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A hybrid programming methodology was proposed to derive a simple empirical formulation for the estimation of the required pipe diameter in the sizing problems (Type 3) of pipe distribution systems. The model was derived based on multiple regression-based analysis using the Richardson's extrapolation approach and the Levenberg–Marquardt algorithm with double precision. The proposed formulation was developed using a total of 300,000 different data points within the framework of MATLAB and DataFit scientific software. The application of the model was explored for a wide range of five fundamental pipeline design variables [absolute roughness of the pipe wall (ε=0–9 mm), water temperature (T=5°C–100°C), pipe length (L=5–500 m), flow rate (Q=0.01–1 m3/s), and head loss (Δh=1–30 m)] and tested against a total of 10,000 additional computational scenarios and the available models reported in the literature. The uncertainty prediction of the proposed formula was quantified and compared with those of existing prediction models. For the new empirical equation, the mean prediction errors between the estimated and the theoretical diameter values (calculated from the numerical solution of the Colebrook–White equation) were significantly smaller than those of existing models. Moreover, the narrowest uncertainty bands, the lowest 95% confidence prediction error intervals, and the lowest amounts of expanded uncertainty (U95) were achieved for the proposed model. Other statistics (e.g., mean absolute relative error, absolute relative error, coefficient of variation of root mean squared error, determination coefficient) also corroborated that the proposed empirical model produced realistic estimations that were superior to those obtained from other well-known explicit models in the literature. The findings of this study concluded that the computational analysis yielded a simple mathematical structure to be easily and accurately used for educational and practical purposes. [ABSTRACT FROM AUTHOR]

• ANN outnumbered other standalone AI models (single models) applied to WWTPs. • Hybrid models were relatively more accurate than the standalone models. • Most of hybrid models were classified as CI-metaheuristic models. • FL was the most suitable model for the incomplete data sets. • Despite recent developments, industrial deployment is lacking. Increasing energy efficiency in wastewater treatment plants (WWTPs) is becoming more important. An emerging approach to addressing this issue is to exploit development in data science and modelling. Deployment of sensors to measure various parameters in WWTPs opens greater opportunities for exploiting the wealth of data. Artificial intelligence (AI) is emerging as a solution for automation and digitalization in the wastewater sector. This review aims to comprehensively investigate, summarize and analyze recent developments in AI methods applied to the modelling of WWTPs. The review shows that among the standalone models, Artificial Neural Networks (ANN) was the most popular model followed by, in descending order: Decision Trees (DT), Fuzzy Logic (FL), Genetic algorithm (GA) and Support Vector Machine (SVM). In the case of incomplete data, FL was the most frequently used method as it uses linguistic expert rules to find an approximation for the missing data. Regarding accuracy and precision, hybrid models demonstrated relatively better performance than the standalone ones. Among these models, the Machine Learning (ML)-metaheuristic, which integrates an AI model with a bioinspired optimization method, was the most preferred type as it was used in more than 45% of the hybrid models. Correlation coefficient (R), Correlation of Determination (R2) and Root Mean Square Error (RMSE) were the frequently used metrics for model performance evaluation. Finally, the review shows that despite recent developments, industrial deployment is still lacking. The industrial application requires close interaction of interested parties, among which research institutes, private sector and public sector play an inevitable role. The future research should focus on mitigating the barriers for more in-depth collaboration of interested parties and finding new paths for more cooperative and harmonized activity of them. [ABSTRACT FROM AUTHOR]

The large consumption of natural gas, one of the most important energy sources in the world, necessitates reliable, precise, and accurate calculation of gas flow rate and amount in order to use this resource in an efficient and sustainable way. The present computational study investigates the possibilities of several soft-computing strategies in estimating the mass flow rate of natural gas flow stream (kg/h) (output variable) based on four input variables of orifice plate diameter ratio, differential pressure of orifice plate (kPa), operating pressure of the natural gas (bar), and operating temperature of the natural gas (°C). A genotype/phenotype genetic algorithm (gene expression programming (GEP) technique), two decision tree-based methods (random forest (RF), random tree (RT) models), and two kernel-based approaches (Gaussian process regression (GPR) and support vector machines (SVM) methods) were applied for the first time to predict gas mass flow rate. Coefficient of correlation (CC), mean absolute error (MAE), root mean square error (RMSE), Scattering index (SI), Nash–Sutcliffe efficiency (NSE), and mean absolute relative error (MARE) were computed as the statistical performance evaluators to determine of the best-performing soft-computing approach. The performance assessment indices corroborated the superiority of the Pearson VII universal kernel function-based GPR model (GPR-PUKF) model (CC = 0.9997, MAE = 64.8091 kg/h, RMSE = 248.7584 kg/h, SI = 0.0237, and NSE = 0.9993 for the testing dataset) over other data-intelligent models in predicting the gas mass flow rate. In addition, statistical results revealed that the predictions of the RF method were better than those of the GEP- and RT-based models, but the GEP approach showed the lowest performance among all applied models. Although the CC values of all models were satisfactory (>0.993), the percentile deviation of GPR model (1.7325%) from the actual values showed competitive lower values, indicating its superior performance than other models (GEP = 15.1436%, RF = 6.5403%, RT = 9.5576%, and SVM = 3.2107%). This study highlighted the significance of employing advanced soft-computing approaches in determining the mass flow rate of natural gas, a vital source of energy, as well as its value to the gas sector. [Display omitted] • Soft-computing implementation for prediction of natural gas mass flow rate. • Benchmarking of genetic/decision tree/kernel-based data-intelligent models. • Lower deviations (1.73% and 0.36%) of GPR-PUKF over other methods. • Superiority of RF (6.54%) over GEP (15.14%) and RT (9.56%) in terms of errors. • Flexibility of soft computation in highly nonlinear real-world gas measurement. [ABSTRACT FROM AUTHOR]

