Weir velocity formulation for sharp-crested rectangular weirs

Discharge in open channels can be measured by sharp-crested rectangular weirs. Generally, measured head over the weir crest is substituted into an empirical formula derived from energy considerations to calculate the discharge. Assumptions made on the derivation are taken into account by defining a discharge coefficient that fits into the experimental data. In this study, a physical quantity, the average velocity over the weir section defined as ‘weir velocity’ is directly formulated as function of weir geometry and head over the weir. Weir velocity plotted against the weir head has a universal behavior for constant weir width to channel width ratio independent of the weir size. This unique behavior is described in terms of weir parameters to calculate the discharge without involving a discharge coefficient. Combining weir velocity data for variable weir widths provides a basis for direct formulation of discharge. The weir velocity exhibits simpler functional dependency on weir parameters in contrast to the discharge coefficient.     

Discharge formula for rectangular sharp-crested weirs

Sharp-crested rectangular weirs used for discharge measurement in channels and laboratories are experimentally investigated. Height and width of weir plate are the two parameters characterizing the head-discharge relationship. Laboratory experiments are conducted by measuring the discharge and the head over the weir for variable weir heights and widths. Applicability of various formulations for the discharge coefficient are investigated. Experiments indicate that discharge is independent of weir height, when the weir is operated within an appropriate discharge range. Average velocity over the weir plotted against the weir head displays universal characteristics such that it can be used in the expression of discharge over the weir, eliminating the need for a discharge coefficient. The head-discharge relationship for a rectangular weir has distinct features for the partially contracted weirs and for the fully contracted slit weirs.     

Discharge prediction for rectangular sharp-crested weirs by machine learning techniques

The stage-discharge relationship of a weir is essential for posteriori calculations of flow discharges. Conventionally, it is determined by regression methods, which is time-consuming and may subject to limited prediction accuracy. To provide a better estimate, the machine learning models, artificial neural network (ANN), support vector machine (SVM) and extreme learning machine (ELM), are assessed for the prediction of discharges of rectangular sharp-crested weirs. A large number of experimental data sets are adopted to develop and calibrate these models. Different input scenarios and data management strategies are employed to optimize the models, for which performance is evaluated in the light of statistical criteria. The results show that all three models are capable of predicting the discharge coefficient with high accuracy, but the SVM exhibits somewhat better performance. Its maximum and mean relative error are respectively 5.44 and 0.99%, and 99% of the predicted data show an error below 5%. The coefficient of determination and root mean square error are 0.95 and 0.01, respectively. The model sensitivity is examined, indicative of the dominant roles of weir Reynolds number and contraction ratio in discharge estimation. The existing empirical formulas are assessed and compared against the machine learning models. It is found that the relationship proposed by Vatankhah exhibits the highest accuracy. However, it is still less accurate than the machine learning approaches. The study is intended to provide reference for discharge determination of overflow structures including spillways.     

Unified discharge coefficient formula for free and submerged triangular labyrinth weirs

The flow through a triangular plan labyrinth weir is studied for both free and submerged flow conditions experimentally and theoretically. The free flow condition is studied using a new experimental data set collected in this study. For the submerged flow condition, the threshold between free and submerged flow regimes is studied experimentally. Then Buckingham analysis is employed to determine the submerged head-discharge formula of the triangular plan labyrinth weir. Finally, a step by step calibration method is proposed to find the unified discharge coefficient. The proposed discharge coefficient can be used for both free and submerged flow conditions continuously and within the transition zone.     

Radial Basis Neural Network and Particle Swarm Optimization-based equations for predicting the discharge capacity of triangular labyrinth weirs

Conventional weirs are utilized for controlling, measuring and adjusting the flow depth in hydraulic structures, such as those found in irrigation and drainage networks. Various weirs with modified shapes are utilized to increase the discharge capacity. The main goal of this study is to investigate the discharge coefficient (C_d) of triangular labyrinth weirs using soft computing methods. The performance of the Radial Basis Neural Network (RBNN) is compared with that of Multiple Nonlinear and Multiple Linear Particle Swarm Optimization (MNLPSO and MLPSO). Models developments are conducted using published experimental data from the literature. Comparing the RBNN, MLPSO and MNLPSO results obtained through these soft computing techniques with experimental data shows that all models perform well in predicting the discharge coefficient of a triangular labyrinth weir. Performance of the proposed approaches which demonstrated explicit equation given by MNLPSO model provided the discharge capacity with lower error (RMSE=0.0223) is compared with the MLPSO (RMSE=0.0346) and RBNN (RMSE=0.045) approaches.     

