Sphere drag revisited using Shuffled Complex Evolution algorithm

River Flow 2014 - Schleiss et al. (Eds) © 2014 Taylor & Francis Group, London, ISBN 978-1-138-02674-2 Sphere drag revisited using Shuffled Complex Evolution algorithm R. Barati Tarbicil Modares University, Tehran, Iran S.A.A. Salehi Neyshabouri Water Engineering Research Center, Tarhiat Modares University, Tehran, Iran G. Ahmadi Clarkson University, Potsdam, New York, USA ABSTRACT: The present study aims to the improvement of the accuracy of the empirical drag coef¬ ficient correlations using global optimization. Sixteen popular models are considered in two groups based on the range of applicability. The first group covers subcritical region while the second covers Reynolds numbers up to 10®. Shuffled Complex Evolution (SCE) algorithm is used to improve the parameters of the equations through a direct fit to the 486 reliable experimental data points. Furthermore, some parameters are added to drag equations forgiving more degrees of freedom in fitting experimental data. By using this procedure, ten new equations which were improved both in range and in accuracy were developed with the same or different forms of the existing drag correlations. The proposed equations in comparison with the existing correlations substantially (up to almost 96%) improve the fit to experimental data in terms of the Sum of Squared of Logarithmic Deviations (SSLD). 1 INTRODUCTION adequately. Brown & Lawler (2003) reviewed the experimental studies of sphere drag coefficient The motion of particles in fluids is a key subject in for the subcritical region (i.e. Re < 2 x 105). They many fields such as sediment transport. Among dif¬ assembled 606 data points which were originally ferent hydrodynamic forces which act on particles, presented in tabular form. By excluding some the drag due to the pressure and shear forces is one experimental data form various reasons. Brown & of the most important hydrodynamic forces. For Lawler (2003) presented 480 very high quality data example, in studying the motion of the sediment points by considering wall effects. This data set particles close to river beds using the Lagrangian seems acceptable among other researchers for equation of the motion, the drag force dominates developing correlations (Cheng 2009, Mikhailov & the particle motion (Lukerchenko et al. 2006, Lee Freire, in press). On the other hand, Voloshuk & et al. 2006, Nasrollahi et al. 2008, Bialik 201 1). Sedunow (1971) presented the experimental data Based on both theoretical and experimental for higher Re with good quality. This data set were investigations, the drag coefficient of a smooth also used in several studies such as Ceylan et al. sphere in incompressible flow is the function of (2001) and Almedeij (2008). Reynolds number Re. The analytical solution of Many empirical or semi-empirical correlations the drag coefficient by solving the Navier-Stokes that vary somewhat in form have been developed equations is available for Re less than one. In to estimate the standard drag curve of sphere using higher Re, the only way for the estimation of the regression techniques. However, the drag estima¬ drag coefficient is the use of empirical and semi- tion models suffer from the complicated correla¬ empirical models. tion. bounds of applicability, and/or low accuracy. Most of the information pertaining to drag For example, recently, Terfous et al. (2013) and force on the sphere arises from numerous experi¬ Mikhailov & Frcirc (2013) proposed two corre¬ ments with wind tunnels, water tunnels, towing lations with moderate