Boundary conditions, axial dispersion model

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... 

These boundary conditions are really quite marvelous. Equation (9.16) predicts a discontinuity in concentration at the inlet to the reactor so that ain a Q+) if D >0. This may seem counterintuitive until the behavior of a CSTR is recalled. At the inlet to a CSTR, the concentration goes immediately from to The axial dispersion model behaves as a CSTR in the limit as T) — 00. It behaves as a piston flow reactor, which has no inlet discontinuity, when D = 0. For intermediate values of D, an inlet discontinuity in concentrations exists but is intermediate in size. The concentration n(O-l-) results from backmixing between entering material and material downstream in the reactor. For a reactant, a(O-l-) [Pg.332]

The TIS model is relatively simple mathematically and hence easy to use. The DPF or axial dispersion model is mathematically more complex and yields significantly different results for different choices of boundary conditions, if the extent of backmixing is large (small Pe,). On this basis, the TIS model may be favored. 

Figure El2.2a shows the boundary conditions X0 and Yx. Given values for m, Nox, and the length of the column, a solution for Y0 in terms of vx and vY can be obtained Xx is related to Y0 and F via a material balance Xx = 1 - (Yq/F). Hartland and Meck-lenburgh (1975) list the solutions for the plug flow model (and also the axial dispersion model) for a linear equilibrium relationship, in terms of F ...

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,... 

Langmuir [18] first proposed the axial dispersion model and obtained steady state solutions from the following boundary conditions ... 

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

The solutions of the steady-state two-model model given by Eq. (140) (141) should be compared to the parabolic axial dispersion model with Danckwerts boundary conditions (Danckwerts, 1953 Wehner and Wilhelm, 1956) ... 

Although the above method can give a simple evaluation of Peclet number for the system, the tailing in the RTD curve can cause significant inaccuracy in the evaluation of the Peclet number. Michell and Furzer67 suggested that a better estimation of the axial dispersion coefficient is obtained if the observed RTD is statistically fitted to the exact solution of the axial dispersion model equation with appropriate boundary conditions. For example, a time-domain solution to the partial differential equation describing the dispersion model, i.e.,... 

The model for the H-Oil reactor introduces two complications beyond the axial dispersion model. First, the boundary conditions are modified to account for the recycle and second, the catalyst in the reactor means that both thermal and catalytic reactions are occurring simultaneously. The set of equations given in Eq. (18) are solved numerically with a differential equation solver. This allows the reactor size to be... 

The axial dispersion model has a long and honored history within chemical engineering. It was first used by Langmuir (1908), who also used the correct boundary conditions. These boundary conditions are subtle. Langmuir s work was forgotten, and it was many years before the correct boundary conditions were rediscovered by Danckwerts (1953). 

The initial and boundary conditions are given in Chapter 9. The present treatment does not change the results of Chapter 9 but instead provides a rational basis for using pseudohomogeneous kinetics for a solid-catalyzed reaction. The axial dispersion model in Chapter 9, again with pseudohomogeneous kinetics, is an alternative to Equation 10.1 that can be used when the radial temperature and concentration gradients are small. 

As the Figure 8.12 reveals, the flow pattern deviates from plug flow. The residence time distribution function E(l) is calculated from the experimentally recorded responses, after which the F(t) function was obtained from integration of E(t). The experimental functions are compared to the theoretical ones. The expressions of E(t) and F(t) obtained from the analytical solution of the dynamic, non-reactive axial dispersion model with closed Danckwerts boundary conditions were used in comparison. A comparison of the results shown in Figure 8.12 suggests that a reasonable value for the Peclet number is Pe=3. 

A transient solution for step-change in the inlet concentration for the axial dispersion model, using Type A boundary conditions, has been given by Fan and Ahn [L-T. Fan and Y-R. Ahn, Chem. Eng. Progr. Symp. Ser., 46(59), 91 (1963)]. For... 

In particular, equation (7-146) expresses that there is no mass transfer at the wall, since the concentration derivative is zero, and that heat transfer occurs with a constant wall temperature, Tw, and a local heat-transfer coefficient, This heat-transfer coefficient is now appearing in a boundary condition and is not equivalent to the overall heat-transfer coefficient used in nonisothermal axial dispersion models. The radial dispersion coefficient, Z) is, as the name implies, the radial counterpart to the axial dispersion coefficient, and while we expect a different correlation for it there are no new conceptual boundaries set here. The effective bed thermal conductivity, A however, is another matter altogether and we will worry about it more later. 

Many of the available solutions are given in Levenspiel and BischofT [1], Himmelblau and Bischoff [4], and Wen and Fan [2], As a practical matter, these mathematical complexities should probably not be taken too literally since the precise conditions at the boundaries of real equipment cannot usually be exactly defined in any event. That is, if the boundary condition details give significantly different results, be cautious about the axial dispersion model as being a faithful... 

Some of the numerical problems in nonlinear regression seem to be able to be partially avoidable by utilizing special features of a given transfer function, as proposed by Mixon, Whitaker, and Orcutt [ 104] and extended by 0ster-gaard and Michelsen [ 105]. This is best illustrated by considering the transfer function of the axial dispersion model with semiinfinite boundary conditions ... 

C(g, 0) represents the solution to the Danckwarts model Equation 3.329 with the boundary conditions (3.331) and (3.334), then the RTD function for the axial dispersion model is... 

Derive a dynamic model for a packed-bed extraction unit that includes an axial dispersion term for the dispersed phase. Give an appropriate set of boundary conditions for the model. 

The axial dispersion model has been discussed exhaustively in the literature. The reader is referred to Levenspiel (57), Nauman and Buffham (4), Wen and Fan (58), and Levenspiel and Bischoff (91) for numerous available references. The appropriateness of various boundary conditions has been debated for decades (92-95) and arguments about their effect on reactor performance continue to the present day (96). We now know that the Danckwerts boundary conditions make the model closed so that a proper residence time distribution can be obtained from the model equations given below (when the reaction rate term is set to zero) ... 

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

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial differential equation of the second order Fickian model, requires two boundary conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundary condition cf — 0 as NPeC, —> < >. Breakthrough behavior presumes the existence of a bed outlet, and a boundary condition must be applied there. 

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