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An experimental study of methane synthesis by concentration forcing
Ckmicd
Engineering Science, Vol. 45. No.
11, pp. 3221-3226,
ooo%2509/!m $3.00 + 0.00 0 1990 Per~amm F%v.!pepk
1990.
Printed in Great Britain.
AN
EXPERIMENTAL STUDY OF METHANE BY CONCENTRATION FORCING
SYNTHESIS
RAJIV YADAV and ROBERT G. RINKER? Department of Chemical and Nuclear Engineering,University of California, Santa Barbara, CA 93106, U.S.A. (First received 3 July 1989; acceptedin revised
form
2 February
1990)
Abstract-The production of methane from CO and Hz over an Ni-Al,OB catalyst was studied in a gradientlessreactor understeady-stateand pure component cyclingconditions,at a pressure of 1 atm and a temperature of 513 K. It was observed that the time-averagedproduction rate of methaneincreasedas the cycle period decreased.The highest rates were obtained for a cycle period of 12 s, the lowest investigated, and at a cycle split of 0.4 which coincideswith the composition of the optimum steady-state feed mixture. Time-averaged rates up to 7% higher than the experimentally determined optimum steady-state rate on a mole basis were obtained. To the best of our knowledge this is the first report on concentration forcing of the methanation reaction to appear in the open literature, and the results are encouraging. INTRODUCTION Concentration
forcing
In recent years, the traditional notion that the steady state is the best method of operating chemical reactors has been challenged. The first concerted challenges were launched by Bailey and Horn. In a series of pioneering papers, Horn and Bailey (1968), as well as Bailey and Horn (1969,1970,1971), showed that for a hypothetical reaction system, under certain conditions, forced oscillations of reactant concentrations lead to a set of selectivities which surpasses the set of all steady-state selectivities. Operating a reactor with oscillating feed composition is now recognized in the literature under several names: concentration forcing, forced cycling, and periodic forcing, among others. Bailey and Horn’s papers also provided a conceptual framework for interpreting and articulating concentration forcing research. This framework was further elaborated and crystallized in two reviews by Bailey (1973, 1977). One of the most important aspects of this development was the recognition of three distinct cyclic regimes. Thus, very high frequency operation was categorized as the relaxed steady state, very low frequency operation as the quasi-steady state and the in-between region as the intermediate cyclic regime. Bailey and Horn’s papers focused only on the relaxed steady state since it was perceived to be easily amenable to theoretical analysis. It was in this region that selectivities beyond the steady state were obtained. Although there are other important applications of concentration forcing (e.g. validating rate models) improving reactor productivity remains a compelling objective. At this point it is important to note that any improvement must be referential, and in the quest for improving reactor productivity an acceptable basis of comparison must be invoked. One such basis is what we choose to call the optimal steady state, i.e. that steady state which yields the highest attainable rate ‘Author
to whom correspondence
should be addressed.
for a given temperature, pressure, reactor configuration and size as determined by its residence time. The feed composition is completely unconstrained. Thus, such an optimal for a two-component feed can be easily depicted on a two-dimensional plot of rate vs feed composition. A number of investigators have demonstrated that production rates at such an optimutn steady state can be exceeded by time-averaged production rates obtained under concentration forcing conditions. It must be emphasized here that, for the comparison to be “fair”, periodic operation must be constrained exactly like the steady-state operation and again the feed composition must be the only unconstrained variable. Some specific instances where time-averaged rates higher than those at the optimum steady state have been reported are Wilson and Rinker (1982), Jain et al. (1983a) and Chiao et al. (1987) for the ammonia synthesis reaction; El Masry (1985) for the Claus reaction; and Prairie and Bailey (1987) for the ethylene hydrogenation reaction. In this paper results for rate enhancement of the methanation reaction, over a nickel catalyst, using concentration forcing are presented. To the best of our knowledge, this is the first such study of forced cycling of the methanation reaction to appear in the open literature. Methane synthesis The synthesis of methane from CO and Hz over a nickel catalyst according to the reaction CO + 3H, c-, CH, + H,O, is commonly known as the “methanation reaction,” and in this paper all references to methane synthesis or methanation should be construed to refer to it. This reaction is predominantly used to remove CO from hydrogen-rich synthesis gases for reactions like the ammonia synthesis reaction. However, in recent years interest in the synthesis of methane from CO-H, mixtures has been revived due to methane’s potential as a clean energy source from coal. Considerable effort is being expended worldwide to understand and explain this important reaction.
3221
3222
RAJIV
YADAV
and ROBERT
Obviously enhancing the production rate for the methane synthesis reaction is a desirable objective and concentration forcing, which has not been reported previously for the methanation reaction, presents itself as a possible technique for realizing such an objective. With our current understanding of cycled reactor behavior, one cannot establish u priori whether concentration forcing will yield rates higher than the optimum steady-state rate for the methane synthesis reaction. Hence an experimental study is warranted. This paper presents results of an experimental investigation of pure-component cycling for the methane synthesis reaction in a fixed catalyst bed surrounded by a well-mixed gas phase. These results show that time-averaged production rates higher than the optimal steady-state rate can be obtained through pure-component cycling of the methanation reaction. Again to the best of our knowledge, no such instance of rate enhancement beyond the optimal steady state has been reported for the methanation reaction in the open literature thus far. In fact, apparently this is the first report on an experimental investigation of the methanation reaction using concentration forcing. Although, as mentioned above, it appears that methane synthesis has not been the subject of concentration forcing studies, a study using the transient method by Underwood and Bennett (1984) deserves mention. In this study the authors used step response experiments to study the methanation reaction over an Ni-A1203 catalyst, the same as ours, in a oncethrough flow reactor, taking care that the flow rate and hence the residence time through the reactor was constant_ The crucial result for our purpose from this study is shown in Fig. 1. Figure 1 depicts the response of the methanation rate when the feed to the reactor was switched from 10% CO-H, to pure H,. It can be seen from Fig. 1 that, on switching the feed to pure H,, the methanation rate dramatically increased to more than 10 times the steady-state rate observed with the 10% CO-H, mixture. This dramatic rate increase, which can perhaps be ascribed to a hydrogen
G.RINKER
deficiency on the surface during the CO-H, part of the experiment, seems to suggest the methanation reaction as an attractive candidate for concentration forcing experiments. However, a note of caution must be included here: Underwood and Bennett (1984) did not report the optimum steady-state rate for their experiment. Hence even though the rise in rate observed by them was remarkable, the highest rate may not have exceeded the optimum steady-state rate. EXPERIMENTAL
VENT
0
1
2 Time
3 (min)
4
DETAILS
A simplified schematic of the experimental apparatus is given in Fig. 2; a more detailed diagram is given elsewhere (Schack et al., 1989). This apparatus was used for both steady-state and concentration forcing studies. The Whitey four-way ball valve (S) was used to switch between H2 and CO feedstreams, flowing at equal volumetric flow rates, to the reactor. With the valve in one position (A) H, flowed to the reactor while CO was bypassed to the vent via an empty reactor; with the valve in the other position (B) the reverse situation was obtained. A Whitey threeway valve was used to direct the CO stream either to the H, line or directly to S. For steady-state operation the three-way valve was positioned such that CO flowed to the H, line and S was kept in position A. Concentration forcing experiments were conducted by keeping the three-way valve in the other position and allowing S to switch between H, and CO streams. Of course, the same setup can be used for both purecomponent and mixture cycling. The gases used in the experiments were of the following quality: CO = CP grade, 99.5%, and H, = UHP, 99.995%. The gas flows were monitored and controlled with Linde mass flow controllers which were calibrated and their performance checked periodically. The switching valve, S, was driven by a Superior Electric Co. synchronous motor. The cycle split (the fraction of each cycle for which the valve S is
0 K Y E R
5
Fig. 1. Transient behavior of methanation catalyst replotted from Underwood and Bennett (1984). The catalyst surface was swept clean with helium, followed by exposure to a mixture of 10% CO in )Tz for . 2 min, ___ followed ---_-by exposure to pure H, for 1 mm, all done at 220°C.
