عنوان مقاله [English]
نویسندگان [English]چکیده [English]
In this study, two optimization models for cross-sectional design of trapezoidal open channel have been investigated. In one of the optimization models, the lined free board was not considered while it was regarded in the other.. In fact in this study, for the first time in the field of the optimization of open channels, the lined free board was considered as a design variable in addition to the total free board. In the two optimization models mentioned above, the first objective function was determined as a cost. The discharge, Manning's roughness coefficient and the longitudinal bed slope of the channel were random variables, because in real situations, the actual flow may exceed the design flow due to probable fault in the control of flow at the off take point, and the uncertain lateral inflow. The actual Manning roughness values may exceed the assumed design values according to imperfections in fabrication. The physical bed slope which was achieved may differ from the design values because of fabrication faults. These variations can cause the occasional flooding of man-made open channels. To maintain provisions for these possible variations within the designed cross-sectional dimensions, a flooding probability constraint must be incorporated in the optimal design model. The Manning flow equation is also considered as a constraint. Overall in the two models of optimization, the first objective was to minimize the cost of excavation and the lining channel cost and the second objective was to minimize the probability of overflow from the cross section. These models were solved with Wolfram Mathematica software.
Due to the probability of the second objective function, the model is classified into random optimization models and this is a two-objective optimization problem. To obtain the answers of the two-objective optimization problem, the multi-objective constrained programming method was used, which converts the vector optimization into a numerical one. This conversion from vector to numerical formulas was accomplished with regard to the first objective was to minimize the total cost of the channel (as the single objective in this problem) and the second objective which was sought to minimize the overflow (as an additional constraint). In this study, the flooding probability constraint was developed by using the first order analysis that essentially uses the calculus based differentiation of the Manning uniform flow. The numerical and deterministic forms of the previous modeling were solved with Wolfram Mathematica software for a numerical example.
The results of solving models with the Wolfram Mathematica software for a numerical example showed that in both optimization models (with and without lined free board), for the probability of overflow, the total cost was greater and with increasing probability of overflow from 0.225 to 0.3, the total cost, the bottom width of the channel, the total free board and the lined free board were decreased; While,, the depth of flow in the channel and side slope was increased. But in general, the total cost for the construction of a channel in a model with a lined free board is lowered in compare with that in which the lined free board was not considered. In other words, considering the lined free board in the open channel, the optimal section has been created at a lower total cost. These results are presented in the form of tables and probability overflow–cost diagrams. According to the results of this study, the cost of channel construction had increase for both models with decreasing the possibility of overflow. Also, for low values of overflow, it is necessary to increase the total free board (and lined) and make the channel wider. Comparison of the free board model with the model that did not include the free board showed that considering the lined free board or somehow the unlined part reduced the total cost. Besides, in general, the best choice among all the optimized responses obtained with different overflow probabilities in each model is up to the design engineer, who can determine the budget, importance, and usability of the channel and many other factors. Consequently, It is suggested that land acquisition and water loss costs (evaporation, transpiration) included in the objective function to develop the current design in future research; it is also possible to compare the results with current design standards.