sphere drag ver 1doc - ResearchGate球体阻力很1doc -研究之门.doc

Comparison of formulas for drag coefficient and settling velocity of spherical particlesNian-Sheng ChengSchool of Civil and Environmental Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798Tel: 65-6790 6936; Fax: 65-6791 0676; Email: .sgAbstractTwo formulas are proposed for explicitly evaluating drag coefficient and settling velocity of spherical particles, respectively, in the entire subcritical region. Comparisons with fourteen previously-developed formulas show that the present study gives the best representation of a complete set of historical data reported in the literature for Reynolds numbers up to 2105555.Keywords: drag coefficient, Reynolds number, settling velocity, sphere 1. IntroductionEvaluation of drag coefficient and settling velocity is essential for various theoretical analyses and engineering applications. Many empirical or semiempirical formulas are available in the literature for performing such evaluations. Some are simple but only used for limited Reynolds numbers. The others, though applicable for a wide range of Reynolds numbers, may involve application procedure that is tedious. For example, the correlation presented by Clift et al 1, which has been considered the best approximation, consists of ten piecewise functions used for different Reynolds numbers.In this note, two formulas are proposed, one for quantifying the relationship of drag coefficient and Reynolds number, and the other for explicitly evaluating the terminal velocity of settling particle. Comparisons are also made with fourteen similar formulas available in the literature, which were developed for Reynolds numbers from the Stokes regime to about 2105. The results show that the function proposed here, despite its simple form, gives the best approximation of experimental data for Reynolds numbers in the subcritical region.2. Dimensionless parametersSeveral dimensionless parameters are used in this study. They are defined as follows: Reynolds number, (1)where w is the terminal velocity of settling particle, d is the particle diameter, and n is the kinematic viscosity of fluid. Drag coefficient, (2)where D = (rs - r)/r, rs is the particle density, r is the fluid density, and g is the gravitational acceleration. Dimensionless grain diameter, (3) Dimensionless settling velocity, (4)It can be shown that both Re and CD can be expressed as a function of d* and w*, i.e. (5) (6)and d* and w* can be also written in terms of Re and CD, i.e. (7) (8)3. Drag coefficient formula proposed in this studyIn this study, the following five-parameter correlation is proposed to describe the CD-Re relationship, (9)In Eq. (9), CD is predicted with two terms. The first term on the RHS can be considered as an extended Stokes law applicable approximately for Re 100, and the second term is an exponential function accounting for slight deviations from the Newtons law for high Reynolds numbers. The sum of the two terms is used to predict the drag coefficient for any Reynolds number over the entire subcritical region. Altogether six constants are included in Eq. (9). The first constant is taken as 24 following the Stokes law for very low Reynolds numbers. The other five constants were evaluated by minimizing the deviation when comparing Eq. (9) with the experimental data by Brown and Lawler 2. After conducting a critical review of historical experimental data on sphere drag, Brown and Lawler produced a high-quality raw data set of 480 points for Re = 210-3 - 2105. Using this data set, Brown and Lawler also recommended two correlations, i.e. Eqs. (11) and (24), for computing drag coefficient and settling velocity, respectively.Fig. 1 shows the CD-Re relationship plotted using Eq. (9), together with the data provided by Brown and Lawler 2. The two asymptotes, i.e. the two individual terms on the RHS of Eq. (9), and the Stokes law are also plotted in the figure. 3.1. Comparison with other CD-Re relationshipsDozens of drag coefficient formulas, empirical or semi-empirical, have been published in the literature; some examples are reported by Clift et al 1 and Heiskanen 3. In this study, seven of them are selected for comparisons, as listed in Table 1. The selected formulas are different from the others in that they were not only considered of high accuracy but also applicable for the entire subcritical region (e.g., Re 2 105). Among the seven correlations, Eqs. (10), (12), (14) and (15), appear complicated while the rest three are given in the same function but with different constants.20CDReEq. (9)Fig. 1. Standard drag curve represented by Eq. (9) in comparison with experimental data2.Table 1 Previous drag coefficient formulas applicable for the entire subcritical region.No.InvestigatorsCD-Re relationshipEq.1Almedeij 4 where, , , and for Re 106(10)2Brown and Lawler 2 for Re 2105(11)3Clift et al. 1for Re 0.01(12)for 0.01 Re 20for 20 Re 260for 260 Re 1500for 1500 Re 1.2104 for 1.2104 Re 4.4104 for 4.4104 Re 3.38105 for 3.38105 Re 4105 for 4105 Re 106 for 106 Re 4Clift and Gauvin 1(13)5Concha and Brrientos 5 for Re 3 106(14)6Flemmer and Banks 6 for Re 3105, where . (15)7Turton and Levenspiel 7 for Re 2105(16)To assess the goodness of fit of the correlations when compared with data, three statistical parameters are used. They are the average of the relative error, (17)where n is the total number of data points used, the sum of squared errors, (18)and the sum of the deviation defined