Thus the force (f) necessary to maintain the motion of the gas in the pipe is seen to vary (1) as pild, of which pi is a constant; (2) as v^2, where v = the velocity in feet per hour; and (3) as s. Hence, combining these and deleting the constant pi, it appears that
f varies as ldsv^2.
Now the actuating force is equal to f, and is represented by the difference of pressure at the two ends of the pipe, i.e., the initial pressure, viz., that at the place whence gas is distributed or issues from a larger pipe will be greater by the quantity f than the terminal pressure, viz., that at the far end of the pipe where it branches or narrows to a pipe or pipes of smaller size, or terminates in a burner. The terminal pressure in the case of service-pipes must be settled, as mentioned in Chapter II., broadly according to the pressure at which the burners in use work best, and this is very different in the case of flat-flame burners for coal-gas and burners for acetylene. The most suitable pressure for acetylene burners will be referred to later, but may be taken as equal to p_0 inches head of water. Then, calling the initial pressure (i.e., at the inlet head of service-pipe) p_1, it follows that p_1 - p_0 = f. Now the cross-section of the pipe has an area (pi/4)d^2, and if h represents the difference of pressure between the two ends of the pipe per square inch of its area, it follows that f = h(pi/4)d^2. But since f has been found above to vary as ldsv^2 , it is evident that
h(pi/4)d^2 varies as ldsv^2.
Hence
v^2 varies as hd/ls, and putting in some constant M, the value of which must be determined by experiment, this becomes
v^2 = Mhd/ls.
The value of M deduced from experiments on the friction of coal-gas in pipes was inserted in this equation, and then taking Q = pi/4d^2v, it was found that for coal-gas Q = 780(hd/sl)^(1/2)
This formula, in its usual form, is
Q = 1350d^2(hd/sl)^(1/2)