commercial refmiig operations where tixed Inbedmanycatalytic processes are used, such as in catalytic reforming, there is a loss in catalyst activity di to a build-up of a hydrocarbon polymer on the cataly su*ace. Normally, the simplest way tor_egenerate the catalyst is to burn the deposit (usually termed “coke” or “carbon”) from the catalyst, I n many cases, excessive high temperatures which develop during burning CI damage the catalyst or equipment. A mathematical model of the carbon burning process would be a valuable tool in studying catalyst regener tion, serving as a basis for :
-Detmnining conditions which lead to harmful tempcratwes. -Deoeloping the apfarmt best regenerationprocedures. -Minimizing the amount of experimmtnl work ncedcd to J out the regenerationpicture. --Correlation and interpretation of pilot unit andplant operatior The purpose of this article is to illustrate how such model can be developed. Mathematics, physical cher istry, and engineering principles can be combined to s up general equations for this particular problem, to soli the equations, and to interpret the results meaningfull The mathematical model described below was four to simulate a process of this type to a satisfactory degre It improves on previous work as fewer simplifying a sumptions are used. T o fit other regeneration procesa or where more accuracy of prediction is needed, it tlli be necessary to include the effects ofstill other variablc These effects can be included in the general equation Of course, complexity of the model and computer tin needed for the solution of the equations will then be i creased.
3
building mathematical model
Pod Work Based On Over-Simplified Approoches
As far as is known, the fmt analysis of the temperatu runaway problem was made in 1937 by W. I. Thompsc (3) in considering the regenmtion of a fixed-bed cat, lytic cracker. Later studies by Van D o m t e r (4) ar Johnson (2) provided further information on t h i s combustion process. In all three of the cases cited, simplificationswere ma< in order to obtain analytical solutions and the effect more realistic assumptions was left unexplored. Tiapproach has the definite advantage of setting limits ar presenting an analytical solution to the problem. T1 lack of definition of the effects of the assumptions infinite reaction rates and infinite bed lengths leav doubt regarding the value of the solutions. Predicting the theoretical adiabatic maximum temperature does not describe the whole process. Kinetics must be included to answer such questions as: -How nitical are heat losses? -How much of the carbon is burncd off the catalyst? -How can the burning ~ I O C Gbe S Scontrolled so as to obtnin a satisfactory regeneration’ -What inpU... does low reaction rate have on the regeneration process? 44
INDUSTRIAL A N D ENGINEERING CHEMISTRY
6
With the availability of high speed computers, it is possible to set up as general a regeneration model as desired and obtain numerical solutions, maintaining desirable standards of rigor and judgment so as to answer the above types of question. Assuming that there was no heat conduction radially or longitudinally, and that the reaction velocity constant for carbon and oxygen was infinite-i.e., assuming a sharp burning front-Thompson showed that for an infinitely long bed, the maximum temperature rise could be expressed by:
where OB is the time needed to burn the carbon from the bed if oxygen is completely consumed :
P&OM 12UPSO and 8, is the time for the heat front to move the length of the bed:
ea
=
__
Thompson showed that three cases existed for a bed:
usB> e, If 8g
< 8,
If 0, = 8,
T,, T,, T,,
= =
+
T , , ~ AT,, T.,o - ATmm
+ rn
In the latter case, an instability occurs-the temperature in a narrow zone is continually increased by the combustion reaction in the same zone. A key point in this analysis is that the unstable situation can occur even when the amount of carbon on catalyst is low provided the oxygen content of the inlet gas is low also. However, actual kinetics and the actual length of the bed can limit the maximum temperature. A later study by Van Doemter (4) followed a similar line of reasoning. By making rigid assumptions as to the burning rates, he showed the effects of the movement of the heat and carbon fronts. while the existence of high temperatures was predicted, the burning rate assumptions used did not lead to the same temperature rise given by Thompson. Johnson (2) derived approximate solutions to more rigorous differential equations of burning and confirmed that Equation 1 is \ alid and does give the maximum possible temperature; but the kinetics of the system can greatly reduce the actiial temperature. Regenemlion Model kt Up To Include Kindcs and Heat Losses
In order to keep the initial approach to the problem relatively simple, the following assumptions were made: -The temperature in the bed is uniform along the radius and there is no longitudinal conduction of heat. -No temperature gradients exist in the catalyst particle. 4 a s flow is uniform and pressure drop small. -The increase in gas weight due to pickup of carbon from combustion can be neglected. VOL 5 5
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DECEMBER 1963
45
P
