July, 1929
INDUSTRIAL A N D ENGINEERING CHEXISTRY
who are each year more familiar with their subjects and are constantly developing into better and better investigators. In the long run this is z t serious deficiency. The governing body of the Dry Milk Institute is the board of directors under whom is the paid president with his paid assistants. The president is the responsible executive who carries out the work of the institute. From the personnel of the companies making up the membership of the Dry hlilk Institute there are appointed several committees, of which the research committee is one. These committew are advisory only. The president can lean upon them for advice and assistance as much as he sees fit, but the action and also the final responsibility are his. After consultation with research committee, the president makes arrangements with some institution for a fellowship. The money is paid to the institution. The final responsibility for picking the research fellow and for directing his work lies with the institution. I n practice, this is usually done after consultation with the president of the Dry Milk Institute. Occasional conferences are held between the research fellow, the head of the department in the institution where the work is being done, the president of the institute, and the research committee. It would be advantageous for all concerned if these conferences could be more frequent. Progress reports a t monthly
661
or quarterly intervals are required. On the completion of any piece of work a full report is made, which is published with the name of the research fellow as author. So far this plan has worked well, and the results obtained from our researches have been of direct value in developing new outlets for our product and in helping our customers to get the best results. One of the principal difficulties is to keep in sufficiently close touch with the real needs of the members of the institute and to retain their active interest, so that they will make full use of results obtained. Research is of no value unless it is used. The motto of the Quartermaster Corps of the U. S. Army is "Service to the Line." In other words, the Quartermaster Corps does not exist for its onm sake. No matter how smugly and smoothly it may be organized within itself, the sole measure of its value lies in the practical assistance which it renders t o the fighting line forces. The same is true of industrial research. Its value depends on its practical assistance to the production department or to the sales department. For this reason it must work in close touch with these departments and have their full cooperation. This beneficial relationship depends primarily upon those in executive control of the laboratory, rather than upon the research chemist himself.
The Gravitational Flow of Fertilizers and Other Comminuted Solids' W. Edwards Deming and Arnon L. Mehring BUREAUOB CHEMISTRY A N D SOILS, WASHINGTON, D. C.
I
S A previous study on the
An expression giving the rate of gravity flow of comThis study was therefore minuted solids from bins or hoppers has been derived. factors affecting the drillundertaken t o derive an equaThe time required for a given mass of material to flow ability of fertilizers (.2),* tion that will express the law as calculated from this formula agrees closely with it was found that one of the governing the rate of flow of that determined experimentally over a wide range of most important was the rate solid particles. conditions. The experimental materials included of flowof the material through crystallized urea and ammonium phosphate, crushed Theoretical Treatment the distributor. A niathephosphate rock, potassium nitrate pellets, glass beads, matical expression of the laws lead shot, marbles, and several varieties of seed. The governing the flow of solid When the material in a flatrate of flow of such materials is found to depend upon particles would be of value bottomed bin flows from an the average particle size, the kinetic coefficient of friction in designing not only fertilizer opening in the floor, a vacant and apparent density of the material, the diameter of distributors, but a l s o s u c h space in the form of an inthe opening and the vertical angle of the hopper or, other equipment as seed drills, verted truncated cone will be if all of the material does not flow out, the angle bestorage bins, and automatic produced. The vertical angle filling machines. An exprestween opposite slopes of the remaining material at of this conical space will be repose. sion of this kind might also 180" - 2 +, where is the be applied in calculating the internal angle of repose of time- bf discharge of a-bin or hopper filled with grain, coal, the given material. The rate of flow should be equal to or similar material. that of the same material from a funnel of the same angle It might appear that the flow of solid particles is a special as this vacant space and of the same opening. The time case of the flow of viscous liquids. But this is impossible for necessary for a funnel, such as shown in Figure 1, to empty two reasons. First, the rate of flow of a liquid is proportional by gravity flow will depend on the vertical angle 4, the to the head, and pressure is transmitted equally in all direc- diameter B of the opening, the friction of the material, and tions, even in very viscous liquids; however, Major Phillips, the size of the particles. as quoted by Shaxby and Evans ( 3 ) ,showed that pressure is not Direct application of the principles of mechanics is difficult, transmitted any considerable distance through columns of if indeed possible, but a formula that will represent the time powders. Second, no provision is made in the formulas of rate of flow of comminuted solids may be obtained by means hydrodynamics for size of particles, but the size of particles of dimensional analysis (1) and experimental data. A special influences the rate of flow of solids unless this size is infini- case in which the particles are spherical and uniform in size tesimal in comparison with the size of the orifice. will be considered first. The differentialequations of motion for a mass of equal spheres might be expected to contain the 1 Received March 6, 1929. * Italic numbers in parenthesis refer to literature cited at end of article. following physical quantities:
+
Vol. 21, No. 7
INDUSTRIAL AND ENGINEERING CHEMISTRY
662
DIMENSIOKS IN NAME SYMBOL TERMSOF M , L,T Diameter of spheres D L Diameter of orifice B L Density d ML-8 Gravity LT-2 Time for a given mass to flow through an orifice TM-1 Tangent of angle of repose of material (coefficient of friction) I" Xone Angle of cone 6 None
f
Four dimensionless combinations from the seven quantities are possible. D:B, 1, 9 , are three, easily picked out. If D"dug't is another, z, y, 2 must satisfy: Condition on M , Condition on L , Condition on T , 5 = 5/*,
y = 1, z =
1/2.
