Chemical Engineering Science, Vol. 47. No. 8. pp. 1825-1837, 1992. ooo9-2509p2 s5.00 + 0.00 Printed in Great Britain. 0 1992 Pergamon Press Ltd

ANALYSIS OF AN EXOTHERMIC REVERSIBLE REACTION IN A CATALYTIC REACTOR WITH PERIODIC FLOW

REVERSAL

BRUCE YOUNG, DIANE HILDEBRANDT and DAVID GLASSER Department of Chemical Engineering, University of the Witwatersrand. PO Wits 2050, Johannesburg,

South Africa

(Received 19 March 1991; acceptedfor publication 4 October 1991)

Abstract-Periodic flow reversal in a packed catalytic reactor in which a reversible exothermic reaction is occurring is studied using a simplified mathematical model. Model simulations are firstly performed for a single reactor and the influence of cycle time and the use of inert sections on the ends of the reactor, which act as regenerative heat exchangers, are studied. It is shown that the volume of the inert sections can be used to control the temperature profile in the reactor. The performance of this reactor is compared to a conventional steady-state adiabatic reactor in which the feed is preheated using a heat exchanger. The cycled reactor is shown to give a better conversion for the same catalyst volume. In order to approximate the optimal temperature profile more closely, a two-stage cycled reactor, each with inert sections, is studied. This arrangement is shown to have a higher average conversion than an optimally operated two-stage steady-state reactor with interstage cooling. Suggestions are made for future work in this area.

INTRODUCTION The use of cyclic operation of packed-bed catalytic reactors by periodic flow reversal was pioneered by Matros. His work has been summarised in his recent book (Matros, 1989). He studies both reversible and irreversible reactions. The primary advantage of the method is that the thermal capacity of the solid material in the bed causes it to act as a regenerative heat exchanger, allowing autothermal operation with- out the use of heat exchangers. This concept has also been used by Heggs (1986) and Heggs and Abdullah (1986). This problem has been studied for an ir- reversible first-order reaction by Bhatia (1991) and by Eigenberger and Nieken (1988). Bhatia considers the special case of small cycle times and develops a perturbation solution for this case. A simple model for fast cycling has also been presented by Matros (1989) who shows that the system exhibits multiple solutions as is expected with an autothermal system. Eigenberger and Nieken (1988) considered the charac- teristics of a simplified model and investigated a number of factors, including the rate constant, the cycle period and the stability of the model. They also suggested the use of inert end and front pieces on the reactor to control the temperature in the reactor. The problem has also been studied for a reversible first- order exothermic reaction by Gawdzik and Rawkowski (1988). They did not consider the use of inert front and end pieces to improve the reactor performance but they did show that periodic flow reversal is also feasible for reversible reactions.

This paper specifically investigates reversible exothermic reactions. There are a large class of im- portant industrial reactions such as the production of SO,, methanol and ammonia which fall into this class. Reversible reactions represent a much more complicated problem because they have an equilib-

rium limitation and there is a temperature at which the reaction rate is a maximum for a given conversion. The locus of the temperature at which the maximum rate occurs, decreases with increasing conversion, and is the so-called optimum temperature profile. If the reactor could follow this optimum temperature profile the minimum catalyst volume for a given conversion would be obtained. For a steady-state, adiabatic, catalytic reactor the temperature increases with con- version. In order to try and follow the optimum temperature profile as closely as possible interstage cooling or cold shot cooling has traditionally been used. The optimisation of steady-state reactor systems has been studied by Hildebrandt et al. (1990) and Glasser et al. (1991). The principle of periodic flow reversal has been commercially exploited in Russia for SO3 production and has been described by Matros (1986). Experimental results have also been reported by Matros (1989) who successfully modelled the results with a model similar to that presented in this paper. He does not, however, consider the optim- isation of the reactor system.

