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Methanol synthesis in a forced unsteady-state reactor network
Chemical Engineering Science 57 (2002) 2995 – 3004
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Methanol synthesis in a forced unsteady-state reactor network Salvatore A. Velardi, Antonello A. Barresi ∗ Dipartimento di Scienza dei Materiali ed Ingegneria Chimica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Received 22 October 2001; received in revised form 28 November 2001; accepted 10 May 2002
Abstract The feasibility of carrying out the low-pressure methanol-synthesis process in forced unsteady-state conditions, using a network of three catalytic 3xed bed reactors with periodical change of the inlet position, has been investigated; advantages and limitations in comparison with the previously proposed reverse-5ow reactor have been highlighted. The e6ect of the main operating parameters—inlet temperature, switching time, inlet 5ow rate—has been studied. A cyclic-steady-state condition and auto-thermal behaviour are possible; nevertheless, they are attainable only for switching times varying in two narrow ranges. Out of these regions, complex steady-states of high periodicity, where conversion is low, or extinction of the reactors occur. For low values of the switching time, the establishing of optimal temperature pro3les along the network allows higher conversions than in the reverse 5ow reactor. Furthermore, the performances of the network are weakly a6ected by wash-out, the removal of unconverted gas in correspondence of switching, which is in intrinsic disadvantage of reverse 5ow operation. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Reaction engineering; Forced unsteady-state chemical reactor; Ring reactor; Methanol synthesis; Mathematical modelling; Dynamic simulation
1. Introduction Forced unsteady-state operation of catalytic reactors has been discussed in the chemical engineering literature since the mid-1960s. Increased conversion, improved selectivity, reduced catalyst deactivation, auto-thermal behaviour are the main goals of such a mode of reactor operation. Virtually, almost all reactor inputs can be forced periodically in order to realise transient operation, but variations in 5ow rate, feed composition and 5ow direction are generally considered. The last case, usually referred to as periodically reversed 5ow operation, was developed by Boreskov and Matros (1983), who 3rstly described the behaviour of a catalytic 3xed bed reactor under transient conditions forced by periodic reversal of the gas 5ow. Two are the major advantages of reverse 5ow operation: 3rst, the possibility of exploiting the thermal storage capacity of the catalyst bed, which acts as a regenerative heat exchanger; thus allowing auto-thermal behaviour even at low reactants concentration; second, an approach toward optimum temperature distribution, which makes possible the creation of favourable thermodynamic conditions for exothermic equilibrium-limited reactions. ∗ Corresponding author. Tel.: +39-011-5644658; fax: +39-011-5644699. E-mail address: [email protected] (A. A. Barresi).
The reverse 5ow reactor has found a number of industrial applications and continues to be a topic of investigation worldwide, but it is not the only way to achieve the above-mentioned advantages. Vanden Bussche and Froment (1996) proposed the concept of star reactor, which can operate in a transient mode giving practically constant exit concentration and higher conversion than the reverse 5ow reactor. The reactor network, which consists of a closed sequence of two or more catalytic 3xed bed reactors, is another way to attain a sustained dynamic behaviour (Matros, 1985); this con3guration, which has been also called “ring reactor”, corresponds to a simulated moving bed. The unsteady-state condition is realised by periodically changing the order of reactors that form the network, varying the feed position. Fig. 1 shows the working principle of a network made of three reactors: the system is fed through reactor number 1 and the order of the reactors is 1–2–3. After a time period tc , the feed position is shifted acting on a set of valves (not showed in Fig. 1), so that the 3rst reactor of the sequence becomes the third one, thus changing the order to 2–3–1. A further change of the feed position leads to the sequence of reactors 3–1–2. By this way it is possible to create a closed cycle, which prevents the heat front from leaving the system. Contrary to the reverse 5ow reactor, the 5ow direction is maintained, ensuring a uniform catalyst exploitation because temperature and
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pared to steady-state technology; nevertheless, reverse 5ow reactor presents the problem of wash-out, i.e., the drop in product concentration upon 5ow reversal, due to the removal of unconverted gas immediately after the change of the inlet section. Previous works on VOC combustion (Brinkmann et al., 1999; Barresi et al., 1999) suggest that a network of catalytic 3xed bed reactors can be a suitable alternative to reverse 5ow operation, because it can reduce the emissions of unburnt gas related to the phenomenon of wash-out. In this work, we consider the in5uence of the main operative parameters (switching time, inlet velocity, inlet temperature) on the performance of a three reactors network for methanol synthesis; a comparison with reverse 5ow reactor is also presented. Fig. 1. Working principle of a network of three catalytic 3xed bed reactors in series.
concentration pro3les migrates throughout the entire length of the system. The simulated moving bed reactor has received little attention up to now. Haynes and Caram (1994) presented some theoretical results concerning the operation of a two reactors network compared with reverse 5ow operation, showing the applicability to mildly exothermic processes, both for generic reversible and irreversible reactions. Auto-thermal behaviour with a nearly uniform catalyst utilisation are the main advantages of the network; it however presents a small range of switching times tc which allow to reach and maintain a pseudo-steady state of operation. The performance and behaviour of a network of three beds, applied to non-stationary catalytic destruction of volatile organic compounds (VOC), have been investigated by means of numerical simulations by Brinkmann, Barresi, Vanni, and Baldi (1999). Each reactor presented a large inert section for heat exchange followed by the catalytic active part. The e6ect of transport parameters on conversion and maximum bed temperature have been studied as well as the in5uence of the design variables. Good conversion and auto-thermal behaviour can be obtained in certain conditions, even at low VOC concentration, but safe operation is related to a narrow stability range of switching times. This aspect has been investigated in detail and a more robust control policy than the open loop strategy, based on 3xed switching times, has been proposed (Barresi, Vanni, Brinkmann, & Baldi, 1999; Barresi & Vanni, 2002). In this work, the application of this reaction concept to exothermic, equilibrium-limited synthesis reactions has been deeply investigated. Methanol synthesis from CO, CO2 and H2 over a commercial Cu–Zn–Al catalyst has been considered. The process is traditionally carried out in the multi-bed adiabatic reactor, allowing carbon conversion per pass of the order of 30 – 40%. It was shown (Vanden Bussche & Froment, 1996) that the application of reverse 5ow operation to methanol synthesis can be economically attractive com-
