Two-phase multiplier

In the separated flow models presented in Sections 7.7 and 7.8, the method of calculating the frictional component of the pressure gradient involves use of the two-phase multiplier 4 1 2 3 4 defined by 

That is, the two-phase frictional pressure gradient is calculated from a reference single-phase frictional pressure gradient (dP/dx)R by multiplying by the two-phase multiplier, the value of which is determined from empirical correlations. In equation 7.73 the two-phase multiplier is written as % to denote that it corresponds to the reference single-phase flow denoted by R. 

For a gas-liquid two-phase flow there are four possible reference flows  

3 only the liquid in the two-phase flow, denoted by subscript L 

When the reference flow is the whole of the two-phase flow as liquid, then the two-phase frictional pressure gradient is given by 

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Two-phase multiplier, pressure drop for two-phase flow... 

A plot of the two-phase multiplier ratio, 0, as a function of property index at one mass flux (Fig. 3.42)... 

A similar relationship could be written by taking the single phase gas flow as the reference instead of the liquid, i.e., Gq = Gm. This is the basis for the two-phase multiplier method ... 

The two-phase multiplier method is used primarily for separated flows, which will be discussed later. 

The classic Lockhart- Martinelli (1949) method is based on the two-phase multiplier defined previously for either liquid-only (Lm) or gas-only (Gm) reference flows, i.e.,... 

Table 15-2 Values of Constant C in Two-Phase Multiplier Equations...

R two-phase multiplier with reference to single phase R, [—] ip volume fraction of the more dense phase, [—]... 

The terms represent, respectively, the effect of pressure gradient, acceleration, line friction, and potential energy (static head). The effect of fittings, bends, entrance effects, etc., is included in the term Ke correlated as a number of effective velocity heads. The inclination angle 0 is the angle to the horizontal from the elevation of the pipe connection to the vessel to the discharge point. The term bi is the two-phase multiplier that corrects the liquid-phase friction pressure loss to a two-phase pressure loss. Equation (23-39) is written in units of pressure/density. 

The notation used here and in Section 7.7 is standard in the literature on this subject and originates in the pioneering work of Martinelli and co-workers. There are two aspects of the notation that may lead to confusion and error. First, note that LO and GO do not denote liquid only and gas only reference flows, as might be expected. On the contrary, they denote flows in which the whole of the flow rate is liquid or gas. It may help to remember them as liquid overall and gas overall . The second point to note is that 2 denotes the two-phase multiplier. Correlations may present values of but it must be remembered that this is the square root of the two-phase multiplier. 

In contrast to the case of the homogeneous model, the accelerative term cannot be put in a simpler form because the phase velocities differ. It is therefore necessary to carry out the differentiation in the accelerative term. When this is done and the frictional component of the pressure gradient is represented using the wholly liquid two-phase multiplier, the resulting form of the momentum equation is... 

Lockhart and Martinelli (1949) used only liquid and only gas reference flows and, having derived equations for the frictional pressure gradient in the two-phase flow in terms of shape factors and equivalent diameters of the portions of the pipe through which the phases are assumed to flow, argued that the two-phase multipliers and 4>g could be uniquely correlated against the ratio X2 of the pressure gradients of the two reference flows ... 

It was assumed that four flow regimes could occur depending on whether each phase was in turbulent or laminar (viscous) flow. Their empirical correlation is shown in Figure 7.13. The second and third subscripts denote the type of flow of the liquid and gas respectively. Note that and X are the square roots of the two-phase multiplier and the ratio of reference flow pressure gradients. 

Void fraction and square root of two-phase multiplier against Martinelli parameter X Source R. W. Lockhart and R. C. Martinelli, Chemical Engineering Progress 45, pp. 39-46(1949)... 

The value of the square root of the two-phase multiplier is read from Figure 7.13, or calculated from equation 7.85 or 7.86, and the two-phase frictional pressure gradient calculated from... 

Consequently, from the definitions of the Two-Phase Multipliers, equations 7.74 and 7.77... 

Mean value of the two-phase multiplier as a function of absolute pressure... 

In Eq (17), the two-phase multiplier for only liquid flowing ( ) (mass flux = G) is given by... 

Where R2 is known as the Two-Phase Multiplier its value depends on which single-phase flow is chosen as the reference flow. 

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