The primary advantage of using periodic flow re- versal is the elimination of heat exchangers, but a secondary advantage in the case of reversible exo- thermic reactions is the possibility of more closely approximating the optimum temperature profile. As far as could be established this point has not been adequately addressed. In this work we have expanded on the concepts introduced by Matros and try to more closely follow the optimal temperature profile, firstly by considering the role of the temperature conversion relationship in the reactor, including the idea of pinch points, secondly by the use of inert sections and finally by the use of two or more staged cycled reactors-th inert sections. The performance of these systems is then compared to optimal opera-

1826 BRUCE YOUNG et al.

tion of the conventional reactor system with inter- fluid material balance and the gas-phase energy stage cooling. balance. It is therefore assumed that the solid temper-

THEORY ature If. in the bed changes slowly in comparison to

Reaction kinetics the fluid temperature T# and the bed composition

The purpose of this work is to study the cycling of profile. This has been discussed by Matros (1989).

the feed to a packed catalytic reactor in which a The rate of reaction over the bed is thus

0 v< v, W

r A= + k?p,(l - X) exp V, d Vs V, + v, (3b)

0 v> v, + v,. (34

reversible exothermic reaction is occurring. The The solid-phase energy balance includes conduc-

model is kept as simple as possible while still retaining tion which is modelled with a single effective thermal the essential features of this system. The model is conductivity. The transient energy balance equation is similar to that of Bhatia (1991) and Gawdzik and Rawkowski (1988). The following simple reaction will

PT, PG.= % = k,, ax2 + ha(T’, - TJ + AH,r,. (4)

be studied:

A=B (1) The fluid-phase energy balance is then

The following rate expression, incorporating the aT,

Arrhenius equation for the rate constants will be used: NCDay = - ha(T, - T,). (5)

The material balance is r A- - k? exp Cg_ ax

(2) N*= -r". (6)

It is assumed that there is no density change with The following dimensionless variables are defined: reaction or with temperature. The specific heat capa- cities of A and B will be taken to be constant and equal. The mass transfer resistance in the catalyst is

T:=$, Tz=$ )n=+x,

neglected and the heat transfer between the fluid and t* =-ELr, rz +.

(7)

the solid is assumed to be determined by film transfer. These assumptions allow very significant simplifica-

d pr

The dimensionless rate equation rt; is:

0 x* < x: (84

rf;= xr’ < x* d 1 - xf (Sb)

x* > 1 - xf. (8~)

tions in the model formulation without changing the characteristics of the problem. These simplifications have been shown to be justified by the work of Matros (1989) who has modelbd experimental results for SO, production with a model which incorporates these assumptions.

Forward Flow

In*rL 1 Volumr

VI

“tyb=” Inrrt Voluac \

“C “I

Simple model for an unsteady-state packed catalytic reactor

The system to be modelled is a packed catalytic reactor In which the above exothermic reaction is occurring. There may be inert sections of equal vol-

Reverse Flow

tnrrt < voluul*

=t”,t:rY=t I nmrt Volumr

ume VI on either side of a central catalyst section of VI VC VI -+

volume V, in the reactor and the direction of flow through the reactor Is periodically reversed (see Fig. Fig. 1. Schematic diagram showing flow reversal and inert 1). Psuedo-steady-state assumptions are made for the sections.

Analysis of an exothermic reversible reaction 1827

The three governing differential equations in di- mensionless form are the following:

Dimensionless solid energy balance:

aT,* 1 a2T* at* =Nu ax*2

z+(T,'- T,*)+ Qr);_ (9)

Dimensionless fluid energy balance:

aTz _ -- -Bi(T,* - T$) ax* (10)

Dimensionless material balance:

ax -=rrl;_ ax* (11)

This system of differential equations is character- ised by seven dimensionless numbers which are de- fined by

hV0 Si = - NC,

(12)

Equations (19) and (20) are similar to the analytical solutions developed by Bhatia (1991) for an irrevers- ible reaction. The problem has now been reduced to one parabolic partial differential equation for the solid temperature profile [eq. (9)] and algebraic equa- tions for the gas temperature profile and conversion profile [eqs (19) and (20)]. It is possible to make significant further simplifications if fast switching is assumed. This has been discussed by Matros (1989). The problem was solved numerically, using IMSL software library routine MOLCH. At each time inter- val eqs (19) and (20) were integrated numerically using IMSL routine QDAG. The problem was solved on an IBM model 3083-J24 mainframe computer.

