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Dual diffusion—reaction coupling in first order multicomponent systems
Chemical Engineering Science, 1965, Vol. 20, pp. 941-951. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Dual diffusion-reaction
coupling in fust order multicomponent systems H. L. TOOR
Carnegie Institute of Technology, Pittsburgh, Pennsylvania (Received 11 February 1965; in revised form 15 March 1965) Abstract-The problem of unsteady, unidirectional diffusion with complex first order reversible chemical reactions in an isothermal, isobaric multicomponent system of constant molar or mass density is investigated. In this system components are coupled by chemical reactions as well as by diffusion and in general the equations cannot be. uncoupled by a transformation of concentration coordinates. Solutions are obtained by a separation of variable matrices technique from which it is found that an ideal system (e.g., a mixture of perfect gases) near equilibrium is stable. If all of the reactants are infinitely dilute, or Knudsen diffusion takes place the system is always stable. A multicomponent solution represents one of the possible matrix generalizations of the solution to the analogous binary problem with a single irreversible reaction. -
WHEN chemical reactions take place in non-homogeneous mixtures of more than two components the possibility of simultaneous reaction and diffusion coupling exists. The former coupling arises through the reaction mechanism and depends upon the local concentrations, while the latter coupling arises because of differences in properties between the various molecular species and depends upon the local concentration gradients. Systems of this sort have not been studied to any great extent although they may be encountered in chemical reactors, biological systems and in combustion processes. Solutions to the linearized equations of multicomponent mass transfer may be obtained for many classes of problems as matrix generalizations of binary solutions [l]. This is frequently not true when there are chemical reactions as well as diffusion so solutions are not necessarily just linear combinations of binary solutions. Hence dual coupling may cause unusual behavior. In this paper the simplest class of unsteady problems of this type will be considered-the isothermal, isobaric, unidirectional, diffusion-reaction problem with no external fields in which all II + 1 components are coupled to each other by diffusion and reversible first order chemical reactions; with the further stipulation that the concentration changes are small and the molar or mass density is constant. This is the simplest system with dual coupling and an understanding of it is an
essential preliminary to an analysis of the more general problem with higher order reactions, etc. Although truly first order systems with no inerts present are likely to show little diffusion coupling because of the similarity among reacting species, the presence of inerts can cause significant diffusion coupling. Futhermore, near equilibrium one may treat higher order reactions as approximately first order. Although there have been many studies of diffusion-reaction systems these are almost all limited to systems with negligible diffusion coupling, usually dilute reactants in a solvent [2, 3, 41. DEGROOT and MAZUR [5] have considered the stationary state with dual coupling but the only study of dual coupling in the unsteady state appears to have been done by ALBRIGHT [6] who solved a transient problem for a ternary system with a single reversible first order reaction. FORMULATION The equations for the concentrations in an n + 1 component mixture at constant temperature and pressure with no external forces acting may be written in matrix notation as [I]. y
where
941
+V(.V(C))
=V.{[D]V(C)}
+(r)
V is taken here as the volume
(1)
average
H. L. TOOR velocity of the mixture. Only n concentrations, 12 diffusion fluxes and n production rates are independent so equation (1) may be taken to represent either n + 1 equations with n of them independent of just the n independent equations. In the former case (C) and (I) are column vectors of order (n -t- 1, 1) and [D] is a matrix of order (n + 1, n + 1) while in the later case (C) and (r) are order (n, 1) and [D] is order (n, n). The formal solutions to be developed later hold for both formulations. If the later formulation is used equation (1) is then a combination of equations (1) and (2) of reference [l] (with the reference velocity specified in this case) and [D] is the diffusivity matrix in the volume reference frame. Its elements are concentration dependent but in a thermodynamically stable system it has real, positive characteristic roots at any composition [7]. It is rarely symmetric and is diagonal only in mixtures of like species or in dilute systems [g, 91. Concentration changes are assumed to be small enough so that [D] may be taken as constant at some average concentration. This appears to be quite a good approximation over reasonably large concentration variations [lo, 11, 121. If there is no volume change on mixing the continuity equations lead to the equation.
The condition V. V = 0 will be used later so it is assumed at the outset that this is valid. It is seen that this implies constant and equal partial molal volumes of the reactants since for the reactions consideredxr, = 0. It is assumed further that the
partial molal volumes of any inerts present are equal to the partial molal volumes of the reactants. Thus the molar density of the mixture is assumed constant with or without inerts. This makes the volume reference frame equal to the molar frame. This is a good assumption in most gas mixtures at low and moderate pressures and in some liquid mixtures. With the above assumptions equation (1) becomes
a(Y) ae +
V.V(y)
=
[D]V(y> +
(2
(3)
where (y) is the vector of mole fractions. THE REACTION MATRIX
The reactions are assumed to be reversible monomolecular first order. The reactions are like those treated by WEI and PRATER,[13] but here non-reactants (inerts) can couple with the reaction through the cross diffusion coefficients, D,, i # j. These inerts introduce into the reaction matrix columns and rows of zeros. There are other differences between the reaction matrix used here and that used by WEI and PRATERand these are best illustrated by an example. Consider three reacting components and two inerts n ,A 2 -
where I;ij is the rate constant for the reaction j to i. Then if (r’) is the production vector inn + 1 concentration space, and (_v’) the concentration vector in this space,
(4)
942
Dual diffusion-reaction
coupling in tirst order multicomponent systems
where [K] is the WEI and PRATER reaction matrix [13]. At chemical equilibrium there is no reaction so if (y:) is any equilibrium concentration vector in n + 1 space,
It is noted that equation (9) can be written as= -PI(y) + Cklbe>
c
(W
When no inerts are present the principle of detailed balancing gives c Note that because species 4 and 5 do not enter into the reaction, y4= and yse may take on any (suitable) values, but yle, yze and y3e are in chemical equilibrium and hence bear a fixed ratio to each other. Adding equations (4) and (5)
(2 = - [k’](y’ C
- y’)e = _
Ck’](-J)
Ckl(y3= -
h+1
&n+l ( Enn+l 1
(12)
and WEI and PRATER [13] shows that there is a unique equilibrium composition in this case. Hence [k] is non-singular and
(6)
fl.+1
This is the desired form for use in n + 1 dimensional space. Since
(~3 = [k]-’
.2”+1
(13)
i L+1 1
If inerts are present equations (4) and (5) lead to
Ckl(y3= 0
[k] is obviously now singular and the equilibrium composition is not unique since the inert mole fractions are not fixed. It is easily shown that the characteristic roots of [k] are minus the roots of [Kj plus q - 1 zero roots. Hence [k] always has real non-negative roots since [K] has only real non-positive roots [13]. Substituting equation (9) into (3) and taking advantage of the fact that (~3 is a constant vector,
Equation (6) may be rewritten with independent quantities only,
(I) -=-
c
0 0
[’‘I -K
o
0
0
0
(4 = - Ckl(x> (9)
0
This is the desired form for use in n dimensional space. [k] is obtained from [k’] by deleting the last row and column and subtracting the last column from the earlier columns. Thus if there is only one inert [k] = -[K] while if there are no inerts the order is reduced by one. For the earlier example, if there are no inerts.
In general it is seen that if there are q inerts, 4 > 0, &I = - [K] with q - 1 rows and columns of zeros aflixed, while if q = 0, [k] is obtained from [K’j by deleting the last row and column of [K], subtracting the last column from each of the others and affixing a minus sign to the entire matrix.
(14)
d(x) ae + I’. V(X) = [D]V’(x) - [k](x)
(15)
This is the starting equation which will be used for the analysis of the first order diffusion-reaction problem. The same form is obtained if the equations are formulated in n + 1 concentration space but unless noted otherwise it is assumed below that the equations are written in n space. Primes will be used when it is necessary to refer to n + 1 space explicitly. Equation (15) and the earlier equations also hold for a system of constant mass density if mole fractions are replaced by mass fractions and other molar quantities are replaced by appropriate mass quantities. Hence solutions obtained subsequently for constant molar density are also valid 943
H. L. TOOR
is not singular equation (15) becomes V(X) - [O] -‘[/C](x) = 0
for constant mass density when the above changes are made.
