Open channel structures are essential to infrastructure networks and expensive to manufacture. Optimizing the design of channel structures can reduce the total cost of a channel's length, including costs of lining, earthwork, and water lost through seepage and evaporation. The present research aims to present various optimization models towards the design of trapezoidal channel cross section. First, a general resistance equation was applied as a constraint. Next, a genetic algorithm (GA) was used to determine the optimal geometry of a trapezoidal channel section based on several parameters, i.e., depth, bottom width, and side slope. Eight different models were proposed and evaluated with no other constraint besides financial cost as well as with a normal depth, flow velocity, Froude number, top width, and by ignoring the cost of seepage. Numerical outcomes obtained by the GA are compared to previous studies in order to determine the most efficient model. Results from a single application indicate that the restriction of depth, velocity, and Froude number can increase the total cost, while restriction of the top width can decrease the cost of the construction. Also, the solution for various example problems incorporating different discharge values and bed slopes caused increase and decrease in cost, respectively.

## NOTATION

*A*flow area of channel (m

^{2})*b*bed width of channel (m)

*C*cost per unit length of canal ($/m)

*C*_{e}cost of earthwork per unit length of channel ($/m)

*C*_{L}cost of lining per unit length of channel ($/m)

*C*_{w}capitalized cost of water lost per unit length of channel ($/m)

*c*_{e}cost per unit volume of earthwork at ground level ($/m

^{3})*c*_{L}cost per unit surface area of lining ($/m

^{2})*c*_{r}increase in unit excavation cost per unit depth ($/m

^{4})*c*_{w}cost per unit volume of water ($/m

^{3})*E*evaporation discharge per unit free surface area (m/s)

*Fr*Froude number

*Fs*seepage function (dimensionless)

*g*gravitational acceleration (m/s

^{2})*k*hydraulic conductivity (m/s)

*m*side slope of channel (dimensionless)

*P*flow perimeter of channel (m)

*p*penalty parameter (dimensionless)

*Q*discharge (m

^{3}/s)*R*hydraulic radius (m)

*S*_{0}bed slope of channel (dimensionless)

*T*_{w}top width (m)

*V*average velocity (m/s)

*y*_{n}normal depth of flow in channel (m)

*ɛ*average roughness height of canal lining (m)

*λ*length scale (m)

*ν*kinematic viscosity (m

^{2}/s)*ϕ*equality constraint (dimensionless)

*Ψ*augmented function (dimensionless)

- $
monetary unit

## INTRODUCTION

Irrigation channel networks are very substantial and are utilized as water supply and conveyance facilities, flood control and for use in other fields. Due to the high cost of channel construction, investigating the optimal and economic cross-sectional designs of these channels is a necessity. A channel in the network may be subjected to lining. The supported cost of a lined channel is less in comparison to unlined channels, since the lining protects against bed and bank erosion. Channels in alluvium are typically lined and capable of decreasing seepage. The seepage loss from channels has been estimated for different sets of special conditions. The analytical form of these solutions, which contain complex integrals and unknown implicit state variables, is not convenient in designing or estimating seepage from the existent channels (Morel-Seytoux 1964; Garg & Chawla 1970; Subramanya *et al.* 1973; Sharma & Chawla 1979).

Swamee *et al.* (2000) considered seepage loss in the objective functions and applied nonlinear optimization techniques to design explicit equations. Wachyan & Rushton (1987) used empirical measurements to reveal that main losses occurring from channels are related to whether they are lined or unlined. Kacimov (1992) developed a complex-variable method to optimize the shape design problems of channel beds. In order to minimize the cost function, the seepage losses and cost of lining was constrained by specified hydraulic characteristics. Additionally, Aksoy & Altan-Sakarya (2006) studied the optimum values of section variables for different channel types by minimizing their cost.

Usually, evaporation from a channel is only a small proportion of the total loss, but it becomes substantial for long channels running through arid regions. Warnaka & Pochop (1988) and Ikebuchi *et al.* (1988) compared the capabilities of different evaporative sorts of estimation models and found that the mass transfer-based models presented the most accurate results (Fulford & Sturm 1984).

The primary factor affecting the channel design is the channel surface forming material which determines the roughness coefficient, the minimum permissible velocity to avoid deposition of silt or debris, the constrained velocity to prevent erosion of the channel surface, and the topography of the channel route which indicates how much the section is hydraulically and/or economically efficient (Chow 1973). The choice of hydraulic parameters is a vital task in the hydraulic design of channels since they are entitled to high uncertainty. Dimitriadis *et al.* (2016) employed extended sensitivity analysis by simultaneously varying the input discharge, longitudinal and lateral gradients and roughness coefficient.