Experimental study of discharge formulas for rectangular sharp-crested weirs under free flow condition

Laboratory experiments were carried out to investigate the discharge characteristics of rectangular sharp-crested weirs under free flow condition. The performances of available discharge formulas have been evaluated by using the experimental data sets of present and previous studies. Error statistics of our experimental data indicate that the recent stage-discharge relationships show satisfactory performances. Discharge formula in terms of weir Reynolds number proposed by Vatankhah gives the highest accuracy among the existing slit weir equations, with $E_{\pm 4}=100.00\text{\%}$ (i.e. percent error less than or equal to $\pm 4$) and a mean absolute error ${\left|E\right|}_m=0.88\text{\%}$. The full-range discharge equation presented by Bijankhan and Mahdavi Mazdeh shows the highest accuracy among the relationships in terms of weir contraction ratio, with $E_{\pm 4}=100.00\text{\%}$, ${\left|E\right|}_m=0.91\text{\%}$ for slit weirs and, $E_{\pm 4}=94.64\text{\%}$, ${\left|E\right|}_m=1.60\text{\%}$ for partially contracted weirs, respectively. The weir velocity formulae suggested by Gharahjeh et al. exhibit the relatively better performance, with $E_{\pm 4}=98.41\text{\%}$, ${\left|E\right|}_m=1.34\text{\%}$ for slit weirs and, $E_{\pm 4}=91.07\text{\%}$, ${\left|E\right|}_m=1.91\text{\%}$ for contracted weirs, respectively. Statistical results of this study confirm the weir velocity approach presented by Aydin et al. and show that, the weir velocity is a predominant quantity for rectangular sharp-crested weirs, unique characteristics of the weir velocity curves make it more suitable for expressing the discharges. Moreover, it is important to point out that the performance of weir velocity formulae can be further improved.     

A novel approach for estimation of discharge coefficient in broad-crested weirs based on Harris Hawks Optimization algorithm

Accurate determination of the discharge coefficient play a very important role in estimating the flow discharge over the weirs. As a result, it is significant to estimate the discharge coefficient correctly. The aim of this study is simulation and estimation the discharge coefficients (C_d) over the broad-crested weirs with cross section rectangular and suppressed. Hence, numerical simulation of hydraulic characteristics of these weirs were performed by ANSYS FLUENT software and results were obtained. Then two intelligent models of ANN, GPR and hybrid both of models namely ANN-HHO, GPR-HHO were used to determine the discharge coefficients using the efficient parameters and the results of these models were compared. Assessment of the results were performed using the statistical metrics: coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MAE), scatter index (SI) and Kling-Gupta efficiency (KGE) and graphical diagrams including violin plot, percent relative error (RE%) plot and probability density function (PDF) plot of residuals. It was found that hybrid artificial neural network and gaussian process regression with Harris Hawks optimization (ANN-HHO and GPR-HHO) could improve ANN and GPR models performance in estimating the C_d in broad-crested weirs. Overall, results indicated that a combination of the HHO with the ANN (ANN-HHO) model performs better than GPR-HHO model for the estimation of the C_d.     

Stage-discharge relationship for triangular and curved-edge triangular weirs

A sharp-crested weir with power-law sides is a general form, which reduces to the wildly used rectangular, parabolic, and triangular weirs. This general form allows modeling the weirs with different shapes. Up to now, the hydraulic performance of the power-law weirs with the exponent n in the range of 1 ≤ n ≤ 2 has not been studied. In this study, the triangular (n = 1) and curved-edge triangular (n > 1) weirs are studied experimentally and theoretically. For this, weir and critical flow theories along with Buckingham's theorem of dimensional analysis are used to deduce the stage-discharge relationship of the triangular and curved-edge triangular weirs. A series of laboratory experiments (464 runs) were conducted to calibrate the deduced theoretical stage-discharge relationships under free outflow condition. The proposed general stage-discharge relationship using the weir theory has a mean absolute relative error of 2.51% for 1 ≤ n ≤ 2. For this relationship, the mean absolute relative errors for n = 1, n = 1.5 and n = 2 are respectively as 1.96%, 2.64% and 3.08% whilst for the proposed stage-discharge relationship using the critical flow theory they are as 1.96%, 2.75% and 3.82%, respectively. Thus, the proposed stage-discharge relationship using the weir theory may be preferable due to its accuracy and generality.     

