Engineering Aspects of
Solid Catalysts
OLAF A. HOUGEN Department of Chemical Engineering, University of Wisconsin, Madison, Wis,
T R E MOST CHALLENGING PROBLEM in the unknown fringes of science and chemical technology is to establish scientific principles of catalyst activity which will lead to direct methods of catalyst selection for specific chemical reactions and industrial processes. Despite the enormous amount of research and development undertaken throughout the world in the use of solid catalysts for chemical processing, conclusive scientific principles involved in the selection of a catalyst and factors determining catalyst activity still remain obscure. The selection of a suitable and effective catalyst for a particular process has been arrived at largely in an empirical fashion by endless trials. Over 20,000 experiments with different catalyst compositions and conditions of use were carried out by German chemists in developing a catalyst for the synthesis of ammonia by the Haber process. During the last century the use of solid catalysts has been extended to
thousands of industrial reactions. The major bulk of the world’s gasoline is produced by chemical processing using solid catalysts. In the United States alone, over 30 billion gallons of gasoline are produced annually by catalytic cracking. I n 1959, one American company alone produced solid catalysts for the hydrogenation of 200,000 tons of ethylene, for the synthesis of 3,100,000 tons of ammonia, and for the purification of 180 billion cubic feet of hydrogen. I n recent years, scientists and engineers have become alarmed at the relatively primitive status of catalyst selection. Since 1948 annual progress in this field has been compiled in the publication Advances in Catalysis. The First International Congress on Catalysis was held in Philadelphia in 1956 and the Second in Paris in 1960. I n 1954 the Presidium of the Academy of Sciences of the U.S.S.R. endorsed a national plan for establishing the Scientific Principles of
The ACS Award in Industrial and Engineering Chemistry, sponsored by the Esso Research and Engineering Co., was established “to stimulate fundamental research in industrial and engineering chemistry and in the development and application of chemical engineering principles to industrial processes.” A biographical sketch of Olaf A. Hougen, 1961 recipient of this award, is unnecessary. His name is known internationally to chemical engineers and chemists in academic and industrial circles. This article is a shortened version, prepared by Dr. Hougen, of the award address, which he gave before the Chemical Process Subdivision of the Division of Industrial and Engineering Chemistry, at the 139th ACS Meeting,
the Selection of Catalysts with collaboration of outstanding specialists in the field of catalysis from various parts of Russia. Experts were assigned to research investigations on catalysts under Balandin ( 2 ) at the Zelinski Institute of Organic Chemistry and Moscow State University. I n recent years, the constants of kinetics and adsorption of hundreds of organic reactions catalyzed by solids have been established a t these research centers.
Coordination in Research Needed
Researches on reactions catalyzed by solids as conducted in academic institutions are fragmentary and inconclusive. These investigations have been of chief value in training students but have contributed relatively little toward the selection of catalysts for a given process or toward measuring absolute values of catalyst activity. I n departments of
March 2 1-30, 196 1,in St. Louis, Missouri. It outlines the present status of engineering aspects of solid catalysts and urges the collaboration of engineers, chemists, and physicists in arriving at a systematic method of selecting catalysts. The present fragmentation of experimentation by individual efforts is unavoidable, but general comprehension of all the facets of catalyst activity through collaboration of engineers and scientists i s essential to a scientific selection of solid catalysts, to the interpretation of experimental data and to the attainment of the optimum design and performance of catalytic reactors. This i s the outstanding scientific and engineering challenge in chemical processing in the remaining decades of the twentieth century.
VOL. 53,
NO.
7
JULY 1961
509
chemical engineering, kinetics investigations have been restricted largely to establishing plausible reaction models assuming that a single or two rate-controlling chemical steps prevail, the objective being to establish rate equations useful for reactor design and for optimizing of given processes. In these investigations, little attention has been given to the composition and structure of catalysts or to their method of preparation. Experiments in measuring pore volume, internal surface area, magnetic susceptibility, the work function, adsorption isotherms, ionization potential, and surface mobility are neglected. Likewise scientists have been equally remiss in confining their researches to selected physical properties without coming to grips with problems of internal or external diffusion, mass and heat transfer, effective thermal conductivities, effectiveness factors, pressure drops, mechanical strength, and attrition losses. The lack of collaboration among university departments of engineering and science on these problems is not due to lack of mutual interest or unwillingness to cooperate but rather to the divergence of training and preoccupation in their separate fields of interest. The over-all problem is too vast for individual undertaking. The requirements of experimentation on both kinetics of a given reaction as well as on the properties of the catalyst for any single doctoral thesis is unreasonable. For these reasons, university researches in the field of catalyst selection and measurements of catalyst activity have been fragmentary. Progress has been made a t a discouragingly slow rate. I n contrast, industrial research in this field, at least with some of the larger companies, has necessarily been highly collaborative with scientists and engineers assigned to the numerous separate problems and combining their efforts toward a more complete understanding of catalyst behavior, reaction model, optimum selection, and reactor design. Industrial research is of a proprietary interest, and information on catalyst selection seldom reaches publication except in
fragmentary parts which lack economic interest. This report on the available fragments of progress in the engineering aspects of solid catalysts shows the present status of progress and the unexplored problems. The greatest barrier in the selection of solid catalysts is the lack of a n absolute measure of catalyst activity. At present, no absolute numbers can be assigned to activity; numerical values are all relative based upon some arbitrary property usually ratios of reaction rates among different catalysts when employed under similar conditions. The contributing factors of catalyst activity are numerous with unestablished interrelationships. A hundred different causes may contribute to the activity of a catalyst or to its deactivation. The technical aspects of solid catalysts fall logically into three catagories: Manufacturing Scientific Engineering The technical responsibilities involved in the production, selection, and use of solid catalysts fall conveniently into these three classifications with science common to all three. This paper deals primarily with the engineering aspects of solid catalysts, the others are merely outlined. This very paper is an example of the fragmentary approach to the principles of catalyst activity which I deplore. In attempting to cover all three aspects of solid catalysts I found myself unprepared to deal with the techniques of manufacture and with the electronic properties of the solid state. Scope of Manufacturing Aspects of Solid Catalysts
The manufacture of each solid catalyst is a highly specialized enterprise involving extensive services of chemists, physicists, and engineers. Coverage of the entire scope of manufacture does not permit a generalized academic treatise. A summary of the many technical problems involved in the preparation of solid cat-
Sources of Evaluation Errors Five serious sources of error that arise in the evaluation of kinetics models and interpretation of rate data in flow reactions catalyzed b y solids may be summarized as follows, arranged in order of decreasing importance:
b Variation in catalyst activity (Parf VI/) b Use o f catalyst particles having effectiveness factors differing appreciably from unity (Part VII)
b Neglect o f external resistances to mass and heat transfer (Part /V) b Appreciable deparfure from plug flow (Parf V//I) b Neglect o f pressure drop due to flow (Part I ) Even b y providing all these precautions measurements of reaction rates alone usually afford a crude device for evaluation of a dependable reaction model and should b e supplemented b y measurements of other properties.
5 10
INDUSTRIAL AND ENGlNEERlNG CHEMISTRY
alysts is reported in the Industrial and Engineering Chemistry Symposium of 1957 (75). The general procedure in the preparation of solid catalysts involves the following five steps: selection of raw materials, removal of impurities, conversion to the desired compound, formation of the desired shape, and final activation. All chemical and engineering techniques of dissolution, crystallization, precipitation, coprecipitation, filtration, leaching, washing, sedimentation, drying, calcination, and mixing are involved with the control of density, porosity, internal surface area, pore structure, and mechanical strength of the pellet of paramount importance. Where catalyst carriers and promoters are used, additional problems are involved in the impregnation of the carrier, distribution and dispersion of catalyst, and promoter. Impregnation is followed by drying, calcination, and activation. The bonding of catalysts, shape forming, extrusion, and pelleting impose specialized mechanical problems. The final pellet must possess high mechanical strength and resistance to attrition. Most time involved in developing a catalyst is spent in attaining the proper bonding and desired mechanical strength to withstand handling, loading, and attrition. The manufacture of each type of catalyst requires a wide variety of highly specialized technical skills attained only by years of experience. Many standard catalysts used throughout the world were developed in Germany more than 30 years ago after years of intensive research. For example, the coprecipitated iron oxide-chromia catalyst of high mechanical strength for use in the production of hydrogen by catalyzing the water gas shift reaction was developed by Badische-Anilin und Soda Fabrik before 1930 and is still in general commercial use. Scope of Scientific Aspects of Solid Catalysts
-4number of excellent comprehensive reviews on various scientific aspects of catalysis have appeared recently: on the basic principles of catalysis by Seitz (72) and by Schwab ( 7 7), on chemisorption by Taylor (76), Kwan ( 9 ) , and deBoer ( 4 ) ,on physical structure and texture by deBoer (5)and Emmett ( 6 ) ,on the defect structure by Gray and Darby (8), on electronic factors in catalysis by Garner (7) and by Baker and Jenkins (I), on magnetic susceptibility of metal catalysts by Sellwood (73, 7 4 , on poisoning of solid catalysts by Maxted (70), and on active centers and reaction models by Balandin ( 3 ) . These various scientific aspects will eventually provide the key to unlock the mysteries of catalysis but a n over-all picture from these fragmentary studies has as yet not emerged. The chemical
S O L I D CATALYSTS 17,900
I
1
I
I
I
4
I
I
300
Figure 1-1. Cost contours near the optimum point (3)
280
x = 0.9855.
17,700
17,500
z= 8
8e c n
. u I
3 260
e c
ur i
2
c
-..
2 240
'
17,100
0
>
W
b
tJY v)
a 220
0."
Figure 1-2. Cost vs. catalyst particle size and mass velocity
$0.
%eo
$4. *a0 $4' 200
.I2300
/
I
I
0.020
I
0.025
I
0.030
I
0.035
t6,SOO
16,700
(3) I
0.040
I 0.046
x
= 0.9855.
z
16,500
8
I
0.020
O.Oe5
Catalyst
CATALYST
PARTICLE
I
I
0.030
0.035
Particle
I
0.040
0.045
S l z e , Op,fY.
SIZE, D p , ft.
engineer has concentrated on performance and design, giving little attention to these scientific studies. Scope of Engineering Aspects
A review will be presented here on those aspects of solid catalysts that are of special significance in the engineering problems of process design, such as, pressure drops in the flow of fluids through packed beds, pore structure and internal area, effective diffusivity in voids surrounding the pellets, temperatures and concentrations a t catalyst surfaces, effective thermal conductivity of packed beds, thermal conductivity and effectiveness factor of individual catalyst pellets, internal gradients of temperature and concentration, hold-up in packed beds, and reaction models of the chemical kinetics involved. O n certain aspects of these engineering problems comprehensive treatments have been published such as on physical structure and texture by deBoer (5) and by Emmett (6), and on reaction selectivity in catalyst pellets by Wheeler (77).
I.
