J. J. CARBERRY
DONALD WHITE
On the Role of Transport Phenomena in Catalytic Reactor Behavior Digital Simulation of Naphthalene Oxidation Over V2 0 5 A stea&-state behavior model of ajxed-bed catalytic reactor demonstrates parametric sensitivity and selectivity in naphthalene oxidation upposedly, the behavior of a heterogeneous cataS lytic reactor can be simulated when given reasonably precise chemical kinetic and transport coefficient data properly incorporated into the continuity equations governing the system. I n the general case of the nonisothermal, nonadiabatic reactor environment, analytical solution isn’t possible ; however, advances in numerical analyses and the availability of high speed digital computers invite the hope that unambiguous simulation and, therefore, reactor performance predictions may be realized. Indeed, recent work of Hawthorn and colleagues (73) in which model predictions and experimental reactor data are very favorably compared for a n endothermic catalytic reaction suggests that bases for a priori prediction are a t hand. A more severe test of our simulation powers might be provided by a consideration of a highly exothermic catalytic reaction, involving yield of a n intermediate. Further, transport of heat and mass should affect performance in all regimes of import. The catalytic oxidation of naphthalene over Vz05 in which the desired intermediate, phthalic anhydride, is susceptible to further oxidation to undesired C O Z and HzO admirably fulfills the cited characteristics. I t is a highly exothermic reaction network of such velocity that heat and mass transfer limitations prevail both throughout the reactor (radial interparticle dispersion of heat and mass), about and within the porous catalyst pellets (inter-intraphase diffusion of heat and mass), and, of course, heat transport through the reactor wall to the coolant sink. Earlier studies of nonisothermal, nonadiabatic reactor simulation have focused attention upon the numerical techniques (22), while generally ignoring inter-intraphase transport phenomena. Wendel and Carberry (25) considered these intrusions as they affect yield in a consecutive reaction carried out in a n adiabatic environment, where, of coursej radial interparticle gradients do not exist. Beek ( 7 ) reviews the general problem of packed-bed reactor behavior while a complete though lengthy solution of conversion (not yield) in a nonisothermal, nonadiabatic fixed bed operating in both steady-state and transient conditions is provided by McGuire and Lapidus (75). They employed the mixing cell concept (each void is a CSTR in both radial and axial directions) and inter-intraphase dif-
fusion is anticipated. As a n integral fixed-bed reactor may be characterized by some several hundred axial mixing calls (bed length to pellet diameter ratio), computations rooted in the cell mixing model can be rather time consuming. Froment (70)has recently addressed himself to the comparison of the one-dimensional fixed-bed model (radial transport resistances are lumped a t the wall) and the two-dimensional model which accounts for both radial and axial transport of heat and mass. Xylene oxidation (a yield problem) was considered by Froment; however, inter-intraphase resistances were not incorporated into his model. Parametric sensitivity was found by Froment insofar as his solutions revealed the sensitivity of profiles and overall performance to small variations in the heat transfer coefficient a t the wall of the fixed bed. Froment’s results suggest that the onedimensional model can yield deceptive results in contrast to those revealed by the more realistic two-dimensional model. O u r work was designed to explore a number of reactor parameters to gain insight into those factors which affect reactor performance, and to determine precisely how uncertainties in transport coefficient data and operating variables affect predicted performance. By choosing a n exothermic yield-sensitive reaction as conducted in a steady-state nonisothermal, nonadiabatic fixed-bed environment, the influences of interparticle and inter-intraphase transport of heat and mass could be explored insofar as these events affect both conversion of naphthalene and yield of phthalic anhydride. A complimentary study of adiabatic SO2 oxidation simulation as catalyzed by supported Pt and, also, V Z Orevealed ~ telling effects of inter-intraphase diffusion upon space-time-yield of SOX (77). Kinetics of Naphthalene Oxidation
DeMaria and associates (9), in their report on simulation of fluidized bed isothermal oxidation of naphthalene, present kinetic data for the reaction network visualized to be of the following simplified character (in excess 0 2 ) : ki
Naphthalene \zk
+- Naphthequinone ka&
k4
(1)
Phthalic anhydride --t COz, HzO VOL. 6 1
NO. 7 J U L Y 1 9 6 9
27
TABLE I .
NAPHTHALENE OXIDATION KINETICS
Temp., kl=kz 0.25 3.5
ka
t
KJKz
320 370
0.4 12
0.0095 0.035
50 200
., .
38
50
'C
Catalyst A Activation energy, (kcal/mol) CatalystB Activation energy, (kcal/mol)
320 370
...
0.11 0.42 20
20
0.45 1 12
0.0016 0.03 43
bed length to pellet diameter; m = r / d p , radial distance to pellet diameter and (Pe), = dPU - = radial Peclet number for mass Dr
d,u = radial Peclet number for heat (Pe), = -
...
K
140 28
...