GAS CHROMATOGRAPH
Fie. 2. Simalified schematic of the exuerimental aunaratus. _.
An experimentalstudy of methane synthesisby concentrationforcing in position B) was regulated using Potter and Brumfield time-delay relays. Additional details of the switching mechanism can be found in Wilson (1980). The reactor temperature was controlled using a Techne Tecam TC4D temperature controller. The thermocouple for measuring the reactor temperature was placed inside the catalyst bed. During steadystate experiments the reactor temperature exhibited no measurable variations (it was precise) but during cycling a variation of f 1°C was observed. This variation probably arises from the fact that pure-component concentration forcing entails switching between two feedstreams of quite disparate thermal conductivities. The effluent from the reactor was analyzed using a Hewlett-Packard 5750 gas cbromatograph equipped with a thermal conductivity detector. A 6 ft x _kin. Porapak-Q cohtmn was used. Samples of the effluent were automatically injected with a pneumatically driven eight-way gas sampling valve. Before being injected into the GC, the sample stream was physically integrated in a surge tank to obtain timeaveraged concentrations. The output of the GC was monitored with a Hewlett-Packard 3393A integrator. The GC was calibrated for CO and methane using nitrogen as an external standard. The analysis and measurement of the reactor effluent stream was rendered somewhat problematic by the presence of water. To obviate this difficulty water was removed from the effluent stream by passing the effluent through a bed of CaSO,. Thus, both the GC analysis and the measurement of the exit flow rate were done on a dry basis. Once the moles of methane in the effluent stream were determined, the total outlet “wet” flow rate was calculated assuming that the moles of water formed in the reaction were equal to the moles of methane formed. Carbon balances around the reactor checked to within 1%. The reactor was a Berty-type gradientless reactor placed inside a 1.0 liter autoclave from Autoclave Engineers. Details of the reactor system can be found in Schack et al. (1989). A commercial, proprietary BASF Rl-10 Ni-Al,O, catalyst was used. Before charging to the reactor the catalyst was crushed to 14/18 mesh size so as to eliminate the possibility of intraparticle gradients. Since the reaction is highly exothermic ( - 215 kJ/mol), the catalyst was diluted in a ratio of one volume of catalyst to 20 volumes of 14/18 mesh crushed glass to ensure that no intraparticle heat transfer gradients existed. Calculations showed that gas phase heat and mass transfer gradients were negligible. The reactor was operated at 1 atm pressure and 513 K. The residence time in the reactor was - 9.5 s. The lowest cycle time that was used was 12 s; hence cycling effects were not masked by bulk mixing within the reactor.
EXPERIMENTAL
RESULTS
AND
cp=
optimum steady-state production rate
-
The two parameters which have the greatest effect on cycled reactor behavior are the cycle period and cycle split, y, which for our study was defined as the fraction of total cycle time for which CO was fed to the reactor. Figures 3 and 4 are plots of enhancement factor cp vs cycle period for splits of 0.25 (stoichiometric) and 0.4, respectively. Figures 3 and 4 show that the enhancement factor increases as the cycle period decreases or in other words as the cycle frequency increases. Thus, the reactor performance improves as the reactor moves towards the relaxed steady state. This is an anticipated result because, for pure-component cycling the time-averaged reaction rate would be zero during the quasi-steady state since the steady-state rates for feeds of pure H, and pure CO must necessarily be zero. For pure-component cycling the quasi-steady state would simply be the average of those steady states (for mixture cycling the quasi-steady state will, in general, not have a zero rate). Therefore as the system moves from low to high frequency, the rate must increase. However, it is not necessary that the rate will increase monotonously with frequency since, due to effects like resonance, in the sense of Unni et al. (1973), and other manifestations of the interaction between surface dynamics
, .
I.lc..
0.51 13 “.
10
,
. ,
,
‘~“~~‘~~~~~‘.~‘.~*“~~.
16
I9
22 Cycle
,
.
_ ,
25 period.
28
,
,
31
34
37
40
s
Fig. 3. Enhancement factor vs cycle period for cycle split = 0.25. The curve indicating the trend of the data points is not a model fit.
10
DISCUSSION
The results of concentration forcing studies are presented in terms of a rate enhancement factor, cp, which is defined as
3223
time-averaged production rate during cycling
12
14
16
18 Cycle
20 22 24 period, s
26
28
30
Enhancement factor vs cycle period for cycle split = 0.40. The curve indicating the trend of the data points is not a model fit.
Fig. 4.
RAJIV YADAV and
3224
and oscillations in the gas phase, a maximum rate in the intermediate cycling region may be obtained. As is evident from Figs 3 and 4 our data do not exhibit a resonance maximum. This, of course, does not imply that the methanation reaction is bereft of resonancelike behavior. It is plausible that resonance occurs at higher frequencies for the system we studied. Figures 5 and 6 show the effect of cycle split on the enhancement factor for periods of 12 and 20 s, respectively. From Figs 5 and 6 it can be seen that the enhancement factor increases as the split increases to 0.4. This effect is most pronounced at a period of 12 s. It is interesting to note that the optimum steady-state rate was obtained when the mole fraction of CO in the feed was equal to 0.4. Thus, for this system, the best split for pure-component cycling seems to coincide with the optimum steady-state feed mixture. However, this observation does not necessarily hold for all binary reacting systems (Wilson and Rinker, 1982). The most salient point to emerge from the data presented in Figs 3-6 is that enhancement factors greater than 1 were obtained_ This means that timeaveraged rates higher than the optimum steady-state rate were obtained during pure-component cycling for some periods and splits. The maximum enhancement was 7% (cp = 1.07) at a period of - 12 s and a split of 0.4.
