intra-particle diffusion limitations in

Chemical Enyineeriny Science, Vol. 45, No. 4, pp. 773 783, 1990.

Printed in Great Britain.

CKKW 2509/90 s3.00 t 0.00

((3 1990 Pergamon Press plc

INTRA-PARTICLE DIFFUSION LIMITATIONS IN LOW-PRESSURE METHANOL SYNTHESIS

G. H. GRAAF,+ H. SCHOLTENS,* E. J. STAMHUIS and A. A. C. M. BEENACKERSO

Department of Chemical Engineering, The University of Groningen, Nijenborgh 16,9747 AG Groningen, The Netherlands

(First received 21 July 1988; accepted in revised form 20 June 1989)

Abstract-The dusty gas diffusion model was applied for the description of intra-particle diffusion limitations in methanol synthesis over a commercial Cu-Zn-Al catalyst. Experimental kinetic data were obtained at p = l&50 bar and T = 210-275°C using a spinning basket reactor and a fixed bed catalytic reactor, starting from carbon monoxide, carbon dioxide and hydrogen. The results, obtained from experiments with two different catalyst particle sizes, show that commercial size catalyst particles exhibit intra-particle diffusion limitations. Combining the dusty gas diffusion model with kinetic rate expressions for methanol formation from CO, methanol formation from CO, and the water-gas shift reaction, good agreement with experimental data was obtained.

INTRODUCTION LITERATURE

In a previous paper (Graaf et al., 1988), we presented the kinetics of low-pressure methanol synthesis in the temperature range 20&245”C based on the following reactions:

Several authors of publications on the subject of catalytic methanol synthesis have mentioned intra-particle diffusion effects. A short summary of these publications is given in Table 1.

A: CO+ 2H, = CH,OH (1)

B: CO, + H, = CO + H,O (2)

C: CO, + 3H, = CH,OH + H,O. (3)

At temperatures above 245°C intra-particle diffusion limitations were observed. It is well known, however, that considerably higher temperatures occur in indus- trial methanol reactors. Maximum temperatures of 280°C are reported for the ICI process (Macnaughton

et al., 1984) and for the Haldor Topsee process (Dybkjaer, 1985). These processes are based on adiabatic multibed reactors with cooling between the catalyst beds. The Lurgireactor (Supp, 1981) yields maximum temperatures in the range 260-265”C, which are realized by indirect cooling with boiling water.

Therefore, it is clear that a complete description of the kinetics should include proper modelling of the intra-particle diffusion limitations. In this paper we will present the results of the application of the dusty

gas diffusion model (Mason et al., 1967; Mason and Evans, 1969) for describing the intra-particle diffusion limitations in low-pressure methanol synthesis in the temperature range 21&275”C.

Although Brown and Bennett (1970) did not study low-pressure methanol synthesis, their work is of importance because they were able to correlate experimentally obtained effectiveness factors with realistic diffusion coefficients. However, their study concerns the high-pressure catalytic methanol synthesis. There- fore, the results cannot be applied directly to the experimental conditions described in this paper. Seyfert and Luft (1985) observed diffusion limitations for low-pressure methanol synthesis, based on experiments with different sizes of catalyst used. They applied a modified Thiele modulus to describe these limitations. In this Thiele modulus these authors in- troduced an adaptation parameter instead of a diffusion coefficient. From their results we calculated an effective diffusion coefficient of about low9 m’s_‘. However, this value is very low, even for high-pressure gas diffusion in this type of catalyst. A similar problem was encountered by Berty et al. (1983), who used the Weisz-Prater criterion in combination with realistic values for the diffusion coefficients. Although Berty et al. did observe diffusion limitations in experiments with different particle sizes, the Weisz-Prater criterion predicted the absence of diffusion limitations.

Dybkjaer (1985) reported (calculated) effectiveness factors for low-pressure methanol synthesis ranging from 80% to 40%. These effectiveness factors were obtained with the use of a multicomponent diffusion-reaction model based on pore diffusion. Dybkjaer did not report whether this model was validated by experimental results. Recently, Skrzypek et al. (1985) applied the dusty gas model for the

description of methanol synthesis in a single iso-

‘Present address: N. V. Nederlandse Gasunie, Laan Corpus den Hoorn 102, 9728 JR Groningen, The Netherlands.

fPresent address: DSM, Postbus 18, 6160 MD Geleen, The Netherlands.

#To whom correspondence should be addressed.

773

774 G. H. GRAAF~~ al.

thermal catalyst pellet. However, these authors used simplified kinetic rate expressions which are not likely to result in a complete description of the low-pressure methanol kinetics.

From this literature survey it will be clear that the use of simple methods to calculate the effectiveness factor is actually not suitable for describing the kinetic behaviour of a complex system such as low-pressure methanol synthesis.

