S = component S S1 = adsorbed S
Conclusions
Obtaining precise kinetic data for catalytic reactions is an even more formidable task than has been previously thought. The need for the consideration of more than one rate-controlling step can be revealed only by extensive experimentation, covering extremely wide ranges of variables.
c
Acknowledgment
D i
SUBSCRIPTS
A, H? S, Al, H1, S1 = components A? H, S:‘41: H1, SI, respectively
I
= chemical reaction = mass transfer = at solid surface
= vacant active sites = over-all
One of the authors (K.B.B.) gratefully acknowledges assistance from the National Science Foundation (U.S.A.) in the form of a postdoctoral fellowship.
R
= surface reaction
t
= total active sites
Nomenclature
literature Cited
A = component A A‘ = defined by Equation 32b A1 = adsorbed A B‘ = defined by Equation 32c G = concentration, moles per cu. meter C’ = defined by Equation 32d E = activation energy, kcal./gram-mole H = component H HI = adsorbed H k = rate coefficient k* = effective mass transfer coefficient K = over-all equilibrium constant. or with subscripts, adsorption equilibrium constants 1 = active site n = order of reaction p = partial pressure, atm. r = rate
(1) Boudart, M., A.Z.Ch.E. Journal 2, 62 (1956). (2) Chou, C., IND.ENG.CHEM. 50, 799 (1958). (3) Frank-Kamenetzki, D. A., “Stoff- und Warmeiibertragung in
o
der chemischen Kinetik,” Springer Verlag, Berlin, 1959. (4) Froment, G., Znd. Chim. Belge 25, 245 (1960). (5) Hofmann. H.. Bill. W.. Chem. Iner. Tech. 31. 81 11959). (6) Hougen, 0.A., Watson. K. M., “Chemical Prockss P&iples,” Vol. 111, Wiley, New York, 1947. ( 7 ) ,Jungers, J. C., et al., “Cinttique chimique appliqute,” Technip, Paris, 1958. (8) Thaller, L. H., Thodos. G., A.Z.Ch.E. Journal 6, 369 (1960). ( 9 ) Walas. S. M.. “Reaction Kinetics for Chemical Encineers.” McGraw-Hill, New York, 1959. (10) Weller. S., A.I.Ch.E. Journal2, 59 (1956). (11) Yang. K. H., Hougen, 0. A.. Chem. Eng. Progr. 46, 146 >
,
v
(1950).
RECEIVED for review March 20, 1961 A C C E P T E D August 31, 1961
STABILITY OF ADIABATIC PACKED BED REACTORS. A N ELEMENTARY TREATMENT -
SH EA N L I N L I U A ND NEA L R
.
A M U N D S 0 N, University of .Winnesota, .Minneapolis 14, .Minn.
The problem of stability of a packed bed adiabatic catalytic reactor is considered for a simple model in which mass and heat transfer resistances are lumped at the particle surface and the only intraparticle effect is that of chemical reaction. As has been shown, such a catalytic particle may exist in more than one state. A fixed bed reactor may contain particles whose states are determined by their past histories. The transient equations for the reactor are written and solved by the method of characteristics. The solutions show that under certain conditions, amenable to a priori prediction, there will be nonunique temperature and concentration profiles. In fact, there may be a multiply infinite set of profiles, depending upon the initial state of the reactor. Calculations are made for a series of initial temperatures.
