FUZZY REGRESSION APPROACH TO ESTIMATING THE SETTLING VELOCITY OF SEDIMENT PARTICLES

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1 D- Fourth International Conferene on Sour and Erosion 008 FUZZY REGRESSION PPROCH TO ESTIMTING THE SETTLING VELOCITY OF SEDIMENT PRTICLES Seyed Morteza SDT-HELBR, Mohammad Hossein NIKSOKHN and Ebrahim MIRI-TOKLDNY M.S. Student Department of Irrigation and Relamation Engineering, University of Tehran, Iran. Ph.D. Candidate Shool of Civil Engineering, University of Tehran & Member of Yekom Consulting Engineers, Iran ssistant Professor Department of Irrigation and Relamation Engineering, University of Tehran, Iran. Fall veloity has a strong influene on river morphology, suspended sediment transport, and beah profile shape. In this researh a fuzzy regression analysis is performed to study the fall veloity of non ohesive sediment partiles (sand and gravel). Estimating the fall veloity of natural partiles is related to some parameters. Espeially, size of the partile plays an important role in the estimation of fall veloity. In addition, suh parameter is diffiult to preisely evaluate for natural partiles and many unertainties in measurement are existent. Deviations between the observed values and the estimated values and existent unertainties are regarded as the fuzziness of the system's parameters and oeffiients. fuzzy regression model and definition of fall veloity as a fuzzy number might be very onvenient and useful for finding a fuzzy struture in estimating the fall veloity of natural partiles. In this paper, the fuzzy regression onept and its appliation in estimation of the fall veloity of natural partiles by using experimental data are developed. Keywords: Settling Veloity, Sediment Partile, Fuzzy Regression, fuzzy number.. INTRODUCTION The settling veloity of sediment is one of the key variables in the study of sediment transport (Jimenez and madsen 00). Hallermeier (98) explains Settling veloity this way: a sediment grain in a less dense, visous fluid attains a terminal settling veloity as the gravitational fore is balaned by the hydrodynami drag fore on the grain. Fall veloity of a partile, depends on the density and visosity of the fluid, and the density, size, shape, spherially, and the surfae texture of the partile. Many attempts to predit the partile fall veloity have been arried out by researhes, started by Stokes in 85 [ited in Graf 97] and followed by Oseen (97), Rubby (9), Rouse (98), Interageny Committee (957), Zanke (977), Yallin (977), Hallermier (98), Dietrih (98), Van Rijn (989), Conharov [ited in Ibad-zadeh 99], Julien (995), Cheng (997), Jimenez and Madsen (00), Brown and Lawler (00), She et al. (005), and Wu and Wang (006) among others, who developed empirial or semi-empirial relations for estimating the settling veloity of sediment partiles. Most of above mentioned investigations, however, have some limitations when it omes to applying them to engineering works. For instane, the relations developed by Stokes [ited in Graf 97], Rouse (98), Brown and Lawler (00), are appliable only to spherial partiles. Of ourse, it is well known that the shape of natural sediment partiles departs from a sphere. This departure will have some onsequenes, one being that the settling veloity will be lower than that of a sphere with the nominal diameter. Due to the pratial impliations of this differene, several formulas have been proposed to alulate the settling veloity of natural sediments. ll of these 580

2 have been empirially derived, and in this sense they fit very well the data set employed. However, beause new relationships have been proposed they have not inorporated previously published data to hek the general auray of the respetive formulas, and in this sense there is onsiderable unertainty about whih formula is the most aurate [ited in Jemenez and Madsen 00]. Otherwise, an anillary problem related to alulating the fall veloity is using the orret value of kinemati visosity (hrens 000). It an be seen that it is very inonvenient to make a deision on seleting an optimal fall veloity relation when several empirial or semi-empirial formulas give different answers to the same problem. In response to this unertainty, in this researh these