y - l = O x-3y+z=o --22+1=0
down glass plates. Crystallized ammonium phosphate and urea and crushed phosphate rock were prepared by screening only. Examination with the naked eye or with a microscope showed that the rolled pellets of all sizes and kinds were very nearly perfect spheres. It was found later, however, that the time required for ellipsoids and other non-spherical grains of the same material and sieve interval to flow from a given funnel deviated from that consumed by the same mass of spheres only when their shape was sufficiently different from spherical to change the apparent density or the angle of repose. 70
so
tD5"d g1/2 = f ( D : B , p, +) or t
t
=
D-5/2d-1g-"y(D:B,
p,
=
F ( D : B , p, 4) or t
=
-
mtnutes per 100 grams
+) (1)
wherein f denotes some function whose form will be investigated. By putting (D:B)"Y(D:B,p , +) = F ( D : B , p, +), t B 6 / * dg"'
Sieved Potassium NLtrate Pellets
d B-IO
B-"*d-1g-'/*F(D:B, p, +) (2)
When h, 4, g are constant, tB'/z d should be a function of D:B. I n order to keep p constant experimentally and yet vary D:B, different-sized pellets of the same substance can be tested. By using the same funnel in a series of tests to determine t (the time in minutes for 100 grams of pellets t o flow through a funnel), 4 and B are kept constant. And if B can be changed on a particular funnel by screwing on different-sized openings, then D:B is further varied while 4 remains unaltered. Now if tB5/' d is plotted against D:B, the points should determine a curve of some sort. Other values of q5 would give other curves, thus establishing a family of curves with 4 as parameter. By repeating this process with other materials having different p, different families with p as a second parameter would be obtained. From this reasoning it seemed possible to determine empirically the form of the function F(D:B, p, 4). Such curves will be presented after the experiments wherein the necessary values were obtained have been described. Preparation of Materials
The materials used in these experiments may conveniently be divided into two classes: (1) those in which the particles were spherical such as sprayed potassium nitrate and m a r b l e s ; and (2) those in which t h e p a r t i c l e s were angular or irregular in shape. T h e s e t w o classes will be called pellets and granules, respectively. Potassium n i t r a t e pellets were prepared LB -4 by fusing the salt and Figure 1 then spraying it under pressure from a nozzle at an altitude-of 40 feet (12.2 meters) from the ground, The droplets congealed into pellets of varying sizes as they fell. The entire mass was next separated into fractions containing particles with diameters falling within certain narrow limits by screening with high-grade metal sieves calibrated by the U. S. Bureau of Standards. Each portion was then rolled down glass plates, as is done in the manufacture of lead shot, to eliminate the non-spherical ones. Sprayed urea, unperforated glass beads, lead shot, and mustard, vetch, and kale seeds were also carefully sieved and rolled
40
Jo /
,' /' , , ' . *, ''0 , , ' /'
/
r
.05
.IO
.I5
.20
.25
Figure 2
Experimental Methods