Monadjemi (1994) and Froehlich (1994) modeled optimized channel design using a Lagrangian undetermined multiplier method. Alternatively, for non-linear, non-convex optimization problems, the compound and implicit construction of the cost function and/or constraints makes the employment of customary gradient-based techniques very difficult, so that the optimization process cannot be applied in many locally optimized processes. This has caused the wide use of heuristic approaches, e.g., genetic algorithm (GA) (Goldberg 1989), particle swarm optimization (PSO) (Kennedy & Eberhart 1995), genetic programming (GP) (Koza 1992), gene expression programming (GEP) (Azamathulla 2012), and charged system search (Kaveh & Talatahari 2010), among many others. Different optimization algorithms are applied to open channel section design problems.

The GA has been successfully utilized for optimizing the design of open channels (Jain *et al.* 2004; Bhattacharjya & Satish 2007), as well as irrigation scheduling with flow of water through channel networks (Nixon *et al.* 2000), along with other hydraulic problems (Wu & Simpson 2002; Roushangar & Koosheh 2015). The GP has been applied in different engineering optimization problems (Sharifi *et al.* 2011). The GEP approach has also been used to solve engineering problems by deriving a new predictive model (Azamathulla & Ahmad 2013; Azamathulla 2013). PSO also has been successfully applied for solving water resources management problems (Janga Reddy & Nagesh Kumar 2009). Also Nourani *et al.* (2009) optimized composite channels using ant colony optimization.

To the authors’ knowledge, despite considerable investigations into providing a protocol for minimizing cross-sectional area of channels, there are no investigations dealing with constrained hydraulic parameters as well as the role of seepage cost on optimal channel sections. Therefore, in this study, six different models of optimum design for open channels were established. The first model was evaluated with no additional constraint equation, while in the second model, the channel top width was considered as an additional constraint. The third and fourth models had additional constraints of different values of velocity and Froude numbers, respectively. The fifth model utilized additional constraint of different depth values. The sixth model was determined for no seepage cost state. Finally, the applications section shows the effect of discharge (seventh model) and longitudinal bed slopes (eighth model) on optimum design of channels. We applied all models to a real case scenario and showed that the constrained hydraulic parameters have a great influence in the design of the trapezoidal channel section.

## FORMULATION OF OPTIMAL DESIGN OF OPEN CHANNELS

Selection of the geometric variables, e.g., side slope, bottom width, and flow depth for open channel sections varies according to the designer's perspective. Also, the longitudinal bed slope of the channel is influenced by topography that is considered as a constant amount in seven models. One of the important objectives is minimizing the total cost of a channel section, that it has the capability of passing the channel distance safely, which must be considered. Generally, the cost per unit length of a lined open channel section is defined as the summation of three terms, namely, the depth-dependent earthwork cost, the cost of lining, and the cost of water lost as seepage and evaporation. These terms are explained in the following sections.

### Earthwork cost, lining cost, and cost of water loss

*C*($/m) was obtained as: The earthwork cost (monetary unit per unit length, e.g., $/m) is given as: where = cost per unit volume of earthwork at ground level ; = the additional cost per unit volume of excavation per unit depth ;

*A*= flow area ; = depth of the centroid of the area of excavation from the ground surface (m).

*P*= flow perimeter (m). The capitalized cost of water lost ($/m) might be expressed as: where

*r*= rate of interest ($/$/year) and = cost per unit volume of water ($/m

^{3}). The volumetric cost of water may differ for evaporation and seepage losses, depending upon the side effects caused by the seepage loss.

*k*= coefficient of permeability (m/s);

*=*normal depth of flow in the channel (m); and

*Fs*= seepage function (dimensionless), which depends on channel geometry.

*T*= width of free surface (m); and

_{w}*E*= evaporation discharge per unit surface area (m/s) (Swamee

*et al.*2000). The

*/*and

*/*ratios were obtained as listed in Table 1 (Schedule of rates 1997). These ratios can be obtained for various kinds of linings, soil strata, and climatic conditions by utilizing appropriate unit rates.

Dimensionless variables | ||||||
---|---|---|---|---|---|---|

Dimensionless variables | ||||||
---|---|---|---|---|---|---|

*V*= average flow velocity (m/s);

*g*= gravitational acceleration (m/s

^{2});

*R*= hydraulic radius (m); = longitudinal bed slope (dimensionless); = average roughness height of the channel lining (m); and = kinematic viscosity of water . Utilizing the continuity equation, the discharge Q was obtained as: Combining Equations (1) and (8) forms the general optimization algorithm for a minimum cost section of open channel. The terms of these models are all in dimensional forms. In order to facilitate the detection of the effects of variables on the models, the aforementioned equations are transformed to dimensionless forms, through defining a length scale, as follows: The following dimensionless variables were then obtained in Table 1. The subscript * denotes the corresponding dimensionless parameters of each hydraulic parameter.

where = equality restriction function.

*ψ*distributed by: where

*β*= exponent of equality constraint function ; α = penalty function parameter with a high positive value;

*i*= index representing restriction; and

*I*= total number of restrictions imposed on a particular nonlinear optimization programming (NLOP). The optimum design was applied on trapezoidal channel cross sections (Figure 1). According to Figure 1, the bottom width, flow depth, and side slope are represented by , , and

*m*, respectively. Consequently, the equations are as follows:

## PROCEDURES FOR OPTIMAL DESIGN

### Validation with using empirical equations

Analysis of the optimal channel sections for a number of input variables led to the following generalized empirical equations in clear form for trapezoidal channel section (Swamee *et al.* 2000).

