Optimization of convergent angle of the Venturi meter for best coefficient of discharge Open Access

ASME

American Society for Mechanical Engineers

SCDM

Space Claim Design Modeller

β-ratio

Diameter ratio

CA

Convergent Angle of Venturi Meter

DA

Divergent Angle of Venturi Meter

RCA

Recovery Cone Angle/Divergent Angle

C_d

Coefficient of Discharge

CFD

Computational Fluid Dynamics

PDEs

Partial Differential Equations

Re

Reynolds Number

FEA

Finite Element Analysis

Q_act

Actual Discharge

Q_theo

Theoretical Discharge

Accurate flow measurement remains as one of the biggest concerns in many industries because variations in flows of the product can cost these organizations considerable profits. With the increase in demand for highly accurate flow measurement meters, the use and application of Venturi meters has increased many-fold. In general, there are different meters used to compute the fluid flow: the turbine-type meter, rotameter, orifice meter, and the Venturi meters are only a few. Mubarok et al. (2020) conducted experimental and CFD study on six pressure differential flow meters, namely concentric orifice, bottom eccentric orifice, segmental orifice, top eccentric orifice, nozzle and Venturi. CFD analysis indicated that nozzle and Venturi flow meters showed the lowest net pressure-drops among all the instruments and are most suitable for use in geothermal industries.

These meters work on their ability to alter/affect a certain physical property of the flowing fluid and then this alteration is accurately measured. The measured change in that physical property is then related to the flow, for example the physical property being pressure-drop in orifice and Venturi meters.

Convergent Inlet Cone: This is the region where the cross-section of the inlet pipe reduces to a conical shape for its connectivity with the throat such that its cross-sectional area decreases from start to end. On one side it is attached to the inlet pipe and its other side is attached to the cylindrical throat. As per the manual of ASME (ASME MFC-3M-2004 2004), the angle of convergence is generally 20–22° and its length of flow is 2.7(D−d). Here D is the diameter of the inlet section and d is the diameter of the throat. The converging region is attached to the inlet pipe throat region at the downstream end. Due to the decrease in the cross-sectional area, the fluid accelerates and static pressure decreases. The maximum cone angle of the converging area is limited to avoid the vena contracta so the flow area will be minimum at the throat. The convergent angle is believed to be a function of β-ratio, as well as the Reynolds number. The ratio of throat diameter to the diameter of inlet pipe is often termed the β-ratio. The β-ratio acts as a very important physical parameter in designing a Venturi meter. Any change in Reynolds number or the β-ratio affects the most efficient convergent angle for that specific Venturi. The accuracy of a Venturi for discharge measurement is well established and documented through various research, however, the design of convergent cones and the associated head loss is not. This is the area under study in the current research. From earlier studies on gases, Venturi tubes with a convergent angle of 10.5° are much smoother than Venturi tubes with the standard or higher convergent angle (Reader-Harris et al. 2001; Tukimin et al. 2016; Demir 2021).

Throat Section: This is the central part of the Venturi meter and has least cross-sectional area. The length is generally equal to the diameter of the throat. Normally, the diameter of the throat is 0.25–0.75 times the diameter of the inlet pipe, but mostly it is 0.5 times the diameter of the inlet pipe. Throughout its length, the diameter of the throat remains unchanged. Experiment and CFD analysis can be conducted to determine the large energy losses occurring in the throat region for an inhomogeneous flow condition and these losses can be minimized by taking large diameter-ratios (β) (Kim et al. 2015). For two-phase flow experiments which include an oil–water phase, the Venturi pressure-drop is found to be inversely proportional to the Venturi β (Elobeid et al. 2018).

Divergent/Recovery Cone Section: The diverging section is the last part of this instrument. On one side, it is attached to the throat cylinder and on the other to the outlet pipe. The diameter of this section gradually increases. As per the manual from ASME, the diverging section has an angle of 5–13°. The diverging angle is less than the converging angle and the reason for the small diverging angle is to avoid any flow separation from the walls and prevent the formation of eddies. Earlier research has been done on optimizing the Venturi recovery cone angle to minimize the head loss. Sharp et al. (2018) optimized the ASME Venturi recovery cone angle to minimize head loss. The paper mentioned that in many instances where highly accurate flow measurement and low head loss are required, Venturi flow meters are a viable option. This study used computational fluid dynamics and laboratory data to demonstrate the relationship between recovery cone angle in the classical Venturi meter design and associated head loss. Nithin et al. (2012) used the computational fluid dynamics (CFD) software ANSYS FLUENT as a tool to perform modelling and simulation of a Venturi meter and reported that the results from ANSYS FLUENT are in reasonable agreement with experimental values.