Values of the rate coefficients a t 320" and 370°C are reported for two catalysts ( A , B ) , by DeMaria et al. ( 9 ) )and in Table I. As k l = k:! a t both temperatures and k3 > k l , then as suggested by Westerterp (26)) the above network might be further simplified to
while 8 0 = ZO/U; and pCp is the volumetric heat capacity of the gas stream. Term a represents the axial gradient, while b is the radial gradient, and c the generation function. D , and K are the radial dispersion coefficients for mass and heat, respectively. Axial dispersion is neglected in the continuity equations since bed length to pellet diameter ratio is so great that plug flow residence time can be anticipated (4, 25). a, the species generation term is, for mass transfer to a particle of external surface to volume ratio, a ,
Kz
K1
Yaphthalene + phthalic anhydride + COz, H2O (2) where K1 and K2 are reported in Table I. Activation energies, estimated from the rate coefficients provided a t the two temperatures, are also in Table I. Kinetic studies reported by DeMaria (for small, fluidizable catalysts) suggest all reactions to be pseudo-first order in naphthalene and anhydride and zero order in oxygen (always present in vast stoichiometric excess). Inspection of the data in Table I suggests that catalyst A (a V205 formulation) is preferred over catalyst B (also a VzOb formulation) insofar as an expected teniperature increase over feed conditions would increase K1/K2 for catalyst A to the benefit of anhydride yield, while K1/K2 will decrease with increasing temperature for catalyst B. This point has been discussed elsewhere ( 3 ) . If the industrial reactor could be maintained isothermal, at 320°C) then catalyst B would provide the better yield. While a fluidized bed might seem to promise such as isothermal environment, Westerterp's (26) stability analyses actually suggest that an isothermal fluidized bed of catalyst B can be dangerously unstable. A seemingly attractive alternative is the fixedbed network (a multiplicity of packed tubes) filled with catalyst A . Apparently, excursions from isothermality can then be expected to increase K1/K2 to the benefit of anhydride yield. In fact, the expected benefit is partially nullified by reason of inter-intraphase diffusional instrusions.
CR
= ?&Cs
= k g a ( C g - C,)
=
4
- a, -
Z=L
for heat
(-AH)CR
=
h a ( T , - Tg)= - A
(4)
9 1 -
l=L
where k , and h are the interphase mass and heat transport coefficients, assumed to be defined by (6).
a relationship rooted in a developing boundary layer concept ( 7 9 ) ) which correlates a vast body of interphase transport coefficient data for flow of fluids through fixed beds. The coefficients h and k , are average values in contrast to the detailed reality which reveals considerable variation of these coefficients with position about a particle immersed in a fixed bed (72). Intraphase diffusion is taken into account through the catalytic effectiveness factor, 7 , due to Thiele (27). Q represents the ratio of the diffusion-affected catalytic reaction to that which would prevail in the absence of intraphase concentration-temperature gradients. For an isothermal pellet, of characteristic dimension, L, we have tanh cp 17=-
4;
P
0-
where
p =
L
for first-order reaction
The Reactor Model
The governing continuity equations for the steadystate cylindrical fixed-bed reactor are,
(a) =
at
(6)
+
(3)
(c>
0oCR ( -A H )
pCPT o n
~~~
~
AUTHORS J . J . Carberry is Professor of Chemical Engineering
for heat
where f = C/C,, the reduced naphthalene concentration; t = T/To, the reduced gas temperature; n = Z / d p , 28
Nonisothermality within a porous catalyst complicates description of 17 (7, 20, 23). However, for a gas-porous solid system, nature proves to be considerate. An inspection of the mass and thermal Biot numbers (derived from Equation 4) defined as
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
at the University of Notre D a m e . Donald W h i t e , a graduate student in Chemical Engineering at the University of Rochester, received fiis M.S. at hrotre D a m e . T h i s paper wasjrstpresented at the 752nd meeting, ACS, AVew York City, September 1966.
