Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 12-19
12
-try NO.Nylon 6,2503854-4;polyethyleneterephthalate (homopolymer), 25038-59-9; polypropylene (homopolymer), 9003-07-0. Literature Cited Albano-Mudler, L. Powder M e t e U . Int. 1982, 74(2), 73. Bird, R. 8.; Armstrong, R. C.; Hassager. 0. "Plnamlcs Of Polymeric LiquMs", Vol. 1. "Fluld Mechanics": Wlbv: New York. 1977: ChaDter 5. Bird, R. 6.; Stewart, W. E.; Lightfdot, E. N. "Transpoi Phenomena"; Wlley: New York, 1060 pp 196-207. Christopher, R. H.; MMleman, S. Ind. Eng. Chem. Fundem. 1985, 4 , 422.
Ergun, S. Chem. Eng. frog. 1952, 48(2). 80. @egorY, D. R.; @iskey, R. G. AIChE J . 1987. 73(1), 122. Kemblowskl, 2.; Dziubinskl, M. Rheol. Acta. 1978, 77, 178. Marshall. R. J.: Metzner. A. B. Ind. €no. Chem. Fundam. 1987. 6. 393. Morland,' C. D. Fiber producer 1980, 8 ( $ , 32. Sadowski, T. J. Trans. Soc. Rheol. 1985, 9(2), 251. Sadowskl, T. J.; Bird, R. B. Trans. Soc.Rheol. 1985, 9(2), 243 . Chem. 1989, 67, 18. Savins, J. G. ~ n dEng.
Received for review October 3, 1983 Accepted February 17, 1984
Synthesis of Aicohots from Carbon Oxides and Hydrogen. 1. Kinetics of the Low-Pressure Methanol Synthesis Plerlulgl Vllla, Pi0 Forzattl, Guldo BuzzCFerrarlo, Guldo Garone, and Halo Paoquon Dipartbnento di Chimica Industriale ed Ingegneria Chimica "G.Natta '' del Poiitecncico, Piazza Leonard0 da Vinci 32, 20 13f Milano, Italy
The kinetics of the low-pressure synthesis of methanol from carbon monoxide and hydrogen and of the reverse shift reaction of carbon dioxide over a commercial Cu/tnO/AI,O3 catalyst are studied. The investigation is performed under typical commercial conditkns with a Betty CSTR reactor. Reliable kinetlc models which conveniently describe the results are derived for the two reactions. The analysis of the data is carried out considering the outlet mole fractions as the variables subject to experlmental error.
Introduction As a first step in the long-term research effort for the synthesis of higher alcohols from carbon oxides and hydrogen, the kinetics of low-pressure methanol synthesis was studied. The synthesis of methanol on a commercial basis is currently carried out from carbon oxides and hydrogen at reduced pressures and temperatures over Cu/ZnO/A1203 or Cu/ZnO/Cr203 catalysts. Typical reaction conditions are T = 200-270 OC, P = 50-100 atm, and gas composition as follows: CO, &lo%; C02, 5-6%, the balance being hydrogen. An appropriate, complete kinetic description of the reacting system requires the knowledge of the kinetics of two independent reactions. The couple of reactions chosen in this work were the synthesis of methanol from carbon monoxide and the reverse shift reaction CO + 2H2 F= CH30H (1) C02 + Hz +== CO
+ HzO
(2)
Previous developments in the field of methanol kinetics are mainly concerned with the older zinc chromite catalysh (Natta et al., 1953). Kinetic investigations on copper-based catalysts so far published in the literature are scarce and somehow unsatisfactory. Natta et al. (1955), during the study of methanol synthesis over a few Cu/ZnO/A1203 catalysts, proposed a Langmuir-Hinshelwood-HoughenWatson (L-H-H-W) type equation derived under the assumption that the rate-determining step involves the reaction between one adsorbed CO molecule and two adsorbed H2 molecules. The investigation was carried out in the absence of C02 and at high temperatures where catalyst deactivation is remarkable, as was also noted by the authors. Also, the catalyst was prepared according to a method that today appears to be obsolete (Ruggeri et al., 1982). Later, the same data were interpreted by 0196-430518511 124-0012$01.5010
Pasquon (1960) assuming dissociative adsorption of H2. Leonov et al. (1973) studied the kinetics of methanol synthesis on a Cu/ZnO/A1203 catalyst and developed a kinetic expression under the assumption that methanol desorption is rate determining. The investigation was limited to a narrow pressure interval and did not consider the presence of any reaction accounting for the consumption of carbon dioxide. A kinetic model for the synthesis of methanol over a Cu/ZnO catalyst referring to a wide range of C02/C0 ratio was recently presented by Klier et al. (1982). The model was developed by taking that a redox equilibrium involving active copper sites is readily established and that the rate of methanol synthesis is determined by the rate of surface reaction involving CO and Hzmolecules to produce methanol. An empirical term for the slow hydrogenation process of C02 was also included. The analysis of the data was presented in terms of total carbon conversion to methanol. Table I summarizes the most noticeable features of published studies of methanol synthesis kinetics over copper-based catalysts. Some kinetic models proposed for the reverse shift reaction (RSR) of carbon dioxide and for the forward shift reaction (FSR) of carbon monoxide are listed in Table 11. None of these models was obtained under experimental conditions close to those employed in the low-temperature methanol synthesis. Most rate equations in the table contain both the forward and the reverse terms, which are required to satisfy the equilibrium conditions of reaction 2. In this paper we report a kinetic study of methanol synthesis reaction over a commercial Cu/ZnO/A1203catalyst at temperatures, pressures, and gas compositions typical of industrial operation. A Berty gradientless reactor (Berty, 1974) is used which ensures isothermal conditions and allows the use of catalyst in industrial size. Both reactions 1and 2 are considered. Preliminary com0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24,
No. 1, 1985 13
Table I. Kinetic Models Proposed for Methanol Synthesis Reaction T, "C
cat.
