The Danckwerts Boundary Conditions

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates 

It is assumed that no reaction takes place outside the reactor, as would be the case if the reactor were packed with catalyst, and that the concentrations are therefore uniform outside the reactor. Thus, the inlet condition is merely the continuity of flux, which is vAcin just before entering and A vc - D(dcl dz) f just after. Thus, at the inlet, there is the condition 

The question can be answered by noting that, as the value of D goes to infinity, the tubular reactor becomes more and more completely mixed until in the limit it is a stirred tank. We should therefore be able to get the equations for the stirred tank as a limiting case. At this point, we should really work in dimensionless variables. = zIL is a natural way of reducing the length and, because the residence time is Llv, the dimensionless time is r = tvIL. Note that, by comparing the two models, 8 = Vlq = Llv, Da = kd, and we need the dimensionless dispersion coefficient Pe = vLID. The limit we want is then Pe 0. With u( ) = c(z)/cin and U= cproduct/cin 

Integrating the differential equation from 0 to 1 and using the two boundary conditions, we get 

The beauty of this argument is that it can be used for the transient case (we only gave up the time derivative in the interests of clarity), for nonlinear 

The axial dispersion model has a long and honored history within chemical engineering. It was first used by Langmuir, who also used the correct boundary 

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 

FIGURE 9.8 The axial dispersion model applied to a closed system. 

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-) a, . 

The concentration is continuous at the reactor exit for all values of D and this forces the zero-slope condition of Equation (9.17). The zero-slope condition may also seem counterintuitive, but recall that CSTRs behave in the same way. The reaction stops so the concentration stops changing. 

The marvelousness of the Danckwerts boundary conditions is further explored in Example 9.3, which treats open systems. 

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The boundary conditions for a closed-vessel reactor are analogous to those for a tracer in a closed vessel without reaction, equations 19.4-66 and -67, except that we are assuming steady-state operation here. These are called the Danckwerts boundary conditions (Danckwerts, 1953).1 With reference to Figure 19.18,... 

That the Danckwerts boundary conditions still elicit reasearch is demonstrated by the reference J. A. Barber, J. D. Perkins, and R. W. H. Sargent. Boundary conditions for flow with dispersion. Chem. Eng. Sci. S3,1463-1464 (1998). 

A one-dimensional one-phase dispersion model subject to the Danckwerts boundary conditions has been used for a description of the dynamics of a nonisothermal nonadiabatic packed bed reactor. The dimensionless governing equations are ... 

The Danckwerts boundary conditions are used most often and force discontinuities in both concentration and its gradient at z = 0. 

The solution to this equation using the Danckwerts boundary conditions of ... 

For a first-order reaction, such as we have in Equation (14-36), Da = kLIU.] We shall consider the case of a closed-closed system, in which case we use the Danckwerts boundary conditions... 

Figure 2.5 Illustration of the Danckwerts Boundary Condition. 1 Rectangular pulse injection. 2, 3,4 Danckwerts injection conditions for the same sample amoimt 2 D = 0.04 cm /s 3 D = 0.08 cm /s 4 D = 0.12 cm /s. Reproduced with permission from G. Guiochon, B. Lin, Modeling for Preparative Chromatography, Academic Press, San Diego, CA, USA, 2003 (Fig. III-2).

In order to solve Equation (11-22) we need to specify the boundary conditions. In this chapter we will consider some of the simple boundary conditions, and in Chapter 14 we will consider the more complicated boundary conditions, such as the Danckwerts boundary conditions. 

For the closed-closed system, the Danckwerts boundary conditions in dimensionless form are... 

Dispersion model For a first-order reaction, use the Danckwerts boundary conditions... 

Dispersion with Reaction using the Danckwerts Boundary Conditions ft wo cases)... 

The zero-slope condition may seem counterintuitive. CSTRs behave in this way, but PFRs do not. The reasonableness of the assumption can be verified by a limiting process on a system with an open outlet as discussed in Example 9.2. The Danckwerts boundary conditions are further explored in Example 9.2, which treats open systems. The end result is that the boundary conditions are somewhat unimportant in the sense that closed and open systems behave identically as reactors. 

These two boundary conditions have become known as Danckwerts boundary conditions 16], but they were derived at least 45 years prior to Danckwerts in a classic paper by Langmuir [161. Further discussion of the applicability and some alternatives to the Danckwerts boundary conditions are given by Bischoff [3], Levenspiel [17, p.272, and Pamlekar and Ramkrishna [21]. 

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... 

In 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 (Danckwerts, 1953) at the reactor inlet is... 

The proper boundary conditions to use in solving Eq. 12.5.a-16 have been extensively considered in the literature. For a closed reactor, consideration of flux balances at the entrance and exit provide what are usually termed the Danckwerts boundary conditions [6] ... 

The Danckwerts boundary conditions are assumed in the liquid phase... 

These tw o boundary conditions, equations (7.1.3) and (7.1.4), are known as the Danckwerts boundary conditions (Danckwerts, 1953). We will rewrite the problem in terms of dimensionless variables... 

For tubular reactors with the Danckwerts boundary conditions, the backward integration method gives stable convergence properties, while the forward integration procedure is unstable. 

Due to the Danckwerts boundary conditions, we anticipate stability via backward integration. We therefore introduce a new length variable z — 1 — A. The differential equation and boundary conditions now become... 

For the bounded flow system of (dimensionless) reactor length L the Danckwerts boundary conditions apply ... 

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) ... 

The Danckwerts boundary condition should be applied where there is appreciable axial dispersion. For transient problems, initial conditions must also be specified. Equation (16) can be applied to gas mixing experiments without reaction by dropping the final term. Equations analogous to Equation (16) can be developed to describe reactions of solid species. 

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