determination of constrained lumping schemes for ... - Science Direct

Chemical Engmeering Science, Vol. 46, No. 2, pp. 583 596, 1991. Printed in Great Britain.

cm9 zsw,91 53.00, 0.00 Q 1990 Per,wnon Press plc

DETERMINATION OF CONSTRAINED LUMPING SCHEMES FOR NONISOTHERMAL FIRST-ORDER REACTION SYSTEMS GENYUAN Department

LI and HERSCHEL

of Chemistry, Princeton University,

(First received 22 January 1990; accepted in

RABITZ’ NJ 08540, U.S.A.

Princeton,

revisedform 11 April 1990)

Abstract-The direct approach to determining the constrained lumping schemes presented in a previous paper is applied to nonisothermal first-order reaction systems. The constant basis matrices of the transpose of the Jacobian matrix for the kinetic equations are replaced by a set of rate constant matrices at different temperatures, which properly cover the desired temperature region. The Mobil “lo-lump cracking model” is used as an example to illustrate this approach.

2. THE DIRECT APPROACH

1. INTRODUCTION

Our previous paper (Li and Rabitz, 1991) presented a direct approach to determining the constrained lumping schemes for an arbitrary reaction system. When the system is isothermal, the transpose of the Jacobian matrix of the kinetic equations can be readily decomposed as a linear combination of a set of constant matrices. They are viewed as a basis of the transpose of the Jacobian matrix. Using the concept of the simultaneous minimal invariant subspace to all these basis matrices over a given subspace, the direct approach will supply the best constrained lumping matrices with different dimensions. For a nonisotherma1 first-order reaction system the transpose of the Jacobian matrix is the transpose of the rate constant matrix, which is a function of temperature and also has a set of constant basis matrices. Therefore, the direct approach can, in principle, be employed to determine the constrained lumping matrices for this system if one can find the basis matrices. Unfortunately, the rate constants are generally exponential functions of temperature and then it is not easy to determine the constant basis matrices of the transpose of the rate constant matrix. However, the basis matrices can simply be replaced by a set of rate constant matrices corresponding to different fixed temperatures in the desired temperature region. When the number of chosen constant matrices in the set is large enough and the temperature region is properly covered by the chosen temperature points, the results will be the same or close to those obtained by using the basis matrices. In Section 2 the theoretical basis of the direct approach for application to nonisothermal first-order reaction systems is presented. The Mobil “lo-lump cracking model” is used as an example to illustrate this method in Section 3. Finally, Section 4 presents a conclusion and discussion.

‘Author

to whom

correspondence

should

be addressed. 583

FIRST-ORDER

FOR NONiSOTHERMAL

REACTION

SYSTEMS

Our previous papers (Li and Rabitz, 1989, 1990) presented a general analysis of exact and approximate lumping for a reaction system in a desired region R of the composition x-space. The original reaction system with n-components can be described by dyldt = f(y)

(1)

where y is an n-composition vector; f(y) is an arbitrary n-function vector, which does not contain t explicitly. If the system can be exactly lumped by an d x n real constant matrix M with rank A (ii C n), then for ?=My the lumped

(2)

system can be described djr/dt = Mf (I@)

as (3)

where the subspace _& spanned by the row vectors of M is a fixed invariant one to the transpose of the Jacobian matrix Jr(y) of f[y) for any value of yea, and @ is one of the generalized inverses of M (BcnIsrael and Greville, 1974) satisfying MM=

Iti.

(4)

If Jr(y) does not have a fixed invariant subspace which has a given dimension li or satisfies some desired restriction, then eq. (3) can still be used to describe the lumped system approximately. In this case, one needs to find a subspace .& which meets the requirements and is as nearly Jr(y)-invariant as possible. This lumping matrix is the best one for the given dimension fi and under the required restriction. The accuracy may not be satisfactory if A is too small. When R is the whole n-dimensional composition space and M has orthonormal rows, MT is the best choice of n? for approximate lumping (Li and Rabitz, 1990). Considering this we will choose orthonormal rows for M and consequently h? = M'.

584

GENYUANLI and

For a nonisothermal first-order reaction kinetic equations are the following:

system the

dy/dt = K( T)y

(5)

HERSCHEL

RABITZ

In order to determine Im (BAB . . As-’ B) we can first determine the kernel by solving the following equation

where K( T) is the rate constant matrix, which is a function of temperature T. According to eq. (3) the lumped system can be represented as df/dt

x = 0.

(12)

= k( T)f = MK( 7-)MTi.

(6)

For the constrained lumping problem matrix M can be represented as

the lumping

M=

Suppose the dimension of ImX is n - 1. After the determination of X the matrix representation M’ of the smallest A-invariant subspace _N with dimension I over Im B can be determined by solving the equation

(7)

where M, is given and also required to satisfy M,Mz = Ie _ r; M, will be determined and satisfy M,Mg = I, (where r is the row number of M,) as well. The direct approach to determine the constrained lumping schemes with different ri has been presented in our previous paper (Li and Rabitz, 1991). This approach is based on the concept of the minimal JT(y)invariant subspace over Im Mi. Again following the previous work on exact lumping, Jr(y) can be decomposed into a linear combination of appropriate constant matrices A, (k = I, 2, . . . , m), i.e.

XTMT

= 0.

1

BT BTA;

BT(A;)“‘-

’ 1

JT(Y) = f %(Y)A, k=1

(8)

where m is less than n2 and the A,s are viewed as a basis of J’(y). When R is the whole n-dimensional space, the minima1 simultaneously all A,-invariant subspace over Im Mg is the minimal J’( y)-invariant one over Im ME. In order to understand the basic idea of the direct approach in the application of the nonisothermal first-order reaction system, we will briefly draw from our previous paper about the basis of this method. It is well known that the minimal invariant subspace A? for an n x n matrix A over a given subspace ImB coincides with A-? = f Im (MB) = “2’ Im(AjB) ,=o j=O

(9)

(13)

It is straightforward to determine the minimal simA,, (k = 1,2, . , m)-invariant subspace ultaneously A over the subspace Im B. We only need to determine X first by solving the following equation:

Y=O

(14)

B’ B’A’ m

BT(A;)“-’