Pressure Drop and Size of Pellets in Packed Beds
The evaluation of pressure drops through packed beds has been well established for beds of uniform particle size and for the flow of single fluids where the packing is in random dense arrangement without channeling. For the laminar flow of a single fluid through a packed bed the volumetric rate of flow is given by the Blake-Kozeny equation where in order to fit the experimental data the constant 150 replaces the value 72 obtained from theoretical considerations.
where Zo is the average linear velocity of fluid = (volumetric rate of flow)/(total cross sectional area), L is the depth of bed, Po is the inlet pressure, p L is the exit pressure, E is the external void fraction of the bed, and p is viscosity of the fluid. The term Dp is the equivalent diameter of the particle defined as
D,
(1-2)
= 6/av
where a,, is the total external surface per unit volume of bed and in turn is equal to the external surface a, of a single particle divided by its volume up, thus a, = a,/v,. For a sphere of diameter D,D, =
D. For turbulent flow of a single fluid through a packed bed pressure drop is given by the Burke Plummer equation L
€8
( 1-3 )
as for values of (D,G/p)(I - e) >IO00 where G is the mass velocity of fluid based on total cross section of the bed and p is the fluid density. Ergun (4) has combined the above equations for both laminar and turbulent flows in terms of dimensionless groups to give
150 (1 - 6) DoGol~
+ 1.75
(1-4)
Equations 1, 3, and 4 are based on packed beds with particles of similar size and shape throughout, with uniform packing, no channeling and negligible wall effect. A detailed development of these relations is given by Bird, Stewart, and Lightfoot (7).
Effect of Pellet Size. Aside from the influence of size on the effectivenessfactor of a catalyst pellet to be discussed later, the diameter of pellet has a n important effect on pressure drop. Increase in catalyst size is accompanied by a decrease in effectiveness factor but with a n increase in pressure drop for a given mass velocity of fluid flow. I n a recent study by Chu (2, 3) in optimizing the design and performance of a catalytic reactor for the oxidation of nitric oxide in dilute concentrations in a fixed bed of activated carbon, the significant influence of catalyst size in minimizing operating costs is shown in Figures 1-1 and 1-2. The effect of pressure drop is not shown directly but is included in the cost for power requirements. The minimum cost in blowing gas through the bed of catalyst pellets is obtained at a mass velocity of 254 pounds/(sq. ft.) (hour) with a particle diameter of 0.032 foot (0.384 inch) as seen from Figures 1-1 and 1-2. With larger sizes, the over-all costs increase owing to increase in the required thickness of bed and to a decrease in reaction rate per unit mass; with smaller diameters, costs increase owing to increase in pressure drop with corresponding increase in power for blowing the gas through the bed. Figure 1-2 shows that the parameters of mass velocity fall with increasing mass velocity until anoptimum mass velocity of 254 is reached ; exceeding this mass velocity results in an increase in costs. Cost optimizations were obtained statistically by the Box method of steepest radial descent. II.
Pore Structure and Surface Area The pore structure and surface area of catalyst pellets have marked influences on their catalyst activity, effectiveness VOL. 53,
NO 7
0
JULY 1961
51 1
10 0
5.0
3.0
N r 1.0
a
N
.n
05
0.3
01 10
30
100
50 Pore
300
Figure 11-1. Values of p/po, k12, and increments o f pore diameter (2)
Rlz
factor, permeability, effective diffusivity, thermal conductivity, and mechanical strength. The standard procedure of measuring the internal surface of a porous pellet is by the method of the Brunauer, Emmett, and Teller (3) and of measuring pore diameter and pore size distribution by the mercury penetrationpressure test of Ritter and Drake (4) for pore sizes down to 20A. at 100,000 p.s.i. For smaller pore sizes, gas adsorption methods are required. Recently, Cranston and Inkley (2) developed an ingenious method of calculating the pore structure of a porous particle from measurements of the physical adsorption or desorption isotherms of nitrogen. By this method, the internal surface area, internal void volume, average pore size, and pore size distribution of a porous solid may be calculated. This method is a n improvement over that of Barrett, Joyner, and Halenda (7). Both methods are based on Wheeler's theory of multilayer physical adsorption and capillary condensation. From measurements of physical adsorption on 15 nonporous substances, the thickness, t , of the adsorbed layer as a function of relative pressure has been found to be nearly the same for each with average values as shown in Table 11-1. By combination of the adsorption isotherm data for a particular substance with thickness of adsorbed layer, the diameter of the largest pore filled can be established a t a given relative pressure. The method is illustrated by Cranston and Inkley from the adsorption isotherm data of nitrogen on a silicaalumina catalyst where the volume of gas adsorbed, u, is given in milliliters (NTP) per gram of solid us. relative saturation pressure of N2. At any given relative pressure, pipo, between 0 and 1 all pores with diameters
5 12
SO0
1000
D i a m e i e r , A,
for standard 10-A.
Figure 11-2. Function (d-2t)/d2for mean values of d and t in each standard increment of pore diameter (2)
larger than d contain an adsorbed layer of thickness, t, while all pores of diameter, d, or smaller are filled. Let Vd be the volume of pores having diameters between d and dd. Let p a be the pressure at which pores of diameter (d dd) are about to fill and p ( d + ( ) d l be the pressure a t which this pore is filled. During the pressure change from fld t o p ( d + d d ) all pores in the range d to (d d d ) are being filled whereas all pores of greater radius remain unfilled but increase in thickness of the adsorbed layer from td to (td d t ) . The incremental volume of gas adsorbed u d in filling pores and increasing film thickness i s given as
+
+
+
(11-4)
and tl2
=
+2
t_ l _ tz
(11-5)
For any small increment of pore diameter from dl to dz values of R12 and k12 can be calculated since for filled pores film thickness becomes equal to pore radius. Values of Rlz and k12 are plotted in Figure 11-1 us. pore diameter for intervals of 10A. in pore diameters. Values of the d
- 2t13
fraction ___ d2 are plotted against pore (filling pores)
+
diameter at different parameters for various ranges of pore diameters in Figure 11-2.
increasing film where V , is the filled volume in pores of diameter d. Equation 1 is solved by numerical integration taking small increments of d, where u d = u12, and assuming that Vd is constant for a small increment of diameter, where VI^ is the incremental volume of pores having diameters between dl and d2. Integration of Equation 1 is carried out to a maximum pore diameter, d,,,, rather than to infinity. I n terms of finite increments Equation 1 reduces to Viz = Riz
[
a12
Table layer
I!-I. Thickness of
Adsorbed Angstrb'rns vs. Relative Pressure" Based on 15 nonporous solids Rel. Pressure, P/P" Thickness, A. in
0 0.05
0.1 0.2 0.3 0.4 0.5 0.6
0.7
z
- kiz dz
0.8
x
0.9 0.95
f Ad/Z a
INDUSTRIAL AND ENGINEERING CHEMISTRY
5.67 6.35 7.0
7.5 8.6 10.0 12.2 14.0
From figure by Cranston and Inkley
(6,p. 148).
where
0 3.39 4.12 4.85
SOLID CATALYSTS I i
!
more than one product is formed, to decrease in reaction selectivity. Convection in the external voids of a packed bed is defined in terms of an external effective diffusivity D,. The effective diffusion coefficient is related to the Peclet number
1
as Pe =
Du -E particle diameter. u is the D,
interstitial fluid velocity. From a theoretical analysis of motion within the void spaces of a rhombohedral packing of spheres, Ranz (8) found the Peclet number for radial diffusion (Pe,,) to be 11.3. Actually the Peclet number for radial diffusion increases toward the wall and with the ratio of particle size to tube diameter. Fahien and Smith (4) evaluated the mean value of the Peclet number for radial diffusion as follows,
(111-1) 10
30 50
100
300 500
1000
PORE DIAMETER, A.
Figure 11-3. Distribution of surface area, volume adsorbed, and relative pressure vs. pore diameter (2)
From the experimental isotherm u us.
p / p , for a specific silica-alumina catalyst combined with the thickness of adsorbed layer t us. p / p o (Table 11-1) the relationship u us. disestablished (Figure 11-3). T o convert the volume of gas adsorbed (NTP) to the volume occupied by the condensed gas, VI^ should be multiplied by the factor 1.854 X the ratio of the density of gaseous N B (NTP) to that of liquid Nz. For pores of diameter d, the surface area S i 2 in square meters per gram of porous solid is given by the relation (11-6)
The total area per gram of solid for all pores is the summation of values of SI* for all pore sizes. By a stepwise procedure for a specific silica alumina catalyst to a pore size of 14A. the total internal area, S = 194.6 square meter/gram as compared with 197.5 by BET method. The total pore volume V = 0.165 ml./gram and the average pore diameter, d = 33.9 A. The method of Cranston and Inkley requires highly accurate measurements of u us. p / p . for a specific catalyst.
111.
Effective External Diffusivities in Packed Beds
Where radial temperature gradients occur in a packed bed because of lateral cooling or heating, convection currents are established which contribute to holdup, decreased reaction rates, and, where
A summary of the status of experimental data on the effective external diffusivjty of fluids in packed beds is presented by Froment (5). For axial diffusion, McHenry and Wilhelm (6) obtained a value for the Peclet number of 1.88 f 0.15 for gases. For water, Ebach and White ( 3 ) reported lower values for the Peclet number than for gases, indicating that in liquid systems greater retention of liquids occurs in the void spaces of the packing than with gases. Strang and Geankopolis (9) obtained lower values of Peclet numbers with Raschig rings than with spheres. I n general, the effective diffusion coefficients for axial diffusion are five times greater than the values for radial diffusion for the same values of D,u. Further data on radial diffusion are given by Plautz and Johnstone (7) and Bernard and Wilhelm (7) and on axial diffusion by Ebach and White ( 3 ) and Carberry and Bretton ( 2 ) . External a n d Internal Temperature a n d Concentration Gradients of Pellets. The interpretation of reaction rate data and establishment of a kinetics model of fluid reactions catalyzed by solid surfaces go astray owing to neglect of the gradients of temperature and concentration from the catalyst surface to the ambient stream and inside the catalyst pellet. T h e external gradients of temperature and concentration can be made negligible by diluting the fluid with an inert fluid to reduce the heat evolved per unit volume and to operate at a high mass velocity to reduce the resistance to the transfer of heat and mass. The internal gradients of temperature and concentration can be made negligible by using fine catalyst pellets or by confining the catalyst to the exterior surface of an impervious pellet. Where
surface temperatures, surface concenrrations, and gradients of temperature and concentration within catalyst pellets are unknown, it becomes impossible to evaluate a reliable kinetic model from experimental data on reaction rates. These two sources of error will be discussed below.
IV.