(18)
and the equation may be uncoupled by diagonalizing the matrix [D]-‘[k]. WEI [16] has analyzed problems of this sort when [D] is diagonal. Straightforward solutions may also be obtained when V is a constant if the problem is unidirectional.
DLUONABLEPROBLEMS Solutions to equation (15) are obtained simply as linear combinations of binary solutions when the equation can be put in diagonal form. Solutions of special cases of the equations have been obtained in this manner; they are summarized briefly below.
[D] and [k] commute
It has been seen that when diffusion takes place alone the equations can be uncoupled, hence certain linear combinations of components diffuse No d@ision If there are no concentration gradients in the independently of each other and when reaction takes place alone other linear combinations react system equation (15) becomes independently of each other. Only when both linear combinations are the same are there linear do = -[k](x) (16) combinations in the diffusion-reaction system d0 which diffuse and react independently of each other. The equations can be uncoupled [13] by diag- The linear combinations are the same only when onalizing [k] and are equivalent to a set of first [D] and [k] have the same characteristic vectors order irreversible reactions with rate constants which is equivalent to saying that they can be equal to the characteristic roots of [k]. The solution diagonalized by the same similarity transform. is formally [D] and [k] may be diagonalized by the same (x) = e-tkl’(XO) (17) similarity transform when they commute and equation (15) may then be reduced to a set of uncoupled where (x,,) is the initial value of(x). This problem diffusion-reaction problems, each of which correswith no inerts has been thoroughly analyzed by ponds to a binary system with a single irreversible WEI and PRATER[13]. Since the roots are real and reaction or with none. This system is clearly stable non-negative the system is stable. Since the since the diffusion coefficients and reaction velocity inert concentrations are unaffected by the reaction constants which appear are the roots of [D] and under these conditions the inerts may be ignored. [k] and hence are real non-negative numbers. If neither [D] nor [k] is a scalar times the identity No reactions matrix the commuting of [D] and [k] represents an If there are no reactions equation (15) reduces exceptional case which does not appear to have to the form treated by KIRKALDY [7, 14, 151 much physical significance. Also, since the case in (for V = 0), TOOR, [I, 121 and STEWART and which [k] is a scalar times the identity matrix has PROBER [Ill. The equations are uncoupled by physical significance only in a binary system, the diagonalizing [D] so the problem is equivalent to a only unexceptional case of interest in which [D] set of binary diffusion problems with diffusivities and [k] commute is that system in which [D] is a equal to the characteristic roots of [D]. Since these scalar times the identity matrix.? But this is really are real positive numbers in a thermodynamically stable system [7] this system is also stable. The t This represents the general system of like components stability of these two limiting cases does not ensure with no diffusional coupling [8,9]. The only uncoupled svstems which do not fall in this class are those in which all the stability of the general case. Steady
state
Another problem which can be uncoupled is the steady state problem with V = 0 [5, 161. Since [D]I
but one of the components are infinitely dilute in ordinary diffusion or those in which Knudsen diffusion takes place. In these systems [D] is diagonal, but the diagonal elements need not be equal. Then [D] and [/cl do not in general commute and the more general methods developed in the next section are required.
944
Dual diffusion-reaction couplingin first order multicomponentsystems a singly coupled system and PAO [17] has already developed a generalization Of DANCKWERT’S method [3] which yields solutions in terms of the solution to the problem of diffusion without reaction. Since the procedure outlined above could do no better than this it will not be pursued further.
has a unique matrix generalization which is the correct solution to equation (19) while if [D] and [k] do not commute there is no unique matrix generalization. To solve equation (19) let (x) = e-w@) So (19) becomes
THE NON-DIAGONABLE DUAL PROBLEM In this section a method is developed for solving the non-diagonable diffusion-reaction problem when V is zero and the diffusion is unidirectional. The slab is treated explicitly, but spheres and cylinders follow by the same method. Consider a quiescent it + 1 component medium extending from z = - a to z = + a and infinite in the other two directions. The initial composition is given by (y (z, 0)), it may or may not be an equilibrium composition. At time zero the concentrations at the boundaries z = - a and z = + a are changed to some equilibrium value (y,). (Obviously this problem is trivial if the initial composition is the equilibrium composition when there are no inerts present for then there is only one equilibrium composition. The case in which the boundries are not at equilibrium will be treated later in the paper.) V is assumed to be held at zero at one boundary (or at the centreline which can be a solid wall). Since V.V = 0 from equation (2) and the earlier assumptions, and the problem is one dimensional, this implies V = 0 everywhere. With the above conditions equation (15) reduces to
$ =[M(e)]
(20)
z
(21)
with [M(O)] = ec’le[D]e-we
(2la)
and from (19a) and (19b), (u) satisfies 8 =o,
(u) = (x(z, 0))
(2lb)
z=+a (v)=O (2lc) Assume the variables may be separated as follows (v) =
cWI
E&m%)
(22)
with [T(6)] and [Z(z)] variable matrices of order (n, n) and (Q) a column of constants. Substituting equation (22) into (21), collecting terms and cancelling (Q) gives
cn%wl
431 dZCzl - de = dz2 [Z]-’ = -[A]
(23)
and since the left side depends only upon 8 and the right upon z, they have both been set equal to the constant matrix - [A]. The equations to be solved then are
d2IZI
--j-g- + c4cz1 = 0 z=*u,
[Z] = 0
(244
and and since (x) is (y - y,), (yJ is for the time being taken as the equilibrium composition at z = If: a. Hence the boundary conditions may be written as 0 =o,
(x) = (~(z, 0) - ~3 = (x(z, 0)) z=+u,
(x) = 0
(19a) (19b)
Since it is now assumed that [D] and [k] do not commute the above equations cannot be uncoupled. It is interesting to note, however, that if [O] and [k] do commute the solution to the scalar analog of equation (19) (this is the binary diffusion-reaction problem with a single irreversible reaction [18])
43 7
+ C~UOICTlE4 =0
(25)
and of course (21b) must be satisfied. Equation (24a) comes from (21~) and (22). The solutions to equation (24) which satisfy (24a) are
ca = 4/&NcJ~
p = 0, 1,2 . . . (26)
with
945
(26a)
H. L. TOOR
and the [I’,] are constant matrices of order (n, n). Hence the characteristic matrices of equation (25) are just constants times the identity matrix. Since [A] takes on the values, &I, A, I. .., Equation (25) becomes
42’1 -=
-A P eckl”[D]e-Ckle[T]
df?
(27)
spheres. The characteristic functions and characteristic values of the boundary value problem, cos &z and JA,, respectively, are merely replaced by the forms corresponding to the geometry at hand. When the initial concentrations are independent of z equation (33) takes on the, simple form (Y - Y,) = Pzojp cosIJ&zIe-cLple(y~
where equation (21a) has been used. To solve equation (27) let (28)
4( - 1)” fP = (2p + 1)7L
which reduces the equation to (29) with
(294
[LPI = A,Pl + WI
Equation (29) has the well known solution [19] [U] = e-ctP1e[U P]
(30)
[T] = eCklee-CL~le[Up]
(31)
Combining equations (20), (22), (26) and (31) and collecting constants, (x,) = cos{~z}e-~Lple(b$
(32)
where (b,) is a column vector of IZconstants. Since each (x,> solves equation (19), one can write as the complete solution, (x) = pzO cos{J~>e-rLple(bp)
(33)
and from equation (19a) (x(z, 0)) = F cos&&IzI(b,)
xi = T cos{~‘Ap~}e-~A~Dia+kia)e~pi (37) p=o
One can write the exponential in any of the three ways, e-ApD,0Be-kio.9 e - (ApD,O + kio)B ,
which is just a set of scalar equations for the bpi which can be treated by usual Fourier analysis to give dz’
(35)
Direct substitution of equation (33) into equation (19) with the (b,,) given by (35)coni%ms the solution. Similar solutions are obtained for cylinders and
in equation (37) ,
e
-k,%9e-ApD$9
Hence when [D] and [k] do not commute replacing I$’ by [II] and ky by [k] in the above expressions (and xi and bpi by (x) and (b,), respectively) gives three possible generalizations. The first is the correct one ! This suggests a procedure for generalizing various binary solutions. As an example consider the semiinfinite slab with, for simplicity, uniform initial concentrations-Equation (19) with (19a) and (19b) replaced by,
(34)
p=o
(364
where (yo) is the initial concentration vector. The nature of the roots of the [LPI, which determine the behavior predicted by equations (33) and (36), will be considered later. First consider the scalar solution analogous to equation (33), the solution to the binary problem with an irreversible reaction, [18]
with the [Up] constant matrices of order (n,n). Substituting into equation (28) and solving for [T],
-a
(36)
with
[U] = e-Ckle[T]
(b,) = ; I= (x(z’, O))cos{&-z’}
- Y,)
8 =o,
(x) = (x0)
(3ga)
z= 0,
(x) = 0
(3gb)
Taking the scalar solution and using the matrix generalization of the first exponential form considered earlier gives (x) = flom
F
,-k%l+IW
de@,)
and direct substitution confirms this solution. 946
(39)
Dual diffusion-reaction
coupling in first order multicomponent
and if (yl) is some particular equilibrium vector, (y,), then again (y(z)) is equal to this particular (yJ everywhere since the entire matrix on the right times (ye) is (yJ. Hence when (yJ is not unique it may be set equal to zero in equation (43). The result that the steady state concentration varies with z if the boundaries are not at equilibrium and is everywhere equal to the boundary concentration when the boundaries are at a constant equilibrium concentration is clearly valid irrespective of the type of reaction or the geometry. The specific solution for the slab with equal and constant boundary concentrations is obtained from equations (42) and (43). The result is
Returning to the finite problem theresultis now generalized to include boundaries not at chemical equilibrium. Thus Equation (19) with (19a) and the boundary conditions, z = a, uw (x> = (Y, - Ye) z=-a,
(4
= (Y-0 - YeI
is to be solved with (y,) and (y_,) constant.