For a given set of data, a channel can be designed by minimizing Equation (10) and subjected to constraint given by Equation (11). Swamee's method is used to verify the results.

### GA-based optimization procedure

The GA is employed to solve the formulated nonlinear models. The GA is a search technique based on the genetics concept of natural selection, which combines an artificial survival of the fittest with genetic operators abstracted from nature (Holland 1975). The important difference between GA and classical optimization search techniques is that the GA generates a population of possible solutions, whereas the classical optimization techniques lead to a single solution. An individual solution in a population of solutions is analogous to a biological chromosome. While a natural chromosome specifies genetic characteristics of a human being, an artificial chromosome in GA indicates the values of varied decision variables representing a decision or a solution. For most GAs, candidate solutions are represented by chromosomes coded using a binary number system (Goldberg 1989). The GA that employs binary strings as its chromosomes is named binary-coded GA. The binary-coded GA contains three basic operators: selection, crossover or mating, and mutation. The selection function chooses parents for the next generation based on their scaled values from the fittest scaling function. In this study, the stochastic uniform selection is used in conjunction with elitism. The stochastic uniform selection lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation.

#### Model I

This model is determined by Equations (10) and (11), representing no additional constraint when consisting of three variables.

#### Model II

*T*: where additional equality constraint function which limits the total top width of the channel to

_{wmax}*T*.

_{wmax}#### Model III

For the designed section, the average flow velocity *V _{av}* could be achieved by Equation (8). However, in order to safely convey the required discharge through a channel, it is necessary to ensure that the velocity of the channel does not exceed the corresponding maximum velocity which is related to the roughness coefficient of that segment.

#### Model IV

#### Model V

#### Model VI

## RESULTS AND DISCUSSION

Let us suppose that the channel should be designed to transport a discharge of 100 on a longitudinal bed slope of 0.001. The channel passes through a stratum of typical soil, in which and (Table 2). Further, it is proposed to supply concrete lining with . The climatic condition of the channel area satisfies the . For the purpose of design, it is assumed that , (water at 20 °C), and *ɛ* = 1 mm (concrete lining) (Table 3).

Types of data | cl/ce (m) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Concrete tile lining | Brick tile lining | Brunt clay tile lining | ce/cr (m) (11) | |||||||

With LDPE film | Without film (4) | With LDPE film | Without film (7) | With LDPE Film | Without film (10) | |||||

100 m (2) | 200 m (3) | 100 m (5) | 200 m (6) | 100 m (8) | 200 m (9) | |||||

Ordinary soil | 12.75 | 13.02 | 12.24 | 6.39 | 6.67 | 5.88 | 6.08 | 6.35 | 5.57 | 6.96 |

Hard soil | 10 | 10.22 | 9.60 | 5.01 | 5.23 | 4.62 | 4.77 | 4.99 | 3.37 | 8.86 |

Impure lime nodules | 8.9 | 9.10 | 8.55 | 4.47 | 4.66 | 4.11 | 4.25 | 4.44 | 3.89 | 9.96 |

Dry shoal with shingle | 6.56 | 6.71 | 6.30 | 3.29 | 3.43 | 3.03 | 3.13 | 3.27 | 2.86 | 13.50 |

Slush and lahel | 6.40 | 6.54 | 6.14 | 3.21 | 3.53 | 2.95 | 3.05 | 3.19 | 2.79 | 13.86 |

Types of data | cl/ce (m) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Concrete tile lining | Brick tile lining | Brunt clay tile lining | ce/cr (m) (11) | |||||||

With LDPE film | Without film (4) | With LDPE film | Without film (7) | With LDPE Film | Without film (10) | |||||

100 m (2) | 200 m (3) | 100 m (5) | 200 m (6) | 100 m (8) | 200 m (9) | |||||

Ordinary soil | 12.75 | 13.02 | 12.24 | 6.39 | 6.67 | 5.88 | 6.08 | 6.35 | 5.57 | 6.96 |

Hard soil | 10 | 10.22 | 9.60 | 5.01 | 5.23 | 4.62 | 4.77 | 4.99 | 3.37 | 8.86 |

Impure lime nodules | 8.9 | 9.10 | 8.55 | 4.47 | 4.66 | 4.11 | 4.25 | 4.44 | 3.89 | 9.96 |

Dry shoal with shingle | 6.56 | 6.71 | 6.30 | 3.29 | 3.43 | 3.03 | 3.13 | 3.27 | 2.86 | 13.50 |

Slush and lahel | 6.40 | 6.54 | 6.14 | 3.21 | 3.53 | 2.95 | 3.05 | 3.19 | 2.79 | 13.86 |

LPDE = Low density polyethylene.

