HYDRAULICS

HYDRAULICS. In addition to the general principles of Hydraulics, as they have been detailed in the article HYDRODYNAMICS, of the Encyclopædia, it will be necessary to give some account of the later attempts that have been made to improve the theory of this department of science, especially those of Mr De Prony, contained in his Recherches Physico-mathématiques sur la Théorie des eaux courantes, 4to. Paris, 1804; and those of Dr Thomas Young, in his

VOL. V. PART I.

Hydraulic Investigations, published in the Philosophical Transactions for 1808. Hydraulics.

Ingenious and important as the Chevalier Dubuat's theory of the friction of fluids is admitted to have been, it cannot be denied, that it is extremely deficient both in the distinct elucidation of the physical grounds of the phenomena, and with respect to the neatness and simplicity of the methods of calculation. Mr Girard, according to Mr De Prony, was the first who entertained the fortunate idea of applying the theory of Mr Coulomb to the resistance of water flowing in pipes and canals; and in his two memoirs on the Theory of Running Waters, he proposes a formula depending on the sum of the velocity and its square, for the expression of the friction; and deducting a constant coefficient from twelve experiments of Chezy and Dubuat, he obtains a formula equally correct with that of Dubuat, and far more simple.

Mr De Prony has obtained an expression founded on the same theoretical principles that Mr Girard had adopted, but much more perfect and accurate, and agreeing very sufficiently with all those of Mr Dubuat's experiments to which he has applied it. But Mr De Prony's formula, as well as Mr Dubuat's, fails altogether, when we attempt to employ it for the computation of the discharge through very slender tubes, and though his work was printed a year or two earlier than Dr Young's investigations, it would have been impossible for this philosopher to make any use of it in his physiological inquiries, even if it had not been wholly unknown to him, since the resistances which he had occasion to compute were principally such as occurred in tubes considerably less than a thousandth of an inch in diameter.

The most material deviation of the experiments from the theory of Mr Coulomb and Mr De Prony appears to arise from the great comparative increase of that portion of the resistance which varies simply as the velocity, when the tubes become extremely small: and it is upon this circumstance that Dr Young has founded the principal characteristic of his theory of the circulation of the blood; the resistance, notwithstanding the extreme slowness of the motion of the blood through the capillary vessels, being almost entirely confined to those vessels; and in order to comprehend in one formula the whole of Dubuat's experiments, together with those of Gerstner, and his own, on very small pipes, he has found it necessary to introduce, into the determination of the coefficients for expressing the two kinds of resistance, a quantity depending on the diameter of the tube, and capable of being readily computed from it, although by a formula which is founded on empirical evidence only. The method of calculation, however, appears to be equally general and convenient, and it will, therefore, not be superfluous to copy the particulars of the investigation in the words of the author.

Dr Young "found," he observes, "that the friction could not be represented by any single power of the velocity, although it frequently approached," for the same pipe, "to the proportion of that power, of which the exponent is 1.8; but that it appeared to consist of two parts, the one varying simply as the

Hydraulics. velocity, the other as its square. The proportion of these parts to each other must, however, be considered as different, in pipes of different diameters, the first part being less perceptible in very large pipes, or in rivers, but becoming greater than the second in very minute tubes, while the second also becomes greater, for each given portion of the internal surface of the pipe, as the diameter is diminished."

"If we express, in the first place, all the measures in French inches, calling the height employed in overcoming the friction $f$, the velocity in a second $v$, the diameter of the pipe $d$, and its length $l$, we may

make $f\; =\; a\; \backslash frac\{l\}\{d\}\; v^2\; +\; 2c\; \backslash frac\{l\}\{d\}\; v$; for, it is obvious, that the friction must be directly as the length of the pipe; and since the pressure is proportional to the area of the section, and the surface producing the friction to its circumference or diameter, the relative magnitude of the friction, "or the friction for any given portion of the section," must also be inversely as the diameter, or nearly so, as Dubuat has justly observed.

