Axial dispersion flow model

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

As discussed in Sec. 4.3.6, the axial dispersion flow model is given by... 

From the axial dispersion flow model the component balance equation is 9Ca 9Ca 3 Ca, p... 

The washing of filter cake is carried out to remove liquid impurities from valuable solid product or to increase recovery of valuable filtrates from the cake. Wakeman (1990) has shown that the axial dispersion flow model, as developed in Sec. 4.3.6, provides a fundamental description of cake washing. It takes into account such situations as non-uniformities in the liquid flow pattern, non-uniform porosity distributions, the initial spread of washing liquid onto the topmost surface of the filter cake and the desorption of solute from the solid surfaces. 

In this section, we apply the axial dispersion flow model (or DPF model) of Section 19.4.2 to design or assess the performance of a reactor with nonideal flow. We consider, for example, the effect of axial dispersion on the concentration profile of a species, or its fractional conversion at the reactor outlet. For simplicity, we assume steady-state, isothermal operation for a simple system of constant density reacting according to A - products. 

The washing of filter cake is carried out to remove liquid impurities from the valuable solid filter cake or to increase the recovery of valuable filtrate. Wake-man (1990) has shown that the axial dispersion flow model, as developed in... 

The axial dispersion flow model can be valid when we do not have the gradient of the property with respect to the normal flow direction. In other words, for this direction, we have a perfect mixing state. When this last condition is not met, we have to consider a flow model with two dispersion coefficients a coefficient for the axial dispersion and another one for the radial dispersion. In this case, the flow model equation becomes ... 

For the unitary impulse signal (relation (3.100)) the axial dispersion flow model has an analytical solution ... 

Axial Dispersion. Enthusiastic modelers sometimes add axial dispersion terms to their two-phase, piston flow models. The component balances are... 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

Dispersed flow model. To calculate the actual quality, vapor temperature, and wall temperature, or heat flux, as functions of axial position beyond dryout... 

Equation (9.27) defines the so-called axial dispersion coefficient Dax as a model parameter of mixing. Nd is the dispersion flow rate, c the concentration of the tracer mentioned earlier, and S the cross-sectional area of the column. The complete mole flow rate of the tracer consists of an axial convection flow and the axial dispersion flow. The balance of the tracer amount at a cross section of the extractor leads to second-order partial differential equations for both phase flows at steady state. For example, for continuous liquids ... 

The models of flow dispersion are based on the plug flow model. However, in comparison with the PF model, the dispersion flow model considers various perturbation modes of the piston distribution in the flow velocity. If the forward and backward perturbations present random components with respect to the global flow direction, then we have the case of an axial dispersion flow (ADF). In addition, the axial and radial dispersion flow is introduced when the axial flow perturbations are coupled with other perturbations that induce the random fluid movement in the normal direction with respect to the global flow. 

The liquid flowing inside a MWPB can be described with a one-parameter dispersion flow model. As we show in Section 3.3, the axial mixing coefficient or, more correctly, the axial dispersion coefficient is the specific parameter for this model. Relation (3.112) contains the link between the variance of the residence time of liquid elements and the Peclet number. We can rewrite this relation so as to particularize it to the case of a MWPB. Here, we have the possibility to compute the variance of the residence time of the liquid through the stochastic model for the liquid flow developed previously in order to obtain the value of the axial dispersion coefficient ... 

In Fig. 6.15 two different models for parameter estimation are used and the resulting simulated concentration profiles are compared with the measurements. In one case ideal plug-flow (Eq. 6.116) and in the other axial dispersive flow (Eq. 6.117) is assumed for the pipe system, while both models use the C.S.T. model (Eq. 6.121) to describe the detector system. Figure 6.15 shows that the second model using axial dispersion provides an excellent fit for this set-up, while the other cannot predict the peak deformation. Because of the asymmetric shape a model without a tank would also be inappropriate. 

One of the most important cases is when there are two (or more) distinct regions within the reactor. This might be a packed bed of porous solids, two fluid phases, partially stagnant regions, or other complicated flows through a vessel that can basically be described by an axial dispersion type model. Transport balances can be made for each phase, per unit reactor volume ... 

The model is based on the standard dispersion model that describes the gas phase mass and energy transport as convective flow with superimposed radial and axial dispersion. The model equations in two-dimensional axi-symmetrical cylindrical coordinates and the boundary conditions are listed below. The following assumptions have been made in this model (although the model could be extended to include these phenomena) ... 

The design, scaleup and performance prediction of slurry reactors require models which must consider not only the hydrodynamic and mixing behavior of the three phases, but also the mass transfer between the phases along with the intrinsic kinetics. In the DCL and FTS processes, an axial dispersion model is applicable, with the solid phase assumed to follow sedimentation or dispersed flow model. However, in the CCC, where the solid particles take part in the reaction, dispersion model is no longer applicable. 

Modelling both the continuous countercurrent and simulated moving bed processes has been considered by a number of authors. The continuous countercurrent separation process has been addressed by Ching and Ruthven (1984), who assumed axially dispersed flow of fluid and counter-current plug flow of solids in a column. The fundamental differential equation describing the steady state operation of such a system is, for each component. 

A dense-bed center-fed column (Fig. 22-li) having provision for internal crystal formation and variable reflux was tested by Moyers et al. (op. cit.). In the theoretical development (ibid.) a nonadiabatic, plug-flow axial-dispersion model was employed to describe the performance of the entire column. Terms describing interphase transport of impurity between adhering and free liquid are not considered. 

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. 

The axial dispersion model also gives a good representation of fluid mixing in paeked-bed reaetors. Figure 8-34 depiets the eorrelation for flow of fluids in paeked beds. 

Comparison of solutions of the axially dispersed plug flow model for different boundary conditions... 

The axial dispersion plug flow model is used to determine the performanee of a reaetor with non-ideal flow. Consider a steady state reaeting speeies A, under isothermal operation for a system at eonstant density Equation 8-121 reduees to a seeond order differential equation ... 

Both the tank in series (TIS) and the dispersion plug flow (DPF) models require traeer tests for their aeeurate determination. However, the TIS model is relatively simple mathematieally and thus ean be used with any kineties. Also, it ean be extended to any eonfiguration of eompartments witli or without reeycle. The DPF axial dispersion model is eomplex and therefore gives signifieantly different results for different ehoiees of boundary eonditions. 

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