In equation (36) the first term is again the most important, and we can therefore proceed by successive approximations. First write T2 for T; we then have as the first approximation for 9,

g' = C..

The corresponding temperature t' can be found from the table, and from this we can easily obtain the absolute temperature T". Write T" for T, in equation (36); then we have

3

This gives T". Similarly we can obtain the equation

T"

g"" =g" + a log

T"

(376),

which again gives T"; and so on. Here again a very few approximations will give a value which very closely agrees with the actual value of T.

§ 20. Determination of c and r.

It now only remains to determine the quantities c and r, before proceeding to the numerical application of equations (32).

If

The quantity c, or the specific heat of the liquid, has hitherto been treated as constant. This is not perfectly correct, since it increases slightly as the temperature increases. however we take as its general value the correct value for the mean of the temperatures which occur in the investigation, the divergencies may be neglected. This mean temperature in the case of engines driven by steam may be taken at 100o, which for an ordinary high-pressure condensing engine is about a mean between the temperature of the boiler and that of the condenser. Taking therefore the value of the specific heat which Regnault gives for water at 100°, we may put

To determiner we start from the equation which Regnault gives for the whole quantity of heat required to heat a unit-weight of water from 0° to the temperature t, and to transform it at that temperature into steam. This equation is

λ = 606·5 + 0.305t.

In the integral we must use for c the function of temperature exactly determined by Regnault*, if we are to obtain for r the precise values which Regnault gives. For the present purpose it is perhaps sufficient here also to use for c the constant above-mentioned. This gives

cdt = 1.013t;

and the two terms involving t in the last equation can now be combined into one, viz. — 0.708t. We must at the same time make some change in the constant term of the equation. This we will so determine, that the same value of r which is probably the most accurate of all given by observation, shall also be correct as given by the formula.

Now

at 100° Regnault found for A, as the mean of 38 observations, the value 636 67. If from this we subtract the quantity of heat required to heat the unit-weight of water from 0° to 100°, which according to Regnault is 100-5 heat-units, there remains to one place of decimals only

100=536.2+.

Applying this value we obtain for the formula

r=607-0-708t..

....

(39).

This formula is already laid down in Chapter VII., § 3 ; and a short table is there given, which exhibits the close accordance between the values of r as calculated by this formula, and as given by Regnault in his tables.

§ 21. Special Form of Equation (32) for an Engine working without expansion.

In order to distinguish the effects of the two different kinds of expansive action, to which the two latter of equa

* Relation des expériences, Vol. 1. p. 748.

+ Regnault himself used in his tables the number 536.5 instead of the above; this simply arises from the fact that he has used the round number 637 for X at 100°, instead of 636.67 as given above.

tions (32) refer, it seems desirable first to consider an engine in which only one of the two occurs. We will therefore begin with an engine that works without expansion. In this case we may substitute for e, or the ratio of the volumes before and after expansion, the value 1; and may also put T. The equations (32) are thus greatly simplified. The last becomes an identity, and is therefore useless. In the second the right-hand side remains unaltered, whilst the left-hand side becomes (V-lo) Tg. Finally the first equation takes the form

W k

=

{r ̧ + lc (T ̧ − T2)} − ( V — lo) (T291⁄2 — P2+Po)

If we here replace (V-lo) T2g, by the expression on the right-hand side of the second equation, all the terms conE taining as factor cancel each other, as do two terms conk

taining lo as factor, and the terms which remain can be collected together as two products. Then the two equations

v

and =

(E

k

lc

le (T, − T)}

(40).

The first of these equations is exactly the same as is given by Pambour's theory, if in equation (18) we put e=1, and then by means of equation (12) (having first put e=1 V in that equation) replace B by the volume V. The only difference therefore is in the second equation, which replaces the simple relation between volume and pressure assumed by Pambour,

m

§ 22. Numerical Values of the Constants.

The quantity e which occurs in these equations, and represents the ratio of the waste space to the whole space left open to the steam, may be taken as 0.05. The quantity of water which the steam carries over with it into the cylinder is different in different engines. Pambour considers that it averages 0-25 of the whole mass entering the cylinder in the case of locomotives, but much less, 0.05, in the case of stationary engines. We will here take the latter value, which gives 1:095 as the ratio of the whole mass entering the cylinder to that part of it which is in the form of steam. Further, let the pressure in the boiler, or p,, be five atmospheres, to which corresponds the temperature 152.22°; and let it be supposed that the engine has no condenser, or, which is the same thing, a condenser at the pressure of one atmosphere. The mean back-pressure in the cylinder is then greater than one atmosphere. With locomotives this excess of backpressure may be considerable, for the special reasons already mentioned; with stationary engines it is insignificant. Pambour, in his tables for stationary non-condensing engines, has altogether neglected this excess; and since our present object is to compare the new formula with his, we will follow his example and put p, one atmosphere.

=

We have then in this case the following values to insert in equations (40):

We may also fix once for all the following values:

k = 13.596,

σ = 0·001.

There now remain only the quantities V and p, undetermined in equations (40), in addition to the quantity W, of which we are seeking the value. Of these Vis the volume of steam in the cylinder per unit-weight, and p, is the final pressure within the cylinder.

§ 23. The least possible Value of V, and the corresponding amount of Work.

We must first enquire what is the least possible value of V. This value corresponds to the case in which the pressure in the cylinder is the same as in the boiler, and we have therefore only to substitute p, for p, in the last of equations (40). Thus we obtain

In order to give an example of the influence of the waste space, two values have been calculated from this expression; the first that which would exist if there were no waste space, and therefore e = 0; the second that which exists upon our assumption that e=0.05. These two values expressed in cubic metres per kilogram of steam passing out of the boiler are 03637 and 0.3690.

The reason why the second value is greater than the first is that the steam rushes at first into the waste space with great velocity; the vis viva of this motion is then transformed into heat, and this heat again evaporates some part of the priming water. A second reason is that the steam already existing in the waste space before the admission goes to increase the whole quantity of steam during the rest of the process.

If we substitute these two values of V in the first of equations (40), at the same time putting e=0 in the one case, and e=0.05 in the other, the corresponding quantities of work, expressed in kilogrammetres, are respectively 14990 and 14450.

According to Pambour's theory it makes no difference with regard to the volume, whether part of it is waste space or not; in both cases it is given by the same equation (116), if for p is substituted the special value p1. The value thus obtained for the volume is 0.3883. The fact that this value is greater than the value 03637 found above for the same quantity of steam, is explained by the circumstance that it has been until now usual to assume for steam at its maxi