HE PURPOSE of this work was to consider the stability Tof adiabatic, fixed bed catalytic reactors. The stability of continuous, well agitated homogeneous reactors was examined by van Heerden (6) and others (2, 4). The latter problem is not difficult because the transient behavior is described by first-order ordinary differential equations. The packed bed reactor is much more difficult, even for the simplest nontrivial physical model, since the analogous mathematical justification is not available for partial differential equations. Barkelew (3) made an extensive numerical study for a packed bed system and arrived a t some empirical generalizations. Wagner (7) studied the
200
ILEC F U N D A M E N T A L S
stability of a single catalytic particle and showed the existence of multiple steady states in an analysis very similar to that used for stirred tanks. Cannon and Denbigh (5) examined the thermal stability of a single reacting particle. Wicke and Vortmeyer (8-7 7) considered the packed bed reactor, and the purpose of the work described here was to extend their results and examine the problem in somewhat more detail. Their treatment was in terms of the steady state, while this work considered the transient behavior and its consequences. A very simple model for a packed bed reactor is considered. Because it is adiabatic there is no radial transport of heat or mass, and, in addition, it is assumed that axial transport, save
that by forced convection, is negligible. The velocity is assumed to be uniform over the cross section and length, and temperature effects on the velocity are neglected. Focusing attention on the particle, the heat and mass transfer resistances are lumped at the particle surface, with the reaction taking place on the porous surface of the particle. Although it is not necessary, the reaction is assumed to be first order and irreversible, A .-f B. More complicated reactions can be treated, but no new features are introduced. With these restrictions. the partial differential equations for the system are written and solved by the method of characteristics ( 7 7 ) . Because of the existence of multiple steady states for a single particle, the steady-state temperature and concentration profiles for the reactor are not uniquely determined. T h e effect of inlet concentration and temperature is superficially examined.
The transient behavior of the system is given by the solution of Equations 2 through 7. Steady States
The steady-state equations for the bed are:
H dp = p
- p
~~
"dx
S,k -j7
a P;.
3
t p - t =
PP(
-AH)
Note that: .k = ko exp( -AE/Rt,)
Development of Equations
With the assumptions made above, the mass conservation equation for the interstitial fluid is:
so that these equations are highly nonlinear. A heat balance taken over any section including the bed entrance gives:
It is assumed that as a first approximation G/PM is a constant. The continuity equation for the interstitial fluid is:
so that:
G = rP/u
If one defines the H T U for mass transfer as:
Equation 4a may be written:
G H - - MPa,k,
(9)
then Equation 1 may be \vritten: This may be substituted into Equation 5a to give: Q, =
The heat balance may be written: HI,
at + I1I at (& %)
t - t p _=
6
+
QII
(10)
PP =
t,
-t
(3)
where Q I and Q I I are related to the heat rejection and heat generation functions for a single particle and where
where and if Equation 8 is used, the result is: and tis the fluid temperature while t , is the particle temperature. If a is the radius of a single particle, S, the catalytic surface area per unit mass of catalyst, and kpP the rate of disappearance of A in moles per unit time per unit surface area of catalyst, the transient mass balance on a single particle is: (4)
S
\Vith the usual assumptions. the transient heat balance on a particle is:
where -AH is the heat of reaction for A B. I t is also necessary to state the conditions a t the entrance to the bed : --f
R 0 0
0"
t = t,(O)
p
=
fie(@),
x = 0,
e>o
(6)
and the initial condition of the solid and interstitial fluid:
p
= Pi(.)
t
=
1,
= t,*(x)
p p
= P,i(X),
L
ti(X)
max
PARTICLE TEMPERATURE
e
=
0, x
>0
(7)
Figure 1.
Graphical illustration of VOL. 1
steady
states
NO. 3 A U G U S T 1 9 6 2
201
I t is well known that for a given value of t, and with the other parameters specified, this equation may have one or three solutions for t,. This may be seen as follows: QII when plotted as a function of t, gives the sigmoidal curve shown in Figure 1, while for fixed t , Q I is a linear function of t,, as in lines Lo, L I , L z , and so on, showing one or three intersections with QII. These are the steady states of the particle. I t is necessary to analyze these steady states for local stability, as was done in previous work (2, 4, 6)-stability to small perturbations. Equations 4 and 5 may be considered as ordinary differential equations for fixed ambient conditions for the transient behavior of a single particle. Their linearization about a steady state will describe the transient behavior in the neighborhood of that steady state so long as the particle is definitely stable or unstable. Equations 4 and 5 may be written:
