, we have followed the Butterworth idea, improved his equation, and proposed a new and more general correlation for void fraction which also includes the effect of mass flux on the aforementioned quantity. Then, we calculated the corresponding coefficients with genetic algorithm, one of the most recent and robust methods for optimizing problems.
Abstract Void fraction models, which are currently used in thermal-hydraulic analysis codes in the nuclear and other industries, are based on correlations to compute void fraction distribution and slip ratio in two-phase flow. The accuracy of such correlations has a decisive role in determining the correct transport of phases and, subsequently, in the prediction of the correct response of nuclear or industrial systems. The present study aims at presenting a unique and accurate void fraction correlation that could satisfactorily match most of the available experimental data, without working with complex expressions that require iterative schemes or multiple solutions. In order to obtain such a correlation, we have followed the Butterworth idea [1], improved his equation, and proposed a new and more general correlation for void fraction which also includes the effect of mass flux on the aforementioned quantity. Then, we calculated the correlation coefficients with the aid of genetic algorithm, one of the most recent and robust methods for optimizing problems. The presented model for the void fraction is shown to well fit the available empirical counterparts in the literature. Keywords: Two-phase flow, Slip ratio models, General void fraction correlations, Genetic algorithm, Void fraction correlation. Introduction Two-phase flow, theoretically, is the simultaneous flow of two of any three discrete phases (solid, liquid or gas) of any substance or combination of substances. These two phases may have different components or there could be a phase change due to evaporation and condensation of a single fluid. Practical applications of a gas–liquid flow, of a single substance or two different components, are commonly encountered in petroleum, nuclear and process industries. In industrial applications, where two phase flow exists, the task of sizing the equipment for gathering, pumping, transporting and storing such a two phase mixture requires the formidable task of predicting the phase distribution in the system from given operating conditions. Hence, developing simple and efficient methods to predict the system behavior is crucial to fulfill industrial demands. Considerable effort has been devoted to understand the underlying physics of two phase flow in the past. Theoretical and empirical correlations have been developed to predict the various 1
Void Fraction Correlations In general, the published void correlations can be grouped into four, viz. (a) slip ratio models, (b) k models, (c) drift flux models and (d) general void fraction correlations. [7] A general expression for “slip ratio” correlation family was investigated by Butterworth [1] as a function of the ratios between the ‘‘dryness fraction” x, and “wetness fraction” (1 - x); the ratios of densities of the gas and liquid phases ( G and L); and the ratios of the viscosities of the liquid and gas phases ( LandG). “k” models calculate the void fraction by multiplying the homogenous void fraction, , by a constant, k. The “drift flux” model has the following general form jG (1) C 0 j U GU where UGU is the drift velocity, j is the mixture superficial velocity ( j=jG+jL) and C0 is the distribution coefficient. The “general” void fraction correlations are mostly empirical in nature with the basic physical principles into the different physical parameters when developing them [2]. Thom [8], Chisholm [6], Tandon et al. [9], Geraham et al. [10] and Huq and Loth [11] correlations belong to this category. The Lockhart and Martinelli 0.9
0.5
Figure 1: Schematic flowchart of genetic algorithm
where subscripts L and G stand for liquid and gas, respectively, x is mass quality, is the dynamic viscosity and is the density. Butterworth has proposed his own coefficients for A, , , We have employed genetic algorithm to achieve the best possible coefficients for Butterworth correlation. We used 1000 initial population, crossover fraction of 0.8, migration fraction of 0.2 and 500 generations. The fitness function used in the current work is infinity norm of deviation of correlation from the desired correlation or empirical data. Therefore, genetic algorithm would result in the best possible answers for the unknown coefficients. The coefficients derived in our analysis for Butterworth correlation are shown in table (1).
0.1
1 x G L [12] parameter, X tt , is x L G used in all of aforementioned models correlations. El-Boher et al. [13] definied the Froud number (Fr) and Weber number (We) based on the superficial liquid velocity, while Geraham et al. [10] included a Froud rate parameter (Ft).
Genetic Algorithm Genetic algorithms strongly differ in conception from other search methods, including traditional optimization methods and other stochastic search methods. By analogy with Nature, potential solutions of the problem live to reproduce, and the weak individuals, which are not so fit, die off. New individuals are created from one or two parents by mutation and crossover, respectively (unary and binary operators). They replace old individuals in the population and they are usually similar to their parents. In other words, in a new generation there will appear individuals that resemble the fit individuals from the previous generation. The individuals survive if they are fitted to the given environment. [13] A schematic flowchart of genetic algorithm is shown in figure 1.
