Orifice Discharge Coefficient Research Articles

Equations and methodologies of inlet drainage system discharge coefficients: A review

Abstract Accurate determination of grate inlet discharge coefficients is crucial in reducing modeling uncertainties and mitigating urban flooding hazards. This review critically examines the methods, equations, and recommendations for determining the weir/orifice discharge coefficients, based on the inlet parameters and flow conditions. Reviewing previous studies for inlets showed that the discharge coefficient of rectangular inlets under subcritical flow ranges from 0.53 to 0.6 for weirs and from 0.4 to 0.46 for orifices, while in grated circular inlets, it falls between 0.115 and 0.372 for weirs and between 0.349 and 2.038 for orifices. For circular non-grated inlets under subcritical flow, the weir and orifice coefficients are in the range of 0.493–0.587 and 0.159–0.174, respectively. However, the orifice discharge coefficients of grated and non-grated inlets with unknown Froude number range between 0.14–0.39 and 0.677–0.82, respectively. For supercritical flow, the weir and orifice discharge coefficients of grated and non-grated rectangular inlets are 0.03–0.47 and 1.67–2.68, respectively. Previous studies showed that it is recommended to correlate the discharge coefficients with the approaching flow and Froude number under subcritical and supercritical flows, respectively. Yet, additional studies are recommended for a better understanding of the limits and parameters governing the flow transitional stage between weir and orifice and between subcritical and supercritical conditions. Moreover, further research is required to determine the weir and orifice discharge coefficients of circular inlets under supercritical flow as well as the orifice discharge coefficient range of rectangular non-grated inlets under subcritical flow. Finally, it is recommended to increase the road surface roughness to reduce Froude number, and thereby, increase discharge coefficients of street inlets. The aim of this review is to help inlet designers and authorities promote sustainable cities with resilient urban drainage systems and reduce the environmental, economic, health, and social impacts of urban drainage failure.

Analysis of upper and lower nappe profiles of large orifice for the design of bottom and roof profiles of high head orifice spillway

Abstract Large orifices are constructed for dams to release water and sediments from reservoirs. Such structures are called submerged spillways. Numerous studies have investigated discharge coefficient, velocity coefficient, and head loss coefficient of large orifices; however, the literature lacks data on the upper and lower nappes of the jets from these orifices. In the present experimental study, the upper and lower nappes are investigated up to 80 m head at different gate openings. The observed minor deviation between the lower nappe profile and trajectory profile equation suggests sensitivity to different factors. The significant role of the coefficient of velocity, averaging at 0.926, highlights its impact on minor deviation. Subsequently, the impact of the solid bottom profile on the discharge coefficient and upper nappe profile are also examined. The results show improvement in discharge coefficient of a sharp-edged large orifice, which increased from 0.69 to 0.74. The results also indicate that the upper nappe profiles and United States Bureau of Reclamation (USBR) profiles are similar. The improvement in the upper nappe profile indicates the significant role of the solid bottom profile, which consequently was found to be helpful in defining the roof profile of an orifice spillway. .

Discharge coefficients to be used in inlet hydraulics

Urban pluvial floods can be produced or exacerbated by an insufficient density of inlets or by their poor hydraulic efficiency. Therefore, proper consideration of the hydraulic performance of inlets is essential to guarantee the correct functioning of urban drainage systems during heavy storm events. Recent advances in computational analysis in the field of hydrodynamics modelling allow the use of the well-known concept of dual drainage for design and planning purposes by simulating flow transfers between the surface layer (streets) and underground layers (sewers) through a proper hydraulic characterisation of inlet efficiency. Powerful commercial software packages allow simulations of flow transfer and use different approaches and formulas. Many of these approaches include the possibility of treating a sewer inlet as an orifice. In this context, this paper presents a methodology to obtain orifice discharge coefficients for three inlets previously tested at the hydraulics laboratory of the Technical University of Catalonia. Discharge coefficients of 0.18–0.58 were obtained for Froude numbers of 1.12–4.40, quite far from the usual recommended values. The proposed procedure can also be applied to non-tested grates.