A I

= Voki”Dp, N

(13)

A 2

= F/,kFp, N (14)

g, ui = RT, (1%

E2

OL2 = RT, (16)

Mu+ (17) eq

AW,N Q=p.

ha T,, V, (18)

Equation (10) is a first-order linear differential equation. If the dimensionless gas temperature at the reactor entrance is T.5. then the analytical solution to this equation is

T,*(z) exp (Biz) dz 1 . (19)

Equation (11) is also a first-order linear differential equation and if the conversion at the reactor entrance is taken as X, the analytical solution to this equation is

Cycling of the feed In order to find the cyclic steady state it is necessary

to start from a given initial catalyst temperature profile and then to cycle the feed a sufficient number of times to achieve cyclic steady state. For the pur- poses of this work an initial catalyst temperature profile of T,* = 2 was used. The feed was then cycled until an approximation of cyclic steady state was achieved. As the system can have multiple solutions a low-conversion cyclic steady state will be achieved if the initial guess for T,’ is too low. This has been discussed by Matros (1989).

The average conversion over one cycle can then be computed so that the performance of this reactor system can be compared to conventional reactor sys- tems. The average conversion is defined as follows:

(21)

The average exit gas temperature is defined in an analogous fashion. Since the reactor is adiabatic it follows from an overall energy balance that the aver- age conversion is related to the average gas temper- ature, at cyclic steady state, as follows:

i?z = Tz, + BiQR. (22)

This equation serves as a useful check of the numerical procedure and the ultimate equilibrium limitation is easily determined_

Wa)

xf<x*<l-xxf (2Ob)

x* > 1 - x: PW

where

f(z)=exp rr--41exp(+$) RESULTS AND DISCUSSION

Values of the dimensionless numbers used for this study

t20d) In order to investigate a particular problem it is necessary to specify the seven dimensionless numbers,

1828 BRUCE YOUNG etal.

eqs (12)-(18). The kinetic parameters were chosen to be analogous to those of Hildebrandt and Glasser (1990) who selected kinetic parameters which dis- played the typical reversible exothermic type of beha- viour. This behaviour is determined by the ratio of the forward and the reverse reaction rate constants and activation energies. The forward reaction was there- fore taken to have a rate constant a thousand times smaller than the reverse reaction and the activation energy of the reverse reaction was taken as twice that of the forward reaction. The magnitude of the rate was adjusted to produce reasonable conversions but the above ratios were always maintained. Rough estim- ates of the remaining three parameters, the Biot num- ber, the Nusselt number and the dimensionless heat of reaction were obtained from using reasonable values for the various physical properties included in their definition. The system was initially studied, assuming no inert sections, using the following values of the seven dimensionless parameters:

A, = 250

A, = 2.5 x 10s

ccl = 13.33

x2 = 26.66 (23)

Bi = 18.5

Nu = 1620

Q = 0.12.

In all the subsequent work in this paper the only things that were changed were the inert volume and the catalyst volume. If the catalyst volume is increased by a factor y and the fraction of inert volume is 2x:, a new set of dimensionless numbers is obtained as follows:

YA,

A1N = (1 - 2x?)

A YA, 2N = t1

- 2x7)

yBi BiN = (1 _ TX:)

y2Nt4 NuN = (1 - 2x:)2

QN = (I -2xf)Q Y .

The value of y is used as a measure of the catalyst volume, with a value of one representing the base case.

Characteristics of the model In order to investigate the model a base case was

considered using the values of the dimensionless para- meters specified in eq. (23). The only remaining para- meter is the dimensionless cycle time. If the cycle time is very fast then the catalyst temperature profile does not have time to change very much over one cycle.

Four cycle times were investigated, the dimensionless cycle times were one, two, four and seven. The results for cycle times of one and seven are shown on Figs 2 and 3. Each figure contains three graphs. The top graph represents the solid temperature profile at vari- ous stages through a cycle. The solid curve represents the temperature profile at the beginning of a cycle, the middle curve is in the middle of the cycle and the final curve, which is a mirror image of the first curve, is at the end of the cycle. The middle graph shows the same results plotted on temperature conversion axes. Superimposed on this graph, are the equilibrium lim- itations and the optimal temperature-conversion pro- file (Levenspiel, 1972). If the optimal temperature- conversion profile were followed then the maximum conversion for a given volume of catalyst would be achieved. It is obviously desirable from an optimisa- tion point of view to try to keep as much of the reactor as possible close to the optimum profile. In the conventional steady-state reactor there is a straight- line relationship between temperature and conver- sion. The bottom graph shows the conversion profile in the reactor over one cycle. The solid curve repres- ents the beginning of a cycle, the centre curve repres- ents mid-cycle and the final curve the end of the cycle. It is interesting to note that the conversion does not vary much over one cycle.

