Page:Lowell Hydraulic Experiments, 4th edition.djvu/69

 Calling $$P'$$ the total power of the turbine on the double fall, we have $$P'=0.085 \sqrt{2}Q2h,$$ or

Thus, the total power of the turbine is increased 2.8284 times, by doubling the fall; on the double fall, therefore, in order to preserve the effective power uniform, the regulating gate must be shut down to a point that will give only $1⁄2.8284$ total power of the turbine.

In experiment 15, the fall acting upon the wheel was 12.888 feet, and the total useful effect of the turbine was 85625.3 pounds raised one foot per second; $1⁄2.8284$ is is 30273.4 lbs.; consequently, the same opening of gate that would give this last power, on a fall of 12.888 feet, would give a power of 85625.3 lbs. raised one foot per second, on a fall of $$2 \times 12.888\ \mathrm{feet} = 25.776\ \mathrm{feet}$$. To find this opening of gate, we must have recourse to some of the other experiments.

In experiment 73, the fall was 13.310 feet, the height of gate 5.65 inches, and the useful effect 58830.1 pounds. In experiment 83, the fall was 13.435 feet, the height of gate 2.875 inches, and the useful effect, 27310.9 pounds. Reducing both these useful effects to what they would have been, if the fall was 12.888 feet, —

the useful effect in experiment 73, $$58830.1 ({{12.888} \over {13.310}})^{3 \over 2} = 56054.5,$$

83, $$27310.9 ({{12.888} \over {13.435}})^{3 \over 2} = 25660.1.$$

By a comparison of these useful effects with the corresponding heights of gate, we find, by simple proportion of the differences, that a useful effect of 30273.4 pounds raised one foot high per second, would be given when the height of the regulating gate was 3.296 inches.

By another mode : — $\mathrm{as}\ 25660.1 : 2.875 :: 30273.4 : 2.875 \times {30278.4 \over 25660.1} = 3.392\ \mathrm{inches,}$ a little consideration will show, that the first mode must give too little, and the second, too much; taking a mean of the two results, we have for the height of the gate, giving $1⁄2.8284$ power of the turbine, 3.344 inches. Referring to table II., we. [sic] see that, with this height of gate, in order to obtain the best coefiicient of useful effect, the velocity of the interior circumference of 7