Longitudinal dispersion model

We can characterize the mixed systems most easily in terms of the longitudinal dispersion model or in terms of the cascade of stirred tank reactors model. The maximum amount of mixing occurs for the cases where Q)L = oo or n = 1. In general, for reaction orders greater than unity, these models place a lower limit on the conversion that will be obtained in an actual reactor. The applications of these models are treated in Sections 11.2.2 and 11.2.3. 

The Longitudinal Dispersion Model in the Presence of Chemical Reaction... 

In Section 11.1.3.1 we considered the longitudinal dispersion model for flow in tubular reactors and indicated how one may employ tracer measurements to determine the magnitude of the dispersion parameter used in the model. In this section we will consider the problem of determining the conversion that will be attained when the model reactor operates at steady state. We will proceed by writing a material balance on a reactant species A using a tubular reactor. The mass balance over a reactor element of length AZ becomes ... 

Illustration 11.6 indicates how the longitudinal dispersion model may be used to predict reactor performance. 

The influence of dispersion on the yield of an intermediate produced in a series of consecutive reactions has also been studied. When Tt /uL is less than 0.05, Tichacek s results (22) indicate that the fractional decrease in the maximum amount of intermediate formed relative to plug flow conditions is approximated by T) /uL itself. Results obtained at higher dispersion numbers are given in the original article. Douglas and Bischoff (23) considered the influence of volumetric expansion effects on the yields obtained with dispersion. Illustration 11.6 indicates how the longitudinal dispersion model may be used to predict reactor performance. 

Determine the conversion predicted using the longitudinal dispersion model. Use open-open boundary conditions. 

How does the value of L/u corresponding to the maximum concentration of species B correspond to the space time determined in part (a) What is the ratio of the maximum effluent concentration of species B predicted using the longitudinal dispersion model to that predicted using the PFR model Which conversions of species A correspond to the maxima for the two reactor models Comment. 

This model is referred to as the axial dispersed plug flow model or the longitudinal dispersed plug flow model. (Dg)j. ean be negleeted relative to (Dg)[ when the ratio of eolumn diameter to length is very small and the flow is in the turbulent regime. This model is widely used for ehemieal reaetors and other eontaeting deviees. 

It is clearly visible that longitudinal cross sections of the spots are very similar to the peak profiles shown in Figure 2.1, Figure 2.2, and Figure 2.3 and calculated with the equihbrium-dispersive model (Equation 2.21) ... 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

Equations 12.7.48 and 12.7.39 provide the simplest one-dimensional mathematical model of tubular fixed bed reactor behavior. They neglect longitudinal dispersion of both matter and energy and, in essence, are completely equivalent to the plug flow model for homogeneous reactors that was examined in some detail in Chapters 8 to 10. Various simplifications in these equations will occur for different constraints on the energy transfer to or from the reactor. Normally, equations 12.7.48 and 12.7.39... 

A more general one-dimensional model of tubular, packed bed reactors is contained within equations 12.7.38 and 12.7.47. These equations include all of the elements of the simple model discussed above and, in addition, account for the longitudinal dispersion of both thermal... 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

Rivers are close to the perfect environmental flow for describing the flow as plug flow with dispersion. The flow is confined in the transverse and vertical directions, such that a cross-sectional mean velocity and concentration can be easily defined. In addition, there is less variation in rivers than there is, for example, in estuaries or reactors - both of which are also described by the plug flow with dispersion model. For that reason, the numerous tracer tests that have been made in rivers are useful to characterize longitudinal dispersion coefficient for use in untested river reaches. A sampling of the dispersion coefficients at various river reaches that were... 

Note The travel time calculated by this simple model is not quite correct. More realistic results are obtained by including longitudinal dispersion as discussed in the next section. 

We have seen that the basic P model has the form of a first-order partial differential Eq. (22) describing each narrow slice as a little batch reactor being transported through the reactor at constant speed. This equation was so elementary that it could be solved at sight in Eq. (30). When we added a longitudinal dispersion term governed by Fick s law and took the steady state, Eq. (40), we had a second-order o.d.e. with controversial boundary conditions. This is the model with ( ) = c(z)lcm and Pe = vLID, Da = kL/v,... 

However, there is a severe disadvantage with one-dimensional models they do not take into account the dilution due to the transversal dispersion. Consequently a mass M, that is not susceptible to any chemical reaction, occurs blurred at a point x downstream from xo (the location of M input) due to longitudinal dispersion. The dispersion leads to a smaller maximum concentration, however, and the mass integral equals the mass added at xo. Thus, the impulse of mass remains constant along any simulated one-dimensional distance. 

Johnson T. M. and DePaolo D. J. (1996) Reaction-transport models for radiocarbon in groundwater the effects of longitudinal dispersion and the use of Sr isotope ratios to correct for water-rock interaction. Water Resour. Res. 32, 2203-2212. 

Latinen, G. A., Stockton, F. D., Application of the Diffusion Model to Longitudinal Dispersion in Flow Systems, A. I. Ch. E. National Meeting, St. Paul, Minn., September 1959... 

A gas bubble column is taken here as a model equipment undergoing longitudinal dispersion of the continuous phase. The theory obtained is equally applicable to a fluidized catalyst bed of good fluidity exhibiting similar flow properties. The following procedure is from Miyauchi (M27). 

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