1.50 



THE CIVIL ENCINKKK AM) \ K( 1 1 11 ICC r S .lOLKX AL 



[M.' 



A V 



To show nimieiically liy !io\v iinic.li (lie pressuio p of tlie sleam in 

 the cvlinder may iail sliuVt of P, whiili is its pressure in the ste.iin 

 iLissnees, we shall apjilv these forinn'a- to one or two examples, when 

 we shall also show thai the error introdnccil by neglecting the (lilferencc 

 hetween r' aiid r iloes not ;xinount to so much as one Inmilreclth part 

 i)f a pound, whether the steam be used at a high or low pressure, pro- 

 \ ided the area of the steam passages be not excessively small, nor the 

 velocity of the piston verv great. 



As a first exam]i!e let P = H-71, >ii — ia, and V = 210. The 

 temperature of the steam in the passages is in this case 212 degrees, 

 which gives T = (JilU, and r' — 1700. 



Having substituted lliese values, we find 



V—lj = -OOliya P = •0'J751I)., 

 whence 



;; = •99337 P = H-l>125 lbs. 



The relative vuluinc of steam of this elastic force i> 1711, which 



makes 



= •yb7"J, and if we imillijily the abu\e value of P — /"by 



this fraction, we shall obtain 



P— y/= •(>9i3lb., 

 w hicli gives 



J, = n-:il37lbs., 



which (liU'crs from tin' former value by no more than ■iiiil:7 lh>., uhich 

 is a negligeable (|iraiitity. 



As an exani|ile of exces.rively high jiressure steam, let P - 130'93, 

 and III and V the same as in the lormcr example. In tliis case we 

 have T = 7;N and r' = 230-9. 

 From formula (/i) we obtain 



V—p = -005484 P = -7 isi lb. 

 whence 



;;= 130-2 12 lbs. 



Tlie relative volume of steam of this elastic force is 2321, so that 



(' - 

 r 



-C897, and, multiplying liy this fraction the value of — y just 



obtained, the latter becomes 



P — 7; = -711 lbs., 

 wlieice 



;; = 130-2 19 lbs. 



V ' 



The error introduced by neglecting the fraction , is therefore also 



ill this case too small to be worth taking account of, so that we may 

 alwavs content ourselves witli formula (6), when we wish to ascertain 

 llie loss of ))rcssure which the steam sutli-rs in passing through the 

 steam port into the cylin<ler of a steam engine. 



( )n referring to ecpiation (6), it will be seen that the loss of pressure 

 which the steam suiters in passing through the port into the cyliniler 

 Varies dirictly as the square of the velocity of the piston, and as the 

 s(|uare of the ratio of the area of the piston to that of the steam port, 

 and /;H-(;sf /y as the number of degrees by which the temperature of 

 the steam in the steam jia'^s ages exceeds — US degrees Kalir., which 

 shows that, the higher the pressure of the steam used, the less is the 

 comparative loss in passing through the port, and, the greater the 

 \elocitv of the ])iston, (he larger the steam port must be in the same 

 proportion, that llie loss of pressure may be the same. 



Wc assumed a rather considerable value for V in the alxive calcula- 

 tions, in order to show more satisfactorily how trilling is the error 

 which can be committed in deducing the elastic tone of the steam in 

 the cylinder from that in the steam jiassages. By making V ^^ 210 

 Icct per minute, which is t!ie spt ed usually given to the pi>lon of an 

 engine, instead of 2 10, which we assumed above, the value of P — ji 

 will be reduced in the ntio of 210- to -240', or 19 to 114. When 

 therefore the area of the steam ] ort is one 2.')tli part cd' that of the 

 piston, and the mean velocity of the piston is about 210 feet per 

 minute, we may assume, as an average for low pressure engines, 



P —p = -005 P, 



;;= -995 P; 

 and for high pressure engines, 



P— ;; = •004(; P, 

 o.- 



J, - -9954 P. 



It is a very good ]ilaii In lix a sti am guage on to the slide box, or 

 steam pipe very near the cylii der, as that dispenses with the calcula- 



tion of the loss of elastic force esperienccil by the steam during its 

 passage llirough the steam pipe, before it arrives at the slide box. 



\Vhen speaking of the lead of the s'ide necessary to allow the waste 

 space at tin- end of the c)iinder to be filled with steam before the be- 

 ginning of the stroke of the |iiston, we said we should prove it to be 

 excessively small. The calculation of the exact lead re(]nired for that 

 purpose is verv long anil dillicult, involving integrals of a very com- 

 plicated nature; but it will an-wer our juirpose ciiually well to prove 

 it for a greater lead than necessary, for it wiU tlien be proved a fortiori 

 for the necessary lead. 



Let P be the elastic force and D the density of the steam in the 

 steam pipe, and let H = (he height of a column of the same steam 

 whose weight is equivalent to its pressure. Also let// be the elastic 

 force, and S the density of »he steam in the waste space when the 

 port is open to a certain degree, a the area of the orifice at that mo- 

 ment, rtlie velocity of the steam through it, aiuWytlie volume of steam 

 of the density D which has passed through the pott, and let d be the 

 density of the steam in the condenser, and consequently a'so in the 

 waste space before the ])ort has begun to open. In the ease of non- 

 condensing engines d is equal to the density of atmospheric steam, or 

 1. Also let c be the contents of the waste space, A the area of the 



piston, L the length of the stroke, and - the ratio of the area of the 



steam port to that of the jiiston. 



The height of the column of steam ecjuivalent to the pressure P — 1>, 

 to wliich the flowing of the steam througli the port is due, is eijual to 



H ( 1 — - y the velocity will therefore be eipial to 



V-^^-(-'p)- 



But this formula would lead to very complicated calculations, as we 

 have already observed, for which reason we shall substitute the frac- 

 tion for ^\, which w ill render the case less favourable ; for the former 



being greater than the latter, the factor ('—,)) '» '•''■s 'h.m 



(.-;;). . 



herefore also the value of r will be less after the sub- 



stitution than before, and consequently the lime re(|uirpd to ra 

 pressure of the steim in (lie waste space to the maximum \\\ 

 attains in the cvlinder w ill appear greater than it really is. If 

 fore we can prove this (o be exceedingly slior(, it will be demons 

 a fordori lor (he true time. We shall therefore assume, in p 

 the above equation 



ise (he 

 licli it 

 (hcie- 

 .( rated 

 ace of 



r 



a/-^^'"\/^ 



S 



V-'.V"-" "• 



We have also bclwecn (he vaiiible (juantities q and 5 (he fi>llo«ing 

 rel.idon 



whence 



And, ly diirrientiation, 



But we have also 





dif. 7= ' dif. 5. 



dif. (y =r a (1 dif. I, 



where ilif. / is the infinitely small space of time during which the in- 

 finitely sm.dl (piantity (d' steam dil. 5 of the density D passes through 

 the orifice o. These (Wo cipiations, having their first ineinbcis cipial, 

 give 



-'" dif. S^- av dif. /. (2.) 



Let 6 c r -present the area of the steam port when bill open, 6 being its 

 constant length and ;■ the greatest width to which it is opened by the 