P, atm
kinetic equation
reference Natta e t al. (195 5)
Cu/ZnO/Al,O, Cu/ZnO/Al,O,
300-330 220-260
225-250
CulZnO
200-315 40-55
r (1+ K f H , ) =
r= k(
(A
PcH,oH'-~
"CH,OH
PCOo'SPH2Keq
-
+
pco
(PC$H,Z
-
Leonov et al. (1973)
1 PCH,OH/Keq)
-)-3
K'PCO,
Pasquon ( 1 9 6 0 )
BfCO t c f H Z l i z+ D f C H , 0 H ) 3
+
PCO~.~PH,
r = k(1
75
(fH ,'fC 0 - fCH ,OH / K e q
+
Klier e t al. (1982)
( F + Kco;Pco,)"
Table 11. Kinetic Models for Reverse Shift Reaction (RSR) and Forward Shift Reaction (FSR) reaction RSR
cat. Fe/Cu
T , "C 540
P,atm
1
reference
kinetic equation
r=
FSR
Fe/Cr
330-500
1
r=
RSR
Fe/Cr
305-335
1
r=
k(pCOpH 2O- pCO ,pH,
(l
KAPCO,
f
lKeq
Barkley et al. (1952)
KRPCO)
k P c o a P H ,o bPc0
Bohlbro (1961)
kPH,apC02Q
Hulburt and Srini Vasan (1961)
P C O ~ P H , O ~ - '+ ( ~KPH,O/PH,)
FSR
Fe/Cr
315-482
30
r=
FSR
Cu/ZnO
170-280
13
r=
k @ c o P H , o - PC02PH,/Keq) k@COPH,O - PCO,PH,/Keq)
(1 + RSR
Cu/ZnO
195-225
1-6
r=
KjPCO
+ K3Pco, +
Cu/ZnO
172-228
1
kPC0,PH2
kPCOPH,O
r=
1+
KCPCOPH,O + KDPCO
putations are presented which confirm the absence of heat and mass transfer limitations. The analysis of the kinetic data is performed by considering the outlet mole fractions as the variables subject to significant experimental error, while all the other measured variables are taken to be deterministic in nature. This approach contrasts with the usual treatment of kinetic data which assumes that the reaction rate r has random error associated with the parameters @ in r = r(& x),while x, the vector of independent variables, is known exactly. The usual treatment is clearly not correct in the present case since r is calculated by r = Fy/ W,and errors in r are due to errors in F, y, and W,.Although it is generally used for reason of practical convenience, this treatment is suspected to be very critical when rate equations contain both the forward and the reverse terms, as is the case, especially if the experimental conditions are close to the thermodynamic equilibrium or the equilibrium constant is either small or great. Indeed, assuming an L-H-H-W type equation, the sign of the rate equation will be very sensitive to errors in the "independent" variables so that they can no longer be considered error-uncorrupted variables. According to the proposed statistical approach, the estimates of the parameters are determined by using an algorithm which involves the maximization of the likelihood function, developed along the lines provided by Anderson et al. (1978) for the case of multicomponent gas-liquid equilibria. It is worth pointing out that the effective application of the statistical treatment used in the present paper requires good convergence of the iterative technique associated with
Campbell (1970)
K,PH2)2
+ KAPCO,PH, + K@H2
FSR
Moe (1962)
Van Herwijnen and De Jong (19 80) Van Herwijnen and De Jong (1980)
the parameter estimation algorithm. In the present case we could prove that a maximum is indeed reached. Fugacity coefficients and equilibrium compositions are calculated by using the Redlich-Kwong-Soave state equation (Soave, 1972).
Experimental Section The experimental apparatus used in this study is schematically shown in Figure 1. The gaseous feed composed of calibrated mixtures of CO, COz,H2,and He was supplied from cylinders and monitored and controlled by means of FRC 1. Helium was added to the reagents as an internal gas chromatographic standard in order to perform the material balances. The flow was measured before starting the experiment on the gas counter GC 1, after bypassing the reactor. The pressure in the gas line was controlled with a dome-loaded regulator manufactured by Grove Regulators Co., G 2, and measured with manometers PI 1 and PI 3. Another dome-loaded regulator G 1 with pressure set at 110 atm was placed before the reactor to prevent uncontrolled pressure rise in the system. The kinetic study was carried out with a Berty gradientless reactor manufactured by Autoclave Engineers. The reactor contains a stationary draft tube catalyst basket with gas mixture internally recirculating over the catalyst bed. Mass velocity in the equipment can be evaluated so that superficial linear velocity values can be selected that are close to those in production units. Catalyst pellets of commercial size were used together with pellets of inert material to obtain a uniform flow distribution in the bed.
