250 ON THE MECHANICAL THEORY OF HEAT. Let the mass μ, which is not forced back into the boiler, be first cooled in the liquid condition temperature let the part μ, be from T, to T, and then at this transformed into steam; the piston receding so far, that this steam can again occupy its original space. Thus the mass M+μ will have passed through a complete cyclical process, to which we may now apply the principle that the sum of all the quantities of heat taken in during a cyclical process must be equal to the total external work done. In this case the following quantities of heat have been taken in: (1) In the boiler, where the mass M has been heated from T to T, and at the latter temperature the part m transformed into steam, we have the quantity of heat m1p1 + MC (T, − T). (2) During the condensation of m, at temperature T2, we have (3) During the cooling of the part μ from T to To, we have -μC (T2-T). (4) During the vaporization of the part μ at temperature To, we have Mo Po The whole quantity of heat taken in, which we may call Q, is thus given by Q=m ̧p1− m ̧p2+ MC (T, − T2) + μ。。 — μ С (T2 — T.).....(19). The quantities of work are obtained as follows : (1) To determine the space swept through by the piston during the entrance of the steam, we must remember gaseous condition at the end as at the beginning, we need only assume that the water forced back into the boiler is not only in quantity, but also in its actual molecules, the same as that which left the boiler previously; and that when this water takes up the temperature T1, the quantity m1 which was formerly vapour, is again vaporized, whilst an equal quantity of the existing steam is condensed. For this purpose there is no need that the whole mass in the boiler should take in or give out any heat, because that required for the vaporization, and that generated by the condensation, exactly balance APPLICATION TO THE STEAM-ENGINE. 249 before the entrance of the steam; for the sake of generality we will assume this to be partly in the liquid, partly in the gaseous condition, and will call the latter part μ. The pressure of this steam may for the moment be denoted by po, and the corresponding absolute temperature by Te, without implying thereby that these are exactly the same values as hold for the same quantities within the condenser. Let p1 and T be, as before, the pressure and temperature in the boiler, M the mass which flows from the boiler into the cylinder, and m, the part of M which is in the condition of steam. The pressure exerted on the piston during the entrance of the steam need not, as already explained, be constant. We will call p, the mean pressure, by which the space swept through by the piston during the entrance of the steam must be multiplied, in order to obtain the same amount of work as is actually done by the varying pressure. Let p, be the actual pressure in the cylinder at the moment when the steam is cut off, T, the corresponding temperature. Lastly let m, be the magnitude we have to determine, viz. the part of the whole mass M+μ within the cylinder, which is in the condition of steam. be To determine this quantity, let us suppose the mass M+μ to be brought back to its initial condition in any way whatever, e. g. the following. Let the gaseous part m condensed in the cylinder by the fall of the piston, it being assumed that the piston can force itself even into the waste space; at the same time let the mass have such a quantity of heat continuously imparted to it, that its temperature T, remains constant. Then let the part M of the whole liquid mass be forced back into the boiler, where it again takes up its original temperature T We have now within the boiler the same condition of things as before the flow of the steam, since every part of the boiler has its original temperature; and therefore the proportions of liquid and steam must be the same as at the commencement. Whether the individual molecules, which are in the gaseous and in the liquid condition, are exactly the same as at the commencement does not concern us; we make no distinction between these, and never enquire what molecules, but simply how many molecules, are in each of the two conditions.