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46I6 IJAET0612637 46I6 IJAET0612637 Document Transcript

  • International Journal of Advances in Engineering & Technology, Jan 2012.©IJAET ISSN: 2231-1963 MINIMUM LINING COST OF TRAPEZOIDAL ROUND CORNERED SECTION OF CANAL Syed Zafar Syed Muzaffar1, S. L. Atmapoojya2, D.K. Agarwal3 1 Assistant Professor (Civil Engg), Anjuman College of Engg. & Technology, Nagpur, India 2 Principal, S.B Jain Institute of Technology & Management Nagpur, India 3 Ex Dean, Faculty of Engg. & Tech., RSTM & Research University Nagpur, IndiaABSTRACTEconomics and Environmental pressure require efficient use of water for irrigation purpose. To satisfy theEconomics constrains in canal construction the total cost of section should be minimum which include cost oflining for minimizing the total cost of lined canal trapezoidal shape the Lagrangian multiplier technique is used.The total cost of lining is the sum of cost of lining on sides on curves and in the base. Radius of rounding can beprovided in two ways 1) unbound radius case i.e. the radius of rounding remain constant and it predecided 2)Bound radius case i.e. the radius rounding depend on depth of flow. These two cases are only for designpurpose of canal section. It is seen that cost of lining is less in case of bound radius case.KEYWORDS: Unbound radius, Bound radius, side lining, Base lining. I. INTRODUCTIONTrapezoidal round cornered section is more efficient than sharp cornered section. Attempts have beenmade by researches to derive the optimal sharp cornered section. The paper deals with minimize thetotal cost of lining canal system which is provide to convey water from a source to required point. It isfound that about 45% to 50% water is lost due to seepage from canal system during its journey fromhead works to the field. The seepage also enhances water logging of the adjacent area to the canalwhich causes in reduction of crop production. Hence it required how to control loss of water due toseepage. Then lining of canal system is one of the majors which overcome this problem. Lining ofcanal prevent soil erosion due to high velocity which increases cost of canal system. A section ofunlined canal system does not remain in trapezoidal shape for longer times there is considerableseepage loss through it. Whereas the lined canal section is provided with a hard surface coatingaround the perimeter of canals which prevent the seepage loss. Hence canal may be designed by usingflow equations. To get economy in canal construction the section should be design in such a way thatits total cost of lining should be minimum.The radius of rounding may be taken as constant and independent of depth of flow, which is termed asunbound radius case. In second case radius of rounding is taken as a function of depth of flow whichis termed as bound radius case.The cost of lining material for the base, curves and sides are considered as different. The curvedportion need special type of form work and labour cost is also more, similarly on the sides theperformance of work will be different from the base, and hence the cost of lining per unit area isconsidered as different for base, curve and sides.LetCb = Cost of Lining per unit area for base.Cc = Cost of Lining per unit area for the curved portion. 433 Vol. 2, Issue 1, pp. 433-436
