Acknowledgment
P
Steven Greenwald assisted in the digital calculations.
U
= density, gram/cc. = surface tension, dynes/cm.
literature Cited
Nomenclature
Ca = capillary number, pu,/u Ca*, CaO = lower speed limit on Ca, Equations 6 and 7 F = F function, Equation 3 g = acceleration of gravity Go = Goucher number, R(pg/2u)W = film thickness, cm. = flow thickness, Equation A-4, cm. r = radial coordinate, cm. R = radius of cylinder, cm. T = dimensionless flow thickness, k(pg/pu,)’/* U = local velocity in film, cm./sec. u4 = surface velocity at liquid-gas interface, cm./sec. UtC = wire withdrawal velocity, cm./sec. V = volume flow rate, cc./sec. W = dimensionless velocity, u,(p/R2pg) W* = lower speed limit on W Y = Y function, Equation 2
2
Chien, S.-F., J . Appl. Mech. 33, 222 (1966); Deryagin, B. V., Levi, L. N., “Film Coating Theory,” Chaps. 2 to 4, Focal Press, New York, 1964. Deryagin, B. V., Titiyevskaya, A. S., Dokl. Akad. Nauk USSR 50, 307 (1945). Gutfinger, Chaim, Tallmadge, J. A., Znd. Eng. Chem. 59, No. 11, 18 (1967); 60,No. 2, 74 (1968). Jeffreys, H., Proc. Cambridge Phil. SOC.26, 204 (1930). Matsuhisa, Seikichi, Bird, R. B., A.Z.Ch.E. J . 11, 588 (1965). Rajani, G. R., M. S. thesis (chemical engineering), Drexel Institute of Technolow. June 1968. Tallmadge, J.A., A.fCh.E. J . 12,810 (1966). Tallmadge, J. A,, Labine, R. A., Wood, B. H., IND.ENG.CHEM. FUNDAMENTALS 4,400 (1965). Tallmadge, J. A., Soroka, .4.J., Chem. Eng. Sci., in press. Tallmadge, J. A., Tlrhite, D. A,, Znd. Eng. Chem. Process Design Develop. 7, 503 ( 1 968). White, D. A., Ph.D. dissertation, Yale University, April 1965. White,D. A,, Tallmadge, J. A., A.I.Ch.E. J . 12, 333 (1966). White, D. A., Tallmadge, J. A , , A.Z.Ch.E. J . 13, 745 (1967) RECEIVED
GREEKLETTERS
Work supported by the National Science Foundation Grant GK1206.
= viscosity, poises
Fc
for review March 25, 1968 ACCEPTED October 24, 1968
A LUMPING ANALYSIS IN MONOMOLECULAR REACTION SYSTEMS AnalJysis of the ExactlJy Lumpable &stem J A M E S WE1 A N D J A M E S C. W . KUO Research Department, Central Research Division, Mobil Research and Development Corp., Princeton, N .J. 08540 Because of the enormous number of chemical species encountered in many chemical reaction systems (especially those related to the petroleum industry), it is often expedient to lump all the species into a few groups for practical purposes. This paper presents a theoretical study on the exact lumping of a monomolecular reaction system. One can obtain the necessary and sufficient conditions under which the kinetics of the lumped classes can b e exactly described b y a complex first-order reaction scheme. Many implications of these lumpability conditions are investigated. The lumpability of a monomolecular reaction system coupled with diffusion has also been studied, The analysis for the system with diffusion is similar to analysis without diffusion. N MANY
chemical processes, particularly those related to the
I petroleum industry, the number of molecular species involved often runs into the thousands. For instance, in a naphtha-reforming process, one would like to investigate the reforming kinetics, the feedstock characterization, and the product characterization. I n these investigations, one is unable to deal with each chemical species separately. One may, however, partition the species into a few equivalence classes (or lumped classes), and then consider each class as an independent entity. Such a lumping process is by no means strange to us. We often consider, for example, all oxygen molecules as “oxygen” even though the kinetic energies of the individual oxygen molecules are different. Such lumping also gave petroleum processing the PONA analysis, in which all species are divided into four classes: paraffins, olefins, naphthenes, and aromatics. 114
I&EC FUNDAMENTALS
There are many reasons to lump-for example, in the simulation of a petroleum reformer, we may lump in the chemical analysis of the feedstock, such as a PONA analysis, in describing the kinetics of the system and in determining the quality of the products, such as the octane number. A reformer system is “perfectly lumpable” if one can make a single lumping scheme that is good at all of these stages. Here, however, we concentrate entirely upon describing the kinetics of the system, and search only for the kinetic lumpability conditions. As we proceed, we must realize that lumping always leads to a loss of information, and that the information lost in the kinetic lumping may be important, for example, in the characterization of the product. Among all the conceivable chemical reaction systems, we particularly consider the monomolecular reaction system (complex first-order kinetic system). For this system, we
lump all the chemical species into a few classes, and examine the necessary and sufficient conditions under which the kinetics of these lumped classes may be described by a complex first-order reaction scheme. Many implications of the concept of lumpability are also examined. For example, one may design a set of simple experiments to test whether a particular system is lumpable, without its being necessary to investigate the detailed kinetics of the whole system. The lumpability of a monomolecular reaction system coupled with diffusion has also been studied. There the analysis is similar to that in the nondiffusion system. We need only replace the quantities in the nondiffusion system by the corresponding diffusion-disguised quantities. For convenience, all mathematically exact lumping can be divided into three categories : proper lumping, semiproper lumping, and improper lumping. Proper lumping possesses the following physical meaning: The chemical species of the system are partitioned into several classes that may be considered independent entities for kinetic purposes. Proper lumping has been considered by Kemeny and Snell (1960) for a Markov chain. For semiproper and improper lumping, each chemical species is not necessarily assigned to a unique class-for instance, species A1 may belong to both classes A1 and &. In semiproper lumping, the corresponding lumped system follows a monomolecular reaction scheme. The lumped system resulting from an improper lumping will not follow a monomolecular reaction scheme.
