The principle behind measuring wet gas with a combination of an orifice and a vortex flow meter is that the gas phase volume flow is measured by the vortex flow meter and brought into the over-reading model of the orifice flow meter. The liquid phase volume flow is then calculated from the over-reading flow, as shown in Fig. 1.

Orifice flow meters have been studied for many years, and their application to single-phase flow measurements was defined in ISO 5167. Between 1967 and 1977, Chisholm studied the characteristics of orifice flowmeters for two-phase flow measurements and put forward the ratio of the flow rate by weight during two-phase flow to the flow rate during single-phase flow with a two-phase pressure drop [10,11]. Until 2007, the experimental data of Skea et al. [12] and Steven et al. [13] confirmed the correctness of the Chisholm over-reading model.

Over-reading is defined as the positive deviation between the calculated value of the gas phase flow and the actual value caused by the liquid phase. Wet gas has the typical flow pattern of a gas-liquid two-phase flow. The key parameter for wet gas is the Lockhart Martinelli parameter X_LM, which is defined as [14]:

(1) XLM=mlmgρgρl

The deviation between the calculated gas mass flow rate and the actual value is caused by the velocity difference and the density difference between the gas and the liquid. An uncorrected gas mass flow rate is called the over-reading flow rate, expressed as m_{g,approximate}. The over-reading, OR, is the ratio of the over-reading flow rate to real gas mass flow rate, expressed as:

(2) OR=mg,apparentmg

The over-reading equation established by Chisholm defines it as the ratio of the flow rate by weight during two-phase flow to the flow rate during single-phase flow with a two-phase pressure drop [10]. Chisholm believed that the ratio of an orifice flow meter is a function of the Lockhart–Martinelli parameter and the ratio of the gas-liquid density for other flow patterns. He also stated that the ratio can be calculated in terms of the gas flow rates using the following formula:

(3) OR=1+CXLM+XLM2

(4) C=(ρgρl)0.25+(ρlρg)0.25

The over-reading formula can be used to modify the gas flow rate. Further mathematical derivation shows that the over-reading formula and the gas flow rate can be used to predict the liquid flow rate.

As shown in Fig. 2, part of the liquid flows in the form of a liquid film along the pipe wall, and the other part of the liquid is entrained in the form of droplets in the gas core area. The droplet diameter ϕ_d is calculated by simplifying the empirical formula proposed by Yoon [15].

(5) ϕd=1.67D(2.45We)0.12

(6) We=ρlul2Dσ

The droplet flow rate and gas flow rate are different. Azzopardi’s experimental results showed that the droplet flow rate is 20% lower than the gas flow rate [16], and Yoon also pointed out that the droplet flow rate is 12–15% lower than the apparent gas flow rate [15]. It is necessary to consider the influence of the velocity difference. The slip velocity ratio is the ratio of the droplet flow rate to the gas flow rate, which can be calculated using the Rahman model:

(7) s=11+stk

The calculation formula for the Stokes number is [17]:

(8) stk=ρdϕd2ug18μgD

(9) Ad=Qdud

(10) ud=S∗ug

(11) Ac=Ag+Ad=αA+Ad

Due to the fluid viscosity, the flow rate in the center of the pipeline is high, and the flow rate near the wall is zero. The velocity distribution is shown in Fig. 3. The velocity profile follows a power exponential distribution related to the Reynolds number, so the average flow rate u_c can be calculated with the velocity u detected by the vortex flow meter.

(12) uumax=(yR)n

Using a geometric derivation, the relationship between the velocity at the edge of the generator and the average velocity is given as follows:

(13) uc=ug2n+2(DW)n

n is calculated using the empirical formula:

(14) n=0.314Re−0.077

(15) α=QgQg+Ql

(16) μ=αμg+e(1−α)μl

(17) ρ=αρg+e(1−α)ρl

(18) Re=Ducρμ

The relationships between the average gas flow rate, the droplet flow rate in the center of the gas core area in the pipeline, and the gas volume flow rate are as follows:

(19) Qg=ucAc−Qd

(20) Qd=e ∗ Ql

The entrained liquid fraction can be calculated using Cioncolini’s formula [18]:

(21) e=(1+13.18We−0.655)−10.77

The equations are coupled by the parameters ϕ_d, S, A_d, e, α, Re, u_c, and u_g. These equations cannot be solved analytically but can be solved using an iterative algorithm. The solution flow chart is shown in Fig. 4. The condition for the end of this calculation process is:

(22) |ugn−ugn−1|<ξ

The relationship between the uncorrected gas flow rate and the output of the vortex flow meter is shown in Eq. (24). The gas flow rate can be obtained by correcting the output of the vortex flow meter according to the algorithm given in Section 3.1.

