Development of empirical models with high accuracy for estimation of drag coefficient of flow around a smooth sphere: An evolutionary approach
Graphical abstract
Introduction
The motion of particles in fluids is a key subject in many problems in the fields of chemical and metallurgical engineering as well as mechanical and environmental engineering. The solution of these problems generally involves determining the local behavior of flow and the interaction between solid and liquid phases through the knowledge of hydrodynamic forces such as drag. The drag force is the combination of the normal (i.e. pressure) and tangential (i.e. wall shear stress) forces on the body in the flow direction. However, the distributions of the pressure and wall shear stress are often very difficult to achieve, so the magnitude of the drag force can be determined only through the knowledge of drag coefficient. Analytical determination of the drag coefficient such as Stokes' law is only valid for Reynolds number, Re, less than 0.1 (Flemmer and Banks [1], Kreith [2]), although the drag coefficient can be ascertained using empirical and semi-empirical correlations based on experimental data when inertial effects are significant (i.e. higher Reynolds numbers).
The drag coefficient of a smooth sphere in incompressible flow is a function of Re based on both theoretical investigations and numerous experimental data (Kreith [2]). The main classes of the dependence of drag coefficient on Reynolds number are (1) very low Reynolds number flow (i.e. creeping flow), (2) moderate Reynolds number flow (i.e. laminar boundary layer), and (3) very large Reynolds number flow (i.e. turbulent boundary layer) (Munson et al. [3]). In the first class (Re < 1), the flows reflect entirely the viscous effect of flow with no separation results. By increasing Reynolds numbers (i.e. increasing the particle size or flow velocity for a given Kinematic viscosity), the separation region can be observed at Re ≈ 10, and the region increases until Re ≈ 1000, where most of the drag is due to pressure drag rather than frictional drag. Parenthetically, it should be noted that the value of the drag coefficient decreases, as wake area becomes larger. At a sufficiently high Reynolds number (103 < Re < 105), the drag coefficient is relatively constant (Munson et al. [3]). When transition from laminar to turbulent flow occurs, a dramatic dip (up to almost 80%) in the drag coefficient appears at critical value Re ≈ 2 × 105 since the turbulent boundary layer travels further along the surface into the adverse pressure gradient on the rear portion of the sphere before the separation, so the wake is smaller, causing less pressure drag. After this abrupt descent, the value of the drag coefficient increases by increasing Reynolds numbers. Finally, for Re > 106 a constant value of the drag coefficient (≈ 0.2) is acceptable (Potter et al. [4]).
Most of the information pertaining to drag force on the sphere arises from numerous experiments with wind tunnels, water tunnels, towing tanks, and other ingenious devices (Munson et al. [3]). Experimental data of the drag coefficient of spherical particles have been presented in the literature having a wide range of Re. However, some of the available experimental data are not accurate, adequately. Brown and Lawler [5] reviewed the experimental studies of sphere drag coefficient for Re < 2 × 105. They assembled 606 data points which were originally presented in tabular form. By excluding some experimental data for various reasons, Brown and Lawler [5] presented 480 very high quality data points by considering wall effects. This data set seems acceptable among other researchers for developing correlations (Cheng [6], Mikhailov and Freire [7]). On the other hand, Voloshuk and Sedunow [8] presented the experimental data for higher Reynolds numbers with good quality. This data set was also used in several studies such as Ceylan et al. [9] and Almedeij [10]. Fig. 1 illustrates all of the mentioned data. The variations of the drag coefficient with Reynolds numbers can follow as explained in the previous paragraph by considering Fig. 1.
In the previous studies, the regression analyses were applied to obtain a correlation for the estimation of the drag coefficient of spherical particles. Several forms and procedures such as multi-segment polynomial, exponential function, piecewise matched, power function and rational fraction were used in these studies. These forms of the correlations were developed by imposing general arithmetic operations (i.e. plus, minus, multiplication and division), and/or some function set (e.g. logarithm, and exponential functions) without considering other popular functions such as sin, cos, tan, tanh, and natural logarithm. Therefore, the performance of existing correlations is less than perfect. In the present study, multi-gene Genetic Programming (GP) is adopted to develop high accurate models for the estimation of the drag coefficient of the free falling smooth sphere. Unlike traditional regression analysis in which the structure of the model must be specified, GP automatically evolves both the structure and parameters of the drag coefficient estimation model. Therefore, both parameters and structure of the model will be optimized. The experimental data points of Voloshuk and Sedunow [8], and Brown and Lawler [5] will be used to develop empirical models. Seventeen popular correlations will be reviewed for comparison purposes, and experimental data points of Morsi and Alexander [11] along with analytical solution of Stokes regime will be considered for the validation of the developed models.