. . ) m) is greater than or equal to the where s, (k = rank of At, and then solve eq. (13) to determine M. In the current problem B = M& 32” = x and the resultant M is the exact lumping matrix containing M, with the smallest row number 1. When we want to proceed further to find goodquality approximate lumping matrices with A less than 1, we need first to determine higher-dimensional Im X which are as nearly as possible orthogonal to

for every integer s greater than or equal to the rank or the degree of a minimal polynomial for A in particun-1 lar, .& = 1 Im(AjB) (Gohberg et al., 1986). We know tha(=O

2 Im (AjB) = Im(B Al3 . . A”-

* 5)

UO)

(15)

j=O

and the orthogonal decomposition siona1 real space 5e” is

of the n-dimen-

I-

(11) Then

the

resultant

J/s

will be as nearly

all A,-

585

Determination of constrained lumping schemes

invariant as possible. The corresponding MS are good approximate lumping matrices containing M, with 6 less than 1. This consideration is equivalent to finding the subspace Im X, which is simultaneously as .., nearly orthogonal to ImMz, Im(M,AT)T, Im(M,AT)T,. .., Im ME, Im [M,(AT)“l-‘]T, Im [M,(A,T)“--’ ] ’ as possible. This X can be readily determined by using the concept of the degree of coincidence between two subspaces given in our previous paper (Li and Rabitz, 1990). Let Q(G)&) (k = 1, 2, . . . , m; i = 91, . , sk - 1) be the orthonormal matrix representation of Im [M,(A:)‘IT. Using the Schmidt orthogonalization method one can transform [MG( A;)‘]’ to Q(G),&. First we define a matrix m

If we choose

MC

w-1

’ = C C Q(G)Gi,QtG)(ki,. k-1

ing to different temperatures in the desired temperature region. When the number of the rate constant matrices is large enough (i.e. some of these constant matrices compose a basis) and the temperature region is covered properly by the chosen temperature points (i.e. the different regions of temperature are appropriately weighted), the results should be the same or close to those obtained by using the basis matrices. Since this is easy to realize, the approach above is very useful for those systems whose Jacobian matrix cannot readily be decomposed to a linear combination of constant matrices. Let K( T) be the rate constant matrix at temperature T. then eq. (15) becomes

M,K(T,)

(16)

i=O

an orthonormal

basis for Im X, i.e.

X’X=I,_fi, then the problem becomes the determination which gives the smallest trace min xTx=I,Ps

MGK(G)Si_’

(17)

(20)

of X, MC M,K(T.J

tr XT YX.

(18)

The solution can be readily obtained by determining the eigenvalues and eigenvectors of Y (Bellman, 1970). The n - ri eigenvectors with the smallest sum of their eigenvalues are X and the rest of the eigenvectors compose MT. When all the eigenvalues are distinct, the solution for M with a specified ri is unique. If there exist multiple eigenvalues, the sets of eigenvectors with the same sum of eigenvalues are all solutions. When the eigenvectors of Y are arranged according to the nonincreasing order of their eigenvalues, the last n - ri eigenvectors are X and the first h eigenvectors are MT. Therefore, the eigenvector matrix R of Y supplies all the best approximate lumping matrices with different it. There are two further issues we need to consider. is a null matrix. In this case First, sometimes M,A’ the contribution of Ai to the determination of the lumping matrix can be neglected. In order to avoid this situation, we can use the resultant M from other A, with row number 1 higher than M, as a new M, to calculate M, AF. If M,Af for the new M, is still a null matrix, we can use the resultant M with row number 2 higher than the original M, as a new M, to calculate M,A’ and so on. Second, in order to satisfactorily assure that the resultant M, is orthogonal to M,, one can multipIy M, in eq. (15) by a large positive con-

M,K(

first-order

reaction

where Q(y) is an ri x ii matrix. It is easy to demonstrate that A! is also invariant to any analytic function of JT(y). Let f’[ JT(y)] be an analytic function of JT(y). It can be expanded in a Taylor series: f’C

JT(y)]

= f.

where [ JT( y)]O = I, and cis are coefficients. to find that

fCJT(y)l = f

Since the rate constant

is an exponential

(19) function

(22)

It is easy

ciCJ(~)l’-

i=O

Then we have W-C JT(y)l

system

T).

CiCJ’CY)]’

i=O

= ~4 2 ciC J(Y)]’ i=0 ciCQ(Y)l’M

i=o

we have J’ (y) = K”(

(21)

MJ(Y) = Q(Y)M

= f the nonisothermal

’

Thus the constrained lumping matrices with different fi can be obtained by the corresponding eigenvectors of Y. If the subspace YK spanned by the row vectors of M is J+(y)-invariant, we have

stant c. For

T,r”-

=f^lQT(y)lM

(24)

f ciCQTt~)3’ i=o

(25)

where

of

temperature T, it is not easy to determine the basis matrices of KT( T). However, all the A,s can be replaced by a set of rate constant matrices correspond-

f^=CQT(~)l

and

we have

used

=

the relation

of eq. (21) in the

586

GENYUAN

LI and

HERSCHEL

RABITZ

deduction of eq. (24). Equation (24) shows that .& is j”‘[ JT(y)j-invariant. This is very useful for the firstorder reaction system, because the analytic function eK(‘)’ of K( T) can often be determined experimentally. The solution of eq. (5) is eK’T”y(0). Let yI(0), y,(O) be n linearly independent initial Y2(0)> ..., values of y and compose the matrix Y(0). yl(r), y2@), . . , Y,(T) are the corresponding solutions for t = r and compose the matrix Y(z). Then we have Y(r) = eKcT)’Y(0).

Ah

Nh

(26)

Since Y(0) and Y(z) can be determined experimentally and Y(O) is nonsingular, c~(~)’ will be obtained by #‘?‘)I

= Y(z) Y -

’ (0).