2. The model A one-dimensional heterogeneous model has been considered. The pressure loss inside the network of adiabatic reactors is neglected. The plug 5ow condition is assumed for the gas phase with dispersive transport of mass and energy; the ideal gas law has been adopted. The transient term is taken into account in the gas phase equations and in the energy equation for the solid phase, while the solid catalytic surface is considered in pseudo-steady-state condition. The e6ect of the intraparticle mass transport has been included in the model by estimation of the e6ectiveness factors, using the linearization method proposed by Gosiewski, Bartmann, Moszczynski, and Mleczko (1999). Thus, the dynamics of the process can be described by the following set of algebraic-di6erential equations. Continuity equation for the gas phase: n
r @cG kG; i av @ + cG v = (yS; i − yG; i ): @t @x
(1)
i=1
Continuity equation for component j in the gas phase: kG; j av @yG; j @yG; j @2 yG; j = De6 + (yS; j − yG; j ) −v 2 @t @x @x cG −yG; j
nr kG; i av i=1
cG
(yS; i − yG; i )
with j = 1 : : : (nr − 1):
(2)
Energy balance for the gas phase @TG @TG ke6 @2 TG hav (TS − TG ): −v = + 2 @t %cˆP; G @x @x %cˆP; G
(3)
Mass balance for the solid phase: kG; j av (yS; j − yG; j ) =[%S (1 − )]
NR k=1
k j; k Rk
with j = 1 : : : nr :
(4)
S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
Energy balance for the solid phase: @TS S @2 T S hav = (TS − TG ) − @t %S cˆP; S @x2 %S cˆP; S (1 − ) N nr R 1 k i; k Rk (−QH˜ f; i ): + cˆP; S i=1
(5)
k=1
Danckwerts boundary conditions are assumed for the gas phase in each reactor and the continuity of gas temperature and concentration pro3les are imposed between the reactors of the network, i.e. in sections at x = K‘, with K = 1; 2. In the following, we refer to L as the total length of the network and to ‘ = L=3 as the length of a single reactor of the sequence. The origin of the x-axis corresponds with the inlet section to the network; consequently it translates from the 3rst reactor of the sequence to the second one when switching time is reached. At this point the boundary conditions are switched in order to simulate the variation of the inlet position. Initially, gas phase and solid temperature are considered equal and constant along the reactors; the initial reactants concentration is null. Boundary conditions De6 @yG; j yG; j |x=0+ − = yin; j ; v @x x=0+ TG |x=0+
ke6 @TG − = Tin %cˆP; G @x x=0+
@TS = 0; v|x=0 = vin @x x=0 @yG; j @yG; j @TG @TS = = = @x x=K‘− @x x=K‘− @x x=K‘− @x x=K‘+
S
@TG @TS = = =0 @x x=K‘+ @x x=K‘+ with K = 1; 2 yG; j |x=K‘− = yG; j |x=K‘+ ; v|x=K‘− = v|x=K‘+ @yG; j De6 = 0; @x x=L
Switching conditions yG; j (x)|t + = yG; j (x + ‘)|t − ; TG (x)|t + = TG (x + ‘)|t − ; x ∈ ]0; 2‘[ TS (x)|t + = TS (x + ‘)|t − ; yG; j (x)|t + = yG; j (x − 2‘)|t − ; TG (x)|t + = TG (x − 2‘)|t − ; x ∈ [2‘; 3‘] TS (x)|t + = TS (x − 2‘)|t − : The model is completed by the kinetic equations by Graaf, Stamhuis, and Beenackers (1988), corresponding to a dual-site Langmuir–Hinshelwood mechanism, based on three independent reactions: methanol formation from CO, water–gas-shift reaction and methanol formation from CO2 : (A) CO + H2 CH3 OH;
(6)
(B) CO2 + H2 CO + H2 O;
(7)
(C) CO2 + 3H2 CH3 OH + H2 O:
(8)
The ideal gas law was chosen as equation of state rather than a more complex equation, in order to save computational time. However, this choice leads to better results than the Redlich–Kwong equation of state and the virial equation truncated after the second virial coeTcient (Graaf, Sijtsema, Stamhuis, & Joosten, 1986). The reaction rates for methanol and water from reactions (A) – (C) are given by the following equations, according to Graaf et al. (1988), where partial pressure are used instead of partial fugacity because of the assumption of ideal gas behaviour: RCH3 OH;A =
=
with K = 1; 2
Initial conditions t = 0; ∀x
0 yG; j |t=0 = yG; j (x);
t = 0; ∀x
TG |t=0 = TG0 (x);
t = 0; ∀x
TS |t=0 = TS0 (x):
3=2 1=2 kps; A KCO [pCO pH2 − pCH3 OH =(pH2 Kp; A )]
; 1=2 (1 + KCO pCO + KCO2 pCO2 )[pH + (KH2 O =KH1=2 )pH2 O ] 2 2 (9)
RH2 O;B
TG |x=K‘− = TG |x=K‘+ ;
@TG ke6 = 0; @x x=L
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@TS S = 0: @x x=L
kps; B KCO2 (pCO2 pH2 − pH2 O pCO =Kp; B ) 1=2 (1 + KCO pCO + KCO2 pCO2 )[pH + (KH2 O =KH1=2 )pH2 O ] 2 2 (10)
RCH3 OH;C = RH2 O;C =
3=2 3=2 kps; C KCO2 [pCO2 pH2 − pCH3 OH pH2 O =(pH2 Kp; C )]
: 1=2 1=2 (1 + KCO pCO + KCO2 pCO2 )[pH + (K =K )p ] H O H O 2 2 H 2 2 (11)
Transport and dispersion parameters have been evaluated similarly to previous works, adopting the same correlations of van de Beld (1995). Concerning the gas–solid heat
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transfer coeTcient, the following correlation has been adopted:
Table 1 Conditions adopted in the simulations (base case)
hdP = 1:6(2 + F ReP0:5 Pr 1=3 ) G
Total length Void fraction Catalyst density Catalyst void fraction Pellet diameter Total pressure Super3cial inlet 5ow rate
with
F = 0:664
1+
0:0557 ReP0:3 Pr 2=3 1 + 2:44(Pr 2=3 − 1)ReP−0:1
(12)
2 (13)
and a similar correlation has been considered for the mass transfer coeTcient, according to Chilton–Colburn’s analogy. The prediction of the axial heat dispersion coeTcient has been carried out adopting a correlation by Dixon and Cresswell (1979): ke6 0:5 0:73 + st =G + ; = %vcˆP; G dP ReP Pr 1 + 9:7=(ReP Pr)
(14)
where the term st =G represents the stagnant zone contribution. According to Edwards and Richardson (1968), a correlation of the same general form as the (14) can be used for the prediction of mass dispersion: 0:73 De6 0:5 = + : vdP ReP Sc 1 + 9:7=(ReP Sc)
(15)
L
%S dp Fin
0:5 m 0.4 1750 kg m−3 0.5 0:0054 m 5 MPa 32:65 mol m−2 s−1
Feed composition (mol%) CO CO2 CH3 OH H2 O H2
4.5 2.0 0.0 0.0 93.5
Kinetic and equilibrium constants kps; A kps; B kps; C KCO KCO2 1=2 KH2 O =KH2 log10 Kp; A log10 Kp; B log10 Kp; C
2:69 × 107 exp[ − 109; 900=(RT )] 7:31 × 108 exp[ − 123; 400=(RT )] 4:36 × 102 exp[ − 65; 200=(RT )] 7:99 × 10−7 exp[58; 100=(RT )] 1:02 × 10−7 exp[67; 400=(RT )] 4:13 × 10−11 exp[104; 500=(RT )] 5139=T − 12:621 −2073=T − 2:029 3066=T − 14:650
The PDE system (1) – (5) has been solved by discretising the domain of the spatial variable x thus obtaining a DAE system; for the algebraic part, given by the mass balances (4) for the solid phase, the non-linear equations solver HYBRID1 from FORTRAN package MINPACK has been used, while the routine LSODE from ODEPACK library has been implemented in order to solve the di6erential part of the system. The conditions adopted in the simulations are given in Table 1; they are the same previously considered for the reverse 5ow operation of methanol synthesis by Vanden Bussche, Neophytides, Zolotarskii, and Froment (1993).
3. Results 3.1. Temperature and concentration pro1les Fig. 2 shows an example of the temperature pro3les of the catalyst along the reactors, taken at various times between two consecutive switches. The synthesis gas mixture is fed ◦ to the network at Tin = 100 C while the catalytic beds are ◦ uniformly pre-heated at TS0 = 220 C, well above the ignition ◦ temperature (∼ = 180 C), in order to have a safe start-up. The required feeding temperature is related to switching time, as will be shown in the following; this low Tin value has been selected according to the results of Vanden Bussche et al. (1993) and in order to show the possibility to operate with low feeding temperatures. The cold syngas progressively extracts heat from the catalyst which gradually
Fig. 2. Temperature pro3les of the catalyst at the beginning, in the middle and at the end of the cycle after cyclic steady state has been reached; ◦ tc = 170 s: Tin = 100 C.
cools down; at the ignition temperature the exothermic synthesis reaction starts with heat generation. After a certain number of switches of the inlet position, a cyclic steady state is reached, in which the same temperature pro3les are obtained in subsequent cycles. In this way auto-thermal processing is realised because no energy supply is needed ◦ except the moderate feed preheating at Tin = 100 C, required in order to prevent the possible condensation of the products.
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Fig. 4. Periodic evolution of methanol outlet concentration; tc = 162 s: ◦ Tin = 100 C.
Fig. 3. Methanol concentration and solid temperature pro3les at various ◦ times in the 3rst part of the cycle; tc = 170 s: Tin = 100 C.
