= h depth in feet area of cross-section in square feet divided by wetted perim. eter in lineal feet; s= fall of water-surface (h) in any distance (1) divided by that distance, = sine of slope; n = the coefficient of rugosity, depending on the nature of the lining or surface of the channel. If we let the first term of the right-hand side of the equation equal c, we have Chezy's formula, vc Vrs = cx √r × √s. Values of n in Kutter's Formula.-The accuracy of Kutter's formula depends, in a great measure, on the proper selection of the coefficient of roughness n. Experience is required in order to give the right value to this coefficient, and to this end great assistance can be obtained, in making this selection, by consulting and comparing the results obtained from experiments on the flow of water already made in different channels. In some cases it would be well to provide for the contingency of future deterioration of channel, by selecting a high value of n, as, for instance, where a dense growth of weeds is likely to occur in small channels, and also where channels are likely not to be kept in a state of good repair. The foliowing table, giving the value of n for different materials, is compiled from Kutter, Jackson, and Hering, and this value of n applies also in each instance, to the surfaces of other materials equally rough. VALUE OF N IN KUTTER'S FORMULA FOR DIFFERENT CHANNELS. n = .009, well-planed timber, in perfect order and alignment; otherwise, perhaps .01 would be suitable. n = .010, plaster in pure cement; planed timber; glazed, coated, or enamelled stoneware and iron pipes; glazed surfaces of every sort in perfect order. n = .011, plaster in cement with one third sand, in good condition; also for iron, cement, and terra cotta pipes, well joined, and in best order. n = .012, unplaned timber, when perfectly continuous on the inside; flumes. n = .013, ashlar and well-laid brickwork; ordinary metal; earthen and stoneware pipe in good condition, but not new; cement and terra-cotta pipe not well jointed nor in perfect order, plaster and planed wood in imperfect or inferior condition; and, generally, the materials mentioned with n = .010, when in imperfect or inferior condition. n = .015, second class or rough-faced brickwork; well-dressed stonework; foul and slightly tuberculated iron; cement and terra-cotta pipes, with im perfect joints and in bad order; and canvas lining on wooden frames. n = .017, brickwork, ashlar, and stoneware in an inferior condition; tu berculated iron pipes; rubble in cement or plaster in good order; fine gravel, well rammed, to inch diameter; and, generally, the materials mentioned with n = .013 when in bad order and condition. n = .020, rubble in cement in an inferior condition; coarse rubble, rough set in a normal condition; coarse rubble set dry ruined brickwork and masonry coarse gravel well rammed, from 1 to 1% inch diameter; canals with beds and banks of very firm, regular gravel, carefully trimmed and rammed in defective places; rough rubble with bed partially covered with silt and mud; rectangular wooden troughs, with battens on the inside two inches apart; trimmed earth in perfect order. n = .0225, canals in earth above the average in order and regimen. n = .025, canals and rivers in earth of tolerably uniform cross-section; slope and direction, in moderately good order and regimen, and free from stones and weeds. n = .0275, canals and rivers in earth below the average in order and regi men. n = .030, canals and rivers in earth in rather bad order and regimen, having stones and weeds occasionally, and obstructed by detritus. = .035, suitable for rivers and canals with earthen beds in bad order and regimen, and having stones and weeds in great quantities. n = .05, torrents encumbered with detritus. Kutter's formula has the advantage of being easily adapted to a change in the surface of the pipe exposed to the flow of water, by a change in the value of n. For cast-iron pipes it is usual to use n = .013 to provide for the future deterioration of the surface. Reducing Kutter's formula to the form v=cX Vr× √s, and taking n, the coefficient of roughness in the formula = .011, .012, and .013, and s = .001, we have the following values of the coefficient c for different diameters of conduit. Values of c in Formula v = c × √r × √s for Metal Pipes and Moderately Smooth Conduits Generally. By KUTTER'S FORMULA (s = .001 or greater.) Diameter. n = .011 n = .012n = .013 Diameter. n = .011n = .012n = .013 For circular pipes the hydraulic mean depth r equals 14 of the diameter. According to Kutter's formula the value of c, the coefficient of discharge, is the same for all slopes greater than 1 in 1000; that is, within these limits c is constant. We further find that up to a slope of 1 in 2640 the value of c is, for all practical purposes, constant, and even up to a slope of 1 in 5000 the difference in the value of c is very little. This is