k,L
- interphase
a
intraphase
(Biot), = - -
transport
hL
(Biot)heat = X
and indicates (3) that
h , is the local wall heat transfer coefficient, given by the
(Bi), =
-c (Biot)massS c,
co
(BiJ = (Biot)heat
- cs
(7)
Nu =
T, - T To - T ,
is the intraphase diffusivity and X the pellet thermal conductivity. A comparison of the Biot numbers reveals that for the gas-porous solid system, as first noted by Hutchings ( 7 4 , (Bi), >> (Bi)* a finding confirmed by experimental measurements of Fulton and Crosser ( 7 7 ) and Maymo and Smith (76). I n consequence, one can expect the major concentration gradient to be within (intraphase) the porous catalyst, while the principal seat of thermal resistance will be in the “film” surrounding the catalyst pellet. As a simplifying assumption, then, we can treat the catalyst as isothermal a t a temperature dictated by heat transfer in the boundary layers bathing the pellet. Thus q can be simply computed using the rate coefficient value determined by the pellet surface temperature, T,. While the above assumption is rooted not in a universal, but a generalization, some specific support is provided by computations. Catalytic effectiveness under nonisothermal intraphase conditions 71 can be analytically described relative to the isothermal value 7 0 for Thiele moduli greater than about 3 by expression ( 7 ) . 71/qo = exp
[;I
(-A”DCo(&)
where
correlation (27)
(8)
CY=
To
At feed conditions of ‘/2% naphthalene, assuming a we compute pellet conductivity of 1O-* cal/crn--’C a value of 71/70 of 1.05; hardly a significant influence of internal temperature gradient upon effectiveness. Further in a typical run (Re = 360, d , = ‘/zcm., u = 100 cm/sec) the mass Biot number is about 25 compared to the thermal Biot value of about 4. Finally, in this highly demanding exothermic reaction, catalytic effectiveness assumes such small values at critical points in the reactor that reaction is virtually confined to the external surface of the catalyst pellet so that the assumption of pellet isothermality at a surface temperature governed by interphase transport seems most reasonable. Equations 3 and 4, in which Equations 5 and 6 are incorporated would then describe conversion of naphthalene, and gas-solids temperature profiles in the radial and axial directions in the fixed bed, subject to the boundary conditions : Bed entrance z=o f =1 (9) t = l Centerline df/dm, dt/dm = 0 Reactor wall
h,d,= K,,,
C (Re)0.8
(11)
I n Equation 10 h,d,/pC,K is also a Biot number, expressing the ratio of thermal transport at the reactor wall to that in the core of the bed. A large value of this Biot number indicates that the major temperature profile will be within the packed bed, while a small value indicates that the major temperature gradient will be confined to within the “fluid film” at the inner wall of the reactor. This Biot number can also be expressed as a product of the wall coefficient Stanton number, ST, and the thermal radial Peclet number, Pe, defined earlier (Biot) wall = (ST) (Pe)h Hence a large value of (ST) and/or (Pe)h places the major transverse temperature gradient within the bed. I n terms of Reynolds number, aspect ratio (tube radius to pellet diameter), Peclet, and Prandtl (Pr) numbers, we have
So while the wall biot number is but slightly altered by Reynolds number, it varies linearly with aspect ratio and, the radial Peclet number for heat transport in the body of the bed. Yield of Anhydride
An additional continuity equation is formally required to describe the axial and radial concentration distribution of the desired intermediate, phthalic anhydride. However, if radial dispersion of mass (but not heat) is relatively unimportant in this nonisothermal system, then the point yield of anhydride is related to the local consumption of naphthalene by a simple relationship which reflects the inter-intraphase diffusion-affected KI
reaction, namely (8) for the system A
Ka
+ B --c
C
9 2 tanh 9 2 tanh 91. , m 2 = 1 + (Bi) m (Bi), 7 = DB/D* The results of our inquiry rather tellingly suggest that overall reactor performance and detailed profiles are virtually independent of the radial Peclet number for mass, thus the use of Equation 13 to compute point concentrations of the intermediate anhydride is justified.
m l = l +
CPI
Summary of Reaction-Reactor Model
I n sum, we assume a nonisothermal, nonadiabatic cylindrical reactor, packed with a porous VZOScatalyst formulation. Axial and radial gradients in concentration and temperature are anticipated with heat being VOL. 6 1
NO. 7 J U L Y 1 9 6 9
29
transferred a t the reactor wall as governed by a n overall wall coefficient, h,, a function of Reynolds number. Local reaction rates which determine conversion of naphthalene and yield of phthalic anhydride are governed by first-order consecutive reaction, with due account for inter-intraphase diffusion of mass and interphase heat transfer. The pellets are, therefore, assumed to be isothermal a t a particular point in the bed, said pellet temperature being dictated by interphase heat transfer. T h e model thus invoked permits computation of con20
1
I
Scope of Investigation
18
16
Tm, To
14
12
10
01
03
07
05
REDUCED AXIAL DISTANCE, 2
Figure 7 . Normalzzed catalyst temperature profile at bed centerlineinfluence of radial thermal Peclet number on pro@ 7/27', naphthalene i n feed: radial mass Peclet number = 70 Conditions as cited in Figure 4 2.0
18
-
version, yield, catalytic effectiveness, and gas and catalyst temperatures throughout the axial and radial confines of the reactor. These valuable results are secured via solution of two continuity equations coupled with inter-intraphase relationships so fashioned that solution of local continuity equations which describe events within the porous catalyst at all points need not be undertaken. Instead local rates are concisely described in terms of the chemical kinetics (first order), and the yield-determining rate coefficient ratio a t catalyst surface temperature, T,. Rate modifications by external and internal mass diffusion are explicitly defined by overall catalytic effectiveness factors for each step of the consecutive networks (8).
