MECHANICAL ANALYSES 



73 



horizontal layer of the suspension is removed from the 

 depth h, at the end of time t, the weight of suspended 

 material in it will be a measure of the amount of parti- 

 cles in the suspension having settling velocities less than 

 h/1- Similar measurements of the proportion of parti- 

 cles of various settling velocities will be obtained as 

 often as the process of sampling is repeated. 



The fact that particles settling through a medium 

 such as water soon attain constant velocities proportion- 

 al to the sizes of the particles is expressed in Stokes' 

 law which, in terms of diameters in microns and veloc- 

 ities in microns per second of spherical particles set- 

 tling in pure water, may be expressed as 



V = 10-*xli5L=iiD2 



187) 



in which g= the acceleration of gravity, a = the density 

 of the settling spherical particles, and t; = the coeffi- 

 cient of viscosity of water in dyne seconds/cm2. The 

 diameters of the spherical particles may be obtained 

 from the simplified expression D=C/v', where C is a 

 constant whose value is essentially dependent on the 

 temperature and the grain densities of the particles. 

 Values of C for differences in density of the settling par- 

 ticles and water from 1.0 to 2.0, and for temperatures 

 between and 30° C are shown in figure 10. The equa- 

 tion gives only the particle diameters of spherical par- 

 ticles, and Stokes' law is not precise for particles which 

 are not spherical. Waddell (1934) has shown, however, 

 that the actual size of even a very flat particle, that is, 

 the radius of a sphere of the same volume as the parti- 

 cle, can be determined with fair accuracy from Stokes' 

 law for particles less than about 0.1 micron in size. 

 Oden (1915, p. 222) has proposed that the difficulty at- 

 tending the determination of actual particle sizes may 

 be eliminated by the concept of the equivalent radius of 

 a settling particle which is defined as "the radius of a 

 perfect sphere of the- same material which will sink at 

 the same average rate as the particle in question, the 

 latter being supposed to retain during its fall a certain 

 orientation with respect to its line of motion." Various 

 other workers, including Atterberg (1905), Mohr (1910), 

 Holmes (1921), Wentworth (1926), and Galliher (1932), 

 have set up empirical relations for the conversion of 

 settling velocities into particle diameters, and these are 

 shown graphically in figure 9, together with the diameter- 

 velocity relation calculated from Stokes' law for parti- 

 cles of density 2.6 settling in pure water at a temperature 

 of 20° C. 



Rubey (1930) statesthat in the sediments investigated 

 by him, the constituent clay particles have thicknesses 

 only about one -seventh of their length and breadth, and 

 that, consequently, the settling velocities of the particles 

 should be equal to those of spheres of diameters between 

 one -third and two -thirds of the maximum diameters of 

 the clay particles. There is shown in figure 9 a curve of 

 the average relation between settling velocity and ob- 

 served maximum diameters of the particles in the ana- 

 lyzed Carnegie sediments. Although the individual 

 observed points on which the curve is based are rather 

 widely variable, it may be seen that the average maxi- 

 mum diameters of these particles are approximately 

 five-thirds as great as those of perfect spheres of the 

 same settling velocities, a fact which is in close agree- 

 ment with Rubey' s findings. 



Stokes' law is applicable only for particles less than 

 about 0.1 mm in size. Beyond this diameter the more 



complex relation for particles and settling velocities 

 given by Oseen must be used. In the present investiga- 

 tion, however, only particles less than 0.07 mm were 

 analyzed by the pipette method, the coarser materials 

 being analyzed by sieving. The applicability of Stokes' 

 law to extremely fine-grained so-called colloidal parti- 

 cles less than 1 micron in size has been the subject of 

 considerable controversy, but Mason and Weaver (1924) 

 and Johnston and Howell (1930), among others, have 

 shown that the Brownian movement of extremely small 

 particles takes place equally in all directions, and that 

 particles subject to Brownian movement have a residual 

 downward velocity which is given by Stokes' law. This 

 holds true except for the extremely narrow zone just 

 above the bottom of a colloidal suspension. In this zone 

 there is a concentration gradient, as shown by the experi- 

 mental work of Perrin (1908, 1911), which maintains a 

 steady state. The fact that in most colloidal solutions 

 the suspended particles do not settle out is owing prima- 

 rily to the presence of slight convection currents caused 

 by inequalities of temperature. When the temperature is 

 maintained constant, it has been found that settling does 

 occur. 



Presentation of Results. The data reduced to stand- 

 ard grade sizes obtained in the pipette analyses are 

 shown in table 23, which gives the interpolated percent- 

 ages for the various size grades of the classification of 

 Udden (1914) and Wentworth (1922). The data are also 

 illustrated graphically in figures 13 to 27, in which the 

 observed settling velocities, as well as the calculated 

 equivalent diameters, are plotted as histograms and cu- 

 mulative curves. In table 24, the values for the statisti- 

 cal constants employed by Trask (1932), namely the 

 median and quartile diameters, the coefficients of sort- 

 ing and of skewness, the logarithm of the skewness, and 

 the parts of sand (greater than 50 microns), silt (50 to 5 

 microns), and clay (less than 5 microns) are shown. The 

 median diameter indicates the mid-point by weight in the 

 size distribution, whereas the quartiles indicate the min- 

 imum particle diameters of the parts making up less 

 than 25 per cent, and less than 75 per cent of the cumu- 

 lative distribution of the samp le. The c oefficient of sort- 

 ing, which is defined as Sq =V(Ql/Q3), indicates the 

 degree of spreading or sorting of the particles in a sedi- 

 ment. The better a sediment is sorted, the smaller is 

 the coefficient of sorting. Trask found in the one himdred 

 and seventy sediments analyzed by htm that the average 

 coefficient of sorting was close to 3.0, whereas in 25 per 

 cent of his samples So was smaller than 2.5, and in 75 

 per cent it was smaller than 4.5, the extremes being 

 1.26 and 9.4. 



The dissymmetry of the size distribution curve, that 

 is, the position of the mode or peak of the distribution 

 and its distance from the median, is measured by the co- 

 efficient of skewness, which is defined by the equation 



Sk = 012103 

 m2 



If the mode lies on the fine side of the median diameter, 

 Sk wUl be greater than 1.0 and log Sk will be positive; if 

 the mode is coarser than the median, Sk will be less 

 than 1.0 and log Sk will be negative. For sediments of 

 symmetrical size distribution, Sk will vary about 1.0 and 

 log Sk will be close to 0; whereas the more Sk diverges 

 from 1.0 and log Sk from 0, the farther the mode lies 

 from the median. In fifty per cent of the samples ana- 



