The method presented here is based on a homogeneous flow theory which assumes that liquid and gas move with the same velocity and are homogeneously intermixed. The method can be applied in the following two cases of two-phase flow:
Case 1: Single-component, two-phase flow (e.g. steam and liquid water)
Case 2: Two-component, two-phase flow (e.g. crude oil and natural gas).
Homogeneous flow theory is based on average properties such as the density and velocity of a two-phase mixture. After necessary average properties have been determined, a valve can be sized with equations similar to the standard equations for single-phase flow.
The density of a two-phase flow is calculated using separate densities for both phases on the upstream side of a valve. In addition, gas expansion must be taken into account when gas flows through the valve. The density of the mixture, something that we call effective density, can then be formulated as in equation (77).
There are not enough experimental studies of choked two-phase flow to allow exact terminal pressure drops to be determined. In control valve sizing, the terminal pressure drop approximation is thus performed using a combination of the terminal pressure drop values of pure gas and pure liquid. Pure gas flow chokes when the pressure drop reaches the value given by equation (79).
In a case where all the fluid is in the liquid-phase flow, choking will start when the pressure drop in equation (80) is reached. When a small fraction of gas is added to the flow, the choked flow pressure differential changes, but it is still quite close to the value in equation (80). An increased mass fraction of gas will further change the terminal pressure drop that produces choking, but it is not clear how this happens in control valves. Finally, when all the fluid is in the gas phase, choking will start with a pressure drop as in equation (79). Theoretical calculations and flow tests have been performed to determine the terminal pressure drop for ideal nozzles. These results show that the linear relationship between the terminal pressure drop in the liquid and gas phase as a function of gas mass fraction is accurate enough to describe the terminal pressure drop in choked two-phase flow. Therefore, it can be formulated as in equation (81).
When the actual pressure drop exceeds the value of equation (81), the two-phase flow is considered to be a choked flow, and the pressure drop in equation (81) is used for the valve sizing equation (78), and, to calculate the expansion factor Y. Note that the minimum value for expansion factor Y is Ymin(= 0.667).
Due to the nature of the two-phase flow of liquid and gas, it is impossible to adequately describe the various possible forms of flow using a single mathematical formula. The method presented here is based on what we call homogeneous flow theory, an approach which assumes that the velocities of liquid and gas are the same and that they are completely intermixed. This is a fairly common flow type, so the method described can be utilized in many two-phase flow applications.
The accuracy of sizing decreases if the form of the flow deviates from the type described. The following forms of flow are possible in a pipe:
Phase changes in the liquid and gas, vaporization of the liquid, or condensation of the gas into liquid make it difficult to calculate mass fractions and effective density. These factors in sizing accuracy are especially evident with single-component two-phase flow. When the pressure decreases and the temperature is almost constant, the liquid tends to evaporate whereby the mass fraction of vapor and the required valve capacity increase. On the other hand, something called the metastability phenomenon tends to 'slow down' the phase change. In this phenomenon, the liquid does not vaporize at once in the vena contracta although the thermodynamic equilibrium of the substance would indicate this. Instead, vaporization occurs only after the vena contracta.
The effect of mass fractions on sizing accuracy is especially visible with small mass fractions of vapor. For example, a change in the mass fraction of a saturated 7 barA water and steam mixture from 1% to 2% causes a 73% change in the specific volume of the mixture. This means that the required capacity increases by about 30%. On the other hand, if the mass fraction of the flow changes from 98% to 99%, the specific volume of the mixture changes by 1%. If the mass fractions with single-component two-phase flow are not exactly known, the sizing can be checked by assuming the whole mass flow as vapor flow. This guarantees that the valve capacity is adequate in all situations.
There are no methods for predicting noise in two-phase flow. In practice, estimation of noise in two-phase flow is very difficult. It is known from experience that cavitation noise in pure liquid flow is lower if, for instance, air is mixed with the liquid. Air bubbles attenuate the pressure waves created by the collapse of cavitation bubbles.