Ind. Eng. Chem. Process Des. Dev. 1980, 19,
148
148-153
Steam Reforming of n-Heptane at Low Concentration for Hydrogen Injection into Internal Combustion Engines Krister Sjostrom Department of Chemical Technology, The Royal Institute of Technology, S- 100 44 Stockholm 70, Sweden
The reaction of n-heptane with steam over a nickel-alumina catalyst was investigated at partial pressures of 0.15- 1.75 kPa for n-heptane in the temperature range of 632-679 K. The reaction rate was proportional to (C,H,6)-o,23(H2)0,22 and the activation energy for the reaction was 83.6 kJ/mol. The reactant concentrations did not satisfy the equilibrium equations for the water-gas shift and methane-steam reactions at low conversions. Agreement was good at high conversion, however. Simultaneously with the steam reforming reaction a small benzene production was detected. This was used to confirm the reaction order of n-heptane in the main reaction. Furthermore, the benzene yield provided information of value in the preparation of Ni catalysts.
Introduction The purpose of this project has been to develop a method for generating hydrogen gas for injection into an internal combustion engine (Lindstrom, 1970). Hydrogen injection makes it possible to run the engine with a lean fuel/air ratio. Lean running conditions reduce hazardous emissions and give good fuel economy (Sjostrom, 1977). Partial oxidation and steam reforming of hydrocarbons are conventional processes for hydrogen generation. Steam reforming was chosen because of the possibility of supplying the energy for the endothermal reaction by heat exchange with the exhaust gas. Water vapor for the reforming reaction is provided from recirculated exhaust gas. Rostrup-Nielsen (1975) gives an excellent review of steam reforming of hydrocarbons on nickel catalyst. The overall reaction can be described as an initial breakdown of the hydrocarbon C,H, + n H 2 0 nCO + ( n + (rn/2))H2 (1) followed by establishment of the equilibria CO + HzO t+ CO2 + H2 (2) CH4 + HZO t+ CO + 3H2 (3) As can be seen from Table I, widely different results have been reported on the kinetics of reaction 1. The reaction sequence formulated by Rostrup-Nielsen (1975) and the expression for the reaction rate deduced from this sequence seem to explain the different results. The simplified power expression given is 1-2x pH*02(x-z) 2(t-x)-iy/2)(1-2x) (4) -
-
rC,H,
=
kp PC,H,
PHz
where 0 < x < 1 , 0 < z < 1, and 0 < y < m. It is, however, doubtful if reactions 2 and 3 reach equilibrium in practice. Bhatta and Dixon, for example, have shown that these equilibria are established for nbutane reforming on Ni/y-A1203(1967) but not on Ni/aA1203U02(1969). Phillips et al. (1970) and Rogers and Crooks (1966) have also shown that the equilibria are not established. The investigations of Kikuchi et al. (1975a) on steam reforming of n-heptane on an Rh catalyst showed that the extent of equilibria establishment depends on the conversion. The equilibria are established only at a high conversion. For Rh the steam-methane reaction is on the opposite side of the equilibrium compared to Ni catalyst. Experimental Apparatus The experiments were performed in a flow system with two different reactors (one integral fixed bed reactor and 0019-7882/80/1119-0148$01.00/0
one gradientless fixed bed reactor) as shown in Figure 1. The gases were taken from gas cylinders with the flow rates controlled by flow meters. The addition of water and n-heptane into the evaporators was controlled with motor burets as shown in Figure 2. The liquids were fed through capillaries in the top, flowing down onto an internally heated glass rod. The gas flows entering the bottom of the evaporators were externally heated by electric furnaces. This design gives a uniform and stable flow of the evaporating liquids. The gas flow is then mixed and preheated to approximately 675 K. The integral reactor was made of glass surrounded by a heating coil. A copper block was mounted on the reactor to ensure isothermal conditions. The axial temperature difference was about 3 K. The inlet section of the reactor was empty and served as a preheater. The catalyst was mixed with quartz grains to a weight of 50 g. This also ensured isothermal conditions. Quartz grains were packed above and below the bed (10 g, respectively). The reactor had an internal diameter of 33 mm with a central 4-mm thermowell. The gradientless reactor, made by Autoclave Engineers, is described by Berty (1974). The reactor is of the modified type with water cooling (Mahoney, 1974). The recycling ratio was greater than 20 as proposed by Kuchcinski and Squires (1976) for atmospheric pressure. As the partial pressure of n-heptane is low and the reaction order is less than 1, there is a gradient due to the external mass diffusion. The gradient was calculated with the expression given by Kuchcinski and Squires (1976) and was found to be 5-1070 of the n-heptane partial pressure. The conversion at impeller speeds of 1000 and 1500 rpm was measured and corresponding n-heptane pressure gradients were calculated. From this and the reaction rate expression the relative error in the gradient calculations was found to be about 5%; 40 g of quartz grains was placed at the bottom of the sample bag, then 10 g of catalyst quartz grain mixture followed by 10 or 20 g of quartz on top. The diameter of the sample bag was 89 mm and the height about 10 mm. The catalyst was a Ni catalyst on an alumina carrier (Girdler G 56), well described by Satterfield (1970). The exit gas flow was water cooled and dried in Dehydrite. The gas samples of 1 or 5 cm3 were analyzed with gas chromatographs. H, and CO were separated in a molecular sieve 5A column and C02in a Purapac Q column and detected with a thermal conductivity detector. All hydrocarbons were separated in a silicone oil column and 0 1979
American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19,
Table I.