Flow measurement using a triangular broad crested weir theory and experimental validation

The self-cleaning and semi-modular triangular broad-crested weir without crest height was firstly subjected to a rigorous theory. The main objective was to establish the discharge relationship as well as that of the resulting discharge coefficient. For this, both energy equation and momentum equation applied between two judiciously chosen sections were necessary and proved to be essential. Contrary to previous studies related to flow metering, the relationship governing the flow rate was established by taking into account the approach flow velocity. Secondarily, the device was subjected to an intense experimental program to confirm the validity of the proposed theoretical relationships. It was observed an excellent agreement between the experimental and theoretical values of the flow rate. It has been found that the experimental and theoretical flow rates are related by a linear relationship such that $Q_{Exp}=1.0057Q_{Th}$. The constant clearly indicates that the flow rate theoretical formula only needs a slight correction. The theoretical stage-discharge formula was then very accurate even no calibration parameter was employed. The theoretical development has shown that the discharge coefficient C_d only depends on the dimensionless parameter M₁ that reflects the effect of the contraction of the cross-section of the approach channel. The variation curve of C_d(M₁) showed that C_d increases in the range [0.233; 0.277] with the increase in M₁.     

Support vector regression for modified oblique side weirs discharge coefficient prediction

Accurate determination of discharge coefficient is one of the major concerns in the process of the designing of side weirs. Relation between the modified side weirs discharge coefficient to various geometric and hydraulic situations leads to a high flow complexity around the weirs. In this study, two types of support vector regression (SVR) methods were employed to model the discharge coefficient of a modified triangular side weir. Two types of SVR are obtained by using the radial basis and polynomial as the kernel functions. Six different non-dimensional input combinations with different input variables were used to find the most appropriate one. The results show that both SVR-rbf and SVR-poly methods perform better when the number of input variables is higher, and there is no compaction in the non-dimensional input variables. Comparison between the investigated models shows that the SVR-rbf by RMSE of 0.063 performs much better that SVR-poly by RMSE of 0.084.     

Experimental and numerical simulation of arced trapezoidal piano key weirs

Piano key weirs are high performance labyrinth weirs particularly at lower heads (water height over the weir). These weirs are considered a recent development in terms of application and research. This study was conducted to numerically and experimentally investigate the hydraulic performance of arced trapezoidal piano key weirs ATPKW under different hydraulic and geometric conditions. The main purpose was to investigate the effect of the planform arc angle on the weir's hydraulic performance. To this end, four LRPKW models as well as four ATPKW models were studied. Due to the complexities associated with flow over piano key weirs, numerical models were used to simulate the flow pattern over the weir. Comparison of the results obtained for the ATPKW and LRPKW models indicated that, at upper heads, the ATPKW models offered a higher hydraulic performance than the LRPKW models. On the other hand, LRPKW model indicated better performance at lower heads. In addition, it was found that reducing the planform arc angle from 90° to 45° caused an initial reduction in the discharge coefficient, after which it dramatically increased. Also, LRPKW models indicated a more effective length and a more appropriate performance.     

Discharging capacity of rectangular side weirs in straight open channels

A side weir is a hydraulic control structure used in irrigation and drainage systems and combined sewer systems. A comprehensive laboratory study, including 843 tests for the discharge coefficient of a sharp-crested rectangular side weir in a straight channel, was conducted in a large physical model under subcritical flow conditions. The discharge coefficient is a function of the upstream Froude number, the ratios of weir length to channel width, weir length to flow depth, and weir height to flow depth. An equation was developed considering all dimensional parameters for discharge coefficient of the sharp-crested rectangular side weir. The average error of the proposed equation is 4.54%. The present study data were compared with ten different discharge coefficient equations developed by several researchers. The study also presents water surface profile and surface velocity streamlines.     

Discharge forecasting using an Online Sequential Extreme Learning Machine (OS-ELM) model: A case study in Neckar River,Germany

