Parametric sensitivity of temperature at the regeneration of a single catalyst pellet

Pergamon

Chemical En~lineeriny Science, Vol. 50, No. 4, pp. 685-694, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00

0009-2509(94)00435-8

P A R A M E T R I C SENSITIVITY OF T E M P E R A T U R E AT T H E R E G E N E R A T I O N O F A SINGLE CATALYST P E L L E T HSIN-AN H W A N G and SHU-MIN CHIAO t Department of Chemical Engineering, Tunghai University, Taichung, Taiwan 40704, ROC (Received 29 January 1994; accepted in revised form 9 September 1994)

Al~traet--Coked catalysts may often be regenerated by in situ oxidation. Being a highly exothermic process, the high regeneration temperature that may result is of ultimate concern. The possibility of a thermal runaway, however, has traditionally been ignored. In this study, the theory of parametric sensitivity was extended to investigate the regeneration of a single coked catalyst pellet. Our results showed that the sensitivity of the pellet temperature varies with different operating conditions. The sensitivities become critical when the burning rate and the migration rate of oxygen inside the pellet become equivalently important. It is found that the system sensitivities with different operating parameters behave similarly if these sensitive parameters affect the system through a similar path. For example, the gas concentration and the film transfer-ability affect the regeneration behavior through the pellet boundary. The initial temperature and coke contents, on the other hand, change the reaction condition uniformly throughout the entire catalyst. As a result, maximum sensitivities were observed to occur at different operating conditions.

INTRODUCTION

Catalysts employed in the hydrocarbon processing industry eventually become deactivated by the deposition of a condensed carbonaceous material. For coked catalysts, the activity can usually be restored by oxidation at an elevated temperature of 650 to 900 K, using a gas stream containing a small percentage of oxygen. The sole concern in this regeneration process is to limit the excess temperature rise due to the exothermic oxidation reaction so as to prevent sintering of the catalyst. Because of its industrial importance, regeneration of coked catalyst pellets has attracted a great deal of attention. Numerous investigations, which include both theoretical and experimental works, have been conducted in their attempts to predict the variation of catalyst temperature (Walker et al., 1959; Luss and Amundson, 1969; Ishida and Wen, 1968; Bowen and Cheng, 1969; Shettigar and Hughes, 1972; Massoth, 1967, Ramachandran et al., 1975; Szekely and Evans, 1971; Sampath et al., 1975; Hashimoto et al., 1983, 1984; Hwang et al., 1993). From their studies, the early stage of the contact between oxygen and coke is of crucial importance since the catalyst temperature rises fast, and a maximum temperature usually results. It is of interest to understand how this maximum temperature responds to small perturbations of various operating variables. The runaway phenomenon, which is also referred to as parametric sensitivity (Bil~ms and Amundson, 1956), represents the phenomenon that a small change in certain physicochemical parameters of the system

+Author to whom correspondence should be addressed.

causes a large variation in the system's behavior. For a chemical reaction system, thermal ignition can usually be characterized by the existence of a stability change. However, even for a single-valued system where no bifurcation exists, there may still exist a narrow range of values of the system parameters in which the temperature varies significantly, which is similar to ignition (Balakotaiah, 1989). A number of studies have attempted to provide bounds of the parametric sensitive region for various reaction systems based on some geometric property of a characteristic temperature profile (Barkelew, 1959, 1984; Adler and Enig, 1964; van Welsenacere and Froment, 1970; Morbidelli and Varma, 1982, 1987) or the explosion theory (Lacey, 1983; Boddington et al., 1983; Morbidelli and Varma, 1986, 1988, 1989; Bauman et al., 1990; Vajda and Rabitz, 1992). Considering the regeneration of coked catalyst pellets, very few of them can be related directly. These studies include the stability analysis of single catalyst pellet (Burghardt and Berezowski, 1991) and char combustion (Hsuen and Sotirchos, 1989), and the experimental work of Furimsky (1988). In this study, we extend the generalized sensitivity analysis developed by Morbidelli and Varma (1988) to the system of coke burning in a single spent catalyst pellet. Since the burning of coke within a catalyst pellet is a dynamic process, bifurcation analysis is not considered here. Moreover, most physicochemical parameters of a reaction system, e.g., heat of reaction, thermal conductivity, etc., are of known values here. Therefore, instead of looking for stability bounds on system parameters, the sensitivities of the catalyst regeneration temperature with respect to the operating variables such as the ambient temperature, oxygen concentration and gas flow rate are investigated. 685

