Reaction/diffusion model mixing theory

Let us consider a shallow fluidized bed combustor with multiple coal feeders which are used to reduce the lateral concentration gradient of coal (11). For simplicity, let us assume that the bed can be divided into N similar cylinders of radius R, each with a single feed point in the center. The assumption allows us to use the symmetrical properties of a cylindrical coordinate system and thus greatly reduce the difficulty of computation. The model proposed is based on the two phase theory of fluidization. Both diffusion and reaction resistances in combustion are considered, and the particle size distribution of coal is taken into account also. The assumptions of the model are (a) The bed consists of two phases, namely, the bubble and emulsion phases. The voidage of emulsion phase remains constant and is equal to that at incipient fluidization, and the flow of gas through the bed in excess of minimum fluidization passes through the bed in the form of bubbles (12). (b) The emulsion phase is well mixed in the axial... 

Several physicochemical models of ion exchange that link diffuse-layer theory and various models of surface adsorption exist (9, 10, 14, 15). The difficulty in calculating the diffuse-layer sorption in the presence of mixed electrolytes by using analytical methods, and the sometimes over simplified representation of surface sorption have hindered the development and application of these models. The advances in numerical solution techniques and representations of surface chemical reactions embodied in modem surface complexation mod-... 

A key idea of theories and algorithms to extend timescales is the use of short time trajectories to construct a probabilistic model of rate computed at much longer timescales [4-8]. From a physical perspective, long-time dynamics can come in many flavors however, it is typically a mix of two kinds of fundamental processes (1) diffusive and (2) activated dynamics. One class of slow processes typical to chemical reactions (activated dynamics) was the focus of theoreticians in the last half a century or so [9]. 

Another representative two-phase model is the one proposed by Partridge and Rowe (1966). In this model, the two-phase theory of Toomey and Johnstone (1952) is still used to estimate the visible gas flow, as in the model of Davidson and Harrison (1963). However, this model considers the gas interchange to occur at the cloud-emulsion interphase, i.e., the bubble and the cloud phase are considered to be well-mixed, the result being called bubble-cloud phase. The model thus interprets the flow distribution in terms of the bubble-cloud phase and the emulsion phase. With the inclusion of the clouds, the model also allows reactions to take place in the bubble-cloud phase. The rate of interphase mass transfer proposed in the model, however, considers the diffusive mechanism only (i.e., without throughflow) and is much lower than that used in the model of Davidson and Harrison (1963). 

In order to derive a good model for the reactor, kinetic data free from mixing, heat transfer and diffusion effects must be available. In contrast with thermodynamic data, which can now be found in data banks, or calculated from existing theories, kinetic data are very scarce in the literature. When they exist, they are mainly concerned with mechanistic studies performed in dilute media. Phenomenologic laws, expressing the rate of reaction as a function of observable quantities are lacking in most cases such as... 

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