Ind. Eng. Chem. Process Des. Dev. 1984, 23, 851-854
85 1
is well to the right. The overall reaction can be written approximately as CO + UHz a(CHz), + bC02 + cHzO where U is the usage ratio and n is typically in the neighborhood of 3 for a representative average molecular weight distribution. At constant apace velocity CO conversion will increase with temperature. At low conversions, c = 0. As conversion increases, c increases and therefore b must decrease (from an oxygen balance) and consequently a must increase (from a carbon balance). Acknowledgment This study was supported by the Office of Fossil Energy, U.S.Department of Energy under Grant No. DE-FG2281PC40771.
borhood of C8 cannot be ascertained from their results. Their data from Cll to about C43on a Flory plot with a sulfur-poisoned catalyst could be fitted with two straight lines with a slight break at about Czo-Cz5. From their Figure 5 we estimate a = 0.84, based on the Cll-Cz0portion of the distribution and a = 0.90, based on the C25-C40 portion. The data for the unpoisoned catalyst yield the same value of a for the Cl1-Cz0 portion but above about C25the data are too scattered to be meaningful. COz Formation. Madon and Taylor reported information on the percentage of CO reacted that appeared as C02in the products or as CH4. These studies covered the range of 230-277 “C, at constant space velocity and (H,/CO) feed ratio of 1.5. They observed in the range of 240-250 “C a marked drop in the percentage of CO reacted that appeared as COP and a marked increase in the CO reacted that appeared as CHI. The C2-C4 selectivity increased in proportion to that of CHI but the C5+product did not. They suggested that a rapid change in the catalyst surface occurred above 240 “C. We do not have similar data to compare the C144 product group to the C5+group but in our slurry reactor studies the percentage of CO reacted that appeared as C02decreased steadily with increasing temperature and that appearing as CHI increased. These are in the same directions as observed by Madon and Taylor but there was no marked change in the 240-250 “C or in any other limited temperature region. The above trends come about as a natural consequence of the stoichiometry of the overall reaction and the equilibrium of the water gas shift reaction. The primary Fischel-Tropsch reaction is the formation of hydrocarbons (and some oxygenates) and water. On an iron catalyst the H20 in turn reacts rapidly with CO to generate Hz and C02 by the water gas shift reaction, the equilibrium for which
-
Registry No. Fe, 7439-89-6; Cu, 7440-50-8; K,7440-09-7; CO, 630-08-0.
Literature Cited Felmer, J. .; Sllveston, P. L.; Hudglns, R. R. Ind. Eng. Chem. Prod. Res. -Dev. 1981, 2 0 , 609. Huff, 0. A., Jr. W.D. Thesis, M.I.T., 1982. Huff, G. A., Jr.; Satterfleld, C. N. Ind. Eng. Chem. Fundam. 1982. 21, 479. Huff, 0. A., Jr.; Satterfleld, C. N.; Wolf, M. H. Ind. Eng. Chem. Fundam. 1983, 22, 258. Huff, G. A., Jr.; Satterfield, C. N. J . Catal. 1984, 85, 370. Madon, R. J.; Taylor, W. F. J. Catal. 1981, 69, 32. Satterfleld. C. N.; Huff, 0. A., Jr. J. Catal. 1982, 73, 187. Stenger. H. G.; Johnson, H. E.; Satterfield, C. N. J. Catal. 1984, 86, 477.
Department of Chemical Charles N . Satterfield* Engineering Harvey G. Stenger Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Receiued for review May 5, 1983 Reuised manuscript received November 21,1983 Accepted December 9, 1983
Some Kinetic Design Considerations in the Fischer-Tropsch Synthesis on a Reduced Fused-Magnetite Catalyst The results of calculations, based on a newly developed kinetic model, show here the extent to which the simple first-order kinetic expression commonly used will be in error, as a function of degree of conversion and other variables. The effect of conversion on HJCO usage ratio is also presented.