accuracy for the range of tanks, and other ingenious devices (Munson et al. 0.1 < Re < 5 x 104 and Re < 118300, respectively, 2009). Experimental data of the drag coefficient of while some engineering problems have Re outside spherical particles have been presented in the lit¬ of these ranges. Meanwhile, some of the existing erature having a wide range of Re. However, some correlations were presented based on poor and/ of the available experimental data are not accurate, or scatter experimental data points. For example, 345 Yen (1992) presented a popular equation that was space. The population is partitioned into a number used in many researches (Lukerchenko et al., 2006; of complexes. The partition of the population Bombardelli et al., 2008). However, this model that facilitates an extensive exploration of the solution covers subcritical region yields considerable errors space in different directions, thereby reducing the especially for higher Reynolds numbers, as will be search getting trapped in local optima. Each com¬ shown. plex evolves based on a competitive evolution tech¬ Most researchers have focused on using regres¬ nique that uses the downhill simplex method for a sion techniques to generate a correlation for the set number of evolutions. Then, the complexes are estimation of the drag coefficient by fitting the shuffled and reassigned into new complexes to ena¬ experimental data. However, the performance of ble information sharing. A new set of evolutions the existing models is less than perfect. This fact for each complex is performed if the convergence motivated the authors to challenge other ways, is not reached. These multiple complex shuffling including global optimization as a new point of and complex evolution provide a reasonably good view in this field, to improve fitting performance. balance between the exploration and exploitation. In the present study, sixteen standard correlations The exploration that probes new and unknown were reviewed in order to improve range of appli¬ areas in the search space and the exploitation that cability and/or accuracy of them using the global uses the important information found at previously optimization rather than the regression analysis explored points are the two techniques utilized by only. Shuffled Complex Evolution (SCE) algo¬ any efficient algorithm in the search for optimal rithm was adopted to improve the parameters of solution. Therefore, it can be said that SCE algo¬ the equations through a direct fit to a complete set rithm is one of the best available algorithm for of reliable historical data reported in the literature. global optimization of the correlation of the drag Furthermore, the structure of the some correla¬ coefficient. For further details of the procedure of tions were revised by considering additional either SCE algorithm, the readers can refer to Duan et al. constants or exponent parameters to increase (1993). degrees of freedom in fitting experimental data [i.e. 486 data points of Brown & Lawler (2003), and Voloshuk & Sedunow (1971)]. Furthermore, 3 REGRESSION-OPTIMIZATION although the standard drag equation of the PROCEDURE smooth sohere has a uniaue curve. cXDerimental data points of Morsi & Alexander (1972) will be Shuffled Complex Evolution algorithm was used for validation of the developed models. applied to improve the parameters of the empiri¬ cal drag equations. From previous studies of the same or different applications, it can be concluded 2 SHUFFLED COMPLEX EVOLUTION that the sum of the squared deviations is a good ALGORITHM objective function to minimize the