No. 1, 1980 149
Results from Some Kinetic Studies of Steam Reforming of Higher Hydrocarbons
authors Balashova et al. ( 1 9 6 6 ) Bhatta and Dixon ( 1 9 6 7 ) Bhatta and Dixon ( 1 9 6 9 )
catalyst system
Phillips e t al. ( 1 9 6 9 )
Ni/SiO, Ni/r-Al,O, Ni/r-Al,O, Ni/y-Al,O,UO, (0.3% K ) Ni/ y -A1,0
Rostrup-Nielsen ( 1 9 7 5 )
Ni/MgO, (A1,0,)
hydrocarbon
temp, K
cyclohexane n-butane n-butane n-butane
673-733 698-748 693-753 677-764
(101) 3039 3039 3039
0 0 0 1
n-hexane n-heptane ethane
633-723
1520
0.3
733
3140
0.54
8 Figure 1. Apparatus for studies of n-heptane steam reforming at atmospheric pressure: 1,gas cylinder; 2, flowmeters; 3, motor burets; 4, evaporators; 5, preheater; 6, reactor; 7, coolers; 8, separator; 9, drier. 1
1 I Figure 2. Evaporator: 1, liquid inlet; 2, gas outlet; 3, gas inlet; 4, capillary; 5, internally heated glass rod; 6, furnace.
detected with a flame ionization detector. Experimental Procedure Integral Reactor. The experiments with the integral reactor were performed a t 670-1000 K, the higher temperatures with a steel reactor. The inlet feed consisted of n-heptane, water, carbon dioxide, and nitrogen. Preliminary runs showed that diffusion limitations, both external and internal, influenced the reaction rate. In addition, the conversion observed in the experiments was not reproducible. For the same conversion, however, the product gas composition was the same. The reason for this was probably that the nickel catalyst oxidized near the reactor inlet. The length of the oxidized zone gradually increased during the run. To avoid this oxidation, a flow of H2 was maintained during the experiment. This made the experiments reproducible with respect to the conversion with a relative error of 1-3%. I t was recognized that greater relative accuracy could be obtained without carbon dioxide in the inlet feed. The errors in the mass balances over the reactor were 1G-1570 with carbon dioxide and 1-5% without. For this reason carbon dioxide was excluded in the later runs.
