§ 14. Substitution of the Volume for the corresponding Temperature in certain cases.

In the above equations it is assumed that, besides the masses M, m1, μ, and μ, of which the two first must be found by direct observation, and the two latter can be approximately determined from the size of the waste space, we have also given the four pressures p1, P2, P., and P., or, which is the same thing, the four temperatures T,, T,, T, and T. In practice however this condition is only partially fulfilled, and we must therefore bring in other data to assist us.

Po

Of the four pressures here mentioned, two only, p1 and may be taken as known of these the first is given directly by the gauge on the boiler, and the latter can be at least approximately fixed by the gauge on the condenser. The two others, p, and p,, are not given directly; but we know the dimensions of the cylinder and the point of cut-off, and can thence deduce the volume of the steam at the moment of cut-off and at the end of the stroke. We may then take these volumes as our data in place of the pressures p2 and P3. For this purpose we must throw the equations into a form, which will enable us to use the above data for calculation.

Let us now, as before in the explanation of Pambour's theory, denote by the whole space within the cylinder which is open to the steain during one stroke, including the waste space; by ev' the space opened to the steam up to the moment of cut-off; and by ev′ the waste space. Then in the same way as before we have the following equations:

The quantities μ and σ are both so small that their product may be at once neglected; whence we have

Further, we have by equation (13) of Chapter VI.

p = Tug,

dp dT'

.(29).

which will occur very

writing the single letter g for frequently in what follows. We can therefore replace p, and Pa in equations (28) by u, and u,. Then the masses m, and m, occur only in the products mu, and m,u,, for which we may substitute the values given in the two first of equations (29). Again, by the last of these equations we may eliminate the mass ; and as regards the other mass μ, though this may be somewhat larger than μ, yet, since the terms which contain u are generally insignificant, we can without serious error use the value found for μ in other words we may drop for the purposes of calculation the assumption made for the sake of generality, that the original mass in the waste space is partly liquid and partly gaseous, and consider the whole of it to be in the gaseous form.

The above substitutions may be effected in the more general equations (27), as well as in the simplified equations (28). This substitution presents no difficulty, and we will here confine ourselves to the latter set, in order to have the equations in a form adapted for calculation. With these changes they read as follows:

W' = m ̧p1+MC (T, — T ̧) — (v' — Mo) (T391⁄2 −P3+Po)

+ev' Po-C(TT)

ευ

Ио

(ev′

Ио

+P2-Po+Mo (P1 — P2)

(e' — Me) 9, = (er' — Me) 9, + (M + Clog

(30).

§ 15. Work per Unit-Weight of Steam.

To adapt the above equations, which give the work per stroke or per quantity of steam m,, to determine the work per unit-weight of steam, we must apply the same method as before in transforming equation (17) into (18): viz. to divide the three equations by m,, and then put

W=p2+1C (T1 - T3) — (V—lo) (T393 — P3 + Po)

2

(eV−lo) T12 = P1 + IC (T, — T2)

-

+ € Vo― C (T, − T)

И

Ио

(31).

The application of the above equations to calculate the work done may be effected as follows. Assuming the pressure of steam to be known, and also the speed at which the engine works, we can thence determine the volume V of one unit-weight of steam. By help of this value we first calculate the temperature T, from the second equation, then T from the third, and finally apply these to determine the work done from the first equation.

3.

Here however we are met by another difficulty. In order to calculate T, and T, from the two latter equations, these must be solved for those temperatures. But they contain those temperatures not only explicitly but also implicitly, since p and g are functions of temperature. If to eliminate these quantities we use for p one of the ordinary empirical expressions for the pressure as a function of temperature, and for g the differential coefficient of the same expression, then the equations become too complicated for

further treatment. It would probably be possible to adopt the same method as Pambour, and to form new empirical formulæ, more convenient for the present purpose, and sufficiently accurate, if not for all temperatures at least within certain limits. This however will not here be attempted, but another method will be adopted instead, which makes the calculation somewhat extended, but easily performed in its individual parts.

§ 17. Determination of dp or g, and of the Product Tg.

dt

If the range of pressure of the vapour at different temperatures is known with sufficient accuracy for any liquid, then the values of g and Tg can also be calculated for the same temperatures, and collected in tables, as usually done with the values of p. For steam, which has hitherto been the only vapour used for the steam-engine, the author has performed such a calculation, by help of Regnault's tables, for temperatures from 0° to 200°. For this purpose he differentiated according to t the formula used by Regnault to calculate the values of p under and above 100°; and then calculated g by means of the new formulæ thus obtained. But since these formulæ did not seem to answer the purpose so well as to repay the great labour involved, and since the forming and calculating of other more suitable formulæ proved still more tedious, the author was contented to use the numbers already calculated for the pressure to determine approximately the differential coefficient of that pressure: e.g., if the pressure for the temperatures 146° and 148° were P148 P146 denoted by P and Pig, it was assumed that P146 P148 146 would 2 represent with sufficient accuracy the value of the differential coefficient for the mean temperature 147°.

Above 100° the same numbers were employed as were used by Regnault*. With regard to values under 100° it has been recently pointed out by Moritz † that the formula used by Regnault is somewhat inaccurate, especially in the neigh

* Mém. de l'Acad. des Sciences, Vol. xxi. p. 625.

+ Bulletin de la Classe physico-mathématique de l'Acad. de St Pétersbourg. Vol. XII. p. 41.

bourhood of 100°, owing to his having employed for the calculation of the constants logarithms to seven places of decimals only. Moritz has therefore recalculated these constants from the same observed values, but using ten places of logarithms, and has published the values of p derived from this improved formula, so far as they differ from Regnault's values, which first takes place at 40°. These are the values used by the author*.

When g has been calculated for the various temperatures, the product Tg can be calculated without further difficulty, since T is given by the simple equation

T= 273+t.

The values of g and Tg thus found are given in a table at the end of this Chapter. To complete this the corresponding values of p are added, those from 0° to 40° and above 100° being Regnault's numbers, and those from 40° to 100° being Moritz's. By the side of each of these three columns are given the differences between each two successive numbers, so that this table enables the values of these three quantities to be found for any given temperature, and conversely the temperature corresponding to any given value of one of these three quantities.

§ 18. Introduction of other Measures of Pressure and Heat.

One other remark is to be made as to the mode of using this table. In equations (31) it is assumed that the pressure p and its differential coefficient g are expressed in kilograms per square metre, whereas in the table the same unit of pressure is employed as in Regnault's tables, viz. a millimetre of mercury. Hence, if in the formulæ which follow we are to consider p and g as expressed in these latter units, we must alter equations (31) by multiplying p and g by the number 13.596 which expresses the specific weight of mercury; inasmuch as this, by Chapter VI., § 10, is the ratio between the units. If we denote this number by k, we must substitute kp and kg for p and g, whenever these occur in the above equations. Similarly for the quantities C and P, which express the specific heat and the heat of vaporization in

* See note at end of chapter.