Q=discharge in cubic feet per second, l= length of weir in feet, effective head in feet, measured from the level of the crest to the level of still water above the weir. If Q' discharge in cubic feet per minute, and l' and h' are taken in inches, the first of the above formulæ reduces to Q = 0.41'h'. From this formula the following table is calculated. The values are sufficiently accurate for ordinary computations of water-power for weirs without end contraction, that is, for a weir the full width of the channel of approach, and are approximate also for weirs with end contraction when at least 10h, but about 6% in excess of the truth when l = 4h. Weir Table. GIVING CUBIC FEET OF WATER PER MINUTE THAT WILL FLOW OVER A WEIR ONE INCH WIDE and from % TO 20% INCHES deep. For other widths multiply by the width in inches. For more accurate computations, the coefficients of flow of Hamilton Smith, Jr., or of Bazin should be used. In Smith's hydraulics will be found a collection of results of experiments on orifices and weirs of various shapes made by many different authorities, together with a discussion of their several formulæ. (See also Trautwine's Pocket Book.) Bazin's Experiments.--M. Bazin (Annales des Ponts et Chaussées, Oct., 1888, translated by Marichal and Trautwine, Proc. Engrs. Club of Phila., Jan, 1890), made an extensive series of experiments with a sharp-crested weir without lateral contraction, the air being admitted freely behind the falling sheet, and found values of m varying from 0.42 to 0.50, with variations of the length of the weir from 1934 to 7834 in., of the height of the crest above the bottom of the channel from 0.79 to 2.46 ft., and of the head from 1.97 to 23.62 in. From these experiments he deduces the following formula: Н Q = [0.425 +0.21 (#1)]LH √29H, in which Pis the height in feet of the crest of the weir above the bottom of the channel of approach, L the length of the weir, H the head, both in feet, and the discharge in cu. ft. per sec. This formula, says M. Bazin, is entirely practical where errors of 2% to 3% are admissible. The following table is condensed from M. Bazin's paper: VALUES OF THE COEFFICIENT M IN THE FORMULA Q = mLH √29H, FOR A SHARP-CRESTED WEIR WITHOUT LATERAL CONTRACTION; THE AIR BEING ADMITTED FREELY BEHIND THE FALLING SHEET. Head, Height of Crest of Weir Above Bed of Channel. Feet...0.66 0.98 1.31 1.64 1.97 2.62 8.28 4.92 6 56 88 m m m m m m m m m 0.453 0.451 0.450 0.449 0.449 0.449 0.448 0.448 0.4481 0.448 0.445 0.443 0.442 0.441 0.440 0.440 0.439 0.4391 0.447 0.442 0.440 0.438 0.436 0.436 0.435 0.434 0.4340 0.448 0.442 0.438 0.436 0.433 0.432 0.480 0.430 0.4291 0.453 0.444 0.438 0.435 0.431 0.429 0.427 0.426 0.4246 0.459 0.447 0.440 0.436 0.431 0.428 0.425 0.423 0.4215 0.465 0.452 0.444 0.438 0.432 0.428 0.424 0.422 0.4194 0.472 0.457 0.448 0.441 0.433 0.429 0.424 0.422 0.4181 0.478 0.462 0.452 0.444 0.436 0.430 0.424 0.421 0.4168 0.483 0.467 0.456 0.448 0.438 0 432 0.424 0.421 0.4156 0.489 0.472 0.459 0.451 0.440 0 433 0.424 0.421 0.4144 0.494 0.476 0.463 0.454 0.442 0.435 0.425 0.421 0.4134 0.480 0.467 0.457 0.444 0.436 0.425 0.421 0.4122 0.483 0.470 0.460 0.446 0.438 0.426 0.421 0.4112 0.487 0.473 0.463 0.448 0.439 0.427 0.421 0.4101 0.490 0.476 0.466 0.451 0.441 0.427 0.421 0.4092 ..... ..... ..... A comparison of the results of this formula with those of experiments, says M. Bazin, justifies us in believing that, except in the unusual case of a very low weir (which should always be avoided), the preceding table will give the coefficient m in all cases within 1%; provided, however, that the ar rangements of the standard weir are exactly reproduced. It is especially important that the admission of the air behind the falling sheet be perfectly assured. If this condition is not complied with, m may vary within much wider limits. The type adopted gives the least possible variation in the coefficient. WATER-POWER. Power of a Fall of Water-Efficiency.