Curves 2 and 3 are very similar to those observed in the reverse 5ow reactor, while pro3le 1, concerning the 3rst instants after the switch, presents a positive slope in the terminal part of the heat wave. This behaviour can be understood if we consider that the 3rst reactor of the sequence takes the place of the last one when the inlet position is changed. Since in the 3rst reactor, before the switch (curve 3, 0 ¡ x=‘ ¡ 1), the temperature pro3le has a positive slope, this behaviour will also be maintained after the switch (curve 1, 2 ¡ x=‘ ¡ 3) for a short time. The presence of a positive slope at the end of the heat wave gives some advantages, as showed in Fig. 3. In the third reactor, immediately after the switch, there is the unconverted syngas which was fed to the system when this reactor was the 3rst of the sequence. While the concentration pro3le moves along the third bed, the high-temperature “tail” of the heat wave (2:5 ¡ x=‘ ¡ 3) brings the gases above the ignition temperature, thus allowing the partial conversion of the fresh reactants that otherwise would leave the network unconverted. Thus, the waste related to wash-out in the 3rst instants after the switching can be strongly limited (the wash-out time is about 4 s). Curves 2b and 3b of Fig. 3 clearly shows two plateaus: in the 3rst one methanol concentration reaches the equilibrium value, corresponding to the maximum temperature. When the temperature decreases, methanol equilibrium concentration is shifted towards higher values, but it remains constant once the temperature falls below the ignition value. Therefore, the 3rst plateau corresponds to an equilibrium limitation, while the second one to a kinetic constraint be◦ cause reaction rate is negligible below 180 C. Nevertheless, 3nal conversion in the network can be higher than the second plateau limit (curve 2b, x=‘ = 3), because the high-temperature “tail” of the heat wave can re-ignite the
partially converted gases coming from the 3rst two beds at a concentration below the equilibrium value, thus giving an extra push in conversion. Wash-out occurs in the 3rst few seconds of the cycle, and can be signi3cantly reduced as a consequence of the particular temperature pro3le established in the last reactor, as discussed above; but the increasing temperature pro3le at the end of the reactor lasts about 10 times longer, and is responsible for the increasing outlet conversion in the 3rst part of the cycle. When the high-temperature end of the heat front is driven out of the system and the temperature of the “tail” falls below the ignition value (curve 3a), the outlet conversion decreases returning to a value equal to that of the second plateau. As a consequence, the time evolution of the exit concentration of methanol exhibits a maximum in the 3rst part of every cycle (Fig. 4), followed by a drop when the high-temperature end of the heat wave leaves the third bed. 3.2. In4uence of the switching time tc The performances of the network versus the switching time tc are plotted in Fig. 5. It appears that two relatively limited ranges of tc exist in which the system can operate adiabatically and a cyclic steady state of period tc is obtained; it can be observed that the e6ect of an increase in tc is opposite in the two ranges. In these ranges the solutions of the problem are given by periodic functions of period tc . The example of Section 3.1 refers to a cycle time falling in the range of high tc values. Fig. 5 shows that in the high tc operating region better performances correspond to lower switching times. This is due to an optimal exploitation of the decreasing temperature pro3le in the third reactor, as it can be seen in Fig. 6: if tc is high (curve 4a), the end of the heat wave leaves the system and the increment in conversion due to the 3nal diminution of temperature is very small (curve 4b); on the contrary, if tc is low (curve 1a and 1b) higher conversion can be obtained
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Fig. 5. In5uence of the switching time tc on the average methanol outlet ◦ concentration. Dashed zones correspond to extinction. Tin = 100 C.
Fig. 6. In5uence of the switching time tc on the temperature pro3les of the catalyst and on methanol concentration pro3les along the network. High ◦ tc operating range. Pro3les taken at the end of the cycle. Tin = 100 C.
thanks to the 3nal decrease in the temperature pro3le, as the heat front has not yet abandoned the system at the end of the cycle. Better performances compared to the high tc range can be obtained at low tc ; in this case the conversion increases with tc . This is related to the particular morphology of the temperature pro3les along the reactors, showed in Fig. 7 for various tc values. In the second and third reactor of the sequence the heat waves initially have a negative slope; so the result of the switching strategy realised in the network is analogous to an intermediate cooling, which allows for a further increase of conversion by shifting the reaction far from equilibrium. For very low switching times (curve 1a
Fig. 7. In5uence of the switching time tc on the temperature pro3les of the catalyst and on methanol concentration along the network. Low tc ◦ operating range. Pro3les taken at the end of the cycle. Tin = 100 C.
Fig. 8. Solid bed temperature pro3les during the phase of extinction. ◦ Pro3les taken in the middle of the cycle. Tin = 100 C; tc = 160 s, below the lower limit of the high tc operating range.
and 1b) the cooling e6ect is small and it is vanished by the subsequent increase of temperature. For higher tc values (curve 3a and 3b), the initial sections of the second and third beds show a deeper temperature drop which allows for a continuously increasing conversion inside the system. A narrow range also exists, close to the zone of maximum conversion, in which complex behaviours (cyclic steady state of high periodicity) are observed of limited practical interest because the average conversion is lower. The triangles plotted in Fig. 5 show some values of the conversion that can be obtained in this region. In the range between the high tc and low tc operating region (25 ¡ tc ¡ 162 in Fig. 5) the reaction is led to extinction, as auto-thermal behaviour is not possible for a wide intermediate range of cycle times. Fig. 8 shows the phase
S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
Fig. 9. In5uence of the switching time tc on the average methanol outlet concentration for various values of inlet 5ow rate. Logarithmic scale in ◦ the low tc operating region. Tin = 100 C.
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Fig. 10. In5uence of the inlet temperature on the average methanol outlet concentration; two representative cycle times are chosen: tc = 40 s (in the low tc range) and tc = 176 s (in the high tc range).
of extinction of the reactor for tc = 160 s, just below the lower limit of the high tc operating region. It can be seen that the temperature wave becomes progressively narrow while the extension of the non-ignited end of the network increases. As long as the second reactor of the sequence is “hot” (curves 1 and 2) the network does not extinguishes because the amount of heat trapped in the second bed allows for the ignition of the feed in the subsequent cycle. As a consequence the maximum temperature of the solid bed remains constant. When the cold zone of the network reaches the second bed (curves 3–5), the maximum temperature decreases and the reaction rapidly extinguishes. 3.3. E6ect of the inlet 4ow rate Fin Fig. 9 shows the in5uence of the inlet 5ow rate on conversion and limit switching time; a variation of 50% in both directions with respect to the base case has been considered. When the inlet 5ow rate increases, the velocity of the heat front increases too, as a consequence of the more effective heat transfer, so that a lower switching period has to be chosen in order to keep the thermal wave inside the network. The 3nal result is that the useful operating range is decreased proportionally to the inlet 5ow rate. For example, if Fin increases by 50%, the region of stable operation turns into 110 ¡ tc ¡ 138 which is about 23 (162 ¡ tc ¡ 206). 3.4. In4uence of the inlet temperature Tin At high switching time, the average methanol concentration increases with the inlet temperature before a maximum is reached. This behaviour, shown in Fig. 10 for tc = 176 s, can be explained by comparing the temperature pro3les in the third reactor (for di6erent inlet temperatures) with that corresponding to maximum generation of methanol along the reactor (Fig. 11). It can be seen that the temperature pro-
Fig. 11. Temperature pro3les in the third reactor of the network for di6erent inlet temperatures (tc = 176 s); comparison with the temperature pro3le which gives maximum CH3 OH generation.
◦
3le obtained with Tin = 200 C, corresponding to the maximum in Fig. 10, is the closest to the optimal one. By operating at low tc values the performances of the network become worse when increasing the inlet temperature, as it can be seen in Fig. 10 for tc = 40 s. In fact, the larger is Tin , the smaller is the e6ect of intermediate cooling and the higher is the temperature level of the catalytic bed. Fig. 12 shows that the pro3le closest to the optimal one cor◦ responds to an inlet temperature Tin = 130 C, which is the minimum required temperature for this tc ; for larger Tin values, instead, the 3nal temperature increase shifts back the equilibrium with a diminution of methanol concentrations in the 3nal sections of the network (curves 2b and 3b). Fig. 13 shows the e6ect of the switching time on average methanol concentration for di6erent values of the inlet temperature. Both for large and small tc , the extension of the operating ranges increases with Tin . Above the ignition ◦ temperature (curve at Tin = 180 C), the distinction between
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Fig. 12. Temperature pro3les (above) and CH3 OH concentration pro3les (below) in the third reactor of the network for di6erent inlet temperatures: comparison with the temperature pro3le which gives maximum CH3 OH generation; tc = 40 s.
