Axial dispersion definition

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

The model developed below is for dispersion in the axial direction only. Because the underlying mechanisms producing axial dispersion are complex as the discussion above shows, equation 2.12 is best regarded as essentially a mathematical definition... 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

However, the definitions related to specific processes have to be kept in mind. In chromatography the plate height is a measure that lumps together the contribution of the fluid dynamic non-idealities (axial dispersion) and the mass transfer resistance... 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

Convergence is obtained when the appropriate guess for d p./di at the reactor inlet predicts the correct Danckwerts condition in the exit stream, within acceptable tolerance. To determine the range of mass transfer Peclet numbers where residence-time distribution effects via interpellet axial dispersion are important, it is necessary to compare plug-flow tubular reactor simulations with and without axial dispersion. The solution to the non-ideal problem, described by equation (22-61) and the definition of Axial Grad, at the reactor outlet is I/a( = 1, RTD). The performance of the ideal plug-flow tubular reactor without interpellet axial dispersion is described by... 

The approximate additivity of the effects of axial dispersion and mass transfer resistance was first deduced by van Deemter et al. by considering the asymptotic form of model lb. The same conclusion may be reached in a simpler way from- moments analysis and leads to Eq. (8.42) as the definition for an overall effective rate coefficient incorporating both the effects of axial dispersion and finite mass transfer resistance. 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

Values of the radial dispersion coefficient, or the corresponding radial Peclet number, udp/Dr), in packed beds have been determined for both liquids and gases by a number of researchers they are definitely not the same as those in the axial direction. These results are shown in Figure 5.10a and b for liquids and gases, respectively. The corresponding empirical equations fitting these data are... 

Here v and d represent dimensionless convection velocity and dimensionless thermal dif-fusivity, resp. (axial mass dispersion is neglected) B is dimensionless reaction enthalpy. Da is the Damkohler number and Le is the Lewis number. We have avoided the conventional introduction of the Peclet number since we would like to examine the effects of convection and thermal diffusion separately. Otherwise, the scaling of the variables and definitions of the dimensionless quantities are conventional, see e.g. (Nekhamkina et al. (2000)). 

Dispersion of heat can be described in a similar manner as dispersion of mass if we use an effective thermal conductivity in the axial and radial direction (kax, 2-rad)- The corresponding dimensionless Pedet numbers are Pch ad (= MsCpp oidp/Xrad) and Pch,ax (= WsCpPmoi p/ ax)- Note that the superficial fluid velocity, Us, and not the interstitial velocity, Ws/e, is used in the definition of Pejj, as the effective heat conduction (reflecting both the effective heat conduction in the gas and solid phase) is not limited to the empty space of the packed bed as in the case of dispersion of mass (see also differential equations of a fixed bed reactor in Section 4.10.7). As a rule of thumb, we can approximately use the same values for the Pedet number for dispersion of heat for high Rep numbers (>100) as for the corresponding numbers for dispersion of mass, that is, Pe r d 10 and Pe 2. Details on the radial heat dispersion, which is important for wall-cooled reactors, are given in Section 4.10.7.3. 

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