686

H.-A. HWANGand S.-M. CHIAO where

KINETICS OF COKE BURNING

To simplify our computation, the following reaction stoichiometry suggested by Nascimento (1982) is used: CH. +

1 + ~ 0 2 ~ CO2 +

H20

(1)

where n represents atomic ratio of hydrogen to carbon in coke. In general, its initial value no varies from 0.4 to 2, and decreases as regeneration proceeds. A seriously coked catalyst will have a smaller value of n since more hydrogen are replaced by cyclic carbon chains (Haideman and Botty, 1959). Due to the fact that the generation of carbon dioxide produces more heat than carbon monoxide, the system temperature rises higher consequently. The neglect of the latter therefore serves our purpose of studying the maximum temperature sensitivity adequately. The consumption rate of oxygen based upon eq. (1) is no dXh'~,

{-

E~_

ro= l +---~--~)roexp[~T)t~ot~c

dp2 = R2ko exp ( - E/Rg To) Coo Deo

(2)

where first-order reaction kinetics with respect to both oxygen and carbon is applied. The ratio dXddX, is the instantaneous conversion rate ratio between hydrogen and carbon components in the coke. The purpose of employing this conversion rate ratio is to include the effect of faster burning rate of hydrogen, which has been shown to have critical effect on the scales of maximum temperature rises (Ramachandran et al., 1975; Hashimoto et al., 1984; Hwang et al., 1993).

r=

Le = p~CpsD~o ges DeoCco( - AHc) Ke~ To(1 -- %)

are the Thiele modules, Lewis number and thermicity factor, respectively. The subscripts o, c, and s represent oxygen, carbon component of the coke, and the catalyst support, respectively. The following dimensionless variables have also been used: t Deo

r

C~

Xc=cc ° Co T Xo=-~ooo, 0 = E , AH -

- AH~ - AHc'

D~(Xc) 2 = De---~

MASS AND ENERGY CONSERVATION EQUATIONS

0

-~ Xc = -- d)2XoXc~P a

(3)

1 O/ 220Xo'X ¢ 2 X o X t F ( l + ~ d X n ) (4)

Le~O=bO/

200`

dXJ

(5)

(7)

~P = exp( - E/RoT ) exp( -- g/RgTo)"

Notice that the decreasing of the hydrogen-to-carbon ratio n is implicitly included in the hydrogen-to-carbon conversion ratio dXh/dXc, which is predetermined as will be discussed in the results section. The solid thermal conductivity Ke, is kept constant throughout the study. The effective diffusivity De, however, is defined as a function of coke content to recover the effect of coke-dependent mass transfer resistance in the particle pellet model. The following simple expression was used (Hwang et al., 1993): Cc

De(Cc)=De°[2min+(1--2min)( 1 - C In this study, we intend to determine the sensitivity of the pellet temperature to various operating variables. The first step is to predict the variation of pellet temperature correctly. Instead of the more popular particle-pellet model (Szekely and Evans, 1971; Hashimoto et al., 1983, 1984), we chose to use the traditional reaction-diffusion model. Detailed derivation of the model can be found elsewhere (Hwang, 1993). Considering the conservation of mass of both coke and oxygen, and the conservation of energy, the following dimensionless equations were derived:

(6)

.....

t, ) ]

(8)

where 2mi n represents the portion of coke-free diffusivity left when the catalyst pellet is deposited with maximum amount of coke allowed (coke concentration equals C ..... ). In this study, 2mi, is set to be zero for simplicity. Its effect is minimized by employing a relatively light initial coking condition. The value of exponent P determines types of coke plugging which can occur (see Fig. 1). Since we are concerned with the mobility of oxygen rather than the effects of possible reaction surface area studied by Mo and Wei (1986), the correlation for P is different here as discussed hereafter: When an intracrystalline channel system is uniformly occupied by coke, a higher coking level is required for obtaining a smaller transport rate. The effective diffusivity is therefore decreased appreciably only when the coking level is relatively high. This bulk pore blocking .situation, which decreases the constriction factor, may be described with a small value of P. On the other hand, a large value of P represents the situation where border blocking occurs. In which case, coke mainly plugs the pore entrances. The pore tortuosity increases, and the mobility of reactant significantly drops with only a small amount of coke deposited (Hwang, 1993).

