Vol. 20, No. 11
I-VDC’STRIAL AND ELVGINEERING CHEMISTRY
1196
Gas Flow through Packed Columns’ S. P. Burke and W. B. Plummer COMBUSTION UTILITIESCORPORATIOK, LISDEN, N. J.
HE factors determining
T
the energy losses (pressure drop) accompanying the flow of fluids through chambers packed with s p h e r e s , granular material, e t c . , a r e not susceptible to complete and exact mathematical discussion. As a simple method for arriving at a useful expression correlating these factors, we may base our premises on the general expression governing the flow of fluids through pipes,2 viz., p = kV”Dn-3p2-npn-11 (I)
Results are reported for numerous tests on the pressure drop for air flowing at various rates through systems packed with spheres. Since both the exponent and the coefficient of the general expressions connecting gas flow and pressure drop vary continuously over the range from viscous to turbulent flow, it has been found desirable to adopt the procedure used by various authorities in treating the subject of gas flow in pipes, and to represent all cases by one general expression containing a variable coefficient, which may be representedby a smooth curve. This general expression for gas (or other fluid) flow through packed spaces is p/l =
=
kVndn-3p2-npn-ll(l
- f)af-b
(2)
Now V is the velocity based on the gross area, whereas the actual velocity in the packing is evidently the governing factor and hence V / f should be substituted for V . Note-It is unquestionably true that in any packed system a certain fraction of the “free area” of any given cross section is not available for fluid flow, owing t o the formation of dead-end spaces or pockets in the packing. Moreover, the flow channels must be ordinarily of various sizes and a wide range of fluid velocities must therefore exist in any given packing. Since mathematically the statistical sum of the squares (or other power) of a number of terms is not identical to the square of the statistical sum of the terms, in the ultimdte analysis pressure drop through a packing cannot be defined exactly in terms of any “average velocity through the packing” term. Therefore, since a t best there is no way of estimating the actual average percentage of cross-sectional area available for fluid flow, the simplest assumption has been used-i. e., that V / f be taken as the average velocity in the packing.
Further, in order to make the expression general and applicable to non-spherical packings, it is desirable to replace the diameter term by a term X,representing the total surface per unit of packed volume. Evidently for spheres S = (I-f) (nd2)/(d3/6) 7 6(1 -f)/d. Making these substitutions, (2) becomes
p
Rg = 67rprV
(4)
The variable coefficient is a function of, and is plotted against, p S j p V as shown by Figure 2. In use it is only necessary to evaluate p S / p V for the given case, read off C, and apply the general equation, without consideration as to the character of the flow, since this is taken care of by the plot of C.
Considering for the moment the simple case of spherical packings, the packing diameter, d, may immediately be substituted for the pipe diameter. If, however, we introduce the concept of the packing constant, f (free volume per unit of packed volume), it is evident that our equation (1) is and when f = deficient, since obviously whenf = 0, p = 1, p = 0. Therefore, we may set up the factor (l-f)”/P &s bemg the simplest exponential function which will fulfil this requirement. Hence (1) becomes
p
c .-pv2s f3
a value of 1 for stream-line flow conditions to 2 for comp l e t e l y turbulent flow. A priori it would appear that the exponents a and b were indeterminate save by experiment, but these can be fixed as follows by consideration of the energetics of such systems. Under stream-line flow conditions the force R (grams) acting on an isolated spherical particle of radius r suspended in a fluid stream is4
= k VnS3-np2-n P n-ll(1- f ) ” +a -8f
-b-n
(3)
The exponent n is, of course, dependent on the conditions of fluid flow in a given case, and is usually found3 to vary from 1 Presented in part before the Division of Gas and Fuel Chemistry a t the 72nd Meeting of the American Chemical Society, Philadelphia, Pa,, September 6 t o 11, 1926. Received June 1, 1928. Gibson, e t al., “The Mechanical Properties of Fluids,” p. 192, D. Van Nostrand Co.. Inc., 1924.
