literature Cited (1) Cnhen. W. C.. Johnson. E. F.. Ind. Ene. Chem. 48. 1031 11956) : \-, - - - - - \
I
,
Chem. Eig. Prog,. Symf. Sir. No.’36, 57,u86 (1961). ’ (2) Fanning, R. J., Ph. I>.thesis, University of Oklahoma, 1958. (3) Fanning, R. J., Sliepcevich, C. M., A.I.CI2.E. J . 5 , 240 11959). (4)’ Fuchs, A. M., Control Eng. 6, NO. 5, 125 (1959). (5) Koppel, L. B., IND.END.CHEM.FUNDAMENTALS 1, 131 (1962). (6) McAdams, W. H., “Heat Transmission,” 3rd ed., McGrawHill, New York, 1954. (7) Norwood, K. W., Metzner, .4.B., A.I.Ch.E. J . 6 , 432 (1960).
(8) Stewart, W. C., Sliepcevich, C. M., Puckett, T. H., Chem. Eng. Progr. S y m f . Ser. No. 36, 57, 119 (1961). (9) Wilson, E. E., Trans. A . S. M . E. 37, 47 (1915). (10) Weber, T. W., Ph. D. thesis, Cornell University, Ithaca, N. Y., 1963. RECEIVED for review September 25, 1963 RESUBMITTED September 16, 1964 ACCEPTEDDecember 16, 1964 19th Southwest Regional Meeting, ACS, Houston, Tex., December
1963.
STRUCTURE OF COMPLEX CHEMICAL REACTION SYSTEMS J A M E S W E I , Princeton University and Socony Mobil Oil Co., Inc., Princeton, h’. J .
A mathematical theory of complex chemical reaction systems of polynomial order (mass action kinetics) is formulated in terms of multilinear (or tensor) algebra. The canonical forms are discussed. The “porcupine theorem” which asserts the existence of straight-line reaction paths in homogeneous open subsystems is demonstrated.
system of chemical reactions consists of many For a closed system, the rates of reaction can be given by a set of nonlinear differential equations (74) COMPLEX
A chemical species and many independent reactions. dx*/dt = f*(xl, 2, . . , P ) i = 1, 2, . . . n
(1)
where the superscript is an index, not a power, and x’ is the concentration of molecular species i. One then attempts to solve for x ( t ) when a n initial value x(0) is given, or to determine how the composition of a system evolves with time when the initial composition is known. In general, such nonlinear differential equations cannot be integrated by analytical methods. It is our aim to discover as much structural information as we can about such systems. In principle, there exists a nonlinear transformation to a new coordinate system in which the equivalent differential equation takes the form dyl/dt = gb’, yz, . . . yn)
dy‘/dt = 0
i
=
(2)
2, 3, . . n
An example is shown in Figure 1, This is just another way of saying that there must be n - 1 integrals of motion or invariants for Equation 1. These first, second, and subsequent
integrals are related to the Lie group of Equation 1. It is often just as difficult to find them as to find the solution to Equation 1 by direct methods (9). The invariant subsets of the n-dimensional vector space, V,, generated by Equation 1 give a less detailed structural information that is often useful. An invariant subset S of V , has the property that if x(0) is in S, then x(t) will remain in S for all positive values of t . An invariant subset has the intuitive meaning that once a point moves into S, it will stay within S forever. T h e whole vector space, V,, and the equilibrium point are trivial examples of invariant subsets. Each reaction path considered as a set of points is also an invariant subset. The positive orthant of V , (all x i 2 0) is a particularly important invariant subset in chemical kinetics, since negative concentrations are not considered. The set of all invariant subsets of Equation 1 forms a lattice (6) under the operations of set union and intersection. If x(0) is in both invariant subsets S a n d T , then x ( t ) must remain in the union of S and T as well as in their intersection. An example is shown in Figure 2. If a large class of invariant subsets is known, one can qualitatively trace the destination of any reaction path x ( t ) . Mass Action Kinetics and Multilinear Algebras
