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Optimal Design of Channel Having Horizontal Bottomand Parabolic Sides

Amlan Das1

Abstract: The cost of open channels can be minimized by using �1� the optimal design concept; �2� a new geometric shape to substitutefor the trapezoidal channels, and/or �3� a composite channel. The channels in which the roughness along the wetted perimeter becomedistinctly different from part to part of the perimeter are called composite channels. The feasibility of a new cross-sectional shape that hasa horizontal bed and two parabolic sides and lined as a composite channel is investigated to substitute for the trapezoidal cross section.The optimal design concept is used to establish the efficacy of the proposed new cross-sectional shape, because it gives the best andunique design of open channels. In optimal design concept, the geometric dimensions of a channel cross section are determined in amanner to minimize the total construction costs. The constraints are the given channel capacity and other imposed restrictions ongeometric dimensions. The Lagrange multiplier technique is used to solve the resulting channel optimization models. The developedoptimization models are applied to design the proposed and trapezoidal channels to convey a given design flow considering various designscenarios which include unrestricted, flow depth constrained, side slopes constrained, and top width constrained design. Each of thesedesign scenarios again takes into account fixed freeboard, and depth-dependent freeboard cases of design. An analysis of the optimizationresults establishes the cost-saving capability of the proposed cross-sectional shape in comparison to a trapezoidal cross section.

DOI: 10.1061/�ASCE�0733-9437�2007�133:2�192�

CE Database subject headings: Open channels; Optimization; Design.

Introduction

The practical channels which carry the water for long distancesare designed in trapezoidal shape. The cost of these open channelscan be minimized by using �1� the optimal design concept, �2� anew geometric shape to substitute for the trapezoidal channels,and/or �3� a composite channel. The channels in which the rough-ness along the wetted perimeter become distinctly different frompart to part of the perimeter are called composite channels. Thistechnical note presents the feasibility of a new cross-sectionalshape to substitute for the trapezoidal cross section. The proposednew cross-sectional shape is made of a horizontal bed and twoparabolic sides and lined as a composite channel.

The geometric dimensions of a cross section will be optimalwhen all geometric parameters defining the cross-sectional shapeare optimized. The physical implementation of true or unre-stricted optimal cross section may get constrained by the limits onslope stability which leads to the development of side slope con-strained optimal design. To reduce the land acquisition costs, thetop width constrained optimal design is used �Froehlich 1994�.Freeboard, being part of the practical cross section of the chan-nels, was incorporated into the objective function by Guo andHughes �1984�. The flow depth constrained optimal design is used

1Professor, Dept. of Civil Engineering, National Institute ofTechnology, Durgapur - 713209, West Bengal, India.

Note. Discussion open until September 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this technical note was submitted for reviewand possible publication on April 19, 2005; approved on July 6, 2006.This technical note is part of the Journal of Irrigation and DrainageEngineering, Vol. 133, No. 2, April 1, 2007. ©ASCE, ISSN 0733-9437/

2007/2-192–197/$25.00.

192 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE

J. Irrig. Drain Eng. 200

to handle the flow depth maintenance problems in open channeldesign �Froehlich 1994�. The trapezoidal and the proposed cross-sectional shape are the two shapes for which the above-mentionedvarieties of optimal designs are possible.

Mironenko et al. �1984� introduced formal design of parabolicchannels considering the fact that: �1� river beds, unlined canals,and irrigation furrows tend to approximate a stable parabolicshape after a long period of unattended service �Chow 1959�;�2� parabolic sides of the cross section have improved slope sta-bility as compared to trapezoidal cross sections because theygradually increase from horizontal at bed level to maximum at thetop of the cross section; �3� parabolic channels can be excavatedby using parabolic-shaped buckets, bulldozers, tractors, andscrapers; and �4� parabolic sides do not lead to formation of sharpangles where cracks may occur because of stress concentration.Loganathan �1991� developed procedures for optimal design ofparabolic channels. Das et al. �2001� developed procedures fordesign of nonsilting and nonscouring parabolic channels usingLacey’s �1929� and Kennedy’s �1895� regime concepts. Anwarand de Vries �2003� developed procedures for design of hydrau-lically efficient power-law channels. However, other than triangu-lar and parabolic channels �which also belong to the class ofpower-law channels�, the computation of perimeter for power-law channels requires numerical evaluation of elliptic integrals byapproximate series expansion. In case of constrained optimal de-sign, the parabolic shape results in nonoptimal cross section.Babaeyan-Koopaei et al. �2000� combined parabolic and triangu-lar shape to introduce the parabolic bottom triangular channels.The mention of circular bottom triangular channels is available inChow �1959�. This technical note is intended to describe the de-sign of the proposed cross section.

