-
that for slow reactions - c ~ g is a t least not appreciably greater than the value calculated for very slow reactions. The extrapolated data for y = 21.9 (Vassilatos and Toor, 1962) are compared to the predictions of Equation 35 in Figure 2, where the homogeneous conversion curve is also shown. This is a rapid reaction by the authors’ definition. Equation 35 falls below the extrapolated data, and -ZB is greater than the values predicted for very slow reactions. Thus, apparently -GB goes through a maximum as the reaction velocity constant (or y ) varies from very small to very large values, but further data with stoichiometric mixtures are needed to settle this point.
u
Acknowledgment
A B
The author is grateful to the National Science Foundation for support of this work and to Kao-Wen Mao for the computations. Nomenclature
C = concentration c = concentration fluctuation d ( 2 ) = decay law D = diffusivity = dimensionless concentration, Equations 8 and 9 f k = reaction velocity constant 6: = operator, Equation 5 n = stoichiometric number 7 = rate of chemical reaction per unit volume R = correlation coefficient
V u)
2
y 2
= velocity in 2 direction = velocity vector
= mass flow rate = dimensionless concentration = dimensionless distance = distance
GREEKLETTERS y e
0
= first Damkohler number = distance normal to surface = time
SUBSCRIPTS AND SUPERSCRIPTS
0 1
2
= species A
I
= species B = inlet to reactor = jet set 1 = jet set 2
03
= very rapid reaction
m
= mixing distance
- = time average Literature Cited
Kattan, A., Adler, R. J., A.Z.Ch3.J. 13, 580 (1967). Keeler, R. N., Petersen, E. E., Prausnitz, J. M., A.Z.Ch.E.J. 11, 221 (1965).
Mao, K. W., Toor, H. L., A.Z.Ch.E.J. in press, 1969. Toor, H. L., A.I.Ch.E.J. 8, 70 (1962). Vassilatos, G., Toor, H. L., A.Z.Ch.E.J. 11, 666 (1965).
RECEIVED for review June 27, 1968 ACCEPTEDApril 10, 1969
EQUILIBRIUM COMPOSITION WITH MULTIPLE REACTIONS H. P. M E I S S N E R Massachusetts Institute of Technology, Cambridge, Mass.
C.
L. K U S I K
Arthur D. Little, Inc., Acorn Park, Cambridge, Mass.
W.
O,t?l,$O
H. D A L Z E L L
Massachusetts Institute of Technology, Cambridge, Mass. The reactor series method is proposed for calculating the equilibrium composition of a system in which many reactions occur. Any system of multiple reactions, homogeneous or heterogeneous, is reduced to a series of individual reactions occurring separately. The rnaior advantage lies in the simplicity of the calculations required, since only one reaction occurs at a time, with simple algebra involved per step. This method is suited to either hand or computer calculation.
IT IS often necessary to calculate the equilibrium composition of a system of materials undergoing chemical reaction. Regardless of the number of reactions occurring, there is no difficulty or novelty in formulating the simultaneous equations needed for such calculations-the equilibrium constant and the mass balance equations. With only a single reaction, solution of these equations is relatively simple, but calcula-
tions become more involved with two reactions, and are often extremely complex and laborious when three or more reactions occur. Several techniques for determining the equilibrium composition in such complex systems have been published, as recently reviewed by Zeleznik and Gordon (1968). These methods generally involve iterative solution of the full set of equations, and are most readily applied with the aid of VOL.
8
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Table 1.
Mole Table for Reaction 1, Example 1 1.0 mole feed, 657, Nz, 35% H2S
Basis.
Feed Component
Gas, Mole
HzS
0.350
S?
