# 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  
For gas-liquid two-phase stratified flow, there is relative motion between the gas and liquid phases. Each phase medium has a different average flow velocity and independent physical property  
Figure 5: The first six order mode shapes of the micro-bend measuring tube. (**a**) First-order vibration mode; (**b**) Second-order vibration mode; (**c**) Third-order vibration mode; (**d**) Fourth-order vibration mode; (**e**) fifth-order vibration mode; (**f**) Sixth-order vibration mode. parameters [24]. The expression of the total density of fluid with slippage loss is different from bubble flow [25]:  
\[\rho_{s}=\rho_{g}\delta+\rho_{l}(1-\delta) \tag{14}\]  
where the cross-section void fraction \(\delta\) is expressed as follows:  
\[\delta=\frac{A_{g}}{A}=\frac{A_{g}}{A_{g}+A_{l}} \tag{15}\]  
where \(A_{g}\) and \(A_{l}\) represent the gas-phase and the liquid-phase cross-sectional area, respectively. In the stratified flow, the ratio of gas volume flow through the same section to the total volume flow of fluid is not equal to the gas volume fraction, which is caused by the velocity difference between gas and liquid phases. The ratio between these two velocities is called slip ratio:  
\[S_{p}=\frac{u_{g}}{u_{l}} \tag{16}\]  
where \(u_{g}\) and \(u_{l}\) refer to the real velocities of the gas and liquid phases, respectively. The slip ratio is expressed in Equation (17), which is also a function of the gas mass fraction:  
\[\delta=\frac{1}{1+\left(\frac{1-x_{g}}{x_{g}}\right)\left(\frac{\rho_{g}}{ \rho_{l}}\right)S_{p}} \tag{17}\]  
Based on the research results of [PERSON] and [PERSON] [26], the slip ratio can also be calculated by the following expression:  
\[S_{p}=0.28\left(\frac{1-x_{g}}{x_{g}}\right)^{-0.36}\left(\frac{\rho_{g}}{ \rho_{l}}\right)^{-0.64}\left(\frac{\mu_{l}}{\mu_{g}}\right)^{0.07} \tag{18}\]  
where \(\mu\) is the dynamic viscosity under the standard condition. The combination of Equations (15) and (16) can be obtained:  
\[\delta=\frac{1}{1+0.28\left(\frac{1-x_{g}}{x_{g}}\right)^{0.64}\left(\frac{ \rho_{g}}{\rho_{l}}\right)^{0.36}\left(\frac{\mu_{l}}{\mu_{g}}\right)^{0.07}} \tag{19}\]  
The expression of gas mass fraction is as follows:  
\[x_{g}=\frac{\dot{m}_{g}}{\dot{m}_{s}}=\frac{\dot{m}_{g}}{\dot{m}_{g}+\dot{m}_ {l}} \tag{20}\]  
where \(\dot{m}_{s}\) is the total mass flow of stratified flow. Considering the influence of bubble effect and resonance effect, the transfer function of the vibration system when the two-phase fluid flows through the measuring tube is [13]:  
\[H(j\omega)=-\omega^{2}\left[m_{2}+m_{1}\frac{1}{1-k_{f}^{2}}-jm_{1}\frac{k_{ f}^{3}}{Q_{0}(1-k_{f}^{2})^{2}}\right] \tag{21}\]  
where \(k_{f}=\omega/\omega_{0}=f/f_{0}\), \(\omega\) and \(\omega_{0}\) are angular frequencies corresponding to \(f\) and \(f_{0}\) respectively; \(Q_{0}\) is the mass factor of resonator; \(m_{1}\) is the active mass of asynchronous oscillation with the measuring tube in the tube; \(m_{2}\) is the inactive part of synchronous oscillation with the measuring tube. The transfer function (19) is divided into the real part and imaginary part: the real part defines the mass flow measurement effect of gas-containing fluid; the imaginary part expresses the damping effect of two-phase fluid. This expression shows the relationship between the excitation and the reaction force when the two-phase fluid is still in the measuring tube. The gas and liquid phases flow asynchronously in stratified flow. The unit mass of gas and liquid phases contained in the total mass of the fluid in the pipeline is expressed by the cross-section void fraction as follows:  
\[m_{sg}=\rho_{g}A_{F}\delta \tag{22}\]  
\[\rho_{s}=\rho_{g}\delta+\rho_{l}(1-\delta) \tag{14}\]0  
Of course, this also corresponds to the active mass part and the inactive mass part of the measured fluid in the pipeline. Therefore, substituting Equations (22) and (23) into the real part of Equation (24), it can be obtained:  
\[