The open-channel cross-section design is generally based on

one-dimensional analysis of steady flow in which the composite

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roughness resistance is conventionally expressed in the equivalentManning n form. Yen �2002� listed 17 equivalent roughness for-mulations out of which the equivalent roughness expression ofHorton �1933� and Einstein �1934� present a case in which thechannel having uniform roughness becomes a special case ofchannels with composite roughness. Das �2000� used Horton’s�1933� and Einstein’s �1934� equivalent roughness equation foroptimal design of trapezoidal cross section with composite rough-ness. Jain et al. �2004� used Lotter’s �1933� equation again fortrapezoidal cross section. The optimal designs presented in thistechnical note use Horton’s �1933� and Einstein’s �1934� equiva-lent roughness equation.

The choice and use of more than one lining material for con-struction of channels depends on seepage protection offered bythe particular lining material, cost of construction, effective life ofthe lining material, local availability, aesthetic requirements, andmany other regional factors. The Manning’s n values of liningmaterials are inversely proportional to their respective costs. Also,from a construction viewpoint, the number of lining materialsshould be kept as small as possible. Therefore, one may considerto use either two different lining materials, or a common liningmaterial for the two side faces and a third lining material for thehorizontal bed �Das 2000�. For example, a trapezoidal cross sec-tion may be constructed using boulder pitching and rough ashlermasonry in the two side faces, and neatly finished cement con-crete in the horizontal bed. Given a decision that a compositechannel is to be made, the trapezoidal cross section has two dis-tinct side faces and a horizontal bed and provides scope for use ofthree distinctly different lining materials. The parabolic, circular,and triangular cross sections also have two side faces but nohorizontal bed and provide a scope for use of two lining materi-als. For channels having composite roughness, the comparisonof performance of cross-sectional shapes having two side facesand a horizontal bed, with respect to that having only two sidefaces may not lead to a meaningful decision because these twotypes of cross sections may possess their own readily intangibleimplications. Therefore, the present study compares the resultswith respect to trapezoidal channels only.

During periods of ceasing of flow, the channels become moreprone to bank stability problems caused by seepage water thatremains stored in the porous banks of the channel. The channelshaving parabolic sides are safe in this regard because the sideslopes start from horizontal at the base and become steepest atthe top of the cross section. Using this particular considerationand also aiming to improve the economy in construction ofchannels running for long distances, a new cross-sectional shapehaving two parabolic side faces and a horizontal bed is attemptedin the following. This attempted cross-sectional shape inherentlycaptures all good properties of trapezoidal and parabolic shapes.Because it has two distinct side faces and a horizontal bed, itsperformance is compared with that of a trapezoidal cross section.

Mathematical Formulation

Fig. 1 shows a definition sketch of the proposed cross sectionhaving two parabolic sides and a horizontal bottom. Four optimaldesign problems are formulated by using assumptions identical toDas �2000�. The objective function seeks to minimize the totalconstruction cost that is obtained by summing up the costfor cross-sectional area, the cost for lining the two side facesusing two different lining materials, and the cost for lining the

bed. These costs apply for the total cross section that includes

JOURNAL OF IRRIGATION A

J. Irrig. Drain Eng. 200

wetted and freeboard parts. The overall objective function isexpressed as

C = C1��b +4

3�Z1 + Z2��y + f���y + f��

+ C2��Z12 + 1 + Z1

2 log� 1

Z1+�1 +

1

Z12��y + f�

+ C3��Z22 + 1 + Z2

2 log� 1

Z2+�1 +

1

Z22��y + f� + C4b

�1�

in which C1�per unit area cost for cross-sectional area; C2�perunit length cost for the side having a slope Z1�H� :1�V� at the topof the channel cross section; C3�per unit length cost for the sidehaving a slope Z2�H� :1�V� at the top of the section; C4�per unitlength cost for the horizontal bed; b�width of the horizontal bed;y�depth of flow above the horizontal bed; and f�freeboardabove the free water surface which may be expressed as

f = k1 + k2yk3 �2�

where k1, k2, and k3= fixed coefficients. Here, k2=0 gives a fixedfreeboard, and k1=0 gives depth-dependent freeboard. The flowarea is expressed as