1.6
1.4
Parameters which dictate reactor performance for a given reaction-catalyst system are (a) operational--i.e., feed composition, temperature, and coolant temperavarious transport ture, (b) intrinsic-physical-i.e., coefficients such as Peclet numbers, (c) intrinsicchemical-i.e., reaction activation energies as they might be modified by catalytic promoters or aging. I n this study, the influence of Peclet numbers, feed composition, and coolant temperature upon overall performance and such local factors as maximum radial temperature difference, interphase temperature difference, apparent (diffusion modified), and intrinsic rate coefficient ratios have been assessed. T h e radial Peclet number for mass dispersion has been experimentally established a t a value of about 10 for turbulent flow through fixed beds. The radial thermal Peclet number is more difficult to specify as modes of transport (conduction, radiation) exist in addition to radial thermal transport via enthalpy of the dispersed mass. Therefore the thermal Peclet number may be identical in value to that of mass (about 10) when particulate conduction and radiation are negligible or a value of the thermal Peclet number of as low as 5 can be anticipated when indeed radiation and conduction are important means whereby heat is transported radially across the bed. Beek ( 7 ) discusses this issue a t some length and recommends a modified form of the Argo-Smith equation as best describing radial heat transport in packed beds. Here, the authors have varied both mass and thermal Peclet numbers over a wide range of numerical values, said range including anticipated values between 5 and 10 for thermal transport. Numerical Solution
1.2 1 .o 1.0 0.8
0.4
0
0.4
0.8
1.0
r -
Ro
Figure 2. Computed radial catalyst temperature profiles at Z and Z = 0.5
=
(Peclet)heat = 70 5 Conditions identical to those cited in Figure 4
_____. (Peclet)heat =
30
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
0.7
T h e continuity equations are cast into a different form and solved numerically for the specified boundary conditions. Essentially, an explicit marching technique is employed in the axial direction while radial profiles are obtained using the Thomas implicit technique (25). Calculations indicate that 50 axial increments suffice to generate the reactor profiles while only five radial increments are required, An effect of increment size is evident when 10 axial steps are used. Solutions presented here are for 100 axial increments and 5 radial
1 .o -b
steps. Computing time for each case is about a minute on the Univac 1107.
0.4 0 . 1 cm2/sec cal 7mol 'C -AH1 = 43 kcal/mol -AH, = 87 kcal/mol
Bed void fraction Intraphase (pore) diffusivity Heat capacity of gas Reaction enthalpies
AT WALL
I 0.8
Constants Employed in Reactor-Reaction Model
0.6
-
=
0.4
Results
I n Figures 1-3, profiles of temperature and effectiveness are presented for thermal radial Peclet numbers of 5 and 10 (mass Peclet number of 10). The computed profiles in each figure are for two cases in which all conditions are fixed, save for the thermal Peclet numbers. Catalyst temperature a t the bed centerline is displayed as a function of reduced bed length in Figure 1. Note the telling influence of the value of Peheat upon the axial temperature profile. At two axial positions (20% and 50% of bed length), radial catalyst temperature profiles are shown in Figure 2; again for two values of the thermal radial Peclet number. Figure 3 reveals values of the overall (interintraphase) catalytic effectiveness factor for the naphthalene reaction step leading to anhydride; 71 is defined
0.2
0 0.1
0.2
0.3 0.4 0.5 REDUCED REACTOR LENGTH, 2
0.6
0.7
0.8
Figure 3. Inter-intraphase catalytic eflectiveness us. reactor length at reactor wall and centerline for (Peclet)hest = 70 5 Conditions identical to those cited i n Figure 4
-----_(Peclet)heat = 70
71 =
+
tanh
+
cp
(o
1
(14)
(m7n tanh ('
T h e axial 71 profiles are shown in Figure 3 a t bed centerline and a t the wall for (Peclet)h,,+ values of 5 and 10. Recalling that low values of 71 prove detrimental to yield of intermediate (anhydride) in the consecutive reaction network, the lessons revealed in Figures 1-3 are apparent: a limitation in radial heat transport invites low effectiveness since high local temperatures increase the Thiele modulus. Influence of Radial Peclet Values
The justification for relating point rate of disappearance of naphthalene to net generation of intermediate product anhydride by Equation 13, assumes that behavior of the system is rather insensitive to the radial mass Peclet number. I n other words, the mass continuity equation need not contain term ( b ) in Equation 3 if variations in Pemassvirtually uninfluence the resulting solutions. If such be the case, a n additional mass continuity equation for anhydride is not required. T o explore this assumption, solutions were generated for a range of values of both the thermal and mass radial Peclet numbers. Figure 4 reveals the results for conditions cited. Both conversion and yield are essentially independent of the value of the mass Peclet number for radial transport. I n contrast, a dramatic sensitivity of the results (yield) to the value ascribed to the radial thermal Peclet number is evident. I n essence, then, plug flow mass continuity equations
