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Maximum catalyst temperature (hot spot) in adiabatic fixed-beds
MAXIMUM
CATALYST TEMPERATURE (HOT SPOT) IN ADIABATIC FIXED-BEDS
HONG H. LEE Department of Chemicd Engineering,University of plorida, Gainesville, FL 32611,U.S.A. (Rcceikd 8 October 1981;occejbd 22 March 1982) Ab&aet-XohIitions are derived under.which the reactor proEk of catalyst temperaturegoes througha maximum at u point other than outlet in an adiabatic reactor. Relatively simple methods are developed for the cahdation of this maximum temperature.It is shown that the maximum catalyst temperaturecan increase with time for a certain
time period when the catalyst undergoes deactivation. Conditiens under which this can happen are also derived. The maximum temperature always increases with time ia the co9esof uniform sintering and dependent poisoning nrovided that a certain ccndition is met in the latter case. if the initial maximumoccurs at a point other than the butlet. INIXODUCTION
Design of a heterogeneous fixed-bed reactor is often subject to a constraint on the maximnm temperatrue of the bed. If the temperature is too high, the catalyst as well as the support can undergo a chemical/physical transformation, which either disables the catalyst or results in break-up and dusting of catalyst pellets. For nonadiabatic reactors, this maximum temperature is known as hot spot. Considerable attention has been directed to the studies concern&this hot spot because of its importance in the design of nonadiabatic reactors. Much less attention has been paid to the maximum temperature in an adiabatic reactor mainly because. the maximum temperature is often tahen as that correspoading to the adiabatic temperature rise. This maximum temperature, which always occurs at the reactor outlet, is not that of catalyst but rather that of bulk fluid. In view of the fact that the temperature ditkence between bulk tluid (fhrid phase) and catalyst (solid phase)’ can be in excess of 50°C for some reactions, it is more prudent to use the maximum catalyst temperature rather than the maximum fluid temperature for the design. Much more important than this is the fact that the maximum catalyst temperature does not necessarily occur at the reactor outlet as opposed to the maxiplum 5uid temperature which *always does. This has been verified experimentally [ l] ‘and the s@lation of an adiabatic reactor undergoing catalyst deactivation has also shown[2] the cxixtence of the maximum catalyst temperatnre at a point other than the reactor outlet. Fnrthermore, this simulation has also shown that the maximum catalyst temperatrue can increase with time when the catalyst undergoes deactivation. In this paper, we present a method of cafculating the maximum catalyst temperature and develop a criterion that can be used to determine whether the maximum catalyst temperature occurs at a point other than the reactor outlet. We also offer an explanation a8 to why the maximum catalyst temperature increases with increasinrg level of catalyst deactivation under certain conditions and determine what these conditions arc. These results
enable one to avoid pitfalls one can come across and put the design of an adiabatic reactor on a iirmer basis. FIXED-BE&3WlTSOUT
CATALYST DEM3lVATtON
A necessary condition Assuming constant physical properties, conservation equations for an adiabatic reactor are:
(2) where 7 =
Z(1 - +!)lfJj u.
(3)
Here the global rate 92 is based on pellet volume and the subscript b denotes bulk fluid. Combining these two equations leads to:
where C, and T,,, are inlet concentration and temperature respectively. A heat balance across the pelletbulk fluid interface yields: h( T, - Tb) = BL( - AH)
(5)
where L is the characteristic *Ret length and the subscript s denotes pellet surface. We assume here that the external mass transfer resistance is negligible and that the pellet is isothermal. These assumptions are quite valid under realistic reaction conditions[3]. Under these assumptions, the catalyst temperature is at T, and the pellet surface concentrauon is equal to bulk fluid concentration. For a general form of intrinsic hinetics r,:
1539
r, = kf
(6)
H.
1540
use of the generalized effectiveness factor[41 yields the following expression for the global rate:
H. LEE by: Tb= Ti,+o(Ci.-Cd)
W = [ZZXlr, Lb f(c) dc]“‘/L
(7)
where the effective diffusivity D. has been assumed constant, C, is the concentration at pellet center and k, the rate constant evaluated at T,. In order for a maximum catalyst temperature to exist, it is necessary that dZ’,./dz= 0. Let z,,,( # 1) be the point at which the maximum occurs. Differentation of eqn (5) with respect to z yields: k(!&l$)
= L(-AH)%
which reduces at z, to: (8) where eqn (2) has been used for dTJdz. Consider two extreme cases: diffusion-free and difIusion-limited reactions. In the case of diffusion-limited reactions, the pellet center concentration C, can be set to zero and the differentiation of eqn (7) with respect to z yields:
where eqn (1) has been used for dCddz. Using eqns (7) and (9) in (8) and rearranging, one gets:
B=
[&f&J” f
at z = z,
(10)
where z(cb)
=
I,”f(C) dc,
(11)
In the case of diffusion-free reactions, W =cE( = kfl and similar procedures as in the case of diffusion-limited reactions yield: /9= SL-$-.
L
In order to determine the maximum catalyst temperature, the solution(s) of eqn (10) or (12) and T, at z = 1 has to be obtained, the largest of which is the maximum catalyst temperature. Solution procedures
Consider the solution(s) of eqn (IO) (or eqn 12) and the calculation of T, at t = 1. Consider lirst the case of diffusion-limited reactions. If we let C,, be the reactor outlet concentration corresponding to the desired coversion, the bulk fluid temperature at the outlet Tb is given
(13)
a = ( - AZZ)/& which follows from eqn (14). The pellet temperature at the outlet z is a solution of eqn (5), which upon inserting eqn (7) into (5) for B becomes: h( T, - Fbb)= ( - AH)[ZD.k, exp ( - Z?/RT,)Z(Cd)]“’ = Y (14)
where the Arrhenius relationship for k. and Z(G) are given by: k.. = ko exp ( - EIRT,)
(1W
I( C,) = led f(c) dc.