14
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
5
B Figure 1. Schematic diagram of the apparatus.
The effluent from the Berty reactor was heated up to the sampling valve SV to prevent condensation of the products. Aliquots of the effluent stream were analyzed by gas chromatography on a 5750 G Hewlett-Packard gas chromatograph with hydrogen as carrier gas. A Porapak (80-100 mesh) column 5 m long was used and operated isothermally at 40 "C for the analysis of the reagents and with a proper temperature program for the analysis of the products. The effluent stream, after the pressure had been reduced down to 1atm and after analysis, passed through a stainless steel cylinder where liquid methanol was separated and collected. The gas continued through the rotameter FI 4 and the gas counter to the outside atmosphere. Before starting the kinetic study, the catalyst was pretreated as follows. The catalyst was charged into the reactor. After purge with N2, it was desiccated by raising the temperature to 160 "C in 16 h and soon afterward cooled down to room temperature, with nitrogen always flowing at 50 mL/min. The catalyst was then reduced by slowly raising the reactor temperature to 235 "C while a dilute hydrogen stream (0.8% vol. H2 in N2 at 200 mL/ min) flowed through the bed at a total pressure of 1atm. Finally, the hydrogen content was raised up to 100% vol. by keepting the temperature constant in a time interval of 10 h. After reduction the catalyst was always kept under reaction mixture or inert atmosphere until the kinetic study had been completed. The main characteristics of the catalyst are listed in Table 111. The weight of catalyst employed, as oxidized precursor, was 3 g. Results and Discussion 1. Results. Forty kinetic runs were performed at pressures ranging from 30 to 95 atm and temperatures in the range 215-245 O C . The experimental conditions for each run are listed in Table IV. H2 outlet mole fraction and outlet total flow are not quoted in the table because
Table 111. Catalyst Specifications Chemical Composition, wt % CUO ZnO A1203
co2 loss at 900 "C: individuated phases (in catalyst precursor) Cu,(OH),CO,; CuO; ZnO (C03)o.87 mole composition: Cu, .,,Zn, dimensions of crystallites of reduced copper as determined by X-ray diffraction : as determined by N 2 0 decomposition : surface area:
54.6 19.0 9.1 8.9 17.6
80 a
100 a
61 mz/g
they could not be measured according to the experimental setup. The reaction conditions were chosen to roughly provide a uniform exploration of the variable space, as it is required for obtaining representative models. It is worth noticing in this respect that the Berty reactor does not allow for a well-defined statistical design of experiments because the outlet mole compositions cannot be known in advance. Inlet mole compositions were assumed to be known exactly and to agree with the specifications of the supplier, as also checked directly. Inlet total flow, F'", temperature, and pressure were also taken to be known exactly based on the accuracy of the corresponding metering devices and on their precise calibrations. The outlet mole fractions were treated as random variables and their variances were estimated from genuine replicated experiments, as given below (number of degrees of freedom 60): a2He = 2.25 X lo4; s2co a 2 C 0 = 3.2 X 10"; s2Co, s2He $2 = 1.6 X lo4; s2cH30H a2cH30H= 3.75 X s2H,0 a H20= 1.5 X All the different covariances were assumed to be zero based on the absence of any coupling between measurements, according to good separation of
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985 15 Table IV. Exoerimental Conditions for Kinetic Runs 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
220 217 215 215 214 215 214 214 214 214 234 234 234 234 234 234 234 234 234 234 234 234 235 236 234 235 238 236 247 244 247 244 245 245 246 246 246 248 246 246
30 30 90 90 92 62 60 60 29 30 50 50 50 65 65 65 65 30 30 30 30 90 94 86 94 88 63 32 89 60 56 85 89 63 59 31 84 79 33 35
1.102 2.204 3.111 1.036 1.976 3.306 2.171 0.955 3.306 2.009 3.177 1.524 0.940 3.240 2.204 1.557 0.955 3.240 2.268 1.653 0.973 0.940 3.118 1.199 3.095 1.816 0.983 3.111 0.929 3.144 2.009 3.029 0.978 3.337 0.983 3.078 3.179 1.701 3.111 2.057
1.68 1.68 1.73 1.79 1.79 1.68 1.68 1.68 1.68 1.68 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.68 1.85 2.21 1.85 1.85 1.85 1.85 1.72 1.85 1.71 1.86 1.86 1.86 1.86 1.97 1.72 1.72 1.73 1.73
5.52 5.16 5.06 7.20 7.21 4.57 4.64 4.59 4.67 4.64 8.69 7.72 8.96 9.90 8.91 8.78 8.62 8.44 9.02 8.81 8.87 4.39 8.55 9.21 12.90 13.24 14.14 13.62 6.93 7.98 4.76 16.56 16.21 16.57 16.34 17.57 6.94 7.03 7.44 7.35