XL
A first-order reaction system can always be completely uncoupled into n independent one-dimensional differential
y13
y12
y’= I
Figure 1, Nonlinear transformation where Y2 i s the invariant and (0,O) is a point of singularity
Figure 2. Union and intersection of invariant subsets S and T VOL. 4
NO. 2
MAY
1965
161
equations-which are easily solved (75). This is accomplished with the aid of linear algebra and the principle of detailed balancing; the result is a single canonical form or formal equivalence to n independent chemical reactions The structure of such a system is completely understood; we know all the qualitative behaviors of all reaction pathssuch as the absence of oscillations and inflection points, the existence of straight-line reaction paths, and the convergence effect of all reaction paths. For reaction systems obeying the law of mass action, the kinetic equations can be written as
dx/dt
=
(3)
P(x)
where P(x) is a polynomial in the n-vector x. reactions of up to third order can be written as
dx’/dt =
ujax3
f bt3txjxk f
A system of
Glkp‘.dXkXP
(4)
Here the summation convention is used. The first term on the right hand side represents all the reactions of first order. T h e coefficients, a, represent a tensor of covariant rank one and contravariant rank one, which is of course the same thing as a square matrix ( 7 7 ) . It is more useful to define a as a linear operator, A, that accepts a vector x as argument, and yields another-the rate of change of x-as the value. T h e coefficient al‘ can be interpreted as the rate of change of species 2 due to the action of species J. The second term on the right-hand side of Equation 4 represents all reactions of the second order. The tensor b is then a bilinear form that accepts two vectors as argument, and yields one vector as the value ( 7 ) . The coefficient blki can be interpreted as the rate of change in species i due to the interaction of species 1 and k . This is equivalent to a bilinear operator B which is linear with respect to each vector argument. Let x, y, and z be three vectors, and Q and B two scalars; then
B(Qx, By) = aPB(x, y ) or
+ y, B(x, y + B(x
bjx‘(ax)’(bY)k
2)
f B(y,
2)
z) = B(x, y )
f Bb,
z)
z) = B(x,
= a@jt‘x’Yk
(5)
bjktX5Yk
= b5ety5xk
When both arguments are the same vector x, it is convenient to write
B ~ ( x )2 B(x, X)
(6)
For a chemical reaction system, any polynomial differential equation of order s can be upgraded to a homogeneous (mathematical, not chemical) equation of order s. One method is to use the invariant due to mass conservation and let zxp =
1
(7)
Substituting Equation 7 in Equation 4, one obtains
dx’/dt =
(Ujkpi
f
bfkDf
(0)
P(X
(b)
+ Y)
f
Gjkpt)X5XkXP
(8)
=
P (X I
+ P(y)
P(C4 = f b ) P ( X )
where f(a) is a continuous monotonic function of a. This amounts to the decomposition to n one-dimensional ideals that are disjoint, or to the existence of a basis {zl, 2 2 , . . . z n } to the vector space, V,, such that P(z1) is a scalar multiple of
z1
P(z2) is a scalar multiple of
22,
etc.
An example of this system is the set of independent reactions +B
2A
3C -t D Another interesting canonical form is called “solvable” and has a basis such that P(z1) is a multiple of z1
P(z2) is a linear combination of zi and z 2 P(z3) is a linear combination of zl,z2, and z3,etc.
