in which _R_ is the hydraulic radius, _S_ the slope ratio of the hydraulic gradient, and _C_ a factor similar to _f_ in the Darcy formula.
Kutter’s formula was derived by the Swiss engineers, Ganguillet and Kutter, as the result of a series of experimental observations. It was introduced into the United States by Rudolph Hering and its derivation is given in Hering and Trautwine’s translation of “The Flow of Water in Open Channels by Ganguillet and Kutter.” In English units it is,
_V_ = { (1.81/_n_ + 41.67 + .0028/_S_)/(1 + (_n_/√_R_)(41.67 + (.0028/_S_)))}√(_RS_),
in which _n_ is a factor expressing the character of the surface of the conduit and the other notation is as in the Chezy formula. _V_ is the velocity in feet per second, _S_ is the slope ratio, and _R_ the hydraulic radius in feet. The values of _n_ to be used in all cases are not agreed upon, but in general the values shown below are used in practice.
Smooth masonry or brickwork, or concrete sewers under ordinary 0.013 conditions.
0.015 Vitrified pipe or ordinary brickwork.
0.030 } Rough channels overgrown with grass. 0.050
Kutter’s formula is of general application to all classes of material and to all shapes of conduits. It is the most generally used formula in sewerage design.
The cumbersomeness of Kutter’s formula is caused somewhat by the attempt to allow for the effect of the low slopes of the Mississippi River experiments on the coefficients. The correctness of these experiments has not been well established and the slopes are so flat that the omission of the term 0.0028⁄_S_ will have no appreciable effect on the value of _V_ ordinarily used in sewer design. The difference between the value of _V_ determined by the omission of this term and the value of _V_ found by including it is less than 1 per cent for all slopes greater than 1 in 1,000 for 8 inch pipe (_R_ = 0.167 feet). As the diameter of the pipe or the hydraulic radius of the channel increases up to a diameter of 13.02 feet (_R_ = 3.28 feet), the difference becomes less and at this value of _R_ there is no difference whether the slope is included or not. For larger pipes the difference increases slowly. For a 16 foot pipe (_R_ = 4 feet) on a slope of 1 in 1,000 the difference is less than 0.2 per cent, and on a slope of 1 in 10,000 the difference is approximately 1 per cent. Flatter slopes than these are seldom used in sewer design, except for very large sewers where careful determinations of the hydraulic slope are necessary. It is therefore safe in sewer design to use Kutter’s formula in the modified form shown below in which the term (.0028)⁄_S_ has been omitted.