110 PROFESSOR A. H. GIBSON ON THE RESISTANCE TO FLOW OF WATER 
From these results it appears that by the use of curved pipes of this form the loss 
with a given length of pipe may be reduced considerably as compared with the loss in 
a straight taper pipe between the same limits of area. The proportional saving is 
oreater the greater the ratio of enlargement and also the shorter the pipe. 7 
While no direct evidence that this is the best possible curve has been obtained, the 
great reduction in loss accompanying its use is evidence that no great improvement by 
a modification in its equation is probable, or indeed possible. 
For values of @ less than about 15° in the case of a rectangular pipe and less than 
about 7°5° in the case of a circular pipe, evidently little is to be gained by introducing 
the curve. The proportional saving following on its introduction varies in these experi- 
ments from 62 per cent. in the case of a rectangular pipe having an enlargement ratio 
of 9: 1 and corresponding to a straight taper pipe with 0= 26°, to 22 per cent. in the 
case of a circular pipe with a ratio of 2°25:1 and corresponding to a straight pipe 
with 6=10°. As might be expected, the gain is more marked in a rectangular pipe, 
in which the enlargement of section takes place in one plane, than in a circular pipe. 
The effect of modifying the outlet end of the pipe as indicated in fig. 4 (A and B) is 
somewhat surprisingly large, the mere cutting away of the curved boundary to form 
a sudden enlargement reducing the percentage loss by about 7 per cent. on the 
average. ‘The pipe as thus modified is, in effect, moreover, shorter than the original 
curved pipe giving the same change of. section, and this led to the examination of a 
further series of pipes designed from considerations based on the following reasoning. 
The loss of head in a pipe whose section increases gradually from A, to Ag, and 
which then suffers a sudden enlargement of area to A,, might be expected, on 
theoretical grounds, to be equal to the sum of the separate losses which would be 
experienced in the taper portion of the pipe and at the sudden enlargement, if these 
were independent of each other. By reducing the angle of divergence of the first 
portion of such a tube, the sudden enlargement of section and the accompanying Joss 
is made greater, but the loss in the diverging portion is reduced in a double degree, 
since not only is the numerical coefficient expressing such loss as a percentage of 
(v,—3)/2g diminished, but A; is diminished at the same time, and thus the factor 
V,— 3 is also diminished. A diminution in the angle of divergence therefore causes 
a rapid diminution in this portion of the loss, which may, or may not, be counter- 
balanced by the loss at the sudden enlargement of section. Owing to the comparatively 
low velocities at the large end of the pipe, however, except in pipes whose length is 
comparatively very short, and whose ratio of enlargement is small, this latter loss may 
be expected to be comparatively small and the total loss to be a minimum with a pipe— 
straight or curved—whose angle of divergence—actual or effective—is little greater 
than that giving minimum loss in the diverging portion of the pipe alone. 
Furthermore it would appear possible, although at first sight paradoxical, that in 
a fairly long pipe having a uniform divergence from A, to A, at the best angle (about 
11° in a rectangular pipe) the loss of head might even be reduced by reducing the 
a Se 
