The general expression for the flow of sewage would then be,
in which _h_ is the head or energy lost between any two sections, and _V_ is the average velocity of flow between these sections. It is to be noted in this general expression that the quantity and rate of flow past all sections is assumed to be constant. This condition is known as _steady flow_. Problems are encountered in sewerage design which involve conditions of unsteady flow, and methods of solution of them have been developed based on modifications of this general expression. The average velocity of flow is computed by dividing the rate (quantity) of flow past any section by the cross-sectional area of the stream at that section. This does not represent the true velocity at any particular point in the stream, as the velocity near the center is faster than that near the sides of the channel. The distribution of velocities in a closed circular channel is somewhat in the form of a paraboloid superimposed on a cylinder.
The laws of flow are expressed as formulas the constants of which have been determined by experiment. It has been found that these constants depend on the character of the material forming the channel and the hydraulic radius. The _hydraulic radius_ is defined as the ratio of the cross-sectional area of the stream to the length of the wetted perimeter, or line of contact between the liquid and the channel, exclusive of the horizontal line between the air and the liquid.
=35. Formulas.=—The loss of head due to friction caused by flow through circular pipes flowing full as expressed by Darcy is,
in which _h_ is the head lost due to friction in the distance _l_, _V_ is the velocity of flow, _g_ is the acceleration due to gravity, and _f_ is a factor dependent on _d_ and the material of which the pipe is made. A formula for _f_ expressed by Darcy as the result of experiments on cast-iron pipe is,
in which _d_ is the diameter in feet. In using the formula with this factor the units used must be feet and seconds.
Another form of the same expression is known as the Chezy formula. It is an algebraic transformation of the Darcy formula, but in the form shown here, by the use of the hydraulic radius, it is made applicable to any shape of conduit either full or partly full. The Chezy formula is,