The average conversion is tabulated in Table 1. From these results it can be seen that the average conversion decreases very slightly with cycle time. There is effectively no difference between a cycle time of one and two. From Fig. 2 it can be seen that at a cycle time of one the solid temperature profile does not change very much over one cycle. At a cycle time of seven the maximum temperature in the bed moves a considerable distance over one cycle. If the cycle time is increased to 10 the reactor “blows out”. This means that the solution is a low-temperature low- conversion one. The cycling process is started by assuming a catalyst temperature of two. As the cycling is continued the average bed temperature drops until the entire bed is effectively at the inlet temperature because the reaction rate at the inlet temperature is negligibly small. It therefore appears that below a certain critical cycle time there are two stable solu- tions and above this cycle time there is only one.

If the volume of the reactor is increased there is only a small increase in conversion because the conversion becomes limited by the equilibrium limitation (“pinch point”). It is interesting to note that unlike a conven- Conal reactor where the equilibrium limitation occurs at the end of the reactor a cycled reactor is limited by equilibrium in the centre where the temperature is at its highest. If the volume is decreased the reactor becomes too short to support the high-temperature high-conversion solution and “blows out”.

The performance of this reactor was compared to a conventional steady-state reactor where the feed is preheated. In order to obtain the best conversion for the given volume of catalyst the feed must be heated to a specific temperature. The optimisation of this

Analysis of an exothermic reversible reaction 1829

1.0

0.0 0.2 D.4 0.6 0.e 1.0

DIMENSNDM_ESS VOLUME

2.4 - 2.2 __EquYbium LimitaUon

---_ ---_ - 2.0 - ----_

----_ r.e - OlmmalTsmperahrre

Prafile

1.6 _

1.0 +_ T 0.00 0.10 0.29 0.30 0.40

CONVERSFON

0.0 02 0.2 0.. 0.5 0.e 0.7 0.8 1.0

DIMENS~ SS VOLUME

Fig. 2. Results for the base case, eq.. (U), and a dimensionless cycle time of one.

1830 BRUCE YOUNG et al.

0.0 0.2 0.4 0.s 0.1) 1 .o DIMENSIONLESS VOLUME

-. --._

---.__ A,..

--\_ EquUm.-i”m IA”

--- -- ---__

‘, ---

‘\ \ ----_ Dptimal Te-2.

Protile

_._.l ____.....,_._. ,,l._._.__.,.___.__..,_._._._._,......__., 0.10 0.20 0.30 0.40

CONVERSION

00 0.2 0.4 0.6 0.8

OlMENSlOhlLESS VOLUME

Fig. 3. Results for the base case, eq. (231, and a dimensionless cycle time of seven.

Analysis of an exothermic reversible reaction 1831

Table 1. Average conversion as a function of cycle time

Dimensionless Average cycle time conversion

: 0.2177 0.2175

4 0.216 7 0.205

problem was studied by Hildebrandt and Glassor

(1990) and their technique was used to evaluate the optimum performance. The optimum steady-state one-stage reactor of the same catalyst volume gives a conversion of 0.2175 and the feed gas must be pre- heated to a temperature of TB+ = 1.83. The pcr- formance of the cycled reactor can be seen to be comparable to this conventional reactor.

Inert front and end pieces

From an examination of Figs 2 and 3 it can be seen that the reaction rate at either end of the reactor is very low because the temperature is very low. This means that this catalyst is being wasted. By adding inert front and end pieces the feed-gas temperature to

the catalyst can be increased thus increasing the reaction rate in the catalyst section. This means that it should be possible to follow the optimal temperature profile more closely, thus utilising the catalyst more efficiently.

Model simulations were performed for a reactor with a smaller catalyst volume equivalent to y = 2/3 and the size of the inert sections at the end of the reactor were varied. A dimensionless cycle time of one was used for all the simulations_ The results are tabulated in Table 2. There are only a few results presented here because the calculations require a lot of computer time. If there are no inert sections on the reactor then there is no high-temperature solution. It was found that each inert section must comprise at least about 13% of the reactor volume in order for a high-temperature solution to exist.

The results are plotted for an inert volume fraction of 0.167 and 0.218 in Figs 4 and 5, respectively. As the inert volume is increased the conversion increases slightly and then starts to decrease-The average con- version has only decreased slightly even though the catalyst volume has been decreased by 33%. From Fig. 4 it can be seen that the reactor now operates closer to the optimal temperature profile. This means that nowhere in the catalyst volume is the reaction rate very low and the catalyst is therefore being used more efficiently. In Fig. 5 the effect of increasing the inert volume beyond the optimum amount can be seen. The average temperature in the bed rises and the reaction-becomes limited by equilibrium in the centre of the reactor causing a drop in the average conver- sion.