We shall then find that $a$ must be $.0000001\; \backslash left(\; 430\; +\; \backslash frac\{75\}\{d\}\; -\; \backslash frac\{1440\}\{d+12\}\; -\; \backslash frac\{180\}\{d+1\}\; \backslash right)$, and $c\; =\; .0000001\; \backslash left(\; \backslash frac\{9000d\}\{dd+1000\}\; +\; \backslash sqrt\{\backslash frac\{1\}\{d\}\}\; \backslash cdot\; \backslash left[\; 1050\; +\; \backslash frac\{12\}\{d\}\; +\; \backslash frac\{9\}\{dd\}\; \backslash right]\; \backslash right)$. Hence it is easy to calculate the velocity for any given pipe or river, and with any given head of water. For the height required for producing the velocity, independently of friction, is,

according to Dubuat, $\backslash frac\{v^2\}\{478\}$, or rather, as it appears from almost all the experiments which I have compared, $\backslash frac\{v^2\}\{550\}$; and the whole height $h$ is therefore equal to $f$

$+\; \backslash frac\{v^2\}\{550\}$, or $h\; =\; \backslash left(\; \backslash frac\{al\}\{d\}\; +\; \backslash frac\{1\}\{550\}\; \backslash right)\; v^2\; +\; \backslash frac\{2cl\}\{d\}\; v$; and making $b\; =\; \backslash frac\{1\}\{al/d\; +\; .00182\}$, and $c\; =\; \backslash frac\{bcl\}\{d\}$, $v^2\; +\; 2ce\; =\; bh$, whence $v\; =\; \backslash sqrt\{bh\; +\; c^2\}\; -\; c$. In order to adapt this formula to the case of rivers, we must make $l$ infinite; then $b$ becomes

$\backslash frac\{d\}\{al\}$, and $bh\; =\; \backslash frac\{d\}\{a\}\; \backslash cdot\; \backslash frac\{h\}\{l\}\; =\; \backslash frac\{ds\}\{a\}$, $s$ being the sine of the inclination, and $d$ four times the hydraulic mean depth; and since $c$ is here $=\; \backslash frac\{c\}\{a\}$, $v\; =\; \backslash frac\{\backslash sqrt\{ads\; +\; ce\}\; -\; c\}\{a\}$; and in

most rivers, $v$ becomes nearly $\backslash sqrt\{20000ds\}$.
"In order to show the agreement of these formulae with the result of observation, I have extracted," says Dr Young, "as indiscriminately and impartially as possible, forty of the experiments made and collected by Dubuat; I have added to these some of Gerstner's, with a few of my own; and I have compared the results of these experiments with Dubuat's calculations, and with my own formulae, in separate columns." "It appears from this comparison, that in the forty ex-

periments, extracted from the collection, which served as a basis for Dubuat's calculations, the mean error of his formula is $\backslash frac\{1\}\{24\}$ of the whole velocity, and

that of mine $\backslash frac\{1\}\{25\}$ only; but if we omit the four experiments, in which the superficial velocity only of a river was observed, and in which I have calculated the mean velocity by Dubuat's rules, the mean error of the remaining 36 is $\backslash frac\{1\}\{35\}$, according to my mode of calculation, and $\backslash frac\{1\}\{37\}$, according to Mr Dubuat's; so

that "the accuracy of the two formulae may be considered as precisely equal with respect to these" forty "experiments. In "six experiments which Dubuat has wholly rejected," "without any very sufficient reason," "the mean error of his formula is about $\backslash frac\{1\}\{24\}$, and that of mine $\backslash frac\{1\}\{45\}$. In fifteen of Gerstner's ex-

periments, the mean error of Dubuat's rule is $\backslash frac\{1\}\{3\}$, that of mine $\backslash frac\{1\}\{4\}$; and in the three experiments which I made with very fine tubes, the error of my own rules is $\backslash frac\{1\}\{15\}$ of the whole, while in such cases Dubuat's formulae completely fail." "It would be useless to seek for a much greater degree of accuracy, unless it were probable that the errors of the experiments themselves were less than those of the calculations; but if a sufficient number of extremely accurate and frequently repeated experiments could be obtained, it would be very possible to adopt my formulae still more correctly to their results.

"If any person should be desirous of making use of Dubuat's formula, it would still be a great convenience to begin by determining $v$ according to this method; then, taking $b\; =\; \backslash frac\{l\}\{h\; -\; v^2\; \backslash cdot\; 478\}$, or rather, as Langsdorf makes it, $b\; =\; \backslash frac\{l\}\{h\; -\; v^2\; \backslash cdot\; 482\}$, to proceed in calculating $v$ again "by the formula $v\; =\; 148.5\; (\backslash sqrt\{d\}\; -\; .2)\; \backslash cdot\; \backslash left(\; \backslash frac\{1\}\{\backslash sqrt\{b\; -\; HL\}\; \backslash sqrt\{(b+1.6)\}\}\; -\; .001\; \backslash right)$, since this determination of $b$ will, in general, be far more accurate than the simple expression $b\; =\; \backslash frac\{l+45d\}\{h\}$, and the

continued repetition of the calculation, with approximate values of $v$, may thus be avoided. Sometimes, indeed, the values of $v$, found by this repetition, will constitute a diverging instead of a converging series; and in such cases, we can only employ a conjectural value of $v$, intermediate between the two preceding ones."

