# Each-Phase Metering with Gas-Liquid Stratified Flow Based on the Multi-Frequency Coriolis Principle
## 4 Deviation Correction Model Based on Multi-Frequency Coriolis Principle
### The Correction Model with Stratified Flow Based on the Multi-Frequency Coriolis Principle
\begin{array}{ll}\mathrm{Re}(j\omega)&=-\omega^{2}\bigg{[}m_{sl}+m_{sg} \frac{1}{1-k_{f}^{2}}\bigg{]}\\ =&-\omega^{2}\bigg{[}\rho_{l}A_{F}(1-\delta)+\rho_{g}A_{F}\delta\frac{1}{1-k_{ f}^{2}}\bigg{]}\end{array}  \tag{24}\]  
Without considering the bubble effect and resonance effect, the transfer function is expressed as:  
\[H(j\omega)=-\omega^{2}\rho_{l}A_{F}(1-\alpha) \tag{25}\]  
Therefore, by comparing the Equations (24) and (25), the measurement deviation expression in the stratified flow is obtained as follows:  
\[E_{\rho}=\frac{\rho_{g}\delta(1+k_{f}^{2})}{\rho_{l}(1-\delta)}=\frac{\rho_{g }\delta}{\rho_{l}(1-\delta)}\Bigg{(}\frac{1}{1-\frac{f^{2}}{f_{0}^{2}}}\Bigg{)} \tag{26}\]  
The relative deviation of density in the gas-containing fluid measured by a Coriolis flowmeter is expressed as follows:  
\[E_{\rho}=\frac{\rho_{app}-\rho}{\rho} \tag{27}\]  
Combining Equations (24) and (25) can be obtained:  
\[\rho_{app}=\rho_{g}\delta\Bigg{(}\frac{1}{1-\frac{f^{2}}{f_{0}^{2}}}\Bigg{)}+ \rho_{l}\cdot(1-\delta) \tag{28}\]  
where the resonance frequency \(f_{0}\) is a variable parameter of gas content and liquid density [27]. Two vibration frequencies corresponding to the first and third modes with similar mode shapes are selected, and different apparent characteristics of fluid are used to establish two expressions in the same form as Equation (28):  
\[\rho_{appi}=\rho_{g}\delta\Bigg{(}1-\frac{1}{1-\frac{1.179\pi^{2}R_{0}^{2} \delta(1-\delta)\rho_{l}f_{l}^{2}}{c_{g}^{2}\rho_{g}}}\Bigg{)}+\rho_{l}(1- \delta) \tag{29}\]  
where the radius of the measuring tube \(R_{0}\) is equal to 25.4 mm. Given the gas density, two expressions the same as Equation (29) are combined to solve two unknown parameters: \(\delta\) and \(\rho_{l}\). Then the corrected density \(\rho_{s}\) in the measured liquid is determined. Finally, the gas mass fraction \(x_{g}\) can be obtained from Equation (19) by using the obtained cross-section void fraction and liquid density. In addition, the GVF in the stratified flow can be calculated from Equation (30):  
\[\alpha=\frac{1}{1+\frac{1-\delta}{\delta}\frac{1}{S_{p}}} \tag{30}\]  
Stratified flow is the flow of gas and liquid-phase in the pipeline, in which the two phases are in a stable flow state. Therefore, the mass flow deviation of stratified flow is compensated by the gas mass fraction \(x_{g}\). So the expression of the corrected mass flow in the stratified flow can be obtained as follows:  
\[\dot{m}_{s}=\frac{1}{x_{g}}\dot{m}_{g} \tag{31}\]  
However, the gas entrained by the measured fluid accounts for a certain amount of total fluid mass. Similarly, the apparent value of total mass flow in gas-containing fluid is expressed as follows:  
\[\dot{m}_{\rm app}=\frac{1}{x_{\rm ag}}\dot{m}_{g} \tag{32}\]  
where \(x_{ag}\) is the apparent value of the gas mass fraction, which is calculated as follows:  
\[x_{ag}=\frac{1}{\frac{1}{a_{a}}\frac{\rho_{l}}{\rho_{g}}+\left(1-\frac{\rho_{ l}}{\rho_{g}}\right)} \tag{33}\]  
where \(a_{a}\) is the original measurement value of the GVF entrained by the measured fluid, which is obtained by using the density apparent value obtained by a single-frequency Coriolis flowmeter in the numerical simulation on the premise of the mass conservation law in three-phase flow. Therefore, it is known that a single-frequency Coriolis flowmeter can directly calculate the apparent value of the total mass flow \(\dot{m}_{app}\) in the gas-containing fluid by using the simulation results of the phase difference of the measuring tube.