Temperature and Partial Pressures at the Surface of Catalyst Particles
Where resistances to mass transfer are negligible, partial pressure drops from surface to ambient stream approach zero, whereas when mass transfer is completely controlling, the ratio of Apj/pj for the limiting reactant becomes unity. Under intermediate conditions this ratio ranges from 0 to 1.0 for the limiting reactant. For the reaction products, however, this ratio can assume any valuefor example, where mass transfer of reactants is rate controlling, the value of this ratio for the product becomes infinite at the entrance to a reactor where no product exists in the feed. Where resistances to mass transfer become appreciable, neglecting the pressure drop for the product becomes a serious source of error that is frequently neglected. I n examining the experimental data of 15 different gaseous reactions catalyzed by solid surfaces from 14 different groups of investigators, Yoshida, Ramaswami, and Hougen (7) found that values of A+j/pf for reactants covered a range from 0.0004 to 0.5 and for products from 0.001 to 8.0. For these same experimental runs, temperature drops from catalyst surface to ambient stream covered a range from 0.05' to 250'C. I n the hydrogenation of propylene values of Ap for hydrogen, propylene, and propane are shown in Figure IV-1 as related to mass velocity a t 47' and 103' C ; values of At are similarly related in Figure IV-2. Figure IV-1 shows that at a mass velocity of 4 gmoles/(sq. cm.)(hr.) the catalyst surface is at 393' C. with the ambient stream at 103' C. ( A t = 290' C.) and a t 443' C. with the ambient stream a t 47' C ( A t = 396' C). T o render the resistances to mass and heat transfer negligible would require increasing the mass velocity 100 fold. I n studying the kinetics of the hydrogenation of propylene Fair ( 2 ) controlled temperatures of the ambient fluid at three different levels. However, the calculated surface temperature for these three levels of ambient temperature covered a wide temperature range from 100" to 400" C. higher than that of the ambient stream, as shown in Figure IV-3. To establish the reaction model under these conditions required calculations of surface temperatures and concentrations. VOL. 53, KO. 7
JULY 1961
51 3
+a16
70
I
I
I
I
I
+0.12
+0.08
-
-
-
-
p.
a
+0.04
-0.04
I
I
0.o
I
I
I
I
I
I
I I
t
1
+-- SURFACE TEMPERATURES
ITEMPERATURES
-0.08
-012
C3H6
-0.16
I
0
2
I I I 6 8 4 G M = ( g rnolesl/(cm.‘)(hour)
J
I IO
C.”
12
Figure IV-3. Number of runs at given temperature levelscatalytic hydrogenation of propylene
Figure IV-1. Pressure drops from catalyst surface to ambient gas stream in the hydrogenation of propylene at 1 atm. Ambient gas stream P C , H ~= 0.33; PH* = 0.67 atm.
the model accepted was a surface reaction between adsorbed isobutylene and atomic hydrogen with the following rate equation ka K H K u P H P u
I n so doing, only few data became available at a given level of surface temperature, hence in order to use data of all runs it became necessary to accept surface temperature as a simultaneous variable. Fair ( 2 ) recognized the uncertainties and difficulties involved in calculating surface temperatures and concentrations and accordingly designed a new reactor that permitted operation at sufficiently high mass velocities to render negligible values of A$ and At and thus to facilitate and improve the correlation of data. I n the hydrogenation of isobutylene,
‘ = (1 + d%& +
+ K8pda(IV-1)
K8 = exp ( A & / R ) exp (-AH,/RT)
where k, = exp ( A S * / R ) exp ( - A H * / R ) (IV-2)
KR = exp ( A S a / R ) exp ( - A H = / R T ) (IV-3) Ku = exp ( A S , / R ) exp ( A H J R T ) (IV-4)
-
AH* As* AH= ASH AH,, A&
400
-
360
-
-I
\
0 0 .. I1
4-
Q
320
-
I
X
200
514
I
I
I
I
INDUSTRIAL AND ENGINEERING CHEMISTRY
I
1 I
Figure IV-2. Temperature drops from catalyst surface to ambient gas stream in the hyJrogenation of propylene at 1
(IV-5)
By correlation of data under both ambient and under surface conditions without correction for Ap and At the following values were obtained. Hydrogenation of Isobutylene Using Ambient Conditions Using Surface (Uncorrected for Conditions A p and A t ) +I431 - 1547 should be 6.2245 f1.32 -6555.1 -3112 -7.5504 -2.66 - 4295 1364 should be -7.17 -5.4129 AH, = - 1062 should show no adsorption of isobutane AS, = 0.525
+
+
I n using ambient conditions, without correction for Afl and At, the accepted model was not confirmed and the constants were all in error. The correlations were faulty in three respecnamely : The energy of activation AH* becomes negative (-1547) whereas it should be positive (+1431). The heat of adsorption of the isobutylene 4H,, becomes positive (f1364) whereas it should be negative (-4295). Strong adsorption of isobutane is indicated whereas when corrections are made for 4~and At the adsorption constant for isobutane K, was negligible.
S O L I D CATALYSTS In this recorrelation of data, corrections were made only for external gradients and not for internal gradients. However, from estimated values of internal diffusivity and thermal conductivity, it was calculated by the method of Schilson and Amundson that the internal gradients were only 2% of the external gradients and hence considered negligible for the given size catalyst. Estimation of Temperatures and Partial Pressures at the Surface of Catalyst Particles. To facilitate the estimation of partial pressures and temperatures at the surface of catalyst particles, Yoshida and Ramaswami (7) prepared dimensionless charts for the evaluation of partial pressure drops Ap,/pi for any component j in terms of a Reynolds number for various parameters of Schmidt numbers, Sc, and pressure factors P f A , and rate numbers, R, as shown in Figure IV-4, where R E -
TmA
am
+ GM
(IV-6)
is the moles of A reacting per unit time per unit mass of catalyst, a, is the external surface area of the catalyst per unit mass, GM is the molal mass velocity of gas flowing based upon the total cross sectional area; 4 is a shape factor, equal ymA
to 1.O for spheres, 0.91 for cylinders, and 0.90 for irregular grains. Similarly, a chart (Figure IV-5) was prepared for evaluating At in terms of a Reynolds number for various parameters of Prandtl number and of a heat transmission number, Q, where (YmA)
= am
AHA
(IV-7)
+ C,GM
where A H A is the heat of reaction per mole of A at temperature, T. The rate of mass transfer per unit mass may be defined as rmA
= kara,
+ ( f a - fat)
(1V-8)
where pAi is the partial pressure of component A a t the catalyst surface and k Q A is the mass transfer coefficient for component A . Similarly, the rate of heat transfer q m A is defined as q m = ~
rmAL\Ha
= ham @ ( t
- ti)
(IV-9)
where ti is the surface temperature of the catalyst, t is the temperature of the ambient fluid, and h is the heat transfer coefficient per unit exterior surface of catalyst particle. Values of j D were related to k g A by the following relation for gases
where Dan, is the mean diffusion coefficient, f i / A is the pressure factor for component A . Similarly, for heat transfer j h is defined as
where k, is the thermal conductivity of the fluid. RECORRELATION OF VALUES OF j,. In the low range of Reynolds numbers encountered in laboratory reactors, additional experimental data on jD were needed. These were supplied from the recent work of Wakao, Oshima, and Yagi ( 5 ) . When combined with the data of Wilke and Hougen (6) and Gamson and Thodos ( 3 )a t low Reynolds numbers, the following correlation was obtained jD =
0.84 Re*J1 when 0.01
< Re < 50 (IV-12)
For high Reynolds numbers based on published data, formulation for jD gave jD =
0.57 Re-O.*l where 50
< Re < 1000 (IV-13)
and where Re = G/(a,+p), a, is the external surface area of catalyst pellets per is a shape unit volume of catalyst bed; factor equal to 1.0 for spheres and 0.91 for cylinders.
+
Re: G/o,jifv Figure IV-4. Evaluation of partial pressure gradients between a flowing fluid and the exterior surface of catalyst particles in a packed bed (7) VOL. 53, NO. 7
JULY 1961
515
A t O C .
0.01
0.2
0.4 0.6 0.8 1.0
0.02
0.04 0.06 0.1
IO
I .o
100
1000
lu
I30
particles
From Equations 8 and 10 the partial pressure drop for component A becomes
which accounts for the bulk flow of the fluid. For a general gaseous reaction aA bB --t rR sS,the pressure factor fila is defined as
+
+
(U
,@/A
Figure IV-4 is based upon Equation 14. Similarly from Equations 9 and 11 the temperature drop becomes
=
?r
where x is the direction of transfer, DmA is the average diffusion coefficient of A in the mixture, T is the total pressure. The term fifA is the pressure factor
51 6
~
+ b _-
7
-
J)
_
(IV-17) _
Assuming DAm constant across the gas film gives NA = -
Figure IV-5 is based upon Equation 15. Pressure Factor pf. T h e rate of mass transfer of a component, A, through unit area of a multicomponent gas film for combined diffusion and bulk flow is:
-$'A
D,A
T
(PjA)fm
Ap R T z
(IV-18)
where is the logarithmic mean value ofpf across the film. Effective Diffusion Coefficient in Multicomponent Mixtures. T h e average diffusion coefficient for component A in a multicomponent gas mixture was calculated from Stewart's ( 4 )equation in terms of binary diffusion coefficients as,
where ym = @fA/ir)DAi is the diffusion coefficient for a binary mixture of A and
INDUSTRIAL AND ENGINEERING CHEMISTRY
(7)
j , NA and N i are the molal rates of diffusion per unit area of components A a n d j . For a given chemical reaction the ratios of Ni/N,, are integral stoichiometric values. External Surface Area of Catalyst Particles. The surface area uy of particles per unit volume of packed bed can be readily calculated for geometric shapes of uniform size and shape if void fraction or number of particles per unit volume are known. For estimating the surface area of irregular shaped granules the following relation applies :
where D, is the diameter of a sphere having the same volume as the particle in question and S is the sphericity of particle-i.e., area of sphere having the same volume as the particle divided by the area of the particle. Brown and others (7) give an empirical relation for estimating sphericity from porosity. Most granular particles have a sphericity in the range of 0.7 to 0.8.
S O L I D CATALYSTS V. Effective Thermal Conductivity of Beds of Pellets I n the flow of a fluid through a bed of pellets, the radial loss or gain of heat can be calculated by treating the problem as one of ordinary heat conduction, employing a n effective thermal conductivity in place of the ordinary thermal conductivity of a solid. This effective thermal conductivity has been resolved into its various static and dynamic components by Yagi and Kunii (8) as shown in Figure V-1 for spherical pellets in a packed bed. The static components consist of the following five mechanisms, numbered in agreement with Figure V-1.