-Good heat transfer exists between the solid and gas, so that they are at the same temperature. Initial studies incorporating a separate differential equation for gas temperature showed that the maximum temperature difference encountered was 3 F. Simpler calculations by Johnson confirmed that the difference is usually small. -The weight of the gas contained in the voids is small compared to the weight of the catalyst in the bed. -The carbon burning rate is proportional to the fraction of carbon and the oxygen partial pressure. -Combustion produces only CO2-no C O or H20 is formed. This assumption is made in order to set up the equations. Production of CO and HzO can be handled as shown later. -The reaction velocity constant follows an Arrhenius function of temperature. With these assumptions, the partial differential equations for temperature, oxygen, and carbon become :
Thus, Equation 11 gives an exact solution to the initial oxygen distribution and suggests that a general form of oxygen disappearance over any segment of bed, where carbon and temperature do not vary appreciably, can be calculated as an exponential decay. This exponential decay is far superior to the usual straight line extrapolation inasmuch as it can provide an exact solution and, even under extremely high values of k , does not allow calculated values of oxygen to become negative. T o change the differential equations to the best finite difference form, it is necessary to relate the time increment, AO, to the length increment, AZ, chosen:
With this relation, Equations 2-5 can be written :
+
T ~ , sA@
=
+
T n - l , ~ a1 ( k x ~ ) , ,-~
(3) (4) Ink = F -
G
+ 460
___-
T
(5)
where : n = 1 , 2 , ...-I’
For this study the boundary conditions chosen were: At 6 = 0
T
=
T,,o for all values of 2
y = yo for all values of 2 At 2 = 0 T = T,,o for all values of 0 x = x o for all values of 6
(6)
(7)
so that
B . L . Schulman is Senior Chemical Engineer in the Process Research Division, Esso Research and Engineering Co., Linden, A’. J . I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
~
a2
= pshfPL/12 pQuAY
a3
=
014
= 4 h L , ~~poC3A17d
(9)
Initial attempts to handle the numerical solutions by conventional finite difference procedures resulted in excessive amounts of computer time to control truncation errors at high values of k . This occurred when assuming that the rate over a small interval of time and length remained constant. Too small an increment of time and length was needed to be practical. A technique called partial integration was developed to overcome this. Referring to Equation 3, at 0 = 0, the value of carbon on catalyst, y , and catalyst temperature are uniform. and k = ko, Equation 3 can Hence, fory = yo, T = TS,o, be integrated directly to give:
46
A H p 8C,PL _ _ UP,COAI‘
(8)
Numerical Solutions Speeded Up By Use Of “Partial Integration”
AUTHOR
a1 =
PAS
The general computational scheme becomes :
(1) Select operating and initial conditions. ( 2 ) Calculate AO, a1, a2, as, and ad. (3) Set the values ofy and T for each bed increment to the initial values and set time equal to zero. (4) Compute the oxygen values at time 0 for each bed when n = 1. increment by Equation 15 using xn-l = (5) Having the oxygen distribution down the bed at the time 0, calculate the temperature and carbon profiles Ae) by Equations 13 and 14, using Tn-l = at time (6 T,,o when n = 1. (6) Having the new profile of temperature and carbon, go back to step 4 and repeat for as long as required. The above computational scheme proved to be easily handled on high speed computers. Using a variable number of subdivisions, it was found that usually 20length subdivisions were sufficient to obtain convergence of solutions-material balance on carbon was in the range of 99.0-101.0~0 and heat balances were in the range 98-1027G. At extreme conditions-either high levels of oxygen removal or temperature rises above 1000° F.-the nurnber of subdivisions would increase to 100-200 to give good heat and material balances. T h e
+
I
1050 I
T
1
1000
i
I
W
I
R
Simple Correlation Predicts Critical Effects in Catalyst Regeneration
Since the key problem in regeneration is to prevent damage to the catalyst, it is desirable to have a simple or rule-of-thumb way to estimate the peak temperatures occurring in the catalyst bed a t any point. Numerical solutions were obtained for commercial and laboratory units covering the following range of conditions :
20% 10% BED LENGTH
750
I
~
0
'
I
TM I E, HOURS
2
xg
yo pa
C,
C,
0.07 0.066 = 55.2 Ib./cu.ft. = 0.25 = 0.25 = =
I
L = 7.5f t . u = 2700 ft./hr. pQ = 0.272 lb./Cu.ft. P = 75p.s.i.a. M = 30 ko = 0.703
computer time needed for such calculations was quite reasonable. For a typical calculation using 20-length subdivisions, a n IBM 650 required 30 minutes per case, a n IBM 704 required about 15 seconds. Even the extreme cases required only 15-20 minutes of IBM 704 time. Only a few such cases were needed to confirm the aspects of the model. Numerical Solutions Tie In With limiting Solutions
4
Inlet oxygen Carbon on catalyst Gas velocity Pressure Calculated temp. rise
3
Figure 7 . In adiabatic regeneration, sharp temperature fronts develop. For these curves, variables had the following values: Te,o = 780' E. T,,O = 780' E.