(194
Let
(x(z, 0)) = (Q, 0)) + (x(z)) (40) where (x(z)) is the steady state solution which satisfies (19c) and (19d). (R) then satisfies equations (19) and (19b) and the condition e =o,
(2) = (x(z, 0) - x(z))
(Y(z, 0)) = (j(~, 0)) +
(41) -
cosh[B]z[cosh[B]a]-’
systems
pgpcos{&;z)B-[~“~ ['+11-J
(yr-
ye)
P (424
The solution follows much as before and the result is (y(z,
ej -
y(~)) =
where (j(z,O)) is the concentration when (yr) = (~3 as given by Equation (33). If there are no inerts
2 COS{~~~ z}~-~~J~ i [”(~(40) -
where (y(z)) is the steady state concentration profile. As before this is one of the possible generalizations of the solution to the equivalent binary problem. Equation (42) is in accord with equation (36) only if (y(z)) = (y,) when the boundaries are at equilibrium. This is confirmed by the steady state solution. For example, when (y,,) = (y_,) = (yr) standard methods give the steady state solution (y(z) - Y,) = cosh(C~lz}Ccosh(C~la}l-‘(y~ - Y,) (43) where [B-J = [[D]-‘[k]]i’2 (43a) As noted earlier the steady state problem is diagonable. Equation (43) is the matrix generalization of the scalar problem. When no inerts are present (ye) is the unique equilibrium composition vector and when (yr) = (ye), (y(z)) = (y,) everywhere as required. When inerts are present expansion of the matrix functions in equation (43) and use of equation (14) reduces equation (43) to (Y(Z)) = cosh{CBlz)Ccosh{CBla}l-’ (A
Y(z'))
c-J& z’ dz’
UJ -a
p=o
(42)
(y,) is given by equation (13) while use of equation (44) in place of (43) shows that when inerts are present (y,) may be set equal to zero in the above equation. The above solution may also be used to construct solutions for time dependent boundary concentrations by use of the method of superposition, while the solution for constant but unequal boundary concentrations is obtained simply by modifying equation (43). Explicit forms for particular cases are obtained by combining equation (48) below with the above solutions. TOTAL RATE OF REACTION AND RATE OF DIFFUSION
The total rate of reaction in the slab per unit volume is given by (R) = - &I
j”,(y(z.
0) - Y,) dz
(45)
Substitution of equation (42) into this equation and integrating gives (Rj = (R3 - ‘+ [k] pzo El
(44 947
e-tLple(hp)
(46)
H. L. TOOR
where (RJ is the total rate of reaction per unit volume in the steady state and (h,) is just the normalized integral on the right of equation (42). The diffusion rates in the z direction with respect to a fixed coordinate may be obtained readily from the concentration profiles and the relation [I] (47) DISCUSSION
The nature of the solutions obtained depend upon the nature ofthe matrix exponentials which appear in all the transient solutions. Assuming the usual case of no repeated roots SYLVESTER’S theorem [19, 201 gives emcLpl’ = j~le-LpJe[Zo(Lpj)]
cz
(L
0
.),
P,
=kg(Lpkz - [Lpl) kgcLPk - Lpj)
(484
Repeated roots may be handled readily with the confluent form of this theorem [20] but show no essential change in behavior so far as the present discussion is concerned. The behavior of equation (48) and hence equation (42), etc. clearly depends upon the nature of the characteristic roots of [LPI, Lpj. If these are real and positive the concentration at any point decays to the steady state value as time increases, but since each matrix exponential is a linear combination of n scalar exponentials each term in equation (42), etc. may have n - 1 extreme values. Even though [L,] is simply related to [D] and [k] through equation (29a), the fact that these matrices have real positive roots does not ensure that [L,] will have real positive roots except for the exceptional circumstances in which [D] and [k] commute or in which both are symmetric. It is shown in the Appendix that the second law of thermodynamics and the principles of detailed balancing of chemical reactions and of microscopic reversibility lead to the conclusion that in a thermodynamically ideal system near equilibrium all the [L,] have real positive roots. A stronger statement is not available even though the above two principles separately can be used to make much stronger statements about [k] and [D]. Hence it is not yet clear whether or not complex or negative roots, which could indicate instability, can occur.
It is important to note, however, that the assumptions made in the derivation of equation (19), small displacement from equilibrium and constant molar density, imply a thermodynamically ideal system near equilibrium (e.g., a mixture of perfect gases near equilibrium); hence the nature of the roots in non-ideal systems or in systems with a large displacement from equilibrium may not necessarily fully determine the stability as in the case of ideal systems near equilibrium. Nevertheless it is shown in the Appendix that there are circumstances under which the roots of [L,] are found to be real and positive without explicitly using the above restrictions. The one most likely to be encountered is the situation in which [D] is diagonal. This corresponds to the system in which n dilute components are diffusing and reacting in a solvent [8] or to the case of Knudsen diffusion with reaction. As discussed earlier, when all n components are of a similar nature [D] is a scalar times the identity matrix and this is really a singly coupled system. However when the components in the dilute system are not similar the elements of the diagonal diffusivity matrix are not all equal and even though there is no diffusion coupling the system can be considered to exhibit a form of dual coupling,in the sense that [D] and [k] do not commute. It is seen, however, that with this pseudo dual coupling the [LPI always have real positive roots. But if one takes the viewpoint that a dilute system is equivalent to an ideal one near equilibrium then one is back to the previous restrictions. Equation (29a) can be written in the form
7 +[k]
CL,1= (P + tYn2
(49)
So if a is small enough the first term in equation(49) will dominate for all values of p. The diffusion is then rapid with respect to the reaction and stability is ensured for [D] has real positive roots. If n is large the converse is not true since there will always be some Fourier components with a large enoughp so that the two terms in equation (49) will be of similar magnitude, even though the coefficients of these terms (e.g.,f, in equations (36)) will be small. 948
Dual diffusion-reaction
coupling in first order multicomponent
WEI [22] has shown that the reaction velocity
matrix in the general linearized mass action system has the same structure as the reaction velocity matrix in first order systems. This indicates that the above results also apply to any mass action system near equilibrium. APPENDIX-NATURE OF THE RENTS OF Lp The proof is carried out most readily in iz + 1 space. Although the former solutions are valid in 12+ 1 space, the use of dependent quantities adds a certain arbitrariness to the matrices [5, 211. However, if it can be shown that one of the possible constructions of the [Lb] have real positive roots for all values of p then stability is clearly ensured since all formulations are equivalent. WEI and PRATER [13] show from the principle of detailed balancing of reversible chemical reactions that one may write [K,=
-@]/y.i
Yr2._.
1
1’ Yenfl-_q
II.o. 3 :
[k’] =
0.. 0... .... .... 0...
