A LUMPING ANALYSIS IN
MONOMOLECULAR REACTION SYSTEMS Analysis of Approximately Lumpable Qstem J A M E S C. W . KUO A N D J A M E S WE1 Research Department, Central Research Division, Mobil Research and Deuelopment Corp., Princeton, N . J. 08540
A lumping analysis of an approximately lumpable monomolecular reaction system is given. One method of constructing the kinetic rate constant matrix for the lumped system is proposed and a method for establishing the magnitude of the errors that accompany lumping is given. ability concept.
EI and Kuo (1969) analyzed the exactly lumpable W m o n o m o l e c u l a r reaction system. As most monomolecular reaction systems are not exactly lumpable by a proper lumping matrix, we now examine systems that are approximately lumpable (with a proper lumping matrix), and give a method for establishing the magnitude of the errors that accompany lumping. Numerical examples of systems that are approximately lumpable with a proper lumping matrix illustrate the lumpability concept. Because we consider only proper lumping from now on, we simply call it “lumping.” When equations and theorems in the preceding paper (Wei and Kuo, 1969) are referred to, their number is preceded by the letter I.
Error Matrix E and a Method for Determining It
Since most monomolecular reaction systems will not be exactly lumpable by the lumping matrix M, we examine the systems that are approximately or poorly lumpable. Definition 1. For a given K,the following matrix
E=MK
-kM
Here A E MAMT is given by Equation 116. Equation 4 implies that for any arbitrary reversible monomolecular reaction system there always exists an E such that Relation 2 holds. We do, however, need to show that matrix R satisfies the three conditions for the monomolecular reaction scheme given by Equation 12. To do so, we have: THEOREM 1. The matrix k given by Equation 3 possesses all the properties of a monomolecular reaction scheme. The proof for this theorem exactly follows the proof for Theorem 15. We illustrate these concepts with the system of butene isomerization of Lago and Haag (1968). This system is nearly lumpable, in that one may lump I-butene and cis-2butene. 0.7245 -0.5327 -0.1918
EAMT = 6
(2)
k
k = MKAMTA-~
from Equation 1 (3)
-0.2381 0.5273 -0.2892
M =
(1)
is called the error matrix of the lumping matrix M . For a reaction system exactly lumpable by matrix M, matrix E is unique and is a null matrix. For a reaction system approximately or poorly lumpable by matrix M, matrix E will depend on matrix k and will not be unique. The choice of matrix k is not, however, completely arbitrary. We want to choose a R that satisfies all the properties of a monomolecular scheme and possesses 6*(= Ma*) as an equilibrium point of the lumped system. The following shows a convenient way of determining k. The same k is used in the following four sections. The most convenient way to generate a k that possesses all the properties of a monomolecular reaction scheme is to choose an E such that
By using Relation 2, we may obtain a that is
Numerical examples illustrate the lump-
-0.0515 -0.1736 0.2251
(A A)
______,___
0 1918 0 2892’ -0 2251
) ’*
__________ 1_ _ _ _ _ _____ _ I 1_____.
_____ -0.1918
0.2892; 0.2251 /0.1436 0.4649 0.3213 0.4649
0.4649 = (0.5351)
3 0
1
g
MKAMTA-’
=
- AMTA-’M)
=
=
E = MK(1
(
(
0.25905 -0.25905
-0.06731 0.06731
0.12044 -0.12044
0.2251) 0.2251
0.03009 -0.03009
0) 0
-0.12044 0.12044
Choosing an E such that EAMT = 8 has an important physical implication. From Equation 18, by using Equation 1, we obtain
and an E
E = MK(1 124
- AMTA-’M)
I&EC F U N D A M E N T A L S
(4)
(5)
R
then-according to Theorem 18, which states that subspace A; is spanned by the row vectors of matrix MA--we have, for any aeA;,
a
=
0.25905 -0.25905
fi
fi
AMT;
where i? is an arbitrary constant vector.
-
-0.2251 0.2251
1) (0 0 -1 0 0.48415
(0.4649 0.5351
Therefore, we have
N=fi-'MX Le., there is no error in computing d i / d t for all aeA;. Kevertheless, one should keep in mind that here A;, is no longer a subspace spanned by the A noncollapsing eigenvectors, but is defined as a subspace composed of the vectors of the form AMTA-% for all This concept is illustrated by a simple three-component system with lumping A1 and A2 as given in Figure 1.
;EA.
NA
(:
=
- liN =
(:
a-1
) (I
i.5351
:.003879
0.5351
-0.4649
0.0288 0.01486
Error Estimation for a Nearly or Poorly Lumpable System
There are two ways to compute i(t).The correct way is by lumping a(t), as first to compute a(t) and then obtain i7(t)
i r ( t ) = Me-"a(O)
A-SPACE
&SPACE
(loa)
The wrong way is to lump a(0) first and then compute & ( t ) by using i(0) according to the kinetics given by Equation 16, as
iW(t) = ePBtMa(0)
(1Ob)
i7(t)does not equal iW(t), since Me-Kf # e-'tM The error of lumping is then given by
&(t) Figure 1 . Illustration of the A2-space of an approximately lumpable system
-
&(t) = (Me-Kf - e-"M)a(O)
This error may also be written in several other forms. One form is to use matrix N defined by Equation 6. tion l l a can then be converted to the form
i7(t)- i,(t)
We may define
N
k--'MX
=
Equa-
(Ilb)
L.
The matrix Ne-A' e-hw in Equation l l b possesses the following properties (compare with matrix NA - fiN in Equation 9) :
- ,-firN
S(NA- AN)x-~
- e-xtN)X-la(0)
=
-
For an exactly lumpable matrix, N becomes a null matrix. With Equation 6, the error matrix can be rewritten as
E
(114
=
(7)
i i
=
1orj = 1 2, . . . , ;1; j = 2,
..., n
(1 2)
We shall study some properties of matrix N. Without loss of generality, we may assume that matrices X and fi are arranged in such a way that their first columns are the equilibrium composition vectors for the original and for the lumped system. Then the elements of the first row of matrix N and the elements of the first column of matrix N must all be zero. This property is shown as follows:
From Equation 12, we can see that there will be no error when t approaches infinity. Another form is to use the error matrix E. Since
(N)l, = GITNej= GIT'x-lMXej = GITL-lgFA-lMxj =
by integrating this equation with respect to t , we obtain
(fiGl)f-lMxj = ii*TA-lMxj = iTMxj = lTxj = j = 1, 2,
da
-'
dt
=
+ E)a = -Ka *
=
-MKa = -(KM
A
- Ea
61j
. . ., n
(8a)
and
or
i7(t)- i w ( t )
(N),l = &TE-IfiTA-lMXel = ;,Te-1kTA-lMa* =
;i~t-i*~A-~;* = ; , j ~ L - ~= x ~et~f,-i;l i
=
i = 2, 3,
;pl=
. . .,
0
(8b)
By using Relations 8a and 8b,
(NA - h N ) t j
- e-'
