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.     

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 Parabolic Canal Section

Optimal design equations for a parabolic canal section are presented in this paper. The design equations for a minimum earthwork cost section and a minimum cost lined section are in explicit form and result in optimal dimensions of a canal in single step computations. These have been obtained after applying the Fibbonaci search method on a nonlinear unconstrained optimization problem. The study also addresses the bounds on the canal dimensions and the velocity of flow. A nondimensional parameter approach has been used to simplify the analysis, and a set of graphs for nondimensional parameters are presented as an alternative for design. Design procedures for different cases have been presented to demonstrate the simplicity of the method.     

Optimal Spacing in an Array of Fully Penetrating Ditches for Subsurface Drainage

A methodology for optimal spacing in an array of ditches fully penetrating into homogeneous and isotropic porous medium of finite depth over an impervious layer is presented. The cost function includes the depth-dependent earthwork cost and the capitalized cost of pumping of drain discharge. Essentially, it is a problem of minimization of a nonlinear objective function of single variable. The input variables consist of rainfall intensity, hydraulic conductivity of the porous medium, width and depth of ditches, earthwork cost, cost of pumps and pumping energy cost, efficiency of pumping unit, and rate of interest. Using nonlinear data fitting method an explicit equation has been proposed for computing the optimal spacing between the ditches.     

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.     

Optimal Design of a Settling Basin for a Small-Scale Drainage Area

This paper develops a nonlinear programming model to optimally design a settling basin for a small-scale drainage area with a minimum total cost. It is assumed that the shape of the settling basin is rectangular parallelepiped, and it is connected to an open channel at both ends. Therefore, the decision variables include the scales of the settling basin (i.e., length, width, and height) and the scales of the channel (i.e., width and height). The design trap efficiency requirement, which must be greater than or equal to the required one of the considered watershed, makes up the main constraint. Other constraints consist of the upper and lower bounds of the decision variables, the equations for computing the trap efficiency, and the average flow velocity in the settling basin. The objective function is to minimize the total annual cost, which is the sum of the land, capital, and maintenance-operation cost. The developed model is solved by using a genetic algorithm. This model is applied to a subwatershed of the Wu-She Reservoir watershed in central Taiwan. The obtained results effectively demonstrate the applicability and practicability of the model.     

Design of Sewer Line

Sewerage involves the major portion of the cost of a wastewater system. In the design of a sewerage system the sewer line is the basic unit occurring repeatedly in the design process. Any savings during the design of this unit will affect the overall cost of the sewerage system. A survey of the literature showed that the present status of sewer line design algorithms use linear programming and dynamic programming. The linear programming, using piecewise linearization of the objective function and constraints in every cycle, requires substantial computer time. On the other hand, dynamic programming algorithms are subjected to the “curse of dimensionality,” thus requiring large amounts of computer memory. In this paper, using a dimensionally consistent resistance equation, the sewer line design problem is formulated as a minimization of the cost function subjected to tight and loose constraints. The problem is solved by iterative application of the Lagrange-multiplier method.     

Design of Minimum Seepage-Loss Nonpolygonal Canal Sections

Analytical solutions exist for the problem of determination of seepage loss from polygonal sections such as a triangle, a rectangle, and a trapezoid. In this investigation, seepage from circular and exponential sections has been calculated by a finite-difference-based numerical solution of the differential equation governing the seepage flow. The phreatic boundaries of the flow domain were described in terms of two parameters that are estimated by a minimization process. Such seepage computations were performed for a large number of independent dimensionless variables of the section geometry. Subjecting the computed seepage to regression analyses explicit equations for seepage loss from canals having circular and exponential cross sections have been obtained. Using these seepage-loss equations, the design variables for minimum seepage loss of circular and exponential canal sections have been obtained. The use of the design equations has been illustrated by design examples.     

Optimal Design and Operation of Irrigation Pumping Stations

A methodology based on solving a large-scale nonlinear programming problem is presented for the optimal design and operation of pumping stations. Optimum design and operation refers to the selection of pump type, capacity, and number of units as well as scheduling the operation of irrigation pumps that results in minimum design and operating cost for a given set of demand curves. The design criteria for such pumping stations are based fundamentally on some important and critical parameters, such as pump capacity, number of units, types of pumps, and civil works. The optimization process consists of three main steps: (1) determination of minimum yearly consumed energy; (2) minimization of the total cost for all sets of pumping stations; and (3) selection of the least-cost set among the feasible sets of pumping stations, recognizing a combination of the cited criteria. The computational analysis is based upon one major objective function and a computer program, which is developed to solve the generated equations. Application of the model to the Farabi Agricultural and Industrial Project, Iran, shows considerable savings, about 25% in total annual cost of the pumping station.     