1.1 I .o 0.9 0.8 0.7 0.6
0.5F........,“‘.“....‘.-..“‘-.‘.-.-’-...l 0.1
0.15
0.2
0.25 0.3 0.35 Cycle split, y
0.4
0.45
0.5
5. Enhancement factor vs cycle split for cycle period = 12 s. The curve indicating the trend of the data points is not a model fit.
ROBERT G. RINKER
An analysis of random errors in our experimental data is presented in the Appendix. Using the confidence intervals for a Gaussian distribution, the maximum standard error, for a 99.5% confidence limit, in the time-averaged cycling rate was less than 1%. For the same confidence limit the maximum standard error in the optimum steady-state rate was 1.23%. Thus, the 7% enhancement observed in the timeaveraged cycling rate compared to the optimum steady-state rate is more than the random error by at least a factor of 5. Cycled feedstream results are difficult to analyze quantitatively. For example, the difficulty of using ordinary adsorptionaesorption-type models to predict production rate improvements over the optimal steady state was highlighted by Jain et al. (1983b) who simulated 22 such models but were unable to find a model which could yield rate improvements over the optimal steady state. The complexity involved in describing concentration forcing results was also underlined by Prairie and Bailey (1987) who asserted that such “results are difficult to analyse in terms of a mathematical mode1 because of the inherently complicated system of differential equations which must be solved numerically”. However, a qualitative interpretation of our results can be obtained. A mode1 to explain production rate improvements over the optimal steady state for the reaction of acetic acid and ethylene was recently proposed by Truffer (1986). This model relied on educt inhibition where it was assumed that preadsorbed acetic acid inhibited the adsorption of ethylene. It appears that the methanation reaction may follow a similar mechanism. It has been suggested that under methanation conditions the catalytic surface is Hz-deficient (Underwood and Bennett, 1984), or that the catalytic surface is covered by inactive or less reactive surface species (Dalmon and Martin, 1985). Thus, it is likely that, during steady-state methanation, preadsorbed CO inhibits the adsorption of Hz. In that case, it is conceivable that under cycling conditions the catalyst surface is periodically “cleaned” during the Hz-rich part of the cycle, leading to high methanation rates. This may explain the observation of rates higher than the optimal steady-state rate during pure-component cycling in the methanation reaction. This is only a qualitative interpretation of our results and a more rigorous treatment would involve estimation of model parameters. At present we are in the process of developing such a rigorous model for forced cycling of the methanation reaction.
CONCLUSIONS
been shown that a 7% improvement in the production rate of methane, compared to the optimum steady-state rate, is possible through purecomponent cycling. For the range of periods and splits investigated, resonance-type behavior was not observed but it is conceivable that, for the conditions of this study, such behavior exists at frequencies It has
0.5 -....,....,....1.........1....1....’....7 0.3 0.35 0.2 0.25 0.1 0.15 Cycle split, Y
0.4
0.45
0.5
Fig. 6. Enhancement factor vs cycle split for cycle period = 20 s. The curve indicating the trend of the data points is not a model
fit.
An experimentalstudy of methane synthesisby concentrationforcing
higher than the maximum frequency used here. The time-averaged rates under pure component cycling conditions displayed a trend with frequency which can be intuitively reconciled with our current understanding of cycled reactor behavior, e.g. time-averaged rates under pure-component cycling must appreach zero at very low frequencies which corresponds to quasi-steady-state behavior. It is suggested that the inhibitory effect of adsorbed CO on the methanation reaction can be relieved under cycling conditions. Acknowledgement-Support for this work by the Division of Chemical Sciences (Office of Basic Energy Sciences) of the Department of Energy (U.S.A.) under grant No. DE-FGO3-184ER13300 is gratefully acknowledged.
V, W x, y. . . _ x,, XCH4,,,
rp
rp =
enhancement factor defined as time-averaged production rate during cycling optimum
rator
output function
in variables
x, y etc.
L
partial derivative of the function f with
A
partial derivative of the function f with respect to y molar flow rate at the reactor outlet, grnol~-~ mean outlet molar flow rate for n readings, gmol s-l number of readings of measured variables for any given data point area of the nitrogen peak in the GC integrator output production rate of methane, gmol -Is-’ g mean production rate of methane for n readings of the outlet molar flow rate and m readings of the mole fraction of methane in the reactor outlet, gmoi g - ’ s- ’ best estimate of the precision of time measurements for flow rate best estimate of the standard error in the measurement of molar flow rate, s best estimate of the standard error in the production rate of methane, gmoi -1 s-1 g best estimate of the standard error in the measurement of time, s best estimate of the standard error in the measurement of volumetric flow rate, ml s-l best estimate of the standard error in the measurement of the outlet mole fraction of methane best estimate of the standard error in a general function z =f(x, y, . _ _ ) or 2 = axayb . . . . measured time for flow rate (subscript i denotes the ith reading), s mean of n measurements of time for flow rate, s
respect
FO F 0” n, m . . . N 2arca rcli4
RzEi.nm
s,(t)
WF,)
%A~CB.)
S,(t)
S”(K)
wkli,)
Snm
t, t, T.
. . (-4
steady-state
production
rate
REFERENCES
area of methane peak in the GC integ-
_f(x, y, _ _ . ) general
reactor outlet volumetric flow rate, mls-’ weight of catalyst in the reactor, g measured variables mole fraction of methane in the reactor outlet mean methane mole fraction in the reactor outlet
Greek letters cycle split, fraction of total cycle time for Y which CO was fed to the reactor
NOTATION CI-Lre.
3225
to x
Bailey, J. E., 1973, Periodic operation of chemical reactors: a review. Chem. Enana Commun. 4. 111-124. Bailey, J. E., 1977, in Chemical Reaction Engineering: a Review (Edited by L. Lapidus and N. R. Amundson),
Chap. 12. Prentice-Hall, Englewood Cliffs, NJ. Bailey, J. E. and Horn, F. J. M., 1969, Catalyst selectivity under steady-state and dynamic operation. Ber. Bunsenges. phjts. Chem. 73, 274279.