carbon, assuming 100% utilization of the oxygen, is approximately 30 hours so that a t temperature breakthrough the regeneration is only 6% complete.
Numerical solutions were obtained for the adiabatic cases of a commercial catalytic reformcr using reaction velocity constants estimated from initial oxygen breakthroughs. Literature data ( I , 2) and plant experience have shown differing levels of reaction constants, therefore a study of the effect of this parameter was included in the solutions. Typical temperature patterns for a catalytic reformer resulting from the solutions are shown in Figure 1. T h e sharp temperature rises are typical of the "heat fronts" which move down the bed. The temperatures reach a maximum at a given bed position and slowly drop as the carbon burning rate drops. T h e maximum temperature of the front continually rises as the front moves down the bed since heat generated by the reaction continually adds to the temperature of the gas coming into the reaction zone. For this particular case, the maximum theoretical temperature rise calculated from Equation 1 is 290' F., while the maximum temperature rise calculated for this case was 260' F. This represents about 90% approach to the theoretical maximum. The calculations point out that the temperature front breaks through the end of the bed a t a time OH, about 1.8 hours for the case shown, well ahead of complete combustion of the carbon. T h e amount of time to burn all the
0.5-5.0 4-8 0.5-10 0-GO 200-3OO'
mole yo wt.% ft./sec. p.s.i.g. F.
From these calculated runs it was possible to develop a satisfactory generalized correlation for peak temperatures. A parameter, +, can be defined as the ratio of the peak temperature rise attained in calculations for the particular case to the maximum theoretical temperature rise possible :
A dimensionless reaction velocity group, y , can be defined as: 7 = koxo pes ( z / L ) (22) Then 1c, becomes a function of y and the energy of activation of the burning reaction. This +, y relation derived from the various numerical solutions is shown in Figure 2. T o calculate the peak temperature rise from Figure 2, it is necessary to obtain the initial burning rate constant -i.e., ko at Tg30--and the energy of activation either from experimental or literature data. With this information and the operating conditions, y and ATm5, can be calculated from Equations 22 and 1. From Figure 2 the Figure 2 . For rapid reaction rates, peak temperatures approach the theoretical maximum value
r
-
0.8
-
ENERGY
/
"
OF ACTIVATION,
B.T.U./LB. M O L E OR.
7
CrE I/ Y
0
0.5
VOL. 5 5
1.o
1.5
2.0
NO. 1 2 D E C E M B E R 1 9 6 3
2 >
47
predicted by Equation 1. A fairly good fit of the data can be obtained if an average value of 0.8 is used for $ in Equation 21, as suggcsted above. (See Figure 3.)