Hydraulics. "Having sufficiently examined the accuracy of my formula, I shall now reduce it into English inches, and shall add a table of the coefficients, for assisting the calculation. In this case, $a$ becomes .0000001

or, "for a river, " $=\; \backslash sqrt\{\backslash left(\backslash frac\{ds\}\{a\}\; +\; \backslash frac\{cc\}\{aa\}\backslash right)\; -\; \backslash frac\{c\}\{a\}\}$, as before;

and in either case the superficial velocity of a river may be found, very nearly, by adding to mean velocity $v$ its square root, and the velocity at the bottom by subtracting it.

"Coulomb's experiments on the friction of fluids, made by means of the torsion of wires, give about .00014 for the value of $c$, which agrees as nearly with this table, as any constant number could be expected to do. I have, however, reason to think, from some experiments communicated to me by Mr Robertson Buchanan, that the value of $a$, for pipes about half an inch in diameter, is somewhat too small; my mode of calculation, as well as Dubuat's, giving too great a velocity in such cases."

It must, indeed, be confessed that, notwithstanding the convenience of this theory for calculation, with the assistance of the tables of coefficients, their determination from the diameters of the pipes is somewhat too complicated either for elegance or for probability, if considered as representing the law of nature. The formula of Prony, though it fails for small pipes, and would, therefore, have been useless for Dr Young's purpose, has the advantage of great simplicity, and even of superior accuracy, within certain limits of the magnitude of the pipes, although it seems to be indebted for this accuracy to accidental causes only.

If we take Dr Young's equation $v\; =\; \backslash sqrt\{\backslash frac\{(ads\; +\; cc)\; -\; c\}\{a\}\}$,

or $v\; =\; \backslash sqrt\{\backslash left(\backslash frac\{ds\}\{a\}\; +\; \backslash frac\{cc\}\{aa\}\backslash right)\; -\; \backslash frac\{c\}\{a\}\}$, and assume for $a$ the con-

stant mean value .0000377, and for $c$ .00003466, we shall have $v\; =\; \backslash sqrt\{(26520ds\; +\; .845)\; -\; .919\}$, which is equivalent to Prony's formula reduced to French inches.

For pipes Mr De Prony merely substitutes $\backslash frac\{h\}\{l\}$ for $s$, ne-

glecting entirely the height $\backslash frac\{ev\}\{550\}$ due to the velocity.

He also gives a still simpler approximation for common purposes, $v\; =\; \backslash sqrt\{\backslash left(26513\; \backslash frac\{dh\}\{l\}\backslash right)\}$, or $\backslash sqrt\{(26513\; ds)\}$,

which differs very little from Dr Young's rule published in his Lectures, that "the velocity is a mean proportional between the hydraulic mean depth and the fall in 2800 yards;" for this, in French measures, would be $\backslash sqrt\{(27000\; ds)\}$. It is obvious, however, that in many cases these formulas must require considerable modification, since, when the velocity is great, the height due to it may become considerable, and since the friction in small pipes is certainly increased beyond its mean value: nor can these opposite causes of error be expected always to compensate each other even in pipes of moderate dimensions.

If we had a greater number of very accurate experiments, we might construct a sufficiently complete table of the values of $a$ and $c$ from their results alone, without any general formulas. Thus, taking $f\; =\; h\; -$

$\backslash frac\{ev\}\{550\}$, we readily obtain the equation, $a\; =\; \backslash left(\backslash frac\{f\; \backslash cdot\; d\}\{v\; \backslash cdot\; l\}\; -\; \backslash frac\{f\text{'}\; \backslash cdot\; d\text{'}\}\{v\text{'}\; \backslash cdot\; l\text{'}\}\backslash right)$,