In determining t , a weighed mass of the material, sufficient in quantity t o be timed accurately with a stop watch, was allowed to flow through a funnel by gravity. The axis of the funnel was kept vertical. The flow was started by one operator and timed by another. The stop watch was started when the first, and stopped when the last, of a batch issued from a funnel. Stop-watch readings were made by a t least two observers in each test, and the means of the results obtained by each of several workers usually varied from each other less than the individual determinations of any one observer deviated from his mean. This method eliminates the effect of reaction time in different individuals. Head was shown to have no practical effect upon the value of t as obtained in these experiments by widely varying the depth while other factors remained the same. Three heavy copper funnels were constructed with angles of 30, 60, and 90 degrees. Screw caps having openings of about l, 2, 3, 5, 10 mm. diameter were made for each funnel and two openings of larger diameter were provided for the 90-degree funnel. These caps could be screwed on and off as depicted in Figure 1. Thus B could be varied while was kept constant. The insides of funnels and screw caps were reamed carefully, so that was precise and there were no apparent seams nor joints on the interiors when any cap was in place. A large funnel with an angle of 60 degrees and an opening of 73 mm. was used for the marbles. The openings were measured with a spectral comparator, the use of which was kindly permitted by Dr. I. C. Gardner at the Bureau of Standards, and the values of B thus obtained are used in the calculations. I n order to determine the coefficient of friction of the
+
+
I S D C S T R I A L A S D ESGIiYEERIA-G CHEMISTRY
July, 1929 Table I-Time
Required for 100 Grams of Various-Sized Potassium Nitrate Pellets to Flow from Funnels with Different Openings a n d Angles Temperature, 20‘ C. Relative humidity, 307, p = 0434 Q
lpening
B
Jf m 8-10
16-20
1 85 1 16
1 26 0 840
20-30
Mm.
2 46
1 s3 10-16
0 840
A! m .
2 16 1 56
2 20 1.61
1.203 1,216
1 05
1.09
1.243
0 663
0.708
1.265
0 428
0,438
1,244
30-40
40-50
279
50-60
0
60-80
0 218 0 178
80-100
0 218
0 178 0 143
10:06
O.OS03
10106 LO2
0.0724 0.483
10:66
5.02 2.94 2,06 0 324
0 331
1.270
0 248
0.252
1.274
0 198
0.200
1.248
0 161
0 163
tobad.
10:06
101 @6 5.02
....
....
0.0647 0.440
....
0.0580
0.375 1.71 5.05
....
10:06 5.02 2 . !I4 2.06
0.0539 0.321 1.44 4.04
10:06 5.02 2 , !I4 2, 0 6
0,0527 0.314 1.37 3.79 .... 0,0528 0,321 1.335 3.608 .... 0.052 0.301 1.262 3.410
10:b6 5.112 2 ,134 2.06 1,290 le’66 5.02 2 94
2.06
__-
....
lcalcd.
+
3pening B
Min.
....
0.0943
....
tobsd.
Mm.
Min.
10:06
1.31 .... 0,0989
10:06
0.0832
....
....
....
Min.
....
1.31
....
0.109
....
10:06 4.97
0,0849
0 0910
0.735
0.737
0.0621 0,435
10:06 4.97
0.0780
10:06 4.97 2.99 2.04
0,0700
0 ’ 0700
0,500
0.487 2.12 6.74
....
....
0.0579 0.377 1.71 4.81
....
....
0.595
....
2.13 6.95
....
0.0547 0,347 1.52 4.20
10:06 4.97 2.99 2.04
0.0625 0 445 1.78 5,36
0.0529 0.328 1.40 3.80
l0:06 4.97 2.99 2.04
0.0613 0.420 1.66 4.89
l0:06 4.97 2.99 2.04
0.0618 0,404 1.61 4.67
....
....
....
.... 0.0630 0.325 1.358 3.63 .... 0.051 0,303 1,257 3.31
.... 10:06 4.97 2.99 2.04
0 0607
0.394 1.49 4 30
d = 90 DEGREES
tcilcd.
