various diameters under various heads of pressure. The quantity discharged at the outlet of any pipe depends on the velocity of flow, and is affected materially by the length and diameter of the pipe, by the number and kind of bends or other changes of the direction of flow, and by the effective head of water, and in a slight degree by the shape of the entry or orifice. The effect produced by the friction of the flowing water on the surface of the pipe or channel is so very great as to form the chief factor in hydraulic calculations for long pipes. It increases in a less ratio than the square of the velocity. If the velocity of the water be doubled the friction is increased nearly four times; if the velocity be increased three times the effect of friction is increased nearly nine times, and so on. The velocity of water flowing through long pipes of any given diameter depends not on the inclination of any particular length, but on the ratio between the head of water and the length of pipe, otherwise the hydraulic mean gradient. Except for the effect of friction at bends, velocity is independent of the direction of the pipes, whether laid at uniform gradient or not throughout its length, provided that the contour of the pipe is below the hydraulic mean gradient along the whole line. Velocity is also independent of the question whether the entry and outgo are at or below their respective upper and under surfaces, the measure of the head being taken from surface of water above to surface of water below, if not discharging freely in air. The effective power of any head of water flowing through pipes is used or worked up under the two opposing forces. One is the initial check given to velocity at the entry irrespective of friction, and the other force is the constant check of friction alone along the whole length of the conduit. The relative importance of these two factors in all hydraulic computations varies with the length and diameter of the pipe, the velocity and quantity of water discharged. For instance, in a pipe 8 inches in diameter, 10,000 feet long, discharging 300 gallons a minute, the relative value of the head necessary to overcome the check to velocity by friction, and the head to overcome the check to velocity at the entry, is as 40 feet to 2 inches. The relative importance of the two factors would be reversed in the case of the pipe being only 10 feet long. Two feet head of water on an 8-inch pipe, 1,000 feet long, will yield a velocity of 179 feet per second, and discharge 234 gallons per minute. Two feet head of water on an 8-inch pipe, 100 feet long, will yield a velocity of 6J feet per second, and discharge 840 gallons per minute ; 10 feet long, a velocity of 22 feet per second, and discharge 2,874 gallons per minute. <Callout type="important" title="Important">Understanding these principles is crucial for ensuring efficient water flow in plumbing systems.</Callout> 72 DOMESTIC SANITARY DRAINAGE AND PLUMBING. Ey telwein's rule for finding the delivery of water in pipes gives a somewhat lower result than these figures, and is as follows : Find the fifth power of the diameter of the pipe in inches ; multiply it by the head of water in feet ; divide the product by the length of pipe in feet, find the square root of the quotient, and multiply that by the constant number 471, which gives the cubic feet of water discharged per minute. Hawksley's formula for the same result in practice is to multiply the diameter of the pipe in inches by the constant number 15, find the fifth power of the product and multiply by the head of water in feet, divide the product by the length of the pipe in yards, and find the square root of the quotient, which gives the number of gallons discharged per hour. Neville's general formula to find the velocity in feet per second is to divide the head in feet by the length of pipe in feet, multiply the quotient by the hydraulic mean depth in feet, find the cube root of the product, and multiply this by the constant number 11 ; note the result. Now proceed to find the square root of the same product already ascertained, of which the cube root has been just taken, and multiply this by 140, and note the result. Deduct the lesser result noted from the greater, and the remainder is the velocity in feet per second. Manning's formula, first published in December, 1889, provides a simple and accurate method of determining the mean velocity of the flow of water in open channels and pipes. It has been carefully tested and compared with the results obtained by actual experiment, and is here given by kind permission. V = CsVr^ + ?-05^. V represents the mean velocity in feet per second; C, a co-efficient which varies with the nature of the conduit ; S, the sine of the angle of inclination of the surface found by dividing the head in feet by the length of conduit in feet ; E, the mean radius, or mean hydraulic depth, found by dividing the area of the section of water actually flowing by the length of the wetted perimeter. The values of the co-efficients given are — Old cast-iron pipes . . . . 85 New cast-iron pipes .114 New drawn-lead pipes . . .165 The formula is worked out as follows : — Find the square root of the mean radius or mean hydraulic depth; divide the mean radius by the constant number 7, and add this quotient to the square root already found; deduct from this sum the decimal '05; multiply the remainder by the co-efficient appertaining to the nature of the conduit in question, and multiply that product by the square root of the sine of the angle of inclination of the surface. This final product of the formula gives the actual velocity in feet per second. It is necessary to bear in mind that new pipes will become old, and therefore that it will be safe to use the co-efficient for old pipes in all calculations, as the values of the co-efficients vary according to the roughness of the pipe or channel. The experience and judgment of the engineer must be called on to determine wisely what co-efficient to adopt between, the extremes of 85 and 114, according to the inner surface of the pipe. In cylindrical pipes the velocity in feet per second, multiplied by the square of the diameter of