  • International Journal of Advances in Engineering & Technology, Jan 2012.©IJAET ISSN: 2231-1963Cs = Cost of Lining per unit area for the sides. C = bCb + 2rz2Cc + (2y √(1+z2) – 2 rz1) Cs -------------------------(1)Round corned section can be • Unbound radius section • Bound radius section Figure1 Trapezoidal round cornered lined canal section II. UNBOUND RADIUS CASERadius of rounding r = constantA = by + 2ryz1+y2z – 2r2 z1 + r2z2 ---------------------------------------(2)A = y2 (b/y + 2 r/y z1 + z – 2(r/y) 2 z1 + (r/y) 2 z2)y2 = A /(b/y + 2 r/y z1 + z – 2( r/y)2 z1 + (r/y)2 z2)y = √ [ A /(b/y + 2 r/y z1 + z – 2( r/y)2 z1 + (r/y)2 z2)]y = [1 x √A] /√ [(b/y + 2 r/y z1 + z – 2( r/y)2 z1 + (r/y)2 z2)]y = φ1 × √ (Q / V ) ------------------------------------ (3)Putting the value of b from equation (2) in equation (1) and we getC = (A/y - 2rz1 - yz + 2r2/y z1 - r2/yz2 ) Cb + 2rz2Cc + (2y √ (1 +z2 )– 2rz1 ) Cs ------------ (4)For minimum value of C differentiating with respective to y and equating to zero we getb/y = 2 [√ (1 +z2) Cs/Cb - z ] – 2(r/y)z1 ----------------------------------- (5)III. BOUND RADIUS CASERadius of rounding r = f (y)And let r = KyWhere K = constant its value is less than unityThe flow areaA = by + 2ky2z1+y2z – 2k2y2 z1 + k2y2z2 ---------------------------------------------(6)Wetted perimeter P = b + 2 y √ (1 +z2) - 2 ky z1 + 2 kyz2 ---------------------------------------------(7)Cost of lining is given byC = bCb + 2kyz2Cc + [2y √ (1 +z2 ) – 2 kyz1 ] Cs ---------------------------------- (8)For minimum value of CDifferentiating C with respective to y and equating to zero we getb/y = 2 (√ (1 +z2 ) Cs/Cb – z ) - 2kz1 ( 2 + Cc/Cb ) + 2 kz2 Cc/Cb + 4k2(z1 – z2/2 ) ----- (9)The value of y can be determined by equation (6) asy = √ [ A /(b/y + 2 k z1 + z – 2 k2 z1 + k2 z2)]y= [1 x √ A ] / √ [2 (√1 +z2 )Cs/Cb – z )+z - 2kz1 (1 + Cs/Cb )+2 kz2 Cs/Cb+ 2k2(z1– z2/2 )] Ory = φ2 × √ (Q/V) -----------------------------------------(10)Whereφ2 = 1 / √ [2 (√1 +z2 )Cs/Cb – z )+z - 2kz1 (1 + Cs/Cb )+2 kz2 Cs/Cb+ 2k2(z1– z2/2 )]φ2 = f ( Cs/Cb, k, z)After obtaining the value of y for a given value of b can be obtained discharge and velocity of flow. 434 Vol. 2, Issue 1, pp. 433-436
  • International Journal of Advances in Engineering & Technology, Jan 2012.©IJAET ISSN: 2231-1963Example: Q = 90 m3/Sec K = 0.7 V = 1.8 m/Sec Z = 1.5 Cb = Rs. 200 /Sq M Cs = Rs 500 / Sq M r/y = 0.7 Cc = 225 / Sq M Z1 = √(1 +z2 ) - z √ (1 +z2 ) = 1.8027 = √ (1 +1.52 ) - 1.5 Z1 = 0. 30277 As Q/v = 90/1.8 = 50 Z2 = tan-1 (1/z) x /180 Z2 = tan-1 (1/1.5 ) x /180 Z2 = 0.588Unbound Radius Case:b/y = 2 (√(1 +z2) Cs/Cb – z ) – 2 r/y z1 = 2 (√(1 +1.52 ) 500/200 – 1.5) – 2 x 0.7 x 0.30277b/y = 5.59The value of factor, φ1 for calculated value of b/y and given value of r and zΦ1 = 1 / √(b/y + 2z1 r/y + z – 2 r2/y2 z1 + r2/y2 z2)Φ1 = 1 / √ (5.59+2x0.30277x0.7+1.5 - 2 x 0.72 x 0.30277+ 0.72 x 0.588) = 1 / √ (5.59 + 0-42387 + 1.5 – 0.2967 + 0.288) = 1 / 2.739Φ1 = 0.365The depth of flow yy = Φ1√ (Q/V) = 0.365 √(90/1.8) = 2.581 meter and r = r/y × y = 0.7 × 2.58 r = 1.8067and b = 2.58 × 5.56 b = 14.428 meterCost of lining C C = b Cb + 2 rz2Cc + (2y √ (1 +z2 ) – 2 rz1) Cs = 14.428 x 200 + 2 x 1.8067 x 0.588 x 225 + (2 x 2.581 x √(1 +1.52)– 2 x 1.8067 x 0.30277) x500C = 7469.602 per meterBound Radius Case:b/y = 2 (√(1 +z2 )Cs/Cb – z ) – 2kz1 (2 + Cs/Cb )+2 kz2 Cc/Cb+ 4k2(z1– z2/2 ) = 2 (1.8027 x 500/200 – 1.5) – 2 x 0.7 x 0.30277 (2 + 500/200) +2 x 0.7 x 0.588 x 225 /200 + 4 x 0.72 (0.30277 – 0.588/2) b/y = 5.054 2 = 1 / √ [2(√(1 +z ) Cs/Cb – z )+z - 2kz1 (1 + Cs/Cb )+2 kz2 Cc/Cb+ 2k (z1– z2/2 )] 2 2 = 1 / (√ [2(1.8027 x 500/200 – 1.5)+ 1.5 – 2 x 0.7x0.3027 x ( 1 + 500/200) +2 x 0.7 x 0.588 x 225/200 + 2 x 0.72 (0.30277 – 0.588/2)] 2 = 0.2949 y = 2√ (Q/V) = 0.2949 x √ (90/1.8 ) y = 2.085 meter b = 10.54 meterC = bCb + 2kyz2Cc + (2y √(1 +z2 ) – 2 kyz1 ) Cs = 10.54 x200 + 2 x 0.7 x 2.085 x 0.588 x 225 + (2 x 2.085 x 1.8027 – 2 x 0.7 x 2.085 x0.30277) x 500 435 Vol. 2, Issue 1, pp. 433-436