the products provided by the analytical system with no overlapping peaks on the gas chromatograph. 2. Evaluation of Heat and Mass Transfer Limitations. Prior to undertaking the analysis of the kinetic data, computations were performed to establish whether diffusional processes were interfering with the catalytic reactions (Santacesaria et al., 1981). Few experiments performed a t different revolution speeds of the blower together with some pulse response runs allowed us to regard the reactor as a perfect mixed CSTR at the investigated operating conditions. The evaluation of temperature and concentration gradients between the catalyst surface and the bulk gaseous stream was performed for run 25 where the rate of methanol synthesis reaction is the largest. In Table V the data necessary for such an evaluation are presented. By confronting the characteristics of the blower with the pressure drop through the catalyst bed estimated by Leva's equation, as reported by Berty (1974), a superficial gas linear velocity V = 91 cm/s was derived. This led us to estimate Rep = 10333, thus confirming that a turbulent regime is present. Maas and heat transfer coefficients were evaluated from the following relationships (Treybal, 1975)
4.10 3.56 3.51 6.26 6.38 3.23 3.25 3.05 3.17 3.12 8.17 7.30 8.36 8.57 8.43 8.25 8.04 7.93 8.66 8.35 8.34 2.81 7.09 6.79 9.48 9.63 9.71 9.69 5.87 6.86 3.49 10.10 9.95 10.10 10.07 10.80 5.82 6.07 6.39 6.24
88.7 89.6 89.5 84.75 84.62 90.52 90.43 90.68 90.46 90.56 81.43 83.27 80.97 79.82 80.95 81.26 81.63 81.92 80.61 81.13 81.08 91.12 82.51 81.79 75.77 75.28 74.30 74.84 85.48 83.31 90.04 71.48 71.98 71.47 71.73 69.66 85.52 85.18 84.44 84.68
1.78 1.73 1.80 1.93 1.90 1.73 1.76 1.82 1.70 1.72 1.84 1.89 1.95 1.84 1.88 1.92 1.92 1.77 1.79 1.80 1.85 1.85 2.08 2.66 2.15 2.25 2.31 1.94 2.04 2.01 1.82 1.94 2.09 1.91 2.02 1.99 1.91 1.93 1.79 1.82
3.94 4.39 4.05 4.71 5.45 3.71 3.28 1.97 4.45 3.94 6.51 4.55 4.55 7.14 5.62 4.73 3.46 7.66 7.77 7.15 6.09 1.15 5.55 4.10 9.89 8.42 8.25 12.98 1.71 5.20 2.51 15.15 11.39 15.77 13.33 17.41 3.82 3.30 6.38 5.57
3.71 3.29 3.08 5.56 5.89 2.95 2..94 2.45 3.20 3.03 7.99 7.15 8.16 8.35 8.10 7.97 7.76 8.03 8.50 8.30 8.37 1.63 6.46 5.95 9.41 9.77 10.24 9.85 4.12 6.39 2.88 10.28 10.56 10.36 10.53 10.97 5.09 5.13 6.25 5.98
2.96 1.42 2.04 3.57 2.81 1.22 2.10 3.74 0.56 0.96 3.28 4.96 6.29 2.64 4.08 5.46 7.28 1.57 1.74 2.29 3.41 4.61 6.52 11.00 9.12 11.52 13.02 2.69 8.85 3.71 2.70 1.85 4.74 1.41 3.39 0.59 5.35 5.71 1.32 2.03
0.94 0.70 0.90 1.53 1.21 0.70 0.78 1.16 0.32 0.48 1.10 1.21 1.65 1.17 1.47 1.60 1.86 0.46 0.84 0.81 0.94 1.78 1.84 2.58 1.80 2.25 2.17 0.61 3.62 1.38 1.15 0.61 0.95 0.38 0.73 0.29 1.69 2.01 0.68 0.89
Table V. Data for evaluation of Heat and Mass Transfer Limitations for Run 25 dp,c= 0.495 em 7 = 7.3 w, = 0.35 0 = 0.27 pp,c = 1.98 g/cm3 dp,i= 0.65 cm kp,c = 0.01 cal/cm s K W ; 0.65 ~h = 0.67 d, = 5 em M f= 11.5 Dm,CO = 1.6 X cm2/s ctot = 0.0022 mol/cm3 D'k,co = 0.351 cmz/s pf = 0.025 g/cm3 DeSCO = 5.67 X 10"' em2/s pf = 1.33 X 10"' g/cm s SCCO= 0.328 kf = 3.84 X lo4 cal/cm s K Dm,COz = 1.16 X cm2/s (-AH,") = 23460 cal/mol DtcOz = 0.28 cmz/s (-AHz") = 9510 cal/mol De,co2= 4.11 X 10"' cm2/s R1 = 13.4 X 10"' mol/min g SCCO,= 0.455 Rz = 2.35 X 10"' mol/min g Pr = 0.242
ku,cGwere calculated under the simplified assumption that only methanol synthesis reaction or revewe shift reaction, respectively, would be operating. The difference between the temperature of the catalyst surface and the bulk gaseous stream is then AT = (Rl(-AHR,l)
+ R,(-AHR,Z))%dP = 1.1"C
where R = p , j / 6 0 , and similarly for the gradients of concentration of carbon monoxide and carbon dioxide Ayco = d$,/6ky,co = 5.2 X 1 0 ~ 4 ; A ~ c o / = ~ c0.5% o From eq 3 it follows h = 0.072 cal/(cm2 s K), k ,co = 0.007 = 0.0056 mol/(cm2 s;: and mol/(cm2 s), and ky,kCOZ
= d$R,/6k,co2 = 1.15 x
~Ycoz/Yco2= 0.1% Clearly, temperature and concentration gradients in the AYCO,
16
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
film appear to be negligible. Therefore, also for the other runs, where the experimental conditions are less severe, temperature and concentration gradients across the film could be neglected. Also, the isothermality of the catalyst pellet in gas-solid reactions is well known in the literature (Carberry, 1970). In the case under examination this has been checked following the criterion developed by Mears (1971). The film heat transfer resistance was found to dominate the intraparticle resistance since Bih = hdp/kp,c = 3.6 is below 10. For the presence of concentration gradients inside the particle, the analysis reported by Carberry (1976) for intraphase isothermal effectiveness was followed. From the equation