0.0711 0.527
materials, a convenient quantity of each was poured into a heap upon a level surface. The opening from which the niaterial was poured was kept just above the apex of the pile. The angle o f repose included between the sloping side of the pile and its base was measured a t several points and the results were averaged for each substance. The tangent of the angle thus determined is the kinetic coefficient of friction, p,. All materials were stored and run in a room of constant temperature and humidity t o insure p remaining constant throughout a series of tesw The angle of repose may vary slightly with the size of the particles, but the variation, if any, was so small that it could be neglected. D in Equations 1 and 2 stands for the diameter of the spheres flowing through the funnel, and thus implies that all have the same diameter. The experimental particles, however, were not all of exactly the same diameter. Each lot was composed of spheres with diameters between certain limits. Therefore, we must deal with the average diameter D of each batch An average can never be taken until the method of averaging is specified. Since time of flow is the most important consideration in this case, 5 for a mass of particles that flows in a given time t is defined as that diameter necessary to give the same t if they were uniform in size. In dealing with particles of odd shapes the term “average diameter” is replaced by “average size,” but it is D z ) or 5, where D1 denoted by the same symbol, f ( D l and D2 are the smaller and larger sieve openings. Thus D = 1.56 mm. may refer to pellets having diameters ranging from 1.26 to 1.85 mm., or simply to granules that will pass the S o . 10 sieve but not the No. 16. The sieve openings used in this work were those determined by the Bureau of Standards and not the values stated on the screens. If the diameters of a batch of spheres range from I ) , to Dz with a flat frequency distribution, +(Dl Dz) is :t fair approximation to 5, improving as the interval Dz -- D, decreases.
+
4 = 60 DEGREES
30 DEGREES
....
0 370
0 370 0 279
-
Min. .... 0,0994
.Urn.
G./cc.
0 485 0 485
663
0 0772 0 581
....
0.0654 0.442 1.85
5.71
....
0.0627 0.412 1.68 5.04 . ,. . 0.0624 0 401 1 60 4.71
....
0,0582 0.373 1.47 4.27
M m. 15 09 10.14 15.09 10 14 15.09 10.14 4 98 15.09 10.14 4.98
13.09 10.14 4.98 3.00 2.05 15.09 10.14 4.98 3 00 2.05 15.09 10.14 4.98 3.00 2.05 15.09 10.14 4.98 3.00 2.05
15.09 10.14 4.98 3 00 2.05
Mzn 0 0481 0 180 0 0371 0 130 0 0327 0 109 0 0 0 0 0 0 0
888
0293 0890 702 0274 0800
567 2 64 8 14 0 0243 0 0730 0 483 2 18 6 79 0 0239 0 0714 0 449 1 96 5 86 0 0245 0 0696 0 454
1 87
5 49 0 0234 0 0698 0 415 1 7.2 4 94
Min.
0.0461 0.155 0.0396 0 129 0.0336 0 105
0.892 0.0292 0,0884 0.693 0.0270 0.0788 0.510
2.53 8.02
0,0254 0.0726 0.511 2.19 6.74 0,0245 0.0696 0.474 1.96 5.89 0.0238 0.0692 0.457 1.86
5.46 0.0233 0.0652 0,424 1.68 4.90
Later (Equation 8) it will be shown how to refine the calculation of 0. An attempt Tvas msde to find out how closely f(0, Dz) agrees with- the average diameter that one would get by measuring a large number of pellets and taking the mean. They were measured with a microscope, a micrometer, or by Laking one-thousandth of the total length of a thousand pellets laid in a groove side by side and in contact. In another method it was computed from the absolute density, volume, and number in a given sample. The last method theoretically should yield a value a little greater than $(Dl D2). ,411 samples were taken by quartering. Each size was measured by one or more of these methods and the results showed that the sieving had been done well. One might use either absolute or apparent density in Equations 1 and 2 if they varied in the same ratio. However, apparent density varies with the physical form and heterogeneity of size of the particles composing a mass, while absolute density remains the same. Both measures were tried in the equations and excellent results were obtained only when apparent density was used.
+
+
Results
The results with sprayed potassium nitrate are shown in Table I. The columns for t&d. are worked out from Equation 4,t o be derived later. As explained previously, tB5I2d should be a function of 5 : B . Accordingly, tB5I2 d 3 y was plotted as ordinates with i(D1 Dz):BE 5 as abscissas, for the potassium nitrate data (Figure 2). The points determine three straight lines corresponding t o the three values of q5. Similar experiments were performed using crystalline monoammonium phosphate, and the data are presented as Table 11. tB5/%d and $(Dl D2):B computed therefrom also determine three straight lines corresponding to the three values of 4. On these charts two adjacent points may repre-
+
+
INDUSTRIAL A N D ENGINEERING CHEMISTRY
664 Table 11-Time
Required for 100 Grams of Various Sized A m m o n i u m Phosphate Crystals t o Flow from Funnels with Different Openings and Angles p = 0.725 Relative humidity, 50% Temperature, 20° C.
