Ancient and modern engineering and the Isthmian canal

183. Gravity Supplies.—When investigation has shown that a sufficient quantity of water may be obtained for a required public supply from any of the sources to which reference has been made, and that a sufficient storage capacity may be provided to meet the exigencies of low rainfall years, it will be evident if the water can be delivered to the points of consumption by gravity, or whether pumping must be employed, or recourse be made to both agencies.

If the elevation of the source of supply is sufficiently great to permit the water to flow by gravity either to storage-reservoirs or to service-reservoirs and thence to the points of consumption, a proper pipe-line or conduit must be designed to afford a suitable channel. If the topography permits, a conduit may be laid which does not run full, but which has sufficient grade or slope to induce the water to flow in it as if it were an open channel. This is the character of such great closed masonry channels as the new and old Croton aqueducts of the New York water-supply and the Sudbury and Wachusetts aqueducts of the Boston supply. These conduits are of brick masonry backed with concrete carried sometimes on embankments and sometimes through rock tunnels. When they act like open channels a very small slope is employed, 0.7 of a foot per mile being a ruling gradient for the new Croton aqueduct, and 1 foot per mile for the Sudbury. Where these conduits cross depressions and follow approximately the surface, or where they pass under rivers, their construction must be changed so that they will not only run full, but under greater or less pressure, as the case may be.

184. Masonry Conduits.—In general the conduits employed to bring water from the watersheds to reservoirs at or near places of consumption may be divided into two classes, masonry and metal, although timber-stave pipes of large diameter are much used in the western portion of the country. The masonry conduits obviously cannot be permitted to run full, meaning under pressure, for the reason that masonry is not adapted to resist the tension which would be created under the head or pressure of water induced in the full pipe. They must rather be so employed as to permit the water to flow with its upper surface exposed to the atmosphere, although masonry conduits are always closed at the top. In other words, they must be permitted to run partially full, the natural grade or slope of the water surface in them inducing the necessary velocity of flow or current. Evidently the velocity in such masonry conduits is comparatively small, seldom exceeding about 3 feet per second. The new and old Croton aqueducts, the Sudbury and Wachusetts aqueducts of the Metropolitan Water-supply of Boston, are excellent types of such conveyors of water. They are sometimes of circular shape, but more frequently of the horseshoe outline for the sides and top, with an inverted arch at the bottom for the purpose of some concentration of flow when a small amount of water is being discharged and for structural reasons.

The interiors of these conduits are either constructed of brick or they may be of concrete or other masonry affording smooth surfaces. In the latest construction Portland-cement concrete or that concrete reinforced with light rods of iron or steel is much used. Bricks, if employed, should be of good quality and laid accurately to the outline desired with about ¼-inch joints, so as to offer as smooth a surface as possible for the water to flow over. In special cases the interiors of these conduits may be finished with a smooth coating of Portland-cement mortar. If conduits are supported on embankments, great care must be exercised in constructing their foundation supports, since any sensible settlement would be likely to form cracks through which much water might easily escape. When carried through tunnels they are frequently made circular in outline. They must occasionally be cleaned, especially in view of the fact that low orders of vegetable growths appear on their sides and so obstruct the free flow of water.

185. Metal Conduits.—Metal conduits have been much used within the past fifteen or twenty years. Among the most prominent of these are the Hemlock Lake aqueduct of the Rochester Water-works, and that of the East Jersey Water Company through which the water-supply of the city of Newark, N. J., flows. When these metal conduits or pipes equal 24 to 30 or more inches in diameter they are usually made of steel plates, the latter being of such thickness as is required to resist the pressure acting within them. The riveted sections of these pipes may be of cylindrical shape, each alternate section being sufficiently small in diameter just to enter the other alternate sections of little larger diameter, the interior diameter of the larger sections obviously being equal to the interior diameter of the smaller sections plus twice the thickness of the plate. Each section may also be slightly conical in shape, the larger ends having a diameter just large enough to pass sufficiently over the smaller end of the next section to form a joint. Large cast-iron pipes are also sometimes used to form these metal conduits up to an interior diameter of 48 inches. The selection of the type of conduit within the limits of diameter adapted to both metals is usually made a matter of economy. The interior of the cast-iron pipe is smoother than that of the riveted steel, although this is not a serious matter in deciding upon the type of pipe to be used.

Steel-plate conduits have been manufactured and used up to a diameter of 9 feet. In this case the pipe was used in connection with water-power purposes and with a length of 153 feet only, the plates being ½ inch thick. The steel-plate conduits of the East Jersey Water Company’s pipes are as follows:

The diameters and lengths of the metal pipes or conduits of the Hemlock Lake conduit of the Rochester Water-works are as follows:

All metal conduits or pipes are carefully coated with a suitable asphalt or tar preparation or varnish applied hot and sometimes baked before being put in place. This is for the purpose of protecting the metal against corrosion. Cast-iron pipes have been used longer and much more extensively than wrought-iron or steel, but an experience extending over thirty to forty years has shown that the latter class of pipes possesses satisfactory durability and may be used to advantage whenever economical considerations may be served.