A hybrid ensemble machine learning model for discharge coefficient prediction of side orifices with different shapes

Side orifices are widely applied for flow control and regulation in channel systems. Accurate estimation of the discharge coefficient of the side orifice is significant for water management. The main objective of current research is to accurately predict the discharge coefficients of circular and rectangular side orifices. Considering that traditional empirical regressions are hard to estimate the discharge coefficient precisely due to the complex nonlinear relationship between the discharge coefficient and relevant parameters, a new hybrid boosting ensemble machine learning model, BO-XGBoost, is developed, which combines the advantages of the boosting ensemble model (XGBoost) and Bayesian Optimization. To further evaluate the proposed hybrid model, it is also compared with other tree-based machine learning models, including standalone XGBoost, Random Forest (RF) and Decision Tree (DT). Literature experimental data of the flow and geometric parameters relevant to the discharge coefficients of circular and rectangular side orifices are collected and applied to develop the models. Four dimensionless parameters of the relative channel width (B/L), the relative bottom height (W/L), the relative upstream depth (Y/L) and the upstream Froude number (Fr) are taken into consideration for the prediction of discharge coefficient (Cd). Furthermore, four different input combinations are designed and then compared to determine the best one on the basis of RMSE. By using the optimal input combination, our results demonstrate that BO-XGBoost provides the best comprehensive performance among all the involved machine learning models in the discharge coefficient prediction for both types of side orifices. Besides, the uncertainty analysis also reveals that BO-XGBoost shows the narrowest uncertainty bandwidth and gives the highest prediction reliability.

Predicting Discharge Coefficient of Triangular Side Orifice Using LSSVM Optimized by Gravity Search Algorithm

Side orifices are commonly installed in the side of a main channel to spill or divert some of the flow from the source channel to lateral channels. The aim of the present study is the accurate estimation of the discharge coefficient for flow through triangular (Δ-shaped) side orifices by applying three data-driven models including support vector machine (SVM), least squares support vector machine (LSSVM) and least squares support vector machine improved by gravity search algorithm (LSSVM-GSA). The discharge coefficient was estimated by utilizing five dimensionless variables resulted from experimental data (570 runs). Five different scenarios were applied based on the input variables. The models were evaluated through several statistical indices and graphical charts. The results showed that all of the models could successfully estimate the discharge coefficient of Δ-shaped side orifices with adequate accuracy. However, the LSSVM-GSA produced the best performance for the input combination of all variables with the highest coefficients of determination (R2) and Nash–Sutcliffe efficiency (NSE), equal to 0.965 and 0.993, and the least root mean square error (RMSE) and mean absolute error (MAE), equal to 0.0099 and 0.0077, respectively. The LSSVM-GSA improved the RMSE of the SVM and LSSVM by 26% and 20% in estimating the discharge coefficient. Furthermore, the ratio of orifice crest height to orifice height (W/H) was identified as having the highest influence on the discharge coefficient of triangular side orifices among the various input variables.

Study on Discharge Coefficients of Drain Orifices in Closed Cavities

The numerical method was used to simulate the discharge flow, and discharge coefficients were obtained. The main factors affecting the discharge coefficient of orifices were analyzed by dimensional analysis, and the empirical formulas for calculating discharge coefficient were fitted. The results show that when the water head height is less than 200mm, the discharge coefficient decreases with the increase of head height. When the water head height is more than 200mm, the discharge coefficient is stable around 0.61. The discharge coefficient with different thickness diameter ratio shows two different forms: the orifices with small thickness diameter ratio show thin orifice flow characteristics, and the discharge coefficient is about 0.6; the orifices with large thickness diameter ratio show thick orifice flow characteristics, and the discharge coefficient is about 0.8.

