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Accessible composition domains for monomolecular systems
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ACCESSIBLE COMPOSITION DOMAINS MONOMOLECULAR SYSTEMS
FOR
F. J. KRAMBECK Research Department, Mobil Research L Development Corp., Paulsboro, NJ 08066. U.S.A. (Received 20 June 1983; accepted 14 November 1983) Abstract-Constraints on possible chemical changes which are sharper than those obtainable from thermodynamics alone can be obtained by adding rather weak assumptions about the form of the kinetic equations. Assuming only that the kinetics are monomolecular, a very simple ellipsoidal bound on allowable composition is derived that is considerably sharper than the thermodynamic bound. A still sharoer bound can be constructed but it is considerably more complicated and doesn’t reduce the acc&ible region much more.
INTRODUCTION
where the diagonal terms of the matrix coefficients k, have been defined by
A very useful tool in the development of chemical processes is a mapping of the chemical changes thermodynamically allowed under various conditions of temperature and pressure. This can be done with only rudimentary knowledge of catalyst kinetic&t suthces only to know which reaction pathways take place at reasonably high rates-and with data that is often readily available. The utility of this approach could, of course, be increased if the allowable range of chemical changes could be narrowed down still further. Thus it is interesting to consider how more knowledge, or assumptions, about the kinetics could be used to sharpen this mapping. A general approach to this problem has recently been described by Shinnar (1982). It was shown that knowledge of the set of kinetically important elementary reactions can be used to establish sharper constraints than thermodynamics alone. In what follows we will derive still sharper constraints for the special case of monomolecular kinetics.
k;; = -
An important constraint on the rate coefficients results from the principle of detailed balance, which states that at equilibrium the rate of each individual reaction is zero: kucf = kjicfVi,
(Kx, Y) = 6, Kyb'x, YER”
rate rU given by (2)
where c, is the concentration of the ith species. The net effect of all the reactions occurring simultaneously is then given by dc.
”
$=jz,k,c,;
i=l,...,
n
(5)
where x and y denote arbitrary vectors in R”. The detailed balance constraint (5) is then equivalent to
(1)
rii = kiici - k,c,
j
where cf is the equilibrium concentration of species i. The mathematical properties of the system (3j(5) have been analyzed in detail by Wei and Prater (1962). Their basic result can be expressed in a simplified form by defining a nonconventional inner product as follows:
MONOMOLXCULARSYSTEMS
with reaction
c kji
j*i
A monomolecular reaction system is a set of chemical species-whose chemical symbols are here denoted as A,, A,, . . . , Am-which undergo only reactions of the form Ai&Aj
of rate
(3)
(7)
where K is the matrix of rate coefficients, k,. Equation (7) expresses the fact that the matrix K is Hermitian (or self-adjoint) and thus implies that it possesses a complete set of eigenvectors that are mutually orthogonal with respect to the inner product (6) and have real eigenvalues. These eigenvalues are nonpositive. Wei and Prater showed how the reaction paths for a monomolecular system are easily visualized by considering the projections of the composition vector onto these eigenvectors. Let x,, x2,. . . ,x, represent the eigenvectors of the K-matrix. The projections of the composition c onto the xi will be denoted bi, so that c = Z&&x,.
1181
1182
F. J. KRAMBECK
molecular system and the starting composition, a, are specified. Consider an arbitrary reaction path of a 3-component system:
PARA
c = C* + bra e+‘x,
db,_--Ibi; dt
eqn (3) becomes
i=l,...,n
(x -c*, where - Ii is the eigenvalue corresponding the reaction pathways are given by
c* + b2”e-“l’xp
(10)
When plotted in composition space this will be a straight line, since all the points are in the single direction xr from the equilibrium point. This is illustrated in Fig. 1, which shows reaction paths for a typical monomolecular system, that of xylene isomerization over a particular catalyst (Aly el al., 1965). Since this is a three-component system there are three eigenvectors: the equilibrium point and two others. Thus there are two straight line reaction paths for this system. The one whose eigenvalue has the greatest magnitude is called the fast path and the other the slow path. Note that except for the fast straight-line path, all the reaction paths become tangent to the slow straight-line path at the equilibrium point. BOUNDARY OF
x - a) = 0.
