112 PROFESSOR A. H. GIBSON ON THE RESISTANCE TO FLOW OF WATER 
and the above expression becomes 
loss ay A s,) 1S Great) sisters a | mati > inci 
The corresponding expression for circular pipes is 
ip Orey “4 es a m2 mP\? ; | 
Tet ley Gr MI] 
Differentiating these expressions with respect to 0, the value of @ and hence of b, or 
73, giving minimum loss of head, may be obtained. As, however, K varies with 9, the 
resulting expression becomes extremely cumbersome, and the best value is more easily 
obtained by trial of a few values of @ and the corresponding values of K. Handled 
in this way a solution is readily obtained. 
To test the validity of equations (3) and (4), a series of experiments were carried out 
on rectangular pipes of width 1°329 inch and with an initial breadth of 1°329 inch, 
this increasing uniformly to 2°99 inches, after which the breadth suddenly increased to 
5°315 inches, thus making b;=2°25b, and b,=4b,; on rectangular pipes of the same 
length and with the same initial and final areas, but with boundaries made to the 
curved templates already indicated; and on circular pipes having a gradual enlarge- 
ment of section from ‘50 inch diameter to 1°50 inch diameter, followed by a sudden 
enlargement to 3°0 inches diameter. In the latter pipes the full ratio of enlargement 
was thus 36:1. The results were as follow :— 
| emerine = aa Percentage Loss of Head, 
| Taper Pipe Value of 6. 3 a . 
/ (inches). Sonia 4 
| Experimental. Calculated. 
/ 
| 9°50 10° 1746 19-2 17°6 
/ 6°30 15° 2620 23°0 23°3 
Rectangular 3°10 30° *3492 60°7 | 58'8 
Pipes. —- _ 
4:70 Curved 23°4 22°6 
3°10 D 39-0 36°3 
Circular | 3°80 Gumel 21:2 21:1 
Pipes. 1°87 * 31°4 32°5 
From these it appears that the experimental and calculated losses are in sufficiently 
close agreement to amply justify the general adoption of the formule, and these have 
been used to determine the value of @ giving the minimum loss of head in a number 
of typical cases. The following table shows how the loss in such a pipe compares with 
the loss in a straight taper pipe or in a curved pipe of the same length and of the same 
initial and final sections :— 