Parametric sensitivity of temperature

687

where ~Oi are arbitrary system parameters, which are the operating variables of the regeneration process, e.g., To, Coo, etc. Notice that in the definition of S0(~b,), the volume-averaged temperature {0} is employed instead of the distributed pellet temperature 0. The reason for this is to reduce computing effort. From the distributed model, six partial differential equations will have to be solved simultaneously to render the system sensitivities. The formula with {0}, on the other hand, contains only four PDEs and two ODEs. The validity of this simplification is presented in the discussion section. The volume averaged temperature is defined as

1.0 0.9 "~ 0.8

i5 0.7 =o 0.6 m 0.5 0.4 .,~ ~ 0.3

o.2 0.1 0.0

I

I

10

I

I

20 30

40

I

I

~

50 60

I

70

80

{o} = fOdV.

90 100

(13)

% Fraction of Blockage

Fig. 1. Effect of coke blockage on the effective diffusivity.

The energy balance equation based on the volumeaveraged temperature now becomes

In deriving the conservation equation of oxygen, the time-varying porosity is related to the concentration of coke by the following equation: - - ~0 ~*

--

- 1 - Xc

(9)

~o

where e* is the coke-free porosity and eo is the initial porosity. The initial and boundary conditions that we used are z~<0,

0~<~<1.

Xo=O,

Xc=l,

OXo

80

0=1

(10)

and

• >0,

(=1,

~Xo - - - = S h ( 1 - Xo)

(11)

~0

a~= Nu(1 -- O)

where Sh and Nu are the Sherwood and Nusselt numbers, respectively.

× {XoX~} {V}

where {~P} is the dimensionless Arrhenius law evaluated at {0}, and the conduction term in eq. (5) is replaced by the film heat transfer effect of eq. (11). It should be noted that the use of an averaged temperature in the Arrhenius law for the computation of the reaction rate may have a severe effect on the system temperature. Our results, however, showed that less significant effects on the sensitivities were identified as will be discussed later. To further simplify our computation, a constant hydrogen-to-carbon conversion ratio q is employed in the above equation (Hwang et al., 1993). The parentheses { } again represent a volume-averaged value. On differentiating eqs (3), (4) and (14) with respect to ~Oi,the following sensitivity equations result:

~S: =-¢:(S:X~{~t'}+XoS~{$}+XoX:So~) - xoxc{°"}(~k,~---~4~0

SENSITIVITY ANALYSIS

The runaway reaction usually accompanies a large variation in the system variables with a small perturbation of some system parameters. In the regeneration process of coked catalyst, the sole concern is the runaway of temperature. However, it is impossible to obtain the sensitivity of system temperature without computing the sensitivities of oxygen concentration and coke content simultaneously. Following Morbidelli and Varma (1988), the following dimensionless local sensitivities are defined:

so(0,) = 0i

(lS)

O (eSc + exScX°)= 1 OF 2f2 S c3X° A t ' - - ) 1

-

2(sox¢{v} + xosc{v} +XoXcSo

{o} }

2 no dXh -ok XoXc{~}-~S~(-d-~). x - XoXc{~}

{o}, sM,,) = ~ 0 , ~ x ~ ,

×I 1

So(~b,) = ~b,-T-;-,Xo vqJ~

04)