Fig. 13. In5uence of the switching time tc on the average methanol outlet concentration for various values of the inlet temperature Tin .
high and low tc operating region does not exist anymore and cyclic steady-state condition is achievable for any switching time. The dashed line indicates the limit of useful periodic operation. Extinction occurs out of the range when operating at high switching time; for low tc values the transition to a non-ignited state is not well de3ned, but it takes place through a region where highly periodic solutions together with decreasing performances are obtained. The best choice of operating parameters corresponds to ◦ tc = 40 s and Tin = 130 C and leads to an average carbon conversion zCH3 OH = 58%. As it is evident from Fig. 14 (left-hand side), the temperature pro3le in the case of optimal conversion approximates the curve corresponding to the maximum methanol generation. Only for x=‘ = 2:6 the high-temperature level slightly decreases the conversion, but the 3nal temperature reduction leads to a further increase
Fig. 14. On the left: comparison between the solid temperature pro3le in the case of maximum CH3 OH generation (continuous line) and in the optimal network con3guration corresponding to tc = 40 s and ◦ Tin = 130 C (dashed line). On the right: phase plane comparing the temperature-conversion trajectory in the case of maximum CH3 OH generation (continuous line) and in the optimal network con3guration (dashed line). Optimal pro3les are taken at the end of the cycle.
Fig. 15. Comparison between the reverse 5ow reactor and the network: evolution of the average methanol outlet concentration from start-up to ◦ periodic steady-state condition. Reverse 5ow: tc = 114 s; Tin = 100 C. ◦ Network (high tc ): tc = 162 s; Tin = 100 C. Network (low tc ): ◦ tc = 40 s; Tin = 130 C.
in the 3nal sections of the network. On the right-hand side, Fig. 14 shows the temperature-conversion phase plane in the case of maximum methanol generation compared to the network optimal con3guration. The curves show that the temperature-conversion trajectory followed in the network is always close to the ideal trajectory which ensures maximum generation of methanol along the reactors. 3.5. Comparison with the reverse 4ow reactor Fig. 15 shows the time evolution of the average methanol outlet concentration for the reverse 5ow reactor and for the network. During the start-up, curves 2 and 3 exhibit a maxi◦ mum because the initial solid bed temperature (TS0 =220 C)
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Fig. 16. Comparison between the reverse 5ow reactor and the network: time evolution of the methanol outlet concentration (above) and of the outlet gas temperature TG; out (below) during a cycle. Reverse 5ow: ◦ ◦ tc = 114 s; Tin = 100 C. Network (high tc ): tc = 162 s; Tin = 100 C. ◦ Network (low tc ): tc = 40 s; Tin = 130 C.
leads to a high conversion in the initial cycles of operation. At low tc values (curve 1) simulations have been carried out ◦ with a higher solid bed pre-heating temperature (TS0 =290 C) in order to prevent the extinction in the start-up phase, even if in this way methanol generation is not favoured in the 3rst cycles of operation. When a pseudo-steady condition is reached the performances of the high tc network are only slightly better than the reverse 5ow reactor at the same feeding temperature; conversely, the network operating at low tc values shows much higher conversion, but a cyclic steady-state regime is obtained after a larger number of cycles. Another comparison between the reverse 5ow reactor and the network is shown in Fig. 16. At the beginning of the cycle in the reverse 5ow reactor methanol conversion drops to zero because of wash-out. The switching strategy realised in the network of three reactors allows for a strong reduction of wash-out, as it is evident from the curve related to high and low tc values. Furthermore, both for the reverse 5ow reactor and the network operating at high switching times, the out◦ let gas temperature exhibits a wide variation (about 200 C) between the beginning and the end of the cycle; such a variation could disturb the operation of downward equipment. On the contrary, the network shows a quite smaller QTG ◦ at the exit, of the order of 10 C, while operating at low tc values. 4. Conclusions An heterogeneous mathematical model, featuring dispersive transport of mass and energy, has been developed in
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order to simulate the dynamic behaviour of a three reactors network for the low-pressure methanol synthesis process. Numerical simulations have shown that a cyclic unsteady-state regime and auto-thermal behaviour are possible only for two relatively small ranges of switching time values. In the high tc operating region the temperature pro3les along the beds are similar to those attainable in the reverse 5ow reactor and the performances are only slightly better. For low tc values the switching strategy leads to quite di6erent pro3les, presenting a series of temperature drops which simulate a sequence of intermediate coolings. These pro3les well approximate the optimal trajectory which ensures maximum methanol generation in every section of the network. In the range between the high tc and low tc region the reaction is led to extinction. Transition between stable periodic operation and extinction occurs at a precise switching time value while operating in the high tc range. The low tc region, instead, is divided from extinction by a narrow range in which complex steady-states of high periodicity are observed. A given variation of the super3cial velocity leads to an inversely proportional variation of the stability ranges, while the higher is the inlet temperature, the wider is the region of operation. Thus, the reactor network can be a suitable alternative to traditional and periodically reversed 5ow reactors not only for VOC destruction, but also for equilibrium limited synthesis reactions. The network of three reactors allows for higher conversion than the reverse-5ow reactor and it is not signi3cantly a6ected by wash-out at the beginning of the cycle. Furthermore, if the switching time belongs to the low tc operating region, the outlet gas temperature varies only slightly along the cycle, di6erently from the reverse 5ow reactor, thus reducing potential disturbances of the equipment downward. Notation av c cˆP dP De6 Fin h QH˜ f ke6 kG kps K Kp
external particle surface area per unit volume of reactor, m−1 molar concentration, mol m−3 speci3c heat at constant pressure, J kg−1 K −1 pellet diameter, m e6ective mass dispersion coeTcient, m2 s−1 super3cial inlet 5ow rate, mol m−2 s−1 gas–solid heat transfer coeTcient, J m−2 K −1 s−1 molar enthalpy of formation, J mol−1 e6ective heat dispersion coeTcient, J m−1 K −1 s−1 gas–solid mass transfer coeTcient, mol m−2 s−1 reaction rate constant adsorption equilibrium constant, bar −1 chemical equilibrium constant based on partial pressure
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‘ L nr NR Pr R ReP Sc t tc T v x y zCH3 OH
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single reactor length, m total network length, m number of components in the mixture number of reactions Prandtl number reaction rate, mol s−1 kg−1 particle Reynolds number Schmidt number clock time, s switching time, s temperature, K interstitial velocity, m s−1 axial reactor coordinate, m molar fraction carbon to methanol conversion
Greek letters
%
void fraction of the catalytic bed e6ectiveness factor thermal conductivity, J m−1 K −1 s−1 stoichiometric coeTcient density, kg m−3
Subscripts and superscripts A B C G S st in out 0
indicates CH3 OH from CO reaction indicates CH3 OH from CO2 reaction indicates water–gas-shift reaction gas phase solid phase or solid surface stagnant inlet condition outlet value initial condition
Acknowledgements The help of Prof. Marco Vanni and Davide Manca in developing and optimising the simulation code is gratefully acknowledged.
References Barresi, A. A., & Vanni, M. (2002). Dynamics and control of forcedunsteady-state catalytic combustors, In: G. Continillo, S. Crescitelli, & M. Giona (Eds.), Nonlinear dynamics and control in process engineering—Recent advances (pp. 73–88). Milano: Springer. Barresi, A. A., Vanni, M., Brinkmann, M., & Baldi, G. (1999). Control of an autothermal network of nonstationary catalytic reactors. AIChE Journal, 45(7), 1597–1602. Boreskov, G. K., & Matros, Y. S. (1983). Unsteady-state performance of heterogeneous catalytic reactions. Catalysis Reviews—Science and Engineering, 25(4), 551–590. Brinkmann, M., Barresi, A. A., Vanni, M., & Baldi, G. (1999). Unsteady state treatment of very lean waste gases in a network of catalytic burners. Catalysis Today, 47(1– 4), 263–277. Dixon, A., & Cresswell, D. L. (1979). Theoretical prediction of e6ective heat transfer parameters in packed beds. AIChE Journal, 25(4), 663–676. Edwards, M. F., & Richardson, J. F. (1968). Gas dispersion in packed beds. Chemical Engineering Science, 23(2), 109–123. Gosiewski, K., Bartmann, U., Moszczynski, M., & Mleczko, L. (1999). E6ect of the intraparticle mass transport limitation on temperature pro3les and catalytic performance of the reverse-5ow reactor for the partial oxidation of methane to synthesis gas. Chemical Engineering Science, 54(20), 4589–4602. Graaf, G. H., Sijtsema, P. J. J. M., Stamhuis, E. J., & Joosten, G. E. (1986). Chemical equilibria in methanol synthesis. Chemical Engineering Science, 41(11), 2883–2890. Graaf, G. H., Stamhuis, E. J., & Beenackers, A. A. C. M. (1988). Kinetics of low-pressure methanol synthesis. Chemical Engineering Science, 43(12), 3185–3195. Haynes, T. N., & Caram, H. S. (1994). The simulated moving bed chemical reactor. Chemical Engineering Science, 49(24B), 5465–5472. Matros, Y. S. (1985). Unsteady processes in catalytic reactors. Amsterdam: Elsevier. van de Beld, L. (1995). Air puri1cation by catalytic oxidation in an adiabatic packed bed reactor with periodic 4ow reversal. Ph.D. thesis, University of Twente, Enschede, The Netherlands. Vanden Bussche, K. M., & Froment, G. F. (1996). The STAR con3guration for methanol synthesis in reversed 5ow reactors. The Canadian Journal of Chemical Engineering, 74(5), 729–734. Vanden Bussche, K. M., Neophytides, S. N., Zolotarskii, I. A., & Froment, G. F. (1993). Modelling and simulation of the reversed 5ow operation of a 3xed-bed reactor for methanol synthesis. Chemical Engineering Science, 48(19), 3335–3345.