the measurements w i(i = 1, ...,N) %s having been made and the parameters, (GI, ..., t@N,#I as)the , variables. Therefore, the likelihood function is L(W,, ..., Wjq,B) =
(8) where (el, ..., wN,6 ) is restrained to those points which satisfy the equality constraint conditions (i = 1, ..., N) (9) &Xi, w i , 4, 6 ) = 0
The maximum likelihood estimates of_(wl,..., wN; zl, ..., zN, j3), namely (wl ..., WN; k1, ..., &, @),can be obtained by maximizing eq 8 subject to the constraint eq 9. This is clearly equivalent to minimizing N
it was possible to estimate effectiveness average values of 0.912 and 0.911, respectively, for the methanol synthesis reaction and for the reverse shift reaction of carbon dioxide to carbon monoxide. Based on thesc results we assumed for both reactions an effectiveness of unity. It should be noted, however, that for operating conditions on an industrial scale, where equilibrium is approached, both reactions are expected to be much more pore-diffusion limited. 3. Theoretical Background for the Analysis of Kinetic Data. In the following the problem of parameter estimation in a kinetic model is presented in a general context, i.e. when one has to deal with stochastic, deterministic, and unmeasurable variables as well. Suppose kinetic data have been obtained in a reactor where a set of reactions is taking place and that experiment i involves the measurement of X deterministic variables xi and W stochastic variables wi, whereas 2 variables zi cannot be measured. Also suppose that N independent experiments are available and that the kinetic models are in the form (4) r = Ax, w, 2, B) where j3 is a P'vector of unknown parameters. The vectors 8, xi,wi, and ziare related through a Y vector of functions g(xi,wi, zi, 8) with the property (i = 1, 2, ..., N ) (5) g (xi, wi0, zj0,j3) = 0 where wio and zioare the expected values of wi and zi in experiment i. The measurement of wio, wi contains random experimental error, ei, which is assumed to be normally distributed with mean 0 and a known positive definite covariance matrix V independent from 5 wj = wjo + ei
E(ei)= 0 E(eieF)= V
(i = 1, 2, ..., N )
(6)
To estimate the parameters an estimation algorithm based on the maximum likelihood principle can be used, along the lines provided by Anderson et al. (1978) for the problem of parameter estimation in nonlinear thermodynamic models. Then, under the above assumptions, one obtains for the joint probability density function of all the NW readings w l , w2, ..., wN
S(W1,
..a,
(7)
a) = c ( w j -
(wj- wi) (10)
wj)TV1
i=l
with eq 9 as equality constraints. Under the null hypothesis that the model given by eq 4 is correct, S is the sum of squares of unit normal variables N(0, 1) and presents a x2 distribution with v = NW - P' degrees of freedom (Bard, 1974). Then tests on the reliability of the results can be accomplished according to standard statistical procedures. 4. Analysis of the Kinetic Data for the Synthesis of Methanol. Kinetic data obtained during the synthesis of methanol from syngas over copper-based catalyst have been analyzed on the basis of the reaction scheme illustrated by eq 1 and 2. The following L-H-H-W models have been considered
Equation 11 closely resembles the model proposed by Natta et al. (1955) and partially retained by Klier et al. (1982). A term associated with COz adsorption was introduced to account for the strong adsorption of this gas at high concentrations of COz, as reported by Klier et al. (1982). Equation 12 was derived under the assumption that the reaction between one adsorbed Hz molecule and one adsorbed COzmolecule is rate determining, in sympaty with previous assumptions for methanol synthesis reaction. The procedure outlined in the previous section was used to estimate the parameters. In this case we have: number of experiments, N = 40; number of stochastic variables, W = 5 (outlet mole fractions: yg:, y g , y g , , yEb, yyi OH); number of deterministic variables, X = 8 (inlet molar hux: weight of catalyst: W,, T, P, inlet mole fractions y$e, yko, yEo2,yEJ; number of variables that cannot be measured, 2 = 2 (outlet molar flux,Put and outlet mole fraction y$). Also the following equality constraint equations (equation of congruence; He, C, H, and 0 atomic balances; material balances for reaction 1' and 2) hold for each experiment
sw,"+ y g + 988%+
% O ,!-H !