deviations are regarded as the fuzziness of the system's parameters and oeffiients. Thus, these deviations are refleted in a fuzzy definition of settling veloity and fuzzy regression model. Lotfi Zadeh introdued the simple and intuitive onept of a fuzzy set in his paper in 965 (Zadeh, 965). Sine then, fuzzy sets have been applied to a vast number of areas inluding environmental sienes: soil, forest and air pollution, meteorology, water resoures, et. (Bezdek, 999). Fuzzy regression analysis was first proposed by Tanaka et al. (980, 98). They onsidered a regression model in whih the relation of the variables is subjet to fuzziness, i.e., the model with risp input and fuzzy parameters. In fuzzy regression, the relationship between the dependent and the independent variable is not as preise as the relationship in a onventional linear regression. In other words, fuzzy regression represents a phenomenon that is impreise and vague by nature. In traditional statistial inferene, as we know, the regression models are used frequently in the researhes of the relations among several variables in a system. Observing some of the variables, we an make estimates and preditions for the others. If a system under onsideration is not governed by random variables and/or risp observation but is governed by possibility variables and/or impreise observation, it is more natural to seek a fuzzy regression analysis for suh a system. Fuzzy regression, in a general way, an be lassified into two ategories: I) Fuzzy regression when the relations of the variables are subjet to fuzziness, II) Fuzzy regression when the variables themselves are fuzzy. It should be mentioned that, in some approahes more than one ase is onsidered to provide a fuzzy regression model [ited in Taheri 00]. In this paper, estimation of settling veloity is implemented by using fuzzy tehniques rather than the usual and non-fuzzy tehniques used in the past. There is a set of risp observed data as input/output variables. Thus, the oeffiient of fuzzy regression will be fuzzy and by using this model, it is possible to estimate fuzzy number of settling veloity for the related partile diameter. The reasons are () estimation of size, shape and the surfae texture of the partiles and visosity of the fluid is diffiult to preisely evaluate in natural onditions; () the deviations of the data are suitably refleted in fuzzy parameters; () fuzzy regression is reommended for partially available data where human estimation is influential; (4) fuzzy regression model is useful in areas of deision making where there is a great deal of unertainty as well as vague phenomena (Wen and Lee 999) (5) the relation among parameters and effetiveness of them on eah other and on estimated funtion are vague. Therefore, using fuzzy tehniques tends to redue the magnitude of errors and as a result, yields better foreasts and fitting for estimating the settling veloity of natural partiles. One assumption of fuzzy regression is that the residual or deviation of the estimated value from the real value of the dependent variable is due to the fuzziness of the system's parameters (Wen and Lee 999). This researh ould be treated as the first work to investigate fuzzy approah to estimating the settling veloity of the natural partiles. The method of developing the new model is presented in the following setions. In the two next setions, basi theory of settling veloity and fuzzy regression are defined. fter it, developing of fuzzy regression model for settling veloity and related results are explained... SETTLING VELOCITY In 85, Stokes by using the Navier-Stokes equations along with a ontinuity equation expressed in polar oordinates, investigated the oeffiient of drag applied by fluid flow upon a spherial partile (Graf 97). Based on Stokes results, the fall veloity of spherial partiles in the region of partile Reynolds number (Re) less than, an be alulated using (Cheng 997): g ( s )d w 8 υ () 58