errors between experimental data and calculated results (Turton & Shuffled Complex Evolution algorithm that incor¬ Levenspiel 1986, Barati 201 1, Barati 2013). There¬ porates the best features from several determinis¬ fore, the Sum of Squared of Logarithmic Devia¬ tic and stochastic methods have been successfully tions (SSLD) was selected as the objective function proven for global optimization using only function for the parameter estimation of the empirical drag values. SCE algorithm has been extensively used in equations water engineering problems (e.g. Eusuff and Lansey 2004, Zhijia et al. 201 1). From previous studies, it can be concluded that SCE algorithm is able to cope Minimize SSLD = ÿT (jogCB -logCn) (1) very well with rough, insensitive, and highly non- smooth objective function surfaces and is relatively unaffected by the occurrence of a local optima. where CB = the experimentally reported drag coef¬ In essence, SCE algorithm works on the basis ficient of a sphere, = the corresponding drag of following concepts (Duan et al. 1993): (1) blend coefficient that is estimated by the empirical equa¬ of deterministic and stochastic approaches; tions, and N= the total number of the data. (2) systematic evolution of a complex of points; It should be noted that negative values of (3) competitive evolution; and (4) complex shuf¬ drag coefficient can be obtained if infeasiblc fling. All of these concepts were successfully values of coefficients and exponents of the drag proven for global optimization. equations are selected in the optimization proce¬ SCE algorithm begins with an initial popula¬ dure of the model. Therefore, in order to avoid tion of points sampled randomly from the feasible negative drag coefficient, a penalty function 346 approach is imposed to the optimization model 4. Sum of Squared Relative Errors (SSRE) as follows: >rf4 '/ (2) SSRE = £ (6) where /= the penalty constant and Co = the penal¬ ized drag coefficient which is positive but unrealis¬ tic value. 5 EXISTING MODELS It is notable that four stopping criteria are used in SCE algorithm at each generation. The calculation Sixteen available drag estimation models which are is immediately terminated if any of them is satis¬ listed in two groups based on range of applicabil¬ fied. These criteria consist of: (1) difference between ity are presented in Tables 1 and 2. The first group best and worst function evaluation in population is covers Reynolds numbers up to 2 x 10s while the smaller than the tolerance (i.e. 0.001); (2) maximum second covers Reynolds numbers up to 10®. difference between the coordinates of the vertices in For the correlations of the first group, the simplex is less than the tolerance (i.e. 0.001); (3) max¬ models proposed by Rubey (1933), Graf (1984) imum number of function evaluations or iterations and Rouse (1938) estimated the drag coeffi¬ has been reached (i.e. 5000); and (4) maximum dura¬ cient far from experimental values, especially in tion of optimization has been reached (i.e. 30 sec). higher Reynolds numbers. For other correlations, For each of the drag equations, some coefficients the largest differences with experimental values and exponents arc considered as the design param¬ occur when the value of Re is beyond 4 x 102. It eters which will be optimized using SCE algo¬ should be noted that some correlations such as rithm. On the other hand, some coefficients and Graf (1984) and Yen ( 1992) have a monotonically exponents will be added in some drag equations decreasing function while the experimental data through a trial and errors procedure to provide decrease until a Reynolds number value around more degrees of freedom in fitting experimental 5000, and, then increase by growing Reynolds data. By using this regression-optimization proce¬ numbers, so these models yield a systematic error dure, the accuracy of the empirical drag equations trend. On the other hand, some rest of them can can be increased, as will be shown. capture a minimum for the drag coefficient in this region although the accuracy of these models is less than perfect. 