104 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[April. 



ON ATMOSPHERIC RAILWAYS. 



Resistance from the Ileal developed in the Air Pump. 



By F. Bashfokth, B.A., Fellow of St. John's College, Cambridge. 



There are considerable differences between the mean theoretical and ex- 

 perimental resistances to the air pump piston, for given heights of the 

 barometer, as stated in Mr. Stephenson's Report on the Atmospheric Sys- 

 tem. The discrepancies, for low heights of the barometer, are easily ex- 

 plained by a reference to the diagrams from which the numerical results 

 were deduced. It here appears that the exit valve by its flapping through 

 a very considerable space, caused an increased and useless resistance. In 

 addition to this, it will be seen that at the very commencement of the stroke, 

 the indicator frequently gave a pressure of nearly 1 lb. per square inch 

 greater than the pressure in the tube. This no doubt was caused by the 

 stream of air rushing into the cylinder in tlie same manner as water into 

 the hydraulic ram. Thus the mass of air enclosed and expelled wae 

 greater than what was supposed by theory. 



When the barometer rose to about 19 inches, although the above two 

 disturbing causes diminish with the increased rarefaction in the tube, the 

 ditferences of the theoretical and experimental results increased. Thig 

 was so considerable that Mr. Stephenson was driven to suppose that in 

 addition to the heat developed, a considerable leakage in the air pump 

 existed. It appears, however, that the first cause is quite sufficient, and 

 that the connecting tube must have admitted 206 cubic feet of air per 

 minute at the density of the atmosphere. Let 



a = A B = length of stroke of air 

 pump piston. 



k = area of air pump piston. 



n= pressure of the atmosphere on 

 a unit of surface. 



P, P, P„ the pressure of the 



air in the tube after 1, 2 n 



strokes. 

 Pi Pj p^ the density of the air 



in the tube after 1, 2,....7i 

 strokes. 

 V = volume of the receiver, 

 c =: volume of the pump cylinder. 

 Suppose that at the nth stroke 

 when he pump piston has described a space B M = o — x, that the den- 

 sity of the air is p and pressure P. Then 



-^ _/_£_^K AliUer'a Hydrostatics (38.) J 



, suppose 



Also P" = n R , for the rarefaction of the air goes on slowly, and it is 



in contact with so large a surface of metal, that we may consider the tern, 

 perature to be uniform, and . • . the pressure and density. The tend, 

 ency however would be in favour of the atmospheric traction, k a p , 



= mass of air inclosed at the commencement of nth stroke =. k x p. 



2=_i-. AisoP = P Y-^-^''=nr'*-'^''^''' 



when the exit valve opens P = n, and suppose i = a, 



,K 



=__!.. Aisop=p r-^-^''=^R'-Y''^•'' 



pn—1 n—l\Pn~lJ \x/ 



"/•a a /^a K oft 



then,=R"-'(^) 

 Work done in compressing the air = 



_ n g fc ( i- n— 1 J 



~K^^R — R ^ 



Work done in expelling the air = n fc x = 11 R a k. 

 Work done by the assistance of the air in the tube 



y'a 

 V d X „n— 1 V 



V+(a-x) " 



k log, K 



Hence work done during the nth stroke, 



= .^=nak[ r"-' ;- log, U+^,(K k"-=1 -k"-' ) } 

 Work done at the end of n strokes. 



n 

 /i_R>»v 1 / 1— R" i_R"\ > 



l-R' 



Ifw 



n =a. maximum, 



dn 



This gives n = 100 nearly corresponding to a rarefaction of between 91 

 and 22 inches, which agrees exactly with experiment. If we do not allow 

 for the heat developed, theory gives 19 inches. 



K-1 



In addition to the above test we may construct theoretical diagrams, and 



compare them with those found by experiment. 



The equations to the part described before the exit valve opens 



a') 

 is </ = h' (- when heat is neglected (a.) 



.„„ , — - -) in the otb*r case 



(3.) 



where A' denotes the length of the line between Gib. and the dotted line in 

 the figures representing th« pressure of the air in the tube, and a' the length 

 of the diagram, (a') evidently represents a series of rectangular hyper- 

 bolas referred to the same axes as asymptotes. 



The curves (j3) are found to agree very nearly with experiments, whilst 

 (a, are very different when the height of the barometer is about 20 inches. 



Both theories have their use. It has been seen that when heat is neg- 

 lected the work done by the engine, and the useful effect may be considered 

 equal. The other by its close agreement with experiment enables us to 

 estimate the necessary loss of different arrangements on the supposition 

 that the machinery is perfect. 



Between 10 and 2S inches the loss arising from this source is IS or 24 

 per cent, of the useful effect, and is another reason why the rarefaction In 

 common instances ought nut to be pushed beyond IS or 19 inches. 