For this kind of triangular structure, the differential equations are not difficult to solve. One solves for z1 first, independent of the others. Then one solves for 2 2 , while z1 is already a known function of time. This form is called solvable in analogy to the solvable group of Galois in the theory of algebraic equations and the solvable Lie groups in differential equations. An example of this system is the reactions 2A
We shall deal only with symmetric bilinear forms, so that
B(x, y ) = B(y, X) or
One then faces the problem of discovering the structure of P(x). The multilinear operator, P, will seldom have the canonical form of the completely uncoupled first-order system, which reduces the problem to n independent chemical reactions. A complete uncoupling must satisfy the axioms of “semilinear transformations” (7) :
-+
2B
+- 2C
These two forms are the exceptions rather than the rules, as pointed out by Aris (3). Most of the other forms are not as amenable to analysis. One would like to have a complete classification scheme of all the possible canonical forms, and the qualitative behavior of the solutions. One would also need a simple test that will place a given differential equation among one of the possible forms. The best established system is of order two, corresponding to quadratic differential equations. It was initiated by Markus (73) and further developed by Aris ( 3 ) . They reduce the differential equation
dxi/dt =
b,t’xjx’
(10 )
to an equivalent problem of nonassociative algebra defined by the equation
where Uk’U5
b l k p t = b j k i for any value of p and a f k p z= a j t for any values of k and p
Alternative methods for homogenizing Equation 4 were discussed by Markus (73) and Aris (3). We can then write Equation 8 in the form
dx - = P,(x) dt 162
l&EC FUNDAMENTALS
(9)
=
b3ktui
(11)
This is a very fine and promising beginning, but a great deal of work needs to be done here. Their formulations as yet contain none of the properties of detailed balancing, nonnegativity of concentrations, and mass conservation. Systems containing up to two molecular species are completely worked out. But for systems with more species, Markus has reduced the quadratic differential equation to an equivalent nonassociative algebraic problem, just as in the previous reduction of the
The equilibrium vector is xe = (1, 1 , l ) , and the vector y = (-1, 2, -1) satisfies the conditions
B(x,, y) = -3y
B(y, Y)
=
-3y
Thus Equation 14 becomes
y(da/dt) = ~ ( - 6 a - 3a2) The straight-line reaction path is shown in Figure 3. For a homogeneous third-order reaction system, the necessary and sufficient conditions for a straight-line reaction path become Figure 3.
C(x,, xg, y) is a scalar multiple of y
Phase portrait of second-order system
C(x,, y, y) is a scalar multiple of y C(y, y, y) is a scalar multiple of y problem of linear difl'erential equations to the problem of linear algebra. Many important theorems are known in linear algebra that greatly facilitate the solution of linear differential equations. But, alas, the number of known theorems in nonassociative algebra are very few. For these we must wait.
This condition is even harder to meet in general. Thus in a closed system enjoying mass conservation, straightline reaction paths are a rarity for polynomial kinetics. However, for a reaction system containing steps that are nearly irreversible, we can define "open subsystems" where straightline reaction paths are not rare. Consider, for example, the system
Straight-line Reaction Paths and the Porcupine Theorem
2A
One useful feature of the first-order system is the straightline reaction paths given by (75)
A+B+.D
~ ( t= ) xe
+ aft)Y
(12)
where x e is the equilibrium vector and y is a time-independent vector. The scalar coefficient, a ( t ) , decreases to zero as time approaches infinity. This section is concerned with the existence of such straight-line reaction paths in nonlinear systems. Consider a homogen'eous second-order system
dri/dt =
( 1 3)
bjkiX'Xk
dxldt = B(x, x)
or
If this system has a straight-line reaction path, substitution of Equation 12 into Equation 13 yields
dfxe
+ ay)
=
B(xe f ay, xe
+ ay)
y d a l d t = B(xe, xe) f 2aB(xe, y) f a2B(y,y) or
da,.'dtyf
=
bjkixJxek
+ 2abjkfx2ykf a2bjkiy$k
(14)
The first term on the right of Equation 14 is identically zero since it represents the r,ate of change of an equilibrium composition. The left-hand term of Equation 14 is a scalar times the vector y . If Equation 14 is to be true for all time, it is necessary and sufficient that
B(x,, y) be a scalar multiple of y and
B(y, y ) be a scalar multiple of y
(15)
T h u s the existence of a straight-line reaction path implies the existence of a vector y that satisfies Equation 15. This requirement is seldom met, but not impossible. Consider, for instance
2A
e 2B e 2C
d[A]/'dt = - [ A ] '
+
2B
C
+ .
+ .
E
This system contains an open subsystem ( A , B ) . The rates of change of A and B depend only on the concentrations of A and B , and after a long time the concentrations of A and B become zero. In the open subsystem ( A , B ) , the equilibrium point is situated at the origin of the coordinate axes in a two-dimensional space. In general, we reserve indices 1 through rn for the species in the open subsystem, and indices rn f 1 to n for the species in the terminal subsystem. Then for the open subsystem, the rate equations assume the form
dxi/dt = f i ( x l , x2, . . . x")
i
=
1, 2, . , . rn
(16)
At time infinity, the equilibrium point is situated a t (0, 0, . . . 0; xern+l,x , " + ~ , , . . x e n ) . If one chose to center the discussion on the truncated m-dimensional space, the equilibrium point is situated a t the origin ( 0 , 0, . . . 0 ) . For open subsystems, straight-line reaction paths are not rare. Since x, = 0, the requirements of Equation 15 are not as difficult to satisfy. B(x,, y) is zero for any y, and can be considered as zero times y . Thus we have to satisfy only the condition
B(y, y ) is a scalar multiple of y
In a homogeneous kinetic system, the rate expression can be written as d x l d t = P,(x)
(17)
where subscript s signifies the order of the reaction. 17 is, of course, shorthand for
dx/dt = P(x, x, . . , x ) where x is repeated
s
Equation
times.
