A Wavelet-Galerkin Method applied to Separation Processes

  • R. v. Watzdorf
  • K. Urban
  • W. Dahmen
  • W. Marquardt


Many fluid mixtures encountered in chemical and hydrocarbon processing industries are ill-defined in the sense that they contain far too many components for a detailed compositional analysis and subsequent modeling in terms of pure component mass balances. Common examples of such mixtures are frequently related to processes of high economic relevance and include petroleum and reservoir fluids as well as polymer solutions and polyreaction systems [6]. The concept of continuous thermodynamics is a well-established approach for the modeling of these mixtures [2, 6]. The compositional complexity of the mixture is represented in terms of a time-dependent continuous distribution function t) of some characterizing fluid property £ (e.g. molecular weight, natural boiling point or Single Carbon Number). The relation between the discrete mole fraction of any component and the distribution function is given by
$${x_l}\left(t \right) = {\smallint _{\Delta \xi l}}F\left({\xi,t} \right)d\xi $$
.Using definition eq. (1.1), a complete framework of thermodynamic relations analogous to the discrete case can be derived [6, 13].


Trial Solution Fast Wavelet Continuous Thermodynamic Liquid Phase Composition Fast Wavelet Transform 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersson, L., N. Hall, B. Jawerth, G. Peters (1994): Wavelets on closed subsets of the real line. In L.L. Schumaker and G. Webb, editors, Topics in the Theory and Applications of Wavelets, 1 - 61, Academic Press, Boston.Google Scholar
  2. 2.
    Aris, R., G.R. Gavalas (1966): On the theory of reactions in continuous mixtures. Roy. Soc. London Phil. Trans. 260, 351–393CrossRefGoogle Scholar
  3. 3.
    Chimowitz, E.H., S. Macchietto, T.F. Anderson, L.F. Stutzman (1993): Local models for representing phase equilibria in multicomponent, nonideal vaporliquid and liquid-liquid systems. 1. Thermodynamic approximation functions. Ind. Eng. Chem. Process Des. Dev. 22, 217–225CrossRefGoogle Scholar
  4. 4.
    Cohen, A., I. Daubechies, and P. Vial (1993) Wavelets on the Interval and Fast Wavelet Transforms. Appl. Comput. Harmon. Anal., 1 (1): 54–81.CrossRefGoogle Scholar
  5. 5.
    Chui, C.K. (1992): An Introduction to Wavelets. Academic Press, New YorkGoogle Scholar
  6. 6.
    Cotterman, R.L., R. Bender, J.M. Prausnitz (1985): Phase equilibria for mixtures containing very many components. Development and application of continuous thermodynamics for chemical process. Ind. Eng. Chem. Process Des. Dev. 24, 194–203CrossRefGoogle Scholar
  7. 7.
    Dahmen, W. (1995): Multiscale Analysis, Approximation, And Interpolation Spaces. In C.K. Chui and L.L. Schumaker, editors, Approximation Theory VIII, 1–23. World Scientific Publishing Co.Google Scholar
  8. 8.
    Dahmen, W., S. Pröβdorf, and R. Schneider (1993): Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution. Advances in Computational Mathematics, 1: 259–335.CrossRefGoogle Scholar
  9. 9.
    Daubechies, I. (1992): Ten Lectures on Wavelets. SIAM, Philadelphia, Penns.CrossRefGoogle Scholar
  10. 10.
    Fröhlich, J., K. Schneider (1994): An adaptive Wavelet-Galerkin algorithm for one- and two-dimensional flame computations. Eur. J. Mech.,B/Fluids 13 (4), 439–471Google Scholar
  11. 11.
    Mallat, S.G. (1989): Multiresolution Approximations and Wavelet Orthonormal Bases of L2(ℝ). Trans. Amer. Math. Soc., 315(1), 69 –87Google Scholar
  12. 12.
    Marquardt, W., E.D. Gilles (1988): DIVA - A powerful tool for dynamic process simulation. Comp. Chem. Eng. 12, 421–426CrossRefGoogle Scholar
  13. 13.
    Ratsch, M.T., H. Kehlen (1982): Kontinuierliche Thermodynamik von Vielstoffgemischen. Akademie-Verlag, BerlinGoogle Scholar
  14. 14.
    Villadsen, J., M. Michelsen (1982): Solution of Differential Equation Models by Polynominal Approximation. Prentice-Hall, Englewood Cliffs, New JerseyGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • R. v. Watzdorf
    • 1
  • K. Urban
    • 2
  • W. Dahmen
    • 2
  • W. Marquardt
    • 1
  1. 1.Lehrstuhl für ProzeßtechnikRWTH Aachen University of TechnologyGermany
  2. 2.Institut für Geometrie und Praktische MathematikRWTH Aachen University of TechnologyAachenGermany

Personalised recommendations