A-SPACE
A-SPACE
Figure 1. Lumping of a three-dimensional A-space into a two-dimensional A-space
Here, the sizes of the rectangles symbolize the dimensions of vectors f and a , and matrix M. (For convenience, call the original n species composition space the A-space and the lumped 2 species space the A-space.) As an example, three species are lumped into two classes: (4)
or 21
= a1
+
a2
A
a2 = aa
Conditions under Which a Monomolecular Reaction System I s Exactly Lumpable
(Two trivial lumping matrices are: identity matrix I, which leaves a unchanged; and row vector lT, which maps vector
We consider a system of n chemical species described kinetically by a monomolecular reaction scheme da -= dt
-Ka
where K is a monomolecular reaction rate matrix, which has the following characteristics (Wei and Prater, 1962) : 1. Nonnegative rate constant: (K),, = -ktj
5 0,
+(kjt
-Le.,
1TK =
OT)
j=1
3. There exists an equilibrium composition ai* 0, i = 1, 2, . . ., n, such that
.
I
(5)
t
> I
., n; i
(2)
I
#j
By lumping we mean the linear transformation of an ntuple composition vector, a, into an ;-tuple vector, i, of smaller dimension by means of an n^ X n matrix M of rank ^n
a
This equivalence is illustrated in Figure 1.
We have
and detailed balance
i,j = 1, 2,
of a space of higher dimension into a space of smaller dimension means a loss of information, since more than one vector a will correspond to the same vector 8 . For instance, a vector (1,O;O) is easily distinguished from a vector (0,l ;O) in the Aspace, but both correspond to the same vector (1,O) in the A^space. We say that two vectors a and a ’ are “M-equivalent” if and only if the equality
holds.
Ka* = 0
ktjaj* = kjiai*
This homomorphism
M a = Ma’
n j
at.)
i=l
i #j
2. Mass conservation: (K)tt =
n
a in the A-space into a constant,
(3)
-
Ma = M a ’ =
2
Equality 5 is symbolized by a a’. For exact lumping in general, M may be any A X n constant matrix of rank ^n. For a lumping to possess the property of dividing all the species into a few classes, however, each column of M must be a unit vector, e t , such as the one given in Equation 4. Lumping by such an M is called “proper lumping.” Lumping by any other M is called “semiproper lumping” or “improper lumping,” depending on the behavior of the lumped system. Here, we define and discuss theorems valid for all exact lumpings, proper or otherwise. We have the following definition of an exactly lumpable system. Definition 1. A system described by Equation 1 is exactly lumpable by a matrix M if and only if there exists a matrix K such that the kinetic behavior of the lumped system can be described by
VOL. 8
NO. 1
FEBRUARY 1969
115
This definition implies that the two vectors, a and a', that give rise to the same lumped vector f may or may not have the same rates in the A-space, but will have the same rates in the 2-space. We observe that, for a system to be lumpable, there are two ways to compute the rate d&/dt for any vector a. The first is to lump first and then use the kinetic equation (Equation 6) to obtain
The other is to definef(K) as Cauchy's integral
f(K) =
1 -,
2RZ
$ f ( ~(21 )
- K)-'dz
and notice that since M(zI
- K)
=
(d - k ) M
we have
d2 - z -Ki = -KMa dt
(74
The second is to use the kinetic equation (Equation 1) first and then lump to obtain
( ~-i K)-'M
= M(zI
and 1
- K)-'dz
Mf(K) = - $f(z)M(zI 2ni Equations 7a and 7b must be equal; therefore, we have Theorem 1. THEOREM 1. For a monomolecular reaction system to be exactly lumpable, it is necessary and sufficient that
- K)-'
- -1
2ni
$ f ( z ) (zi
- K)-'dzM
= f(K)M
Since the exponential function, eKt, is an analytic function, Theorem 2 assures us that when a(0) is given, the following two ways to evaluate P ( t ) give the same result: 1. We first integrate Equation 1 for a(t) and then lump a(t) to give
6(t) = Me-Kta(0)
(loa)
2. We first lump a(0) and then integrate Equation 6 to give This theorem is illustrated by the following three-component system: 3
10
M
i(t) = e-'!Ma(O)
According to Theorems 1 and 2, we have the following alternative definition for lumpability. Definition 2. A system described by Equation 1 is exactly lumpable if and only if each pair of M-equivaIent initial vectors a(0) and a'(0) remains M-equivalent under the motion of Equation 1-i.e., for all t
a(0)
K implies
a(t) \-lo
-10 A-SPACE
This system is repeatedly used as an illustration in this paper. The following theorem is useful. THEOREM 2. For an exactly lumpable monomolecular reaction system and for any analytic function of matrices, f, we have
MfW) = f ( K ) M
(9)
There are two ways to prove this theorem. f(K) in a Taylor series
cjK'
(KO = I)
j=O
and note that for each term
MKn = (MK)Kn-I
= d ~ ~ n = - 1 K(MK)K"-~=
K(KM)Kn-2 by repeated use of Equation 8.