S ' 65 u-
L c*
n
z U Y
0
9
E
60 MASS,
PE
=
du J-
D,
55 5
6
7 8 RADIAL PECLEf NUMBER
9
10
Figure 4. Influence of radial Peclet number upon overall yield of anhydride, conversion = 99%
TO= 32OoC, T , = 37OoC, YO= 7/2%, Re = 784 d, = 7/2 cm, DT = 5 cm, 8 0 = 2 sec could be invoked, with the consequence that a separate continuity equation for intermediate anhydride is irrelevant; Equation 13 therefore accurately describes anhydride generation as a simple function of naphthalene conversion where both steps may be affected by inter-intraphase diffusional intrusions. O n the other hand, the temperature continuity equation must contain term ( b ) to give due account of radial thermal transport. The one-dimensional model of the nonisothermal, nonadiabatic fixed bed most adequately describes species distribution, but the two-dimensional model must be VOL. 6 1
NO. 7 J U L Y 1 9 6 9 31
invoked in describing temperature distribution. As temperature more drastically affects chemical reaction rate than does composition, these findings hardly prove surprising insofar as local and thus overall conversion are concerned; yield will be far more sensitive to temperature and its distribution than to species concentration. Broad support of the above-cited contentions are in ._ Table 11, where, for various operating conditions, the influence of both mass and thermal Peclet numbers upon conversion, X, and anhydride yield, Y , are noted. Peclet values of 0.1 (last column) represent conditions of zero radial temperature and concentration gradients (true plug flow). Table I1 suggests that a one-dimensional (axial) model for both temperature and concentration distribution (all radial resistances being lumped a t the wall) gives rise to meaningless predictions of overall conversion and yield in contrast with a two-dimensional thermal model in which the one-dimensional mass continuity model is implicitly invoked. I n sum, the exothermic catalytic reaction system reveals great parametric sensitivity to the thermal Peclet number and virtually total insensitivity to the radial mass Peclet number. AS for the axial Peclet numbers, earlier work has demonstrated their irrelevance in reactors characterized by length to particle diameter ratios greater than 100 (25). dther Forms of Parametric Sensitivity
'
Amundson and his colleagues (2) first revealed the susceptibility of plug flow homogeneous reactor performance to small variations in operating parameters such as feed conditions and wall temperature. I n the case of the distributed parameter, heterogeneous reactor (the case herein considered), overall perform-
ance (conversion, yield) is obviously affected by such local conditions as gross radial temperature difference (AT,) and interphase temperature gradient (AT,) (conveniently assessed at the axial centerline poiGt of temperature maximum). Therefore we have explored the influence, upon conversion and yield, of feed composition, wall temperature (physical parameters), and activational energy (chemical parameter). In so doing, ATT and ATD at T,,, are also noted as these local gradients manifestly affect local effectiveness factors and therefore net conversion and yield. Table 111 reveals the effects of feed composition upon conversion X,yield, Y , ATr and ATp for specified values of wall temperature, T,, pellet diameter, and Reynolds number. Parametric sensitivity to feed composition variations between 1/2 and 3/47, naphthalene is clearly shown in Case A and between 3/4 and 1% naphthalene in Case B. I n Case A a significant catalyst particle size effect is apparent insofar as yield is significantly higher for the smaller (1/4 cm) catalyst due to a lower A T r and AT, (and therefore higher effectiveness factors) in contrast with the 1/2-cm catalyst. Since Reynolds number is constant in Case A, the higher interstitial fluid velocity in the 1/4-cm subcase accounts for enhanced interphase thermal transport to the benefit of yield. For, in spite of the fact that E1 > Ez, the higher temperature benefits which might be anticipated are not realized primarily because of mass diffusional modifications, as we shall note shortly. Further evidence of parametric sensitivity is set forth in Figure 5, where computed overall and local performance factors are plotted as a function of coolant (wall) temperature.
TABLE 1 1 1 . INFLUENCE OF FEED COMPOSITION (yo; 7 0 NAPHTHALENE IN AIR) ON CONVERSION, X, AND YIELD OF ANHYDRIDE, Y TABLE II. INFLUENCE OF RADIAL PECLET NUMBERS ON PERFORMANCE (d,
=
I
P e ,-Pe h Conversion, X Yield, Y
Ir Pe,-Pe,+ Conversion, X Yield, Y
Yo% 1/2
1/2 cm, D r = 5 cm in both cases) Re = 784
eo = 2sec
T, = 583'K
10-10 99 8 59 7 Re = 730
5-10 99 2 59 7
10-5 98 5 68 5
10-5 97 2 93 3
eo =
T, =
5sec
623'K
0 1-5 97 6 93 5
10-0 1
90 86 3
7;2g
3/4 1
Case A Re = 360 Tw = 599' K dp = 1 / 2 cm y, 70 A Tr x,72 10 0.7 9 770 96.5 74 66 1000 97.6
0 1-0 1
$4; 0 1-0 1 90 Plug 86 Flow
t
1/2 3 /4 1
Case B Yo%
112 3 /4
1
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
1.7 47 0 590
d, = 1 / 4 cm
36
32
. . I
t . .
97
Re
=
x,72 5 8 98.5
80
184
12
... 590
2.4
... 240
T , = 588' K d, = 1/4 cm AT T A TP y , 76 0 1.5 55
7 65 650
A T , = M a x i m u m radial temperature dlyerence, O C A T P = Inlerphare temperature dlference, 'C, at Tmax Tube diameter = 5 cm in both cases.
