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 8% 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 business, 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 3.00 lbs. = .75 ** 66 66 66 66 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 = 6503.42 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. 2.06" $8 71 2.00 2 16 80 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 = $11 70 66 $13 70 "To this has to be added the amount of steam required for heating purposes, 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 ís, per H. P. of the the total plant per year; Fixed expenses 12% on $7.50....... Coal.... ...................... $0.94 3.26 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 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 estimated 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. Qvolume of water passing through the wheel per second, h1: = head in the supply chamber above the entrance to the buckets, hhead in the tail-race above the exit from the buckets, Z1 = fall in passing through the buckets. H = h2+21h, the effective head, "1 = coefficient of resistance along the guides, coefficient of resistance along the buckets, r radius of the terminal rim, velocity of the water issuing from supply chamber, v1 = initial velocity of the water in the bucket in reference to the bucket, terminal velocity in the bucket, = angular velocity of the wheel, a = terminal angle between the guide and initial rim = CAB, Fig. 132, Y1 = angle between the initial element of bucket and initial rim EAD, Y2GFI, the angle between the terminal rim and terminal element of the bucket. a=eb, Fig. 133 = the arc subtending one gate opening, a1 = the arc subtending one bucket at entrance. (In practice a1 than a,) = gh, the arc subtending one bucket at exit, is larger bf, normal section of passage, it being assumed that the passages and buckets are very narrow, k1 = bd, initial normal section of bucket, = gi, terminal normal section, velocity of initial rim, velocity of terminal rim, 6 HFI, angle between the terminal rim and actual direction of the water at exit, Y depth of K. y, of an and Y2 of K,, then Three simple systems are recognized, r<,lcalled outward flow; r,> T2, called inward flow; r, r, called parallel flow. The first and second may be combined with the third, making a mixed system. Value of y (the quitting angle).-The efficiency is increased as y de creases, and is greatest for Y20. Hence, theoretically, the terminal element of the bucket should be tangent to the quitting rim for best efficiency. This, however, for the discharge of a finite quantity of water, would require an infinite depth of bucket. In practice, therefore, this angle must have a tinite value. The larger the diameter of the terminal rim the smaller may be this angle for a given depth of wheel and given quantity of water discharged. In practice y, is from 10° to 20°. In a wheel in which all the elements except y, are fixed, the velocity of the wheel for best effect must increase as the quitting angle of the bucket decreases. Values of a +-y, must be less than 180°, but the best relation cannot be determined by analysis. However, since the water should be deflected from its course as much as possible from its entering to its leaving the wheel, the angle a for this reason should be as small as practicable. In practice, a cannot be zero, and is made from 20° to 30°. The value r1 = 1.4r, makes the width of the crown for internal flow about the same as for r1 =r, V for outward flow, being approximately 0.3 of the external radius. Values of and pg.-The frictional resistances depend upon the construc tion of the wheel as to smoothness of the surfaces, sharpness of the angles, regularity of the curved parts, and also upon the speed it is run. These values cannot be definitely assigned beforehand, but Weisbach gives for good conditions μ1 = μ2 = 0.05 to 0.10. They are not necessarily equal, and μ, may be from 0.05 to 0.075, and μg from 0.06 to 0.10 or even larger. Values of y1 must be less than 180° -α. To be on the safe side, y1 may be 20 or 30 degrees less than 180°-2a, giving Y1 = 180° - 2a - 25 (say) = 155- 2a. Then if a 30°, Y1 = 95°. Some designers make y, 90°; others more, and still others less, than that amount. Weisbach suggests that it be less, so that the bucket will be shorter and friction less. This reasoning appears to be correct for the inflow wheel, but not for the outflow wheel. In the Tremont turbines, described in the Lowell Hydraulic Experiments, this angle is 90°, the angle a 20°, and y1⁄2 10°, which proportions insured a positive pressure in the wheel. Fourneyron made y1 = 90°, and a from 30° to 33°, which values made the initial pressure in the wheel near zero. Form of Bucket.-The form of the bucket cannot be determined analytically. From the initial and terminal directions and the volume of the water flowing through the wheel, the area of the normal sections may be found. The normal section of the buckets will be: The changes of curvature and section must be gradual, and the general form regular, so that eddies and whirls shall not be formed. For the same reason the wheel must be run with the correct velocity to secure the best effect. In practice the buckets are made of two or three arcs of circles, mutually tangential. The Value of w.