Optimal Design of Channel Having Horizontal Bottom and Parabolic Sides

The cost of open channels can be minimized by using (1) the optimal design concept; (2) a new geometric shape to substitute for the trapezoidal channels, and/or (3) a composite channel. The channels in which the roughness along the wetted perimeter become distinctly different from part to part of the perimeter are called composite channels. The feasibility of a new cross-sectional shape that has a 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 and unique design of open channels. In optimal design concept, the geometric dimensions of a channel cross section are determined in a manner to minimize the total construction costs. The constraints are the given channel capacity and other imposed restrictions on geometric dimensions. The Lagrange multiplier technique is used to solve the resulting channel optimization models. The developed optimization models are applied to design the proposed and trapezoidal channels to convey a given design flow considering various design scenarios which include unrestricted, flow depth constrained, side slopes constrained, and top width constrained design. Each of these design scenarios again takes into account fixed freeboard, and depth-dependent freeboard cases of design. An analysis of the optimization results establishes the cost-saving capability of the proposed cross-sectional shape in comparison to a trapezoidal cross section.     

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 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.     

倾斜运输的板式给料机牵引阻力解析解

大型重型板式给料机的发展急需出现精确的设计理论,文献[6][7]分别研究了具有矩形断面和斜向梯形断面裙板的水平布置给料机牵引阻力。引用上述文献料斗底部仓压的基本方程,推导出工程上常出现的具有矩形断面裙板的倾斜布置板式给料机倾斜仓颈剪切阻力,再导出倾斜物料对侧板和底部托辊机构的摩擦阻力。这些结果和专业手册[1][3]是不同的。     

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.     

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.     

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.     

Exact Equations for Critical Depth in a Trapezoidal Canal

Critical depth is an important parameter in the analysis of varied flow in canals and natural streams. For triangular, rectangular, and parabolic channel sections it is possible to express critical depth analytically. However, for many practical sections, including the trapezoidal section, the governing equations are implicit in the critical depth. For these sections the critical depth is presently obtained either by trial and error procedure, or by using empirical equations based on curve fitting. In this Technical Note exact analytical solutions of critical depth for the trapezoidal open channel section have been obtained in the form of fast converging infinite series.     

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.     

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.     

Least-Cost and Most Efficient Channel Cross Sections

It has been a long-standing concern to decide if a channel should be designed to have the highest hydraulic efficiency or the least cost. In this study, a large amount of channel construction costs were reviewed and analyzed to derive the channel construction cost function as the sum of the costs for the land acquisition of the channel’s alignment, lining material for the channel’s cross section, and earth excavation for the channel’s depth. Case studies conducted in this technical note indicate that the differences between the least-cost and most efficient cross sections are closely related to the channel lining to land acquisition cost ratio. When the lining to land unit cost ratio vanishes, the difference between these two cross sections is diminished. As revealed by the cost data, the least-cost channel section tends to be deeper if the land cost is much higher than the lining cost. This trade-off was incorporated into the normalized equation to provide direct solutions to the least-cost channel cross section. The normalization of the least-cost equations allows this approach to be transferred to other regions when the local cost data are available.     

Quick Method for Open-Channel Discharge Measurements Using Air Bubbles

An economical methodology is proposed by which distinct air bubbles released at the bottom of a channel may be utilized for determining the local flow discharge q per unit width. Simple theoretical analysis shows that q is linearly dependent on the rise length L of bubbles released at the bottom. This length is the horizontal displacement of the bubbles between the release cross section and the cross section where they emerge. The theoretical findings were compared with measurements in three laboratory flumes and in an irrigation canal. Based on the above, a relationship between L and q has been established. The empirically proposed relationship is very useful for fast discharge measurements in channels and natural streams.     

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.     

Dimensionless Curves for Normal-Depth Calculations in Canal Sections

Normal depth is an important parameter for the design of irrigation canals. Direct analytical solution of normal depth is not possible because of the implicit nature of governing equations. The solution usually requires tedious techniques of trial and error. Tabular and graphical methods are only available for more common cross sections. This problem has been previously addressed for some cross sections (i.e., rectangular, triangular, trapezoidal, and circular) and the calculation of normal depth in these cross sections has been reported in the literature. Reported herein are graphical solutions for normal depth in other cross sections (i.e., round-bottomed triangular, parabolic, and round-corner rectangular).     

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.     

Broad-Crested Weirs with Rectangular Compound Cross Sections

A series of laboratory experiments was performed in order to investigate the effects of width of the lower weir crest and step height of broad-crested weirs of rectangular compound cross section on the values of the discharge coefficient, the approach velocity coefficient, and the modular limit. For this purpose, nine different broad-crested weir models with rectangular compound cross sections and a model with a rectangular cross section were tested in a horizontal laboratory flume of 11.0 m length, 0.29 m width, and 0.70 m depth for a wide range of discharges. The compound cross sections were formed by a combination of three sets of step heights and three sets of lower weir crest widths. The sill-referenced heads at the approach channel and at the tailwater channel were measured in each experiment. The dependence of the discharge coefficient, approach velocity coefficient, and modular limit values on model parameters was investigated, and these quantities were compared with those of the broad-crested weir models with a rectangular cross section.     

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.     

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.     

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.     

Semianalytical Model for Shear Stress Distribution in Simple and Compound Open Channels

Semianalytical equations were derived for distribution of shear stress in straight open channels with rectangular, trapezoidal, and compound cross sections. These equations are based on a simplified streamwise vorticity equation that includes secondary Reynolds stresses. Reynolds stresses were then modeled and their different terms were evaluated based on the work of previous researchers and experimental data. Substitution of these terms into the simplified vorticity equation yielded the relative shear stress distribution equation along the width of different channel cross sections. In compound channels the effect of additional secondary flows due to the shear layer between the main channel and the flood plain were also considered. Comparisons between predictions of the model and experimental data, predictions of other analytical or three dimensional numerical models with advanced turbulent closures, were made with good agreement.