  • International Journal of Advances in Engineering & Technology, Jan 2012.©IJAET ISSN: 2231-1963C = Rs 5810.83 per meterIV. CONCLUSIONThe condition for optimal design of trapezoidal canal is developing by considering the cost of lining.The cost of lining which is the sum of cost of base lining, cost of side lining and cost of curve lining isminimise with respect to variable for given area of flow . The optimal cost obtained by the proposedmethod by considering radius as bound and unbound. In unbound radius case we take radius as aconstant. In bound radius case we take radius as function of depth of flow. It is seen that the costobtained by bound radius is less than unbound radius.REFERENCES [1] Prabhata K Swamee, Govinda C, Mishra and Bhagur R, Chahur “Minimum Cost Design of Lined Canal Section” Water Resource Management 14 1-12-2000. [2] M.Riaz and Z Sen European “Aspects of Design and benefits of Alternative lining Systems” European Water 11/12, 17-27, 2005 @ 2005. EW Publications. [3] S.K Garg “A text book of Irrigation Engineering and Hydraulic Structures” member of Indian Water Resource Society. [4] Dr. B.C Punmia and Dr. Pandey B.B Lal “A text book of Irrigation and Water Power Engineering”. [5] IS 10430-1982 “Criteria for Design of Lined Canal and Guide for selection of type of Lining” Bureau of Indian Standards, Manak Bhavan, New Delhi. [6] IS 4515-1993 “stone Pitched Lining for Canals- Code of Practice” Bureau of Indian Standards, Manak Bhavan New Delhi. [7] V. T. Chow “Open Channel Hydraulics” The McGraw Hill book company New York Nu,1959. [8] P. N. Modi and S. M. Seth “Design of Most Economical Trapezoidal Section of open Channels” Journal of Irrigation and Power, New Delhi, 1968, pp 271-280. [9] C. Y. Guo and W. C. Hughes “ Optimal Channel Cross Section With Free Board” Journal of Irrigation and Drainage Division ASCE Vol. 110, no.8, 1984, pp 304-314. [10] A Das “Optimal Channel Cross Section with composite roughness” Journal of Irrigation and Drainage Division ASCE Vol. 126, no.1, 2000, pp 68-71. [11] G. V. Loganathan “Optimal Design of Parabolic Canals” Journal of Irrigation and Drainage Division, ASCE Vol.117 no.5, 1991, pp 716-735. [12] P. Monadjemi “General Formulation of Best Hydraulic Channel Section” Journal of Irrigation and Drainage Division ASCE Vol.120, no.1, 1994, pp 27-35. [13] V. L. Streeter and E. B. Wylie “Fluid Mechanics” Mc Graw Hills Inc, New York Ny, 1979. [14] S. L. Atmapoojya and R. N. Ingle “The Optimal Canal Section With Consideration of Free Board” IE (I) Journal- CV, Vol.83, Feb-2003. Author Syed Zafar Syed Muzaffar was born on 08-07-1964. He has completed his B.E. (Civil Engg.) in 1990 from B. N. College of Engg, Pusad Dist. Yavatmal, M. Tech. (Hydraulic Engg.) in 1999 from V.R.C.E., Nagpur & Pursuing Ph.D. Civil Engg. from Nagpur University. He is working as Assistant Professor in Anjuman College of Engineering, Sadar, Nagpur (India) S.L. Atmapoojya has completed B.E. (CIVIL) from Government College of Engineering, Jabalpur, M.Tech (Hydraulic) from VRCE, Nagpur & Ph D - From RSTM, Nagpur university, Nagpur. He is presently working as Professor in KITS, Ramtek, Nagpur. Dinesh Kumar Onkarnath Agrawal was born on 15.08.1960. He has completed his B.Sc. from Bhopal University (M.P.) in 1980, M.Tech in 1983 from Dr. Harisingh Gour University, Sagar (M.P.) & Ph.D. in 2007 from RTM Nagpur University Nagpur (M.S.). He is working as Professor in Faculty of Engineering & Technology, RSTM & Research University Nagpur, India. 436 Vol. 2, Issue 1, pp. 433-436