Let us consider that, in a g i v e n s y s t e m of p a c k e d spheres, this force is acting on each particle, the number of particles per unit of packed volume being 3(1 -f)/(4.r3). To determine the rate of doing work, W , it is evident that the actual velocity in the packing equals V / j , and that this is in effect the distance through which the force R acts in unit time. Hence W = (6wV/gf)(V/f)(3)(1 - f)/(4*r3) whereas also evidently
w=
V(P/l)
Combining these expressions
PI1 = (9/2g)~V(1- f)/jzr2 and by introducing S = 3(1-f)/~, this becomes
-f)P
P/1 = ( ~ / 2 d ( P v m / u
(5)
We may apply similar reasoning to the case of completely turbulent flow, the only difference being that now for spheres5
6)
R = 0.00084pr2V2 so that our final equation, similar to (j),becomes
Examination of equation (3) shows that the conditions of both (5) and (6) are fulfilled when a = b = 1. Rewriting (3), i t becomes
p
= k V n S a - n P 2-npn-11(l
- f)n-Zj-n-l
which may also, in accordance with the usual treatment6J of the similar pipe-flow expressions, be written 8 Gibson, Phil. Mag., [6] 60, 199 (1925), has recently found that for turbulent flow through a circumferentially corrugated pipe, a = 2.135. As required by the dimensional theory (equation l), in this special case increasing the viscosity of the fluid decreased the pressure drop under otherwise comparable conditions. Lamb, “Hydrodynamics,” Cambridge University Press, 1924. Burke and Plummer, IND. ENG.CHSM.,90, 1200 (1928). Stanton and Pennell, Trans. Roy. Sac. London, 114A, I99 (1914). Walker, Lewis, and McAdams, “Principles of Chemical Engineering,” p. 87, McGraw-Hill Book Co., 1923; see also Wilson, McAdams, and Selzer, J. IND. SNG. CHEM.,14, 105 (1922).
‘
It will be evident that when n = 1 this reduces to the form of (5), while if n = 2 it becomes equivalent to (6). In plotting the results of the present studies, equation (7) has been simplified by the omission of the fraction f/(l -f) in the power term. Since f is dimensionless. this does not affect the dimensional homogeneity of the expression and since the valueof fisusually0.4 to0.5, the termf/(l-~f) isclose tounity. The agreement of the present results would not be improved by the retention of this term. so that its omission has seemed justifiable for the sake of simplifying the eupression, which thus becomes :
(These velocity and pressure units are purely arbitrary.
For the purpose of graphical representation of data, this is rewritten
PI1
= C(PV2S/f3)
(9)
where C is defined as +(PS/,JV)or specifically as equal to k ( p ~ Y / p V ) * - ~ and , is therefore plotted against ( p L Y / p V ) as abscissas. As noted subsequently. this type of expression has been previously used by Blake.8 In this formulation there has been no attempt to consider the possible effect of the geometrical character of the packing, Tvans A m . I n s ! . Cheni. E n g , 14, 415 (1922).
Figure 1 Figure 1 merely illustrates the range of the data gdthered tojether in Figure 2.)
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I N D U S T R I A L AND EiVGILVEERING CHEMISTRY
Vol. 20, No. 11
Figure 2
although this presumably should enter as some dinieiisioriless function. For example, even in a system packed with uniform perfect spheres so as to have the theoretically minimum free volume (25.95 per cent), each sphere being in contact with twelve others, the geometrical symmetry of the packing may be either hexagonal close-packed or of the cubic close-packed type.g A priori there might well be a difference in the pressure drop in these two cases or, in other words, the pressure drop might be a function of the geometrical nature of the packing as well as of the actual free space therein. Fortunately, this is apparently not the case, within the limits of the experimental error, save in certain limiting cases. Blakej8 using a number of different diameters of crushed pumice and various mixtures thereof, and expressing his results in an equation similar to (9) but in different units, found all his results to fall along a single broken line (represented by curve I1 of Figure 3). I n attempting to apply this equation to columns packed with glass cylinders of various sizes or with Raschig rings, he found that the data for various sizes of the same type of special packing material were brought on the same line, but that different curves were obtained for the various types of materials. I n other words, for the limit cases of differing geometrical symmetry which are presented by these specialized packing materials, some factor involving the shape of the material and symmetry of the packing must be introduced into equation (8) to make it generally applicable. Mention may be made here of an article by Zeisberg’O in which friction factors are given for columns packed with quartz lumps up to 6 inches (15.2 cm.) diameter, and for various other materials. I n this work the fluid was assumed to be completely turbulent and the friction factor to be a coilstant. The results are therefore not capable of general application, but are of interest in that the variation of the friction factor for a given packing material, respectively dry. moist, and with actual water circulation, was determined. The present investigation was confined to the study of the e W. H. and W L Bragg, “X-Rays and Crystal Structure,” p 160, Harcourt-Brace Co , London, 1924 l o Trons A m Inst Chem E n g , 13, 231 (1919)