0 0 0.650 -1.000
HP NZ Total
Product Gas Partial Product Equilibrium, Pressures, Gas, Moles Moles Atm. 0.156 0,114 0.350 x 0.071 X/2 0.097 0.194 0,141 X 0.650 0.474 0.650
-
1.0 -t 4 2
-
___
1.097
0.800
high-speed computers. The object here is to propose an alternative procedure, called the reactor series method, for making such calculations. This method is applicable to any system regardless of the number of components and reactions involved, and is algebraically simple because in each calculational step only one reaction a t a time is allowed to occur. Equilibrium with a Single Reaction
It is convenient to introduce discussions of the reactor series method with a brief restatement of the usual procedure for determining the equilibrium composition in a system involving only one reaction. At the outset, a table is usually prepared listing the moles of each constituent in the feed, after picking any convenient quantity of feed as a basis. One of the chemical species present is arbitrarily chosen as the key component, and the moles of this component formed or consumed are designated as x. The moles of the other components present in the equilibrium gas are then expressed in terms of 2, after which an equilibrium constant relation is formulated and solved for x. The use of such a mole table implicitly maintains the atom ratios in the feed equal to those in the equilibrium mixture, and satisfies the criterion that the system's total pressure is the sum of the partial pressures. Example 1. What is the equilibrium composition a t 1500" K. and 0.8 atm. of a system initially containing 35y0 H2S and 65% Nz? It is assumed that only the following reaction occurs: HzS = Hz
+ 3S2,
Kp(1500" K.) = 0.334
(1)
ASSWER. Choosing one mole of feed as a basis, the number of moles of the feed constituents are as listed in Table I. Selecting hydrogen as the key compoiient, the nioles of HZ formed by Reaction 1 are designated as 2 . The moles of the other constituents in the product gas, expressed in terins of 2, are as listed in the third column of Table I. The equilibriuiii constant equation for Reaction 1expressed as partial pressures or in terms of 2 and T , the total pressure, is then:
extended directly to the case in which R independent reactions occur. Thus, for a system in which the independent reactions 1, 2, 3, , . ., R are all significant, the following imaginary steady-state process is operated: Cause the feed to enter the first of a row of isothermal reactors, through which flow is in series. Each reactor contains a selective catalyst which causes one of the reactions to proceed to equilibrium but prevents all other reactions from occurring. Kumbering the reactors in sequence, Reaction 1 occurs in reactor 1; 2 in 2; 3 in 3; etc., and R in R. Next this reaction cycle is repeated, again with Reaction 1 occurring in reactor R I ; 2 in R 2; 3 in R 3; and R in R R. The same cycle of reactions is repeated in the next R reactors, and then repeated again, until all of the independent equilibrium expressions are simultaneously satisfied within a specified accuracy. In each reactor, all species, except those involved in the single reaction occurring, act as inerts.
+
+
+
+
The reactor series method is most simply demonstrated in a system in which only two independent reactions occur. Such a case is obtained by replacing some of the NZwith SO2 in the feed of Example 1, as illustrated in Example 2. Example 2. A gas mixture initially containing 35% HzS, 15y0SOz, and 50% Nz is to attain equilibrium a t 1500" K. and 0.8 atm. The two independent reactions assumed to occur are Reaction 1 followed by Reaction 3. 2H2S
+ SO2 = 2H20 + 3/2 Sz;
(K,)150O0 K. = 30.2 (3)
What is the composition attained when the equilibrium constants are satisfied to within 2%? AXSWER. Pass the feed gas through reactor 1, where only Reaction 1 occurs, with both S2 and SO2 acting as inerts. Since this feed gas contains a total of 65% inerts, the calculations and composition of the off-gases from this reactor are the same as for the reactor of Example 1. This off-gas, whose composition is listed in Table 11, now flows through reactor 2 where only Reaction 3 occurs. The moles of SO2 consumed in reactor 2, per mole of feed to reactor 1, are designated as y, and the moles of each constituent in the product gas from reactor 2 expressed in terms of y are shown in the third column of Table 11. The equilibrium constant equation for Reaction 3 is:
30.2 =
+
( 2 ~(0.097 ) ~ 3 ~ / 2 ) (0.8)1'2 ~'~ (0.156 - 2y)'(0.15 - y ) (1.097 $y)'"
+
(4 1
Solving, y is found to equal 0.066, and the moles of the other constituents are as listed in the last column of Table 11. The gases leaving reactor 2 have completed one cycle; all independent reactions have been used. The degree to which this gas is now a t equilibrium with respect to Reaction 1 can be tested by substituting the composition of this product gas Table II. Mole Table for Reactor 2, Example 2
Solving when R, equals 0.334 and T equals 0.8 atm., x is found to be 0.194 and the equilibrium partial pressure of H2 is therefore x ~ / ( l q ' 2 ) or 0.141 atm. The moles of the various components a t equilibrium in this system are listed in the fourth column of Table I resulting in the partial pressures shown in the final column. The same numbers of significant figures are retained in 1: as are present in K,.