$\backslash frac\{1\}\{v\; -\; v\text{'}\}$, in which $f$ and $f\text{'}$, $v$ and $v\text{'}$, are the correspond-

ing values of $f$ and $v$ for the same pipe with different pressures, or for two pipes so nearly alike in diameter, that the coefficients may be considered as the

same for both; and then $2c\; =\; \backslash frac\{f\}\{v\}\; \backslash cdot\; \backslash frac\{d\}\{l\}\; -\; av$; or for a

river, $a\; =\; \backslash frac\{ds\}\{v\}\; -\; \backslash frac\{d\text{'}s\text{'}\}\{v\text{'}\}$, and $2c\; =\; \backslash frac\{ds\}\{v\}\; -\; av$. But the results

of twelve double experiments, compared in this manner, afford a very irregular series only; thus:

It is, therefore, hopeless to attempt to deduce a good formula for the coefficients from these results alone, and we must be satisfied for the present either with Dubuat's formula, or Prony's, or with Dr Young's table. For a single instance of the comparative accuracy of these modes of computation, we may take an example from Mr Jardine's observation of the pipe supplying Edinburgh with water, as published by Dr Brewster. Here $d$ was 4.5, $h$ 612, $l$ 179160, and $v$ 20.52 English inches. We have, therefore,

20.71, exceeding the observation by .19. Dubuat's formula falls short by .13 only, Prony's table by .96. It must, however, be observed, that, in so great an extent of pipe, it is scarcely possible that some partial obstruction should not always occur, which must make the actual velocity somewhat less than the theory ought to give, so that the slight difference between the formulae of Dubuat and Dr Young can scarcely be said to be in favour of the former.

It is, however, remarkable that in this instance, as well as in some others, a single formula, which Dr Young had published in an abstract of Professor Eytelwein's book, is still more accurate than the more refined calculations. This formula is $v\; =$

English feet; thus, in the example here computed, this formula gives us $v\; =\; 20.56$ inches. It is

implies that $c$ may be neglected in such pipes at least, and $a$ may be taken $.1^7\; \backslash times\; 333$, which does not greatly differ from the tabular value for the given diameter. It has happened, from a combination of accidental circumstances, that Dubuat has been deprived of a considerable portion of his just merits,

in favour of Mr Eytelwein, without any kind of voluntary plagiarism on the part of this very respectable Professor. The author of the English abstract of his work was acquainted with Dubuat's book, only from having read the extracts, copied from it by Professor Robison, in his contributions to the Encyclopaedia Britannica, and he was hence led into the mistake of supposing Professor Eytelwein the author of many improvements, which he had no more idea of claiming, than he could have had of several modifications introduced into the "Summary of Hydraulics," chiefly extracted from his book, which appeared in the Journal of the Royal Institution, and which have been quoted as Eytelwein's.

The resistance occasioned by the flexure of a pipe, must not be neglected when any great accuracy is required in the determination of the water conveyed by it. Dr Young has pointed out the inadequacy of Dubuat's method of calculating this resistance, and his remarks may again be copied in his own words.

"Mr Dubuat," he continues, "has made some experiments on the effect of the flexure of a pipe in retarding the motion of the water flowing through it; but they do not appear to be by any means sufficient to authorise the conclusions which he has drawn from them. He directs the squares of the sines of the angles of flexure to be collected into one sum, which, being multiplied by a certain constant coefficient, and by the square of the velocity, is to show the height required for overcoming the resistance. It is, however, easy to see that such a rule must be fundamentally erroneous, and its coincidence with some experiments merely accidental, since the results afforded by it must vary according to the method of stating the problem, which is entirely arbitrary. Thus it depended only on Mr Dubuat to consider the arc of a pipe bent to an angle of $144^\backslash circ$ as consisting of a single flexure, as composed of two flexures of $72^\backslash circ$ each, or of a much greater number of smaller flexures, although the result of the experiment would only agree with the arbitrary division into two parts, which he has adopted. This difficulty is attached to every mode of computing the effect, either from the squares of the sines or from the sines themselves; and the only way of avoiding it is to attend merely to the angle of flexure as expressed in degrees. It is natural to suppose that the effect of the curvature must increase, as the curvature itself increases, and that the retardation must be inversely proportional to the radius of curvature, or very nearly so; and this supposition is sufficiently confirmed by the experiments, which Mr Dubuat has employed in support of a theory so different. It might be expected that an equal curvature would create a greater resistance in a larger pipe than in a smaller, since the inequality in the motions of the different parts of the fluid is greater; but this circumstance does not seem to have influenced the results of the experiments made with pipes of an inch and of two inches diameter: there must also be some deviation from the general law, in cases of very small pipes having a great "radius of" curvature; but this deviation cannot be determined without further experiments. Of the twenty five which Dubuat has made, he has rejected ten