For such a system, the rate of change of ax is
P,(ax) = P(ax, ax, . . . ax)
=
aSP(x, x, . . . x)
=
a",(x) ( 18 )
[BIZ
Thus along a ray (ax), all the direction vectors dxldt will be pointed in precisely the same direction. Now we can offer a lemma. VOL. 4
NO. 2
MAY 1965
163
Lemma. For an open subsystem in a homogeneous reaction system, the necessary and sufficient condition for the existence of a straight-line reaction path of the form a ( t ) y that terminates a t the origin at time infinity is the existence of a nonnegative vector y which satisfies the equation
P J Y ) = By Proof. Substitution of a ( t ) y into Equation 17 gives d [ a ( t ) y ] / d t= y(da/dt) = P,(cyy) = a"Ps(y) = aspy Thus upon solving the scalar equation da/dt = as& we obtain a solution to Equation 17. In Equation 19, if the scalar constant = 0, we have a ray of equilibrium points that is equivalent to the nilpotents of Markus. If /3 # 0, we have a ray solution that is equivalent to the idempotents of Markus. In either case, the ray ( a y ) is in invariant subset that is also a vector space. If once the trajectory turns towards the origin of the coordinate axes, it will stay on that straight line. For example, the reaction system
t6
A + R
where the rate equations for the open subsystem ( A , B ) are
d [ A ] / d t = -4[A]'
+
d[B]/dt =
- 3[BI2 - 6 [ A ] [ B ]
[B]' - 6 [ A ] [ B ]
has no less than three ray solutions shown in Figure 4. Ray solutions can exist only in the open subsystem of an irreversible system, since it is implied that a t time infinity, the points move to the origin. An open subsystem in a n inhomogeneous reaction system may cease to be an open subsystem after homogenizing with the loss of ray solutions. For instance, in the kinetic system in the open subsystem ( A , B )
any homogenizing scheme will introduce a n invariant, and thus bar the reaction paths from reaching the origin of the coordinate axes. A ray solution is a straight-line reaction path through the origin, but a straight-line reaction path that does not go through the origin is not a ray solution. Straight-line reaction paths are very interesting features of such nonlinear systems. Markus found that for any twocomponent second-order system, there is either a nilpotent (or equilibrium point away from zero) or a t least one idempotent (or ray solution). But from his theorem, one cannot tell whether they will occur in the positive quadrant-where a chemical reaction system should be. We offer here a theorem that can be considered as a generalization of the Perron-Frobenius theorem for positive matrices
(5). Theorem. For a chemical reaction system of any number of components and any homogeneous order, there is either an equilibrium point not identically zero or a t least one ray solution in the positive orthant of the vector space (or both). The key to this theorem is the positiveness of chemical species concentrations. A chemical reaction system must have the characteristics that if xi(0) 2 0 for all i, then x ' ( t ) 2 0 164
l&EC FUNDAMENTALS
Figure 4. Upper. lower.
Ray solutions
Three ray solutions for subsystem (A,B) Three straight-line reaction paths for system (A,B,C)
X2
'i
kx
Figure 5.