On the other hand, the results obtained from rigor- ous diffusion models, as used by Dybkjaer and Skrzypek et al., are promising. However, such a soph- isticated diffusion model has not yet been applied in combination with the really complex reaction kinetics that, according to experimental evidence, control low-pressure methanol synthesis. Such an application is the subject of this paper.

THEORY

As has been pointed out by Skrzypek et al. (1984), the dusty gas flux relations can be presented as follows (only the relation for component i is given):

YjNi - YiNj >

P dyi D!?!fr

= --- U RTdx

-$(I +-$&)g. (4)

In this equation effective diffusion coefficients are used, defined by eqs (5) and (6) (Jackson, 1977; Froment and Bischoff, 1979):

~2 8RT f D?=ca__ _ 1

i 1 23 nM,

DE" =:D. li

t IJ’

order differential equations, yielding expressions for dy,/dx. This has been done by Skrzypek et al. (1984). For a system containing M components only (M - 1) differential equations need to be solved by using: cy, = 1.

The material balances and the stoichiometric relations have to be added to the flux relations in order to be able to describe the multicomponent reactiondif- fusion problem. Since three kinetically independent reactions take place in low-pressure methanol synthesis (Graaf et al., 1988) three material balances are needed as well. For a spherical and isothermal particle this yields the following set of differential equations:

da, dx = x=rk (k = 1, 2, 3) (8)

dX_ 1 --- dx x2 (kiI RljEl ("jkFQ))

with

(i = 1, 2, . . . , M

(5) and for the pressure gradient:

The dusty gas flux relations contain three parameters: the mean pore radius, a, the ratio of porosity and

- 1) (9)

(10)

(11)

(12)

(13)

The kinetic rate expressions have been taken from a previous study (Graaf et al., 1988):

k’ ps,A3&Oc_L2o.fiy -fcw,od(fx2KP?)) rb-IaoH* A3 = (1 + Kc0 fco + K co* Scoz) (Sti: + (R ~,olR8/2~)fHzo)

(14)

(= &o,c3 . )

tortuosity factors, E/Z, and the permeability, B, . The dusty gas flux relations are reduced from a three-parameter model to a two-parameter model using ‘WArcy’s Law” (Jackson, 1977; Froment and Bischoff, 1979):

B, = a2/8. (7)

The flux relations for the distinct compounds as described by eq. (4) can be transformed into a set of first

These reaction rate expressions are referred to as model A3B2C3, because of the rate-controlling steps in the proposed reaction mechanism (for details the reader is referred to the previous paper). The kinetic parameters, estimated on the basis of experimental kinetic data at T = 483.5 K, 499.3 K and 516.7 K, were also taken from that study.

The multicomponent reaction-diffusion problem has now been defined by eight coupled non-linear first

Low-pressure methanol synthesis

Table 1. The role of intra-particle diffusion limitations in methanol synthesis as reported in the open literature

775

Authors

Diffusion limitations observed?

Diffusion limitations described?

Diffusion model or criterion

used

Brown and Bennett (1970) Berty et al. (1983) Seyfert and Luft (1985) Dybkjaer (1985) Skrzypek et al. (1985)

yes Yes yes

yes no

yes yes yes

pore diffusion model W&z modulus (adapted) Thiele modulus pore diffusion model dusty gas model

Table 2. Experimental conditions in the present study (catalyst: Cu-Zn-Al)

Feed Feed composition no. yco yco> YHz

P (bar)

1cJ3 4,/w (mS S- 1 kg-‘)

0.065 0.261 0.674 15 30 50 483.5 499.3 516.7 532.4 547.8 l-6 0.053 0.047 0.900 15 30 50 483.5 499.3 516.7 532.4 547.8 l-6 0.220 0.155 0.625 15 30 50 483.5 499.3 516.7 532.4 547.8 l-6 0.120 0.02 1 0.859 15 30 50 483.5 499.3 516.7 532.4 547.8 l-6 0.179 0.067 0.754 15 30 50 483.5 499.3 516.7 532.4 547.8 l-6 0.131 0.023 0.846 20 483.5 499.3 516.7 532.4 547.8 4-18 0.131 0.023 0.846 lo-15 483.5 499.3 516.7 532.4 547.8 5-15 0.131 0.023 0.846 20 483.5 499.3 516.7 532.4 547.8 2-27

Feeds l-7: catalyst particles are cylindrical with d, = 4.2 mm (height z diameter). Feed 8: catalyst particles are assumed to be spherical with d, = 0.154.20 mm.

order differential equations, which have to be solved numerically. The boundary conditions for these differential equations are:

0, = 0 (k = 1, 2, 3) at x = 0

(centre of the catalyst) (17)

yi = yf (i = 1, 2, 3, 4) at x = rp (18)

P = PS at x = rr. (19)

The effectiveness factors (defined as the observed rate divided by the rate based on surface conditions) for methanol and water follow from eq. (8) and the fact that methanol is formed from reactions (1) and (3) while water is formed from reactions (2) and (3). The result is:

3 (0: + @) tlCHsoH = 2 (5 + 6)

3 (a”2 +n:, ? H20 = a (4 + 4) .

