# 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
### Analysis of Correction Results  
The frequencies of the basic working mode (first-order mode) and the phase differences of the measuring tube under their vibration frequencies obtained by the numerical simulation method are used to determine the original measurement values of the total density and mass flow in gas-containing fluid by a single-frequency Coriolis flowmeter through Equations (8) and (9), respectively. Then, the deviations of the apparent density and mass flow are obtained from Equations (36) and (37). Furthermore, the influence of flow patterns on the total density and mass flow deviations in gas-containing fluid with a single-frequency Coriolis flowmeter is analyzed. The results are shown in Figure 7.  
\[E_{\rho_{1}}=\frac{\rho_{app1}-\rho_{0}}{\rho_{0}} \tag{38}\]  
\[E_{\dot{m}_{1}}=\frac{\dot{m}_{app}-\dot{m}_{0}}{\dot{m}_{0}} \tag{39}\]  
where \(\rho_{0}\) and \(\dot{m}_{0}\) represent the actual values of density and mass flow of two-phase fluid, respectively. The corrected relative deviations of total density and mass flow in the measured fluid are calculated from Equations (40) and (41), which can analyze the correction effect of the stratified flow measurement model based on the multi-frequency Coriolis principle. The correction results are shown in Figure 8.  
\[E_{\rho_{2}}=\frac{\rho_{s}-\rho_{0}}{\rho_{0}} \tag{40}\]  
Figure 6: The block diagram of the compensation method with stratified flow based on the multi-frequency Coriolis principle. Figure 7: The original deviations of stratified flow relative to GVF before correction. (**a**) Relative deviations of density. (**b**) Relative deviations of mass flow. The comparison of the results in Figures 7 and 8 shows that the correction effect of the stratified flow metering compensation model based on the multi-frequency Coriolis principle is significant. Before the correction, the maximum relative deviation of the total density with a single-frequency Coriolis flowmeter is \(-37.3\%\), and the total mass flow is not more than \(-9.3\%\). The results of the stratified flow correction model based on the multi-frequency Coriolis principle show that the relative deviations of total density are less than \(\pm 4\%\) from \(92\%\) measured fluid, the relative deviations of mass flow are less than \(\pm 3\%\) from \(90\%\) measured fluid. In the stratified flow (\(\dot{m}_{0}=0.2280\sim 0.3420\) kg/s), there are different coupling forces between the gas and liquid phases and the pipeline, which leads to more significant deviations in the vibration frequencies of the pipe. Of course, in the uniform bubble flow, the distribution of bubble size also plays an essential role in the measurement accuracy of the fluid. The low flow (\(\dot{m}_{0}=0.09120\)-\(0.1368\) kg/s) and light mass of stratified flow in the straight pipe section make the existence of small suspended bubbles have little influence on the vibration response of the pipe, so the correction result of small suspended bubble flow is better than that of uniform bubble flow. Taking the two vibration frequencies of the measuring tube, and apparent densities of the measured fluid as the input parameters, the calculated gas and liquid contents can be converted to the gas-phase and liquid-phase mass flows in the stratified flow. The corrected relative deviations in the gas-liquid two-phase flow are obtained from Equations (42) and (43), respectively. The calculation results are shown in Figure 9.  
\[E_{\dot{m}_{g}}=\frac{\dot{m}_{g}-\dot{m}_{sg}}{\dot{m}_{sg}} \tag{42}\]  
\[E_{\dot{m}_{l}}=\frac{\dot{m}_{l}-\dot{m}_{sl}}{\dot{m}_{sl}} \tag{43}\]  
where \(\dot{m}_{sg}\) and \(\dot{m}_{sl}\) represent the actual values of gas and liquid mass flow in the two-phase fluid (inlet setting values of gas and liquid phases in numerical simulation). \[E_{\rho_{1}}=\frac{\rho_{app1}-\rho_{0}}{\rho_{0}} \tag{38}\]0 The deviations relative to GVF after correction by the proposed compensation model. (**a**) Relative deviations of density. (**b**) Relative deviations of mass flow.