100 PROFESSOR A. H. GIBSON ON THE RESISTANCE TO FLOW OF WATER 
From these results it appears that the total loss in the pipes having = 180° is in 
every case in close agreement with that given by BouLancrr’s formula, eed The 
actual loss is, in the majority of cases, slightly greater than that given by this formula. 
Experiments on larger pipes having a sudden change in diameter from 3 inches to 6 
inches and from 4 inches to 6 inches respectively * show an experimental loss some- 
what less than that given by the formula. The percentage loss increases slightly with 
the area-ratio, and in pipes with the same area-ratio is greater the smaller the pipe. 
Denoting the ratio of enlargement by m, and the smaller diameter by d, the loss 
at a sudden enlargement for values of m between 2 and 12, and for pipe diameters 
ranging from ‘50 inch to 6 inches, can be expressed, within narrow limits, by the 
relationship— 
loss of head = 
102°5 + :25m — 2:0d (0, — U_)” 
25n feet. 
100 29 | fee 
The following table shows the results obtained by the use of the formula, against 
those experimentally obtained :— 
: ay y2 
Loss expressed as Percentage of eo ; 
Size of Pipes. Value of m. Pare 
Experimental. By Formula. 
65 to 2°15 inches 10°96 103°5 103°9 
Oia, “NID 55 9-0 102°8 103°7 
1:0) FeSO. | 3 9:0 102:1 102°8 
DO: pew. te, 4:0 101°7 100°5 
a sae ee 2°25 99°2 Suu 
JOS Seow. Fo, 4:0 97°5 97°5 
£02 G:0" 2°25 92 (approx.) 95:0 
As @ is diminished from 180° the percentage loss in every case increases, attains 
a maximum value for some value of @ in the neighbourhood of 65°, and afterwards 
diminishes rapidly with @ until @ is about 5° 30’. This value gives a minimum loss of 
approximately 13°5 per cent. Any further diminution in @ is accompanied by an 
increased loss owing to the large value of the wall friction in pipes of the comparatively 
great length accompanying such small values of 6. The value of 6 which coincides 
with the maximum loss, varies somewhat, both with the size of pipe and with the 
area-ratio, increasing slightly with the dimensions when the latter is constant, and 
with the area-ratio when the mean pipe diameter is constant. Over the range of 
pipe diameters and area-ratios examined in these experiments its value lies between 
63° and 70°. 
* BricHrTmore, Proc. Inst. Civil Engineers, vol. clxix., 1906-7, Pt. iii. p. 322. Here the loss of head was 975 per 
cent. of the theoretical for an enlargement of area 1 to 4, and was about 92 per cent. of the theoretical for an 
enlargement of 1 to 2°25. 