gP = Ap - A p p - Bkp, dB -dtp_ dB
-
Ct - Ct,
p
=
pe -
(t
- t,) MPc -AH
f
’
and for the case of irreversible reaction when p = 0, t = t,,,
:
Subtracting these two gives: PP
t,
- PP. = x
- tps
=y
wherep,, and t,, are steady-state values ofp, and t,:
-
which is the same as the first condition. Hence, for gaseous systems, and perhaps for others, the slope condition is necessary and sufficient for local stability. If there is only one intersection. the corresponding state will be stable. For three intersections, the two outside states will be stable while the intermediate state is unstable. The outside states are stable to small perturbations but not to large perturbations. All of the lines L, for an adiabatic reactor in which the parameters are constant pass through the same point S. Note that Equation 8 can be written in the form:
+ Dkpp
Defining new variables:
=
is equivalent to :
(A
+ Bk,)x-
( B .b.ksRRt,?)y E
dB
\$‘hen p = 0, t,,, = tPmsx, and this is a vertical line at t , = t,,. The intersection of LO and L,,, has a value for QI of: tmax
- t,
PPe The conditions, necessary and sufficient, that these two equations have asymptotic solutions tending toward zero are :
AC
+ A2 + A B k , > ADP k
E Rt,,l
where
I n general, the first condition implies the second, since at least for gaseous systems it may be shown that: A2
+ A B k, > BC k ,
hi P d - ~ H ~ k> ,
since A2
+ A B k, -- - _ _ _
BC k ,
H,
PfP
c/
ff
For gaseous systems, the predominate factor is ps p f P , since this is very large. The ratio H,,’H, is of order one, while is about 2. The first condition is that the slope of the straight line must be greater than the slope of the sigmoidal curve a t the point of intersection. Now:
and so the condition:
202
I&EC FUNDAMENTALS
but any other line L, is (see Figure 1) QI = (t, - t ) / & which From Equations 11 and has a value (t,,, - t ) / @ at t , = t,,,. 12 above this is equal to (t,,, - te)//3pe,the value of QI a t S, and hence all L, must pass through the common point S. In Figure 1, to is the same as t,. Thus, one can obtain considerable insight into the stability problem from Figure 1, since LOand L,,, may be constructed from steady-state considerations only. I t can be determined from the diagram that there may be particles whose state is not determined uniquely by steady-state conditions alone. In Figure 1, for example, the particles in the entrance region will have a single unique state. Following this section, there will be particles which could exist in three different states. This region may be determined from a temperature point of view by drawing the appropriate tangent lines from S (not shown). The later sections of the bed will have particles with unique states. If: dQII
( dt,
Point
each particle in the bed has a unique state, and there are unique temperature and partial pressure profiles through the bed. The unstable state for a particle will never be attained in a natural way, so that any profile through the bed made up of stable particle states will probably be stable to small perturbations but not to large perturbations. The analysis of stirred tank reactors showed that the final state of a system depends on the perturbed state. I n any event, this analysis shows that the profile of temperature and concentration will depend upon how the reactor was started up, since approaches to a steady state from widely different initial conditions could lead to widely different steady states. Also, there is the possibility that the point S in Figure 1 will fall below QII as in S’. I n this case, the maximum temperature of the particles can be greater than the maximum temperature attained by the fluid.
Numerical Solution of Transient Equations
These are integrations along the characteristic x = 0 to obtain tn(e) andpp(-8) for x = 0. Initial values to be used are those for t, and p p a t x = 0 and -8 = 0. General methods of solution for the four equations are not described here, since the methods used depend on the accuracy desired and expenditure of computer time possible. -4 possible method for the problem is discussed. Consider the following numerical example :
The steady state obtained when starting with given specified initial states can now be determined. Analyses similar to the phase plane studies fclr the stirred tank should be made. However, here the problem is much more difficult. Whereas the phase plane was a real geometrical plane of two (or a t least a finite number of dimensions), the initial state of this system is not specified by four functions: p , ( x ) . t l ( x ) , and t p 1 ( x ) , bpi(.). Clearly, no finite number of dimensions will suffice for the geometrical illustration. Further, the steady states are given by four functions: t ( x ) , t,(x), p ( x ) , and fin(.). The necessary calculations are illustrated by the method of characteristics for Equations 2 through 5 . Equations 2 and 3 have the same characteristics, the straight lines:
600 lb./hr./sq. ft. 40.8 ft./min. 0.167 in. 0.35 1 atm. = 0.196 B.t.u./lb./" F. = 0.07 lb./cu. ft.