Basis. 1 mole of feed to reactor 1
+
Component
- 2y
0.156 0.194
so2
0.097 0.150 0 .oo
0.097 t 3y/2 0.150 y 0 2Y
SZ
The method of solving for the equilibrium composition when only a single reaction is involved, as just outlined, can be
Total
FUNDAMENTALS
Gas, Moles
0.156 0.194
HzO
I&EC
Reactor 2 Product
HzS H? (inert for reactor 2)
Reactor Series Method
660
Reactor 2 Feed Gas, RIoles
Nf (inert)
0.500
1.097
+ 0.50
_-___1.097
+ y/2
Equilibrium Product Gas, Moles 0.023 0.194
0.196 0.084 0.132
0.500 -1.129
Table 111.
Reaction of Hydrogen Sulflde with Sulfur Dioxide, Composition leaving Reactors, Example 2
Basis. 1 mole of feed to reactor 1 Cycle
2
1 ______-
3
-
_
4
5
-___--
6
Reactor
1
2
3
4
5
6
7
8
9
10
11
12
12
Equation
1
3
1
3
1
3
1
3
1
3
1
3
3
T = AH2
0.194
...
-0.086
0.066
y = AS02
...
-0.035
.
0.033
...
0.012
...
-0.009
...
...
-0.007
0.003
...
-0.002
...
0.001
0.001 Partial
Feed,
HZS
H? 82
so2
H20 N?
Pressure,
Moles
hIoles Moles
Moles
Moles Moles
Moles
Moles
Moles
Moles
Moles
Moles
Atm.
0.35 0 0 0.15 0 0.50
0.150 0.023 0.194 0.194 0.097 0,196 0.15 0.084 0 0.139 0.50 0.50
0.108 0.044 0.108 0.108 0.153 0.202 0.084 0.051 0.132 0.197 0.50 0.50
0.080 0.073 0.184 0.051 0.197 0.50
0.061 0.064 0.201 0.036 0.228 0.50
0.065 0.057 0.199 0.036 0.228 0.50
- - --
0.500
0.063 0.055 0.201 0.034 0.231 0.50
0.0465 0 ,0406 0.1483 0.0251 0.1706 0.3690
1.088
1.090
1.085
0.063 0.057 0.201 0.035 0.230 0.50 ___ 1.086
0.064 0.055 0.200 0.035 0.230
- --
0.069 0.064 0.195 0.039 0.221 0.50
1.084
1.084
0.8001
___
Moles
- -__
0.056 0.073 0.202 0.039 0.221 0.50
Total
1.097
1,129
1.085
1.102
1.085 1.091
(Kpr)sumrcut
0.334
3.14
0.334 0.939 0.914 30.4
0.334 0.500 8.04 31.1
(Kn)auparent
. . . 28.8
as listed in t,he last column of Table I1 into the equilibrium constant equation for Equation 1 : Ps21"Pn2 - (0.196)1/2(0.194) (0.8)1'2 = 0.334 PH2S (0.023) (1.129)1'2
~-
0.352 0.404 19.5 29.9
0.334 0.350 26.1 29.4
0.334 0.337 28.5 30.4
positive for all possible changes. In passing through each successive reactor, therefore, the system will always approach equilibrium more closely, but never move away from it.
(5)
Equilibrium for Reaction 1 is clearly not attained, since the value of 3.14 is far greater than 0.334, the actual equilibrium constant a t 1500" K. Additional reactors are needed; consequently, the gas leaving reactor 2 is fed to reactor 3 where only Reaction 1 can occur, then to reactor 4 where only Reaction 3 occurs, then to reactor 5 where only Reaction 1 occurs, etc. The results of calculations for reactors 1 through 12 are tabulated in Table 111, along with the associated values of 2 and y. Segative valueh indicate that the particular reaction in the converter under consideration proceeded to the left to reach equilibrium. Also listed in Table 111are the apparent values for the equilibrium constants of Reactions 1 and 3 in the product gases leaving each reactor. Inspection shows that after completion of the sixth cycle (12 reactors), these apparent equilibrium constant values are within 2% of the true values of 0.334 and 30.2, respectively. The specified criterion for attainment of equilibrium has therefore been fulfilled. Convergence
The reactor series method outlined above and illustrated by Example 2 for two reactions can be applied to systems involving many reactions. For each additional independent reaction, another isothermal reactor is added to each cycle of calculations. Irrespective of the number of reactors in the series, calculations remain simple because only one reaction occurs per reactor stage. When a larger number of reactions are involved, of course, time will be saved by using a digital computer, and an example of a computer solution is presented in the Appendix. Regardless of the particular series of independent reactions chosen for calculation, the composition of the reacting systems will more nearly approach the final equilibrium composition in passing through each successive reactor. This follows from the second law of thermodynamics, which states that for a closed or steady-state system undergoing an irreversible (GF)T,,-is change, the change in free energy-namely, always negative until equilibrium is reached. As has been pointed out (Gihbs, 1948), ( ~ F ) T ,at , equilibrium is zero or
Alternative Reaction Schemes
There is clearly wide latitude in the choice of independent equations for calculating the equilibrium composition of a system. For instance, hydrogen can be produced by reaction of methane with steam a t 1-atm. pressure and 1100" F. The species expected a t equilibrium are CH4, HzO, CO, COz, Hz, and CZH4. The following three independent homogeneous gas-phase reactions might be used. K,(llOOO F.)