116
K2(MK)Kn-a = * . . = KnM
=
Thus,
I&EC F U N D A M E N T A L S
__ Figure 2.
-
a'(0) a'(t)
A-SPACE
__---
Lumpability of a system
One is to expand
m
Mf(K) = M
(lob)
This definition implies that, for the IumpabIe system of Equation 4, when we feed (1,O;O) and (0,l ;O) to two identical reactors, the resulting reaction paths a ( t ) and a'@) will of course be different; but both vectors will lead to an identical path a ( t ) in the 2-space. The point of lumping is that we do not care about the precise amounts of AI or A2 in the feed or product; we care only about the sum of their concentrations in the feed or product-that is, whether or not two feedstocks lumped into the same composition in the A -space will produce the same lumped product, such as the system given in Figure 2. Reaction 11 is equivalent to the statement: " i ( t ) is only a function of i(O)."
Let us examine in detail one of the consequences of Equation 1 on an exactly lumpable system-lumped by a proper lumping matrix. Consider the following scheme:
.iAlc+A?I - - -- -- - -- I-----------,
-1
5XI
j
Ai
3
Az
r---------.
_____-_-__
iA3eA4j
b
4
matrix K of an exactly lumpable system, corresponding to the eigenvalue A,. Then vector Mx, either vanishes or is an eigenvector of the reaction rate matrix f with the same eigenvalue h,-i.e.,
K x ~= X ~ X ,
5
implies
Mx, = 0 or K(Mxt) = X,(Mx,)
We have
This theorem is easily demonstrated by considering
k(Mx,) = MKxi = Mhix, = hr(Mx$) We designate the eigenvector of matrix f by Gi = Mxi. Now, two eigenvectors, x, and xl, cannot correspond to the same nonvanishing Mx, unless they have the same eigenvalue. Matrix K has n eigenvectors, and matrix f has only n^ eigenvectors. So there must be a lot of attrition in the lumping process, and as many as n - n^ eigenvectors xt will have Mxf =
0. Without loss of generality, one can arrange M and X so that
k(Pl0)
MX =
All intraclass rate constants, such as k12, k21, k34, and k43, do not appear in MK. This property is not surprising to us, since internal shufflings between A1 and A2 and between A3 and A4 have no effect on the collapsed A-space. But, on the other hand, according to Equation 8, matrix MK must equal KM, which is
where f is an n^ X n^ nonsingular matrix and 0 is an n^ X (n - n^) null matrix. The inverse of Theorem 3 is given as Theorem 4. THEOREM 4. If, for a given K, which is diagonalizable as K = XAX-I, Mx, = 0 for its n - n^ eigenvectors, the reaction system is lumpable by matrix M. As proof, let
MX = f ( f [ O )
or
(i;O)A = A(il0)
or
f
= MX(P1O)T
(12)
= (i/O)A(i/O)T
(13 )
and f =
jipp
(14)
Then
MK = MXAX-' = (filO)AX-l = k(ilO)AX-l =
k(filO)X-l = %iX-l(frlO)X-l = KM These two theorems are illustrated as follows:
Thus, we require that kai
f kai
=
k32
k13
+
=
k14
k23
+ +
k42
=
Lzl
k24
=
LIZ
=
(
13 -3
-2 12
-10
AZ
The sum of the rate constants from A1 to class must be the same as the sum of those from A2 to class A,. A similar relation exists between A3 or A4 and class A,. By extending this relation to an arbitrary monomolecular reaction system, we will have the result that, for any two species A, and Al in the same class At, the sum of all rate constants from A, to another class must be the same as the sum of the rate constants from A l to class A,. This relation is exceedingly reasonable, since a2 without it means that if we know the lumped value of a i knowing the separate values of a1 and a2 we can uniquely determine
-10
(a1
+
az) = -Lz,(a,
+ az) +
hl = 0
A2
1 1
(;---b---l--:) ha = 15
= 20
xi =
-0.5 Mxl =
(:::)
Mxz =
(
0'5) -0.5
10 = (-10
K(Mx1) = 0 L12(a*
M =
10
+
d dt -
-4 -6)
+ a4)
Mx3 =
(3
-10
10)
K(Mx2) = 20(Mxz)
no matter whether the mixture is rich in a1 or in a2. This unique determination is possible only if A1 and A2 go to the combined a3 a4 a t the same rate. We now examine the consequences of lumpability on the eigenvectors and eigenvalues of the system. We have: THEOREM 3. Let xi be the eigenvector of the reaction rate
