526 



Motion of 



Water in 



Pipes and 



Canals. 



Method of 

 using the 

 preceding 

 Tables. 



HYDRODYNAMICS. 



Method of Using the preceding Tables. 



EXAMPLE I. Water is brought into Edinburgh by 

 several pipes, one of which is 5 inches in diameter. 



9.6587213, and the sum is the logarithm of the Scotch 

 pints discharged in a minute. 



The facts in the preceding example respecting the 

 supply of Edinburgh with water were taken from Dr 

 Robison's article on Waterworks already quoted. We 

 are informed, however, by James Jardine, Esq. civil 



Metion of 

 Water in 

 Pipes and 



Canels. 



several pipes, one i are informed, nowever, uy jai a jar 



This pipe is 14,367* feet long, and the reservoir at engineer> t h at the facts are wholly erroneous, and we 



Comiston is 44 feet higher than the reservoir on 1 : haye been m d e bted to the kindness of this gentleman 



j- .1 i_ rii : A. ...UC..U t-\\o wat^r is clplivercfi. It IS 1*C* /_ . i_ _ <* 11 .* ..4-,, , .(' + K.. I.'liilii-/-T 



l^UUUSLUll 10 1-1? **-w fc ^ . 



Castle-hill into which the water is delivered. 



quired to know how many Scots pints the pipe should 



deliver in a minute. 



1. In this case we have d = = 1.25 inches. 



2. We have s = 



- = 326.36. 



Now, by entering Table III. with 1.25 as the value of 

 d, and Table I. with 326 as the value of s, we obtain 

 2 49490 as the logarithm for the numerator, and 

 1 18065 as the logarithm corresponding to 326.36. 

 . _ the difference of which logarithms is 

 1 31425 the logarithm of 20.618, or the value of 



01) 



for the following state of the Edinburgh water pipes, to 

 which we shall apply the formula of Du Buat. 



EXAMPLE II. An excellent cast leaden pipe was laid 

 from the fountain head at Comiston to the reservoir on 

 the Castlehill of Edinburgh in the year 1720. The in- 

 terior diameter of the pipe was 4^ inches, the foun- 

 tainhead was 51 feet above the point of delivery, and 

 the length of the pipe was 14,930 feet. Its maximum 

 discharge during the years 1738, 1739, 1740, 1741, and 

 1742, was 11^ cubic feet, or 189.4 Scotch pints per 

 minute. 



4i 

 In this example we have d.= -^ = 1.125 



y s Hyp. Log. </s + 1.6 



In order to find the value of the negative quantity 0.3 

 ( </-d0.l ) enter Table 1 1 1. col. 1. with 1.25, and in col. 

 5. will be found, by taking proportional parts, 0.305 ; 

 hence we have the velocity V=20.618 0.305=20.313, 

 the velocity of the water in inches per second. 



The whole of the preceding operation may be saved 

 by Table II. ; for, by entering col. 1. of this Table with 

 5 inches as the diameter of the pipe, we obtain at once 

 2.49490, and 0.305 as the values of the numerator an( 

 the negative quantity. In order to obtain the numbe 



s = -^ ; = 292.745 



51 



Log. of numerator 



Log. of denominator . 



Log. of .... 20.673 

 Subtract negative quantity .288 



2.46971 

 1.15431 



1.31540 



Remains 

 Hence 



Log. of 20.385 



the negative quantity. 111 uruer w ww* L O f 4!' O r 20.25 . . . . 



of Scotch pints per minute, each of which contains iw.* for reducing to Scotch pints 

 cubic inches, we must multiply the velocity by 60 , and 

 this product by 5" or 25, and then by 0.7854, the area 

 of a circle whose diameter is 1, and then divide by 1 OJ.4. 



Thus, 



1.3077741 

 1.778151S 

 1.3979400 

 9.8950909 



20.385 the velocity per second. 



1.3093107 

 1.3064250 

 9.6587217 



Log. of 20.313 

 Log. of 60" 

 Log. of 5 2 or 25 

 Log. of 0.7854 



From 4.3789563 



Subtract Log. of J03.4 2.0145205 



Remains Log. of 231.44 2.3644358 

 Scots pints, which should be delivered by the pipe. 

 Now, the pipe, when in its best order, yielded * 

 pints in a minute, * w,e have learned from a MS. note 

 of Dr Robison. . 



Since the logarithm of 60, of .7854, and of 103.4 is 

 constant, we may take 1.7781513 + 9-8950909 

 2.0145205 = 9.6587217, and the operation will stand 

 thus: 



Log. of 20.313 -1.3077741 



Log. of 5* 1.3979400 



Log. for reducing to Scotch pints 9.6587217 



Log. of 188.13 Scotch pints .... 2.2744574 

 A result which agrees in a very wonderful manner 

 with 189.4, the quantity actually delivered by the 



EXAMPLE III. A flanch cast iron pipe is laid from 

 Swanston cistern to the reservoir on the Castlehill, E- 

 dinburgh. Its diameter is seven inches ; the cistern at 

 Swanston is 222 feet higher than the point of delivery, 

 the length of the pipe is 21,350 feet, and in its best 

 state it delivers 3| cubic feet, or 593.3 Scotch pints 

 in a minute. 



7 

 In this case we have d = 1.75 



Log. of 231.44 as before. 



2.3644358 



; -222- 



Log. of numerator 2.57452 



Log. of denominator .. 0.87595 



Log. of .... 49-964 1.69857 

 Subtract negative quantity .367 



Remains 49-597 the velocity per second. 



Log. of 49.597 - - 1-695447 



Log. of .... 7 s or 49 .. 1.690196 



Log. for reducing to Scotch pints . . 9-658721 



Hence we have the following Rule : Add together 

 the logarithm of the velocity in inches per second, as 

 found by the formula, the logarithm of the square of 

 the diameter of the pipe, and the constant logarithm Log. ol 1 



- Dr Robison, who applied his tables to this exaznpl., makes the length of the pipe 14,637 by mistake, and has corrected it to 14,367 

 in his MS, notes. 