(20)

(21)

EXPERIMENTAL

Equipment and analysis The equipment used for this kinetic study was the

same as described in the previous paper (Graaf et al.,

1988). The experiments with commercial size (dp = 4.2 mm) catalyst particles were carried out in a spinning basket reactor as described by Tjabl et al.

(1966). For experiments with small catalyst particles

(dp = 0.15-0.2 mm) a fixed bed reactor was used, similar to the one used for determining the chemical

equilibria of the reactions involved in methanol synthesis (Graaf et al., 1986). Properties of the catalyst used (Haldor Topsee MK 101) are reported by Dybkjaer (1981). The GLC analysis of the compounds of interest is described in the previous paper (Graaf et al., 1988).

Measurements The kinetic experiments were always carried out

under steady state conditions and influences of exter- nal mass-transfer and heat-transfer limitations were negligible. The Appendix shows in more detail that all possible falsifying effects can be reasonably neglected. A brief summary of the experimental conditions is given in Table 2. The calculation of reaction rates of methanol and water are based on mixed flow material balances for the spinning basket reactor, as described in the previous paper on the kinetics of methanol synthesis (Graaf et al., 1988). The kinetic results from the fixed bed reactor experiments were evaluated by solving the mass balance equations numerically, assuming plug flow for the gas phase.

RESULTS

Demonstration of the presence of intra-particle diffusion limitations

It turns out that the methanol formation kinetics can be simplified for feed compositions with a high CO/CO, ratio together with a relatively high content of carbon oxides:

YcolYco2 > 3; Yco + Yco, > 0.15.

776 G.H. GRAAF~ al.

For these specific feed conditions in combination with the temperatures and pressures as used in this study (see Table 2) methanol is primarily formed from CO

(4~~0~ e t(-H30H,A3), water adsorption is almost negligible (fgz > (Ku,o/K$~)j&o) and adsorption of CO predominates (Kcofco > 1 + KCo2fco2).

Incorporating these simplifications in eq. (14) the methanol production rate becomes first order with respect to hydrogen:

&,OH = 4s f&OH fnz -fcofH,Kpy >

. (22)

In order to check whether intra-particle diffusion limitations occur, the results of the experiments with the commercial size catalyst particles (Table 2, feed no. 6) are compared with the results of experiments with small particles (Table 2, feed no. 8). The Arrhenius diagrams for k;, values are presented in Fig. 1. For small size catalyst particles ( l in Fig. 1) an activation energy of 67 kJmol_’ (based on methanol) is obtained. Furthermore, the pseudo-first order simplifi- cation is confirmed by the straight line in the Arrhenius diagram. For full size particles (0 in Fig. 1) the apparent activation energy decreases to 34 kJ mol- ’ above 500 K, while at lower temperatures a value corresponding to the one obtained from the experiments with the small particles results. Such a re- duction of the observed activation energy by a factor of 2 is a well known indication for the presence of intra-particle diffusion limitations (Froment and Bischoff, 1979; Westerterp et al., 1987). This is further supported by the results obtained using partially deactivated (aged) catalysts (A and 0 in Fig. 1). As might be expected, the temperature at which diffusion limitation becomes noticeable increases with decreasing catalyst activity. Finally, it should be noted that the activation energy increases with decreasing catalyst activity. This can be seen in Fig. 1: the results of feed no. 5 yield an activation energy of 83 kJmol-’ (T< 515 K).

Fig. 1. Arrhenius diagrams of the pseudo-first order reaction rate constant with respect to H,. 0: results of feed 8 (see Table 2); 0 : results of feed 6; A : results of feed 7; 0 : results

of feed 5.

Parameters of the dusty gas model Binary diffusion coefficients for each combination

of CO, CO,, H,, CH,OH and H,O were calculated according to Fuller et al. (1966), because this method gives the best agreement with experimental data for these components (Marrero and Mason, 1972).

The mean pore diameter is assumed to be 20 nm, taken from Petrini and Schneider (1984), who studied the water-gas shift reaction on a similar catalyst.