Table 1: Butterworth improved coefficients
Correlation Lockhart-Martinelli Thom Baroczy
Improved Butterworth Coefficients
Butterworth proposed his general correlation based on the similarity between models and correlations which have been proposed for predicting void fractions in cocurrent gas-liquid flows [1]. Butterworth correlation is shown in Eq. 2.
1
1 x G L 1 A x L G
0.81 1.01 0.50
1.01 0.94 0.49
0.60 0.21 0.16
Figure 2 shows the comparison of Butterworth correlation based on his own coefficients, the counterpart with genetic algorithm coefficients presented here, and the data from Lockhart and Martinelli [12] for isothermal steam-water flows. Figure 3 illustrates a similar comparison including Thom correlation [8] instead of Lockhart and Martinelli [12] data. In order to demonstrate the advantage of the predicted coefficients by genetic algorithm in our study over Butterworth coefficients, infinity norm of the error of these coefficients are shown in Figure 4. It is obvious that genetic algorithm reduces the error by at least 50% in these comparisons, which may lead to much more satisfactory results.
Results and Discussions 1.
A 0.92 0.97 0.72
(2)
2. New void fraction correlation Despite the simplicity and efficiency of the format of 2
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G 1 A 1 x G *
(3) 1 x G L 1 B x L G where G is the mass flux and G* is the reference mass flux set as G*=5000 kg/m2 s. In the above equation, there are seven coefficients that are to be computed by genetic algorithm as described in the previous chapter. Investigating the general void fraction correlation, the basic underlying physics must be considered. One may deduce from Eq. 3 that as x approaches unity, the limiting value of =1 is obtained. Moreover, while x0, the denominator of the equation goes to infinity and becomes zero. These two criteria and limiting cases have previously been used to verify the proposed void fraction correlation. In addition, G is incorporated in Eq. 3 in a way that with the value of this quantity approaching zero, the Butterworth correlation is recovered. Also, the considered value of G* in our study is greater than any applied value of G. In order to validate our correlation and find the corresponding coefficients, we used four other correlations proposed by Baroczy [5], Chisholm [6], Huq and Loth [11], and Graham et al. [10].
Figure 2: Comparison of the basic Butterworth [1], GA and Martinelli [12] correlations at different pressures (steamwater)
Table 2: New correlation coefficients
Figure 3: Comparison of the basic Butterworth [1], GA and Thom [8] correlations at different pressures (steam-water)
Inf inity Nor m E rr o r
0.3 GA Coefficients
0.2
Butterworth [1]
A
B
Baroczy Chisholm Graham Huq and Loth
-0.6 -2.1 0.03
2.42 5.76 0.01
1.00 2.86 0.17
1.09 1.79 0.29
0.71 0.88 0.58
0.05 0.84 0.40
1.50 0.51 0.25
0.00
0.80
-
-
0.87
0.87
0.45
For example, working with the Chisholm correlation, one must deal with two different complex correlations depending on whether the mass flux is more or less than 2000 kg/m2 s [6]. Figures 5 and 6 show the comparison of Chisholm data and that of the new proposed void fraction correlation for G=800 kg/m2s and G=3000 kg/m2s and at different values of pressure. As seen, the new correlation with the specified coefficients is absolutely effective in predicting both high and low values of mass flux obtained from the various complex
0.35
0.25
Correlation
0.15 0.1 0.05 0 Martinellli
Thom
Baroczy
Figure 4: Error comparison of the new coefficients and Butterworth coefficients to counterpart database [1]
Butterworth formula, this correlation states that the void fraction depends only on x, and associated with each phase. But in many studies such as those of Baroczy [5] and Chisholm [6], it is shown that the mass flux (G) is also an important influential parameter affecting the void fraction. Hence, a general correlation must include this parameter as well. Accordingly, we have added the corresponding term into Butterworth equation to obtain
Figure 5: Comparison of the new model, and Chisholm [6] data at G=800 kg/m2s and different pressures (steam-water)
3
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G
Gas viscosity