equate and the extent to which this assumption is in error as higher degrees of conversion are encountered. In previous modeling it has also been generally assumed that the (Hz/CO) usage ratio was independent of conversion. The adequacy of this assumption is also analyzed. Chemistry of the Process In general, the stoichiometry for the Fischer-Tropsch reaction may be expressed as nCO ( m + n)Hz C,H,, + nH20 (1) In our kinetic study the average organic product composition was “C,H,”; i.e., n = 3 and m = 3.5 in eq 1 (Satterfield and Huff, 1982). Huff and Satterfield (1984) measured the rate of carbon monoxide plus hydrogen conversion (-RH,+CO) in a semicontinuous slurry reactor at 232 to 263 OC, 445 to 1480 kPa, H2/C0 molar feed ratios of 0.55 to 1.8, hydrogen conversions of 17 to 68%, and carbon monoxide conversions of 16 to 98%. The inhibiting effects of carbon monoxide and
Introduction In recent proposed models for the Fischer-Tropsch synthesis on iron-based catalysts in slurry reactors or bubble columns (Satterfield and Huff, 1980; Deckwer et al., 1981,1982), the rate of synthesis gas consumption has generally been taken to be dependent only on the hydrogen concentration to the first power, based on Anderson’s report (1956)that this was valid for synthesis-gas conversions below about 60%. Although it has been known that at higher conversionsthe rate is inhibited by H20,the paucity of quantitative information has militated against trying to incorporate a more accurate, kinetic expression into such models. We (Huff and Satterfield, 1984) recently conducted a systematic kinetic study on a reduced fused-magnetite catalyst in a well-stirred slurry reactor over a wide range of reaction conditions. Based on those results, we have calculated and present here graphs showing sets of conditions for which the simple first-order expression is ad0196-4305/84/1123-0851$01.50/0
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American Chemical Society
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852
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
water were quantitatively determined and the following rate expression was developed. -RH2+C0
= ab'PCflH,2/(PHz0 +
bFC8Hz)
(2)
+
Here a = 2.39 X 108e(-19700/R" pmol of H2 CO converted/g of unreduced cat.-min-kPa, b' = 9.50 X 106e(-24000/Rn Pa-', T i s temperature (K), and R = 1.987 cal/K. At low conversions &e., b'Pc$H, >> PH~o), eq 2 reduces to the simplified first-order form proposed by Anderson (1956) -RH~+CO
=
(3)
Water, formed as a primary product of the FischerTropsch reaction (eq l),reacts with carbon monoxide by the water gas shift reaction CO + H2O = COZ + H2 (4)
Keg = 0.0102e(4730/n
1.3
A 263'C
0''
HzICO = 0 5 5 790 k P o
(5)
(6)
The equilibrium is far to the right at the temperatures of interest in Fischer-Tropsch synthesis. Based on the reaction stoichiometry of eq 1 and 4, the partial pressures of hydrogen, carbon monoxide, and water and the (H2/CO) usage ratio, U , can be calculated by the following procedure. Take as a basis 1mole of CO and F moles of H2entering. F is then the inlet feed ratio of H2/C0. Assume that the water gas shift reaction proceeds to equilibrium. Let z be the number of UCnH2mn moles formed by eq 1and y be the number of COz moles formed by eq 4. Material balances result in the following ( m + n)z - y U= (7) nz + y
H 'Z
10
ti2
pC0zpH2
PCflHzO
The equilibrium constant, Kq, as a function of temperature T (in K) is given by (Rossini et al.)
- (zn + m)z
-
C O N V E R S I O N OF CO
Figure 1. Deviation from first-order expression as a function of syngas conversion; well-mixed reactor.
Ronga C o l c u l o l Q d f o r
i; 0 2
XCO+H2
08
06
04
FRACTIONAL
This reaction proceeds essentially to equilibrium on an alkalized iron catalyst under most conditions, in which case the product composition corresponds to that predicted by
Keq=
02
0
( F + 1)
T ( F - mz - nz + y) = ( F + 1 + z - mz - nz) T ( l - nz - y) = ( F + 1 + z - mz - nz)
(10) (11)
where P is the total pressure and XH+ c and ~ XCOare the fractional conversions of hydrogen p k ~ carbon s monoxide or of carbon monoxide, respectively. The equilibrium relationship given by eq 5 becomes y ( F + y - mz - nz) (13) Keq = (nz - y)(l - 71.2 - y) For our specifk product composition, n = 3 and m = 3.5, so z can be calculated from eq 8 given the fractional conversion of carbon monoxide plus hydrogen and the feed ratio. Substitution of z into eq 13 allows y to be calculated for a specified temperature.