Bailey, J. E. and Horn, F. J. M., 1970, Catalyst selectivity under steady-state and dynamic operation: an investigation of several kinetic mechanisms. Ber. Bunsenges. phys. Chem. 74,611-617. Bailey, J. E. and Horn, F. J. M.. 1971, Improvement of the nerformance of a fixed-bed catalvtic reactor bv relaxed steady state operation. A.2.Ch.E. i. 17, 550-553: Chiao. L.. Zack. F. K.. Rinkar. R. G. and Thullie. J.. 1987. Cod&ratio& for&g in ammonia synthesis: & flow experiments at -high temperature and pressure. Chem. Engng Commun. 49,27>289. Dalmon, J. A. and Martin, G. A., 1983, The kinetics and mechanism of carbon monoxide metbanation over silicasupported nickel catalysts. J. Catal. 84, 45-54. El Masry. H. A., 1985, The Claus reaction: effect of forced feed composition cycling. Appl. Catai. 16, 301-313. Horn, F. J. M. and Bailey, J. E., 1968, An application of the theorem of relaxed control to the problem of increasing catalyst selectivity_ J. Optimization Theory Applic. 2, 44149. Jain, A. K., Hudgins, R. R. and Silveston, P. L., 19834 Influence of forced feed composition cycling on the rate of ammonia synthesis over an industrial iron catalyst. Part I-effect of cycling parameters and mean composition. Can. J. them. Engng 61, 824-832. Jain, A. K., Hudgins, R. R. and Silveston, P. L.. 1983b. Adsorption/desorption models: how useful for predicting reaction rates under cyclic operation. Can. J. them. Engng
61, 4649. Prairie, M. R. and Bailey, J. E.. 1987, Experimental and modelling investigations of steady-state and dynamic characteristics of ethylene hydrogenation on Pt/AI,O,. Chem. Engng Sci. 42,2085-2102. Schack. C. J., McNeil, M. A. and Rinker, R. G., 1989. Methanol synthesis from hydrogen, carbon monoxidh and carbon dioxide over a CuO/ZnO/Al,O, catalyst. I Steady-state kinetic experiments. Appl. Catal. 50,247-2&l. Truffer, M. A., 1986, Transient behavior of heterogeneous catalytic reactions with educt inhibition. A.Z.Ch.E. J. 32,
1612-1621. Underwood, R. P. and Bennett, C. 0.. 1984, The CO/H2 reaction over nickel-alumina studied by the transient method. J. Catal. 86, 245-253. Unni, M. P., Hudgins, R. R. and Silveston, P. L., 1973.
RAJIV YADAV and ROBERT G. RINKER
3226
Influence of cycling on the rate of oxidation of SO, over a vanadia catalyst. Can. J. them. Engng 51.623-629. Wilson, H. D., 1980, Ph.D. thesis, University of California, Santa Barbara. Wilson, H. D. and Rinker, R. G., 1982, Concentration forcing in ammonia synthesis-I. Controlled cyclic operation. Chem. Engng Sci. 37,343-355. APPENDIX:
ERROR ANALYSIS
An analysis of random errors in our experiments is presented below. Although, in general, experiments are prone to systematic errors, care was taken to minimize the effect of such errors on our system and we have assumed that the effect of such errors on our data was negligible. The rate of production of methane is given by r CH.
=
F2hi. W
It is assumed that the weight of the catalyst, W, for any given experiment is known with an accuracy of 100.0%. Thus, random errors in the rate of production of methane arise only from random errors in F,, the molar gow rate out of the reactor, and x,-n,, the mole fraction of methane in the outlet stream. Let us now examine the errors in each of these variables. The molar flow rate was inferred from measurements of the volumetric flow rate, V., which was measured using a bubble flow meter or a wet test meter. The volumetric flow rate is given by, for example for the bubble flow meter: 32.9
t
where 32.9 is the volume of the flow meter tube in ml and t is the time, in s, that a bubble takes to negotiate that volume. Hence, the variable directly measured was the time, t. If T. is the mean of the time measurements, the adjusted root mean square of these measurements, s.(t), which is the best estimate of the precision of the flow meter, is then found to be
%(a =
J--& 5oi -
T.F-
(A3)
1
The adjusted standard error which is the best estimate of the standard error is S,(t) = s,(t)/n0.5.
S,,,,..(z)
(A4)
Once the standard error in time measurements is known, the standard error in volumetric measurements can be calculated as follows: For a general function of the form r = ax’yb . . . the
= [a=S:(x)/X:
+ b2S:(y)/Y:
+
. . . ]l” WI
where X, is the mean of x. Y,,, is the mean of y, etc. For our case the volumetric flow rate V, is given by eq. (A2) and hence the standard error in volumetric flow rate from eqs (A2) and (AS) is given by S,(V,)
= 32.9S&)/T”.
(A6)
Using the ideal gas law, the molar flow rate is given by F, = V, x 273.16/(300
x 22,400).
Again, using eqs (AS) and (A7), the standard molar flow rate is given by
(AlI
-0
v, = -
standard error for n measurements of x, m measurements of y, etc., is given by
S,(F,)
= S,(V,)
(A7) error in the
x 273.16/(300 x 22,400)
(A8)
where it is assumed that volumetric flow rates were measured at 300 K and 1 atm. The next step is to calculate the standard error in the measurement of mole fraction of methane in the exit stream. The composition of the exit stream was analyzed using a gas chromatograph which was calibrated using an external nitrogen standard. The following equation was found to fit the mole fraction of methane to the areas of the nitrogen and methane peaks: x CH+ = 0.008513 + 0.860219N,,,,,/CH,,,.,. Again it is known variables, z =f(x, y, imated by S.,...(r)
(A9)
that, for a general function in several _), the standard error can be approx-
= CSIS,2(x)
+f:gf(y)
+ . . . 1”’
(Al’3
wheref, represents the partial derivative off with respect to x.& the partial derivative offwith respect to y, etc., and S,(x), S,(y) etc. are the standard errors in the variables x, y etc., respectively. Thus, using eqs(A9) and (AlO), the standard error in mole fraction of methane can be found easily. Once the standard errors in the molar flow rate and in the methane mole fraction in the outlet stream are known, the standard error in the production rate of methane is found simply from eqs (Al) and (AS) to be Sn&cn,)
= CS:(F,)IF&
+
S:(XCH.)/X~H.~I”~RCH.~~~ (All)
where F,, is the mean molar flow rate, Sn(xCH4) is the standard error in the outlet methane mole fraction, XcHl,,, is the mean methane mole fraction, and lQHInm is the mean production rate. Thus, the random error in the methanation rate can be calculated.
