Fig. 3.—Apparatus Used for Measuring the Friction of Water in Sands.
Another important point in regard to a material is its degree of uniformity—whether the particles are mainly of the same size or whether there is a great range in their diameters. This is shown by the uniformity coefficient, a term used to designate the ratio of the size of the grain which has 60 per cent of the sample finer than itself to the size which has 10 per cent finer than itself.
The frictional resistance of sand to water when closely packed, with the pores completely filled with water and in the entire absence of clogging, was found to be expressed by the formula
v = cd2(h/l)(t Fah. + 10°)/60,
where v is the velocity of the water in meters daily in a solid column of the same area as that of the sand, or approximately in million gallons per acre daily;
c is an approximately constant factor;
d is the effective size of sand grain in millimeters;
h is the loss of head (Fig. 3);
l is the thickness of sand through which the water passes;
t is the temperature (Fahr.).
| TABLE SHOWING RATE AT WHICH WATER WILL PASS THROUGH EVEN-GRAINED ANDCLEAN SANDS OF THE STATED GRAIN SIZES AND WITH VARIOUS HEADS AT ATEMPERATURE OF 50°. | ||||||||
|---|---|---|---|---|---|---|---|---|
| h l | Effective Size in Millimeters 10 per cent finer than: | |||||||
| 0.10 | 0.20 | 0.30 | 0.35 | 0.40 | 0.50 | 1.00 | 3.00 | |
| Million Gallons per Acre daily. | ||||||||
| .001 | .01 | .04 | .10 | .13 | .17 | .27 | 1.07 | 9.63 |
| .005 | .05 | .21 | .48 | .65 | .85 | 1.34 | 5.35 | 48.15 |
| .010 | .11 | .43 | .96 | 1.31 | 1.71 | 2.67 | 10.70 | 96.30 |
| .050 | .54 | 2.14 | 4.82 | 6.55 | 8.55 | 13.40 | 53.50 | |
| .100 | 1.07 | 4.28 | 9.63 | 13.10 | 17.10 | 26.70 | 107.00 | |
| 1.000 | 10.70 | 42.80 | 96.30 | 131.00 | 171.00 | 267.00 | ||
The above table is computed with the value c taken as 1000, this being approximately the values deduced from the earliest experiments. More recent and extended data have shown that the value of c is not entirely constant, but depends upon the uniformity coefficient, upon the shape of the sand grains, upon their chemical composition, and upon the cleanliness and closeness of packing of the sand. The value may be as high as 1200 for very uniform, and perfectly clean sand, and maybe as low as 400 for very closely packed sands containing a good deal of alumina or iron, and especially if they are not quite clean. The friction is usually less in new sand than in sand which has been in use for some years. In making computations of the frictional resistance of filters, the average value of c may be taken at from 700 to 1000 for new sand, and from 500 to 700 for sand which has been in use for a number of years.
The value of c decreases as the uniformity coefficient increases. With ordinary filter sands with uniformity coefficients of 3 or less the differences are not great. With mixed sands having much higher uniformity coefficients, lower and less constant values of c are obtained, and the arrangement of the particles becomes a controlling factor in the increase in friction.