Acknowledgment
The authors are indebted to the Yale Computer Center for the support of computation costs, and W. G. Hunter for his assistance and advice.
t Y
=
k 8,
- CY)
xm2(1
A , E, 0, Lt‘ = fractional pressure of alcohol, ether, olefin, and water respectively, with respect to initial partial pressures c, s = concentration of complex and unoccupied sites, respectively - AE = activation energy j , I , u, u = summing indices k = vector of parameters to be estimated K1/, Klv = forward and reverse rate constants, alcoholsite reactions Kz,, Kzl = forward and reverse rate constants, alcoholcompkx reaction K3 = forward rate constant, complex decomposition reaction K4/ = forward rate constant, ether-site reaction Ke, = equilibrium constant, Equation 8 m = degrees of freedom, (r - 1) n = number of experimental points P, = initial pressure of reactant R1 . . . Rd = rate terms defined in Equations 4 to 7 r = number of responses So = surface site concentration
computed value corresponding to y determinant defined by Equation 2 fraction of surface coverage by complex with respect to So = fraction of vacant sites with respect to So = abscissa value below which is 100 (1 - CY) % ’ of area under the chi-squared distribution curve with m degrees of freedom
= = =
I VU
ea
Nomenclature
time
= measured datum
literature Cited
Box, G. E. P., Ann. N . Y. Acad. Sci.86, 792 (1960). Box, G. E. P., Draper, N. R., Biometrika 52, 355 (1965). Cope, C. S., A.I.Ch.E. J . 10, 277 (1964). Hougen, 0. A., Watson, K. M., “Chemical Process Principles,” Wiley, New York, 1947. Hunter, W. G., IND.ENC.CHEM.FUNDAMENTALS 6, 461 (1967). Hunter, W. G., personal communication, 1966. Kittrell, J. R., Hunter, W. G., Watson, C. C., A.Z.Ch.E. J . 11, 1051 (1965). Lapidus, Leon, Peterson, T. I., A.I.Ch.E. J . 11, 891 (1965); Chem. Ene. Sci. 21. 655 (1966). Pine: H.. HaaP. W. 0..J.’Am. Chem. SOC.82. 2471 (1960). Soboi, M.,IBM Share Program, No. 155i; NU‘ Scoop Linear Surface Minimization Routine, 1963. Solomon, H. J., D. Eng. - dissertation, Yale University, New Haven, Conn., 1966. Solomon, H. J., Bliss, Harding, Butt, J. B., IND.ENC. CHEM. 6, 325 (1967). FUNDAMENTALS RECEIVED for review March 13, 1967 ACCEPTEDNovember 6, 1967
KINETICS OF COUPLED FIRST-ORDER REACTIONS WITH TIME-DEPENDENT RATE COEFFICIIENTS IN TERNARY SYSTEMS LOTHAR R I E K E R T AND J A M E S WE1 Research Department, Central Research Division,“Mobil Rcsearch and Development Cor)., Princeton, N . J . 08540
A method for calculating the kinetic behavior of coupled first-arder reactions in a ternary system from any given set of time-dependent rate coefficients i s based on Lie algebra. The method reduces the integration of a set of simultaneous differential equations with variable coefficients to successive integrations of ordinary differential equations. The application is illustrated by an example.
HE
kinetics of coupled first-order reactions can be handled
T by linear algebra (as described by Wei and Prater, 1962) if the rate coefficients are: constant. I n many cases of practical importance, however, the rate coefficients are functions of time ; two important examples are reactions proceeding under nonisothermal conditions and catalyst deactivation. T h e problem of optimization of a temperature profile in systems of coupled reactions has been treated by Bilous and Amundson (1956) and by Mah and Aris (1964). Froment and Bischoff (1962) have considered the effect of catalyst deactivation on the selectivity of irreversible para.llel reactions.
A method is described for calculating the behavior (reaction paths and time dependence) of any system of coupled firstorder reactions that have time-dependent rate coefficients and involve three components. A mathematically equivalent problem was encountered by Sargent (1963) in connection with the dynamics of distillation columns; he obtained numerical solutions with the help of power series. T h e present method is based on the theory of Lie algebras (Jacobson, 1962), which allows reduction of the integration of a set of simultaneous linear differential equations with variable coefficients to successive integrations of ordinary differential equations. VOL 7
NO.
