# 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
### Multi-Frequency Coriolis Principle  
The multi-frequency Coriolis principle refers to the multi-frequency response characteristics of Coriolis flowmeters and the different apparent characteristics of the same measured fluid based on the basic working principle of single-frequency Coriolis flowmeters to adequately compensate the deviations of bubble effect and resonance effect caused by the entrained gas.  
The selection of an additional vibration mode in the metering method based on the multi-frequency Coriolis principle is based on the vibration parameters and dynamic characteristics generated by the fluid flow in tube. It can be seen from the first six modes of the micro-bend measuring tube (Figure 5) that the basic working mode is characterized by reciprocating vibration in the horizontal direction. As the vertical direction of driving coil is taken as the central axis, the vibration mode is symmetrical, the second-order vibration mode is in an antisymmetric shape, the third-order mode is similar to that of the first-order, but its vibration frequencies increase obviously; other higher-order ones are more complicated.  
Figure 4: The simulation results of flow patterns relative to basic working mode vibration frequencies of the measuring tube.  
In addition to the basic working mode, the third-order vibration mode of Coriolis flowmeters can also measure the fluid density (Equation (12)), which provides additional information for the phase-separation metering of gas-containing fluids. The third-order mode shape is similar to the first-order, but its corresponding vibration frequency is about five times higher than that of the first order. Such two modes can be driven by the same driver installed at the bottom of the measuring tube.  
\[\rho_{\text{app3}}=-1064.57+5.69\times 10^{9}\frac{1}{f3^{2}} \tag{12}\]  
where \(f_{3}\) is the third-order vibration frequency obtained by modal analysis of the Coriolis flowmeter measuring tube.  
The same two-phase fluid has different effects on the vibration response of the Coriolis flowmeter measuring tube. The qualitative relationship between the gas-containing fluid (GVF, flow patterns) and the vibration response (first and third-order vibration frequencies) of the instrument measuring tube is obtained:  
\[\begin{array}{cc}(\rho,\alpha,\dot{m})&=f(f_{1},f_{3},\dot{m}_{app})\\ &=f(\rho_{app1}(f_{1}),\rho_{app3}(f_{3}),\dot{m}_{app})\end{array} \tag{13}\]  
where \(\rho\) and \(\dot{m}\) represent the corrected values of the total density and mass flow in the gas-liquid two-phase flow, respectively.  
Therefore, for the phase-separation metering of the gas-liquid two-phase flows, an additional independent measurement parameter (\(\rho_{app_{3}}(f_{3})\)) can be added to on the premise of the multi-parameter measurement (\(\rho_{app_{1}}(f_{1})\), \(\dot{m}_{app}\)) of the traditional single-frequency Coriolis flowmeters to calculate the total density of liquid-phase, volume void fraction (or cross-section void fraction) and total mass flow. Then the gas-phase and liquid-phase mass flows are obtained. In order to improve the metering accuracy of Coriolis flowmeters for gas-containing fluids and intricate flow patterns, it is a more feasible phase-separation metering method to establish a deviation compensation model for different flow conditions in actual working conditions. The stratified flow is used as a research object to develop a compensation model for phase-separation metering in this paper.