Boundary conditions, axial dispersion model reactors

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 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,... 

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 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 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) ... 

This simply assumes that axial dispersion (D m. s ) is superimposed onto plug flow. Axial dispersion may be caused by a velocity profile in the radial direction or statistical dispersion in a packing or turbulent diffusion or by any physicochemical process which delayes some particles with respect to others. The model parameter is the axial PECLET number, Pe = uL/D, or its reciprocal, the dispersion number, D /uL. Depending on the boundary conditions assumed at the reactor inlet and outlet (which are different from those of the simple assumptions above), a lot of mathematical formulae can be found in the literature for the RTD [3]. This is often academic as in the range of usefulness of the model (small deviation from plug flow, say Pe > 20) all conditions lead to res-... 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

Inclusion of axial dispersion in the plug-flow model makes the model equations a boundary-value problem, so that conditions at both the reactor inlet and outlet need to be specified. The commonly used boundary conditions are the so-called Danckwerts type (12), although their origin goes back to Langmuir (13). 

This model can be reduced if isobaric conditions are considered and/or axial dispersion is neglected. Most of the times, simulations are carried out in steady state conditions, so that the whole model is reduces neglecting the time derivative. Generally, standard Danckwerts boundary conditions at the reactor inlet and outlet can be assumed for the gas phase balance. ... 

According to the idealized flow patterns (Hlavacek and Kubicek, 1972), the slice of the bed cut in the radial direction can be considered as a fixed-bed with the inlet condition at r = Rx and the outlet condition at r — Rz- Conservation equations written for this slice should be applicable to the whole length of the bed. Therefore, the conservation equations of l s. 9.27 through 9.36 are applicable to radial flow reactors with the convection terms 9c/dz and bT/bz replaced by bc/br and bT/br. This model is then identical to the one-dimensional model with axial dispersion as for the usual fixed-bed reactor. The same boundary conditions as those of Eqs. 9.10 and 9.11 apply, but this time Eq. 9.10 applies at r = Ri and Eq. 9.11 at r = R2. with z replaced by r for the flow pattern (a) in Figure 9.12. Unless the bed depth is quite shallow, the dispersion term can be neglected, resulting in a plug-flow model in the radial direction. 

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