flow of air through columns packed with spherical materials. for the purpose of confirming and extending the results of Blake.* Equation (9) has been found to bring all data, both for uniformly sized spheres and for mixtures of various sizes. on the same smooth curve, the method of plotting being that previously outlined. Experimental Procedure
The packing materials used were commercial lead shot of three sizes, their diameters being respectively 1.48 * 0.04. 30.08 * 0.03, and 6.34 * 0.07 mm. As shown in Figure 1. three mixtures of these shot were also used, there referred to as mixtures Kos. 1, 2, and 3, in which the proportions by weight of the above three sizes were, respectively, 2:1:1. 1:2:1, and 1:1:2. Four different columns were used, of the following inside diameters: 1.794 (glass), 3.41 (glass), 5.25 (standard “2-inch” galvanized iron pipe), and 7.81 (standard “3-inch” galvanized iron pipe) em. The fluid in all tests was air saturated with water vapor and a t an approximately constant temperature of 27” C. Air-flow rates up to 20 cubic feet per hour (0.57 cubic meter per hour) were measured by a Sargent wet test meter [0.1 cubic feet (0.00283 cubic meter) per revolution], from 20 to 80 cubic feet per hour (0.57 to 2.26 cubic meters per hour), two such meters being used in parallel. For higher rates of flow a set of calibrated Venturi meters was used, whose throat diameters were, respectively, ‘/3*, $/37, 7/16, and 6 / 8 inches (0.555, 0.714, 1.11,and 1.59 cm.). The wet test meters were adjusted to zero water-level correction against a standard copper 0.5 cubic foot (0.0142 cubic meter) displacement test bottle, and their rate correction determined by the same means. This correction is additive in all cases; up to 10 cubic feet per hour (0.283 cubic meter per hour) the correction is zero, it being 1 per cent a t 24 cubic feet per hour (0.68 cubic meter per hour), 2 per cent a t 34 (0.95))and 3 per cent a t 41.5 (1.17). The Venturi meters were standardized against a dry gas meter equipped witha special6-inch (15-cm.) dial reading 1 cubic foot (0.0283 cubic meter) to hundredths,
ISDCSTRIAL d A DENGINEERI,YG CHEMISTRY
November, 1928
the rated capacity of the meter being 600 cubic feet per hour (17.0 cubic meters per hour). The meter was new and immediately preceding its use in this calibration had been proved and adjusted by its makers to “I per cent error or less” a t its rated capacity and a t two lower rates. The probable error of the Venturi calibrations is therefore less than 2 per cent. I n all cases the weight of shot used in filling a given column was recorded, so that from their density, the column diameter. and the length of the packed section the value o f f for the given case could be determined. Wherever possible-to wit, whenever an iron pipe column was being used-two run. were made with the same size ball and the same column hut different values of f. If, for example, a 5.25-em. column wabfilled with 6.34 mm. shot merely by pouring them in, f = 0.415, whereas by hammering the column for some time it could be reduced to 0.375. Pressure drops were measured, according to their magnitude, by suitable multiplying manometers, or ordinary water or mercury manometers. In all cases the resistance of the supporting screen plus 1or 2 cm. of the size of balls in question was determined and subtracted from the total. No comment seems necessary with regard to the actual procedure during runs. Results The original data of the individual runs are represented in Figure 1, using the arbitrary coordinutes of velocity in terms of cubic feet per hour per square centimeter of cross section. and pressure drop in centimeters of water per centimeter length of column. The velocities have been corrected to average conditions within the packing a t the existing pressure and pressure drop. Figure 1 thus gives an over-all picture of the runs which are combined in the final curve. In the final calculations p(centipoises) for air has been taken 0.00005 (t - 27) for air a t t o C. and saturated as 0.0182 with water vapor; in few cases was the deviation of the air temperature from 27 O C. sufficient to warrant correction. The air density p was taken as 0.001172 - 0.000004 (t - 27) and was corrected to the basis of average density throughout the column. The surface per unit