(15b)
The choice of Cd and operating conditions should be such that the reactor ‘is stable. This means that the r.h.s. of eqn (14) which is the heat generation rate, should not be greater than the 1.h.s. of eqn (14). which is the heat removal rate, for the chosen values of C, and operating conditions. Otherwise, heat is accumulated in the pellet, eventually leading to runaway situationl The solutiou(s) of eqn (14) for T, in the interval, T. 2 T,, is the catalystemperature at z = t, or c since physically T. 2 Tb for the exothermic reactions b&g considered. This is shown in Fig. 1 when there exists at least one solution. As shown iu the ligure, two different situations can arise for a given value of h and fb’,: the r.h.s. of eqn (14) either touches the straight line (point D) given by the 1.h.s. or crosses the straight line at two points (points A and B). Because of the exponential term of T, in the r.h.s., the curve RHSz will always cross the straight line at two points if it crosses at one point, say A. Any initial sening of operating conditions in start-up that leads to a point on the arc A-B of the RHS2 curve will eventually settle to point A since the rate of heat removal is higher than that of heat generation in this region. Any initial setting that leads to a point above B on RHS,, on the other hau% will cause runaway situation.Thtrefore, the value of T, under steady-state operating COQditiOQS is that corresponding to either point A or point LX In this regard, we note that given C,,. all quantities in eqn (14) are known except for T,. For the solution(s) of eqn (IO), we rewrite eqn (14) for any point in the reactor as follows: T. - T,, = ; [2D.k,r(c,)]“*.
(16)
At an extremum, eqn (10) is satisfied. Therefore, the following should also be satisfied at that point: 7-s= Ti. + a(‘% -
cb)
+
2af(cb)/f(cb)
(17)
which follows from eqn (16) with the aid of eqns (4) and (10). Since the value of Cb at the extremum is still
Maximumcatalyst temperature (hot spot) in adiabatic &d-beds
/Left eqn
hand (14)
side
of
1541
and (6) and from the fact that R = rs. Equation (C’) in the table follows from eqns (4) and (12) and eqn (B’) in the table. The approximate solution procedures given in the table for the intermediate case are based on the approximation of the pellet center concentration C, by: C, = Cdcosh$
(18)
where 4= = L2kf(CrXD&).
I
TL- Tb Fig. I. Solutionsof eqn (4).
unknown, eqns (10) and (17) constitute a set of equations for the solution(s) for the extremum of T,. For a given value of Cb in the interval, Cb E (C,, Ci& the corresponding T. can readily be calculated from eqn (17). This comprises pairs of (G, T.) that can be used in eqn (10) for the determination of the extreme values of T,. This solution is shown in Fig. 2. If no solution is found or F8 is the larger than the solution for T., the maximum occurs at the outlet. If the solution for T, is larger than F=,.a maximum may exist in the reactor which is not at the outlet. Tbis is due to the fact that there exists only one solution for T, when eqns (1) and (17) are solved but there can exist two solutions for eqn (16). Therefore, one must check whether the solution of T. from eqns (10) and (17) corresponds to point A in Fig. 1. If it does, a maximum exists in the reactor which is not at the outlet. If the solution corresponds to point J3 in Fii. 1, the maximum is at the outlet since the point corresponds to operating conditions that can lead to unstable conditions, which will not be chosen as the operating condition. Summarized in Table 1 are the solution procedures for the maximum catalyst temperature. Equation (B’) in the table for the diffusion-free case follows from eqns (5)
(1%
It has been shown[5] that the value of the integral in eqn (7) is rather insensitive to an error in the estimated value of C,. When eqn (18) is used for C, in eqn (7), the same procedures as those used in arriving at the results for the diffusion-limited case lead to the eqns (B”), (C”) and @“) in Table 1. The question of diifusion Limitation and hence that of which entry in Table 1 to use should be resolved at the point of interest. This should not pose any problem since a calculation of T, can be made initially on the assumption of diffusion-limitation, for instance, which can be corrected based on the calculated value of T,. Determination of whether a maximum T, occurs at a point other than the outlet can be made by simply checking the solution(s) of eqa (D)‘s in Table 1. If there is no solution of eqn @‘) for the case of diKusion-free reactions, for instance, the maximum is at the outlet. If a solution exists, then, further calculations are necessary as described by the item (b) in tbe table. In this regard, it is helpful to note tbat the solution of eqn (C)‘s necessary for eqn @)‘s does not involve any trial and error procedures: T. can be calculated in a straightforward manner by inserting values of Cbr thus obtaining pairs of (T,, C,) to be used in eqn (D)‘s. An example
Consider as an example the reaction system given in Table 2, which is for a one-half order reaction. Since f(G) = Cc’, eqns (10) and (17) for this system become: T. = Ti, + a&, + i aCb
(21) In the interval, C, E (Cd, Gin), a unique solution exists which is 667°K for T.. The corresponding value of Cb is 0.5 x 10P5moljcm’. Use of tbis value of Cr, and a temperature slightly higher than 667°K in eqn (16) reveals that the 1.h.s. of eqn (16) is higher than the r.h.s., meaning that the solution of T, (667°K) corresponds to point A in Fig. 1. In order to compare this value of T. with F,;, the following has to be solved (eqn 16) for T.:
Fig. 2. Solutionfor a maximumcatalyst temperatureat 2 = 1.
