_V_ = (1.81 + 41.67_n_)_R_√_S_/_n_(√_R_ + 41.67_n_).
in which α and β are constants for different classes of material. For cast-iron pipe α is 0.00007726 and β is 0.00000647. This formula is seldom used in sewerage design.
Exponential formulas have been developed as the result of experiments which have demonstrated that _V_ does not vary as the one-half power of _R_ and _S_ but that the relation should be expressed as,
in which _p_ and _q_ are constants and _C_ is a factor dependent on the character of the material. The various formulas coming under this classification have been given the names of the experimenters proposing them. Examples of these formulas are: Flamant’s, in English units, for new cast-iron pipe, which is,
and Lampé’s for the same material which is,
These formulas are useful only for the material to which they apply, but they can be used for conduits of any shape. A. V. Saph and E. W. Schoder have shown[31] that the general formula for all materials lies between the limits,
_V_ = (93 to 142)_S_^{ .50 to .55}_R_^{ .63 to .69}.
Hazen and Williams’ formula is in the form,
in which _C_ is a factor dependent on the character of the material of the conduit. The values of _C_ as given by Hazen and Williams are,
_C_ CHARACTER OF MATERIAL 95 Steel pipe under future conditions. (Riveted steel.) Cast iron under ordinary future conditions and brick 100 sewers in good condition. 110 New riveted steel, and cement pipe. 120 Smooth wood or masonry conduits under ordinary conditions. Masonry conduits after some time and for very smooth pipes such as glass, brass, lead, etc., when old, and for new 130 cast-iron pipe under ordinary conditions.
This formula is of as general application as Kutter’s formula and is easier of solution, but being more recently in the field and because of the ease of the solution of Kutter’s formula by diagrams it is not in such general use. Exponential formulas are used more in waterworks than in sewerage practice.
in which _n_ is the same as for Kutter’s formula. Charts for the solution of Manning’s formula are given in Eng. News-Record, Vol. 85, 1920, p. 837.
=36. Solution of Formulas.=—The solution of even the simplest of these formulas, such as Flamant’s, is laborious because of the exponents involved. Darcy’s and Kutter’s formulas are even more cumbersome because of the character of the coefficient. The labor involved in the solution of these formulas has resulted in the development of a number of diagrams and other short cuts. Since each formula involves three or more variables it cannot be represented by a single straight line on rectangular coordinate paper. The simplest form of diagram for the solution of three or more variables is the nomograph, an example of which is shown in Fig. 13 for the solution of Flamant’s formula. A straight-edge placed on any two points of the scales of two different vertical lines will cross the other line at a point on the scale corresponding to its correct value in the formula. Such a diagram is in common use for the solution of problems for the flow of water in cast-iron pipe.