32
ATP
0.5 18 590
TABLE IV. INFLUENCE O F ACTIVATION ENERGY VARIATION ON X , Y , AT,, AND AT, 1
Re
= 360,
2
N-+A+C T, = 599'K,
d p = 1/4 c m
Ez = 20 kcal yo = 3/470 yo
Figure 5. Fixed bed parametric sensitivity influence of coolant temperature upon conversion, X,yield Y , and AT, and ATp R e = 730; y o =
As Froment (70) has demonstrated parametric sensitivity of fixed bed (homogeneous) behavior to small variations in the wall coefficient, the general contention that the interparticle, inter-intraphase heat-mass diffusion influenced reaction system exhibits unique physical parametric sensitivity now seems to be well established. Indeed in view of expected variations in feed and coolant conditions and the rather drastic spread evident in thermal Peclet and wall coefficient data, nonisothermal, nonadiabatic fixed bed catalytic reactor performance predictions of a n absolute nature must be viewed with considerable caution. Chemical parameter sensitivity. Errors in the estimation of activation energies are more prevalent than is often anticipated. Indeed a 10% or 20% error is not unusual, particularly in catalytic systems in which seemingly small additions of promoters and unanticipated impurities can markedly affect activation energies. The system simulated herein, for catalyst A, is characterized by estimated activation energies of 38 and 20 kcal for the two consecutive steps, respectively. It is now a well-known fact that in those instances in which mass intraphase diffusional intrusions are invited by reason of reaction vigor within a limitedly accessible porous catalyst, the true activational energy is apparently reduced to one-half that value a t values of the Thiele modulus greater than about 3. That is, Equation 6 assumes the limiting form 171
= l/Pl
(15)
so that the effective rate within the catalyst becomes proportional to the square root of K1 and thus E1 apparently assumes a value of E1/2. The benefits then of positive temperature excursions depend not upon the positive difference E1 - E2 (a benefit to yield of intermediate) but in the diffusion-affected limit, upon E1/2 - E2. In spite of a favorable intrinsic activational energy difference (38 - 20), the apparent diffusion-affected difference can be unfavorable (19 -
1/27"
4
x,%
42 32 42 32
97.4 9 37 7.4
Y,% 75.6 3.7 32 0
AT,
ATP
770 Nil 110
470 Nil 2.4 Nil
Nil
20). This situation logically invites one to consider simulated behavior in the light of highly probable errors associated with activation energy estimates. Table I V sets forth predicted performance and local maximum gradients ( A T , and A T p )for the case in which, for a fixed value of E2 = 20 kcal, reasonable variations in E1 are entertained. The apparent chemical parametric sensitivity revealed by these computations is most instructive. As in the instances of physical parametric sensitivity cited earlier, these results clearly dramatize the need for more precise parameter determinations. Factors Affecting Yield
The simulation undertaken in this work, providing detailed information on local gas-solids temperatures, concentration of key species, and effectiveness factors, permits a detailed analysis of those factors which affect conversion and yield. The importance of such detailed inquiries cannot be overemphasized, for phenomenological correlations (conversion, yield as a function of feed conditions, coolant temperature) can be disastrously misleading. T o be specific, yield may either increase or decrease with a variation in, say, tube-topellet diameter ratio; said trend depending upon a number of other factors which affect local events within the bed such as the Reynolds number which determines the Biot numbers a t the pellet and thus A T p and, at the wall the Biot number which governs AT,, thus altering the locale and magnitude of both short- and long-range temperature gradients. Table V illustrates the virtues inherent in a detailed inquiry. At a particular axial distance (the point of maximum AT,), reduced gas and catalyst temperatures are tabulated for the bed centerline, at an intermediate radial location and a t the reactor wall. Corresponding values of the intrinsic rate coefficient ratio and the apparent, diffusion-affected ratio are also tabulated. The apparent ratio is simply the intrinsic ratio modified by the respective effectiveness factors, so that the promising diffusion-uninfluenced ratio is drastically modified by VOL. 6 1
NO. 7
JULY 1969
33
TABLE V.