(12)

nodXh]f,

688 L e ~ Ss = 3 I -

H.-A. HWANGand S.-M. CmAO NuSs +

- {O})~l, ~

Table 1. Initial and boundary conditions for the dimensionless sensitivities

Nu]

ff~

r~<0, 0~<(~<1

To

So= 1 Sc = 0

r>0,(=l

---

---

~So

So = 0 Coo

r>0,(=0

~So

--=0

S O ~ 0

--

-0

--

S~ - 0 So = 0 Nu

where subscript ,x represents total differentiation with respect to Xc. Since operating variables are involved in the dimensionless groups, proper differentiations appear. Notice that this set of sensitivity equations must be solved together with eqs (3), (4) and (14). The initial condition for {0} is the same as 0. The corresponding initial and boundary conditions for eqs (15)-(17), however, vary along with the selection of 0~. The parameters that we studied include ambient (initial) temperature To, ambient oxygen concentration Coo, initial coke concentration Coo and the film heat transfer coefficient h~. The last parameter is chosen to represent the effect of gas velocity in the regenerator. It is clear that the ambient temperature will affect mainly the reaction rate. The oxygen concentration will enhance effect of oxygen diffusion. The heat transfer coefficient represents energy dissipation effect. The initial coke concentration is not determined by the operating parameters of the regenerator, it is included, however, to demonstrate the character of the sensitivity as will be discussed. The initial and boundary conditions that we used are listed in Table I.

RESULTS

AND

S 0 ~ 0

So = 0 Coo

OSo

--

Sc = 0

- Sh(So - 1) --

-~So -- = 0

S o l O

-8So --= ShSo

--

__

S,=I 3So

So = 0

8So

--=0

---=0

Table. 2. Physicochemical data used in the computation De0 = 10- 6 m2/s ko = 1.33 x 107 mS/(gmol K) AHh = 468 KJ/gmol Coo = 3 gmol/m 3 Cc* = 1365 gmol/m 3 p~ = 1.33 × 10 6 gmol/m 3 R = 0.003 m

Kes = 0.4 J/(m/K) E = 157.2 KJ/gmol AHc = 393 KJ/gmol Coo = 780 gmol/m 3 Rg = 8.3143 J/(gmol K) Cp~ = 1.5 J/(g K) Sh = 175

Le = 5

Nu=0.25

eo = 0.15 ~/= 50

e* = 0.35 ni = 0.5

120

10o

DISCUSSION

The PDEs in our model were solved by the finitedifference method in the spatial domain. The timedomain integration for both the P D E s and the O D E s are accomplished by a fourth-order R u n g e - K u t t a scheme with adaptive step-size control. The volumeaveraged values of { X o X e } , {XoS~} and {SoX~} are computed numerically by a Gaussian quadrature type of scheme. Basic physicochemical parameters used in our calculation are listed in Table 2. Typical computing time on a PC-486-33 machine is about 5 h. Since the system temperature is the most important index of a runaway reaction, only So will be discussed in this paper. S e n s i t i v i t y a n a l y s i s a n d the a v e r a g e - t e m p e r a t u r e approach

aSo -- = 0

ap-

By solving eqs (3)-(5), we are able to resolve the history of temperature distribution within the catalyst pellet. The variation of the maximum system temperature rises (ATma, = Tmax - To) with respect to To is plotted in Fig. 2, where Tm~ is referred to the highest temperature that is observed within the catalyst pellet

0~ " o

80

~ 60 ~ 4o 20 0 600

650

700

750 800 T0(K)

850

900

Fig. 2. Effect of ambient temperature TO on the maximum temperature rise AT,,,x. through one complete regeneration. There are two regions where ATmax increases more drastically with T o. The first one is between To = 640 and 720 K, and the second one is between To = 750 and 820 K. Indeed, the system temperature is nonuniformly sensitive to To under different operating conditions.

689

Parametric sensitivity of temperature Before further discussing the physics of this sensitive behavior, we would like to verify the validity of using the average-temperature formulation. Shown in Fig. 3 is the variation of So(To) along the regeneration process. Notice that the material coordinate a is used, which is nothing but the averaged conversion of the coke. The broken curve which represents the distributed model [eq. (5)] is derived by taking the finitedifference approximation of two temperature histories with a small difference between their To. It is clear that the average-temperature approach produces smaller sensitivities. The trend of these two curves, however, is very much alike. Due to the fact that the system temperature must equal To before and after the regeneration, the dimensionless sensitivities are always unity at t~ -- 0 and ~ = 1. A single maximum value occurs in between. The positions of the maximum sensitivities are almost exactly the same for both the distributed and the average-temperature models. By differentiating the curve in Fig. 2 with respect to To, the broken curve in Fig. 4, which represents the temperature sensitivity at Tmax[S~'(To)], is obtained. The solid line is derived from the average sensitivity formulation. In general, the distributed model predicts slightly higher sensitivities than the averaged approach. However, the peaks and valleys of these two curves locate at the same places. The average sensitivity model is thus shown to be solid, and our following discussion is based upon this approach.