Condillon5 G i v e n ( n F i g 1
c
\A4 01
32
03
FRACTIONAL
34
35
CONVERSION
06 OF
07
08
09
12
CO
Figure 2. Deviation from first-order expression as a function of carbon monoxide conversion; well-mixed reactor.
Results and Discussion The ratio of rates predicted by eq 2 and 3 is calculated as detailed rate (eq 2) 1 (14) first-order rate (eq 3) (1 + PHzO/b'Pc&'HJ Figure 1shows the ratio calculated from eq 14 as a function of conversion of H2 + CO, for various combinations of pressure, temperature, and inlet gas H2/C0 ratio. A set of representative data points at 263 "C is also shown for three different inlet (Hz/CO) ratios and various degrees of conversion. There is close agreement between the kinetic model and experimental data obtained in a wellmixed reactor over a wide range of conditions. Over this range, the simplified first-order rate expression (eq 3) is applicable at synthesis gas conversions below 40 to 70%. The simplified first-order rate expression becomes increasingly less accurate as carbon monoxide is depleted. This is illustrated in Figure 2 where the ratio of rates (eq 14) is plotted against carbon monoxide conversion as such for the same range of parameters shown in Figure 1. Individual curves are not depicted in Figure 2 to avoid confusion since the range of extremes is small when CO conversion is used as the variable. The lowest line of the envelope of curves corresponds to 445 kPa, 263 "C,and H2/C0 feed ratio of 1.8. The highest line of the envelope occurs at 790 kPa, 263 OC, and H2/C0 feed ratio of 0.55. The other combinations of temperature, pressure, and H2/C0 feed ratio shown in Figure 1 give results lying within the envelope. Usage Ratio. A theoretical plot of H2/C0 usage ratio (calculated by eq 7) as a function of syngas conversion is shown in Figure 3. Over a range of conditions of interest (232-263 O C , 445-1480 kPa, and 0.55-1.8 H2/C0 feed ratio), the curves vary significantly only with H2/C0 feed ratio because the water gas shift reaction, which alters the usage ratio, is independent of total pressure (eq 13) and is weakly dependent on temperature (eq 6). The usage
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
X 263’C.H2ICO = 1.8, 7 9 0 k P O . Z ~ ~ ’ C , H ~ / C: aO9 0 , 7 9 0 k p o A 2 6 3 . C . Hz/CO :0.55,790 kP0
853
”
445-1480 k
K
w
I
v) 4
2
3
0
2 I N I 01
0.51
A
I 0.2
0.3
0.4
0.5
I 0.7
0.6
F R A C T I O N A L CONVERSION
I QS
1 0.9
I
I
I
I
I
I
0.2
0.3
FRACTIONAL
OF H2.CO
Figure 3. Deviation of Hz/CO usage ratio from a constant value as a function of syngas conversion. 1.51
0.1
1.0
I
0.4
0.5
0.7
0.6
CONVERSION
0.8
OF H2
0.9
1.0
CO
Figure 6. Effect of syngas conversion on H2/C0 usage ratio for selected product compositions: 263 O C , 790 kPa, (H,/CO)j,, = 1.8.
