Dispersion model Subject

A one-dimensional one-phase dispersion model subject to the Danckwerts boundary conditions has been used for a description of the dynamics of a nonisothermal nonadiabatic packed bed reactor. The dimensionless governing equations are ... 

Simpler optimization problems exist in which the process models represent flow through a single pipe, flow in parallel pipes, compressors, heat exchangers, and so on. Other flow optimization problems occur in chemical reactors, for which various types of process models have been proposed for the flow behavior, including well-mixed tanks, tanks with dead space and bypassing, plug flow vessels, dispersion models, and so on. This subject is treated in Chapter 14. 

Even with this enormous number of scientific papers on the subject, mathematically rigorous results on the subject are rare. Let us mention just ones aiming toward a rigorous justification of Taylor s dispersion model and its generalization to reactive flows. We could distinguish them by their approach... 

Although it also is subject to the limitations of a single characterizing parameter which is not well correlated, the Peclet number, the dispersion model predicts conversions or residence times unambiguously. For a reaction with rate equation rc = fcC , this model is represented by the differential equation... 

The two-phase theory of fluidization has been extensively used to describe fluidization (e.g., see Kunii and Levenspiel, Fluidization Engineering, 2d ed., Wiley, 1990). The fluidized bed is assumed to contain a bubble and an emulsion phase. The bubble phase may be modeled by a plug flow (or dispersion) model, and the emulsion phase is assumed to be well mixed and may be modeled as a CSTR. Correlations for the size of the bubbles and the heat and mass transport from the bubbles to the emulsion phase are available in Sec. 17 of this Handbook and in textbooks on the subject. Davidson and Harrison (Fluidization, 2d ed., Academic Press, 1985), Geldart (Gas Fluidization Technology, Wiley, 1986), Kunii and Levenspiel (Fluidization Engineering, Wiley, 1969), and Zenz (Fluidization and Fluid-Particle Systems, Pemm-Corp Publications, 1989) are good reference books. 

A useful description of mixing in bubble columns is provided by the dispersion model. The global mixing effects are generally characterized by the dispersion coefficients El and Eq of the two phases which are defined in analogy to Fick s law for diffusive transport. Dispersion in bubble columns has been the subject of many investigations which have recently been reviewed by Shah et al. (45). Particularly, plenty of data are available for liquid-phase dispersion. 

Several models have been suggested to simulate the behavior inside a reactor [53, 71, 72]. Accordingly, homogeneous flow models, which are the subject of this chapter, may be classified into (1) velocity profile model, for a reactor whose velocity profile is rather simple and describable by some mathematical expression, (2) dispersion model, which draws analogy between mixing and diffusion processes, and (3) compartmental model, which consists of a series of perfectly-mixed reactors, plug-flow reactors, dead water elements as well as recycle streams, by pass and cross flow etc., in order to describe a non-ideal flow reactor. 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

As mentioned earlier, the dispersion model, like other models, is subject to two errors inadequate representation of the RTD and improper allowance for the extent of micromixing. We can evaluate the first error for a specific case by using the dispersion model to obtain an alternate solution for Example 6-5. The second error does not exist for first-order kinetics. However, the maximum value of this error for second- and halforder kinetics was indicated in Sec. 6-8. 

As has been stressed throughout this book, risk has both objective and subjective elements. The objective part of the work means that those working in the area of risk management need to be numerate they need to be comfortable with a variety of quantitative topics such as gas dispersion modeling, the development of F—N curves, and the use of Boolean algebra. 

Flash fire models are also subject to similar dispersion model errors present in VCE calculations. 

Models of BCR can be developed on the basis of various view points. The mathematical structure of the model equations is mainly determined by the residence time distribution of the phases, the reaction kinetics, the number of reactive species involved in the process, and the absorption-reaction regime (slow or fast reaction in comparison to mass transfer rate). One can anticipate that the gas phase as well as the liquid phase can be either completely backmixed (CSTR), partially mixed, as described by the axial dispersion model (ADM), or unmixed (PFR). Thus, it is possible to construct a model matrix as shown in Fig. 3. This matrix refers only to the gaseous key reactant (A) which is subjected to interphase mass transfer and undergoes chemical reaction in the liquid phase. The mass balances of the gaseous reactant A are the starting point of the model development. By solving the mass balances for A alone, it is often possible to calculate conversions and space-time-yields of the other reactive species which are only present in the liquid phase. Heat effects can be estimated, as well. It is, however, assumed that the temperature is constant throughout the reactor volume. Hence, isothermal models can be applied. 

Identify the flow pattern of the prototype system by subjecting it to an impulse, step, or sinusoidal disturbance by injection of a tracer material as reviewed in Chapter 8. The result is classified as either complete mixing, plug flow, and an option between a dispersion, cascade, or combined model. 

The electrodes in the direct methanol fuel cell (DMFC) (i.e. the anode for oxidising the fuel and the cathode for the reduction of oxygen) are based on finely divided Pt dispersed onto a porous carbon support, and the electro-oxidation of methanol at a polycrystalline Pt electrode as a model for the DMFC has been the subject of numerous electrochemical studies dating back to the early years ot the 20th century. In this particular section, the discussion is restricted to the identity of the species that result from the chemisorption of methanol at Pt in acid electrolyte. This is principally because (i) the identity of the catalytic poison formed during the chemisorption of methanol has been a source of controversy for many years, and (ii) the advent of in situ IR culminated in this controversy being resolved. 

A critical property of minimum protocells in the prebiotic environment would be their ability to sequester other molecules, including macromolecules. [142] In 1982, Deamer and Barchfeld [143] subjected phospholipid vesides to dehydration-rehydration cycles in the presence of either monomeric 6-carboxyfluorescein molecules or polymeric salmon sperm DNA molecules as extraneous solutes. The experiment modeled a prebiotic tidal pool containing dilute dispersions of phospholipids in the presence of external solutes, with the dehydration-rehydration cydes representing episodic dry and wet eras. They found that the vesides formed after rehydration... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

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