108 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Mav, 



In the secotad case, ll'.e quantity of water actually spent, is marked 

 A BEN; and the iinanlity saved is marked S M E N. As the area 

 of each reservoir is Mipposcd to be equal to twice the ar.'a of the cham- 

 ber, the space r.hich a certain quantity of water occupies in tlie cham- 

 ber will be twice as high, or detqi, as the space required for the same 

 quantity of water in one of the reservoirs. Hence it follows, that 

 BP=-2l'N = i\K = 2.i- 

 and CQ = 2QO = OS = 2y 

 The whole lift A S, or 



L^AC+CQ + QO + OS 

 or L z= 2.r + -J ?/ + V + 2;/ ^ 'ix + 5y 



and likewise is L = BP+PN + Nk+ KM 



or L = •2x + -r + -■'■• + '2>/ = 53C + 2y 

 Hence ix + 5(/ = 5x 4- 2y 



or Sr/ = 3x 



consequently II =x 



The quantity and stage of water ii 

 required to be equal. 



Fis-. 2. 



each reservoir are thctcfore 



Let the ratio which indicates how many times tlie area of the lock- 

 chamber is contained in the area of each reservoir, be denoted by the 

 letter E, so that whea the area of the chamber is equal to !)0 X I'i = 

 13.50 square feet, let the area of each reservoir be expressed by K x 

 1.3.50 square feet. 



By examining the diagram fig. 2, it follows now, as a matter of 

 course, that O.S'= R.r = Km = K« = QC= PB = AC 



By adding' the diiferent altitudes, which constitute the lift, we find 

 L = Rx+x + R.r + Hi = 3R.r + X- (?, R + 1)£ 



an therefore, .r = — -:-- — - 



^ K+ 1 



which expression gives the stage of water in the reservoirs, provided 



there are two. 'Without anv further examination we can employ the 



following expressions as formula- for the other required dimensions : — 



1. The elevation of the bottom of the lower reservoir H. L 



above the lower canal level, or OS = Ka;= 3~Rr+~l 



1. The elevation of top-water line of the lower rcsi'rvoirfR + 1) L 

 above the lov.er canal level, QS (R -\- \)x = 3 R 4- 1 



3. The elevation of the bottom of the upper reservoir o R. L 

 above the lower canal level, or NRI = 2 lix = g l{ i j 



2 R. L 



4. The water saved, marked by the space MSEN, or 

 BAHQ = AQ = W N = 2 Rr = all +1 



5. The water used is marked by MSQH or ABEN = SQ R+1)L 



= BN = (R + l).r = 



By e.camining the formula No. 4 for the water saved 

 2 R L 



3 K + 1 



2 RL 



L 



3 R + 1 3 o) -f 1 

 As the quantity 1 does not increase an infinitely great quantity, 

 2 X L 2 oo L _^ _2JL 

 3 



it follows, 



L 



The foregoing result of the maximum of water-saving will become also 

 visible by mere examinatiun of the diagrams, fig. 2. We see that 

 when the stage of water in the reservoirs, or :r = PN = QO, becomes, 

 by bting spread over an infinitely treat surface, reduced to an infinitely 

 small height, the points P and N, and Q and O, will be brought so 

 near t(]gether, that they m.iy be regarded as being leduced to the sin- 

 gle points N and O, and therefore is 



SO = OE = EA= iL 



and the water saved = SE or BH = I L. 



3R+r 



we find that the saving increases with the ratio K, though not as 

 fast. When we suppose R = x , that is, the area of each reservoir to 

 be infinitely great, so that x, or the stage of water in each reservoir, 

 will be almost reduced to nothing, the formula will then bo 



r, + \ 3 * 3 3 



The greatest saving of water by two reservoirs is therefore equal to 

 two-thirds of the lift of the loci;. However, this much can never be 

 gained in roalily, though we can come near to it, without extending 

 the reservoir too much, which would imply other inconveniences, as 

 increase of cost, loss of time, and loss ol water by greater evaporation. 



For a given lift L = 14 feet, and R = 4, or the area of each of the 

 two reservoirs to be equal to 5400 square feet, where the lock-chamber 

 is supposed to be 90 X 1 J in the clear, we find 



L 14 14 



13 



1-077 feet 



3R + 1 "" 3X 4-M 



The elevation OS = R.r = 4 X 1-077 = 4-308 



The elevation NM = 2R.r = 8-61C 



AVater saved = 2Rj- = 8-616 



Water used = ( R _)- 1 ) .;• = 5-3S.5 



By means of two reservoirs of 5400 square feet area each, a boat may 

 therefore pass a lock of 14 feet lift, and not use more than 5-385 feet 

 water, drawn from the upper level, i' here formerly, without reservoirs, 

 a body of water of 14 feet height liad to be used. 



The following tabic shows how the quantity of water saved increases 

 with the area of the reservoirs, supposing two reservoirs attached to the 

 lock : — 



For R = ^ the water saved, or %?-^.= %^\ ^ = 0.285 L 



3R + 1 3.i+l 



0.400 L 



0.461 L 



0.500 L 



0.545 L 



„ ■ „ 0.571 L 



0.600 L 



„ „ 0.615 L 



„ „ 0.625 L 



0.644 L 



„ „ 0.664 L 



„ 0.6664 L 



0.6666... L 



the formnlrc for all 



R = 2 



-f*- — A it S» J» 



II = li 



R = 2 



R=4 



R — 5 „ „ „ 



R=io 



R^^lOO 



R= lono 



R = ^ .. » •. 



When only one reservoir is attached to the lock 

 the required dimensions will be found : — 



1. The water stage in the reservoir, or x = 



L 



2R-f i 



The elevation of the bottom of the reservoir above" p ^ 

 the lower level, is expressed by Rt = 



3. The height of the water saved is = Rx = 



4. The height of the water saved is = (R -f 1 ) a' =: 



2R -f 1 



R.L 

 2 R + I 

 (R+jj-L 



211+1 



5. The maximum of water saved by one reservoir is fo\md 



= ^L_-XL= =°^ 



2ao -f 1 2 00 



XL = 



-1 L 



By means of one reservoir, therefore, nearly one-half of the lockage 

 water may be saved in reality. 



Fig. 3. 