1
I
(H~ICO),,=l.8 263.C
263.C 7 9 0 kP0
\0,90
I
1
0.5
I
1
I
0.1
0.2 Q3 0.4 FRACTIONAL
I I 0.7 0.8 C O N V E R S I O N OF C O 0.5
I 0.6
I
0
0.9
1.0
1
0
1
I
a2 FRACTIONAL
Figure 4. Deviation of Hz/CO usage ratio from a constant value as a function of CO conversion. 1.0
I
0.4
I
I
‘ I
I
0.6
0.8
I
1.0
CONVERSION O F C O * H2
Figure 7. Deviation from first-order expression as a function of syngas conversion: well-mixed reactor; no water gas shift reaction; product. 263 “C, 790 kPa, YCgH7n
0.8 0.6 0.4
““_
0.4
0.2
0
0
01
02
03
04
FRACTIONAL
0 5
06
CONVERSION
07
OF
OB
09
10
H2+CO
Figure 5. Deviation from first-order expression aa a function of syngas conversion for selected product compositions: 263 O C , 790 kPa, (H,/CO)h = 1.8.
ratio is essentially constant and similar for the different H2/C0 feed ratios at low syngas conversions; however, the usage ratio increases greatly at high percent conversions as the consumption of CO approaches completion. This is emphasized by Figure 4. No change in the usage ratio is seen in Figure 3 and Figure 4 for an H2/C0 feed ratio of 0.55 as carbon monoxide and hydrogen are being consumed a t about the same rate (0.59 H2/C0 usage ratio). The data plotted in Figure 3 are for the same runs reported in Figures 1 and 2 and agree well with the curves theoretically predicted. Effect of Product Composition. Depending upon operating conditions, the product may vary in net H/C ratio and in average molecular weight. For present purposes the latter may be characterized by a single value of a in the Flory molecular weight distribution. Figures 5 and 6 show some representative effects of these two variables on the ratio of detailed rate to first-order rate and on the usage ratio. The choice of “C3H7”as a representative product corresponds to a = 0.67. Figure 5, which may be compared to Figure 1, and Figure 6, which may be compared to Figure 3, show the effect of changing the net H/C ratio for a value of a = 0.67, expressed as ‘C&” and “C3HSm.The effect of a change in a is shown by UC3H6n
O2I O
0
l 0.2 0.4 0.6 FRACTIONAL CONVERSION OF CO * H2
0.8
Figure 8. Deviation from first-order expression as a function of syngas conversion; plug flow reactor.
(a = 0.52) and “C4Hg” ((u = 0.75).
Effect of Water Gas Shift, The above calculations assume that the water gas shift proceeds to equilibrium. If this is not so, the firsborder rate expression will be much more in error. This is illustrated by Figure 7 for the limiting case in which the water gas shift is assumed to be negligible. Plug Flow Reactor. All of the above results pertain to a differential reactor or a continuous stirred tank reactor (CSTR). The same type of analyses can be applied to a plug flow reactor (PFR) in which eq 14 is integrated along the reactor. Figure 8 shows representative results, that may be compared to Figure 1. As would be expected, the degree of error associated with the use of the first-order expression is less for a PFR than for a CSTR for a specified degree of conversion. Figure 9 compares the CO conversion to be expected in a CSTR with that in a PFR at the same contact time, for a representative set of operating conditions and three inlet H2/C0 ratios. Conclusions and Summary From our own data and analyses of previous kinetic studies it appears that the form of eq 2 gives the best representation presently available of the rate of Fischer-
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 854-857
854
kinetic expression, eq 3. The assumption of a simple first-order rate equation and constant usage ratio is valid to high degrees of conversion if the initial (H,/CO) ratio is quite low (e.g., 0.55). When this ratio is high (e.g., 1.8) the rate deviates increasingly from first order at high conversions and the usage ratio likewise changes significantly.
c
Registry No. CO, 630-08-0; magnetite, 1309-38-2.
//
HzKO a 0 5 5
Literature Cited
iI
/ 0
1
I
42
0
CO
I
I
I
l
~
0.4 0.6 0.0 CONVERSION I N A PFR
!
I
1
1.0
Figure 9. Comparison of CO conversion in a CSTR and a PFR for the same contact time: 263 O C , 0.79 MPa, T3H7* product.