Engineering Science, Vol. 45. No.
11, pp. 3221-3226,
ooo%2509/!m $3.00 + 0.00 0 1990 Per~amm F%v.!pepk
1990.
Printed in Great Britain.
AN
EXPERIMENTAL STUDY OF METHANE BY CONCENTRATION FORCING
SYNTHESIS
RAJIV YADAV and ROBERT G. RINKER? Department of Chemical and Nuclear Engineering,University of California, Santa Barbara, CA 93106, U.S.A. (First received 3 July 1989; acceptedin revised
form
2 February
1990)
Abstract-The production of methane from CO and Hz over an Ni-Al,OB catalyst was studied in a gradientlessreactor understeady-stateand pure component cyclingconditions,at a pressure of 1 atm and a temperature of 513 K. It was observed that the time-averagedproduction rate of methaneincreasedas the cycle period decreased.The highest rates were obtained for a cycle period of 12 s, the lowest investigated, and at a cycle split of 0.4 which coincideswith the composition of the optimum steady-state feed mixture. Time-averaged rates up to 7% higher than the experimentally determined optimum steady-state rate on a mole basis were obtained. To the best of our knowledge this is the first report on concentration forcing of the methanation reaction to appear in the open literature, and the results are encouraging. INTRODUCTION Concentration
forcing
In recent years, the traditional notion that the steady state is the best method of operating chemical reactors has been challenged. The first concerted challenges were launched by Bailey and Horn. In a series of pioneering papers, Horn and Bailey (1968), as well as Bailey and Horn (1969,1970,1971), showed that for a hypothetical reaction system, under certain conditions, forced oscillations of reactant concentrations lead to a set of selectivities which surpasses the set of all steady-state selectivities. Operating a reactor with oscillating feed composition is now recognized in the literature under several names: concentration forcing, forced cycling, and periodic forcing, among others. Bailey and Horn’s papers also provided a conceptual framework for interpreting and articulating concentration forcing research. This framework was further elaborated and crystallized in two reviews by Bailey (1973, 1977). One of the most important aspects of this development was the recognition of three distinct cyclic regimes. Thus, very high frequency operation was categorized as the relaxed steady state, very low frequency operation as the quasi-steady state and the in-between region as the intermediate cyclic regime. Bailey and Horn’s papers focused only on the relaxed steady state since it was perceived to be easily amenable to theoretical analysis. It was in this region that selectivities beyond the steady state were obtained. Although there are other important applications of concentration forcing (e.g. validating rate models) improving reactor productivity remains a compelling objective. At this point it is important to note that any improvement must be referential, and in the quest for improving reactor productivity an acceptable basis of comparison must be invoked. One such basis is what we choose to call the optimal steady state, i.e. that steady state which yields the highest attainable rate ‘Author
to whom correspondence
should be addressed.
for a given temperature, pressure, reactor configuration and size as determined by its residence time. The feed composition is completely unconstrained. Thus, such an optimal for a two-component feed can be easily depicted on a two-dimensional plot of rate vs feed composition. A number of investigators have demonstrated that production rates at such an optimutn steady state can be exceeded by time-averaged production rates obtained under concentration forcing conditions. It must be emphasized here that, for the comparison to be “fair”, periodic operation must be constrained exactly like the steady-state operation and again the feed composition must be the only unconstrained variable. Some specific instances where time-averaged rates higher than those at the optimum steady state have been reported are Wilson and Rinker (1982), Jain et al. (1983a) and Chiao et al. (1987) for the ammonia synthesis reaction; El Masry (1985) for the Claus reaction; and Prairie and Bailey (1987) for the ethylene hydrogenation reaction. In this paper results for rate enhancement of the methanation reaction, over a nickel catalyst, using concentration forcing are presented. To the best of our knowledge, this is the first such study of forced cycling of the methanation reaction to appear in the open literature. Methane synthesis The synthesis of methane from CO and Hz over a nickel catalyst according to the reaction CO + 3H, c-, CH, + H,O, is commonly known as the “methanation reaction,” and in this paper all references to methane synthesis or methanation should be construed to refer to it. This reaction is predominantly used to remove CO from hydrogen-rich synthesis gases for reactions like the ammonia synthesis reaction. However, in recent years interest in the synthesis of methane from CO-H, mixtures has been revived due to methane’s potential as a clean energy source from coal. Considerable effort is being expended worldwide to understand and explain this important reaction.
3221
3222
RAJIV
YADAV
and ROBERT
Obviously enhancing the production rate for the methane synthesis reaction is a desirable objective and concentration forcing, which has not been reported previously for the methanation reaction, presents itself as a possible technique for realizing such an objective. With our current understanding of cycled reactor behavior, one cannot establish u priori whether concentration forcing will yield rates higher than the optimum steady-state rate for the methane synthesis reaction. Hence an experimental study is warranted. This paper presents results of an experimental investigation of pure-component cycling for the methane synthesis reaction in a fixed catalyst bed surrounded by a well-mixed gas phase. These results show that time-averaged production rates higher than the optimal steady-state rate can be obtained through pure-component cycling of the methanation reaction. Again to the best of our knowledge, no such instance of rate enhancement beyond the optimal steady state has been reported for the methanation reaction in the open literature thus far. In fact, apparently this is the first report on an experimental investigation of the methanation reaction using concentration forcing. Although, as mentioned above, it appears that methane synthesis has not been the subject of concentration forcing studies, a study using the transient method by Underwood and Bennett (1984) deserves mention. In this study the authors used step response experiments to study the methanation reaction over an Ni-A1203 catalyst, the same as ours, in a oncethrough flow reactor, taking care that the flow rate and hence the residence time through the reactor was constant_ The crucial result for our purpose from this study is shown in Fig. 1. Figure 1 depicts the response of the methanation rate when the feed to the reactor was switched from 10% CO-H, to pure H,. It can be seen from Fig. 1 that, on switching the feed to pure H,, the methanation rate dramatically increased to more than 10 times the steady-state rate observed with the 10% CO-H, mixture. This dramatic rate increase, which can perhaps be ascribed to a hydrogen
G.RINKER
deficiency on the surface during the CO-H, part of the experiment, seems to suggest the methanation reaction as an attractive candidate for concentration forcing experiments. However, a note of caution must be included here: Underwood and Bennett (1984) did not report the optimum steady-state rate for their experiment. Hence even though the rise in rate observed by them was remarkable, the highest rate may not have exceeded the optimum steady-state rate. EXPERIMENTAL
VENT
0
1
2 Time
3 (min)
4
DETAILS
A simplified schematic of the experimental apparatus is given in Fig. 2; a more detailed diagram is given elsewhere (Schack et al., 1989). This apparatus was used for both steady-state and concentration forcing studies. The Whitey four-way ball valve (S) was used to switch between H2 and CO feedstreams, flowing at equal volumetric flow rates, to the reactor. With the valve in one position (A) H, flowed to the reactor while CO was bypassed to the vent via an empty reactor; with the valve in the other position (B) the reverse situation was obtained. A Whitey threeway valve was used to direct the CO stream either to the H, line or directly to S. For steady-state operation the three-way valve was positioned such that CO flowed to the H, line and S was kept in position A. Concentration forcing experiments were conducted by keeping the three-way valve in the other position and allowing S to switch between H, and CO streams. Of course, the same setup can be used for both purecomponent and mixture cycling. The gases used in the experiments were of the following quality: CO = CP grade, 99.5%, and H, = UHP, 99.995%. The gas flows were monitored and controlled with Linde mass flow controllers which were calibrated and their performance checked periodically. The switching valve, S, was driven by a Superior Electric Co. synchronous motor. The cycle split (the fraction of each cycle for which the valve S is
0 K Y E R
5
Fig. 1. Transient behavior of methanation catalyst replotted from Underwood and Bennett (1984). The catalyst surface was swept clean with helium, followed by exposure to a mixture of 10% CO in )Tz for . 2 min, ___ followed ---_-by exposure to pure H, for 1 mm, all done at 220°C.
