1839.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



423 



ON THE THEORY OF THE STEAM-ENGINE. 



By Aeistides Mornay, Esq. 



No. IV. 



In our last paper, published in June, we proposed the following 

 ibrimila for calculating the elastic force of steam at dillcrent tempe- 

 ratures : — 



log. ^=: log. (^/+41S) -f 



5(1 — 212) 



1.3124227. 



ail). 



t+U8 



This equation coincides, as we have shown, more geni'rally tliiui 

 any other which has been proposed witji the results of exi)erimenl ; 

 besides whicli i( possesses the advantage, that ue can obtain froni it 

 the elastic force of steam in fernis of its density. The ecpialion (IIj 



^■. d = 



can be put imder the form 



log. rf=5 — 

 whence we find 



< + 448 = 



5 (< — 212) 

 ^+448 ~ 



3300 



I + 4 IS 

 3300 



5 — log. d' 



Substituting this value in the equation (I), wliich is 



il (/+44S) 



we obtain 



P = 



p — 



22 

 150^ 



(IV). 



■ log. d 



Thus we can, at any time, deduce the elastic force of steam from 

 its density by a very simple method. Or it may be more convenient 

 to use the volume occupied by a given volume of water, when con- 

 verted into steam, instead of its density. For this purpose we must 

 substitute for (/ some function of (lie volume V, occupied by one cubic 

 foot of water converted into steam. Now, when the density of steam 

 is 1, it occupies a volume =:; 1700 cubic feel, so that the value of d 



would be (/ ;= —,,--, and the above equation would become 



V 

 P = 



255,000 



V (5 — log. 1700-1- log. V.) 

 235,000 



(V). 



'' V(log. V-f 1-7G95511) 

 If P be any other elastic force, and V the corresponding volume, 

 we shall have 



p _ V'(log . V + 1-7(59 5511) 

 P "~ V {logTv + l-70y5511) 



(VI). 



This equation will furnish us with tlie means of calculating the 

 mean pressure of the steam on the ])iston of an engine, in wliich the 

 steam is used expansively ; but we shall return to this subject when 

 we treat of the action of the steam in that variety of engine. 



Uii the Action of the Steam in the Cytiiukr of a Sttam-tngine. 



The whole resistance overcome by the steam acting on tlie piston 

 of an engine, may be divided into the use/id effect and the mcidtiitai 

 7-esistaiicts, the latter comprising the friction of the various parts of 

 the engine, and the resistance of the steam on the opposite side of the 

 piston. When we mention the resistance simply, it signifies the tutal 

 nsisla)ice. 



It is self-evident that the pressure of the steam againist the pi«lon 

 must be precisely e((ual to the resistance on the opposite side. In a 

 critical notice of the Coimt do Pambour's theory of the steam-engine, 

 inserted in the September number of this Journal, it was stated tliat 

 the resistance ove*pome by the force exerted by the steam against the 

 piston of an engine in motion is not, as ivsserted by Pambour, strictly 

 ei|ual to its whole elastic force. We shall show that the dilf'erence 

 is too small to be regarded in calculating the eHV-cts of steam-engines. 



Let jLi be the clastic force of the steam in the cylinder, or the pres- 

 sure in tts. which it exerts on each square inch of the interior surface 

 of the cylindei-, and » the velocity of the piston in feet per minute. 



We must, in tlie first place, determine the height of a column of 

 steam of the given clastic force, whose weight is equivalent to its 

 pressure, in order to deduce from it the velocity with which it would 

 flow into a vacuum, or tree from any resistance. Now we know that 

 the height of a column of atmospheric steam, (steam generated under 



the ordinary pressure of the atmosphere, or 14-7 His. per square inch,) 

 whose weight is equivalent to its pressure, is about 58,000 feet, and 

 that the height of the column increases uniformly with the tempe- 

 rature, when the steam is in the saturated state. Thus, if t is the 

 temperature of ste.uii whose elastic force 

 [Kuiding height of the column wiU be 



tiUO 



H r= 58,000 



or, putting Ibr <-l-41S its value fomid above, 



290,000 



is equal to p, the corres- 

 t+U8 



II — 



log. V -i- 1-70955 1 r 

 Now the pressure exerted by the steam against the piston is equal 

 to its whole elastic force, or (he weight of the column II, nanus the 

 weight of the column whose height, which we will call //, is equal to 

 that due to the velocity w of the piston ; for it would require the 

 pressure of that column to give the steam the velocity c, which it 

 must assume in order to follow the piston. Thus, if we call r the re- 

 sistance referred to a square inch of the piston, we shall have 



h 

 r—P—p^ 



But we have also, 



or, substituting for 2 ^ 



3000 X 2 g* 

 its value 04-38, 



231,708 

 Substituting this value, as well as that of H, in the expression of the 

 loss of pressure, which is 



h 



X =z p 



H 



it becomes 



-P 



"' (log. V-}- 1-7(3955 11) 



(VII). 



07,212,720,000 ' 

 To ascertain the mean loss through the whole stroke of the piston, 

 let /J be the length of the crank, V the velocity of the crank-pin, and 

 L the mean loss of pressure ; \ the loss at any given instant, c the 

 velocity of the jiiston, and a the angle described by the crank from its 

 deail centre at (hat instant. 



If we suppose, to simplify llie calculation, that the length of the 

 connecting- rod is infinitely long in comparison with that of the crank, 

 and that tlie latter moves with an uniform velocity, we shall have 



:= V sin. a. 

 Substituting this value in the equation (VII), it becomes 

 V- sin, g' (log. V 4 - 1-7095511) 

 07,212,720";00() 



The distance travelled by the piston iluring an infinitely short 

 period of time is p sin a iio, and the amount of power consumed in 

 producing the motion of the steam during that element of time is, 



, • , p V (log. V-f 1-7G95511) 



A p sin. ada-^z p '- ^--s. — J i „:„ ,, j „ 



' ^ 07,21-2,720,000 "■" '^'^- 



The whole loss during one single stroke is therefore equal to 



9 T - . /' V^ (l og- V+ 1-7095511) /'TT . 



df,L — p -^—- ^-^~—~ J sm.crda. 



II) 



:p- 



— p 



07,212,720,000 



4 p V= (log. V-1- 1-70955 



201,038,160,000 

 whence we deduce 



L _ V- (lo g. V-t- 1 -7 695511) 

 p ~ roO,8 19,080,000 



If V were used to represent the mean velocity of the piston, w; 



(VIII), 



should have to muUi|)ly V- by - by which the last equation would 

 become 



L_ _ V^(lo g. V+l-7(i95511) 

 p ~ 40,860,400,0110 ■ *^'*^'^ 



It is evident that, the greater the velocity of the piston, and the 



• It will be oliscrvcil that we cmiiloy Liic letter r/ to represent the uniform 

 aocclciatiou \iei seaond, wludi a body receives when soUcited by the force of 

 gravity tree from the uitlueuce of any distiubiiig forces. 



2N 