packed volume, X, is numerically equal to (6/d) (1- f ) , while for a mixture whose fractional composition (by weight) was X , Y , 2, of balls of diameter d, d’, d”, the surface S = 6 (1 - f ) ( X / d Y/d‘ Z/d”). The velocity was converted to V , the linear velocity in centimeters per second based on the total column area. These various values were then substituted in equation (9) and pf3/b V2S plotted as ordinates against p S / p V as abscissas. both on a logarithmic scale, the points for all observations falling as shown in Figure 2. By the use of this method of plotting, C having been read from Figure 2 , equation (9) may be applied directly without knowing whether the flow is viscous or turbulent. In the viscous-flow range equation (8) is numerically defined, sincc ?i = 1 and C (Figure 2) is a straight-line function. From the curve it is evident that in this range (7 = 0.000050 (p,S’/pT~~, which on substitution in (9) gives
1199
For convenience in application t o engineering problems these data may be expressed in English units, as has already been done by Blake.* I n this case the general equation is
d i e r e the friction factor C’ is defined by the relations
and ii plotted in Figure 3 as a function of .M/pX’. The agreement between Blakes’ results and those of the present tests is excellent. I t is interesting to compare the constants of the empirically derived equations ( 5 ) and (6) with those determined from the curve, Figure 2. Equation ( 5 )has the constant (1/2g) (1/100) = 0.000005 (converting fi from poises to centipoises, as used in the curve), whereas from the curve or equation (10) the constant is 0.00005, or ten times B S great. Similarly, equation (6) has the constant 0.000067, whereas the curve of Figure 2 appears to approach a slope of zero, equivalent to equation (6), a t a constant of about 0.00023. Since in both limiting cases the observed constant is much higher than that calculated from the reactions of a single sphere in a fluid stream, it may be assumed that this represents the disturbing effect of adjacent spheres on the conditions surrounding an individual one. The greater difference (tenfold 21s.fourfold) in the case of the stream-line flow region is in logical accord with this assumption.
+
+
p/I
= 0.000050 p
vs=/1s
+
(10)
In the application of (IO) it must be made certain that the given case is within the viscous flow range; it is evident from Figure 2 that this range extends to values of H,S’/~TT = 40; i. e., that the critical velocityT-, is defined by
vo = ps/4op
(11)
and that (IO) must not be applied for values higher than this. It will be evident that for completely turbulent conditions. could such ever be assumed, C in equation (9) approaches a constant value of 0.00025 (approximate).
?
Figure 3
.4pplication of Equations
It will be evident on inspection of the form of the general equations (9 et seq.) that there are certain difficulties or uncertainties in their application to many practical cases, but it will also be obvious that these are unavoidable. Considerable error may be introduced in the estimation off, particularly since this enters as the cube, it being in many cases difficult to measure directly. Wherever the actual density and total weight of the granular or other material filling a given packed volume are known, f may be readily calculated. Otherwise it must be estimated from experience; uniform spheres “tightly” packed in vertical columns ordinarily have’ a free space of 37 to 38 per cent, provided the diameter of the column is a t least ten times that of the spheres, the actual diameter of the spheres having little effect on the packing. Irregular material having comparatively smooth surfaces, such as coal or crushed quartz, if of uniform size will pack in vertical columns with about 45 per cent free space, again irrespective of the actual size of the material provided the diameter of the column is ten times as great. Irregular material with rough
1200
IiYDGSTRIAL A S D ENGIXEERING CHEMISTRY
Vol. 20, KO. 11
surfaces, such as coke, of uniform size will give about 50 per packing material in question (for further discussion of this cent free space. With spheres or any irregular material of point, cf. Blakes). mixed sizes the fractional free space will evidently be deTable of Symbols creased owing to the fine material fitting into the voids beC = Constant of equation (9), c. g. s. units, except p in tween the larger. Generalizations cannot be made for this case, however, it being necessary to evaluate f from weight, centipoises C' = Constant of equation (lo), English units except p in density, and volume for the given case or for a closely analogous one. The estimation of the surface factor S offers no centipoises d = ,Ball diameter, centimeters further difficulties for the case of uniformly sized material, , T is the radius of since it is defined as equal (1-f) ( 3 / ~ )where f = Packing factor or fractional free space I = Length of packing, centimeters the material particles. For mixtures of different sized ma*M = Mass velocity, thousands of pounds per square foot per terial it may be calculated readily (see previous section) if the proportions of the various sizes of material present are hour based on gross column area known. P = Absolute viscosity of the fluid, centipoises, or relative I n any given case where it is thought probable that the viscosity based on water a t 20" C. n = Exponent of general equation (I) fluid flow is in the viscous or stream-line range, it may be more convenient to determine from equation (11) for critical flow P = Pressure drop, cm. of H;O or grams per square centivelocity that the flow is truly in the stream-line range, and meter P = Pressure drop per unit length, English units, inches of then to apply equation (10) directly, without reference to Figure 2. water per foot r = Radius of particle, centimeter These equations have been successfully checked against a 7' = Radius of particle, feet number of cases taken from the practical operation of gas machines, furnaces, or scrubbing systems, although obviously P = Density of fluid, grams per cubic centimeter p' = Density of the fluid, pounds per cubic foot little can be told of the former two cases if considerable coking, coal fusion, or clinkering are taking place. In the case of s = Surface of packing per unit packed volume, square centimeters per cubic centimeter = (1 - f ) ( 3 / r ) scrubbers Zeisberg'O found that, compared with dry lump S' = Same as S, square feet per cubic foot = (1 - j)(3/r') packings, water circulating a t a rate of I1 pounds per square v = Average linear velocity, centimeters per second in foot per minute (3.0 cc. per sq. cm. per minute) gave an increased back pressure of 3 per cent with 6-inch (15.2-cm.) column based on total area of column lumps, and 63 per cent with 0.5- to 1.0-inch (1.3- to 2.5-cm.) vc = Same as V a t the minimum critical point lumps. These data indicate the magnitude of the effect to Acknowledgment be expected in such cases. Application of the general equaThe authors wish to express their thanks to the Combustion tions derived herein to the case of scrubbing systems filled with special-shaped packing material. particularly if these Utilities Corporation for permission to publish the results bodies be hollow as is frequently true, is not possible without herein presented, and to T. E. Schumann for his assistance in empirical compensation for the characteristics of the special connection with the mathematical discussion.
Suspension of Macroscopic Particles in a Turbulent Gas Stream' S. P. Burke and W. B. Plummer COMBUSTION CTILITIES CORPORATION, LINDEN,h-.J.
HE general laws governing the motion of particles
T
through fluids, and more particularly the important rase of the free fall of particles, have been the subject of much theoretical analysis and considerable experimental investigation. The purpose of the present paper is to summarize existing information, present certain additional data, and to discuss practical applications of this information. The experimental studies have been confined to the reactions between macroscopic particles and a turbulent fluid stream. This case is the one usually encountered in practice, but is not necessarily identical with the case most thoroughly studied by previous investigators-via., the free fall of a macroscopic particle through a large quiescent body of fluid, turbulent conditions existing only in the layer immediately surrounding the particle. As derived by application of the principles and methods of dimensional analysis,e the general equation for the resist1 Presented in part before the Division of Gas and Fuel Chemistry a t the 72nd Meeting of the American Chemical Society, Philadelphia, Pa., September 6 to 11, 1926. 2 "The Mechanical Properties of Fluids," A Collective Work, p. 198, D.Van Nostrand Co , Inc , 1924.
ance R (grams) acting between a sphere in uniform irrotational rectilinear motion through s fluid possessing viscosity and the fluid is Rg = k p n - l p 2 - n 7 n V n (1) where P is the velocity, T the radius, p and p, respectively, the density and viscosity of the fluid, and g the acceleration due to gravity-all in c. g. s. units. For the important special case of free fall through the fluid this becomes u n =
4T
g(u--p)r3--npL"-*
3k
p"-'
(2)
where C is the terminal velocity of free fall, and u the density of the particle. For the ideal case of slow stream-line motion through a viscous liquid of infinite extent and a t rest a t infinity, R = 6 and n = 1. equation (1) becoming Rg = 6 ~ p 7 V (3) This equation was originally derived by Stokes and is discussed a t length by Lamb.3 Provided the stated conditions a
"Hydrodynamics," p. 338, Cambridge University Press. 1024.