GAS CHROMATOGRAPH
Fie. 2. Simalified schematic of the exuerimental aunaratus. _.
An experimentalstudy of methane synthesisby concentrationforcing in position B) was regulated using Potter and Brumfield time-delay relays. Additional details of the switching mechanism can be found in Wilson (1980). The reactor temperature was controlled using a Techne Tecam TC4D temperature controller. The thermocouple for measuring the reactor temperature was placed inside the catalyst bed. During steadystate experiments the reactor temperature exhibited no measurable variations (it was precise) but during cycling a variation of f 1°C was observed. This variation probably arises from the fact that pure-component concentration forcing entails switching between two feedstreams of quite disparate thermal conductivities. The effluent from the reactor was analyzed using a Hewlett-Packard 5750 gas cbromatograph equipped with a thermal conductivity detector. A 6 ft x _kin. Porapak-Q cohtmn was used. Samples of the effluent were automatically injected with a pneumatically driven eight-way gas sampling valve. Before being injected into the GC, the sample stream was physically integrated in a surge tank to obtain timeaveraged concentrations. The output of the GC was monitored with a Hewlett-Packard 3393A integrator. The GC was calibrated for CO and methane using nitrogen as an external standard. The analysis and measurement of the reactor effluent stream was rendered somewhat problematic by the presence of water. To obviate this difficulty water was removed from the effluent stream by passing the effluent through a bed of CaSO,. Thus, both the GC analysis and the measurement of the exit flow rate were done on a dry basis. Once the moles of methane in the effluent stream were determined, the total outlet “wet” flow rate was calculated assuming that the moles of water formed in the reaction were equal to the moles of methane formed. Carbon balances around the reactor checked to within 1%. The reactor was a Berty-type gradientless reactor placed inside a 1.0 liter autoclave from Autoclave Engineers. Details of the reactor system can be found in Schack et al. (1989). A commercial, proprietary BASF Rl-10 Ni-Al,O, catalyst was used. Before charging to the reactor the catalyst was crushed to 14/18 mesh size so as to eliminate the possibility of intraparticle gradients. Since the reaction is highly exothermic ( - 215 kJ/mol), the catalyst was diluted in a ratio of one volume of catalyst to 20 volumes of 14/18 mesh crushed glass to ensure that no intraparticle heat transfer gradients existed. Calculations showed that gas phase heat and mass transfer gradients were negligible. The reactor was operated at 1 atm pressure and 513 K. The residence time in the reactor was - 9.5 s. The lowest cycle time that was used was 12 s; hence cycling effects were not masked by bulk mixing within the reactor.
EXPERIMENTAL
RESULTS
AND
cp=
optimum steady-state production rate
-
The two parameters which have the greatest effect on cycled reactor behavior are the cycle period and cycle split, y, which for our study was defined as the fraction of total cycle time for which CO was fed to the reactor. Figures 3 and 4 are plots of enhancement factor cp vs cycle period for splits of 0.25 (stoichiometric) and 0.4, respectively. Figures 3 and 4 show that the enhancement factor increases as the cycle period decreases or in other words as the cycle frequency increases. Thus, the reactor performance improves as the reactor moves towards the relaxed steady state. This is an anticipated result because, for pure-component cycling the time-averaged reaction rate would be zero during the quasi-steady state since the steady-state rates for feeds of pure H, and pure CO must necessarily be zero. For pure-component cycling the quasi-steady state would simply be the average of those steady states (for mixture cycling the quasi-steady state will, in general, not have a zero rate). Therefore as the system moves from low to high frequency, the rate must increase. However, it is not necessary that the rate will increase monotonously with frequency since, due to effects like resonance, in the sense of Unni et al. (1973), and other manifestations of the interaction between surface dynamics
, .
I.lc..
0.51 13 “.
10
,
. ,
,
‘~“~~‘~~~~~‘.~‘.~*“~~.
16
I9
22 Cycle
,
.
_ ,
25 period.
28
,
,
31
34
37
40
s
Fig. 3. Enhancement factor vs cycle period for cycle split = 0.25. The curve indicating the trend of the data points is not a model fit.
10
DISCUSSION
The results of concentration forcing studies are presented in terms of a rate enhancement factor, cp, which is defined as
3223
time-averaged production rate during cycling
12
14
16
18 Cycle
20 22 24 period, s
26
28
30
Enhancement factor vs cycle period for cycle split = 0.40. The curve indicating the trend of the data points is not a model fit.
Fig. 4.
RAJIV YADAV and
3224
and oscillations in the gas phase, a maximum rate in the intermediate cycling region may be obtained. As is evident from Figs 3 and 4 our data do not exhibit a resonance maximum. This, of course, does not imply that the methanation reaction is bereft of resonancelike behavior. It is plausible that resonance occurs at higher frequencies for the system we studied. Figures 5 and 6 show the effect of cycle split on the enhancement factor for periods of 12 and 20 s, respectively. From Figs 5 and 6 it can be seen that the enhancement factor increases as the split increases to 0.4. This effect is most pronounced at a period of 12 s. It is interesting to note that the optimum steady-state rate was obtained when the mole fraction of CO in the feed was equal to 0.4. Thus, for this system, the best split for pure-component cycling seems to coincide with the optimum steady-state feed mixture. However, this observation does not necessarily hold for all binary reacting systems (Wilson and Rinker, 1982). The most salient point to emerge from the data presented in Figs 3-6 is that enhancement factors greater than 1 were obtained_ This means that timeaveraged rates higher than the optimum steady-state rate were obtained during pure-component cycling for some periods and splits. The maximum enhancement was 7% (cp = 1.07) at a period of - 12 s and a split of 0.4.
