and knowledge of rates is important in design and planning of operations, but many other factors must be carefully balanced for optimum design. Ease of separation of catalyst and vapor, ease of catalyst circulation, isothermal operation, etc., may far outweigh the limitations of the mixed bed from an efficiency standpoint. SUM M A R Y
-.
A basis for determining the approximate extent of conversion of fractions obtained in the catalytic cracking of petroleum has been termed the “20f conversion.” Using this conversion basis, the basic reaction appears to be about first order, but because of dilution of reactant ~1ith cracked products, the reaction in a constant pressure system approaches second order with respect to unconverted feed. Catalyst fouling during a cracking cycle due to carbon deposition reduce! initial activity very rapidly; instantaneous catalyst activity after 1-hour contact with a petroleum gas oil is less than l/laaothat of the freshlg regenerated catalyst. The rate and extent of fouling vary appreciably, depending on operating conditions. Proper separation of basic catalytic cracking reaction rates requires knowledge of the extent of the concurrent effects of fouling. This fact has usually been overlooked in the past. Reaction rate constants for catalytic crarking of a light East Texas gas oil are presented for a wide range of operating condi-
tions. Methods for using the constants to predict eRects of actual catalytic operating conditions and to predict conversions for practical situations are discussed. ACKNOWLEDGMENT
The extensive work of many individuals associated with the Standard Oil Development Co. and affiliated companies in obtaining the data used in this study is acknowledged. Particular acknowledgment in this regard is due C. L. Brown, A. Voorhies, Jr., and associates, and H. G. M. Fischer and R. M. Shepardson and associates. Some of the data were contributed by and used with permission of the Universal Oil Products Co., Chicago, 111. LITERATURE CITED
(1) Archibald, R. C., May, N. C., and Greensfelder, B. (2)
(3) (4) (5)
(6)
S.,IND. ENG.
CHEM.,44,1811 (1952). Conn, M. E., and Connolly, G. C., Ibid., 39, 1138-43 (1947). Houpen, 0. A., and Watson, K. M., “Chemical Process Principles,” p. 906, New York, John Wiley & Sons, 1950. Sachanen, A. N., “Conversion of Petroleum,” p. 324, New York, Reinhold Publishing Corp., 1948. Shankland, R. V., and Schmitkons, G. E., American Petroleum Institute, Division of Refining, Symposium on Cracking Catalysts, 27th annual meeting, Chicago, Nov. 10, 1947. Voorhies, A., Jr., IND.ENG.CHEM., 37, 318-22 (1945).
RECEIVED for review January 8, 1Q63.
ACCEPTED March 30, 1953.
Filtration of Fluid Catalyst Fines from D.6 . PALL MICRO METALLIC CORP.. GLEN COVE. N. Y .
The use of porous stainless steel filters for removing fluid catalyst fines is becoming of major industrial importance, with two full scale installations scheduled t o go on stream in 1953. Since other pilot scale installations point t o further expansion, t h e conditions surrounding the use of porous stainless steel and for scaling up from laboratory t o large plants have become of considerable technical importance. Filtration accomplishes 100% removal of solids. Filters can be used, with automatic blowback, for years without increase i n pressure drop. Conditions necessary t o achieve such continuous automatic operation have been delineated. A mechanism of filtration and of cleaning of t h e filter by blowback was developed and successfully applied t o diagnosing t h e cause of occasional excessive pressure drop i n pilot installations. The conditions for preventing excessive pressure drop are outlined. T h e necessary information and formulas are also given for designing pilot plants employing filters and for scaling up pilot data t o full scale plant conditions. c
D
URING 1953, two full scale fluid catalyst plants will go on stream using porous stainless steel filters for removal of catalyst fines. One of these plants uses an expensive catalyst to obtain high conversions; losses incident to the use of the usual cyclone-electrostatic precipitator combination cannot be tolerated; furthermore, since the reacted vapor condenses directly to a solid, wet recovery of catalyst cannot be practiced. A second plant uses filters on the regenerator side; this plant operates a t high pressures; the filters obviate difficulties involved in throttling a gas flow bearing abrasive solids. Numerous pilot plant and laboratory installations have been made in the last 4 years, using porous stainless steel for filtration as well as for dispersing gases into solids. Results have indicated that the porous material can be successfully used over a wide range of conditions. June 1953
It is anticipated that future full scale plants will be installed in place of conventional units for the following reasons:
.