X = b + 43 �Z1 + Z2��y + f�y �y �3�

where the notation X, representing the flow area, is used for easeof expression. The Manning’s roughness raised to the power 1.5weighted perimeter, denoted here as Y, is expressed as

Y = n11.5�y�Z1

2 y + f

y+ 1 + �y + f�Z1

2

�log 1

Z1�y + f

y

+�1 +1

Z12 y + f

y��

+ n21.5 y�Z2

2 y + f

y+ 1 + �y + f�Z2

2

Fig. 1. Definition sketch for the cross section having horizontal bedand parabolic sides

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�log 1

Z2�y + f

y

+�1 +1

Z22 y + f

y�� + n3

1.5b �4�

where n1�Manning’s roughness for side having a slopeZ1�H� :1�V� at the top of the section; n2�Manning’s rough-ness for side having a slope Z2�H� :1�V� at the top of the section;and n3�Manning’s roughness for horizontal bed. The Manningequation for flow capacity of the channel is expressed as

q�S0

−X5/3

Y2/3 = 0 �5�

where q�design flow and S0�longitudinal bed slope. Forn1=n2=n3=n, Eq. �5� reduces to the uniform roughness case inall respects. The top width of the total cross-sectional area isgiven as

Tf = b + 2�Z1 + Z2��y + f� �6�

The first optimization model is used to optimize all geometricparameters. It seeks to minimize the objective function defined byEq. �1� subject to an equality type constraint given by Eq. �5� andthe restriction that the variable values can never be less than zero.It is formally expressed as follows.

Minimize

C �7�

Subject to

q�S0

−X5/3

Y2/3 = 0 �8�

b,y,Z1,Z2� � 0 �9�

To solve this constrained optimization problem, an unconstrainedLagrangian objective function, L, is minimized. Here, L isformulated by using the Lagrangian multiplier, �1 �Das 2000� asfollows:

L = C + �1� q�S0

−X5/3

Y2/3� �10�

To minimize L, the necessary conditions of unconstrained mini-mization, i.e., �L /��1=�L /�b=�L /�y=�L /�Z1=�L /�Z2=0 areapplied to obtain Eq. �5� and

�C

�b�5

�X

�y− 2

X

Y

�Y

�y� −

�C

�y�5

�X

�b− 2

X

Y

�Y

�b� = 0 �11�

�C

�Z1�5

�X

�y− 2

X

Y

�Y

�y� −

�C

�y�5

�X

�Z1− 2

X

Y

�Y

�Z1� = 0 �12�

�C

�Z2�5

�X

�y− 2

X

Y

�Y

�y� −

�C

�y�5

�X

�Z2− 2

X

Y

�Y

�Z2� = 0 �13�

Eqs. �5� and �11�–�13� are solved iteratively �Das 2000� to obtainthe optimal geometric dimensions of the proposed cross section.Here, �1 indicates the shadow price, i.e., how the total cost willchange because of a unit change in q /S0.

The second is the side slope constrained optimization model. Itseeks to determine the optimal values of b and y for given valuesof Z1 and Z2. Therefore, Eqs. �5� and �11� are solved iteratively to

obtain the optimal values of b and y in this model.

