Separated Flow Model for

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 model (for more details, see Collier and Thome [54]) considers that the phases are artificially segregated into two steams one liquid and one vapor, and has been continuously developed since 1949 when Lockhart and Martinelli [56] published their classic paper on two-phase gas-liquid flow. The main goal in this approach is to find an empirical correlation or simplified concept to relate the two-phase friction multiplier, ( ), to the independent variables of the flow. For example, the... 

Crystallization-based separation of multi-component mixtures has widespread application. The technique consists of sequences of heating, cooling, evaporation, dilution, diluent addition and solid-liquid separation. Berry and Ng (1996, 1997), Cisternas and Rudd (1993), Dye and Ng (1995), Ng (1991) and Oyander etal. (1997) proposed various schemes based on the phase diagram. Cisternas (1999) presented an alternate network flow model for synthesizing crystallization-based separations for multi-component systems. The construction... 

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. 

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... 

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. 

Predictions of the column height required for any given separation can be obtained by using either a staged approach or a transfer unit approach. The plug flow models for determining the height of a column are of limited value due to the effect of axial dispersion, which is caused by... 

Fici. l.S. Flow model for combined reaction-separation targeting. 

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. 

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. 

Assuming ID isothermal flow, steady-state, constant cross-sectional area, negligible mass transfer between the gas and liquid phases, and constant properties in a cross section, Merchuk and Stein (1981) used a separated flow model of Wallis (1969) for vertical gas-Uquid cocurrent flows to determine gas holdup in gas-liquid bubble columns and airlift reactors ... 

Two-phase pressure drop can typically be correlated with two models, i.e. homogeneous or separated. Homogeneous fluid models are well suited to emulsions and flow with negligible surface forces, where the two-phase mixture can be treated as a single fluid with appropriately averaged physical properties of the individual phases. Separated flow models consider that the two phases flow continuously and separated by an interface across which momentum can be transferred (Angeli and Hewitt 1999). The simplest patterns that can be easily modelled are separated and annular flow (Brauner 1991 Rovinsky et al. 1997 Bannwart 2001). In this case, momentum balances are written for both phases with appropriate interfacial and wall friction factors. 

The cross-flow model for reverse osmosis is similar to that for gas separation by membranes discussed in Section 13.6. Because of the small solute concentration, the permeate side acts as if completely mixed. Hence, even if the module is designed for countercurrent or cocurrent flow, the cross-flow model is valid. This is discussed in detail elsewhere (HI). 

Even for the maximum mixedness model, the calculation of an extreme concentration value is possible. This extreme value, in comparison with the previous one for the segregation model, represents the opposite end of the scale. The maximum mixedness model is relevant to microfluids and yields, even in this case, results that do not differ considerably from those that can be obtained by means of direct utilization of separate flow models. The model is, however, of importance in cases in which experimentally determined residence time functions are available for a reactor system. The maximum mixedness model is more difficult to visualize than the segregation model. However, its underlying philosophy can be described as follows ... 

Solving this flow model for the velocity the pressure is calculated from the ideal gas law. The temperature therein is obtained from the heat balance and the mixture density is estimated from the sum of the species densities. It is noted that the viscous velocity is normally computed from the pressure gradient by use of a phenomenologically derived constitutive correlation, known as Darcy s law, which is based on laminar shear flow theory [139]. Laminar shear flow theory assumes no slip condition at the solid wall, inducing viscous shear in the fluid. Knudsen diffusion and slip flow at the solid matrix separate the gas flow behavior from Darcy-type flow. Whenever the mean free path of the gas molecules approaches the dimensions of pore diameter, the individual gas molecules are in motion at the interface and contribute an additional flux. This phenomena is called slip flow. In slip flow, the layer of gas next to the surface is in motion with respect to the solid surface. Strictly, the Darcy s law is valid only when the flow regime is laminar and dominated by viscous forces. The theoretical foundation of the dusty gas model considers that the model is applied to a transition regime between Knudsen and continuum bulk diffusion. To estimate the combined flux, the model is based on the assumption that the combined flux can be expressed as a linear sum of the Knudsen flux and the convective flux due to laminar flow. 

P. Sukitpaneenit, T.S. Chung, and L.Y. Jiang. (2010). Modified pore-flow model for pervaporation mass transport in PVDF hollow fiber membranes for ethanol-water separation, J. Memb. Sci. 362 393-406. 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

---