(23) Qsingle= fdASt

(24) γ=QcorrQsingle

Q_corr is the gas volume flow rate, which is calculated using the algorithm provided in Section 3.1. γ is the ratio of the corrected value to the uncorrected value. The gas volume flow rate of a gas-liquid two-phase flow needs to be corrected to account for a deviation in its measurement with a vortex flow meter due to the liquid effect.

At a fixed flow rate, when the cross-section gas content is low, the gas flow area and gas volume flow rate are both small shown in Fig. 5. The value of γ is less than one when the void fraction is small. With increasing void fraction, the two-phase flow tends toward a single-phase flow, so γ approaches one. The smaller the droplet radius, the more uniform the mixture in the gas core region. That is, the velocity of a droplet is closer to the velocity of the gas, so the velocity measured by the vortex flow meter is closer to the real velocity of the gas. With increasing droplet radius, the velocity difference between the gas and the droplet increases so that the velocity measured by the vortex flow meter is smaller than the real velocity of the gas phase. Therefore, as the droplet radius increases, γ decreases. Similarly, when the entrainment is small, the proportion of gas is large in the gas core area, so the velocity measured by the vortex flow meter is closer to the real velocity of the gas phase. On the contrary, when the entrainment is large, the flow velocity measured by the vortex flow meter deviates from the real gas velocity, so the correction ratio is small. The smaller the slip velocity ratio of the gas phase and droplets in the gas core area, the greater the velocity difference between the droplets and gas, and the greater the deviation between the measured velocity from the vortex flow meter and the real velocity of the gas. As the droplet velocity approaches the gas velocity, the velocity measured by the vortex flow meter becomes closer to the real gas velocity.

Experiments were carried out using the horizontal atmospheric two-phase flow equipment shown in Fig. 6 and using a DN50 vortex flow meter as shown in Fig. 7. The measurement accuracy of the vortex flow meter is ±0.5% for a single-phase flow. The fluid media were air and water, and three pressure points of 200 kPa, 300 kPa, and 400 kPa were considered, along with five gas-phase flow rates of 80 m³/h, 100 m³/h, 120 m³/h, 125 m³/h, and 130 m³/h. The liquid-phase flow rate range was 0–0.09 m/s. The gas flow rate was corrected after being output from the vortex flow meter according to the algorithm in Section 3.1. The liquid flow rate was obtained by calculating the standard value output from the orifice according to the algorithm provided in Section 2.2. The experimental results are shown in Table 1.

Using the algorithm to correct the gas flow rate, the result from the model agrees well with the experimental data, as shown in Fig. 8. The relative error of the corrected gas flow rate is within 5% for most of the experimental data. This result is acceptable because the error associated with the void fraction model is 5%, and the relative error associated with the entrainment model is 30%.

The corrected and uncorrected errors are the ratios of the calculated flow rates to the real flow rate, where the real flow rate is obtained from the flow meter in the standard equipment shown in Fig. 6.

When the liquid velocity is small, the relative error is small, so the correction effect is not obvious. The relative error increases with increasing liquid velocity, as does the error associated with the calculation formula for the void fraction. It is possible that the main reason for the error lies in the calculation errors of the models, such as for the void fraction, entrainment, sliding speed ratio, and droplet diameter. The percentage of data points within a 5% error for the uncorrected gas flow rate is 78.5%, but the percentage of data points within a 5% error for the corrected gas flow rate is 88.6%. When the liquid velocity is large, the error on the uncorrected data is large, and the correction effect is significant. This shows that the modified algorithm can improve the measurement accuracy of vortex flow meters for gas-liquid two-phase flows.

Unlike Hussein’s method [7], this correction effect can also be applied when the dryness is less than 95%. We can now compare this model with those of other researchers. Although Li’s method makes more accurate predictions, it uses image recognition technology, which requires more complex equipment and has limited adaptability [9]. The accuracy of the method proposed in this paper is somewhat weaker, but it has a wider range of applicability.