4 PERFORMANCE EVALUATION For the correlations of the latter group, the CRITERIA main drawback of these correlations is that some of these models suffer from low accuracy in mod¬ The performance of the proposed drag equations erate Re values while rest of them cannot predict are evaluated by using the following measures: the experimental values in large Re values, accu¬ rately. In other words, none of them follow closely 1. Root-Mcan-Squarc of Logarithmic Deviation the experimental values in both moderate and large (RMSLD) Reynolds numbers, simultaneously. For example, although the proposed model by Clift et al. (1978) has good accuracy in moderate Reynolds numbers RMSLD = J£(iogCD- log CD) N (3) region, the model yields considerable errors for large Re values. On the other hand, the model of Almedeij (2008) has inverse style. The statistics for the various correlations of the 2. Average Relative Error (ARE) first and second groups together with the range of using of the models in the calculation of the cri¬ teria are presented in Tables 3 and 4, respectively. ARE- x 100 (4) It should be noted that the models are ranked in the increasing order of SSLD. Other comparative statistics (i.e. RMSLD, ARE, SRE, and SSRE) 3. Sum of the Relative Errors (SRE) almost have similar trend. The equation proposed by Cheng (2009) has best accuracy among the existing correlations of the suberitical region (i.e. SRE = fj (5) the first group) while the equation of Almedeij (2008) has lowest error in second group. 347 Table 1. Summary of some empirical relationships for the subcritical region. Equation Reference Model and Reynolds number range no. 24 , 5 Rubey (1933) Cn = — + 2 Jor Re < 2 x 10 (7) ° Re 24 3 Rouse ( 1938) Re<2x 10 r7+rÿ-+0.34 for (8) Morsi & 24 (9) Alexander for Re < 0.1, (1972) Re 22.7300 0.0903 -+ -— + 3.6900 for 0.1 < Re < 1, Re Re2 29.1667 3.8889 - + 1.2220 for 1 < Re < 10, Re Rc2 46.5000 1 16.6700 - + 0.6167 for 10 < Re <100, Re Re2 98.3300 2778 - + 0.3644 for 100 < Re < 1000, Re Re" 148.6200 4.75 X 104 - + 0.3570 for 1000 < Re <5000, Re Re 490.5460 57.87x10* - + 0.4600 for 5000 < Re <10000, Re Rc" 1662.5000 5.4167x10" - + -;-+ 0.5191 for 10000 < Rc < 50000 Re Re- 24 / — (l — +O.OI7Rc)\>-- i 0.208 --— ReV + O.I5VRe for Yen (1972) C„= Re<2xlO (10) " 1+10 Re 7.3 Graf (1984) C„= — + -ÿ— + 0.25 for Re < 2x 10 (11) ° Re 1+ Re ?4 o 143 Flemmer & CD = — Re 10£ where E = 0.383 Rc0356- 0.207 Re0 396 -- ---for Re < 2 x 105 l (logRe)" (12) Banks (1986)* + Khan & illdll Ol. . .3 |g (13) Richardson C„ = (2.49Re"°32s+0.34Re0067) for Re<2xl0: (1987)* 1(aider & ->* 0.407 (14) CD = — (l + 0.150 Re" 6X1 )+ - ÿ Levenspiel for Re < 2 X 10 (1989)* Rc I + 87 10 Re" |2.S Swamee & for Re < 1.5 x 10s (15) Ojha (1991) Cheng (2009) CD = — (l + 0.27Re)"4' + 0.47f 1 - exp(-0.04Re" L v W)1 /J for Re<2x 105 (16) Re Terfous et al. (2013) Mikhailov and 21.683 0.131 C„ = 2.689 + -+ -; --- 10.616 12.216 — +- — 3808[(l6 17933/2030) + (178861/l063)Re+ (l219/1084)Re2] , „, „ , . for 0.1<Re<5xl04 for Re <118300 (17) (18) Freire (2013) 681 Re[(7753 1/422) + (13529/976)Rc- (1/7 1 1 54)Re2] *These models were improved by Brown & Lawler (2003). 