F'"yR, = p , t
+ 3?& + m : =1
jjg
+ sg2+ YPG3OH) FnybP= @'@Rt + mb + 29F4,OH) F n ( y g O + 2y80510,)= fit@8% + 29882+ sEb + YE%,OH) P(yg0
The likelihood function is derived from eq 7 by considering
whl,
+ YgO,)
= fit@#
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
Constants Ci in rl and r2 were expressed as follows, after reparameterization Ci = exp(Pii- Piz(l/T - l/O) (14) T = 506 K is the mean temperature of all the observations. The reparameterization of the models was adopted to increase the convergence of the regression procedure. The last two constraints contain the dependence of the expected outlet mole fractions from @.Also, considering that the off-diagonal elements of matrix V are zero (see Results), eq 10, which has to be minimized, reduces to
17
0.lSt
c I
A
010
ycO
I
i”
YCO
Figure 2. Estimated vs. experimental yco values in the kinetic runs.
5. Parameter Estimation Procedure. The estimates of the outlet mole fractions and outlet molar flux,gyt and Put, were obtained for each experiment by solving 40 systems of 7 X 7 equations in 7 variables, given @ and x. An iterative procedure was employed which uses the Newton and the quasi-Newton algorithms. The updating of the Jacobian matrix in the quasi-Newton algorithm was performed by an orthogonal procedure (Buzzi-Ferraris and Mazzotti, submitted). The system was considered to be solved only when very small errors were obtained, in order to get reliable estimates of the variables. Then the conditions for a minimum of S are applied and the constraint equations are solved for new estimates of @ and x. The minimization of the objective function S was carried out using a general purpose optimization program, making use of several direct search procedures combined (Buzzi-Ferraris, 1970), which ensures quick convergence. One step of resolution of the 40 constraint systems was followed by one step of minimization of S. When the program did not succeed in solving the system of constraint equations for experiment i, within the given accuracy and with the assigned number of iterations, the contribution of experiment i to the objective function was automatically amplified so as to force the new parameter estimates to solve the system. The fugacity coefficients pi were taken constant during the whole procedure. They were recalculated only before the last iterations for minimizing S. The following initial guesses were used for the parameters: Oil = 5.0; Pi2 = 5000 (i = 1, 2, ..., 9). The method outlined above involves a two-step procedure which may be subject to oscillation. Alternatively, a penalty function method can be used. This, however, does not allow us to overcome convergence problems, if any, since, in case Lagrange multipliers are used, the equations obtained still represent a large system of nonlinear equations which must be solved iteratively. As a matter of fact, comparison of parameter estimates obtained by direct search method and by the Gauss-Newton method confirmed that a minimum was reached in the present case. The computer time required for the entire procedure was 1 min and 30 s on a Univac 1100 computer. In the course of the analysis, constants C5,C,, C8, and C9 turned out to be negligible, so that the following models were derived
012 I
1
0O08 l0I
Oo2t
-m O
/
0.00 002
004 006 008
010
012
ycop
Figure 3. Estimated vs. experimental ycq values in the kinetic runs.
The following parameter estimates were obtained (where T = 506 K) C1 = exp(3.49 + 4883(1/T - 1/0) C2 = exp(2.53 - 39060(1/T -
l/n)
C3 = exp(3.70 + 15948(1/T - 1/0) C4 = exp(1.54 + 8229(1/T - l/O) c 6
= exp(5.18 + 9380(1/T- 1/0)
6. Error Analysis. The goodness of fit of the estimated outlet mole fractions to the experimental data was checked according to the standard x 2 test. Indeed, under the null hypothesis that models given by eq 16 and 17 are correct, S is the sum of squares of unit normal variables N(0, 1) and presents a x2 distribution with v = NW - P’ = 190 degrees of freedom. The realization of S ( S = 192 with v = 190) is to be confronted with the value x + = ~ 224.1, obtained from critical tables for P ( X + ~=) 0.95. Consider that in this case a single-sided test is appropriate. Based on this comparison (192 < 224.1) we concluded that the models provide a satisfactory representation of the experimental data. The goodness of fit is also shown in Figures 2-5, where the estimated values of the outlet mole fractions @Yt) are plotted against the experimental values byt)for CO, COz, CH30H, and H20, respectively. The agreement between and y z i is very satisfactory too (mean percent error = 1.05). A figure is not given because the data are concentrated in a very narrow range. Conclusions A complete kinetic investigation of methanol synthesis from syngas over a commercial Cu/ZnO/Al2O3catalyst was performed. Both the methanol synthesis reaction from
18
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985
2, = diameter of the equivalent particle, cm O
1
4
0.121
/ I
m
OlOt
YCH30H
F i g u r e 4. Estimated vs. experimental yCH3OH values in the kinetic runs.