L
Heat Losses Are Critical To Regeneration Simulation I
Predictions of a detailed temperature history for the regeneration of a commercial catalytic reformer showed ACTUAL the severe effect of heat losses on the temperature pat~Tmr tern. The actual temperature history of a regeneration is shown in Figure 4, with the predicted profiles. It is evident that the adiabatic condition does not represent the actual case in two important respects: the actual temperature does not develop as quickly or rise as high as the prediction. Since the time at which the front develops is determined by a heat balance, it is evident that the actual bed is far from adiabatic. If the temperaL L - ~ture profiles are calculated using an estimated external 200 400 600 MAXIMUM THEORETICAL TEMPERATURE RISE, ATmox overall heat transfer coefficient of 12 B.t.u./hr. sq. ft. F., the initial portion of the regeneration cycle is Figure 3. Simple correlation checks pilot unit data predicted well. The effect of the heat loss, equivalent to only about 15y0 of the heat generated, is to lower the value of $ can be obtained and the peak temperature rise temperature of the bed ahead of the burning frontIf no kinetic data are calculated from Equation 21. often below the initial temperature. This heat loss available, a value of = 0.8 is a useful approximation. delays the buildup of the heat front and lowers the peak If adiabatic regeneration data are available, Figure 2 temperature attained. This sharp response of the temcan be used to evaluate an approximate value of the rate perature profiles to heat losses makes the inclusion of their constant, ko. The observed peak temperature rise, expressed as +, can be plotted against the term xOPOB(Z/L). effects mandatory if a detailed simulation of regeneration is desired. This point was also noted by Johnson in Since $ is not a large function of the energy of activation, examining his pilot unit data, where it was known that a value is assumed (usually 5,000-10,000 B.t.u./lb. the unit was nonadiabatic. R.) and the corresponding values of y selected mol. The ability of the model to predict the plant data at experimental values of t,b. The values of ko can then gives confidence in use of the model for extreme conhe calculated and a suitable mean value computed : ditions if the basic data needed are available. Broadly, the model predicts the commercial case well and ties in with the solutions developed for extreme cases, such as high reaction rates. Hence, more complexities of reIt is evident from Figure 2 that for y > 3, the detailed generation behavior, such as cracking of the carbon model is very close to the simpler model of Thompson. Thus, for y > 3, the oxygen front becomes very steep and Figure 4. Heat losses must be included for accurate prediction of the assumption of an infinite reaction rate, or a sharp temperatures. Conditions arc the same as for Figure 7 oxygen-carbon interface, becomes a good one. This blending of the two cases is valuable since for the case of high reaction rates, the time for machine computation C-ALCULATED ADIABATIC CASE becomes very high. The model actually reduces to a simpler form requiring only a slide rule for evaluation. 4cc -
~
-.I
+
1
Pilot Unit Regeneration Data Well Handled
by Simple Correlation
Studies of regenerations in small laboratory fixed beds, roughly adiabatic, showed that the simple correlation given by Equation 21 can match experimental data. Regeneration conditions covering a range of 0.5-5.0 mol.yo oxygen in inlet gas and 0.1-1.5 wt.% carbon on catalyst produced maximum temperature rises up to 410 F. Carbon-hydrogen balances showed the original deposit to have approximately one atom of carbon to one atom of hydrogen. Predictions of the theoretical maximum temperature rise from Equation 1, modified for the hydrogen content of the coke (as described later), show that the actual peak temperature rises are close to, but usually less than, those 48
INDUSTRIAL A N D ENGINEERING CHEMISTRY
I/
CALCULATED NON-ADIABATIC CASE (h=12 B.t.u./hr. ft.2 OF.)
800
L 2 4
0
I
I
6
8
TIME, HOURS
1
J
*
+
deposit, can be added to the numerical model if desired. Even without the extensive elaborations, the most important aspects of regeneration behavior can be derived.
