Axial dispersion flow

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... 

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 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 ... 

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. 

Show that for an axial dispersion flow system with variable density, area, velocity, and dispersion coefficients between two adjacent sections, equating total fluxes leads to the boundary condition ... 

The analysis we carried out earlier resulting in solutions, e.g. equation (8.1.360), was carried out for plug flow of hoth phases. However, there is usually some axial dispersion at least in the fluid phase (e.g. a liquid phase). The result of existing analyses as described by Ruthven (1984) will be provided here for downward plug flow of the solid phase 7 = 1 and upward (z > 0) axially dispersed flow of fluid phase j = 2. The governing balance equation from equation (6.2.34) is therefore... 

Table 6.3 Summary of available numerical solutions for breakthrough curves in isothermal systems with plug flow or axially dispersed flow and Freundlich, Langmuir or other non-linear isotherms...

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. 

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