Experimental and numerical study on wet steam flow metering overreading characteristics in venturi tube

The precise measurement of wet-steam flow is of great importance in various industrial systems, including nuclear reactors, oil and gas transportation, and cogeneration systems. However, the measurement accuracy of wet-steam flow can be affected by overreading (OR) when using venturi meters. This paper presents a systematic numerical simulation using the DPM model and an experimental study on the characteristics of overreading in wet-steam flow metering. The experiments are carried out on a two-phase flow liquid and steam platform. Firstly, the OR characteristics of a venturi structure with fixed dimensions (a throttling ratio of 0.6, a divergence angle of 15°, and a venturi diameter of 20 mm) are quantitatively analyzed at different flow rates (0.14-0.32 kg/s) and qualities (ranging from 0.14 to 0.90). It is found that the Lockhart-Martinelli parameter (XLM) plays a more significant role than the gas Froude number (Frg) in determining the overreading (OR) characteristics of a venturi under a given temperature, pressure, and venturi structure. Secondly, a venturi overreading prediction model based on XLM is proposed, and its ability to predict overreading for larger mass flow rates and venturi diameters is verified by numerical simulation data. As evaluated through numerical simulation data, the proposed model’s maximum relative error and average deviation are below 15% and 10%, respectively. This study provides valuable insights into developing an overreading model for industrial applications under large flow and venturi diameter conditions.


Introduction
Accurately metering wet steam flow is critical in various industrial applications, including nuclear reactors and oil and gas transmission [1].The venturi meter has been identified as a promising solution for wet steam flow measurement due to its simple design and high reliability [2].However, overreading occurs during wet steam flow measurement when a venturi is used, leading to distortion of the measured value [3].Therefore, developing a reasonable venturi overreading correction model is essential for achieving accurate wet steam flow metering.
The factors contributing to overreading include the Lockhart-Martinelli parameter XLM, gas Froude number Frg, pressure P, and throttling ratio β [4]- [6].Among the existing studies on venturi overreading models, theoretical models such as the Homogeneous Model [7], Separated Model [8], and Chisholm Model [9] are deemed less accurate due to their oversimplification, which makes them unable to reflect the actual situation of two-phase flow.Therefore, these models must be revised with experimental data before being applied to engineering wet steam metering.To this end, semi-empirical and semi- [19] As a result of the obstruction effect of the liquid on the gas flow in wet steam, the venturi pressure drop in wet steam flow is higher than that of the gas singly flowing through at an equal mass flow rate.Generally, the degree of this overestimation is reflected by utilizing the overreading coefficient,  [20] Current studies have indicated that several factors, including the Lockhart-Martinelli parameter XLM, gas Froude number Frg, pressure P, and throttling ratio β, influence the overreading of venturi.

Lockhart-Martinelli parameter XLM
. XLM is closely related to the gas-liquid density ratio and quality, and it is a dimensionless parameter that characterizes the gas and liquid superficial inertia forces, g 1 As the XLM parameter increases, the influence of the liquid phase on the gas phase becomes more pronounced, leading to an increase in overreading.

Gas froude number
Frg. Frg is a dimensionless similarity criterion number characterizing the ratio of the gas inertial force to the gravitational force, Assuming the decrease in Frg is equivalent to the decrease in gas superficial velocity, which reduces friction between the gas and liquid.As a result, the overreading decreases.

Wet steam pressure P.
As the pressure decreases, the gas density also decreases, while the liquid density remains relatively constant due to its low sensitivity to pressure changes.Consequently, the gas velocity increases, while the liquid velocity increases only slightly (compared to the gas velocity).It results in the sliding velocity's increase, and the liquid increasingly hinders the vapor.Finally, the overreading becomes greater.

2.2.4
Throttling ratio β.When the inlet diameter of the venturi remains constant, a smaller throttling ratio reduces the effective flow cross-sectional area of the throat, resulting in a more pronounced obstruction of the liquid to the gas.So, a smaller throttling ratio will lead to an increase in overreading.

Control of experimental variables
This paper investigates the characteristics of wet steam overreading for a specified pressure and venturi structure.In Eqs. ( 2) and ( 3), for a given gas and liquid density and venturi structure, the XLM parameter is solely dependent on the quality, while the Frg parameter is solely dependent on the gas superficial velocity vsg, which is a variable coupled with the steam mass flow rate.The vsg can be calculated using the following equation,  3) and (4) that the Frg parameter is also dependent solely on the quality and mass flow rate.The variables of quality and mass flow rate can be considered for studying the venturi overreading characteristics on the wet steam pressure and venturi structure conditions.The working fluid follows a specified process.Firstly, deionized water is pumped from Tank 1 by High-pressure Piston Pump 3 to reach the required pressure for the experiment.The water is then split into two paths, the bypass, and the main road.The bypass controls the mass flow rate, and the main road passes through the Mass Flow Meter 7 for flow rate measurement.The main road enters Heat Exchanger 5 to recover the heat of the working fluid at the outlet of the test section.The water then passes through Preheater 6 and Test Section 8.The resulting steam-water mixture from the test section travels through Heat Exchanger 5 and Condenser 9 before going through a series of pressure reduction devices to lower the pressure and ultimately returns to the water tank.

