Multiphase control volume

As mentioned, the pseudocontinuum approach can be used for the multiphase flow analysis only when the control volume contains enough particles that the volume average is of statistical significance, and at the same time, the control volume is small enough to ensure the validity of the continuum assumption. Thus, in order to have a statistically meaningful phase average, there exists a minimum averaging volume. 

The main approach for modelling multiphase flows has been through solving conservation equations described in terms of Eulerian phase-averaged mean quantities - the two-fluid approach [665], The Eulerian mean velocity in a control volume V (such as the volume within the perimeter S of the cloud of particles) is defined as the velocity ux (for each component) averaged over the volume occupied by the fluid (ie the fluid space between the bodies),... 

The physical meaning of the terms (or group of terms) in the entropy equation is not always obvious. However, the term on the LHS denotes the rate of accumulation of entropy within the control volume per unit volume. On the RHS the entropy flow terms included in show that for open systems the entropy flow consists of two parts one is the reduced heat flow the other is connected with the diffusion flows of matter jc, Secondly, the entropy production terms included in totai demonstrates that the entropy production contains four different contributions. (The third term on the RHS vanishes by use of the continuity equation, but retained for the purpose of indicating possible contributions from the interfacial mass transfer in multiphase flows, discussed later). The first term in totai arises from heat fluxes as conduction and radiation, the third from diffusion, the fourth is connected to the gradients of the velocity field, giving rise to viscous flow, and the fifth is due to chemical reactions. 

But neither the number of phases nor the number of reactions in the control volume affect the number of material and energy conduits. Recall, this insensitivity of V to internal constraints also occurs for closed systems (9.1.1) for V in closed multiphase systems is the same as (3.1.3) for closed one-phase systems. 

The interface segments are then determined based on the volume fraction in each control volume. In the VOF method, the interface is usually simulated as linear segments or planers in control volumes. In this way. the VOF method conserves the volume of the fluid and describes the interfaces within the multiphase flow. Once the interface has been determined, the pressure and velocity can be updated with constitutive equations. The evolution of the topology can represent multiphase phenomena, such as breaking up and joint of the flow front. 

The boundary conditions are essentially the same for all the methods reviewed above. However, the multiphase and multi-domain methods need extra boundary conditions due to the additional equations solved. Therefore the general boundary conditions employed in control volume methods and FEMs are discussed together and additional boundary conditions required for other methods are given in subsequent sections. 

Why are the CSTRs worth considering at all They are more expensive per unit volume and less efficient as chemical reactors (except for autocatalysis). In fact, CSTRs are useful for some multiphase reactions, but that is not the situation here. Their potential justification in this example is temperature control. BoiUng (autorefrigerated) reactors can be kept precisely at the desired temperature. The shell-and-tube reactors cost less but offer less effective temperature control. Adiabatic reactors have no control at all, except that can be set. 

Multiphase catalytic reactions, such as catalytic hydrogenations and oxidations are important in academic research laboratories and chemical and pharmaceutical industries alike. The reaction times are often long because of poor mixing and interactions between the different phases. The use of gaseous reagents itself may cause various additional problems (see above). As mentioned previously, continuous-flow microreactors ensure higher reaction rates due to an increased surface-to-volume ratio and allow for the careful control of temperature and residence time. 

Ruy et al. have performed a similar reaction under microreactor conditions in a multiphase solvent system containing an ionic liquid as the catalyst carrier and reaction promoter [35]. Their system consisted of two T-shaped micromixers (i.d. 1,000 and 400 pm) and a capillary stainless steel tube as an RTU (1,000 pm i.d. and 18 m length, giving a 14.1 ml volume), equipped with pumps and control valves. Under the optimized conditions, Pd-catalysed carbonylation of aromatic iodides in the presence of a secondary amine provided only the double carbonylated product, ot-ketoamide, while the amide obtained by the single carbonylation was observed in high quantities only when the reaction was performed in batch (Scheme 13). 

A variety of factors affect the horizontal and vertical migration of PAHs, including contaminant volume and viscosity, temperature, land contour, plant cover, and soil composition (Morgan Watkinson, 1989)- Vertical movement occurs as a multiphase flow that will be controlled by soil chemistry and structure, pore size, and water content. For example, non-reactive small molecules (i.e., not PAHs) penetrate very rapidly through dry soils and migration is faster in clays than in loams due to the increased porosity of the clays. Once intercalated, however, sorbed PAHs are essentially immobilized. Mobility of oily hydrophobic substances can potentially be enhanced by the biosurfactant-production capability of bacteria (Zajic et al., 1974) but clear demonstrations of this effect are rare. This is discussed below in more detail (see Section 5 5). 

As will be discussed in detail in Chapter 20, the properties of multiphase materials are generally controlled mainly by the continuous component(s). The discontinuous component(s) mainly cause a shift of each property relative to this baseline. For example, if a rigid component and a much softer component are each present at 50% by volume, then their composite will have a significantly higher tensile modulus if the rigid component is continuous and the soft component is discontinuous than vice versa, while it will have a tensile modulus that is intermediate between these two extremes if the two phases are co-continuous. 

The morphology of latex particles is controlled by the thermodynamic and kinetic factors. The thermodynamic factors determine the ultimate stability of the multiphase system, inherent in the production of a composite latex particle, while the kinetic factors determine the ease with which such a thermodynamically favored state can be achieved. The parameters affecting the thermodynamics of the system include the particle surface polarity, the relative phase volumes, and the core particle size. The parameters affecting the kinetics of the morphological development include the mode of monomer addition (monomer starved or batch) and the use of crosslinking agents. Of course, crosslinked core/shell latexes constitute IPNs, see Section 6.4.1. 

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