Michelsen method

One of the most successful methods is the Michelsen method, obtained as a small variant on the Caillaud-Padmanabhan method (Chan et al., 1978) ... 

The Michelsen method is a third-order method and is strongly A-stable. It requires, at each iteration, only one calculation of J, two calculations of f, and a single factorization for the solution of the three systems (since they have the same matrbc of coefficients, see Buzzi-Ferraris and Manenti, 2010b). 

The Michelsen method has an additional advantage Chan et al. (1978) demonstrated that the same values of Iq can be used to form, in addition to the third-order Michelsen method, another first-order method, which is hence embedded, that also turns out to be strongly A-stable ... 

The semi-implicit Michelsen method has the advantage of being third order and strongly A-stable without requiring, as per the implicit methods, the solution of a nonlinear system. Moreover, the control of the error can be performed with an embedded first-order method, also strongly A-stable. 

Repeat Problem 7.6, but instead of a constant step size, use the step size control described in Section 7.10 for the Michelsen method. 

The Michelsen method [68] proceeds in a similar manner, but the nonionic surfactants are determined by means of thin-layer chromatography. 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

Fredenslund, A., Gmehling, J., Michelsen, M. L., Rasmussen, P. and Prausnitz, J. M. (1977a) Ind. Eng. Chem. Proc. Des. and Dev. 16, 450. Computerized design of multicomponent distillation columns using the UNIFAC group contribution method for calculation of activity coefficients. 

In addition to the three methods described above, nonlinear regression methods or other transform approaches may be used to determine the dispersion parameter. For a more complete treatment of the use of transform methods, consult the articles by Hopkins et al. (15) and Ostergaard and Michelsen (14). 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

The formulation of Heidemann and Khalil (14) is used here, with some modifications proposed by Michelsen (15, 16) which improve computational speed. Reference (14) contains adequate detail to permit the computations to be reproduced. The elements of the method are given here. 

The evaluation of these elements and the underlying theoretical support for the method can be found in Villadsen and Michelsen [38] who also provided subroutine listings that were used in this study. The boundary condition for the adsorbent particles is 0li>L+1 = collocation points that corresponds to a particular L-th-order polynomial approximation. The boundary condition for the capsule core is 0ci> M+1 = 0mi>o where M is the number of internal collocation points that correspond to a particular M-th-order polynomial approximation, and the boundary condition for the hydrogel membrane is bl where N is the number of internal collocation points that corresponds to a particular N-th-order polynomial approximation. Since the boundary conditions for the adsorbent and capsule core are coupled, and that of the capsule core and hydrogel membrane are also coupled, the boundary... 

Michelsen and Ostergaard68 subsequently proposed three additional methods for calculating mean residence time and Peclet number, based upon numerical evaluation of the transfer function and its derivative for a number of values of the Laplace transform parameter sfm. They defined... 

A plot of Cf 2 versus s thus gives a straight line of slope 4/[tm Pe) and intercept 2 on the ordinate axis. The main advantages of the above methods, compared with the normally-used method of central moments, are (a) the validity of the model may be easily assessed, and (b) the sensitivity to experimental errors in the determination of transient response is greatly reduced, provided suitable s-values are used. Michelsen and Ostergaard68 showed that the last method can also be applied to the N-tanks-in-series model. Very recently, Pham and Keey,82 by working with the general definitions of t/g, Uu and U2 as ... 

Solution The calculations for the method of moments can be carried out using the procedure outlined in the previous illustration. The calculations of mean residence time and Peclet number by the method of Ostergaard and Michelsen are carried out as follows. 

The values of tm and Pe obtained by the method of moments and the method of Ostergaard and Michelsen are as follows. 

Wasylkiewicz et al. (1993) present a method to find regions where mixtures partition in two or more liquid phases. They base it on the Gibbs tangent plane test (Michelsen, 1982, 1993) to decide if a current composition resides in a single- or a multiple-liquid phase region. 

McDonald and Floudas (1994) and Michelsen (1994) also present methods to find the phase conditions for a given mixture. Both methods discover the number of phases and the compositions of these phases. McDonald and Floudas globally minimize the Gibbs free energy of the mixture using a computer pack-... 

This equation, together with eqn. (3.3.8), is known as MHV2 model (Dahl and Michelsen 1990). The MHVl and MHV2 models are considered here for coirela-tion and prediction of the VLE of various mixtures, and computer programs that use these methods are provided in the accompanying disk. Further details are given later in this section. 

For highly nonideal and polar mixtures of organic compounds, the IPVDW model is inadequate, and the various forms of the 2PVDW model have limitations as a correlative method and suffer from computational problems such as the dilution effect and the Michelsen-Kistenmacher syndrome mentioned in Section 3.5. Such models should only be used with caution as semipredictive methods, and they have little utility as... 

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