The value of i;, is obtained from eqn (13) to he WK. The solution of eqn (22) corresponding to point A (FJ in
H. IL LEE
1542
Table 1. Determination of maximum catalyst temperature
Diffusion-limited reactions (e6 2 3) a)
Catalyst temperature at z = 1 (ts)
Tb= T,, + oKin
's - 'b
- Cd)
- ~12DeksI(C,,)]1'2
(Ts 27,)
(Bl
A smaller value of Ts satisfying the equality of eqn (6) >s 7,. b)
Extramum of Ts at z # 1 (?,I
T, = Tin + dCin
5-
(C)
- cb) + Za l~c,,)/f(cb)
112
I 1 gf&
UC,) IC,
E
tc,.
C,,))
(D)
A value of Ts satfsfying the equalfty of eqn (D) is a candidate for the maxim
catalyst temperature.
If a velue
slightly htgher than this
value of Ts and the.corresponding Cb value glva a higher value for
left
the
hand sldc of eqn (16) than the rtght hand side, the value of Ts obtaIned
Is 7,. c)
If is S Ts. a maxbmm
exists ate
f 1.
If not. or no solution can be
found for cqn (0). the maxtmum Is at z ,= 1. Diffusion-free reactions (eg .x0.3) Equations corresponding to eqns (A). (6). (C) and (D) are:
Tb = T,, + a&,,
TsTS-
T,,
l
t,- 3
u(&,
- cd)
ksf(Cb)
- c,) * uf(Cb)/(df/dCb)
dfKb) 8=
(C')
(D’1
ksLT
Intennedlate case (0.3 5 $6 c 3):
(8:)
approximate solution
Equations'corresponding to eqns (A). (6). CC), and (D) are:
t, = Tin
TS- Tb=
+
U(c,,
; [PD,ks
- cd)
% I
f(C)dC]"2
(D")
Cbfcosh $
Ts - Ttn + u(cjn - cb) + 2u
-'b c ,~~;;/[(1-&$'(c,)l
I
b
(C")
Maximum catalyst temperature(hot spot) in sdiabstk Bxcd-bsd8
1543
Table 1. (COnr8f)
l/2
es 6=c
OG -
(1 - &$f(Cb)
CD”)
f(C)dC [
Lf(CJ
I
r
1
l/2
&
;
I(CJ =
'b
I
f&MC
0
9' = L2k,f(Cb)/(DeCb)
Tabk 2. An example reaction system ac = kf(c)
= kC10
k - 2.65 I(10' CXP(-lo.~/T)
Cl”
=
10'5mo\/cm3,
(1 =
(-AHnc,)
Tl"
= 550°K.
= 1.0 x 107,
oc - 0.05 m*/sec,
At the mexlmm
potnt:
-5 Cd = 0.1 x 10 VROl/Ul3
E - (h/tCp) - 0.37
L-lcm
At the outlet:
T, - 667'~
Ts * 65VK
Tb = 600-K
Tb = 640-K
Cb = 0.5 x 10-5 IWCm3
Cb - 0.1 x 10'5 Irrl/CJ
Fig. 1 was found to be 658°K. It is seen that a maxinium exists, which is some 10°K h@t than the catalyst temperature at the outlet (Fs)., The example system has also been solved numtrically using taps (l), (2) and (5). The reactor profiles thus obtained are shown in Fig. 3. It is seen that the maximum T, shownin tht ligurt agrees quitt well with the calculated value of 667’K. FIXED-BEIEiWlTIICATAL~DltACl¶A~N
A catalyst can be deactivated either chemically or physically: poisoning is an example for chemical deactivation and sinteriog is for physical deactivation. Sinteriog can be considered to take place uniformly throughout the catalyst pellet under usual reaction conditions[6]; poisoning cao be uniform, shell-proflessivt or intermediate between the two extremes.
Considtr a uniformly deactivated catalyst pellet. For the general form of intrinsic l&~&s of cqn (a), the global rate can k writttn as
where 7 is the fraction of catalyst diffusion-free reactions, we have:
Jo = k(l - MC).
deactivated.
For
(24)
Difller~otiating eqn (23) or tqn (24) with respect to z and substituting the resulting expression along with the
H. H. LEE
1544
lowered. Let us examine eqn (25). If we note that the last term in the bracket is the r.h.s. of eqn (10) (representing the curve in Fig. 2). it is seen that the r.h.s. of eqn (25) will always decrease with increasing y (or equivalently with time), resulting in lowering of the curve, if (dyldz) is negative. If (d$dz) is positive, it may or may not decrease with time. The same is true with eqn (26) for diffusion-free reactions. Therefore, the maximum catalyst temperature can increase with time for a certain period of time when (dyldz) is negative for some interval of z. The slope dyfdr is always negative for some interval of z if there exists an extremum of y at a point other than the outlet. Indeed, the simulation by Lee and Butt cited earlier has shown that the maximum catalyst tem-
7”
40 Nomalired
reactor
len@h
Fii. 3. Reactorprofiles of temperaturesfar the reaction system in Table 1.