.. .. .. ..
I
CD’1 = C~‘lrdl
[D’] may
(A4)
where [Y] is a matrix of coefficients and the elements of [$I are a~,/LYC,. The n + 1 fluxes and chemical potentials are linearly related and HOOYMAN and DEGROOT[21] show that this leads to an arbitrariness in [Y] and hence in [D’]. However, they show that when the Onsager reciprocal relations hold for properly defined fluxes and forces symmetric [_%“Imatrices may always be constructed. It can be shown that within their most general symmetric form one can find [U’] matrices which are positive definite by use of the second law requirement that entropy production must be positive. When the system is ideal and the molar density is constant ~~‘1 = 7
n/y’,
(A5)
ri/y:l
(A@
and near equilibrium
I
1
~~~‘1= y
Combining Equations (29a) (written space), (A3), (A4) and (A6), [I&]
-1
(A21
since the inert mole fractions which have been added do not affect the reaction matrix. Using primes as usual to represent n + 1 space the above equation may be rewritten as Ck’l = CS’lrl/~h
Following irreversible thermodynamics be written as [S]
(Al)
where [ - s] is a symmetric matrix with real nonnegative roots-a positive semi-definite matrix. The above matrices are of order (n + l-q, n + l-q) where n + l-q is the number of reacting species. If there are no inerts [k’] = - [K] while if inerts are present one may write by virtue of equations (4) and (Al) 0.
systems
(A3)
where [S’] is first matrix on the right of equation (A2) and Q/y:, is.the second. The former is positive semi-definite and the latter positive definite.
=
pp!gL-v’1 + Cs’l] rUy:,
in n + 1
(A7)
Since the A, are positive, [Y] is positive definite and [S] positive semi-definite, the first matrix must be positive definite. Hence each [L;] is the product of two positive definite matrices and by a proof given by RIRKALDY[7] each [L;] has real positive characteristic roots. Thus if a solution is written in the redundant IZ+ 1 space the exponents which appear when e-tLple is expanded by Sylvester’s theorem are all negative and the system is stable. The solution must also be stable when written in n space so the n independent characteristic roots for each value of p must also be real and positive. It is noted that there are exceptional cases in which it is easy to prove that the [L,] have real positive roots without the restrictions used above.
949
_.
I
n. L .TOOR
When [D] and [k] commute and hence may be diagonalized by the same similarity transform it follows by diagonalizing equation (29a) that Lpi
= A,Di + ki
(A@
and since A,, Di and ki are real and positive, the Lpi are also. But this case was shown earlier to be stable. Again if [D] and [k] are symmetric they are positive definite, so [L,] is positive definite, hence has real positive roots. Finally equations (A3) and (29a) (written in n space) can be put down in the form,
CL,1= [A,CDYY,I+ C~ll’lI~e~
(A%
If there is at least one inert [S] is positive semidefinite and if [D] is diagonal [D] rye1 is positive definite. Hence by earlier arguments [L,] has real positive roots. This corresponds to n dilute reactants in a solvent.
14 Matrix of constants AZ, Constant defined by equation (26a)
4 D{ DP Di3
ID1 fP c.i: kr” . kc3 krr El
rp; 4 &?I (Rd
ra (r) rs1 WI 1; IV1 WPI v
fiii (0) (00) Xt
Characteristic roots of [LPI Matrix defined by equation (29a) Matrix defined by equation (A4) Matrix detined by eauation (21a) One less than the number of components Matrix of constants Number of inerts Gas constant Column vector whose ith element is the total rate of production of species i by reaction per unit volume Steady state value of(R) Rate of production of species i by reaction per unit volume Column vector whose elements are rg Matrix defined by equation (Al) Matrix defined by equation (A3) Absolute temperature Matrix defined bv eauation (22) Matrix defined by equation (28j Matrix of constants Vector velocity Partial molal volume of species i Column vector defined by equation (20) Column vector of constants YVYiC
Column vector whose elements are xr Column vector defined bv eauation (40) . . Steady state value of(x)_ Mole fraction of species i (Z Column vector whose elements are yr rye1 Diagonal matrix with elements yl, ‘l/Y1 Diagonal matrix with elements l/yr ‘l/Ye{ Diagonal matrix with elements l/vie ._ (Y(Z)) Steady state value of(y) Distance : Variable of integration [Zl Matrix defined by equation (22) Greek Letters Variable of integration f Time Clr Chemical potential of species f [$I Matrix whose elements are +t/aC3 Subscripts a z=a -a z=-a e Equilibrium value I Boundaries i, j Species i, j p Index running from 0 to co Superscripts ’ 12-t 1 space :; (x(z))
NOTATION
I; b,t (bp) c
lkl Reaction velocity matrix LPl LPI WI [Ml
Half width of slab Matrix defined by equation (43a) Constant Column vector of constants Molar density Moles i per unit volume Column vector whose elements are Cr Characteristic root of p] Binary diffusion coefficient Multicomponent diffusion coefficients Matrix whose elements are Dt3 Constant defined by equation (36a) Identity matrix Column vector whose elements are fluxes Reaction velocity constant for reaction in binary system Reaction velocity constant for reaction of species jtoi 4 element of matrix [k] WEI and PaATER reaction velocity matrix
REFERENCES IOOR H. L., Amer. Inst. Chem. Engng. J. 1964 10,460. CRANK J., The Mathematics of Difision, Oxford University Press, Oxford, 1956. 1DANCKWERT~ P. V., Trans. Farad. Sot. 1951 47, 1014. PRAGER STEPHEN,Chem. Engng. Progr. Symp. Ser. No. 25, 1959 55, p. 11. DEGROOT S.R. and MAZUR P., Non-Equilibrium Thermodynamics, Interscience, New York, 1962. ~LBRIGHTJ. G., J. Phys. Chem. 1963 67,2628. KIRKALDYJ. S., Canad. J. Phys. 1963 41,2166.
950
Dual diffusion-reaction
coupling in first order multicomponent
[8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] 1191 [20]
systems
TOOR H. L. and ARNOLDK. R., Industr. Engng. Chem. Fund. 1965 4, 363. Toot H. L., SESHADRIC. V. and ARNOLDK. R., Amer. Inst. Chem. Engng. J., to be published. ARNOLDK. R., Ph.D. Thesis, Carnegie Institute of Technology, Pittsburgh, Pa. 1965. STEWARTW. E. and PROBERRICHARD, Zndustr. Engng. Chem. Fund. 1964 3,224. TOOR H. L., Amer. Inst. Chem. Engng. J., 1964 10,448. WEI Jm and PRATERC. D., Advances in Catalysis, 1962 13, p. 203. KrRKALDYJ. S., Canad. J. Phys., 1957 35,435. KIRKALDYJ. S., Canad. J. Phys. 1959 37, 30. WEI, JAMES,J. Catal. 1962 1,526. PAO YIH-Ho, Chem. Engng. Ski., 1964 19,694. YOST W., Diffusion in Solids, Liquids and Gases, Academic Press, New York, 1960. PIP= L. A., Matrix Methods&w Engineers, Prentice Hall, Englewood Cliffs, New Jersey, 1963. FRAZER R. A., DUNCAN W. J. and COLLAR A. R., Elementary Matrices, Cambridge University England 1952. [21] H~~YMAN G. J. and DEGR~~T S. R., Physica, 1955 21,73. [22] WEI JAMES,Socony Mobil Oil Co., Inc., Princeton N.J., private communication, 1965.
Press, Cambridge,
R&.mn~L’article presente une etude de la diffusion non-stationnaire et uni-directionnelle avec reaction chimique complexe du ler ordre et reversible dans un systeme isobare et a multiples composants de densite molaire ou massique constante. Dans ce systeme les composants sont couples par des reactions chimiques et par la diffusion et on ne peut generalement pas Biminer le couplage par une transformation des coordonnees de concentration. Des solutions obtenues ii l’aide dune technique de separation de matrices variables il apparait qu’un systeme ideal (p.e. un melange de gaz ideaux) proche de l’equilibre est stable. Lorsque tous les composants, a l’exception dun seul, sont dilues a l’infini, ou lorsque la diffusion est du type Knudsen, le systeme est toujours stable. Une solution du cas a multiples composants represente tgalement une des generalisations matricielles possibles de la solution du probleme analogue avec une reaction irreversible seulement.