1
FEBRUARY 1968
125
This method has been described in detail by Wei and Norman (1963). Below it is shown how the present problem leads to the application of Lie algebra if it is presented in the framework of the analysis of coupled first-order reactions as given by Wei and Prater (1962). This treatment should serve only as a brief survey of the mathematical background; a more vigorous treatment is given by Wei and Norman (1963). Survey of General Method
as compared to a system of first-order reactions with constant rate coefficients-but the reaction path (selectivity) is not affected. If, however, K is not represented by Equation 6, so that the time-dependent factors that multiply the individual rate constants in K are different, the situation will be much more complex. This case will be encountered in nonisothermal systems with different energies of activation, or where selective deactivation affects individual rate constants in different ways. T h e rate “constant” matrix may be written in the form
A system of first-order reactions between n components is described by the rate equation
which relates the rate of change of the n-dimensional composition vector, a, to the rate coefficient matrix, K. [Throughout this paper the terminology and notation of Wei and Prater (1962) are used unless otherwise stated.] T h e solution of Equation 1 can be written
a (t) =
u a,
(2)
I n Equation 2, U is a time-dependent n X n matrix U(t), such that U(0) = I. I t is called the matrizant (Amundson, 1966). Since
da - _ -- du dt
dt
a, = K U a,
This tentative solution may be written
u
=
nul 1
where Ut is given by
u 1 --- e (4)
combined with Equation 2, gives the solution to Equation 1. If K is a scalar function of time
(5)
the solution of Equation 4 is U = exp[q(t)K,I
where q 2 is defined as
(3)
we see that a solution of the matrix equation
K(t) = p ( t P 0
where m is equal to or less than n2 - n (n being the number of components), and the H 2 are linearly independent matrices with entry 1 a t some position i j and -1 at j j . All other entries are zero. If a solution similar to that in Equation 6 is attempted, one obtains
(6)
where
(7) Expanding Equation 6 to show that it is a solution of Equation 4, we obtain
giHz
Since the different operators, HI, do not commute, Equations 11 and 12 will not be unequivocal. When Equation 11 is expanded into a series (as in Equation 8) and differentiated, we see that it is not a solution in this case. Although the arbitrary order of summands does not matter in Equation 10, it will be important in Equations 11 and 12 if UtUl # UjUt, or if H,Hl # HjHt. Only if all the H I commute will the U l commute, and only then will Equation 12 be an exact solution. Wei and Norman (1963) have shown that if the HZdo not commute, the general form (Equation 12) can be made an exact solution if it is expanded over the whole Lie algebra ( L } generated by the HI. This Lie algebra { L } is the set of operators spanned by the Hi and all their commutators [H,,H,] H,Hj - HfHt. I n other words, Equation 12 may not be a correct solution because it neglects elements resulting from a different ordering (commutation) of the H l ; therefore, all these elements must be included in the solution, which may then be written
and
= PKo
(I+
= pK,U = K
qK,
Here the Hi form a basis of the Lie algebra generated by the HI, and n may be greater than or equal to m. Equation 13
q2 +K,2 + . . . 2
yields
U
If K, is diagonalized as XAX-’, U may be written in the form
u
=
x ,dOAX-’
(9)
Equations 6 and 9 cover the nonselective deactivation of a catalyst that affects all rate constants in K in the same way. I n this case, which corresponds to the pseudo-monomolecular system (Wei and Prater, 1962), the time behavior is changed126
IbEC FUNDAMENTALS
Multiplying Equation 4 from the right by UW1and using Equation 10 gives
du -
dt
m
U-’ =
Pl(t)Ht 1-1
As in Equation 10, this matrix can be written
Hence, from Equations ’14 and 15, rn
c n
CPlHl =
1-1
1-1
1
1-1
or
rI ‘exPfg,H,lHtJ C rI 1-1 I-1
(16)
exPI -g,H,l
Equation 16 gives a relation between the p l ( t ) and the oi. The right-hand side is of the form eXYe-X. Therefore, Equation 16 can be evaluated by applying repeatedly the BakerHausdorff formula eXYe-X
=
Y
where
+ [X,Yl -t51 [X,(X,Y)I + 3!1 x -
+ ...