The dynamic viscosity of the gas mixture is taken as 1.3 x 10e5 Nsm-‘. The assumption of a constant dynamic viscosity is justified by the fact that the model calculations are very insensitive to this parameter, because the factor w [eq. (12)3 is much larger than unity (lOC-1000) under typical methanol synthesis conditions (p = l&100 bar, T = 20&3OO”C). Consequently, the pressure gradient [eq. (13)] in the catalyst particle is negligible: the pressure in the centre of the catalyst is calculated to be 0.03-0.3 bar lower than the bulk pressure.

In the reaction-diffusion model the reaction rates have to be based on (pellet) volume of catalyst. Since the kinetic rate expressions are based on weight of catalyst, the value of the apparent density of the catalyst is needed. Based on dimensions and weight the density of the unactivated catalyst is estimated to be 1950 kg rnm3.

Furthermore, the catalyst particles are assumed to be spherical with a radius of 2.1 nm. Although the actual catalyst particles were cylindrical (height x diameter = 4.2 mm) this assumption is reasonable, because it is known that the geometry of the catalyst particle is of minor importance, provided the proper characteristic length is used (Aris, 1957; Rester and Aris, 1969). Since both cylinders (with height = diameter) and spheres yield the same characteristic length (volume divided by exterior surface = d,/6), no correction for the actual size is necessary in the math- ematical model.

This leaves one parameter still unknown: the ratio between porosity and tortuosity factor, ~/7. The value of this parameter has to be determined from experimental data. For this purpose the experiments of feed 6 from Table 2 (21 experiments) were used, because these results exhibit severe diffusion limitations due to the high catalyst activity. Another consequence of this high catalyst activity is the necessity to correct the kinetic rate constants for the difference in catalyst activity. For this reason the kinetic rate constants were multiplied by a temperature-dependent correction factor:

E A. PC, is estimated from the results of feed 8 (small catalyst particles) to be - 20.0 kJ mol- ‘, assuming this value also holds for the results of feed 6. This

assumption seems plausible, since freshly activated

catalyst was used for both series of experiments. The pre-exponential factor, kacteo, is estimated to be 2.33 x 10m2 from the results of feed 6 at T = 483.5 K.

Low-pressure methanol synthesis 777

At this temperature the diffusion limitations are negligible (this was checked afterwards).

For each experiment of feed 6 the overall production rate of methanol was calculated using the dusty gas model in combination with the (corrected) kinetic rate expressions. An optimal value for E/T was found by minimizing the following objective function:

OFSSR = i$l ( &iJOH - f&-l+Xi)*. (24)

The differential equations were solved by means of a standard computer library routine (NAGLIB, 1982) based on a finite difference technique with deferred correction, allied with a Newton iteration to solve the finite difference equations. A description of this technique is given by Pereyra (1979). Initial estimates for the remaining boundary values were calculated as follows.

1 Q =-gp

k 6 P k at x = rp (25)

Yi = Yi. EQ atx=O (26)

p = 0.99ps at x = 0. (27)

The computation of the equilibrium composition was carried out numerically in a similar way as has been presented by Chang et al. (1986). The results of the calculations yield an optimal value of 0.123 for E/Z, which will be used in further calculations.

Parameter estimation for the kinetic rate expressions Effectiveness factors, both for the methanol and the

water production rates, were calculated for the results of the kinetic experiments carried out with the spinning basket reactor. At 532.6 and 547.8 K the effectiveness factors vary in the range 7&40%. At 483.5, 499.3 and 516.7 K these effectiveness factors vary in the range loO-SO%, indicating that the kinetic parameters given in the previous paper (Graaf et al., 1988) might be affected to some extent by internal diffusion effects. For this reason all experimental data of feeds l-5 were corrected with the obtained effectiveness factors and used for the parameter estimation procedure as described in the previous paper (Graaf et al., 1988). Next a new set of effectiveness factors was calculated using the corrected kinetic parameters. This procedure was repeated until consistent values of the kinetic parameters and the effectiveness factors were found. The same procedure was carried out for the other kinetic models presented in the previous paper, but it was found that model A3B2C3 still gives the best results. The following expressions for the kinetic parameters were obtained:

kd,.,3 = (4.89 f 0.29) x 10’

- 113,000 + 300 x exp

RT > (28)

kb,B2 = (9.64 f 7.30) x 10”

~ 152,900 +- 1800 x exp

RT > (29)

kd,ca = (1.09 f 0.07) x 105

- 87,500 f 300 x exp

RT > (30)

Z&, = (2.16 + 0.44) x 1O-5

(

46,800 f 800 x exp

RT >

KC02 = (7.05 + 1.39) x 10-7

61,700 + 800 x exp

RT >

(31)

(32)

K,,,/K,!/; = (6.37 f 2.88) x lo-’

(

84,000 f 1400 x exp

> RT . (33)

The confidence intervals in these equations were calculated from:

