And for larger drains, including the main drains, their cross-sections at any point should be at least 1⁄6000 of the area drained, giving a velocity of 0.55 foot per second with the rate of filtration mentioned above.
Fig. 4.—Plan of one of the Hamburg Filters, Showing Frictional Resistance of the Underdrains.
The total friction of the underdrains from the most remote points to the outlet will be friction in the gravel, plus friction in the lateral drains, plus the friction in main drain, plus the velocity head.
Constructing the Underdrainage System of a Filter, Hamburg.
[To face page 42.]
I have calculated in this way the friction of one of the Hamburg filters for the rate of 1,600,000 gallons per acre daily at which it is used. The friction was calculated for each section of the drains separately, so that the friction from intermediate points was also known. Kutter’s formula was used throughout with n = 0.013. On the accompanying plan of the filter I have drawn the lines of equal frictional resistance from the junction of the main drain with the last laterals. My information was incomplete in regard to one or two points, so that the calculation may not be strictly accurate, but it is nearly so and will illustrate the principles involved.
The extreme friction of the underdrains is 11 millimeters = 0.036 foot.
The frictional resistance of the sand 39 inches thick, effective size 0.32 mm. and rate 1.60 million gallons per acre daily, when absolutely free from clogging, is by the formula, page 21, 15mm., or .0490 foot, when the temperature is 50°. Practically there is some matter deposited upon the surface of the sand before filtration starts, and further, after the first scraping, there is some slight clogging in the sand below the layer removed by scraping. We can thus safely take the minimum frictional resistance of the sand including the surface layer at .07 foot. The average friction of the underdrains for all points is about .023 foot and the friction at starting will be .07 + .023 = .093 foot (including the friction in the last section to the effluent well where the head is measured, .100 foot, but the friction beyond the last lateral does not affect the uniformity of filtration). The actual head on the sand close to the outlet will be .093 and the rate of filtration .093⁄.070 · 1.60 = 2.12. The actual head at the most remote point will be .093 - .036 = .057, and the rate of filtration will there be .057⁄.070 · 160 = 1.30 million gallons per acre daily. The extreme rates of filtration are thus 2.12 and 1.30, instead of the average rate of 1.60. As can be seen from the diagram, only very small areas work at these extreme rates, the great bulk of the area working at rates much nearer the average. Actually the filter is started at a rate below 1.60, and the nearest portion never filters so rapidly as 2.12, for when the rate is increased to the standard, the sand has become so far clogged that the loss of head is more than the .07 foot assumed, and the differences in the rates are correspondingly reduced. Taking this into account, it would not seem that the irregularities in the rate of filtration are sufficient to affect seriously the action of the filter. They could evidently have been largely reduced by moderately increasing the sizes of the lower ends of the underdrains, where most of the friction occurs with the high velocities (up to .97 foot) which there result.