G
= = = = =
u a y
P cs pI
H, -AH r'
dx
h:
= = = =
dB=u
a
=
cj ps
= = =
while the characteristics of Equations 4 and 5 are the lines The partial differential equations may then be written as ordinary differential equations along the characteristics shown in Figure 2,A.
A4
x = constant.
0.10 ft. 1.2 X 106 B.t.u./lb. mole 40 A. 20 B.t.u./hr./sq. ft./" F.
0.40 0.25 B.t.u./lb./" F. 60 lb./cu. ft. 48 lb./lb. mole 0.06 ft. 1.00 mole/hr./atm./sq. ft.
Hh k,
=
k
=
aa
=
2.085 exp 125 sq. ft./cu. ft.
6
=
exp (12.98
(- 'y) mole/hr./atm./sq. ft.
=
- 'Eo) tP
In this example, the initial and influent conditions are not specified! and some reasonable approximations are made. I n general, the gas velocity in a fixed bed reactor is so great that the time lag from entrance to any position x for the fluid is negligible compared with other transient effects. This has the effect of assuming that u - l is zero, and the characteristics are therefore not of slope U P but are parallel to the x-axis, as shown in Figure 2,B--8 = constant. A second approximation used here is to neglect the partial pressure capacity term. With these slight changes the equations now become, with time in minutes:
+
where I and 11 refer to the characteristics e = x ' u constant and x = constant, respectively. These must be solved with the conditions: Along e
=
0, x
> 0; p
.Along x
=
:=
p z ( x ) , t = t t ( x ) , p p = p p t ( x ) ,t, = t p l ( x )
0, 0
> 0; p
=
pe(8j, t
=
t,(ej = 10 (P, - P
A point of procedure about the method of characteristics
(2)
should be brought out here. The first two equations along the characteristics cannot be integrated, since p p and t, are not specified along x = 0, so that the solution cannot be started. However, this can be overcome by integrating the second two equations:
(2)
=
)
=
= 16.67 ( t P - t ) = f z
5.450 X
11
p exp (12.98 (t
fl
- t u ) -t (6 X 103)
y)] 22,000
=
1 f exp (12.98 - tP >)
f3
(19)
50
40
41
42
k 30
31
32
21
22
43
44
33
34
W
r
f'
20
X
, REACTOR
Figure 2.
LENGTH
35
36
25
26
a-
Q)
23
24
A
Schematic diagram of real ( A ) and approximate (6) set of characteristic curves VOL 1
NO. 3
AUGUST 1 9 6 2
203
PP
P
=
1
+ exp (12.98
-
(20)
'Xo) tP
where the network of characteristics is shown in Figure 2,B. It is assumed that:
> 0; p , = 0, pp2 = 0, t,, = t = t , (a constant) > 0; p = p e , t = t, (constants) The values used for t,, t,, and p , are given on the graphs. The e
0, x
=
based on Euler's formula and the trapezoidal rule. The calculations are illustrated for the point ( l , l ) , and the others are obtained similarly. As a first approximation : P(IY1,l) = P(l,O)
v ~ 1 , 1 = ) tN(o,i)
x = 0, 0
numbers at the intersections indicate the number of space and time increments, Ax and A@ to the right and above (O,O), and therefore (7,k) indicates a position in the two families of characteristics-the intersection of the j t h horizontal ILith the kth vertical characteristic. Let the notation P ( l , k ) be the value of p at ( I $ ) and similarly for the other variables. The procedure here uses a simple set of predictor-corrector formulas
p,=O.O7 te
I
0.25
-----
PARTICLE, t p
I
0.5
I
0.75
I
I
I .o
1.25
J
1.5
L E N G T H , FEET
Figure 3. Transient temperature profiles for the case in which each particle has a unique state
GAS, P
p,
_ _ - - _PARTICLE, pp -
0.0' 0
'
0.25
0.5 X I REACTOR
0.75
1,
= 0.07 atm. = 1250 R.