Flow factors | Average roughness of height | Cost of terms | ||||||
---|---|---|---|---|---|---|---|---|

ɛ (mm) | ||||||||

100 | 0.001 | 9.79 | 1 | 2 | 12 | 7 | 10 |

Flow factors | Average roughness of height | Cost of terms | ||||||
---|---|---|---|---|---|---|---|---|

ɛ (mm) | ||||||||

100 | 0.001 | 9.79 | 1 | 2 | 12 | 7 | 10 |

*Note:* For water at 20 °C.

Using Equation (9), the was determined to be 15.9 m, so the following parameters were identified using Equation (10): ɛ_{*} = 6.3 × 10^{−5}, ϑ_{*} = 1.75 × 10^{−7}, *C*_{l*} = 0.75, *C*_{r*} = 2.27, *C*_{ws*} = 0.63, and *C*_{wE*} = 0.125.

Table 4 compares the Swamee method (Swamee *et al.* 2000) with the proposed GA-based Model I of the present study (no additional constraint model). The sum total construction costs in both methods for Model I (no additional restriction) are 429.27 and 417.26, respectively. It can be noted from Table 4 that the proposed GA-based model shows better results compared to the Swamee method, with relatively less expensive values.

Parameters | Method I (Swamee et al. (2000), Model I) | Method II (GA, Model I) |
---|---|---|

b (m) | 5.829 | 5.159 |

y (m) | 3.878 | 3.784 |

m | 0.512 | 0.540 |

A (m^{2}) | 30.304 | 27.253 |

Fr | 0.6 | 0.68 |

V (m/s) | 3.3 | 3.66 |

Cost | 429.27 | 417.26 |

Parameters | Method I (Swamee et al. (2000), Model I) | Method II (GA, Model I) |
---|---|---|

b (m) | 5.829 | 5.159 |

y (m) | 3.878 | 3.784 |

m | 0.512 | 0.540 |

A (m^{2}) | 30.304 | 27.253 |

Fr | 0.6 | 0.68 |

V (m/s) | 3.3 | 3.66 |

Cost | 429.27 | 417.26 |

*Note:* Cost equivalent ; .

*T*) effect on dimensionless area (

_{w*}*A*) and total cost (

_{*}*C*) values. From Figure 4 it is clear that the channel area (

_{*}*A*) presents a direct linear relation with

_{*}*T*: the higher the

_{w*}*T*, the greater the

_{w*}*A*value. In the case of

_{*}*C*variations, however, there is no direct linear relationship, where

_{*}*C*tends to yield a constant value for

_{*}*T*values of approximately 0.45. Compared with no additional constraint model (Model I) in Table 5, it is clear that decreasing the channel top width would decrease the total cost by 30%.

_{w*}Parameters | When top width is: | |||
---|---|---|---|---|

b (m) | 4.388 | 3.752 | 3.100 | 2.575 |

y (m) | 4.086 | 4.324 | 4.070 | 3.625 |

m | 0.337 | 0.300 | 0.300 | 0.300 |

A (m^{2}) | 23.655 | 21.895 | 17.718 | 13.374 |

V (m/s) | 4.222 | 4.58 | 5.636 | 7.466 |

Cost | 391.855 | 381.743 | 345.338 | 298.821 |

Parameters | When top width is: | |||
---|---|---|---|---|

b (m) | 4.388 | 3.752 | 3.100 | 2.575 |

y (m) | 4.086 | 4.324 | 4.070 | 3.625 |

m | 0.337 | 0.300 | 0.300 | 0.300 |

A (m^{2}) | 23.655 | 21.895 | 17.718 | 13.374 |

V (m/s) | 4.222 | 4.58 | 5.636 | 7.466 |

Cost | 391.855 | 381.743 | 345.338 | 298.821 |

*Note:* Cost equivalent *C* = *C*_{e} × k; *k* = *C*_{*} × *λ*^{2}.

*C*, while there is an inverse relation between the depth variations vs.

_{*}*C*.

_{*}Parameters | When velocity is: | ||||
---|---|---|---|---|---|

V ≤ 3.5 | V ≤ 3 | V ≤ 2.9 | V ≤ 2.5 | V ≤ 2.1 | |

b (m) | 5.399 | 5.135 | 3.434 | 8.792 | 10.478 |

y (m) | 3.699 | 3.084 | 3.545 | 3.434 | 3.657 |

m | 0.632 | 1.804 | 1.766 | 0.823 | 0.690 |

A (m^{2}) | 28.625 | 33.291 | 34.439 | 39.949 | 47.559 |

Fr | 0.66 | 0.67 | 0.63 | 0.48 | 0.38 |

Cost | 429.194 | 505.620 | 511.181 | 537.979 | 570.086 |

Parameters | When velocity is: | ||||
---|---|---|---|---|---|

V ≤ 3.5 | V ≤ 3 | V ≤ 2.9 | V ≤ 2.5 | V ≤ 2.1 | |

b (m) | 5.399 | 5.135 | 3.434 | 8.792 | 10.478 |

y (m) | 3.699 | 3.084 | 3.545 | 3.434 | 3.657 |

m | 0.632 | 1.804 | 1.766 | 0.823 | 0.690 |

A (m^{2}) | 28.625 | 33.291 | 34.439 | 39.949 | 47.559 |

Fr | 0.66 | 0.67 | 0.63 | 0.48 | 0.38 |

Cost | 429.194 | 505.620 | 511.181 | 537.979 | 570.086 |

*Note:* Cost equivalent ; .