194 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE

J. Irrig. Drain Eng. 200

The third is the depth constrained optimization model. Thismodel is used when the optimal depth of the first optimizationmodel appears to be on the higher side of values. This modelseeks to minimize Eq. �1� subject to Eq. �5� and a specified valueof depth. It is intended to obtain the optimal values of b, Z1, andZ2 for given values of y. The governing equations to solve thismodel are obtained in a manner identical to that of the first model.In this model, Eq. �5� and the following two equations are solvedto obtain the optimal dimensions:

�C

�Z1�5

�X

�b− 2

X

Y

�Y

�b� −

�C

�b�5

�X

�Z1− 2

X

Y

�Y

�Z1� = 0 �14�

�C

�Z2�5

�X

�b− 2

X

Y

�Y

�b� −

�C

�b�5

�X

�Z2− 2

X

Y

�Y

�Z2� = 0 �15�

The fourth is the top width restrained optimization model. It isformally expressed as follows.

Minimize

C �16�

Subject to

q�S0

−X5/3

Y2/3 = 0 �17�

Tf − T1 = 0 �18�

where T1�given value of top width. The Lagrangian objectivefunction, L, for this fourth optimization model is formulated as

L = C + �1� q�S0

−X5/3

Y2/3� + �2Tf − T1� �19�

where �2�second Lagrangian multiplier associated with Eq. �18�.The necessary optimality conditions for unconstrained minimiza-tion are again applied in a manner identical to that of the firstmodel to obtain Eqs. �5� and �18�, and the following two govern-ing equations:

�C

�Z1��5

�X

�y− 2

X

Y

�Y

�y� �Tf

�b− �5

�X

�b− 2

X

Y

�Y

�b� �Tf

�y�

− �5�X

�Z1− 2

X

Y

�Y

�Z1�� �C

�y

�Tf

�b−

�C

�b

�Tf

�y�

+�Tf ��5

�X− 2

X �Y� �C− �5

�X− 2

X �Y� �C� = 0 �20�

�Z1 �b Y �b �y �y Y �y �b

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�C

�Z2��5

�X

�y− 2

X

Y

�Y

�y� �Tf

�b− �5

�X

�b− 2

X

Y

�Y

�b� �Tf

�y�

− �5�X

�Z2− 2

X

Y

�Y

�Z2�� �C

�y

�Tf

�b−

�C

�b

�Tf

�y�

+�Tf

�Z2��5

�X

�b− 2

X

Y

�Y

�b� �C

�y− �5

�X

�y− 2

X

Y

�Y

�y� �C

�b� = 0 �21�

Eqs. �5�, �18�, �20�, and �21� are solved iteratively to obtain theoptimal solution for the top width restrained optimal design prob-lem. The expression for derivative terms in the above-mentionedoptimization models is easily obtained.

Performance Evaluation

To evaluate the performance of the proposed cross section againstthe trapezoidal cross section, the developed optimal designmodels of the present study are applied to design channelsconsidering various design scenarios. The scenarios that areconsidered include unrestricted, flow depth constrained, sideslopes constrained, and top width constrained design. Each ofthese design scenarios again takes into account fixed freeboardand depth-dependent freeboard cases of design. Two sets of opti-mal designs for the above-mentioned design scenarios areperformed. The first set is for trapezoidal channels, and the

Table 1. Optimal Dimensions for Trapezoidal Cross Section

y�m�

b�m� Z1 Z2

4.496 4.720 0.340 0.344

4.000 5.570 0.337 0.344

3.500 6.660 0.334 0.343

3.000 8.137 0.330 0.342

2.500 10.310 0.326 0.339

2.000 13.902 0.319 0.333

1.500 20.927 0.307 0.320

1.000 38.909 0.280 0.286

4.496 4.459 0.400 0.400

4.532 4.756 0.320 0.324

4.671 4.893 0.250 0.254

4.823 5.041 0.183 0.185

4.988 5.203 0.117 0.119

4.051 5.479 0.336 0.342

3.500 6.610 0.348 0.358

3.000 8.050 0.358 0.371

2.500 10.199 0.368 0.384

2.000 13.776 0.378 0.397

1.500 20.794 0.391 0.409

1.000 38.770 0.411 0.424

4.005 4.935 0.500 0.500

4.218 5.516 0.260 0.265

4.376 5.570 0.195 0.199

4.544 5.645 0.132 0.135

4.722 5.743 0.071 0.072

Note: ur�unrestricted; ffb�fixed freeboard; dc�depth constrained; scfreeboard.