yH,O
act. energy, kJ/mol
0-1
92-100
reaction orders
press., kPa
oC,H,
pH,
1 1
- 0.6
0-0.07 0.2
-0.33
54 100 88 75.8
Gradientless Reactor. The experiments with the gradientless reactor were performed a t 632-679 K. A flow of hydrogen was maintained through the reactor all the time the reactor was a t reaction temperature. The experiments started with adjustment of the flows of nitrogen and hydrogen. The water buret was started and after 15 min the system reached stable conditions. The n-heptane buret was then started. When no hydrogen was used as reactant the hydrogen flow was turned off after 5 min. This hydrogen turnoff was allowed since the reactor is a recirculation reactor and enough hydrogen is produced from the reaction to avoid oxidation of the catalyst. Analysis of the products was done 1 h after the introduction of n-heptane. Mathematical Model In heterogeneous catalysis the measured reaction rate is governed by several different steps: the intrinsic reaction step and internal and external diffusion steps for reactants and products. Knowledge of these reaction steps is essential for the design of the catalytic system. The gradientless reactor is superior for reaction rate measurements, especially when the rate equation can be expected to be complex. The intrinsic reaction rate was assumed to be represented by an equation similar to eq 4 r1
=
-kpPlaP2hY
(5)
where index 1 = C7H16,2 = H2, and 3 = HzO. Characteristic features of our system are low partial pressure of n-heptane and a catalyst with very high activity for steam reforming. Therefore, it is necessary to calculate the influence of diffusion on the overall reaction rate to be able to distinguish the intrinsic reaction rate. The relation between the measured rate and the intrinsic reaction rate can be expressed by means of an effectiveness factor (7) r1,m
= 91r1
(6)
The external mass transport resistance was calculated and the partial pressures were corrected according to Kuchcinski and Squires (1976). The pressure drop of n-heptane was thus estimated to about 5-10%. The determination of q1 requires a macrokinetic model. Taking into account Knudsen diffusion and reaction in a spherical catalyst pellet, the model gives six simultaneous differential equations, one for each component (Satterfield, 1970). This would give the fairly complex model presented in the Appendix. In this case some changes are valid which greatly simplify the model: (1) By assuming equilibrium a t every point in the pellet, with regard to the water-gas shift and steam-methane reactions, the number of differential equations can be reduced. (2) The variations of hydrogen and water in the pellet do not influence the effectiveness factor. (3) The intrinsic reaction rate is so fast that the reaction is diffusion limited and the effectiveness factor
150
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
Table 111. Integral Reactora
Table 11. Integral Reactora conversion 11.0 30.3 67.0 75.0 91.7
Ql
T I ,K
a,,(MPa)*
T,, K
conversion
1.09 1.82 3.18 3.50 4.21
1062 934 835 816 793
16.8 62.1 128 177 234
735 763 780 786 794
12.0 18.2 36.6 71.9 97.0
is inversely proportional to & (Satterfield, 1970). The validity of these simplifications is shown in the Appendix. The first simplification replaces two differential equations with the two equilibrium equations.
K1 = PCO’PHzO
T I ,K
o,,(MPa),
T,, K
704 730 752 771 783
25.1 18.9 117 382 380
743 738 778 805 805
8.83 6.86 5.72 4.94 4.53
a Reactant pressurepC-H,, = 0.493 kPa;PH,O = 16.0 k P a ; p c o , = 1 4 . 8 kPa; a n d p ~ =, 73.8 kPa; reaction temperature, 783 K.
Reactant p r e s s u r e p c , ~ , ,= 0.493 kPa;pH,o = 1 6 . 0 kPa; and p N , = 88.6 kPa; reaction temperature, 7 8 3 K.
PCOP.PH2
QI
(7)
Table IV.
Differential Reactora
conversion
aI
T I ,K
az,(MPa)*
3.9 28.5 43.9
0.63 5.84 6.7
1227 750 731
1.67 X lo-* 1.11X 10.’ 1.67 X 10.’
T,, K 619 646 654
a Reactant pressurepC7Hl6= 0.493 k P a ; P H 2 0 = 16.0 kPa; and p ~ = ,88.6 kPa; reaction temperature, 653 K.
The second assumption reduces the variations of concentration within the catalyst particle to one variable equation rl = -kp’-plo (9) where kp’
=
kp’P2,,8*P3,sY
(10)
The third assumption gives a simple, explicit equation for the effectiveness factor (Aris, 1975)
Together with eq 6 this gives
kP = Ae-EIRT
(13)
Equations 12 and 13 can be used as a kinetic model after the parameters have been estimated by fitting these equations to the experimentally measured reaction rate. Equations 12 and 13 can, however, first be transformed to a linear equation by logarithmation if the first a is included in the constant. This LY is therefore decoupled from the exponent a in the estimation. Results Calculations were made on the basis of 100 mol of N2. The content of water in the product stream was calculated from the mass balances of H and 0 and the mean values were taken as product value. The conversion of n-heptane was calculated as (CHJ + (CO) + (COJ X = (14) (CH,) + (CO) + (CO2) + 7 (C7H16) with (i) = mol of i/100 mol of N,. The deviation from equilibrium was calculated as cy1
=
PCOPH2 ~
PCOPHzO
(15)
Table V. Differential Reactora
31.0 36.3 39.2
7.22 8.50 7.96
725 706 713
1.35 1.68 5.18
687 689 712
7.86 9.66 18.64
Reactant p r e s s u r e p c , ~ , ,= 0.493 kPa;pH,O = 1 6 . 0 kPa; and p ~= ,( 8 8 . 6 - p s 2 )kPa; reaction temperature, 6 5 3 K.