Section snippets
Literature review
Many empirical or semi-empirical correlations that vary somewhat in form have been developed to estimate the standard drag curve of smooth sphere using regression techniques. Seventeen of them which are allocated in two groups based on range of applicability are presented in Table 1, Table 2. The first group covers Reynolds numbers up to 2 × 105 while the second covers Reynolds numbers up to 106. A critical discussion about these models will be presented in the next paragraph.
Rubey [12] suggested
Genetic Programming
Genetic Programming (GP) is a random-based procedure for automatically learning the most “fit” computer programs by means of artificial evolution (Johari et al. [24]). Recently, GP has been successfully applied in many applications such as the prediction of the soil–water characteristic of soils (Johari et al. [24]), the estimation of the bridge pier scour (Azamathulla et al. [25]), and the prediction of the outflow hydrograph from earthen dam breach (Hakimzadeh et al. [26]).
GP, which is a
Developed models based on multi-gene GP
The sum of the squared deviations is a good objective function to minimize the errors between experimental data and calculated results by considering the previous studies (Turton and Levenspiel [18]; Haider and Levenspiel [19]; Brown and Lawler [5]; Barati [34], Barati [35]). Therefore, the fitness function of multi-gene GP is to minimize the sum of squared of logarithmic deviations (SSLD) between the estimated drag coefficient and experimental data pointswhere CD
Test of the developed equations
In order to examine the developed models, the results of the drag coefficient obtained from the proposed models were compared with those from models developed by other researchers in terms of SSLD, RMSLD and SRE.
Performance evaluation criteria for the correlation of the first and second groups together with the corresponding values of the developed models are listed in Table 4, Table 5, respectively. It should be stated that the models are ranked in the increasing order of SSLD in Table 4,
Discussion
In this section, two issues about the multi-gene GP procedure will be discussed: 1) Compatibility of multi-gene GP approach with the natural of the problem and 2) evaluation of the level of the accuracy of the developed models.
For the first issue, as mentioned previously, over 100 models were developed with different forms of the equations using multi-gene GP procedure. Although Eqs. (22), (23) have the lowest errors than the others, most of the developed models are better than the best
Conclusions
In the present study, the existing correlations of drag coefficient were discussed, critically. Then, a reliable and complete set of historical data was collected for the development and validation of correlations for the estimation of the smooth sphere drag coefficient. An effective procedure (i.e. multi-gene Genetic Programming) was used to develop drag coefficient models through optimizing both parameters and structure of models. Because the procedure is stochastic, the multi-gene GP was run
References (37)
- et al.
On the drag coefficient of a sphere
Powder Technol.
(1986) Comparison of formulas for drag coefficient and settling velocity of spherical particles
Powder Technol.
(2009)- et al.
The drag coefficient of a sphere: an approximation using Shanks transform
Powder Technol.
(2013) - et al.
A new model for estimation of drag force in the flow of Newtonian fluids around rigid or deformable particles
Powder Technol.
(2001) Drag coefficient of flow around a sphere: matching asymptotically the wide trend
Powder Technol.
(2008)- et al.
A short note on the drag correlation for spheres
Powder Technol.
(1986) - et al.
Drag coefficient and terminal velocity of spherical and nonspherical particles
Powder Technol.
(1989) - et al.
Predicting the drag coefficient and settling velocity of spherical particles
Powder Technol.
(2013) - et al.
Prediction of strain energy-based liquefaction resistance of sand–silt mixtures: an evolutionary approach
Comput. Geosci.
(2011) Mechanical Engineering Handbook
(1999)
Fundamentals of Fluid Mechanics
Mechanics of Fluids, SI Version
Sphere drag and settling velocity revisited
J. Environ. Eng.
The Processes of Coagulation in Dispersed Systems
An investigation of particle trajectories in two-phase flow systems
J. Fluid Mech.
Settling velocity of gravel, sand, and silt particles
Am. J. Sci.
Monograph on Sediment Transport in Alluvial Streams
Fluid Mechanics for Hydraulic Engineers
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