951
Dual diffusion-reaction
coupling in fust order multicomponent systems H. L. TOOR
Carnegie Institute of Technology, Pittsburgh, Pennsylvania (Received 11 February 1965; in revised form 15 March 1965) Abstract-The problem of unsteady, unidirectional diffusion with complex first order reversible chemical reactions in an isothermal, isobaric multicomponent system of constant molar or mass density is investigated. In this system components are coupled by chemical reactions as well as by diffusion and in general the equations cannot be. uncoupled by a transformation of concentration coordinates. Solutions are obtained by a separation of variable matrices technique from which it is found that an ideal system (e.g., a mixture of perfect gases) near equilibrium is stable. If all of the reactants are infinitely dilute, or Knudsen diffusion takes place the system is always stable. A multicomponent solution represents one of the possible matrix generalizations of the solution to the analogous binary problem with a single irreversible reaction. -
WHEN chemical reactions take place in non-homogeneous mixtures of more than two components the possibility of simultaneous reaction and diffusion coupling exists. The former coupling arises through the reaction mechanism and depends upon the local concentrations, while the latter coupling arises because of differences in properties between the various molecular species and depends upon the local concentration gradients. Systems of this sort have not been studied to any great extent although they may be encountered in chemical reactors, biological systems and in combustion processes. Solutions to the linearized equations of multicomponent mass transfer may be obtained for many classes of problems as matrix generalizations of binary solutions [l]. This is frequently not true when there are chemical reactions as well as diffusion so solutions are not necessarily just linear combinations of binary solutions. Hence dual coupling may cause unusual behavior. In this paper the simplest class of unsteady problems of this type will be considered-the isothermal, isobaric, unidirectional, diffusion-reaction problem with no external fields in which all II + 1 components are coupled to each other by diffusion and reversible first order chemical reactions; with the further stipulation that the concentration changes are small and the molar or mass density is constant. This is the simplest system with dual coupling and an understanding of it is an
essential preliminary to an analysis of the more general problem with higher order reactions, etc. Although truly first order systems with no inerts present are likely to show little diffusion coupling because of the similarity among reacting species, the presence of inerts can cause significant diffusion coupling. Futhermore, near equilibrium one may treat higher order reactions as approximately first order. Although there have been many studies of diffusion-reaction systems these are almost all limited to systems with negligible diffusion coupling, usually dilute reactants in a solvent [2, 3, 41. DEGROOT and MAZUR [5] have considered the stationary state with dual coupling but the only study of dual coupling in the unsteady state appears to have been done by ALBRIGHT [6] who solved a transient problem for a ternary system with a single reversible first order reaction. FORMULATION The equations for the concentrations in an n + 1 component mixture at constant temperature and pressure with no external forces acting may be written in matrix notation as [I]. y
where
941
+V(.V(C))
=V.{[D]V(C)}
+(r)
V is taken here as the volume
(1)
average
H. L. TOOR velocity of the mixture. Only n concentrations, 12 diffusion fluxes and n production rates are independent so equation (1) may be taken to represent either n + 1 equations with n of them independent of just the n independent equations. In the former case (C) and (I) are column vectors of order (n -t- 1, 1) and [D] is a matrix of order (n + 1, n + 1) while in the later case (C) and (r) are order (n, 1) and [D] is order (n, n). The formal solutions to be developed later hold for both formulations. If the later formulation is used equation (1) is then a combination of equations (1) and (2) of reference [l] (with the reference velocity specified in this case) and [D] is the diffusivity matrix in the volume reference frame. Its elements are concentration dependent but in a thermodynamically stable system it has real, positive characteristic roots at any composition [7]. It is rarely symmetric and is diagonal only in mixtures of like species or in dilute systems [g, 91. Concentration changes are assumed to be small enough so that [D] may be taken as constant at some average concentration. This appears to be quite a good approximation over reasonably large concentration variations [lo, 11, 121. If there is no volume change on mixing the continuity equations lead to the equation.
The condition V. V = 0 will be used later so it is assumed at the outset that this is valid. It is seen that this implies constant and equal partial molal volumes of the reactants since for the reactions consideredxr, = 0. It is assumed further that the
partial molal volumes of any inerts present are equal to the partial molal volumes of the reactants. Thus the molar density of the mixture is assumed constant with or without inerts. This makes the volume reference frame equal to the molar frame. This is a good assumption in most gas mixtures at low and moderate pressures and in some liquid mixtures. With the above assumptions equation (1) becomes
a(Y) ae +
V.V(y)
=
[D]V(y> +
(2
(3)
where (y) is the vector of mole fractions. THE REACTION MATRIX
The reactions are assumed to be reversible monomolecular first order. The reactions are like those treated by WEI and PRATER,[13] but here non-reactants (inerts) can couple with the reaction through the cross diffusion coefficients, D,, i # j. These inerts introduce into the reaction matrix columns and rows of zeros. There are other differences between the reaction matrix used here and that used by WEI and PRATERand these are best illustrated by an example. Consider three reacting components and two inerts n ,A 2 -
where I;ij is the rate constant for the reaction j to i. Then if (r’) is the production vector inn + 1 concentration space, and (_v’) the concentration vector in this space,
(4)
942
Dual diffusion-reaction
coupling in tirst order multicomponent systems
where [K] is the WEI and PRATER reaction matrix [13]. At chemical equilibrium there is no reaction so if (y:) is any equilibrium concentration vector in n + 1 space,
It is noted that equation (9) can be written as
c
(W
When no inerts are present the principle of detailed balancing gives c Note that because species 4 and 5 do not enter into the reaction, y4= and yse may take on any (suitable) values, but yle, yze and y3e are in chemical equilibrium and hence bear a fixed ratio to each other. Adding equations (4) and (5)
(2 = - [k’](y’ C
- y’)e = _
Ck’](-J)
Ckl(y3= -
h+1
&n+l ( Enn+l 1
(12)
and WEI and PRATER [13] shows that there is a unique equilibrium composition in this case. Hence [k] is non-singular and
(6)
fl.+1
This is the desired form for use in n + 1 dimensional space. Since
(~3 = [k]-’
.2”+1
(13)
i L+1 1
If inerts are present equations (4) and (5) lead to
Ckl(y3= 0
[k] is obviously now singular and the equilibrium composition is not unique since the inert mole fractions are not fixed. It is easily shown that the characteristic roots of [k] are minus the roots of [Kj plus q - 1 zero roots. Hence [k] always has real non-negative roots since [K] has only real non-positive roots [13]. Substituting equation (9) into (3) and taking advantage of the fact that (~3 is a constant vector,
Equation (6) may be rewritten with independent quantities only,
(I) -=-
c
0 0
[’‘I -K
o
0
0
0
(4 = - Ckl(x> (9)
0
This is the desired form for use in n dimensional space. [k] is obtained from [k’] by deleting the last row and column and subtracting the last column from the earlier columns. Thus if there is only one inert [k] = -[K] while if there are no inerts the order is reduced by one. For the earlier example, if there are no inerts.
In general it is seen that if there are q inerts, 4 > 0, &I = - [K] with q - 1 rows and columns of zeros aflixed, while if q = 0, [k] is obtained from [K’j by deleting the last row and column of [K], subtracting the last column from each of the others and affixing a minus sign to the entire matrix.
(14)
d(x) ae + I’. V(X) = [D]V’(x) - [k](x)
(15)
This is the starting equation which will be used for the analysis of the first order diffusion-reaction problem. The same form is obtained if the equations are formulated in n + 1 concentration space but unless noted otherwise it is assumed below that the equations are written in n space. Primes will be used when it is necessary to refer to n + 1 space explicitly. Equation (15) and the earlier equations also hold for a system of constant mass density if mole fractions are replaced by mass fractions and other molar quantities are replaced by appropriate mass quantities. Hence solutions obtained subsequently for constant molar density are also valid 943
H. L. TOOR
is not singular equation (15) becomes V(X) - [O] -‘[/C](x) = 0
for constant mass density when the above changes are made.
(18)
and the equation may be uncoupled by diagonalizing the matrix [D]-‘[k]. WEI [16] has analyzed problems of this sort when [D] is diagonal. Straightforward solutions may also be obtained when V is a constant if the problem is unidirectional.
