Modeling of coefficient of contraction of differential pressure flowmeters

The article provides a solution to the problem of contraction process modeling when measuring the flow of fluids and gases using differential pressure flowmeters. The article shows the feasibility of obtaining the coefficient of contraction from the mathematical model of the reduced/contracted flow of fluids or gases. An analytical expression is obtained for the coefficient of contraction, which agrees well with the results of the experiment. Analysis of this expression is performed and dependencies on the geometrical parameters of the pipeline and orifice plate are obtained.     

A review of studies on estimating the discharge coefficient of flow control structures based on the soft computing models

The discharge coefficient (Cd) plays a vital role in the accurate design and safety of weirs, spillways, and dams. In the last decade, Soft Computing(SC) models, which showed excellent capabilities for non-linear mapping between parameters, were widely used to estimate the discharge coefficient of flow control structures. This study provides a comprehensive review of the application of SC models for estimating Cd of different flow control structures such as ogee spillways, orifices, side weirs, etc. In addition, the most common empirical relations which are obtained from laboratory experiments are discussed briefly. The findings revealed that weir length/flow depth ratio, weir length/channel width ratio, weir height/flow depth, and Froude number are widely used to estimate Cd in the side weirs. Besides, the ratio of orifice crest height to height of side orifice, the ratio of main channel width to length of side orifice, ratio of main channel width to height of side orifice, and ratio of the height of side orifice to upstream flow depth were extensively employed to calculate Cd of orifice structures. The common parameters for measuring Cd of labyrinth weirs are, discharge over a labyrinth cycle, weir height, channel width, apex constant, upstream head, discharge over the weir, effective length, convergence constant, sidewall angle, and Froude number. In the weir-gate structure, some factors such as contraction coefficient of the gate, head loss, and weir height are key parameters for the accurate evaluation of Cd. The trends of SC models, features of popular models, and the background of models are discussed briefly in this paper. Also, research gaps and possible directions for new studies are suggested.     

The calculation of the uncertainty of the orifice-plate discharge coefficient

The uncertainty of the orifice-plate discharge coefficient given by the Reader-Harris/Gallagher (1998) Equation has been calculated taking account of the uncertainty of the data on which it is based and of the variability in manufacture permitted by ISO 5167–2. This work has shown that using the correct method to determine the uncertainty in ISO 5167-2 has made an insignificant difference to the value given in the standard. However, in other similar cases where the uncertainty for an artefact is based on data from other similar artefacts the uncertainty values obtained by the correct method may be significantly different from those by the incorrect method.     

A novel comprehensive correlation for discharge coefficient of square-edged concentric orifice plate at low Reynolds numbers

Among various differential pressure flow meters, the orifice meter has gained its publicity in applications where cost, space, and ease of maintenance are of high priority. A major problem associated with the use of orifice flow meters at low Reynolds number flows is the significant variation of discharge coefficient (C_d) as a function of orifice geometry and the Reynolds number. In this work, a two-dimensional axisymmetric numerical model was applied to the investigation of viscous, incompressible flow through square-edged concentric orifice plate for the purpose of studying the performance of discharge coefficient consequent to variations of Reynolds number (Re), orifice/pipe diameter ratio (β), and orifice thickness ratio (t*). The analysis of numerical results by means of multiple regression method has yielded a new correlation incorporating the effect of the parameters under study on orifice meter discharge coefficient for orifice bore Reynolds numbers (Re_o) < 250. Results of relevant investigations from the literature are used in the present work as references for the validation of the numerical model as well as the proposed correlation for discharge coefficient.     

Natural gas flow computer with open architecture using intelligent instrumentation and field bus

A new approach to natural gas flow computer design is presented in this paper. The developed system runs on a personal computer and employs the state-of-the-art mathematical models for corrections of some aspects of fluid flow dynamics, as well as for compressible behavior of gaseous fluid considerations. Orifice plates were used as primary elements. Measurements were performed through intelligent sensors. Results of the system metrological tests are also presented.     

