Modelling and parameter estimation in reactive continuous mixtures: the catalytic cracking of alkanes - part II

Modelling and parameter estimation in reactive continuous mixtures: the catalytic cracking of alkanes - part II

F. C. PEIXOTO ¹ and J.L. de MEDEIROS²

¹Instituto de Pesquisa e Desenvolvimento (IPD), Av. das Américas, n° 28705, Guaratiba, Rio de Janeiro - RJ, Brazil,

Phone: [phone] Ramal: 354,

E-mail: [e-mail]

²Depto. de Eng. Química, Escola de Química, Bl. E, Centro de Tecnologia, UFRJ, Cidade Universitária 21949-900,

Rio de Janeiro - RJ, Brazil, Phone: [phone],

E-mail: [e-mail]

(Received: August 25, 1998; Accepted: April 6, 1999)

Abstract - Fragmentation kinetics is employed to model a continuous reactive mixture of alkanes under catalytic cracking conditions. Standard moment analysis techniques are employed, and a dynamic system for the time evolution of moments of the mixture's dimensionless concentration distribution function (DCDF) is found. The time behavior of the DCDF is recovered with successive estimations of scaled gamma distributions using the moments time data.

Keywords: Modelling, parameter, estimation, catalytic cracking of alkanes.

INTRODUCTION

The modelling of catalytic cracking reactions, which considers the continuity of the reactive mixture, has recently been intensively studied (Laxminarasimhan et al., 1996; McCoy, 1996). The association of this idea with fragmentation kinetics, by analogy with reduction of particle size, is helpful and has still been made frequently (McCoy and Wang, 1994). The mathematical description is based on population or mass balance equations for the space-temporal behavior of a dimensionless concentration distribution function (DCDF), g(x,z,t). The Variable x is associated with a property or index (size, molecular weight, etc.) which characterizes a given element of entities (particle, molecule, etc). Therefore, g(x,z,t)dxⁿ stands for the number of entities or their dimensionless molar concentration in the index domain [x, x + dx]. Integration of the DCDF in a given domain quantifies the entities present in it, and when applied in the modelling of hydrocarbon fractions, this operation generates a "lump." The spatial coordinates are described by vector z and the time by t. In the catalytic cracking case, this time scale is warped ("crazy clock") in order to take into account the inherent effects of catalyst deactivation (Nace et al., 1971; Weekman, 1979).

The DCDF moment analysis technique has been proposed for solving the resulting balance equations in fragmentation kinetics (McCoy, 1996) because analytical solutions for these equations can rarely be found unless some simplifying hypotheses are used (McCoy and Wang, 1994).

On the other hand, classic approximate strategies may encounter convergence difficulties as a consequence of the inherent nonlinearities of the problem. This was the case of the model studied in the first part of this research, presented by Peixoto and Medeiros (1998a), which henceforth will be referred to as Part I.

In Part I it was considered the numerical solution of a classic fragmentation kinetics problem (McCoy and Wang, 1994) applied to the description of the catalytic cracking of alkanes. The mathematical model was

(1)

with the initial condition:

(2)

where g(x,t ) is the DCDF after contact time t of the mixture with the catalyst. In Part I the following standard models were used:

K(x, t) = k0.K(x).D(t) ; K(x) = x (3)

where K(x) and v(x,y) are called the dimensionless kinetics constant distribution function (DKCDF) and the stoichiometric coefficient distribution function (SCDF), respectively. In Equation (3), D(t) represents the fraction of active catalytic sites present in the catalyst with age t (reactor run time or "Time on Stream" – TOS). Due to the deactivation phenomenon, D(t) is a decreasing function of t, which may be presented in many forms such as (Kobolakis and Wojciechowski, 1985):

(4)

Since t << t, D(t) is approximately constant in the interval (0 , t), once the fraction of active sites depends on catalyst age t (TOS), and contact time t has little influence on it. This assumption leads to the transformation:

(5)

Problem (1) + (2) is then finally expressed in the following form:

(6)

In Part I, a perturbational approach to the solution of Equation (6) was proposed, given by

(7)

The truncated coefficient vector was expanded in an orthonormal basis functions within (0 , ¥ ), which are the normalized Laguerre polynomials weighted by exp(-x/2). This can be expressed as

(8)

where f _i(x) is the normalized and weigthed Laguerre polynomial of degree i.