(12)
to x+ Thus
where b/ is the starting value of bi. One of the eigenvectors of the X-matrix is the equilibrium composition c*, and it will have an eigenvalue of zero. Let us assume that xi = c* and I, = 0. If all the remaining bo except one, say bra, are zero, the reaction path becomes c=
(11)
As the ratio of the fast eigenvalue to the slow eigenvalue, &/Jr, becomes large, the reaction path will approach two straight line segments. It will initially move along the direction of the fast path, xr, until it reaches the slow path. It will then move along the slow path, x2, toward the equilibrium point. Thus given a starting point, a, and the eigenvectors x2 and xj, one can draw a line through at with direction xj and one through c* with direction x2 to give a limiting path for that particular choice of x, and x,. This is shown in Fig. 2, where the intersection of these two lines is labeled x. Now because the eigenvectors are orthogonal with respect to the inner product (6), we can calculate x, for arbitrary choices of xr. Thus we may construct a locus of such limiting points, x, by assuming all possible directions for x,. As shown in Figure 2, the equation for this locus is
Fig. 1. Reaction paths for xylene isomerisation. In terms of these projections
+ b,’ e+‘x,.
Equation (12) thus described an ellipsoid which cannot be crossed by any reaction path starting at a no matter what values are assumed for the reaction rate coefficient k, There are other constraints on the reaction paths. Clearly they cannot leave the reaction triangle, which would imply negative values for some components of the concentration vector. The fact that the reaction rate coefficients are positive implies that each individual reaction must drive toward equilibrium. Shinnar (1982) showed that this limits the allowable directions in various parts of the reaction triangle, as shown in Fig. 3. Thus the dotted tine in Fig. 3 must also contain any reaction path beginning at a, even if the kinetic parameters change along the path (by changing catalyst, say). Both of these bounds are shown in Fig. 4.
ACCESSIBLE DOMAIN
The above properties can be used to determine a boundary for the accessible composition domain when only the equilibrium point, c*, of a mono-
Fig. 2. Method of analysis.
Accessible composition
domains
for monomolecular systems
1183
INITIAL COMPOSiTlON
Fig. 3. Implication of positive rate coefficients.
Fig. 6. Bounds on accessible domain.
INITIAL COMPOSiTlON
Fig. 4. Bounds on accessible domain.
INmAL COMPosmON
Fig. 7. Bounds on
accessible domain.
ORIENTATION OF THE ELLIPSE The
shape and orientation of the ellipse depends only on the equilibrium point c* and not on the initial composition. This can be seen in Figs. 4-7. This is because the quadratic terms in (12) do not involve a but do involve C* through the inner product (6). Analysis of this equation in the plane of the reaction triangle gives the result shown in Fig. 8. Here the line r is drawn from the center of the triangle to the equilibrium point c*. The angle it makes with the horizontal is denoted 4. The angle made by a principal axis of the ellipse is then 0, where INITIAL COMPOSITION
tan20=
cos4 .
+r
sin24
sin4 +rcos2$
(13)
Fig. 5. Bounds on accessible domain. SHARPER BOUNDS
Figures 5-7 give examples for different starting compositions but the same equilibrium point. Note that the smallest region occurs in Fig. 6, where the initial composition has components A, and A, already in equilibrium. In this case the ellipse is tangent to the A,-A, side of the triangle.
An important question is whether the bound given by eqn (12) can actually be realized by a physically meaningful set of rate coeficients or whether still sharper bounds can be obtained. We note in Fig. 7 that the dashed line bound that is based simply on positivity of rate coefficients lies within some parts of
1184
F. J.
bAM3ECK
vector has been normalized (&x,)=;
so that
. ;;; . .
eqn (14) becomes kii= ,+?$ n=l
i,j=l,...,
(16)
n.