Flow measurement based on the combination of swirler and differential pressure under slug flow

The alternating appearance of elongated bubbles and liquid slugs of slug flow in the pipe causes severe pressure fluctuation. As a result, measuring the flow rate of the slug flow with the throttling unit based differential pressure method is difficult. This paper investigates a new swirler-based flow measurement method in slug flow. The swirler converts the slug flow into a swirling annular flow, and the differential pressure method is used to measure the flow rate. The influences of gas and liquid flow rates on the differential pressure ΔP_X across the swirler as well as its downstream axial differential pressure ΔP_Z are investigated. ΔP_X^0.5 increases linearly as the liquid mass flow rate increases, and the slope of the curve increases as the gas mass flow rate increases. The influence of gas mass flow rate on ΔP_X^0.5 is comparable to that of liquid mass flow rate on ΔP_X^0.5. ΔP_Z^0.5 increases linearly with increasing gas/liquid mass flow rate, and the slope of the curve of ΔP_Z^0.5 with m_l differs slightly from the slope of the curve in single-phase water conditions. Based on the research presented above, new empirical correlations of mass flow rate based on ΔP_X and ΔP_Z are established respectively. The superficial liquid velocity ranges from 0.6 to 2 m per second, while the superficial gas velocity ranges from 2 to 6 m per second. If the gas mass flow rate and ΔP_X are known, the relative error of liquid mass flow is less than 3%. The relative error of the gas mass flow rate is less than 10% if the liquid mass flow rate and ΔP_X are given. The calculation accuracy of the flow measurement model using ΔP_X is better than the calculation accuracy of the flow measurement model using ΔP_Z.     

Experimental investigation of discharge coefficient over novel kind of sharp-crested V-notch weir

Thin-plate weirs are the simplest and least expensive devices, which frequently have been employed in many water projects. In this research, a unique type of Sharp-Crested V-notch weir, entitled SCVW, has been introduced. The hydraulic characteristics of the present weir were investigated theoretically and experimentally under free aerated and non-aerated flow conditions in an open channel for large physical models. To investigate the variations of the discharge coefficient of SCVW versus weir height and vertex angle, a comprehensive laboratory experiments were conducted by measuring the discharge and the water head over the crest of weir. Possibility of different formulations for the head- discharge relationship of SCVW was examined and suitable analytical equation was proposed. The computed discharge using the suggested equation was within 0–10% of the observed ones. According to the experimental observations, the SCVW showed better performance in comparison with normal wire.     

Measurement of high-water-content oil-water two-phase flow by electromagnetic flowmeter and differential pressure based on phase-isolation

When the oil field has been exploited by long-term water-flooding, it will be in high water-content stage of production. However, it is a great challenge for high-water-content measurement due to oil droplets extremely dispersed in the water. In this paper, we developed a phase-isolation based method for high-water-content oil-water two-phase flow measurement. Phase-isolation was realized by axial-flow swirler to concentrate scattered and random oil droplets into the pipe center and change the inlet flow pattern into a particular annular flow before measuring. Owing to the axisymmetric velocity and phase distribution, the electromagnetic flow meter avoided the effect of random distribution of insulating phase, and then had a good measurement performance for total volume flow rate. Furthermore, we respectively studied using axial pressure drop, radial pressure drop and the ratio of the two pressure drops to measure water content. The results showed that the ratio of the two pressure drops not only improves the resolution of oil and water, but also effectively reduces the impact of error transfer. In the dual-parameter measurement experiment, the relative errors of total volume flow rate and water content were almost within ±5%.     

A study of a pressure differential flow meter that is insensitive to inlet conditions

Orifice meters are a type of differential pressure flow meter widely used in industry and their behaviour is very well understood. The standard discharge coefficient can only be used if the flow approaching the meter is perfectly settled and fully developed. Thus the installation of these meters is subject to many constraints. Normally the required flow regime is established by the use of a combination of a flow conditioner and a settling length.

This paper describes the initial work carried out to show the concept of an orifice meter whose performance is independent of the inlet conditions by the introduction of a standard swirl before the meter. These results show that a swirler renders the measurements independent of the influence of upstream disturbance. This is for both a partial blockage in the pipe and also upstream swirl, and is likely to be universal.

A proposal for a new calibration equation is also made for the particular swirler employed. However, more data need to be acquired over a wide range of geometries, flow and operating conditions.     

Performance evaluation of piezoelectric and differential pressure sensor for vortex flowmeters

Piezoelectric and transient differential pressure sensors are two among the most widely employed sensors for vortex flowmeter application. The present study evaluates the performance of these two techniques under fully developed and disturbed flow conditions. Firstly, the location of the transient differential pressure sensor is optimized to obtain high amplitude signals and good linearity in Strouhal number. Empirical mode decomposition method in combination with autocorrelation decay is successfully employed at high Reynolds numbers to identify the vortex shedding frequency in presence of hydrodynamic noise. The performance of the differential pressure sensor deteriorates significantly under disturbed flow conditions at low Reynolds number due to the presence of low frequency components. This deterioration in the signal quality limits the lower operating range of the flowmeter with differential pressure sensor. The output signals of the piezoelectric sensor and differential pressure sensor under no flow condition are compared to obtain the background noise due to piping vibrations and electrical interferences. These results will help a designer to suggest robust signal processing algorithms for vortex frequency detection.     

