Ε

the mechanical power of a cubic foot of water, when is the additional power gained by employing the expansion of the steam.

On the value of being inserted in the equation it becomes

E

the mechanical power of a cubic foot of water, when the expansive force of the steam is employed.

308.-This equation has a maximum, which will be when

Consequently, we shall have the greatest possible quantity of mechanical power when

And where a table of hyperbolic logarithms cannot be conveniently referred to, the result may be obtained by multiplying the logarithm of

f f'F'

found from the common tables of logarithms, by 2.302585, which will give the corresponding hyperbolic logarithm.

309.—In the best constructed engines, the waste of steam is not less than one-tenth; and, to get the extreme power of the steam of a given quantity of water at this rate of waste, we have 1 — w = 9; and the equation becomes

the greatest possible power the steam of a cubic foot of water can afford, when acting expansively.

310.—In like manner taking the same loss by waste, we have from (art. 304.)

the greatest possible power of the steam of a cubic foot of water, when the expansive

power of the steam is not used. Consequently,

311.-Though these equations shew us the limits of steam power, and are fittest for illustrating the advantages or disadvantages of difference of temperature, and elastic force, clearly exhibiting the economy of using steam of considerable elastic force, yet they still require to be applied to engines of different species.* This will be done in Sect. V. and VI. but before I quit the illustration of general principles, it will be desirable to investigate the rotary action of steam.

Of computing the Power of Steam to produce Rotary Motion.

312.—In a great variety of the cases where steam is employed, a continuous circular motion is to be produced, and it is very generally imagined that a great advantage would be gained if the rotary motion were produced by the direct action of steam, instead of being obtained by the intervention of moving parts, for converting the rectilineal motion produced by steam into a rotary one.

But the fact of every person who has attempted to produce an engine acting by the rotary power of steam, having in a greater or less degree failed in rendering it as effective as a reciprocating engine, makes the theoretical principle of rotary action, an interesting subject of investigation.

* A series of tables calculated by these formula were published in my Treatise on Rail Roads, p. 161–166.

313.-Conceive a piston, DE, to be fitted to a regularly curved vessel A B, so that it may move round C, the centre of curvature of the vessel, and consequently the centre of motion. Now whether the piston be moved by the force of high pressure steam, or otherwise, the pressure on an inch of area of the piston, will be equal on all its parts; that is,

the pressure on an inch, at the most distant part D from the centre of motion, is the same as the pressure on an inch at the part E, nearest to that centre. But since the piston is constrained to move in a circle, the effects of these equal pressures are as their distances from the centre of motion, and limited by the effect of the pressure at the most distant part D. Hence, if the effective pressure of the steam be ten pounds on the inch, we have

the effect at E, that at D being 10. If the centre of curvature C, were nearer to the side of the vessel, the effect at E would be less; therefore, the effect of the pressure to produce motion, is less than in a straight vessel, having the same base; and if the bases be the same, the space the pressure acts through will be as the quantity of steam. Consequently, the quantities of steam being equal, the power of rotary action will be less than that of rectilineal action.

314.—If a rectangular piston, D C, revolve round a centre C, then nearly half the power of the steam will be lost.

This rough inquiry will be sufficient to shew that much is lost by attempting to employ the rotary action of steam, besides the various other objections arising out of the excess of friction, and the difficulties of executing the parts so as to act properly; usually called practical difficulties.

315.-To conduct the inquiry so as to reduce the effect to more accurate measures; put

d = D E the diameter of the piston;

r = E C, the radius of the interior circle;

x = any variable portion of the diameter of the piston counted from E; y = the breadth of the piston, and

f = the force of the steam on an inch of area.

Then r+ d = DC; and as

the fluxion of the pressure at that point; and the space described being 2 p ( r + x, ) we

and making this equation nothing, when, x = o, that is when the power is nothing, we have

and when x = d,

the

power of the steam acting in a rotary direction, the piston being a rectangle d y. 316.-If the piston D C revolve on an axis in the centre C, then r = o, and

the rotary power.

2 2 pfy dr

But the space occupied by the steam is p d (2 r + d) y, and its rectilineal power is pfd (2 r + d) y. Hence the rectilineal is to the rotary effect of the steam as

When r = o, or the piston revolves on a centre, then the ratio becomes 3: 2, or onethird of the power is lost; the same conclusion resulting however the steam acts.

317.-We have supposed the piston to be of parallel width, but in some schemes it has been made circular; and in such a case the valve of y is ✔d x-x2. Consequently,

:

This is a little less than the effect of a rectangular piston. When the piston revolves round an axis in its edge the rectilineal power of a given quantity of steam is to its rotary power as 3.2 2. In the rectangular one it was as 3: 2. Hence, we see there is no possibility of applying steam with the same advantage in a rotary, as in a rectilineal engine; and, even to approximate to it, the radius of the circle described must be great in comparison with the diameter of the piston, and consequently difficult to execute. To employ any other than a circular form for the piston, would cause more friction, and expose a larger portion of surface to the cooling effect of the atmosphere. These are radical objections to the rotary action of steam that cannot be removed by art.

318.-It is so obvious, that it is not necessary to shew, that the impulse of steam cannot be employed without great loss of fuel, we may, however, take a general view of the modes in which the action of steam may be applied.

Modes of applying the Power of Steam.

319.-The arrangement being presented in a tabular form will be more clear than in