-
SIEVE OPENINGS 0 2
DI
1
Mm.
Mm.
Mm.
1.345
1.51
G./cc. 0.835
0.663
0.706
0.840
0.427
0.435
0.847
40-60
0.37 0.218
0,294
0.313
0.854
60-80
0.218 0.178
0.198
0.200
0.892
80-100
0.178 0.143
0.161
0.163
0.906
100-125
0.143 0.120
0.131
0.134
1.926
20-30 30-40
a
0 30 DEGREES
APPARENT
( D I T D ~ )Eq.8
1.85 0.84 0.84 0.485 0.485 0.37
10-20
VOl. 21, No. 7
1
OPegning
lobsd.
0
tcalcd.
=
60 DEGREES
Opening B
tobsd.
4 = 90 DEGREES
Opening B
toslcd.
tobsd.
toaled.
Mm.
Min.
Min.
Mm.
Min.
Min.
Mm.
Min.
Min.
10.06
0.158
0.174
10.06
0.215
0.222
10.14
0.251
0,256
10.06 5.02 10.06 5.02 2.94 10.06 5.02 2.94 2.05 10.06 5.02 2.94 2.05 10.06 5.02 2.94 2.05 1
0.135 0.935 0.119 0,790 3.68 0.115 0.700 2.98 9.06 0.103 0.640 2.75 7.33 0.102 0,620 2.62 6.37 45
0.133 0.956 0.119 0.798 3.68 0.113 0.725 3.23 8.98 0.103 0.635 2.69 7.28 0.099 0.606 2.53 6.73 48
10.06 4.97 10.06 4.97 2.99 10.06 4.97 2.99 2.04 10.06 4.97 2.99 2.04 10.06 4.97 2.99 2.04 1
0.168 1.31 0.150 1.00 5.78 0.129 0.885 3.70 12.4 0.120 0.740 2.97 9.44 0.113 0.720 2.68 8.33 56
0.158 1.25 0,136 0,990 4.44 0,126 0,872 3.76 11.9 0.112 0,738 3.03 9.16 0,107 0,694 2.79 8.28 60
10.14 4,98 10.14 4.98
0.182 1.50 0.150 1.14
0,178 1.46 0.146 1.13
10.14 4.98 3.00 2.05 10.14 4.98 3.00 2.05 10.14 4.98 3.00 2.05 1
0.129 0.960 4.37 16.4 0.115 0,800 3.30 12.6 0.108 0.740 2.99 10.5 68
0.133 0.974 4.33 16.6 0.117 0.806 3.39 13.1 0.112 0.752 3.10 12.0 71
... a
...
...
Will not flow.
s
sent data obtained with widely different sizes of material B-5l2d-1 (a D : B f b)(krD3d/6)dD and funnel openings. As the points approach the intercept Ds t = on the y axis, D becomes less and less important. Disregarding molecular forces, the intercept on the y axis represents the (kirDSd/6)dD point where size of particles is no longer a factor in the rate of flow. I n both figures the lines intersect in a point (alp). O.S(Dz' The coordinates cy, p of this point are functions of p which are = a 'I5) :B b = a5:B b (7) Dz4 - D14 assumed to be linear. CY = - 0.130 0.161 ,u and (3 = 3 4 . 6 ~ give exactly the points of intersection (-0.06, 15) in Figure so D = 0.8(Dz6 - D15)/(D24- D14) (8) 2 and (-0.013, 25) in Figure 3. The slope m of each line When 5 is computed and Figures 2 and 3 redrawn using 5 : B is a function of both p and 4. m = ~ ( 6 7 . 4 444 sin :+) fits very closely. Using these values of a , p, m, in y - /3 = in place of +(DL 0 2 ) : B , the points, on the whole, lie closer o the lines. D in the tables-was computed from Equation 8. m (z - cy), which is the equation of a straight line passing tThe more accurate formula for D ought to be used when DZ:D1 >2. through ( a , p) and having slope m,
+
+
+
y
+
- 34.6 p = ~ ( 6 7 . 4+ 444 sin i + ) ( x + 0.130 - 0 . 1 6 1 ~ )
(3)
whence
t
=
y/Bs/zd = &d[34.6
- 0.161 p ) ]
INGS
(4)
where t is in minutes per 100 grams, B is in millimeters, and d is in grams per cubic centimeter; or = - /*
B5/2d
[0.201
+ (0.392 + 2.58
sin $+)(D:B
0.130
+
- 0.161 p)]
(5)
where t is in hours per ton (2000 pounds), B is in feet, and d is in pounds per cubic foot.