(27)

In many realistic problems, the rate constant matrix K(T) is usually unknown in advance. Therefore, tak-

ing advantage of this situation we can use &*I)’ in eq. (20) instead of X(K) to determine the constrained lumping matrices with different ii. Let G(q) = eX(Ti)r. Then we have

MG

-

M,G(T,)

Fig. 1. IO-Lump cracking model kinetic scheme: P, = wt % spectroscopy analysis), molecules (mass paraffinic 430-650°F; N, = wt % naphthenic molecules (mass spectrascopy analysis), 430-650°F: C,, = wt % carbon atoms among aromatic rings (n-d-M method), 43O-650°F; A, = wt % aromatic suhstituent groups, 43M5O”F, P, = wt % paraffinic molecules (mass spectroscopy analysis), 650+“F; IV, = wt % naphthenic molecules (mass spectroscopy analysis), 650+“F; C,, = wt % carbon atoms among aromatic rings (n-d-M method), 650+“F; A, = wt % aromatic substituent groups, 650+“F; G = G-lump (C,, 430°F); C = C-lump (C,-C, + coke); C,, + P, + N, + A, = LFO (430-650°F); C,, + P, + N, + A, = HFO (650+“F).

MG M,G(T,)

cess (Weekman, 1979; Jacob et al., 1976). The scheme of this model is shown in Fig. 1. The composition vector is

This approach will be illustrated by the Mobil “lolump cracking model”. The best constrained further lumped systems with ri = 36 valid in a given temperature region will be given.

3. THE

MOBILE

“IO-LUMP

CRACKING

MODEL”

The method proposed above will be illustrated by the Mobil “IO-lump model” of catalytic cracking pro-

K(W)

=

I - 83.55 0.00 0.00 0.00 20.70 0.00 0.00 0.00 55.00 7.85

0.00 - 122.07 0.00 0.00 0.00 22.50 0.00 0.00 84.70 14.87

0.00 0.00 - 166.20 0.00 0.00 0.00 19.00 50.00 63.00 34.20

0.00 0.00 0.00 - 20.49 0.00 0.00 0.00 5.86 0.00 14.63

Y=

(PhN, -4 CA,P, N, 4 CA,G QT.

The corresponding rate constant matrix K( T) is given in Fig. 2. The sum of P,, N,, A,, and C,, is called the heavy fuel oil (HFO) and the sum of P,, N,, A, and CAl is called the light fuel oil (LFO). The data of K( T) for T = 900°F and the activation energies derived from temperatures of 900, 950 and 1000°F are available (Gross et al., 1976). Using these data and weight % units for the concentration of the species, we obtain the K(T) for T = 900, 950 and 1000°F as follows (in units of lo3 h- ‘):

0.00 0.00 0.00 0.00 -33.29 0.00 0.00 0.00 23.85 9.44

0.00 0.00 0.00 0.00 0.00 - 74.33 0.00 0.00 66.15 8.18

0.00 0.00 0.00 0.00 0.00 0.00 -22.13 0.00 18.50 3.63

0.00 0.00 0.00 0.00 0.00 0.00 0.00 ~ 1.00 0.00 I.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 - 4.40 4.40

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 O.OO_

Determination of constrained lumpingschemes

K(950)=

K(loOO)=

-83.86 0.00 0.00 -.122.51 0.00 0.00 0.00 0.00 20.80 0.00 0.00 22.60 0.00 0.00 0.00 0.00 55.17 84.97 7.89 14.94

.84.15 0.00 0.00 0.00 20.89 0.00 0.00 0.00 55.34 7.92

587

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

167.38 0.00 0.00 0.00 0.00 0.00 --20.67 0.00 0.00 0.00 0.00 0.00 --33.42 0.00 0.00 0.00 0.00 0.00 --74.58 0.00 19.09 0.00 0.00 0.00 -22.32 50.23 5.89 0.00 0.00 0.00 63.52 0.00 66.36 18.65 23.93 34.54 8.22 3.67 14.78 9.49

0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00 -4.45

-1.01 0.00 1.01

0.00 0.00 0.00 0.00 0.00 0.00 0.00 22.93 0.00 0.00 O.aO 0.00 0.00 0.00 0.00 -168.51 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -20.83 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -33.53 22.70 0.00 0.00 0.00 --74.81 0.00 0.00 0.00 0.00 --22.50 19.17 0.00 0.00 0.00 0.00 50.45 5.91 0.00 0.00 0.00 -1.02 85.22 66.55 64.02 0.00 24.00 18.80 0.00 15.01 8.24 3.70 1.02 34.87 14.92 9.53

0.00 OSQ

4.45

O.OO_

0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 o.cnI 0.00 0.00 .

0.00 0.00

0.00 0.00

-4.50 4.50

0.00 0.00 I

The G( T) = eKtrijr were computed with 7 = lo-’ which was chosen because the significant dynamics occurred within 102:

G(900)=

G(950) =

0.4337 O.oooO 0.0000 O.OQOO 0.1166 0.0000 0.0000 0.0000 0.3803 0.0694

0.4323 O.OCOO 0.0000 OSK00 0.1169 O.CNIOO 0.0000 0.0030 0.3810 0.0698

0.4311 O.OCNI 0.0000 0.0000 0.1172 0.0000 G(lOOO)= o.oooo 0.0000 0.3816 _ 0.0701

O.oooO O.oooO

O.oooO O.CNMl O.OOOO

O.WCNl

0.0000

O.OCKNJ

O.OOCJO 0.0000O.oooO O.oooO

O.oooO

O.M)oo

O.oooO

O.WCQ

0.2950 0.0000 0.0000 0.0000 0.0851 0.0000 O.WOO 0.5157 0.1042

OSXKNJ

O.OOW

0.1898 O.oDoo O.WOO 0.0000 0.0807 0.2422 0.3085 0.1788

0.0000

0.8147 O.OCHlO O.oooO O.oooO 0.0527 O.OODO 0.1326

O.oooO O.OC@O O.oooO

0.7168 O.oooO 0.0000 0.0000 0.4755 O.OWJ 0.0000 O.oooO 0.8015 O.oooO 0.0000 O.OWO 0.1982 0.4554 0.1622 0.0849 0.0691 0.0363