With n-heptane and water as reactants cyl and cy2 are smaller than K1 and K2, respectively, but tend toward equality a t higher conversion. Hydrogen fed as reactant gives a higher cy1 but still less than K1 and a higher cyz than K2with increasing values for increasing hydrogen pressure. If carbon dioxide is fed as reactant cy1 is higher than K1 and cy2 is less than K,. The partial pressures of the product gases in the bulk never seemed to exceed those in the catalyst, which can be calculated from eq A2-A7 in the Appendix with pc,Hls = 0 (Figure 3). The only exception was carbon monoxide when hydrogen was used as a reactant. When the conversion is high and the gradients in the catalyst are to be small, the gas composition comes closer to the equilibrium values. The product gas composition can be misinterpreted by assuming that methane is a primary product because of the much higher methane concentration than the equilibrium value, especially a t low conversions. The primary products are more likely to be carbon monoxide and hydrogen as concluded above. The gas composition inside the catalyst should then control the gas composition through the catalyst and in the bulk. Thus, the high methane partial pressure is probably due to the high methane partial pressure in the interior of the catalyst. A total of 151 experiments with the differential gradientless reactor were used for the estimation of the kinetic parameters cy, 0,y,and E. In this case the overall reaction rate is simple to calculate
X
rl,m= F -
and corresponding equilibrium temperatures were calculated. Typical results are presented in Tables I1 and I11 for the integral reactor and in Tables IV and V for the differential reactor.
w ps
The experimental results with the integral reactor were not very suitable for parameter estimations. The overall reaction rate has to be integrated over the bed. The partial
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
2 0 0.L
T
1 0 0.c
K kPa
PC7H16 160 'H20
kPa
88.6
kPa
P N ~
5 0.C
783
0493
151
Inlet feed
v) Y
a
v)
K Y
a
g -
10.0
+
3 5.0
1.0
0.5 0.0
I
I
I
0.1
0.2
0.3
I
1
1
1
1
I
0.4
0.5
0.6
0.7
0.8
0.9
PC, H1 .Pa
Figure 4. Correlation between benzene production and n-heptane partial pressure a t different temperature.
1.0
CONVERSION
Figure 3. The calculated ratio between the steady-state equilibrium ~ pressure in the interior of the catalyst where ~ c = 0~ andHin the bulk as a function of the actual overall conversion (solid lines) compared to experimental results in the bulk: A, CH,; V, CO; 0,COz; 0,Hz.
pressures of water and hydrogen, included in the reaction rate expression, changed in an uncontrolled manner, since equilibria were not established at every point along the reactor. Estimation of C Y , P, y,and E in eq 12 and 13 were made by a stepwise linear regression method (Draper and Smith, 1966, BMDP 2R, 1974). Optimal fit with the experimental data gave the following values for the kinetic parameters and their standard errors: a = -0.23; P = 0.22; s, = 0.05; sg = 0.04; y not significantly different from zero; E = 83.6 kJ/mol; SE = 5.0 kJ/mol. Individual 95% confidence intervals for the various parameters were: CY = 4 . 2 3 f 0.09; P = 0.22 f 0.08; and E = 83.6 f 9.9 kJ/mol. The respective maximum and minimum values for the experimental parameters were: 1.75 and 0.15 kPa; pHz, 25.2 and 2.52 kPa; pHz?, 40.5 and 11.9 kPa; and T,679 and 632 K. No significant influences of pco, pco2,and pCHl were found. These parameter values inserted in eq 4 give x = z = 0.6 and y = 2. Benzene Production A small amount of benzene was produced during nheptane reforming. At 633 K traces of toluene were also detected. The production of benzene increases with decreasing temperature. The effectiveness factor for the benzene production was calculated in order to check whether this might change the picture. I t was assumed here that the reaction rate for benzene was so slow that it would not affect the n-heptane partial pressure and that the reaction rate for formation of benzene was of the mth order with respect to n-heptane. The model for flat slab geometry was used. This effectiveness factor depends on the effectiveness factor for n-heptane reforming and is in accordance with Aris (1975).