3.0

2.5

S0~(To)2.0

1.5

1.0 600

650

700

750

800

850

900

T0(K) Fig. 4. Variation of the temperature sensitivity at the maximum temperature S~(To) with respect to the ambient temperature To calculated from the distributed model ( . . . . ) and the average-temperature approach ( ).

t2of 100

80

(6)

Burning of the hydrogen component Two maxima are apparent in Fig. 4, the smaller one occurs at about 680 K, and the other one with much greater magnitude is positioned at about 770 K. Different causes for these peaks can be observed from the temperature histories plotted in Fig. 5. When the ambient temperature is kept below 750 K, the temperature history curve always has a sharp front and a long tail. The fast temperature rising rates in the

5.0

4.0

~(~)3.0

2.0

1.0

0.0

0.2

0.4

0.6

0.8

1.0

O

Fig. 3. Histories of the temperature sensitivity So(To) calculated from the distributed model ( - - - ) and the average-temperature approach ( ).

0 0

50

100

150

200

250

300

350

400

Time (sec)

Fig. 5. Histories of the system temperature rise ( T - To) with (1) To=723K, (2) To=743K, (3) To=763K, (4) To = 783 K, (5) To = 813 K, (6) To - 833 K.

early stage of the regeneration are known to be a result of the burning of hydrogen in coke. Increasing To not only increases the peak temperature, but also reduces the time needed to reach the maximum temperature. The system temperature starts to fall as soon as the hydrogen component is consumed completely, a long tail therefore results. By increasing the operating temperature over 760 K, the burning of the carbon component becomes fast enough to support system temperature, and a second peak temperature results. Due to the fact that the total heating value of the hydrogen component is smaller than the carbon component, a greater value of Sg'(To) is expected. Notice that as To increases, the second peak temperature becomes higher, the time required to reach the maximum temperature is again reduced, which indicates that the burning of hydrogen and carbon are very much alike.

690

H.-A. HWANGand S.-M. CH1AO

Recall that in our formulation, the hydrogen-tocarbon conversion rate ratio v/is set to be a constant value for simplicity. The value of 50 was chosen so as to provide good approximation to the results of Ramachandran et al. (Hwang et al., 1993). To further verify that the smaller peak is caused by the burning of hydrogen, a series calculations were carried out with different values of v/. The ambient temperatures employed in these computations were set to be 680 K, where local ultimate sensitivity occurs. As shown in Fig. 6, S~(To) decreases almost linearly with respect to ~/. The effect of the coke composition is presented in Fig. 7. S~(To) at To -- 680 K increases with no until a maximum occurs, in which case the coke is composed of equal amounts of hydrogen and carbon atoms. The initial increase of S'~(To) when more hydrogen appears is obviously induced by the enhanced burning. However, when we have more hydrogen than carbon, the sensitivity decreases. It is suspected that when we have hydrogen concentration increases, the oxidation rate becomes high enough to consume almost all oxygen imported in a short time. Further increase of hydrogen thus has little effect on the system temperature, the sensitivity drops. In practical operations, no is usually below 0.5, and decreases with increasing time on stream in the catalytic reactor. A seriously coked catalyst therefore possesses little low-temperature parametric sensitivity.

/o-o\

2.2

2.0

1.8

S o ( T 0 ) 1.6

0 1.4

1.2

1.0 0.0

I

I

1

I

0.5

1.0

1.5

2.0

Fig. 7. Effect of the initial c o m p o s i t i o n of the c o k e with

To = 680 K.

5.0

(3)

,~x

1.6

/ O

1.4

sOm(TO)

/

1.2

/ 1.0 0

o/

O

I

I

I

I

I

10

20

30

40

50

60

Fig. 6. Effect of the hydrogen to carbon conversion rate ratio r/with To = 680 K.