Existence of ray solution
1
ATTRACTIVE
X
I I
/
I /’
dx’/dt
2
0 whenever x i = 0
For a homogeneous second-order system, this means
b j R Z 2 0 if both] and k are different from z We can give several proofs for this theorem, centering our discussions on either the. positive orthant of the projective plane PflP1or the sphere S,-,. The projective plane P,+I consists of all rays ( a x ) in the vector space V,; in other words, a n element in P,-I is a ray. The positive orthant of the sphere consists of all points in V , satisfying the condition that x E x a == 1 and all x i
2
0
I n Figure 5 , consider arc J , which is the positive sector of the circle for a two-component system. O n each point x on J , there is a vector d x / d t defined on that point; the sum total of all the vectors d x l d t creates a vector field. If the vector field vanishes identically on a point x , this point is a n equilibrium point or a nilpotent. If the vector field does not vanish on a point of J , the direction vector dx/dt will make an angle cp with the vector x . The rate equation stretches as well as rotates a vector x , but the rotation aspect is singled out in a consideration of cp, We use the convention that the angle is positive if it is counterclockwise from line x . At the extreme right side of J , the direction vector may not point downward and create negative compositions ; thus angle cp must be negative At the extreme left side of J , the direction vector may not point to the left and creative negative compositions; thus angle p must be positive. An appeal to the intermediate value theorem ( 8 ) will convince one that somewhere along J , there is a t least one point x where angle cp is This is only an appeal to continuity. At the point zero where p is zero, the direction vector d x l d t must point toward the origin; a t this point, the rate equation causes only a stretching of vector x and does not cause a rotation. We have now proved that there is a t least one nonnegative vector, y, that satisfies the equation
P,(y) = By where may be zero Together with the lemma, we have shown that if the system
\-J
/
Figure 7.
Figure 6. Phase portrait of system with three ray solutions
for all positive values of t . This can be ensured by the relatively easy criterion (74) that the value of
/
\
Mapping
M ( * )of J into J
does not have a vector y with /3 = 0, then it must have a ray solution. Furthermore, this ray solution is “attractive” in the sense that all neighboring reaction paths have arrows pointed toward it. One anticipates that all neighboring paths will converge toward it, and make it possible to locate such ray solutions by a series of experiments. This ray solution forms the backbone of the phase portrait. There may be, of course, more than one such ray solution. Consider the phase portrait in Figure 6, where there are three ray solutions; it is evident that two of them are “attractive” and one is “repulsive.” This theorem is easily generalized to any number of dimensions and components. The most elegant proof is based on Brouwer’s fixed point theorem. For a system of any dimension n, we confine our attention to the manifold, J : which is the positive sector of the sphere S,- 1. J consists of the points of V , where xixi = 1 (unit length) and all x 1 2 0 (nonnegative). For each point x on J , one defines the mapping M ( x ) by
M(x) =
+ ( d x l d t ) 6t I + ( d x l d t ) 6t x x
~
Geometrically, this mapping joins a ray from the origin to the ( d x / d t ) 6t and extends it until it meets manifold J . point x This is illustrated in Figure 7 . For each point x on J , point M ( x ) is also on J because of the nonnegativity of the mapping of chemical reactions. Thus we have the mapping of a closed (n - 1) cell J into itself, and Brouwer‘s fixed point theorem guarantees that there exists a t least one point x so that M ( x ) = x . At this point, the direction vector dx,’dt must point toward the origin. This procedure is based on the Alexandrov-Hopf ( 7 , 2, 70) proof of the Perron-Frobenius theorem for positive matrices, and can be called the “porcupine theorem” because on the back of a porcupine, curled into a ball, a t least one quill will be perpendicular to its skin.