I.o
1.25
I&EC FUNDAMENTALS
wherefl(ll(1,l) is the value o f f l ( l , l ) using the values from the first approximation. Higher order approximations, p("'(1 , l ) , can be found in an obvious way, and the iterations on these formulas are continued until the desired accuracy is obtained. Obviously, any of the variables at ( I & ) can be found if the values a t (j-l,k) and (1.k-1) are known. I n this work, the values of all the variables Ivere computed along the time characteristics-complete profiles were obtained for each time increment, and then the time increment was increased and new profiles were obtained. The values of t, along the characteristic x = 0 were obtained by using the Runge-Kutta-Gill routine on Equation 19 with t = t,, and p P was obtained directly from Equation 20. For this calculation, a time increment of 2 - 6 minutes was used. For the predictor-corrector calculations along the characteristics, a good deal of trial and error work was done to determine the best set of increments. For these calculations : Ax = 2-5 ft. and A0 = 2-3 min.
seemed to give satisfactory accuracy without excessive machine time. Smaller increments showed little improvement in profiles. The steady state was obtained when 0 = 20 minutes, and the computing time was about 1 hour and 20 minutes (Univac Scientific hlodel 1103).
Discussion of Calculations
1.5
LENGTH, F E E T
Figure 4. Transient partial pressure profiles for case in Figure 3 204
For a second approximation, use the trapezoidal rule :
I
X , REACTOR
-"-I
+ f 3 ( o , i ) Ae
- GAS, t
atm
=I250 OR.
1.1' 0
t f*(I,O)
t ( l ) ( l , l ) = t(1,O) + f ? ( l , O ) Ax
Transient Behavior of Gas Temperature. The transient behavior of gas temperature and particle temperature is showm in Figure 3. At the early section of the bed, the steady states are reached relatively faster compared with the region near the bed outlet. The corresponding concentration profiles are plotted in Figure 4. At low temperatures, the differences in temperature and concentration betkveen the gas phase and solid are almost negligible. This is the region of chemical reaction controlling, since the chemical reaction is very slow at these temperatures. TYhen the particle temperature is relatively higher, the chemical reaction is fast and heat is generated rapidly ; the whole process becomes diffusion controlled. .4t this stage, there are large differences in concentration and temperature between the gas and solid. Figure 5 sholvs the effect of changes in the inlet gas temperature upon the steady-state gas and solid temperature profiles, all other parameters being fixed for a set of parameter:
pe 20.07 a t m .
tpi = 1500 ,t = 1100
OR.
I
O R .
1.25 X I REACTOR
15
LENGTH, FEET
Figure 6. Steady-state partial pressure profiles for the case in Figure 5 pe=0.12 a t m . 0
0.25
0.5
X , REACTOR
0.75 LENGTH
1.0
1.25
I .5
, FEET
Figure 5. Steady-state temperature profiles for different influent temperatures for the case in which each particle has a unique state
for \vhich each particle has a single state? as in Figure 9. Figure 6 shoivs the corresponding concentration profiles. These profiles Lvere obtained by integrating the transient equations ; only one steady-state profile existed. When the inlet gas temperature is lolv. for example, 1100' R. in Figure 5. almost no chemical reaction takes place throughout the bed, and the profiles are flat. TYhen the inlet gas temperature is 1400" R.? the chemical conversion of component A is almost complete at x = 0.75 ft.. and the temperature gradient from that point to the bed outlet is almost zero. The effect of changes in the inlet partial pressure of component '4 on the temperature and concentration profiles is shoivn in Figures 7 and 8, respectively, for the case in which each particle has a single state, as in Figure 9. The higher the inlet concentration, the higher is the outlet gas temperature? and the chemical reaction is completed in a shorter length of bed. For the same inlet temperature and concentration, the possibility of nonunique profiles depends upon the existence of multiple steady states for some particles. For the case shown in Figure 9> all the straight lines LO:L1, L,,, intersect \rith the S-curve a t just one point (points U O , U ~ ,. . . amax)? and the system is absolutely stable. From different initial particle temperatures, unique temperature and concentration profiles will be obtained, as shown in Figure 10. HOWever, if any particle in 1:he bed has multiple steady states, the system will be unstable for a certain range of initial conditions. and nonunique profiles will be obtained. As shown in Figure 11> the first particle at the bed entrance may have three steady states, U O , 60:and GO, corresponding to an inlet gas temare stable, while perature of 1200' R. Obviously, a0 and C O is unstable. As soon as the straight line passes the tangential line, L?. only the high steady states: such as 6 3 and 64 in Figure 11 are possible. If the initial particle temperature is lower than 1204' R., the steady-state profiles are as shown by curves A and A' in ~