(4)

4.o

So(To) 3.0

,,\

Maximum temperature and maximum sensitivity To accomplish an efficient regeneration, the regeneration temperature is usually between 700 and 900 K, our following discussion concentrates on the peak which occurred at about 770 K only. Histories of S~(To) with different ambient temperatures are plotted in Fig. 8. Also shown in the figure are circles which mark the positions where maximum temperatures within each run occur. The maximum temperatures always appear earlier than the maximum sensi-

2.5

Initial H/C Molar Ratio (gmol/gmol)

1.0

~ 0.0

0.2

0.4

0.6

0.8

1.0

(J

Fig. 8. Histories of the temperature sensitivity So(To ) with (1) To = 743 K, (2) To = 763 K, (3) To = 770K, (4) To = 783 K, (5) To = 795 K.

tivities So.m~.x(To), which are the peak points of these curves. Before the temperature reaches its maximum, the regeneration is dominated by the burning reaction. When the energy lost through the catalyst boundary becomes more significant, the system temperature starts to fall. The phenomenon that So .... (To) comes later than Tma~indicates that the latter is more sensitive to the operating temperature. The average coke conversion at which the maximum sensitivity (solid line) and the maximum temperature (broken line) occur are presented more clearly in Fig. 9. With To varying from 720 to 800 K, the occurrence of the maximum sensitivities delayed almost linearly. When To is further increased, the curve levels up with the maximum sensitivity occurring almost at the end of the regeneration process. The maximum temperature curve behaves similarly except that the delay begins from To = 750 K, where the carbon component burns fast enough to sustain temperature increases. The plateau is observed to occur at tr = 65%. Inter-

691

Parametric sensitivity of temperature 7.0

1.0 0.9

6.0

-

0.8 0.7

5.0

0.6 c

S0(T0 ) 4.0

0.5 0.4

30

0.3 0.2

/

/

2.0

0.1

/

0.0

I 720

/

I I

740

~

I

I

760

I

I

780

I

I

800

I

I

820

1.0

I

840

T0(K)

2.0

I 4.0

t 6.0

I 8.0

I 10.

12.

Coo(8mol/rfl)

Fig. 9. Effect of the ambient temperature To on the conversions at which the maximum temperature T,,(. . . . ) and the maximum sensitivity So.... (To) ( ) occur.

Fig. 10. Effect of the ambient oxygen concentration Coo on the temperature sensitivity at the maximum temperature ST(To) ( - - - ) and the maximum sensitivity So.... (To) ( ) with To = 770 K.

estingly, the lag between these two curves remains almost unchanged for To > 750K. On the other hand, the difference between the maximum sensitivity and the sensitivity at the maximum temperature varies with To. Both So.... (To) and S~'(To) approach unity at high and low ends of the temperature spectrum. The former is caused by the saturation of the burning rates of coke (limited by the supply of oxygen), and the latter can be explained by the extinguished low reaction rate. The maximum So.... (To) is expected to occur when the diffusion and the burning rates of oxygen are equally important.

through the boundary. The latter, however, is neglected in this study. Results are presented in Fig. 11 with To = 770 K. Smaller Nu allows less heat loss through the boundary, a high catalyst temperature, and a high reaction rate result. The diffusion of oxygen becomes the rate-determining mechanism, sensitivity decreases. The situation is reversed on a fast heat-losing condition, in which case low temperature makes the regeneration system reaction-rate limited. It is therefore clear that the oxidation reaction and the diffusion of oxygen are of equivalent strength at Nu = 0.2 when To = 770 K. Moreover, the magnitude of the maximum values of So,rex(To), and also Sg'(To), are about equal in Figs 10 and 11. This similarity again verifies our conjecture that the parametric sensitive behavior is caused by the competition between the two rate-limiting mechanisms. Due to the fact that the decreasing of So.... (To) is always caused by the domination of either the diffusion-rate limitation or the reaction-rate limitation, a comparison between these two mechanisms would be interesting. By examining Figs 10 and 11 more carefully, we can see that the variation of the sensitivity is always more gentle at the reaction-rate controlling stage (low To, high Coo and high Nu) rather than the diffusion-rate controlling stage, which indicates that the latter is more crucial on So.... (To).