-So far as analysis indicates, the wheel may run at any speed; but in order that the stream shall flow smoothly from the supply chamber into the bucket, the velocity V should be properly regulated. If μMg0.10, r2+r1 = 1.40, a = 25°, Y1 = 90°, Y2 = 120, the velocity of the initial rim for outward flow will be for maximum efficiency 0.614 of the velocity due to the head, or wr1 = 0.614 √/2gH. The velocity due to the head would be V2yH=1.414 VgH. For an inflow wheel for the case in which r2 = 2r,2, and the other dimen sions as given above, wr, = 0.682 √2gH. The highest efficiency of the Tremont turbine, found experimentally, was 0.79375, and the corresponding velocity, 0.62645 of that due to the head, and for all velocities above and below this value the efficiency was less. In the Tremont wheel a = 20° instead of 25°, and y2 = 10° instead of 12°. These would make the theoretical efficiency and velocity of the wheel some what greater. Experiment showed that the velocity might be considerably larger or smaller than this amount without much diminution of the efficiency. It was found that if the velocity of the initial (or interior) rim was not less than 44% nor more than 75% of that due to the fall, the efficiency was 75% or more. This wheel was allowed to run freely without any brake except its own friction, and the velocity of the initial rim was observed to be 1.335 1/2gH, half of which is 0.6675 V/2gH, which is not far from the velocity giving maximum effect; that is to say,when the gate is fully raised the coefficient of effect is a maximum when the wheel is moving with about half its maximum velocity. Number of Buckets.-Successful wheels have been made in which the distance between the buckets was as small as 0.75 of an inch, and others as much as 2.75 inches. Turbines at the Centennial Exposition had buckets from 41⁄2 inches to 9 inches from centre to centre. If too large they will not work properly. Neither should they be too deep. Horizontal partitions are sometimes introduced. These secure more efficient working in case the gates are only partly opened. The form and number of buckets for com mercial purposes are chiefly the result of experience. 1 Ratio of Radii.-Theory does not limit the dimensions of the wheel. In practice, for outward flow, r2+r, is from 1.25 to 1.50; It appears that the inflow-wheel has a higher efficiency than the outwardflow wheel. The inflow-wheel also runs somewhat slower for best effect. The centrifugal force in the outward-flow wheel tends to force the water outward faster than it would otherwise flow; while in the inward-flow wheel it has the contrary effect, acting as it does in opposition to the velocity in the buckets. It also appears that the efficiency of the outward-flow wheel increases slightly as the width of the crown is less and the velocity for maximum efficiency is slower; while for the inflow-wheel the efficiency slightly increases for increased width of crown, and the velocity of the outer rim at the same time also increases. Efficiency. The exact value of the efficiency for a particular wheel must be found by experiment. It seems hardly possible for the effective efficiency to equal, much less exceed, 80%, and all claims of 90 or more per cent for these motors should be discarded as improbable. A turbine yielding from 75% to 80% is extremely good. Experiments with higher efficiencies have been reported. The celebrated Tremont turbine gave 794% without the "diffuser," which might have added some 2%. A Jonval turbine (parallel flow) was reported as yielding 0.75 to 0.90, but Morin suggested corrections reducing it to 0.63 to 0.71. Weisbach gives the results of many experiments, in which the efficiency ranged from 50% to 84%. Numerous experiments give E = 0.60 to 0.65. The efficiency, considering only the energy imparted to the wheel, will ex ceed by several per cent the efficiency of the wheel, for the latter will include the friction of the support and leakage at the joint between the sluice and wheel, which are not included in the former; also as a plant the resistances and losses in the supply-chamber are to be still further deducted. The Crowns.-The crowns may be plane annular disks, or conical, or curved. If the partitions forming the buckets be so thin that they may be discarded, the faw of radial flow will be determined by the form of the crowns. If the crowns be plane, the radial flow (or radial component) will diminish, for the outward flow-wheel, as the distance from the axis increases -the buckets being full-for the angular space will be greater. Prof. Wood deduces from the formulæ in his paper the tables on page 595. It appears from these tables: 1. That the terminal angle, a, has frequently been made too large in practice for the best efficiency. 2. That the terminal angle, a, of the guide should be for the inflow less than 10 for the wheels here considered, but when the initial angle of the bucket is 90°, and the terminal angle of the guide is 5° 28', the gain of efficiency is not 2% greater than when the latter is 25°. 3. That the initial angle of the bucket should exceed 90° for best effect for outflow-wheels. 4. That with the initial angle between 60° and 120° for best effect on inflow wheels the efficiency varies scarcely 1%. 5. In the outflow-wheel, column (9) shows that for the outflow for best effect the direction of the quitting water in reference to the earth should be nearly radial (from 76° to 97°), but for the inflow wheel the water is thrown forward in quitting. This shows that the velocity of the rim should somewhat exceed the relative final velocity backward in the bucket, as shown in columns (4) and (5). 6. In these tables the velocities given are in terms of V2gh, and the coefficients of this expression will be the part of the head which would produce that velocity if the water issued freely. There is only one case, column (5), where the coefficient exceeds unity, and the excess is so small it may be discarded; and it may be said that in a properly proportioned turbine with the conditions here given none of the velocities will equal that due to the head in the supply-chamber when running at best effect. 7. The inflow turbine presents the best conditions for construction for producing a given effect, the only apparent disadvantage being an increased first cost due to an increased depth, or an increased diameter for producing a given amount of work. The larger efficiency should, however, more than neutralize the increased first cost. |