COMPUTED RADIAL PROFILE A T
2 = 0.2 FOR NAPHTHALENE OXIDATION yo = 1/2%, d p = 1/2 cm, To= 31OoC, X = gg%,
Re = 5 cm,
0 (center-
line)
0.6 1 (wall)
y = 84%
1.3 1.6 1 . 5 X 104 7 . 3 X 10-4 1 x 104 1 . 5 x 10-3 1.25 1.53 0.98 1 . 0 1.005 50
13 17 50
virtue of diffusional intrusions. The seemingly attractive merits of high temperature actually are totally nullified insofar as the apparent yield-determining ratecoefficient ratio proves to be less favorable in the hotter (centerline) region of the bed than a t the lower temperature wall region. Given catalyst A , its favorable activational energy character, and the exothermic nature of naphthalene oxidation, one could reasonably expect a 100yo yield of anhydride in plant operationif diffusional intrusions of heat and mass were ignored. I t is no secret that 100yo yield of anhydride is not to be found in commercial operation-a fact which can only be rationalized by the recognition that, in spite of favorable intrinsic dispositions of the catalyst, diffusional factors intrude to the detriment of promised yield. A now classic experimental example of mass diffusional modification of yield in a consecutive reaction system has been provided by Weisz and Swegler (24). Merits of Simulation
T h e persuasive evidence of chemical and physical parametric sensitivity would certainly suggest that, pending the availability of far more precise parameter data, simulation can hardly be expected to provide results of absolute meaning. For example, overall performance figures (conversion, yield) are far too sensitive to assumed values of key parameters such as the thermal Peclet number and, as Froment’s work (70) shows, the heat transfer coefficient at the wall. This work and that of Froment therefore declares that the wall Biot number (see Equation 10) is not known with the precision required to allow one to be confident of computed results. O n the other hand, while absolute simulation is not to be realized there are merits in what might be termed relative simulation. For given reasonable values of key parameters, the influences of variations in operational parameters (pellet size, Reynolds number) will reveal trends of value to the analyst and designer. Consider, for example, the computed data in Table V I a t a fixed contact time, wall temperature, feed composition, and temperature, and radial Peclet numbers of 10 and 7 for mass and heat, respectively. Conversion and yield ( X , Y ) are tabulated for various particle sizes, tube-to-particle-diameter ratios, tube-surfaceto-volume ratios, and Reynolds numbers. Inspection of these calculated performance figures, 34
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
TABLE VI. COMPUTED OVERALL CONVERSION AND YIELD FOR NAPHTHALENE OXIDATION
T , = 583 OK, yo = 1/2%, (Pe), = IO, (Pe)h = 7, e = 2 sec Run d,, cm DT,cm S / V , cm-1 Re X , % Y,% a b c d e
1 112 1/2 1/2 1/2
10 5 2.5 5 10
0.4 0.8 1.6 0.8 0.4
365 365 184 184 184
99.9 98 93 99 100
42 74 74 65 47
interpreted in the light of interparticle, inter-intraphase diffusion of heat and mass with catalytic reaction, certainly suggests a detrimental effect of increase in particle size on yield, a benefit in increase in flow rate a t fixed Reynolds number upon yield (runs a and b , b and d), and a benefit in increasing tube surface to volume ratio (runs c, d, and e ) . An analysis of the detailed structure of the profiles in each of these cases points to yield deterioration owing to interphase temperature gradients, low catalytic effectiveness, and severe interparticle radial temperature gradients. While performance predictions are sensitive to assumed values of transport and chemical parameters, certain variables affect yield, and therefore the locale of remedial engineering is specified. I n the reaction system herein treated, relative simulation results would suggest that pellet size is critical. As Table V reveals, low catalytic effectiveness erases the benefit expected of high temperatures. A design specifying very small particle size is not often economically justified because of resulting high pressure drop and low Reynolds number. I n this case of naphthalene oxidation, in fact, a particle of about 1/100-cm diameter would be required to realize high effectiveness factors (approaching unity). There is an attractive alternative, namely partial impregnation of the catalytic ingredient upon the support. If then a shell of catalytic agent confined to the outer regions of the support can be fabricated, the Thiele modulus becomes
where F is the fraction of support radius actually occupied by the catalytic agent. If the original quantity of catalytic agent, VaOj, were imposed and confined to the outer 10% of the support, then in view of the limiting form of the 9 - p function (Equation 14), effectiveness at the bed centerline (Table V) is increased by one order of magnitude to the distinct benefit of anhydride yield. The merits of partial impregnation are discussed elsewhere (78) for SO2 oxidation. Comparison with reality, While we do not have operating data to compare with our relative simulation trends, phthalic anhydride-fixed beds are characterized by low tube-to-pellet diameter ratios, a fact in conformity with Equation 12 and our findings that large
E
temperature gradients prove deleterious to yield, not to mention stability.
