note on basics of hydraulics

7/28/2019 Note on Basics of Hydraulics

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PrefaceHydraulics is as old as books of history are available; the science was inexistence before the engineering books came into existence.This is the note and shall not be treated as book as I am not the author of the subject but the basics are required to be put into one paper so that thefresh comers in water supply sectors can learn easily.The fresher now appointed are bright and can carry forward the study inhydraulics, hence for their guidance some case study are included in thisnote.The old engineers who had lost their theoretical touch can also refresh byreading this note.Once I was also fresher initially I stated to work in maintenance sector and also task for re-commissioning the portion of Banni water supply

scheme was given, without proper knowledge of hydraulic and pumpsmany months , efforts and money were wasted in the given task. Also ingovernment sectors we have sufficient number of bosses but restrictednumber of Gurus. Every engineer is supposed to rise on its capabilities.Hence these notes shall serve the purpose of guiding the fresher in water supply sector.This note does not cover the design part of gravity main, rising main or networking and this could only be learnt after basics of hydraulics isclear.

In preparing notes use of internet and some books is made.

Adipur Deepak Ramchandani


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IndexSr.No Description Page

Number 1 History of Hydraulics 4

2 Basic requirement of water supply 93 Definition of hydraulics 124 Hazen Williams formula 135 Quotes from different experts for HW formula 176 Modified Hazen-Williams Formula 197 History of Chezy formula 198 History of Darcy Weisbach equation: 209 Manning formula & Moodys diagram 2210 Hydraulics radius 2311 Discharge through v- notch 29

12 Velocity of free flow pipes 3013 Bernoullis equation & applications 3114 Continuity & conservation of matter 3815 Design of thrust block 4516 Minor head losses 4717 Analysis of rising main design 5518 Case study of valves on reduced section 6719 Case study of importance of minor losses 7120 Affinity law 7521 Water losses 8722 Reasons of failure of pipe network 9023 Calculation of % of valve opening in RCC ESR 9324 Correction to be applied in rising mains where intermediate

station is at higher elevation then terminal point95


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History of HydraulicsIf the word hydraulics is understood to mean the use of water for the benefit of mankind, then its practice must be considered to be even older than recordedhistory itself. Traces of irrigation canals from prehistoric times still exist in

Egypt and Mesopotamia; the Nile is known to have been dammed at Memphissome six thousand years ago to provide the necessary water supply, and theEuphrates River was diverted into the Tigris even earlier for the samepurpose. Ancient wells still in existence reach to surprisingly great depths; andunderground aqueducts were bored considerable distances, even throughbedrock. The observation of excavation of Harappa & Mohejo-daro of Sindhuvalley civilization in Dhora vira in Kachchh , which is about 5000 years oldshows that , houses were provided with ceramic conduits for water supply anddrainage; and legend tells of vast flood-control projects in China barely amillennium later. All of this [1] clearly demonstrates that men must havebegun to deal with the flow of water countless millennia before these times.

Though both the art and the science of hydraulics treat of such flows, theyobviously differ significantly in time and substance. Hydraulic practicenecessarily originated as an art, for the principles involved could beformulated only after long experience with science in general and water inparticular. However necessary the conduct of the art thus was to the eventualdevelopment of the science, it is almost exclusively with the science of hydraulics that the present article will deal. As a matter of fact, the subjectmatter of the traditional college course in hydraulics -- particularly as it wastaught in the not-too-recent past -- provides a framework on which the historyof the science can conveniently be based.

Such a course usually began with the topic of hydrostatics -- thecharacteristics of liquids at rest. Instructors then proceeded to the principle of continuity (the conservation of fluid mass ) and a form of the work-energyprinciple known as the Bernoulli theorem. In passing, note was taken of means of measuring velocity, pressure, and discharge, including the use of small-scale models to simulate flow conditions in themselves too large to test.These principles were then applied to the study of flow from orifices, over weirs, through closed and open conduits, and past immersed bodies. Simpleas such matters now seem when taught, they actually took centuries tounderstand. Particularly noteworthy is the fact that many such principles werefirst clarified by men like Isaac Newton whose interests extended far beyondhydraulics itself.

The Greek who made the most lasting contribution to hydraulics was theSicilian mathematician Archimedes (287-212 B.C.), who reasoned that afloating or immersed body must be acted upon an upward force equal to theweight of the liquid that it displaces. This is the basis of hydrostatics and alsoof the apocryphal story that Archimedes made this discovery in his bath andforthwith ran un clothed through the streets crying "Eureka!" Nevertheless,even though Archimedes' writings, like those of his fellow Greeks, were

faithfully transmitted to the West by Arabian scientists, further progress inhydrostatics was not to be made for another 18 centuries.