expression for B into eqn (8), one gets:
for diffusion-limited reactions
perature increases with time when a maximum y exists in the reactor. Suppose that there exists a maximum catalyst temperature (z~,Z 1) at time zero. Consider the reactor profile of 7. If catalyst deactivation is due to uniform sintering, the profile is almost entirely determined by temperature since sintering is almost entirely dependent on temperature. Therefore, the initial reactor profile of y has a maximum at z,. This in turn means that for z > zmr (dy/dz) is negative: the maximum temperature must be at a poiat between z,,, and outlet. It is seen then that the maximum temperature a& well as the maximum y moves toward the outlet as deactivation progresses. In the case of poisoning, 7 depends not only on temperature but also on the concentration of poisoning species. If the concentration is relatively constant throughout the reactor, the same conclusion can be made as in the case of uniform sintering. For dependent poisoning, a simple check can be made as to whether a maximum y exists by calculating the following ratio: R = ‘%X4, c)~Z=,,,,I%.(kP, C’LO
for ditIusion-free reactions. For fresh catalysts of uniform activity, y = 0 and dyldz = 0, and these expressions reduce to eqns (10) and (12) respectively. Common sense tells ug that tbe maximum catalyst temperature should decrease with increasing deactivation (or with time) since lower activity caused by deactivation means less heat evolved by the reaction. That this is not necessarily true has been demonstrated[2] for uniformly poisoned pellet. Our major concern, therefore, has to do with the conditions under which the maximum catalyst temperature increases with increasing level of deactivation since then the usual design practice, which is based on fresh catalyst (at time zero), will fail in meeting the constraint on the maximum temperature. Consider the conditions under which the maximum temperature increases with time. Referring to Fig. 2, which is for fresh catalyst (no deactivation), we see that the effect of deactivation is to raise or lower the curve for the r.h.s. of eqn (10). If the curve is raised as a result of deactivation, the maximum catalyst temperature will decrease with time since the curve wilI cross the constant p line at a lower value of T,. On the other hand, the maximum temperature will increase with time if the curve is
at t = 0
(27)
where Se, is the global rate for the poisoning reaction and k, is the corresponding rate constant. If this ratio is greater than unity, a maximum y should occur at around I,,, since i&ally the temperature is the highest at z,,, and consequently k, is the highest at this point. Therefore, dyldz is negative for some interval of z in the initial stage of deactivation and thus the maximum temperature
must increase with increasing time for a certain time period. For independent poisoning, a maximum y can exist whether or not the initial maximum occurs at the outlet if the poisoning species is almost depleted toward the outlet. In such a case, the maximum temperature can increase with time. We have so far limited our discussion
to uniform
deactivation.
While sintering usually takes ph~ce uniformly, poisoning is not. In fact, poisoning is usually severer at the surface of pellet thau the interior. For this general case of poisoning, the global rate for the main reaction can be written with the aid of an approximate generalized modulus developed[7] for nonuniform catalyst distribution as follows: R = [ 2D.k.(1-
9) Icy f(c) dC]1’2/L
(28)
Maximum catalyst temperature(hot spot) in adiabatic fixed-beds which is eqn (23) with y replaced by the value of y evaluated at pellet surface T. This approximation is quite accurate when f is not close to unity, say 7 <0.5. Since the increase of the maximum temperature with time usually occurs in the initial stage of deactivation during which time ? is relatively low, the global rate of eqn (28) can be used to examine the time progression of the maximum temperature. It is readily seen that the same conclusions arrived at for uniform deactivation apply to this general case of poisoning since the behavior of y in the case of uniform deactivation is the same as that of 7. !WMMMY A necessary condition (eqns 10 or 12) for the existence of a maximum catalyst temperature leads to the solution procedures for the maximum given in Table 1. Determination of whether a maximum catalyst temperature exists at a point other than the outlet can be made by simply checking the solution of eqn (D)‘s in Table 1. If there is no solution, the maximum is at the outlet. If a solution exists, further calculations are necessary as described by the item (b) in the table. Perhaps, more important than this determination is the use of the relatively simple relationships obtained and summarized in Table 1 for the choice of inlet conditions (F,,, Ci.) and h(B) for the highest conversion possible within the constraint placed on the maximum catalyst temperature. When a fixed-bed is affected by catalyst deactivation, the maximum catalyst temperature always increases with time for a certain period of time if the initial reactor profile of the catalyst temperature goes through a maximum in the cases of uniform sintering and dependent poisoning provided that eqn (27) is satisfied in the latter. In such cases, a maximum y exists such that d_Adr is negative for some interval of I The fact that the initial maximum occurs at the outlet does not necessarily mean that the maximum temperature cannot increase with time. In particular, a maximum y can exist in the case of independent poisoning when the poisoning species is depleted toward the outlet. NOTATION
concentration of key species bulk fluid concentration pellet center concentration outlet concentration correspondingto desire con-
version specific heat capacity
1545
reactorinlet concentration effective diffusivitv activation energy _ concentrationdependence of intrinsic rate of reaction film heat transfer coefficient heat of reaction integrals defined by eqns (11) and (15b) respectively rate constant for main reaction preexponential factor rate constant for ooisonina reaction k evaluated at peileetsurfa& temperature characteristicdimension of pellet (volume/external surface area)
ratio defined by eqn (29) global rate of main reaction based on pellet volume &ovyu;e of poisoning reaction based on pellet intrinsic rate of main reaction based on pellet volume time b+dktfiti Fmperatwe p&et surface temperature (catalyst pellet temperature) T, at z= I T. at 2 = 2,
&actor inlet temperature intestitial Euid velocity normalizedreactor length * at which the maximum catalyst temperature occurs (2, z 1) reactorlength Greek rvmbols
r
I-J NpCp 7, f fraction of catalyst deactivated, 7 at pellet surface respectively p !Iuiddensity 7 Z(1 - 9Y(,v) 4, & Thile and generalized modulerespectively defined in Table 2
Price T. H. and Butt J. B., C&em.i%gng Sci 197732 393. Lee H. H. and Butt I. B., A.LCAE.J. 198228 410. Carberry J. J., Chemical Catalytic Reaction Engineerbtg.
McGraw-Hii, New York 1976. Fromeat G. F. and Bischoff K. B., Chemical Reactor Anafysis and L&Sian Wiley. New York 1979. Lee H. H., A.I.ChE.L 198127 558. Lee H. H., Ckenr. Engng Sci 198136 950. Lee H. H., Ckem. Engng Sci 193136 1921.
