The Derivation of Energy Dissipation Equation for Adverse-Slopped Stepped Spillway

Transcription

1 World Applied Sciences Journal 15 (5): , 011 ISSN IOSI ublications, 011 Te erivation of nerg issipation quation for Adverse-Slopped Stepped Spillwa 1 Medi Fuladipana and Reza Jafarinia 1 epartment of Civil ngineering, Ramormoz Branc, Islamic Azad niversit, Ramormoz, Iran epartment of Agriculture and Natural Resources, Arak Branc, Islamic Azad niversit, Arak, Iran Abstract: Te analsis of energ dissipation rate plas a major role in te problem of inclined stepped spillwas. In tis paper, te derivation of relative energ dissipation equation on adverse-slopped stepped spillwas as been considered. Step lengt (), step eigt (), te adverse-slope of step ( ), ratio c ( ), ratio ( ) and energ dissipation rate for eac step ( ) were used to derive te equation. sing Bukingam pi teorem and dimensional analsis, five dimensionless parameters in te form of a dimensionless function was derived. Te multiple linear regression (using SSS software) was used to calculate unknown coefficients of te equation. Te equation was applied to simulate relative energ dissipation. Te results b predictions were in good agreement wit te measurements. Ke words: Adverse-Slopped Stepped Spillwa nerg issipation quation Bukingam i Teorem imensional Analsis Multiple inear Regression INTROCTION 0.54( c ) ( ) = c 0.55 Contrar to te belief tat te use of stepped cannels for energ dissipation purposes is a new c concept (developed along wit te introduction of new construction tecniques, e.g., roller compacted concrete, Fratino et al. analzed energ dissipation at te base gabions), stepped cutes ave been used since antiquit. of a stepped spillwa under nappe flow as follows []: Stepped cannels were designed to contribute to te stabilit of structures (e.g. overflow weir) and to dissipate flow energ. In fact, te tecnique of stepped cannels = 1 () was developed independentl b several ancient c civilizations. Some 16 dams wit stepped spillwas were built during ancient times, ranging in eigt from 1.4mto 50 m, widt from 3.7m to 150 m, maximum discarge up to = 3 (3) about 9000 cumec, and wit step eigt varing from c 0.6 m to 5 m wile te number of steps varied from to 14. In tese spillwas, steps increase energ dissipation Camani and Rajaratnam investigated energ rate and decrease necessit of stilling basin for energ dissipation for nappe flow on inclined steps. Teir dissipation wic will decrease dissipaters dimensions at relationsip is as following [3]: downstream. Studing of te effect of different parameters N 1 on energ dissipation rate as been subject of various i [(1 a)( ) + (1 a) i= 1 researces. Canson developed te following expression = 1 C (4) valid for free flow spillwas and nappe flow wit full N developed draulic jump [1]: C (1) Corresponding Autor: Fuladipana Medi, epartment of Civil ngineering, Ramormoz Branc, Islamic Azad niversit, Ramormoz, Iran. 637