SSR,,.,,, = SSR,i” + SSR,i” & Fc,.~--m. 0.991-

(34)

In this equation F,,. N-m, o.ggl is Fisher’s F-value with [m, N - m] degrees of freedom at a 99% significance level (Fisher, 1958). The confidence intervals were obtained by varying one parameter at a time and keeping all the other parameters at their optimal values. The Arrhenius diagrams of the kinetic parameters are presented in Figs 2-7. The shaded areas were calculated from the confidence intervals. These graphs show that the differences between the uncor- rected kinetic parameters and the corrected parameters are maximal at T = 516.7 K. This seems logical, because at this temperature the results of the previous paper were most significantly influenced by diffusional effects. The mean relative deviations for the production rates of methanol and water, based on 144 kinetic experiments, are 9.6% and 22.5%, respect- ively. In our opinion these results are quite satisfac-

100

80

60

20 104kbs,A3

mol <“kg boP’

1.0

08

06

18 1.9 20 m

T/K

Fig. 2. Arrhenius diagram for I&,,. Dashed line: result of the previous paper (Graaf et ai., 1988). Solid line: this work

[eq. (24)]. Shaded areas: confidence intervals.

778 G. H. GRAAF et al.

tory considering the uncertainty coupled with eq. (7) and the estimated value of the mean pore diameter. In fact it shows that conversion data obtained under diffusion-limited conditions are almost as suitable for the determination of the intrinsic reaction kinetics as data obtained in the so called “kinetic regime”, provided an appropriate mass transport with reaction model is applied for taking into account the diffusion limitations.

Aspects of intra-particle d$Lsion limitations in low-pressure methanol synthesis

The results presented below are based on calculations with the dusty gas model in combination with the corrected kinetic model.

Typical methanol and water concentration profiles in the catalyst particle are presented in Figs 8 and 9. As these figures show, chemical equilibrium is almost reached in the centre of the catalyst at the highest

18 1 .Q 2.0 1000 T/K

Fig. 3. Arrhenius diagram for PP.,,. Dashed line: result of Fig. 6. Arrhenius diagram for KCo2. Dashed line: result of the previous paper (Graaf et al., 1988). Solid line: this work the previous paper (Graaf et al., 1988). Solid line: this work

[eq. (25)]. Shaded areas: confidence inervals. [eq. (28)j. Shaded areas: confidence intervals.

I”.”

8.0 -

6.0 -

2.0 - 10‘kbs.C~

mo, c'k$boi'

1.0 - 0.8 -

0.6-

1.8 19 2Kl 21 a T/K

Fig. 4. Arrhenius diagram for k;,co. Dashed line: result of Fig. 7. Arrhenius diagram for K,,,/KA’:. Dashed line: re- the previous paper (Graaf et al., 1988). Solid line: this work sult of the previous paper (Graaf et al., 1988). Solid line: this

[eq. (2611. Shaded areas: confidence intervals. work [eq. (29)]. Shaded areas: confidence intervals.

20 KCO ba?'

1.0

08

06

Fig. 5. Arrhenius diagram for K,. Dashed line: result of the previous paper (Graaf et al., 1988). Solid line: this work

[eq. (27)J Shaded areas: confidence intervals.

,.O- L.0 -

2.0 -

K% bar’

1.0 - 0.8-

0.6 -

0.L -

I

1.8 1.9 2.0 1000 T/K

10.0

8.0

6.0

4.0 I

2.0 %iZ~+i;'2

boT"' 1.0

08

06

!I L I I

1.8 1.9 2.0 2 1000 T/K

Low-pressure methanol synthesis

Fig. 8. Methanol concentration profiles in the catalyst particle. p = 50 bar, d, = 4.2mm. Bulk gas composition: y, = 0.15; y,, = 0.05; JJH2 = 0.80. A:T=483.5K; B:T=499,3K: C:T=516,7K; D:T=532.4K; E:T=

547.8 K.

1.0

*

YH,O,EO

0.5-

0 0 05 1.0 15 2.1

1

Fig. 9. Water concentration profiles in the catalyst particle. p = 50 bar, d, = 4.2 mm. Bulk gas composition: yco = 0.15; yco, = 0.05; y,,, = 0.80. A: T = 483.5 K; B: T = 499.3 K;

C:T=516.7K;D:T=532.4K;E:T=547.8K.

temperature at which experimental kinetic data were collected (547.8 K).

