Improved Channel Cross Section with Two-Segment Parabolic Sides and Horizontal Bottom

A channel cross section with parabolic sides and horizontal bottom has been recently published and proved to be more economical (provide lesser construction cost per unit length) than the trapezoidal cross section. This paper presents a new and improved cross section with two-segment parabolic sides and horizontal bottom. Each side of the cross section consists of two parabolic segments smoothly connected. Closed-form relationships for the cross-sectional area and perimeter are developed. For specific parameter conditions, the new cross section produces most of the common cross sections, including the parabolic sides—horizontal bottom and trapezoidal cross sections, as well as new cross-sectional shapes. It provides an additional degree of freedom in determining the optimal cross-sectional design. A spreadsheet-based optimization model for the new cross section that minimizes the total construction cost (excavation and composite linings) is developed. The constraints of the model include channel discharge and physical requirements, such as flow depth, top width, and side slope with fixed or depth-dependent freeboard. The model was validated and the cross-sectional performance was evaluated using different design scenarios. The optimization results show that the new cross section is more economical and more flexible than a cross section with (one-segment) parabolic sides. As such, it should be of interest to the irrigation and drainage engineers.     

Optimal Design of Composite Channels Using Genetic Algorithm

In the past, studies involving optimal design of composite channels have employed Horton’s equivalent roughness coefficient, which uses a lumped approach in assuming constant velocity across a composite channel cross section. In this paper, a new nonlinear optimization program (NLOP) is proposed based on a distributed approach that is equivalent to Lotter’s observations, which allows spatial variations in velocity across a composite channel cross section. The proposed NLOP, which consists of an objective function of minimizing total construction cost per unit length of a channel, is solved using genetic algorithm (GA). Several scenarios are evaluated, including no restrictions, restricted top width, and restricted channel side slopes, to account for certain site conditions. In addition, the proposed NLOP is modified to include constraints on maximum permissible velocities corresponding to different lining materials of the composite channel cross section, probably for the first time. The proposed methodology is applied to trapezoidal and triangular channel cross sections but can be easily extended to other shapes or compound channels. Optimal design graphs are presented to determine the channel dimensions of a composite trapezoidal channel cross section. The results obtained in this study indicate that cost savings up to 35% can be achieved for the unconstrained velocity case and up to 55% for the limiting velocity case when the proposed NLOP is solved using GA as compared with the existing NLOP solved using either the classical optimization solution technique or GA.     

Best Hydraulic Section of a Composite Channel

The half-circle is the ideal shape of the best hydraulic section of open channels, but semicircular channels are not practical to construct. The semicircular section is approximated by a composite section that is composed of a trapezoidal section at the bottom and a rectangular section at the top. Such a composite channel can be easily constructed in rocks. This technical note presents an analysis for determining the channel proportions that yield a minimum wetted perimeter for a given flow area of the composite channel. The results of the analysis show that the best hydraulic section has the shape of a half-octagon for composite channel and a composite section is more efficient than a trapezoidal section.     

Simplified Design of Hydraulically Efficient Power-Law Channels with Freeboard

A power-law channel section is very versatile. It can model a wide range of familiar man-made or natural channel shapes. However estimating the wetted perimeter of a power-law channel section is difficult. The problem gets complicated further when considering the freeboard in the design process. In this paper, the wetted perimeter is estimated using the isoperimetric theorem which results in a simple and accurate expression for the wetted perimeter that does not lead to discontinuity in the optimal solution. For unconstrained optimum power-law channels, it is shown that the exponent and side slope at the water surface have values of 0.314 and 0.352, respectively, independent of either the maximum side slope or the relative freeboard. The analyses have also shown that the most hydraulically efficient power-law channel sections come closest to the semicircle. They tend to be U shaped and narrow with small relative freeboard ( ? 0.30). A design procedure and two design charts are presented together with two illustrative examples to demonstrate the simplicity of the method.     

Design of Hydraulically Efficient Power-Law Channels with Freeboard

A power-law channel is a generalized form of the parabolic channel. The exponent of the governing equation is a variable that for certain maximum permissible side slopes can be determined by maximizing the cross-sectional area of flow (or minimizing the wetted perimeter). Using this exponent rather than the constant allows a hydraulically more efficient open channel section to be designed. In earlier work on power-law channels freeboard was neglected to simplify the analysis. However as pointed out by several authors, a channel without freeboard is of academic interest only and not practical. All open channels are in practice designed and constructed with freeboard as a factor of safety. In this paper freeboard has been introduced as an additional parameter to be taken into account when designing a power-law channel. The work from this paper is applied to an earlier example of a parabolic channel to demonstrate a practical design.     