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

Analytical solutions exist for the seepage discharge from polygonal and nonpolygonal canal sections underlain by a drainage layer at a hydraulically infinite depth. These solutions lead to underestimation of the seepage discharge if a drainage layer occurs at a shallow depth. This paper presents solutions for seepage discharge from circular and exponential sections overlying a shallow drainage layer. The discharge has been calculated by a finite-difference-based numerical solution of the differential governing the seepage flow. The phreatic boundaries of the flow domain were described in terms of two parameters that were estimated by a minimization process. Such seepage computations were performed for a large number of independent dimensionless variables of the section geometry. Subjecting the computed seepage to regression analyses, explicit equations for seepage discharge loss have been obtained. Using these seepage loss equations, the design variables for minimum seepage loss have been obtained. The use of the design equations has been illustrated by design examples.     

Hydraulic Analysis and Optimum Design of Multidiameter Irrigation Laterals

A new analytical continuous-uniform outflow approach that takes into account the effect of the number of outlets on the multidiameter lateral hydraulics is presented. The pressure head profile along the multidiameter pipeline is described by a simple analytical function providing direct calculation of the outlet pressure head along the pipeline. The method is significantly improved by introducing an adjusted spatially variable outflow equation—of power function form—for the errors caused by the assumption of equal outlet discharge. The effect of ground slope on hydraulic computation is also considered. Simple equations are derived for the direct calculation of the maximum, minimum, and inlet pressure head along the multidiameter pipeline. The optimum design problem for two-diameter laterals is also solved analytically. For specified total length of a two-diameter pipeline, a simple algebraic equation is derived to calculate directly the appropriate lengths of the reaches of different diameters in such a way that the total cost of the pipeline is minimized. Comparison tests with an accurate numerical stepwise method indicate that the proposed analytical approach is sufficiently accurate.     

Simple Approximation for Flowing Well Problem

The discharge and yield of a flowing well can be computed by existing complex solutions or by Duhamel's technique using kernel coefficients. The mathematical complexities of these methods can be very much simplified through numerical methods without loss of accuracy. In the present study simple equations for both the well discharge and well production functions are presented. The equation for the well discharge function has been used to find aquifer constants for known well discharge and drawdown through the error minimization method. The results are compared in order to demonstrate the relative simplicity of the proposed equations.     

镀锌机组带钢C翘缺陷综合治理技术的开发

 为了研究带钢热镀产品C翘板形缺陷导致其镀层不均匀性严重问题,优化热镀锌机组连退工艺过程,开发带钢C翘缺陷综合治理技术。首先考虑热镀锌连退过程的设备与其工艺特点,分析C翘板形发生机理,明火烧嘴状态、辐射管状态、风机状态、带钢重力、上游工艺段遗传板形等影响因素,使得带钢表面各处存温度差异性较大,致使带钢横向残留塑性变形在其厚度方向分布均匀,当其超过其临界失稳条件而发生C翘。然后,从系统学的观点出发建立兼顾C翘缺陷影响因素的连退炉系统函数,考虑到连退炉多变量耦合、复杂非线性特点,以工艺段为单位将连退过程划分子系统,以各工艺段板形翘曲最小为目标,基于带钢C翘预报模型分别建立加热段温度优化目标函数、均热段温度优化目标函数、冷却段冷却风机状态优化目标函数,并以各工艺段目标函数相叠加的形式建立系统函数。得出了连退过程各工艺段工艺参数综合优化方法,制定其工艺优化流程,并形成加热段温度优化设定技术、均热段温度优化设定技术、冷却段冷却风机状态优化设定技术等关键技术。最后,将该综合治理技术应用于国内型钢铁机组取得良好效果。结果表明,典型规格带钢在连退过程各工艺参数优化前后对比,其最大翘曲量由17.0 mm降至5.9 mm;该类型带钢的镀层厚度生产统计数据显示,带钢C翘板形改善后,带钢上下表面平均镀层厚度降至用户要求的范围内。     