{X,[X,(X,Y>IJ
(17)
where
[X,W] =
x Y - Yx
As Equation 16 contains only basis elements of the Lie algebra ( L ) multiplied by some scalars, a relation between the scalars gt, p l ) can be found. T h e resulting scalar differential equations can then be integrated to give the gt’s needed in Equation 13. Only if the Hz commute are the solutions in Equations 12 and 13 identical. Two cases may result, as described in detail by Wei and Norman (1963). If the subsets of ( L ] resulting from repeated commutation
(st,
[L,L] = L’
[L’,L’]= L”
For the other schemes given above, Equation 18 is simplified because several ki are zero. T h e result of an examination of the Lie algebras associated with the eight different types of rate constant matrices for schemes I to VI11 is that only the algebras associated with schemes I to I V are solvable; those related to schemes V to VI11 are not. Only the solutions for the most complicated schemes I V and VI11 of each group are given here, since they also contain the solutions for all other schemes of each group.
[L”,L”]= L’” Solvable Algebra
decrease and finally zero, the algebra is said to be solvable. If the algebra is solvable, the series resulting from the combination of Equations 16 and 17 will contain elements of ( L } multiplied by exponential functions of the g:s. T h e solution of the system of differential equations for the gt can then be found by making succesiive quadratures. If the algebra is not solvable, the relation between thegi and p z resulting from Equations 16 and 17 will not be as simple. Under certain conditions, however, after an appropriate linear transformation of the coordinate space, a decomposition of the algebra is possible. T h e problem then presents itself as a system of differential equations that can be solved for the problem that follows.
T h e most complicated scheme of the first group is
with the rate constant matrix
K = klH1
[HI,HZ]= H I - HI [Hi,H-z] = 0
There are only eight kinetic schemes for a three-component system that do not viola.te the principle of microscopic reversibility: k
(I)
k
A-?.B&C
k.a
k3H3
T h e commutators are
Application to Kinetics of Three-Component Systems
Akt‘BkC
+ kzHz + k-zH-z +
[ H I , H ~= ] H3
- Hi
[Hz,H-z] = Hz
- H-2
[Hz,Ha] = 0 [H-z,Ha] HI
(11)
- H3
The form of the commutator relations suggests that another basis for { L ] be taken :
{ L } = {Hi, H2, Mi, Mz} with
M I = HI
- H3
Mz = Hz
- H-2
T h e commutators are then T h e rate constant matrix, K, of the general scheme VI11 is
-(ki
K = (
+ k3)
kl k3
k-
- (k-
I
I
+ kz)
kz
-(k-z
::: ) + k-3)
[HI,HZ] = [HI,MI]= [HI,Mz]= Mi [Hz,Ml]
=
-MI
(19 )
[Hz,Mzl = -Mz [Mi,Mz]
= 0
VOL 7
NO. 1
FEBRUARY 1968
127
T h e subsets of { L ] ,resulting from repeated commutation (the so-called “derived algebras”), are
{L’}
=
{ M ~ , M ~ ]{ ;L ” } =
and the algebra is solvable.
{o}
Using the notation
[eadx]y =
exYe-x
Nonsolvable Algebras
General Scheme. VIII. For schemes V to VIII, no solution can be found by the procedure used in the above example. As soon as the operator H-1 associated with the reactor B + A appears, the algebra is no longer solvable. However, after decomposition of the matrix K = z p i H i into 1
two linearly independent parts,
K = Ks
+ KR
(which are, of course, themselves some linear combinations of the different pLHJ, a solution is possible if this decomposition meets the following requirements: The basis of K R produces a solvable Lie algebra LR, so that the commutators of all elements of LR, with all the elements of L, are elements of LR or linear combinations thereof. A solution to the equation
can be found. Then Equation 6 becomes
and has the solution
u = UsUR where UR is the solution of
g,
=
T h e result is easily verified by differentiation:
-k-ze92
du S - -- U s - +m +R U U Rm
From Equation 21, scalar functions f through i can be found by making successive quadratures. T h e matrix U is now constructed according to Equation 13. T h e ordering of the factors must, of course, be the same as was assumed in Equation 20.
