Reduced-Order Models Based on the Orthogonal Collocation Method 66 n+

0 - - ^ i j (3.65) k=: df y = 2,3, -,n + 2 I = X2,'",nc n + 2 n+2 (3.66) k=^ k=^ j = 1,2, -,n + 1 n + 2 nc 0 = - l A , « ' ^ X - n y ' Z N . j H i (3.67) /f= 1 /= 1 y = 2,3, " , n + 2

Now the boundary conditions to be satisfied are:

(3.68) (3.69) ~^n+2 (3.70) ^ L - o = ^ i (3.71) r | = T / (3.72) l z „ = 0 ^ ’ y\zfj=o~y^ (3.73)

The overall enthalpy balance [equation (3.66)] can be reduced to an algebraic equation adopting a similar procedure to that presented in equations (3.33) to (3.35).

Using the model represented by the set of equations above Srivastava and Joseph (1984) simulated some multicomponent distillation and absorption columns. The integration in time was accomplished using a semi-implicit Runge-Kutta integration method and the agreement between steady state calculated and reported values were good. No comparison between dynamic results was presented.

Chapter 3 - Reduced-Order Models Based on the Orthogonal Collocation Method 67 Wang and Cameron (1991) extended the approach presented by Srivastava and Joseph (1984) to include the possibility to simulate a complete column with intermediate feeds and/or sidestreams. They also introduced in the model a simple hydrodynamic equation for packed columns developed by Jificny and

Stanek (1990).

Wang and Cameron (1991) simulated an industrial depropanizer that uses a Mellapak 250Y structured packing. It is important to stress that the mass and heat transfer coefficients they used were evaluated using a more rigorous procedure than the one adopted by Srivastava and Joseph (1984). The mass transfer coefficients for binary systems were determined either using the Onda correlation [see equation (5.75)] or the Spiegel and Meier correlation [Spiegel and Meier (1987)] written as follows:

k_{'i,i -}- K 0.8r \ r [ v V3 [ p v & ï , j (3.74)

with Kg in the range 0.018 to 0.040.

For the computation of an effective mass transfer coefficient for a component in the multicomponent system an equation suggested by Wilke was used (1950)

{ K o o \ =

.0

0

The vapour phase heat transfer coefficient was approximated by using a Chilton-Colburn type analogy:

h ' ' (3.76)

In their studies the vapour phase mass and heat transfer coefficients were affected not only by the vapour flowrates but also by the temperature and compositions. Since the widely used simplified one variable correlations gave

Chapter 3 - Reduced-Order Models Based on the Orthogonal Collocation Method 68

poor approximations they concluded that the rigorous computations of mass and energy transfer coefficients and liquid holdup based on the physicochemical properties of the streams are essential to the entire study of packed column dynamics. Their results indicate, as expected, that the fluid dynamics are much faster than the composition transients.

Wardle and Hapoglu (1992) solved the same example problem presented by von Rosenberg and Hadi (1980) but, instead of using a polynomial approximation in the solution of the equations of the model as Srivastava and Joseph (1984) did, they opted for employing the technique of orthogonal collocation on finite elements [Finlayson (1980)]. They obtained a very close agreement with the results obtained using the finite difference scheme as proposed by von Rosenberg and Hadi (1980) but the orthogonal collocation on finite elements approach requires far fewer discretization points. Some polynomial families were tested (Jacobi, Legendre and cubic Hermite) leading to the same results but requiring different CPU times. When more than three collocation points are required the cubic Hermite polynomial leads to a smaller number of equations (less GPU time) because continuity relationships are not required. Wardle and Hapoglu (1992) employed the same simplified correlations for evaluation of the overall mass transfer coefficient that were presented by von Rosenberg and Hadi (1980).

There are in the open literature some references to the application of the reduced order models originated by the utilisation of the method of orthogonal collocation to some control studies. Karlstrom and Breitholtz (1990) obtained a linearised large state space model that was further reduced by means of the optimal Hankel norm method. The dynamic results obtained using this model are compared with experimental data for a binary system in a full scale industrial distillation column with structured packing. They reported good agreement between experimental and simulated data.

Wang and Cameron (1992) introduced some simplifications (e.g., considered a binary system, approximated Kqg as a pointwise linear function of V,

Chapter 3 - Reduced-Order Models Based on the Orthogonal Collocation Method 69 approximated H r as a pointwise linear function of L, etc.) in the model previously presented by Wang and Cameron (1991) in order to develop some optimal control studies. They concluded that the model provides a good basis to carry out optimal control studies but recommended the investigation of more rigorous methods for the computation of mass and heat transfer rates.

3 .4 - Co n c l u s io n s

From the results reported in the literature it is clear that the orthogonal collocation technique is a powerful option for the order reduction of the models for separation columns.

Almost all the performance comparisons of the reduced-order models have been made against ‘rigorous’ models based on the equilibrium stage concept and/or with the interphase mass transfer rates evaluated in a simplified way. The reduced-order models were developed on the same basis.

It would be worthwhile to develop reduced-order models based on the nonequilibrium stage concept as well as to incorporate in the model some of the rigorous matriciel methods for the computation of the interphase mass transfer rates.

This is the chosen direction and a review on the rigorous treatment of multicomponent mass transfer is presented in the following chapter.

Chapter 4 - Multicomponent Mass Transfer 70

4- MULTICOMPONENT MASS TRANSFER

4 .1 - In t r o d u c t io n

When dealing with the problem of designing a new separation process a chemical engineer will normally be looking at concentrated systems involving the simultaneous diffusion of several components as well as simultaneous heat transfer.

The most traditional approaches to solve the problem above assume that the diffusion flux of a component in a multicomponent mixture is a function of its concentration gradient only. It is also assumed that the heat flux depends only on the temperature gradient. There are few situations of practical relevance where these simplifications are strictly correct.

During the last two decades it was proved without doubt that multicomponent systems can display transport characteristics completely different from those of a binary nature. Several different formulations (rigorous and approximate) to tackle the problem of simultaneous mass and heat transfer in multicomponent systems were developed [Krishna and Standart (1979)].

In this chapter a review on these different formulations is presented. Greater emphasis is given to the methods that are going to be incorporated in the reduced order models for separation processes to be presented on Chapters 6, 7 and 8.

In the first part of the chapter [sections 4.2 to 4.5] the basic equations to analyse multicomponent mass transfer processes are presented. In section 4.6 a number of different solutions (exact and approximate) to these equations, based on the film model to describe the multicomponent mass and heat transfer, is presented. Finally, in section 4.6.7, some brief comments are made on the applicability of the methods presented in this chapter.

Chapter 4 - Multicomponent Mass Transfer 11

4 .2 - The Ma x w e l l-Ste f a n Eq u a tio n s

The starting point for a rigorous treatment of the diffusion in a system with nc

components may be the Maxwell-Stefan equations [Taylor and Krishna (1993)] that can be expressed as:

(4.1)

where the diffusion fluxes, J„ are functions of the Maxwell-Stefan diffusivity,

-Bjj , and d, is the driving force for the diffusion of component / in a

multicomponent mixture at constant temperature and pressure.