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1、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: cnschengntu.edu.sgAbstractTwo formulas
2、 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 dat
3、a 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 sem
4、iempirical 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 present
5、ed 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 evaluatin
6、g 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 bes
7、t 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,
8、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 expresse
9、d 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
10、 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
11、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. (
12、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,
13、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 ar
14、e 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,
15、 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 relatio