Flood forecasting in natural rivers is a complicated procedure because of uncertainties involved in the behaviour of the flood wave movement. This leads to complex problems in hydrological modelling which have been widely solved by soft computing techniques. In real time flood forecasting, data generation is continuous and hence there is a need to update the developed mapping equation frequently which increases the computational burden. In short term flood forecasting where the accuracy of flood peak value and time to peak are critical, frequent model updating is unavoidable. In this paper, we studied a new technique: Online Sequential Extreme Learning Machine (OS-ELM) which is capable of updating the model equation based on new data entry without much increase in computational cost. The OS-ELM was explored for use in flood forecasting on the Neckar River, Germany. The reach was characterized by significant lateral flow that affected the flood wave formation. Hourly data from 1999–2000 at the upstream section of Rottweil were used to forecast flooding at the Oberndorf downstream site with a lead time of 1–6 h. Model performance was assessed by using three evaluation measures: the coefficient of determination (R²), the Nash-Sutcliffe efficiency coefficient (NS) and the root mean squared error (RMSE). The performance of the OS-ELM was comparable to those of other widely used Artificial Intelligence (AI) techniques like support vector machines (SVM), Artificial Neural Networks (ANN) and Genetic Programming (GP). The frequent updating of the model in OS-ELM gave a closer reproduction of flood events and peak values with minimum error compared to SVM, ANN and GP.     

Experimental study of effect of creating an additional cycle along the lateral crest of the labyrinth weirs

One of the practical and economical ways to enhance the discharge capacity is to use labyrinth weirs. The longer crest length in labyrinth weirs than in linear weirs has caused these weirs to have both a higher discharge coefficient and water discharge capacity than a linear weir. In the present study, the discharge coefficient of trapezoidal and triangular labyrinth weirs was investigated by creating an additional cycle along the lateral crest of the weir. By constructing 10 physical models of labyrinth weirs, tests were performed in the hydraulic and sediment laboratory of the Khuzestan Water and Power Authority (KWPA). Dimensional analysis by the Buckingham method revealed the discharge coefficient (Cd) as a function of variable parameters such as the total hydraulic head to weir height ratio (Ht/P) and weir shape factor (Sf). The results of experimental tests showed that at the hydraulic head ratio (Ht/P) of 0.1, the TP weir had a higher discharge coefficient of 3.5% than the TPTPO weir and 2.5% than the TPTRO weir. However, at a hydraulic head ratio of 0.12, the TR weir had a lower discharge coefficient of 4.6% than the TRTPO weir and 6.9% compared to the TRTRO weir. For the hydraulic head ratio of 0.14, the TRTPI weir was 5.8% and the TRTRI weir was 9.4% higher than the TR weir. Statistical analysis using SPSS indicated that TRTPO and TPTRO weirs had the highest correlation with the cubic model.     

改进的多变量极限学习机在滚动轴承故障预测中的应用

传统滚动轴承故障预测仅对单个故障特征频率做时间序列预测,而滚动轴承故障由多个故障频率共同表征.为了全面的表征整个频谱的结构,并且不破坏各个频率间的内部联系,提出奇异值分解和极限学习机相结合的多变量时间序列预测方法.首先通过全矢谱方法得到振动信号频谱,然后以整个频谱的各个频率作为输入变量,构建多变量时间序列.最后通过多变...     

Application of hybrid neural network in discharge coefficient prediction of triangular labyrinth weir

Discharge coefficient (C_d) is an important parameter of triangular labyrinth weir. It is of great significance to predict the discharge coefficient accurately. In this research, in order to more accurately predict the C_d, in view of the traditional BP neural network is easy to fall into the local minimum in the training process, genetic algorithm (GA) and particle swarm optimization (PSO) are employed to optimize the traditional BP neural network's initial weights and thresholds. Nonlinear regression analysis (NLR) is also added to compare with these intelligent methods and four discharge coefficient prediction models are built, namely the NLR, the BPNN, the GA-BPNN and the PSO-BPNN. After the completion of the model construction, in order to objectively evaluate the performance of these models, the prediction results of these models are compared with the experiment results, and the determination coefficient (R²), the mean absolute error (MAE) and the root mean square error (RMSE) are introduced as the performance indicators to quantify the model performance. The results show that the accuracy and stability of the NLR are much worse than that of the intelligent models. The prediction results of the GA-BPNN and the PSO-BPNN are quite accurate with a higher decision coefficient than the BPNN. Moreover, the MAEs and the RMSEs of the GA-BPNN and the PSO-BPNN were significantly reduced by 25 and 40% compared with BPNN, respectively, and the maximum prediction errors were 4.4% and 2.6%, severally. Meanwhile, the width of error uncertainty band of GA-BPNN and PSO-BPNN is narrower than BPNN. By comparing GA-BPNN and PSO-BPNN with the discharge coefficient prediction models of triangular labyrinth weir in previous literatures, it is found that the mean absolute percentage error (MAPE) values of GA-BPNN and PSO-BPNN are 1.504% and 1.225% respectively, which are lower than other existing models. At the same time, the other performance indexes are better than most existing models, indicating that the genetic algorithm and PSO algorithm are more effective than the traditional BP algorithm in adjusting BP neural network parameters, easier to find the global optimal value, and improve the prediction accuracy and applicability of the model.     