References [1] Butterworth, D., 1974. “A Comparison of Some Void-Fraction Relationships for Co-Current GasLiquid Flow”. Int. J. Multiphase Flow, (1), March, pp. 845-850. [2] Woldesemayat, M. A., Ghajar A. J., 2007. “Comparison of Void Fraction Correlations for Different Flow Patterns in Horizontal And Upward Inclined Pipes”, Int. J. Multiphase Flow, (33), pp. 347–370. [3] Osmachkin, VS. Borisov, V., 1970. “Pressure Drop and Heat Transfer for Flow of Boiling Water in Vertical Rob Bundles”, IV International heat transfer conference Paris-Versailles. [4] Mochizuki, Y. Ishii, Y., 1992. “Study of Thermal Hydraulics Relevant to Natural Circulation in ATR”, 5th international topical meeting on reactor thermal hydraulics, pp. 127-134. [5] Baroczy, C., 1963. “Correlation of Liquid Fraction in Two-Phase Flow with Application to Liquid Metals”, NAA-SR-817 I. [6] Chisholm, D., 1973. “Pressure Gradients Due to Friction During the Flow of Evaporating TwoPhase Mixtures in Smooth Tubes and Channels”, Int. J. Heat Mass Transfer, (16), pp. 347–358. [7] Vijayan, P.K., Patil, A.P., Pilkhawal, D.S., Saha, D., Venkat Raj, V., 2000. “An Assessment of Pressure Drop and Void Fraction Correlations with Data from Two-Phase Natural Circulation Loops”, Int. J. Heat and Mass Transfer (36), pp. 541–548. [8] Thom, J.R., 1964. “Prediction of Pressure Drop During Forced Circulation Boiling Water”, Int. J. Heat Mass Transfer, (7), pp. 709-724. [9] Tandon, T.N., Varma, H.K., Gupta, C.P., 1985. “A Void Fraction Model for Annular Two-Phase Flow”, Int. J. Heat Mass Transfer (28), pp. 191– 198. [10] Graham, D.M., Yashar, D.A., Wilson, M.J., Kopke, H.R., Chato, J.C., Newell, T.A., 2001. “An Investigation of Refrigerant Void Fraction in Horizontal, Micro Fin Tubes”, HVAC&R Res. (7), pp. 67–82. [11] Huq, R.H., Loth, J.L., 1992. “Analytical Two-Phase Flow Void Fraction Prediction Method”, J. Thermo Phys, (6), pp. 139–144. [12] Lockhart, R.W., Martinelli, R.C., 1949. “Proposed Correlation of Data for Isothermal Two-Phase, Two Component Flow in Pipes”, Chem. Eng. Progr, (45), pp. 39–48. [13] El-Boher, A., Lesin, S., Unger, Y., Branover, H., 1988. “Experimental Studies of Two Phase Liquid Metal Gas Flows in Vertical Pipes”, Proceedings of the 1st World conference on Experimental Heat Transfer, Dubrovnik, Yugoslavia [14] Renner, G., Ekart. A., 2003. “Genetic Algorithms in Computer Aided Design”, Computer-Aided Design, (35), pp. 709-726. [15] Huq, R.H., Loth, J.L., 1992. “Analytical Two-Phase Flow Void Fraction Prediction Method”, J. Thermo Phys., (6), pp. 139–144.
Figure 6: Comparison of the new model, and Chisholm [6] data at G=3000 kg/m2s and different pressures (steam-water)
correlations presented by Chisholm study [6]. Furthermore, it should be noted that comparing the new correlation with the counterparts presented in Table 2, the maximum values of error (the absolute difference between the values obtained from the new correlation and the original data) for the data given by Baroczy [5], Chisholm [6], Graham et al. [10], and Loth and Huq [15] are 0.12, 0.082, 0.018 and 0.011, respectively. Conclusion This paper includes two distinct sections. In first section, genetic algorithm (GA) was used to improve the void fraction correlation proposed by Butterworth [1] and achieved the new coefficient for his correlation. The results showed that the correlation using new coefficients from genetic algorithm reduces the error by at least 50% in comparisons with original coefficients. Despite the simplicity and efficiency of the format of Butterworth formula, in his correlation the mass flux effect on void fraction is ignored. In the second section, the new void fraction correlation for two phase isothermal steam-water flows was proposed. Proposing the new format of correlation, the basic underlying physics was considered. For example, when x approaches unity and zero, the limiting value of =1 and =0 is obtained, respectively. Also, the mass flux independency from the quality was considered when x approaches unity. The genetic algorithm was used in order to obtain new correlation coefficients. The presented model for the void fraction was shown to well fit the available empirical counterparts in the literature. List of Symbols x Quality G Mass flux G* Reference mass flux Drift Velocity UGU j Mixture superficial velocity Greek symbols Void fraction Homogenous void fraction L Liquid density G Gas density Liquid viscosity L 4
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