exemplified in the following: Value of c for Different Values of Formula, with n = rand s in Kutter's .013. The reliability of the values of the coefficient of Kutter's formula for pipes of less than 6 in. diameter is considered doubtful. (See note under table on page 564.) Values of c for Earthen Channels, by Kutter's Formula, for Use in Formula v = c√rs. 55.7 80.2 94.8 107.9 17.1 62.5 80.3 89.2 99.9 19.7 37.6 59.3 69.2 59.4 69.4 59.5 69.8 35.3 51.6 60.5 75.4 34.3 51.6 62.5 78.6 Mr. Molesworth, in the 22d edition of his "Pocket-book of Engineering Formulæ," gives a modification of Kutter's formula as follows: For flow in cast-iron pipes, v = c √rs, in which in which d= diameter of the pipe in feet. 00281 8 (This formula was given incorrectly in Molesworth's 21st edition.) Molesworth's Formula.-v = √krs, in which the values of k are as follows: In very large channels, rivers, etc., the description of the channel affects the result so slightly that it may be practically neglected, and k assumed = from 8500 to 9000. Flynn's Formula.-Mr. Flynn obtains the following expression of the value of Kutter's coefficient for a slope of .001 and a value of n = .013: The following table shows the close agreement of the values of c obtained from Kutter's, Molesworth's, and Flynn's formulæ : Mr. Flynn gives another simplified form of Kutter's formula for use with different values of n as follows: In the following table the value of K is given for the several values of n: .009 .010 .011 245.63 .012 195.33 .015 If in the application of Mr. Flynn's formula given above within the limits of n as given in the table, we substitute for n, K, and r their values, we have a simplified form of Kutter's formula. For instance, when n = .011, and d =3 feet, we have Bazin's Formulæ : For very even surfaces, fine plastered sides and bed, planed planks, etc., For even surfaces such as cut-stone, brickwork, unplaned planking, mortar, etc.: v = 1+.000018 (4.354 + 1) × √ñs. For slightly uneven surfaces, such as rubble masonry: v = 1 + .00006 (1.219 + 1) × √rs. For uneven surfaces, such as earth: v = 1+.00085 (0.2438 + 1) × √TS. A modification of Bazin's formula, known as D'Arcy's Bazin's: For small channels of less than 20 feet bed Bazin's formula for earthen channels in good order gives very fair results, but Kutter's formula is superseding it in almost all countries where its accuracy has been investigated. The last table on p. 561 shows the value of c, in Kutter's formula, for a wide range of channels in earth, that will cover anything likely to occur in the ordinary practice of an engineer. D'Arcy's Formula for clean iron pipes under pressure is in which d = diameter in feet. D'Arcy's formula, as given by J. B. Francis, C.E., for old cast-iron pipe, lined with deposit and under pressure, is Flynn's modification of D'Arcy's formula for old cast-iron pipe is 70243.9d% v = .12d +1 For Pipes Less than 5 inches in Diameter, coefficients (c) in the formula v = c Vrs, from the formula of D'Arcy, Kutter, and Fanning. Mr. Flynn, in giving the above table, says that the facts show that the coefficients diminish from a diameter of 5 inches to smaller diameters, and it is a safer plan to adopt coefficients varying with the diameter than a constant coefficient. No opinion is advanced as to what coefficients should be used with Kutter's formula for small diameters. The facts are simply stated, giving the results of well-known authors. Older Formulæ.-The following are a few of the many formulæ for flow of water in pipes given by earlier writers. As they have constant coefficients, they are not considered as reliable as the newer formulæ. In these formulæ d = diameter in feet; h head of water in feet; 1 = length of pipe in feet; s = sine of slope h = で ; mean hydraulic depth, wet perimeter = for circular pipe. Mr. Santo Crimp (Eng'g, August 4, 1893) states that observations on flow in brick sewers show that the actual discharge is 33% greater than that calculated by Eytelwein's formula. He thinks Kutter's formula not superior to D'Arcy's for brick sewers, the usual coefficient of roughness in the former, viz., .013, being too low for large sewers and far too small in the case of small sewers. VELOCITY OF WATER IN OPEN CHANNELS. Irrigation Canals.-The minimum mean velocity required to prevent the deposit of silt or the growth of aquatic plants is in Northern India taken at 1% feet per second. It is stated that in America a higher velocity is required for this purpose, and it varies from 2 to 3% feet per second. The maximum allowable velocity will vary with the nature of the soil of the bed. A sandy bed will be disturbed if the velocity exceeds 3 feet per second. Good loam with not too much sand will bear a velocity of 4 feet per second. The Cavour Canal in Italy, over a gravel bed, has a velocity of about 5 per second. (Flynn's "Irrigation Canals.") Mean Surface and Bottom Velocities.-According to the formula of Bazin, |