the pipe in feet multiplied by the constant number 47124, gives the discharge in cubic feet per minute. It was found by experiment with the flow of water through the pipes of the Vartry water service to Dublin that the actual delivery and flow considerably exceeded the theoretical flow calculated from formula. Some hydraulic engineers calculate the loss of head through friction in pipes as follows : To find the due delivery in gallons per minute multiply the diameter of the pipe in inches by the constant number 3, and find the fifth power of the product ; multiply this by the head of water in feet, and divide the product by the length of the pipe in yards ; find the square root of the quotient, which gives the number of gallons discharged per minute. To find the head of water in feet necessary to discharge a given number of gallons through a given length and diameter of pipe, square the number of gallons and multiply by the length of pipe in yards, and note the result ; multiply the diameter of the 'pipe in inches by the constant number 3, and find the fifth power of the product ; divide this into the result already noted, and the quotient will be the requisite head of water in feet. To find the diameter of a pipe of any given length which will deliver a given number of gallons per minute, square the number of gallons per minute, multiply by the length in yards, divide the product by the head of water in feet, find the root of the fifth power, and divide that root by the constant number 3. To find the length of any given diameter straight pipe which will deliver a given number of gallons per minute under a given head of water, multiply the diameter in inches by the constant number 3, find the fifth power, multiply by the head of water in feet, and divide the product by the square of the number of gallons per minute. ELEMENTARY SCIENCE FOR PLUMBERS. 75 To find the head of water necessary to overcome friction due to change of direction of flow in bends, first find the radius of the circle drawn through the centre line of the bend in inches, then the radius or half of the diameter of the pipe in inches, and divide the former into the latter. The quotient will be the ratio of the radius of the bend to radius of bore. Now, the co-efficients for the curvature thus found in pipes of circular section are as follows : — -1 -2 -3 -4 -5 -6 -7 -8 -9 10 •131 138 158 -206 '294 44 -66 98 14 20 Now find the angle of the bend with the forward line of direction, divide by the constant 180, and multiply the quotient by the co-efficient for curvature as found in the table; multiply the product by the square of the velocity in feet per second, and the product by the constant '0155. The final product gives the head of water in feet required to overcome the extra friction caused by the bend. Box gives some general idea of the head of water lost by right-angle quick bends. In a 2-inch diameter pipe, with the velocity necessary to discharge 45, 65, 80, 95, 110, 130, and 160 gallons per minute, the loss of head in inches for each bend will be respectively 0'5, 10, 15, 20, 30, 40, and 60 in. Overflow pipes are essential for all tanks, but are frequently fixed too small, and are sometimes neglected altogether. When they are provided as stand-pipes, having brass ground-in wash-out valve-washers, they should be formed with trumpet-shaped tops, as greatly increased efficiency is secured. Three inches margin of safety should be given from the top of stand-pipe to top of tank. If this loss of useful depth of tank cannot be afforded, recourse must be had to Mr. Appold's contrivance. He places a hollowed copper cover over the trumpet mouth, fixed on brackets, so that the lip of the inverted hollow cover is level with the lip of the trumpet mouth. The water does not immediately flow over, but rises a little above the lip, suddenly overflows, causes partial vacuum, and the maximum quantity which the stand-pipe can take is discharged. The stand-pipe overflow may be within one Fio. 13. — Overflow stand-pipes. inch of the level top of cistern when this contrivance is adopted. Stand overflow pipes three feet long, with ordinary trumpet mouths, will discharge the following quantities of water in gallons per minute : — Diameter: 1 in. 18 40 2 in. 2i in. 80 140 Sin. 200 Si in. 300 4 in. 400 5 in. 600 6 in. 900 It is necessary to bear in mind that overflows taken in short pipes from the sides of tanks will not give anything like the results shown by the stand-pipes, and increased provision for safe overflow must be made. Side overflow pipes will only discharge the following quantities of water in gallons per minute: — Diameter : 1 io. 1} in. 2 in. 2^ in. 'S in. 3} in. 4 in. 6 in. 6 in. 3 8 12 30 50 70 110 170 280 This extraordinary difference in discharging power will surprise many plumbers. Aerometry. Aerometry brings us to consider another class of phenomena which plumbers require some acquaintance with in their trade. Pumping water from wells, etc., syphonage in connection with flushing-tanks, steam heating, ventilation of drains, pipes, buildings, are all dependent on this branch of the laws of nature. Failure must follow the workmen and masters who know not these laws. There are three layers of matter composing this earth: the solid ground, the liquid water, and the gaseous air. Fishes living in the liquid ocean are provided with contrivances suitable to the liquid medium in which they live ; they possess a natural instinct which tells them where best they can exist, and warns them of dangers they must avoid. Fishes accustomed to shallow waters dare not go down to the depths of the sea, where deep-sea fishes alone may live ; the increased pressure of sea water at those depths would crush them. Divers cannot work long under water owing to this constant pressure of the water on their bodies, and beyond a certain moderate number of feet depth they cannot exist at all. At the bottom of the great oceans, where the depth extends to thousands of feet, nothing can live: strong metal vessels are crushed in by the overpowering pressure. 