0.0000

0.0000

O.WOO

O.oooO

O.oooO

0.2937 O.C0DO O.OXQ O.oooO 0.0852 o.OOoO 0.0000 0.5164 0.1047

0.0000

0.0000 0.0000

O.OCKKl

0.0000

O.oooO

O.OOCNl

O.CMlOO

0.0000

0.0000

O.OCJCQ

0.0000

0.7159 0.0000 0.0000 O.OlXW 0.1987 0.0854

O.OOOO 0.4744 OSlODO O.OOCNI 0.4562 0.0694

O.CKWO

O.oooO

O.oooO O.oooO

0.0000 0.0000

o.oLloo

0.0000 0.0000

O.OOCO

0.0000

0.0000 O.OOW

O.OOW 0.2925 0.0000 O.OCKJO 0.0000 0.0853 o.c000 0.0000 0.5171 0.1051

0.1875 O.oooO 0.8133 0.0000 O.C!OOO 0.0000 00000 0.0806 0.0000 0.2423 0.0529 0.3097 0.0000 0.1799 0.1338

0.0000 O.OOCQ

o.oooo

o.oooO

0.0000

0.1854 0.0000 0.0000 O.OOC@ 0.0805 0.2423 0.3108 0.1810

0.0000

0.0000 O.CU3OO

0.8120 O.OOOD 0.0000 O.OC00 0.0531 0.0000 0.1350

O.OWQ O.OOOQ

O.oooO O.oooO O.oooO

O.OOOO

O.oooO

O.OWJ 0.0000

0.0000 O.WlOO

0.0000 0.0000

O.OOOU O.OMXl

0.0000

O.OOGU

0.0000 O.oooO

O.KJOO

0.0000

0.9570 O.OOGU 0.0430 l.OOGQ

O.oooO

O.CJWl

O.oooO

O.oooO 0.0000

0.0000

O.MWO

0.0000 0.8000 O.OOW O.oooO O.OLWO 0.9900 0.0000 0.1634 0.0000 0.9565 0.0367 0.0100 0.0435

O.oooO 0.0000 0.0000

0.0000

O.OCHXl

O.OOW

0.0000

0.0000

O.COOO

0.0000 O.OWO

O.oooO

0.7151 0.0000 0.0000 O.OOOO 0.1991 0.0857

0.0000

0.0000

O.WOO

O.OCNXl

0.4733 0.0000 0.0000 0.4569 0.0698

0.0000

0.0000

0.0000

0.7985 0.0000 0.1645 0.0370

O.MOO

0.0000

0.0000 o.olxo O.CQW

0.9899 O.ODOO 0.0101

O.OCW

0.0000

.

0.9560 O.OCW 0.0440 1.MJOO I

588

GENYUAN LI and HERSCHEL RABITZ

0

0

0

0

0

0

0

0

of constrained lumping schemes

Determination

The goal of the catalytic cracking process is the production of gasoline. The C-lump (H,, H, S, C ,-C, and coke) is the undesired by-product. These two species correspond to y, and y,, of y. Therefore, we keep them unlumped and lump the other species to simplify this system. Hence the given part of the lumping matrix M is M,=

0000000010

(

0000000001’

This information tions.

>

will be used in the following

0.71

Y(K)=

1.80

0.07

0.21 8.67 0.46

-0.16 0.46 0.30

0.14 0.21 0.07

-0.11

0.07 0.78

0.14 1.78

0.21 -0.11

0.07 -0.19

0.05 0.08

0.08 0.87

0.02 0.11

0.14

0.02

- 0.02

0.02

0.02 0.11

0.01 0.02

0.11 -0.01

0.02

0.03 0.13

-0.01 -0.06

_ 0.00

0.00

0.00

0.00

3A. The lumping schemes in the isothermal regime In order to find the difference between K( T) and ti”)’ in the determination of constrained lumping

104,

8.7822,

-0 0

0.0695

-0.2395

00

0.3426

-0.8695

00

0.9329 0.0411

00 00 R(K)=

0.09 0.14

-0.14

7.33 0.27 -0.16

0.09

rli= 104,

Here we choose s = n = 10. Then using eq. (16) the symmetric matrices Y and their eigenvector matrices R are determined. In order to force Ma to be located on the first two columns of R and the lumped species to be composed of the other eight original species (correspondingly the last two elements of each column of M, are zero), M, in the first row of eqs (29) and (30) are multiplied by 100. Let Y(K), Y(G) and R(K), R(G) represent the corresponding symmetric matrices and their eigenvector matrices for using K(900) and G(9CO), respectively. The eigenvalues are also given right above the corresponding eigenvector matrix. In the case of using K(900) the resultant Y(K) is the following:

1.a0 -0.11 -0.14

-0.01 - 0.05

-0.11

set-

589

8.2302,

0.3537 0.0484

0.6857,

I.

0.00

0.13 0.11 0.02 - 0.07 - 0.01 0.01 104 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 10*_

0.2433,

0.0167,

00007,

0.0001,

O.OOCO

0.2937 - 0.0504

- 0.7059 0.0302

0.1875 -0.0083

0.0736 -0.0023

0.0463 -0.3379

0.0485 - 0.8708

0.0102 0.2554

- 0.0052 0.0531

-0.0006 -0.1717

0.0001 -0.1613

and

_M, G;900)9 :

0.M)

-0.06

-0.2174 0.0887

(29)

MC

- 0.07

0.00 -0.01

-0.05

0.5151 -0.3395

schemes we first determine the constrained lumping schemes at 900°F by using K(900) and G(900), respectively. In this case eqs (20) and (28) become

M, G(9W

0.02 -0.02

-0.19

-0.01 -0.01 0.03 0.02 0.01 -0.01 0.00 0.00 0.01 0.00

The eigenvalues li of Y(K) arranged in nonincreasing order and the eigenvector matrix R(K), to the whose eigenvectors are arranged according order of their eigenvalues, are given below:

0.0292 -0.0105 0.0481 0.0699 -0.2424 0.6969 0.0096 -0.0197 0.1210 0.0024 0.0025 -00.0203 0.0000 O.OONl 0.0000 0.0000 0.0000 OSWQ

00 00 00 10 -0 1

0.78 1.78

-0.3758 -0.6799 0.0522 o.s602 0.2768 -0.1852 0.0365 0.6173 -0.1684 -0.0750 -0.0699 -0.6176 -0.3377 -0.6279 -0.3004 -0.0616 0.0277 -0.o.0099-0.4460 0.8922 0.0000 o.coOo o.coOO O.oooO O.oooO OSXQO 0.0000 0.0000 O.oooO O.WOO_

According to the direct approach the first three columns on the left of R(K) compose the best constrained approximate lumping matrix with ri = 3, the first four columns compose the best constrained approximate lumping matrix with li = 4 and so on. Since the last three eigenvalues are equal to or almost equal to zero, the first seven columns of R(Y) compose an almost exact lumping matrix. From eq. (6) we know that r?( T) = MK( T) MT.