2 = ( ) C cos C.0 = ( ) Fr cos ( Ra ) Fr World Appl. Sci. J., 15 (5): , 011 C a = og og (5) including eras et al., Rice and Kadav, Yasuda and Otsu, Canson and Toombes, Boes and Hager, Gonzalez, Yasuda et al. ave derived a general expression for Takaasi et al., Hunt and Kadav, and Felder and te energ loss at te base of a stepped spillwa wit Canson ave examined flat-sloped ((H):1(V) or flatter) skimming flow, regardless of weter te flow is uniform stepped spillwas [10-18]. In above mentioned papers, or not [4]: equation for effect of adverse-slopped steps asn't been w ( ) + ( w)cos presented. In tis researc, te effect of tis parameter C = 1 C (6) as been studied. 3 + ( ) C MATRIAS AN MTHOS Boes and Minor suggested te following equation for te non-uniform flow in skimming regime [5]: In tis researc, dimensional analsis and multiple liner regression as been used. resented equations are = exp[( 0.045[ ] (sin ) ] ] (7) based on a psical model of stepped spillwa wit C adverse-slopped. Applied parameters in te experiments Boes and Hager introduced a relationsip for te rate of are as following: step number (n), step lengt (l), spillwa energ dissipation in skimming regime as following [6]: eigt (), adverse-slope of eac step( ). Te spillwa model was 871(mm) to 3594.(mm) ig and covering flow 1 f 3 Φ f 3 3 ( ) cos + ( ) 3 cm cm of = 1 8 sin? 8 sin (8) ( ) to 300( ). Steps ranged from 8 to 30 in cm.s cm.s C numbers. Te Ratio ( ( ) was 0.41, 0.56, 0.631, 0.736, 1 Yazdani assumed uniform flow all over te spillwa and te rate of energ dissipation was suggested as a function of tis drag coefficient as follow [7]: (9) (10) Anoter relationsip as been presented for C as following [8]: ( ) (tan ) C = cos (11) Kavianpour and Masoumi presented an expression wic was derived to determine te rate of energ dissipation for non-uniform flow regime as follow [8]: 0.84 in amount. Five different values were selected for : 5, 10, 15 and 0 in degrees. Neglecting te effect of air entrainment on energ dissipation, relative energ dissipation is function of step numbers (n), step eigt (), step lengt (l), critical dept on spillwa crest ( c), adverse-slopped of step ( ), and energ dissipation coefficient of eac step (a). On te oter and, it can be written: = f(,n,,, a, Φ) (13) Wic is energ loss between downstream and upstream of spillwa and is energ amount just upstream of spillwa. is called relative energ dissipation. According to Bukingam pi teorem, five dimensionless parameters must be derived. Tese dimensionless parameters are as following: = 0.047( s ) Re Fr (tan ) ( ) u s (1) Hunt and Kadav studied te effect of spillwa slope on energ dissipation [9]. Te found tat man of researces studies were performed on steeper ((H):1(V) or steeper) stepped spillwas, but a few researcers, 1 3 a C 4 s p = n, p =Φ, p =, p =, p = (14) At lost, equation (13) can be written following relation: kn x1 x x 3 ( C ) x = Φ 4 ( ) x5 a (15) Multiple liner regression tecnique was applied to calculate k, x 1, x, x 3, x 4, and x5exponents wic was done using SSS software. xtending multiple linear regression 638

3 World Appl. Sci. J., 15 (5): , 011 tecnique to dependent variable i and independent variables x 1, x,, x k ields te following relation equation: i = 0 + 0x i +... k xik i=1,,...,n(n>k) (16) Wic ( 0) is a constant value and ( 1,... k) are coefficients of independent variables. According to equation (16), equation (15) sould be liner b taking logaritm from bot side of equation (15): software. Final result of software output is presented in table 1. According to table 1, equation (17) can be written as follow: og( ) = og(n) -3 - (18) og( Φ ) og( a) og( c ) og( ) Final form of equation (15) can be derived from equation (18) as following: og( ) = og(k)+x1og(n)+x og( Φ) 0.83n ( c 0.04 = Φ ) ( ) a (17) (19) +x3og(a)+x4og( c ) + x5og( ) Measured and calculated (from eq. (19)) values of relative energ dissipation are compared in figures 1 to 4. Relation between te logaritm of dependent variable Final results of comparison are presented in and independent variables was derived using SSS tables and 3. Table 1: SSS output for coefficient of equation (17) Model Amount of coefficient Standard rror Sig. og (k) Table : Comparison of measured and calculated values of for ( ) 0.84,n =10 and 30, = 5,10,15, 0 = Φ n=10 n= Measured Calculated Measured Calculated Correlation Correlation µ s µ s coefficient µ s µ s coefficient = = = = Table 3. Comparison of measured and calculated values of for ( ) 0.84,n =10 and 30, = 5,10,15, 0 = Φ n=10 n= Measured Calculated Measured Calculated Correlation Correlation µ s µ s coefficient µ s µ s coefficient = = = =

4 World Appl. Sci. J., 15 (5): , 011 Fig. 1: Comparison of measured and calculated of relative energ dissipation for (/l)=0.41 and n=10 Fig. : Comparison of measured and calculated of relative energ dissipation for (/l)=0.41 and n=30 Fig. 3: Comparison of measured and calculated of relative energ dissipation for (/l)=0.84 and n=10 640