1. Radiation and conduction from pore to pore involving a thermal conductivity kf of the fluid and a heat transfer coefficient for radiation h,. 2. Heat transfer by conduction across contact areas between particles involving a heat transfer coefficient h,. 3. Heat transfer by conduction across the film of fluid near contact surfaces involving a thermal conductivity k , of the fluid. 4. Heat transfer by radiation from surface to surface involving a heat transfer coefficient hT8. 5. Heat transfer through the solid particles by conduction involving a thermal conductivity k, of the solid. The dynamic component 6 represents the heat flow by convection in the radial direction due to the mixing of the fluid stream as it passes through the bed with transfer of heat from the solid surface to the fluid stream and involves a n effective thermal conductivity kart. As depicted in Figure V-1, heat fluxes 2, 3, and 4 are in parallel; the combination of these is in series with flux 5. Heat fluxes 1, 5, and 6 are in parallel. From theoretical considerations of the components 1 to 5, Kunii and Smith (5) formulated the following expression for the total static effective thermal conductivity of a packed bed
ratio of thermal conductivities k , / k , as shown in Figure V-2. Values of k , ~vary from 0.07 to 3.0 kcal ./ (meter) (hr.) ( C .) for particle diameters below 0.8 cm. depending chiefly on the separate thermal conductivities of the fluid and solid as shown in Figure V-3 by Kunii and Smith (5). From the average results of many investigators (7, 2, 4, 6, 7, 8), the dynamic contribution to effective thermal conductivity by Yagi and Kunii (8) has been evaluated in (k-cal.)/(meter2)(hr.)(OC.) as (ha?)$= 0.00225 Re
(V-3)
where Re = D,G/p. I n the axial direction, the effective thermal conduc-
tivity for dynamic component (keU)' is (kea)t = 0.00957 Re (V-4) The total static contributions ',k to effective thermal conductivity are highly complex but fortunately become small relative to the dynamic contribution in the operation of commercial catalyst beds. Heat transfer in the axial direction is provided by the flowing stream; the relative flow of heat by conduction in the axial direction is negligible. Where the size of pellets becomes large, for example, exceeding one tenth the diameter of the bed, a separate allowance must be made for temperature gradients and thermal resistance at the wall surface. A number of investigators
FLUID FLOW 4. RADIATION BETWEEN SURFACES
w-
n n f Y -
I.RADIATION BETWEEN PORES
Fiaure V-1. Model for t h i radial transfer of heat in a bed of solid pellets through which a fluid is flowing
\5, CONDUCTION THRWW SOILDS
6. CONVECTION BY FLOWING STREAM
4
I
FLOW OF FLUID
4
c
1
where y = = =
P P
2/3 1.0 (loose packed particles) 0.895 (close packed spheres)
where 0.1952
hm 2(1
- E)
P
I
w-2)
P
=
emissivity
The term, 6,has been evaluated for close and loose packing as a function of the
-01
.02 .04.06 0.1
Figure V-2.
1.0
I
I
IO
,
100
I 1000
4 value vs. thermal conductivity ratio k,/k, (5) VOL. 53, NO. 7
JULY 1961
517
For simple molecules, u is about 3 X 10-8 cm. and a t 1 atm. pressure and 20' C. c is about 3 X l O I 9 p molecules per cc., hence, approximately
10
I
I
I
:
I I I
8 -
I
, ,
8
I
I
/9
,
j ;
%
,
6 -
I
I
1
! .
I
I
t I
!
I I
I,
10-6 X = - cm.
--oAS
wherep is in atmospheres. Knudsen diffusion results in the flow or diffusion of gases through a capillary tube or porous solid where the diameter of the tube or pore becomes much less than the mean free path between molecules. At small pore diameters, diffusion or flow is retarded by impact with pore surfaces rather than by bombardment with other molecules. I n Knudsen flow, the diffusion coefficient D k is proportional to pore diameter, to the square root of temperature, and is independent of total pressure, thus:
e l
.e
-
,4
-
.8
2 -
0.1 .08
-
,04
-
.02
-
.01
-
.06
(VI-4)
P
4 -
I
,002
,001
I
I
,004 ,008
,
.01
.02
.04
.OS.OB .I
,
,2
4
.6
8 1.0
where ii is the average molecular velocity, r is the pore radius, kb is the Boltzmann constant per molecule. For diatomic gases at 20' C. and 1 atm. pressure
Figure V-3. 0.42 (5)
D k = 0.01 cm.2/second for a pore radius have evaluated mass transfer coefficients a t the wall surface in packed beds (2, 3, 7). For spherical pellets Yagi and Wakao (9) recommend the relation
For cylinders Hanratty (3) recommends,
where h, is the heat transfer coefficient, and k, is the thermal conductivity of the fluid.
VI.
Effective Diffusion Coefficient inside Porous Pellets
where N A is the moles of A diffusing through unit area per unit time, D, is the effective diffusion coefficient; and dcA/dx is the concentration gradient in the direction of diffusion x . Bulk Diffusion. The bulk coefficient D, varies with temperature to the 1.75 power, is inversely proportional to total pressure, and in binary mixtures is nearly independent of composition at pressures of 1 atm. or below. I n a binary gas mixture of A and B, Bird, Stewart, and Lightfoot (2) give the following equation for the bulk diffusion coefficient at low pressures in terms of critical temperature and critical pressure
e b y bulk diffusion in large pores e by Knudsen diffusion in small pores ( < I O 3 A., at 1 atm.) e b y hydrodynamic flow under a n external pressure gradient 0 by capillary flow of adsorbed gases or liquids, and o b y surface diffusion of adsorbate over the interior surface of the pellet
For gases, diffusion is the most significant medium of interior transport; for liquids, capillary transport becomes important. The rate of transport of a gas by either bulk or Knudsen diffusion is expressed as NA
dca
-D a
51 8
dx
Db
=
10 crn.%/secondfor a pore radius of 10,000 A.
Combined Bulk a n d Knudsen Diffusion, A summary of the effects of temperature, pressure, and pore radius on Knudsen and bulk diffusion coefficients follows: Knudsen
Bulk
Proportional to 4 T Proportional to pore radius Independent of pressure
Proportional to T1*7' Independent of pore radius Inversely proportional to pressure
Wheeler (6) recommends the following equation to take care of conditions for either or both bulk and Knudsen diffusion,
The transport of a gas inside a porous pellet may take place
(>IO00 A., at 1 atm.)
of 10 A.
where DABis in cm.%/sec.,p is in atmospheres, A4 is the molecular weight, and T is in OK. For nonpolar gases, a = 2.745 X IO-' and b = 1.823. Knudsen Diffusion. For diatomic gases, at one atmosphere pressure and 20" C., the linear velocity of a gas molecule is IO5 cm. per second with a mean free path of low5cm. Because of random molecular motion resulting from molecules bombarding one another, gaseous diffusion is relatively slow compared to the linear velocity of a molecule. The mean free path of a gas molecule is given as
(VI-1)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
x = - 0.707 ,OSC
(VI-3)
where 8 is the average molecular velocity, cm./sec. X is the mean free path, r is the pore radius. Equation VI-6 reduces to bulk diffusion a t high values of r and to Knudsen diffusion at low values of r . Based upon Equation VI-6 the general relation of diffusion coefficient to pore radius is shown in Figure VI-1 for a diatomic gas having a bulk diffusion coefficient of 0.33 cm.2/sec. at 20' C. and 1 atm. where i = IO5 cm./sec. and X = cm. At low pore radii, Knudsen
S O L I D CATALYSTS where k, k,, kd are the thermal conductivities of the pellet, catalyst, and carrier, respectively, and a is volume fraction of the supported catalyst. This formulation is based on the assumption that the individual particles of catalyst are small compared to the pellet, that uniform dispersion exists, and that kd and k, do not differ widely.
VII.
Effectiveness Factor
of
Solid Catalysts I
I
I 1 1 1 1 1
PORE RADIUS,
I
I
I
l l l l l
A:
Figure VI-1. Effect of pore radius on diffusion coefficient of gases in pores at various pressures-bulk diffusion coefficient = 0.33 cm.2/sec. (6)
diffusion prevails and is independent of pressure. At pore radii exceeding 1000 A., bulk diffusion coefficient prevails, is independent of pore radius, and decreases with increase i n pressure. A few values of the ratios of effective diffusivity to bulk diffusivity are as follows:
Material
DJDb
Fritted glass disks and active carbon Compressed Ni turnings 100 to 500 mesh, external void fraction = 0.75 Compressed ZnO powder external void fraction = 0.05 Dried oxide gels
0 . 1 to 0.03 0.15 0.07
Clay pellets Fritted glass Charcoal Silica gel
VI-I. Thermal Conductivity of Individual Catalyst Pellets. Sehr ( 4 ) established experimental values of the thermal conductivities of powders and of individual catalysts in pellet form. For mixed catalysts, where two catalysts appear in the same pellet or where the catalyst is supported on a carrier, the following relation was used by Sehr ( 4 ) in calculating the thermal conductivity of the pellet
0.001 to 0.03
Table VI-1.
Catalyst
Types of Diffusion Encountered in Commercial Catalysts. A brief summary of effectiveness factors and the types of diffusion encountered in industrial catalysts is reported in Table
Properties of Commercial Catalysts
Pore Diameter, A.
Types of Diffusion
< 1000 Knudsen >10,000 Bulk Both types with surface migration 30 Knudsen(retarded by physical adsorption when d < 10 A. 200 Knudsen
Porous nickel film (6000 atoms thick) 16 m.2/g.. Silica-alumina l/g inch 20 to 100 Knudsen DK = lo4 r diameter cm.a/sec. Iron-alumina (ammonia synthesis) 1 atm.; 387-467' C. 11.03 m.2/g. Iron-alumina (ammonia synthesis) 6000 atm., 500' C. 5 mm. diam.
Bulk Da = see.
Effectiveness InvestiFactor and Modulus gator (9) (9) (9)
(7)
m = 0.69 E = 0.98
m = 0.82 E = 0.78 m = 0.75 E = 0.98 (not affected by size in range) 10-14 to 35-40 mesh 0.674 crnB2/ E < 0.8 at high rates E > 0.8 at low rates (little affected by size in given range)
(1)
(7) (3)
(7)
The effectiveness factor is a convenient number for use in reactor design where rate data obtained for a catalyst particle of given size and porosity is applied to the same catalyst differing only i n size and porosity. However, the use of this factor in experimental work in establishing a reaction model and in establishing energies of activation is a source of considerable error especially when high gradients occur within the catalyst particle. Under these conditions, the use of a catalyst with a n effectiveness factor differing from nearly unity should be avoided. Solid catalysts are used commercially in the form of cylindrical, spherical, or granular pellets with the active material supported on a porous carrier and distributed throughout the pellet. The catalyst material on the inside of the pellet is less available and less effective than that supported on the exterior surface. The effectiveness factor of a catalyst pellet is the ratio of the actual reaction rate per unit mass of pellet when the catalyst is distributed throughout a porous pellet to the rate where the available catalyst is restricted to the exterior surface of the pellet when in both cases the conditions of temperature, pressure, and composition of the reacting fluid remained the same as a t the exterior surface of the pellet. The effectiveness factor may also be defined as the ratio of the reaction rate per unit area of catalyst when distributed throughout a porous pellet to the rate per unit area of a surface where the catalyst covered only the exterior surface of a n impervious pellet when under both cases the reacting system at the exterior surface of the pellet is under the same conditions of temperature, pressure, and composition. These two definitions are equivalent. The catalyst dispersed throughout a pellet is less effective than when exposed directly to the reacting system at the exterior of the pellet because of the progressive reduction in concentration of the reactants toward the center of the pellet as the reactants diffuse inward and products diffuse outward. Where isothermal conditions prevail throughout the catalyst, the effectiveness factor is always less than unity and diminishes VOL. 53, NO. 7
JULY 1961
519
1 .o
reciprocal of the average pore diameter c. Also, in Thiele's modulus, k' is replaced by the reaction velocity constant k of the uncatalyzed homogeneous reaction. By use of an effectivenessfactor, the reaction rate of a first order irreversible reaction A R catalyzed by a porous pellet per unit volume of bed then becomes
0.6 8.6
0.4
-
0.2
0.1
w .08 .06
.04
wherek" = k' (1 - E). The term (1 e) is introduced to include only the reaction catalyzed by the pellet. To retain its dimensionless form, the modulus for a second order reaction must be altered to m2 = R l/(k'acAi)/Dc. Effect of Different Shapes in the Effectiveness Factor. By use of the
.O 2
.o I 0506 0810
Figure VII-1.
