###
Art. XLIX.—*Resistance to the Flow of Fluids Through Pipes*.

[*Read before the Technological Section of the Wellington Philosophical Society*, 10*th November*, 1915.]

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

The present work is the result of an effort to express the resistance offered by rough pipes, as distinct from smooth pipes, to the flow of fluids in terms of *v*d/ν where *v* is the mean velocity of flow, *d* the diameter of the pipes, and *v* the kinematic viscosity of the fluid.

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

The investigation was prompted by the publication in the “Philosophical Transactions of the Royal Society A,” vol. 214 (1914), of Stanton and Pannell's experiments upon smooth pipes, and more particularly by the publication in the “Proceedings of the Royal Society A,” vol. 91 (1914), of Professor Lees's discussion of those experiments. The latter work contains an admirable historical *résumé* of the subject, so there is no need to dwell further upon that aspect than to state that the resistance offered by a viscous fluid of any kind flowing in a circular pipe is proved by Stoker, Helmholtz, Rayleigh, and Reynolds from dynamical considerations to be a function of the expression *v*d/ν.

Below a certain critical value of the velocity the resistance is purely viscous, and the function mentioned is a simple one, viz.,—

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

R/ρv^{2}=(ν/*vd*) (1)

where R is the resistance per unit of surface, ρ is the density of the fluid, *a* is a coefficient, the other values having the same significance as before.

Above the critical value of the velocity the resistance is apparently partly viscous and partly an mertia effect, and the relation between the elements a complex one; so much so that the probability is that it cannot be exactly expressed by any formula; neither is a formula an absolute necessity, though an approximate formula, if obtainable, is undoubtedly a convenience.

In this work the resistance curve is regarded somewhat in the same manner as a stress strain or magnetic force and magnetization curve is regarded—that is to say, it is a complicated function obtained by experiment and expressed by means of a diagram which shows the relation between two chosen functions.

Professor Lees, on examination of Stanton's experimental results, finds that within the limits of experimental error the law of resistance above the critical value of a smooth pipe can be expressed very approximately in the form

R/ρ*v*^{2}=*a* (ν/*vd*)^{χ} + *b* (2)

where *a, b*, and χ are experimental coefficients. When the equation is expressed in absolute units he gives the following values to the coefficients, viz.:—

*a* = .0765, *b* = .0009; *χ* = 0–35,

so that the equation becomes

R/ρv^{2} = (ν/*vd*)^{0.35} + .0009 (3)

Converting to foot pound units, and also transforming from resistance per unit area and per unit of mass to energy per pound of fluid per foot of length or “head” per foot of length, we have

ι*d*/4/*v*^{2} = .00801 (ν/*vd*)^{0.35} + .000028 (4)

where ι is the hydraulic gradient or slope, or, in other words, the resistance head per foot of pipe. This form is more convenient for practical purpose, and conforms to engineering practice. The formula most commonly in use is that known as Chezy's formula, or, rather, a modification of the same, viz.,—

*v* = C√*ri* (5)

where C is an experimental coefficient, ι the hydraulic gradient, and *r* is the hydraulic mean radius which for round pipe is equal to *d*/4. It will be seen by comparing equation (4) and (5) that for smooth pipe

1/C^{2} = .0081 (ν/*vd*)^{0.35} + 000028 (6)

Returning to equation (4), this is plotted in Fig 1, curve with *a*, *i**d*/4/*v*^{2} as ordinates and log *vd*/ν as abscissae. The reason why it is necessary to adopt logarithmic values is that the range is so great that a comprehensive diagram could not otherwise be drawn within the limits of ordinary sheet. The range covered by experiment and observation lies between the values 5.2 and 7.2 of the expression log *vd*/ν, and it will be seen later that certain observations on a wood stave pipe he on an extension of the curve up to a value of 7–7, which tends to confirm the law as expressed by Professor Lees.

Accepting curve *a*, Fig. 1, as a sort of datum-line from which to gauge experimental results, the sequel is an account of an examination of experiments carried out by different observers on the resistance to the flow of water in pipes with the object of determining whether it is possible in the light of present knowledge to systematize them and to deduce a law for their behaviour.

In order to comprehend the whole range satisfactorily both below and above the critical velocity it is necessary to take logs of both sides, and the standard curve is set out again in Fig. 2, curve *a*, with log 4*v*^{2}/*id* as ordinates and log *vd*/ν as abscissae

The diagram Fig. 2 shows the first of stream-line stage as a straight line, and as the index χ is equal to unity in this region the line is inclined at an angle of 45°. The equation of the line is

ι*d*/4/ *v*^{2} = *a*(ν/*vd*)

where *a* has the value 8, an experimental value obtained from an examination of Stanton's experiments already referred to. Thus straight portion of the characteristic in Fig. 2 intercepts the abscissa at log 8.

The curved portion of the characteristic marked *a* is Professor Lees's equation already referred to, which represents the behaviour of fluids in smooth pipes above the critical velocity in the eddy state of motion.