Tropsch synthesis on iron-based catalysts. We cannot expect it to apply precisely quantitatively to other types of iron-based catalyses or even necessarily to reduced fused magnetite catalysts of different composition or reduced in a different manner. However, in the absence of specific and precise data on an iron catalyst of interest, eq 2 provides a basis for design estimates and interpretation of experimental data. Figures 1,2, and 8 show representative sets of conditions for which eq 2 should be used instead of the first-order
Anderson, R. B. I n "Catalysis", Voi. 4; Emmett, P. ti., Ed.; Reinhold: New York, 1956. Deckwer, W.-D.; Serpemen, Y.; Ralek, M.; Schmldt, 8. Chem. Eng. Sci. 1981, 38,765. Deckwer, W.-D.; Serpemen, Y.; Ralek, M.; Schmidt, B. I d . Eng. Chem. Process Des. Dev. 1982, 21, 231. Huff, G. A., Jr.; Satterfleld, C. N. Ind. Eng. Chem. Process Des. Dev. 1984, in press. Rossinl, F. D.; Pltrer, K. S.; Taylor, W. J.; Ebert, J. P.; Kllpatrlck, J. E.; Beckett, C. W.; Williams, M. G.; Werner, H. G. A.P.I. Research Project 44. Satterfield, C. N.; Huff, G. A., Jr. Chem. Eng. Sci. 1980. 35, 195. Sanedieid. C. N., Huff, G. A,, Jr. Can. J . Chem. Eng. 1982, 6 0 , 159.
Department of Chemical Engineering George A. Huff, Jr. C h a r l e s N.Satter€ield* Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Received for review September 27, 1983 Accepted January 24, 1984
Multivariable Control of Non-Square Systems A ternary distillation column is used as an example of how Rosenbrock's Direct Nyquist Array method can be used to design a multivariable control scheme for processes with Don-square transfer function matrices. A pre-compensator, which is analogous to the inverse gain array for square systems, is used to square and decouple the system. The form of the compensator also permits constraint of actuator movement.
Introduction The problem of controlling a process with an unequal number of manipulated and controlled variables arises fairly often. The usual solution when designing a feedback control scheme is to square the system by eliminating or adding variables. This can lead to an increase in hardware expense if an actuator is to be added or poor performance if a controlled variable is to be dropped. Among the formal approaches to solution of this problem is to apply linear programming if a steady-state control scheme is adequate or schemes such as Dynamic Matrix Control due to Cutler and Ramaker (1979). These methods require considerable computing power and may be inappropriately complex for many problems. MacFarlane et al. (1978) and Kouvaritakis and Edmunds (1978) developed a method to design a compensator which squares down systems with more measured than manipulated variables. The method is based on frequency domain and state space design, and focuses on placement of the multivariable zeros generated in the squaring down process. The method developed here avoids the need to knock out controlled variables or square up by adding actuators. In contrast to the work of Kouvaritakis and Edmunds (1978) it retains the relative simplicity of square multivariable control schemes, using only a proportional controller for each controlled variable and a constant gain pre-compensator derived from the quadratic inverse or pseudo-inverse of the non-square gain matrix. A ternary
distillation control problem suggested by Doukas and Luyben (1978) is used here to illustrate the method and simulation results are presented as support for the technique. Process Description The process, shown in Figure 1, was suggested by Doukas and Luyben (1978) as the most economical configuration for separating a mixture of benzene, toluene, and xylene. The problem posed was to control the concentration of four impurities in the three product streams. Since there were only three manipulated variables, reboil duty, reflux ratio, and side stream flow rate, they decided to square the system by adding the position of the sidedraw to the set of manipulated variables. The distillation column was configured so that the side stream could be taken from any one of five trays. This is a very common configuration for side draw columns; however, side draw position is normally controlled by manually operated valves. It would be necessary to increase the number of remotely actuated valves, from 3 to 8, in order to realize this design in the field. This is clearly a very expensive scheme. As an alternative the scheme that will be proposed here controls the four impurities with only three manipulated variables. Compensator Design The transfer function matrix shown in Table I is that given by Doukas and Luyben except that their column
Ol96-43O5l84Ill23-O8548Ol.5OlO 0 1984 American Chemical Society