Setting k,= 0 gives a set of homogeneous linear relations in the nonzero eigenvalues. In the threecomponent case there are just two of these. Setting the slow one arbitrarily to one leaves a set of linear equations in one unknown. Each of these is solved and whichever gives the smallest solution determines the maximum eigenvalue ratio.
tcln20=-6+rsin24 sin4
(-1,);
+rws2+
Fig. 8. Orientation of elliptical bound. NOTATION the ellipse, so the ellipse is clearly not the sharpest possible bound. This occurs because for each choice of eigenvectors there is a finite maximum ratio of the two eigenvalues above which one or more of the reaction rate coefficients becomes negative, while attainment of the elliptical bound requires this ratio to approach infinity. Thus we may develop a minimal bound as follows: we choose a set of arbitrary directions for the fast eigenvector through the initial composition consistent with the directional constraints of Fig. 3. For each of these we calculate the direction of the slow eigenvector from the orthogonality relationship. We then calculate the maximum eigenvalue ratio for these eigenvectors, as explained below, and use this eigenvalue ratio to compute a reaction path. Repeating this for all the chosen directions and drawing the envelope then gives the bound. To determine the maximum eigenvalue ratio we make use of the relationship K =
x.4xmt
(14)
A
symbol for ith chemical species initial concentration vector projection of concentration vector onto xi initial value of bi concentration vector concentration of itb chemical species equilibrium concentration of ith species rate coefficient matrix rate coefficient for reaction A,+A, number of chemical species net rate of reaction A, + A, time matrix formed with eigenvectors as columns ith eigenvector of rate coefficient matrix negative of ith eigenvalue of rate coefficient matrix diagonal matrix of eigenvalue (-&)
REFERENCES Shinnar, R. and C. A. Feng, 1982, Thermodynamic cons~oints in cufu!ytic reucfionr, AIChB Meeting, Los Angeles. Aly, A. F., B. W. Rope and J. J. Wise, 1965, Kin&s o/ xylem isomerizufion, 74th AlChE National Meeting. Pap& No. 566. Wei, J. and C. D. Rater, 1962, The structure and analysis of complex reaction systems. Aduunces in Catalysis, Academic Press, New York, Vol. 13, pp. 203391. I_
where X is a matrix formed by arranging the eigenvectors as columns, X-’ is its inverse, and A is the diagonal matrix of eigenvalues, -&. If each eigen-
ACCESSIBLE COMPOSITION DOMAINS MONOMOLECULAR SYSTEMS
FOR
F. J. KRAMBECK Research Department, Mobil Research L Development Corp., Paulsboro, NJ 08066. U.S.A. (Received 20 June 1983; accepted 14 November 1983) Abstract-Constraints on possible chemical changes which are sharper than those obtainable from thermodynamics alone can be obtained by adding rather weak assumptions about the form of the kinetic equations. Assuming only that the kinetics are monomolecular, a very simple ellipsoidal bound on allowable composition is derived that is considerably sharper than the thermodynamic bound. A still sharoer bound can be constructed but it is considerably more complicated and doesn’t reduce the acc&ible region much more.
INTRODUCTION
where the diagonal terms of the matrix coefficients k, have been defined by
A very useful tool in the development of chemical processes is a mapping of the chemical changes thermodynamically allowed under various conditions of temperature and pressure. This can be done with only rudimentary knowledge of catalyst kinetic&t suthces only to know which reaction pathways take place at reasonably high rates-and with data that is often readily available. The utility of this approach could, of course, be increased if the allowable range of chemical changes could be narrowed down still further. Thus it is interesting to consider how more knowledge, or assumptions, about the kinetics could be used to sharpen this mapping. A general approach to this problem has recently been described by Shinnar (1982). It was shown that knowledge of the set of kinetically important elementary reactions can be used to establish sharper constraints than thermodynamics alone. In what follows we will derive still sharper constraints for the special case of monomolecular kinetics.
k;; = -
An important constraint on the rate coefficients results from the principle of detailed balance, which states that at equilibrium the rate of each individual reaction is zero: kucf = kjicfVi,
(Kx, Y) = 6, Kyb'x, YER”
rate rU given by (2)
where c, is the concentration of the ith species. The net effect of all the reactions occurring simultaneously is then given by dc.