*Fr*) values. As a predecessor, it is clear that for better stability of the designed channel, the flow regime must be sub-critical, for which

*Fr*should be less than unity (

*Fr*< 1). From Figure 6, it can be seen that total area and the cost of channel construction increase with the reduction of the maximum

*Fr*number value, due to the reduction in velocity of the channel and subsequent increase in the cross-sectional area. It is clear from the table that the cost reduction with increasing the smaller

*Fr*number is more significant than those observed for larger

*Fr*values increasing. The total cost of construction obtained by Model IV for

*Fr*≤ =0.15 is approximately 1.93 times more than that obtained from Model I (Fr = 0.68). In Table 7, it is clear that for

*Fr*≤ =0.3 side slopes (

*m*) tend to zero , for which the trapezoidal cross section would transform to a rectangular cross section. Comparison between Model IV for

*Fr*= 0.9 (Table 6) and Model I with

*Fr*= 0.68 (no additional restriction) shows that total cost of construction decreases by approximately 21%.

Parameter | When Fr is: | ||||||||
---|---|---|---|---|---|---|---|---|---|

Fr ≤ 0.15 | Fr ≤ 0.2 | Fr ≤ 0.3 | Fr ≤ 0.4 | Fr ≤ 0.5 | Fr ≤ 0.6 | Fr ≤ 0.7 | Fr ≤ 0.8 | Fr ≤ 0.9 | |

b (m) | 2.368 | 2.573 | 3.554 | 3.175 | 4.532 | 4.554 | 4.404 | 4.927 | 4.675 |

y (m) | 20.070 | 15.678 | 9.648 | 7.743 | 5.430 | 4.496 | 4.078 | 3.717 | 3.553 |

m | 1.9 × 10^{−5} | 4.44 × 10^{−5} | 9.09 × 10^{−5} | 0.121 | 0.172 | 0.408 | 0.471 | 0.309 | 0.313 |

V (m/s) | 2.102 | 2.477 | 2.914 | 3.138 | 3.368 | 3.479 | 3.873 | 4.425 | 4.860 |

A (m^{2}) | 47.503 | 40.314 | 34.263 | 31.823 | 29.650 | 28.707 | 25.784 | 22.568 | 20.546 |

P (m) | 42.48 | 33.909 | 22.837 | 18.765 | 15.543 | 14.260 | 13.416 | 12.702 | 12.115 |

Cost | 1224.326 | 976.2052 | 669.768 | 558.526 | 469.243 | 435.686 | 406.570 | 381.325 | 362.526 |

Parameter | When Fr is: | ||||||||
---|---|---|---|---|---|---|---|---|---|

Fr ≤ 0.15 | Fr ≤ 0.2 | Fr ≤ 0.3 | Fr ≤ 0.4 | Fr ≤ 0.5 | Fr ≤ 0.6 | Fr ≤ 0.7 | Fr ≤ 0.8 | Fr ≤ 0.9 | |

b (m) | 2.368 | 2.573 | 3.554 | 3.175 | 4.532 | 4.554 | 4.404 | 4.927 | 4.675 |

y (m) | 20.070 | 15.678 | 9.648 | 7.743 | 5.430 | 4.496 | 4.078 | 3.717 | 3.553 |

m | 1.9 × 10^{−5} | 4.44 × 10^{−5} | 9.09 × 10^{−5} | 0.121 | 0.172 | 0.408 | 0.471 | 0.309 | 0.313 |

V (m/s) | 2.102 | 2.477 | 2.914 | 3.138 | 3.368 | 3.479 | 3.873 | 4.425 | 4.860 |

A (m^{2}) | 47.503 | 40.314 | 34.263 | 31.823 | 29.650 | 28.707 | 25.784 | 22.568 | 20.546 |

P (m) | 42.48 | 33.909 | 22.837 | 18.765 | 15.543 | 14.260 | 13.416 | 12.702 | 12.115 |

Cost | 1224.326 | 976.2052 | 669.768 | 558.526 | 469.243 | 435.686 | 406.570 | 381.325 | 362.526 |

*Note:* Cost equivalent ; .

*b*versus

_{*}*y*, with different Froude numbers around the critical flow regime (

_{n*}*m*= 0.5). Results indicate that for near critical state condition, as

*b*increases

_{*}*y*decreases.