Owing to its importance in extraction of natural gas from underground gas storage as well as its crucial role in determination of final gas mixture in the production facilities of gas/oil industry, the dry content of wet gas mixture needs to be calculated precisely. The present study explores the potential of different soft-computing techniques in estimation of the dry gas flow rate (kg/h) (output variable) of wet gas mixture based on two input variables of wet gas flow rate (kg/h) and absolute gas humidity (g/m3). Decision tree-based methods (M5P tree, random forest (RF), random tree (RT), and reduced error pruning tree (REPT) models), kernel function-based approaches (Gaussian process regression (GPR) and support vector machines (SVM)), and non-parametric regression-based technique (multivariate adaptive regression splines (MARS)) were implemented for the first time to estimate the dry gas flow rate, and their respective prediction performances were analyzed statistically. Coefficient of correlation (CC), Nash–Sutcliffe efficiency (NSE), root mean square error (RMSE), mean absolute error (MAE), Legates and McCabe's index (LMI), and Willmott's Index (WI) were used as the statistical indicators for validating the performance of each soft-computing model. While M5P model (MAE = 122.2382 kg/h, RMSE = 580.5626 kg/h, CC = 0.9875 for the testing data set) was better than other tree-based models (MAE = 363.2802–542.6119 kg/h, RMSE = 871.9363–1025.3444 kg/h, CC = 0.9587–0.9706 for the testing data set) and MARS model (MAE = 128.0083 kg/h, RMSE = 622.9515 kg/h, CC = 0.9852 for the testing data set), the statistical indicators approved the superiority of the radial basis kernel function-based GPR model (GPR-RBKF) model (MAE = 163.3266 kg/h, RMSE = 483.1359 kg/h, CC = 0.9915 for the testing data set) over other implemented models in predicting the dry gas flow rate. The findings highlighted the potential of soft-computing methodologies in precise estimation of dry gas flow rate in wet gas mixture, particularly, in situations where the measurement of such parameters with traditional deterministic models is practically not possible. [Display omitted] • Soft-computing methods for estimation of dry content of wet gas mixture. • Performance evaluation of M5P, RF, RT, REPT, GPR, SVM, and MARS. • Superiority of GPR-RBKF over other models in terms of accuracy. • Superiority of M5P over other tree decision tree models in terms of error. • Higher precision of Gaussian processing with kernel-based regression vector. [ABSTRACT FROM AUTHOR]

Due to its importance in flow measurement and instrumentation, as well as its frequent application in differential pressure flowmeters, orifice discharge coefficient (C d) needs to be estimated precisely. In this study, different soft computing models (including multiple linear regression (MLR), group method of data handling (GMDH), multivariate adaptive regression splines (MARS), M5P tree model, and random forest (RF)) were employed for the first time in estimation of the C d value, and their respective prediction performances were analyzed statistically. Coefficient of correlation (CC), mean absolute error (MAE), root mean square error (RMSE), scattering index (SI), and Nash–Sutcliffe model efficiency coefficient (NSE) were used as the statistical indicators for validating the performance of each soft computing model. The statistical indicators approved the superiority of the RF model over the other models, while the MARS model also showed a competitive prediction potential over M5P, GMDH, and MLR models. The findings of this computational study clearly demonstrated that the implemented soft computing strategy had the capability to be used in precise estimation of the C d of the orifice meter, specifically, in situations where the measurement of the parameters in deterministic equation is not practically feasible. [Display omitted] • Differential pressure flowmeters were simulated using soft computing methodology. • MLR, GMDH, MARS, M5P, and RF were used to estimate C d for the first time. • Effectiveness of models was tested by CC, MAE, RMSE, SI, and NSE metrics. • RF outperformed other models, and MARS showed competitive prediction accuracy. • RF-based soft computation was of potential to be used in precise estimation of C d. [ABSTRACT FROM AUTHOR]