evidences in its favor. On any other basis high pressure engines could not work as economically as they are now known to do, and in looking over some results from the Dixwell experiments as reported by Mr. Barrus,* to prove another point, the writer discovered that the difference between the quantity of heat lost in the performance of work on a steam piston and that lost by blowing steam through the same cylinder and over the same surfaces without doing work was not sufficient to account for the heat required in the performance of the work if that for displacing the atmosphere were included. The number of pounds of feed water required to furnish 2,565 heat units between the actual limits of temperature will vary with the pressure, and for strict accuracy should be calculated for each case. The illustrative table at the

end of this paper has been calculated by taking from the total heat of the steam due to the pressure the total heat of water due to a pressure of 17 lbs. steam pressure absolute, and dividing 2,565 by the result. On this basis it will be seen that the condensation of 3 pounds of water is more than sufficient to furnish the heat transmuted into work within the limits considered or even when the lower limit is at atmospheric pressure, and this may be used in approximate calculations. Hence

(17) C2 = 3 approximately.

3

The cost due to cylinder condensation represented by C's in formula refers to the great losses occasioned by the changes of temperature of the walls of a steam cylinder fully discussed in a previous paper by the writer. It is entirely independent of external refrigeration which, by proper arrangements, can be rendered quite insignificant. The actual cost of an indicated horse power in pounds of feed water is increased as compared with the calculated cost not only by cylinder condensation as above set forth and represented by C, in formula (5), but by causes which produce deficiency in power, such as contracted passages or ports and imperfect steam distribution, as well as the minor losses due to external refrigeration, all represented by C1 in formula (5). The cylinder condensation could apparently be most completely ascertained by calculations based on the surfaces exposed and the differences of temperature at different points of the stroke on the * See discussion of the writer's papers, CCXLVIII. and CCXLIX., at the Washington meeting, Vol. VIII., page 472; Vol. IX. Trans. A. S. M. E. See Paper by the writer on VIII., Trans. A. S. M. E.

་་

Cylinder Condensation," etc., CCXLVII., Vol.

basis of the laws of the transmission and radiation of heat, but this elaborate study would not include nearly all the elements necessary to make a complete analysis of this complex problem, so it has been considered practically as accurate to attempt merely to produce formula which will show approximately the experimental results and therefrom deduce the sum of C3 and C4 rather than attempt to separate C, which will form the larger proportion of the whole.

The ordinates included between curves G and H and the experimental curve D in diagram No. 2 represent the sums of all the losses above stated, and it will be shown that these losses are expressed by the formula.

(18) C3 + C1 = (E−(C1 + C'2)n;

that is, the difference between the experimental cost E in small engine and the sum of the fixed costs C, and C, due to filling the cylinder and the performance of the work (which difference is represented by the portion of the full stroke ordinate intercepted between the curves D and G), is to be reduced for engines of greater power by a fractional multiplier n, to cause a change of value corresponding to the change of condition. The first step is to formulate the experimental value of E. The engine was provided with a lap valve and so did not at any time operate at exactly full stroke, but the costs due to full stroke are readily found by producing the curves A, B, C, D and E, Diagram No. 1. From the conditions of the problem the cost should be infinite with no pressure, and the pressure infinite with no cost, so each branch of the curve sought should be an asymptote, and equation (4) of the hyperbola is the natural one to use. It however contains but three constants, a, b and d, but from these the full stroke results due to curves B, C and D, corresponding to pressures of 40, 60 and 80 pounds, were used, since, as has been stated, curves A and E were in portions incomplete and developed to correspond with those first named. Making 4 equal the cost and the corresponding steam pressure, and substituting in the three values named, we have

which will be found to give accurately full stroke values for the curves B, C and D, to wit: for 40 pounds pressure, a cost of 63 pounds of feed water; for 60 pounds pressure, 55.3 pounds cost,

and for 80 pounds pressure, 51 pounds cost. For 25 pounds pressure the calculated cost is 74.8 pounds, and the cost as plotted about 75.5 pounds, and for 100 pounds pressure the calculated cost is 48.2 pounds, and the same as plotted, 48 pounds. It is probable that the equation more accurately represents the true values than was possible with the free hand work necessary in extending the curves last named.

