Let a certain volume of water, s, be converted into steam of the pressure p, and let x represent the volume of gears produced, then SI = ; if se and p' stand for the volume and pressure of steam from the same volume of water s, under other conditions, then = + q p" and therefore the ratio of the volumes of steam produced under these different conditions from the name volume of water will bo n + q (I) )n + that is, the volumes will not be inversely as the pressures simply, according to Ttlariotte's law ; but inversely as the pressures augmented by a constant.
From the above equation we get N178 n r si + P ) - . . . (E) Let t = pressure of the steam in the boiler.
r'= pressure of the steam in the cylinder ; r'
7 = pressure at any instant when acting expansively in the engine.
/ = length of the stroke.
the length of that part of stroke performed before the cons munie.ation between the boiler and cylinder is emit oft: a = the length of that portion of the stroke performed when tit pressure is become w.
a = area of piston.
c = cicarage, or space in the cylinder at each end left between the piston and the ends of the cylinder, including thi part of the steam-pipe between the slide-valve and tin cylinder, which space is necessarily filled with steam a each stroke.
When the piston has performed 1, of its course tinder the expansics force of the steam, let d . A be the differential of this length, then Us corresponding force or effect will be wad. A: and at the same install the space a (1' +c) occupied by the steam before the expansion wil become a (A+ e). Hence from (E) n /'+ c it = + r' 1t+ cc ' q n d. A n and wad .h= a (l' + e)I + c + r') q ad . A.
Integrating between tho limit 1' and 1, we obtain ) I+c it a 0' + e) (-7 C+ r' log for the value of the total effect produced by expansion from th moment when the communication with the boiler is cut off, to the em of the stroke. By adding to this therefore the effect eat', produce previously, we get / a + 7 LC +c 1 + c + r') + log q al= ant . . (F) If in this expression, /'=/, which is equivalent to suppozzing th engine to he working without expansion, we get r'= It, as it ought to he.
8 Resuming the equation which expresses the volume of stealat the pressure r' furnished by the boiler in the unit of tint:, an +0 being the volume of this steam expended at each stroke then if there are K strokes in that time, the expenditure of stoat will be xa (/'+ c); and if r be put for the velocity of time piston, we have r=s1, K = / ; hence, by substitution, the expenditure will be est (/' + c) n+qr (G) by equating the expenditure to the volume furnished by the boile which, as has been above stated, must be the condition when th motion is uniform. Eliminating ri between (F) and (G), we get for
the final general equation s 1 r/ 2+ci a ' n+qn + l+o_1 (H) The resistance expressed by n in this formula is the total pressure on each unit of surface of the opposite aide of the piston, and is composed of three parts. First, of the load, or work to be moved or done, which we will denote by r. Secondly, of the resistance arising from the friction of the engine, which may be expressed by f +ar ; f being the friction when there is no load, and ar the increment due to the additional friction for each unit of the load r. And, lastly, of the pressure on the opposite surface of the piston, which will be the atmospheric pressure in non-condensing engines, or that of the uncon densed steam and residue of air in condensing ones; this we shall call p. All these, r, f+ ar, and p, refer only-to each unit of surface of the piston, or n = (1 + 8) r + p + f, by substituting this value for a in (H), and putting k for rl l L + we obtain s k v . . . (K) a . n+q[(1 +8) r-Fp+f] Now the quantity it will be seen (C) is the total space occu pied by the steam (in contact with the water), under the pressure hence to deduce the velocity v, the volume of steam corresponding to the volume of water s, supposed to be converted into steam under a pressure equal to R, must be calculated; and this volume being divided by a, the area of the piston must be multiplied by k.
The equation thus deduced shows the relation between all the quantities, known or sought, that enter into the mechanical theory of the engine in its most general form : it should be observed how ever that to preserve homogeneity, the dimensions a, 1, l', should be expressed in the same unit as the volume s of the water evaporated ; and the pressures r, r, and p referred to the same unit When this formula is used for computation, it must be understood that the quantity s expresses the effective evaporation ; that is, the volume of water which really passes to the cylinder in the form of steam, and which acts on the piston, and does not allow for any loss by leakage or from any peculiarity in the structure of the engine.