As evident in different studies, a lot of analysis has already optimized the divergent angle of Venturi meters in order to bring the value of C_d closer to 1. However, little information is available on that value of convergent angle (CA) which results in the least drop in pressure and the best C_d. Therefore, this study focuses on optimizing the CA to minimize the losses in pressure and thereby results in greater discharge value. This sequentially will lead to better instrument performance.

The software package used in this study is ANSYS FLUENT. Often known as Analysis System, ANSYS is a software package that is widely renowned for its engineering simulations. FLUENT, which is an ANSYS solver, contains wide physical modelling abilities needed to model the flow, turbulence, heat transfer and reactions for different industrial applications. It is based on computational fluid dynamics (CFD). CFD is a way of using applied mathematics, physics and computational software to visualize how a fluid flows – as well as how the gas or liquid affects objects as it flows past. CFD has the Navier–Stokes equations as its main working principle. These equations give a description of how the pressure, velocity, temperature, and density of a moving fluid are related. The very foundation of CFD is the Navier–Stokes equations, which are a set of partial differential equations (PDEs) that describe the flow of fluid. With CFD, the study area/model is subdivided into many control volumes, which can also be called cells. In each of these cells, the Navier–Stokes PDEs can be rewritten in the form of algebraic equations that relate the temperature, velocity, flow rate, pressure, and other variables, such as species concentrations, to the values of these parameters in the neighboring cells. Numerical simulation can then be used to solve these equations, yielding a complete picture of the flow domain, down to the resolution of the grid. By solving the resulting set of equations iteratively, a complete description of the flow throughout the domain is yielded. In Venturi meters, instead of costly experimental methods, a computational model can be prepared in ANSYS FLUENT and the coefficient of discharge can be easily calculated (Tamhakar et al. 2014). Ameresh et al.(2017) analyzed a varying inlet diameter of a Venturi meter of 25, 30 and 35 mm and theoretical calculations were performed to evaluate mass flow rate of air and the results further successfully validated with ANSYS FLUENT. From the previous studies, Sanghani et al. (2016) used CFD to investigate the effect of different geometrical parameters like divergent cone angle, diameter ratio and throat length on pressure drop in a Venturi meter. By varying one parameter and keeping the other three constant, the effect of each parameter was found. The results showed fluctuation in pressure drop with increase in convergent cone angle, divergent cone angle and throat length. Also, a reduction in pressure drop with increase in diameter ratio was observed. Furthermore, the employability of CFD tools can be extended for wet gas flow through a Venturi tube by utilizing the discrete phase mode (DPM) and standard k-ε model (Jing et al. 2019). Further research suggests that cavitating flow in a Venturi can be successfully modelled utilizing the Singhal cavitation model in ANSYS FLUENT (Dastane et al. 2019).

Numerical simulations and experimental measurements have shown good agreement in studying the influence of convergent angles on cavitating Venturi tubes (Shi et al. 2019). Not only have Venturi meter instruments shown reasonable agreement between experimental and numerical simulation results, but Venturi nozzles that are specifically used for gas metering have also revealed good agreement between numerical and experimental results (von Lavante et al. 2000). Several studies have been conducted on Venturi-related systems in recent years such as the analysis performed by Li et al. (2022), which depicts enhanced understanding of the mechanism of bubble break in a Venturi-type bubble generator using a volume-of-fluid method and large eddy simulation. Similar studies such as optimization of the geometry of a vane-type pre-swirl nozzle and wet gas flow measurement have also been successfully performed with CFD (Lee et al. 2021; Weise et al. 2021). CFD also has the potential to analyze non-Newtonian fluid in two-phase flow (flow of water–kerosene) in a Venturi meter and results have suggested that coefficient of discharge increases with increments in velocity of fluid and decreases continual with increments in gas volume fraction, as indicated in Khayat & Afarideh (2021). The pressure recovery phenomenon has also been studied by Caetano & Lima (2021) using CFD tools in a Venturi meter utilizing three different k-ɛ turbulence models. Chen & Chen (2022) combined the CFD method and machine learning technique to forecast the radial maximum wall shear stress of an ultra-high pressure (UHP) water jet nozzle. Das et al. (2022) designed a Venturi-type mixture to modulate the flow of producer and bio-gases in a diesel engine. A 2D CFD-based model to study the influence of different concentrations of solid mass on the characteristics of the flow and cavitation generation has also been studied for the Venturi meter by Shi et al. (2020). Hence a sufficient review of the literature reveals that ANSYS FLUENT can be used as a model for simulating the flows in flow-measuring devices.

