440 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[January, 



the inventors themselves, sanguine men — we have tried that already 

 to onr cost. Dr. Arnott, though not as he would have us believe, the 

 first in the field, was the first to get the public ear— taking his advice 

 too readily, we had several explosions, many accidents, and much 

 vexation. Next came Mr. Joyce, and set us all a-gape, but in very 

 little time his stove, too, had to be set aside— Chunk, Vesta, Olm- 

 steed, Solar, and a score more, have been successively tried, and in 

 many instances have caused bitter disappointment. And in connexion 

 with all these, have undoubtedly been many accidental fires, some of 

 a very serious character. In all these cases, possibly, a little more 

 perseverance or a little more information, would have taught us that 

 the fault was not in the stove but in its unsuitable application. 



Will you then, Sir, endeavour to induce some clear-sighted prac- 

 tical man to give us such a handbook as shall prevent the repetition 

 of such mistakes, and at the same time bring into notice such stoves 

 as are best suited for each situation. Not only shall we, the public, 

 be benefited thereby, it would prove, I doubt not, a valuable boon to 

 architects, and to the stovemakers themselves. 

 I am, Sir, 



London, Dec. 1843. Your obedient servant, 



J. B. 



ON THE EFFLUX OF GASEOUS FLUIDS UNDER 

 PRESSURE. 



By Charles Hood, Esq., F.R.S., F.R.A.S., &c. 



Read be/ore the Institution of Civil Engineers. 



The theoretical determination of the velocity with which gaseous 

 fluids are discharged through tubes and apertures under pressure, has 

 often been submitted to mathematical investigation; and the subject 

 being of importance in various branches of practical science, it is to 

 be regretted that considerable differences exist in the results of the 

 several formula; which have been propounded for its elucidation. Dr. 

 Papin, 1 in 1G8G, first showed that the efflux of all fluids follows a 

 general law ; and that the velocities are inversely as the square roots 

 of the specific gravities. Dr. Gregory 2 has likewise given various 

 formulae for calculating the velocities of air in motion, under different 

 circumstances ; and Mr. Davies Gilbert, 3 Mr. Sylvester," Mr. Tred- 

 gold, 5 and many other writers of equal authority, have also investi- 

 gated the subject. 



The hydrodynamic law of spouting fluids has, by all writers, been 

 applied in the calculations for the determination of this question. 

 This law, it is well known, is the same as that of the accelerating 

 velocity of falling bodies; and is proportional to the square root of 

 the height of the superincumbent column of homogeneous fluid. But 

 although the various writers all agree in this fundamental principle, 

 they differ materially in the mode of applying it, and in the several 

 corrections introduced in their theorems ; and the results they have 

 arrived at are of a very contradictory character. 



Dr. Gregory's formula for calculating the velocity with which air 

 of the natural density will rush into a place containing rarer air, is 

 based upon the velocity with which air flows into a vacuum. This is 

 equal to the velocity a heavy body would acquire by falling freely 

 from a height equal to that which a homogeneous atmosphere would 

 have, whose weight is equal to 30 inches of mercury. The height of 

 this homogeneous atmosphere is 27.S1S feet; and the velocity which 

 a body would acquire by falling from this height (and consequently 

 the velocity with which air will flow into a vacuum) is i/(2?818x 

 64-36} = 1339 feet per second. The density of the rarefied air, 

 divided by the density of the natural atmosphere, and this number 

 subtracted from unity, represents the force which produces motion; 

 and the square root of tins number multiplied by 1339 feet (the ve- 

 locity with which air rushes into a vacuum) is the velocity with which 

 the atmosphere will rush into any place containing rarer air." 



The method employed by Mr. Davies Gilbert is also based upon the 

 velocity with which air rushes into a vacuum, when pressed by a 

 homogeneous atmosphere, equal to the weight of the natural atmo- 



1 Phil. Transactions, 1686. - Gregory's Mechanics, Vol. II, p. 513. 



* Quarterly Journal of Science, Vol. XIII, p. 113. 

 •* Annals of Philosophy, Vol. XIX, p. 408. 



* Tredgold on Warming Buildings, p. 76. 