Examination of disturbed pipe flow and its effects on flow measurement using orifice plates

In a great number of measurements the influence of a disturbed flow on the flow coefficient of a standard orifice plate was investigated. Single bends and double bends out of plane with and without spacer tubes were used as typical disturbances. Experiments were also performed using a combination with a star-shaped flow straightener. The necessary correction factors of the flow coefficient were determined for upstream straight length shorter than detailed in ISO 5167. The flow velocity profiles produced by the disturbances were examined and on this basis profile numbers were calculated. The examinations presented here show that the existing standard should be revised as regards the definition of the fully developed turbulent flow profile and the selection of the required upstream straight lengths.     

Study and application of discharge calibration for submerged radial gates

Sophisticated stage-discharge rating curves for radial gates are essential to modernization efforts in water conservancy and irrigation projects. However, it is a challenging hydraulic issue to accurately determine discharges through radial gates, especially when subjected to highly submerged flow conditions. According to the variation trend of jet Froude number and the mechanism of energy dissipation subjected to submerged flows, a new criterion was proposed to subdivide submerged flow into partially submerged and totally submerged flows. Thus, the three stage-discharge equations were developed for free flow, partially submerged flow and totally submerged flow, respectively. With the aid of the experimental data of single radial gate and the field data of multi-radial gates, the proposed discharge calibration method, referred to as the identification method, was tested and verified. The results indicated the discharge prediction accuracies were satisfactory, in which the mean absolute percentage errors were less than 10%. The proposed method is feasible and easily programmed.     

脉动流对差压式流量计测量误差影响的研究

基于脉动流场中差压式流量计的计量特性，本文分析了差压式流量计在各种脉动流频率和幅度下的流量测量误差。针对不同脉动频率和脉动幅度的脉动流，在自行设计的实验装置中使用响应时间不同的差压式流量计进行流量测量的研究，获得了流量计响应时间、流体的脉动幅度和频率与流量测量误差间的变化对应关系。研究结果表明，通过减少脉动幅值和缩短流量计的响应时间可有效地减少流量测量误差。     

Research on the measuring characteristics of a new design butterfly valve flowmeter

Flowmeters and control valves are important components of flow measurement and control in heating, ventilating, and air conditioning (HVAC) system, which directly or indirectly impact building room comfort and energy costs. Valves as resistance components produce differential pressure which in turn can be used for flow measurement. This paper studies the function among valve opening position, pressure difference and flowrate of a new designed butterfly valve. The flow model of the butterfly valve is established based on the Bernoulli equation, the discharge coefficient C under different valve opening conditions are studied by CFD simulations and verified by experiments. The simulation results show that the discharge coefficient C reached a stable value of 0.67–0.70 as Reynolds number exceeded 5000, and the permanent pressure loss ratio is range from 0.95 to 0.37 corresponding to opening range from 10° to 70°. The correctness of the simulation results of C is verified by experiments, in which C is about 0.60. With the corrected values obtained from experiments, the simulation results are instructive to practice. The new designed butterfly valve flowmeter can be used efficiently in HVAC system, especially in variable air volume (VAV) air conditioning system. And the work of this paper offers a reference for other types of valve flowmeters in fluid control processes.     

Discharge coefficient relationship for the sharp-edged width constriction new theory and experiment

In the field of flow measurement, what is needed in general rule is to design devices that are inexpensive, fairly accurate, and easy to implement. Some devices are more accurate than others but are very expensive.The present study examines a device for measuring flow in rectangular open channels that combines the three abovementioned advantages. It is made up of two separate vertical thin plates arranged in such a way that they form a central opening of width b less than the rectangular channel width B in which they are inserted. It is the simplest device ever imagined and designed for flow measurement in open channels.It is planned to derive the stage-discharge relationship by borrowing a rigorous theoretical development based on classical hydraulic formulas, accounting for the effect of the approach flow velocity. The intended stage-discharge relationship will allow us to derive the resulting discharge coefficient equation of the device.It is established that the resulting theoretical discharge coefficient is formally identified as being exclusively dependent on the contraction rate β = b/B, and this finding is predicted by dimensional analysis. Both dimensional and experimental analyses show no effect of the upstream water depth on the discharge coefficient for the selected values of β. The derived theoretical discharge coefficient relationship is straightforward, although it contains trigonometric functions that are somewhat cumbersome when the designer needs to perform a rapid field calculation.The theoretical and mean-experimental discharge coefficients of the eight tested devices are carefully compared using the one hundred and fifty-seven experimental values collected. An excellent agreement, even perfect, is observed since the maximum deviation worked out over the tested range of β is only 0.05%. This confirmed the validity of the theoretical relationship governing the discharge coefficient, which does not need any correction.     