~
This also holds for reforming with a = -0.5, P = 0.5, and y = 0.5 which was checked by calculating v2 with m = 1 and 2 by means of numerical integration of the n-heptane partial pressure profile obtained from the orthogonal collocation method. The value of the effectiveness factor did not alter the picture of the temperature dependency for benzene production. It can be shown that the benzene production should give a correlation to the n-heptane partial pressure as follows
ifm=1
with
CY
= -0.23
The partial pressure of benzene is proportional to the benzene production in this type of reactor. The constant of proportionality between the benzene production and the n-heptane partial pressure is the same for experiments with the same particle diameter and the same amount of catalyst (eq 21). According to eq 22 it can be seen that if the partial pressures of benzene and n-heptane are plotted in a log-log diagram the slope should be 1.615 with m = 1. In Figure 4 this has been done for three different temperatures with one catalyst weight and one particle diameter. Obviously the agreement is very good and thereby confirms the value of CY from the regression estimation. The reason for the decrease in production of benzene with increasing temperature is presumably some competing reaction. One possibility is that the benzene structures formed on the catalyst surface polymerize and gradually give carbon. These assumptions agree with the published data that carbon formation increases with increasing temperature
152
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
to about 830 K (Rostrup-Nielsen, 1975). Furthermore, unbranched hydrocarbons with 6 or 7 carbon atoms yield more carbon than smaller or longer unbranched hydrocarbons. Reforming of benzene and toluene gives a large amount of carbon. Unsaturated hydrocarbons give far more carbon than saturated ones. This may depend on their tendency to form polymer structures. The polymer fraction from a reforming nickel catalyst is of the aromatic type (Balashova et al., 1966; Bhatta and Dixon, 1967). For comparison, reforming on Rh catalyst can be used (Rabinovich et al., 1973). At temperatures below 780 K there is substantial production of benzene and toluene. On the other hand, no deactivation of the catalyst is observed and presumably no carbon was formed. On Rh the reforming rate is higher for benzene than for n-heptane (Kikuchi et al., 1975b), while the opposite is true for Ni (Rostrup-Nielsen, 1975). This indicates that benzene is more strongly bonded to the Ni surface than to the Rh surface and the possibility of polymerizing is therefore greater on Ni. This in turn should indicate that an Ni catalyst with a lower ratio between n-heptane and benzene reforming rates should have a lesser tendency toward polymerization and following carbon formation.
Conclusions A method was derived to estimate the intrinsic reaction rate for a complex reaction where experimental conditions such as mass transport influence the measured reaction rate. The estimated reaction expression is in agreement with the model postulated by Rostrup-Nielsen (1975). A small production of benzene was observed simultaneously with steam reforming. A parallel slow reaction has an effectiveness factor that depends on the effectiveness factor for the main reaction. This parallel reaction was used to confirm the reaction order of the main reaction. The establishment of the equilibria of the water-gas and the methane-steam reactions depended both on the conversion and on the reaction feed composition. Theoretical investigation of the gas composition in the catalyst pellets showed that the product partial pressures in the bulk never exceeded the equilibrium values in the middle of the pellets where the partial pressure of n-heptane was zero, with the exception of carbon monoxide. This indicates that the establishment of the equilibria in the bulk is governed by the gradients in the reaction zone and that carbon monoxide and hydrogen are the primary products. The low benzene production in this Ni catalyst system can be compared with the benzene formation on an Rh catalyst. This comparison suggests possibilities for the preparation of Ni catalyst with less of a tendency for carbon formation. Hydrogen production in the reactor inlet, depending on steam reforming of n-heptane, was not sufficiently large to retain the catalyst in reduced condition. Consequently, a gradually increased oxidation of the catalyst in the reactor inlet occurred. This problem could be solved in two ways, either by modification of the catalyst or by modification of the reactants. In the laboratory experiments hydrogen addition was used in the reactant feed. This solution is not feasible in our engine system. The solution we are working further on is to use methanol as a reactant instead of n-heptane or gasoline. In the engine system (Lindstrom, 1970; Sjostrom, 19771, where recirculated exhaust gas is used for the water supply, there is always some degree of oxygen in the inlet feed to the reactor. This oxygen gives no trouble when methanol is used as feed. The oxidized zone can easily move to and fro during the run time.