348 Table 2. Empirical relationships for Reynolds numbers up to 106. Reference Model and Reynolds number ranee Equation no. Clift et al. 24 3 (19) (1978) — +— Re 16 for Re <0.01 -ÿ-(l + 0.1315ReO82"oosll°sRc0 for 0.01 < Re <20 Re — (\ + 0. 1935 Re""05 ) ' for 20 < Re < 260 Re 10I MJ5-IUtttloiR.l+o.iswtoR.f jor 260 < Re <1500 |0-2 457U15558[logR«}-09295[loeRe]! ÿ0.I049[|o8 R.f r |5Q0 < Rc < 1.2 X 10J 10-..9I.,*o.6)7o[i,r.]-oo6*[wr.]! /ot 1.2 x 104 < Re < 4.4 x 104 10-I33*)»| SKflliojReJ-o iMelkjgRcJ5 4 4 x 104 < Re < 3.38 x I05 29.78-5.3[logRe] for 3.38 x 10' < Re S 4 X 105 0.l[logRe]-0.49 for 4 X 105 < Re £ 10® Ceylan et al. CD = 1 -0.5exp(0.182) + 10.1 IRe~2/3exp(o.952Re~l/4) (20) (2001) - 0.03859 Re exp (l .30 Re~l/2 ) + 0.037 X 10J Rc exp (-0.125 X 10 Re) J - 0.1 16 x 10"'° Re2 cxp(-0.444 X 10"' Re) for 0.1</fe<106 . 1°' Almcdeij (21) (2008) m v>2) ' + W ' M + V:) w j where </>, = (24 Re"')'" + (2 1 Re"067]!" + (t Re"0 33)'" + (0.4)'° , ip, = ÿJ148Re" 'J'" + (0.5)"'° 1 , (l = .57 x 10s Re"' 6-5)".<p4 = (6 x 10"'7 Re2'3)""' + (0.2)",oJ for Re < 10* Morrison 24 (t) f_5i_ , Re08 ' 2.C 152 0.41 V 263000 r -sum ' for Rc < 10 (22) (2013) Re '*(t) 'ÿ(———1 461000ÿ 2630(Xy 6 RESULTS AND DISCUSSION in the original forms, respectively, are applicable in the whole of the subcritical region. Other improve¬ Ten new equation along with their range of appli¬ ment of the range of the applicability of the equa¬ cability are listed in Table 4. Except one of these tions can be investigated by comparing Tables 1, correlations which is for the higher Reynolds num¬ 2 and 5. bers (Re < 10*); others are applicable in subcritical Comparative statistics values of the devel¬ region. In addition to higher accuracy, the range oped models together with the percentage of the of applicability of the equations are extended for improvement of them (in terms of SSLD) than some of the developed models. For example, two the original forms are presented in Table 6, where recent developed models by Terfous et al. (2013) the equations are ranked in the increasing order of and Mikhailov & Frcirc (2013) which are valid for SSLD. The results indicate that most of the new the range of 0.1 < Re < 5 x 104 and Rc < 1 18300, correlations arc substantially improved (between 349 Table 3. Comparative statistics of existing drag coefficient correlations for subcritical region. Order of accuracy Reference SSLD RMSLD ARE SRE SSRE Range of using 1 Cheng (2009) 0.1051 0.0148 2.4701 11.8567 0.5764 Re < 2 x I05 2 Morsi and 0.1314 0.0176 3.0770 13.0465 0.7063 Re < 5 X I04 Alexander ( 1972) 3 Clift ctal.( 1978) 0.1457 0.0174 3.0188 14.4902 0.8157 Re < 2 X 10s 4 Brown and Lawler 0.1510 0.0177 3.2346 15.5260 0.8095 Re < 2 X 10s (2003) Eq. (11) 5 Brown and Lawler 0.1978 0.0203 3.6404 17.4739 1 .0439 Re<2x I05 (2003) Eq. (9) 6 Brown and Lawler 0.2091 0.0209 3.6347 17.4464 1.1318 Re < 2 X I05 (2003) Eq. (10) 7 Swamee and 0.2778 0.0242 4.3152 20.4107 1.5497 Re < 1.5 X 10s Ojha (1991) 8 Almedeij (2008) 0.3158 0.0256 4.7622 22.8585 1.5989 Re < 2 x I05 9 Terfous et al. (2013) 0.4670 0.0322 5.1659 23.1950 2.9002 0.1 <Re<5x 10' 10 Mikhailov and 0.4703 0.0319 5.6105 25.9766 2.8755 Re < 118300 Freire (2013) 11 Morrison (2013) 0.5237 0.0330 5.9316 28.4716 2.5406 Re < 2 X 10s 12 Yen (1992) 0.8475 0.0420 8.0672 38.7227 4.7571 Re < 2 X 10s 13 Ccylan ct al. (2001) 0.8616 0.0438 7.2010 32.3325 4.2597 0.1 <Rc<2x 105 14 Rouse ( 1938) 2.1313 0.0666 10.7011 51.3652 8.9876 Re < 2 X 10' 15 Graf (1984) 5.2163 0.1042 13.1639 63.1865 18.0420 Re < 2 X 10' 16 Rubey ( 1933) 95.9657 0.4471 168.3168 807.9209 