Hydraulics. as irregular, because they do not agree with his theory; indeed, four of them, which were made with a much shorter pipe than the rest, differ so manifestly from them that they cannot be reconciled; but five others agree sufficiently, as well as all the rest, with the theory which I have here proposed, supposing the resistance to be as the angular flexure, and to increase besides, almost in the same proportion as the radius of curvature diminishes, but more nearly as that power of the radius of which the index is $\backslash frac\{7\}\{8\}$. Thus if $p$ be the number of degrees subtended at the centre of flexure, and $q$ the radius of curvature of the axis of the pipe in French inches, we shall have $r\; =\; \backslash frac\{pv^2\}\{200000q\}$ nearly, or more accurately $r\; =\; \backslash frac\{.0000045pv^2q^\{\backslash frac\{1\}\{3\}\}\}\{q\}$. These calculations are compared with the whole of Dubuat's experiments in the following table. The mean error of his formula in fifteen experiments, and of mine in twenty, is $\backslash frac\{1\}\{25\}$ of the whole."

The methods of computation here proposed may be illustrated by a practical example, which is of a nature very likely to be of frequent occurrence. Let the length of a pipe $l$ be 1000 feet, let it have five flexures, each equal in extent to a right angle with a radius of curvature $q$ of six inches; let the height of the head of water $h$ be ten feet, and let the quantity of water to be delivered be one ale hogshead, or fifty-four gallons in a minute; required the diameter of the pipe $d$.

Neglecting the slight difference between French and English inches, with respect to the effect of curvature, we have $p\; =\; 450$, $q^\{\backslash frac\{1\}\{3\}\}\; =\; .2085$, and $r\; =\; .0004222v^2$, which must be added to $\backslash frac\{al\}\{d\}$, the coefficient of $v^2$ in the general formula. Now, since an ale gallon fills 10 yards of inch pipe, the velocity discharging a gallon in a minute by such a pipe would

be 6 inches, and for 54 gallons, 324 inches; consequently, for any other pipe, of the diameter and the velocity discharging a hogshead in a minute must be

$\backslash frac\{324\}\{dd\}$. Hence we might obtain, by means of the for-

mulae expressing the coefficients, a regular equation in terms of $d$ and its powers, but it would be of too high an order to be at all manageable; the very general method, termed by arithmeticians the rule of double position, affords much the readiest solution. Thus, supposing $d\; =\; 4$, we have $a\; =\; .0000306$ and $c\; =\; .0000556$, and since $\backslash frac\{l\}\{d\}\; =\; 3000$, the coefficient $al/d$

becomes .0918, to which if we add .00171, according to the formula, and $r\; =\; .0004222$, we have $b\; =$

$\backslash frac\{1\}\{.0939322\}$, and $c\; =\; \backslash frac\{bc\}\{d\}\; =\; \backslash frac\{.1668\}\{.0939322\}\; =\; 1.776$, whence $e$

$=\; (bh\; +\; c^2)\; -\; c\; =\; 8.70$; but $\backslash frac\{324\}\{16\}\; =\; 20.25$; therefore,

the pipe is too small, and we must try five inches, $a$ being .0000312, and $c\; =\; .0000507$; hence $al/d\; =$

.07488, and $b\; =\; \backslash frac\{1\}\{.0770122\}$, $c\; =\; \backslash frac\{.1468\}\{.0770122\}\; =\; 1.901$

and $v\; =\; 9.65$, while $\backslash frac\{324\}\{25\}\; =\; 12.96$. Here the differences of the values of $v$ are $+11.55$ and $+3.31$,

their difference 8.24 answering to a difference of an inch in the diameters. We must, therefore, add

$\backslash frac\{3.31\}\{8.24\}\; =\; .402$, and the diameter required will be 5.40

inches; and if we wish for very great accuracy, we may repeat the calculation once more with this value of $d$.

The remainder of the investigations contained in Dr Young's two papers are more immediately connected with Physiology than with Hydraulics; and the interesting experiments of Girard, published in the Memoirs of the Academy of Sciences for 1818, relate almost entirely to the effects of temperature on the mechanical properties of bodies possessed of different chemical characters.

(R. T.)