Model parameters k0, Gd and Nd were adjusted in a classic procedure for appropriate cracking data (Kobolakis and Wojciechowski, 1985; see Part I), leading to a good fit of the experimental data but with large uncertainties regarding estimated parameters. Simultaneously, another model developed by the authors (Population Balance Model - Peixoto and Medeiros, 1998c) also obtained a good fit for the above-mentioned cracking data with additionally very low uncertainties for the estimated parameters.

This model applied an analytical one-dimensional populational balance (PB-1) theory with a p-order dependence on the DKCDF:

K(x) = x^p (p > 1) (9)

This suggested a new application of the perturbational scheme to Equation (6), with the adoption of the generalized kinetic model (Equation 9) in order to reduce the previous drawbacks of the statistical analysis conducted in Part I.

Unfortunately, this did not result in any progress and revealed an increasing difficulty in the convergence of Equation (7) as p increases. In order to quantify this difficulty, a convergence measure is defined by

(10)

where the norm is given by

(11)

The numerical problems that motivated the present work are shown in Table (1). These results were calculated with n = 20 in (10).

Table 1:
Order of magnitude of r(p)

P	r(p)
1	O(101)
1.5	O(102)
2	O(103)
2.5	O(104)

This argument strengthens the thesis that the solution procedure employed must be modified to accept the additional parameter p > 1, with Equation (9) applied to Equation (6).

The present work contemplates an alternative approach to the catalytic cracking problem modelled with Equation (6). More general hypotheses are now used: a nonlinear model for K(x), a generalized v(x,y) and a semi-infinite domain, and an explicit solution for the DCDF moments was found, despite all this assumptions. This technique was first presented by the authors in (Peixoto and Medeiros, 1998b), where a similar problem without catalyst deactivation was solved.

MODEL RESOLUTION WITH MOMENT ANALYSIS TECHNIQUES

Anti-Volterra integro-differential modelling for the catalytic cracking of a continuous mixture (Equation 6) is applied again, employing the already mentioned model for the KCDF, K(x) = x^p (p>1). For the SCDF, the generalized model proposed by McCoy and Wang (1994) is used as follows:

with

(12)

where B stands for the normalization condition

(13)

This model is able to reproduce random breakage (only requiring m = 0 for this). In this case, a y-sized molecule can be broken at any point of its structure with uniform probability. If a Gaussian distribution is desired, m must be large. Similarly other breakage models can be set with the choice of m within (0 , ¥). Therefore, Equation (6) becomes

(14)

The monotonous character of K(x) is used in the transformation g(x,q ) ® g(k,q ). This change is the essence of the works of Laxminarasimhan et al. (1996) and Ho (1987), used by these authors for other purposes. Then, Equation (14) becomes

(15)

The moment operation (Abramowitz and Stegun, 1964) is carried out, which is nothing more than a Laplace transformation, where (-s) was substituted for s, as follows

(16)

Expanding the exponential series, it can be found that

(17)

(18)

(19)

where

(20)

is the DCDF n^th moment in warped time scale q ("crazy clock") and

(21)

is the beta function (Abramovitz and Stegun, 1964).

In Equation (19), the change in the order of integration in the operation domain (Peixoto, 1997) is made (Figure 1) with the integral definition of the beta function.