fraction of hydrocarbon (C H) on the catalyst. For example, if equal molar ratios of CO2 and CO are produced, a general stoichiometric equation may be written :
Model Shows Way To Optimum Method of Regeneration
2CnHm
Using the concepts of the regeneration model it is possible to predict directly the effects of many of the operating variables on regeneration. Such predictions are particularly valuable when data on actual regenerations are available, and improvement is desired. T h e concept of a maximum temperature rise immediately suggests a method of limiting bed temperatures. For the usual case of low inlet oxygen concentrations, where OB > OH, a simple and effective way to keep bed temperatures to a limit is to set the inlet gas temperature to a value equal to the maximum catalyst temperature level desired minus the expected maximum temperature rise. I n this way, the temperature range of the bed is set and can be controlled easily by inlet oxygen. Control of bed temperatures is easily achieved by control of oxygen. At low oxygen rate the ratio OB/OH is much greater than one. Equation 1 can be reduced to: (23)
I n this case, the maximum temperature rise is directly proportional to the oxygen concentration and is independent of the initial carbon level. If the temperature level begins to rise above that desired, the oxygen can be scaled back accordingly. Adding a diluent of high specific heat, such as metal shot, to a bed has often been suggested as a method of lowering the temperature rise. For the base case where OB > OH, this method is incorrect and can cause a n increased temperature rise. Adding diluent in effect increases OH since the diluent serves to add heat capacity to the bed. Rearranging Equation 1 to give: Tmaz
- T,?O =
AHYOOH/C* OB
- OH
AHyOPs UPocg(eB
- 6,)
(24)
shows that as OH and p s are increased, holding other conditions constant, temperature rise will increase. I n order to keep the temperature from rising it is desirable to remove the heat from the bed-adding the diluent keeps the heat in, and leads to higher temperatures. Increasing gas flow rate at constant carbon and oxygen levels tends to lower maximum temperatures depending on reaction kinetics-AT,,, is unchanged with higher gas rates, but y decreases. Depending on the level of 7, the temperature change may be slight. Effects of other operating variables can be deduced similarly. Model Handles Various Types Of Stoichiometry
Various types of stoichiometry of the carbon-oxygen reaction can be handled via the general Equations 2-5 or the simplified correlations in Equations 21-22. While the numerical model assumed that the reaction 0 2 + COZ, the effects of C O and occurring was C HzO formation can be handled by the appropriate recalculation of OB and A H , and use of y o as the weight
+
+ 3n +2 m ~
nCO2
0 2 -+
+ nCO + m H z O
(24)
If n and rn are known, AHcan be calculated as
AH =
n(AHco,
+
+
AHGO) mAHH,o 2(12n 2m)
+
(25)
where AHco,, AH,,, and A H H ~ oare the molar heats of formation of COz, CO, and H20, respectively. T h e burning time, OB, becomes : OB=-
+ +
pSLyo
4M(12n 2m) (3n m)
UPOX0
Summing Up-Generalized
(26)
Model Explains Behavior
A general approach to the problem of fixed bed catalyst regeneration has been shown to be feasible and worthwhile. When coupled with the use of high speed computer, the general burning model developed can explain and predict experimental data. -The use of a computer to handle the partial dzyerential equations is economical and serves to tie in the general model with previously simfil$ed analytical solutions. -The results of the numerical solutions can be correlated to provide a simple basis for predicting operating effects on the peak temperature rise. -Good simulation of regeneration temperature patterns requires good knowledge of heat loss effects, reaction kinetics, and the heat conductivity of catalyst pills at extreme cases. -The numerical methods used are general, therefore the model can be extended to include such additional ejects as heat conduction (both axial and radial), cracking, or str$ping of the hydrocarbon from the catalyst. NOM ENCLATU RE = specific heat of gas = specific heat of catalyst = bed diameter = fraction voids in bed = constant for reaction velocity constant k defined in Equations 5 and 15 G = constant for reaction velocity constant k defined in Equations 5 and 15 and equal to the energy of activation divided by the universal gas con: stant h = external heat transfer coefficient for loss to surroundings AH = heat of burning of carbon k = reaction velocity constant for C f 0 2 + COz M = gas molecular weight N = number of length subdivisions P = system pressure T = temperature u = superficial gas velocity x = mole fraction of Oe in gas = weight fraction of carbon on catalyst %/L = variable and total bed length O = time = time for complete burning of carbon from catalyst, assuming complete OB utilization of inlet oxygen. OH = time for temperature front to move through the bed, defined as C, CS
d E F
OH
= p&Ca uPQcO
0. pa I
= gas density = catalyst bulk density = temperature of bed surroundings
Subscripts s = refers to solids g = refers to gas = refers to inlet or initial conditions o n = index for position in bed
BIBLIOGRAPHY (1) Dart, J. C., Savage, R. T., Kirkbridge, C. G., Chcm. Eng. Prog. 45, 102 (1949). (2) Johnson, B. M “Temperature Profiles I n Packed Beds Of Catalyst During Regeneration,” P2.D. thesis, University of Wisconsin (1956). (3) Thompson, W I “Calculation of Flame Front Temperatures in Catalytic Cracking Unit D&g Regeneration,” Memorandum, Aug. 13, 1937. ( 4 ) Van Doemter, J. J., IND.END.CHEM.45,1227 (1953).
VOL
55
NO. 1 2
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49