Experimental system
The experimental parameters vary within 1 MPa for pressure, 0.10-0.32kg/s for mass flow rate, and 0.14-0.90for quality.The venturi structure employed in the experiments is presented in table 1, figure 2  The size of the venturi structure is as follows The experiment involves measuring several parameters, including the temperature and pressure at the venturi inlet, venturi pressure drop, venturi mass flow rate, heating current, and voltage.The pressure and pressure drop are measured using Rosemount 3051 intelligent pressure or pressure drop transmitter, while the temperature is measured using NiCr-NiSi, K-type, and Φ3 mm stainless steel sheathed thermocouples.The approximate arrangement of temperature and pressure sensors is depicted in figure 2. The flow rate passing through the test section is measured using the RHEONIK-RHE08 Coriolis mass flow meter, as shown in figure 1 (Mass Flow Meter 7).The measurement uncertainties of these devices are detailed in table 2.

Experimental procedure 1)
Adjust the pressure stabilization system to maintain the required pressure of the experimental system at the desired level.
2) Control the flow of the experimental section by adjusting the main and bypass flow regulating valves.
3) Adjust the power of the preheating section to achieve the desired level of quality in the working fluid.
4) Once the experimental system has stabilized, record the relevant experimental data.5) Repeat the above steps to complete the data collection process for all experimental trials.

Experimental Data Processing
In previous studies, OR, a parameter used to assess the deviation of two-phase flow through the venturi, employs the pressure drop when the gas flowed singly through the venturi as a benchmark (all-gas-phase benchmark).However, a significant amount of energy is required to heat the water from the subcooled state to dry saturation due to the large latent heat of the vaporization.As a result, it is difficult to obtain an all-gas-phase benchmark.This paper redefines OR based on the pressure drop of liquid flowing alone through the venturi (all-liquid-phase benchmark), In addition, the gas Froude number Frg and Lockhart-Martinelli parameter XLM can be calculated according to Eqs. ( 2) and (3).

Effect of XLM parameter on the OR
Figure 4 illustrates the relationship between the OR coefficient and XLM.The figure shows a monotonically decreasing trend between XLM and OR.This is because the higher the liquid content in the wet steam, the larger the obstruction effect of the liquid on the gas phase and the smaller the frictional pressure drop of the gas due to the acceleration of the liquid phase.As a result, the OR is lower.It is worth noting that this conclusion seemingly contradicts the one presented in Section 2.2.1, as the former definition of overreading is based on all-liquid-phase, while the latter is based on all-gas-phase.So, these two conclusions are consistent.
On the other hand, it can be observed that the trend of OR relative to XLM is consistent under different mass flow rates, suggesting that the mass flow rate factor does not affect the establishment of the overreading model.

Effect of Frg parameter on the OR
The relationship between Frg and OR is presented in figure 5.It is evident from the figure that Frg is not the primary factor affecting the OR, as there exist multiple Frg values corresponding to the same OR value.Thus, Frg does not significantly impact the reproduction of OR.Consequently, the construction of the overreading model can disregard the influence of Frg.

Overreading model construction
The momentum equation for the steady-state wet steam flow is as follows, ( 1 ) sin ( 1) x A (6) where z, c, A, τ, ρws, θ, M, x, and α represent the length of the channel, wetted perimeter, crosssectional area, friction force, the density of the gas-liquid mixture, pipe inclination angle, mass flux, quality, and void fraction, respectively.
The first term on the right-hand side of Eq. ( 6) represents the frictional pressure drop generated by the friction between the fluid and the pipe wall.The second term represents the gravity pressure drop caused by the change in gravitational potential energy.The third term represents the acceleration pressure drop caused by the density variation of the fluid and the change in flow area along the flow direction.When wet steam flow passes through a horizontally installed venturi tube, the change in gravitational potential energy is negligible, and the frictional losses in the venturi can also be neglected.Therefore, the main component of the venturi pressure drop is the acceleration pressure drop caused by the significant reduction in flow area.In summary, Eq. ( 6) can be simplified as follows, Assuming that the density of the fluid along the pipe diameter direction remains constant and the void fraction of the fluid passing through the venturi does not change, the venturi pressure drop for wet steam flow can be obtained by integrating Eq. ( 7 Similarly, using Eq. ( 7), the pressure drops for the gas and liquid phases separately passing through the venturi tube can be obtained, Liquid phase, x M (10) Combining Eqs. ( 1), (8), and ( 9), the calculation method for OR can be obtained, x OR (11) The continuity equation for the steady-state wet steam flow is as follows, Liquid phase, Gas phase, Combining Eqs (11), (12), and ( 13), we can further obtain, where Cch is the Chisholm coefficient, calculated according to the following equation, where s represents the slip velocity ratio, vg/vl.Based on the Chisholm model, De Leeuw proposed that Cch can be calculated by the following equation [10], The n in Eq. ( 16) is the overreading index related to the XLM and Frg.Similarly, using the same method as deriving Eq. ( 14), the OR based on the all-liquid-phase benchmark in this paper can be obtained, Following Eq. ( 17), this paper intends to construct an overreading model in the following form, After fitting the experimental data, the overreading correction model proposed in this paper can be obtained as follows, 55 2 11 The fitting result is presented in figure 6 The Fitting Result of Overreading.