 " Gregory's Mechanics, Vol. I, p. 515. 



sphere at the earth's surface. This supposed homogeneous atmo- 

 sphere is, according to Mr. Davies Gilbert's calculation, 26058 feet; 

 and the velocity with which air would rush into a vacuum, when 

 pressed by this weight, will be V(2o058)X8= 1295 feet per second. 

 When this calculation is applied to two columns of air of unequal 

 density — as, for instance, the discharge of air through a chimney 

 shaft— the height of the heated column of air divided by the height 

 of this homogeneous atmosphere, and the square root of this number, 

 multiplied by the velocity with which air flows into a vacuum, and 

 this product again multiplied by the square root of the number repre- 

 senting the expansion of the heated air, will give the velocity in feet 

 per second. The expansion of air when heated is found, (by Mr. 

 Gilbert's method) by raising the decimal 1-0020S3 (which represents 

 a volume of air expanded by 1° of Fahrenheit) to the power whose 

 index is the number of degrees which the temperature of the air is 



4s7)" 

 raised ; or it is equal to the fraction - | n being the number of 



degrees of Fahrenheit, which the temperature of the ascending 

 column exceeds that of the external atmosphere. 7 



Mr. Sylvester's method of calculation proceeds upon the suppo- 

 sition that the respective columns of light and heavy air represent 

 two unequal weights suspended by a cord hanging over a pulley ; and 

 this mode of calculation gives a result very much less than by any 

 other method. 



The unequal weight of two columns of air is found by Mr. Sylvester 

 nearly in the same manner as by Mr. Gilbert. The volume of air 

 expanded by 1° of heat, is equal to 1-00208: and this number, when 

 raised to the power whose index is the excess of temperature of the 

 heated column, gives the expanded volume of the air ; and assuming 



the atmospheric density to be unity, we have 1 — i,,,onci g = ^' 



(1*001.08) ' 



e being the excess of temperature of the heated column, and d the 

 difference of density between the two columns. This difference of 

 density, multiplied by 8 times the square root of the height of the 

 tube or shaft containing the heated air, gives the velocity in feet per 

 second. 6 



In Mr. TredgoM's theorem for calculating the efflux of air, the force 

 which produces motion is assumed to be the difference in weight of a 

 column of extern. d and one of internal ;\ir, when the bases and heights 

 are the same. The difference of temperature of the two columns by 

 Fahrenheit's scale, divided by the constant number 150 plus the tem- 

 perature of the heated column, and this quotient, multiplied by the 

 height of the tube or shaft, gives the difference in weight. Then by 

 the common theorem for falling bodies, 8 times the square root of this 

 number will give the velocity in feet per second ; or accurately, 



V= A /■ A . (l ~ z \ h being the height of the tube, t the tem- 

 perature of the internal, and x the temperature of the external air." 



The method of calculation proposed by Moutgollier, appears, how- 

 ever, by recent experiments, to be the most accurate, as it is also the 

 most simple, of all the modes of determining this question. The diffe- 

 rence in height must be ascertained which two columns of air would 

 assume when the one is heated to the given temperature, the other 

 being the temperature of the external air; and the rate of efflux is 

 equal to the velocity that a heavy body would acquire by falling freely 

 through this difference of height. 



The space which a gravitating body will pass through in one second 

 we know to be 16-09 feet; but by the principle of accelerating forces, 

 the velocity of a falling body at the end of any given time, is equal to 

 twice the space through which it has passed in that time; or, the 

 velocity is equal to the square root of the height of the fall, multiplied 

 by the square root of 64-36 feet ; or, again, to the square root of the 

 number obtained by multiplying 6 i-3t> feet by the height of the fall 

 in feet. 



When the via rira is the difference in weight between two columns 

 of air caused by the expansion of one of these columns by heat, the 

 decimal -00208 which represents the expansion of air by T of Fah- 

 renheit must be multiplied by the number of degrees the temperature 

 is raised, and this product again by the height of the heated column. 

 Thus, if the height of the column is 50 leet, and the increase of 

 temperature 20°, we shall have 20 x -00208 X 50 = 2-08 ft., or 

 52-OS ft. of hot air will balance 50 ft. of the cold air ; and the velocity 

 of efflux of the heated column when pressed by the greater weight 



7 Quarterly Journal of Science, Vol. XIII, p. 113. 



8 Annals of Philosophy, Vol. XIX, p. 408. 

 8 Tredgold on Warming Buildings, p. 76. 



