3.64 21800 slope will therefore be represented by the fraction or 0.000167 feet per foot. Taking the formula as given by Eytelwein, somewhat simplified for practical use, in which A is the area of cross-section, S the slope in feet, per foot, and p the wetted perimeter, and, making the proper substitution, we have— Mean velocity, per second. 3.892 or 2.65 miles per hour. Discharge, per second.... 17,085 cubic feet. At Dam No. 9, on the Monongahela River, which is 400 feet long, the results of a number of observations taken indicated that for a rise of 10 feet in the lower pool above low water the water gauge in the upper pool gave generally a rise above the crest of the dam of 5.50 feet. If the regimen of the Kentucky River is considered similar to that of the Monongahela, and that a dam 400 feet long and 12 or 14 feet high at Beattyville would have approximately the same effect upon the flow in the Kentucky River as Dam No. 9 has upon the Monongahela, the discharge can be obtained at a 10-foot stage above low water, by calculating the discharge over a weir or dam 400 feet long with a head of water 5.50 feet. The formula for discharge over a weir or dam resulting from the experiments made by J. B. Francis, civil engineer, is D=3.331√H3 in which H=head and l-length of weir in feet and 3.33 a constant. If the water approaching the weir has a material velocity the head becomes H' and we have, H' = [(1+h?) — h3] = h=theoretical head due to the velocity of approaching current theoretical head due to a velocity of 2.5 feet per second=0.10 feet, hence H'-5.6 feet. D=5.63 x 400' x 3.33=17648.9 cubic feet. De Lagrené's formula for discharge over a dam or weir is, D=mlH√2gH m=coefficient=0.45; g=32'.2; and H=5'.5 In presenting this formula De Lagrené remarks that he has not known of any case in which the velocity of approach was inappreciable where the constant was as great as 0.45. It is therefore presumed that this value allows for the velocity of approach. Substituting the proper values in the last equation it becomes D= 18,660 cubic feet per second. Three different results have therefore been obtained for the amount of discharge, as follows: The second and third results compared with the first are evidently too great, though it is not improbable that the latter is too small. In order to give them their proper weight in the final result a double mean is taken of the three values, which gives an approximation sufficiently near for the purposes of this discussion. The discharges at a stage of 6, 7, 8, 9, 11, 12, 13, and 14 feet are then calculated in the same manner. The ratio of rise in the upper pool at these different heights to that in the lower pool is approximately 1 to 2; the pour over the dam being about 0.5 of a foot at ordinary low water. Designating by a the rise in feet above low water in the lower pool the rise in the upper pool will be 2 + 0.5 foot. It is now proposed to ascertain the width of a navigable pass, which is so constructed as to give a fall of 2 feet in its length of 300 feet, or a slope of 1 in 150 or 0.00666 of a foot per foot. The bottom of the entrance to the pass is 2 feet above low water, and it is supposed that the sill of the closing gates is placed midway between the ends. The mean velocities through the pass will also be determined. Taking the 6-foot stage above low water, when the discharge is 8,898 cubic feet per second, and the depth of water over the sill 5 feet, and using Eytelwein's formula as before and substituting, we have: V mean velocity in feet per second, x=width of pass, x+14=wetted perimeter, 5x=area of water-way, 0.00666-slope, 8975-coefficient, and 0.1089 a constant. The latter may be omitted without sensibly affecting the results. The mean velocity also is equal to the volume voided per second, divided by the area of the section, thus: equating these values of V, and their results, V=16.54 feet per second; x=107.6 feet. Proceed in the same manner for the other heights of water given in the preceding table and Table 2 is formed, as follows: From the above table it can readily be seen that for a given fall between the two pools distributed over a length of 300 feet, or a slope of 130 0.00666, the width of the pass required decreases as the river rises, and increases as it falls. It may also be noticed that the velocity through the pass increases as the height of the river increases. = The width required for the navigable pass is determined by the velocity created, as upon that velocity the amount of disturbance at the outlet of the pass below the dam depends. If a velocity of 12.66 miles an hour is adopted as the maximum velocity consistent with the safety of