Heat transfer heterogeneous model

Mathematical models of packed bed reactors can be classified into two broad categories (1) one-phase, or pseudohomogeneous, models in which the reactor bed is approximated as a quasi-homogeneous medium and (2) two-phase, or heterogeneous, models in which the catalyst and fluid phases and the heat and mass transfer between phases are treated explicitly. Although the... 

Effective thermal conductivities and heat transfer coefficients are given by De Wasch and Froment (1971) for the solid and gas phases in a heterogeneous packed bed model. Representative values for Peclet numbers in a packed bed reactor are given by Carberry (1976) and Mears (1976). Values for Peclet numbers from 0.5 to 200 were used throughout the simulations. 

De Wasch and Froment (1971) and Hoiberg et. al. (1971) published the first two-dimensional packed bed reactor models that distinguished between conditions in the fluid and on the solid. The basic emphasis of the work by De Wasch and Froment (1971) was the comparison of simple homogeneous and heterogeneous models and the relationships between lumped heat transfer parameters (wall heat transfer coefficient and thermal conductivity) and the effective parameters in the gas and solid phases. Hoiberg et al. (1971)... 

Heat Transfer by Conduction. In the theoretical analysis of steady state, heterogeneous combustion as encountered in the burning of a liquid droplet, a spherically symmetric model is employed with a finite cold boundary as a flame holder corresponding to the liquid vapor interface. To permit a detailed analysis of the combustion process the following assumptions are made in the definition of the mathematical model ... 

It is important to have the correct set of variables specified as independent and dependent to meet the modeling objectives. For monitoring objectives observed conditions, including the aforementioned independent variables (FICs, TICs, etc.) and many of the "normally" (for simulation and optimization cases) dependent variables (FIs, TIs, etc.) are specified as independent, while numerous equipment performance parameters are specified as dependent. These equipment performance parameters include heat exchanger heat transfer coefficients, heterogeneous catalyst "activities" (representing the relative number of active sites), distillation column efficiencies, and similar parameters for compressors, gas and steam turbines, resistance-to-flow parameters (indicated by pressure drops), as well as many others. These equipment performance parameters are independent in simulation and optimization model executions. 

Previous one-phase continuum heat transfer models (1), (5), (10), (11), which are all based upon "large diameter tube" heat transfer data, fail to extrapolate to narrow diameter tubes. These equations systematically underpredict the overall heat transfer coefficient by 40 - 50%, on average. When allowance is made in the one-phase model for the effect of tube diameter on the apparent solid conductivity (kr>s), Eqn. (7), the mean error is reduced to 18%. However, the best predictions by far (to within 6.8% mean error) are obtained from the heterogeneous model equations. 

We consider methods for describing how molecules interact with aerosol particles and how to obtain molecular properties and rate constants of relevance when studying the molecular level mechanisms for the formation of aerosol particles and how these provide the basis for heterogeneous chemistry. For understanding mass and heat transfer to and from aerosol particles we need to focus on the processes related to a gas molecule as it approaches the surface of an aerosol particle. A macroscopic property related to these processes is the sticking probabilities/ mass accommodation coefficients that are used when modelling evaporation. 

The simplest heterogeneous model is one with plug-flow in the fluid phase, mass and heat transfer between the fluid and solid phases, and surface catalytic reaction on the solid — if the catalyst is indeed deposited near the pellet external surface. 

The preceding sections show that catalytic fuel combustion is a process in which complex kinetics for heterogeneous and homogeneous reactions are combined with mass and heat transfer effects. This leads to difficulties in predicting the behavior of combustion catalysts under real conditions. Therefore, mathematical modeling is a powerful tool to assist experimental work, to interpret results, and to aid in the design of catalytic combustors. 

Finally, it should be noted that one of the other approaches being examined for heterogeneous systems relies on physically based representations for reaction and mass and heat transfer, which are commonly treated as aggregated models. Also, Friedler et al. (1991, 1992, 1993) have provided graph-theoretic algorithms to systematically derive superstructures for process nkworks, given a list of unit operations and process streams. 

The modeling of heterogeneous polymerization systems is generally more complicated than that of the homogenous systems because mass and heat transfer effects between two or more immiscible phases must be considered. Industrially important heterogeneous polymerization reactions include emulsion polymerization, suspension polymerization, precipitation polymerization, and solid-catalyzed olefin polymerization. The general polymerization rate equation is represented simply as... 

We may first divide tubular reactors into those designed for homogeneous reactions, and therefore basically just an empty tube, and those designed for a heterogeneously catalyzed reaction, and hence to be packed with a catalyst. Both types can of course be operated adiabatically, and it was the simplest model of these that we discussed in the last chapter. If the temperature of the reactor is to be controlled this is through the wall, and the associated problems of heat transfer now arise. These include transfer at the wall and subsequent radial diffusion across the flowing reactants. In the empty tubular reactor there may be considerable variations in flow rate across the tube. For example, in the slow laminar flow the fluid... 

For the heterogeneous one particle model, we see from (11.27) that the source term equals the conventional particle-bulk gas phase heat transfer term, defined by ... 

For the two-particle heterogeneous model the source term in the bulk gas phase equation is given as the sum of the conventional particle-bulk phase heat transfer terms for both the catalyst and C02-acceptor particles. 

The simplest heterogeneous model is that with plug flow in the fluid phase and only external mass and heat transfer resistances between the bulk fluid and the catalyst surface. More complex fluid phase behaviour can be accommodated by including axial and radial dispersion mechanisms into the mode). If tJie reactor is non-adiabatic, radial dispersion is usually more important. 

Attempts have been made to develop two-dimensional heterogeneous models (McGreavy and Cresswell, 1968, 1969, Deasch and Froment, 1971). McGreavy and Cresswell proceded by adding to the one-dimensional heterogeneous model the terms accounting for radial heat and mass transfer in the bed. 

This result shows the order of magnitude of the improvement of reactor yield for catalyst decay problems when the more realistic heterogeneous model is used in the optimization of the reactor performance. This improvement in performance depends upon the values of the mass and heat transfer parameters of the system. For systems with high degree of diflfusional limitations for mass and heat transfer, the use of a heterogeneous model in optimization is a must in order to obtain a true optimal performance of the reactor. 

In the heterogeneous model for the high temperature shift converter, the effectiveness factor accounts for the external and the intraparticle mass and heat transfer resistances (e.g. Satterfield and Roberts, 1968 Hutchings and Carberry, 1966 Petersen et ai, 1970 Chu and Hougen, 1972) and is multiplied by the rate of reaction at bulk conditions to get the actual rate of reaction. 

The heterogeneous model developed in this section takes into account interphase as well as intraparticle mass and heat transfer resistances. The following classical simplifying assumptions are used for the modelling of the reactor. 

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