It will be observed that, with the exception of Kutter's formula, the results above given, though not equal, run approximately parallel. Kutter's formula gives much smaller values for sewers of small diameter, and much larger values for sewers of large diameter. When n=15, the values given by Kutter and Latham are approximately equal for a sewer five feet in diameter. This value of n, however, is not applicable to vitrified pipe sewers, well constructed, unless it be on very sharp curves, where ordinarily the work is less perfect. Loss of Head on Curves. -An increase of inclination should be given around curves, both to overcome the increased friction due to angular change in direction, and also for the reason that, as ordinarily laid, there is a slight opening of the joints in the outward circumference and greater liability to stoppage from articles lodging crosswise. The allowance indicated by theory for the increase of friction on curves is not sufficient, for the reason that pipes are not usually laid so truly to line or grade as when laid in straight lines, and, aside from the increased friction due to the angular change in direction, we may properly increase the coefficient of resistance to flow in the pipe. The following is Baldwin Latham's modification of Weisbach's formula for loss of head due to angular friction: h=head necessary to overcome angular friction. =.131 .138 .158 .206 .294 .440 .661 .977 1.408 1.978 Assuming a 6-inch sewer laid with a curve of fifty feet radius, the angle of the curve being sixty degrees, or its length 52.4 feet, and the velocity above the curve to be five feet per second, the increased head necessary to overcome friction due to angular change in direction is, according to equation six, less than one-eighth of an inch. In no case which could possibly occur in curves of a proper radius will the formula give more than a small fraction of an inch as the value of h. This is too small to be considered in work of this class. In proposing a formula for the increased head or inclination required for curves as ordinarily laid in sewer work, we may, therefore, disregard the effect of angular change in direction. If, however, we assume that the increased roughness of the pipe would increase the coefficient n, as given in Kutter's formula, from =.011, its value as given for plaster of cement with 3 sand, to n=.013 and n=.015, its value as given for brick work and terra cotta pipes with imperfect joints and in bad order, we have from Kutter's formula, by computation, the following table: TABLE XX. SHOWING INCREASED FRICTIONAL HEAD REQUIRED FOR CURVES IN VARIOUS CASES. In the columns of loss of grade per 100 feet can be found the increased fall necessary, under the supposition that, on ordinary curves, n is increased from .011 to .013, and on sharp curves from .011 to .015. It will be observed that when n=.013 the value of Cis decreased to about eighty per cent. in all cases. Recurring to the general formula of Chezy, which for the ordinary range of diameter and velocity, becomes, approximately, v=100/RS We may write for ordinary curves, v=80/RS' h' in which S'==slope required. From the preceding equations we have z? S= (100) R z? S'= (80)2 R Or, since in each case the hydraulic mean radius when In pipe sewers, however, the roughness is somewhat dependent on the ratio of the radius of the curve to diameter of the pipe. Empirical Formula. The following formula will give good results in pipe sewers: in which velocity in feet per second. 7-length of curve in feet. d diameter of sewer in feet. r-radius of axis of curve in feet. H-loss of head for curve in feet due to increased roughness. Hin the above formula does not represent the total fall required for the curve, but the excess of fall necessary over that if the sewer were straight, and the flow had an equal velocity. |