Danckwerts boundary condition

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). 

This result assumes concentration to be continuous at z = T, a fact that is obvious. Obvious Langmuir was a Nobel laureate and Danckwerts is regarded by many as the father of chemical engineering science. 

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. 

Equation 9.18 is a linear, second-order ODE with constant coefficients. An analytical solution is possible when the reaction is first order. The general solution to Equation 9.18 with 3 A = —ka is 

The constants Ci and C2 are evaluated using the boundary conditions, Equations 9.20 and 9.21. The outlet concentration is found by setting z = L. Algebra gives 

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

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 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... 

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). 

Besides the deposition on the wafers, the radial boundary conditions (equations 38a and b) account for deposition on the reactor wall and the support boat. The first term in equation 38b represents the deposition on the wafers, and the second term gives the deposition on the boat, a is the boat area relative to the tube area. The axial boundary conditions (equation 38c) are the usual Danckwerts boundary conditions. 

Closed end vessel One in which the inlet and outlet streams are completely mixed and dispersion occurs only between the terminals. At the inlet where z = 0, uC0 = [uC - De>10C/3z)]z=o at the outlet where x = L, (dC/dz)z=0. These are referred to as Danckwerts boundary conditions. 

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 ... 

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. 

We now present the solution of the hyperbolic model defined by Eqs. (52) and (53)-(54) and compare the solution to that of the classical parabolic model with Danckwerts boundary conditions. We use the axial length and convective time scales to non-dimensionalize the variables and write the hyperbolic model in the following form ... 

We now compare the solution of the hyperbolic model with that of the parabolic model used widely in the literature to describe dispersion in tubular reactors. The parabolic model with Danckwerts boundary conditions (in dimensionless form) is given by... 

The traditional parabolic model with Danckwerts boundary conditions is also used in the literature to describe dispersion effects in packed beds (and porous media). However, unlike the case of capillaries and straight tubes, the flow field in packed beds is more complex and is three-dimensional. However, for many cases of interest, the average velocity in the transverse directions is zero. In such cases, dispersion in the flow direction can be described by the... 

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) ... 

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 ... 

These two boimdaiy conditions, Equations (14-22) and (14-23), first stated by Danckwerts, have become known as the famous Danckwerts boundary conditions. Bischoff has given a rigorous derivation of them, solving the differential equations governing the dispersion of component A in the entrance and exit sections and taking the limit as D, in entrance and exit sections approaches zero. From the solutions he obtained boimdary conditions on the reaction section identical with those Danckwerts proposed. The initial condition is... 

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).

At the origin (the inlet node, Ng = 1, z = 0), we have, assuming Danckwerts boundary conditions (Chapter 6, Section 6.2.1.2),... 

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... 

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