+
Enumeration of Homothetic Rays
Ray solutions and equilibrium points away from zero have one feature in common, the direction vector dx di at those points-has no component normal to the vector x (this statemenr is vacuously true for the equilibrium point where d x / d t is VOL. 4
NO. 2
MAY
1965
165
zero and has no component whatever). The rate equation causes no rotation to a ray solution or an equilibrium point, O n the projective plane P,-1 where the elements are rays (ax),two vectors x and y are considered the same element of P,-1 if they lie on the same ray-that is, if x = py. A transform P , will in general carry a ray into another ray. When a transformation carries a ray into itself, it is called "homothetic," since all components of the ray are magnified or reduced proportionately; the ray may be stretched, but not rotated. Each homothetic ray in P,- 1 represents either a ray of equilibrium points or a ray solution. For a homogeneous polynomial differential equation, one would like to know the total number of homothetic rays so that
induction that a two-component system of order s has any1 homothetic rays. where from 1 to rn A similar study for systems with more than two components is much more difficult. For a three-component second-order system, there appear to be at most eight homothetic rays. A homothetic vector may be of the form (0, 0, l), which implies that
+
b33'
bii' f bii2 f
For a first-order system, the results are well known. There are a t most n distinct eigenvectors in real space or one may have degenerate eigenvalues and an infinite number of eigenvectors (75). For a system of any order but with only two components, complete results are available. T h e homothetic vector may have the first coordinate missing; and one can write the vector as (0, 1) ; or the first coordinate is not missing, and one can write the vector as (1, E). If the vector (1, E) is a n eigenvector, we have
+
2612'5 2b1z2€
=
622'
Ax implies P,(ax) = asAx
dx'ldt 611' f dx2/dt = bii2
=
0
There can be only one such vector. A homothetic ray may be of the form (0, 1, E), which implies that
Since the system is homogeneous, if x is a homothetic vector, so is the entire ray (ax),since =
633'
b231
=
b33l
=
0
There are a t most three such vectors. The remaining homothetic vectors are of the form (1, (, 1) and must satisfy
dxi/dt = Ax'
Ps(x)
=
+
+
b113
f
2b12'E
f
4- 2 W t l
2 6 1 ~ ~ 5
2b12'E
2bi321
+
+ 2bdT A + f 2 6 ~ 3 ~ 5 1 A€ =
=
b22'(E)'
b2z3(02
+
2h3E9
+
b333(t)2
=
io
Eliminating A, one obtains two quadratic forms that must be simultaneously zero. The zero of each quadratic form is a
X
b22'5' b2z2E2
= XE
After elimination of A, one obtains
Q(E)
=
Q3(EI3
+ QdO' +
Qi(E)
+
I /
Qo = 0
XI
where
Figure 8.
The number of eigenvectors of the form (1, E ) corresponds to the number of roots to the algebraic equation Q([) = 0. One may have 4 3 = QZ = Q1 = Qo = 0 or P(5) = 0 for all values of 5. This represents the degenerate system where all rays are ray solutions illustrated in Figure 9. If one of the P's is not zero, one has at most three distinct real solutions; thus one has a t most three distinct ray solutions of the form (1, E ) . The vector (0, 1) is a n eigenvector if and only if 4 3 = 0. The remaining algebraic equation Q(E) = 0 is now a secondorder equation, and one may have a t most two other eigenvectors of the form (1, 6). In summary, for any two-component second-order reaction system, we may have anywhere from one to three homothetic rays in the positive quadrant (which may be ray solutions or equilibrium points). An example of this is shown in Figure 10. For a two-component system which is in the third order, this argument is repeated easily. We consider the polynomial
Q(E)
=
Qat4
+
Q3E3
+ Q z P + Qit +
Qo =
0
Quotient space of Pn-]
2A-C
Figure 9. Phase portrait of degenerate where all reaction paths are straight lines
4 2
3Cii2'
3Ci2z1
4 3
~ 2 2 2 ~
=
- 3C1ZZ2 Qo
Qi
= ciii'
B Figure 10. Phase portrait of system with one ray solution ( 1 , l ) and two equilibrium rays (1,O) and
- ~222' - 3Ciiz2
166
(0,l)
-~111~
Thus for this system, there are a t most four, and a t least one homothetic ray in the positive quadrant. One can prove by l&EC FUNDAMENTALS
system
AtB-c
where Qa
dA/dt=-A'-AB
A
~~~
systems a t a neighborhood close to the equilibrium point. One may utilize irreversible thermodynamics and linearize near the equilibrium ( 4 ) . A later paper will show that based strictly on mass action kinetics, one can arrive a t the same conclusion with the help of microscopic reversibility (76). But the Ross malaria equation (72), without the benefit of microscopic reversibility, does oscillate indefinitely. The strongest feature that we possess a t the m’oment is the straight-line reaction paths for homogeneous open subsystems, and they can be found experimentally just as in a first-order system.
Maximum Number of Homothetic Rays
Table 1.
