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Keywords:

  • mathematical modeling;
  • mixing;
  • multiscale modeling;
  • reactor analysis

The spectral properties of the discrete and continuous convection and convection-diffusion operators with loop or recycle boundary condition are analyzed. It is shown that the spectral properties of these nonsymmetric operators are closely related to the theory of circulant (Toeplitz) matrices and the complex Fourier series, respectively. Although there may be many complex eigenvalues, the smallest eigenvalue is real and approaches zero as the loop circulation or recycle ratio increases. This property is used to simplify nonlinear diffusion-convection-reaction models of loop and recycle reactors to obtain two-mode low-dimensional averaged models that are accurate in the limit of large recycle ratio. Explicit expressions for the two mixing coefficients that relate the two concentration modes and their dependence on various inlet conditions are also derived. Finally, the application of the low-dimensional models to determine the impact of macromixing on the conversion, yield, and selectivity for the case of nonlinear kinetics is illustrated. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3365–3377, 2013