The primary goal of the investigation was to decide the ideal convergent angle required to limit the pressure drop for various Venturi meters with differing β proportions. The goals are elaborated as:

• to correctly determine the optimum angle of convergence for a standard ASME Venturi meter out of the given range of 19–22°;

• testing the β-ratios of 0.4, 0.5 and 0.6 for different angles of convergence and finding the ones yielding the best coefficient of discharge and the least pressure drop;

• finding the variation of coefficient of discharge at different Reynolds numbers for each convergent angle;

• to use computational fluid dynamics and laboratory data (for β = 0.6) to demonstrate the effect of the angles of the convergent cone on the coefficient of discharge on common Venturi meter designs.

The laboratory test was done on the Venturi model available in the laboratory. The Venturi model in the laboratory had a β-ratio of 0.6. The equipment had two different pipe dimensions with one pipe of inlet pipe diameter as 20 mm and the other as 35 mm. The pipe material was galvanized iron. Other dimensions are elaborated in the Procedure section.

Once the geometry of each model was constructed in SCDM, meshing was applied to determine the points within the model where numerical computations would occur. After testing of multiple meshing schemes, it was decided that the best size function would be curvature so as to capture the conical transitions in the Venturi geometry, while keeping the element size as 0.003 m. The mesh was made even finer by introducing refinement of the mesh for different elements in the geometry. Once the models had been meshed, there were up to 500,000 computational elements in a model.

Defining boundary conditions is one of the most important processes for each of the models created in SCDM. The flow inlet on the 30-mm-diameter upstream pipe was defined as the velocity inlet, while the exit of the flow meter was defined as a pressure outlet. All other faces in the geometry were identified as walls. The velocity inlet boundary condition was applied, so that the measured flow from the laboratory data could be used for comparison purposes. The pressure outlet condition was employed to ensure that with different fixed outlet pressures the differential pressure through the meter would be unaffected.

The pressure velocity coupling used was the simple consistent algorithm. The under-relaxation factors were set to the default values ranging from 1 to 0.7 for all the variables besides the pressure factor, which was set to 0.3. For this study, standard pressure was used, while the second-order upwind method was applied for momentum and the first-order upwind scheme for kinetic energy and the turbulent dissipation rate. Residual monitors were used to determine if a solution converged to a point where the results had very little difference between successive iterations. For the k-ɛ model application, six different residuals were being monitored: continuity, x, y, and z velocities, k, and ɛ. The study aimed to ensure the utmost iterative accuracy by requiring all the residuals to converge to 0.001, before the model runs were complete. The models were initialized with hybrid initialization as the main initialization schemes with the input velocity to ensure that the initial iterations gave plausible results. During the initialization, there were usually at least 100 iterations performed on the computer while the residuals were plotted and monitored for any unusual increases. Most of the solutions converged within these iterations. The values of pressures obtained at the two tap locations were used to compute the values of discharges, which were then compared with the theoretical values of discharge to obtain an estimation of C_d. To notice the effect of Reynolds number on the convergent angle, three Reynolds numbers were selected, and each convergent angle was tested for those three Re values.

As the study included both laboratory and computational experiments, results were found for both after following the methodology followed in section 3. Both the results were in form of pressure drops between the two pressure taps in the Venturi meter. The head difference thus obtained had to be further analyzed in order to obtain theoretical as well as actual discharge values. These discharge values are further used to obtain the coefficient of discharge. This section presents those values and the calculations as well.

The ANSYS computations were done for three β-ratios (0.4, 0.5 and 0.6). For each β-ratio, 21 models, corresponding to different convergent cone angles and variable Reynolds numbers, were made. The Q_theo is computed by simply multiplying the value of the area of the input pipe with the input velocity (Table 1). We have taken a standard 30 mm pipe.

