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evaporated by such form of boiler, to know what extent of heating surface should be given to the boiler of the engine, in order to obtain the proposed effects.
As the quantity S represents here the effective evaporation disposable by the engine, it is understood that, if the usual construction of the engines under consideration give rise to a certain loss of steam, either by safety-valves or otherwise, account of this must be taken, with as close an approximation as possible, by first adding that loss to the quantity S deduced from the preceding equation, then by estimating the heating surface suitable to the production of the useful steam augmented by the lost steam. Section V-Of the different expressions of the useful effects of the engines.
1. The useful effect produced by the engine in the unit of time at the velocity, is evidently a rv, since the velocity v is at the same time the space traversed by the piston in a unit of time. Consequently, by multiplying both members of equation (2) by v, we shall have the useful effect;
u. E.=ar v=
This may be expressed in terms of the load, by multipying the two members of equation (1) by a r. We have then for the useful effect the
engine may produce with a given load,
n+q} (1 + d) r+p+f}
. (4 bis)
It will be remarked, that in a given engine this useful effect does not depend on the pressure at which the steam is generated in the boiler, since the quantity P does not appear in the above equations; but that it depends essentially on the evaporation S effected by the boiler in a unit of time.
2. If it be required to know the horse-power which represents the effect of the engine, when working at the velocity v, or when loaded with the resistance r, it suffices to observe that what is called one-horse-power represents an effect of 33,000 lbs. raised one foot per minute. All consists then in referring the useful effect produced by the engine in the unit of time, to the new measure just chosen, that is, to the power of one horse; and consequently it will suffice to divide the expression already obtained in equation (4) by 33,000.
Thus the horse-power of the engine, at the velocity v, or with the resistance r, will be
We will here observe, that what is designated by horse-power would, with much more propriety, be termed horse-effect, since it is an effect and not a force. It should then be said, that an engine is of so many horse-effect, instead of saying that it is of so many horse-power.
3. In the two preceding questions, we have expressed the power of the engine from the total effect it is capable of developing, without regard to its consumption of fuel or water. We are now about to express the same, either from the effect it produces per unit of fuel or of water expended; or from its consumption while performing a given work.
The useful effect obtained in equation (4), is that which is produced by the volume S of water transformed into steam; and as that volume of water S is evaporated in a unit of a time, the result is, as has been said, the useful effect produced by the engine in a unit of time. But if it be supposed that during the unit of time, there be consumed N pounds of fuel, the useful effect produced by each pound of fuel will plainly be the Nth. part of the above effect.
Hence the effect arising from the consumption of 1 lb. of fuel will be
To apply this formula, it suffices to know the quantity of fuel consumed in the furnace per minute, that is to say, while the evaporation S is taking place. This datum may be determined by a direct experiment on the boiler itself, or by analogy with other boilers similarly disposed. And the datum once obtained, may be used for every other case, and for every supposition of velocity of the engine.
4. We have seen above, that the effect indicated by u. E. is that which is due to the volume of water S transformed into steam. If then it be required to know the useful effect arising from each cubic foot of water, or from each unit of the volume S, it will obviously suffice to divide the total effect u. E. by the number of units in S. Thus, for the useful effect due to the evaporation of one cubic foot of water in the engine, we have
u. E. 1 ft. wa.=
5. we have obtained above the useful effect produced by one pound of fuel. It consequently becomes easy to know the number of pounds of fuel which represent any given useful effect, as, for instance, one horse-power. A simple proportion is, in fact, enough, and we have for the quantity, in weight, of fuel requisite to produce one horse power,
6. By a simple proportion will also be found the quantity of water that must be evaporated, in order to produce one horse-power, viz.:
Q. wa. for 1 hp.=
7. It may yet be required to know what horse-power will be produced by a pound of coal; which will evidently be
u. HP. for 1 lb. co.
u. Ē. 33000 N
8. Finally, the horse-power produced by the evaporation of 1 cubic foot of water will likewise be
u. HP. for 1 ft. wa.=
u. E. 33000 S
Substituting, then, in these several equations for u. E. its value determined by the formula, (4,) we immediately deduce the numerical solution of the proposed problems.
Sec. VI.- Table for the numericte solution of the formula (rotative engines.)
As the formulæ we have just obtained, and those which are about to follow, contain hyperbolic logarithms, the use of which is inconvenient, we here subjoin a table which gives, without calculation, the principal elements of the equations, and will greatly simplify the matter.
In this table we have supposed the clearance of the cylinder c=05 1, as is the case in rotative steam-engines, of which we are now treating. In single-acting engines the clearance of the cylinder, including the adjoining passages, amounts to 1 of the stroke, because the motion of the piston not being limited by a crank, is it more liable to strike the bottom of the cylinder.
