226 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



unit time by the acceleration of gravity. As 

 the fall of the stream in unit time is equal to 

 the mean velocity multiplied by the slope, the 

 measure of the particle's work of flow is Ms V m g. 

 This is a measure of the particle's contribution 

 to the stream's energy, while 0.63 MV s g is a 

 coordinate measure of that factor of its draft 

 on the stream's energy which depends on the 

 direct action of gravity. The other factor of 

 draft, the factor depending on tangential 

 forces, varies with the violence of vertical 

 movements and is of comparable importance 

 only in the case of torrents. If we leave it out 

 of account, the result of the comparison is that 

 when 0.63 of the rate at which the particle is 

 pulled through the water by gravity is greater 

 than the rate at which it falls by reason of the 

 general descent of the stream, its tax on the 

 stream's energy is greater than its contribution 

 thereto. Any allowance for the neglected 

 factor would be equivalent to increasing the 

 fraction 0.63. 



To illustrate this relation by a concrete 

 example: Mississippi River between Cairo and 

 its mouth has, at flood stage, such velocity and 

 slope that any suspended particle of silt which 

 would sink in still water faster than half an 

 inch a minute retards the current more through 

 the work of suspension than it accelerates the 

 current through the addition of its mass to the 

 mass of the stream. 



The velocity of subsidence has been elabo- 

 rately studied. So far as river problems are 

 concerned, it depends chiefly on the size of the 

 suspended particles. For particles below a 

 certain magnitude, which is controlled in part 

 by impurities in solution, the velocity is zero. 

 Between two critical diameters, which for 

 quartz sand are about 0.02 and 0.5 millimeter ' 

 (0.00007 and 0.00016 foot) the velocity varies 

 with the square of the diameter. Below the 

 lower diameter the variation is more rapid, 

 and above the upper it is less rapid, becoming 

 for large particles as the square root of the 

 diameter. 



It follows that the consumption of energy 

 involved in the suspension of the suspended 

 load is an increasing function of the size of the 

 particles into which the load is divided; in 

 other words, it is a decreasing function of the 

 degree of comminution. On the other hand, 



' Richards, R. H., Textbook on ore dressing, pp. 2C2-2C8, 1909. 



the contribution which it makes to energy by 

 adding its mass to that of the water is inde- 

 pendent of the degree of comminution. 



The viscosity factor is not easily compared 

 in a quantitative way with those just consid- 

 ered, but something may be said of the laws 

 by which it is related to comminution. At- 

 tending first to that part which depends on 

 interference with internal shearing of the water, 

 let us conceive of a particle with center at C, 

 figure 71, surrounded by water which is sub- 

 jected to uniformly distributed shearing along 

 planes parallel to A^AA^BJiB^ the direction 

 of shearing being parallel to the line ACE. 

 Conceive a right cylindroid figure tangent to 

 the particle and parallel to AB, its bases being 

 AJIA 2 G and B 1 EB 2 F. Motions being re- 

 ferred to as origin, the cylindroid body of 

 water would assume after a time, but for the 

 presence of the particle, the form of the oblique 

 cylindroid with bases A 1 U 1 A 2 G 1 and B 1 E 1 B 2 F 1 . 



f, f 



FIGURE 71. Interference by suspended particle with freedom of shearing. 



Because of the obstruction by the rigid par- 

 ticle the simple shearing motions thus indi- 

 cated are replaced by other motions, compo- 

 nents of which are normal to the shearing 

 planes. It is assumed that the sum of the 

 transverse elements of motion measures the 

 action occasioned by the presence of the par- 

 ticle. The actual movements caused in the 

 water doubtless affect regions within and with- 

 out the cylindroid, but their nature need not 

 be considered. The necessary transverse move- 

 ments are equivalent to the transfer of a lunate 

 wedge of water, HAJI 1 A 2 to a symmetric 

 position on the opposite side of the plane 

 A l A 2 B 2 B l and a similar transfer of the wedge 

 FB 1 F i B 2 . Linear dimensions of the first- 

 named wedge, in the direction of rectangular 

 coordinates, are A^A^, HA, and HII^. A^A 2 

 equals a diameter of the particle. IIA equals 

 a semidiameter. As the angle HAII lt being 

 given by the general amount of shearing, is 

 independent of the size of the particle, IIH, 



