=186. General Formula for Discharge of Conduits—Chezy’s Formula.=—It is imperative in designing aqueducts of either masonry or metal to determine their discharging capacity, which in general will depend largely upon the slope of channel or head of water and the resistance offered by the bed or interior of the pipe to the flow of water. The resistance of liquid friction is so much more than all others in this class of water-conveyors that it is usually the only one considered. There is a certain formula much used by civil engineers for this purpose; it is known as Chezy’s formula, for the reason that it was first established by the French engineer Antoine Chezy about the year 1775, although it is an open question whether the beginnings of the formula were not made twenty or more years prior to that date. Its demonstration involves the general consideration of the resistance which a liquid meets in flowing over any surface, such as that of the interior of a pipe or conduit, or the bed and banks of a stream. The force of liquid friction is found to be proportional to the heaviness of the liquid (i.e., to the weight of a cubic unit, such as a cubic foot), to the area of wetted surface over which the liquid flows, and nearly to the square of the velocity with which the liquid moves. Hence if _lʹ_ is the length of channel, _p_ the wetted portion of the perimeter of the cross-section, _w_ the weight of a cubic unit of the liquid, and _v_ the velocity, the total force of liquid friction for the length _l′_ of channel will be _F = ζwpl′v²_, ζ being the coefficient of liquid friction. The path of the force _F_ for a unit of time is _v_, and the work _W_ which it performs in that unit of time is equal to the weight _wal′_ falling through the height _h′_, _a_ being the area of the cross-section of the stream. [Illustration: FIG. 2.] Hence _W_ = ζ_wplʹv².v_ = _wal′h′._ (7) _a_ _____________________ ____ ζ_v²v_ = ---- _hʹ_, _v_ = √(1/ζ) (_a/p_) (_h′/v_) = _c_√_rs_. (8) _p_ In this equation _____ _a_ _c_ = √(1/ζ); _r_ = ---- = hydraulic mean radius; _p_ _hʹ_ _s_ = ----- = sine of inclination of stream’s bed. _v_ As the motion of the water is assumed to be uniform, the head lost by friction for the total length of channel _l_ is the total fall _h_, and by equation (8), since _h′ h_ --- = _s_ = ---, _v l_ _v² l_ _h_ = ---- ---------. (9) _c²_ (_a/p_) If, as in the case of the ordinary cast-iron water-pipes of a public supply system, the cross-section _a_ is circular, _a_ (π_d²_/4) _d_ --- = --------- = ---, _p_ π_d_ 4 and 4.2_g l v² l v²_ _h_ = ----- --- ------ = _f_ --- -----, (10) _c² d_ 2_g d_ 2_g_ in which _f_ = 8_g_ ÷ _c²_.