+
VOL. 8
NO. 1
FEBRUARY 1969
117
Theorems 3 and 4 show us that lumpability amounts to nothing more than crossing off a few of the eigenvectors of matrix K, and projecting the rest to a lower dimensional space. I n terms of the pseudo-species, we have as an example
B Bi z A
/,
?TM = 1, and
flcZ=hZ
Bz-0
=)
M = (g[O)X-l
(15)
For the illustrative reaction system, we have
M o ,l .--,
X -1
0.5 (1-____________. 1 j (0.5
o o i
= 0.5
OT
jTij(flo)x--l = ~ T K M = ~ M K =: 1TK =
From Theorems 3 and 4, the necessary and sufficient condition for exactly lumping a monomolecular reaction system is the existence of an M and an 2 that satisfy Equation 12 (provided K is diagonalizable). A suitable choice of M will depend on a suitable choice of 2,and vice versa. In fact, the row vectors of matrix M are given as a linear combination of the new n^ row vectors of matrix X-l, as shown by
:)
1TK =
and Equations 8 and 15 to give
Bs Xa_
M
that matrix K satisfied all three conditions of Equation 2. That K satisfies condition 1 may be easily checked by a direct manipulation of Equation 8. T o prove that K satisfies condition 2, we use the following relation
-0.5
’
:)
(11 \0.6
1 -0.4
which implies that
?Tit2
I
6T
or = 5T
?Ti(
To prove that k satisfies condition 3, we observe from Equation 18 that 2**> 0 for all i = 1, 2, , . , , %. In addition, since the original monomolecular reaction system must have the property of detailed balance or KA is a symmetric matrix (Wei and Prater, 1962), we readily see that Kf =
-1
01
OT
~ M A M T
= MKAMT
which is also a symmetric matrix. that
Arr^a,*
=
,&if*i, j
(19 )
Equation 19 simply implies
= I, 2,
..., ;;
i #j
f ,Equation 18, and Theorem 5 are illustrated as follows: M
A
f
MT
Then, after some simple manipulations, we have = 0,
i # j ; i , j = I, 2,
Le., A is a diagonal matrix. Equation 16, we obtain a = -Ai
A *
= MAMTi
..., ii
Furthermore, by the use of E=
MA1 = Ma*
(18)
Le., C * is the equilibrium concentration vector for the A-space and A is the corresponding diagonal matrix of the equilibrium compositions in the A-space. If the lumping is not a proper one, the A may not be diagonal and may not satisfy Equation 18. We have the following definitions. Definition 3. A lumping is called “semiproper’J if:
1. A is diagonal and satisfies Equation 18. 2. The corresponding 8, given as MKAMTA-‘, satisfies the condition of (K)ij 5 0 for i # j , i,j = 1 , 2, . . ., c. Definition 4. Any lumping that is neither proper nor semiproper is called an “improper lumping.” The existence of semiproper lumping is shown below. The following theorem shows that in a proper lumping the lumped system follows a monomolecular reaction scheme. THEOREM 5. For a proper lumping, the lumped system follows a monomolecular reaction scheme. We need to show 118
I&EC FUNDAMENTALS
For a proper lumping, M is k e d ; therefore k is completely determined. For a semiproper lumping, the restrictions on and K impose some restrictions on the choice of the elements of X, but there are still ^n (^n 1)/2 degrees of freedom in determining them. For improper lumping, there are ^nz degrees of freedom in choosing the elements o f f . Without loss of generality, we may arrange the eigenvector matrix X of any monomolecular reaction system in such a way that
A
-
1TX = elT = (1, 0,
. . .O)
(20)
By using Equation 20, we have the following theorem and corollary for a system of a semiproper lumping. THEOREM 6. For a semiproper lumping the lumped system possesses the following properties : ITf
=
(2W
&T
iTM =
(21b)
We need
A 5MAMT = X ( i I O)X-lA(X-l)?’(i 1 O ) T g T k(flO)L--l(ilO)TkT =
= ki-12T
(22)
is where L is defined as XTA-lX, a diagonal matrix, and defined as (ilO)L(ilO)T. From Equation 22 we may also conclude that must be nonsingular. By substituting Equations 16 and 22 into Equation 18, we obtain
A
Xe-lXT?