2644.5496 Re < 2 X 10s Table 4. Comparative statistics of existing drag coefficient correlations for reynolds numbers up to 10s. Order of accuracy Researchers SSLD RMSLD ARE SRE SSRE Range of using 1 Almedeij (2008) 0.3210 0.0257 4.7756 23.2094 1.6262 Re < 10' 2 Clift etal. (1978) 0.4467 0.0303 3.5803 17.4004 4.0092 Re < 10" 3 Morrison (2013) 0.5672 0.0342 6.0034 29.1768 2.7086 Re < 106 4 Ceylan et al. (2001 ) 0.9186 0.0449 7.3170 33.2925 4.5226 0.1 < Re < 10' 28.31-95.54%) than the old formulations. How¬ As shown in Table 6, Eqs. (29) and (28) which ever, two of them are improved less than 10% (i.e. were originally proposed by Cheng (2009) and 4.27 and 7.94%). For subcritical region, three equa¬ Clift and Gauvin ( 1970) located at the second and tions that have lowest errors will be challenged. third places, respectively. It should be noted that Among the improved equations, Eq. (25) which although Eq. (28) was previously improved by was originally presented by Swamee and Ojha other researchers such as Turton & Levenspiel (1991) using a piecewise matched procedure has (1986), Haider & Levenspiel (1989), and Brown the best accuracy in terms of the SSLD. It should and Lawler (2003), the present equation that has be noted that the original form of this equation is the same form is improved 28.31% in terms of valid for Re < 1.5 x 105 while the improved form is SSLD. This fact uphold that the necessity of the acceptable for Re < 2 x 105. The improvement of applying global optimization procedure for such a the accuracy of this model is 65.48% than the orig¬ problem. Figure 1 depicts a comparison of three inal form. This improvement derived from opti¬ mentioned models [Eqs. (25), (29) and (28)] with mizing the coefficient and exponent parameters of the experimental data. It can be seen that although the original equation and adding a constant coef¬ all of them accurately follow the experimental data. ficient in the last bracket of the right-hand-side Eq. (25) seems to better capture a minimum for of the equation. Similar procedure was used for the drag coefficient than other formulas, and also other correlations, as mentioned in the regression- it can be said that Eq. (25) represents better the optimization procedure section. mean values of the experimental data for higher 350 Table 5. New and improved sphere drag correlations. Improved correlation Range of applicability Equation no. 24 f 7xl07-+8.8253x10 Re < 2 x 105 (23) Re Re C„ -- I 24 xlO3 Rc2 Re 1.2237 2.3070x10' + 9770 -0.0012 Re < 2 X 10' (24) [7 14 y 4,512 0 *( / 1.89318 C„ =0.49311 Re < 2 X 10s (25) \ 40000V + 0.93213 - +1 1 Re J 4.5011 CD = — (l-0.1072VRl+0.2062Re)- Re < 2 X 10' (26) Re 1 + 9.3098 Re" 24 0.143 C„ = — 10' where E = 0.383Re - 0.207 Re Re < 2 x 10' (27) Re [l + (log Re)"] 2.9160 CB =— Re (l + 0.1250 Re1 0I05)- 1 + 8.5126Re Re<2x 10' (28) 24 CD = — (1+ 0.2991 Re)°41" + 0.4739[l - exp(-0.0463Re0365<1)] Re < 2 X 10' (29) 24 999.9887 740.5887 257.0420 Cc-0.6631+ — + - Re„„,7, " Re„,K7« Re<2x 10' (30) [l526.7573 + 233.1930 Re +1.1236Re:] Re < 2 x 10' (31) e[63.6 Re| 63.6199 + 2.7614Re - 3.0982 x 10ÿ Re2] ( Re V" 2374 ( Re \° 4.2798 0.4769 - 24 Cn= — + - V5J \ 0.8961 / V 263000 J \- Re"815"] Re< 10' (32) Re ( n 461000 I+ 1*£\) 5 i+f-5?