In our previous paper (Graaf et al., 1988) we observed that overshoot of the water-gas shift equilibrium occurs at certain reaction conditions due to the methanol from CO, reaction which also yields water. The same phenomenon is calculated to occur in the catalyst particle. This can he seen in Fig. 10, in which the amount of water related to the water-gas shift

equilibrium is plotted as a function of the position in the catalyst particle. Due to this phenomenon the

CO, concentration profile, also presented in Fig. 10, passes through a minimum. In the outer shell of the catalyst particle CO, is consumed by both the water-gas shift reaction and the methanol from CO,

00605-

779

0

Hz0 fCO 0

H,fC02KP,

5

Fig. 10. Overshoot of the water-gas shift equilibrium and the CO, concentration profile in the catalyst particle. T = 547.8 K, p = 100 bar, d, = 4.2mm. Bulk gas composi-

tion: y,, = 0.3125; yco, = 0.0625; yn2 = 0.6250.

04

Yi t

Fig. 11. Concentration profiles in the catalyst particle. T = 547.8 K, p = 100 bar, d, = 4.2mm. Bulk gas composi-

tion: y, = 0.3125; ycol = 0.0625; yn, = 0.6250.

reaction. Near the centre of the catalyst the water-gas shift reaction produces CO,, while the CO, com- sumption by the methanol from CO, reaction is almost zero due to the approach to chemical equilibrium. The concentration profiles of all components are presented in Fig. 11. Despite the occur- rence of the overshoot of the water-gas shift equilibrium the concentration profiles of CO, II, and CH,OH look quite normal, since the water concentration is very low even in the centre of the catalyst.

The influence of the temperature and the pressure on the effectiveness factors is shown in Figs 12 and ‘13. As might be expected, the effectiveness factors decrease with increasing temperature. The pressure dependency, however, is rather small, although not negligible. With first order gas-solid catalytic reactions effectiveness factors decrease with increasing pressure when Knudsen diffusion is negligible. In


Table 3. The influence of the two diffusion mechanisms on the diffusion limitations

%i,OH GO P

(bar) & case a case b case c case a case b case c

10 520 0.74 0.80 0.93 0.69 0.75 0.92 100 520 0.92 1.00 0.94 0.67 0.90 0.71 10 550 0.32 0.36 0.54 0.48 0.53 0.76

100 550 0.64 0.92 0.70 0.45 0.73 0.50

Bulk composition: Y,__ = 0.15; Y,__ = 0.05; YHz = 0.80. Case a: realistic diffusion coefficients; case b: bulk diffusion neglected (Dij = a3 ); case c: Knudsen diffusion neglected (Di = co ).

‘Cl$OH _

0.5 -

o+’ ” I ’ ’ 8 ’ ’ ’ ’ 500 ~ 550

1. K

10

’ CH,OH 05-

;kJ--e-v I K

Fig. 12. The effectiveness factor for methanol as a function of temperature and pressure (d = 4.2 mm). Bulk gas composition: y,-, = 0.15; yco2 = 0.04; yH1 = 0.80. A: p = 20 bar;

B:p = 50 bar; C:p = 100 bar.

Fig. 13. The effectiveness factor for water as a function of temperature and pressure (d = 4.2 mm). Bulk gas composition: yco = 0.15; y,, = 0.65; y,, = 0.80. A: p = 20 bar;

B:p=50bar,C:p=lOObar.

methanol synthesis the pressure dependency of the binary diffusion coefficients is more or less com- pensated by the following two effects.

1. Both Knudsen diffusion and bulk diffusion are

Fig. 14. The effectiveness factor for methanol as a function of temperature and composition. p = 50 bar, d, = 4.2 mm.

A B C Yco = 0.15 0.10 0.05 Ycoz = 0.05 0.10 0.15 YHI = 0.80 0.80 0.80

important (this can be seen in Table 3). The Knudsen diffusion coefficient is pressure-independent. Although methanol formation kinetics can be simplified as first order kinetics, the number of moles decreases by the reaction. Therefore, the methanol equilibrium conversion increases with increasing pressure, while in “real” first order kinetics the equilibrium composition is pressure-independent.

sensitivity analysis was carried out in order to investigate the relative importance of the two diffusion mechanisms. The results of these calculations are listed in Table 3 and show that Knudsen diffusion predominates at lower pressures, while at higher pressures bulk diffusion is the controlling transport mechanism. However, both diffusion mechanisms should be incorporated in the reaction-diffusion model in order to obtain reliable results. Therefore, the dusty gas model is an appropriate choice.

Figures 14 and 15 show the influence of the gas composition on the effectiveness factors. This influence is rather small, aIthough not negligible. The methanol effectiveness factor decreases with increas-


effects of the diffusion limitations on observed conver-

sion rates.

The ratio of porosity and tortuosity factor of the catalyst, E/T, was estimated from experimental data, yielding a value of 0.123.

The kinetic parameters given in the previous study (Graaf et al., 1988) were influenced slightly by diffusion limitations. Corrected parameters were estimated from an extended set of experimental kinetic data.