second set is for proposed channel cross section having horizontal

JOURNAL OF IRRIGATION A

J. Irrig. Drain Eng. 200

bottom with two parabolic sides. The two sets of designsare made by using a common set of input data which are:q=100 m3/s, S0=0.0016, n1=0.020, n2=0.018, n3=0.015,C1=0.6, C2=0.1, C3=0.2, and C4=0.4. The costs are described inthousand Indian rupee units. To describe a fixed value of free-board, k1=0.5 m, and k2=0.0 are specified. To describe a depth-dependent freeboard, k1=0.0, k2=0.25, and k3=0.5 are specified.These freeboard parameters are arbitrarily selected for this par-ticular demonstration.

Twenty-five typical optimal design problems for each oftrapezoidal and the proposed cross section are solved to studythe performance of the proposed cross section. The optimizationresults are presented in a consolidated manner in two tables.Table 1 presents the optimal dimensions for a trapezoidal crosssection, and Table 2 presents that for the proposed cross sectionhaving two parabolic sides and a horizontal bed. In Tables 1 and2, the costs are per meter run of the channel. Tables 1 and 2 helpto compare the optimization results for various design scenarios.The first problem is for unrestricted design considering a fixedvalue of freeboard. The next seven problems are to study theeffect of incorporating depth constraints in the optimal design.The specified flow depth is varied from 4.0 to 1.0 m in steps of0.5 m. Provision of a single valued slope for the two sides iscommon in engineering practice. But, the earlier optimizationsresult in two different side slopes for the two sides because theroughness values for the two side slopes are not identical. There-fore, the ninth problem is to study the effect of incorporating a

Tf

�m�

Total costin thousandIndian rupee

Optimizationscenario

. 8.137 22.742 ur, ffb

8.635 22.830 dc, ffb

9.370 23.167 dc, ffb

10.490 23.921 dc, ffb

12.304 25.426 dc, ffb

15.531 28.425 dc, ffb

22.180 34.864 dc, ffb

39.759 51.432 dc, ffb

8.456 22.754 sc, ffb

8.000 22.744 twc, ffb

7.500 22.781 twc, ffb

7.000 22.869 twc, ffb

6.500 23.006 twc, ffb

8.569 22.829 ur, ddfb

9.410 22.975 dc, ddfb

10.554 23.477 dc, ddfb

12.377 24.617 dc, ddfb

15.600 27.010 dc, ddfb

22.239 32.219 dc, ddfb

39.814 45.384 dc, ddfb

9.440 22.912 sc, ddfb

8.000 22.858 twc, ddfb

7.500 22.933 twc, ddfb

7.000 23.055 twc, ddfb

6.500 23.222 twc, ddfb

constrained; twc�top width constrained; and ddfb�depth-dependent

f�m�

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.503

0.468

0.433

0.395

0.354

0.306

0.250

0.500

0.513

0.523

0.533

0.543

�slope

side slopes constraint and a common side slope value is used for

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this purpose. The next four optimizations are devoted to studyingthe effect of incorporating top width constraints upon the optimi-zation results. In this series, the specified top width is varied from8.0 to 6.5 m in steps of 0.5 m. The remaining optimizations areto study the effect of using depth-dependent freeboard upon theoptimization results. In this series, the fourteenth optimizationconsiders unrestrained optimal design. The next six examples areto investigate the effect of introducing depth constraints. Here,the specified depths range from 3.5 to 1.0 m in steps of 0.5 m.The slope constrained design is presented in the twentieth andtop width constrained cases are presented in the remaining fiveproblems.

The costs are compared for identical conditions of optimiza-tion for trapezoidal and the proposed cross-sectional shape andare given in Table 2. It may be observed that for all the above-mentioned cases of optimal design, the proposed cross sectionreduces the costs. The cost savings are again expressed in thou-sand Indian rupees per meter run of the channel. When thechannel is to be constructed for long distances, say 50 km, thesesaved costs of Table 2 are to be multiplied by 50,000 to calculatethe total saving of expenses that can be realized by the physicalimplementation of the proposed cross-sectional shape. Any savingin the overall project cost because of selecting a new cross-sectional shape as proposed in this study proves its worth. Thecomparative study shows that the proposed cross-sectional shapehas the promise to reduce overall project costs.