-] I* V. 263000 / Table 6. Comparative statistics of proposed drag coefficient correlations. Order of Improved Improved accuracy correlation SSLD RMSLD ARE SRE SSRE (%) Range of using 1 Eq. (25) 0.0959 0.0141 2.3719 11.3852 0.5145 65.48 Re<2x 10' 2 Eq. (29) 0.0968 0.0142 2.3641 11.3475 0.5197 7.94 Re < 2 X 10' 3 Eq. (28) 0.1083 0.0150 2.4593 11.8044 0.5911 28.31 Re < 2 X 10' 4 Eq. (30) 0.1165 0.0156 2.6236 12.5932 0.6304 75.05 Re < 2 X 10' 5 Eq. (32) 0.1360 0.0167 2.6455 12.8571 0.7328 76.02 Re < 10' 6 Eq. (31) 0.1816 0.0194 3.0907 14.8354 1.0018 61.38 Re < 2 X 10' 7 Eq. (27) 0.1894 0.0199 3.4657 16.6356 0.9755 4.27 Re<2x 10' 8 Eq. (24) 0.2323 0.0220 3.5937 17.2497 1.2846 95.54 Re < 2 x 10' 9 Eq. (26) 0.2500 0.0228 4.0740 19.5554 1.3859 70.50 Rc<2x 10' 10 Eq. (23) 0.2832 0.0243 4.4331 21.2787 1.5554 86.71 Re < 2 X 10' 351  10 E 2 8u 10 Experimental data Eq. (25) « Eq. (29) 10 -Eg. (28) 10'2 10° 102 I04 106 Reynolds number Figure 1. Illustration of accuracy of developed models for subcritical region. 6 - 4 10 c .H s 2 8'J 10 0 ° Experimental data 10 J_Eq. (32) _ i 10"2 10 ,9 10' ,2 10',4 10 6 Reynolds number Figure 2. Illustration of accuracy of developed model for Reynolds numbers up to 106. Reynolds numbers. However, for lower Reynolds For verifying the performance of the selected numbers ( Re <0.1), Eq. (29) has better results than models [i.e. Eqs. (25), (29), (28) and (32)], they are both Eqs. (25) and (28). validated using experimental data of Morsi and On the other hand, Eq. (32) which was origi¬ Alexander (1972). The results indicated that the nally developed by Morrison (2013) is suggested for selected models can predict these experimental val¬ Re < 10' (i.e. the second group which covers tran¬ ues, accurately. sition region from laminar to turbulent flow). This Finally, it should be stated that a constant value equation is improved 76.02% than the original form of the drag coefficient (=0.2) is acceptable for in terms of SSLD. In addition to the higher accu¬ Re > 106 (Almedeij 2008). racy, Eq. (32) has simpler form than other models of the second group [i.e. the proposed equations by Clift et al. (1978), Ceylan et a). (200 1 ) and Almedcij 7 CONCLUSIONS (2008)]. The drag coefficient-Re relationship of the developed model is shown in Figure 2. It can be seen After reviewing several empirical models of the that the estimated drag coefficients of the model, drag coefficient of the smooth sphere, ten new closely follow the experimental data in all of the and improved correlations were developed based creeping flow, transitional, and turbulent regions. on a critical examination of available experimental 352  data for spheres, and through a global optimiza¬ Clift, R. Grace, J.R. & Weber. M.E. 1978. Bubbles, drops. tion procedure using shuffled complex evolution andparticles. Academic, New York. algorithm. The proposed models were improved in Duan. Q.Y. Gupta, V.K. & Sorooshian, S. 1993. Shuffled both range of applicability and accuracy by adjust¬ complex evolution approach for effective and efficient global minimization. Journal of Optimization Theory ing of parameters and/or adding some parameters, and Applications 76(3): 501-521 . and using 486 reliable experimental data points of EusufT. M.M. & Lansey, K.E. 2004. Optimal operation Brown & Lawler (2003), and Voloshuk & Sedunow of artificial groundwater recharge systems considering (1971). Furthermore, the experimental data of water quality transformations. Water Resources Man¬ Morsi & Alexander (1972) were used to verify the agement 18(4): 379-405. selected models. The analyses demonstrated that Flcmmer, R .L.C. & Banks, C.L. 1986. On the drag coef¬ the proposed relationships are efficient for accu¬ ficient of a sphere. Powder technology 48(3): 217 221 . rately estimation of the drag coefficient. Graf, W.H. 1984. Hydraulics of Sediment Transport , Water Resources Publications, Littleton, Colorado. 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