Group A
+ H20 ( 9 ) = CO + 3H2 CHI + 2H20 ( 9 ) = C02 + 4Hz CH4
2CH4 = CzH4+ 2Hz
0.41
(6 )
1.09
(7) (8)
1.23 X
Another possible group is: K,(llOOO F.)
Group I3
+ HzO ( 9 ) = CO + 3H2 2CO + 2Hz = COz + CH4 2CO + 4132 = C2H4 + 2H9.0
CHI
0.41
(9 )
6.5
(10)
7.3 X lo-'
(11)
Obviously, reactions in Group B are simple linear combinations of those in Group A. Another possible set, again derived from Group -4,are: Group C
K,(llOOO F.)
2CH4 = C2H4 4-2H2
+ 2Hz0 = 2C0 + 4H2 CO + HzO = C02 + Hz
CZH4
1.23 X le6 (12) 1.4 X lo5
(13)
2.65
(14)
It is apparent that of the different reactions listed in Groups A, B, and C, only three reactions are independent. Therefore, if the reactions in any one of these groups are at equilibrium, the reactions in all other groups are also a t equilibrium. Any one of the above three groups of reactions might be chosen for series reactor calculations, and many other groups could be proposed. VOL.
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To use the reactor series method effectively, that group of reactions should be chosen whereby a close approach to the equilibrium composition is attained with relatively few reactor steps. It will become evident that some groups of reactions allow much more rapid convergence to the equilibrium compositions than others. Before determining the characteristics of reaction groups which favor rapid convergence, it is helpful to define three types of reaction sequences. Reaction 7 is said to be parallel to Reaction 6, since the same reagents appear on the left side of each reaction. Reaction 10 is called sequential to Reaction 9, since the products of the preceding Reaction 9 are reagents for Reaction 10. Reaction 14 is called mixed, since both a product and a reactant of the pieceding Reaction 13 are used as reactants in Reaction 14. Two situations can arise in which convergence during reactor series calculations is slow, one with sequential reactions and one with parallel reactions. Slow convergence to the equilibrium composition with sequential or mixed reactions is illustrated by the reactions of Group C-that is, little C2H4 is formed in the first reactor because Reaction 12 has a very small equilibrium constant. Since little CzH4 is made available, little CO and H2 can form in the second reactor by the sequential Reaction 13, despite its large equilibrium constant. Because of the limited amounts of C o and Hz produced, Reaction 14 is in turn starved of reactants. Because of this starvation, conversion per cycle is minute, and many cycles are required to attain equilibrium. To avoid such reactant starvation with sequential reactions, the equilibrium constant of each reaction (except the last) should preferably be greater than unity. Obviously, the sequential reactions of Group C are ;badly chosen by this criterion, in that the first reaction of the series-namely, Reaction 12-has a very small equilibrium constant. Reactions 13 and 14 have constants of proper magnitude, but reagent starvation has already developed by using Reaction 12 as the.first of the series. The reactions of Group A obviously do not lead to such reagent starvation, and so Group A is to be preferred over Group C. As with sequential reactions, improperly chosen parallel (or mixed) reactions can also cause reactant starvation and consequent slow convergence. Consider the combustion of carbon disulfide with oxygen by the following sequence of parallel Reactions 15 and 16: Kp(800'F.) CS2 4- 302 = Con f 2S02,
CS2
+ 402
= COz
+ 2SO8,
3.82 X
(15)
2.55 X 10"
(16)
When feeding a gas mixture with a stoichiometric excess of 0 2 to the first reactor, little CS2 will survive in the product gas, since the equilibrium constant of Reaction 15 is so large. In consequence, little SO8 can form in the second reaction despite a large equilibrium constant for Reaction 16. Thus, Reaction 16 is starved of reagents. To avoid such starvation with parallel reactions, the earlier reaction should have a small equilibrium constant, preferably less than unity. When such a parallel reaction having a small value for K , cannot be found, a sequential reaction should be sought. Thus, Reaction 15 could be followed by Reaction 17:
With this sequential reaction, no starvation occurs, since Reaction 17 is now well supplied with SOz and excess oxygen from Reaction 15. 662
ILEC
FUNDAMENTALS
In most systems, more than one group of reactions can be found which will allow rapid convergence, and any one of these groups would be suitable for the determination of the system's equilibrium composition. Redundant reactions in a group are permissible, but they add more reactors per cycle, and so are disadvantageous when they do not reduce the number of cycles involved. Rules of Procedure