1.1'
0
I
0.25 X
,
I
I
I
0.5
0.75
I .o
REACTOR
LENGTH
I
1.25
I 1.5
, FEET
Figure 7. Steady-state temperature profiles for different influent partial pressures for the cass in which each particle has a unique state
Figures 12 and 13. The gas mixture is heated up slowly along the low steady-state points, a. and a1 in Figure 11. and then jumps to the high steady states as soon as the gas temperature exceeds about 1380' R. (corresponding to the tangential line Lz in Figure 11). If the initial particle temperature is higher than 1780' R., the final profiles obtained will be the curves F and F' in Figures 12 and 13. These profiles folloi+ the high steady states. 60, 61,.. .6,,, in Figure 11. and the chemical conversion will be completed in about 0 5 ft. of the bed. An interesting result is obtained from the calculations for the initial particle temperature in the range 1204' R < t p , < 1780' R. For example, if t p , = 1250' R., the final profiles are as sho\z n by curves 6 and B ' in Figures 12 and 13. These VOL. 1
NO. 3
AUGUST 1962
205
profiles show that the gas mixture is heated up following the lower steady states, a0 and al, and then jumps to the higher steady states before the gas temperature exceeds 1380’ R. (tangential line L1 in Figure 11). The profiles C, D ,and E show that the higher the initial particle temperature, the earlier the profile jumps from the lower steady state to the higher steady state. Therefore, this system is unstable in the range 1204’ R. < t p , < 1780’ R. to large perturbations but stable to small perturbations. T o show the effect of inlet temperature and partial pressure on the steady-state profiles,
1, = 1250 O R .
1500 O R .
tp
X , REACTOR
, FEET
LENGTH
Figures 14 and 15 were prepared. These show in a dramatic way the sensitivity of the profiles to inlet conditions in the multiple state case. To study the problem of instability further, the behavior of the first particle at the bed entrance is examined in more detail by considering Equation 19 with t = t,. The values of the derivative, dt,/do, are calculated using Equation 19 with t = t, and plotted against t p , as shown in Figure 16. Then three intersections, ao, bo, and GO, are obtained. These points represent the steady-state particle temperatures which coincide with those obtained in Figure 11. The outside states a0 and bo are stable. and the middle state co is unstable. Figure 16 shows that if the initial particle temperature is in the region @ ( t p , < 1204’ R.), the derivative dt,/dO is always positive, and the final steady state will be the point ug. O n the other hand, if the initial particle is in the region @ ( t p t > 1975’ R.), the derivative dt,/dO is negative, and the point bo will be the final steady state. If t p t is slightly smaller than 1770’ R. (at co), the lower steady state a0 will be obtained, while the high steady state, bo, will be obtained if t,, is slightly larger than 1770’ R. The actual behavior of the first particle is obtained by integration of Equation 19 with t = t, and is shown in Figure 17. The curves in Figure 17 are the transient traces of the first particle for different initial temperatures and show clearly the dependence of the final state on the initial state, as in the continuous stirred reactor analysis (2). I t is possible that the first particle to show multiple steady states may be down the bed some distance from the entrance. I n this case, the region of gas temperature in Mhich instability occurs can be determined as follows. If two tangential lines, Ltl’ and Lt2’ are drawn from the point S’ to the QII-curve, as shown in Figure 11, the instability will occur in the region of gas temperature between at,’ and aiz’ (corresponding to Ltl’ and
L,?’). Figure 8. Steady-state partial pressure profiles for the case in Figure 7
I
Conclusions
The problem of stability of an adiabatic fixed bed reactor packed with small catalyst particles has been considered; by analyzing the equations describing the steady state, it was shown graphically that the individual particle may have multiple steady states. The partial differential equations describing the transient behavior of the reactor were solved
S
p e =0.07 otrn. le= l 3 0 O 0 R .