www.elsevier.com/locate/ces
Methanol synthesis in a forced unsteady-state reactor network Salvatore A. Velardi, Antonello A. Barresi ∗ Dipartimento di Scienza dei Materiali ed Ingegneria Chimica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Received 22 October 2001; received in revised form 28 November 2001; accepted 10 May 2002
Abstract The feasibility of carrying out the low-pressure methanol-synthesis process in forced unsteady-state conditions, using a network of three catalytic 3xed bed reactors with periodical change of the inlet position, has been investigated; advantages and limitations in comparison with the previously proposed reverse-5ow reactor have been highlighted. The e6ect of the main operating parameters—inlet temperature, switching time, inlet 5ow rate—has been studied. A cyclic-steady-state condition and auto-thermal behaviour are possible; nevertheless, they are attainable only for switching times varying in two narrow ranges. Out of these regions, complex steady-states of high periodicity, where conversion is low, or extinction of the reactors occur. For low values of the switching time, the establishing of optimal temperature pro3les along the network allows higher conversions than in the reverse 5ow reactor. Furthermore, the performances of the network are weakly a6ected by wash-out, the removal of unconverted gas in correspondence of switching, which is in intrinsic disadvantage of reverse 5ow operation. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Reaction engineering; Forced unsteady-state chemical reactor; Ring reactor; Methanol synthesis; Mathematical modelling; Dynamic simulation
1. Introduction Forced unsteady-state operation of catalytic reactors has been discussed in the chemical engineering literature since the mid-1960s. Increased conversion, improved selectivity, reduced catalyst deactivation, auto-thermal behaviour are the main goals of such a mode of reactor operation. Virtually, almost all reactor inputs can be forced periodically in order to realise transient operation, but variations in 5ow rate, feed composition and 5ow direction are generally considered. The last case, usually referred to as periodically reversed 5ow operation, was developed by Boreskov and Matros (1983), who 3rstly described the behaviour of a catalytic 3xed bed reactor under transient conditions forced by periodic reversal of the gas 5ow. Two are the major advantages of reverse 5ow operation: 3rst, the possibility of exploiting the thermal storage capacity of the catalyst bed, which acts as a regenerative heat exchanger; thus allowing auto-thermal behaviour even at low reactants concentration; second, an approach toward optimum temperature distribution, which makes possible the creation of favourable thermodynamic conditions for exothermic equilibrium-limited reactions. ∗ Corresponding author. Tel.: +39-011-5644658; fax: +39-011-5644699. E-mail address: [email protected] (A. A. Barresi).
The reverse 5ow reactor has found a number of industrial applications and continues to be a topic of investigation worldwide, but it is not the only way to achieve the above-mentioned advantages. Vanden Bussche and Froment (1996) proposed the concept of star reactor, which can operate in a transient mode giving practically constant exit concentration and higher conversion than the reverse 5ow reactor. The reactor network, which consists of a closed sequence of two or more catalytic 3xed bed reactors, is another way to attain a sustained dynamic behaviour (Matros, 1985); this con3guration, which has been also called “ring reactor”, corresponds to a simulated moving bed. The unsteady-state condition is realised by periodically changing the order of reactors that form the network, varying the feed position. Fig. 1 shows the working principle of a network made of three reactors: the system is fed through reactor number 1 and the order of the reactors is 1–2–3. After a time period tc , the feed position is shifted acting on a set of valves (not showed in Fig. 1), so that the 3rst reactor of the sequence becomes the third one, thus changing the order to 2–3–1. A further change of the feed position leads to the sequence of reactors 3–1–2. By this way it is possible to create a closed cycle, which prevents the heat front from leaving the system. Contrary to the reverse 5ow reactor, the 5ow direction is maintained, ensuring a uniform catalyst exploitation because temperature and
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 1 8 1 - 1
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pared to steady-state technology; nevertheless, reverse 5ow reactor presents the problem of wash-out, i.e., the drop in product concentration upon 5ow reversal, due to the removal of unconverted gas immediately after the change of the inlet section. Previous works on VOC combustion (Brinkmann et al., 1999; Barresi et al., 1999) suggest that a network of catalytic 3xed bed reactors can be a suitable alternative to reverse 5ow operation, because it can reduce the emissions of unburnt gas related to the phenomenon of wash-out. In this work, we consider the in5uence of the main operative parameters (switching time, inlet velocity, inlet temperature) on the performance of a three reactors network for methanol synthesis; a comparison with reverse 5ow reactor is also presented. Fig. 1. Working principle of a network of three catalytic 3xed bed reactors in series.