A sensitivity analysis shows that at low pressures (p = 10 bar) Knudsen diffusion is the most important diffusion mchanism, while at high pressures (p = 100 bar) bulk diffusion predominates.

Fig. 15. The effectiveness factor for water as a function of temperature and composition. p = 50 bar, d, = 4.2 mm.

A B C Yco = 0.15 0.10 0.05 Yco, = 0.05 0.10 0.15 YH1 = 0.80 0.80 0.80

Based on model calculations it can be concluded that the effectiveness factors for methanol and water decrease with increasing temperature, as expected. The influences of pressure and bulk gas composition on the effectiveness factors are small, although not negligible.

Acknowledgements-We thank Haldor Topsrae A/S, Lyngby, Copenhagen, Denmark for delivering their methanol synthesis catalyst Mk 101 and N.V. Nederlandse Gasunie, Groningen, The Netherlands for delivering gas mixtures for calibration purposes.

YCHjOH yH20

yCH,OH.EO yH20, EQ

Fig. 16. The effectiveness factors for methanol and water as a function of the approach to chemical equilibrium. T = 550 K, p = 50 bar, d, = 4.2 mm. Initial bulk gas compo-

sition: y, = 0.15; Y,, = 0.05; yHI = 0.80.

ing CO,/CO ratio at temperatures below 555 K. It should be noted that the bulk gas compositions were chosen with zero product concentrations. The influence of the concentrations approaching equilibrium is presented in Fig. 16. The effectiveness factors decrease slightly with the approach to chemical equilibrium.

CONCLUSIONS Kinetic experiments with two different catalyst par-

ticle sizes show the presence of intra-particle diffusion limitations in low-pressure methanol synthesis. The dusty gas diffusion model, in combination with the complex reaction kinetics taken from Graaf er al. (1988), can provide an accurate description of the

a

-R3

%

2

0”

Di

Dij

;

bii, F,

k

kh, A3

k&.cs

Ki

KP

1, M

Mi N

Ni

NUP

P Pe

rk

NOTATION

mean pore radius, m permeability, m2 particle Bodenstein number ((u >d,/D,) specific heat, J kg- ’ K- ’ catalyst particle diameter, m diffusion coefficient, m2 s- ’ Knudsen diffusion coefficient, mz s-l binary diffusion coefficient, m2 s-l axial dispersion coefficient, m* s-l partial fugacity, bar auxiliary variables, m s mol- ’ mass-transfer coefficient, m s- ’ reaction rate constant [reaction (l)], mol s-l kg-’ bar-’ reaction rate constant [reaction (2)], mol s- 1 kg- ’ bar- I/’ reaction rate constant [reaction (3)], mol s-l kg-’ bar-’ pseudo-first order reaction rate constant, mol s-l kg-’ bar-’ adsorption equilibrium constant, bar- 1 equilibrium constant based on partial pressures length of the reactor tube, m number of components molecular weight, kg mol- ’ number of components molar flux, mol s-l m-” particle Nusselt number (&,/A) total pressure, bar P&let number (<u>IR/D1)

reaction rate based on volume of catalyst, mol s- 1 m 3


rv r’

R gas constant (8.314), J mol-’ K-l

Rep particle Reynolds number (p (V >d,/p)

Shv particle Sherwood number (k&/D)

T temperature, K superficial gas velocity, m s- ’ average actual gas velocity, m s- ’ auxiliary variable

u <u> W

X

radius of the catalyst particle, m reaction rate based on weight of catalyst, mol s-l kg-’

coordinate along radius of catalyst particle, m

Y mole fraction

Greek Setters

CL heat-transfer coefficient, & porosity of the catalyst

_ _ _ _

W rn~‘K~’

porosity of the catalyst bed effectiveness factor thermal conductivity coefficient, W m- 1 K-l

&bed

l!

Superscripts

Eff S 0 ^

Subscripts

cat co

CO, CH,OH

HZ Hz0 EQ gas i, j k

1 2 3

dynamic viscosity, N mm2 s stoichiometric coefficient density, kg m 3 tortuosity factor of the catalyst auxiliary variable

effective value at the surface of the catalyst indicates standard pressure (1 .O 13 bar) indicates calculated value

indicates the catalyst indicates component: CO indicates component: CO, indicates component: CH,OH indicates component: H, indicates component: H,O at equilibrium indicates gas phase indicates component indicates reaction (1, 2 or 3) indicates methanol from CO reaction indicates water-gas shift reaction indicates methanol from CO, reaction

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APPENDIX_ ESTIMATION OF THE CONTRIBUTIONS OF FALSIFYING EFFECTS

The results of this paper are based on the following assumptions:

1. The catalyst particles are isothermal. 2. There is no temperature difference between the gas bulk

and the catalyst particles. 3. There are no concentration differences between the gas

bulk and the catalyst particles. 4. The spinning basket reactor behaves as a perfect mixer. 5. The fixed bed reactor behaves as a plug flow reactor.