1. To prevent loss of expensive catalyst,s 2. In high pressure systems, where problems relating to the throttling of otherwise severely abrasive regenerator waste gas are eliminated 3. To eliminate air pollution by escaped catalyst 4. To permit use of waste heat where effluent gases are burned 5. I n certain installations an actual economy in first cost and maintenance costs will be achieved by substituting filters for systems consisting of cyclones, coolers, and electrostatic precipitators in series.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1197
of air of the various grades of porous material are shown in Table 11. Grade Designationa D E F G H Referring to Table I, it will Coarse Medium Fine Extra-fine Superfine be established later that the Partjcle diameter of powder, 20-micron mean pore diameter microns 150-300 75-150 75-160 35-75 20-60 Mean pore opening, microns 65 35 20 10 5 material (Grade F) is satisfacMinimum tensile strength, 15,000 15,000 15,000 9000 15,000 tory for fluid catalyst work and lb./sq. inch l/az '/sa l/ai '/16 1/88 Minimum thickness, inch that a typical throughput rate 3 . 0 x 106 2.7 x 105 1 . 5 x 106 2 . 5 x 108 Approx. modulus of elasticity 1.0x 1 0 6 2 5 4 5 5 Elongation (min.) , % for such process is 10 cubic feet a Grades coarser than D and finer than H are available in stainless steel and certain resistant alloys; sheets per minute per square foot. outside of thickness limits can also be produced. Referring to Table 11, the pressure drop through '/&xh thick Grade F porous material a t this throughput rate is only 0.1 pound per square inch; a t the elevated temperatures which exist in fluid catalyst operations, this pressure drop through the porous material itself would be about 0.2 pound per square inch, which is virtually . negligible compared with other pressure drops in the system. Table 1.
Properties of Porous Stainless Steel
1H
'
COMPRESSED AIR LINE/
Figure 1
Porous stainless steel is a material manufactured in sheet form using stainless steel powder as raw material. The stainless steel powder is made by a "shotting" procedure in which a molten stream of alloy is atomized by means of high pressure water jets impinging on it; these jets are located on the periphery of a wheel rotating a t high speed ( 3 , d ) . The fabrication of the porous sheet is accomplished by laying out the powder in the form of a sheet-for example, by means of a doctor blade on a suitable ceramic base-and thereafter passing it through a furnace in a strongly reducing atmosphere, a t a temperature just below the melting point and in a suitable atmosphere ( 1 , 2). The points of contact between particles develop into bridges of diameter of about one fifth to one third of the particle diameter, thereby bonding the assembly into a sheet of residual porosity approximately equal to the spaces which initially existed between the metal powder particles. Virtually all the pore openings so formed are interconnected, resulting in high flow capacity. This structure has equivalent capillary diameters, as determined by fluid displacement methods, such that no pore is larger than twice the mean, and 95% of the pores are larger than one half the mean. Type 316 stainless steel is used for fluid catalyst service a t temperatures under 950' F. The physical properties of the material 80 obtained are shown in Table I. The flow capacities for passage
Table I I.
Flow Capacity-Air
Flow Capacity of */a-Inch Thick Filtera, Cu. Ft. of bir/Min./Sq. Ft. Pressure Drop, Lb./Square Inch D E F G H 0.01 13 ... .. ... 0.05 60 18 5 3 1.8 0.1 115 35 9 5 3.5 0.2 175 65 10 7 17 l8 24 28 0.5 320 140 44 1.0 475 220 82 41 2.0 685 340 160 66 46 90 5.0 1045 580 320 140 10.0 ... 900 460 250 160 For ' / i s h . thick filters, multiply results given by two. Q
1198
EXPERIMENTAL EQUIPMENT
c FLOWMETER
In order to determine the utility of the porous material in removing fluid catalyst fines from reacted gases, equipment such as shown in Figure 1 was used. A colunin 12 to 18 inches high of fluid catalyst was used in a tube 4 feet long, with a filter cylinder in an enlarged section of the tube a t the upper end. Tubes of about 11/$ and 2'/2 inches inside diameter have been used. Fluid catalysts used have been materials having particles in the range 0.5 to 80 microns, with yeight mean diameter of 40 microns. Range of air rates through the bed is 1 to 3 feet per second. A
aP w
a mm
w a P
-ac
..I
w z
a u. w
t 0
Figure 2.