concentration pro3les migrates throughout the entire length of the system. The simulated moving bed reactor has received little attention up to now. Haynes and Caram (1994) presented some theoretical results concerning the operation of a two reactors network compared with reverse 5ow operation, showing the applicability to mildly exothermic processes, both for generic reversible and irreversible reactions. Auto-thermal behaviour with a nearly uniform catalyst utilisation are the main advantages of the network; it however presents a small range of switching times tc which allow to reach and maintain a pseudo-steady state of operation. The performance and behaviour of a network of three beds, applied to non-stationary catalytic destruction of volatile organic compounds (VOC), have been investigated by means of numerical simulations by Brinkmann, Barresi, Vanni, and Baldi (1999). Each reactor presented a large inert section for heat exchange followed by the catalytic active part. The e6ect of transport parameters on conversion and maximum bed temperature have been studied as well as the in5uence of the design variables. Good conversion and auto-thermal behaviour can be obtained in certain conditions, even at low VOC concentration, but safe operation is related to a narrow stability range of switching times. This aspect has been investigated in detail and a more robust control policy than the open loop strategy, based on 3xed switching times, has been proposed (Barresi, Vanni, Brinkmann, & Baldi, 1999; Barresi & Vanni, 2002). In this work, the application of this reaction concept to exothermic, equilibrium-limited synthesis reactions has been deeply investigated. Methanol synthesis from CO, CO2 and H2 over a commercial Cu–Zn–Al catalyst has been considered. The process is traditionally carried out in the multi-bed adiabatic reactor, allowing carbon conversion per pass of the order of 30 – 40%. It was shown (Vanden Bussche & Froment, 1996) that the application of reverse 5ow operation to methanol synthesis can be economically attractive com-
2. The model A one-dimensional heterogeneous model has been considered. The pressure loss inside the network of adiabatic reactors is neglected. The plug 5ow condition is assumed for the gas phase with dispersive transport of mass and energy; the ideal gas law has been adopted. The transient term is taken into account in the gas phase equations and in the energy equation for the solid phase, while the solid catalytic surface is considered in pseudo-steady-state condition. The e6ect of the intraparticle mass transport has been included in the model by estimation of the e6ectiveness factors, using the linearization method proposed by Gosiewski, Bartmann, Moszczynski, and Mleczko (1999). Thus, the dynamics of the process can be described by the following set of algebraic-di6erential equations. Continuity equation for the gas phase: n
r @cG kG; i av @ + cG v = (yS; i − yG; i ): @t @x
(1)
i=1
Continuity equation for component j in the gas phase: kG; j av @yG; j @yG; j @2 yG; j = De6 + (yS; j − yG; j ) −v 2 @t @x @x cG −yG; j
nr kG; i av i=1
cG
(yS; i − yG; i )
with j = 1 : : : (nr − 1):
(2)
Energy balance for the gas phase @TG @TG ke6 @2 TG hav (TS − TG ): −v = + 2 @t %cˆP; G @x @x %cˆP; G
(3)
Mass balance for the solid phase: kG; j av (yS; j − yG; j ) =[%S (1 − )]
NR k=1
k j; k Rk
with j = 1 : : : nr :
(4)
S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
Energy balance for the solid phase: @TS S @2 T S hav = (TS − TG ) − @t %S cˆP; S @x2 %S cˆP; S (1 − ) N nr R 1 k i; k Rk (−QH˜ f; i ): + cˆP; S i=1
(5)
k=1
Danckwerts boundary conditions are assumed for the gas phase in each reactor and the continuity of gas temperature and concentration pro3les are imposed between the reactors of the network, i.e. in sections at x = K‘, with K = 1; 2. In the following, we refer to L as the total length of the network and to ‘ = L=3 as the length of a single reactor of the sequence. The origin of the x-axis corresponds with the inlet section to the network; consequently it translates from the 3rst reactor of the sequence to the second one when switching time is reached. At this point the boundary conditions are switched in order to simulate the variation of the inlet position. Initially, gas phase and solid temperature are considered equal and constant along the reactors; the initial reactants concentration is null. Boundary conditions De6 @yG; j yG; j |x=0+ − = yin; j ; v @x x=0+ TG |x=0+
ke6 @TG − = Tin %cˆP; G @x x=0+
@TS = 0; v|x=0 = vin @x x=0 @yG; j @yG; j @TG @TS = = = @x x=K‘− @x x=K‘− @x x=K‘− @x x=K‘+
S
@TG @TS = = =0 @x x=K‘+ @x x=K‘+ with K = 1; 2 yG; j |x=K‘− = yG; j |x=K‘+ ; v|x=K‘− = v|x=K‘+ @yG; j De6 = 0; @x x=L
Switching conditions yG; j (x)|t + = yG; j (x + ‘)|t − ; TG (x)|t + = TG (x + ‘)|t − ; x ∈ ]0; 2‘[ TS (x)|t + = TS (x + ‘)|t − ; yG; j (x)|t + = yG; j (x − 2‘)|t − ; TG (x)|t + = TG (x − 2‘)|t − ; x ∈ [2‘; 3‘] TS (x)|t + = TS (x − 2‘)|t − : The model is completed by the kinetic equations by Graaf, Stamhuis, and Beenackers (1988), corresponding to a dual-site Langmuir–Hinshelwood mechanism, based on three independent reactions: methanol formation from CO, water–gas-shift reaction and methanol formation from CO2 : (A) CO + H2 CH3 OH;
(6)
(B) CO2 + H2 CO + H2 O;
(7)
(C) CO2 + 3H2 CH3 OH + H2 O:
(8)
The ideal gas law was chosen as equation of state rather than a more complex equation, in order to save computational time. However, this choice leads to better results than the Redlich–Kwong equation of state and the virial equation truncated after the second virial coeTcient (Graaf, Sijtsema, Stamhuis, & Joosten, 1986). The reaction rates for methanol and water from reactions (A) – (C) are given by the following equations, according to Graaf et al. (1988), where partial pressure are used instead of partial fugacity because of the assumption of ideal gas behaviour: RCH3 OH;A =
=
with K = 1; 2
Initial conditions t = 0; ∀x
0 yG; j |t=0 = yG; j (x);
t = 0; ∀x
TG |t=0 = TG0 (x);
t = 0; ∀x
TS |t=0 = TS0 (x):
3=2 1=2 kps; A KCO [pCO pH2 − pCH3 OH =(pH2 Kp; A )]
; 1=2 (1 + KCO pCO + KCO2 pCO2 )[pH + (KH2 O =KH1=2 )pH2 O ] 2 2 (9)
RH2 O;B
TG |x=K‘− = TG |x=K‘+ ;
@TG ke6 = 0; @x x=L
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@TS S = 0: @x x=L
kps; B KCO2 (pCO2 pH2 − pH2 O pCO =Kp; B ) 1=2 (1 + KCO pCO + KCO2 pCO2 )[pH + (KH2 O =KH1=2 )pH2 O ] 2 2 (10)
RCH3 OH;C = RH2 O;C =
3=2 3=2 kps; C KCO2 [pCO2 pH2 − pCH3 OH pH2 O =(pH2 Kp; C )]
: 1=2 1=2 (1 + KCO pCO + KCO2 pCO2 )[pH + (K =K )p ] H O H O 2 2 H 2 2 (11)
Transport and dispersion parameters have been evaluated similarly to previous works, adopting the same correlations of van de Beld (1995). Concerning the gas–solid heat
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transfer coeTcient, the following correlation has been adopted:
Table 1 Conditions adopted in the simulations (base case)
hdP = 1:6(2 + F ReP0:5 Pr 1=3 ) G
Total length Void fraction Catalyst density Catalyst void fraction Pellet diameter Total pressure Super3cial inlet 5ow rate
with
F = 0:664
1+
0:0557 ReP0:3 Pr 2=3 1 + 2:44(Pr 2=3 − 1)ReP−0:1
(12)
2 (13)
and a similar correlation has been considered for the mass transfer coeTcient, according to Chilton–Colburn’s analogy. The prediction of the axial heat dispersion coeTcient has been carried out adopting a correlation by Dixon and Cresswell (1979): ke6 0:5 0:73 + st =G + ; = %vcˆP; G dP ReP Pr 1 + 9:7=(ReP Pr)
(14)
where the term st =G represents the stagnant zone contribution. According to Edwards and Richardson (1968), a correlation of the same general form as the (14) can be used for the prediction of mass dispersion: 0:73 De6 0:5 = + : vdP ReP Sc 1 + 9:7=(ReP Sc)
(15)
L
%S dp Fin
0:5 m 0.4 1750 kg m−3 0.5 0:0054 m 5 MPa 32:65 mol m−2 s−1
Feed composition (mol%) CO CO2 CH3 OH H2 O H2
4.5 2.0 0.0 0.0 93.5
Kinetic and equilibrium constants kps; A kps; B kps; C KCO KCO2 1=2 KH2 O =KH2 log10 Kp; A log10 Kp; B log10 Kp; C
2:69 × 107 exp[ − 109; 900=(RT )] 7:31 × 108 exp[ − 123; 400=(RT )] 4:36 × 102 exp[ − 65; 200=(RT )] 7:99 × 10−7 exp[58; 100=(RT )] 1:02 × 10−7 exp[67; 400=(RT )] 4:13 × 10−11 exp[104; 500=(RT )] 5139=T − 12:621 −2073=T − 2:029 3066=T − 14:650
The PDE system (1) – (5) has been solved by discretising the domain of the spatial variable x thus obtaining a DAE system; for the algebraic part, given by the mass balances (4) for the solid phase, the non-linear equations solver HYBRID1 from FORTRAN package MINPACK has been used, while the routine LSODE from ODEPACK library has been implemented in order to solve the di6erential part of the system. The conditions adopted in the simulations are given in Table 1; they are the same previously considered for the reverse 5ow operation of methanol synthesis by Vanden Bussche, Neophytides, Zolotarskii, and Froment (1993).
3. Results 3.1. Temperature and concentration pro1les Fig. 2 shows an example of the temperature pro3les of the catalyst along the reactors, taken at various times between two consecutive switches. The synthesis gas mixture is fed ◦ to the network at Tin = 100 C while the catalytic beds are ◦ uniformly pre-heated at TS0 = 220 C, well above the ignition ◦ temperature (∼ = 180 C), in order to have a safe start-up. The required feeding temperature is related to switching time, as will be shown in the following; this low Tin value has been selected according to the results of Vanden Bussche et al. (1993) and in order to show the possibility to operate with low feeding temperatures. The cold syngas progressively extracts heat from the catalyst which gradually
Fig. 2. Temperature pro3les of the catalyst at the beginning, in the middle and at the end of the cycle after cyclic steady state has been reached; ◦ tc = 170 s: Tin = 100 C.
cools down; at the ignition temperature the exothermic synthesis reaction starts with heat generation. After a certain number of switches of the inlet position, a cyclic steady state is reached, in which the same temperature pro3les are obtained in subsequent cycles. In this way auto-thermal processing is realised because no energy supply is needed ◦ except the moderate feed preheating at Tin = 100 C, required in order to prevent the possible condensation of the products.
