494 HYDROMECHANICS [HYDRAULICS. an 30 nation to a straight lino having for abscissa, for Vui c ordinate, and inclined to the axis of abscissa; at an angle the tan gent of which is - . Plotting the experimental values of and /=, the points so c Vra found indicated a curved rather than a straight line, so that j8 must depend on a. After much comparison the following form was arrived at 1 + An Vm where n is a coefficient depending only on the roughness of the sides of the channel, and A and I are new coefficients, the value of which remains to be determined. From what has been already stated, the coefficient c depends on the inclination of the stream, decreasing as the slope i increases.
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I_j6 1 j ^ (t r~ I a -t- Then 1 + the form of the expression for c ultimately adopted by Ganguillet & Kutter. For the constants a, I, p Ganguillet & Kutter obtain the values 23, 1, andO-00155 for metrical measures, or 41 6, I Sll, and 00281 for English feet. The coefficient of roughness n is found to vary from O OOS to O OSO for either metrical or English measures. The most practically useful values of the coefficient of roughness n are given in the following table : Coefficient of Nature of Sides of Channel. Roughness n. Well-planed timber 009 Cement plaster 010 Plaster of cement with one-third sand Oil Unplaned planks 012 Ashlar and brickwork "013 Canvass on frames 015 Rubble masonry "01 7 Canals in very firm gravel 020 Rivers and canals in perfect order, free from stones or weeds Rivers and canals in moderately good order, not quite free from stones and weeds Rivers and canals in bad order, with weeds and detritus 035 Torrential streams encumbered with detritus "050 Ganguillet & Kutter s formula is so cumbrous that it is difficult to use without the aid of tables. Mr Lowis D A. Jackson has published complete and extensive tables for facilitat ing the use of the Gan guillet & Kutter formula (Canal and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form : of the coefficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from 0163 to 0301. For natural channels or rivers n varies from 020 to 035. In Mr Jackson s opinion even Kutter s numerous classes of channels seem inadequately graduated, and after careful examination he pro poses for artificial canals the following classification : 1. Canals in very firm gravel, in perfect order n = 02 II. Canals in earth, above the average in order n = 0225 III. Canals in earth, in fair order ?i = 025 IV. Canals in earth, below the average in order ?i = 0275 V. Canals in earth, in rather bad order, partially over- ) 7 ,_o-o3 grown with weeds and obstructed by detritus Fig 104. 92. Forms of Section of Channels. The simplest form of section for channels is the semicircular or nearly semicircular channel (fig. 104), a form now often adopted from the facility with which it can be 0-025 0-030 In, Bank k- 1 ^^^^^m^. Concrete/ / -^ Fig. 105. executed in concrete. It has the advantage that the rubbing surface is less in proportion to the area than in any other form. In Cizttinq 33,,
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The following table gives a selection of values of M, taken from Mr Jackson s tables : Values of M for n = 0-010 0-012 0-015 0-017 0-020 0-025 0-030 00(101 3-2200 -3-8712 4 -.8:590 0-4842 6-4520 8-0650 9-6780 00002 1-8210 2-1852 2-7315 3-0957 3-6420 4-5525 5-4630 0001)4 1-1185 1-3422 1-6777 1-9014 2-2370 2-7962 3-3555 00006 0-8843 1-0612 1-3264 1-5033 1-7686 2-2107 2-6529 00008 0-7072 0-9-206 1-1508 1-3042 1-5344 1-9180 2-3016 00010 OT,!)70 0-8364 1-0455 1-1849 1 -3940 1-7425 2-0910 00025 0-5284 0-6341 0-79-26 0-8983 1-0568 1-3210 1-5852 00050 0-47-22 0-5660 0-7083 0-8027 0-9444 1-1805 1-4166 00075 0-1535 0-5442 0-6802 0-7709 0-9070 1-1337 1-3605 00100 0-4441 0-5329 0-6661 0-7550 0-8882 1-1102 1-3323 00200 0-4300 0-51RO 0-6450 0-7310 0-8600 1 -0750 1-2900 00300 0-4254 0-5105 0-6381 0-7232 0-8508 1-0635 1-276-2 I i FIG. 106. Scale 20 feet = l inch. "Wooden channels or flumes, of which there are examples on a large scale in America, are rectangular in section, and the same form is adopted for wrought and cast-iron aqueducts. Channels built with brickwork or masonry may be also rectangular, but they are often trapezoidal, and are always so if the sides are pitched with masonry laid dry. In a trapezoidal channel, let b (fig. 105) be the bottom breadth, 1 the top breadth, d the depth, and let the slope of the sides be n horizontal to 1 vertical. Then the area of section is = (b + nd)d = (b - nd)d, and the wetted perimeter One principal difficulty in the use of this formula is the selection X = When a channel is simply excavated in earth it is always originally trapezoidal, though it becomes more or less rounded in course of time. The slope of the sides then depends on the stability of the earth, a slope of 2 to 1 being the one most com monly adopted.
Figs. 106, 107 show the form of canals excavated in earth, the