using the logarithmic drag coefficient, (19)The assessment results are summarized in Table 2 and also plotted in Fig. 2, where the formulas are ranked in the increasing order of the average relative error. It can be seen that the formula, Eq. (9), proposed in this study has an average relative error of 2.47%, which gives the best representation of the experimental measurements. In particular, in spite of its simpler form, Eq. (9) performs even better than the formula by Clift et al1, which has been considered the best approximation, and that derived from the same data set by Brown and Lawler2. The average error for the other formulas varies from 3.02% to 4.76%. The values of s1 and s2 also clearly indicate the best performance of Eq. (9), by noting that those associated with the other equations could be four times higher.Table 2 Comparisons of CD-Re relationships with experimental data No.InvestigatorEquation No.Prediction errorAverage relative error, r (%)Sum of squared relative errors, s1Sum of logarithmic deviations, s21This study(9)2.4690.5760.1052Clift et al. 1(12)3.0190.8160.1463Brown and Lawler 2(11)3.2360.8100.1514Turton and Levenspiel 7(16)3.7421.0420.1885Clift and Gauvin 1(13)4.0011.1980.2156Concha and Brrientos 5 (14)4.2221.4110.2717Flemmer and Banks 6(15)4.6472.4560.5788Almedeij 4 (10)4.7621.5990.316Fig. 2. Goodness of fit of different CD-Re correlations.4. Explicit formula for evaluation of settling velocityA trial procedure is required if the settling velocity is computed using one of the abovementioned CD-Re correlations. In this section, an explicit equation is suggested for computing CD and thus w* when d* is given. Since d* is related to both Re and CD, as given by Eq. (7), CD can also be expressed as a function of d*. This is shown in Fig. 2 with the data compiled by Brown and Lawler 2. It can be seen that CD varies with d* in a similar fashion to that of CD-Re. In particular, at very low Reynolds numbers, the Stokes law applies and thus CD is related to d* in the form (20)With this consideration, a general form of CD-d* correlation, similar to Eq. (9), for the entire subcritical region is proposed here as, (21)where the five constants, 0.022, 0.54, 0.47, 0.15 and 0.45, were obtained by minimizing the deviation from the data. Eq. (21) and its two individual terms, and the Stokes law are superimposed in Fig. 3. With CD computed from Eq. (21) for d* given, w* can be obtained by (22)which is transformed from Eq. (6). CDEq. (21)Fig. 3. Variation of CD with d* predicted by Eq. (21) in comparison with experimental data24.1. Comparison with other explicit formulasSeven explicit formulas for settling velocity, as listed in Table 3, are selected from previous studies for comparisons. All of them were considered applicable for the entire subcritical region. The comparison results are summarized in Table 4 and also shown in Fig. 4, where the formulas are ranked in the increasing order of the average relative error. When applying Eqs. (17) to (19) for computing the three statistical parameters (r, s1 and s2), CD is replaced here with w*. It can be seen that the formula proposed in this study, i.e. Eq. (22) together with Eq. (21), offers the best fit to the experimental measurements, and the associated average error is 1.66%. For the other formulas, the average error ranges from 2.09% to 7.37%. The good performance can also be observed more clearly from the larger differences in the values of s1 and s2. For example, the s2-value for Eq. (26) is about 18 times greater than that associated with the formula proposed here. Table 3 Previous explicit formulas for settling velocity applicable for entire subcritical region.No.Investigatorw* - d* relationshipEq.1Almedeij 4for Re 106where , , , and .(23)2Brown and Lawler 2 for Re 2105(24)3Clift et al 1for N 73(25)for 73 N 580for 580 N 1.55107for 1.55107 N 51010where , and .4Kan and Richardson 8 for Re 3105(26)5Turian et al 9 for Re 1.5105where and .(27)6Turton and Clark 10 for Re 2105(28)7Zigrang and Sylvester 11 for Re 3105(29) Table 4 Comparisons of w*-d* relationships with experimental data No.InvestigatorEquation No.Prediction errorAverage relative error, r (%)Sum of squared relative errors, s1Sum of logarithmic deviations, s21This study(22)1.6600.2180.0412Clift et al. 1(25)2.0940.3600.0703Almedeij 4 (23)3.2841.4090.2114Brown and Lawler 2(24)3.7700.9720.1845Kan and Richardson 8(26)5.3643.2990.7816Turton and Clark 10(28)5.7772.2360.4617Turian et al 9(27)6.0293.5360.6138Zigrang and Sylvester 11 (29)7.3663.4160.720Fig. 4. Goodness of fit of various w* - d* correlations.5. ConclusionsIn spite of its simple form, the proposed function gives the best representation of experimental data available in the literature, not only for quantifying the standard drag coefficient function but also for explicitly evaluating the terminal velocity of settling particle. It is observed that the functional relationship of CD-Re remains similar if the conventional standard drag curve is recast by relating the drag coefficient to the dimensionless particle diameter in place of Reynolds number. This similarity may simplify the procedure in conducting alike correlations, as demonstrated in this study.References1Clift R, Grace JR, Weber ME. Bubbles, drops, and particles. New York: Academic Press 1978.2Brown PP, Lawler DF. Sphere drag and settling velocity revisited. Journal of Environmental Engineering-Asce. (2003) Mar;129(3):222-31.3Heiskanen KI. Particle classi