_{n*}Table 8 sums up the optimization results for Model V, where *y _{max}* is considered a constraint in the event that unfavorable strata depth of channel should not go beyond a certain limit, because the excavation may not be economical or due to some other problem, such as the presence of shallow ground water table. This may require restriction on the maximum permissible flow depth. Therefore, the bottom width of the channel should be significantly increased to provide the necessary cross-sectional area for conveying the required flow discharge, which leads to increasing section area as well as the total cost and evaporation loss.

Parameter | When depth is: | ||||
---|---|---|---|---|---|

y ≤ 1.5 | y ≤ 2 | y ≤ 2.5 | y ≤ 3 | y ≤ 3.5 | |

b (m) | 18.60 | 13.22 | 9.77 | 7.55 | 5.91 |

y (m) | 1.49 | 1.98 | 2.49 | 2.98 | 3.49 |

m | 0.55 | 0.54 | 0.53 | 0.5 | 0.53 |

A (m^{2}) | 29.05 | 28.41 | 27.72 | 25.27 | 27.25 |

Cost of evaporation | 40.29 | 30.55 | 24.72 | 21.35 | 19.21 |

Cost | 597.87 | 417.59 | 449.01 | 425.91 | 417.59 |

Parameter | When depth is: | ||||
---|---|---|---|---|---|

y ≤ 1.5 | y ≤ 2 | y ≤ 2.5 | y ≤ 3 | y ≤ 3.5 | |

b (m) | 18.60 | 13.22 | 9.77 | 7.55 | 5.91 |

y (m) | 1.49 | 1.98 | 2.49 | 2.98 | 3.49 |

m | 0.55 | 0.54 | 0.53 | 0.5 | 0.53 |

A (m^{2}) | 29.05 | 28.41 | 27.72 | 25.27 | 27.25 |

Cost of evaporation | 40.29 | 30.55 | 24.72 | 21.35 | 19.21 |

Cost | 597.87 | 417.59 | 449.01 | 425.91 | 417.59 |

*Note:* Cost equivalent ; , .

According to Table 9, Model VI, which eliminates seepage from the objective function, caused a decrease in total cost (47% in the problem presented in this paper). It clearly shows the significant effect of seepage on the optimal parameter of a channel.

Parameter | Model VI |
---|---|

b (m) | 4.770 |

y (m) | 4.070 |

m | 0.505 |

A (m^{2}) | 27.779 |

Cost | 218.68 |

Parameter | Model VI |
---|---|

b (m) | 4.770 |

y (m) | 4.070 |

m | 0.505 |

A (m^{2}) | 27.779 |

Cost | 218.68 |

The best state and minimum cost of each model are listed in Table 10. It can be observed that Models I, III, and V produce bottom widths of 5.159, 5.399 and 5.917 m, while Models II, IV, and VI produce 2.575, 4.927, and 4.77 m, respectively. The depths of flow are the same, but the side slopes given by Models II and IV have smaller magnitudes, in comparison with the other models. The total cross-sectional area obtained by Models II and IV (13.374 m^{2} and 22.568 m^{2}, respectively) are approximately 50% and 17%, less than I and V models.

Parameters | I (No restriction) | II (T ≤ 0.45) _{*} | III (V ≤ 3.5) | IV (Fr ≤ 0.8) | V (y ≤ 3.5) | VI |
---|---|---|---|---|---|---|

b (m) | 5.159 | 2.575 | 5.399 | 4.927 | 5.917 | 4.770 |

y (m) | 3.784 | 3.625 | 3.699 | 3.717 | 3.497 | 4.070 |

m | 0.540 | 0.300 | 0.632 | 0.309 | 0.535 | 0.505 |

A (m^{2}) | 27.253 | 13.374 | 28.625 | 22.568 | 27.250 | 27.779 |

P (m) | 9.459 | 7.154 | 9.77 | 12.702 | 13.853 | 13.889 |

T (m) | 4.086 | 4.77 | 4.675 | 7.22 | 9.66 | 8.84 |

V (m/s) | 3.66 | 7.466 | 3.5 | 4.425 | 3.665 | 3.579 |

Cost | 417.26 | 298.821 | 429.194 | 381.325 | 417.594 | 218.68 |

Parameters | I (No restriction) | II (T ≤ 0.45) _{*} | III (V ≤ 3.5) | IV (Fr ≤ 0.8) | V (y ≤ 3.5) | VI |
---|---|---|---|---|---|---|

b (m) | 5.159 | 2.575 | 5.399 | 4.927 | 5.917 | 4.770 |

y (m) | 3.784 | 3.625 | 3.699 | 3.717 | 3.497 | 4.070 |

m | 0.540 | 0.300 | 0.632 | 0.309 | 0.535 | 0.505 |

A (m^{2}) | 27.253 | 13.374 | 28.625 | 22.568 | 27.250 | 27.779 |

P (m) | 9.459 | 7.154 | 9.77 | 12.702 | 13.853 | 13.889 |

T (m) | 4.086 | 4.77 | 4.675 | 7.22 | 9.66 | 8.84 |

V (m/s) | 3.66 | 7.466 | 3.5 | 4.425 | 3.665 | 3.579 |

Cost | 417.26 | 298.821 | 429.194 | 381.325 | 417.594 | 218.68 |

*Note:* Cost equivalent .