=
1
(X O)X-'A1 =
(S1 O)L-I(lTX)T
=
(jtL-llo)el =
fiL-1~1
or iTfi
= &T
(21a)
Furthermore, by using Equation 16, we obtain
We will now look into a few more details of exact lumping. We know that matrix M maps the A-space into the A-space. There exists a reversing matrix AMTA-l, which will map the A-space back into the A-space. The mapping by AMTK-', however, is a mapping of the A-space "into" the A-space. Consequently, the combination of mapping by matrices M and AMTA-Li.e., mapping by matrix AMTA--'M-becomes an "endomorphism" of the A-space. The range of this endomorphic mapping is an ;-dimensional subspace of the Aspace spanned by the 6 noncollapsing eigenvectors. To demonstrate this idea, we use Equation 22 to obtain
(AMTA-'M)X = AMT(fi.T)-lifi-lMX
?TM = iT(fr/O)X--1= (?TfilO)X-l = ($lT/O)X-' = elTX-1 = 1T (21b) COROLLARY 6.1. In a semiproper lumping, the lumped system possesses the following properties in addition to those properties given by Equations 21a and 21b :
5a,*
=
iTAi
n
ITA1 =
I
i-1
i=l
ai*
?TG(t) = 1 for all t 2 0 and ; ( t ) d
(234 (23b)
=
x
(-;-{-:;)
= (iTM)A1 = 1TA1
( i l o ) x ~= (234
Furthermore, by using Equation 21b, we have
iT$(t) = (iTM)a(t) = 1Ta(t) = 1
(23b)
From Corollary 6.1, we conclude that, for semiproper lumping, the total mass of the lumped system is conserved-which is not necessarily true for a lumped system with improper lumping. We are now in a position to show that for semiproper lumping the lumped system follows a monomolecular reaction scheme, just as does a lumped system resulting from proper lumping (see Theorem 5). THEOREM 7 . For semiproper lumping, the lumped system follows a monomolecular reaction system. The first condition of Equation 2 is automatically satisfied for a lumped system resulting from a semiproper lumping-according to the definition of semiproper lumping (Definition 3). T o show that the given system satisfies the third condition of Equation 2, let
. . ., x;,
0,
. . .,
0)
If we denote this ;-dimensional subspace by A;, it can be easily shown that all elements of A 6 are invariant under mapping AMTA-lM. Furthermore, since X-' = L-'XTA-' (see Wei and Prater, 1962), we rearrange Equation 16 to give
From Equations 18 and 21b, we readily obtain ITA?
= '(xl,x2,
(eg-1)~~
i.e., the only eigenvectors that survive under the lumping are the linear combinations of the row vectors of matrix MA. Thus, we have the following theorem. THEOREM 8. The A$-space is spanned by the row vectors of matrix MA. The A$-space and mappings AMFA-' and AMTA-lM are illustrated by Figure 3. We need the following theorem.
ii-s PACE
A- SPACE
A t -SPACE
A*-
- - - --
a, A
Then Equations 18 and 22 imply that
since must be nontrivial. The remaining portion of the proof is essentially similar to the corresponding proof given for Theorem 5, except that the relation iTM = 1 will now come from Theorem 6 (Equation 21b). The number of degrees of freedom in determining the semiproper lumping is the same as the number of degrees of freedom in determining matrix fr (and so M). For a twocomponent system, we have the freedom to choose the equilibrium point of the lumped system. Therefore, there is only one degree of freedom in determining the semiproper lumping matrix. For a three-component system, we have the freedom to choose the equilibrium point and the orientations of the two eigenvectors. Therefore, there are three degrees of freedom in determining the semiproper lumping matrix. For a general ;-component system, the degree of freedom can be easily - 1)/2. This situation is in contrast computed to be ;(; with proper lumping, which has no degree of freedom, and with improper lumping, which has degrees of freedom.
Figure 3. Illustration of mappingsAMTA-l and Ah4TA-M and A;-space THEOREM 9. For an exactly lumpable monomolecular reaction system, and for any analytic function of matrices, f, we have
f (K)AMTA--I = AMTA-lf(K)
(26)
One can prove the theorem forf(K) = K. The proof for any analyticf is essentially similar to the proof given for Theorem 2. Since K satisfies the relation of the microscopic reversibility, KA is symmetric. We have
KAMTA-~= AKTMTA-~= A ( M K ) T ~ - ~
- A(KM)TA-I= AMTKTA-I - AMTA-I(KA)TA-~ = AMTA-IK
With the aid of Theorem 9, it can be shown that the A$-space is invariant under the motion of Equation 1. This conclusion is given as Theorem 10. THEOREM 10. With the initial composition given in the A c space, the reaction path will remain in the same space. Since VOL. 8
NO. 1
FEBRUARY 1969