5 World Appl. Sci. J., 15 (5): , 011 Fig. 4: Comparison of measured and calculated of relative energ dissipation for (/l)=0.84 and n=30 RSTS AN ISCSSION RFRNCS relative energ dissipation directl wereas as indirect effect. According to figures 1 to 4, predicted rates are in good agreement wit measured data. Te quantit of fitness is presented in table. Amount of correlation coefficient between ( ) and ( ) are more tan 0.99 for all cases tat sows ver ig capabilit of equation (19) for amount of energ dissipation for adverse-slopped stepped spillwas. CONCSION Te erivation of energ dissipation equation for adverse-slopped stepped spillwa as been studied in tis paper. imensional analsis and multiple liner regression tecnique (using SSS software) were applied to derivate te equation. Five dimensionless parameters ( p1 = n, p =Φ, p 3 = a, p C 4 =, ps = ) were obtained. Finall, te relation between relative energ dissipation and te pis was derived. Tere was good agreement between predicted and measured data. In tis researc, effect of adverse-slope on energ 1. Canson, H., Comparison of energ dissipation dissipation in stepped spillwas was studied. Te results between nappe and skimming flow regimes on sow tat combination of dimensional analsis and cutes. IAHR J. Hdraulic Res., 3(): multiple linear analsis is effective for derivation of. Fratino,., A.F. iccinni and G. Marinis, 000. equation. It can be seen tat n,,, and c affect issipation efficienc of stepped spillwas, Hdraulics of stepped spillwas. Neterlands. 3. Camani, M.R. and N. Rajaratnam, Caracteristics of skimming flow over stepped spillwas. J. Hdrulic ngineering, 15(4): Yasuda, Y., M. Takaasi, and I. Otsu, 001. nerg dissipation of skimming flows on steppedcannel cutes. 9 IAHR Congress, Beijing, Cina. t 5. Boes, R.M. and H.. Minor, 000. Guidelines for te draulic design of stepped spillwas. In te roceedings of te 000 Conference, Swets and Zeitlinger, Neterlands, pp: Boes, M. and W.H. Hager, 003. Two-pase Flow Caracteristics of Stepped Spillwa. J. Hdraulic ngineering, ASC, 19(9): Yazdani, A.R., Investigation of te effect of te slope on energ dissipation in stepped spillwa. M.S Science tesis, Amirkabir universit, Iran. 8. Kavianpour, M.R. and H.R. Masoumi, 008. New Approac for stimating of nerg issipation over Stepped Spillwas. International J. Civil ngineering, 6(3): Hunt, S.. and K.C. Kadav, 010. nerg issipation On Flat-Slopped Stepped Spillwas: art 1. pstream Of Te Inception oint. American Societ of Agricultural and Biological ngineers, 53(1):

6 World Appl. Sci. J., 15 (5): , eras,.,. Roet and G. egoutte, Boes, R.M. and W.H. Hager, 003b. Hdraulic design Flow and energ dissipation over stepped gabion of stepped spillwas. J. Hdraulic ngineering, weirs. Journal of Hdraulic ngineering, ASC, ASC, 19(9): (5): Gonzalez, C.A., 005. An experimental stud of free- 11. Rice, C.. and K.C. Kadav, Model surface aeration on embankment stepped cutes. stud of a roller compacted concrete stepped diss. Queensland, Australia,niversit of spillwa. J. Hdraulic ngineering, ASC Queensland, epartment of Civil ngineering. 1(6): Takaasi, M., C.A. Gonzalez, and H. Canson, Yasuda, Y. and I. Otsu, Flow resistance of Self-aeration and turbulence in a stepped cannel: skimming flow in stepped cannels. In Te Influence of cavit surface rougness. International t roceeding 0f te 8 IAHR Congress, Session B14. J. Multipase Flow, 3(1): International Association for Hdro-nvironment 17. Hunt, S.. and K.C. Kadav, 009. Velocities and ngineering and Res., energ dissipation on a flat-sloped stepped spillwa. 13. Canson, H. and. Toombes, 00. nerg ASAB aper No St. Josep, Mic.: ASAB, dissipation and air entrainment in a stepped storm 53(): waterwa: An experimental stud. Journal of 18. Hunt, S.. and K.C. Kadav, 010. nerg dissipation Irrigation and rainage ngineering, ASC, on flat-sloped stepped spillwas: art. ownstream 18(5): of te inception point. Trans. ASAB, 53(1):