Cfd simulation
In industry, wet steam flow rate measurement often involves large flow rates and pipe diameters, making it challenging to conduct experiments.This section employs the CFD method to assess the model's applicability under large flow rate and pipe diameter conditions.The CFD geometric model is based on the same geometric model used in the experimental setup, which has been scaled up proportionally.

Computational conditions
The venturi diameter is scaled up from 20 mm to 930 mm, and the scaling ratio of the remaining portion is the same as that of the diameter.Eight different quality conditions are set at a mass flow rate of 190.3 kg/s, namely 0.14, 0.16, 0.19, 0.22, 0.23, 0.25, 0.27, and 0.28.The pressure conditions are consistent with the experimental setup in Section 3.

CFD model
When the wet steam qualities range from 0.14 to 0.32, the volume fraction of the liquid phase is sufficiently small, and the liquid phase exists in the form of dispersed mist within the gas phase.Consequently, the Discrete Phase Model (DPM) is deemed an appropriate simulation method.In the DPM, the gas phase is considered as a continuous medium and solved using the Euler method, and the liquid phase is treated as a discrete medium and solved using the Lagrange method.The two phases are solved alternately to couple their interaction.
The constants C, ε, β, and D represent the venturi discharge coefficient, expansibility factor, throttling ratio, and inlet diameter and are taken as 0.984, 1, 0.6, and 930 mm, respectively.

Predicted
Mass Flow Rate Deviation.The relative error E of the predicted mass flow rate is defined as follows to evaluate the difference between the model proposed in this paper and the classic overreading prediction model, where exact G is the exact venturi mass flow rate, which is 190.3 kg/s.

Simulation results and analysis
To evaluate the applicability of this model constructed on the small flow rate and venturi diameter conditions to the large flow and venturi diameter conditions in the industry, the prediction results are compared with those of the Separated Model [8], Chisholm Model [9], and Graham Model [13].More information on these three classic models can be found in their references.The difference in the OR prediction performance among the four models is presented in figure 7, and the comparison of predicted flow deviation is shown in figure 8. To quantitatively explain the performance differences between this model and the above three classic models, the maximum deviation and average deviation are listed in table 3.As depicted in figure 7, the predicted overreading exhibits an approximately linear relationship with the quality.Compared other three models, this model demonstrates the best prediction performance for large mass flow rate and venturi diameter conditions.
As shown in figure 8, the prediction deviation of mass flow for all models tends to decrease as quality increases.From table 3, it is evident that this model exhibits the best performance with a maximum deviation of 0.15 and an average deviation of 0.10 in the predicted mass flow rate among the four models compared.
Based on the numerical simulation results, this model demonstrates a high potential for industrial application in large venturi diameter and mass flow rate conditions.

Conclusion
Firstly, this study conducts wet steam flow experiments to explore the impact of Lockhart-Martinelli parameter XLM and gas Froude number Frg on the venturi overreading characteristics.The experiments are carried out at a small mass flow rate and venturi diameter conditions.Then, an overreading prediction model based on XLM is established.Then, the applicability of this model on the large flow and venturi diameter conditions is evaluated through numerical simulations.Finally, the performance of this model is compared with that of three classical overreading models, represented by the Chisholm Model.Based on the results, several conclusions are drawn, 1 ) For a given pressure and venturi structure, the overreading characteristic of wet steam in venturi flow is mainly influenced by XLM, while the mass flow rate and Frg have a relatively small impact on the overreading.
2 ) The overreading prediction model proposed in this study demonstrates better performance in industrial conditions with a large mass flow rate and venturi diameter.The average deviation and maximum deviation of the predicted wet steam mass flow rate are 0.10 and 0.15, respectively, which are superior to those of the classic Chisholm Model and the other two models.
3 ) This model is applicable for wet saturated steam with a pressure of 1 MPa, a quality range of 0.14-0.90, and a venturi throttling ratio of 0.60.4 ) This study has proved that the overreading model, proposed for small mass flow rate and venturi diameter conditions, has the potential to be extrapolated to large flow rate and venturi diameter conditions.This finding offers an alternative method for flow measurement in industrial pipelines.

and figure 3 Figure 3 .
Figure 3.The Structure of Venturi.

Figure 5 .
Figure 5. the Relationship Between the or and Frg.

Table 1 .
Size of Venturi.

Table 2 .
Device Uncertainty. ), Theoretical Mass Flow Rate.Combining the single-phase flow calculation formula with the overreading prediction model, the theoretical mass flow rate can be obtained,

Table 3 .
Predicted Average and Maximum Deviation.