It is next required to find the value of n in equation (18). It is well known that the cylinder condensation in small engines is much larger than in large ones, which can be explained by the fact that as the linear dimensions of the cylinder are increased, the interior surfaces do not increase as rapidly as the capacities. In connection with the experiments plotted on the curves another series of experiments also showed conclusively that there is economy in high speeds of revolution, independent of speed of piston, which is explained by the fact that, as the speed of revolution is increased, the alternations of temperature in the cylinder take place at more frequent intervals, and there is not time during each single stroke to change the temperature of the metal to as great a depth.* The result is that less weight of metal is heated and cooled, and the loss by condensation becomes correspondingly less. Higher speeds of revolution of an engine with cylinder of the same size, but otherwise operated under like conditions, cause an increase of power; and an increase in the size of the cylinder, with like conditions otherwise, causes also an increase in power. The comparison of various experiments shows that the cylinder condensation reduces as the power is increased either by increased speed of revolution or an increase in the size of the cylinder (at least when cylinders of substantially milar proportions are considered). This deduction is made the oasis of formulating the value of n in equation (18). For the small engine n would necessarily equal unity to produce the experimental results for 3.27 horse power, and this value was compared with n = 0.1843, deduced for 78.8 horse power developed under similar conditions of pressure and expansion in experiments made with a Babcock & Wilcox engine at the American Institute under the direction of the writer, in the year 1869. The equation obtained gives results corresponding well with experiments made at the same time on a Harris-Corliss engine and those made in the following year with a Porter-Allen * See Topical Discussions, page 375, Vol. VII. Trans. A. S. M. E,, where a tab ular statement of the results of these experiments is given.

engine developing 80 horse power, under the directions of a committee of which Prof. Thurston was chairman. By the conditions. of the problem the multipliers can never be minus or less than 0, but may increase as the size of the engines diminish. By substituting the special values above stated in equation (4) with d = 0, there results

which will be found to fulfill all the conditions referred to.

The value of S, expressing the saving in cost due to expansion and the only remaining quantity in equation (5), has been formulated as follows:

Comparing the general direction and position of the various curves with D, or that corresponding to 80 pounds, and the curve D again in No. 2, with the calculated curves, G and H, we find that the experimental curves are practically parallel with each other up to the points of inflection and nearly parallel with the calculated curves. In other words, that the relations between. the experimental curves at different points of cut-off may be expressed by constant arithmetical differences, but that the difference between results shown by any particular curve and the calculated results for the same pressures and points of cut-off would somewhat increase as the expansion increased up to the points of inflection. The parallelism of the curves makes it possible to use a simple formula to show the saving due to expansion, applicable between the limits of full stroke in one direction and the points of inflection in the other. A complete formula which would be applicable to both sides of the mii-mum would require a series of operations similar to those by which the curves G and H were determined. In developing the curves last named a cylinder was assumed with the volume required to develop one horse power at the pressure and point of cut-off considered, based, of course, on the work done during the admission of the steam and that done during free expansion, and the cost was made up from the weight of steam required to fill the cylinder to the point of cut-off and that required to supply the heat required for the performance of the work; a clearance of one-twentieth of the displacement without cushion having been considered, in developing curve G and curve H, showing the results for no clearance.

It will be observed that the experimental curves nearly to the

points of inflection are practically straight lines as well as parallel, so the result at any point of the stroke compared with that at full stroke for the same pressure and the same general conditions can be closely expressed by the formula of a straight line equation (2) with d 0, as follows:

=

The minimum value of h is determined by the consideration that it is not economical to continue the expansion so that the pressure will fall below say 3 pounds, which would at least be necessary to equilibriate the back pressure on the piston and the resistance of the engine itself so the minimum value of h or

By comparison of experimental data it is also found that (23) = 0.64 approximately for engines cutting off with main valve.

A complete formula for the value of C, equation (5), may be constructed by substituting in the values of the several terms, but it will be simpler to make the substitutions in the arithmetical solution. The formula may be illustrated by applying the same to the conditions of the trial of the Babcock & Wilcox engine previously referred to. The steam pressure being 80 lbs. by (16a) C1=38.56. Adding C-3, C1+C, 41.56. For pressure named E (eq. [19])= 51, and for 78.8 HP by (eq. [18])=0.1843. E-(C1+C2)=9.44. and n times this gives 1.74 as the value of the losses C+C1 (eq. [18]) on basis assumed, which added to C+C gives for full stroke cost C1 +C2+C3+C1 =43.30. The cut off in this case being at 0.19 stroke 1-c=0.81. The minimum value by (22) by a coincidence also equals 0.81 in this case, so from (21) S=17.82, which subtracted from the full stroke cost above gives C=25.48, which is the reported cost. The curve for this engine would therefore fall somewhat below the curve F (No. 2), and fixing in mind the value 43.3 at full stroke and 25.5 at two-tenths stroke we can better appreciate the various operations accomplished with the formula. We have reached the cost due to expansion by first finding the full stroke cost under two conditions, viz.: the experimental one, as shown by curve D, and the calculated one, as shown by curve H. The saving by expansion has been deduced by assuming that the difference in cost is constant throughout the stroke, and repre