; or, introducing the discharge per second, Q, which is the usual figure supplied, and which is connected with the velocity by the relation,
, we have
. Generally the pressure, p1, at the entrance is known, and the pressure, p, has to be found; it is then from p1 that the values of Q and Δ are calculated. In experiments where p1 and p are measured directly, in order to arrive at the value of the coefficient b1, Q and Δ would be calculated for the mean pressure ½(p1 + p). The values given to the coefficient b1 vary considerably, because, as stated above, it varies with the diameter, and also with the nature of the material of the pipe. It is generally admitted that it is independent of the pressure, and it is probable that within certain limits of pressure this hypothesis is in accordance with the truth.
D'Aubuisson gives for this case, in his Traité d'Hydraulique, a rather complicated formula, containing a constant deduced from experiment, whose value, according to a calculation made by the author, is approximately b1 = 0.0003. This constant was determined by taking the mean of experiments made with tin tubes of 0.0235 meter (15/16 in.), 0.05 meter (2 in.), and 0.10 meter (4 in.) diameter; and it was erroneously assumed that it was correct for all diameters and all substances.
M. Arson, engineer to the Paris Gas Company, published in 1867, in the Mémoires de la Société des Ingénieurs Civils de France, the results of some experiments on the loss of pressure in gas when passing through pipes. He employed cast-iron pipes of the ordinary type. He has represented the results of his experiments by the binomial formula, au + bu², and gives values for the coefficients a and b, which diminish with an increase in diameter, but would indicate greater losses of pressure than D'Aubuisson's formula. M. Deviller, in his Rapport sur les travaux de percement du tunnel sous les Alpes, states that the losses of pressure observed in the air pipe at the Mont Cenis Tunnel confirm the correctness of D'Aubuisson's formula; but his reasoning applies to too complicated a formula to be absolutely convincing.
Quite recently M. E. Stockalper, engineer-in-chief at the northern end of the St. Gothard Tunnel, has made some experiments on the air conduit of this tunnel, the results of which he has kindly furnished to the author. These lead to values for the coefficient b1 appreciably less than that which is contained implicitly in D'Aubuisson's formula. As he experimented on a rising pipe, it is necessary to introduce into the formula the difference of level, h, between the two ends; it then becomes