”
$=jz,k,c,;
i=l,...,
n
(5)
where x and y denote arbitrary vectors in R”. The detailed balance constraint (5) is then equivalent to
(1)
rii = kiici - k,c,
j
where cf is the equilibrium concentration of species i. The mathematical properties of the system (3j(5) have been analyzed in detail by Wei and Prater (1962). Their basic result can be expressed in a simplified form by defining a nonconventional inner product as follows:
MONOMOLXCULARSYSTEMS
with reaction
c kji
j*i
A monomolecular reaction system is a set of chemical species-whose chemical symbols are here denoted as A,, A,, . . . , Am-which undergo only reactions of the form Ai&Aj
of rate
(3)
(7)
where K is the matrix of rate coefficients, k,. Equation (7) expresses the fact that the matrix K is Hermitian (or self-adjoint) and thus implies that it possesses a complete set of eigenvectors that are mutually orthogonal with respect to the inner product (6) and have real eigenvalues. These eigenvalues are nonpositive. Wei and Prater showed how the reaction paths for a monomolecular system are easily visualized by considering the projections of the composition vector onto these eigenvectors. Let x,, x2,. . . ,x, represent the eigenvectors of the K-matrix. The projections of the composition c onto the xi will be denoted bi, so that c = Z&&x,.
1181
1182
F. J. KRAMBECK
molecular system and the starting composition, a, are specified. Consider an arbitrary reaction path of a 3-component system:
PARA
c = C* + bra e+‘x,
db,_--Ibi; dt
eqn (3) becomes
i=l,...,n
(x -c*, where - Ii is the eigenvalue corresponding the reaction pathways are given by
c* + b2”e-“l’xp
(10)
When plotted in composition space this will be a straight line, since all the points are in the single direction xr from the equilibrium point. This is illustrated in Fig. 1, which shows reaction paths for a typical monomolecular system, that of xylene isomerization over a particular catalyst (Aly el al., 1965). Since this is a three-component system there are three eigenvectors: the equilibrium point and two others. Thus there are two straight line reaction paths for this system. The one whose eigenvalue has the greatest magnitude is called the fast path and the other the slow path. Note that except for the fast straight-line path, all the reaction paths become tangent to the slow straight-line path at the equilibrium point. BOUNDARY OF
x - a) = 0.
(12)
to x+ Thus
where b/ is the starting value of bi. One of the eigenvectors of the X-matrix is the equilibrium composition c*, and it will have an eigenvalue of zero. Let us assume that xi = c* and I, = 0. If all the remaining bo except one, say bra, are zero, the reaction path becomes c=
(11)
As the ratio of the fast eigenvalue to the slow eigenvalue, &/Jr, becomes large, the reaction path will approach two straight line segments. It will initially move along the direction of the fast path, xr, until it reaches the slow path. It will then move along the slow path, x2, toward the equilibrium point. Thus given a starting point, a, and the eigenvectors x2 and xj, one can draw a line through at with direction xj and one through c* with direction x2 to give a limiting path for that particular choice of x, and x,. This is shown in Fig. 2, where the intersection of these two lines is labeled x. Now because the eigenvectors are orthogonal with respect to the inner product (6), we can calculate x, for arbitrary choices of xr. Thus we may construct a locus of such limiting points, x, by assuming all possible directions for x,. As shown in Figure 2, the equation for this locus is
Fig. 1. Reaction paths for xylene isomerisation. In terms of these projections
+ b,’ e+‘x,.
Equation (12) thus described an ellipsoid which cannot be crossed by any reaction path starting at a no matter what values are assumed for the reaction rate coefficient k, There are other constraints on the reaction paths. Clearly they cannot leave the reaction triangle, which would imply negative values for some components of the concentration vector. The fact that the reaction rate coefficients are positive implies that each individual reaction must drive toward equilibrium. Shinnar (1982) showed that this limits the allowable directions in various parts of the reaction triangle, as shown in Fig. 3. Thus the dotted tine in Fig. 3 must also contain any reaction path beginning at a, even if the kinetic parameters change along the path (by changing catalyst, say). Both of these bounds are shown in Fig. 4.
ACCESSIBLE DOMAIN
The above properties can be used to determine a boundary for the accessible composition domain when only the equilibrium point, c*, of a mono-
Fig. 2. Method of analysis.