In dealing with a new system, the chemical species to be recognized as present are first identified. Various possible groups of independent reactions are thus formulated and that group is selected which promises rapid convergence by avoiding reagent starvation. In selection of a suitable group of reactions for operating at atmospheric pressure, the following should be observed :
A. To the extent possible, make the species present by sequential reactions having equilibrium constants greater than about unity. B. One of the reagents in the feed is usually in stoichiometric excess with reference to the first reaction occurring, and this excess reagent should be made to react with a product of a later reaction in a mixed type reaction. The product of a later reaction selected for this mixed reaction should be present in relatively large amounts and, therefore, will usually be made in a reaction having an equilibrium constant greater than unity. C. I t is often not possible to produce all the chemical species present by sequential reactions having equilibrium constants greater than unity. Make such species in parallel reactions having equilibrium constants less than unity, inserting these in the sequential reaction series a t points where the reagents involved in these parallel reactions are not yet reduced to starvation levels. In other than atmospheric pressure operations, the foregoing rules require minor modification. Instead of comparing values for K , to unity in A and C above, the quantity (Kp/+) should be compared to unity, where T is the total pressure in atmospheres, and a is the algebraic sum of the stoichiometric coefficients of the reactions involved. Thus, a values are: 2 in Equation 6, unity in Equation 8, - 3 in Equation 11, etc. If pressures are high enough so that deviations from the perfect gas laws become significant, the use of fugacities in equilibrium calculations in place of pressures is indicated. I n the reactor series method, as in any other calculation procedure, serious errors result if components comprising a significant fraction of the final equilibrium mixture are neglected. Minor components may safely be neglected, however, without affecting the equilibrium concentration of the major components. Method Applicability
The reactor series method for calculating the equilibrium composition of a system of multiple reactions is applicable to heterogeneous as well as to homogeneous systems, as shown in Example 4 in the Appendix. In heterogeneous systems, it is merely necessary to assume that the gas, liquid, and solid phases present travel together without separation through the reactors. The rules listed above for rapid convergence in homogeneous systems apply equally to heterogeneous systems. A reaction sequence chosen for one temperature or pressure may not be suited to another temperature or pressure because of changes in the values of K,/+ with temperature and pres-
sure. Again, a change in the feed composition may make a change in the reaction sequence desirable. I n the calculational procedure described, it has been assumed that the temperature as well as the feed composition and pressure of the reactant system is known. Frequently, the temperature is not specified and it is desired to find the temperature and corresponding equilibrium composition with a parameter such as entropy or enthalpy assigned. I n these cases, the composition is computed by the method described for a series of assumed temperature spaced to permit interpolation to the correct temperature. Comparison with Free Energy Minimization Method
Even though detailed comparison of different calculational techniques is not easy, general statements can be made about the series reactor method relative to the well-recognized free energy minimization method of White, Johnson, and Dantzig (White et al., 1958). Thus, when dealing with two and even three reactions, the series reactor method is normally not too laborious for hand calculations. This is less practicable with the free energy minimization method, since the change in total free energy with a relatively large change in concentrations, especially of the minority components, is sometimes small, so that slide rule accuracy is insufficient. Moreover, the reactor series method appears simpler, since the calculational procedure is easily set up and only a single reaction is considered a t a time, requiring only elementary algebra and making it unnecessary to invert matrices or solve several equations simultaneously. Selection of an initial gas composition is straightforward, in that t.he feed to the first reactor is always the feed actually contemplated. The equilibrium constant for each reaction involved in the reactor series is satisfied t,o the degree of accuracy required, regardless of whether minority or majoritj components are involved. With multiple. reactions, or when many cases must be calculated, it becomes advantageous with all calculational methods