3 ρ s s ρ where: w : partile fall veloity (m/s); g : aeleration due to gravity (m/s); d : partile diameter (m); υ : kinemati visosity (m/s); s : relative density; ρ s : density of the sediment partile (kg/m); ρ : density of ambient fluid (kg/m). For natural sediment partiles, many researhers have attempted to develop similar equations whih are olleted in Table. Due to the extensive variation of natural partile geometry, however, there has been little suess in this regard; therefore a large number of different equations, eah of whih an only be applied within a limited range of sediment and fluid onditions, are now available... FUZZY REGRESSION The fuzzy regression model an be onsidered as follow: n n x i i (7) Y Where Y : output fuzzy number; x : input risp variable; n : the degree of proposed regression funtion that is equal to one at first; i : oeffiients of fuzzy regression model i ( ai, i with enter i denoted as ) a and spread i. Fig. illustrates details of nonlinear fuzzy regression and output fuzzy numbers. To estimate the fuzzy regression model, the required data set must be inluded input and output risp ( x, ( )) variables j Y x j. With these data and applying to fuzzy regression equations, fuzzy oeffiient of it will be obtained. The output of regression is a fuzzy number whih an be reognized by a fulfillment degree. This degree represents the fuzziness of output and varies between of zero and one (Fig.). In this study, fulfillment degree is assumed zero for reahing maximum fuzziness of output. Therefore, the output value Y an be explained as a triangular fuzzy number Y ( a, ) with enter a and spread.. DEVELOPING OF FUZZY REGRESSION MODEL FOR SETTLING VELOCITY s mentioned in the past setions, it is always diffiult to preisely evaluate the settling veloity and there are many unertainties. Fuzzy set theory an be used for handling the existing unertainties in estimating the settling veloity suh as measurement errors, parameters whih an't be alulated (e.g. partile shapes) and variety of effetive parameters on the settling veloity. Fuzzy regression model is developed to evaluate settling veloity by using experimental data set. The fuzzy regression is required to be fed with a x number of j data and related Y ( x j ) data. Thus, to estimate the fuzzy regression model for settling veloity, the required data set inludes partile diameter and settling veloity for orresponding partile. Four data sets, introdued by Jimenez and Madsen (00), were used for validation and training Fuzzy model (Table ). In Table, the number of data points from eah soure, n, Is listed in the third olumn. The data sets, first, were grouped into two groups. The first group, whih was taken from Cheng (997), Engelund and Hansen (97), and Hallermeir (98), orresponded to the settling veloities of natural sediments without an expliit definition of the shape fator, but taking it equal to 0.7, as it is usually taken as the most ommon value for naturally shaped sediments [for example, see Dietrih (98)]. The Cheng (997) data set is a ompilation of Russian quartz sand experiments (original referenes an be found in Cheng s paper) in whih the sediment size was haraterized through the arithmeti average diameter (Cheng 998). Beause the speifi gravity, s, was not given, it was assumed to be.65. The Engelund and Hansen (97) data set was taken from Fredsoe and Deigaard (99). The sediment size was haraterized through the sieve diameter ds (not used here), and the nominal diameter dn. Similar to Cheng s 997 data set, the speifi gravity s was not given, so it was assumed to be.65. The Hallermeir (98) data set is a ompilation of previously published experiments (original refrenes an be found in Hallermeier s paper), in whih the sediment size was haraterized by the sieve diameter. Sine the method proposed here was derived to be used with the nominal diameter, the given sieve diameters were, as previously mentioned, onverted to nominal diameter by using the rule of 58

4 Table List of relations presented for estimating Settling veloity Originator Main relation Comments Zanke (977) Hallermier (98) Van Rijn (989) Conharov (96) Zhang (99) Zhu & Cheng (99) Soulsby (997) υ ( ) s g d w d υ Re 8. Re 6.5 Re.05 gd w ( s ) 8 υ ( ) w. s g d υ w 0 + ( 0.0 d ) d g ( s )d w 4 ν w.068 ( s ) gd υ υ w (.95 ) +.09( s ) gd.95 d d υ w [ 4 os α+ d ( 9 os α+.8 sin α ) gr ( 576 os α+ ( 8 os α+.6 sin α ) D ) ] 0.6 υ ( s ) g d w ( ) 0.5 d 6υ <.4.4< <.54 >.54 d < 0.0m d > 0.m d 0.m d< 0.05 m d> 0.5 m Cheng (997) wd + 5.d* 5 υ ( s ) g d* υ 0. d.5 sf 0.7 Jimenez and Madsen (00) She et al. (005) w ( s ) g d B + S d S * ( s )g d 4υ * ( ) ( ) R.05D exp 0.08D.5. e gr gr R.05D exp 0.5D e gr gr. sf 0.7 D > gr D < gr Wu and Wang (006) Mν w s Nd 4N + 4 M D / n * n M 5.5e N 5.65e 0.65S f.5s f n S f 4 58