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In the course of the millennium following the time of Archimedes, the scienceof hydraulics retrogressed rather than advanced. True, though the Romansdeveloped extensive water-supply and drainage systems, and windmills andwater wheels appeared on the scene in increasing numbers, these

represented the art rather than the science. Paradoxically, although Aristotletaught that knowledge must progress, his teachings eventually came to becrystallized, so to speak, and in the time of Saint Thomas Aquinas (1225-74),they were even adopted as gospel truth by the church. In the same period, onthe other hand, researchers in the early universities -- particularly Paris,Oxford, and Cambridge -- gradually began to establish simple mechanicalrelationships such as that between velocity and acceleration.

Whereas the Greeks tended to reason without recourse to observation, it wasthe Italian genius Leonardo da Vinci (1452-1519) who first emphasized thedirect study of nature in its many aspects. Leonardo's hydraulic observationsextended to the detailed characteristics of jets, waves and eddies, not tomention the flight of birds and comparable facets of essentially every other field of knowledge. In particular, it was Leonardo who first correctly formulatedthe basic principle of hydraulics known as continuity: the velocity of flowvaries inversely with the cross-sectional area of a stream. Unfortunately, notonly were his copious notes written in mirror image (probably for reasons of secrecy), but, in addition, most of them were lost for several centuries after hisdeath. Thus his discoveries had little effect on the growth of the science.

The second essential contribution to hydrostatics was made by the Dutchhydraulic engineer Simon Stevin (1548-1620) in 1586, nearly two millenniaafter the time of Archimedes. Stevin showed that the force exerted by a liquidon the base of a vessel is equal to the weight of a liquid column extendingfrom the base to the free surface. That this force does not depend on theshape of the vessel became known as the hydrostatic paradox.

If Leonardo was the first scientific observer of note, it was Galileo (1564-1642)who added experimentation to observation, thereby throwing initial light on theproblem of gravitational acceleration. In his study of the phenomenon, henoted that a body sliding freely down an inclined plane attained a certainspeed after a certain vertical descent regardless of the slope; it is said that he

hence advised an engineer that there was no point in eliminating river bends,as the resulting increase in slope would have no effect! Whereas Leonardowas a loner, Galileo gathered a small school around him. One of his students,the Abbe Benedetto Castelli (c.1577-c.1644), rediscovered the principle of continuity and delved further into other aspects of the science, though notalways correctly. His younger colleague Evangelista Torricelli (1608-47)applied his mentor's analysis of parabolic free-fall trajectories to the geometryof liquid jets. Torricelli also experimented with the liquid barometer, thevacuum above the liquid column being comparable to the void that Galileofound to develop in a pump whose suction pipe exceeded a certain length; inother words, nature abhorred a vacuum only up to a certain point.


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The French scientist Edme Mariotte (1620-84) is often called the father of French hydraulics because of the breadth of his experimentation; this includedsuch matters as wind and water pressure and the elasticity of the air, a qualitywhich we usually associate with the name of the Englishman Robert Boyle(1627-91) whereas the latter appears to have coined the word hydraulics, in

France Boyle's law bears the name of Mariotte. Only a few years younger than Mariotte, the Italian Domenico Guglielmini (1655-1710) is similarlyconsidered by many to have been the founder of the Italian school. Butwhereas Mariotte was a laboratory experimenter, Guglielmini made extensivefield measurements of river flow. Interestingly enough, Guglielmini eventuallybecame a professor of medicine!

At about the same time, the short-lived French savant Blaise Pascal (1623-62) concerned himself with the same barometric problems as the equallyshort-lived Torricelli (not to mention Mariotte), but it was Pascal who finallycompleted the principles of hydrostatics. Not only did he clarify thetransmissibility of pressure from point to point and its application to thehydraulic jack, but he also showed that the barometric (i.e., atmospheric)pressure must vary with elevation and hence that the barometer would have azero reading in a vacuum.

Daniel's work contained much that was new -- for example, the use of manometers, the kinetic theory of gases, and its propulsion -- but nowhere inthe book (or in his father's either) can one find what is known as the Bernoullitheorem. Just as its source, Leibniz's energy principle consisted of onlypotential and kinetic terms, so too did the Bernoulli equation; thecorresponding pressure term was evaluated separately by means of Newton'smomentum equation.

