. The steam-engine and other heat-motors. ript is used thequantity to which it is added is no longer variable, for it denotesa fixed value for that problem. Pi is the admission pressurein this discussion and could not be used for any value of thepressure during the expansion. On the contrary, take any pointon the expansion curve, the ordinate at that point measures Pand the abscissa measures its volume, V. Hence PiT^ and P2V2are specific values of the general formula PV. As the area ofthe piston is a common multiplier to all parts of the stroke, itis evident that if we take an infinitesimal pa
. The steam-engine and other heat-motors. ript is used thequantity to which it is added is no longer variable, for it denotesa fixed value for that problem. Pi is the admission pressurein this discussion and could not be used for any value of thepressure during the expansion. On the contrary, take any pointon the expansion curve, the ordinate at that point measures Pand the abscissa measures its volume, V. Hence PiT^ and P2V2are specific values of the general formula PV. As the area ofthe piston is a common multiplier to all parts of the stroke, itis evident that if we take an infinitesimal part of the stroke it 78 THE STEAM-ENGINE AND OTHER HEAT-MOTORS. becomes an infinitesimal part of the volume when multipliedby the area of the piston. It may be called dV and when mul-tiplied by P it becomes work. It could be written (PA)dL,where PA is equal to the total pressure and dL is in work done during admission is P{Vi. The work done during expansion is / PdV. J yl But P varies with V, and to integrate we must have but one. Fig. 49. variable and that must be V. The variable P must thereforebe expressed in terms of the variable V. But PlVl=PV, therefore P^-y1- Substitute this value of P and we obtain for the area BCDEunder the expansion curve =PiV1 (log, V2 -log, V,) -P{Vi log, yx =piyi lo& n CURVES AXD THE WORK OF EXPANSION. 79 As r is the ratio of the final absolute volume to the initial ab-solute volume of the gas it is called the ratio of expansion. Log, r is an abstract quantity and the expression PiVi that the work done during expansion is () times thework of admission. The subscript e denotes that a table ofNaperian or hyperbolic logarithms must be used (Table II). The total gross forward work, HABCDG, is then PXVX + PVl log, r=PiV1 (1 + log, r). The mean gross forward pressure would be found by dividingby V2, PiTVl+) PtCl+) l2 r ~Fm The quantity that is actually desired is the mean pressureof the diagram ABCRLMA, which would be t
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