Inasmuch as the change is abrupt from one state to the other, the exact nature of the connection cannot be determined, but the two portions are joined together on the principle that it requires a higher force to change a state of motion than to maintain either the previous or subsequent stage This phenomenon was observed by Reynolds, and is similar in its behaviour to elastic and magnetic phenomena.

Having obtained a standard of reference in this way, the next step is to plot to the same scale all the observations available on some one class of pipe and see how they stand in reference to the standard. Comparing first riveted pipes in general, the dots on the sheet represent 129 observations made on riveted pipes, varying from 12 m. in diameter to 102 in. in diameter recorded by Messrs. Marx, Wing, and Hoskins in the “Transactions of the American Society of Civil Engineers,” vol. 40. This is a

collection of all known experiments on riveted pipe up to the date of the record—viz., 1898—to which is added a series of thirty-one observations made by the same authors on a riveted pipe 72 in. in diameter and recorded in vol. 44 of the same society.

In these experiments the speeds vary from a fraction of a foot per second to 20 ft. per second.

The observations referred to are plotted in Fig 2, and it will be noticed that despite the number and range of observations the range is extremely limited from a law-determining point of view, and one is entitled to say at once on regarding them that no law of friction could possibly be deduced from them, and that every effort in that direction has been a waste effort. The range of log *vd*/v in these experiments is from 7 to 8, whilst Stanton's observations, before referred to, ranged from log *vd*/ν = 5.2 to log *vd*/ν = 7.2, the extent of which enabled Professor Lees to deduce a law for smooth pipes with some degree of certainty. Even this range could with advantage be extended.

In the absence of a similar range for riveted pipe it is necessary to consider what degree of guidance Professor Lees's curve, as set out in Fig. 2, curve *a*, affords towards deducing a resistance-curve for riveted pipe.

Regarding the plotted points by themselves, all that can be said regarding them is that they indicate an inclination towards the left, but whether a straight line or a curved line would best represent the law no definite answer can be given.

Most recent experimenters work on the assumption that the law is of the form given in equation (1), which would give a straight line when the logarithmic values are plotted. It will be seen that if the true law is of the form given in equation (2), this would result in a curved line, and inasmuch as any one series of experiments is, as a rule, of an extremely small range from the point of view under discussion, the points would fall very approximately upon a straight line, though in reality such lines are segments of a curve. It follows that as the inclination is different in different parts of the curve, and the range of each series being small, a different index would result according to the position of the series, and which might vary from

1.7 to nearly 2.0. An examination of experimental results confirm this view. Each experimenter has a different index, which varies, as a matter of fact, between the limits mentioned.

The diagram of Fig. 2 affords a clear indication of the direction in which further experiments upon riveted pipe should take, and it is evident that no addition to our present knowledge can be made; neither can a law be for certain deduced unless the observations are extended so as to cover a greater range.

Regarding the results plotted in Fig. 2, and taking curve *a* as a guide, it is reasonable to assume that the curve will be of the same form as curve *a* as expressed in equation (2); and, further, one might also reasonably assume that the viscosity element is constant, and that the inertia effect is greater because of the disturbing effect of the rivets and joints. Proceeding on this principle, three curves—viz., *c, d*, and *e*—are drawn through the points in Fig. 2, *c* and *e* being drawn through what may be deemed to be the lower and upper limit after neglecting what are obviously random or stray shots,

or errors in observation. Curve *d* is drawn through the thickest part of the cluster, and may be deemed to represent the commonest or most usual value.

It will be seen that these curves are as likely to represent the true form as any other, and there is ground to believe them to be the true representation of the law.

These curves have been transferred to Fig. 1, curve *a* being the standard, whilst *c, d*, and *e* are the same as *c, d*, and *e* in Fig 2. These may be expressed respectively as follows:—

Curvec |
id/4/v^{2} |
=.00801 (ν/id)^{0.35} |
+.000052 | ||

"d | " | " | +.000068 | ||

"e | " | " | +.000098 |

Large numbers of experiments have been carried out on wood stave pipes, and on this account it is instructive to plot observations in the same manner as for riveted steel pipes. This is done in Fig. 3, where *a* is the curve for smooth pipes, as in figs. 1 and 2.

The dots represent a series of sixty-six observations made by Moritz on wood stave pipe recorded in the “Transactions of the American Society of Civil Engineers,” vol. 74 (1911). The pipe-diameters range from 18 m. to 55 ¾ in., and velocities from a fraction of a foot per second to 5.874. These experiments are noteworthy on account of the extent of and care exercised in carrying out the experiments.

Regarding the results, it will be seen that quite a number of the dots are on the standard curve for smooth pipe, and otherwise by their disposition strengthen the evidence in favour of the curve *a* being the true law for smooth pipe.

