Basis for Calculating
... Composition
. . on a Digital Computer E
Q u A n o N s for calculating the equilibrium composition of a gaseous mixture can be established readily, but cannot be solved easily; many unknown quantities that bear nonlinear relationships to each other appear. Common algebraic techniques of solution, such as matrix methods, will not serve. The proposed method involves the use of a sequence of simple adjustments, and can be applied to any equilibrium composition problem, although it is intended for use with a digital computer. In solving a complex composition problem, estimated values of composition are supplied initially. During the process of solution, these estimated values are gradually altered until they fulfill equilibrium-constant specifications. Each problem is treated as a long series of simple equations, rather than as a few complex formulas. T h e simple equations are solved individually, and correction values are obtained which indicate how to approach the correct composition. With this new set of values, the entire process is repeated ; when the adjustment values are infinitesimally small, the correct composition is attained. Being simple, the equations are easily programmed for a digital computer. Most of the systems that have been solved contained only the elements carbon, hydrogen, oxygen, and nitrogen; some contained fluorine, chlorine, aluminum, and mercury. Systems with lithium, barium, magnesium, and other elements will also yield to this approach. The technique applies when liquid or solid phases also appear among the gaseous products.
Basic Approach T o illustrate the sequential-adjustment approach, a simple gaseous system is considered, consisting of three types of molecular species: A + B e C
(1)
When the system is in equilibrium, it contains a moles of A, b moles of B, c moles of C, and n moles of total gas: c
SAMUEL R. GOLDWASSER Aerojet-General Corp., Sacramento, Calif.
Method uses a sequence of simple adiustrnents to arrive at answers not readily obtainable specified pressure, in atmospheres. T h e problem consists, then, in evaluating a, 6. c, and n. When A, B, and C are not in equilibrium, changes in composition occur in a restricted manner. .4s Equation 1 indicates, if species C increases by the amount x : species A and B must each be simultaneously diminished by the quantity x . When equilibrium is attained: K,
=
where K , is the equilibrium constant a t the specified temperature, and P the
+x)
- x ) (bo - X )
(no - x ) p
(3)
Jvhere a,. bo, c, and no signify estimated values that satisfy the material balance. After quantity x is evaluated, the equilibrium composition can be calculated : a=a,,-x
(4)
b=b0-x c = io x
(5)
n = n o - x
(7)
+
(6)
Values of a,, bo, c, and no describe any composition that fulfills material-balance requirements. If adjustments are made in simple proportions, the equilibrium composition will also fulfill materialbalance requirements. T h e problem is to find the value of Y. the quantity needed for adjusting a nonequilibrium composition. Equation 3 may be expanded, and the terms grouped : x2(1
+ P K z ) - x [ P K L ( a O+ b o ) + no - c o l + PKraobo - cono = 0
(8)
T o solve Equation 8 for x , a sqiiare-root value must be calculated and a decision made about arithmetic sign. When the adjustment quantity x is small, higher-order terms make negligible contributions and may be dropped. T h e abridged, approximate solution takes the form :
n
K”=(a)oP
(co
(ao
x =
- con,
PKsaobo f‘Kz(ao bo)
+ + no -
co
(9)
when Y is a small quantity. When x is large, the adjusted composition represents merely an improvement, and serves only in calculating a new adjustment value. In the second round, x will have a smaller value, and the new composition will match equilibrium rrquirements more closely. Itrration can be stopped as soon as the adjustment factor falls below the limit of permissible error. Equation 9 can be written i n the form: x =
K , - c,n,/a,b,P
K, (a,
+ b,)/a,b, + ( n o - c o ) / a o b o P (10)
T h e numerator evidently measures the departure from equilibrium, while the denominator acts as a conversion factor, Lhanging the degree of departure into an adjustment quantity. T h e initial estimates of composition affect the number of iterations required for convergence. Any composition that satisfies the material-balance requirements can serve as the starting point. When composition is estimated poorly, the sequence must be repeated many times. Difficulties arise when certain constituents are estimated to have a value of zero moles. Terms in denominators cancel out, resulting in gigantic adjustment-factor values. Several iterations are then required just to return to the: proper order of magnitude. A small amount, such as 0.0001 mole, rather than zero, should therefore be assigned initially. A system that contains one more constituent, species D (in addition to the molecular species A, B, and C), is considered ncxt : l/pB $
D
(11)
T h e equilibrium composition is related to the equilibrium constant in the following way:
T h e adjusted composition will fulfill equilibrium requirements perfectly only VOL. 51, NO. 4
APRIL 1959
595
where d represents the quantity of substance D a t equilibrium. An increment in D will be accompanied by the appropriate decrement in B. But species D can increase by y moles only if species B is reduced, at the same time. by ' t ' 2 ( 1 ) moles, and the total number of moles of gas undergoes a net change of +"z 0) moles. At equilibrium, the follouing equation applies:
After Equation 13 is solved for y . the following adjustments are made : b = bo
- '/ d
(14)
d=do+y
(15)
n = no
(16)
+ '/~y
To solve for y, Equation 13 is expanded :
+
P(4P K,*)- y ( 2 ) [ K u 2 ( bo no) 4PdoI 4(PdO2 - K,*b,n,) = 0 (17)
+
When higher-order terms are discarded Y is approximated:
After a value of c is obtained from Equation 18, the estimated composition is revised by means of Equations 14, 15, and 16. This sequence should be repeated \\Thenever y is large, before proceeding to the next adjustment factor Quantities b and n contribute to both the K, and the K , reactions. Any changes in b and n. in making the composition conform to the K , equilibrium, upset the rC, equilibrium; any changes in b or n due to the R, equilibrium affect the K , equilibrium. After b and n are revised by adjustment factor y, adjustment factor x must be re-evaluated ; after they are revised by adjustment factor x. adjustment factor y must be re-evaluated. The composition calculated bv means of the x and the 3. adjustments approaches the correct composition as progressively
smaller adjustment values are computed. Finally, the composition is adjusted by trivial amounts, which means the equilibrium composition is established. Convergence proceeds rapidly if each molecular species appears in more than one equilibrium-reaction equation. Convergence is helped, too, where one molecular species appears in all the reactions: and more equations are used than there are unknowns. A four-component system can be solved with only two equilibrium equations. in addition to t\vo material-balance equations. But the system would converge more readily if a third equilibrium equation were added. Adjustments in composition may be made either after each, or after several adjustment factors are calculated. Both methods have been used and both occasionally have led to divergence. On the other hand, both methods of adjustment have been used in solving problems where the initial composition estimates were extremely poor. Convergence is most likely to occur if adjustments are made after each factor is evaluated, providing that an adjustment sequence is repeated immediately whenever that factor is large. Typical Application
+ CO? 5 H20 + CO K e , . .H, + ' / ? O ?e HrO
K,. . . H