including the series reactor method to use high-speed computers, as illustrated in Example 5 of the iippendix. Comparison of met'hods in computer calculation is, of course, uncertain a t best, depending on differences in coding times, debugging times, compiling times, and running times. Even when dealing with a specific problem, David (1969) has pointed out that productivity of programmers with 2 to 11 years' experience varied widely in completing solution of a specific problem. The ratios of worst to best times cited for coding time was 25 to 1, for debugging was 26 to 1, for code size was 5 to 1, and for running times was 15 to 1. Because of its simplicity, the series reactor method would appear to have advantages in coding times and debugging times. When many problems are to be solved, however, computer running time becomes the more important basis for comparison. The relative amount of running time required by the reactor series method and the free energy minimization method was therefore explored wit,h Example 5 of the Appendix. Solving by free energy minimization met'hod on an IBM 704, White, Johnson, and Dantzig reported running time t'o have been a "matter of seconds." This same problem, solved in Example 5 using the series reactor method on a newer CDC 6600 computer, involved a running time of 0.173 second for the eight cycles shown in Table VI. Considering the fact that core memory times on the IBM are one twelfth as fast as on the CDC 6600 (Luhowy, 1968; Rosen, 1969), a rough comparison between the series reactor met.hod and the free energy minimization method
indicates comparable computer running times. With computer charges running about $1000.00 per hour on the CDC 6600, computer running time costs were less than $0.05 for Example 5 . Only when many problems must be solved do computer running time costs become comparable to coding and debugging costs. Appendix
Example 3. Steam-Methane Reaction, No Carbon Formation. Methane and steam in the ratio of l to 5 react a t
1100" E'. and 2-atm. total pressure. What is the equilibrium gas composition, assuming the equilibrium constants are to be satisfied within 4%? SOLUTIOK.The sequence chosen involves Reaction 6 followed by the parallel Reaction 7. As in the previous examples, the feed mixture, consisting of 1 mole of CH4 and 5 moles of HzO (a total of 6 moles), is fed to reactor 1 in which only Reaction 6 occurs. With z moles of CO formed a t equilibrium, the equilibrium constant equation for this reaction becomes:
Solving, z equals 0.71. The moles of the various components in the product gas from reactor 1 are calculated from this value of 2, and are listed in Table IV. This gas mixture is next introduced to reactor 2, where only React,ion 7 occurs, forming y moles of C o nper 6 moles of mixed feed to reactor 1. The equilibrium constant equation for Reaction 7 in the product gas from reactor 2 is:
+
~(2.13 4 ~ ) ~ ( 2 ) ' = 1.09 (0.29 - y ) (4.28 - 2 ~ ) * ( 7 . 4 1 2 ~ ) '
+
(19)
Solving, y equals 0.22. The product stream from reactor 2 passes through reactor 3, where only Reaction 6 occurs; the product stream from reactor 3 passes through reactor 4, where only Reaction 7 occurs, etc. The moles of the various constituents in the equilibrium gases leaving selected reactors per 6 moles of feed to reactor 1 are listed in Table IV. Inspection shows that changes in the moles of the various constituents present become negligible after reactor 16, and that the equilibrium constants are simultaneously satisfied to within 4'% in the product gas from reactor 16. The composition of this gas therefore represents the desired equilibrium composition. The question of whether carbon will deposit from this gas remains to be answered. Carbon could form as follows:
CH4 = C
+ 2H2,
K , = 1.96
The actual value of ( P H , ~ / P ~as H calculated ,) from Table IV for the exit gas from reactor 16 is greater than 1.96. It follows that this heterogeneous reaction cannot occur in the product gas from reactor 16 and that the composition as reported in the last column of Table IV is truly the equilibrium composition. Example 4. Steam Methane Reaction with Carbon Formation. Methane and steam in the ratio of l to l react a t
l l O O o F. and 2-atm. total pressure. What is the equilibrium composition, assuming that the equilibrium constants should be satisfied within 2'%? SOLUTION.The reaction sequence chosen is 6, followed by 7, followed by the decomposition of methane by Reaction 20. As in the previous example, the feed mixture flows VOL.
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663
~
~~
~~
Table IV. Steam-Methane Reactions, No Solid Carbon, Composition leaving Reactors, Example 3
Basis. 6.0 moles of feed to reactor 1 Cycle
1
4
---
8
--
Reactor
1
2
8
15
16
16
Equation
6
7
7
7
7
7
Feed, Moles
Moles
Moles
Moles
Moles
Moles
Pressures, Atm.