f
Conclusions
h AH j
Simulation via digital numerical solution of the continuity and boundary condition equations assumed to govern the exothermic consecutive catalytic reaction of naphthalene with air shows that: (a) The system yield (phthalic anhydride) is highly sensitive to the value of the thermal radial Peclet number, feed composition, and coolant wall temperature. In contrast, results (conversion and yield) are virtually insensitive to extreme variations in the mass radial Peclet number. (b) I n consequence of (a), a one-dimensional mass continuity equation adequately describes species distribution throughout the bed ; however a two-dimensional description of temperature distribution is required. (c) Detailed computations reveal the existence of significant interparticle radial temperature gradients, interphase concentration and temperature differences, and severe intraphase mass diffusional limitations, all of which conspire to influence yield as well as conversion. (d) Inter-intraphase gradients persist even a t low conversion levels, suggesting that even in laboratory catalytic inquiries conducted a t differential conversion levels, recognition must be given to diffusional intrusions which can markedly frustrate interpretation of laboratory data. (See Table 111, a t low feed composition.) (e) In view of the demonstrated sensitivity of the nonisothermal nonadiabatic heterogeneous catalytic reactor to physical and chemical parameters, confidence in simulation must be limited indeed, pending the procurement of physio-chemical data of greater precision. (f) Whereas absolute powers of simulation cannot be enjoyed a t this time, the merits of relative simulation --i.e., exploration of the effects of parameter variation upon relative reactor performance-are not to be underestimated. For the relative influence upon conversion and yield of variations in negotiable parameters, such as Reynolds number, particle size, and tube-to-particlediameter ratio, can suggest to the reactor analyst certain clinical symptoms and thereby prompt remedial measures which may enhance performance, (9) Finally, the model set forth in this work provides the bases whereby interparticle, inter-intraphase diffusion of heat and mass and chemical reaction can be efficiently coupled to permit rapid numerical computation to provide detailed insights into both the shortand long-range gradients which affect overall performance. Nomenclature 4 = external surface-to-volume ratio A, B, C = molecular species or concentration Bi = Biot number, Equation 7 Cp = heat capacity 9 = intraphase diffusivity D, = radial mass diffusivity d p = catalyst pellet diameter D T = tube diameter
F
k, K k, K ICgaB 1
L m1,2
m n (Pe), (Pe)h Pr
R Re
a
r ST Sc T AT, AT, t tanh U
X
Y yo
-2
-
activation energy
= reduced concentration = fraction of catalyst support impregnated
= heat transfer coefficient = reaction enthalpy change = interphase j factor, Equation 5 = rate coefficient = interphase mass transfer coefficient = radial thermal diffusivity in packed bed = thermal conductivity of gas = distance within catalyst pellet = volume-to-external-surface ratio, 1/a = see Equation 13 = tube-radius-to-catalyst-diameter ratio = tube-length-to-catalyst-diameter ratio = radial Peclet number for mass, dpu/U, = radial Peclet number for heat, d,u/K = Prandtl number = tube radius = Reynolds number = reaction rate function = radial distance across tube = wall Stantan number = Schmidt number = temperature, O K or "C = maximum radial temperature difference = interphase temperature difference a t T,,, = reduced temperature, T/To = hyperbolic tangent = fluid velocity in bed = naphthalene conversion = anhydride yield = naphthalene in feed to reactor = reactor length
Greek Symbols y
eo 7
X p Q
= intraphase diffusivity ratio, in Equation 13 = nominal contact time, Zo/u = catalytic effectiveness factor = pellet thermal conductivity = gas density = Thiele modulus
Subscripts 0 R h
m s
w c
= initial condition = gas = heat = mass = surface = wall = catalyst
REFERENCES (1) Beek, John; Adv. in Chem. Ens., Vol. 3, Academic Press, New York, 1962. (2) Bilous, O., and Amundson, N. R., A.I.Ch.E. J.,2,116 (1956). (3) Carberry, J. J., IND. END.CHEM.,5 8 (lo), 40 (1966). (4) Carberry, J. J., Can. J . Chem. E n s . , 36, 207 (1958). (5) Carberry, J. J., IND.ENC.CHEM.,56 ( l l ) , 39 (1964). (6) Carberry, J. J., A . I . C h . E . J . , 6,460 (1960). (7) Carberry, J. J., ibid., 7 350 (1961). (8) Carberry, J. J., Chem. En:. Sci. 17 675 (1962). ( 9 ) DeMaria, F., Longfield, J. E., and Butler, G . , IND.ENG.CHEM.,53, 259 (1961). (10) Froment, G. F., ibid., 5 9 (Z), 18 (1967). (11) Fulton, J. W. and Crosser, O., A.I.Ch.E. J.,11, 513 (1965). (12) Gillespie, B. M., Crandall, E. D., and Carberry, J. J., ;bid,, 14, 483 (1968). (13) Hawthorn, R. D., Ackerman, G. H., and Nixon, A. C., ibid., 14, 69 (1968). (14) Hutchings, J., and Carberry, J. J., ibid., 12, 30 (1966). (15) McGuire, M . L., and Lapidus, L., ibid., 11,85 (1965). (16) Maymo, J. A., and Smith, J. M., ibid., 12,845 (1966). (17) Minhas, S., and Carberry, J. J., Brit. Chem. Eng., 14 (6), 779 (1969). (18) Minhas, S., and Carberry, J. J., J. Calal., in press (1969). and Carberry, J. J., Chcm. Eng. Sci.,13, 30 (1961). (19) Mixon, F. O., (20) Schilson, R., Amundson, N. R., ibid., p 226. (21) Thiele, E. W., IND. ENG.CHEM.,31, 916 (1939). (22) Von Rosenberg, D . U., Durrill, P. L., and Spenser, E. H., B r i f . Chcm. Eng., 7 (3), 186 (1962). (23) Weisz, P. B., and Hicks, J., Chem. Eng. Sci., 17, 265 (1962). (24) Weisz, P. B., and Swegler, E. W., J . Phyr. Chem., 5 9 , 423 (1955). (25) Wendel, M., and Carberry, J. J., A.I.Ch.E. J. 9,129 (1963). (26) Westerterp, K. R., Chem. Eng. Sci., 17, 423 (1962). (27) Yagi, S., and Wakao, N., A.I.Ch.E. J.,5,79 (1959).
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