(30)

(31)

Then we have the rate constant matrix for the lumped system with 6 = 7 at 900°F as follows:

590

GENYUAN Lr and HERSCHELRABITZ -4.4aoo

f+co)=

O.OOOD

97.1113

-81.1857

51.9774

4.4Wo

0.0000

39.0312

-4.1559

2.2088

- 16.8466

O.oooO

0.0000

- 158.9405

- 17.2578

0.4065

- 7.8806

0.0000

O.OOOD

- 17.9688

- 117.5942

- 13.7646

- 0.8830

o.Oooo

OS000

-28.9118

- 80.1459

18.3513

0.0000

OSKKKJ

- i3.9865

14.3366

- 26.7782

0.0000

- 11.1369

- 20.2488

3.8874

_ O.OOOQ

7.2540

3.4880 -1.1708

When we use the first it (fi < 7) columns of R(K) to compose the lumping matrix, the resultant lumped rate constant matrix is the ri x ii submatrix in the top left-hand corner of the above matrix. Therefore, this matrix supplies all k(900) for ri = 3-7. For the initiat composition (yt - y, = $, others are zero) we obtained the evolutions of the concentration of y, by solving eqs (5) and (6) (for fi = 4-7). The results are shown in Fig. 3. One can see that, when ri becomes larger, the solution of the lumped system is closer to that of the original one. For fi = 7 the lumping is almost exact. Following the same procedure we use G(900) instead of K(900) to determine the constrained lumping matrices for different li. The resultant Y(G) is the following:

1 i

Y(G)=

-23.8956

- 12.6995- 2.6924 -0.6869 0.03 76 - 12.7016 -0.8136 - 37.0395_

Similarly this matrix supplies all $900) with F?< 6 by in the top left-hand corner of the ri x ii submatrices the above matrix. The comparison of y, between the exact solution and the solutions given by the lumped models with ri = 3-6 is shown in Fig. 4. When fi = 6 the coincidence between the exact and the lumped models is very good. From the results obtained by using K(900) and G(900) one can find that G( T) gives the better results. The reason is not entirely clear. Possibly the lumping schemes given by K(T) are valid in the whole ndimensional space, while the lumping schemes obtained from G( T) are suitable for the whole composi-

1.30

1.44

0.77

1.46

0.97

0.00

1.74

0.05

1.61 0.86

0.86

0.02 0.03

0.98

1.44 0.77

1.08

1.62

1.05

0.00

1.97

0.09

0.64

0.49

0.68

0.02

0.03

0.49

1.40

0.31

0.82 -0.11

0.54 - 0.06

0.06 0.16

0.93 -0.31

0.98

1.08

0.68

0.3 1

0.82

1.07

0.74

1.46 0.97

1.62 1.05

0.82 0.54

-0.11 -0.06

1.07 0.74

1.64 1.08

1.08 0.75

0.00 1.74

0.00 1.97

0.06 0.93

0.16 -0.31

0.04 1.20

-0.01 2.00

-0.01 1.25

0.05

0.09

I.16

2.96

0.72

-0.21

-0.08

0.0142,

o.ooo5,

o.ooo2,

0.0646

0.0160 -0.4522

0.04 -0.01 -0.01 0.02 -0.03 0.33

1.16 / 2.96

1.20

0.72

2.oa 1.25

-0.21 J - 0.08

-0.03 104

.

0.33 -0.75

-0.75

to4

The eigenvalues li of Y(G) arranged in nonincreasing order and the eigenvector matrix R(G) arranged according to the order of their eigenvalues are given below: 1, =

104,

104,

6.4681, 0.4475 0.4953 0.2769 0.0381 0.3455 0.4976 0.3307 0.0042

1.6352, -0.0623 -0.0625 0.28 11 0.9203 0.1508 -0.1526 - 0.0928

0.0000

0.1080 O.OOOQ

0.0000

O.OGQO

0.0642, -0.0095 -0.4301

-0.2227 -0.0774

-0.2664 0.0128

0.8340 -0.3109

-0.0871

0.4648

0.1052

0.5944

- 0.2024 0.6976

- 0.3090 0.1121

0.2945 -0.5522

-0.0063

-0.1820 O.OCGll

- 0.3075 O.OOC@

O.OOClO 0.0000

Observing the eigenvalues of R(G) we found that the last four eigenvalues are equal to or almost equal to zero. Therefore, the first six columns of R(G) compose an almost exact lumping matrix. Using eq. (31) the resultant rate constant matrix for the lumped system with ri = 6 at 900°F is the following: 0.0000 O.CK@O 0.0000 O.OO@l O.OWl O.OOCO

-0.3854 - 0.0097

131.2834 29.4419 - 73.7046 - 2.2309 41.2758 -20.1729

O.OOOCl

O.oooO

0.2385 -0.1119 -0.1031 0.3770 -0.ooO5 0.7571

0.0000, 0.0217 -0.3865 0.1183 0.1840 -0.4889 0.4580 0.2576 -0.5358

O.oooO

0.0000

O.OOQO

0.0000

O.OCUKl

-0.8616 0.2396 - 0.0825 0.0295 0.1667 0.3925 0.1069 0.0190 0.0000 O.oooO_

tion region (i.e. all yi being nonnegative and xi yi = 1). Considering the results we will determine the lumping schemes validated in the temperature region ~OC-~OOO”F by using G( T).

0.7743 21.6056 - 2.2557 -32.3531 8.9664 -41.2757

-43.1329 - 10.1357 29.03 15 6.1642 - 56.9739 29.5716

17.88021 19.7656 - 12.7847 - 36.2268 33.7715 - 135.6620.