In this Appendix we will show that the assumptions mentioned above are reasonable. The data necessary for the calculations are listed in Table Al. These data were estimated for those experimental conditions which result in the largest contribution of the various falsifying effects. Values for the various enthalpies of formation were taken from Reid et al. (1977) in order to calculate the heat productions of the three chemical reactions.

Assumption 1 In order to test assumption 1 the reaction-diffusion model

was extended with the heat balance. The maximum temperature rise in the full-size catalyst particle was calculated to be 1.2 K and has no significant effect on the calculated effectiveness factors. For small catalyst particles (0.15-0.20 mm), a maximum temperature rise of 0.1 K was calculated.

Assumptions 2 and 3 A further extension of the non-isothermal reaction+lif-

fusion model was realized by adding film resistances between the gas phase and the catalyst particle for heat- and mass-transfer.

Spinning basket reactor. The catalyst particles in the baskets can be regarded as a packed bed in which the gas velocity is determined by the stirrer speed. A superficial gas velocity of 0.35 ms-’ was estimated for our experimental conditions. Using the Sherwood relation presented by Thoenes and Kramers (1958) and a similar Nusselt relation, the following values were calculated for the transport coefficients:

a #as. fat = 650 W m-ZK-’

k wna, Cat = 0.024m s-l.

It turns out that concentration differences between the gas bulk and the catalyst surface are essentially zero. The maximum temperature difference was calculated to be 1.1 K. This results in a relative falsification of the calculated efiect- iveness factors of 2%, which is acceptable in our opinion.

Fixed bed reactor. For the fixed bed reactor Re,-values (based on superficial gas velocities) fall in the range 2-10.

Table Al. Data used in the estimations of falsifying effects

2.8 1.3 x 1o-5 0.23 6.2 x 1O-6 4.7 x 103 0.4 0.4

kg me3 Nsm-’ Wm-‘K-l rn’s-l J kg-‘K-l W m-lK_’

T = 547.8 K; p = 20 bar. Feed composition: feed 6 of Table 2.

Therefore, the conservative approach of Sh, = 2 and Nu, = 2 was chosen. However, even these values result in negligible temperature and concentration differences, since small catalyst particles were used.

Nevertheless, non-isothermal behaviour can occur in the catalyst bed due to limited heat-transfer to the reactor wall and transverse conductivity. The wall heat-transfer coefficient on the bed-side is calculated from the correlation of Dixon and Cresswell (1979), yielding a value of 3760 W me2 K-l. The apparent transverse thermal conductivity coefficient is calculated to be 0.39 W m-l K-’ from Schliinder (1966) and Zehner and Schliinder (1970, 1973). Westerterp et al. (1987) have outlined a method to estimate the relative importance of the transverse conductivity and the wall heat-transfer coefficient. It turns out that the former predominates in the case of our fixed-bed reactor. Evaluating the heat balances results in a maximum temperature rise of 4 K relative to the reactor wall (which is assumed to have the same temperature as the inlet gas). The mean temperature rise in the reactor is 2.7 K. Experimentally, the temperature was measured in the outlet of the reactor near the end of the catalyst bed. Due to cooling of the exit gas and heat conduc- tion in the temperature probe, it is likely that the measured temperatures are slightly too low. A worst case situation results in a falsification of 2.7 K, which yields an error in the pseudo-first order constant of 7.5%. In practical situations, however, the falsification is believed to be much smaller. Therefore, it is plausible to neglect this falsification.

Assumption 4 From the geometry of the basket stirrer and the experi-

mentally applied stirrer speed (20 s- ’ ) we have estimated that approximately 6 x lo-* m3s-’ of gas is set in motion due to agitation. The largest experimentally applied gas flow rate is about 3 x 10T6 ms s-‘, which is 200 times smaller than the pumping action of the stirrer. Therefore, perfect mixing seems a plausible assumption.

Assumption 5 With Re, = 5-25 (based on actual gas velocities) it follows

that Bo n = 2 (Westerterp et al., 1987). For our fixed bed reactor 5,/d, z 400 (the catalyst is located in an approximately 8 cm reactor tube). This results in Pe z 800, which is large enough to neglect deviations from plug flow (Westerterp et al., 1987).

In the case of the spinning basket reactor it should be noted that experimentally varying the stirrer speed (between 8 and 33 s-l) did not change the experimental results notice- ably, which confirms assumptions 2,3 (for this reactor) and 4.

intra-particle diffusion limitations in

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