A P vs. Time-40p (Single Filter)
Catalyst
Cu. Ft./Min./Sq. Ft. A = 21.9 B = 14.4 C = 6.7 D = 4.8
filtration cycle is begun b y passing air a t room temperature through the fluid catalyst bed and measuring the pressure drop a t frequent intervals. .4t the conclusion of an experimental run, the air rate is cut down to a low value, sufficieiit to maintain the collected filter cake on the filter surface but not sufficient to pick up fines from the catalyst bed. Vacuum is then applied to the downstream connection of the filter cylinder, after which valve B is opened and the filter cylinder carefully removed so as not to disturb the catalyst cake. The over-all diameter of the cake may then be measured in order t o determine its thickness, after which the vacuum is released, and air is blown back a t the desired
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, NO. 6
.
rate through the filter cylinder while i t is held over a large piece of paper. The collected catalyst may then be weighed, examined for particle size, etc. The catalyst is thereafter poured back into the apparatus, filter cylinder is replaced, and the cycle repeated. When desired, of course, the catalyst cake need not be removed but can be simply blown back in position; this is accomplished by opening valve C while passing air in the reverse direction through the filter cylinder. Using this type of equipment, information of considerable value can be obtained. Figure 2 shows the rate of pressure drop increase for various flows. Catalyst loading in the gas incident on the filters was 15 grams per cubic foot of gas. The size distribution of the catalyst collected on the filter appeared on microscopic examination to be unchanged from that in the bed.
drop with blowback pressure, the steady initial pressure drop decreased to a minimum as the blowback rate increased t o about one half the throughput rate. Further increase in blowback rate, contrary to what might be expected, causes an increase in steady initial pressure drop.
BLOWBACK RATE
-
1/2
THROUGHPUT
RATE
THROUGHPUT AND BLOWBACK RATES *
The pressure drop increase itself is linear and forms an intercept on the AP axis, the values of these intercepts being approximately proportional to the flow rates used. The reason for this is discussed later. When the square root of the slope of the curves shown in Figure 2 is plotted against A P , a straight line is obtained, showing that a parabolical relationship is maintained between these functions (Figure 3). That this is true is not surprising; it is very easy to see that if a given set of flow rate conditions is maintained and if, further, the filter area is cut in half, then the remaining filter area will not only be passing gas at twice the original rate but will also collect twice the amount of catalyst, thereby increasing the pressure drop by a factor of four after passage of a given time. Thus, the reason for the square relationship is readily established. A relationship which, in practice, is of great importance, and the study of which has helped greatly in determining the mechanism of filtration, is illustrated by the study of the steady initial pressure drop. Figure 2 shows the course of a single cycle, without considering the effect of blowback during successive cycles; Figure 4 shows what may be considered to be normal effect when suicessive cycles are applied to the same filter. Initially, at the beginning of the first cycle, the pressure drop is very low; this pressure drop increases through the first few cycles and then reaches a steady value, beyond which no further increase is obtained. This condition holds when the blowback rate used is about one half the throughput rate. If the blowback rate is substantially increased, the same condition is not observed; instead, the steady initial pressure drop continues to rise a t almost a constant rate for a long period of time-for example, several thousand cycles. This is shown in Figure 5. Referring now to Figure 6, which shows a typical variation of steady pressure initial
Figure 4. Steady I n i t i a l Pressure Drop, Normal Curve for Moderate Blowback Rate
The reason for this apparently anomalous situation was determined in the following manner: Filters were used which had been put through several thousand cycles and which had reached a steady initial pressure drop of 18 pounds per square inch a t 6 cubic feet per minute per square foot throughput. A differential pressure in excess of 15 p u n d s per square inch was maintained across the filter. Blowback rates thus obtained were initially more than thirty times the throughput rate, decreasing with time t o a value equal to the throughput rate. The catalyst used was colored and could readily be distinguished microscopically from the porous material. The porous material of which this filter was composed
E W
* W
a J
5
z W W
'L
TIME
Figure 5.
4 .