Acknowledgment The author wishes to acknowledge financial support for this project provided by Carl Tryggers Stiftelse for Vetenskaplig Forskning (The Carl Trygger Foundation for Scientific Research). This work is being continued in a project financially supported by the National Swedish Board for Technical Development, with engines for the experimental work supplied by AB Volvo, Car Division. The author is greatly indebted to Professor Olle Lindstrom for initiation of the work and valuable discussions and suggestions on the process design, and to Dr. Daniel Simonsson for valuable discussions on the macrokinetics. Appendix Comparison between the Exact and the Simplified Models. In the general porous catalyst theory the effectiveness factor can be determined by solving the concentration equations, taking into account the reactions and the pore diffusions (Satterfield, 1970). In our case with six components we obtain six differential equations of the form
-d2ci + - - +2- =dcio R dR
ri
D;
where index i = 1 4 stands for 1 = C7H16,2 = Ha, 3 = HzO, 4 = C02, 5 = C H I , and 6 = CO. The partial pressure of N2 is constant through the particle as Knudsen diffusion is dominant. Assuming that the water-gas shift and steam-methane reactions are in equilibrium a t every point in the pellet, we can replace two of the differential equations with equilibrium equations. The concentration of three components can be derived from the steady-state diffusion fluxes coupled by the stoichiometry and can replace three differential equations. The following mathematical model for the system can thus be derived
where z = R / R , and A, = D , ( p , - p , , , ) P4Pz
K1 = P3P6 P S P ~ ~ K2 = P3P5 The boundary conditions are
dc 1 -=Oat2 = 0 dz The assumed intrinsic reaction rate rl = -k,.pl".p2d.p3Y
(A10)
is inserted in eq A2. A simplified model can be obtained, if the assumptions leading to eq 6-13 are valid. In order to investigate this,
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1 9 8 0
numerical calculations were performed with eq A2-AlO and eq 6-13, respectively. Numerical Calculations. Numerical calculations were carried out to validate the assumption of the analytical solution. Equation A2 was solved with the orthogonal collocation method presented by Villadsen (1973) and Michelsen (1973). For every n-heptane pressure the algebraic eq A3-A7 were solved by a pattern search program (Hook and Jeeves, 1961). The solution of eq A3-A7 requires fairly good initial values of the partial pressures. This is somewhat troublesome since the gradients of the partial pressures are high. It can be solved by using more collocation points than necessary for the straightforward method. The solutions from the previous point are then used as initial values for the next point. Values of the exponents in eq 5 were assigned to cy = 0.5 and 1.0, ,6 = 0.0 and l..O, and y = 0.0 and 1.0. The effectiveness factors were scarcely affected by the p and y values. This holds for reaction temperatures 633-673 K, particle diameters 2.5-3.5 mm, n-heptane partial pressures 0.1-2.0 kPa, and reaction rates which give 3/+s less than 0.1. From the parameter estimation it can be seen that cy becomes negative. The numerical calculation problem with the reaction rate approaching infinity when the partial pressure of n-heptane approaches zero was solved by using a modified reaction rate expression r' = r / ( l
+ re)
(All)
A small enough value of e was used for the reaction rate to approach a large but limited value. The limiting value should be reached when pC7H,G has decreased to a value where it has no effect on the integral reaction rate in the particle. In order to stabilize the solution the reaction zone or the particle radius was divided into two parts, as in the work of Paterson and Cresswell (1971). The reaction zone expression from Aris (1975) was used for this.
A solution was obtained for the outer zone while the concentration of n-heptane in the inner zone was set equal to zero. The effectiveness factor (7) was calculated from the n-heptane partial pressure gradient a t the surface, approximated with the gradient between the solution in the first outer collocated point and the surface. When 40 collocation points were used in the outer zone a good estimation of the gradient was obtained. Numerical calculations of the effectiveness factor using the reaction rate expression in eq 9 were made and the results were compared with the analytical results from eq 11. For example, with cy = -0.5 eq 11 gives 7 = 6/4bs while the computation gives the same result with a relative error of only 1-270, Comparing numerical calculations of eq 5 and 9 with cy = -0.5, P = 0.5, and y = 0.5 showed a difference of only 3% although the variations of H2 and H20 in the pellets were 26 and 11%, respectively. This shows that eq 12 can be used in the estimation of the intrinsic rate.