The optimum one-stage conventional reactor for this catalyst volume gives a conversion of 0.192 when

CES 47x8-C

Table 2. Inert front and end sections

Ftuction of reactor volume for each inert

section Average

conversion

0.144 0.205 0.167 0.206 0.218 0.190 0.273 0.152

the feed is heated to 1.95. The cycled reactor with inert sections gives slightly better performance than an optimum single-stage conventional reactor and re- quires regenerators but no heat exchangers. In the conventional interstage cooling reactor system an arbitrarily high conversion can be achieved by having enough reactor stages. In a single-stage cycled reactor the conversion is limited. Increasing the catalyst vol- ume does increase the conversion slightly, but the conversion becomes limited by equilibrium. The tem- perature in the centre of the reactor approaches the equilibrium limitation and the reaction rate becomes very low. This is illustrated in Fig. 5 where it can be seen that the conditions in the centre of the reactor get very close to equilibrium and consequently the reac- tion rate is very low. This means that the centre section of catalyst is wasted

Two-stage cycled reactor

In order to achieve higher conversions and more efficient utilisation of catalyst a second cycled reactor can be placed in series with the first one. The cycling in the two stages is independent and the second reactor is not merely added to the first. By having inert sections in the second reactor the feed temper- ature to the catalyst section of the second reactor can be controlled. This means that the second stage can be designed to operate as closely as possible to the optimal temperature profile.

In order to illustrate the use of a two-stage cycled reactor the calculations already performed for yt = 2/3 and x1 = 0.167 were used as the first stage of

the two-stage reactor. In the first stage the inert volume has been roughly optimised to give the highest conversion. Two different second-stage volumes are considered, using y2 = 2/3 and yz = 4/S. The average conversion and exit gas temperature of the first re- actor were used as inputs to the second reactor. This should be a reasonable approximation because the switching in the first reactor is sufficiently rapid that the exit conditions from the first reactor do not effectively change.

A dimensionless cycle time of unity was also used in the second stage and the inert volume in the second stage was roughly optimised to give the maximum overall conversion. These results were then compared to the optimal operation of a two-stage conventionti reactor with interstage cooling. The comparison was made by requiring the conventional reactor system to produce the same conversion and then comparing the

1832

1 .o

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

DIMENSIONLESS VOLUME

::I\ _.__.._._,_.....r_.,..,,,._..,,........,.........,......... I . .._.._._ * . . . .._. _ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

CONVERSION

0.0 0.1 0.2 0.3 0.4 0.5 0.3 0.7 me o.e 1.0

DIMENSIONLESS VOLUME

Fig. 4. Results for Vf = 0.167, y = 2/3 and a dimensionless cycle time of one.

Analysis of an exothermic reversible reaction 1833

1 .o < 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0

DIMENSIONLESS VOLUME

1.6

1.4

I.2 i

1.0 ...

0.00 0.05 0.10 0.15 0.20 0.26 0.30 0.36 0.40 CONVERSION

0.16

0.16

0.14

0.12

0.10

0.06

0.06

0.04

0.02

0.00

0.0 0.1 0.2 0.3 0.4 0.6 0.6 0.7 0.8 0.6 1.0 DlMENSlONLESS VOLUME

Fig. 5. Results for VF = 0.218. y = 2/3 and a dimensionless cycle time of one.

1834 BRIJCEZYOUNG~~~.

catalyst volume required. The results are presented in Table 3 and are represented graphically in Fig. 6. The temperature conversion plots are shown for both the cycled reactor and the conventional reactor on Fig. 7. The results from mid-cycle are used for each reactor. From these results it can be seen that a two-stage cycled reactor requires slightly less catalyst volume than a conventional two-stage reactor with interstage cooling. This result has been achieved without using any heat exchangers. The two-stage cycled reactor has also not been optimised properly. It is probably pos- sible to obtain a better conversion using the two-stage cycled reactor using different volumes for the first and second reactors while maintaining the same total volume. This optimisation was not tackled because the amount of computation involved was prohibitive.