Prediction and parameter quantitative analysis of side orifice discharge coefficient based on machine learning

AbstractIn this paper, the sparrow search algorithm is used to predict the discharge coefficient (Cd) of the triangular side orifice for the first time. Dimensionless parameters influencing the size of Cd of side orifices are obtained as input values and discharge coefficient as output values of the model. The results show that the determination coefficient R2 is 0.973, the root means square error RMSE is 0.0122, and the average absolute percentage error is 0.010% in the testing phase. The model has high forecast accuracy, strong generalization ability and higher accuracy than other models and traditional empirical formulas. Quantitative analysis by Sobol's method shows that the ratio W/H of top orifice height to side orifice height, Fr of upstream Froude number, and ratio B/L of channel width to a bottom edge length of side orifice are the main factors influencing the discharge capacity of triangular side orifice. The first-order sensitivity coefficient and global sensitivity coefficient are 0.23, 0.11, 0.17 and 0.41, 0.39, 0.35 respectively.

Simulation of Discharge Coefficient of Triangular Lateral Orifices Using an Evolutionary Design of Generalized Structure Group Method of Data Handling

Simulation of Discharge Coefficient of Triangular Lateral Orifices Using an Evolutionary Design of Generalized Structure Group Method of Data Handling

Evaluation of discharge coefficient of triangular side orifices by using regularized extreme learning machine

The present paper attempts to reproduce the discharge coefficient (DC) of triangular side orifices by a new training approach entitled “Regularized Extreme Learning Machine (RELM).” To this end, all parameters influencing the DC of triangular side orifices are initially detected, and then six models are extended by them. For training the RELMs, about 70% of the laboratory measurements are implemented and the remaining (i.e., 30%) are utilized for testing them. In the next steps, the optimal hidden layer neurons number, the best activation function and the most accurate regularization parameter are chosen for the RELM model. As a result of a sensitivity analysis, we figure out that the most important RELM model simulates coefficient values with high exactness. The best RELM model estimates coefficients of discharge using all input factors. The efficiency of the best RELM model is compared with ELM, and it is demonstrated that the former has a lower error and better correlation with the experimental measurements. The error and uncertainty examinations are executed for the RELM and ELM models to indicate that RELM is noticeably stronger. At the final stage, an equation is proposed for computing this coefficient for triangular side orifices and a partial derivative sensitivity analysis is also carried out on it.

An Investigation on the Discharge Coefficient of Compound Orifices in Rotating Disks

Abstract In the modern multishaft gas turbine engines, orifice is an important throttling element, and the discharge coefficient of rotating orifices may vary considerably depending on the operating conditions, the geometry, and surrounding environment. The influences of the rotating number and the pressure ratio on the rotating orifices' flow characteristics are investigated in this study. Besides, the effects of confined space, wall inclination angle (α), and the angle between the axis of orifice and the disk wall normal (β) are also analyzed statistically. It is found that the rotating number has a significant effect on the discharge coefficient. As the rotating number increases from 0 to 0.6, the discharge coefficient reduces by about 47.88%. When rotating number is 0.74 and pressure ratio is 1.10, the discharge coefficient can be improved by 16.88% with α changes from 90 deg to 180 deg. The parameter, β, affects discharge coefficient slightly in rotating condition. However, the maximum discharge coefficient is achieved with β = 0 deg in the static condition. The results also show that a confined space weakens the effect of rotation and changes the air flow direction in the inlet chamber, which also has a positive impact on the discharge coefficient. In the current research, it is found that there is a significant difference between the traditional empirical formulas used in the literature and the fitting result. By modifying the incidence angle and taking account of the influence of the angle of inclination, the maximum error was reduced from 56.79% to 3.16%.