CATALYST TEMPERATURE (HOT SPOT) IN ADIABATIC FIXED-BEDS
HONG H. LEE Department of Chemicd Engineering,University of plorida, Gainesville, FL 32611,U.S.A. (Rcceikd 8 October 1981;occejbd 22 March 1982) Ab&aet-XohIitions are derived under.which the reactor proEk of catalyst temperaturegoes througha maximum at u point other than outlet in an adiabatic reactor. Relatively simple methods are developed for the cahdation of this maximum temperature.It is shown that the maximum catalyst temperaturecan increase with time for a certain
time period when the catalyst undergoes deactivation. Conditiens under which this can happen are also derived. The maximum temperature always increases with time ia the co9esof uniform sintering and dependent poisoning nrovided that a certain ccndition is met in the latter case. if the initial maximumoccurs at a point other than the butlet. INIXODUCTION
Design of a heterogeneous fixed-bed reactor is often subject to a constraint on the maximnm temperatrue of the bed. If the temperature is too high, the catalyst as well as the support can undergo a chemical/physical transformation, which either disables the catalyst or results in break-up and dusting of catalyst pellets. For nonadiabatic reactors, this maximum temperature is known as hot spot. Considerable attention has been directed to the studies concern&this hot spot because of its importance in the design of nonadiabatic reactors. Much less attention has been paid to the maximum temperature in an adiabatic reactor mainly because. the maximum temperature is often tahen as that correspoading to the adiabatic temperature rise. This maximum temperature, which always occurs at the reactor outlet, is not that of catalyst but rather that of bulk fluid. In view of the fact that the temperature ditkence between bulk tluid (fhrid phase) and catalyst (solid phase)’ can be in excess of 50°C for some reactions, it is more prudent to use the maximum catalyst temperature rather than the maximum fluid temperature for the design. Much more important than this is the fact that the maximum catalyst temperature does not necessarily occur at the reactor outlet as opposed to the maxiplum 5uid temperature which *always does. This has been verified experimentally [ l] ‘and the s@lation of an adiabatic reactor undergoing catalyst deactivation has also shown[2] the cxixtence of the maximum catalyst temperatnre at a point other than the reactor outlet. Fnrthermore, this simulation has also shown that the maximum catalyst temperatrue can increase with time when the catalyst undergoes deactivation. In this paper, we present a method of cafculating the maximum catalyst temperature and develop a criterion that can be used to determine whether the maximum catalyst temperature occurs at a point other than the reactor outlet. We also offer an explanation a8 to why the maximum catalyst temperature increases with increasinrg level of catalyst deactivation under certain conditions and determine what these conditions arc. These results
enable one to avoid pitfalls one can come across and put the design of an adiabatic reactor on a iirmer basis. FIXED-BE&3WlTSOUT
CATALYST DEM3lVATtON
A necessary condition Assuming constant physical properties, conservation equations for an adiabatic reactor are:
(2) where 7 =
Z(1 - +!)lfJj u.
(3)
Here the global rate 92 is based on pellet volume and the subscript b denotes bulk fluid. Combining these two equations leads to:
where C, and T,,, are inlet concentration and temperature respectively. A heat balance across the pelletbulk fluid interface yields: h( T, - Tb) = BL( - AH)
(5)
where L is the characteristic *Ret length and the subscript s denotes pellet surface. We assume here that the external mass transfer resistance is negligible and that the pellet is isothermal. These assumptions are quite valid under realistic reaction conditions[3]. Under these assumptions, the catalyst temperature is at T, and the pellet surface concentrauon is equal to bulk fluid concentration. For a general form of intrinsic hinetics r,:
1539
r, = kf
(6)
H.
1540
use of the generalized effectiveness factor[41 yields the following expression for the global rate:
H. LEE by: Tb= Ti,+o(Ci.-Cd)
W = [ZZXlr, Lb f(c) dc]“‘/L
(7)
where the effective diffusivity D. has been assumed constant, C, is the concentration at pellet center and k, the rate constant evaluated at T,. In order for a maximum catalyst temperature to exist, it is necessary that dZ’,./dz= 0. Let z,,,( # 1) be the point at which the maximum occurs. Differentation of eqn (5) with respect to z yields: k(!&l$)
= L(-AH)%
which reduces at z, to: (8) where eqn (2) has been used for dTJdz. Consider two extreme cases: diffusion-free and difIusion-limited reactions. In the case of diffusion-limited reactions, the pellet center concentration C, can be set to zero and the differentiation of eqn (7) with respect to z yields:
where eqn (1) has been used for dCddz. Using eqns (7) and (9) in (8) and rearranging, one gets:
B=
[&f&J” f
at z = z,
(10)
where z(cb)
=
I,”f(C) dc,
(11)
In the case of diffusion-free reactions, W =cE( = kfl and similar procedures as in the case of diffusion-limited reactions yield: /9= SL-$-.
L
In order to determine the maximum catalyst temperature, the solution(s) of eqn (10) or (12) and T, at z = 1 has to be obtained, the largest of which is the maximum catalyst temperature. Solution procedures
Consider the solution(s) of eqn (IO) (or eqn 12) and the calculation of T, at t = 1. Consider lirst the case of diffusion-limited reactions. If we let C,, be the reactor outlet concentration corresponding to the desired coversion, the bulk fluid temperature at the outlet Tb is given
(13)
a = ( - AZZ)/& which follows from eqn (14). The pellet temperature at the outlet z is a solution of eqn (5), which upon inserting eqn (7) into (5) for B becomes: h( T, - Fbb)= ( - AH)[ZD.k, exp ( - Z?/RT,)Z(Cd)]“’ = Y (14)
where the Arrhenius relationship for k. and Z(G) are given by: k.. = ko exp ( - EIRT,)
(1W
I( C,) = led f(c) dc.