Optimal Design of Parabolic-Bottomed Triangle Canals

The characteristics of a parabolic-bottomed triangle cross section are introduced. For this geometry the “best” hydraulic cross section is determined by using the undetermined multipliers method of Lagrange. For a given flow, roughness coefficient, and longitudinal slope, the best hydraulic section is the channel section having the least wetted perimeter or cross-sectional area. The cross-sectional parameters of this novel geometry are compared with those of trapezoidal, parabolic, and round-bottomed triangle cross sections. It is shown that for all values of side slopes, the flow area and wetted perimeter in a parabolic-bottomed triangle cross section are less than those of trapezoidal and parabolic cross sections for the same discharge. This indicates that less excavation and linings are involved and therefore implies that the parabolic-bottomed triangle cross section is more economical than trapezoidal and parabolic cross sections.     

Optimal Channel Cross Section with Composite Roughness

For channels with composite roughness, an equivalent uniform roughness coefficient and flow geometric elements are used in an optimal design method using the Manning equation. The optimal design problems are formulated in a nonlinear optimization framework with the objective function being a cost function per unit length of the canal. Constraints are the Manning equation, positive values for design variables, and specified values of side slopes or top width. The constrained problem is transformed into an unconstrained problem using the Lagrangian multipliers. To obtain an optimal solution for the resulting unconstrained problem, the first-order necessary conditions for optima are applied. The resulting simultaneous nonlinear equations are solved using the computational methodology developed. This technique is applied to illustrative numerical examples. The evaluations establish the potential applicability of the developed computational methodology for optimal design of open channel cross sections with composite roughness.     

Optimal Design of Open Channel Section Incorporating Critical Flow Condition

The flow at critical condition of an open channel is unstable. At critical condition, a small change in specific energy will cause abrupt fluctuation in water depth of the channel. This is because the specific energy curve is almost vertical at critical state. Therefore, if the design depth of the channel is near or equal to critical depth of the channel, the shape of the channel must be altered to avoid a large fluctuation in water depth. In the present study, a nonlinear optimization model is presented for designing an optimal channel section incorporating the critical flow condition of the channel. The optimization model derives the optimal channel section at a desirable difference from the critical condition of the channel so that a small change in the specific energy of the channel will not cause an abrupt change in flow depth. The objective of the optimization model is to minimize the total construction costs of the channel. Manning’s equation is used to specify the uniform flow condition in the channel. The developed optimization model is solved by sequential quadratic programming using MATLAB. Applicability of the model is demonstrated for a trapezoidal channel section with composite roughness. However, it also can be extended to other shapes of channel.     

Hydraulically Efficient Power-Law Channels

A power-law channel is a generalized form of a channel and includes parabolic and triangular cross sections. For an exponent m<0.5 in the power law, the relative wetted perimeter has been estimated from a series expansion truncated to four terms. For values of the exponent m ≥ 0.5 the relative wetted perimeter has been estimated using an appropriate non-linear interpolation expression. A table to estimate relative wetted perimeter based on these expressions is presented for design purposes. With these expressions for relative wetted perimeter, and using the Lagrange method of undetermined multipliers, for any given maximum side slope, the area and/or wetted perimeter is minimized subject to the equality constraint of a uniform flow (Mannings) equation. Using this technique, for any given side slope, the exponent of the power-law channel can be determined and hydraulically efficient power-law channels can be designed. Optimized power-law channels are compared with trapezoidal and parabolic channels. The existing parabolic design of the Pehur High Level Canal, Pakistan is compared with an optimum power-law channel.     

Critical Flow Section in Collector Channel

This paper shows how the critical flow section in a collector channel can be located by solving the dynamic equation of spatially varied flow, Manning's equation, and making use of the singular-point concept. In addition to channel length and tailwater elevation, the occurrence of a critical flow section in a spatially varied flow also depends on the combination of channel cross-sectional geometry, roughness, slope, and inflow rate. When the critical flow section is necessary to be developed in a collector channel, the two dimensionless parameters (Fq∕S0 representing the design capacity and N∕S0 representing the channel roughness) derived in this study guide selection of channel cross-sectional parameters. A set of design charts is provided for trapezoidal channels with a side slope of 1V:1H, 0.5V:1H, or 0V:1H.     