+
Using f(D1 D2) for 6 i s , of course, an approximation. I t may now be calculated more exactly. Let m be the total mass of n spheres having diameters ranging from D1 to Dp, dm the mass of the d n spheres having diameters ranging from D to D -IdD (Dl ID ID z ) . Let t ( D ) be the t for spheres all of diameter D. t ( D ) = B-b/*d-l(a D : B b)
+
if D Z- D1 d will be practically constant as D varies from D1 to is not too great. According to the definition for D given in a former paragraph,
D = Dz
t = J
D = DI
/
t(D)dm f d m = a 3 : B
+b
Table 111-Time Required for 100 Grams of Various Materials t o Flow by Gravity from Funnels with Openings a s Given
SIEVE MATERIAL OPEN-
+ (67.4 + 444sin $4) ( B : B f 0.130
+
(6)
Now, knowing how t varies with D, this integral may be evaluated. Assuming that the frequency distribution between D1 and D2 is flat, so that dn = k dD, k being constant, dm = k?rD3d d D / 6 and
Kaleseed Mustard seed
Phosphate rock
lobsd.
b e d .
Mm.
Min.
Min.
10.06 10.06 10.06 10.06 15.09
0.26 0.38 0.148 0.186 0.0697
0.30 0.32 0.146 0.193 0.0697
2.464 1.850
2.15
10.06 10.06 15.09 10.06 10.06 10.14 16.00 19.09
0.161 0.214 0.0798 0.110 0.137 0.150 0.0714 0.0426
0.167 0.224 0.0796 0.105 0.133 0.154 0.0690 0.0403
Vetch seed Phosphate rock
B
0.663 0,442 1,072 30 60 1.724 0.688 0.416 30 60 90
Ureapellets 0.840 0,485
0,485 0.370
Mm.
0
0.840 0.485 1.850 1.260
Mm. Urea crystals
D APPARENT DENSITYp D d
0'
G./cc.
0,745 0,480 30 60 90 0.663 0.757 0.405 30 60 90 3.38 0.818 0.415 90 90 0.427 1.250 0.690 30 60 90
0,1785 0,161 1.280 0,690 30 0.1435 60 90 Marbles 13.5 1.322 0.412 60 Unperforated glass beads 3.54 1.572 0.418 30 60 90 Lead shot 1.78 6.545 0.355 30 h'o. 10 60 90 Lead shot 2.03 6.595 0,355 30 No. 9 60 90
5.02 0.56 4.97 0.68 4.98 0.75 2.94 2.99 3.00 73.025
1.74 1.99 2.07 0.000775
0.53 0.65 0.74 1.74 1.95 2.16 0.000729
10.06 0.0773 10.06 0.104 15.09 0.0382
0.0762 0.104 0.0361
10.06 0.0139 10.06 0.0180 15.09 0.0068
0.0136 0.0181 0.0064
10.06 0.0151 10.06 0,0192 15.09 0.0072
0.0142 0.0192 0.0069
I S D C S T R T A L AA'D ENGINEERIiVG CHEMISTRY
July, 1929
The locations of the lines in Figures 2 and 3 were calculated from Equation 3, and fcalcd in the tnbks was computed from Equation 4. Five constants ioccuring in Equation 3) were sufficient to represent the data on potassium nitrate pellets and on crystalline ammonium phosphate. These materials have such widely different properties with respect to shape of particles, angle of repose. and density, that it was thought possible that the formula might hold for other conditions. Experirnents were therefore conducted in which the conditions were Taried as widely as practicable t o test .____
'01
I
Siebed Amnoilium PPOsphate
Crystals
,
50
40
665
the general applicability of the equations. The size of the particles, 5, was varied from 0.131 to 13.5 mm., the diameter B of the orifice from 1 to 73 mm., the apparent density d from 0.4 to 6.5 grams per cubic centimeter, the coefficient of friction p from 0.380 to 1.070, and the funnel - angle q5 from 30 to 90 degrees. Thus the range of D : B varied from a value approaching zero to one of about 0.25, above which the material would not flow freely through the opening. t in the several tests varied from 0.0007 to 48.0 minutes. In every case, as may be seen in Table 111, the calculated and observed time agreed satisfactorily. The only cases in which the calculated time varied more than a few per cent from the observed were those in which the value of n : B neared its upper limit of 0.15 t o 0.26 (the limit varies with the shape of the particles) above which, the flow,if any, is not perfectly free. Equations 4 and 5 apparently represent the law governing the flow of solid particles of any density, shape, and size, provided the size is not small enough t o introduce cohesion as a factor. I n this case the particles will usually be finer than 200 mesh. It will also probably hold for any funnel angle between 20 and 110 degrees, which should cover most cases met with in engineering practice. Acknowledgment
D,,D,= sieve operiinqs b h for 0- IO m m ,approx.
