104 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[March, 



piston bv tlie induction passage B, the steam at tlie same time having 

 access to the portion of the valve below at D, wliich tends to press the 



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7////'////// 



' lll/Z/fllf, 



/pm^ 



1 7 ///■:). 



TiOjM//, 



^ 



"V> 



)/ 



Fig. I. 



Fig. 



Fig. 3. 



valve outwards from the cylinder, while the vacuum, acting on the in- 

 side of the valve E, draws it towards the cylinder, the valve being so 

 proportioned that the vacuum must so far overbalance the steam, that 

 the valve will keep close to the cylinder. ]n fig. 2, the piston is at 

 half-stroke, ascending, and the position of the valve is reversed from 

 that in fig. 1. Fig. 3 is a front view and section of the valve. 



John Maxton. 

 Ltiih Engine Works. 



TUBES OF LOCOMOTIVE ENGINES. 



Investigation to determine the diameter of the Tubes of a Locomotive 

 Engine Boiler to produce a maximum effect. 



In treating this subject it appears rational to suppose that the effect 

 of the hot air in passing through the tubes is directly in proportion to 

 the extent of surface in contact therewith, and as the time of contact 

 conjointly : that is, denoting the number of tubes by n, their diameter 

 by 5, their aggregate surface by s, their united area by a, and the time 

 of contact by /, supposing the length of the tubes constant, we shall 

 have the following postulates. 



S= A. 



8' B. 



5 C. 



is a n- S^ a maximum D. 



B Let the parallelogram A B C D be the space which is 

 to be occupied by the ends of the tubes, and letAB = a, 

 and AD= 6 ; also let the distance between the inner 

 surfaces of two adjacent tubes be a given interval c, 

 and because the same distance c must exist between 

 the sides of the extreme tubes, and the ends A D, B C 

 of the above figure, then a — c must be divided into a certain number 

 of equal parts, wh'ich will be regulated by the distance between the 

 centres [of two adjacent tubes, ;ind let this distance be (a — c)x, x 



being a function of (a — c), =: -, the number of tubes in 



^ (a — e).-!: x 



one horizontal row A B. 



The tubes are so disposed, that the lines joiniug the centres of any 

 three adjacent tubes form an equilateral triangle, consequently the 

 vertical distance between the horizontal rows will be 



D 



(a - c).rv/ J 



2(6 -c) 



(a — c)x\^ I (a — c)xi>,/ 3 



the number of liorizontal rows in A D, and 



2(6 - c) 



2(6- c) 



« (a—c)xs/3 {a — c)xW9 

 The whole number of tubes in A B C D = n. 



Further the distance (a — c)x is greater than the diameter of a tube 

 by the interval c, therefore 



(a — c)a: — c=^ the diameter of a tube = 8. 

 Let 7r:=3*141B, then the aggregate circumference of all the tubes 

 (and which may be taken to express the internal surface in consequence 

 of the length of the tubes being constant), will be 



i-(b-c)_ i (a-c)x-c ) ^ ^^j 



(a — c) a/3 *■ 



and the area of the tubes will be 



ir(6 — c) S(a — c)x- 



or s 



(C) 



^r 



n i\ or t (A or B) 

 2(a — c)v'3 ^' 



Now by the postulate we have st a maximum, and substituting the 

 values s and t as found above, we have 



2ir(6 — c) S(a—c)x — c\ t(6 — c> !(a — c)x — cy_ 



(a — c)V3* *' 2(a— c)V3 ^ ~ 



T-(6 — c)2 S(a — c)x — cy . „ .. ,„- 



— -i — . li / , a maximum = n- o' (D) 



3(a — c)= X* ' ^ 



All the above quantities, except .r, are constants, and the second 

 factor only contains the variable, therefore we have 



S(a — c) X — cV 



'■- ' =: a maximum. 



Differentiating and equating with zero, rejecting the denominator, 

 we have 



3 (a — c) ^* {(a — c) x — c}' d x — i x-^ {(a — c) x — c}^ d x = o ; 

 whence (a — c) x =^ 4 c. 



But (a — c) X zn the distance between the centres of two adjacent 

 tubes, which is therefore equal to four times the interval between 

 their internal surfaces. 



Further (a — e) x — c =: 3 c = the diameter of a tube, which must 

 be equal to three times the same interval. 



It is obvioHS that the smaller c is taken, the greater will be the 



value of the expression i^^ ~ / ; and, therefore, the tubes 



ought to be placed as near together as possible. 



In order to exemplify the application of the principles herein 

 developed, I have prepared drawings of the tubes of a locomotive 

 engine boiler, such as are very commouly used, and also of one tubed 

 according to the proportions just determined. 



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