Qn
E
0.08 I
mer
-0.02
13-
B
1.1 I
2100 PARTICLE TEMPERATURE IN R.
,
I
2300
,
2500
Figure 9. Steady-state conditions for the case in which each particle has a unique state 206
I&EC FUNDAMENTALS
,“
I
0
0.25
I
0.5 0.75 1.0 X , REACTOR LENGTH , FEET
1.25
0.0 1.5
Figure 10. Temperature and partial pressure profiles for the unique particle state case showing independence of final profiles of initial particle temperatures
numerically by the method of characteristics. These calculations showed that a space interval equal to 2 - 8 feet and a time interval equal to 2 - 3 minutes were satisfactory. The effects of inlet gas temperature and the inlet partial pressure upon the concentration and temperature profiles were examined, and the stability of the abiabatic packed bed reactor was found to depend upon the existence or multiple stationary states for single particles. If every particle along the bed axis had only one steady stcite,then the reactor would be stable, and unique concentration and temperature profiles ~ o u l dresult for all initial particle temperatures and concentrations. If a single particle in the reactor has multiple steady states, then, from continuity of the mathematical model, adjacent particles would undoubtedly have multiple steady states. Thus, the reactor, although stable to small perturbations, would be unstable to large perturbations, and nonunique concentration and temperature profiles would be obtained. depending upon the initial conditions. The results showed that in the unstable region the particle conditions jumped from one state to another state at different points along the bed. As far as the natural behavior of each single particle was concerned, the unstable state was never attained. Although the equations are written for single particles and for the interstitial fluid, once the mathematical model is obtained the equations no longer reflect the idea that there are discrete particles in the bed. That is, when considering particles. the physical model is a discrete one, but when considering interstitial fluid the model is a continuum model. The mathematical system itself. aside from models, is a continuum model. This difficulty is conceptually confounded by the fact that the numerical solution requires space increments of length one eighth the diameter of a particle. The physical mode1 is also unsatisfactory in that each particle is assumed to be immersed in a homogeneous medium Tvithout gradients of concentration or temperature. I n such a reactor, there are large gradients, such gradients perhaps existing over the diameter of one particle. Therefore, the kind of analysis considered here is only a gross one and not correct in detail. The kinds of initial conditions used here
S
3.0k n
_____
p, = 0.15 a t m . tP
t, = 1 2 0 O 0 R .
I
I.o 0
I
0.25
I
I
I
0.75
0.5
I
1.0
I
1.5
1.25
X , REACTOR LENGTH ,FEET
Figure 12. Steady-state temperature profiles for case of nonuinique particle states showing effect of initial particle temperatures on steady-state profiles
p, = 0.15 o t m . t e = 1200
OR.
tpi
, O R .
A
1200,1100,---
E
1250
C
1300
D
1400
E
1500
F
1780, 1800 B. U P
5 X I REACTOR
LENGTH , FEE7
Figure 13. Steady-state partial pressure profiles for the case in Figure 12
t g =1200° R.
tpi = 1300° R. pe * 0.15 alrn. 1, = 1200 ‘R.
.I6
Ltl
I
s $ z
LtZ
.I2
W K
2
w w a a
o# L
0
.00
I-
O
l-
a
h
.04
J 1500
1900 2300 PARTICLE TEMPERATURE IN ‘R.
I 2700
Figure 1 1 . Steady-stlate conditions for the case of nonunique pcirticle states
0
0.25
0.75
0.5
1.0
1.25
0
15
X - REACTOR LENGTH-FEET
Figure 14. Steady-state temperature and partial pressure profiles for nonunique particle state case for different influent partial pressures VOL.
1
NO. 3
AUGUST
1962
207
-GAS _--_-PARTICLE
--I\
p e = 0.15 a i m .
t p i = 1300'R.
C
I
te =
0 8
2
2.6
C
z W
a
2
.I2
2.2
a
VI
W
3
a
5E f
/
.I6
E'
w
O R .
d
x
5
1200
C
c)
I
I
p. = O . I 5 a t m . -
P
1.8
.08
I .4
.04
+ a
5 I-
B
1.0 0
0.5
0.75 X I REACTOR LENGTH
0.25
1.25
1.0 , FEET
025
0
0
1, =I200
R.
1
P
I I
I
1
b,
PARTICLE TEMF!