* * If it be wished that exactly the same molecules should be in the 218 ON THE MECHANICAL THEORY OF HEAT. large, that the pressure in the cylinder is as great as that in the boiler. This then gives the maximum quantity of work. If with an equal admission of steam the speed of the engine is greater, or if with equal speed the admission of steam is less, then in each case a less quantity of work is obtained from the same quantity of steam. § 10. Changes in the Steam during its passage from the Boiler into the Cylinder. Before we pass on to treat the same connected series of processes on the principles of the Mechanical Theory of Heat, it will be advantageous to consider one of them, which requires a special investigation of its own, in order to fix beforehand the results which refer to it. This process is the flow of the steam into the clearance or waste space and into the cylinder, in the case when it has a smaller pressure to overcome than that which forces it out of the boiler. the space The steam as it comes from the boiler passes first into the waste space; there it compresses the steam of less density which remains over from the last stroke, fills up thus obtained, and then acts upon the piston; this, according to the assumption, on account of its comparatively lighter load recedes so fast, that the steam cannot follow it quickly enough to keep the density in the cylinder the same as in the boiler. Under such circumstances, if nothing but saturated steam escaped from the boiler, this would become superheated in the cylinder, inasmuch as the vis viva of the flow would be transformed into heat; but since the steam always carries with it small particles of water, the superabundant heat goes to vaporize a part of these, and the steam thus remains in the saturated condition. We must now consider the following problem: Given the initial condition of the whole mass under consideration, as well that already found in the waste space as that which is newly received from the boiler; given also the amount of work which is done during the entrance of the steam by the pressure which acts on the piston; lastly given the pressure which exists at the moment when the boiler is shut off from the cylinder; then to determine what proportion of the mass within the cylinder is at that moment in the condition of steam. APPLICATION TO THE STEAM-ENGINE. 247 expression is almost too small to be taken into account, à fortiori we may neglect an error which is small even in comparison with that value; and we shall therefore retain the expression in the form given above. Adding these four several quantities of work together, we obtain the following expression for the whole work done during the cyclical e process: W = mB ( ~ ~ 2 + log - ) − v' (1 − e) (b + p.) − Mo (P1 − P.) e .........(17). § 9. Pambour's Value for the Work done per Unit-weight of Steam. If instead of the work done in one stroke, during which the quantity of steam used is m, we prefer to find the work done per unit-weight of steam, all that is needed is to divide the foregoing value by m. Let us denote by the fraction M m , which gives the ratio of the whole mass which passes into the cylinder to that part of it which is in the form of steam, and is therefore somewhat greater than 1; by V the fraction , i.e. the space which on the whole is occupied by the v m unit-weight of steam in the cylinder; and by W the fraction -, i.e. the work done per unit-weight of steam. Then we W m + log 1) − V (1 − e) (b + p.) − lo (P1−P.) ......(18). In this equation there is only one term which involves the volume V, and this contains V as a factor. Since this term is negative, it follows that the work, which can be obtained from one unit-weight of steam, is the greatest, other things being equal, when the volume which that steam occupies in the cylinder is the least possible. The least value of this volume, to which we may continually approximate, but can never exactly obtain, is that which is given by the assumption that either the engine goes so slowly, or the steam-pipe is so 246 ON THE MECHANICAL THEORY OF HEAT. space swept through by the piston up to this moment, we have the following equation for the first part of the work done: The law of variation of the pressure during the expansion is also given by equation (11). If v is the volume and p the pressure at any moment, then This expression we must substitute in fpdv, and then integrate this from v ev to v=v'. Thence we obtain for the second part of the work done = Next, to determine the negative work done by the resistance during the return stroke, we must know the value of that resistance. Without entering at present into the question how this resistance is related to the pressure in the condenser, we will denote the mean pressure by P.; then the work done will be given by Finally there remains the work which must be expended in forcing back into the boiler the quantity of liquid M. Pambour has taken no special account of this work, but included it with the friction of the engine. Since, however, for the sake of completing the cycle of operations, it has been included in the author's formulae, it will be investigated here in order to facilitate the comparison. If p, be the pressure in the boiler, and p, in the condenser, then equations (4) and (5) show, as in the example already considered, that this work is on the whole given by ....(16). For the present case, where p, is not the pressure in the condenser itself, but in the end of the cylinder which is open to the condenser, this equation is not quite exact; but since on account of the smallness of a the value of the whole |