5 Table Data used for training Fuzzy model. [Cited in Jimenez and Madsen 00] d (mm) w (m/s) Code Data n Range Mean Range Mean Cheng (997) a, Engelund and Hansen (97),d Hallermeier (98) b,,d Raudkivi (990) d,e a rithmeti average diameter (d n ) d Nominal diameter (d N ) b ssuming d s /d N 0.9 Shape fator not given, assumed 0.7 Fig. Fuzzy regression omponents and fuzzy output number. thumb ds/dn 0.9 (Raudkivi 990). s an example of the appliability of this approah, the ratio alulated from the data supplied by Engelund and Hansen (97) gives a mean value of 0.9 with a standard deviation of The analysis in this researh is restrited to sands having size in the quartz range. Hene, only experiments with a speifi gravity equal to.65 between.57 and.67 were onsidered. The original ompilation of Hallermeier s data set also inluded the Engelund and Hansen (97) data, but it was onsidered in this researh separately. The seond group of data sets orresponded to the sediment settling veloities reported by Raudkivi (990), originally given by the U.S. Inter-geny Committee. The reported data onsist of settling veloities of sediment haraterized by its nominal diameter and shape fator. mong those, shape fator equal to 0.7 were used here (Table ). With the first try, it was obviously resulted that the fuzzy regression must be fitted on data set in two parts. The best boundary of different parts of fuzzy regression is obtained on d m whih is proposed by Van Rijn (989). Therefore, with available data and applying nonlinear fuzzy e Shape fator0.7 In all data set: Sg.65 regression equations, the fuzzy regression model of settling veloity was obtained. The best degree of proposed regression models is equal to two. 0 and are triangular fuzzy numbers. Table illustrates harateristis of fuzzy nonlinear regression models and the fuzzy regression urves, and observed risp data are shown in Fig.. By using the proposed fuzzy regression model, it is possible to estimate the fuzzy triangular value of settling veloity for orresponding partile diameter. For example, for partile equal to m, the alulated settling veloity is equal to (0.0605,0.0065). This triangular fuzzy number is shown in Fig.. Fig. Nonlinear fuzzy regression urves for estimating the settling veloity of sediment partiles. In order to evaluate the developed models, settling veloity is alulated for some partile diameter by using methods of other researhers olleted in Table and the results are shown in Fig.4. s shown in Fig.4, these methods estimate various settling veloity for one partile diameter. It means that arrangement of settling veloities, alulated by various methods for different partile diameters is not onstant. In other words, 5 584

6 when a method estimates settling veloity for a sediment partile, greater than other methods, it is possible that for other sediment partile it doesn t estimate the largest value of settling veloity. Finally, it an't be said whih method estimates settling veloity exatly. Thus, Fuzzy definition of settling veloity is the best solution of estimation it by using partile diameter. The fuzzy regression shown in Fig.4, overs the most of other methods. lthough proposed method in this paper defines a novel definition of settling veloity, it aepts all of past methods. Fig. Sample of triangular fuzzy value of settling veloity in d0.0005(m) Table Charateristi of fuzzy regression model for estimation of settling veloity 0 Limit of Diameter a 0 0 a a D< 0. 00m D 0. 00m Fig.4 Estimated settling veloity by using other researhers' methods and fuzzy regression urves 6 585