In the aforesaid comparative study, the side slope constrained

Table 2. Optimal Dimensions for the Proposed Cross Section Having Tw

y�m�

b�m� Z1 Z2

f�m�

4.700 2.415 0.269 0.278 0.500

4.000 3.950 0.255 0.270 0.500

3.500 5.293 0.246 0.263 0.500

3.000 7.006 0.235 0.256 0.500

2.500 9.395 0.225 0.247 0.500

2.000 13.184 0.214 0.237 0.500

1.500 20.387 0.202 0.225 0.500

1.000 38.532 0.186 0.204 0.500

4.748 0.726 0.400 0.400 0.500

4.729 2.494 0.259 0.268 0.500

4.893 2.861 0.211 0.219 0.500

5.093 3.205 0.166 0.173 0.500

5.335 3.536 0.125 0.129 0.500

4.201 3.511 0.256 0.269 0.512

3.500 5.224 0.254 0.272 0.468

3.000 6.887 0.252 0.273 0.433

2.500 9.248 0.250 0.274 0.395

2.000 13.022 0.249 0.276 0.354

1.500 20.220 0.252 0.280 0.306

1.000 38.363 0.265 0.289 0.250

4.130 1.088 0.500 0.500 0.508

4.363 3.673 0.216 0.227 0.522

4.565 3.852 0.174 0.183 0.534

4.799 4.039 0.135 0.142 0.548

5.068 4.236 0.098 0.103 0.563

Note: ur�unrestricted; ffb�fixed freeboard; dc�depth constrained; scfreeboard.

optimal design is performed by using particular values of side

196 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE

J. Irrig. Drain Eng. 200

slope for each of the given freeboard conditions. Repeated solu-tions reveal that the higher values for specified side slopes resultin a parabolic cross section. However, the limiting side slope thatgives a pure parabolic shape has only academic interest and isalso a function of all input values used in the design problem.Therefore, further investigation on this particular issue is skipped.A check for sufficiency conditions of optima also has academicinterest only that is not presented here.

Conclusion

Optimal design is used for unique design of open channels. Thecost of channel construction can be reduced by selecting a newgeometric shape and/or using composite lining along the perim-eter. A geometric cross-sectional shape having two parabolic sidesand a horizontal bottom is attempted in this study. Four optimi-zation models are developed. The LM technique is used to obtainthe solution of the optimization models. The developed modelsare applied to design open channels for various design scenarios.The optimization results are compared with a trapezoidal crosssection. The comparative study establishes the cost reduction ca-pability of the proposed cross-sectional shape in comparison to atrapezoidal cross section. This study establishes the applicationpotential of the proposed cross-sectional shape for use in real-life

abolic Sides and a Horizontal Bed

Tf

m�

Total costin thousandIndian rupee

Cost savedin thousandIndian rupee

Optimizationscenario

.103 22.185 0.557 ur, ffb

.676 22.341 0.489 dc, ffb

.362 22.735 0.432 dc, ffb

.443 23.545 0.376 dc, ffb

.229 25.106 0.320 dc, ffb

.441 28.163 0.262 dc, ffb

.093 34.662 0.202 dc, ffb

.702 51.293 0.139 dc, ffb

.123 22.314 0.440 sc, ffb

.000 22.186 0.558 twc, ffb

.500 22.232 0.549 twc, ffb

.000 22.351 0.518 twc, ffb

.500 22.553 0.453 twc, ffb

.458 22.326 0.503 ur, ddfb

.392 22.543 0.432 dc, ddfb

.490 23.103 0.374 dc, ddfb

.283 24.304 0.313 dc, ddfb

.494 26.759 0.251 dc, ddfb

.141 32.030 0.189 dc, ddfb

.747 45.257 0.127 dc, ddfb

.364 22.728 0.184 sc, ddfb

.000 22.352 0.506 twc, ddfb

.500 22.442 0.491 twc, ddfb

.000 22.607 0.448 twc, ddfb

.500 22.855 0.367 twc, ddfb

constrained; twc�top width constrained; and ddfb�depth-dependent

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Notation

The following symbols are used in this technical note:b � width of the horizontal bed;