Accessible composition
domains
for monomolecular systems
1183
INITIAL COMPOSiTlON
Fig. 3. Implication of positive rate coefficients.
Fig. 6. Bounds on accessible domain.
INITIAL COMPOSiTlON
Fig. 4. Bounds on accessible domain.
INmAL COMPosmON
Fig. 7. Bounds on
accessible domain.
ORIENTATION OF THE ELLIPSE The
shape and orientation of the ellipse depends only on the equilibrium point c* and not on the initial composition. This can be seen in Figs. 4-7. This is because the quadratic terms in (12) do not involve a but do involve C* through the inner product (6). Analysis of this equation in the plane of the reaction triangle gives the result shown in Fig. 8. Here the line r is drawn from the center of the triangle to the equilibrium point c*. The angle it makes with the horizontal is denoted 4. The angle made by a principal axis of the ellipse is then 0, where INITIAL COMPOSITION
tan20=
cos4 .
+r
sin24
sin4 +rcos2$
(13)
Fig. 5. Bounds on accessible domain. SHARPER BOUNDS
Figures 5-7 give examples for different starting compositions but the same equilibrium point. Note that the smallest region occurs in Fig. 6, where the initial composition has components A, and A, already in equilibrium. In this case the ellipse is tangent to the A,-A, side of the triangle.
An important question is whether the bound given by eqn (12) can actually be realized by a physically meaningful set of rate coeficients or whether still sharper bounds can be obtained. We note in Fig. 7 that the dashed line bound that is based simply on positivity of rate coefficients lies within some parts of
1184
F. J.
bAM3ECK
vector has been normalized (&x,)=;
so that
. ;;; . .
eqn (14) becomes kii= ,+?$ n=l
i,j=l,...,
(16)
n.
Setting k,= 0 gives a set of homogeneous linear relations in the nonzero eigenvalues. In the threecomponent case there are just two of these. Setting the slow one arbitrarily to one leaves a set of linear equations in one unknown. Each of these is solved and whichever gives the smallest solution determines the maximum eigenvalue ratio.
tcln20=-6+rsin24 sin4
(-1,);
+rws2+
Fig. 8. Orientation of elliptical bound. NOTATION the ellipse, so the ellipse is clearly not the sharpest possible bound. This occurs because for each choice of eigenvectors there is a finite maximum ratio of the two eigenvalues above which one or more of the reaction rate coefficients becomes negative, while attainment of the elliptical bound requires this ratio to approach infinity. Thus we may develop a minimal bound as follows: we choose a set of arbitrary directions for the fast eigenvector through the initial composition consistent with the directional constraints of Fig. 3. For each of these we calculate the direction of the slow eigenvector from the orthogonality relationship. We then calculate the maximum eigenvalue ratio for these eigenvectors, as explained below, and use this eigenvalue ratio to compute a reaction path. Repeating this for all the chosen directions and drawing the envelope then gives the bound. To determine the maximum eigenvalue ratio we make use of the relationship K =
x.4xmt
(14)
A
symbol for ith chemical species initial concentration vector projection of concentration vector onto xi initial value of bi concentration vector concentration of itb chemical species equilibrium concentration of ith species rate coefficient matrix rate coefficient for reaction A,+A, number of chemical species net rate of reaction A, + A, time matrix formed with eigenvectors as columns ith eigenvector of rate coefficient matrix negative of ith eigenvalue of rate coefficient matrix diagonal matrix of eigenvalue (-&)
REFERENCES Shinnar, R. and C. A. Feng, 1982, Thermodynamic cons~oints in cufu!ytic reucfionr, AIChB Meeting, Los Angeles. Aly, A. F., B. W. Rope and J. J. Wise, 1965, Kin&s o/ xylem isomerizufion, 74th AlChE National Meeting. Pap& No. 566. Wei, J. and C. D. Rater, 1962, The structure and analysis of complex reaction systems. Aduunces in Catalysis, Academic Press, New York, Vol. 13, pp. 203391. I_
where X is a matrix formed by arranging the eigenvectors as columns, X-’ is its inverse, and A is the diagonal matrix of eigenvalues, -&. If each eigen-