Figure 3. Relation between Rate of Pressure Increase and Throughput Rate
June 1953
Rising I n i t i a l Pressure Drop, Abnormal Curve for High Blowback Rate
was examined in section. Similar action was taken with the filter which had been used in somewhat the same manner but with only one half the throughput rate used for blowback, so that the stead initial pressure drop never went over 0.5 pound per square i n d . Examination of these specimens in a binocular microscope readily revealed that the material which had been given the high rate blowback contained colonies of the catalyst throughout its volume; there was no particular concentration of this material in any part of the filter so that it was impossible to determine on1 from microscopic examination of the filters what had been d e direction of the flow when the filters were used. By contrast, the filters blown back a t the low rate contained no catal st fines in the body of the filters; instead, colonies of catagst fines were contained on1 on the upstream surface, and these colonies were found t o be gcated in virtually every surface pore opening.
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
1199
For the porous medium,
I= k , a!
and for the cake
and
where
w = catalybt loading per unit volume of gas, p = density of filter cake, T = duration of one complete cycle, A’ = total number of
Figure 6.
Steady Initial Pressure Drop, a Function of Blowback Rate
PRODUCT OUT
Further experiments in which filters were put through a hundred cycles a t low blowback rate and were thereafter blown back only once a t a high rate, showed that the pressure drop across these filters went back to the value for a clean unused filter. The mechanism of filtration derived from these results is as follows:
4 O R M O R E BANKS OF A-84-8
M U L T I P L E BAYONET UNITS
When the stream carrying the suspended solids impinges on the filter, a few particles get into the internal pores of the filter; hoffever, colonies of particles very rapidly congregate a t each pore opening, thereby providing much finer filtration than the porous material itself. When blowback rate is appreciably smaller than throughput rate, these colonies remain fixed in their original locations, and catalyst fines no longer penetrate into the ewe of the filter. By contrast, when a single high rate blowback is used, these colonies are completely removed, thereby restoring the filter to its original condition. 4 t the beginning of each filtration cycle, a small amount of catalyst fines enter into the filter, and since most of these materials are not removed in blowback, ermanently lodge there. After a sufficient number of cycles rseveral thousand in practice), pressure drops will increase to very high values (in the neighborhood of 15 pounds per square inch). The results stated are for Micro hletallic Corp. standard Grade E filter (35-micron mean pore opening). Under the same conditions whereby a pressure drop of 18 pounds is realized .iiith Grade E filter, a pressure drop not exceeding about 2 pounds per square inch will be realized for Grade F filters, under the conditions of operation. When used under ideal conditions of blowback, Grades E and F may be used interchangeably; however, to allow for the occurrence of the blowback effect described, the use of Grade F porous material has generally been recommended. This material has a pore opening of 20 microns. So far as can be determined, complete 1 0 0 ~ catalyst o removal is obtained, including the finest portions of catalyst down to 1 micron or finer.
REACTOR I
11
‘L
CONVERSION FROM TEST RESULTS
Where it is desired t o establish full scale operation, the need for continuous flow makes impossible the use of a single bank of filters. Therefore, multiple banks of filters are used, whereby only one bank of filters is blown back at any one time while the remaining filters remain in action. -4 recommended type of installation is shown in Figure 7. Generally, a t least four banks should be used, and these should be provided with valves on both sides of each filter container 80 that the filter container may be removed for servicing The over-all differential pressure across the filter medium and cake is, according to D’Arcy’s l a x
P
=
[(k
Figure 7.
Fluid Catalyst Filter System
banks, and V = mean gas volumetiic f l o ~rate for A- banks. For a given number of banks, S,p mill fluctuate between blowbacks about a mean value p m . With increasing iYJ these fluctuFOCUSations become negligible and p may be replaced by Q,. sing attention on any one bank of filters and integrating from 0 = 0 t o 8 = T , the corresponding values of Q are 0 and
T iT
7
Rearranging, ’
+ (; $),!$
where subscripts 1and 2 refer to the porous medium and the cake, respectively; t = thickness, A = area per bank, = permeability (includes viscosity of gas), Q = volume of gas, and 8 = time, measured since previous blowback.
1200
REACTANTS I N
or =
total area
=
INDUSTRIAL AND ENGINEERING CHEMISTRY
2%
+
42% + G] k?T
Vol. 45, No. 6
- 2 \ _FLANGE
-
\ E-14
As RESD.
EXPANSION CONTROL
BAYONETS
Section A-A
Figure 8.