S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
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Fig. 4. Periodic evolution of methanol outlet concentration; tc = 162 s: ◦ Tin = 100 C.
Fig. 3. Methanol concentration and solid temperature pro3les at various ◦ times in the 3rst part of the cycle; tc = 170 s: Tin = 100 C.
Curves 2 and 3 are very similar to those observed in the reverse 5ow reactor, while pro3le 1, concerning the 3rst instants after the switch, presents a positive slope in the terminal part of the heat wave. This behaviour can be understood if we consider that the 3rst reactor of the sequence takes the place of the last one when the inlet position is changed. Since in the 3rst reactor, before the switch (curve 3, 0 ¡ x=‘ ¡ 1), the temperature pro3le has a positive slope, this behaviour will also be maintained after the switch (curve 1, 2 ¡ x=‘ ¡ 3) for a short time. The presence of a positive slope at the end of the heat wave gives some advantages, as showed in Fig. 3. In the third reactor, immediately after the switch, there is the unconverted syngas which was fed to the system when this reactor was the 3rst of the sequence. While the concentration pro3le moves along the third bed, the high-temperature “tail” of the heat wave (2:5 ¡ x=‘ ¡ 3) brings the gases above the ignition temperature, thus allowing the partial conversion of the fresh reactants that otherwise would leave the network unconverted. Thus, the waste related to wash-out in the 3rst instants after the switching can be strongly limited (the wash-out time is about 4 s). Curves 2b and 3b of Fig. 3 clearly shows two plateaus: in the 3rst one methanol concentration reaches the equilibrium value, corresponding to the maximum temperature. When the temperature decreases, methanol equilibrium concentration is shifted towards higher values, but it remains constant once the temperature falls below the ignition value. Therefore, the 3rst plateau corresponds to an equilibrium limitation, while the second one to a kinetic constraint be◦ cause reaction rate is negligible below 180 C. Nevertheless, 3nal conversion in the network can be higher than the second plateau limit (curve 2b, x=‘ = 3), because the high-temperature “tail” of the heat wave can re-ignite the
partially converted gases coming from the 3rst two beds at a concentration below the equilibrium value, thus giving an extra push in conversion. Wash-out occurs in the 3rst few seconds of the cycle, and can be signi3cantly reduced as a consequence of the particular temperature pro3le established in the last reactor, as discussed above; but the increasing temperature pro3le at the end of the reactor lasts about 10 times longer, and is responsible for the increasing outlet conversion in the 3rst part of the cycle. When the high-temperature end of the heat front is driven out of the system and the temperature of the “tail” falls below the ignition value (curve 3a), the outlet conversion decreases returning to a value equal to that of the second plateau. As a consequence, the time evolution of the exit concentration of methanol exhibits a maximum in the 3rst part of every cycle (Fig. 4), followed by a drop when the high-temperature end of the heat wave leaves the third bed. 3.2. In4uence of the switching time tc The performances of the network versus the switching time tc are plotted in Fig. 5. It appears that two relatively limited ranges of tc exist in which the system can operate adiabatically and a cyclic steady state of period tc is obtained; it can be observed that the e6ect of an increase in tc is opposite in the two ranges. In these ranges the solutions of the problem are given by periodic functions of period tc . The example of Section 3.1 refers to a cycle time falling in the range of high tc values. Fig. 5 shows that in the high tc operating region better performances correspond to lower switching times. This is due to an optimal exploitation of the decreasing temperature pro3le in the third reactor, as it can be seen in Fig. 6: if tc is high (curve 4a), the end of the heat wave leaves the system and the increment in conversion due to the 3nal diminution of temperature is very small (curve 4b); on the contrary, if tc is low (curve 1a and 1b) higher conversion can be obtained
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S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
Fig. 5. In5uence of the switching time tc on the average methanol outlet ◦ concentration. Dashed zones correspond to extinction. Tin = 100 C.
Fig. 6. In5uence of the switching time tc on the temperature pro3les of the catalyst and on methanol concentration pro3les along the network. High ◦ tc operating range. Pro3les taken at the end of the cycle. Tin = 100 C.
thanks to the 3nal decrease in the temperature pro3le, as the heat front has not yet abandoned the system at the end of the cycle. Better performances compared to the high tc range can be obtained at low tc ; in this case the conversion increases with tc . This is related to the particular morphology of the temperature pro3les along the reactors, showed in Fig. 7 for various tc values. In the second and third reactor of the sequence the heat waves initially have a negative slope; so the result of the switching strategy realised in the network is analogous to an intermediate cooling, which allows for a further increase of conversion by shifting the reaction far from equilibrium. For very low switching times (curve 1a
Fig. 7. In5uence of the switching time tc on the temperature pro3les of the catalyst and on methanol concentration along the network. Low tc ◦ operating range. Pro3les taken at the end of the cycle. Tin = 100 C.
Fig. 8. Solid bed temperature pro3les during the phase of extinction. ◦ Pro3les taken in the middle of the cycle. Tin = 100 C; tc = 160 s, below the lower limit of the high tc operating range.
and 1b) the cooling e6ect is small and it is vanished by the subsequent increase of temperature. For higher tc values (curve 3a and 3b), the initial sections of the second and third beds show a deeper temperature drop which allows for a continuously increasing conversion inside the system. A narrow range also exists, close to the zone of maximum conversion, in which complex behaviours (cyclic steady state of high periodicity) are observed of limited practical interest because the average conversion is lower. The triangles plotted in Fig. 5 show some values of the conversion that can be obtained in this region. In the range between the high tc and low tc operating region (25 ¡ tc ¡ 162 in Fig. 5) the reaction is led to extinction, as auto-thermal behaviour is not possible for a wide intermediate range of cycle times. Fig. 8 shows the phase
S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
Fig. 9. In5uence of the switching time tc on the average methanol outlet concentration for various values of inlet 5ow rate. Logarithmic scale in ◦ the low tc operating region. Tin = 100 C.
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Fig. 10. In5uence of the inlet temperature on the average methanol outlet concentration; two representative cycle times are chosen: tc = 40 s (in the low tc range) and tc = 176 s (in the high tc range).
of extinction of the reactor for tc = 160 s, just below the lower limit of the high tc operating region. It can be seen that the temperature wave becomes progressively narrow while the extension of the non-ignited end of the network increases. As long as the second reactor of the sequence is “hot” (curves 1 and 2) the network does not extinguishes because the amount of heat trapped in the second bed allows for the ignition of the feed in the subsequent cycle. As a consequence the maximum temperature of the solid bed remains constant. When the cold zone of the network reaches the second bed (curves 3–5), the maximum temperature decreases and the reaction rapidly extinguishes. 3.3. E6ect of the inlet 4ow rate Fin Fig. 9 shows the in5uence of the inlet 5ow rate on conversion and limit switching time; a variation of 50% in both directions with respect to the base case has been considered. When the inlet 5ow rate increases, the velocity of the heat front increases too, as a consequence of the more effective heat transfer, so that a lower switching period has to be chosen in order to keep the thermal wave inside the network. The 3nal result is that the useful operating range is decreased proportionally to the inlet 5ow rate. For example, if Fin increases by 50%, the region of stable operation turns into 110 ¡ tc ¡ 138 which is about 23 (162 ¡ tc ¡ 206). 3.4. In4uence of the inlet temperature Tin At high switching time, the average methanol concentration increases with the inlet temperature before a maximum is reached. This behaviour, shown in Fig. 10 for tc = 176 s, can be explained by comparing the temperature pro3les in the third reactor (for di6erent inlet temperatures) with that corresponding to maximum generation of methanol along the reactor (Fig. 11). It can be seen that the temperature pro-
Fig. 11. Temperature pro3les in the third reactor of the network for di6erent inlet temperatures (tc = 176 s); comparison with the temperature pro3le which gives maximum CH3 OH generation.