Analysis of influence of pipeline internal surface roughness on flow rate measured by means of standard pressure differential devices

The paper presents the analysis of scientific and technical sources on the influence of constructive features of pressure differential flowmeters on flow rate measurement error. According to the research results, the significant surface roughness of pipe sections downstream of the orifice plate does not significantly affect the flow measurement result. The influence of the pipe roughness upstream of the orifice plate depends on geometric characteristics of roughness, the pipe diameter, the relative area of the orifice plate throat, and the Reynolds number. The change in the uncertainty of the orifice discharge coefficient under conditions of inhomogeneous measuring pipe roughness in the pipe section with a length of 10D upstream of the orifice plate is analyzed. The authors have determined that even a pipe section length of 1.5D with high roughness might be enough to change the flowmeter readings by more than 1% at a large orifice diameter ratio.

Data-based estimation and simulation of compressible pulsating flow with reverse-flow through an orifice

Highly compressible pulsating flows are often encountered in devices where knowledge of the flow rate is required but elimination of pulsations is not an option. The current work is a continuation of a previous investigation that characterized the orifice discharge coefficient Cd as a function of dimensionless groups based on pulsation characteristics. The experimental apparatus has been rebuilt in the current work to mitigate temperature and vibration problems, allowing pressure and ΔP measurements to be made very close to the test section with 159-mm of nylon tubing. Data was acquired for 77 operating conditions spanning a range of pulsation frequencies, mass flow rates and system pressures. They confirm previously reported low Cd's in 0.20 range (calculated from time-averaged pressures) at some high-pressure low-flow operating conditions. Computational Fluid Dynamics (CFD) simulations of 12 of these data points suggest that the low Cd's result from reverse flow. Flow direction changed several times during each pulsation cycle closely tracking the orifice ΔP. A ‘core-and-sheath’ phenomena was observed for reverse-flow operating conditions: a positive core flow surrounded by a sheath of negative flow transitioned to a negative core and positive sheath several times during each pulsation cycle. The simulations also suggested that velocity profiles at the orifice stay stable and similar to steady-state profiles except for periods of rapid transitions. Based on these results a data-based quasi-steady method of estimating pulsating flow has been proposed. A pair of forward and reverse flow Cd's chosen by the data are used to predict instantaneous forward and reverse flows using the steady-state orifice discharge equation for compressible flow. The instantaneous values are then summed up over the pulsation cycle to estimate average mass flow rate. Average prediction errors were within 6%. A previously proposed method where regression was used to model Cd as a function of dimensionless groupings was shown to produce similar results. Both methods are designed to extract information from experimental data in order to overcome theoretical limitations and experimental error. The data is available upon request for further understanding of the flow physics.

Toward the accurate estimation of elliptical side orifice discharge coefficient applying two rigorous kernel-based data-intelligence paradigms

In the present study, two kernel-based data-intelligence paradigms, namely, Gaussian Process Regression (GPR) and Kernel Extreme Learning Machine (KELM) along with Generalized Regression Neural Network (GRNN) and Response Surface Methodology (RSM), as the validated schemes, employed to precisely estimate the elliptical side orifice discharge coefficient in rectangular channels. A total of 588 laboratory data in various geometric and hydraulic conditions were used to develop the models. The discharge coefficient was considered as a function of five dimensionless hydraulically and geometrical variables. The results showed that the machine learning models used in this study had shown good performance compared to the regression-based relationships. Comparison between machine learning models showed that GPR (RMSE = 0.0081, R = 0.958, MAPE = 1.3242) and KELM (RMSE = 0.0082, R = 0.9564, MAPE = 1.3499) models provide higher accuracy. Base on the RSM model, a new practical equation was developed to predict the discharge coefficient. Also, the sensitivity analysis of the input parameters showed that the main channel width to orifice height ratio (B/b) has the most significant effect on determining the discharge coefficient. The leveraged approach was applied to identify outlier data and applicability domain.