Fluid Catalyst Filter Bayonet Cluster (Stacked)
Max. Diameter
- 7'14 inches
Length Inches One cluste; = 338//la Two clusters = 63'/1e Three clusters = 94S/ia Four clusters = 12418/16
Section
B-B
The evaluation of k , and kz may be accomplished by tests in pilot plant or model scale, or on similar units. Denoting by subscript t the conditions under which these tests are performed, the basic equation is still applicable kz d&t
Pb =
[k +
-&Qt]z
measuring the pressure rise for a constant flow rate Bt (without blowback) such t h a t d0
=
Vi and
Vt.0
kzv: e A:
klVt p t = - +At -
*
&t =
a of p t intercept (po)t
0
produce a straight line, of
Thus
kl = At V, (PO)$and kz = ($)zVt
vi and
For a highly permeable filter (PO)$is negligible and
AN = V d E The assumption that p = p,, introduces very little error. Operating data have shown that between blowbacks the pressure rises almost linearly from a minimum t o a maximum value. For such behavior the equation becomes exact provided pm is chosen so that pm =
Pmin
+2
pmax
DESIGN FACTORS
Aside from the Problem of determining the loading of the fluid Catalyst carried in the effluent gas, the determination of full
June 1953
scale conditions from experimental data can be determined in using the formulas and information contained in this report. Design of the filters usually allows for a factor of safety of about loo%, whereby it is possible to remove for servicing one bank of filters while the remaining filt'ers carry t.he load without interrupting the operation. In order to make such removal possible, it has generally been concluded that the extra expense of providing separate filter containers is warranted, as compared with locating the filters inside the vessel head. As a precaution, particularly in relation to the design of pilot plants where it is common practice to contain the filters in the head to minimize heat losses, it is essential that the filter banks be baffled from each other; otherwise there is %strong tendency for the discharged catalyst to simply travel across from one bank of filters to the other wit.hout ever reaching the catalyst bed. The maximum recommended temperature a t which the porous material can be used is 950" F. for the Type 316 stainless steel composition which is ordinarily used. That this temperature is considerably lower than the temperature a t which solid stainless steel may be used is not surprising when i t is remembered that the "bridge" between particles of the 20-micron porous material (which is the recommended grade) is of the order of magnitude of 0.001 inch. Clearly, a depth of corrosion of 0.005 inch will leave no bonds whatsoever between particles; by contrast, the same depth of corrosion in solid stainless steel is not serious. .4t 1000' F. a service life of 1 to 3 years is predicted for Type 316 stainless steel. Other analyses have recently become available, particularly the so-called L-605 alloy, which is a cobalt-base alloy containing tungsten, nickel, and chromium. This alloy has not as yet been performance tested, and its life cannot be predicted. However, it is probable that it will operate at several hundred degrees Fahrenheit higher than porous stainless steel. Type 309 stainless steel appears to be most suitable in atmospheres containing large amounts of hydrogen sulfide. An important variable is the effect of variation of particle size of the fluid catalyst. Small scale experiments have not indicated any appreciable variation in particle size between the catalyst carried up to the filter surface and that of a catalyst bed. However, i t has been reported in larger plant scale installations that some segregation and tendency to collect finer material in the
INDUSTRIAL AND ENGINEERING CHEMISTRY
1201
filter does occur. Accurate information on this point has not been obtained. However, as a first approximation, which should not be used for full scale plant design, but may be used, for lack of anything better, in designing a pilot plant, the rate of pressure drop increase may be considered to be inversely proportional to the &eight mean particle diameter. This assumption must be confirmed by future experimental work. The actual form of equipment which is used for full scale plant work involves the use of multiple units of the exact type used for pilot plant work. Such a unit is shown in Flgure 8. These units can be provided in multiple lengths to furnish areas up t o 40 square feet per unit. The basic unit itself, however, consists of an oval-shaped assembly nhich is 13/4inches wide by inch thick by 24 inches long. Design is such that a catalyst can fall freely from all surfaces. The porous bayonet itself is permitted to expand freely, as compared with the central core, but is prevented from vibrating hy means of the expansion control deviczs a t the end of each bayonet unit The individual filter units are tested a t 250 pounds prr square inch differential from the outside in; their use in practicc is not recommended a t more than about 5 pounds per square inch differential. Thus, a very substantial amount of weakening of the porous material as a result of oxidation may occur before failure is actually observed. The bayonets themselves arc strong enough that a 200-pound load applied a t the center of one of the baronets will not damage it. The life of such an installation a t 900" F. is estimated t o be adequate for long-term industrial use; installations used for almost 2 years a t 900' F. have shown no serious corrosion and appeared to be suitable for many additional years of service. I n reports obtained on some thirty pilot plant installations, corrosion has not been a problem in any atmosphere, except as 3, result of hydrogen sulfide corrosion in reducing atmosphere. Where a partial pressure of 1atmosphere of hydrogen sulfide exists the maximum temperature of usage must be reduced from 900' to 500" F. The problem of providing a metal or alloy which will successfully withstand hydrogen sulfide a t higher temperatures has not as yet been completely solved; Type 309 stainless steel offers promise in this direction.
