Two-Phase Modeling

Lamellar morphology variables in semicrystalline polymers can be estimated from the correlation and interface distribution fiinctions using a two-phase model. The analysis of a correlation function by the two-phase model has been demonstrated in detail before [30,11] The thicknesses of the two constituent phases (crystal and amorphous) can be extracted by several approaches described by Strobl and Schneider [32]. For example, one approach is based on the following relationship ... 

The two-phase model is used mostly to check very exothermic or endothermic reactions, to calculate the temperature difference between catalyst and gas at extreme conditions, or when accounting for changes in both phases is needed. This model was applied to the two-phase counter-... 

The basic phenomenon was observed in modeling studies by Bjoreskov and Slinko (1965) that sudden increase in inlet temperature caused a transient drop of the peak temperature. The wrong-way response name was given by Mechta et al (1981) after they experienced the opposite a sudden of inlet temperature resulted in an increase of the peak temperature (which may eventually cause a runaway.) The work used a pseudo-homogeneous reaction model and explained the phenomenon by the different speeds of transient response in gas and solid. The example in the last part of Chapter 7.4 explained the speed difference by the large difference in heat capacity of gas and solid phases. For this a two-phase model is needed and spatial and time changes must be followed. 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

Fig. la and b. Principal sections of the Hashin two-phase model and its respective three-layer unfolding model for a typical particulate composite... 

Although the results of this model are satisfactory, the complexity of the numerical solution of a system of seven equations makes the model rather inexpedient and unstable. However, the model presents an intrinsic flexibility and it is appropriate to yield better results than any two-phase model. 

Transition dipole moment 88 Transverse relaxation time 31, 32, 33, 44 Twinning 126 Two-phase model 129 Two-term models 149 ----unfolding model 183,185... 

A well-defined bed of particles does not exist in the fast-fluidization regime. Instead, the particles are distributed more or less uniformly throughout the reactor. The two-phase model does not apply. Typically, the cracking reactor is described with a pseudohomogeneous, axial dispersion model. The maximum contact time in such a reactor is quite limited because of the low catalyst densities and high gas velocities that prevail in a fast-fluidized or transport-line reactor. Thus, the reaction must be fast, or low conversions must be acceptable. Also, the catalyst must be quite robust to minimize particle attrition. 

FIG. 10 The co-ordinate system used to define the two-phase model for SECMIT. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

Agee, L. J., S. Banerjie, R. B. Duffey, and E. D. Hughes, 1978, Some Aspects of Two-Phase Models for Two-Phase Flow and Their Numerical Solutions, in Transient Two-Phase Flow, Proc. 2nd Specialists Meeting, OECD Committee for the Safety of Nuclear Installations, Paris, Vol. 1, pp. 27-58. (3) Ahmadi, G., and D. Ma, 1990, A Thermodynamical Formulation for Disposed Multiphase Turbulent Flows I. Basic Theory, Int. J. Multiphase Flow 16 323. (3)... 

Because of the inadequacies of the aforementioned models, a number of papers in the 1950s and 1960s developed alternative mathematical descriptions of fluidized beds that explicitly divided the reactor contents into two phases, a bubble phase and an emulsion or dense phase. The bubble or lean phase is presumed to be essentially free of solids so that little, if any, reaction occurs in this portion of the bed. Reaction takes place within the dense phase, where virtually all of the solid catalyst particles are found. This phase may also be referred to as a particulate phase, an interstitial phase, or an emulsion phase by various authors. Figure 12.19 is a schematic representation of two phase models of fluidized beds. Some models also define a cloud phase as the region of space surrounding the bubble that acts as a source and a sink for gas exchange with the bubble. 

Basic two-phase model of fluidized bed. Open arrows indicate movement of solids. (Adapted from Fluidization Engineering by D. Kunii and O. Levenspiel. Copyright 1969. Reprinted by permission of John Wiley and Sons, Inc.)... 

Volume and mass-based expressions for the degree of crystallinity are easily derived from the experimentally measured density (p) of a semi-crystalline polymer. The method is based on an ideal crystalline and liquid-like two-phase model and assumes additivity of the volume corresponding to each phase... 

It has also been inferred that differences found between crystallinities measured by density and those from heat of fusion by DSC area determination, as given for polyethylenes in the example of Figure 4 [72], may be related to baseline uncertainties, or not accounting for the temperature correction of AHc. Given that similar differences in crystallinity from density and heat of fusion were reported for isotactic poly(propylene) [43] and polyfaryl ether ether ketone ketone), PEEKK [73], other features of phase structure that deviate from the two-phase model may be involved in the crystallinity discrepancy. 

A two-phase model best describes the half-life of THC. Within the first phase (a), levels of THC fall to 5-10% of initial levels within 1 hour this is because the drug has been metabolised by the liver and removed from the plasma into lipid tissue. Research has... 

The angular position of the peak then remained almost unchanged. No other peak appeared in the 0.5composition domain, where stage 1 is expected to form according to the stage 2 - stage 1 two-phase model. Such a model has been widely used in the literature to explain the flat voltage... 

Van Swaaij, W. P. M., andZuiderweg, F. J., Investigation of Ozone Decomposition in Fluidized Beds on the Basis of a Two-phase Model, Chemical Reaction Eng., Proc. 5th European/2ndInt. Symp. Chem. Reaction Eng., Elsevier, Amsterdam/London/New York (1972)... 

Geesy, G. G., Bremer, P. J., Smith, J. J., Muegge, M. and Jang, L. K. (1992). Two-phase model for describing the interactions between copper ions and exopolymers from Alteromonas atlantica, Can. J. Microbiol., 38, 785-793. 

Adsorption on Inorganic Surfaces. McCarty et al. (55) and Karickhoff (1 ) proposed a two-phase model for sorption onto mineral surfaces and organic matter partitioning. This model hypothesizes that sorption to inorganic surfaces may occur simultaneously with the accepted partitioning into the organic matter. Assuming additivity of these two contributions, Kj can be estimated from... 

In a biphasic system, the same rules as above apply, however, the rate of the reaction and the position of the equilibrium are determined by the concentration of the reactants and products in the phase where the reaction takes place, rather than their overall concentration in the system. Exactly where the reaction actually takes place is still a matter of debate, with two locations proposed, specifically, at the interfacial layer between the two phases (model 1) and in the bulk of the catalyst-containing phase (model 2), as shown in Figure 2.9. 

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