20
30 4 0
= k'acA
(VII-1)
where a is the internal surface per unit volume of catalyst pellet, k' is the reaction velocity constant, and CA is the concentration of A. For the homogeneous irreversible reaction, the rate per unit volume in the absence of a catalyst is given as r = ~ c A ;however, these two rates are unrelated, and k is not equivalent to k' nor to k 'a. Where the surface reaction is rate controlling, the constant k' includes the effect of adsorption of both A and R thus k'a =
M
(b
A
+ KACA + KRCR)
30 $0 50
Stewart, and Lightfoot (2) derived the average reaction rate taking place inside the pellet as r p = 4m R D,C A ~ 1 coth
4%
[
dg ] R
(VII-4)
Where the concentration of A is the same throughout the pellet as at the exterior surface the rate per pellet becomes T*p
4 T R3 k'acai 3
= -
The ratio r,/r*= factor E or E =
where m = R
(VII-5)
is the effectiveness
3 ( m coth m - 1) m2
4% -3
(VII-6)
a dimensionless
group, termed the modulus. The relationship of E to m is shown in Figure VII-1. Equation 6 for the effectiveness factor is similar to that originally developed by Thiele (7),but the modulus differs by using the internal area a to replace the
Table VII-1. Effectiveness Factor for Plates, Cylinders, and Spheres
(VII-3)
No surface coverage
[From Aris ( I ) ] E =
11L 3
where D,is the effective diffusion coefficient, r is the radial position for a spherical pellet ofradius R. For a porous spherical pellet of radius R under the boundary conditions that C A = c A ~a t the exterior surface of the pellet from Equations 1 and 2, Bird, INDUSTRIAL AND ENGINEERING CHEMISTRY
0.1 0.2 0.5 1.0 2.0 5.0 10.0
Fiat
E
4% -
modulus
(VII-2)
where 6 = 1/RT The rate of diffusion of component A inside a porous pellet is defined as
520
20
Effectiveness factor vs. modulus of catalyst pellets
with increase in pellet diameter and with increase in reaction rate and decreasing with increase in the diffusion coefficient of the reactants. In highly exothermic reactions catalyzed by a pellet of low thermal conductivity, a steep temperature gradient is established within the pellet with a corresponding gradient of reaction velocity constants diminishing towards the external surface. Under these conditions, the effectiveness factor may actually exceed unity. For a first order irreversible reaction A + R the rate of reaction of component A per volume may be expressed as 7.4
6 0 8010
E
plate
Cylinder
Sphere
0.997 0.987 0.924 0.762 0.482 0.200 0.100
0.995 0.981 0.892 0.698 0.432 0.197 0.100
0.994 0.977 0.876 6.672 0.416 0.187 0.097
ii
= VJS,
where V, is
the volume of pellet and S, is its external surface area, Aris (7) found that the effectiveness factor was nearly independent of shape, as shown in the Table VII-1 for flat plates, cylinders, and spheres. The modulus is related to the modulus m by the ratio A = m / 3 and for a sphere V p / S p= R / 3 . Importance of the Modulus in Reaction Kinetics. To illustrate the effect of pellet modulus on reaction rates, product selectivity, and catalyst poisoning, Figure VII-1, relating the effectiveness factor E to the modulus m for first order reactions using spherical pellets, is conveniently divided into three ranges. Range I. For values of m < 1, where E is nearly unity Range 111. For values of rn > 10, where E varies inversely with particle size (or with the ratio 3V,/S,) from 0.27 to 0 Range 11. Covers intermediate values of m from 1.0 to 10, where a marked curvature in the E - m relationship prevails In region I where values of rn are below unity ( m < 1.0) the diffusional resistance of the pellet is negligible. As long as operating conditions remain in this region the reaction rate per unit mass of catalyst is unaffected by particle size, and variations in the modulus due to temperature, concentration, reaction order, and volumetric change have little influence on the effectiveness factor. This range is characterized by particles of small diameter, large pore diameters, and low internal surface areas. As long as the modulus remains in this region, product selectivity in reactions proceeding consecutively or in series will be affected by size of pellets or by nonselective poisoning. Weisz and Porter (8) report the following minimum values of effective
S O L I D CATALYSTS diffusion coefficients necessary to maintain m less than unity for given particle sizes where values of k’a vary from 0.05 to 0.50. Radius of Particle, Cm. 1.0 0.5 0.1
Minimum Range of D,, Cm.Z/Sec. 0.05 to 0 . 5 0.01 to 0.1 0.0005 to 0.005
I n region I11 where values of m exceed 10 (rn > lo), the internal resistance to diffusion is rate controlling, and the reaction rate per unit mass of catalyst decreases inversely with particle size. This range is characterized by particles of large size, small pore radii, high internal surface, and Knudsen diffusion prevails. In this region the effectiveness factor is greatly influenced by changes in temperature, concentration, reaction order, and volumetric changes. I n this region, the effectiveness factor is decreased exponentially with increase in temperature, is decreased directly with particle size, and is increased with the square root of the diffusion coefficient. I n general, the modulus increases exponentially with temperature with a corresponding decline in effectiveness factor. I t is thus possible in a given reaction for internal diffusion to be rate controlling a t high temperatures and to offer negligible resistance a t low temperatures, and for the effectivenessfactor to be nearly unity at low temperatures but to become low at high temperatures. This temperature effect results in calculating erroneous activation energies when velocity constants are measured over a wide temperature range without correction for internal diffusion. This effect is illustrated by WheeIer (9, p. 268) in Figure VII-4. When observed reaction velocity constants are plotted against the reciprocal of absolute temperatures on a semilog plot, correct values of the energy of activation are obtained for pellets with large pores where the effectiveness factor is unity. However, with small pores where the effectiveness factor is low and internal diffusion becomes rate controlling, the calculated activation energy drops to one-half its true value. This situation is revealed in curves A , B, and C of Figure VII-4. A true value of the energy of activation (AHA = 11,000 kcal./kg.-moles) is obtained from curve A where the modulus is nearly zero and effectiveness factor is unity. I n curve C, with a high modulus, the slope of the curve gives an apparent energy of activation, one-half its true value. For the intermediate case, curve B, a curved relationship results with an indeterminate value of AHh. The relation of E to m as shown in Figure VII-1 applies to first order reac-
tions where no change of volume occurs within the pellet. Where Knudsen diffusion prevails, a change in number of gaseous moles results in a change of pressure within the pellet rather than in a change of volume so that the relation of E to m is unaltered. However, where volume changes occur as with bulk diffusion the relative of E to m changes. Under these conditions, Weisz and Porter (8) report that the E-to-m relationship of Figure VII-1 can be retained provided for the same value of E, m is multiplied by 0.74 for a 2 to 1 volume expansion and multiplied by 1.35 for a 1-to-2 volume contraction. Effect of Catalyst Structure on Product Selectivity. Where several reactions occur simultaneously in an industrial reaction, high selectivity is primarily obtained by the selection of the catalyst rather than by changing the size and structure of the pellet. However, size and structure do have secondary effects on selectivity. Where two or more reactions proceed simultaneously and in parallel, the ratio of the rates will be independent of pellet size or structure as long as the modulus m is below unity. However, for conditions of high moduli (m > lo), the slowest reactions will be favored by an increase in particle size 01 by an increase in temperature. I n a consecutive reaction A 2 B C where kl is greater than kz, the selectivity in the formation of B is favored by a pellet having large pores and low modulus. This effect is shown in Figure VII-2 from the data of Wheeler (9)* Effect of Structure on Poisoning of Catalysts. The combined effects of catalyst poisoning and pellet modulus on the reduction in catalyst activity are illustrated in Figure VII-3 as reported by Wheeler (9). For a nonporous catalyst where the poison is distributed uniformly over active sites, the catalyst activity decreases linearly with the fraction of sites poisoned as shown in curve A. For a porous solid with uniform distribution of the poison, the decline in activity follows curve B with an initial slow decline, and with a curvature concave to the abscissa. Where the pellet modulus is high (m > 10) diffusion becomes rate controlling, poisoning occurs preferentially near the exterior surface of the pellet with a rapid initial decline as shown in curves Cand D, with a sharp convex curvature towards the abscissa. Figure VII-4 is a semilog plot of the ratio of reaction rates us. reciprocal temperature showing the highly disturbing effects of interior diffusion and poisoning in attempting to establish energies of activation by such plots of experimental data, as taken from the work of Wheeler (9). In this figure, a is the fraction of sites poisoned. The ordinates represent
m‘
O
r
I
P I
SMALL PORES (m> 3 )
96 TOTAL CONVERSION OF A
Figure VII-2. Effect of pore size on selectivity of reaction (9) A
__f
ki
B
__f
C where kl/ka = 4.0
kz
Figure Vll-3. Effect of poisoning on catalyst activity ( 9 ) A. Nonporous catalyst. E. Uniform adsorption of
poison C. Preferential adsorption near surface, mo = 10. 0. Preferential adsorption near surface, rno = 100.