1.

of constrained lumping schemes

Determination

38. The lumping schemefor the nonisothermal regime Since G(T) for different temperatures (90% 1000°F) are very close to one another, it is enough only to choose three matrices G(900), G(950) and G(1000) to determine the lumping schemes for this temperature region. Utilizing eqs (16) and (28) and following the same procedure as that in Section 3A, we obtain the symmetric matrix Y(G), its eigenvalues and eigenvector matrix R(G):

2.9 1 3.16 1.63

-3.91 4.32 2.31 0.05

4.32 4.81 2.60 0.08

2.3 1 2.60 1.93 I .4-l

0.05 0.08 1.47 4.23

2.95 3.24 2.05 0.94

4.37 4.84 2.46 -0.32

2.95

3.24

2.05

0.94

2.47

3.21

-0.19 2.22

4.37 2.91

4.84 3.16

2.46 1.63

-0.32 -0.18

3.21 2.22

4.93 3.25

3.25 2.28

0.01 5.23 -0.16

0.01 5.89 0.27

0.17 2.80 3.49

0.50 - 0.94 8.93

0.11 3.60 2.15

-0.04 5.99 -0.65

- 0.02 3.76

Y(G)=

3 x 104,

&=3x104,

R(G)=

0.1932,

4.9500,

0.4472

- 0.0622

-0.0102

-0.2227

0 0

0 0

0.4948 0.277 1

~ 0.0622 0.2807

-0.4302 -0.2669

- 0.0779

0

0

0.0383

0.9206

0.0138

0 1 0 0

0 0 0

0.3456 0.4973 0.3319

0.1500 -0.1522 - 0.0934

0.4645 -0.2033 0.6973

0

0

0.0042

0.1077

1 0

0 1

O.oooO O.OOCQ

O.COOO O.COOO

O.COOO 0.0000

Comparing the resultant R(G) with that for isothermal condition in Section 3A, one can see that the eigenvector matrices R(G) are almost the same. Since we use three G(T), the eigenvalues should be nearly 3 times of the eigenvalues for the isothermal condition. This is found to be true. Therefore, the first six columns of R(G) will supply an almost exact lumping matrix for 90% 1000°F. Using eq. (3 1) the lumped rate constant matrices for 900, 950 and 1000°F are as follows: -4.4000

K(900)=

o.oooO

0.0000

-20.1893

-4.5000 4.5Oco O.oooO

:

O.oooO O.OOfM O.oooO

0.0000

O.oooO O.OCMl 0.0000

-0.10 3 x lo4 - 2.25

-2.25 3 x 1oq

0.0014,

0.0005,

0.0000,

0.0633 -0.3829

0.0149

0.0110 - 0.3850

-0.0107

-0.4527 0.2381

- 0.0862 0.5952 0.2937 -0.5514 -0.3121

-0.1130 -0.0974 0.3769 -0.0040 0.7575

0.1172 0.1839 - 0.4879 0.4643 0.2594 -0.5321

0.0275 0.1719 0.3857 0.1044 0.0251

O.OCMl

O.OCOO

O.OOC0

O.oooO

0.0000

0.m

_

aromatic = (0.1, 0.1,0.2,0.4,0.05,0.05,0.05,0.05,0,0); and (c) naphthenic = (0.15, 0.4, 0.1,0.08,0.07,0.2,0,0, 0, 0). They represent the basic charge compositions. To save space we do not give all the results for different initial compositions and temperatures. They slightly differ in accuracy for ri = 3 or 4, but they have

-43.2836 _ 10.1565

17.8111

29.0700

- 12.7736 -36.1860

-32.3174 8.9754 -41.2263

6.1810 - 57.0402 29.5970

131.7775 29.6477

0.8610 21.8202

-43.4412 - 10.2328

-73.9510 -2.3185 41.4835 - 20.4050

-2.3267 - 32.5687 9.0604 -41.4819

29.1827 6.2528 - 57.3446 29.8297

19.7626

33.7829 - 135.6848 I

18.1462 19.9808 - 12.9555 -36.4184 34.0363 - 136.5938 I

0.0000

0.9428 22.0337 - 2.3792

-43.5854 - 10.3181 29.2919

18.4713 20.1938 - 13.1329

0.0000 0.0000 O.OtXlO

-2.3706 41.6453 -20.6144

- 32.7989 9.1425 -41.7281

6.3230 - 57.6298 30.0547

-36.6427 34.2775 - 137.4624 I

0.0000

-

0.2458 - 0.084 1

O.COOO

132.2843 29.8362 - 74.2425

O.CQOO

O.CWO -0.8614

0.0000

0.7763

0.0000 0.0000

0.06 -0.10 1.00

215 - 0.65 -0.25 1.00

0.0000

21.5948 - 2.2723

O.OOW O.OOMl

0.11 - 0.c-l - 0.02

0.18 0.27 3.49 8.93

O.CClOO

29.4447 -73.6407 -2.2644 41.3131

O.MCJO 0.0000

0.8340 -0.3101 0.1046 -0.3093 0.1125 -0.1826

131.2420

0.0000 0.0000

0.0000 O.CJOCKl O.OCQO 0.0000

K(1000) =

O.OOLXl

4.4000 O.OOMl

- 4.4500 4.4500 r2(950)=

0.0428,

19.4200,

0

5.23 5.89 2.80 - 0.94 3.60 5.99 3.76

0.01 0.0 1 0.17 0.M

- 0.25

0

-0.0064

591

Simila_rly these matrices supply all the I?(900), k(950) and K(1000) with li < 6 by the li x r? submatrices in the top left-hand corner of the above matrices. The comparisons of y, and y,, between the exact solutions and the solutions given by the lumped models with r’i = 3-6 and T = 900,950 and 1000°F are shown in Figs 5-10. The initial compositions chosen by Coxson and BischolT(l987) are adopted here: (a) paraffinic = (0.3, 0.1, 0.15, 0.15, 0.2, 0.05, 0.03, 0.02, 0, 0); (b)

592

GENYUAN LI and HERSCHEL RABITZ

I

I

I

I

/

I

I

0 0

0

q

0

0

0

o

1

0

0

0

a

0 ~)

q

0

0

0

04

q

0

q

00

0

I

I

0

solid line. original model( 10 species) o solution of 4-dlmensional lumped model o solution of 5-dimensional lumped model + solution of B-dImensional lumped model x solution of 7-dimensional lumped model initial condition: (0.16,0.16.0.1E,~.16,0.16,0.16.0,0,0.0)

I

I

I

I

I

I

f

I

t( 1O-5h- )‘ Fig. 3. Comparison

ofy, (gasoline) for the original model and the isothermal lumped models obtained by using K(!NO).