Effect of ambient oxyyen concentration In addition to TO, there are two other operating variables which are crucial in the regeneration process, namely, the ambient oxygen concentration (Coo) and the flow rate of the gas stream. The effect of Coo on So.... (To) is shown in Fig. 10. To is set to be 770 K, which shall produce a greater value of So(To). Similar to the temperature effect, a concave bell-like behavior is again observed. The competitive rate-determining mechanism is again the cause: at a low oxygen level, the system is under diffusion-limit condition, and the temperature sensitivity is low. When Coo increases, the restriction on oxygen supply is relaxed, more oxygen is available to the inner coke, which makes the diffusion and the reaction mechanisms equally important. The maximum So.... (To) is therefore observed at Coo = 8.6 gmol/m 3. When the oxygen level is further increased, oxidation reaction of the coke becomes the rate-determining mechanism, So.... (To) decreases consequently.

Effect of the gas velocity As stated earlier, effect of the gas velocity is studied by perturbing the film heat transfer coefficient he, and therefore, the Nusselt number. In reality, the gas velocity affects both the heat and mass transfer rates

Coke-dependent effective diffusivity Until now, a constant diffusivity has been used in our simulation. However, reductions of the effective diffusivity are logically assumed to become significant as a higher level of coke deposits itself and blocks the pores of the catalyst pellet (Hwang et al., 1993). Experimental proof has been provided by Suga et al. (1967) and Richardson (1972). Equation (8) is used in our formulation to include this effect. We chose to use the same coke-free effective diffusivity in our calculation such that the initial mobility of the oxygen

692

H.-A. HWANGand S.-M. CmAO 7.0 6.0 5.0

So(To) 4.0 3.0 2.0 / 1.0

J 0.1

I 0.2

a

I 0.3

I

I 0.4

i 0.5

Nu Fig. 11. Effect of the Nusselt number on the temperature sensitivityat the maximum temperature S'~(To)(- - - -) and the maximum sensitivity S o.... (To) ( ) with To = 770 K. decreases. With the same operating conditions (To = 770 K, Coo = 3 gmol/m3), the ambient temperature sensitivity decreases from 4.3 with P = 0 to 2.3 with P = 2. Moreover, So.... (To)s no longer occur at their original conversions, but, rather, shift to lower conversions where high mass transfer rate occurs. It is recognized from Fig. 7 that the elevated So,max(To) usually occur when the conversion of coke is between 0.6 to 0.8. The effective diffusivity at this time with P = 2 is only 50 to 70% of Deo. Smaller oxygen supply has shifted this system from a reaction-diffusion balanced condition to approach diffusion limitation, So. . . . (To) therefore decreases. Since the use of a coke-dependent diffusivity decreases the initial oxygen supply, it is possible to restore the sensitivity with a compensation from other operating conditions. For example, a higher ambient oxygen concentration may shift the regeneration process out of the territory of diffusion-rate controlling mechanism. At To = 770 K, the oxygen concentration at which So,max(To) occurs increases almost linearly from 8.6 to 13gmol/m 3 when P rises from 0 to 2. Sensitivity with respect to other parameters From the above discussions, it is clear that the temperature sensitivity changes with any parameters which can move the system between the diffusiondominating and reaction-dominating situations. In addition to So(To), we are also interested in the behavior of So(Coo) and So(Nu). Oxygen concentration in the gas stream is well known to be one of the most important operating variables in the regeneration process (Byrne et al., 1989), while So(Nu) was calculated in light of the system sensitivity with respect to the gas flow rate. Effect of To on So.m~,,~(Coo) are similar for So(Coo) and So(Nu), which resemble that of So(To). Smaller values are observed at both ends of the temperature spectrum, and are expected to approach unity. The maximum of So,max(Coo) and So(Nu) occur, again, when we have equivalent con-