186. General Formula for Discharge of Conduits—Chezy’s Formula.—It is imperative in designing aqueducts of either masonry or metal to determine their discharging capacity, which in general will depend largely upon the slope of channel or head of water and the resistance offered by the bed or interior of the pipe to the flow of water. The resistance of liquid friction is so much more than all others in this class of water-conveyors that it is usually the only one considered. There is a certain formula much used by civil engineers for this purpose; it is known as Chezy’s formula, for the reason that it was first established by the French engineer Antoine Chezy about the year 1775, although it is an open question whether the beginnings of the formula were not made twenty or more years prior to that date. Its demonstration involves the general consideration of the resistance which a liquid meets in flowing over any surface, such as that of the interior of a pipe or conduit, or the bed and banks of a stream.

The force of liquid friction is found to be proportional to the heaviness of the liquid (i.e., to the weight of a cubic unit, such as a cubic foot), to the area of wetted surface over which the liquid flows, and nearly to the square of the velocity with which the liquid moves. Hence if lʹ is the length of channel, p the wetted portion of the perimeter of the cross-section, w the weight of a cubic unit of the liquid, and v the velocity, the total force of liquid friction for the length l′ of channel will be F = ζwpl′v², ζ being the coefficient of liquid friction. The path of the force F for a unit of time is v, and the work W which it performs in that unit of time is equal to the weight wal′ falling through the height h′, a being the area of the cross-section of the stream.

Hence W = ζwplʹv².v = wal′h′.(7)

In this equation

As the motion of the water is assumed to be uniform, the head lost by friction for the total length of channel l is the total fall h, and by equation (8), since

If, as in the case of the ordinary cast-iron water-pipes of a public supply system, the cross-section a is circular,

and

in which f = 8g ÷ c².

The quantity f is sometimes called the “friction factor.” For smooth, new pipes from 4 feet down to 3 inches in diameter its value may be taken from .015 to .03. An approximate mean value may be taken at .02.

The last member of equation (8) is Chezy’s formula, and it is one of the most used expressions in hydraulic engineering. Some values for the coefficient c will presently be given. The quantity r found by dividing the area of the cross-section of the stream by the wetted portion of its perimeter is called the “hydraulic mean radius,” or simply the “mean radius.” The other quantity, s, appearing in the formula is, as shown by the figure, the sine of the inclination of the bed of the stream.

In order to determine the discharge of any pipe, conduit, or open channel carrying a known depth of water, it is only necessary to compute r and s from known data and select such a value of the coefficient c as may best fit the circumstances of the particular case in question. The substitution of those quantities in Chezy’s formula, i.e., equation (8), will give the mean velocity v of the water which, when multiplied by the area of cross-section of the stream, will give the discharge of the latter per second of time. It is customary to compute r in feet. The coefficient c is always determined so as to give velocity in feet per second of time. Hence if the area of the cross-section of the stream, a, is taken in square feet, as is ordinarily the case, the discharge av will be in cubic feet per second.

187. Kutter’s Formula.—The coefficient c in Chezy’s formula is not a constant quantity, but it varies with the mean radius r, with the sine of inclination s, and with the character of the bottom and sides of the open channel, i.e., with the roughness of the interior surface of the closed pipe. Many efforts have been made and much labor expended in order to find an expression for this coefficient which may accurately fit various streams and pipes. These efforts have met with only a moderate degree of success. The form of expression for c which is used most among engineers is that known as Kutter’s formula, as it was established by the Swiss engineer W. R. Kutter. This formula is as follows:

The quantity n in this formula is called the “coefficient of roughness,” since its value depends upon the character of the surface over which the water flows. It has the following set of values for the surfaces indicated:

• n = 0.009 for well-planed timber;
• n = 0.010 for neat cement;
• n = 0.011 for cement with one third sand;
• n = 0.012 for unplaned timber;
• n = 0.013 for ashlar and brickwork;
• n = 0.015 for unclean surfaces in sewers and conduits;
• n = 0.017 for rubble masonry;
• n = 0.020 for canals in very firm gravel;
• n = 0.025 for canals and rivers free from stones and weeds;
• n = 0.030 for canals and rivers with some stones and weeds;
• n = 0.035 for canals and rivers in bad order.

188. Hydraulic Gradient.—Before illustrating the use of Chezy’s formula in connection with masonry and metal conduits, of which mention has already been made, it is best to define another quantity constantly used in connection with closed iron or steel pipes. This quantity is called the “hydraulic gradient.” If a closed iron or steel pipe is running full of water and under pressure and if small vertical tubes be inserted in the top of the pipe with their lower ends bent so as to be at right angles to its axis, the water will rise to heights in the tubes depending upon the pressures of water in the pipe or conduit at the points of insertion. Such tubes with the water columns in them are called piezometers. They are constantly used in connection with water-pipes in order to show the pressures at the points where they are inserted. A number of such pipes being inserted along an iron pipe or conduit, a line may be imagined to be drawn through the upper surfaces of the columns of water, and that line is called the “hydraulic gradient.” It represents the upper surface of water in an open channel discharging with the same velocity existing in the closed pipe.