2
2
3 4 5
3 4 5
3 (8)
4
5
6
conic section. The two conic sections can meet a t most at four distinct points, or at infinite number of points. Thus we arrive a t the total of eight. When ( 0 , 0, 1) and (0, 1, () are not homothetic vectors. the total number of homothetic vectors is not known. Thus for homogeneous open subsystems, we can summarize our meager knowledge with the table of homothetic vectors (which may be equilibrium rays or ray solutions). Table I represents the maximum number of such vectors, short of degeneracy of the sort in Figure 9 . The study of systems with more than two components and higher reaction orders involves hyperquadrics which are not well understood.
literature Cited
(1) Alexandrov, P., “Combinatorial Topology,” Vol. 3, p. 119, Graylock, 1960. (2) Alexandrov, P.. Hopf, H., “Topologie,” Band I, p. 480, Springer, Berlin, 1935. (3) Ark, K.. IND.END.CHEM.FUNDAMENTALS 3, 28 (1964). (4) Bak, T., “Contributions to the Theory of Chemical Kinetics,” Munksgaard, Copenhagen, 1959. (5) Bellman, R., “Introduction to Matrix Analysis,” p. 278, McGraw-Hill, New York, 1960. (6) Birkhoff, G., MacLane, S., “Survey of Modern Algebra,” p. 282, Macmillan, New York, 1957. (7) Bourbaki, N., “Elkments de MathCmatique,” Livre 11. Chap. 2, “Algebre LinCaire,” Chap. 3, “Algtbre MultilinCaire,” Hermann, Paris, 1958. (8) Courant, R., “Differential and Integral Calculus,” Vol. 1, p. 66, Interscience, New York, 1947. (9) Eisenhart, L. P., “Continuous Groups of Transformations,” Dover, New York, 1961. (10) Fan, K., Monats. h‘fath. 62, 219-37 (1958). (11) Jacobson, N., “Lectures in Abstract Algebra,” Vol. 2, p. 225, Van Nostrand, New York, 1953. (12) Lotka. A , , “Elements of Mathematical Biology,” Dover, New York, 1956. (13) Markus, L., “Contributions to the Theory of Nonlinear Oscillations,” Vol. 5 , p. 185, Princeton University Press, Princeton, N. J., 1960. (14) Wei, J., J . Chern. Phys. 36, 1578 (1962). 115) \Vei. J.. Prater. C. D.. Aduan. Catalvsis 13. 203-392 (1962). (l6j FYei, J.; Zahne;, J. C.’, to be published.
Conclusions
The theory of complex, polynomial order kinetic systems is in its infancy, and one cannot foresee what theorems one may expect at its maturity. It is probable that it will match the first-order theory in elegance, if not in completeness. Many important problems remain unsolved. For instance, the principle of microscopic reversibility plays a central role in the first-order theory, but has not yet appeared in polynomial theory. I t is not clear how this principle can be stated in terms of multilinear algebra alone. Oscillatory reaction paths are impossible in first-order systems, and they are equally impossible for polynomial
RECEIVED for review April 11. 1964 ACCEPTED September 30, 1964
NUCLEATE BOILING OF HYDROGEN Comparison between Experimental and Predicted Data D
.
E
.
D RA Y ER
,
.Vational Bureau of Standards, Boulder, Colo.
Heat flux values predicted by 1 1 nucleate boiling correlations were compared with experimental data for boiling liquid hydrogen. Three correlations-Forster-Zuber, Forster-Greif, and Cryder-Gilliland-predict heat flux values in good agreement with the experimental data at temperature differences in the neighborfair. The correlations of hood of 1 ’ F. Three other correlations-Hughmark, Gilmour, and McNelly-were Levy, Jakob-Linke, Insinger-Bliss, Miyauchi-Yagi, and Nishikawa were poor. VER
the past several years, numerous empirical and
0 theoretical equations for correlating nucleate boiling heat
transfer data have been developed. These equations, which are based on fluid properties, degrees of superheat, vapor pressure differences, surface characteristics, heat flux, etc., have been only partially successful in general application. T o test the validity of their various correlations, many of the originators compared predicted results with experimental I
Present address, Marathon Oil Co.: Littleton, Colo.
data for the boiling of substances which are liquids under conditions of atmospheric temperature and pressure. These correlations, in many instances, were derived through the use of dimensional analyses, with the exponents of the dimensionless groups being assigned by matching with experimental data. I n many instances where the relative importance of a certain variable is determined through the analyses of various boiling systems, the variation of this physical property in the systems analyzed is small. As a result, the exact influence of this variable may be unknown. This author believes that, in order to VOL. 4
NO. 2
MAY 1965
167