Figure 1:
Integration domain in (19)

Equations (17), (18) and (19) are then introduced in Equation (15), resulting in

(22)

Coefficients of the s terms are then matched in order to obtain the final recurrence

(23)

This form shows the independence of moment indexes with m and p due to the transformation used in Equation (15). Therefore, these parameters can be estimated by an external algorithm, which would not be possible without the change x ® k since noninteger values would make no sense. The initial condition for Equation (23) is given by:

(24)

System (23) + (24) can be expressed in matrix representation after truncation at order n as

(25)

subject to the initial condition

(26)

where (with size (n + 1) x (n + 1)) is

(27)

The last row of is the closure hypothesis, applicable to large values of n, given by <gⁿ⁺¹>@<gⁿ><g0¹>. The solution of the system (25) + (26) is then

(28)

For initial distribution, the following two-parameter gamma distribution is employed

(29)

Once the temporal evolution of the DCDF moments is obtained, one can conduct a parameter estimation of an interpolating distribution, which will emulate the real one. This can be, for instance, a scaled three-parameter gamma distribution, whose moments are given by

(30)

In this way, each time the solution (28) is evaluated, a nonlinear parameter estimation of C, h and e must be carried out with Equation (30). This demands an excessively large computational time. Fortunately, it is almost always possible to suppose that g(x,q ) is sufficiently characterized by its first three moments, resulting (with j = 0, 1, 2 in Equation (30)) in a system for the calculation of C, h , and e. It must be said that, even though the first three moments are the only needed, a large number of moments (40 < n < 50) is calculated with Equation (28) in order to minimize the effects of the approximation in Equation (27).

SIMULATION

In Figures (2), (³), (⁴) and (⁵), the warped time scale corresponds to typical values in cracking experiments (the initial distribution parameters were fixed at h = 10 and e = 1).

Figure 2:
Time behavior of DCDF for m = 1

Figures (2) and (³) were generated with m = 1 in Equation (12), while ^{Figures (4)} and (⁵) used m = 2. The conversion in ^{Figures (3)} and (⁵) refers to the mass fractional consumption of hydrocarbon with 18 or more carbon atoms in the molecule (Kobolakis and Wojciechowski, 1985 and Part I).

It can be noticed that the method can easily solve problems with p > 1, unlike the original perturbational approach.

The value p = 2.4988, which was obtained with the one-dimensional population balance model applied to the alkane cracking problem (Peixoto and Medeiros, 1998c), was used with m = 0 (random breakage) in order to generate Figure (6).

Figure 6:
3-D representation of g(x,q ) for p = 2.4988 and m = 0

In Peixoto (1996), a similar development was carried out aiming at comparing exact DCDF moments with the ones obtained in the perturbational approach described in Equation (7). Such results correspond to the particular case p = 1 and m = 1 in Equation (23).

CONCLUSIONS AND DISCUSSION

The mathematical tool of DCDF moments analysis, instead of searching for an explicit (or even numerical) expression for the DCDF itself, has been shown to be successful. Results are physically consistent, and the algebra used can easily be adapted to other continuous models by, for example, the addition of an "aging" term such as in the following problem:

(31)

It must be said that it would be interesting to conduct a new estimation of kinetic parameters (Gd, Nd, k0, p) for the model given by Equations (6) and (9) within the present context. This is, nevertheless, beyond the scope of this work, which is primarily concerned with the formulation and respective algebraic solution of an alternative continuous approach to the modelling of catalytic cracking of alkanes.

NOMENCLATURE

g(x,t) and g0(x) Dimensionless Concentration Distribution Function (DCDF)

v(x,y) Stoichiometric Coefficient Distribution Function (SCDF)

K(x,t) Kinetics Constant Distribution Funtion (KCDF)

K(x) Dimensionless Kinetics Constant Distribution Function (DKCDF)

D(t) fraction of active catalytic sites

k0, kcin, Gd, Nd, model parameters

md, kd, p

C, e , h gamma distribution parameters

t catalyst age

t contact time

B normalization factor

k, K transformation variables

<gⁿ>, <g0ⁿ> n-th moment of the DCDF

b beta function

G coefficient matrix

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Figure 3: Time behavior of the conversion for m = 1

Figure 4: Time behavior of DCDF for m = 2

Figure 5: Time behavior of the conversion for m = 2

Publication Dates

• Publication in this collection
  16 Dec 1999
• Date of issue
  Sept 1999

History

• Received
  25 Aug 1998
• Accepted
  06 Apr 1999