Figure Vll-4. Effect of modulus and poisoning on evaluation of activation energy, AH* 1 1,000 kcal.-true value
(9) A. large pores, no poison, m = 0. E . Fairly large pores, 90% poisoned, m = 0. C. Small pores, no poison, m = 2.0. D. Medium pores, 50% poisoned, m = 0.5. E. Small pores, 50% poisoned, m = 2.0
VOL. 53, NO. 7
0
JULY 1961
521
show the effects of pore structure and internal thermal conductivity of the pellet with establishment of internal temperature and concentration gradients. All these mathematical studies on effectiveness factors, including the original investigations of Thiele (7), have been based on the assumption that the reaction is homogeneous with neglect of the effects of adsorption. However, reactions catalyzed by solids depend upon adsorption of one or more of the reactants. The catalyst does not act as a mirror to hasten the homogeneous reaction. Merely including the internal area of the catalyst with the reaction velocity constant does not correct for surface adsorption. Shindo and Kubota (6) have studied the effectiveness factor of pellets used in the synthesis of ammonia and have shown that this factor varies with the percentage conversion and pressure as
ratios of reaction rates under given temperature conditions and extents of poisoning to the rate obtained a t 200' K. with no poisoning. Only under condition A is a correct value of the energy of activation obtained from the slope of the line. Under other conditions of operation, the mechanism of the reaction is complicated by poisoning and by internal gradients. Effectiveness Factor Considering Adsorption. The effectiveness factor as developed in Equation 6 and shown in Figure VII-1 and Table VII-1 is for first order reactions assuming no temperature gradient inside the pellet and neglecting adsorption of reactants and products. Schilson (5) has calculated the effect of temperature gradients inside the catalyst pellet on reaction rates with corresponding effects on the effectiveness factor, Mingle and Smith ( 4 ) have carried these investigations further to
I
I
1
shown in Figure VII-5. They converged these relationships into a single curve relating the effectiveness factor to a modulus defined as
where x , is the surface concentration of "3. This modulus bears little resemblance to the modulus of first and second order homogeneous reactions. Recently Chu and Hougen ( 3 ) have established the effectiveness factor of a first order reaction catalyzed by a solid surface including the effect of adsorption where the reaction model is expressed by the relation (9)
whereas Equation 6 is based upon the first order homogeneous reaction equation, r = EkapA. For a flat plate catalyst, the modulus for reaction based on Equation 9 becomes
-
m = R 0-5 -
-
where v is the molal volume of the gas phase. The relation of E to this modulus is shown in Figure VII-6 for various parameters of = I / ( K A x ) . A high value of { corresponds to a low surface coverage. Thus, where adsorption terms are considered, the effectiveness factor for a given value of k, and D, is high for low coverage and low for high surface coverage. At high surface coverage ({ < l ) , the effectiveness factor becomes dependent also upon concentrations a t the external surface, decreasing markedly with decrease in external surface concentration. Where the equilibrium adsorption constant becomes zero, the E - m relationship becomes equal to that of the uncatalyzed reaction. For low surface coverage, the flat portion of the curve approaching unit extends over a wide modulus range whereas for high coverage the range is short. A general development of the effectiveness factor for various models of catalytic reactions and for spherical and cylindrical pellets is a timely field for mathematical and experimental study. The great uncertainty in values of this factor is sufficient reason for restricting experimental determinations of reaction models to exterior surfaces of impervious pellets or to small catalyst particles where the reaction rate per unit mass is unaffected by diameter.
r
W
02 -
0.I 01
I
I
I
I
I
I
02
03
0.4
0.6
08
1.0
20
30
Figure Vll-5. Effectiveness factor for NH3 synthesis as function of concentration450' C., 100 atm. (6)
VIII.
Dynamic Characteristics of Catalyst Beds Figure Vll-6.
Effectiveness factor for reaction model J
.
-
EksKAP.4
1 KAPA
522
INDUSTRIAL AND ENGINEERING CHEMISTRY
Properties pertaining to hold-up and distribution of residence times in catalyst beds cannot be evaluated by steady-state experiments on reaction rates. In the
S O L I D CATALYSTS flow of a fluid, a uniform velocity profie across the exit surface of a bed under steady-state conditions may not necessarily indicate plug flow. Even where back mixing is absent, hold-up is encountered a t surface and point contacts between particles and by the adsorption of fluids by the porous pellets. The inventory of fluids adsorbed under static conditions can be calculated from separate equilibrium adsorption studies, assuming that equilibrium adsorption is also attained under steady state operation. The hold-up due to retention a t surface contacts can be arrived a t by dynamic studies involving a step-wise or pulse-wide change in composition of the feed or by similar introduction of a tracer provided adsorption of the tracer does not occur. Such studies reveal not only the time lag in adjusting to a new steady state but also indicate variations in external surface available with change in mass velocity. Where neither adsorption nor contact retention is involved, step-wise and pulse-wise changes in composition reveal the separate effects of hold back due to back mixing. With the flow of liquids as in trickling beds, the hold back due to surface contacts and the time lag in attaining steady state composition of the liquid within the catalyst pellet are much greater than for gases. Studies of such dynamic characteristics represent refinements which are undergoing recent development. For steady, continuous plug flow the volume of a reactor for a given feed rate, F, and conversion x is given by
where r is the reaction rate per unit volume of reactor and varies throughout the reactor. Under conditions of steady state bulk flow reactor volume can be calculated directly without reference to residence time.
v, = FAX
(2)
where r is calculated from the composition of the stream leaving. For incomplete mixing the following interpolation is given as a convenient design procedure for calculating reactor volume, for a given feed rate, given degree of conversion and for similar conditions of feed. V v= ~ Vrp eH IV,c - V,P] (3)
+
where H i s the hold-up in the reactor and subscripts P,I, and C refer to condition of plug flow, incomplete mixing, and complete mixing, respectively. For plug flow, H = 0, and for complete mixing, H = ' / e where e = 2.7183. From an F diagram established, for example, by use of a radioactive tracer, for conditions of incomplete mixing the hold-up H may
I.o COMPLETE
MIXING H =
INCOMPLETE MIXING H < d
H.0
P,LUGFLOW
1.0
0
vr V
Figure VIII-1.
F diagram for a step change
0.8 1
i
P > 2 0 0 ll
X
0
8
4
P = VOWME Figure Vlll-2. terns
12
POSITION IN REACTOR
20
14
litem
Conversion gradients in a flow reactor under different flow pat-
be evaluated and the reactor volume computed from values calculated for piston flow and complete mixing. For plug flow, H = 0, for complete mixing, H = '/e = 0.359, for incomplete mixing, H < ' / E . T o establish hold-up in a given reactor by means of a tracer technique an F diagram is constructed (Figure VIII-1) from experimental data. An F diagram represents a plot of c/c, us. (vT)/Vwhere c,,c are the concentrations of a tracer in the feed and a t a given position in the reactor, respectively, u is the volumetric rate of flow, Vis the total reactor volume, and T is time. In Figure VIII-2, the concentration gradients at different positions in a 20liter flow reactor are shown under four oatterns of flow. x Con-
Type of Flow Plug Incomplete mixing Complete mixing
Hold-up H 0 0.18
0.359 =
version Leaving 0.728 0.'12
0.547
1I n
Channeling
-
o.ii8
0.487
I n these four cases the reaction rate can be obtained from the slope of the curve,
r
=
dx
F dV - only under conditions of plug
flow. Under other conditions, the change in x is due to mixing as well as to reaction. For complete mixing, the slope is zero, and yet a finite reaction rate prevails. It follows that for conditions intemediate to plug flow and complete mixing the reaction rate cannot be obtained by the slope of the curve.
IX.
Reaction Rate Models
Any rigorous approach to the kinetics of reactions catalyzed by solid surfaces through the medium of quantum mechanics is recognized by experts in that field as quite hopeless within the next generation. The Schrodinger equation has been used precisely for predicting the rate of the para-ortho hydrogen conversion . with atomic hydrogen, but extension to complex reactions is hopeless. For example, in the simple free radical reaction of atomic hydrogen with methane to VOL. 53, NO. 7
JULY 1961
523
form methyl radical and molecular hydrogen, six nuclei and eleven electrons are involved requiring 51 coordinates. Hulburt ( 8 ) states a direct numerical integration of the Schrodinger equation for this relatively simple reaction is a physical impossibility within the supply of paper available. The chemical steps are similarly established best where diffusional resistances are made negligible by operation a t high mass velocities and small particle sizes and under conditions of plug flow. Even under such conditions combined with the simplicity of uniform temperature and pressure, rate data alone are insufficiently accurate to establish a consistent chemical model. Measurements of reaction rates alone afford too crude a device for precise evaluation of a reaction model. The numerous constants involved in even a simple mathematical model require machine calculations to establish dependable values where least squares methods are to be applied to nonlinear equations. In the chemical model of reaction rates. adsorption of at least one of the reactants has been verified as a necessary prerequisite. Chemisorption is usually involved and is a requirement for high selectivity although physical adsorption also accelerates reaction rates by providing a localized concentration of reactants. Physical adsorption of one reactant and chemisorption of another is especially favorable due to the high mobility of physical adsorption. The Watson equations (6,7) for chemical reaction models have been widely accepted and in general are the most comprehensive for establishing rate-controlling chemical steps. They have proved reliable despite the assumption of Langmuir adsorption. This assumption seems justifiable when reactions are restricted to a narrow spectrum of highly selective sites and not to total adsorption and also inasmuch as the kinetic equilibrium adsorption constants enter as ratios. Individual kinetic equilibrium constants vary with surface coverage but the ratios vary little, These equations require extension to cases where adsorption is not restricted to competitive sites and where complete surface coverage is impossible because of steric hindrance of a n immobile adsorbate on multiple sites. The objectives of correlating organic structure in a series of homologous reactions with the reaction velocity constant and kinetic adsorption equilibrium constants have not been realized ; only a few experimenters have contributed towards this. I n the catalytic cracking of eight alkylbenzenes Rase and Kirk (70)found that the rate-controlling chemical step is a surface reaction taking place on a single site of the catalyst. I n each case benzene is the only product adsorbed, and the branch radical is split off without
524
adsorption, thus, adsorbed alkyl benzene 4adsorbed benzene RH. For each of eight alkyl benzenes, the values of the adsorption equilibrium of the products are about the same in agreement with the theory that benzene is the only product adsorbed. Similarly, the value of the energy of activation was nearly the same for each alkyl benzene, namely, 14.4 kg.cal./g.-mole, corresponding to the breakage of the same carbon linkage on the benzene side chain whereas the value of the entropy of activation increased with the increased molecular complexity of alkyl benzene. This accounts for the increase of reaction rate with increase in molecular complexity. The apparent equilibrium constants obtained from kinetic studies are related to bond energies between benzene and alkyl radicals as shown in Figure IX-1. The collaboration of many Russian scientists under Balandin ( 7 ) in the course of the last 6 years has made a significant breakthrough in correlating reaction rates with the bond energies involved in reaction and chemisorption. In several hundreds of reactions involving dehydrogenation and dehydration, a single mathematical model was used with consistent results in the prediction of reaction rates with different catalysts. In this method, the energy of activation of the rate-controlling step of a given reaction is calculated from the bond energies involved in adsorption and reaction. The bond energies involved in the rupture of reactants and formation of products are calculated from thermochemical data; the bond energies involved in adsorption of reactants and products on active sites are calculated from kinetic data. The atomsinvolved in the bonding of adsorbate molecule with an active site of the solid are obtained from molecular configuration postulated by the Balandin multiplet theory. Tables of bond energies between atoms
+
2001
QlO'
'
I
Relaline bond strength,
1
I 10
I
I 14
kcollmole
Figure IX-1. Effect of relative bond strength of five alkyl benzenes on their adsorption equilibrium constants
(70)
INDUSTRIAL AND ENGINEERING CHEMISTRY
in molecular compounds have been compiled by Cottrell ( 5 ) ,Sernenov (TI), and Pauling ( 9 ) . Single bond energies involved in chemisorption have been evaluated by Balandin (2). The method of calculating the activation energy of a reaction is illustrated by considering the reaction AB
+ DC-,X+
-+ AD
+ BC
when catalyzed by a solid. The heat of this reaction may be expressed in terms of energies of the bonds undergoing dissociation and association, thus ti
-
= -QAB
QCD
+
QAD
+
QSC
AH
=
(1X-1)
where u < 0 for an exothermic reaction. The summation q of the bond energies of the various atoms and molecular fragments chemisorbed on active sites of the catalyst is given as 4 =
QAK
+
QBK
+
+
QCK
QDK
(IX-2)
The summation s of the bond energies involved in the reaction results in s = QAB
4-
QCD
+
QAD
+
QBC
(IX-3)
The energy of adsorption of the reactants becomes E' =
-QAB
-
QCD
+
+
QAK
QCK
+
QBK+
QDK
(1x4)
and the energy of desorption of the products becomes E" =
+QAD
4
QBC
QCK
- QBK-
QAK
-
QDX
(1x4)
By combination of Equations 2-5
+ + ~ / 2+ ~ / 2
E' = p - ~ / 2 ~ / 2 E" = - 4 E'
+ E"
= u
(IX-6)
(IX-7) (IX-8)
The value of E' is related approximately to the energy of activation by the relation AH* = 0.75 E' where the factor 0.75 accounts for the fact that in the formation of the activated complex deformation occurs rather than complete dissociation. For any given reaction the values of E' and E" depend upon the nature of the catalyst; according to Balandin the minimum value of -E' is reached for an ideal catalyst when E' = E" = 4 2 and q = s/2. Bond Energies with Catalysts. For a particular chromia catalyst the bond energies of different atoms with active sites of the catalyst are given by Balandin as follows Xcal./Atom with hydrogen Q H K = 56.9 with carbon Q C K = 16.3 with oxygen Q D K = 57.9
In the dehydrogenation of cyclohexane the catalyst bonds shift from hydrogen to carbon with the release of hydrogen.