Table 3 shows the variation of the C_d over a range of Reynolds numbers, and it solidifies the fact that 20° is the optimum angle for β = 0.4 as on increasing the Reynolds number further for that CA, the C_d value further increases.

The Reynolds number and variation associated with it for β = 0.5 are depicted in Table 5. The trend stays the same here: with increase in the Reynolds number value the corresponding coefficient of discharge also increases.

As mentioned earlier, the meters present in the DTU laboratory have a β-ratio of 0.6. Therefore, this section shows the comparison of laboratory results for the 0.6 β-ratio with the CFD results. As the convergent angle in the laboratory is constant, the comparison shown is for an inlet convergent angle of 21°, with a recovery cone angle of 5°. The parameter varied here is Reynolds number only.

This is because the CFD computations took a smooth wall under consideration, whereas the laboratory conditions vary a little. But, in any case, it is safe to say that the change in C_d is comparable in both the cases. It can be observed through the graph that in the range of study, that is, for Reynolds numbers 20,000–100,000, the coefficient of discharge is observed to increase.

This section discusses some key findings of this study.

• CFD is capable of producing data predicting pressure loss/drop for many different Venturi designs. As indicated in Gowtham Sanjai et al. (2021) the CFD analysis is the best approach based on accuracy and time constraint. We observed that the convergent angle corresponding to the maximum coefficient of discharge also has the least value of pressure drop through the CFD computation in ANSYS. Table 9 below gives the percentage loss in pressure that can be avoided if the optimum CAs, as pointed out in this research, are used. These tables are collectively derived from the pressure-drop values in Tables 2, 4 and 6.

Therefore, it is inferred that head loss measurements are sensitive to the convergent cone angle. In other words, slight changes in the convergent angle can result in large increase/decrease in pressure loss as depicted in Table 9.

• The convergent angles are proved to be a function of β-ratio and Reynolds number. The best CA angle corresponding to each β-ratio as found is given in Table 10.

• To validate the results obtained by CFD, an equation presented by Reader-Harris et al. (2001) is used:

  

  (1)

This equation is used to relate C_water, that is, the coefficient of discharge of a Venturi tube, having water as the flowing fluid with different β-ratios. This equation has an uncertainty of 0.74% (based on two standard deviations). To check our results, the value of C_d obtained by our optimum CAs were compared with the C_water (Table 11). The error percentage is calculated and checked to see whether it is within the limit or not. It is seen from Table 11 that the error is even less than 1%, therefore the result is very well validated.

• The relationship between Reynolds number and C_d is also very well established here with the help of CFD results and validated through the laboratory results. This falls into line with the results of Yadav et al. (2022), where the researchers reported that the coefficient of discharge increased with increments in rate of flow for the designed Venturi meter.

The current study employs physical and numerical data to determine optimum CAs for different Venturi designs. Three different β-ratios of 0.40, 0.50, and 0.60 were analyzed with constant Venturi wall roughness along with three different Reynolds numbers and multiple CAs. The CFD-based research presents the optimum value of CA for different β-ratios which can be chosen while manufacturing the Venturi instrument. This will prevent pressure loss and help in discharge estimation with an error less than 0.25%. The study also indicates an increasing CA value on increasing the β-ratio. Additionally, a relationship between C_d & Re has been observed through the analysis in CFD and laboratory experiment. The curve obtained through both the data are seen to follow a rising trend. The trend signifies that on increasing the Reynolds number (within the experimental range), the C_d also increases. The validation brings out a good congruence in the obtained CFD results and the research carried out by Reader-Harris et al. (2001).

This study focuses only on certain values of β-ratio, therefore, as a future scope, more rigorous analysis can be done to establish a clearer relationship between the CA and β-ratio. In addition to CFD, machine learning algorithms can be used for shorter computation times and increased prediction accuracy.

The are grateful to the laboratory staff of the Hydraulics Laboratory of Delhi Technological University (DTU) for their cooperation in carrying out the results of the current research.

All the authors confirm contributions to the paper as follows: ZAK initiated the idea of the research. ZAK and NJ made the CFD model of the Venturi meter in ANSYS FLUENT. NJ performed all the experiments on the Venturi meter in the laboratory. ZAK analyzed and interpreted the results regarding the optimization of the convergent angles of the Venturi meter and subsequent losses associated with it and was a major contributor in writing the manuscript. All authors reviewed the results and approved the final version of the manuscript.

No funding was obtained for this study.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.