-The gross power of a fall of water is the product of the weight of water discharged in a unit of time into the total head. i.e., the difference of vertical elevation of the upper surface of the water at the points where the fall in question begins and ends. The term "head" used in connection with water-wheels is the difference in height from the surface of the water in the wheel-pit to the surface in the pen-stock when the wheel is running, If Q cubic feet of water discharged per second, D= weight of a cubio foot of water = 62.36 lbs. at 60° F., H= fotal head in feet; then DQH= gross power in foot-pounds per second, and DQH + 550 =.1134QH = gross horse-power. If Q'is taken in cubic feet per minute, H. P. = Q'HX 62,36 = .00189Q'H. A water-wheel or motor of any kind carnot utilize the whole of the head H, since there are losses of head at both the entrance to and the exit from the wheel. There are also losses of energy due to friction of the water in its passage through the wheel. The ratio of the power developed by the wheel to the gross power of the fall is the efficiency of the wheel. For 75% efficiency, net horse-power = .00142Q'H = Q'H A head of water can be made use of in one or other of the following ways viz.: 1st. By its weight, as in the water-balance and overshot-wheel. 2d. By its pressure, as in turbines and in the hydraulic engine, hydraulic press, crane, etc. 3d. By its impulse, as in the undershot-wheel, and in the Pelton wheel. 4th. By a combination of the above. Horse-power of a Running Stream.-The gross horse-power is, H. P. = QHX 62.36 + 550 = .1134QH, in which Q is the discharge in cubic feet per second actually impinging on the float or bucket, and H theoretvs v2 ical head due to the velocity of the stream = in which v is the 2g 64.4' velocity in feet per second. If Q be taken in cubic feet per minute, H.P..001890'H. Thus, if the floats of an undershot-wheel driven by a current alone be 5 feet X 1 foot, and the velocity of stream = 210 ft. per minute, or 3% ft. per sec., of which the theoretical head is .19 ft., Q = 5 sq. ft. x 210 = 1050 cu. ft. per minute; H.19 ft.; H. P. = 1050 X .19 .00189,377 H. P. The wheels would realize only about .4 of this power, on account of friction and slip, or .151 H. P., or about .03 H.T. per square foot of float, which is equivalent to 33 sq. ft. of float per H. P. Current Motors.-A current motor could only utilize the whole power of a running stream if it could take all the velocity out of the water, so that it would leave the floats or buckets with no velocity at all; or in other words, it would require the backing up of the whole volume of the stream until the actual head was equivalent to the theoretical head due to the velocity of the stream. As but a small fraction of the velocity of the strean, can be taken up by a current motor, its efficiency is very small. Current motors may be used to obtain small amounts of power from large streams, but for large powers they are not practicable. v2 2g w v2 29 f. w Horse-power of Water Flowing in a Tube.-The head due to the velocity is ; the head due to the pressure is ; the head due to actual height above the datum plane is h feet. The total head is the sum of these = +h+ in feet, in which v velocity in feet per second, f= pressure in lbs. per sq. ft., w = weight of 1 cu. ft. of water = 62.36 lbs. If p = pressure in lbs. per sq. in., 2.309p. In hydraulic transmission the velocity and the height above datum are usually small compared with the pressure. head. The work or energy of a given quantity of water under pressure = its volume in cubic feet X its pressure in lbs. per sq. ft.; or if Q quantity in cubic feet per second, and p = pressure in lbs. per square inch, W = 144pQ 144pQ, and the H. P. = = .2618pQ. w 550 Maximum Efficiency of a Long Conduit.-A. L. Adams and R.C.Gemmell (Eng'g News, May 4, 1893), show by mathematical analysis that the conditions for securing the maximum amount of power through a long conduit of fixed diameter, without regard to the economy of water, is that the draught from the pipe should be such that the frictional loss in the pipe will be equal to one third of the entire static head. Mill-Power.-A "mill-power" is a unit used to rate a water-power for the purpose of renting it. The value of the unit is different in different localities. The following are examples (from Emerson): Holyoke, Mass.