Regarding these results, it may be truly said that under favourable circumstances a wood stave pipe may be treated as a smooth pipe, and the resistance truly represented by curve *a* in the figures. The other curves have been drawn in Fig. 3—viz, *b* and *c*—the former a sort of mean curve, and the latter an outside or superior limit after neglecting what appear to be errors of observation. These curves have been transferred to Fig. 1, and may be identified by the same lettering. The conclusion so far as regards wood stave pipe is that the value of 1/C^{2} lies between

.00801 (ν/*vd*)^{0.35} + .000028

and

.00801 (ν/*vd*)^{0.35} + .000052

whilst the mean or commonest value may be taken as

It is reasonable to assume that asphalted pipes of any material will yield the same results as wood stave pipe, provided the joints are well made and even, and that there are no projections of any kind such as rivet-heads or straps, and that the asphalting lies smoothly and evenly. The common defect in asphalted pipes is that the pipe has been immersed in an asphalt mixture having too low a boiling-point, with the result that the asphalt coating is of uneven thickness and corrugated, thereby considerably increasing the resistance. Cast-iron pipe or solid-drawn or welded iron or steel pipes should, if this asphalting is properly done, yield results within the limits given for wood stave pipe.

Whilst no certain conclusion can be drawn as to the correct expression of the law of resistance of rough pipes, we can say the whole range or experience lies bounded by the curves *a* and a curve similar to and not far different in shape and position from curve *e*, and as an instance of the application of the curves of Fig. 1. Suppose that the conditions are such that the value of log *vd*/ν is 7, and that we wish to know the value of 1/C^{2} for riveted pipe, we can be certain it is not less than .000056, whilst the probability is that it is not less than .00008 nor more much than .000126, whether curves *c* and *e* are correctly drawn or not, or whether the equations given for these curves are right or wrong, whilst there is a preponderance of experience in favour of a value of .000096, which is found on curve *d*

Applying the same reasoning to a wood stave pipe, and assuming that the value of log *vd*/ν is 7 as before, the value of 1/C^{2} cannot be less than .000056, and is not likely to exceed .00008, whilst the probable value is .00007.

It is claimed as a result of this investigation that fairly definite limits have been set within which the loss of head for almost any kind of pipe can be ascertained, but that the exact expression for any class of pipe is not

at present ascertainable and cannot be ascertained unless the range of observations is considerably extended. Certain conclusions may also be drawn as to the value of 1/C^{2} for very large values of log *vd*/ν, and that such values tend towards being constant, the value being that of the second

term on the right-hand side of the equations. By modification of the diagrams these may be made to apply to open conduits and canals, and as the cross-sections, generally speaking, are very much larger compared with the largest pipe in use, the value of the expression which would correspond to log *vd*/ν is large, and in consequence the variation of 1/C^{2} is small, and tends to become nearly constant, and will vary roughly with the surface and very little with the viscosity, and consequently with temperature.

It should be noted that whilst the fluid under discussion has been water, the curves, and therefore the equations, are quite general and apply to any fluid, the link being the kinematic viscosity ν. For convenience of reference two curves are shown in Fig 4, showing the viscosity and kinematic viscosity of water and its variation with temperature, which enables the value of the function *vd*/ν to be obtained for any given temperature, whilst the value of log *vd*/ν can be obtained for any value of *v* and *d*, and any temperature between 0° centigrade and 30° centigrade, by reference to the curves in Fig. 5.

The following is a summary of the foregoing and of the present state of knowledge in respect to the friction of fluids in pipes.—

1. The resistance offered to the flow of fluids through round pipes with smooth surface is represented by curve *a*, Fig 1, which may be expressed, according to Professor Lees, by the equation

ι*d*/4/*v*^{2} = .00801 (ν/*vd*)^{0.35} + .000028

2. The value of 1/C^{2} in Chezy's modified formula *v* = C√*ri* is approximately expressed for smooth pipe by

.00801 (ν/*vd*)^{0.35} + .000028.

3. Wood stave pipe under some conditions comes under the definition of smooth pipe, for which 1/C^{2} has the value given in paragraph 2 of this summary.

4. In the present state of knowledge no certainty exists as to the correct expression of the law of resistance to the flow of fluids in rough pipes. Regarding, however, the observations made of the resistance or loss of head in relation to the curve of resistance for smooth pipe, an equation of the same form as for smooth pipe, in which the first expression on the right is constant and of the same value as Professor Lees's equation, viz.,—

.00801 (ν/*vd*)^{0.35}

whilst the second term varies with the roughness of the surface, will fit the facts as well as any other, and there are some indications that this is the correct mode of expression.

5. On the assumption made in paragraph 4, the value of 1/C^{2} for wood stave pipes lies between

.00801 (ν/*vd*)^{0.35} + .000028

and

.00801 (ν/*vd*)^{0.35} + .000052,

whilst the more usual value may be taken as

.00801 (ν/*vd*)^{0.35} + .000042.

6. The value of 1/C^{2} for riveted pipe lies between

.00801 (ν/*vd*)^{0.35} + .000052

and

.00801 (ν/*vd*)^{0.35} + .000098,

whilst the more usual value may be taken as

.00801 (ν/*vd*)^{0.35} + .000068.

7. For large values of *vd*/ν the value of 1/C^{2} for any kind of pipe tends towards a constant value and independent of viscosity and therefore of temperature, but varies with the nature of the surface.

8. For large conduits or canals the same reasoning applies as in the case of large pipes referred to in paragraph 7.