(15b)
The choice of Cd and operating conditions should be such that the reactor ‘is stable. This means that the r.h.s. of eqn (14) which is the heat generation rate, should not be greater than the 1.h.s. of eqn (14). which is the heat removal rate, for the chosen values of C, and operating conditions. Otherwise, heat is accumulated in the pellet, eventually leading to runaway situationl The solutiou(s) of eqn (14) for T, in the interval, T. 2 T,, is the catalystemperature at z = t, or c since physically T. 2 Tb for the exothermic reactions b&g considered. This is shown in Fig. 1 when there exists at least one solution. As shown iu the ligure, two different situations can arise for a given value of h and fb’,: the r.h.s. of eqn (14) either touches the straight line (point D) given by the 1.h.s. or crosses the straight line at two points (points A and B). Because of the exponential term of T, in the r.h.s., the curve RHSz will always cross the straight line at two points if it crosses at one point, say A. Any initial sening of operating conditions in start-up that leads to a point on the arc A-B of the RHS2 curve will eventually settle to point A since the rate of heat removal is higher than that of heat generation in this region. Any initial setting that leads to a point above B on RHS,, on the other hau% will cause runaway situation.Thtrefore, the value of T, under steady-state operating COQditiOQS is that corresponding to either point A or point LX In this regard, we note that given C,,. all quantities in eqn (14) are known except for T,. For the solution(s) of eqn (IO), we rewrite eqn (14) for any point in the reactor as follows: T. - T,, = ; [2D.k,r(c,)]“*.
(16)
At an extremum, eqn (10) is satisfied. Therefore, the following should also be satisfied at that point: 7-s= Ti. + a(‘% -
cb)
+
2af(cb)/f(cb)
(17)
which follows from eqn (16) with the aid of eqns (4) and (10). Since the value of Cb at the extremum is still
Maximumcatalyst temperature (hot spot) in adiabatic &d-beds
/Left eqn
hand (14)
side
of
1541
and (6) and from the fact that R = rs. Equation (C’) in the table follows from eqns (4) and (12) and eqn (B’) in the table. The approximate solution procedures given in the table for the intermediate case are based on the approximation of the pellet center concentration C, by: C, = Cdcosh$
(18)
where 4= = L2kf(CrXD&).
I
TL- Tb Fig. I. Solutionsof eqn (4).
unknown, eqns (10) and (17) constitute a set of equations for the solution(s) for the extremum of T,. For a given value of Cb in the interval, Cb E (C,, Ci& the corresponding T. can readily be calculated from eqn (17). This comprises pairs of (G, T.) that can be used in eqn (10) for the determination of the extreme values of T,. This solution is shown in Fig. 2. If no solution is found or F8 is the larger than the solution for T., the maximum occurs at the outlet. If the solution for T, is larger than F=,.a maximum may exist in the reactor which is not at the outlet. Tbis is due to the fact that there exists only one solution for T, when eqns (1) and (17) are solved but there can exist two solutions for eqn (16). Therefore, one must check whether the solution of T. from eqns (10) and (17) corresponds to point A in Fig. 1. If it does, a maximum exists in the reactor which is not at the outlet. If the solution corresponds to point J3 in Fii. 1, the maximum is at the outlet since the point corresponds to operating conditions that can lead to unstable conditions, which will not be chosen as the operating condition. Summarized in Table 1 are the solution procedures for the maximum catalyst temperature. Equation (B’) in the table for the diffusion-free case follows from eqns (5)
(1%
It has been shown[5] that the value of the integral in eqn (7) is rather insensitive to an error in the estimated value of C,. When eqn (18) is used for C, in eqn (7), the same procedures as those used in arriving at the results for the diffusion-limited case lead to the eqns (B”), (C”) and @“) in Table 1. The question of diifusion Limitation and hence that of which entry in Table 1 to use should be resolved at the point of interest. This should not pose any problem since a calculation of T, can be made initially on the assumption of diffusion-limitation, for instance, which can be corrected based on the calculated value of T,. Determination of whether a maximum T, occurs at a point other than the outlet can be made by simply checking the solution(s) of eqa (D)‘s in Table 1. If there is no solution of eqn @‘) for the case of diKusion-free reactions, for instance, the maximum is at the outlet. If a solution exists, then, further calculations are necessary as described by the item (b) in tbe table. In this regard, it is helpful to note tbat the solution of eqn (C)‘s necessary for eqn @)‘s does not involve any trial and error procedures: T. can be calculated in a straightforward manner by inserting values of Cbr thus obtaining pairs of (T,, C,) to be used in eqn (D)‘s. An example
Consider as an example the reaction system given in Table 2, which is for a one-half order reaction. Since f(G) = Cc’, eqns (10) and (17) for this system become: T. = Ti, + a&, + i aCb
(21) In the interval, C, E (Cd, Gin), a unique solution exists which is 667°K for T.. The corresponding value of Cb is 0.5 x 10P5moljcm’. Use of tbis value of Cr, and a temperature slightly higher than 667°K in eqn (16) reveals that the 1.h.s. of eqn (16) is higher than the r.h.s., meaning that the solution of T, (667°K) corresponds to point A in Fig. 1. In order to compare this value of T. with F,;, the following has to be solved (eqn 16) for T.:
Fig. 2. Solutionfor a maximumcatalyst temperatureat 2 = 1.
