1919 .] 
Parry.—Power from Town Water-mains. 
379 
formula, h = lv 2 /C 2 r. If A be the area of the pipe in square feet, the 
flow is A.v and the gross power-input into the turbine is 
(h 0 — lv 2 /C 2 r) horse-power, or ~ (h 0 v — lv 3 /C 2 r). 
550 
This is a maximum when the rate of change of power is passing from 
positive to negative, or, in other words, when the differential with 
respect to time is zero. Differentiating and equating to zero, we obtain 
h 0 = 3lv 2 /C 2 r, or lv 2 /C 2 r = h 0 / 3; but lv 2 /C 2 r is the friction head in the 
pipe, and we conclude that the maximum power is obtainable when the 
friction head is equal to one-third of the static head. It is not desirable 
to work to the full limit; in fact, in hydro-electric practice the loss of 
head by friction in a pipe-line is of the order of 3 to 4 per cent., a con¬ 
dition imposed by the necessity for good regulation and the conservation 
of the water-supply. Where, however, a town main with a given flow of 
water is available the object should be to make the fullest possible use 
of the facilities already provided ; and having ascertained the maximum 
possible power which is obtainable, it is then necessary to decide which 
proportion of this should be used, having regard to the requirements of 
regulation ; and in this connection the form of the curves in the accom¬ 
panying diagram should be studied. These curves have for abscissae the 
ratio of the pressure at the delivery end of a given pipe to the static head ; 
the ordinates show the power obtainable, corresponding to a given ratio of 
working-pressure to static pressure. It will be seen that the maximum 
occurs when the working-pressure is two-thirds of the static pressure. For 
the purpose of constructing the curve the following data have been 
assumed : Length of pipe, 5,400 ft. ; value of C = 60 ; static head, 250 ft. ; 
diameter of pipe, 1 ft., 2 ft., 2 ft. 6 in., and 3 ft. When it is desired to 
know the maximum power available after providing for a certain flow for 
domestic purposes the calculation is more complex. One way of approxi¬ 
mating to it, and one which is justifiable, especially in view of the fact that 
the flow which is required to satisfy the general domestic supply is small 
compared with the flow required to satisfy the power-supply, is to calculate 
the maximum power on the assumption that the whole flow is for power, 
in accordance with the foregoing principles, and deduct from this the power 
represented by the flow required for domestic purposes and the pressure 
under which it is acting. 
A question which often arises in this case is, What is the equivalent of 
two or more pipes in parallel ? The equivalent single pipe is obviously 
one which will conduct the same quantity of water with the same frictional 
loss. The answer is easily obtained if it be assumed that the value of the 
coefficient of friction is the same for two different diameters of pipe and 
for different velocities, which is not strictly the case, as the value depends, 
amongst other things, upon the product of the diameter and the velocity,* 
but for practical purposes the same value of the coefficient may be taken 
in each case. Let v x and v 2 be the velocities of flow, and q 1 and q 2 the 
flow in two pipes of areas and A 2 respectively. Let v, q, and A be the 
corresponding values for the equivalent pipe. We have, in order to satisfy 
the first condition, q — q t -f- q 2 ; and in order to satisfy the second condi¬ 
tion, h = lv 1 2 /C 2 r 1 =lv 2 2 /Q 2 r 2 =lv 2 /C 2 r, where r, r 1 , and r 2 represent the 
hydraulic mean depth. The value of C is assumed to be the same, whilst 
* E. Parry, Surface Friction of Fluids, N . Z . Journ. Sci 6 s Tech., vol. 1, pp. 154-56, 
1918. 