1.1 I .o 0.9 0.8 0.7 0.6
0.5F........,“‘.“....‘.-..“‘-.‘.-.-’-...l 0.1
0.15
0.2
0.25 0.3 0.35 Cycle split, y
0.4
0.45
0.5
5. Enhancement factor vs cycle split for cycle period = 12 s. The curve indicating the trend of the data points is not a model fit.
ROBERT G. RINKER
An analysis of random errors in our experimental data is presented in the Appendix. Using the confidence intervals for a Gaussian distribution, the maximum standard error, for a 99.5% confidence limit, in the time-averaged cycling rate was less than 1%. For the same confidence limit the maximum standard error in the optimum steady-state rate was 1.23%. Thus, the 7% enhancement observed in the timeaveraged cycling rate compared to the optimum steady-state rate is more than the random error by at least a factor of 5. Cycled feedstream results are difficult to analyze quantitatively. For example, the difficulty of using ordinary adsorptionaesorption-type models to predict production rate improvements over the optimal steady state was highlighted by Jain et al. (1983b) who simulated 22 such models but were unable to find a model which could yield rate improvements over the optimal steady state. The complexity involved in describing concentration forcing results was also underlined by Prairie and Bailey (1987) who asserted that such “results are difficult to analyse in terms of a mathematical mode1 because of the inherently complicated system of differential equations which must be solved numerically”. However, a qualitative interpretation of our results can be obtained. A mode1 to explain production rate improvements over the optimal steady state for the reaction of acetic acid and ethylene was recently proposed by Truffer (1986). This model relied on educt inhibition where it was assumed that preadsorbed acetic acid inhibited the adsorption of ethylene. It appears that the methanation reaction may follow a similar mechanism. It has been suggested that under methanation conditions the catalytic surface is Hz-deficient (Underwood and Bennett, 1984), or that the catalytic surface is covered by inactive or less reactive surface species (Dalmon and Martin, 1985). Thus, it is likely that, during steady-state methanation, preadsorbed CO inhibits the adsorption of Hz. In that case, it is conceivable that under cycling conditions the catalyst surface is periodically “cleaned” during the Hz-rich part of the cycle, leading to high methanation rates. This may explain the observation of rates higher than the optimal steady-state rate during pure-component cycling in the methanation reaction. This is only a qualitative interpretation of our results and a more rigorous treatment would involve estimation of model parameters. At present we are in the process of developing such a rigorous model for forced cycling of the methanation reaction.
CONCLUSIONS
been shown that a 7% improvement in the production rate of methane, compared to the optimum steady-state rate, is possible through purecomponent cycling. For the range of periods and splits investigated, resonance-type behavior was not observed but it is conceivable that, for the conditions of this study, such behavior exists at frequencies It has
0.5 -....,....,....1.........1....1....’....7 0.3 0.35 0.2 0.25 0.1 0.15 Cycle split, Y
0.4
0.45
0.5
Fig. 6. Enhancement factor vs cycle split for cycle period = 20 s. The curve indicating the trend of the data points is not a model
fit.
An experimentalstudy of methane synthesisby concentrationforcing
higher than the maximum frequency used here. The time-averaged rates under pure component cycling conditions displayed a trend with frequency which can be intuitively reconciled with our current understanding of cycled reactor behavior, e.g. time-averaged rates under pure-component cycling must appreach zero at very low frequencies which corresponds to quasi-steady-state behavior. It is suggested that the inhibitory effect of adsorbed CO on the methanation reaction can be relieved under cycling conditions. Acknowledgement-Support for this work by the Division of Chemical Sciences (Office of Basic Energy Sciences) of the Department of Energy (U.S.A.) under grant No. DE-FGO3-184ER13300 is gratefully acknowledged.
V, W x, y. . . _ x,, XCH4,,,
rp
rp =
enhancement factor defined as time-averaged production rate during cycling optimum
rator
output function
in variables
x, y etc.
L
partial derivative of the function f with
A
partial derivative of the function f with respect to y molar flow rate at the reactor outlet, grnol~-~ mean outlet molar flow rate for n readings, gmol s-l number of readings of measured variables for any given data point area of the nitrogen peak in the GC integrator output production rate of methane, gmol -Is-’ g mean production rate of methane for n readings of the outlet molar flow rate and m readings of the mole fraction of methane in the reactor outlet, gmoi g - ’ s- ’ best estimate of the precision of time measurements for flow rate best estimate of the standard error in the measurement of molar flow rate, s best estimate of the standard error in the production rate of methane, gmoi -1 s-1 g best estimate of the standard error in the measurement of time, s best estimate of the standard error in the measurement of volumetric flow rate, ml s-l best estimate of the standard error in the measurement of the outlet mole fraction of methane best estimate of the standard error in a general function z =f(x, y, . _ _ ) or 2 = axayb . . . . measured time for flow rate (subscript i denotes the ith reading), s mean of n measurements of time for flow rate, s
respect
FO F 0” n, m . . . N 2arca rcli4
RzEi.nm
s,(t)
WF,)
%A~CB.)
S,(t)
S”(K)
wkli,)
Snm
t, t, T.
. . (-4
steady-state
production
rate
REFERENCES
area of methane peak in the GC integ-
_f(x, y, _ _ . ) general
reactor outlet volumetric flow rate, mls-’ weight of catalyst in the reactor, g measured variables mole fraction of methane in the reactor outlet mean methane mole fraction in the reactor outlet
Greek letters cycle split, fraction of total cycle time for Y which CO was fed to the reactor
NOTATION CI-Lre.
3225
to x
Bailey, J. E., 1973, Periodic operation of chemical reactors: a review. Chem. Enana Commun. 4. 111-124. Bailey, J. E., 1977, in Chemical Reaction Engineering: a Review (Edited by L. Lapidus and N. R. Amundson),
Chap. 12. Prentice-Hall, Englewood Cliffs, NJ. Bailey, J. E. and Horn, F. J. M., 1969, Catalyst selectivity under steady-state and dynamic operation. Ber. Bunsenges. phjts. Chem. 73, 274279.