S O L I D CATALYSTS
H H H H
K K
F i t
+
q = ~ Q H K
2(56.9)+ 2(16.3) = 146.4
~ Q C R=
I n the dehydrogenation of isopropyl alcohol, the catalyst bonds shift from hydrogen to carbon and oxygen with the release of hydrogen CH3 CHs
CH3 CHj
\/
\ /
H--OH
-+
K q =
K
f
~ Q H K
QCK
C=O+Hz
. .
K
f
K
= 2(56.9)+ 16.3 f 57.9 = 188
QOK
I n the dehydration of isopropyl alcohol, the catalyst bonds shift from hydrogen to carbon with the release of water CHR CHI adsorbates with various chromia catalysts (3)
K 4 =
r------
K QHK
-
24
-k
QOK
-k
56.9 f 2(16.3) = 147.4
~ Q C K=
57.9
+
22 -
Reaction Model. Balandin ( 3 ) assumed the same model for all catalytic processes in the dehydrogenation of hydrocarbons, amines, and alcohols and in the dehydration of alcohols, namely, the chemisorption of the reactant on dual or triple adjacent active sites with the organic product adsorbed on the sites initially occupied by the reactant and with the release of Hz or HzO without adsorption. The variation of the equilibrium constant of chemisorption with surface coverage was avoided by assuming that this effect was Similar for products as for reactants so that a constant ratio of equilibrium constants could be used for each product independent of coverage, thus in the reaction A +R HZ
20
-
++ a
2
7
16-
E? x
*=a I4
-
IP-
+
.. IO1
I
67
where KA, KR, K H are the equilibrium adsorption constants for components A . R,and Hz respectively. O n this assumption and with this model, the following general rate equation was derived for the irreversible reaction,
"P where
€* p A (IX-9) ( - '€T)
AS+ and AH* are the entropies
68
I 69
I I 70 71 CATALYST NO.
I 72
I
73
Figure IX-3. Effect of preparation of zinc oxide catalyst on the energy of activation for the dehydrogenation of isopropyl alcohol at 345' C. Isopropylene '
where
0.60 - 81 00
-
135O
and enthalpies involved in activation. Values of Z R and z n may then be resolved into their corresponding thermodynamic properties,
- AHORA - So%(Ix-lo) RT R AH'aa = AH'R - AH'A
lnzg =
H2
1.19 - 8500 13.4
z AHo
+
ASORA =
MOR
-
-14.2
lnzx = where
A
~
AS'HA (Ix-ll) - -R T +T AH'HA
~
A H O ~- A H O ~ AS'HA = A S O H - ~ S ' A
H= A
The values of AS*, AHh, and z are related to molecular structure of the reactants and products and can be related to energies of the bonds being broken and VOL. 53, NO. 7
JULY 1961
525
I
141
a
I
I
I
I
1
1
.L 0
J- 2 I
2
I
0
Y U
-
-
Ni-Ai,O,catalyst-
d,=I.O
d,=O
I
I
II
10
Pd-Asbestos catalyst -c
A G " = O . AH'=o, AS"=O I
I
12
13
I
9 RUN
I
NO.
Figure 1x4. Dehydrogenation of arornatics-Cr2O3-asbestos 4 4 8 ' C. ( 1 958)(3)
526
INDUSTRIAL A N D ENGINEERING CHEMISTRY
catalyst, 41 5-
formed in producing the activated COIIIplex and products. These values are also related to the nature of the catalyst and for any one particular catalyst are greatly influenced by methods of preparation and sources of raw materials. Balandin and coworkers have established the constants in Equation 10 for hundreds of reactions involving dehydrogenation of aliphatic hydrocarbons, aromatic hydrocarbons, alcohols and amines, and dehydration of alcohols in combination with many catalysts, such as metals supported on carriers and without support (for example, palladium on asbestos, nickel on alumina, platinum on asbestos, and nickel on quartz); and oxides (for example, zinc oxide, chromia, and manganese oxides). Experiments were also made on the same catalysts prepared by different methods. By the aid of this model and the multiplet theory of adsorption, Balandin and Ponomarev ( 4 ) have extended their investigation to complex transformations in the hydrogenation of furan derivatives with good prediction of the sequence of reaction rates and the selection of suitable catalysts. From catalytic rate studies, bond energies of the atoms of hydrogen, oxygen, and carbon with chromia prepared by a variety of methods have been established. In Figure IX-2, energies of activation are related to the summation of the bond energies in adsorption for three reactions-namely, the dehydrogenation of cyclohesane and methylcyclohexane, the dehydrogenation of isopropyl alcohol and formic acid, and the dehydration of isopropyl alcohol. Activation energies vary fourfold depending upon how the catalyst was prepared. Similar effects were observed in the dehydrogenation of isopropyl alcohol using zinc oxide catalysts prepared by different methods of manufacture (Figure IX-3). In Figure IX-4 the relations of AS* and AHi with structure are shown in the dehydrogenation of four aliphatic amines with two catalysts, nickel on alumina, and palladium on asbestos; the latter catalyst producing faster rates for the same systems. I n Figure 1x4, these same properties are related to structure in the dehydrogenation of aromatic hydrocarbons using a chromia-asbestos catalyst at about 430' C, and in Figure IX-6 in the dehydrogenation of alcohols using an oxide catal!st at about 340" C. The correlations or Balandin and coworkers are impressive even though the reaction model lackr the flexibility of the Watson models (6, 7). From the published literature, no indication is given as to whether or not srii-pdce concentrations and surface tempera tures were calculated or whether or not temperature and concentration gradicn t s inside pellets were considered. The rate equation lacks flexibility i n coilsidering the number of adjacent sites involved in chemi-
S O L I D CATALYSTS 18
l6
1
I
I
55
53
I
I 56
I
6
t
I4t
12-
*n
%I= 5
10-
I
I
0
*:
=a 8 -
6l
58 I
I
52
I
lo
RUN NO.
-;
r E 0
-o--uI \ I
1
-4-5A
I
u-
E
4-
0 9-
+-
-Y-
-7-
E
0
-6-
-+-Q
Figure IX-6. Effect of structure on energies of activation AH* entropies of activation AS/fR, and relative adsorption contents ZE, on the dehydrogenation of alcohols--oxide catalyst 340' C. (3)
sorption, considers only the exponent of unity for the denominator term in Equation 10, and omits an addition of 1 to this term. Acknowledgment Credit is acknowledged to the following for permission to reproduce some figures and tables used in this survey: R . W. Cranston and F. A. Inkley, D. Kunii and J. M. Smith, R. S. Kirk and H. F. Rase, A. Wheeler, R. A. Sehr, and M. Shindo and H. Kubota, and also to Esso Research and Development Co. for initiating this project.
Nomenclature = constant
internal surface per unit volume of pellet = external area of pellets per unit mass = external area of pellets per unit volume of bed = external area of pellet = component = constant = 1/RT = component = number of moles of B = concentration, moles per unit volume = concentration, moles of A per unit volume = concentration a t external surface of catalyst, moles of A per unit volume = molal heat capacity a t constant pressure =
heat capacity per unit mass at constant pressure = pore diameter = componentD = bulk diffusion coefficient in a binary system = bulk diffusion coefficient = effective diffusion coefficient = effective diffusion coefficient, radial direction = Knudsen diffusion coefficient = mean diffusion coefficient of A in a multicomponent system = equivalent particle diameter = diameter of cylinder = base of natural logarithms = effectiveness factor = energy of adsorption = energy of desorption = mass velocity based unit area of total cross section = molal mass velocity = heat transfer coefficient = heat transfer coefficient between surfaces of contact = heat transfer coefficient by radiation, solid to solid = heat transfer coefficient a t wall = enthalpy change a t standard state = enthalpy of activation AHA, AH,, AH, = molal enthalpies of adsorption j~ = mass transfer number 3~ = heat transfer number k12 = term in Cranston-Inkley equations k, = surface reaction velocity cnnstant k, kz, k ' , k", kd = reaction velocity constants kb = Boltzmann constant per molecule =
k,,, k,,, k,,, = effective thermal conductivities of bed kf = thermal conductivity of fluid ko = mass transfer coefficient in gas phase kL = mass transfer coefficient in liquid phase = thermal conductivity of solid k, KA, KB, KC,K D , KE, &, Ku = adsorption equilibrium constants L = thickness of bed m = mass per molecule m = modulus of porous pellet M = molecular weight n = number of moles N = Avagadro number NA, N B , N R , N s = molal rates of transfer = Nusselt number Nu p = pressure p = emissivity PA, p B = partial pressures in bulk fluid P A I , pBc = partial pressures a t interface pc = critical pressure p, = pressure factor pL = exit pressure p? = partial pressure in capillary at radius r Pe = Peclet number = (D,u)/D, Pr = (C,p)/k = Prandtlnumber qm = rate of heat transfer q = summation of bond energies of reactants and products Q = ( m A AHA)/(am$'CpG~) Q A B , Q C D , Q A B , QBC = bond energies between atoms in reactants and products Q A K , Q B K , Q C K , Q D K = bond energies between atoms in adsorbates and adsorbents r = number of moles of R r = radial distance r = pore radius = reaction rate for component A rA = reaction rate per unit mass of rm catalyst pellet = reaction rate per pellet *; = reaction rate per pellet when
E = l
R R
= radius of pellet = gas constant
R
=
RlZ
= term in Cranston-Inkley equa-
Re
Rh
= Reynolds number = G/a,+p) = hydraulic radius = e/ (av+)
S
=
component
R
tions
S
summation of bond energies of adsorbates and adsorbents = number of moles of component
S internal surface area = sphericity sc = Schmidt number = p / ( p D ) = external surface area of pellet = change of entropy at standard state ASH, A&, AS8 = molal entropies of adsorption AS* = entropy of activation t = temperature, O C. 1 = thickness of adsorbed layer = thickness of adsorbed layer on 4 pore surface of radius r = temperature, K. T = critical temperature Tc u = interstitial linear velocity of fluid
SI2 S
=
2s.