Evaluation of shape factor impact on discharge coefficient of side orifices using boost simulation model with extreme learning machine data-driven

ABSTRACT In this paper, for the first time, the impact of the shape factor on the discharge coefficient of side orifices is evaluated using the novel Extreme Learning Machine (ELM) model. In addition, the Monte Carlo simulations (MCs) are applied to assess the accuracy of the modelling. Furthermore, the validation is conducted by means of the k-fold cross-validation approach (with k = 5). In other words, the most optimized number of hidden neurons is chosen to be equal to 30 and the results of all activation functions of the extreme learning machine are examined and the sigmoid activation function is selected for simulating the discharge coefficient. Subsequently, two modelling combinati0ons are introduced using the input parameters and five different extreme learning machine models are also developed. The analysis of the modelling results exhibits that the model with the shape factor is more accurate. The superior model is a function of all input parameters reasonably estimating the discharge coefficient. For example, the values of R and MAPE for this model are estimated to be 0.990 and 0.223, respectively. The results of the superior model are also compared with the empirical equations and it was shown that this model has higher accuracy.

Approximation of the discharge coefficient of differential pressure flowmeters using different soft computing strategies

Due to its importance in flow measurement and instrumentation, as well as its frequent application in differential pressure flowmeters, orifice discharge coefficient (Cd) 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 Cd 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 Cd of the orifice meter, specifically, in situations where the measurement of the parameters in deterministic equation is not practically feasible.

Improvement of a multiphase flow model for wellhead chokes under critical and subcritical conditions using field data

The characterization of the multiphase flow through valves and orifices is a problem yet to be solved in engineering design, and there is a need for a prediction model able to simulate the complexity of this kind of flow in relation to fluid thermodynamic behaviour, and applicable to different incoming stream conditions and compositions. The present paper describes the development of a global model for the calculation of the discharge coefficient of orifices and choke valves operating under two- and three-phase flow as well as critical and subcritical conditions. The model generalizes the hydrovalve model developed by Selmer-Olsen et al. (in: Wilson (ed) Proceedings of 7th international conference on Multiphase Production, BHR Group, pp 441–446, 1995) and the Henry–Fauske (J Heat Transfer 93: 179–187, 1971. https://doi.org/10.1115/1.3449782) non-equilibrium model on the basis of an updated definition of the discharge coefficient. The model has been adapted to real choke valve geometries, by fitting the discharge coefficient and model parameters using field data from three production wells. The model developed is a global quartic function with different constants for the different valve geometries. The new discharge coefficient allows to simulate field data with high accuracy.

Influence of compliance, and effective orifice discharge coefficient on performance of a hydroacoustic projector

A hydroacoustic projector is a compact device which converts pressure energy of fluid into low frequency sounds. It uses a direction control valve with time varying opening to periodically reverse the direction of flow. Its performance is strongly influenced by oil’s effective bulk modulus, and coefficient of discharge (Cd) of orifices in the valve. Such dependence is fundamentally different in nature from that in systems with unidirectional oil flow and constant valve opening. Thus, there is a need to study such effects in a hydroacoustic projector. For this we have developed a validated multi-physics model of the system. In most hydraulic systems, direction control valves are on–off arrangements. However, in our case the valve opening varies with time at different frequencies, and amplitudes. Thus, we have developed a method to characterize effective Cd for such orifices. Our work shows that Cd is a strong function of pressure difference across the valve, frequency of valve actuation, and valve motion amplitude. We have accounted for such dependencies in our projector model. We have also investigated the effect of increase in effective compliance of hydraulic oil on projector’s behavior, which can occur due to the presence of compliant pipes or air in the oil. We show that such an increase can drastically degrade projector’s performance. Hence, using steel pipes and reducing air content in oil can significantly improve projector performance, particularly at post-resonance frequencies.