Bailey, J. E. and Horn, F. J. M., 1970, Catalyst selectivity under steady-state and dynamic operation: an investigation of several kinetic mechanisms. Ber. Bunsenges. phys. Chem. 74,611-617. Bailey, J. E. and Horn, F. J. M.. 1971, Improvement of the nerformance of a fixed-bed catalvtic reactor bv relaxed steady state operation. A.2.Ch.E. i. 17, 550-553: Chiao. L.. Zack. F. K.. Rinkar. R. G. and Thullie. J.. 1987. Cod&ratio& for&g in ammonia synthesis: & flow experiments at -high temperature and pressure. Chem. Engng Commun. 49,27>289. Dalmon, J. A. and Martin, G. A., 1983, The kinetics and mechanism of carbon monoxide metbanation over silicasupported nickel catalysts. J. Catal. 84, 45-54. El Masry. H. A., 1985, The Claus reaction: effect of forced feed composition cycling. Appl. Catai. 16, 301-313. Horn, F. J. M. and Bailey, J. E., 1968, An application of the theorem of relaxed control to the problem of increasing catalyst selectivity_ J. Optimization Theory Applic. 2, 44149. Jain, A. K., Hudgins, R. R. and Silveston, P. L., 19834 Influence of forced feed composition cycling on the rate of ammonia synthesis over an industrial iron catalyst. Part I-effect of cycling parameters and mean composition. Can. J. them. Engng 61, 824-832. Jain, A. K., Hudgins, R. R. and Silveston, P. L.. 1983b. Adsorption/desorption models: how useful for predicting reaction rates under cyclic operation. Can. J. them. Engng
61, 4649. Prairie, M. R. and Bailey, J. E.. 1987, Experimental and modelling investigations of steady-state and dynamic characteristics of ethylene hydrogenation on Pt/AI,O,. Chem. Engng Sci. 42,2085-2102. Schack. C. J., McNeil, M. A. and Rinker, R. G., 1989. Methanol synthesis from hydrogen, carbon monoxidh and carbon dioxide over a CuO/ZnO/Al,O, catalyst. I Steady-state kinetic experiments. Appl. Catal. 50,247-2&l. Truffer, M. A., 1986, Transient behavior of heterogeneous catalytic reactions with educt inhibition. A.Z.Ch.E. J. 32,
1612-1621. Underwood, R. P. and Bennett, C. 0.. 1984, The CO/H2 reaction over nickel-alumina studied by the transient method. J. Catal. 86, 245-253. Unni, M. P., Hudgins, R. R. and Silveston, P. L., 1973.
RAJIV YADAV and ROBERT G. RINKER
3226
Influence of cycling on the rate of oxidation of SO, over a vanadia catalyst. Can. J. them. Engng 51.623-629. Wilson, H. D., 1980, Ph.D. thesis, University of California, Santa Barbara. Wilson, H. D. and Rinker, R. G., 1982, Concentration forcing in ammonia synthesis-I. Controlled cyclic operation. Chem. Engng Sci. 37,343-355. APPENDIX:
ERROR ANALYSIS
An analysis of random errors in our experiments is presented below. Although, in general, experiments are prone to systematic errors, care was taken to minimize the effect of such errors on our system and we have assumed that the effect of such errors on our data was negligible. The rate of production of methane is given by r CH.
=
F2hi. W
It is assumed that the weight of the catalyst, W, for any given experiment is known with an accuracy of 100.0%. Thus, random errors in the rate of production of methane arise only from random errors in F,, the molar gow rate out of the reactor, and x,-n,, the mole fraction of methane in the outlet stream. Let us now examine the errors in each of these variables. The molar flow rate was inferred from measurements of the volumetric flow rate, V., which was measured using a bubble flow meter or a wet test meter. The volumetric flow rate is given by, for example for the bubble flow meter: 32.9
t
where 32.9 is the volume of the flow meter tube in ml and t is the time, in s, that a bubble takes to negotiate that volume. Hence, the variable directly measured was the time, t. If T. is the mean of the time measurements, the adjusted root mean square of these measurements, s.(t), which is the best estimate of the precision of the flow meter, is then found to be
%(a =
J--& 5oi -
T.F-
(A3)
1
The adjusted standard error which is the best estimate of the standard error is S,(t) = s,(t)/n0.5.
S,,,,..(z)
(A4)
Once the standard error in time measurements is known, the standard error in volumetric measurements can be calculated as follows: For a general function of the form r = ax’yb . . . the
= [a=S:(x)/X:
+ b2S:(y)/Y:
+
. . . ]l” WI
where X, is the mean of x. Y,,, is the mean of y, etc. For our case the volumetric flow rate V, is given by eq. (A2) and hence the standard error in volumetric flow rate from eqs (A2) and (AS) is given by S,(V,)
= 32.9S&)/T”.
(A6)
Using the ideal gas law, the molar flow rate is given by F, = V, x 273.16/(300
x 22,400).
Again, using eqs (AS) and (A7), the standard molar flow rate is given by
(AlI
-0
v, = -
standard error for n measurements of x, m measurements of y, etc., is given by
S,(F,)
= S,(V,)
(A7) error in the
x 273.16/(300 x 22,400)
(A8)
where it is assumed that volumetric flow rates were measured at 300 K and 1 atm. The next step is to calculate the standard error in the measurement of mole fraction of methane in the exit stream. The composition of the exit stream was analyzed using a gas chromatograph which was calibrated using an external nitrogen standard. The following equation was found to fit the mole fraction of methane to the areas of the nitrogen and methane peaks: x CH+ = 0.008513 + 0.860219N,,,,,/CH,,,.,. Again it is known variables, z =f(x, y, imated by S.,...(r)
(A9)
that, for a general function in several _), the standard error can be approx-
= CSIS,2(x)
+f:gf(y)
+ . . . 1”’
(Al’3
wheref, represents the partial derivative off with respect to x.& the partial derivative offwith respect to y, etc., and S,(x), S,(y) etc. are the standard errors in the variables x, y etc., respectively. Thus, using eqs(A9) and (AlO), the standard error in mole fraction of methane can be found easily. Once the standard errors in the molar flow rate and in the methane mole fraction in the outlet stream are known, the standard error in the production rate of methane is found simply from eqs (Al) and (AS) to be Sn&cn,)
= CS:(F,)IF&
+
S:(XCH.)/X~H.~I”~RCH.~~~ (All)
where F,, is the mean molar flow rate, Sn(xCH4) is the standard error in the outlet methane mole fraction, XcHl,,, is the mean methane mole fraction, and lQHInm is the mean production rate. Thus, the random error in the methanation rate can be calculated.