Evaluating the application of data-driven intelligent methods to estimate discharge over triangular arced labyrinth weir

The use of artificial intelligence based (AI-based) methods has been known as a promising approach for solving engineering problems in order to model systems with high complexity and uncertainty such as the hydrodynamic behavior of flow passing the hydraulic structures. Considering the importance of weirs in regulating the water level and discharge controlling in water transfer channels and dams, it seems that the application of these methods can be considered as a useful tool for the estimation of discharge capacity. The present study examines the precision and use of six data-driven models including Bayesian neural network (BNN), multiple linear regression (MLR), multi-layer perceptron neural network (MLPNN), gene expression programming (GEP), least square support vector machine (LSSVM), and Chi-squared automatic interaction detector (CHAID) for the estimation of discharge passing triangular arced labyrinth weirs compared to two proposed experimental relations. To this end, 212 laboratory test results were used and statistical parameters of coefficient of determination (${\text{R}}^2$), root-mean-square error (RMSE), mean absolute error (MAE), and bias were employed as the criteria for the comparison of the models' performance. Results showed a good agreement between the observed and estimated values using the AI-based models. Among these models, the MLPNN managed to estimate the discharge passing the weir with the highest precision (RMSE = 0.00385 m³/s, R² = 0.999, and Bias<$\left|0.0001\right|$).     

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 (C_d). 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.     

Fluctuant pressure coefficient of the wake behind a bluff body in mist flow

Fluctuant pressure coefficient is presented as an index of vortex energy of the wake behind a bluff body in mist flow. Calculation for vortex energy is obtained from inter-phase force analysis, based on Basset-Boussinesq-Oseen equation and conservation of mechanical energy. Vortex energy is weakened by liquid disturbance, the relationship between the fluctuant pressure coefficient and liquid velocity is set up on theory analysis. An existence criterion for vortex streets is proposed using the relationship. A new algorithm for prediction of liquid velocity is put forward and compared with the method in Higham's patent. The predicted results fit the experimental results well.     

Application of artificial neural networks instead of the orifice plate discharge coefficient

Differential pressure flowmeters are very often used in many industries. Therefore, the improvement of this method of flow measurement is an important task of flow measurement and instrumentation. One of the important characteristics of differential pressure flowmeters is the discharge coefficient of the flow transducers. A large number of studies and publications were devoted to modeling this coefficient. Therefore, in the framework of this research, this coefficient is simulated using artificial neural networks. The neural representation of this characteristic is made in the form of a multilayer perceptron. In this paper, we replace the traditional equation for the discharge coefficient with an artificial neural network. The advantages and disadvantages of such application of neural networks as discharge coefficients are discussed. The analysis of the results of gas flow measurement, where the neural network is used instead of the traditional equation, is presented. The estimation of flow rate measurement errors with such an approach is made; the error of calculation of the discharge coefficient is estimated.     

Experimental and numerical study of the flow characteristics of a novel olive-shaped flowmeter (OSF)

A novel differential pressure flowmeter with an olive-shaped flowmeter (OSF) is proposed and investigated both experimentally and numerically. The streamline, pressure and velocity are obtained and numerically analysed. The results indicate that the proposed OSF exhibits less permanent pressure loss than the orifice plate flowmeter (OPF). The pressure also tends to be more stable in the OSF, which ensures high measurement accuracy and repeatability. The OSF is superior to the OPF in terms of relative pressure loss, streamline distribution, pressure distribution and velocity distribution. In the experiment, an oil pump transported diesel oil into the measurement pipe, through the check valve, filter, pressure-regulating container, and flow-regulating valve, before it was finally returned to the fuel tank. The experimental results showed that the pressure loss of the OSF was only about 14.94% of that of the OPF under the same conditions. The pressure loss curve of the OPF increased rapidly by up to 2,700 Pa with each 1 m³/h increase in the flow rate, whereas that of the OSF increased only slightly.