Flooding Probability-Based Optimal Design of Trapezoidal Open Channel Using Freeboard as a Design Variable

A flooding probability based cost effective design of open channel section has been proposed using freeboard as an additional design variable. The freeboard of the channel is calculated based on the flooding probability value. The proposed model is solved using classical optimization techniques as well as a nondominated sorting genetic algorithm. The results of the model are compared with an earlier reported model to demonstrate its superiority and field applicability.     

Design of Minimum Seepage Loss Canal Sections

The minimum area section is a thoroughly investigated problem in the hydraulics literature. However, because of the complexities of the analysis, the design of a minimum seepage loss section has not been attempted as yet. In this investigation, using previously derived results, simplified algebraic equations for computation of seepage loss from triangular, rectangular, and trapezoidal canals have been presented, which replace accurately the cumbersome evaluation of complex integrals. Using these seepage loss equations and the general uniform flow equation, explicit equations for the design variables of minimum seepage loss canal sections have been obtained for each of the three canal shapes by applying nonlinear optimization technique. The optimal trapezoidal section has the least seepage loss and cross-sectional area among the three optimal sections. A step-by-step design procedure for rectangular and trapezoidal canal sections has been presented. The analysis also includes the sensitivity of the seepage loss to design variables around the optimum value.     

Most Hydraulically Efficient Riprap-Lined Drainage Channels

Dimensions of the most hydraulically efficient trapezoidal and triangular drainage channels (that is, those with the smallest possible cross-section areas) whose banks are lined with loose rock riprap are found along with the stable stone diameter by solving a constrained nonlinear minimization problem. The problem statement is made dimensionless and less complicated by normalizing solution variables and combining parameters into two dimensionless quantities that describe the composite roughness of a channel and the stability of the rock lining. Normalized values of section bottom width, water depth, and rock diameter, along with the channel side-slope ratio, are found numerically. Results of the analysis are presented graphically and, when practical, in the form of empirical expressions. The solutions, which are comprehensive, give cross-section dimensions and the rock size needed to maintain a stable bank lining, greatly simplify design of hydraulically efficient riprap-lined open channels.     

Overtopping Probability Constrained Optimal Design of Composite Channels Using Swarm Intelligence Technique

In this paper swarm intelligence based methodology is proposed for optimal and reliable design of irrigation channels. The input parameters involved in channel design are prone to uncertainty and the solution of deterministic model may result in flooding risk and affect the stability of the channel. To provide reliability in the design, an overtopping probability constrained design is presented in this study. The deterministic equivalent of the probabilistic constraint is derived by following the principle of first order uncertainty analysis. In order to account for the uncertainty of design parameters in the objective function, a modified cost function is proposed. A methodology is propounded to solve it in a metaheuristic environment and solved it using elitist-mutated particle swarm optimization (EMPSO) method. The EMPSO based solutions are found to be quite successful and better than the classical optimization methods. Finally, it is concluded that the proposed methodology has a good potential for reliable design of composite channels for designer specified reliability values.     

Comprehensive Design of Minimum Cost IrrigationCanal Sections

Design of a minimum cost canal section involves minimization of the sum of costs per unit length of the canal, subject to uniform flow condition in the canal. Essentially it is a problem of minimization of a nonlinear objective function subject to a nonlinear equality constraint. In this investigation, the objective function has been expressed as the cost per unit length of the canal for lining, the depth-dependent unit volume earthwork cost, and the cost of water lost as seepage and evaporation losses. A general resistance equation has been used as an equality constraint. Using a nonlinear optimization technique on an augmented function, generalized empirical equations and section shape coefficients have been obtained for the design of minimum cost irrigation canals of triangular, rectangular, and trapezoidal shapes. The optimal dimensions for any shape can be obtained from the proposed equations along with tabulated section shape coefficients. The equation for optimal cost along with the corresponding section shape coefficients is useful during the planning of a canal project. A design example with sensitivity analysis has been included to demonstrate the simplicity of the present method.     