DIscUssION

Although the calculations have been done for a specific example the results as presented are more general. This is because all exothermic reversible reac- tions give rise to the optimal temperature profile which falls with extent of conversion. Thus all the diagrams are qualitatively consistent with the general situation. The idea of trying to follow the optimum temperature profile as closely as possible and not be limited by pinch points will be relevant to all exo- thermic reversible reactions. Thus it is clear that these

ideas are relevant not just for this example but to real practical situations which are of significant industrial importance as discussed in the Introduction.

It is necessary to first summa&e the known results for an exothermic reversible reaction in order to place the present work into context. It is known (Levenspiel, 1972) that if there are no constraints on the system that in order to minimize the volume of catalyst for a given conversion one should follow the so-called “optimum temperature profile”. This is the locus of points such that for each value of the conver- sion we maximize the rate by choice of the temper- ature. Thus this is really an optimum, temperature versus conversion profile and it is drawn on Fig. 7, for example, with the equilibrium temperature versus conversion profile. It is obviously very difficult and expensive to try to achieve such a profile in practice and so various compromises need to be made.

One of the most common ways of dealing with this, called cold-shot cooling, is to use a feed-product heat exchanger for a proportion of the feed to heat it up to some convenient temperature and then mix in the remaining cold feed at various stages to cool the product stream down. The optimisation of this pro- cess has been looked at by Hildebrandt et al. (1990). The overall optimisation of this process where one is allowed to add the bypassed cold feed continuously has been presented by Glasser et al. (1991). In this case one does not follow the optimum temperature profile.

Table 3. Comparison of two-stage cycled reactor with a two-stage reactor with interstage cooling

Volume of Volume of first stage second stage

(VI) (Y2) Average

conversion

Total catalyst required with

interstage, cooling (total y)

0.667 0.667 0.167 0.292 1.41 0.667 0.800 0.101 0.302 1.55

Fig. 6. Graph showing the optimum performance of a single-stage steady-state reactor with preheated feed and a two-stage steady-state reactor with interstage cooling. These reactors are compared with a cycled

reactor whose results also appear on Table 3.

Analysis of an exothermic reversible reaction 1835

2

1.0 --P----...,.......-T

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.36 0.40

CONVERSION

Fig. 7. Temperature conversion plots for the two-stage cycled reactors. The optimum two-stage steady-state reactor with interstage cooling is also shown. Upper graph has y1 = 0.667 and y2 = 0.667. Lower graph has

yE = 0.667 and yz = 0.800.

One can explain the reason as follows. In order to follow this profile one would need to hold back so much feed that the gain in conversion by that material in the reactor is more than balanced by the fact that some feed is only allowed to contact a small portion of the catalyst. Thus such a scheme at best can never equal the optimal temperature profile conversion.

An alternative approach is to use interstage cooling. Here, in theory, by having a large number of interstage coolers one can attempt to approximate the optimal temperature profile. The optimisation of such a system is discussed by Hildebrandt et al. (1990). The crucial thing to note is that while the gross temper- ature profile is a falling one, oscillating around the optimum temperature profile, each reactor is adia- batic and hence has a rising temperature profile which is exactly contrary to what is required_

In the light of this discussion and the appropriate diagram we can now look at the oscillating systems. In this case, in line with what we have found to be best, we shall limit our discussion to the situation where oscillations are rapid relative to the temper- ature time constant of the solid material in the re- actor. The latter is usually of the order of many hours and so the oscillations need not be unreasonably rapid in order for this discussion to be valid.

For the gas phase, between the entering and leaving streams, the cycled reactor is still adiabatic, however the solid does not follow the gas temperature proftle and so we have the opportunity to have temperature profiles at which the reaction takes place (the solid temperature) which is significantly different from the adiabatic one. In both the steady-state interstage cooling and the steady-state cold-shot cooling we

1836 BRUCE Your-m et al.

were limited to follow this adiabatic temperature profile of the solid within each plug-flow section. The effect of the oscillating system, particularly with the inert sections, is to allow an extra degree of freedom to try to tailor our solid-temperature profile to try to achieve a more favourable profile so that we have a higher average rate in the reactor or achieve a higher conversion for a given catalyst volume. That we can indeed achieve this is clearly shown in Table 3. Thus we have sho\ivn that we can in practice achieve higher conversions than for the optimally operated two-stage interstage cooled reactor. Obviously one would want to optimise such a system to see what limits of conver- sion are possible and how they compare with those of the optimal temperature profile.