DLUONABLEPROBLEMS Solutions to equation (15) are obtained simply as linear combinations of binary solutions when the equation can be put in diagonal form. Solutions of special cases of the equations have been obtained in this manner; they are summarized briefly below.
[D] and [k] commute
It has been seen that when diffusion takes place alone the equations can be uncoupled, hence certain linear combinations of components diffuse No d@ision If there are no concentration gradients in the independently of each other and when reaction takes place alone other linear combinations react system equation (15) becomes independently of each other. Only when both linear combinations are the same are there linear do = -[k](x) (16) combinations in the diffusion-reaction system d0 which diffuse and react independently of each other. The equations can be uncoupled [13] by diag- The linear combinations are the same only when onalizing [k] and are equivalent to a set of first [D] and [k] have the same characteristic vectors order irreversible reactions with rate constants which is equivalent to saying that they can be equal to the characteristic roots of [k]. The solution diagonalized by the same similarity transform. is formally [D] and [k] may be diagonalized by the same (x) = e-tkl’(XO) (17) similarity transform when they commute and equation (15) may then be reduced to a set of uncoupled where (x,,) is the initial value of(x). This problem diffusion-reaction problems, each of which correswith no inerts has been thoroughly analyzed by ponds to a binary system with a single irreversible WEI and PRATER[13]. Since the roots are real and reaction or with none. This system is clearly stable non-negative the system is stable. Since the since the diffusion coefficients and reaction velocity inert concentrations are unaffected by the reaction constants which appear are the roots of [D] and under these conditions the inerts may be ignored. [k] and hence are real non-negative numbers. If neither [D] nor [k] is a scalar times the identity No reactions matrix the commuting of [D] and [k] represents an If there are no reactions equation (15) reduces exceptional case which does not appear to have to the form treated by KIRKALDY [7, 14, 151 much physical significance. Also, since the case in (for V = 0), TOOR, [I, 121 and STEWART and which [k] is a scalar times the identity matrix has PROBER [Ill. The equations are uncoupled by physical significance only in a binary system, the diagonalizing [D] so the problem is equivalent to a only unexceptional case of interest in which [D] set of binary diffusion problems with diffusivities and [k] commute is that system in which [D] is a equal to the characteristic roots of [D]. Since these scalar times the identity matrix.? But this is really are real positive numbers in a thermodynamically stable system [7] this system is also stable. The t This represents the general system of like components stability of these two limiting cases does not ensure with no diffusional coupling [8,9]. The only uncoupled svstems which do not fall in this class are those in which all the stability of the general case. Steady
state
Another problem which can be uncoupled is the steady state problem with V = 0 [5, 161. Since [D]I
but one of the components are infinitely dilute in ordinary diffusion or those in which Knudsen diffusion takes place. In these systems [D] is diagonal, but the diagonal elements need not be equal. Then [D] and [/cl do not in general commute and the more general methods developed in the next section are required.
944
Dual diffusion-reaction couplingin first order multicomponentsystems a singly coupled system and PAO [17] has already developed a generalization Of DANCKWERT’S method [3] which yields solutions in terms of the solution to the problem of diffusion without reaction. Since the procedure outlined above could do no better than this it will not be pursued further.
has a unique matrix generalization which is the correct solution to equation (19) while if [D] and [k] do not commute there is no unique matrix generalization. To solve equation (19) let (x) = e-w@) So (19) becomes
THE NON-DIAGONABLE DUAL PROBLEM In this section a method is developed for solving the non-diagonable diffusion-reaction problem when V is zero and the diffusion is unidirectional. The slab is treated explicitly, but spheres and cylinders follow by the same method. Consider a quiescent it + 1 component medium extending from z = - a to z = + a and infinite in the other two directions. The initial composition is given by (y (z, 0)), it may or may not be an equilibrium composition. At time zero the concentrations at the boundaries z = - a and z = + a are changed to some equilibrium value (y,). (Obviously this problem is trivial if the initial composition is the equilibrium composition when there are no inerts present for then there is only one equilibrium composition. The case in which the boundries are not at equilibrium will be treated later in the paper.) V is assumed to be held at zero at one boundary (or at the centreline which can be a solid wall). Since V.V = 0 from equation (2) and the earlier assumptions, and the problem is one dimensional, this implies V = 0 everywhere. With the above conditions equation (15) reduces to
$ =[M(e)]
(20)
z
(21)
with [M(O)] = ec’le[D]e-we
(2la)
and from (19a) and (19b), (u) satisfies 8 =o,
(u) = (x(z, 0))
(2lb)
z=+a (v)=O (2lc) Assume the variables may be separated as follows (v) =
cWI
E&m%)
(22)
with [T(6)] and [Z(z)] variable matrices of order (n, n) and (Q) a column of constants. Substituting equation (22) into (21), collecting terms and cancelling (Q) gives
cn%wl
431 dZCzl - de = dz2 [Z]-’ = -[A]
(23)
and since the left side depends only upon 8 and the right upon z, they have both been set equal to the constant matrix - [A]. The equations to be solved then are
d2IZI
--j-g- + c4cz1 = 0 z=*u,
[Z] = 0
(244
and and since (x) is (y - y,), (yJ is for the time being taken as the equilibrium composition at z = If: a. Hence the boundary conditions may be written as 0 =o,
(x) = (~(z, 0) - ~3 = (x(z, 0)) z=+u,
(x) = 0
(19a) (19b)
Since it is now assumed that [D] and [k] do not commute the above equations cannot be uncoupled. It is interesting to note, however, that if [O] and [k] do commute the solution to the scalar analog of equation (19) (this is the binary diffusion-reaction problem with a single irreversible reaction [18])
43 7
+ C~UOICTlE4 =0
(25)
and of course (21b) must be satisfied. Equation (24a) comes from (21~) and (22). The solutions to equation (24) which satisfy (24a) are
ca = 4/&NcJ~
p = 0, 1,2 . . . (26)
with
945
(26a)
H. L. TOOR
and the [I’,] are constant matrices of order (n, n). Hence the characteristic matrices of equation (25) are just constants times the identity matrix. Since [A] takes on the values, &I, A, I. .., Equation (25) becomes
42’1 -=
-A P eckl”[D]e-Ckle[T]
df?
(27)
spheres. The characteristic functions and characteristic values of the boundary value problem, cos &z and JA,, respectively, are merely replaced by the forms corresponding to the geometry at hand. When the initial concentrations are independent of z equation (33) takes on the, simple form (Y - Y,) = Pzojp cosIJ&zIe-cLple(y~
where equation (21a) has been used. To solve equation (27) let (28)
4( - 1)” fP = (2p + 1)7L
which reduces the equation to (29) with
(294
[LPI = A,Pl + WI
Equation (29) has the well known solution [19] [U] = e-ctP1e[U P]
(30)
[T] = eCklee-CL~le[Up]
(31)
Combining equations (20), (22), (26) and (31) and collecting constants, (x,) = cos{~z}e-~Lple(b$
(32)
where (b,) is a column vector of IZconstants. Since each (x,> solves equation (19), one can write as the complete solution, (x) = pzO cos{J~>e-rLple(bp)
(33)
and from equation (19a) (x(z, 0)) = F cos&&IzI(b,)
xi = T cos{~‘Ap~}e-~A~Dia+kia)e~pi (37) p=o
One can write the exponential in any of the three ways, e-ApD,0Be-kio.9 e - (ApD,O + kio)B ,
which is just a set of scalar equations for the bpi which can be treated by usual Fourier analysis to give dz’
(35)
Direct substitution of equation (33) into equation (19) with the (b,,) given by (35)coni%ms the solution. Similar solutions are obtained for cylinders and
in equation (37) ,
e
-k,%9e-ApD$9
Hence when [D] and [k] do not commute replacing I$’ by [II] and ky by [k] in the above expressions (and xi and bpi by (x) and (b,), respectively) gives three possible generalizations. The first is the correct one ! This suggests a procedure for generalizing various binary solutions. As an example consider the semiinfinite slab with, for simplicity, uniform initial concentrations-Equation (19) with (19a) and (19b) replaced by,
(34)
p=o
(364
where (yo) is the initial concentration vector. The nature of the roots of the [LPI, which determine the behavior predicted by equations (33) and (36), will be considered later. First consider the scalar solution analogous to equation (33), the solution to the binary problem with an irreversible reaction, [18]
with the [Up] constant matrices of order (n,n). Substituting into equation (28) and solving for [T],
-a
(36)
with
[U] = e-Ckle[T]
(b,) = ; I= (x(z’, O))cos{&-z’}
- Y,)
8 =o,
(x) = (x0)
(3ga)
z= 0,
(x) = 0
(3gb)
Taking the scalar solution and using the matrix generalization of the first exponential form considered earlier gives (x) = flom
F
,-k%l+IW
de@,)
and direct substitution confirms this solution. 946
(39)
Dual diffusion-reaction
coupling in first order multicomponent
and if (yl) is some particular equilibrium vector, (y,), then again (y(z)) is equal to this particular (yJ everywhere since the entire matrix on the right times (ye) is (yJ. Hence when (yJ is not unique it may be set equal to zero in equation (43). The result that the steady state concentration varies with z if the boundaries are not at equilibrium and is everywhere equal to the boundary concentration when the boundaries are at a constant equilibrium concentration is clearly valid irrespective of the type of reaction or the geometry. The specific solution for the slab with equal and constant boundary concentrations is obtained from equations (42) and (43). The result is
Returning to the finite problem theresultis now generalized to include boundaries not at chemical equilibrium. Thus Equation (19) with (19a) and the boundary conditions, z = a, uw (x> = (Y, - Ye) z=-a,
(4
= (Y-0 - YeI
is to be solved with (y,) and (y_,) constant.