0.5

B

sohd line

original

* solutron

of 3-dimensional

lumped

model

solution

of 4-dimenslonal

lumped

model

O solution

of 5-dimensional

lumped

model

+ solution

of 6-dimensional

lumped

model

q

initial

condition:

L 00

10

20

30

model{ 10 species)

(0.16.0.16,0.16,0.16.0

16.0.16,0.0.0,0)

I

I

I

I

I

I

4.0

5.0

6.0

7.0

0.0

9.0

10.0

t(10-5h-1) Fig. 4. Comparison

a similar accuracy

of y, (gasoline) for the original model and the lumped models obtained by using G(900).

for fi = 5 or 6. When ii = 6 the solutions for the lumped model in all these conditions are almost exactly the same as those of the original model. When A = 5 the coincidence between the exact and the lumped models is very good. Considering that there exists experimental error in practice the lumped model with ri = 5 is adequate and the lumped model with fi = 4 is acceptable. Even if A = 3, for most conditions the lumped model approximates the original system quite well. All these results show that the direct approach can be employed to determine the best constrained approximate lumping schemes for

the first-order conditions.

reaction

system

under

nonisothermal

4. CONCLUSION AND DISCUSSION

In the present paper, we have shown that the direct approach to determining the constrained approximate lumping schemes for an arbitrary reaction system can be employed to the determination of the lumping schemes for a first-order reaction system under the isothermal and nonisothermal conditions. In the nonisothermal case the rate constant matrix K( T) is a function of temperature T and the constant

of constrained

Determination

solid line: original c

lumped model

soluhon

of 4-dimensional

lumped model

O solution

of 5-dimensional

lumped

+ solution

of B-dimensional

lumped model

, temperature 1.0

model( 10 species)

of 3-dimensmnal

initial condition:

0.0

593

* solution q

/y,

lumping schemes

2.0

3.0

model

(0.3,0.1,0.15.0 15,0.2,0.05.0.03,0.02,0,0)

, 900” F 4.0

,

,

,

,

,

5.0

6.0

7.0

5.0

9.0

10 0

Fig. 5. Comparison of y9 (gasoline) at T = 900°F for the original model and the lumped models obtained by using G(900), G(950) and G(lOO0).

solid line: original model{ 10 species) * solution of 3-dimensional lumped model o.4 -

solution o solution

q

of 4-dimensional of &dimensional

lumped model lumped

t A d

+ solution of 6-dimensional lumped model initial condition: (0.3,0.1,0.15,0.15,0.2.0.05,0.03,0.02,0,~ temperature: 900” F

I

I

,

I

I

L

I

I

I

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

90

10.0

t( 10-5h- )‘ Fig. 6. Comparison of y,,(C-lump) at T = 900°F for the original model’and the lumped models obtained by using G(!BlO), G(950) and G(lQO0). basis matrices of K( T) are not easy to determine. In this case one can use a set of K( T) for different given temperatures, which properly cover the desired temperature region, instead of the basis matrices of K( T). If the subspace .4 spanned by the row vectors of the lumping matrix is invariant to the transpose of the Jacobian matrix P(y) of the kinetic equations, .N is also invariant to any analytic function of Jr(y). For the first-order reaction system Jr(y) is Kr( T) and [eK(T)r]T is an analytic function of KT( T). Therefore, one can use ex(r)’ instead of K( T) to determine the constrained approximate lumping matrices. The re-

sult of the present paper shows that the lumping schemes obtained by using 8rTjt are even better than those by using K(T). Since $r’r)I can be determined experimentally, using eKCT)’is more advantageous. The Mobil “104ump cracking model” was used to illustrate this approach. The results show that this model can be adequately reduced to lumped ones with five or six additionally lumped species. The accuracy of the lumping schemes validated for the temperature range T = 90% 1000°F is almost the same as that for T = 900°F. This is because that the rate constants do not change much in this temperature range. For a

594

GENYUAN LI and HERSCHELRABITZ

solid

original

line:

model( 10 species)

A solution

of 3-dimensional

lumped

model

0 solution

of 4-dimenslonal

lumped

model

o so!ution

of 5-dimensional

lumped

model

+ solution

of 6-dlmenslonal

lumped

model

initial

condihon:

temperature,

(0.1.0.1,0.2.0.4,0

05,0.05.0.05,0.05,0.0)

950” F

Fig. 7. Comparison of yp (gasoline) at T = 950°F for the original model and the lumped models obtained by using G(900), G(950) and G(lOO0).

solid

line: ‘

oiigmat

I

rhodet( 1b speciks)

*

solution

of 3-dimensIona

lumped

model

q

solution

of 4Pdimensional

lumped

model

of 5-dxnensional

lumped

mod

’ 4 - o solution

A *

A

*

A

initial

A

of y,,(C-lump)

condition:

dyjdt = f(y, 7-j = g(y, T).

1

of B-dimensional

lumped

model

-

(0.1.0.1.0.2,0.4,0.05.0.05,0.05,0~05~0,0) 950” F

at T = 950°F for the original model and the lumped models obtained by using G(900), G(950) and G(1000).

wider range of temperature, the difference between the lumping schemes validated in the large temperature range and that for a given temperature in the same range will become larger. The approach presented in this paper is not only applicable to first-order reaction systems but also to other ones under nonisothermal conditions. Let us consider the general case of a nonisothermal reaction system. It can be described as

dT/dt

I

-

+ solution temperature:

Fig. 8. Comparison

1

(32)

Let 2 =

(y'T)T

VY, T) = CfT(Y? T)dY, T)lT. Then eq. (32) can be rewritten

(33)

as

dz/dt = h(z).

(34)

The exact lumping of eq. (34) can be considered in the same way as that of eq. (l), except that the last “species” T is required unlumped (this means that the lumping matrix M must have a given row e,,,).

Determination of constrained

solid

line:

origmal

lumped

model

solution

of 4-dimensional

lumped

model

o solutmn

of 5-dimensmnal

lumped

model

+ solution

of 6-dimensional

lumped

model

initial

condition.