CO2
1.0 5.0 0 0 0
0.29 4.28 0.71 2.13 0
0.07 3.85 0.71 3.00 0.22
0.19 3.65 0.27 2.95 0.53
0.22 3.62 0.19 2.94 0.59
0.22 3.62 0.19 2.94 0.59
0.057 0.957 0.050 0.777 0.157
Total
6.0
7.41
7.85
7.59
7.56
7.56
Partial
CH4' H2O CO
Hz
-
--
- -
energy minimization method and the series reactor method can be used in systems involving multiple simultaneous reactions. The former method was used to calculate the equilibrium composition of product gases formed when burning hydrazine with stoichiometric oxygen at 3500" K. and 750p.s.i.a. Ten components were assumed to exist in the product gas, as listed in Table VI, with moles per mole of feed as in the last column. Hydrazine concentrations were not reported, since this material was assumed to be completely decomposed under the conditions chosen, This same problem can now be solved by the series reactor method, again assuming that no hydrazine survives. The ten components of Table VI can be made by the following seven independent reactions.
1.998
0.5N2 = N
0.011322
(21)
0.0025877
(22)
0.20352
(23)
0.51840
(24)
4.9185
(25)
H2O = 0.5H2 f OH
0.26527
(26)
H20 = H f OH
0.15661
(27)
+ 0.5H2 0.5hTz + 0.502 0.5Nz
through reactor 1 where only Reaction 6 occurs, then through into reactor 2 where Reaction 7 occurs, then through reactor 3 where only Reaction 20 occurs. From mole Table V, the apparent equilibrium constant for Reaction 20-namely, ( P H Z / P C H ~ )in the gas entering reactor 3-is numerically (1.35)2(2)/ (0.578) (2.836) or 2.22, which is larger than the equilibrium value of 1.96 at 1100°F. In consequence, no solid carbon can form; hence the gas composition remains unchanged in reactor 3. This completes the first cycle. The gases now flow through reactor 4 where only Reaction 6 occurs, then through reactor 5 where only Reaction 7 occurs, and then through reactor 6 where only Reaction 20 occurs. I n this second cycle, methane does decompose in reactor 6, since the term ( P H ~ I P c H in , ) the feed gas to reactor 6 is numerically 1.79, which is smaller than the value of K , for Reaction 20. The amount of carbon produced is the difference between the atoms fed and the atoms of carbon appearing as CO, C02, and CHI, and it is assumed that this carbon travels in suspension with the gas through all the remaining reactors. This sequence of reactions is repeated in cycles 4 through 9, after which the change in gas composition per cycle is negligible and the equilibrium constants are satisfied to the desired 27,. The equilibrium gas composition after 9 cycles is presented in Table V. Example 5 . Both the White, Johnson, Dantzig free
Table V.
K p (3500" K.)
Reaction
~
=
NH
= NO
0.502 = 0 H2
+ 0.502
= H2O
These reactions are listed above in the proper sequence for solution by the reactor series method, in accordance with the rules formulated previously, with equilibrium constants and values of K p / f derived from the free energy data published (White et al., 1958). As in the previous examples of the reactor series method, the feed is introduced into reactor 1 in which only Reaction 21 occurs; the products of reactor 1 are fed into reactor 2 in which Reaction 22 occurs, etc. After seven reactors all reactions have been considered and cycle 1 is thereby completed. The gas composition at this point is shown in Table VI. The same sequence of reactions is now considered again: Reaction 21 occurs in reactor 8, Reaction 22 occurs in reactor 9, etc. The gas compositions after cycles 2, 4, and 8 are shown in Table VI. After 4 cycles, the gas compositions are found to satisfy the equilibrium constants of all reactions to within 2% and to within 0.2% after 6 cycles. After 8 cycles, these constants are satisfied to 0.02%, with no
Steam-Methane Reaction with Solid Carbon, Composition leaving Reactors, Example 4
Basis. 2 moles of feed to reactor 1 Cycle
---
1.o 1.o
co H. coz
0 0 0 2.0
Total Solid carbon atoms (K 1 ( KP61nD Darent ,::::K ;O t;:::
3
4
5
6
27
27
6
7
20
6
7
20
20
20
Moles
Moles
Moles
Moles
Moles
Moles
Moles
Atm.