Minimum Weight Design of Fillet Welds Using Convex Formulation

This paper presents a discrete formulation of the nonlinear minimum weight design of welded connections, in which global throat stresses along every fillet are approximated by piecewise linear shape functions. With this formulation it is possible to find the set of throat thicknesses of a welded connection that gives the minimum amount of weld metal necessary to support an arbitrary set of external actions and a stress field in equilibrium with these actions and satisfying a strength criterion. The formulation adopted involves a linear form of objective function and equilibrium equations, which, together with the convex character of the nonlinear constraints, gives a full convex set of constraints and consequently a global optimum of the objective function. Moreover, the structure of the set of constraints makes it possible to demonstrate that stress and throat thickness values are unique at the optimum, which allows the optimum stress distributions to be simplified, to establish practical design recommendations. Realistic conditions can be imposed on the stress field. The problem is solved with a very accessible and well-known scientific library, which allows this method to be implemented by anyone involved in welding practice. The method proposed is developed for the case of a planar connection. Two practical applications using the strength criterion of Eurocode 3 are included, discussing the results obtained for two types of common arrangements and showing that numerical results are consistent and CPU times reasonably low.     

K-OBM-S不锈钢冶炼最优配料的数学模型

用线性规划方法建立了太钢K-OBM-S法冶炼不锈钢的配料模型。模型中包括13个决策变量,最低配料成本作为模型的目标函数,一组自变量的线性等式和不等式作为约束引入,采用单纯形法对该模型进行求解。计算结果表明,在给定条件下,按建立的数学模型对S0Cr18Ni9不锈钢进行配料,可使吨钢总成本降低10%。     

以热损失最小为目标的焦炉工序分析与优化

基于焦炉工序的物料平衡和能量平衡,以工序热损失最小为目标,构建了焦炉工序的优化模型。以入炉煤配比、加热煤气配比、入炉煤含水量和推焦温度等9个参数为优化变量,考虑13个约束条件,利用焦化厂实际生产数据进行了优化,得到了最小热损失和最优变量值,并分析了优化变量对最小热损失的影响规律。研究表明:适当增加1/3焦煤和瘦煤的用量并减少气煤、肥煤和焦煤的用量可以降低焦炉工序的热损失;高炉煤气和焦炉煤气用量约为7∶1时,能够使焦炉工序的热损失达到最小值。提高脱硫工艺水平可提高硫分的约束上限,从而间接降低热损失。优化后,混合煤挥发分含量可降低3.3%,最小热损失可降低13.74%。实际生产过程中,应使各原料的用量处于使最小热损失变化较小的区间内,且应尽量接近最优配比,以保证焦炉工序连续平稳生产的同时降低热损失。     

Optimal Control of Earthquake Response Using Semiactive Isolation

The responses of two, low-rise, 2-degree-of-freedom base isolated structures with different isolation periods to a set of near-field earthquake ground motions are investigated under passive linear and nonlinear viscous damping, two pseudoskyhook semiactive control methods, and optimal semiactive control. The structures are isolated with a low damping elastic isolation system in parallel with a controllable damper. The optimal semiactive control strategy minimizes an integral norm of superstructure absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied and is determined through a numerical solution to the Euler–Lagrange equations. The optimal closed-loop performance is evaluated for a controllable damper and is compared to passive viscous damping and causal pseudoskyhook control rules. Results obtained from eight different earthquake records illustrate the type of ground motions and structures for which semiactive damping is most promising.     

Optimal Furrow Design.?II: Explicit Calculation of Design Variables

The furrow irrigation system design problem (at minimum cost) is significantly simplified by analytically solving it. For a specified furrow length, a simple algebraic equation is derived to directly calculate the appropriate inflow rate (and cutoff time) so that the minimum cost of the furrow system is obtained. The proposed equation is independent of the water and labor cost coefficients. Comparison tests indicated that the optimum inflow rate values obtained analytically were in close agreement to the optimum values obtained using the outcome of the zero-inertia numerical model. The method is extended for furrow design considering the furrow length also as a design variable. The optimum number of distribution lines and widthwise furrow sets are easily determined by a simple calculation procedure.     

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.