I*T, -2200
Illuu;\
15
20
225
atm./sq. ft. = molecular weight of gas mixture, lb./lb. mole = total pressure, atm. = partial pressure of component .4 in the gas phase, atm. = partial pressure of component A inside the particle, atm. = influent partial pressure of component A in the gas phase, atm. = initial partial pressure of component X inside the particle, atm. = abbreviations for Eq. 11 = average radius of pores, A. = surface area per unit mass of particle sq. ft. 4 b . ; p,S = 2a/r = gas temperature, O F. = particle temperature, O F. = influent gas temperature, O F. = initial interstitial gas temperature, O F. = initial particle temperature, 0 F. = average interstitial velocity, ft. min. = axial variable = void fraction of particle = ( -AH) (k/h,.) = fractional void volume of bed = time variable = density of gas mixture. lb., cu. ft. = density of intraparticle gas. lb. cu. f t . = density of particle, lb. cu. ft. = n ps S& 3kg
PP I
125
Ui rnm
= chemical reaction rate coefficient, mole 'hr.1
k
P
400 1
10
,e,
Figure 17. Temperature history of first particle in bed
M pe = 0.15 atm.
075 TIME
Figure 15. Steady-state temperature and partial pressure profiles for nonunique particle state case for different influent temperatures
l
05
1.5
,OR.
2400
I
Pe
1 Q,,
QII
r
so
v
-4001
Figure 16. Direction of temperature movement of the first particle with respect to time
t t, t, tpi
U X
led to particle states in a natural order-the particle temperature increased monotonically, and, therefore, the system might be called regular in the steady state. However, it is conceivable (with this model) that the particle temperature order could be irregular, leading to profiles of a still different type than that envisioned here. Although it stresses credulity somewhat. it might be considered that each particle in an unstable region can have one of two stable states, and if these are of equal probability then there is a multiple infinity of profiles for such a model. Of course, in a real reactor the latter condition is probably not possible, since diffusive and conductive transport would tend to reduce the gradients.
CY
s
Y
e
P, PlP
P. 6
Acknowledgment
The authors are indebted to the Computing Center of the L-niversity of Minnesota for their considerable aid and counsel. literature Cited
Nomenclature a
= radius of catalyst particle, in.
A, E
= =
a,
chemical components total surface area of particles per unit volume of bed, sq. ft./cu. ft. = specific heat of gas mixture, B.t.u.1 lb , F. = specific heat of particle, B.t.u./lb./' F. = average length of pores, A. = fluid mass flow rate, lb.hr., sq. ft. = film heat transfer coefficient, B.t.u hr sq. ft. O
F.
= H T U for mass transfer, ft. = H T U for heat transfer, ft. = heat of reaction, B.t.u.jlb. mole
= mass transfer coefficient, mole hr./atm. sq. ft. 205
l&EC FUNDAMENTALS
(1) Acrivos, A , , IND.ENG.CIIEM.48, 703 (1956). (2) Ark, R., Amundson. N. R . , Chem. Eng. Sci. 7, 121 (1958). (3) Barkelew, C., Chem. Eng. Progr. Symjosium Ser. 5 5 , KO. 25, 37
,.,-, (1 oqC)\
/.
Bilous, O., Amundson, N. R., A . Z. Ch. E. J . 1, 513 (1955). Cannon, K. J.. Denbigh, K. G., Chem. Eng. Sci.6 , 155 (1957). van Heerden, C., IND.ENG.CHEM.45, 1242 (1953). IVagner. C., Chem.-Tech. 18, 28 (1945). IYicke, E.:Alta Tecnologia Chimica, Accadamia Nazionale dei Lincei, lecture course, Varese, Sept. 26-Oct. 8, 1960. ( 9 ) it'icke, E.. Chem.-Zng.-Tech. 29, 305 (1957). (10) IVicke, E., Z. Elektrochem. 65, 267 (1961). (11) IVicke? E., Vortmeyer, D., Zbid., 63, 145 (1959). (4) (5) (6) (7) (8)
RECEIVED for review March 30, 1962 ACCEPTEDJune 15, 1962 iVork supported by a grant from National Science Foundation.