0.668 0.668 0.331 0.994 0
0.578 0.487 0.331 1.35 0.090
0.578 0.487 0.331 1.35 0.090
0.674 0.584 0.234 1.06 0.094
0.630 0.496 0.234 1.24 0.134
0.608 0.496 0.234 1.28 0.134
0.589 0.567 0.126 1.25 0.153
0.438 0.421 0.094 0.932 0.113
2.661
2.836
2.836
2.646
2.734
2.752
2.685
1.998
0 0.420
0 1.44 1.08
0 1.44 1.08 2.22
0 0.410 0.173 1.26
0 0.765 1.09 1.79
0.024 0.856 1.28 1.97
0.132 0.41 1.10 1.97
0
--
0
...
... ~
664
I&EC
-
2
Equation
CH, Hz0
9
1
Reactor
Feed, Moles
2
1
FUNDAMENTALS
-_
...
-_
-
-
--
-
-
Table VI.
Hydrazine Combustion with Stoichiometric Oxygen, Example 5
Basis. 0.5 mole of hydrazine, 0.5 mole of oxygen in feed, 3500”K., 51.02atm. (750 p.s.i.a.). Mole per Mole of Feed, at End of Cycle
H Hz HzO
s
N? SH XO
0 0 2
OH
Feed
Cycle 1
0 1 .O 0 0 0.5 0 0 0 0.5 0
0.034206 0.236344 0,692648 0,001584 0.452193 0.001826 0,092203 0.067114 0.021027 0.105982
Cycle 2
0.039100 0.151878 0,777947 0.001392 0.482890 O.OOO847 0.031981 0.025282 0.032194 0.100400
attempts a t closer attainment appearing to be justified in view of the number of significant figures in the equilibrium constants listed. A comparison of the product moles calculated by the series reactor method with those calculated by the Khite, Johnson, and Dantzig method shows differences of only a few parts per thousand. As pointed out earlier, the computer time required by the series reactor method compares favorably with that of the free energy minimization method of Khite, Johnson, and Dantzig. If desired, other reactions, such as the formation of ethane, acetylene, and the like, can now be considered to
White, Johnson, Dantzig method
Cycle 8
Cylce 4
0.040445 0.147312 0.783418 0.001414 0.485409 0,000691 0,027075 0.017790 0,037158 0.097400
0.040668 0.147730 0.783155 0.001414 0.485247 O.OOO693 0.027399 0.017947 0.037314 0.096872
0.040657 0.147708 0.783178 0.001413 0.485256 O.OOO693 0.027381 0.017941 0,037311 0.096878
determine whether these constituents are present to a significant degree. literature Cited
David, E., Technol. Rev. 71, 6,58-62 (1969). Gibbs, J. W.,“Collected Works of J. Willard Gibbs,” Vol. 1, p. 91,Yale University Press, New Haven, Conn., 1948. Luhowy, G. M., Computer Characteristics Quart. 8 , 3, 12 (1968). Rosen, S., Computing Surveys 1, 1, 7-54 (1969). White, W.H., et al., J . Chem. Phys. 28, 751-5 (1958). Zeleznick, F.J.,Gordon. S., Ind. Eng. Chem. 60,6 , 27-57 (1968). RECEIVED for review August 23, 1968 A C C E P T E D June 12, 1969
GENERAL SOLUTION OF EQUATIONS REPRESENTING EFFECTS OF C A T A L Y S T D E A C T I V A T I O N I N FIXED-BED REACTORS KENNETH B. BISCHOFF Department of Chemical Engineering, University of Maryland, College Park, M d . 20742 Equations used to represent the time-dependent behavior of fixed-bed catalytic reactors with catalyst poisoning occurring have analytical solutions, to within quadratures, for arbitrary poisoning functions. Discussion is presented on earlier results which are special cases of the general solutions, and also ways by which bed-average poison concentration and catalyst activity ratios may be easily evaluated.
T HAS been only fairly recently that a concerted effort has the poisoning of catalysts, whether by coking or other means. One of the earliest contributions was by Voorhies (1945), who used a diffusion-like mechanism toderive that the amount of coke would be proportional to the square root of the time that the catalyst has been in contact with the reacting stream, or process time. Several other investigators experimentally found that this type of relation was approximately observed (Blue and Engle, 1981; Eberly et al., 1966; Panchenkov and Lolesnikov, 1959; Rudershausen and Watson, 1955; Tyuryaev, 1939; Kilson and den Herder, 1958). Other work showed roughly similar results, but seemed to indicate differences between the initial (