Investigation on the Dimensions and Shape of a Submerged Vane for Sediment Management in Alluvial Channels

A submerged vane is a flow-training facility mounted vertically on the channel bed to control the sediment movement in the channel cross section, and has been utilized in various applications, such as prevention of bank erosion, sediment exclusion at water intakes, and deepening channels for navigation. The performance of a submerged vane is related to its dimensions and shape. This study aims to investigate a vane’s sediment control effectiveness as a function of its size and shape, with the expectation of an optimal combination of dimensions and shape. A model for the calculation of the transverse bed profile in a cross section of a straight alluvial channel induced by a single submerged vane is developed. The model is utilized to investigate the performances for three types of vanes: (1) rectangular plates with various height and length; (2) tapered plates with linear decreasing in length from the base to the top; and (3) plates of parallelogram with the top of the plates swept forward or backward. Design guidelines and suggestions on the dimensions and shape of the vane are provided based on the results.     

Most Hydraulically Efficient Standard Lined Canal Sections

Cross-sectional dimensions of the most hydraulically efficient lined canals are evaluated based on an analysis of a generalized trapezoidal shape that reduces to two standard sections with rounded bottom vertices used in India, as well as to the commonly used trapezoidal section with sharp bottom vertices. The method of Lagrange multipliers is applied to find the dimensions of optimal sections when the only constraint imposed is that of uniform flow and normal depth and, in addition, when values of either channel side slope, bottom width, top width, or supply depth are specified as well. The analytic solutions obtained for the generalized trapezoidal section are shown to match known solutions for limiting cases including those for sections in the shape of sharp-cornered trapezoids, rectangles, triangles, and semicircles. Solutions presented will be useful for evaluating standard cross-sectional shapes used for lined canals in India, as well as other sections that can be obtained from the generalized trapezoid with rounded bottom vertices.     

Analysis of Seepage from Polygon Channels

An exact analytical solution for the quantity of seepage from a trapezoidal channel underlain by a drainage layer at a shallow depth has been obtained using an inverse hodograph and a Schwarz-Christoffel transformation. The symmetry about the vertical axis has been utilized in obtaining the solution for half of the seepage domain only. The solution also includes relations for variation in seepage velocity along the channel perimeter and a set of parametric equations for the location of phreatic line. From this generalized case, particular solutions have also been deduced for rectangular and triangular channels with a drainage layer at finite depth and trapezoidal, rectangular, and triangular channels with a drainage layer and water table at infinite depth. Moreover, the analysis includes solutions for a slit, which is also a special case of polygon channels, for both cases of the drainage layer. These solutions are useful in quantifying seepage loss and/or artificial recharge of groundwater through polygon channels.     

Optimal Design of a Stable Trapezoidal Channel Section Using Hybrid Optimization Techniques

A cost effective channel section for a specified flow rate, roughness coefficients, longitudinal slope, and various cost parameters can be determined using an optimization technique. However, the derived optimal channel section may not be feasible for construction because of in situ conditions. The local soil conditions may not support the optimal side slope of the channel and if constructed, the slope may fail. It is therefore necessary to also incorporate the criteria for side slope stability in designing an optimal open channel section. In this paper, a new methodology has been developed to design a stable and optimal channel section using hybrid optimization techniques. A genetic algorithm based optimization model is developed initially to determine the factor of safety of a channel slope for given soil parameters. This optimization model is then externally linked with a separate sequential quadratic programming based optimization model to evaluate the parameters of the stable and optimal channel section. Solution for various example problems incorporating different soil parameters are illustrated to demonstrate the applicability of the developed methodology.     

Design of Minimum Seepage Loss Canal Sections with Drainage Layer at Shallow Depth

This paper presents an analytical solution for the quantity of seepage from a rectangular canal underlain by a drainage layer at shallow depth. The solution has been obtained using inverse hodograph and conformal mapping. Using the solution for the rectangular canal and the existing analytical solutions for triangular and trapezoidal canals, simplified algebraic equations for computation of seepage loss from these canals, when the drainage layer lies at finite depth, have been presented, which replace the cumbersome evaluation of complex integrals. Using these seepage loss equations and a general uniform flow equation, simplified equations for the design variables of minimum seepage loss sections have been obtained for each of the three canal shapes by applying a nonlinear optimization technique. The optimal design equations along with the tabulated section shape coefficients provide a convenient method for design of the minimum seepage loss section. A step-by-step design procedure for rectangular and trapezoidal canal sections has been presented.