navigating craft, as well as the safety of the works themselves, from the table it is seen that a dam 12 feet high and a rise of 10 feet, the width required is approximately 110 feet. We will now obtain the velocities corresponding to the different heights of water, when the pass is 110 feet wide, to ascertain if at any navigable stage the velocity in the pass will be greater than that assumed as the maximum. At the 6-foot stage above low water, giving a discharge of 8,898 feet per second V = 16.30 feet per second = 11.11 miles per hour. The same method is employed to find the values of Y, V, and S for the 7, 8, 9, and 10 foot stages. At a 11-foot stage and a dam 12 feet high a certain amount of the discharge will find its way over the dam. The width of the dam is about 280 feet. Obtaining by approximation the values for the same quantities for this height of water, and also for a 12-foot stage, we have by arranging the results in a table the following: The above table shows that the greatest velocity is found when the upper pool is just commencing to discharge over the dam. Discussing a 14-foot dam under the same conditions of width of pass and height of sill, and tabulating the results, table No. 4 is formed. Feet. Feet. Feet. Feet. 0.00513 1. 54 00493 1.48 Difference of level between pools. With a 14-foot dam this table gives velocities which indicate that the pass has not been given sufficient width. Assuming this to be increased 10 feet, or made 120 feet wide, another table is formed giving the discharges, the depth of water over the sill, velocities, slopes, and differences of level between the pools for the vari ous depths, from a stage of 6 feet to that of 14 feet above low water, for a dam 12 or 14 feet high. From this table it may be observed that for a 12-foot dam and 120 foot pass the highest velocity obtained is 11.82 miles an hour, and for a 14foot dam, under similar conditions, the velocity will be 12.43 miles per hour, when the upper pool is just level with the crest of the dam. HEIGHT OF THE DAM. Within certain limits the height of the fixed dam should be sufficient to form the pool desired for the interest of commerce and navigation. The pool formed by a dam 12 feet high would reach Head of Log Shoal, North Fork.. Foot of Buffalo Shoal, South Fork... Long Shoal, Middle Fork.......... Length of pool for a 12-foot dam..... The pool formed by a dam 14 feet high would reach Head of Laurel Shoal, North Fork. Head of Buffalo Shoal, South Fork. Length of pool for a 14-foot dam....... Miles. 7.0 5.0 2.5 14.5 Miles. 8.0 5.3 2.5 15. 8 If the small gain in length of pool up the forks of a 14-foot dam over that of a 12-foot dam was alone considered, it would hardly seem to justify the difference in cost of the structure. But the extension of the pool into the Stufflebeam Creeks, where a safe and commodious harbor can be formed, for the accumulation of boats and rafts in time of danger, is a matter of such vital importance that the higher dam, 14 feet in height, has been selected as the one best suited for the interest of commerce at this point. NAVIGABLE PASS. In the preceding discussion it will be observed that the bottom of the entrance to the pass was placed at 2 feet above low water. From an engineering point of view it is evident that the higher the position of the sill the greater the facility in the movement of the gates. The width of the gates or leaves will be less, the size of the timbers reduced, and the number and weight of the movable parts diminished. But the raising of the sill reduces the depth of water in the pass, while it is essential for the purpose of navigation that there should be as great a depth in it, at an ordinary rafting or boating stage, as there is over the bars and ripples below Beattyville. To give as much ease as possible in the manipulation of the gates, and at the same time not to interfere with the latter condition, the level of 2 feet above low water was assumed for the sill. The width of the pass depends on the height of the sill and upon the maximum velocity consistent with safety in passing through it when the gates are down. As the higher the fixed dam the greater the maximum velocity in the pass, it follows that the width of the pass is also dependent upon the height of the dam. Again, it is important that the pass should be wide, so that when it is open in high water the fixed dam will form less of an obstacle and consequently there will be a nearer approach to the normal conditions of the river. This will diminish the undermining of the structure, the erosion of the river banks, and also decrease the disturbance below the dam in the shape of broken |