sumption and supplying rates of oxygen. Moreover, the temperatures at which both the maximum So. . . . (Coo) and the maximum So. . . . (Nu) occur are almost the same (780 K), and is slightly higher than the temperature at which the maximum So. . . . (To) appears. More dramatically, effects of Coo and Nu on these two sensitivities are also the same. Instead of Coo = 8.6 gmol/m 3 and Nu = 2.0, where we have the maximum So.... (To) (Figs l0 and l l), the ultimate value of So. . . . (Coo) and So. . . . (Nu) occur at a Coo well above 10 gmol/m 3 and a Nu below 1.0. This similarity in behavior will be discussed in the following section. The sensitivity of system temperature with respect to initial coke content So(Co0) was also calculated. Without considering variable diffusivities, the extra amount of coke increases reaction rate and thus shifts the system towards diffusion-control. When cokedependent diffusivity was involved, the mobility of oxygen is further decreased due to serious plugging. Diffusion limitation, therefore, dominates even earlier. Notice that the initial coke content is not determined by any of the operating variables of the regenerator. In fact, extent of coking usually results from the operation of the catalytic reactor before regeneration. The main reason to include So(Coo) is to support our argument about the similarity in properties. No further discussion will be proceeded. Comparison among So(To), So (Coo) and So (Nu) In their generalized criterion for the parametric sensitivity, Morbidelti and Varma (1988) pointed out that the maximum temperature becomes simultaneously sensitive to small changes of any of the model inputs. Their conclusion was further extended and combined with the self-similarity relation in thermal explosion theory by Vajda and Rabitz (1992). Similar but not exactly the same results are observed in our simulation. Different So. . . . are plotted against To in Fig. 12 on a normalized scale. The maximum values of

So

740

750

760

770

780

790

800

T0(K) Fig. 12. Comparison among four different temperature sensitivitiesSo.... (To), S o.... (Coo), So.m,,x(Cco), and So. . . . (Nu) on a normalized scale.

693

Parametric sensitivity of temperature So. . . . (Coo) and So. . . . (Nu) are observed to occur at the same ambient temperature, So. . . . (To) and So. . . . (Coo), however, start to decrease early at lower temperatures. It is realized that both Coo and Nu affect the system temperature through the catalyst boundary, Variation on Coo, on the other hand, changes the initial reactivity uniformly through the entire catalyst pellet. It is expected different sensitivity behavior may be observed for either case. Most interestingly, perturbation of To not only changes the boundary condition, but also modifies the initial condition. As a resuit, the maximum of So. . . . (To) appears at a temperature between that of So. . . . (Coo) [or So. . . . (Nu)] and So.~a~(Cco). It may therefore be concluded that the parameter sensitivities posses similarity to those parameters which affect the system behavior through the same geometric paths, and may be different for other parameters which affect the system behavior through different schemes.

r S Sh T t X

CONCLUSION The parameter sensitivity behavior of the regeneration of a single coked catalyst pellet was studied. Three different operating variables were discussed, namely, the initial/ambient temperature, the ambient oxygen concentration and the velocity of the gas stream, which is represented by the Nusselt number. Results of our simulation show that the maximum sensitivity within a single run always appears after the maximum temperature. The system sensitivity maximizes when the diffusion and the reaction mechanisms are competitively equivalent in strength. The system sensitivities with respect to the ambient oxygen sensitivity and the gas flow rate sensitivity behave similarly since both of them affect the system temperature through the boundary. When the initial coke content of the catalyst is perturbed, however, the temperature sensitivity maximizes at a different location on the map of the operating variables. Moreover, changing both the initial and boundary conditions, such as To in this study, the maximum sensitivity locates in between the above two cases.

Subscript 0 c max min o s 0

Acknowledgement Suggestions from Dr W. C. Yu and Dr S. Y. Ju are appreciated. This research was funded by the National Science Council of the Republic of China under contract No. NSC 82-0402-E-029-008. NOTATION C Cp De E AH ko Ke Le Nu n R R.

concentration heat capacity effective diffusivity Arrhenius activation energy heat of reaction reaction rate constant thermal conductivity Lewis number Nusselt number hydrogen to carbon atomic ratio in coke radius of the catalyst pellet gas constant

reaction rate sensitivity Sherwood number temperature time dimensionless concentration

Greek letters thermicity factor 13 porosity coke-free porosity e* dimensionless radius ( dimensionless temperature 0 dimensionless diffusivity 2 density p dimensionless time r Thiele modules ~2 dimensionless Arrhenius equation operating variable for sensitivity analysis ~i

initial conditions carbon maximum minimum oxygen catalyst support system temperature REFERENCES

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