Pigford model

Emulsion phase in completely mixed state General solution—bubble phase free of solids (Orcutt model) In the following, the simplified Orcutt-Davidson-Pigford model (Orcutt et al., 1962) is presented. This model assumes in addition that the bubble phase is free of solids, and thus yh = 0. This means that the reaction takes place only in the emulsion (dense) phase. Then, the third term in eq. (3.516) disappears. The gas flow is in the inlet of the bed ,A, the particulate phase wfmA, and the bubble phase wswfmA, where A is the cross-sectional area of the bed and (eq. 3.476)... 

It has been stated that the effect ofyj,ub and yh is greatly dependent on the parameter kR, and tlius basically on the reaction coefficient. For a slow reaction, kR is much less than unity and the effect of/bub and yh is minimal, whereas the opposite holds for a fast reaction, where kR is greater than unity. Furthermore, the Orcutt-Davidson-Pigford model has proved on some occasions to deviate from experimental results, probably due to the choice of the value of /bub (Grace, 1984). At the same time, it is the only model that takes into account the effect of this parameter. 

The resulting conversion under the specified conditions is approximately 25%. Evaluate the rate coefficient of the reaction using the Orcutt—Davisdon Pigford model. Assume that the internal effectiveness factor is unity and the expansion factor zero. 

Model application The parameters of the Orcutt—Davidson—Pigford model (eq. 5.400) are the following ... 

The experimentally determined values of parameters Ki and K2 allow the Pigford model to be used to examine the extent of the spin-up zone on the disk surface. Before this can be done, however, an unambiguous mathematical definition for the spin-up zone is required. This was provided by Burns et al. as the location at which the radial velocity stops accelerating and starts following the decelerating profile described by the synchronized flow model as shown in Eq. (3). ... 

Numerical analysis of the Pigford model, combined with Eqs. (17) and (18), implied that the spin-up zone rs based on this definition should be given by... 

Furthermore, van Baten and Krishna [55] formulated an equation for the description of kia, which is based on the fundamental approach. The authors combined the Higbie penetration model to describe mass transfer from the two hemispherical caps and the Pigford model to describe the mass transfer to the liquid film [56]. Furthermore, they used CFD simulations for studying the separate influence of different parameters on the volumetric mass transfer coefficient and compared these results with the predictions of Equation 12.29. 

Emmert and Pigford (E2) have studied the reaction between carbon dioxide and aqueous solutions of monoethanolamine (MEA) and report that the reaction rate constant is 5400 liter/mole sec at 25°C. If it is assumed that MEA is present in excess, the reaction may be treated as pseudo first-order. This pseudo first-order reaction has been recently used by Johnson et al. (J4) to study the rate of absorption from single carbon dioxide bubbles under forced convection conditions, and the results were compared with their theoretical model. 

The only model that takes into consideration the fraction of the gas that flows through the emulsion phase is the Orcutt-Davidson-Pigford one and for the specific superficial velocity of 0.1 m/s, /bub = 0.86. The results of the models are close due to the fact that the flow... 

The feeder and injector produced a thin pencil-like p.c. stream which passed down through the hot zone. The total combustion air supplied was approximately 3 liters/min for the bituminous coals, giving between 10 and 25 percent excess air for p.c. feed rates of 0.24 to 0.28 g/min. The flow and heat transfer conditions were modeled using the methods described by Pigford (16) for conditions of superimposed natural and forced convection at very low mass flow rates. Particle residence times were calculated by summing the centerline gas velocity and terminal velocity using Stokes s law (17). The error introduced using this method should never have exceeded 10 percent, even when pyrite was tested and particle Reynold s numbers approached one. The residence times thus calculated were found to be between one and two seconds. 

The model and mathematics for the case of adsorbent regeneration, governed by equilibrium effects, is covered in many texts. The discussion by Sherwood, Pigford, and Wilke (1975) is especially thorough. The material balance between the concentration of dissolved material in the CO2 and the concentration in the solid activated carbon during the course of the desorption is given by the relation... 

The experimental evidence, as summarized by Sherwood, Pigford, and Wilke, indicates that the mass transfer coefficients are more nearly proportional to the molecular diffusivity to the square root power. Nevertheless, the film theory is used in the development of the working equations in this chapter, since the physical picture it depicts is simple and adequate. Actually, it is irrelevant, from a pragmatic point of view, what model is used to develop a working equation based on empirical mass or heat transfer coefficients that must, ultimately, be obtained from experimental data. 

An analysis of experimental results on the reaction of SOj with limestone using a grain model similar to the one discussed in this section was published by Pigford and Sliger [15]. 

Higashi, K. Pigford, T. H. (1980) Analytical models for migration of radionuclides in geologic sorbing media. J. Nucl. Sci Technol. 17,700-709. 

Benedict, M., and T. Pigford, Nuclear Chemical Engineering, McGraw-Hill, New York, 1957. Bogacki, M. B., and J. Szymanowski, Modeling of Extraction Equilibrium and Conputer Simulation... 

The authors show that when the first reaction is more nearly irreversible than the second and both reactions are rapid, the reactions occur at successive planes within the liquid. In this case, the first equilibrium constant is in fact greater, by a factor of about 14,000. Using this approach changes the mathematics somewhat from the single-plane model of Sherwood and Pigford [90]. 

The models discussed above will not be applied to the situation of transfer accompanied by reaction. We first use the two-film theory, then the surface renewal theory. The literature on the subject is overwhelming, and no attempt is made to be complete. Instead, the general concepts are synthesized and illustrated. More extensive coverage can be found in several textbooks more oriented toward gas absorption [Sherwood, Pigford and Wilke, 1975 Ramm, 1953 Astarita, 1967 Danckwerts, 1970 and Astarita et al., 1983]. 

Grain Models Snow et al (1988) Borgwardt,Bruce (1986) Marsh,Ulrichson (1985) Pigford,Sliger (1973) Szekelly,Evans (1970) Wen,Ishida (1973)... 

Numerical computation of the simulated moving bed system provides a second alternative description to the continuous countercurrent and equilibrium stage models (Barker et al. 1983, Ching 1983, Hashimito et at. 1983, Carta and Pigford 1986, and Ching etal. 1987). Each bed of total volume V is considered to be equivalent to a number, n, of ideal mixing cells in which the fluid and adsorbed phases, volumes Vl and Vs, respectively, are distributed according to... 

From the results of Section 3.3.3, it may be expected that the = 0 asymptote is valid for <0.1. In this regime the system becomes identical to the grain model presented earlier. On the other hand, when 10 the reaction of an individual grain is expected to be controlled by the intragrain diffusion. For this condition Pigford and Sliger [29] have proposed an approximate method of solution that is valid for low conversion of the solid. 

This insensitivity to details of the basic Pigford/Jackson model was to have a profound effect in cementing views on the nature of the fluidization process. It seemed that however the interaction between fluid and particles is described, the essential conclusion remains unchanged the uniform fluidized state remains intrinsically unstable. So when irrefutable experimental evidence for stable gas fluidization became available in the mid-1970s, the initial reaction was one of disbelief, soon to be followed by an earnest search for a way out of the dilemma. As far as the fluidization research community was concerned, the additional term in Walhs s formulation, which predicted just such an outcome, remained in the shadows... 

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