Separated flow models

Pressure drop in two-phase flow is found in terms of pressure drops of the individual phases with empirical multipliers. The basic relation is 

The last term is the pressure drop calculated on the assumption that the total mass flow has the properties of the liquid phase. 

Some correlations of multipliers are listed in Table 6.8. Lockhart and Martinelli distinguish between the various combinations of turbulent and laminar (viscous) flows of the individual phases in this work the transition Reynolds number is taken as 

1000 instead of the usual 2000 or so because the phases are recognized to disturb each other. Item 1 of Table 6.8 is a guide to the applicability of the Lockhart-Martinelli method, which is the oldest, and two more recent methods. An indication of the attention that has been devoted to experimentation with two phase flow is the fact that Friedel (1979) based his correlation on some 25,000 data points. 

Example 6.14 compares the homogeneous and Lockhart-Martinelli models for the flow of a mixture of oil and hydrogen. 

In these models the phases are treated as if they are separate and flow in well defined but unspecified parts of the cross section. Only the simplest case, in which the phases are allowed to have different but uniform velocities, will be considered here. An overall momentum equation will be given and it will be seen that merely allowing the gas and liquid velocities to differ leads to considerable complexity. Two empirical correlations from the pioneering work of Martinelli and co-workers will then be described. These methods can be used for the churn and annular flow regimes. 

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 

If the only liquid reference flow had been used the frictional term in equation 7.81 would be 2fL(l — w)2G2VL j 2Lldi. 

Integration of equation 7.81 to determine the pressure drop over a length of pipe generally requires a stepwise procedure. As with the homogeneous model, in some cases simplifications may be possible  

1 the denominator of equation 7.81 differs little from unity  

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Methods for determining the drop in pressure start with a physical model of the two-phase system, and the analysis is developed as an extension of that used for single-phase flow. In the separated flow model the phases are first considered to flow separately and their combined effect is then examined. 

Sfglf.r, G. 217. 220. 222, 229 Separated flow model, multiphase flow 187 Separation during flow 47... 

A new approach was developed by Lee and Mudawar (2005a) to improve the accuracy of pressure drop prediction in two-phase micro-channels. Since the bubbly and churn flow patterns are rarely detected in high-flux micro-channel flow, the separated flow model was deemed more appropriate than the homogeneous. 

This map has been checked by many researchers, indicating that it is applicable to a wide range of conditions. Also shown in Figure 3.4 are correlations derived by Mishima and Ishii (1984), which used similar basic principles except for the slug-to-churn transition. These authors pointed out that, in view of the practical applications of the separate-fluid model to transient analysis, flow regime criteria based on the superficial velocities of the liquid and gas may not be consistent with the separate-flow model formulation. A direct geometric parameter such as the... 

A separated flow model for stratified flow was presented by Taitel and Dukler (1976a). They indicated analytically that the liquid holdup, R, and the dimensionless pressure drop, 4>G, can be calculated as unique f unctions of the Lockhart-Martinelli parameter, X (Lockhart and Martinelli, 1949). Considering equilibrium stratified flow (Fig. 3.37), the momentum balance equations for each phase are... 

Fauske (1962) developed a phases in equilibrium but separated flow model for a long pipe, which could be used even at exit conditions, by assuming ... 

The separated flow models consider that each phase occupies a specified fraction of the flow cross section and account for possible differences in the phase velocities (i.e., slip). There are a variety of such models in the literature, and many of these have been compared against data for various horizontal flow regimes by Duckler et al. (1964a), and later by Ferguson and Spedding (1995). 

The homogeneous flow model and the separated flow model may be used to estimate the pressure drop for the churn regime but the former is not recommended for use with annular flow. The separated flow model of Martinelli and Nelson (1948), and developments thereof, may be used for annular flow. 

The approaches adopted in the homogeneous model and the separated flow model are opposites in the former it is assumed that the two-phase flow can be treated as a hypothetical single-phase flow having some kind of average properties, while in the separated flow model it is assumed that distinct parts of the flow cross section can be assigned to the two phases, reflecting what occurs to a large extent in annular flow. 

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... 

Liquid-Gas Row in Pipelines 111 Homogeneous Model 113 Separated Flow Models 114 Other Aspects 114... 

Computational fluid dynamics based flow models were then developed to simulate flow and mixing in the loop reactor. Even here, instead of developing a single CFD model to simulate complex flows in the loop reactor (gas dispersed in liquid phase in the heater section and liquid dispersed in gas phase in the vapor space of the vapor-liquid separator), four separate flow models were developed. In the first, the bottom portion of the reactor, in which liquid is a continuous phase, was modeled using a Eulerian-Eulerian approach. Instead of actually simulating reactions in the CFD model, results obtained from the simplified reactor model were used to specify vapor generation rate along the heater. Initially some preliminary simulations were carried out for the whole reactor. However, it was noticed that the presence of the gas-liquid interface within the solution domain and inversion of the continuous phase. 

Banerjee S, Chan AMC (1980) Separated Flow Models-I Analysis of the Averaged and Local Instantaneous Formulations. Int J Multiphase Flow 6 1-24... 

Several frictional multiplier correlations have been developed using a separated flow model. These correlations are listed in Table 14.2. Hewitt [151] makes the following tentative recommendations ... 

The velocity distribution in the cross section can be simplified by introduction of a limited number of velocity ranges. Each range is characterized by the flow condition and its part of area in the cross section (Fig. 2.8). A separate flow model with a special transfer function

flow range and combined in a simple way if linear models and the Laplace- or Fourier-transformation are used. 

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