A Brief History of Chemical Reactor Models

Irving Langmuir (1908) first replaced the assumption of no axial mixing of the PFR model with finite axial mixing and the accompanying Dirichlet boundary condition ((Cy) = Cyyn at x = 0) by a flux-type boundary condition 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

The study of mixing effects on chemical reactions has been an active area of research since the pioneering papers of Danckwerts (1958) and Zwietering (1959). The topic has become a part of classical Chemical Reaction Engineering and has been discussed in textbooks (Froment and Bischoff, 1990 Levenspiel, 1999 Westerterp et al., 1984) and review articles (Villermaux, 1991). Historically, this study has progressed in two parallel branches, based on the Lagrangian and Eulerian frameworks of description, respectively. 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with 

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