In actuality, the first true Bernoulli equation was derived by Euler, anoutstanding mathematician, from his equations of acceleration for theconditions of steady, irrotational flow under gravitational action. Euler alsodeserved credit for a number of equations of hydraulics and for inventing -- atleast on paper -- a workable hydraulic turbine. Worthy of mention in the samebreath as Euler and the Bernoullis was Jean Lerond d'Alembert ( 1717-83 ),best known for his co editorship of the French encyclopedia but also amathematician in his own right. He proved in 1752 that under steady,

irrotational conditions a fluid should offer no resistance to the relative motionof an immersed body: the d'Alembert paradox. D'Alembert is also known for having been one of three French scientists to have made in 1775 what weresaid to have been the first towing-tank tests of ship-model drag; they were,however, preceded by some nine years by those of our own BenjaminFranklin (1706 90), himself a potential hydraulician!

Two essential measuring instruments came into being at this time, the Pitottube and the rotating arm. The first still bears the name of its inventor, theFrenchman Henri de Pitot (1695-1771), who called it a "machine" for determining the speed of flowing water. [25] It consisted of two vertical glass

tubes connected at their top by a valve, one tube simply being open at thebottom and the other L-shaped with its open end pointing upstream; the


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the mathematics of the phenomenon. Hagen, however, had remarked in an1854 paper that the flow was not always laminar, the efflux jet sometimesbeing clear and sometimes frosty; similarly, sawdust suspended in the water sometimes moved in straight lines and sometimes very irregularly; in the latter instances he noted that his resistance equation no longer applied.

Though countless contributors to hydraulic science of this period are to befound in the ever-growing literature, only a few can be mentioned at this point.These include the Italian Giovanni Battista Venturi (1746-1822) and theGermans Johann Albert Eytelwein (1764-1848) and Julius Weisbach (1806-71). In addition to Bernoulli, the men whose names are now best known inhydraulics were two Englishmen who lived in the latter part of the last century.One was the Manchester professor Osborne Reynolds (1842-1912), who in1873 also experimented with flow through tubes, introducing the viscosity toform a parameter marking the borderline between laminar and turbulent flow,now known as the Reynolds number. Reynolds also showed by the injectionof dye the difference between the two states of motion.

William Froude (1810-79) was a somewhat older contemporary of Reynoldswhose interest lay in the field of naval architecture. Froude built himself atowing tank on his own property and in part with his own funds, for theoperation of which he had formulated a similarity law for flows under theinfluence of gravity. This law has come to be known under Froude's name,although it had actually been announced at least 20 years earlier byFerdinand Reech (1805 80) ,an Alsatian teaching in a naval college at Paris.But Froude was the first to note the development along the hull of ships of what came to be known as the boundary layer, a phenomenon of viscousshear which eventually was shown to be a function of the Reynolds number. Itis hence only fair to note that Reynolds was the first to utilize the Froude lawof similarity in model tests of tidal action in the Mersey estuary.

At the time that hydraulics was becoming an applied science, mathematicianswere developing its theoretical counterpart known as hydrodynamics. Ablybegun by Euler and d'Alembert, the practice was continued by such equallyfamous men as Lagrange (1736-1813 ), Laplace ( 1749-1827 ) [45],Helmholtz ( 1821-94 ) [46], Kelvin (1824-1907), and Rayleigh (1842-1919), asrecorded in the many editions of the treatise Hydrodymimics by the

Manchester professor Horace Lamb (1849-1934). However, althoughpresumably dealing with the same fluids, the two subjects were far apart, for hydraulics still lacked mathematical rigor, and hydrodynamics, sufficientcontact with reality. Thus, when human flight became a likelihood, neither hydraulics nor hydrodynamics could provide a useful scientific basis for theunderstanding of aerodynamic lift if not of drag.

Fortunately, a new science, the mechanics of fluids, came into being at thehands of Ludwig Prandtl (1875-1953), a German mechanical engineer teaching at the University of Gottingen. He reasoned as early as 1904 thatrelative motion between a fluid and a streamlined boundary could be analyzed

in two parts: a thin layer at the boundary providing the viscous resistance tomotion, and the fluid outside the boundary layer providing, in accordance with


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Basic Requirement of Water Supply ( IS:1172-1983)

1. Requirement of water supply, drainage and sanitation for residenceshall be assume that a minimum water supply of 200 liters per day isassured together with a full flushing system

2. The minimum value of water supply of 200 liters per head per day maybe reduced to 135 lpcd for houses for Lower Income Group (LIG) or Economically Weaker Section of society (EWS) depending uponprevailing conditions

3. Out of 200 liters per head per day, 45 liters per head per day may betaken for flushing requirement and remaining quantity for other domestic purpose.