.
30
*,,+
I
?,+
*I
E3= 5
"
*
B=3
''
8=2
"
.
-=,4
, , ,
"
1
"
Credit is due Mr. RIerrill E . Jefferson for the careful nianner in which he made many of the measurements used in this work. Literature Cited (1) Bridgman, "Dimensional Analysis," Yale University Press, 1922 (2) (3)
Figure 3
Mehring and Cummings, U.S. Dept. Agr , Tech. B u l l , in publication. Shaxby and Evans, Trans. Faraday Soc., 65, 60 (1923).
Dispersion of Pigments in Rubber-11' Ernst A. Grenquist THEFISKRUBBERCOMPANY, CHICOPEE FALLS,
A
PREVIOUS paper (IO)* d e s c r i b e d t h e distribution of parti-
MASS.
A theoretical conception of the reenforcement of rubber by pigments has been developed. New experimental evidence has been presented which leads to a better understanding of the final dispersion and reenforcement of a rubber compound. I t is shown that pigment reenforcement is influenced by (a)rubber structure, (b; the state of aggregation of proteins and natural resins, ( c ) the isotropic properties of carbonblack particles, and ( d ) the presence of recrystallized rhombic sulfur at the beginning of vulcanization.
cles in compounded rubber with special reference to agglomeration and flocculation. It was emphasized that a correct understanding of t h e final dispersion a n d r e e n forcement of a rubber compound could only be obtained with a more thorough knowledge of the structure and physicalchemical properties of rubber and pigments themselves and of the nature of the vulcanization process. Kew experimental results in regard to these particular points are presented in the following investigation. Theoretical
Owing t o lack of agreement in regard to terminology, a number of conceptions considered in this paper \\Till first be defined and discussed from a rubber-compounding point of view. Surface energy or tension, free surface energy, interfacial Presented before the Division of Rubber Chemistry a t the 76th Meeting of the American Chemical Society, Swampscott, Mass., September 10 to 14, 1928. * Italic numbers in parenthesis refer to literature cited at end of article.
tension is the specific attrac-
tion existing a t a surface. On account of the unsymmetrical field of force surrounding molecules a t a surface, the molecules adjust themselves in such manner as to give a surface of minimum potential energy. It is e v i d e n t t h a t energy must be added to a system of rubber and undispersed pigment if additional rubber and pigment surface or interface between rubber and pigment is to be formed. The greatest part of this energy is added in form of work. Since the work may be given back in contraction of the surface, this amount of energy is said to be preserved in the surface as free energy. Interfacial tension is a measure of the amount of free energy present in the interface. Vetting, cdhesion tension, attraction (between rubber and Pigment) is the decrease in free surface energy taking place when a rubber surface is brought in contact with a pigment surface forming an interface. Change in the distribution of pigmenis in a system rubberpigment before reaching a state of equilibrium and a final degree of dispersion can be classified as follows: (1) aggregation-by (a) agglomeration or (b) flocculation; (2) disaggregation (dispersion)-by (a) disagglomeration or (b) de-