The value of i;, is obtained from eqn (13) to he WK. The solution of eqn (22) corresponding to point A (FJ in
H. IL LEE
1542
Table 1. Determination of maximum catalyst temperature
Diffusion-limited reactions (e6 2 3) a)
Catalyst temperature at z = 1 (ts)
Tb= T,, + oKin
's - 'b
- Cd)
- ~12DeksI(C,,)]1'2
(Ts 27,)
(Bl
A smaller value of Ts satisfying the equality of eqn (6) >s 7,. b)
Extramum of Ts at z # 1 (?,I
T, = Tin + dCin
5-
(C)
- cb) + Za l~c,,)/f(cb)
112
I 1 gf&
UC,) IC,
E
tc,.
C,,))
(D)
A value of Ts satfsfying the equalfty of eqn (D) is a candidate for the maxim
catalyst temperature.
If a velue
slightly htgher than this
value of Ts and the.corresponding Cb value glva a higher value for
left
the
hand sldc of eqn (16) than the rtght hand side, the value of Ts obtaIned
Is 7,. c)
If is S Ts. a maxbmm
exists ate
f 1.
If not. or no solution can be
found for cqn (0). the maxtmum Is at z ,= 1. Diffusion-free reactions (eg .x0.3) Equations corresponding to eqns (A). (6). (C) and (D) are:
Tb = T,, + a&,,
TsTS-
T,,
l
t,- 3
u(&,
- cd)
ksf(Cb)
- c,) * uf(Cb)/(df/dCb)
dfKb) 8=
(C')
(D’1
ksLT
Intennedlate case (0.3 5 $6 c 3):
(8:)
approximate solution
Equations'corresponding to eqns (A). (6). CC), and (D) are:
t, = Tin
TS- Tb=
+
U(c,,
; [PD,ks
- cd)
% I
f(C)dC]"2
(D")
Cbfcosh $
Ts - Ttn + u(cjn - cb) + 2u
-'b c ,~~;;/[(1-&$'(c,)l
I
b
(C")
Maximum catalyst temperature(hot spot) in sdiabstk Bxcd-bsd8
1543
Table 1. (COnr8f)
l/2
es 6=c
OG -
(1 - &$f(Cb)
CD”)
f(C)dC [
Lf(CJ
I
r
1
l/2
&
;
I(CJ =
'b
I
f&MC
0
9' = L2k,f(Cb)/(DeCb)
Tabk 2. An example reaction system ac = kf(c)
= kC10
k - 2.65 I(10' CXP(-lo.~/T)
Cl”
=
10'5mo\/cm3,
(1 =
(-AHnc,)
Tl"
= 550°K.
= 1.0 x 107,
oc - 0.05 m*/sec,
At the mexlmm
potnt:
-5 Cd = 0.1 x 10 VROl/Ul3
E - (h/tCp) - 0.37
L-lcm
At the outlet:
T, - 667'~
Ts * 65VK
Tb = 600-K
Tb = 640-K
Cb = 0.5 x 10-5 IWCm3
Cb - 0.1 x 10'5 Irrl/CJ
Fig. 1 was found to be 658°K. It is seen that a maxinium exists, which is some 10°K h@t than the catalyst temperature at the outlet (Fs)., The example system has also been solved numtrically using taps (l), (2) and (5). The reactor profiles thus obtained are shown in Fig. 3. It is seen that the maximum T, shownin tht ligurt agrees quitt well with the calculated value of 667’K. FIXED-BEIEiWlTIICATAL~DltACl¶A~N
A catalyst can be deactivated either chemically or physically: poisoning is an example for chemical deactivation and sinteriog is for physical deactivation. Sinteriog can be considered to take place uniformly throughout the catalyst pellet under usual reaction conditions[6]; poisoning cao be uniform, shell-proflessivt or intermediate between the two extremes.
Considtr a uniformly deactivated catalyst pellet. For the general form of intrinsic l&~&s of cqn (a), the global rate can k writttn as
where 7 is the fraction of catalyst diffusion-free reactions, we have:
Jo = k(l - MC).
deactivated.
For
(24)
Difller~otiating eqn (23) or tqn (24) with respect to z and substituting the resulting expression along with the
H. H. LEE
1544
lowered. Let us examine eqn (25). If we note that the last term in the bracket is the r.h.s. of eqn (10) (representing the curve in Fig. 2). it is seen that the r.h.s. of eqn (25) will always decrease with increasing y (or equivalently with time), resulting in lowering of the curve, if (dyldz) is negative. If (d$dz) is positive, it may or may not decrease with time. The same is true with eqn (26) for diffusion-free reactions. Therefore, the maximum catalyst temperature can increase with time for a certain period of time when (dyldz) is negative for some interval of z. The slope dyfdr is always negative for some interval of z if there exists an extremum of y at a point other than the outlet. Indeed, the simulation by Lee and Butt cited earlier has shown that the maximum catalyst tem-
7”
40 Nomalired
reactor
len@h
Fii. 3. Reactorprofiles of temperaturesfar the reaction system in Table 1.
