Dispersion model assumptions

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

Fig. 9.16 Basic assumptions of the one-dimensional dispersion model. The dispersion of the residence time of the phases is modeled by superimposing the plug profile of the basic flow with a stochastic dispersion process in axial direction.

The method of lines and system identification are not restricted in their applicability. System identification is preferred because the order of the resulting state space model is significantly lower. Another advantage of system Identification is that it can directly be applied on experimental data without complicated analysis to determine the kinetic parameters. Furthermore, no model assumptions are required with respect to the form of the kinetic expressions, attrition, agglomeration, the occurence of growth rate dispersion, etc. 

On the assumption that the closed vessel of Example 11.1, Chapter 11, is well represented by the dispersion model, calculate the vessel dispersion number D/uL. The C versus t tracer response of this vessel is... 

The assumption that Cou = 1 in equation (6.43) is really only accurate when Pe > 10. The only way to apply this tracer curve to the plug flow with dispersion model while Cou 1 would be to route each portion of the tracer curve through the reactor. With Pe = 9.4, this solution will be close, although stiU an approximation. 

Completion of emission inventory and modeling assumption improvements to match dispersion model source impacts to CMB results and... 

The development of new source apportionment methods have, for the first time, led to the development of regional particulate control strategies. Source impacts assigned using a chemical mass balance (CMB) model have been used in association with airshed dispersion models to identify emission inventory deficiencies and Improve modeling assumptions. 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

In this report, a kinetic model based on the solid film linear driving force assumption is used. Unlike the equilibrium-dispersive model, which lumps all transfer and kinetic effects into an effective dispersion term, the kinetic model is effective when the column efficiency is low and the effects of column kinetics are significant. 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

To couple the intrinsic coke burning kinetics described in Section II.B with gas and solid flow models, the simplest approach is provided by the one-dimensional solid dispersion model based on following assumptions ... 

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

Early attempts to approximate gas-solid contacting in fluid catalyst beds were based on the assumption either of isothermal plug flow of the fluidizing gas through the bed with the catalyst uniformly distributed or of isothermal complete mixing of the gas within the bed. The simple dispersion model, falling between the above two cases, was also used (G8, R4). Evidence from both large-scale and laboratory observations (G9a, L12),... 

UDM (Urban Dispersion Model) (Hall et al., 2002 [248]) is a widely-used model developed by the UK Defence Science and Technology Laboratory (DSTL) based on assumptions of a Gaussian shape and empirical parameterizations developed from special field and laboratory experiments involving obstacle arrays. 

In the following the most relevant models for liquid chromatography are derived in a bottom-up procedure related to Fig. 6.2. To illustrate the difference between these models their specific assumptions are discussed and the level of accuracy and their field of application are pointed out. The mass balances are completed by their boundary conditions (Section 6.2.7). For the favored transport dispersive model a dimensionless representation will also be presented. 

If it is assumed that the equilibrium between the two phases is instantaneous and, at the same time, that axial dispersion is negligible, the column efficiency is infinite. This set of assumptions defines the ideal model of chromatography, which was first described by Wicke [3] and Wilson [4], then abundantly studied [5-7,32-39] and solved in a number of cases [7,33,36,40,41]. In Section 2.2.2, which deals with the equilibrium-dispersive model, it is shown how small deviations from equilibrium can be handled while retaining the simplicity of Eq. 2.4 and of the ideal model. 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

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