-Each mill-power at the respective falls is declared to be the right during 16 hours in a day to draw 38 cu. ft. of water per second at the upper fall when the head there is 20 feet, or a quantity proportionate to the height at the falls. This is equal to 86.2 horse-power as a maximum. Lowell, Mass.-The right to draw during 15 hours in the day so much water as shall give a power equal to 25 cu. ft. a second at the great fall, when the fall there is 30 feet. Equal to 85 H. P. maximum. Lawrence, Mass.-The right to draw during 16 hours in a day so much water as shall give a power equal to 30 cu. ft. per second when the head is 25 feet. Equal to 85 H.P. maximum. Minneapolis, Minn.-30 cu. ft. of water per second with head of 22 feet. Equal to 74.8 H.P. Manchester, N. H.-Divide 725 by the number of feet of fall minus 1, and the quotient will be the number of cubic feet per second in that fall. For 20 feet fall this equals 38.1 cu. ft., equal to 86.4 H. P. maximum. Cohoes, N. Y.-"Mill-power" equivalent to the power given by 6 cu. ft. per second, when the fall is 20 feet. Equal to 13.6 H. P., maximum. Passaic, N. J.-Mill-power: The right to draw 81⁄2 cu. ft. of water per sec., fall of 22 feet, equal to 21.2 horse-power. Maximum rental $700 per year for each mill-power = $33.00 per H. P. The horse-power maximum above given is that due theoretically to the weight of water and the height of the fall, assuming the water-wheel to have perfect efficiency. It should be multiplied by the efficiency of the wheel, say 75% for good turbines, to obtain the H. P. delivered by the wheel. Value of a Water-power.-In estimating the value of a waterpower, especially where such value is used as testimony for a plaintiff whose water-power has been diminished or confiscated, it is a common custom for the person making such estimate to say that the value is represented by a sum of money which, when put at interest, would maintain a steam-plant of the same power in the same place. Mr. Charles T. Main (Trans. A. S. M. E. xiii. 140) points out that this system of estimating is erroneous; that the value of a power depends upon a great number of conditions, such as location, quantity of water, fall or head, uniformity of flow, conditions which fix the expense of dams, canals, founda tions of buildings, freight charges for fuel, raw materials and finished prod. act, etc. He gives an estimate of relative cost of steam and water-power for a 500 H. P. plant from which the following is condensed: The amount of heat required per H. P. varies with different kinds of busi ness, but in an average plain cotton-mill, the steam required for heating and slashing is equivalent to about 25% of steam exhausted from the highpressure cylinder of a compound engine of the power required to run that mill, the steam to be taken from the receiver. The coal consumption per H. P. per hour for a compound engine is taken at 134 lbs. per hour, when no steam is taken from the receiver for heating purposes. The gross consumption when 25% is taken from the receiver is about 2.06 lbs. 75% of the steam is used as in a compound engine at 1.75 lbs. = 1.31 lbs. 25% high-pressure 8.00 lbs. = 66 66 66 66 .75 2.06" = 6503.42 $8 71 2.00 2 16 80 The running expenses per H. P. per year are as follows: 2.06 lbs. coal per hour = 21.115 lbs. for 104 hours or one day lbs. for 308 days, which, at $3.00 per long ton = Attendance of boilers, one man @ $2.00, and one man @ $1.25 = 66 66 engine, .6 $3.50. Oil, waste, and supplies. 8 13 Gross cost of power and low-pressure steam per H. P. $21 80 Comparing this with water-power, Mr. Main says: "At Lawrence the cost of dam and canals was about $650,000, or $65 per H. P. The cost per H. P. of wheel-plant from canal to river is about $45 per H. P. of plant, or about $65 per H. P. used, the additional $20 being caused by making the plant large enough to compensate for fluctuation of power due to rise and fall of river. The total cost per H. P. of developed plant is then about $130 per H. P. Placing the depreciation on the whole plant at 2%, repairs at 1%, interest at 5%, taxes and insurance at 1%, or a total of 9%, gives: Fixed expenses per H. P. $130 X .09 = (Estimated) $11 70 2.00 $13 70 "To this has to be added the amount of steam required for heating pur poses, said to be about 25% of the total amount used, but in winter months the consumption is at least 37%. It is therefore necessary to have a boiler plant of about 37% of the size of the one considered with the steam-plant, costing about $20 x .375 = $7.50 per H. P. of total power used. The expense of running this boiler-plant is, per H. P. of the the total plant per year: Fixed expenses 12% on $7.50... Coal... Total .... .... ....................... .......... $0.94 3.26 1.23 $5.43 Making a total cost per year for water-power with the auxiliary boiler plant $13.70+$5.43 $19.13 which deducted from $21.80 make a difference in favor of water-power of $2.67, or for 10,000 H. P. a saving of $26,700 per year. "It is fair to say," says Mr. Main," that the value of this constant power is a sum of money which when put at interest will produce the saving; or if 6% is a fair interest to receive on money thus invested the value would be $26.700.06 = $445,000." Mr. Main makes the following general statements as to the value of a water-power: "The value of an undeveloped variable power is usually nothing if its variation is great, unless it is to be supplemented by a steam-plant. It is of value then only when the cost per horse-power for the double plant is less than the cost of steam-power under the same conditions as mentioned for a permanent power, and its value can be represented in the same man. ner as the value of a permanent power has been represented. "The value of a developed power is as follows: If the power can be run cheaper than steam, the value is that of the power, plus the cost of plant, less depreciation. If it cannot be run as cheaply as steam, considering its cost, etc., the value of the power itself is nothing, but the value of the plant is such as could be paid for it new, which would bring the total cost of running down to the cost of steam-power, less depreciation.' Mr. Samuel Webber, Iron Age, Feb. and March, 1893, writes a series of articles showing the development of American turbine wheels, and incidentally criticises the statements of Mr. Main and others who have made comparisons of costs of steam and of water-power unfavorable to the latter. Hesays: "They have based their calculations on the cost of steam, on large compound engines of 1000 or more H. P. and 120 pounds pressure of steam in their boilers, and by careful 10-hour trials succeeded in figuring_down steam to a cost of about $20 per H. P., ignoring the well-known fact that its average cost in practical use, except near the coal mines, is from $40 to $50. In many instances dams, canals, and modern turbines can be all completed for a cost of $100 per H. P.; and the interest on that, and the cost of attendance and oil, will bring water-power up to but about $10 or $12 per annum; and with a man competent to attend the dynamo in attendance, it can probably be safely estiinated at not over $15 per H. P." TURBINE WHEELS. Proportions of Turbines.-Prof. De Volson Wood discusses at length the theory of turbines in his paper on Hydraulic Reaction Motors, Trans. A. S. M. E. xiv. 266. His principal deductions which have an immediate bearing upon practice are condensed in the following: Notation. Q volume of water passing through the wheel per seco::d, h1 = head in the supply chamber above the entrance to the buckets, = head in the tail-race above the exit from the buckets, fall in passing through the buckets. H = h2+z1h, the effective head, M1 = coefficient of resistance along the guides, coefficient of resistance along the buckets, r1 = radius of the initial rim, = radius of the terminal rim, velocity of the water issuing from supply chamber, v1initial velocity of the water in the bucket in reference to the bucket, terminal velocity in the bucket, a = angular velocity of the wheel, terminal angle between the guide and initial rim = CAB, Fig. 132, Y1 = angle between the initial element of bucket and initial rim EAD, Ya GFI, the angle between the terminal rim and terminal element of the bucket. a=eb, Fig. 133 = the arc subtending one gate opening |