Performance evaluation of two different neural network and particle swarm optimization methods for prediction of discharge capacity of modified triangular side weirs

Technical design of side weirs needs high accuracy in predicting discharge coefficient. In this study, discharge coefficient prediction performance of multi-layer perceptron neural network (MLPNN) and radial basis neural network (RBNN) were compared with linear and nonlinear particle swarm optimization (PSO) based equations. Performance evaluation of the model was done by using root mean squared error (RMSE), coefficient of determination (R²), mean absolute error (MAE), average absolute deviation (δ) and mean absolute relative error (MARE). Comparison of the results showed that both neural networks and PSO based equations could determine discharge coefficient of modified triangular side weirs with high accuracy. The RBNN with RMSE of 0.037 in test data was found to be better than MLPNN with RMSE of 0.044 and multiple linear and nonlinear PSO based equations (ML-PSO and MNL-PSO) with RMSE of 0.043 and 0.041, respectively. However, due to their simplicity, PSO based equations can be sufficient for use in practical cases.     

Discharge coefficient of rectangular sharp-crested side weirs,Part I: Traditional weir equation

A comprehensive study was performed to examine the flow characteristics over rectangular sharp-crested side weirs based on the traditional weir equation. To obtain a generally convenient discharge coefficient relationship, series of experiments were conducted according to manipulation of different prevailing parameters. The flow regime was consistently subcritical for upstream Froude numbers ranging from 0.08 to 0.91. Furthermore, experimental data sets of the former investigators were also applied. In order to identify the most important parameters affecting the discharge coefficient of rectangular sharp-crested side weirs, a sensitivity analysis was carried out based upon an artificial neural network modeling. Results of the sensitivity analysis indicated the Froude number to be the most influential parameter on discharge coefficient. Accordingly, a power equation is derived for estimating the discharge coefficient, which is applicable for both sub- and supercritical flow conditions simultaneously. Moreover, considering all the influential parameters, a nonlinear correlation was obtained with the highest precision to determine the discharge coefficient of sharp-crested rectangular side weirs.     

Experimental investigation of flow characteristics over asymmetric Joukowsky hydrofoil weirs for free and submerged flow

Weirs are one of the most common hydraulic structures used to regulate the upstream approach flow depth and measure the flow discharge. The hydrofoil weirs are a type of short-crested weirs that are designed based on the airfoil theory. These weirs have some merits compared to other types, such as a higher discharge coefficient, more stability, better submergence limiting condition, and lower fluctuations of the pressure and the free-surface profile. In the present study, experimental models of hydrofoil weirs with different relative eccentricities, cambers, angles of attack, and upstream slope angles are applied to investigate their hydraulic characteristics under free and submerged flow conditions. The longitudinal profiles of static pressure over different hydrofoil weirs are compared to circular-crested and ogee weirs. The results indicate that the maximum bed negative pressure belongs to the circular-crested weir, and the lowest bed pressure over the hydrofoil and ogee weirs are approximately the same. Applying a hydrofoil weir with an appropriate curvature and angle of attack instead of a circular-crested weir not only increases the structural weir height as well as the upstream water depth but also results in the lowest values of bed negative pressure, thereby reduces the potential of cavitation over the weir body, being safer hydraulic structures. The results also show that the discharge coefficient of hydrofoil weirs is greater than that of the broad- and short-crested weirs for the upstream approach flow depth relative to the weir crest to weir length h₁/L > 0.12 and is greater than that of the ogee weirs for 0.35 < h₁/L < 0.45. Furthermore, the derived relationships for the discharge coefficient, threshold submergence, and the discharge reduction factor due to submergence accurately predict the hydraulic characteristics of hydrofoil weirs compared to the available developed empirical relationships for these weirs and can be used efficiently for design purposes.     

数控机床热误差鲁棒建模新方法及实时补偿

提出数控机床热鲁棒建模的综合极值和优化试验设计法两种新方法。它们不仅使用统计理论,还结合机床结构,工程判断和众多经验等,使热误差数学模型的鲁棒性更强,精确性更高,文章针对研究车削中心,还提出了一个结构紧凑,简单易用和成本低廉的热误差补偿系统。提出的建模方法和补偿系统在某生产厂家５０多台相同类型车削中心的热误差补偿应用中,仅使用一或二种热误差数学模型,几乎所有机床的热误差补偿都得到良好的效果,经补偿