1840.1 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



151 



eccentrif, wliieh is er|ii;il to the eccentrioily of the latter ; we sliull 

 llien linvf, calling 9 tlie angle (leseril)ecl by tile e<:centrie, IVom the nio- 

 ment when the port began to open till its aperture had heeoine eipial 

 to a, 



a=br sin e : (3.) 



whence we obtain by differentiation 



dif. a — 6 )• cos 9 (lif. p, 

 or 



dif. a 



dif. a 

 b r cos a 



Bnt equation (3) gives 



h r cos = /y/A- f' — a'l 

 wliich, being substituted, makes 



dif. a 

 dif. 9 = . 



Vi' »•' — a- 



And, if we call t the duration of a single stroke of the jiiston, or half 

 a revolution of the shaft, we shall also liave 



wlience l)y differentiation 



dif. / = ! dif. 0, 

 It 



which becomes by substituting the value of dif. 9 just found 



dif. t = dif. a. 



T (i-' r" — 0-) " 



Substituting this value, as well as that of v given by equatiow (V>, 

 in equation (2), this latter becomes 



dif. S . , 



(4). 



(D — 5)^ 



T V 2 g H D a dif. o 



(i- /- — o-)- 



The greatest value which S can acquire, being equal to the maxi- 

 nunu <lensity of the steam in the cylinder during the stroke of the 

 piston, cannot be quite equal to D, but will not fall far short of it. On 

 the other hand it is evident that, if we assume D as the maximum 

 value of 5, the hypothesis will be unfavourable to our demonstration; 

 we are therefore permitted to make it; and as the minimum v.due of 

 S is equal to d, we must integrate the first member of precediiig equa- 

 tion between the limits 5 = D and 5 ^ rf. The limits of the value 

 of a in the second number are a=: a, the aperture of the port when 5 

 has attianed its greatest value, and o ^ o. We must therefore have 



D 



' dif. ! 



J cD-5) i 



tV'2, 



2 gHD /* g dif. tt 



(6^r-a=)^ 



(5.) 



lu the tirst integral let D — 5 ^ .r; then dif. S := — dif. .r, and 

 D D — (i 



dif, 5 



": (lif. X. 



/'dif, 5 /• 



a, 



4 



=z--2{V> — d) . 

 In the second integral make 



b- r- — «'- = z. 

 Py difi'ereutiation we obtain 



a dif. a = — i dif. z. 

 We have, therefore, 



a h" r- 



y ^ g dif. g _ _ /•! 



■ dif. z. 



i 



= — [br — (b-.-- - rr-) ]. 



Substituting these values of the integrals in equation (5"), we ob- 

 tain 



2(p-d)^ = LVMii^ [i ,_ f/;: ,.. _ a-:) * ], 



and, subsliluting lor a its value given by equation (3), 



2 (D — (/) = -^—f [br — br(l— I'n. 9- ) ] 



>j^W2gHD^.„„_,^ 



whence we deduce 



suiv. 8 =: 



2 c TT VD • 



6rT^2gHD 



or, putting for b r, which is the area of the steam port when full open, 



, A 

 its value — , 



sinv. -- 



2»ctVD — d 



T A V2gHD" 



Now — '■ '— expresses (he distance passed through by the piston 



while the shaft ilcscribes the angle 9 round its axis, and consequently 

 during the time the waste sjiace is filUug with steam, therefore, if we 



call I that distance, and the ratio of the waste space c to the con- 



S 

 tents of the cylinder between the limits of the stroke, we shall have 



AL, " 

 c = ^ and 



/ = 



« «• L" V D — d 



St /^2gHli 



/ _ M T L V D — i = iittL "y 1 — p' 

 L ~ S T vTg- HET S t V 2gH 



or, since the densities are inversely as their relative volumes, 



(G.) 



As an example for low pressure steam, let L ^ 5, » — ^''vl — ^-,''' 

 p _ 17-7^ lbs., whence V = 1427, T = U70 and H = .)Sl,/ ..•/ ../b ; 

 let the temperature of the condenser be 110 degrees, '"";''.; J'^' ";•; 

 V = 14y.-i2 ; and, if we suppose the piston to move throu^^i -00 tee 

 in a minute, r = 1-5 second. We have besides n = o-UlU and 



° Subrtituliug these values in the second member of equation (b), we 

 obtain 



3-1416 X 



25 X 5 /\/ ^ " 



1427 

 14952 



= -0032. 



Ii 

 tiftl 



L 1-5 X 20 V G4-3S X 58G75-737G 

 u the example chosen the value of / would thus be less than one- 

 h of an inch; and it will be remembered that this value is too great 



in consequence of our having substituted the ratio — for -. 



As an example for high pressure steam, we wiU^ take the^data 

 from locomotive engines, and assume L = 1-5, n — la, s _ ^ , 

 P = 77-05 ; whence V = 3ol)-5, T = 700, and H = 'il>5^>' j'^^b ; also 

 ,- = 1700 ; and, if we suppose the piston to move through 3bO teet 

 in a minute, t = -ij second. 



Substituting these values, we lind 



/ 3-MlOX 25 X 1-5 /y ^' 



3ii'J-5 

 "iTOO 



= •010)7. 



L -25 X 20 V IJ4"38 X GG557-57(; 



In this example, therefore, the value of / is but a trifle more than 

 •is of an inch, or less than 3 of the lead of i of an inch usually allowed 

 in locomotives. 



The ratio - a'so expresses the proportion of the whole area of the 



steam port by which its aperture is diminished at the moment the 

 piston reaches the middle of its stroke, and as this quantity is, as the 

 wo above examples show, exceedingly small, it is unnecessary to 

 make any allowance for it. 