◦
3le obtained with Tin = 200 C, corresponding to the maximum in Fig. 10, is the closest to the optimal one. By operating at low tc values the performances of the network become worse when increasing the inlet temperature, as it can be seen in Fig. 10 for tc = 40 s. In fact, the larger is Tin , the smaller is the e6ect of intermediate cooling and the higher is the temperature level of the catalytic bed. Fig. 12 shows that the pro3le closest to the optimal one cor◦ responds to an inlet temperature Tin = 130 C, which is the minimum required temperature for this tc ; for larger Tin values, instead, the 3nal temperature increase shifts back the equilibrium with a diminution of methanol concentrations in the 3nal sections of the network (curves 2b and 3b). Fig. 13 shows the e6ect of the switching time on average methanol concentration for di6erent values of the inlet temperature. Both for large and small tc , the extension of the operating ranges increases with Tin . Above the ignition ◦ temperature (curve at Tin = 180 C), the distinction between
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Fig. 12. Temperature pro3les (above) and CH3 OH concentration pro3les (below) in the third reactor of the network for di6erent inlet temperatures: comparison with the temperature pro3le which gives maximum CH3 OH generation; tc = 40 s.
Fig. 13. In5uence of the switching time tc on the average methanol outlet concentration for various values of the inlet temperature Tin .
high and low tc operating region does not exist anymore and cyclic steady-state condition is achievable for any switching time. The dashed line indicates the limit of useful periodic operation. Extinction occurs out of the range when operating at high switching time; for low tc values the transition to a non-ignited state is not well de3ned, but it takes place through a region where highly periodic solutions together with decreasing performances are obtained. The best choice of operating parameters corresponds to ◦ tc = 40 s and Tin = 130 C and leads to an average carbon conversion zCH3 OH = 58%. As it is evident from Fig. 14 (left-hand side), the temperature pro3le in the case of optimal conversion approximates the curve corresponding to the maximum methanol generation. Only for x=‘ = 2:6 the high-temperature level slightly decreases the conversion, but the 3nal temperature reduction leads to a further increase
Fig. 14. On the left: comparison between the solid temperature pro3le in the case of maximum CH3 OH generation (continuous line) and in the optimal network con3guration corresponding to tc = 40 s and ◦ Tin = 130 C (dashed line). On the right: phase plane comparing the temperature-conversion trajectory in the case of maximum CH3 OH generation (continuous line) and in the optimal network con3guration (dashed line). Optimal pro3les are taken at the end of the cycle.
Fig. 15. Comparison between the reverse 5ow reactor and the network: evolution of the average methanol outlet concentration from start-up to ◦ periodic steady-state condition. Reverse 5ow: tc = 114 s; Tin = 100 C. ◦ Network (high tc ): tc = 162 s; Tin = 100 C. Network (low tc ): ◦ tc = 40 s; Tin = 130 C.
in the 3nal sections of the network. On the right-hand side, Fig. 14 shows the temperature-conversion phase plane in the case of maximum methanol generation compared to the network optimal con3guration. The curves show that the temperature-conversion trajectory followed in the network is always close to the ideal trajectory which ensures maximum generation of methanol along the reactors. 3.5. Comparison with the reverse 4ow reactor Fig. 15 shows the time evolution of the average methanol outlet concentration for the reverse 5ow reactor and for the network. During the start-up, curves 2 and 3 exhibit a maxi◦ mum because the initial solid bed temperature (TS0 =220 C)
S. A. Velardi, A. A. Barresi / Chemical Engineering Science 57 (2002) 2995–3004
Fig. 16. Comparison between the reverse 5ow reactor and the network: time evolution of the methanol outlet concentration (above) and of the outlet gas temperature TG; out (below) during a cycle. Reverse 5ow: ◦ ◦ tc = 114 s; Tin = 100 C. Network (high tc ): tc = 162 s; Tin = 100 C. ◦ Network (low tc ): tc = 40 s; Tin = 130 C.
leads to a high conversion in the initial cycles of operation. At low tc values (curve 1) simulations have been carried out ◦ with a higher solid bed pre-heating temperature (TS0 =290 C) in order to prevent the extinction in the start-up phase, even if in this way methanol generation is not favoured in the 3rst cycles of operation. When a pseudo-steady condition is reached the performances of the high tc network are only slightly better than the reverse 5ow reactor at the same feeding temperature; conversely, the network operating at low tc values shows much higher conversion, but a cyclic steady-state regime is obtained after a larger number of cycles. Another comparison between the reverse 5ow reactor and the network is shown in Fig. 16. At the beginning of the cycle in the reverse 5ow reactor methanol conversion drops to zero because of wash-out. The switching strategy realised in the network of three reactors allows for a strong reduction of wash-out, as it is evident from the curve related to high and low tc values. Furthermore, both for the reverse 5ow reactor and the network operating at high switching times, the out◦ let gas temperature exhibits a wide variation (about 200 C) between the beginning and the end of the cycle; such a variation could disturb the operation of downward equipment. On the contrary, the network shows a quite smaller QTG ◦ at the exit, of the order of 10 C, while operating at low tc values. 4. Conclusions An heterogeneous mathematical model, featuring dispersive transport of mass and energy, has been developed in
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order to simulate the dynamic behaviour of a three reactors network for the low-pressure methanol synthesis process. Numerical simulations have shown that a cyclic unsteady-state regime and auto-thermal behaviour are possible only for two relatively small ranges of switching time values. In the high tc operating region the temperature pro3les along the beds are similar to those attainable in the reverse 5ow reactor and the performances are only slightly better. For low tc values the switching strategy leads to quite di6erent pro3les, presenting a series of temperature drops which simulate a sequence of intermediate coolings. These pro3les well approximate the optimal trajectory which ensures maximum methanol generation in every section of the network. In the range between the high tc and low tc region the reaction is led to extinction. Transition between stable periodic operation and extinction occurs at a precise switching time value while operating in the high tc range. The low tc region, instead, is divided from extinction by a narrow range in which complex steady-states of high periodicity are observed. A given variation of the super3cial velocity leads to an inversely proportional variation of the stability ranges, while the higher is the inlet temperature, the wider is the region of operation. Thus, the reactor network can be a suitable alternative to traditional and periodically reversed 5ow reactors not only for VOC destruction, but also for equilibrium limited synthesis reactions. The network of three reactors allows for higher conversion than the reverse-5ow reactor and it is not signi3cantly a6ected by wash-out at the beginning of the cycle. Furthermore, if the switching time belongs to the low tc operating region, the outlet gas temperature varies only slightly along the cycle, di6erently from the reverse 5ow reactor, thus reducing potential disturbances of the equipment downward. Notation av c cˆP dP De6 Fin h QH˜ f ke6 kG kps K Kp
external particle surface area per unit volume of reactor, m−1 molar concentration, mol m−3 speci3c heat at constant pressure, J kg−1 K −1 pellet diameter, m e6ective mass dispersion coeTcient, m2 s−1 super3cial inlet 5ow rate, mol m−2 s−1 gas–solid heat transfer coeTcient, J m−2 K −1 s−1 molar enthalpy of formation, J mol−1 e6ective heat dispersion coeTcient, J m−1 K −1 s−1 gas–solid mass transfer coeTcient, mol m−2 s−1 reaction rate constant adsorption equilibrium constant, bar −1 chemical equilibrium constant based on partial pressure
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‘ L nr NR Pr R ReP Sc t tc T v x y zCH3 OH
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single reactor length, m total network length, m number of components in the mixture number of reactions Prandtl number reaction rate, mol s−1 kg−1 particle Reynolds number Schmidt number clock time, s switching time, s temperature, K interstitial velocity, m s−1 axial reactor coordinate, m molar fraction carbon to methanol conversion
Greek letters
%
void fraction of the catalytic bed e6ectiveness factor thermal conductivity, J m−1 K −1 s−1 stoichiometric coeTcient density, kg m−3
Subscripts and superscripts A B C G S st in out 0
indicates CH3 OH from CO reaction indicates CH3 OH from CO2 reaction indicates water–gas-shift reaction gas phase solid phase or solid surface stagnant inlet condition outlet value initial condition
Acknowledgements The help of Prof. Marco Vanni and Davide Manca in developing and optimising the simulation code is gratefully acknowledged.
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