As a precautionary word, the filters cannot be used in reactors where coking may occur to plug the reactor surface. They can be used under all conditions in regenerators, particularly since sulfur dioxide, as contrasted with hydrogen sulfide, does not appear to cause serious corrosion. The filters can be used in the reactors in applications mhere coking does not occur. Khere tempeI atures exceed those permitted, particularly where the filters are used in regenerators, a simple solution to the problem is available By introducing liquid water spray in the upper part of the regenerator, the gas tempwature can be lowered by several hundred degrees. Coincidentally, owing to reduced gas volume and viscosity, pressure drop across the filters is actually reduced. This method is stronglv recommended where temperatures over 900" F. are obtained. The porous stainless steel filters have been used in a large number of pilot plant installations, in nearly all cases with good success. No failures due to breakage, other than those directly traceable to use of excessive use of temperature or to hydrogen sulfide atmosphere, have been reported where the bayonet type of unit was used. Other types of units developed earlier have also been used with almost uniform success. Two 1000-square foot installations are due to go on stream in 1953. one for the catalytic oxidation of a chemical and the second in the regenerating section of a synthesis plant. Another smaller regenerator application is also in process of construction. ACKNOWLEDGMENT
The assistance of Leon Laaare, of the American Cyanamid Corp., in connection with the derivation of the conversion formula, is gratefully acknovledged. LITERATURE C I T E D
(1) Pall, D. B., U. S. Patent 2,554,343 (1951). ( 2 ) Micro Metallic Corp., Glen Cove, N. Y.,Porous Stainless Steel Release, No. 204. (3) Vanadium-Alloys Steel Co., Prealloyed Steel Powder, Bull. No. 2. (4) Ibid., Bull. Xo. 3. RECEIVED for review
March 9,
1953.
ACCEPTEDApril 13, 1953.
Diffusion in a hid Moving at in a Tube ADRIAAN KLINKENBERG N.V. D E B A T A A F S C H E P E T R O L E U M M A A T S G H A P P I J , T H E H A G U E
H.
J. KRAJENBRINK
AND
H. A. LAUWERIER
ROYAL DUTCH S H E L L LABORATORY, AMSTERDAM, HOLLAND
T
HE present study deals with the concentration distributions
caused by diffusion in a fluid moving in a cylindrical tube a t uniform velocity. The solute emerges from a continuous source in a point on the axis of the tube. The amount of solution introduced in this source is so small that it does not upset the uniformity of the velocity. Axial and radial diffusivity are not assumed to be necessarily equal. The stationary state is assumed to have been reached. A general equation has been derived for such nonisotropic diffusion. The solution is obtained with the aid of two-sided Laplace integrals, which are made to fit a t the injection point. Later on D, may be equated either to D, or to zero. The results are computed numerically. Corresponding equations are given for diffusion in the infinite
1202
space (absence of the cylinder wall). These must be applicable when the solute has not yet reached the wall. It is examined when they lose validity. The boundary conditions in this problem correspond to the experimental arrangement used by Bernard and JTilhelm ( I ) in their determination of eddy diffusion constants. RADIAL AND AXIAL EDDY DlFFUSlVlTY
In order to explain the practical importanct of the above mathematical problem, it is necessary t o discuss its phyeical background. It is often stated that a velocity which is uniform over the cross section is realized in the case of flow through a packed bed. This, of course, refers t o macroscopic observations only. This
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 6