119
any composition vector in the A&-space can be given as
AMTL-lS with a d , Theorem 10 is equivalent to a ( t ) = e-KtAMTA--’^a(O)eA&for all t 2 0. The proof for this fact is given as
a(t) =
e - ~ t ~ ~ ~ i=-AlMsT (~ -~- ’ )~ - % ~ ;= (o)
AMTL-~;(~)~A~ Experimental Ways of Determining whether a System I s Lumpable by a Proper lumping Matrix M
SOfar most of our lumpability theorems concern knowledge of matrix K, which one may not possess in advance. We need some easy experimental ways to determine whether a system can be lumped by the proper lumping matrix M . There are several good methods for doing so. The first is based on Definition 2. Consider a system with five components, which we wish to lump as (123) and (45). We should feed an initial composition (l,O,O,O,O) into the a&) 4reactor and obtain the behavior of ; l ( t ) = a l ( t ) a&), and of ;2(t) = a4(t) a&). We next repeat the experiment with a feed of (O,l,O,O,O), and then with (O,O,l,O,O). Of course, we will get different reaction paths in the A-space; but, if they coincide in the A -space, the lumping is good. We must repeat the experiment to make certain the reaction path from a feed of (O,O,O,l,O) coincides with the reaction path from a feed of (O,O,O,O,l). Then the system is lumpable. This method is easy and general, and ensures that Relation 5 will be obeyed for any two initial composition vectors that are Mequivalent. We advocate feeding pure component into the reactor only because pure components form the convex hull of all conceivable feeds. If the natural variation of feedstocks in a reformer is not very great, we need only find the convex hull, and make certain that the reaction paths that start from the vertices of the convex hull behave correctly. Finally, if the feedstock never varies, and it is impossible to make it do so, we have nothing to worry about. The second method is based on Theorems 8 and 10. According to Theorem 10, if a(0) is in the A~-space,the subsequent a(t) will also be in the same space for all t 2 0. According to Theorem 8, the A&-space is spanned by the row vectors of matrix M A , or, for the system we are considering, by vectors
+
+
30
20\3
5
4
I
Sa
-20
/30+0
-0
30
K = ( -20 \-lo
~
\
-5
+ 26
-2 -3 -4a 5 4a/
0
+
5 5a -5a
0 -30
+
i)
2 0 0 A = i21( O 01 0 8
0 0 0 1 0
1
A = - 7(
0 6
2-1
a
2
K
(-;:
)
1 0
-3 (:;; -:)(:
33(-:,7
=
:,7)
Example 11. We lump A1 and A2 of the following system
5
3
(E at*) -1(a~*,a~*,a3*,0,0) and (Eai*) -1(0,0,0,a4*,a~*). i=l i=4
Conse-
quently, to test the lumpability of the system, we simply feed 3
(E at*) -1(al*,az*,a3*,0,0) i=l
5
and
(E ai*)-1(0,0,0,a4*,a~*)and see i=4
if components AI, A*, and As converge to equilibrium compositions while continuously maintaining themselves in constant proportion to one another, and if components 4 and 5 converge to equilibrium compositions while remaining in constant proportion. This method seems to require fewer runs, but is less sensitive to errors. Again, if a pure component is impossible to obtain, we merely work with the convex hull. The final idea is to feed some a(0) that is M-equivalent to a*. If 2(t) = ^a* for all t, the system is lumpable.
2a f 9 -01 -2a a -3 -6
+9
-6 0.1
-1 -2
-1.5 -3.0 11 -6.0 -8 10.5 1 1/3
Numerical Examples of Proper Lumping
We show two examples of proper lumping. In the first, it is intuitively obvious that, if a and p approach infinity, we may lump A I with Az, and A3 with A4. It is not so obvious that the system is exactly lumpable for any values for a and p. In fact, we can lump A1 with A2, and A3 with A4 even when there is no direct reaction between A1 and Az, and between 120
l&EC FUNDAMENTALS
0.4
-1
0
A=(!
-4
0
0
1;0
9+3a
)
B
MK 9 -3 -6
9 -3 -6
-3 11 -8
-4.5 -6.0) 10.5
9 =
A4
-3
11
(-3 -6
-8
-4.5 -6.0) 10.5
A4
x M
(: ,:) 0
=
0
Figure 4. Reaction simplex and lumped system of hypothetical four-component system given in Example II of proper lumping
li.5
0.3 0.3 0.4
1 ' 3
3 -2 -1
1 1
-4
:)
Positions of straight-line reaction paths shown approximately
0
Let us lump away the last eigenvector; then we need
3
M = (%)O)X-l -1
0.4
-4
The eigenvector, which has been lumped away, could correspond to the slowest or the fastest straight-line reaction path. A reaction simplex for the system is given in Figure 4, where the shaded plane is subspace A ; and the triangle on the right is space A^. The positions of straight-line reaction paths are approximately shown. The straight-line reaction path LS disappears when we lump A I and Az.
