Calculation of Complex Equilibrium Relations IIAKOLD J. KANDINER .4ND STUART R. BRINKLET, JR. U. S. Bureau of Mirtes, Pittsburgh, Pa.
E
s>
slematic detailed directioris are given for analyzing QUILIBRIA involvt.quilihrium is discussed, complex equilibria existing among any number of chemical ing a single equation and then detailed methods species in a single (gas) phase with or w-ithout the presence [elating the product and for solving the resulting of a single pure solid phase. The principle of the method reactant concentrations are equations are considered. i s outlined using a gas combustion reaction at 40 atmosreadily treated by using the FORMULATION OF WORKpheres pressure giving rise to an equilibrium mixture of "degree of reaction" as a I Y G EQUATIONS ten gaseous constituents, with or without the formation of parameter. Where two or solid carbon. Deviations from ideal behavior, due to presthree equations are necesSISGLE-PHASE IDEAL Gas sure, temperature, or concentration, can be accommosary to describe the conSYsTmfs. It will be condated in formulation of the necessary working equations. venient to study a single (witration relationships in Several methods for the numerical solution of these equa$1 system, this procedure (em) phme system obeytions are included; each of these methods is particularly ing ideal relations. To inmay still be used, but the applicable to certain typical types of working equations. dicate the simplicity and development and solution mechanical nature of the The working equations, regardless of the complexity of of the algebraic equations steps involved, a fairly the problem, can be set up very rapidly for any equilibrium encountered are considersj stem by what is effectively a mechanical almost routine complex cornbustion probably more difficult. In procedure and are in a form which is exceptionally amenleni has been selected M really complex systems, the able to numerical solution by personnel engaged in making an illustrative example. initial problem of selecting This particular problem rribprofessionalcalculations. a consistent set of suitable has been discussed by chemical equilibria becomes Damkohler and Edse (6) very involved, and the labor in conjunction with their trial-and-error method for solution of of mathematical solution is almost prohibitiw. complex equilibria. With the method discussed in this paper, the necessary eyuaConsider the single-phase reaction system containing propane tions can be deduced readily without extensive st,udy of the sysand the stoichiometric equivalent, for complete combustion, of tem; also, these equations can be solved easily. An operator, air at 2200' K. and 40 atmospheres pressure. Per mole ef propane who need not have any chemical background, can handle all of the the system composition initially is l(laHs, SO2, and 20Si calculatians. In a typical though comparatively simple problem, (assuming air is 80% nitrogen for numerical simplicity in this CHI 2Hz0 -+- CO, C/O2,CH,, H2,and H20 a t 1400" C. and 1 example). The principal species present in 6he gas mixture at atmosphere, the actual working time, for slide-rule accuracy (excquilibriuni certainly must be COZ, HaO, and Sz. Some CO and clusive of set,-up), is less than 1 hour using t,his method. To ob€12 (from the wat'er-gas shift reaction) and 13, OH, 0, and NO (by tain results of equivalent accuracy, the "degree of reaction" thermal dissociation) together with possibly some frre 0 2 will also method requires some 3 to 4 hours for calculations alone. be present. These species are selected a priori by a reasonable The method is completely a general one, applicable to systems analysis of what is likely to be present. The choice would, of cmontaining any number of chemical species coexisting a t equilibcourse, vary with temperature, pressure, and feed selected. It is rium in any number of phases. Most industrial equilibria, howiissumed for the present that solid carbon is absent from the ever, involve either only single-phase gaseous systems or a single equilibrium mixture. Systems containing a pure mlid phase in purr solid phase in contact with a gas phase. This discussion ici equilibrium with a gas phase are treated later. restricted to these systems to simplify the algebraic notations. Next det,ermine the number of independent components, the The formal mathemat8icdbasis and rigorous proof of the proccspecification of whose concentrations for a given system feed dedure are contained in t,wo recent papers (1, a), and the emphasis tines and fixes the system composition at equilibrium. The