VOL. 53, NO. 7
JULY
1961
527
&‘a
UP 0,
V, X
A’* Y v,
AH
= heat of reaction average linear velocity volume of pellet incremental volume of gas adsor bed = volume of pores of radii bedr tween r and r = direction of mass transfer = activated complex = mole fraction = mole fraction of pressure factor = = = =
U
+
PI
z
= axial distance
zR
=
Z”,
ratio of equilibrium adsorption constants = KF!KA = mean compressibility factor
Greek Symbols
1961
- I - - .
a = fraction of active sites poisoned @ = constant y = constant A = finite difference E = external void fraction 4 = terms in Kunii-Smith equation 4 = shape factor X = molecular mean free path .i = pellet modulus = m / 3 p = viscosity P = total pressure ?r = 3.1416 p = density u = molecular diameter 2 = summation
Bubscripts
A , B, C, D = components a = axial D = diffusion e = effective = fdm f = fluid f = gas G z = surface of particle j = component L = liquid lm = logarithmic mean rn = mean m = per unit mass 0 = a t saturation = particle P = at constant pressure P r = radial S = solid t = dynamic effect U = radiation V = per unit volume Literature Cited General
(1) Baker, M. McD., Jenkins, G. I., Advances in Catalysis 7, 1-43 (1953). (2) Balandin, A. A., Itvest. Akad. Nauk. Otdel. Khim.
Nauk.
No. 4,
624-38 (1955). (3) Balandin, A. A., Advances in Catalysis I O , 96-130 (1958). (4) deBoer, J . H., Zbid., 8, 17-161 (1956). (5) Zbid., 9, 131-42 (1957). (6) Emmett, P. H., Ibid., 1, 65-89 (1948). (7) Garner, W. E., Ibid., 9, 169-86 (1957). (8) Gray, T. J., Darby, P. W., J . Phys. Chem. 60, 201 (1956). (9) Kwan, T., Advances in Catalysis 6 , 67120 (1954). (10) Maxted. E. B., Ibid.,. 3,. 129-77 ‘ (1951). (11) Schwab, G. M., Ibid., 2, 251-66 (1950). ’
528
I. Pressure Drop a n d Size of Pellet (1) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, 1960. (2) Chu, Chieh, Ph.D. thesis, University of Wisconsin, Madison, Wis., 1961. (3) Chu, Chieh, Hougen, 0. A., Am. Inst. Chem. Engrs., Cleveland, Ohio, May
(4) Ergun, A,, Chem. Eng. Progr. 48, 89-94 (19 52).
a = constant
S.S.S.R.
(12) Seitz, F., Zbid., pp. 1-20. (13) Sellwood, P. W., Zbid., 3, 27-105 (1951). (14) Zbid., 9, 93-106 (1957). (15) “Sy2posium on Preparation of Catalysis, IND. ENC. CHEM.49, 240-85 (1957). (16) Taylor, H. S., Advances in Catalysis 1, 1-27 (1948). (17) Wheeler, A., Zbid., 3, 249-325 (1951).
11. Pore Structure and Surface Area
(1) Barrett, E. P., Joyner, L. G., Halenda, P. P., J . Am. Chem. SOC.73, 373 (1951). (2) Cranston, R. W., Inkley, F. A., Advances in Catalysis 9, 143-54 (1957). (3 Emmett, P. H., Zbid., 1, 65 (1948). (41 R itter, H. L., Drake, L. C., IND.ENG. CHEM.,ANAL.ED. 17, 787 (1945). Effective External Diffusion Coefficients (1) Bernard, R . A., Wilhelm, R. H., Chem. Eng. Progr. 46, 233 (1950). (2) Carberry, J. J., Bretton, R. H., A.I.Ch.E. Journal 4, 367 (1958). (3) Ebach, E. A., White, R. R., Ibid., p. 161. (4) Fahien, R. W.: Smith, J. M., Ibid., 1, 28 (1955). (5) Froment, G. F., Ind. chim. belge 34, 620-33 (1959). (6) McHenry, K. W.,Wilhelm, R. H., A.I.CI1.E. Journal 3. 83 (1957). f7) Plautz. D. A..’ Johnstohe. H. F.. Zbid., 1, 193 (1955). (8) Ram, W. E.. Chem. Eng. Progr. 48, 247 (1952). (9) Strang, D. A., Geankopolis, C. J., IND.ENG.CHEM.50, 1305 (1958).
111.
\
I
Temperatures a n d Partial Pressures at Catalyst Surfaces (1) Brown, G. G., and others, “Unit Operations,” p. 214, Wiley, New York, 1950. (2) Fair, J. P., Jr., Ph.D. thesis, University of Texas, Austin, Tex., 1955. (3) Gamson, B. W., Thodos, G., Hougen, 0. A,, Trans. Am. Inst. Chem. Engrs. 39, 1 (1943). (4) Stewart, W. E., unpublished analysis, Drivate communication. (5j Wakao, N., Oshima, T., Yagi, S., Chem. Eng., Japan 22, 780 (1958). (6) Wilke, C. R., Hougen, 0. A., Trans. Am. Inst. Chem. Engrs. 61,445 (1945). (7) Yoshida, F., Ramaswami, D., Hougen, 0. A., Am. Inst. Chem. Engrs., Convention, Washington, D. C., December 1960.
IV.
V.
Effective Thermal Conductivity
(1) Argo, W.B., Smith, J. M., Chem. Eng. Progr. 49, 443 (1953). (2) Coberly, C. A., Marshall, W. R., Zbid., 47, 141 (1951). (3) Hanratty, T. J., Chem. Eng. Sci. 3, 209 11954). \ - - - . I -
(4) Hougen, J. O.: Piret, E. L., Chem. Eng. Progr. 47, 295 (1951). (5) Kunii, D., Smith, J. M., A.Z.Ch.E. Journal 6, 71 (1960). (6) Kwong, S . S., Smith, J. M., IND.ENG. CHEM. 49, 894 (1957).
INDUSTRIALAND ENGINEERINGCHEMISTRY
(7) Plautz, B. A., Johnstone, H. F., A.Z.Ch.E. Journal 1, 193 (1955). (8) Yagi, S., Kunii, D., Chem. Eng., Japan 18, 576 (1954). (9) Yagi, S., Wakao, N., A.I.Ch.E. Journal 5 , 79 (1959). Effective Diffusion Coefficients Inside Pellets (1) Beeck, O., Smith, A. E., Wheeler, A., Proc. Roy. Soc. (London) A177, 62 (1940). (2) Bird, R. B., Stewart, W.E., Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, 1960. (3) Love, K. S., Emmett, P. H., J . Am. Chem. Soc. 63, 3297 (1941). (4) Sehr, R. A., Chem. Eng. Sci. 9, 145 (1958). (5) Weisz, P. B., Prater, C. D., Advances in Catalysis 6 , 144-95 (1954). (6) Wheeler, A., Zbid., 3, 266 (1951). (7) Zbid., pp. 280-328. (8) Wicke, E., Br$tz, W., Chem. Zng. Tech. 21, 219 (1946). (9) W‘icke, E., Kallenback, R., Kolloid 2. 97, 135 (1941).
VI.
VII.
Effectiveness Factor
(1) Ark, R., Chem. Eng. Sci. 6, 262 (1957). (2) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Wiley,
New York, 1960. (3) Chu, Chieh, Hougen, 0. A., Unpublishedreport, University of Wisconsin, Madison, Wis. (4) Mingle, J. 0.: Smith, J. M., Am. Inst. Chem. Engrs., Convention, New Orleans, March 1961. (5) Schilson, R. E., Ph.D. thesis, University of Minnesota, Minneapolis, Minn., December 19 57. (6) Shindo, H.; Kubota, H., Chem. Eng., Japan 20, 11 (1956). (7) Thiele, E. W., IND.ENG. CHEM.31, 916 (1939). (8) Weisz, P. B., Porter, C. D., Advances in Catalysis 6, 144-65 (1954). (9) Wheeler, A., Ibid., 3, 268, 280-328 (1951).
Reaction Rate Models (1) Balandin, A. A., Zzvtst. Akad. Nauk. S.S.S.R. Otdel Khim. Nauk No. 7, 624 (1955). (2)’ Balandin, A. A., Proc. Acad. Sci. S.S.S.R. 110, 133 (1956). (3) Balandin, A. A., Advances in Catalysis 10, 96-129 (1958). (4) Balandin, -4. A., Ponomarev, A. A,, J . Gen. Chem. S.S.S.R. (Eng. Transl.) 26, 1301-16 (1956). (5) Cottrell, T. L., “The Strength of Chemical Bonds,” Academic Press, New York, 1954. (6) Hougen, 0. A., Watson, K. M., IND. ENG.CHEM.35, 529 (1943). (7) Hougen, 0. A., Watson, K. M., “Chemical Process Principles I11 Kinetics and Catalysis,” Wiley, New York, 1947. (8) Hulburt, H. M., Am. Inst. Chem. Engrs., Convention, Philadelphia, Pa. June 1958. (9) Pauling, L., “The Nature of the Chemical Bond,” Cornel1 Cniv. Press, Ithaca, N. Y., 1960. (10) Rase, H. F., Kirk, R. S., Chem. Eng. Progr. 50, 34 (1954). (11) Semenov. N. N.. “Some Problems in ‘ Chemical Kinetics and Selectivity,” Princeton Univ. Press, Princeton, N. J., 1958.
IX.
RECEIVED for review April 17, 1961 ACCEPTED April 26, 1961 Division of Industrial and Engineering Chemistry, 139th Meeting, ACS, St. Louis, Mo., March 1961.