where X is to be determined. To satisfy the requirement that A be diagonal, we observe Equation 22, =
(ilO)t(ilO)T
i
Equation 22 requires that
1-butene (A,)
J \
cis-2-butene
trans-2-butene
(Ad
(Ad
(
1
-0.0515 -0.1736 0.2251
X 0.1436 0.3213 0.5351
(
A
0.2056 0.3295 -0.5351
(; )p ( i.4769
(22)
1.0000 (0
2 have
0 1.1674
the form 1.080501 -1.0805a
cy2
where cy is any real number, which is the degree of freedom referred to before. Therefore, we have
1
1
1
+ 1 . 3 2 5 2 ~ ~ 1 + 0.94911a
+ a2 - 1.3252a cy2
cy2
- 0.94911~~ 1 a2
and
K ,. = - (0.4769 1 cy2
K -0.2381 0.5273 -0.2892
=
a = 1- +1 a2 (1 M=-(
The following data for the A-space are chosen from Silvestri and Prater (1964) :
0.7245 -0.5327 -0.1918
YL-lXT
where
Nontrivial Example of Semiproper lumping of a Butene Isomerization System
The interconversion of 1-butene, cis-2-butene, and trans-2butene can be described as
(15)
+
a2
- 0.92556~~
+ 0.92556~~
-:)
-a2
The resulting lumped system is monomolecular and has an eigenvalue
=
K z = 0.4769 -0.1436 0.1724 -0.0288
with an equilibrium composition
) X -1
1.oooo 1.oooo 1.2265 0.8784 -4.2077 2.2579
0.1436 0 A =( 0 0.3213 0 0 0 0.5351 O ) and
0.2377
1.oooo -0.8566 -0.2264
The specification of a is equivalent to specifying the position of the equilibrium point in the lumped system. When a goes to zero or infinity, the system becomes an irreversible system in the A-space. lumping of a Monomolecular Reaction System Coupled with Diffusion
In considering a chemical reaction system coupled with diffusion, we may study either the transient or the steady-state problem. For practical purposes, we are more interested in the latter than the former. Therefore, the analysis given here is for steady-state problems only. I t may, however, be easily extended to include the transient problem. VOL. 8
NO.
1
FEBRUARY 1969
121
For a system coupled with diffusion, we obtain an intrinsic rate constant matrix, K+, which plays a role similar to that of K for the nondiffusion system. The lumpability study will then be completely based on this K+. When there are equal diffusivities for all species-Le., D = DI-and when matrices K and D are both lumpable by the same lumping matrix M, the lumpability analysis given above can be immediately applied (see Theorems 11 and 12). Let V be the interior of a catalyst particle or the interior of a reactor, and let bV be the boundary of V across which the mass transfer occurs. At steady state, the composition vector, a(r), at any point rev, can be described by the following differential equation (Wei, 1962)
DV2a(r) - Ka(r) = 0
rev
(274
Here D is the diagonal effective diffusivity matrix, which has D,, i = 1, 2, . . . , n, as its ith diagonal element. For simplicity, we consider the simplest boundary condition :
rebV
a(.) = ae
(27b)
The meaning of “lumping” for the present system is the same as that given earlier. The meaning of “an exactly lumpable system,” however, differs somewhat from the one given by Definition 1. The new definition of an exactly lumpable system is : Definition 5. A system described by Equation 27a is exactly lumpable by a matrix M if and only if there exists a matrix k+such that for any a,
MKfae = &+Mae
(334
With this definition, the lumpability analysis of the present system is expected to be essentially similar to that given in the preceding discussion for a nondiffusion system. All we must do is replace matrices K, X, and A by the corresponding diffusion-disguised matrices; most of the analysis will be carried through without difficulty. The lumpability of the present system has two special features. The first occurs when the diffusivities of all species are equal-i.e., when D = DI, P = D-lK, and
Since the effective diffusivities of all species must be nonzero, D must be nonsingular and Equation 27a can be converted to
V2a(r) - Pa(r) = 0
rev
(274
=
where P D-IK. According to Wei (1962), matrix P possesses real and nonnegative eigenvalues. Therefore, if we let Y be the eigenvector matrix of P, with Q, the diagonal eigenvalue matrix of P, and
b
i = 1, 2,
. . .,n
bt = b,,=
(Y-1a,)2:
rdV
raV
(28a)
= f(@, r)Y-la,
a = Y~(Q,,r)Y-’a,
(29b)
Intrinsic reaction rate = -K+a , _= - - ~ D1 ~ p y ( l l . v ) a d ~ =
V
v)f(Q,,r)dSY-la,
(30)
V)f(*,r)dS
(n
(31)
V
122
=
l&EC FUNDAMENTALS
X+A+(X+)-’
Theorem 11 implies that when D = DI, the lumpability analysis given above can be immediately applied to the present system. The second feature of the present system is given as another theorem. THEOREM 12. If there exist matrices D (a diagonal matrix with positive diagonal elements) and K such that (35)
and
MK = KM the system described by Equation 34a is exactly lumpable. As proof, we must show the following:
MYcpy-1 = MP = MD-IK
=
fi-1KM
after using Equations 35 and 8. This expression can be extended to include any analytic functions of matrices,f, as
From Equations 30 and 31, we have 1 K + = -DYeY-l V
MK = KM
MD = fiM
