THROUGH PIPES OR PASSAGES HAVING DIVERGENT BOUNDARIES. 101 
The maximum percentage loss varies in these experiments between 105 per cent. 
and 122 per cent. It increases with the mean diameter of the pipe for the same area- 
ratio, and also increases as the ratio of enlargement is reduced. 
As @ is diminished below 60° the percentage loss curves rapidly converge, and for 
values of @ less than 25° these are very sensibly the same for all the pipes. With @=20° 
the percentage losses due essentially to the divergence of the walls, the friction losses 
being calculated as explained in the former paper.* are as follows :— 
Pipe diameters (inches) . : ‘50 to 1°50 | 1:0 to 3-0 1°5 to 3:0 2-0 to 3:0 
Percentage loss . . : 41:0 41°7 40°5 40°3 
For a given ratio of enlargement, for values of @ less than 60°, the percentage loss 
inereases slightly as the mean diameter diminishes, and, for a given mean diameter, on 
the whole increases slightly although irregularly as the ratio enlargement is reduced. 
The value of 6 giving rise to the same loss as is experienced with a sudden change of 
section, varies within fairly wide limits from 41° to 60°, being slightly greater, for a 
given area-ratio, the larger the pipes, and, for a given mean diameter, increasing in an 
irreoular manner with the area-ratio. Its mean value over the range of ratios con- 
sidered is approximately 50°, and where, in the design of hydraulic machinery, it is 
necessary for this value to be exceeded, a sudden enlargement of section will give a more 
efficient transformation of energy than will a uniformly tapering pipe. For values of 0 
between 7°5° and 35° the loss may be expressed with a fair degree of accuracy by the 
relationships— 
loss ="0110 6” Meer feet, where 6 is in degrees, 
or 
22 : 
16s 8-50 (tan $y (%=%)" fet, 
2 29 
This latter relationship becomes of importance in the design of trumpet-shaped 
pipes to give a minimum loss of energy. Values obtained by calculation from these 
formulz are compared in the following table with those obtained experimentally :— 
Percentage Loss. 
Nee By Calculation. 
ie Experimental g\re 
Values. 110 6, 350(tan ¢ ee 
75 14:5 12°8 12°8 
10 17:5 18°2 18°1 
15 28:0 29°7 29-9 
20 43-5 42:5 42-4 
25 58:0 55°5 56:2 
30 71:0 70°5 73°0 
35 80°8 §3°8 85°7 
* Ibid. p. 370. 