In case Chezy’s formula is used to determine the velocity of discharge in a closed pipe running under pressure, the sine of inclination s must be that of the hydraulic gradient and not the sine of inclination of the axis of the closed pipe. In the determination of this quantity s by the use of piezometer tubes, if a straight pipe remains of constant section between any two points, it is only necessary to insert the tubes at those points and observe the difference in levels of the water columns in them. That difference of levels or elevations will represent the height which is to be divided by the length of pipe or conduit between the same two points in order to determine the sine s.

The hydraulic gradient plays a very important part in the construction of a long pipe-line or conduit. If any part of the pipe should rise above the hydraulic gradient, the discharge would no longer be full below that point. It is necessary, therefore, always to lay the pipe or the closed conduit so that all parts of it shall be below the hydraulic gradient. Caution is obviously necessary to lay a pipe carrying water deep enough below the surface of the ground in cold climates to protect the water against freezing. At the same time if the pipe-line is a long one it must follow the surface of the ground approximately in order to save expensive cutting. There will, therefore, generally be summits in pipe-lines, and inasmuch as all potable water carries some air dissolved in it, that air is liable to accumulate at the high points or summits. If that accumulation goes on long enough, it will seriously trench upon the carrying capacity of the pipe and decrease its flow. It is therefore necessary to provide at summits what are called blow-off cocks to let the air escape. At the low points of the pipe-line, on the contrary, the solid matter, such as sand and dirt, carried by the water is liable to accumulate, and it is customary to arrange blow-offs also at such points, so as to enable some of the water to escape and carry with it the sand and dirt.

189. Flow of Water in Large Masonry Conduits.—In order to apply Chezy’s formula first to the flow of the masonry aqueducts of the New York and Boston water-supplies, it is necessary to have the outlines of those conduits so that the wetted perimeter and hence the mean radius may be determined for any depth of water in them.

The figure shows the desired cross-sections drawn carefully to scale. Table XIV has been computed and arranged from data taken from various official sources so as to show the depth, mean velocity, discharge per second and per twenty-four hours, and the coefficient used in Chezy’s formula, together with the coefficient of roughness n in Kutter’s formula for the conduits shown in the figure.

This table exhibits in a concise and clear manner the use of Chezy’s formula in this class of hydraulic work.

TABLE XIV.

• LEGEND:
• (A) = Depth, in Feet.
• (B) = Hydraulic Radius r, Feet.
• (C) = Coefficient c.
• (D) = Mean Velocity, Feet.
• (E) = Cubic Feet per Second.
• (F) = Gallons per 24 Hours.

190. Flow of Water through Large Closed Pipes.—The masonry conduits to which consideration has been given in the preceding paragraphs carry water precisely as in an open canal, but the closed conduits or pipes of steel plates and cast-iron, like the Hemlock Lake conduit at Rochester and the East Jersey conduit of the Newark Water-works, are of an entirely different type, as they carry water under pressure. Hence the slope or sine of inclination s belongs to the hydraulic gradient rather than to the grade of the pipe itself. Where the pipe-line is a long one its average grade frequently does not differ much from the hydraulic gradient, but the latter quantity must always be used. As in the case of the masonry conduits, the coefficient c in Chezy’s formula will vary considerably with the degree of roughness of the interior surface of the pipe, with the slope s, and with the mean radius r. An important distinction must be made between riveted steel pipes and those of cast-iron, for the reason that the rivet-heads on the inside of the former exert an appreciable influence upon the coefficient c. The rivet-heads add to the roughness or unevenness of the interior of the pipe. Table XV gives the elements of the flow or discharge in the two pipe-lines which have been taken as types, as determined by actual measurements; it also exhibits similar elements for timber-stave pipes, to which reference will be made later.

As would be expected, the velocity of flow in these pipes may be and generally is considerably higher than the velocity of movement in masonry channels. Both Tables XV and XVI give considerable range of coefficients computed and arranged from authoritative sources, and the coefficients c for Chezy’s formula represent the best hydraulic practice in connection with such works at the present time. In using the formula for any special case, great care must be taken to select a value for c which has been established for conditions as closely as possible to those in question. This is essential in order that the results of estimated discharges may not be disappointing, as they sometimes have been where that condition so necessary to accuracy has not been fulfilled.

TABLE XV.

VALUES OF COEFFICIENT  c.

• LEGEND:
• (A) = Hydraulic Radius r.
• (B) = Hydraulic Gradient.
• (C) = Mean Velocity.

• LEGEND:
• (A) = Coefficient c.
• (B) = Cubic Feet per Second.
• (C) = Gallons per 24 Hours.

TABLE XVI.

VALUES OF c IN v = c√rs.