It is also clear that just as for the interstage cooling case we may use multi-staging to improve our overall temperature profiles and this has been shown in Fig. 7, but has again not been optimised.

We haue not in this paper looked at all the degrees of freedom made available to us by the switching and the multi-staging but in the light of the discussion above we can examine some of these possibilities in principle. Firstly we have assumed that we have switching with equal fractions of time in each direc- tion and this gives rise to a symmetric temperature versus distance profile. We could, by varying the fraction of time in each direction and the sizes of inert sections, make an apparently much more favourable profile in the one direction, but only at the expense of a much worse one in the other direction. In our opinion this is not likely to give rise to any significant improvements of performance.

A much more interesting set of results appear to be possible by having much more complex switching patterns. In the adiabatic solid-temperature profile we effectively reach a “pinch point” at the end of a reactor as we approach equilibrium. In contrast to this the “pinch point” in the switched reactor is approached at the centre of the reactor where again we approach equilibrium. What we want is to be able to design a temperature profile that does not cause this to happen, that is one that is flatter in the middle of the reactor. One way of achieving this would be to have a

Stage 1

I I I I

more complex three stage switching arrangement as shown in Fig. 8. The effect of this arrangement is for part of the cycle to cool the centre part of the bed, which in the two stage switching does not occur. This will help make a better temperature profile during the first two stages, but over and above this during the third stage there will be a much better falling temper- ature profile so that the overall result should be much more favourable. In this case the degrees of freedom become quite large (ratio Vr1/Vr2, V,,/V, and times for the third stage relative to the first and second) and so the optimisation becomes difficult. On the other hand the large number of degrees of freedom gives promise of being able to achieve temperature profiles which approximate those of the optimal temperature profile and so give significant improvements over presently used steady-state systems. Since we have shown that relatively rapid switching appears to pro- duce the best results, a much simpler model, discussed by Matros (1989), can be formulated for this case. This would make solving this optimisation problem more feasible.

One might think that recycling might help achieve such a flatter temperature profile. This is indeed the case but only at the expense of having a higher average conversion in the reactor. Calculations we have performed seem to indicate that there is no advantage in having such a recycle.

In conclusion we can see that periodic flow switching might prove a very useful way to avoid having heat exchangers for exothermic reversible systems. This might provide significant savings for high-pressure systems such as ammonia or methanol synthesis. Over and above this it appears to offer the possibility of significantly increased yields relative to those in conventional equipment. Whether these ad- vantages outweigh the increased complexity of oper- ating such switched systems must be left to com- mercial enterprises to decide.

a

-4,

NOTATION

catalyst bed specific area, m2/m3 of bed dimensionless pre-exponential factor for the forward reaction defined by eq. (13)

Stage 2

Stage 3

Fig. 8. Schematic representation of the proposed three-stage cycling arrangement.

Analysis of an exothexmic reversible reaction 1837

A*

Bi

CA CB CP =I- El

E2

h k -=I

k” 2

N NU Q

rA R t

t,

r, r, T e+ To V

V0 VI V, x

XI X

Y

dimensionless pre-exponential factor for the reverse reaction defined by eq. (14) Biot number defined by eq. (12) concentration of A, mol/m3 concentration of B, mol/m3 specific heat capacity of the fluid, J/mol K specific heat capacity of the solid, J/kg K activation energy for the forward reaction, J/mol activation energy for the reverse reaction, J/mol heat transfer coefficient, W/m2 K equivalent thermal conductivity of the cata- lyst bed, W/m K pre-exponential factor for the forward reac- tion, s-l pre-exponential factor for the reverse reac- tion, s-l molar flow rate, mol/s Nusselt number defined by eq. (17) dimensionless heat of reaction defined by eq.

(18) reaction rate, mol/m3 of bed s universal gas constant, 8.314 J/mol K time, s cycle time, s gas temperature, K solid temperature, K feed gas temperature, K reference temperature, K total volume coordinate, m3 total reactor volume, m3 volume of inert section, m3 catalyst volume, m3 distance coordinate, m length of inert section in reactor, m conversion

dummy variable

z dummy variable

Greek letters

a1 dimensionless activation energy for the for- ward reaction defined by eq. (15)

a2 dimensionless activation energy for the re- verse reaction defined by eq. (16)

AH, enthalpy of reaction, J/mol

Pnr fluid density, mol/m3

P. solid density, kg/m3

Superscripts Average value

* dimensionless variable

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