numhew will be on the application of the met,hod t o typical problem. ber of independent components usually may be taken equal to the Details of the derivation are contained in the original papers. number of different chemical clenients present in the reaction sysBriefly, the following steps are required: tem. For this problem then, there are four coniponents, because 1. Select chemical species present at equilibrium the system contains four eleinents--.calbon, oxygen, hydrogen, 2. Determine number of independent components 3. Select a set of these components and regard t.he remitining arid nitrogen. ;pecies as derived constituents 4. Write, by inspection, a set of chemical equations by which .Uthough the number of components can never be greater each derived constituent is produced individually from the conithan the number of chemical elements present, on occasion it may popents alone be less. A general analytical criterion for determining the nuni3 . Make a table of the coefficients in this set of chemical ecjualw of independent components is available in the following protions cedure. Construct a double-entry table, having for column head6. Following the rules, write the necessary working equations: ings the va,rious chemical elements in the system and list t'lie two sets are required-one derived from chemical equilibria m d various species in the left column. The entries are the subthe other from mass balances scripts to each element in the various chemical formulas of the 7 . Solve the working equations constituents. Performing this operatiem on the system under discussion gives: First the application of steps 1 through 6 to t,ypical system at
+
850
May
1950
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY Ha H
C 0
O 0
0 0 1 1 0 0 0 0
1 1 1 2 0 1 2 1
n
OH HzO
cot co N2
NO 09 0
H 2
n
i
1 2
0
0 0 0 0 0
N 0
n 0 0
0
0 2 1 0 0
From this table, form determinants by trial, to ascertain the largest-ordered nonzero determinant that may be written from this array of table entries. The number of independent components in the system is equal to the order of the largest-ordered nonzero determinant that can be written from the table-for example, regarding the table as a matrix, the number of independent components is equal to the rank of this matrix. Obviously, in this case, one would start with fourth-order determinants, using all four columns and any four rows. Should no nonzero fourthorder determinant be obtainable from any set of four rows, the system would necessarily contain less than four components, and the possibility of forming a nonzero third-order determinant (and hence establishing that there are three independent components) would next be investigated. In the example, several nonzero fourth-order determinants can be obtained readily-using the second, fifth, seventh, and tenth rows: 0 1
0 0
0
1
1 0 1
0 0 0
Thus, since a nonzero fourth-order determinant can be written for this system, there are exactly four independent components. Having fixed the number of independent compounds, a selection must be made. If this choice were completely unrestricted, there would be many different ways to select four components from the species presumed to exist in the equilibrium mixture. In practice, the number of suitable alternative sets is quite small. In order t o speed up the actual numerical calculations, it is convenient to pick as components those species which have the greatest probable concentrations at equilibrium, subject to the following limitations: I . The coniponents must be stoichiometrically independent of one another (If the system contained, say, both NO2 and N20, among other constituents, it would not be permissible to select both of these as components-only one could be taken.) 2. ,111 the chemical elements present in the system must also be present in the group of components selected. (These two limitations are equivalent to the requirement that the determinant formed by the rows in the table Corresponding to the selected components must not equal zero.) 3. Any constituent whose concentration is not expressible by an equilibrium constant relation-for example, a pure solid phase-must be selected as a component.
Select carbon dioxide, water, and nitrogen as three of the coinponents, as these undoubtedly compose the bulk of the equilibrium nlixture at the tcmperature and pressure *conditions of this problem and also satisfy the given limitations. For the fourth component, almost any othcr constituent can be selected. Choose carbon monoxide for purposes of illustration; the reniaining six constituents of the equilibrium mixture are regarded as derived constituents. Construct a set of six chemical equations, one for each derived constituent, employing as participants in each reaction only the independent components and the corresponding derived constituent, as follows:
+ CO ri Con+ Hz, for Hn '/zHzO + '/zCO ri '/zCOZ + H, for H '/zHzO + I/zCOZri l/&O + OH, for OH COS ri CO + 0, for 0 CO + NO, for NO '/zNz + COz 2C02 2CO + for HzO
02,
There is only one possible equation by which any specific derived constituent can be formed from the components alone. For each of these equilibria, a mass-action expression can be written, and the value of the mass-action constant can be obtained from reference sources. Introduce the symbols n, 0' = 1 , 2 , 3 , 4 )to denote the number of moles of the components COZ, H20, Na, and GO,respectively, in the equilibrium mixture. Also, let ni (i = 5, 6, 7, 8, 9, 10) represent the number of moles at equilibrium of the derived constituents Hz, H, OH, 0, NO, and 0 2 , respectively. Construct the following double-entry coefficient table where each entry, denoted by vit,is the appropriate coefficient (on thejth component) taken from the equilibrium equation for that derived constituent when all terms involving components are transposed to the left side of the equation; or, stated otherwise, each column of this table corresponds exactly to one of the equations of the previous set:
j j j j
= 1co* = 2Hz0 = 3N2 = 4co
.4
0
0 2
851
-1
1
L/1
-1/2
1 0 1 0
0 0 -1 -1
01/2
1/1
0 -112
L/s
-I/%
-1/2
1
0 -1
'/a
-I/¶
2 0 0
-2 -1
The final row of entries, labeled Ai, is obtained by summing each column and subtracting 1. The first set of working equations, based on equi1ibria;is then written directly by substitutingin to the fundamental equation : C
N~ = /ci(p/n)Ai.n(ni)v*, j = l , 2 , 3,
. . .> c ;
j - 1
i=(c+l),(c+2)
(1)
,,,., s
Where c is the number of independent components; s, the total number of species present at equilibrium; ki, the mass-action constant in partial pressure units for the reaction forming the ith constituent; p , the total pressure, atmospheres; and n, the total number of moles of gas in the equilibrium mixture. For a syst,em obeying the ideal gas equation of state and exhibiting ideal solution behavior, the mass-action constants, IC,, are equal t,o the corresponding thermodynamic equilibrium constants, Kj. Writing Equation 1 in expanded form for the problem under consideration, taking exponent,sfrom the above table, and regarding the system as ideal, for the present, the series I working equations are obt,ained: Kb.(,nl)-l.nq.nq
n6
=
n,6
=
K,.(p/n)-'
n7
=
K.r.(p/n)-' /s.(nl)l '2.(nz)l '"(nd)-'
n?,
=
Ks.(p/n)-',(nl)(n4)-'
ng = Kg.(p/n) -1
'Z.(
nl)-l /Z.(nz)'/2.(
na)l
'za(
n4)l'2 '2
n,) -1
Series I
n1o = Klo~(p/n)-'~(nl)"(n,)~
The second set of working equations, based on mass balances, is developed from the defining equation:
where z a n d j take all values tbs given for. Equation 1, and Q, are constants, dependent on the gross composition of the system feed. The physical interpretation of constants q j may be inferred by observing that they represent the composition of the system if all of the derived constituents are absent-that is, by Equation 2, 721 = q l , nz = 92, n3 = q 3 , and n4 = q 4 , if 125 = ne = . = nlo = 0. In expanding Equation 1 into series I, a vertical traverse was made of each column of the coefficient table. Analogously, Equation 2 is expanded into series I1 by a horizontal traverse of each row of the same table, as follows:
..
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
852 n1 = p1 n2
= pz
+ n5 + 2-1 - n b - 21-
n3 = q 3
- 2-1 nng
m
- 125
= q4
- 2-1 n7 - n8 - n9 - 2n10
12.9
equilibrium mixture as well as the other constituents previously selected. The modifications necessary to account for this possibility when formulating the norking equations are discussed in this section. If solid carbon is assumed present, it must be selected as one of the components; CO, H,, and Sz are here an appropriate choice for the remaining components. Equilibria for the formation of each of the derived constituents individually from these components alone are:
- 2-1 n7
+ 51 + + n9 + 27210
1 - 2 n6
728
7~7
Series I1
e C($)+ coz + Hz S C t HzO
2co
It remains to evaluate constants ql, q2, qa, and q4, which is con-
CO
veniently done by solving the equations:
('/z)Hz (3) j = 1 , 2,
. . .) e ;
x = l
K = 2
n = 3
x - 4
C 1
O 2
H 0
X 0
0 0 1
i 0 1
j = lCOz j = 2Hz0 j = 3N2 j = 4C0
2 0
0
Traversing the table vertically by columns, the series from Equation 3 is:
+ p4 + qz -t.