153
Nomenclature A = preexponential factor in rate equation D, = effective diffusivity species i, m2/s E = activation energy, kJ/mol F = molar inlet flow of rz-heptane, kmol/s K 1 = equilibrium constant water-gas shift reaction K 2 = equilibrium constant methane-steam reaction, (MPaY R = gas constant R, = catalyst radius, m T = temperature, K X = conversion W = catalyst weight, kg c, = concentration species i, kmol/m3 ci,s = concentration species i at catalyst surface, kmol/m3 k , = rate constant based upon concentration k , = rate constant based upon partial pressure k,' = rate constant based upon partial pressure defined by eq 10 m,n = numbers p , = partial pressure species i, kPa p i , s = partial pressure species i at catalyst surface, kPa r, = reaction rate species i, kmol/m3 s = measured reaction rate species i, kmol/m3 s s, = standard error s, y , z = numbers in eq 4 t = dimensionless radius cy, p, y = kinetic coefficients v, vi = effectiveness factor reaction i cy1 = values of the equilibrium function of the water-gas shift reaction cyp = values of the equilibrium function of the methanesteam reaction, (MPa)2 ps = catalyst density, kg/m3 w = dimensionless point where p 1 = 0 & &,, = Thiele modulus for sphere reaction i Literature Cited Aris, R., "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts", Voi. I, pp 144-150, 362-368, Clarendon Press, Oxford, 1975. Balashova, S. A,, Siovokhotova, T. A,, Balandin. A. A,, Kinet. Cafal., 7, 273 (1966). Berty, J. M., Chem. Eng. Prog., 70(5), 78 (1974). Bhatta, K. S. M., Dixon, G. M., Trans. faraday Soc., 63,2217 (1967). Bhatta, K. S. M., Dixon, G. M., Ind. Eng. Chem. Prod. Res. Dev., 8, 324 (1969). BMDPPR-Stepwise Regression, Health Sciences Computing Facility University of California, Los Angeies, Program Revised Oct 7, 1974; Writeup Revised April 1974. Draper, N. R., Smith, H., "Applied Regression Analysis", pp 163-195, Wiley, New York, London, Sydney, 1966. Hooke, R., Jeeves. T. A., J . Assoc. Comput. Mach., 8(2), 212 (1961). Kikuchi, E., Yamazaki, Y., Morita, Y., Bull. Jpn. Pet. Inst., 17(1), 3 (1975a). Kikuchi, E., Ito, K., Morita, Y., Bull. Jpn. Pet. Insf., 17(2), 206 (1975b). Kuchcinski, G.R., Squires, R. G., J. Cafal., 41, 486 (1976). Lindstrom, O., Swedish Patent No. 349549 (1970); 360062 (1971); 770301 1-2 (1977); U S . Patent No. 3918412 (1975). Mahoney, J. A,, J. Cafal., 32,247 (1974). Michelsen, M. L., "Algorithms for Collocation Solution of Ordinary and Partial Differential Eauations", DD . . 13-15, Denmarks Tekniske Hoiskole. CoDenhagen, 1973. Paterson, W. R . , Cresswell, D. L., Chem. Eng. Sci., 28,605 (1971). Phillips, T. R., Mulhaii, J., Turner, G. E., J. Catal., 15, 233 (1969). Phillips, T. R., Y a r w d , T. A,. Mulhall, J., Turner, G. E., J . Cafal., 17, 28 (1970). Rabinovitch, G. L., Treiger, L. M., Maslyanskii, G. N., Pet. Chem. USSR, 13, 199 (1973). Rogers,'M. C: F., Crooks, W. M., J , Appi. Chem., 18, 253 (1966). Rostrup-Nielsen, J. R., "Steam Reforming Catalysts", Teknisk Forlag A/S, Copenhagen, 1975. Satterfield, C. N., "Mass Transfer in Heterogeneous Catalysis", pp 64-73, 129-141, MIT Press, Cambridge, Mass., 1970. Sjostrom, K., International Symposium on Alcohol Fuel Technology-Methanol and Ethanol, Woifsburg, Federal Republic of Germany, Nov 21-23, 1977. Viiladsen, J., "Application of the Collocation Method", Denmarks Tekniske Hojskole, Copenhagen, 1973.
Receiced f o r review March 27, 1979 Accepted July 30, 1979