C1 � per unit area cost for cross-sectional area;C2 � per unit length cost for the side having a slope

Z1�H� :1�V� at the top of the channel crosssection;

C3 � per unit length cost for the side having a slopeZ2�H� :1�V� at the top of the section;

C4 � per unit length cost for the horizontal bed;f � the freeboard above the free water surface;

k1, k2, k3 � fixed coefficients;L � Lagrangian objective function;

n1 � Manning’s roughness for side having a slopeZ1�H� :1�V� at the top of the section;

n2 � Manning’s roughness for side having a slopeZ2�H� :1�V� at the top of the section;

n3 � Manning’s roughness for horizontal bed;q � design flow;

S0 � longitudinal bed slope;Tf � top width of the total cross-sectional area;T1 � specified value of top width of the total

cross-sectional area;X � flow area;Y � Manning’s roughness raised to the power 1.5

weighted perimeter;y � depth of flow above the horizontal bed;

Z1 and Z2 � side slopes; and�1, and �2 � Lagrangian multipliers.

References

Anwar, A. A., and de Vries, T. T. �2003�. “Hydraulically efficient power-

law channels.” J. Irrig. Drain. Eng., 129�1�, 18–26.

JOURNAL OF IRRIGATION A

J. Irrig. Drain Eng. 200

Babaeyan-Koopaei, K., Valentine, E. M., and Swailes, D. C. �2000�. “Op-timal design of parabolic-bottomed triangle canals.” J. Irrig. Drain.Eng., 126�6�, 408–411.

Chow, V. T. �1959�. Open channel hydraulics, McGraw-Hill, Singapore.

Das, A. �2000�. “Optimal channel cross section with composite rough-ness.” J. Irrig. Drain. Eng., 126�1�, 68–72.

Das, A., Gaur, Y. K., Chaubey, A., Veena, S. N., and Khati, J. S. �2001�.“Parabolic channel design.” J. Inst. Eng. (India), IE�I�, 81�CV4�,174–181.

Einstein, H. A. �1934�. “Der hydraulische oder profile-radius �The hy-draulic or cross-section radius�.” Schweizerische Bauzeitung, Zurich,103�8�, 89–91 �in German�.

Froehlich, D. C. �1994�. “Width and depth constrained best trapezoidalsection.” J. Irrig. Drain. Eng., 120�4�, 828–835.

Guo, C. Y., and Hughes, W. C. �1984�. “Optimal channel cross sectionwith freeboard.” J. Irrig. Drain. Eng., 110�3�, 304–314.

Horton, R. E. �1933�. “Separate roughness coefficients for channel bot-tom and sides.” Eng. News-Rec., 111�22�, 652–653.

Jain, A., Bhattacharjya, R. K., and Sanaga, S. �2004�. “Optimal design ofcomposite channels using genetic algorithm.” J. Irrig. Drain. Eng.,130�4�, 286–295.

Kennedy, R. G. �1895�. “The prevention of silting in irrigation canals.”Min. Proc. Inst. Civil Engineers, CXIX.

Lacey, G. �1929�. “Stable channels in alluvium.” Proc. Inst. Engineers,229, 259–384.

Loganathan, G. V. �1991�. “Optimal design of parabolic canals.” J. Irrig.Drain. Eng., 117�5�, 716–735.

Lotter, G. K. �1933�. “Soobrazheniia k gidravlicheskomu raschetu rusel arazlichinoi sherokhovatostiiu stenok �Considerations on hydraulicdesign of channels with different roughness of walls�.” IzvestiiaVsesoiuznoga Nauchno-Issledovatel’skogo Instituta Gidrotekhniki�Trans. All-Union Sci. Res. Inst. Hydraulic Eng.�, Leningrad, Vol. 9,238–241 �in Russian�.

Mironenko, A. P., Willardson, L. S., and Jenab, A. S. �1984�. “Paraboliccanal design and analysis.” J. Irrig. Drain. Eng., 110�2�, 241–246.

Yen, B. C. �2002�. “Open channel flow resistance.” J. Hydraul. Eng.,128�1�, 20–39.

ND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2007 / 197

7.133:192-197.