A
0 020 0°*’1
ooloi I
A
0015
0000 0005 0010 0015 0020 0025 0030 YHzO
Figure 5.
Estimated vs. experimental y H I Ovalues in the kinetic
runs.
carbon monoxide and hydrogen and the reverse shift reaction of carbon dioxide and hydrogen were considered. The study was accomplished in the mass flow regime typical of industrial operations and the kinetic data were found not to be significantly contaminated by diffusional processes. Rate equations were derived which describe adequately the experimental results and can be conveniently used for design purposes. The statistical treatment of the kinetic data used in the present paper, by considering the outlet mole fractions as the variables subject to experimental error, appears to be fundamentally more correct than the one currently employed based on random errors associated only with the parameters @ in r = r(B, x). Because of this, it is believed that it will be more and more extensively employed in future kinetic investigations, as soon as the efficiency of the parameter estimation algorithm will be proved on a more general basis. Acknowledgment The authors thank Assoreni for financial support and Drs. G . Manara, V. Fattore, and B. Notari for valuable assistance and discussion. The authors also wish to thank one of the referees for critical suggestions which helped to improve the original manuscript. Nomenclature Bib = thermal Biot number ( d , h / k , , ) , dimensionless C = constant relative to adsorption equilibrium terms in the models cpPz = specific heat of the fluid mixture, cal/g K cbt = total concentration of the fluid mixture, mol/cm3 De = effective diffusivity in porous catalyst, cmz/s Dk = Knudsen diffusivity, cmz/s D, = diffusivity through the fluid mixture, cm2/s
dp,in= diameter of the inert particle, cm d, = diameter of the reactor, cm
ei= vector of random errors in experiment i Put= outlet molar flux, mol/h F‘“ = inlet molar flux, mol/h f = fugacity, atm h = fluid-particle heat transfer coefficient, cal/cm2 s K JD = mass transfer Colburn factor, dimensionless JH = heat transfer Colburn factor, dimensionless k f = thermal conductivity of the fluid mixture, cal/cm s K kp,c = thermal conductivity of the catalyst particle, cal/cm sK k, = fluid-particle mass transfer coefficient, mol/cm2 s (mole fraction) L_ = likelihood function Mf = average molecular weight of the fluid mixture, g/mol N = number of experiments P = pressure P’ = number of parameters p = partial pressure, atm Pr = Prandtl number ( t p , f p f / k fdimensionless ), r = reaction rate, mol/g min Ri = reaction rate i for unit volume of catalyst particle, mol/cm3 s Re, = Reynolds number (pfVd /kf), dimensionless S = sum of weighted squared Jeviations si2 = estimate of ai2 Sc = Schmidt number (pf/Dmpf),dimensionless T = temperature T = mean temperature, K V = empty cross section velocity, cm/s V = error covariance matrix W = number of stochastic variables W , = weight of catalyst, g w = vector of stochastic variables w,,~, = weight fraction of catalyst win = weight fraction of inert material X = number of deterministic variables x,= vector of deterministic variables y’” = inlet mole fraction yoUt= outlet mole fraction 2 = number of variables that cannot be measured z = vector of variables that cannot be measured Greek Symbols j3 = vector of parameters -Mr= heat of reaction, cal/mol q, = bed void fraction, dimensionless 7=
effectiveness, dimensionless
0 = catalyst porosity, dimensionless
wf
= viscosity of the fluid mixture, g/cm s
v = number of degrees of freedom pf = density of the fluid mixture, g/cm3
density of the catalyst particle, g/cm3 3’3catalyst variance of the outlet mole fraction of component i tortuosity, dimensionless =
r = 4 = Thiele modulus, dimensionless p=
fugacity coefficient, dimensionless
Superscripts ^= maximum likelihood estimate O
= expected value
T = transposition matrix Regietry No. CH,OH, 67-56-1; CO, 630-08-0; COz, 124-38-9; Cu, 7440-50-8; ZnO, 1314-13-2.
Literature Cited Anderson, T. F.; Abrams. D.S.;Grens, E. A. AIChEJ. 1878. 2 4 . 20. Bard, Y. “Nonlinear Parameter Estimation”; Academic Press: New York, 1974; p 21. Barkley. L. W.; Corrigan, T. E.: Wainwright, H. W.; Sands, A. E. Ind. Eng. Chem. 1052. 44. 1066. Berty. J. M. Chem. Eng. Prog. 1974, 7 0 , 7%. Bohlbro, H. Acta Chem. S a n d . 1961, 15, 502. Buzzi-Ferraris, G. Working party on routine computer programs and the use of computers in chemical engineering, Florence, 1970. Buzzi-Ferraris, G.; Mazzotti, M. Submitted to Compuf . Chem . Eng .
Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 19-30 Campbell, J. S. Ind. Eng. Chem. Process Des. Dev 1070, 9 , 588. Carberry, J. J. Catel. Rev. 1070, 3 , 61. Carberry, J. J. ”Chemical and Catalytic Reaction Engineering”; McGraw-HIII: New-York, 1976; p 217. Huiburt, H. M.; Srini Vasan, C. D. AIChE J . 1061, 7 , 143. Kiier, K.; Chatikavanu. V.; Herman, R. G.; Simmons, G. W. J . Catal. 1082, .74. . , 343. - .-. Leonov, V. E.; Karabaev, M. M.; Tsybina, E. N.; Petrishcheva. G. S. Kinet. Katal. 1073. 14. 970. M a r s , J. Ind.’Eng: Chem. ProcessDes. Dev. 1071, 1 0 , 541. Moe, J. M. Chem. €ng. Prog. 1082, 58, 33. Plno, P.; Mazzanti, G.: Pasquon, I. Chim. Ind. (Mllan) 1053, 35, Natta. 0.; 705. Natta, G.; Mazzanti, G.; Pasquon, I. Chim. Ind. (Milan) 1055, 3 7 , 1015.
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Pasquon, I . Chim. Ind. (Mlan)1060, 42, 352. Ruggeri, D.; Vaccari, A.; Trifir6, F.; Gtwrardl, P.; Del Piero, G.; Notari, 6.; Manara, G. 3rd International Congr. “Sclentiflc Bases for the Preparation of Catalysts”, Louvaln-Le Neuve, Oct 1982. Santacesaria, E.; Morbideiii, M.; C a r 6 S. Chem. Eng. Sci. 1081, 36, 909. Soave, G. Chem. Eng. Scl. 1072, 27, 1197. Treybal, R. E. “Mass Transfer Operations”, 3rd ed.; McGraw-Hill: New York, 1975; p 75. Van Herwijnen, T.; De Jong, W. A. J . Catal. 1080, 63, 83.
Received for review July 8, 1982 Revised manuscript received December 29, 1983 Accepted January 18, 1984
Synthesis of Optimal Heat and Power Supply Systems for Energy Conservation Masatoshl Nishlo, Ichiro Koshljlma, Katsuo Shlroko, and Tomlo Umeda Chiyo& Chemical Engineering 8 Construction Co., Ltd., Tsurumi, Yokohama, Japan
The synthesis problem of energy supply systems includes the selection of proper heating and power generating
units as well as energy integration. Such problems have been generally formulated as a linear programming (LP) problem. The structural analysis of the problem and solutions will be focussed on rather than merely solved. As a practical consideration for preliminary selection of technologies, a problem regarding the selection of heating devices such as electric heaters, furnaces, steam heaters, or heaters with a heating medium under given loads is defined as a synthesis problem of optimal heat and power supply systems and formulated into an LP form with an objective function of minimum fuel. The results of analysis of the LP problem clarify essential structures of heat and power supply systems under arbitrary heat and power demands.
Introduction A number of studies on energy conservation have been made by the process industries in the wake of the need to save energy. For example, aiming a t effective use of energy, energy management and saving have been strengthened, efficiency of equipment has been improved, and an increase in overall efficiency has been sought by means of an appropriate combination of equipment. When an energy conservation project is carried out, economics is an important factor that dominates whether the project should be carried out or not. It is expected to have a rational way of preliminary choice of energy conservation technologies. There are a variety of ways to utilize heat and power for process industries that consume great amounts of energy. Various methods for efficient use of heat and power have been applied ranging from the energy consuming system where process systems are the central part to the energy supply system where a steam-power system is the central part. Efforts at energy utilization have been made independently from each other in individual subsystems, namely, process systems and steam-power systems. Among them are studies in energy conservation technologies for process systems (Umeda et al., 1978; Umeda and Shiroko; 1980, Umeda et al., 1981) and the steam-power system (Nishio et al., 1980). Nishio et al. (1982) have presented a method for an optimal use of steam and power through the coordination of process systems and the steam-power system. In that paper a qualitative approach has been proposed for the coordination problem on energy use and practical candidates of energy conservation technologies 0196-4305/85/1124-0019$01.50/0
to be chosen have been shown based on necessary steam and power demands. On the other hand, Nishio et al. (1981) have taken up the problem of determining an optimal structure for fuel supply systems as an example of a quantitative approach for the energy coordination problem. The problem has been formulated in terms of an LP form and general feasible solutions have been derived on the basis of heat and power demands. The motivation of this approach has been described in detail. Some explanation is repeated later in this paper. Furthermore, Nishio et al. (1983) have modified the formulation of the same problem, obtained general optimal solutions, and proved them to be optimal. This paper expands upon the previous paper in determining a structure for an optimal supply system of heat and power. First, the problem is formulated as an LP problem, and next general optimal solutions are derived by applying the method for solution given in the previous paper (Nishio et al., 1984). Finally, numerical studies are carried out as to the effects of steam header pressure level, i.e., high and medium pressure levels which are design parameters for problem, to determine the optimal system structure. Problem Formulation In general a number of heating loads are required to generate fine products in process industries where various types of heating devices are employed, depending on the extent of the heating loads as well as the heating levels. While there are cases where furnaces or electric heaters are used for a heating load at a high temperature level, and steam heaters or heaters with heating mediums such as hot oil used for heating loads at medium temperature levels, 0 1984 American Chemical Society