expression for B into eqn (8), one gets:
for diffusion-limited reactions
perature increases with time when a maximum y exists in the reactor. Suppose that there exists a maximum catalyst temperature (z~,Z 1) at time zero. Consider the reactor profile of 7. If catalyst deactivation is due to uniform sintering, the profile is almost entirely determined by temperature since sintering is almost entirely dependent on temperature. Therefore, the initial reactor profile of y has a maximum at z,. This in turn means that for z > zmr (dy/dz) is negative: the maximum temperature must be at a poiat between z,,, and outlet. It is seen then that the maximum temperature a& well as the maximum y moves toward the outlet as deactivation progresses. In the case of poisoning, 7 depends not only on temperature but also on the concentration of poisoning species. If the concentration is relatively constant throughout the reactor, the same conclusion can be made as in the case of uniform sintering. For dependent poisoning, a simple check can be made as to whether a maximum y exists by calculating the following ratio: R = ‘%X4, c)~Z=,,,,I%.(kP, C’LO
for ditIusion-free reactions. For fresh catalysts of uniform activity, y = 0 and dyldz = 0, and these expressions reduce to eqns (10) and (12) respectively. Common sense tells ug that tbe maximum catalyst temperature should decrease with increasing deactivation (or with time) since lower activity caused by deactivation means less heat evolved by the reaction. That this is not necessarily true has been demonstrated[2] for uniformly poisoned pellet. Our major concern, therefore, has to do with the conditions under which the maximum catalyst temperature increases with increasing level of deactivation since then the usual design practice, which is based on fresh catalyst (at time zero), will fail in meeting the constraint on the maximum temperature. Consider the conditions under which the maximum temperature increases with time. Referring to Fig. 2, which is for fresh catalyst (no deactivation), we see that the effect of deactivation is to raise or lower the curve for the r.h.s. of eqn (10). If the curve is raised as a result of deactivation, the maximum catalyst temperature will decrease with time since the curve wilI cross the constant p line at a lower value of T,. On the other hand, the maximum temperature will increase with time if the curve is
at t = 0
(27)
where Se, is the global rate for the poisoning reaction and k, is the corresponding rate constant. If this ratio is greater than unity, a maximum y should occur at around I,,, since i&ally the temperature is the highest at z,,, and consequently k, is the highest at this point. Therefore, dyldz is negative for some interval of z in the initial stage of deactivation and thus the maximum temperature
must increase with increasing time for a certain time period. For independent poisoning, a maximum y can exist whether or not the initial maximum occurs at the outlet if the poisoning species is almost depleted toward the outlet. In such a case, the maximum temperature can increase with time. We have so far limited our discussion
to uniform
deactivation.
While sintering usually takes ph~ce uniformly, poisoning is not. In fact, poisoning is usually severer at the surface of pellet thau the interior. For this general case of poisoning, the global rate for the main reaction can be written with the aid of an approximate generalized modulus developed[7] for nonuniform catalyst distribution as follows: R = [ 2D.k.(1-
9) Icy f(c) dC]1’2/L
(28)
Maximum catalyst temperature(hot spot) in adiabatic fixed-beds which is eqn (23) with y replaced by the value of y evaluated at pellet surface T. This approximation is quite accurate when f is not close to unity, say 7 <0.5. Since the increase of the maximum temperature with time usually occurs in the initial stage of deactivation during which time ? is relatively low, the global rate of eqn (28) can be used to examine the time progression of the maximum temperature. It is readily seen that the same conclusions arrived at for uniform deactivation apply to this general case of poisoning since the behavior of y in the case of uniform deactivation is the same as that of 7. !WMMMY A necessary condition (eqns 10 or 12) for the existence of a maximum catalyst temperature leads to the solution procedures for the maximum given in Table 1. Determination of whether a maximum catalyst temperature exists at a point other than the outlet can be made by simply checking the solution of eqn (D)‘s in Table 1. If there is no solution, the maximum is at the outlet. If a solution exists, further calculations are necessary as described by the item (b) in the table. Perhaps, more important than this determination is the use of the relatively simple relationships obtained and summarized in Table 1 for the choice of inlet conditions (F,,, Ci.) and h(B) for the highest conversion possible within the constraint placed on the maximum catalyst temperature. When a fixed-bed is affected by catalyst deactivation, the maximum catalyst temperature always increases with time for a certain period of time if the initial reactor profile of the catalyst temperature goes through a maximum in the cases of uniform sintering and dependent poisoning provided that eqn (27) is satisfied in the latter. In such cases, a maximum y exists such that d_Adr is negative for some interval of I The fact that the initial maximum occurs at the outlet does not necessarily mean that the maximum temperature cannot increase with time. In particular, a maximum y can exist in the case of independent poisoning when the poisoning species is depleted toward the outlet. NOTATION
concentration of key species bulk fluid concentration pellet center concentration outlet concentration correspondingto desire con-
version specific heat capacity
1545
reactorinlet concentration effective diffusivitv activation energy _ concentrationdependence of intrinsic rate of reaction film heat transfer coefficient heat of reaction integrals defined by eqns (11) and (15b) respectively rate constant for main reaction preexponential factor rate constant for ooisonina reaction k evaluated at peileetsurfa& temperature characteristicdimension of pellet (volume/external surface area)
ratio defined by eqn (29) global rate of main reaction based on pellet volume &ovyu;e of poisoning reaction based on pellet intrinsic rate of main reaction based on pellet volume time b+dktfiti Fmperatwe p&et surface temperature (catalyst pellet temperature) T, at z= I T. at 2 = 2,
&actor inlet temperature intestitial Euid velocity normalizedreactor length * at which the maximum catalyst temperature occurs (2, z 1) reactorlength Greek rvmbols
r
I-J NpCp 7, f fraction of catalyst deactivated, 7 at pellet surface respectively p !Iuiddensity 7 Z(1 - 9Y(,v) 4, & Thile and generalized modulerespectively defined in Table 2
Price T. H. and Butt J. B., C&em.i%gng Sci 197732 393. Lee H. H. and Butt I. B., A.LCAE.J. 198228 410. Carberry J. J., Chemical Catalytic Reaction Engineerbtg.
McGraw-Hii, New York 1976. Fromeat G. F. and Bischoff K. B., Chemical Reactor Anafysis and L&Sian Wiley. New York 1979. Lee H. H., A.I.ChE.L 198127 558. Lee H. H., Ckenr. Engng Sci 198136 950. Lee H. H., Ckem. Engng Sci 193136 1921.