# Each-Phase Metering with Gas-Liquid Stratified Flow Based on the Multi-Frequency Coriolis Principle
## 2 Theoretical Background  
### Measuring Principle of Single-Frequency Coriolis Flowmeters  
A single-frequency Coriolis flowmeter is considered to be one of the high-precision flowmeters for measuring single-phase medium, as shown in Figure 1[13]. Its main components include: measuring tube, drive system, electromagnetic sensors, and secondary containment.  
A single-frequency Coriolis flowmeter relies on the vibration of the measuring tube to measure the measured medium. For the gas or liquid single-phase, single-frequency Coriolis flowmeters can directly measure the density and mass flow of the measured fluid by using the output values of vibration frequency and time difference of the measuring tube [14]. The measurement principle between them is independent.  
\[\rho=c_{0}+c_{1}\frac{1}{f^{2}} \tag{1}\]  
\[\dot{m}=K_{f}\Delta t \tag{2}\]  
where \(f\) is the vibration frequency of the measuring tube; \(c_{0}\) and \(c_{1}\) are the density coefficients, which are related to the inherent properties of the measuring tube; \(K_{f}\) is mass flow coefficient; \(\Delta t\) is the vibration time difference between two monitoring points when the measuring pipe vibrates.  
For liquid-liquid two-phase flow (oil-water two-phase flow), the decoupling between the two phases is relatively small [15]. Under conditions of known single-phase densities of oil and water, there are only two independent unknown variables. It is considered that two-phase fluid flows synchronously. The phase holdup and phase-separation flow in the two-phase fluid can be calculated by using two measurement values of total density and total mass flow provided by a single-frequency Coriolis flowmeter.  
For gas-liquid two-phase flow, it can be seen from Equations (3) and (4) that the measured density of fluid provided by a single-frequency Coriolis flowmeter is affected by the gas [16]. Measuring results cannot be determined, which may reduce the measurement accuracy. There are three unknown  
Figure 1: The structure diagram of a micro-bend Coriolis flowmeter.  
independent variables in the measurement of gas-containing fluid: liquid flow, gas flow, and volume void fraction (or cross-section void fraction) [17]. The accurate measurement of gas-containing fluid needs to be realized by adding at least one additional measured independent parameter (or a principle formula) based on two independent variables of single-frequency Coriolis flowmeters [18].  
\[\rho_{mixture}=\alpha\rho_{g}+\beta\rho_{l} \tag{3}\]  
\[\alpha+\beta=1 \tag{4}\]  
where \(\alpha\) is the gas volume fraction (GVF); \(\rho\) is the density. The subscripts \(g\) and \(l\) are gas and liquid, respectively.  
Measurement principle of single-frequency Coriolis flowmeters shows that influences of gas entrainment on the application performance of Coriolis flowmeters are still a challenging work. Therefore, for multiphase fluid, it is an ideal method to solve the measured variables by referring to the number of unknown quantities, combining with other variables, or using equations of different measurement principles.