78 DOMESTIC SANITARY DRAINAGE AND PLUMBING. We can see this ocean of water and can feel its power, so we learn by experience how best to control and utilise it. There is another ocean, at the bottom of which we live and move. It is about a hundred times greater in height above us than the sea in depth below us. Fortunately it is also much lighter, or we would be unable to sustain the crushing pressure. Water is composed of a combination of oxygen and hydrogen gas. Pure air is composed of a mixture of oxygen (20'97), nitrogen (7900), and carbonic acid gas (03) ; aqueous vapour and sometimes ammonia, in varying proportions, are also found. Air containing 2096 oxygen, 7900 nitrogen, and 04 carbonic acid is a fairly attainable pure air. The oxygen and nitrogen are always proportioned as nearly 21 parts oxygen to 79 parts nitrogen by volume in 100 parts. We cannot see the air, but occasionally we can both feel and see its effects when it moves, as in winds and storms, and we are constantly in subjection to the pressure of the atmosphere caused by the attraction of gravitation of the earth upon its particles of gaseous matter. The immense volume of the ocean of air above us presses down upon the earth in exact proportion to its mass. We have already considered some of the properties of gaseous matter of which the atmosphere is formed. We know now that it is elastic and expansive; for if a small volume be admitted to an empty vessel it will expand and fill it ; if compressed it will contract in volume, recovering and expanding when the pressure is removed. In 1640 an Italian plumber in Florence, unacquainted with the science of aerometry, received an order to make, supply, and fix a pump over a well at the palace of the grand duke. He made a first-class pump, and fixed it with great care. The well was deeper than usual, and the water excellent, no doubt ; but the water would not pump out of the well, nor rise in the suction-pipe beyond a height of thirty-four feet from the surface, although the workmen pumped as rapidly and strongly as possible. The engineers of Florence were consulted, but were unable to explain the matter; the poor plumber, therefore, could hardly be blamed for his ignorance. Galileo, then seventy-six years of age, was applied to; but he gave an unsatisfactory reply. However, the question set him to study the problem. He believed that the pressure of the atmosphere on the surface of the water in the well forced the water to rise as far as it did rise, and he saw at once that the answer, 'Nature abhors a vacuum,' was not a proper answer, for this gave no reason why Nature should abhor a vacuum up to a certain height, and then abhor it no longer. His experiments proved beyond doubt the weight of air, but before he could solve the problem he died, and the plumber's pump had to be taken away, and put in the window for some other customer! Galileo's pupil, Torricelli, born in 1608, took up the question. He made a glass tube, which he closed at one end and then filled with mercury, which is thirteen and a half times heavier than water; he then reversed the tube, covering the open end with his finger, and plunged the end into a vessel of mercury. He watched the mercury descend in the tube and settle itself about thirty inches from the level of the mercury in the vessel, where it remained nearly invariable. Now, 30 inches of mercury multiplied by 13J, the difference between the weight of mercury and water, gives 34 feet nearly; so Galileo was right in his theory that the thirty-four feet of water in the suction-pipe was supported by the pressure of the atmosphere, and the atmosphere being capable of supporting not more than thirty-four feet of water or thirty inches of mercury, it was useless waste of time to endeavour to make more of it. To Galileo and Torricelli, therefore, plumbers are indebted for this knowledge. We need no longer try, like the Florentine plumber in 1640, to force our pumps to do impossibilities.
Key Takeaways
- Understanding hydraulic principles is essential for efficient water flow in plumbing systems.
- Various formulas can be used to calculate the velocity and discharge of water through pipes.
- The head of water required to overcome frictional losses varies with pipe diameter, length, and velocity.
Practical Tips
- Use Manning's formula for accurate calculations of mean velocity in open channels and pipes.
- Ensure that overflow pipes are properly sized to prevent tank overflows and ensure safety.
- Regularly inspect and maintain plumbing systems to minimize frictional losses and maximize efficiency.
Warnings & Risks
- Improper sizing or placement of overflow pipes can lead to water damage and potential hazards.
- Ignoring hydraulic principles can result in inefficient water delivery, leading to increased energy consumption.
- Incorrect application of formulas may lead to miscalculations and system failures.
Modern Application
While the specific formulas and methods described in this chapter have evolved over time, the fundamental principles of hydraulics remain crucial for modern plumbing systems. Understanding these principles helps ensure efficient water delivery, proper sizing of pipes, and safe operation of plumbing fixtures. Modern tools and software can assist with more precise calculations but the basic concepts outlined here provide a solid foundation.
Frequently Asked Questions
Q: What is Manning's formula used for in plumbing systems?
Manning's formula is used to determine the mean velocity of water flow in open channels and pipes. It provides an accurate method that has been tested against experimental results, making it a reliable tool for hydraulic calculations.
Q: Why are overflow pipes important in plumbing systems?
Overflow pipes are essential as they prevent tanks from overflowing when the water level exceeds safe limits. They ensure safety by allowing excess water to be discharged without causing damage or hazards, and maintain proper tank functionality.
Q: How does the head of water affect pipe flow velocity?
The head of water significantly affects the velocity of water in pipes. A higher head results in a greater velocity, as the pressure from the height of the water column drives more water through the system. This relationship is crucial for designing efficient plumbing systems.