U =
~~IHI~OZHZ~OSMI~~~MZ (22)
Since
HiHi = -HI;
HiHz = 0; HzMi = -MI;
MiMz = 0 ;
HzHz = -HI!; HIM1 = 0; H2M2 = -Mz; MIMI
0;
HIM*
0;
dt
dt
dt
-
As Us-’ . H R Us must be a linear combination of the elements of the solvable algebra LR, a solution for Equation 28 can be obtained by the procedure used in the above example of a solvable algebra. T h e operators H 1 in the matrix K corresponding to scheme VI11 are the same as those of the general, continuous-time, three-dimensional Markov process. As outlined by Wei and Norman (1963), and using their notation, its operator space can be decomposed into a simple subalgebra, S,
MzMz = 0; expanding the exponentials into series and multiplying gives
+ Hl(1 -
U =I
e-”1)
f Hz(l - e - 0 8 ) f Mlg3e-”@f M~g4e-*2 (23)
Hence, and a solvable radical, R,
(-;-p i), (-; -: li), -1
R
=
{A
=
(24) This solution also contains those of the simpler schemes, I, 11, and 111. Thus, the set of rate equations can here be integrated by four successive quadratures. 128
I&EC FUNDAMENTALS
-1
B =
C =
(: -1
-1
;
I);
-1
s 1 1 SI2
us = (s*l \s31
s22 s32
1)
(34)
1/
where =
SI1
b = k-2/2 c = k-2/2
+
(1
-p-
ep)e-'J
k-3
T o obtain UR, one must first form Us-'KRU,. Us gives
Ius/
=
SllS22
- s12s21 =
Inversion of
1
(35) T h e solution will be of the form
+ bB + cC = aU,-'AU, + bU,-'BU, + CU,-~CU,
K R = aA T h e scalar functions e, $7, and 9 are found from an expression analogous to Equation 16. The differential equations defining e, p, and q have been given (Wei and Norman, 1963). For completeness, the derivaltion is outlined here. T h e commutator table is
x
I
Y = [X,Y]=
A
B -B
0
C -C
o
0
C
0
E
E
F
0
0 0
0
-B
0
H
c
0
H
U,-'KRU,
As A commutes with all elements of { S},
0
From
U,-'BU,
dt
U,-'CU,
+ (1 - ep)e-'JB
= e'JC
- UR-l = Us-'KRUS = aA dt
1~ + , j ( p d E ) ~+ ,j(tadE ew d F)H
(32)
- pe-'JB
(38)
+ [b(1 - ep) - crp]e-'JB + (be + c)enC (39)
From
uR= ea'ePe7'
and by means of Equatilons 17 and 19, one obtains
+ hH = i E+. G(F + eH + ?E) + $(H + 2 EE - 2 pF - 2 peH - 2 pe2E)
(37)
Then
= eE + j F $: hH
-
eE +fF
= ee'JC
m R
m* - U8-1
(36)
Although the calculation of the two remaining terms is rather lengthy, it is straightforward; the result is
-B C 2E -2F 0
= A
U,-'AU,
(40)
Equation 16 gives
Eliminating E,F, and H yields the differential equations
+ 2 hlosiue-like Violence A.
H. M A S S 0 A N D D . F. R U D D
Department of Chemical Engineering, The University of Wisconsin, Madison, Wis. 53706 The spread of violence through weakly connected systems by a probabilistic mechanism was studied. Probability theory and simulation are used to develop preliminary criteria for the safe design of storage systems through which disaster may spread by random series of detonations. Parametric results are qualitatively compared to tlhose given by data suggested for the specification of explosives storage. This is an initial report on studiies in disaster propagation.
occasional destruction of a major processing complex to the need for further investigation into the anatomy of disaster. i4pparently, the inherent instability of certain systems to the more violent modes of operation is not known until after the system has been racked by violence. Investigations in disaster propagation should yield recommendations for the design of safer systems. Tracing the path of fire and explosion through a system indicates that disasters often consist of complex chains of events, each event triggering one or more other events. After the fact, it is often obvious how a slight change in the design of a system might have broken a critical chain of events and prevented the sweep of disaster through the system. I t is also obvious that the element of chance enters into disaster HE
T points
propagation, and that the violence moves through the system by mechanisms foreign to the common principles of transport. Some deterministic chains of events are easi!y detected and broken by the competent engineer. For example, the engineer may inspect the design of a system and observe that the chance failure of a pump will lead necessarily to a sequence of events involving the overflow of a storage tank, the flow of flammable liquid into an area of ignition, and the final explosion of a processing component which will be surrounded by the burning liquid. This chain of events, while triggered by the chance failure of the pump, is completely deterministic in the sense that each event in the chain will occur with certainty once the chain is initiated. Often a slight modification in the design will break the chain and improve the inherent safety of the VOL 7
NO. 1
FEBRUARY 1968
131