Estimation of triangular side orifice discharge coefficient under a free flow condition using data-driven models

In irrigation channel system, diverse side orifice shapes (e.g., circular, rectangular, or triangular) are widely designed for flow control and regulation. Hence, accurate estimating the discharge diverted from the main channel to the side one is an essential for water management. The main objective of this research is to estimate the discharge coefficient (Cd) for a sharp-crested triangular side orifice under a free flow condition. Three linear data-driven models including locally weighted learning regression (LWLR), multiple linear regressions with interaction (MLRI), and multivariate linear regression (MLR) were developed for this purpose. 570 experimental datasets were used to build the predictive models. Two modeling scenarios (with and without incorporating the upstream flow Froude number) were investigated for estimating Cd. The best input combinations for both scenarios were identified by applying the Gamma test approach. The performance of the models was assessed by using various graphical analysis and statistical metrics. The modeling results indicated that LWLR and MLRI had similar good performance for both modeling scenarios and provided accurate estimation of the Cd values. Overall, this study demonstrated the capacities of the data-driven models in estimating the discharge coefficient of triangular side orifice under a free flow condition.

Experimental and numerical modeling of sidewall orifices

The discharge coefficient of an orifice is important for the outflow through the orifice. While it cannot be quantified theoretically as the outflow through an orifice depends on a number of parameters such as the pipe pressure, the liquid velocity and the shape and the area of the orifice. In this study, experiments and computational fluid dynamics (CFD) simulations were performed to find out the factors influencing discharge coefficient and the corresponding mechanism. The CFD simulation is based on Navier–Stokes equations combined with RNG k − e turbulence model. The results show that a negative exponential function could fit the relationship between orifice discharge coefficient, pipe pressure, and orifice area more accurately. The relationship between the discharge coefficient of the orifice and the velocity was linear. In general, the simulation results fit well with the experimental results, which indicates that CFD simulation could be used to study pipeline leakage.

A Hybrid Intelligent Model and Computational Fluid Dynamics to Simulate Discharge Coefficient of Circular Side Orifices

In this paper, a hybrid meta-heuristic artificial intelligence (AI) method, named “ANFIS–PSOGA,” is developed for modeling the discharge coefficient of circular side orifices for the first time. For this purpose, the adaptive neuro fuzzy inference system (ANFIS) network is optimized by means of two robust optimization algorithms: the genetic algorithm (GA) and the particle swarm optimization (PSO) algorithm. Furthermore, the Monte Carlo simulations are utilized to enhance the performance of the AI hybrid model. Besides, the k-fold cross-validation method is employed in order to verify the numerical results. Initially, the performance of three fuzzy inference system (FIS) generations of the ANFIS model including grid partitioning, subtractive clustering and fuzzy C-means clustering are evaluated. After that, using the parameters affecting the discharge coefficient, eleven ANFIS models are defined and then the best ANFIS model and the most influencing input parameters are identified through the conduction of a sensitivity analysis. For the superior ANFIS model, mean absolute percentage error, root-mean-square error and the correlation coefficient (R) are obtained 0.021, 0.020 and 0.871, respectively. Also, the Froude number (Fr) is detected as the most effective input parameter. Moreover, the discharge coefficient of the circular side orifices and the flow field within main channels along the side orifices are simulated through the adoption of a computational fluid dynamics (CFD) model. Additionally, the flow field at three different cross sections (beginning, middle and end of orifice) along the side weir is evaluated. For instance, by advancing the flow toward the side orifice, the Froude number is reduced. Besides, at each cross section, the maximum shear stress is estimated to occur in the vicinity of the crest of the circular side orifice. According to the CFD results, the maximum lateral velocity is estimated near the circular side orifice. In addition, the results yielded by the ANFIS, ANFIS–PSO, ANFIS–PSOGA and CFD models are compared and an uncertainty analysis is carried out for the models. Based on the comparison, ANFIS–PSOGA has the best performance in simulating the discharge coefficient. Ultimately, a simple and practical computer code for the best model (ANFIS–PSOGA) is proposed which can be easily used by scholars and engineers in practical usages.