(194
Let
(x(z, 0)) = (Q, 0)) + (x(z)) (40) where (x(z)) is the steady state solution which satisfies (19c) and (19d). (R) then satisfies equations (19) and (19b) and the condition e =o,
(2) = (x(z, 0) - x(z))
(Y(z, 0)) = (j(~, 0)) +
(41) -
cosh[B]z[cosh[B]a]-’
systems
pgpcos{&;z)B-[~“~ ['+11-J
(yr-
ye)
P (424
The solution follows much as before and the result is (y(z,
ej -
y(~)) =
where (j(z,O)) is the concentration when (yr) = (~3 as given by Equation (33). If there are no inerts
2 COS{~~~ z}~-~~J~ i [”(~(40) -
where (y(z)) is the steady state concentration profile. As before this is one of the possible generalizations of the solution to the equivalent binary problem. Equation (42) is in accord with equation (36) only if (y(z)) = (y,) when the boundaries are at equilibrium. This is confirmed by the steady state solution. For example, when (y,,) = (y_,) = (yr) standard methods give the steady state solution (y(z) - Y,) = cosh(C~lz}Ccosh(C~la}l-‘(y~ - Y,) (43) where [B-J = [[D]-‘[k]]i’2 (43a) As noted earlier the steady state problem is diagonable. Equation (43) is the matrix generalization of the scalar problem. When no inerts are present (ye) is the unique equilibrium composition vector and when (yr) = (ye), (y(z)) = (y,) everywhere as required. When inerts are present expansion of the matrix functions in equation (43) and use of equation (14) reduces equation (43) to (Y(Z)) = cosh{CBlz)Ccosh{CBla}l-’ (A
Y(z'))
c-J& z’ dz’
UJ -a
p=o
(42)
(y,) is given by equation (13) while use of equation (44) in place of (43) shows that when inerts are present (y,) may be set equal to zero in the above equation. The above solution may also be used to construct solutions for time dependent boundary concentrations by use of the method of superposition, while the solution for constant but unequal boundary concentrations is obtained simply by modifying equation (43). Explicit forms for particular cases are obtained by combining equation (48) below with the above solutions. TOTAL RATE OF REACTION AND RATE OF DIFFUSION
The total rate of reaction in the slab per unit volume is given by (R) = - &I
j”,(y(z.
0) - Y,) dz
(45)
Substitution of equation (42) into this equation and integrating gives (Rj = (R3 - ‘+ [k] pzo El
(44 947
e-tLple(hp)
(46)
H. L. TOOR
where (RJ is the total rate of reaction per unit volume in the steady state and (h,) is just the normalized integral on the right of equation (42). The diffusion rates in the z direction with respect to a fixed coordinate may be obtained readily from the concentration profiles and the relation [I] (47) DISCUSSION
The nature of the solutions obtained depend upon the nature ofthe matrix exponentials which appear in all the transient solutions. Assuming the usual case of no repeated roots SYLVESTER’S theorem [19, 201 gives emcLpl’ = j~le-LpJe[Zo(Lpj)]
cz
(L
0
.),
P,
=kg(Lpkz - [Lpl) kgcLPk - Lpj)
(484
Repeated roots may be handled readily with the confluent form of this theorem [20] but show no essential change in behavior so far as the present discussion is concerned. The behavior of equation (48) and hence equation (42), etc. clearly depends upon the nature of the characteristic roots of [LPI, Lpj. If these are real and positive the concentration at any point decays to the steady state value as time increases, but since each matrix exponential is a linear combination of n scalar exponentials each term in equation (42), etc. may have n - 1 extreme values. Even though [L,] is simply related to [D] and [k] through equation (29a), the fact that these matrices have real positive roots does not ensure that [L,] will have real positive roots except for the exceptional circumstances in which [D] and [k] commute or in which both are symmetric. It is shown in the Appendix that the second law of thermodynamics and the principles of detailed balancing of chemical reactions and of microscopic reversibility lead to the conclusion that in a thermodynamically ideal system near equilibrium all the [L,] have real positive roots. A stronger statement is not available even though the above two principles separately can be used to make much stronger statements about [k] and [D]. Hence it is not yet clear whether or not complex or negative roots, which could indicate instability, can occur.
It is important to note, however, that the assumptions made in the derivation of equation (19), small displacement from equilibrium and constant molar density, imply a thermodynamically ideal system near equilibrium (e.g., a mixture of perfect gases near equilibrium); hence the nature of the roots in non-ideal systems or in systems with a large displacement from equilibrium may not necessarily fully determine the stability as in the case of ideal systems near equilibrium. Nevertheless it is shown in the Appendix that there are circumstances under which the roots of [L,] are found to be real and positive without explicitly using the above restrictions. The one most likely to be encountered is the situation in which [D] is diagonal. This corresponds to the system in which n dilute components are diffusing and reacting in a solvent [8] or to the case of Knudsen diffusion with reaction. As discussed earlier, when all n components are of a similar nature [D] is a scalar times the identity matrix and this is really a singly coupled system. However when the components in the dilute system are not similar the elements of the diagonal diffusivity matrix are not all equal and even though there is no diffusion coupling the system can be considered to exhibit a form of dual coupling,in the sense that [D] and [k] do not commute. It is seen, however, that with this pseudo dual coupling the [LPI always have real positive roots. But if one takes the viewpoint that a dilute system is equivalent to an ideal one near equilibrium then one is back to the previous restrictions. Equation (29a) can be written in the form
7 +[k]
CL,1= (P + tYn2
(49)
So if a is small enough the first term in equation(49) will dominate for all values of p. The diffusion is then rapid with respect to the reaction and stability is ensured for [D] has real positive roots. If n is large the converse is not true since there will always be some Fourier components with a large enoughp so that the two terms in equation (49) will be of similar magnitude, even though the coefficients of these terms (e.g.,f, in equations (36)) will be small. 948
Dual diffusion-reaction
coupling in first order multicomponent
WEI [22] has shown that the reaction velocity
matrix in the general linearized mass action system has the same structure as the reaction velocity matrix in first order systems. This indicates that the above results also apply to any mass action system near equilibrium. APPENDIX-NATURE OF THE RENTS OF Lp The proof is carried out most readily in iz + 1 space. Although the former solutions are valid in 12+ 1 space, the use of dependent quantities adds a certain arbitrariness to the matrices [5, 211. However, if it can be shown that one of the possible constructions of the [Lb] have real positive roots for all values of p then stability is clearly ensured since all formulations are equivalent. WEI and PRATER [13] show from the principle of detailed balancing of reversible chemical reactions that one may write [K,=
-@]/y.i
Yr2._.
1
1’ Yenfl-_q
II.o. 3 :
[k’] =
0.. 0... .... .... 0...
.. .. .. ..