(0 15.0.4,0.1,0.08,0.07,0.2,0,0,0,0)

1000° F

I

I

I

I

I

I

1.0

2.0

3.0

4.0

5.0

6.0

t(10

Fig. 9. Comparison

of y,(gasoline)

solid 0.4 2 2 0

10 species)

of 3-dimensional

temperature:

0.0

model(

n solution q

0.0

line:

595

lumping schemes

-5h-

I

I

70

8.0

I

90

10.0

1)

at T = 1000°F for the original model and the lumped models obtained by using G(900), G(950) and G(lOO0).

original

model(

10 species)

n solution

of 3-dimensmnal

lumped

model

O solution

of 4-dimensional

lumped

model

0.3

.k z x E

0.2

2 A 0.1

’ solutmn

of 5-dimensIona

lumped

model

+ solution

of B-dimensional

lumped

model

initial

condition:

temperature

(0.15,0.4.0.1,0

08,0.07.0

2,0,0,0,0)

1000” F

0.0 0.0

1.0

Fig. 10. Comparison

2.0

30

40

IF%(YlM n.

k=l

to find a fixed invariant subspace containing at least the unit vector e,, , simultaneously for all A,( T) in the desired region of T. If the constant basis matrices for every A,(T) are known, the fixed inWe need

6.0

7.0

of y,,, (C-lump) at T= 1000°F for the original model obtained by using G(900), G(950) and G(1000).

Considering that the rate constants are exponential functions of temperature, the constant basis matrices of the transpose of the Jacobian matrix JT(z) of h(z) cannot generally be determined. However, for most reaction systems it may be decomposed as

Jr(z) =

5.0

8.0

90

10 0

and the lumped

models

variant subspaces of Jr(z) are just the common fixed invariant subspaces to all these constant matrices. However, the A,( T)s are like K( T) and their constant basis matrices are not easy to determine. Therefore, the approach to determine the fixed invariant subspaces of K ( T) presented in this paper can be employed to A,(T). We only need to properly choose a sufficient number of temperaure T in the desired region and then to calculate the corresponding A,( r). Using equations similar to eqs (15) and (16) one can determine the constrained lumping matrices with different dimensions for any nonisothermal reaction sys-

GENYUAN LI and HERSCHEL RABITZ

596

tern. In order to obtain a good result the number of

constant matrices for different temperatures may be quite large, but the computational effort is not very expensive, bccausc the computation only contains matrix multiplication and determination of the eigeovalues and eigenvectors for a symmetric matrix. In conclusion, this approach is an easy way to determine constrained lumping schemes for any reaction system under nonisothermal conditions. Acknowledgement-The authors acknowledge support from the Office of Naval Research and the Air Force Office of

Scientific Research.

NOTATION

C Ci

ify* 1

m &4Y n ii r 9” s Sk T Ti t Y, Yk

T,

M MD MG M

QW;i, Q(Y)

R(K) R(G) X Y

9

Scalars ah)

KCT)

kth coefficient of the decomposition JT(y) constant coefficient derivative function of temperature integer integer

of

integer

subspace spanned by the row vectors of M dimension of vector y dimension of vector f row number of M, n-dimensional real space integer rank of A, temperature temperature time n-dimensional composition space kth element of vector y

Vectors and matrices Capital letters represent matrices, bold-face iowercase letters represent vectors. A constant matrix basis matrix of JT(y) A, B constant matrix e IIt1 unit vector with 1 as its n + 1 entry, 0 for others analytic function of J’(y) analytic function of Q’(y) n-dimensional function vector &dimensional function vector W) defined as eK(‘)’ G( T) defined as [F(z) g (@IT h(z) I identity matrix Jacobian matrix of f(y) J(Y) Jacobian matrix of h(z) J(z) rate constant matrix at temperature T K( T)

Y

Y(K) Y(G) Y(O)

rate constant

matrix of the lumped system at temperature T lumping matrix dctermincd submatrix of M given submatrix of M generalized inverse of M satisfying MG = 16 matrix representation of Im 1 MG(A:)‘lT with orthonormal columns ii x fi function matrix eigenvector matrix of Y(K) eigenvector matrix of Y(G) n x (n - A) matrix n-dimensional variable vector ii-dimensional variable vector symmetric matrix symmetric matrix determined by K( T) symmetric matrix determined by G( r)

Y(7)

defined as tulW) ~~(0). . defined as Cyl(d ~~(4 .

Z

defined

Y,WI y,(7)]

as (yT T)*

Greek letters 4

;

ith eigenvalue of matrix Y(K) or Y(G)

time desired region of the composition space

Symbol any property related to the lumped system REFERENCES Bellman, R., 1970, Introduction McGraw-Hill, New York.

to Matrix Analysis.

A. and Greville, T. N. E., 1974, Genernlized Inverse: Theory and Applications. John Wiley, New York. Coxson, P. G. and Bischofl, K. B., 1987, Lumping strategy. 1. Ben-Israel,

Introductory techniques and applications of cluster analysis. Ind. Engng Chem. Res. 26, 1239-1248. Gross, B., Jacob, S. M., Nate, D. M. and Voltz, S. E., 1976, Simulation of catalytic cracking process. US Patent 3,960,707. Gohberg, I., Lancaster, P. and Rodman, L., 1986, Invariant Subspaces ofMatrices with Applications. John Wiley, New York. Jacob, S. M., Gross, B., Voltz, S. E. and Weekman, V. W., Jr., 1976, A lumping and reaction scheme for catalytic cracking. A.I.Ch.E. J. 22, 701-713. Li, G. and Rabitz, H. 1989, A general analysis of exact lumping in chemical kinetics. Chem. Engng Sci. 44, 1413-1430. Li, G. and Rabitz, H., 1990, A general analysis of approximate lumping in chemical kinetics. Ckem. Engng Sci. 45, 977-1002.

Li, G. and Rabitz, H., 1991, New approaches to determination of constrained lumpingschemes for a reaction system

in the whole composition space. Ckem. Engng Sci. 46, 95-111. Weekman, V. W., Jr., 1979, Lumps, models, and kinetics in practice. A.1.Ch.E. Monogr. Ser. 75(11), 3-29.