## APPLICATIONS

^{3}/s) compared to the discharge of 80 (m

^{3}/s), the cost has been increased by approximately 79%. The depth and width of the channel for the discharge of 200 (m

^{3}/s) are equal to 6.92 m and 4.98 m and for the discharge of 100 (m

^{3}/s) are equal to 5.159 m and 3.784 m, which represents an increment in channel dimensions. However, the amounts of side slopes are close to each other and equal to about 0.5. For

*Q*= 250 (m

^{3}/s), the Froude number is 0.27, which has been decreased by 60% compared to the discharge of 100 (m

^{3}/s). In the next model, different longitudinal bed slopes ranging from 0 to 0.0016 are examined for discharges of 100 m

^{3}/s and 250 (m

^{3}/s) and the optimum size obtained. It is also seen that there is a reduction in optimal cross section area and cost of channel construction and an increased Froude number and velocity by increasing longitudinal bed slope (Tables 11 and 12). For example, for

*S*= 0.0013, depth, bottom width, and slope are 4.89 m, 3.60 m and 0.53, respectively. For the discharge of 100 (m

_{0}^{3}/s) and

*S*= 0.0016 compared to

_{0}*S*= 0.0001, it can be seen that there is a significant decline in the cost by 45%. Comparison between the discharge of 100 (m

_{0}^{3}/s) with bed slope of 0.001 (Table 2) and discharge of 250 (m

^{3}/s) with bed slope of 0.0016 shows that the cost and area cross section are raised, respectively, by approximately 36% and 41%, however the Froude number has been decreased by 48%. Here, the final results are shown in Table 13. It can be concluded from Table 13 that cost change for other values of discharge with bed slopes might be the same as the models (II to V) with discharge of 100 m

^{3}/s and

*S*= 0.001.

_{0}S_{0} | Cost ($/m) | m | Fr | V (m/s) | A (m^{2}) | b (m) | y (m) | T (m) |
---|---|---|---|---|---|---|---|---|

0.0001 | 702.90 | 0.55 | 0.22 | 1.48 | 67.55 | 8.27 | 5.86 | 14.77 |

0.0003 | 551.94 | 0.54 | 0.37 | 2.25 | 44.33 | 6.49 | 4.85 | 11.78 |

0.0005 | 492.10 | 0.53 | 0.47 | 2.76 | 36.13 | 5.92 | 4.36 | 10.63 |

0.0007 | 456.81 | 0.53 | 0.56 | 3.16 | 31.58 | 5.55 | 4.07 | 9.93 |

0.0009 | 432.16 | 0.53 | 0.64 | 3.50 | 28.57 | 5.29 | 3.87 | 9.45 |

0.0011 | 413.45 | 0.53 | 0.71 | 3.79 | 26.35 | 5.06 | 3.72 | 9.08 |

0.0013 | 398.67 | 0.53 | 0.77 | 4.05 | 24.65 | 4.89 | 3.60 | 8.78 |

0.0014 | 392.15 | 0.53 | 0.80 | 4.17 | 23.93 | 4.82 | 3.55 | 8.65 |

0.0016 | 381.06 | 0.54 | 0.85 | 4.40 | 22.69 | 4.69 | 3.45 | 8.43 |

S_{0} | Cost ($/m) | m | Fr | V (m/s) | A (m^{2}) | b (m) | y (m) | T (m) |
---|---|---|---|---|---|---|---|---|

0.0001 | 702.90 | 0.55 | 0.22 | 1.48 | 67.55 | 8.27 | 5.86 | 14.77 |

0.0003 | 551.94 | 0.54 | 0.37 | 2.25 | 44.33 | 6.49 | 4.85 | 11.78 |

0.0005 | 492.10 | 0.53 | 0.47 | 2.76 | 36.13 | 5.92 | 4.36 | 10.63 |

0.0007 | 456.81 | 0.53 | 0.56 | 3.16 | 31.58 | 5.55 | 4.07 | 9.93 |

0.0009 | 432.16 | 0.53 | 0.64 | 3.50 | 28.57 | 5.29 | 3.87 | 9.45 |

0.0011 | 413.45 | 0.53 | 0.71 | 3.79 | 26.35 | 5.06 | 3.72 | 9.08 |

0.0013 | 398.67 | 0.53 | 0.77 | 4.05 | 24.65 | 4.89 | 3.60 | 8.78 |

0.0014 | 392.15 | 0.53 | 0.80 | 4.17 | 23.93 | 4.82 | 3.55 | 8.65 |

0.0016 | 381.06 | 0.54 | 0.85 | 4.40 | 22.69 | 4.69 | 3.45 | 8.43 |

S_{0} | Cost ($/m) | m | Fr | V (m/s) | A (m^{2}) | b (m) | y (m) | T (m) |
---|---|---|---|---|---|---|---|---|