where K + is the diffusion-disguised, over-all rate constant matrix based on the surface composition a,. Following the notation of Wei (1962), X + and A + are the diffusion-disguised and the diagonal eigenvalue matrices, and a matrix function e(@) is L
(kl0)
This expression implies that there exists a matrix K so that, according to Theorems 4 and 1,
We are particularly interested in the intrinsic reaction rate of the system, which may be evaluated by computing the total flux across bV from the gradient of a on bV. The intrinsic reaction rate is
e(@)
MX =
(294
Converting b back to the a system, we have
(n
according to Theorem 2 :
To prove the reverse, we may use an approach similar to the proof for Theorem 3. When Equation 33b holds, matrices M and X may be arranged into the following form :
In vector and matrix form, this is simply
sa,
(8)
(28b)
bi = f(%, r)bi.,
-V DY
k such that
If Equation 8 holds, the following K + satisfies Equation 34,
The solution for Equations 28a and 28b can be given as
1
if and only if there exists a matrix
(33b)
MK = KM
- pibt = 0,
b
MK+ = K+M
Y -la
Equations 27c and 27b can be decoupled as
V2bt
Therefore, we have the following theorem: THEOREM 11. If D = DI, there exists a matrix K + such that
(32)
MYf(Q,)Y-’ = f(D-’K)M
(36)
Now, from Equation 32, we obtain
h
1 MK+ =: - MDYB(@)Y-' V 1
== --
V
A = diagonal equilibrium composition matrix with ut*, i = 1, 2, . . . , n, as its ith diagonal elements A = defined as MAMT b = defined as Y-la effective diffusivitv matrix with Di. i = D = the diagonal " 1, 2, . . . , n, as its ith diagonal element D = diagonal matrix with positive diagonal elements et = unit vector with 1 as its ith element, and 0 for the rest of the elements I = identity matrix K = kinetic rate constant matrix L = defined as XTA-lX, a diagonal matrix L = defined as (ilO)L(ilO)T M = lumping matrix, an ^n X n matrix of rank e, (n 2 h) n = outward normal vector of bV P = defined as D-lK r = position vector X( = ith eigenvector of matrix K (i = 1, 2, . . ., n) x = eigenvector matrix of matrix K, defined as (XI, x2, . . .,
DMYe(Q,)Y-'
h
or
(37) Therefore, the system is exactly lumpable. With proper lumping, lielation 35 has an important implication: The diffusivities of :species that are lumped together must be equal. For lumping other than the proper one, the implication is different: If all the elements of matrix M are nonzero, it may easily be shown that
D = DI
and
D =
bi
(38)
Also, if some of the elements of matrix M are zero, the implication is much more complicated and a separate analysis must be done for each individual case. Nomenclature
SCALARS ai = composition of species A d or ith element of vector a (i = 1, 2, , . ., n]' ai* = equilibrium composition of species Ad (i = 1, 2, . . ., n)
A
Ai
= = =
A; Bi = bt
=
ci
=
Dt D
=
f
=
bi, =
=
kij =
Lt = n =
S V
= = =
z
=
t
space composed of vector a ith species in A-space subspace of A-space, totality of vectors AMTA-'1(&~) pseudo-species in characteristic composition space (i = 1, 2, . . ., 7L) ith element of vector b (i = 1, 2, . . ., n) defined as (Y-laJ;; coefficient of Taylor series expansion of function f, (i = 1, 2, . . . ) effective diffusivity of Ai (i = 1, 2, . . ., n) defined as D i(i = 1, 2, . . ., n) analytic function of matrices, or an analytic function of matrix Q, and vector r, as given in Equation 36a reaction rate constant of reaction A j 4Ai straight-line reaction paths (i = 1, 2, 3) number of elements in vector a surface area of V time interior of a catalyst particle, or interior of a reactor, or their volume dummy variable
VECTORS AND MATRICES Bold-face capital letters represent matrices; bold-face lower case letters represent vectors. a = composition vector a, = composition vector on bV I = defined as Ma
Xn )
= 1, 2, . . ., 2) Y = eigenvector matrix of matrix P, defined as (yl, y2, . . .,
91: = ith column vector of matrix t-lI2XT (i Yd GREEK LETTERS positive real number positive real number
fi
= =
e
= defined as
a!
Lv
(n . v)f(@, r)dS
. . ., n) diagonal eigenvalue matrix of matrix K with 1, 2, . . . , n, as its diagonal elements pi = ith eigenvalue of matrix P (i = 1, 2, . . . , n) Q, = diagonal eigenvalue matrix of matrix P with 1, 2, . . . , n, as its diagonal elements hi = ith eigenvalue of matrix K (i = 1, 2,
A
=
At,
i
=
pt,
i
=
SYMBOLS =
* +
= =
1
=
aV = 0 =
any property related to the lumped system (for a vector it has 6 elements, and, for a matrix, a dimension of ii x &) any property related to equilibrium state diffusion-disguised properties column vector with (1) as its elements (1, 1, 1, . . ., 1) boundary of V null vector or null matrix
literature Cited
Kemeny, J. G., Snell, J. L., "Finite Markov Chains," p. 123, Van Nostrand, New York, 1960. Silvestri, A. J., Prater, C. D., J . Phys. Chem. 68,3268 (1964). Wei, J., J . Catalysis 1, 526 (1962). Wei, J., Prater, C. D., Advan. Catalysis 13,203 (1962). RECEIVED for review April 9, 1968 ACCEPTED July 29, 1968
First in a series on Lumping Analysis in Monomolecular Reaction Systems. Paper prepared with the editorial assistance of J. A. Gorog.
VOL. 8
NO. 1 F E B R U A R Y 1 9 6 9
123