Qi
=
pi
Qz
=
2 ~ 1
&3
=
292
&d
= 2Qd
= Qz -
(14
1
Qi
(l/2)H2 ;=t
C(s)
+ OH
CO
+
('/2)K2
e C a)
+ NO
t
0 2
j=aC(sr j = 1CO j = 2Hz j 3N2
-
Ai
C02
i = 5
HzO
t = 6
= 7
t = 8
(-li 1 (]/si
(-1)
0I-I
H
(-1)
2
(-1) 1
0
0 0
1 0
('/d 0
1
1
0
0
NO
I 0
(-1) 1 0
0
'/2
0
-lir
t = 9
'/?
t = l O 02
(-2)
2 0 0
1
It is well known that a mass-action expression for a heterogeneous gas-pure solid equilibrium does not, explicitly contain tmhe composition variable for the solid phase and that the resulting expression is valid only in the presence of the solid material. For these reasons the t,able entries in the C($) row have been placed in parentheses. These values are to be ignored both in summing columns to find the entries, Ai, and in writing the series I working equations from Equation 1. They are to be used, however, in formulating the series I1 working equations from Equation 2 . Thus, a typical member of series I is:
Solving these equations for each q3 in terms only of t8heQ N : ~1
+
2co e 2C(,)
LII
0
CO
Denote by symbol n, the number of moles of solid carbon in the equilibrium mixture (use a letter subscript for the solid phase anti reserve numerical subscripts for gaseous species), and let nil n2, na, . ,, nlo represent the number of moles at equilibrium of GO, Hz, Kz, CO,, HzO, H, OH, 0, S O , and 02, respectively. The table of coefficients o,! then is: t = 4
0 2
e H:
co * C@)+ 0
~ = 1 , 2 , ..,m
QK is the number of gram-atoms of the xth element in the system feed; CY,': is the subscript occurring on the symbol of the K~~ element in the chemical formula of the jtbcomponent; and m is the total number of elements. For this problem Equation 3 is solved by setting up a table of a 3 Kvalues:
CZ!K
Vol. 42, No. 5
- 2 Qa -I
ne = Iand (ni)' the best, values for these unknowns a.ith n hirh to Legin t h e next trial, are given by: (n)' = ( n )
+g
(for j # j ' )
G
-it [&+ , = I
(8) i = c f l
In order to evaluate t.Iie coefficients and constants defined by Equation 8, it will he convenient t,o derive from the coefficient t,able previously employed the following double-entry table of products of t.he form V , ; V , ~ ' where for any value of i, j and j ' take all values up to c, the number of components, which is four here. In this table, as follows, t,he previous coefficient table has been repeated in thr first four lines, to clarify t,he operation:
U'.
. u,,,
Within any column all the entries in the lower pal t of the table are obtained by multiplying together all possible pairs of entries in the same column in the uppei part of the table. Using this table and Equation 8 all of the nurnerical values required to prepare Equation 7 foi solution ale obtained by horizontal traverses of the appropriate row-for example,
AS3 = ns
The quantities g, h ~ 112, , ha, and ha are obtained by solution of the following set of simultaneous linear nonhomogeneous equat,ions:
= 1
+ 41
n9
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
May 1950
Fl =
91
- n1
+ n6 + 2-1 ne - 21- nr - nn - ns - 2n10
- n2 - n6 - a1
Fz
= q2
G
=1-
1 [nl n
1
n6
- 5 n’
+ + na + n4 + ni + + nl + ns + ng + nlo1 7La
n2
The values of n, n1, n2, n3,and n4 are the trial values chosen for the primary unknowns in any trial and the values of na, n6, , . , ; nlo, are those obtained for the secondary unknowns in the same trial. Having established numerical coefficients and constants for each part of Equation 7 , the corrections g, hl, h ~ha, , and ha may be obtained by direct solution. It will be found convenient to solve Equation 7 by the Crout ( 4 ) method, which is particularly suitable for routine computing machine techniques. The corrections so obtained define, from Equation 6, the starting values of the next trial calculation. Although the above description of this method may be somewhat cumbersome, the actual operations me quite simple in practice, and the application of the method to any system to set up the specific equations required is almost mechanical. As an illustration of the utility of this last procedure in speeding the calculations, one problem involving sixteen constituents waa brought to convergence after four trials by the Newton-Raphson modification whereas after thirteen trials of simple iteration, six of the unknowns were considerably “out of tolerance.” For systems containing a pure solid phase in equilibrium with a single-gas phase, Equation 7 becomes, for the illustrative problem under discussion:
+ A a z h + . l u h + 9. Ailhl + A&L + + Blg Azihi + Anhz + Azhr + Bzg Aahi + Aanhz + AIrhd + Bsg Blhl + Blh2 + Bdhl A&
9 1 h r
Fa = F1 = Fz
=
Fs
= G
4
Brown (3) and by Damkohler and Edse (6) on explosion and combustion equilibria, respectively. Brown’s calculations are based on the use of the CO/COz ratio at equilibrium as a single parameter by means of which a relatively complex set of algebraic equations is solved by trial. The calculations reported in Table I required less than 2 hours, exclusive of set-up and evaluation of mass-action constants, and the results obtained are identical with those obtained by Damkohler and Edse. These latter authors reported a method based on the simultaneous use of two trial parameters, which, although ingenious, requires considerable time for its initial set-up. Actual numerical solution of the same problem by the Damkohler and Edse method required some 12 hours. Usually, each system, to which either the Damkohler and Edse method or some modification of Brown’s procedure is to be applied, must be independently analyzed to establish an applicable, consistent set of independent mathematical relationships. Workers who have occasion to study many different syst e m will fmd the methods of this paper very economical of set-up time and labor because the necessary equations can be written sensibly by inspection, once appropriate symbols have been assigned to the several unknowns. NOMENCLATURE
number of independent components in gas phase = number of distinct chemical species present a t equilihrium m = number of chemical elements in system
c s
=
Subscripts
Evaluation of these coefficients is performed in an entirely analogous manner to that in the previous discussion and need not be detailed here. The solution to Equation 7A, a set of values of h,, h,ha, g, and go gives a new set of trial values for the primary unknowns using the relations given in Equation 6 for (n~)’, (nZ)’* (ma)’, and (n)’ together with: ha)’= (n,) 9..
+
DISCUSSION
The methods outlined above are valid for all types of equilibrium calculation. Their application to a specific problem is very straightforward and requires a minimum amount of set-up labor. Those who have tried the classical “degree-of-reaction” method on a complex system will find the present procedure surprisingly simple. It will be of interest to contrast the final equations and .directness with whieh they can be written with the treatments by
+
i reiers to derived constituents and takes value+ ( c I), (C+2), s j refers to independent components and takes values, 1,2, . , . , c K refers to chemical elements and takes values, 1, 2, , , , m a refers to a single solid phase, if present n = equilibrium number of moles of material identified by subscript-ni, nl, n3, n,, etc; without any subscript, n = total number of moles in gas phase oii = coefficient on chemical formula of jthcomponent in the equilibrium equation forming the ith derived constituent
...,
.
(7Af
Coefficients and constants in Equation 7A having exclusively numerical subscripts are defined by Equation 8 rn before; the other coefficients are given by:
855
’4, =
(2
Uti)
-1
1 ” 1
k,, K,, Gi = mass-action constant, thermodynamic equilibrium constant, and correction factor for all deviation from ideality, respectively, for the reaction forming the ithderived constituent from the components alone; partial pressure units, where applicable q1 = stoichiometric constants defined by Equation 3 & K = gram atoms of the dhelement present in the system O ~ , K = subscript on K~~ element in the chemical formula of the j t h component p = total pressure, atmospheres LITERATURE CITED
( 1 ) Brinkley, Stuart R., Jr., J. Chem. Phys., 14, Ne. 9, 583-4 (1946). (2) Ibid., 15, NO. 2, 107-10 (1947). (3) Brown, F. W., U.S. Bur. Mines Tech. Paper 632 (1941). (4) Crout, P. D., Trans. Am. Inst. Elec. Engrs., 60, 1235-41 (1941). (5) Damkohler. Gerhard. and Edse. Rudolf. 2. Elektrochem.. 49 178-86 (1943). ( G ) Scarborounh, J. B., “Numerical Mathematical Analysis,” p. 1 8 i , Baltimore, Johns Hopkins Press, 1930. (7) Ibid., p. 193. RECEIVEDKovember 9, 1949. Pzesented before the Division of Industrial and Engineering Chemistry at the 116th Meeting of the AMERICAN CHEMICAL SOCIETY, Atlantic City, N. J. Contribution from Central Experiment Station, U.6. Bureau of Mines, Pittsburgh, Pa.