0.0001 | 1082.93 | 0.58 | 0.09 | 1.81 | 137.86 | 11.84 | 8.25 | 21.55 |

0.0003 | 832.71 | 0.60 | 0.15 | 2.78 | 89.71 | 8.71 | 6.95 | 17.08 |

0.0005 | 738.90 | 0.56 | 0.19 | 3.39 | 73.57 | 8.55 | 6.11 | 15.49 |

0.0007 | 683.05 | 0.56 | 0.23 | 3.89 | 64.25 | 8.06 | 5.70 | 14.45 |

0.0009 | 645.14 | 0.56 | 0.26 | 4.28 | 58.31 | 7.52 | 5.49 | 13.70 |

0.0011 | 618.72 | 0.53 | 0.29 | 4.62 | 54.05 | 7.52 | 5.24 | 13.09 |

0.0013 | 594.69 | 0.56 | 0.31 | 4.94 | 50.57 | 6.91 | 5.15 | 12.70 |

0.0014 | 585.17 | 0.54 | 0.32 | 5.08 | 49.14 | 6.99 | 5.03 | 12.52 |

0.0016 | 567.83 | 0.55 | 0.35 | 5.36 | 46.61 | 6.65 | 4.94 | 12.18 |

S_{0} | Cost ($/m) | m | Fr | V (m/s) | A (m^{2}) | b (m) | y (m) | T (m) |
---|---|---|---|---|---|---|---|---|

0.0001 | 1082.93 | 0.58 | 0.09 | 1.81 | 137.86 | 11.84 | 8.25 | 21.55 |

0.0003 | 832.71 | 0.60 | 0.15 | 2.78 | 89.71 | 8.71 | 6.95 | 17.08 |

0.0005 | 738.90 | 0.56 | 0.19 | 3.39 | 73.57 | 8.55 | 6.11 | 15.49 |

0.0007 | 683.05 | 0.56 | 0.23 | 3.89 | 64.25 | 8.06 | 5.70 | 14.45 |

0.0009 | 645.14 | 0.56 | 0.26 | 4.28 | 58.31 | 7.52 | 5.49 | 13.70 |

0.0011 | 618.72 | 0.53 | 0.29 | 4.62 | 54.05 | 7.52 | 5.24 | 13.09 |

0.0013 | 594.69 | 0.56 | 0.31 | 4.94 | 50.57 | 6.91 | 5.15 | 12.70 |

0.0014 | 585.17 | 0.54 | 0.32 | 5.08 | 49.14 | 6.99 | 5.03 | 12.52 |

0.0016 | 567.83 | 0.55 | 0.35 | 5.36 | 46.61 | 6.65 | 4.94 | 12.18 |

The variation of cost for proposed models (Q = 100 m, S = 0.001)_{0} | |||
---|---|---|---|

Optimization models | Increases range | Cost decrease | Cost increase |

Top width (m) Model II | (4.78–7.17) | * | |

V (m/s) Model III | (2.1–3.5) | * | |

Froude number Model IV | (0.15–1) | * | |

Depth (m) Model V | (1.5–3.5) | * |

The variation of cost for proposed models (Q = 100 m, S = 0.001)_{0} | |||
---|---|---|---|

Optimization models | Increases range | Cost decrease | Cost increase |

Top width (m) Model II | (4.78–7.17) | * | |

V (m/s) Model III | (2.1–3.5) | * | |

Froude number Model IV | (0.15–1) | * | |

Depth (m) Model V | (1.5–3.5) | * |

## CONCLUSIONS

In the present research, the effect of parameter restrictions on optimal design of trapezoidal channels was investigated. The proposed non-linear optimization formulation consists of minimizing the cost of lining, the depth-dependent unit volume earthwork, water lost by seepage, and evaporative losses of the open channel that are constrained by uniform flow conditions and the resistance equation. For the aforementioned optimization issue, six different models were proposed. These non-linear models were evaluated as including no restriction, constrained normal depth, constrained velocity of flow, constrained Froude number, and constrained top width. The optimization formulations corresponding to all of the models are investigated in the present research and solved using GA. The results indicate that for Models III and V (with constrained velocity and normal depth)**,** the total cost of construction is high and for Models II and IV (with constrained top width and Froude number (*Fr* > 0.7)), the cost of construction in the open channels is smaller. Also, the results indicate that a model which disregards seepage in the objective function causes a considerable decrease in the cost. The obtained result suggests that seepage cost plays an important role on optimal design of open channels. Also, the different applications in the last two models for different discharge with constant bed slope (*S _{0}* = 0.001) and varying longitudinal bed slopes with constant discharge (

*Q*= 100 m

^{3}/s,

*Q*= 250 m

^{3}/s) was evaluated. It was concluded that with increasing discharge and bed slope, cost of channel construction increases and decreases, respectively.

It is to be noted that the present study is conducted for a specified set of input values and it can be easily extended to any other combination of input design parameters. Also the proposed models for design of open channels are simpler to implement and effective for practical applications, thus it can be used for reliable design of irrigation channels.