Mixtures, continuous

One major question of interest is how much asphaltene will flocculate out under certain conditions. Since the system under study consist generally of a mixture of oil, aromatics, resins, and asphaltenes it may be possible to consider each of the constituents of this system as a continuous or discrete mixture (depending on the number of its components) interacting with each other as pseudo-pure-components. The theory of continuous mixtures (24), and the statistical mechanical theory of monomer/polymer solutions, and the theory of colloidal aggregations and solutions are utilized in our laboratories to analyze and predict the phase behavior and other properties of this system. 

Figure 3. Molecular weight distributions of asphaltenes before and after flocculation predicted by our continuous mixture model.

It is critical that the mixture of hybridization solution and radioactive-labeled probe touches the whole surface of the array and that the movement of the container provides for continuous mixture. 

If the number of components is very large, a mixture can be regarded as continuous and sharp distinctions between individual components are not made. Methods for dealing with stoichiometry, thermodynamics and kinetics for continuous mixtures are discussed by Aris and Gavalas [33]. An indication is given that rules for grouping in such mixtures depend on the nature of the reaction scheme. Wei and Kuo [34] considered ways in which species in a multicomponent reaction mixture could be lumped when the reaction network was composed of first-... 

Example 17. Continuous Mixtures and Parallel Gray-Scott Reactions... 

The parametric approach, which is not strictly needed for a single Gray-Scott reaction, works very well for an arbitrary number of parallel reactions and for continuous mixtures. Figure 16 shows a case of two parallel reactions for which an isola and a mushroom coexist. Because the notions of continuous mixtures and reactions will be treated in Chapter 8, G H and in the group of papers listed in the Index of Subjects in Publications under the heading Continuous mixtures, we can be very brief and start with the nondimensional equations. Let x be the index of the mixture whose species are /4(x). The steady-state concentration of the material with index in (x, x + dx) is V(x)dx, the feed concentration a(x)dx and the conversion U(x) = 1 - V/(x)/a(x), the last being defined only for values of x for which a(x) is not zero. B, the autocatalytic agent, forms itself as an undifferentiated product whose concentration is W. The rate of the first reaction, and hence p,(x), depends on the... 

Thus, the parametric approach works well for continuous mixtures and we have only to calculate integrals, which, except in particular cases, we may have to calculate using finite sums. 

I am sorry that there is not space for the paper [62], which was based on Gavalas dissertation, for it laid the foundations of continuous mixture kinetics of the first order. When Astarita and Ocone revived the subject again in the late 1980s, the emphasis was on underlying kinetics other than first order. Reprint G is a paper from the second period, the first being represented only by the last section of Reprint E. Papers on continuous mixtures may be traced through the Index of Subjects in Publications. 

Reprint H is a late paper written as a festschrift paper in honor of Davidson s retirement. It combines the continuous mixture techniques with the model reduction method given in Chapter 2, the section entitled Scaling and Partial Solution in Linear Systems. A closely similar paper was requested... 

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

It is better to define the continuous mixture in terms of an index variable (Aris and Gavalas, 1966 Astarita and Ocone, 1988) rather than to use the dimensional rate constant k. Let x be an index variable, which without loss of generality we can take to be in the interval [0, °°). The initial concentration of material characterized by indices in the interval (x, x + dx) is... 

Ho and Aris (1987) argued that any formulation of reaction in continuous mixtures must satisfy the single-component identity (SCI), namely that it should reduce to the kinetics of a single component when the mixture is pure. This is true of Eq. 29, for with/(x) = S(x - x0), U(t) = V(x0t). The corresponding H(x, y) = discrete component each satisfying the kinetic law given by G. We see that this is... 

Lumping Coupled Nonlinear Reactions in Continuous Mixtures, AIChE J. 35, 533... 

The simplest model of a bubbling fluidized bed, with uniform bubbles exchanging matter with a dense phase of catalytic particles which promote a continuum of parallel first order reactions is considered. It is shown that the system behaves like a stirred tank with two feeds the one, direct at the inlet the other, distributed from the bubble train. The basic results can be extended to cases of catalyst replacement for a single reactant and to Astarita s uniform kinetics for the continuous mixture. 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

We wish to see what the overall conversion of a continuous mixture will be, but, first, we have to ask which parameters will depend on jc, the index variable of the continuous mixture. Clearly k the rate constant in the Damkohler number will be a function of jc, and, if monotonic, can be put equal to Da.x. The parameter /3 is clearly hydrodynamic and so, for the most part, are the terms in the Davidson number. The only term in the equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity,... 

If the decay cannot be expressed solely as a function of age, the linearity is lost and the generalization to continuous mixtures is no longer possible. 

A good example is afforded by Langmuir kinetics of a continuous mixture,... 

One of us (RA) is indebted to the PRF of the ACS for continued support of an ongoing investigation of reactions in continuous mixtures (PRF25133-AC7E). The figures and the calculations that lie behind them were done by Paolo Cicarelli. 

Gavalas, G. and Aris, R., 1966, On the theory of reactions in continuous mixtures. Phil Trans Roy Soc, A260 351. As far as I know this is the first use of continuous mixtures in the context of chemical reaction. They had been used in distillation by Amundson and Acrivos, 1955, Chem Eng Sci, 4 29, in froth flotation by Loveday, 1966, Inst Min and Metall Trans. C75 219 and in thermodynamics by de Donder, 1931, L Afflnite (Gauthier-Villars, Paris). 

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