We have not inserted in the table a column to represent the fraction
ľ + c Τ
because it is evident that being known by the first column, the fraction
will be equal to the former augmented by, that is, by 05.
Table for the numerical solution of the formula (rotative engines.)
We limit ourselves to the preceding problems, because they are those which are most commonly wanted; but it is obvious that, by means of the same general analogies, any one of the quantities which appear in the problem may be determined, in case that quantity should be unknown, and that it were desired to determine it according to a given condition. Thus, for instance, might be determined the area of the piston, or the pressure in the boiler, or the pressure of condensation, &c, corresponding to given effects of the engine, as we have done for locomotive engines, in a preceding work (TREATISE ON LOCOMOTIVES.) But as these questions rarely occur, and as they offer no difficulty, we deem it sufficient to indicate here the manner of obtaining their solution.
OF THE MAXIMUM OF USEFUL EFFECT WITH A GIVEN EXPANSION.
Section I-Of the velocity of maximum useful effect.
The preceding problems have been solved in the most general way, tha is to say, supposing the engine to set in motion any load whatever at any velocity whatever, with the single condition that the load and velocity be compatible with the power of the engine. In constructing an engine for a determined object, or to move a certain load with a given velocity, it must not be planned in such manner as to require the greatest effort of which it is capable, to perform that task which is to be its regular work; for in that case, it would have no power in reserve, to meet whatever emergencies may occur in the service. On the other hand, since the maximum effort of the engine with a given expansion, corresponds, as we shall presently see, to its maximum useful effect, it follows that we are not to expect regularly from the engine its maximum of useful effect, nor can the engine be constructed with such pre-intention. It is necessary, however, when an engine is constructed, or to be constructed, to know what is the velocity at which it will produce its maximum useful effect, and what this maximum useful effect will be; for it is evidently that knowledge which must decide the regular working load of the engine, and mark the possible limits of its effects in case of emergency.
What is that velocity or that load, most advantageous for the work, and what are the divers effects which will then be produced by the engine? This is what now remains to determine, first, in supposing the expansion of the engine fixed a priori, that in making that expansion itself to vary, in order to obtain a further increase of effect.
To know the velocity corresponding to the greatest useful effect, it suffices to examine the expression of the useful effect produced by the engine under any velocity whatever, namely, (equa. 4):
It is observable here, at the first glance, that since the velocity enters only into the negative terms, the less that velocity is, for a given expansion, the greater will be the useful effect of the engine. On the other hand, referring to the expression of the velocity of the engine under a given load, before having substituted for P' its numerical value, viz. (equa. B):
we perceive that the velocity is the smallest possible, without loss of steam, when P' is the greatest; and as P', which is the pressure of the steam in the cylinder, can in no case exceed P, which is the pressure in the boiler, the condition of the minimum velocity, or of the maximum useful effect, will be given by the equation P'=P, or
Expressing by m the volume of the steam under the pressure P, referred to the volume of the same weight of water, this formula may, from equation (a) take the form
In this manner the calculation of the term (n+qP) is avoided, since the quantity m is given by the tables of Chapter II., and may thence be taken with greater accuracy than from its approximative value
This observation will equally apply to all the following formula, wherein the quantities n and q recur united under the form (n+qP).
It is to be remarked, with respect to the preceding formula, that, mathematically speaking, the pressure P' can never be quite equal to P. In fact, since there exist pipes between the boiler and cylinder, through which the steam must pass, and that the passages of those pipes form an obstacle to the free motion of the steam, there must necessarily be, on the side of the boiler, a small surplus of pressure equivalent to the resistance of the obstacle in question; otherwise the motion of the steam could not take place. This surplus of pressure, then, on the side of the boiler, prevents P' from becoming mathematically equal to P, and thus the real volocity will always be rather greater than v'. The difference between P' and P (we mean the difference merely arising from the obstacle just mentioned) will be by so much the less, as the area of the passages is larger and their way more direct; but as, with the dimensions of ordinary use in steam-engines, that difference is very trifling, we shall not notice it here. Seeking it, in fact, by known formulæ for the flowing of gasses, we find that it is hardly appreciable by the instruments used for measuring the pressure in the boiler; consequently, to introduce them into the calculation would only complicate the formulæ, without rendering them more exact.
To return to the inquiry before us, the maximum useful effect will be given by the condition P'=P, or
This is, then, the veiocity at which the engine must work, in order to obtain the greatest effect possible; and the equation P'P shows reciprocally. that, when that velocity takes place, the steam enters the cylinder at full pressure, that is, nearly at the same pressure which it had when in the boiler.
(To be continued.)