I
CD’1 = C~‘lrdl
[D’] may
(A4)
where [Y] is a matrix of coefficients and the elements of [$I are a~,/LYC,. The n + 1 fluxes and chemical potentials are linearly related and HOOYMAN and DEGROOT[21] show that this leads to an arbitrariness in [Y] and hence in [D’]. However, they show that when the Onsager reciprocal relations hold for properly defined fluxes and forces symmetric [_%“Imatrices may always be constructed. It can be shown that within their most general symmetric form one can find [U’] matrices which are positive definite by use of the second law requirement that entropy production must be positive. When the system is ideal and the molar density is constant ~~‘1 = 7
n/y’,
(A5)
ri/y:l
(A@
and near equilibrium
I
1
~~~‘1= y
Combining Equations (29a) (written space), (A3), (A4) and (A6), [I&]
-1
(A21
since the inert mole fractions which have been added do not affect the reaction matrix. Using primes as usual to represent n + 1 space the above equation may be rewritten as Ck’l = CS’lrl/~h
Following irreversible thermodynamics be written as [S]
(Al)
where [ - s] is a symmetric matrix with real nonnegative roots-a positive semi-definite matrix. The above matrices are of order (n + l-q, n + l-q) where n + l-q is the number of reacting species. If there are no inerts [k’] = - [K] while if inerts are present one may write by virtue of equations (4) and (Al) 0.
systems
(A3)
where [S’] is first matrix on the right of equation (A2) and Q/y:, is.the second. The former is positive semi-definite and the latter positive definite.
=
pp!gL-v’1 + Cs’l] rUy:,
in n + 1
(A7)
Since the A, are positive, [Y] is positive definite and [S] positive semi-definite, the first matrix must be positive definite. Hence each [L;] is the product of two positive definite matrices and by a proof given by RIRKALDY[7] each [L;] has real positive characteristic roots. Thus if a solution is written in the redundant IZ+ 1 space the exponents which appear when e-tLple is expanded by Sylvester’s theorem are all negative and the system is stable. The solution must also be stable when written in n space so the n independent characteristic roots for each value of p must also be real and positive. It is noted that there are exceptional cases in which it is easy to prove that the [L,] have real positive roots without the restrictions used above.
949
_.
I
n. L .TOOR
When [D] and [k] commute and hence may be diagonalized by the same similarity transform it follows by diagonalizing equation (29a) that Lpi
= A,Di + ki
(A@
and since A,, Di and ki are real and positive, the Lpi are also. But this case was shown earlier to be stable. Again if [D] and [k] are symmetric they are positive definite, so [L,] is positive definite, hence has real positive roots. Finally equations (A3) and (29a) (written in n space) can be put down in the form,
CL,1= [A,CDYY,I+ C~ll’lI~e~
(A%
If there is at least one inert [S] is positive semidefinite and if [D] is diagonal [D] rye1 is positive definite. Hence by earlier arguments [L,] has real positive roots. This corresponds to n dilute reactants in a solvent.
14 Matrix of constants AZ, Constant defined by equation (26a)
4 D{ DP Di3
ID1 fP c.i: kr” . kc3 krr El
rp; 4 &?I (Rd
ra (r) rs1 WI 1; IV1 WPI v
fiii (0) (00) Xt
Characteristic roots of [LPI Matrix defined by equation (29a) Matrix defined by equation (A4) Matrix detined by eauation (21a) One less than the number of components Matrix of constants Number of inerts Gas constant Column vector whose ith element is the total rate of production of species i by reaction per unit volume Steady state value of(R) Rate of production of species i by reaction per unit volume Column vector whose elements are rg Matrix defined by equation (Al) Matrix defined by equation (A3) Absolute temperature Matrix defined bv eauation (22) Matrix defined by equation (28j Matrix of constants Vector velocity Partial molal volume of species i Column vector defined by equation (20) Column vector of constants YVYiC
Column vector whose elements are xr Column vector defined bv eauation (40) . . Steady state value of(x)_ Mole fraction of species i (Z Column vector whose elements are yr rye1 Diagonal matrix with elements yl, ‘l/Y1 Diagonal matrix with elements l/yr ‘l/Ye{ Diagonal matrix with elements l/vie ._ (Y(Z)) Steady state value of(y) Distance : Variable of integration [Zl Matrix defined by equation (22) Greek Letters Variable of integration f Time Clr Chemical potential of species f [$I Matrix whose elements are +t/aC3 Subscripts a z=a -a z=-a e Equilibrium value I Boundaries i, j Species i, j p Index running from 0 to co Superscripts ’ 12-t 1 space :; (x(z))
NOTATION
I; b,t (bp) c
lkl Reaction velocity matrix LPl LPI WI [Ml
Half width of slab Matrix defined by equation (43a) Constant Column vector of constants Molar density Moles i per unit volume Column vector whose elements are Cr Characteristic root of p] Binary diffusion coefficient Multicomponent diffusion coefficients Matrix whose elements are Dt3 Constant defined by equation (36a) Identity matrix Column vector whose elements are fluxes Reaction velocity constant for reaction in binary system Reaction velocity constant for reaction of species jtoi 4 element of matrix [k] WEI and PaATER reaction velocity matrix
REFERENCES IOOR H. L., Amer. Inst. Chem. Engng. J. 1964 10,460. CRANK J., The Mathematics of Difision, Oxford University Press, Oxford, 1956. 1DANCKWERT~ P. V., Trans. Farad. Sot. 1951 47, 1014. PRAGER STEPHEN,Chem. Engng. Progr. Symp. Ser. No. 25, 1959 55, p. 11. DEGROOT S.R. and MAZUR P., Non-Equilibrium Thermodynamics, Interscience, New York, 1962. ~LBRIGHTJ. G., J. Phys. Chem. 1963 67,2628. KIRKALDYJ. S., Canad. J. Phys. 1963 41,2166.
950
Dual diffusion-reaction
coupling in first order multicomponent
[8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] 1191 [20]
systems
TOOR H. L. and ARNOLDK. R., Industr. Engng. Chem. Fund. 1965 4, 363. Toot H. L., SESHADRIC. V. and ARNOLDK. R., Amer. Inst. Chem. Engng. J., to be published. ARNOLDK. R., Ph.D. Thesis, Carnegie Institute of Technology, Pittsburgh, Pa. 1965. STEWARTW. E. and PROBERRICHARD, Zndustr. Engng. Chem. Fund. 1964 3,224. TOOR H. L., Amer. Inst. Chem. Engng. J., 1964 10,448. WEI Jm and PRATERC. D., Advances in Catalysis, 1962 13, p. 203. KrRKALDYJ. S., Canad. J. Phys., 1957 35,435. KIRKALDYJ. S., Canad. J. Phys. 1959 37, 30. WEI, JAMES,J. Catal. 1962 1,526. PAO YIH-Ho, Chem. Engng. Ski., 1964 19,694. YOST W., Diffusion in Solids, Liquids and Gases, Academic Press, New York, 1960. PIP= L. A., Matrix Methods&w Engineers, Prentice Hall, Englewood Cliffs, New Jersey, 1963. FRAZER R. A., DUNCAN W. J. and COLLAR A. R., Elementary Matrices, Cambridge University England 1952. [21] H~~YMAN G. J. and DEGR~~T S. R., Physica, 1955 21,73. [22] WEI JAMES,Socony Mobil Oil Co., Inc., Princeton N.J., private communication, 1965.
Press, Cambridge,
R&.mn~L’article presente une etude de la diffusion non-stationnaire et uni-directionnelle avec reaction chimique complexe du ler ordre et reversible dans un systeme isobare et a multiples composants de densite molaire ou massique constante. Dans ce systeme les composants sont couples par des reactions chimiques et par la diffusion et on ne peut generalement pas Biminer le couplage par une transformation des coordonnees de concentration. Des solutions obtenues ii l’aide dune technique de separation de matrices variables il apparait qu’un systeme ideal (p.e. un melange de gaz ideaux) proche de l’equilibre est stable. Lorsque tous les composants, a l’exception dun seul, sont dilues a l’infini, ou lorsque la diffusion est du type Knudsen, le systeme est toujours stable. Une solution du cas a multiples composants represente tgalement une des generalisations matricielles possibles de la solution du probleme analogue avec une reaction irreversible seulement.
951