Pressure drop separated flow model

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. 

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. 

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... 

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. 

Foam films are usually used as a model in the study of various physicochemical processes, such as thinning, expansion and contraction of films, formation of black spots, film rupture, molecular interactions in films. Thus, it is possible to model not only the properties of a foam but also the processes undergoing in it. These studies allow to clarify the mechanism of these processes and to derive quantitative dependences for foams, O/W type emulsions and foamed emulsions, which in fact are closely related by properties to foams. Furthermore, a number of theoretical and practical problems of colloid chemistry, molecular physics, biophysics and biochemistry can also be solved. Several physico-technical parameters, such as pressure drop, volumetric flow rate (foam rotameter) and rate of gas diffusion through the film, are based on the measurement of some of the foam film parameters. For instance, Dewar [1] has used foam films in acoustic measurements. The study of the shape and tension of foam bubble films, in particular of bubbles floating at a liquid surface, provides information that is used in designing pneumatic constructions [2], Given bellow are the most important foam properties that determine their practical application. The processes of foam flotation of suspensions, ion flotation, foam accumulation and foam separation of soluble surfactants as well as the treatment of waste waters polluted by various substances (soluble and insoluble), are based on the difference in the compositions of the initial foaming solution and the liquid phase in the foam. Due ro this difference it is possible to accelerate some reactions (foam catalysis) and to shift the chemical equilibrium of some reactions in the foam. The low heat... 

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 regenerator (Figure 4-80) is represented by a simplified model that ineludes the total volume and mass balanee ealeulation. The regenerator exit temperature is assumed eonstant for the duration of the transient. The third-stage separator is handled as a fixed volume and assoeiated pressure drop. Blow-down (bypass) flow is subtraeted from the input flow. 

Probably the most widely used method for estimating the drop in pressure due to friction is that proposed by LOCKHART and Martinelli(,5) and later modified by Chisholm(,8 . This is based on the physical model of separated flow in which each phase is considered separately and then a combined effect formulated. The two-phase pressure drop due to friction — APtpf is taken as the pressure drop — AP/, or — APG that would arise for either phase flowing alone in the pipe at the stated rate, multiplied by some factor 2L or . This factor is presented as a function of the ratio of the individual single-phase pressure drops and ... 

A major complication, especially for separated flows, arises from the effect of slip. Slip occurs because the less dense and less viscous phase exhibits a lower resistance to flow, as well as expansion and acceleration of the gas phase as the pressure drops. The result is an increase in the local holdup of the more dense phase within the pipe (phase density, pm), as given by Eq. (15-11). A large number of expressions and correlations for the holdup or (equivalent) slip ratio have appeared in the literature, and the one deduced by Lockhart and Martinelli is shown in Fig. 15-7. Many of these slip models can be summarized in terms of a general equation of the form... 

For many applications, like chemical-vapor-deposition reactors, the semi-infinite outer flow is not an appropriate model. Reactors are often designed so that the incoming flow issues through a physical manifold that is parallel to the stagnation surface and separated by a fixed distance. Typically the manifolds (also called showerheads) are designed so that the axial velocity u is uniform, that is, independent of the radial position. Moreover, since the manifold is a solid material, the radial velocity at the manifold face is zero, due to the no-slip condition. One way to fabricate a showerhead manifold is to drill many small holes in a plate, thus causing a large pressure drop across the manifold relative to the pressure variations in the plenum upstream of the manifold and the reactor downstream of the manifold. A porous metal or ceramic plate would provide another way to fabricate the manifold. 

Without too much difficulty, we can extend the model to any screw design consisting of constant depth channels, and moderate tapers by using the taper correction factors of Eq. 6.4-4 separately for each section, and adding up the pressure drops (rises) according to Eq. 9.2.2. Thus, for a tapered channel, the drag and pressure flow terms are multiplied by the expressions 2/(1 + 0) and 2/ 0(l + CoX respectively, where 0 = Hq/H, with Hq... 

---