7 . SUMMRY ND CONCLUSIONS nonlinear Fuzzy model was developed for evaluating the settling veloity of non ohesive sediment partiles (sand and gravel). With the use of the 86 sets of experimental data and results, fuzzy regression approah was introdued to develop the proess models for settling veloity. This model estimates the triangular fuzzy number of settling veloity for eah sediment diameter. Comparison with past published methods shows that the proposed model has an innovation in definition of settling veloity with fuzzy set theory and estimation by using fuzzy regression model. lthough the proposed model applies past methods by using a fuzzy definition, unertainties are inorporated in predition of settling veloity. Future researh will be onerned with applying fuzzy set theory to estimation of other parameters in sediment studies and to developing sediment model. Sine, there is more unertainties in sediment issues, fuzzy set theory is very useful method for inorporating them into the models. 4. CKNOWLEDGMENT This work was done under a grant from the Yekom Consulting Engineers Company, whih is appreiated. The writers would like to thank Dr. J.. Jimenez for his useful omments and his assistane to ollet the relevant data. Editing of Mr.. Effatti hereby is aknowledged. 5. REFERENCE ) hrens, J.P fall veloity equation. J. Waterw., Port, Coastal, Oean Eng., SCE, ) Bezdek, J.C., and Kunheva, L.I Fuzzy pattern reognition. In J.G.Webster, editor, Wiley Enylopedia of Eletrial and Eletronis Engineering, 7-8. John Wiley and Sons, In. ) Brown, P. P., and Lawler, D. F. 00. Sphere drag and settling veloity revisited. J. Environ. Eng., 9(), -. 4) Cheng, N. S Simplified settling veloity formula for sediment partile. J. Hyraul. Eng., SCE, (8), ) Dietrih, W.E. 98. Settling veloity of natural partiles. Water Resoure. Res., 8(6), ) Graf, W. H. 97. Hydraulis of sediment transport. MGraw-Hill, New York. 7) Engelund, F., and Hansen, E. 97. monograph on sediment transport in alluvial streams. rd Ed., Tehnial Press, Copenhagen, Denmark. 8) Fredsoe, J., and Deigaard, R. 99. Mehanis of oastal sediment transport. World Sientifi, Singapore. 9) Hallermeier, R. J. 98. Terminal settling veloity of ommonly ourring sand grains. Sedimentology, 8(6), ) Ibad-zadeh, Y Movement of sediment in open hannels. S. P. Ghosh, translator, Russian translations series, 49,.. Balkema, Rotterdam, The Netherlands. ) Interageny Committee Some fundamentals of partile size analysis: study of methods used in measurement and analysis of sediment loads in streams. Rep. No., Subommittee on Sedimentation, Interageny Committee on Water Resoures, St. nthony Falls Hydrauli Laboratory, Minneapolis. ) Jimenez, J.., and Madsen, O. S. 00. simple formula to estimate settling veloity of natural sediments. J. Waterw., Port, Coastal, Oean Eng., SCE, 9(), ) Julien, Y. P Erosion and sedimentation. Cambridge University Press, Cambridge, U.K. 4) Komar,P.D. and Reimers, C.E Grain shape effets on settling rates. J. Geol., 86, ) Oseen, C. 97. Hydrodynamik. kademishe Verlagsgesellshaft, Leipzig, Germany. 6) Raudkivi,. J Loose boundary hydraulis. rd Ed., Pergamon Press, Oxford, U.K. 7) Rouse, H. 98. Fluid mehanis for hydrauli engineers. Dover, New York. 8) Rubey, W. 9. Settling veloities of gravel, sand and silt partiles. m. J. Si., 5, ) She. K., Trim, L., and Pope, D Fall veloities of natural sediment partiles: a simple mathematial presentation of the fall veloity law. J. Hydraul. Res., 4(): ) Soulsby, R. L Dynamis of marine sands. Thomas Telford, London. ) Taheri, S.M. 00. Trends in Fuzzy Statistis. ustrian journal of statisti, (),9-57. ) Tanaka, H., Uejima, S., and sai, K Fuzzy linear regression model. IEEE Trans. Systems Man Cybernet., 0, ) Tanaka, H., Uejima, S. and sai, K. 98. Linear regression analysis with fuzzy model. IEEE Trans. Systems Man Cybernet.,, ) Van Rijn, L.C Handbook: Sediment transport by urrents and waves. Rep. No. H 46, Delft Hydraulis, Delft, The Netherlands. 5) Wen, C.G. and Lee, C.S Development of a ost funtion for wastewater treatment systems with fuzzy regression. Fuzzy Sets and Systems, 06, ) Wu, W., and Wang, S. S. Y Formulas for sediment porosity and settling veloity. J. Hydraul. Eng., (8), ) Yalin, M.S Mehanis of sediment transport. Pergamon Press, Oxford, ) Zadeh, L Fuzzy sets. Information and Control, 8, ) Zanke, U Berehnung der sinkgeshwindigkeiten von sedimenten. Mitt. Des Franzius-Instituts fuer Wasserbau, Heft 46, Seite 4, Tehnial University, Hannover, Germany. 0) Zhu, L. J., and Cheng, N. S. 99. Settlement of sediment partiles. Res. Rep., Dept. Of River and Harbor Engrg., Nanjing Hydr. Res. Inst., Nanjing, Chi