4. Water supply requirement for building other than residence.

Sr.No Type of structure Requirement

aFactories where bath rooms are requiredto be provided 45 lpcd

bFactories where no bath rooms arerequired to be provided 30 lpcdHospital ( Including laundry)

c Beds not exceeding 100 340 per bedd Beds exceeding 100 450 per bede Nurses home and medical quarters 135 lpcdf Hostel 180 lpcdg Offices 45 lpcdh Restaurant 70 per seati Cinema/ Theaters 15 per seat

j Schools Day school 45 lpcdk Schools Boarding 135 lpcd


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5. Water requirement for railway stations

Sr.No Nature of station

Withbathingfacalities

Without bathingfacalities

aIntermediate stations( excluding main andexpress stops) 45 lpcd 25 lpcd

b Junction station with mail and express stops 70 lpcd 45 lpcdc Terminal stations 45 lpcd 45 lpcd

dStations where international and domesticports 70 lpcd 70 lpcdThe number of persons shall be determinedby the average number of passenger

handled by the station daily, dueconsideration may be given to the staff andvendor likely to use facalitiesConsideration shall be given for seasonalaverage peak requirement

6. Public parks and private parks 20-30 lit / sqmt

7. For washing cars 20 lit/car/day

8. Water requirement of animals, public purpose, SP etc 10% of total

demand

9. Road watering 140 lit/ 100 sqmt

10. Sewer flushing 4.5 lit/ head

11. Road side trees 28000 lit/ km


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Definition of Hydraulics:-

"a branch of science that deals with practicalapplications of liquid in motion."

The science started over thousands of years ago with Aristotle (384-322 B.C.)and Archimedes (287-212 B.C.).

Many European scientists also contributed to hydraulics, including da Vinci,Mariotte and Boyle.

The scientist who discovered the main principle I will be talking about isPascal

A French scientist who studied math and physics. His work included studyingatmospheric pressure, conic sections and the principles of hydrostatics.

After Pascal there is Bernoilli, Franklin and Froud, who all also contributed tothe science of hydraulics.

How it works (the physics part)

The basic rule of using hydraulic power is Pascal's Principle.Pascal's Principle: pressure exerted on a fluid is distributed equally throughout the fluid.

Hydraulics uses incompressible liquids so the applied pressure from one end(small arrow) is equal to the desired pressure on the other end (big arrow).

The big arrow is pointing toward a piston that is free to move, and issometimes connected to a rod. When the force is applied, the piston moves


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y in this empirical equation are 0.63 and 0.54. Equations for U.S. customaryand the S.I. units are shown below.

S.I. Units

V= Velocity in mt./second D= Diameter in mt. S=Slope i. e. head loss per kilometer C=Is constant, value is 100

THIS HAZEN -WILLIAMS EQUATION IS OBTAINED FOR V=0.9 mt/second AND R=0.3 BUT USED FOR WIDE RANGES OF VELOCITIES ANDDIAMETERS

The formula is calculated considering the reference velocity as 0.9mts/second , the Hazen -Williams equation correct if the velocity of water inthe pipe is 0.9 mt/second. As velocity changes from that value there is a smallerror. To eliminate the error Walski in 1984 gave the below mentioned formulafor the correction in value of C.

C= C0 ( V0/V)^0.081

Where

C= Derived constant at actual velocity

C0= Constant at reference velocity

V0= Reference velocity

V= Actual velocity

For gravity mains where velocity is not much more such minor correction invalue of C can be neglected , where as in rising main where pipe is designedfor considerable velocity and length is long, then the correction factor isadvisable to be applied. The increase in head shall decrease the discharge of the pump.


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Value of Hazen Williams coefficient for different type of pipes

Type of pipe Value of Constant " C

Cast Iron pipe 100

Mild Steel Bare pipe 100

A.C. Pipe 130

PSC pipe 130

D.I.Pipe 130

P.V.C Pipe 140

CORRECTION FACTOR FOR HAZEN WILLIAMS COEFFICIENT

C Value Velocity below 0.9mt/sec

Below 100 Add 5% to C value

100-130 Add 3% to C value

130-140 Add 1% to C valueReference:- Book of Pramod R.Bhave P/41


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EFFECT IF AGE ON HAZEN WILLIAMS COEFICIENT FOR COATED CIPIPES

Degree of corrosion Diameter in mm75 300 600 1200

30 Years ageSlight 100 112 117 120Moderate 83 97 102 107

Appreciable 59 78 83 89Severe 41 58 66 73

60 Years ageSlight 90 102 107 112

Moderate 69 85 92 96 Appreciable 49 66 72 82Severe 30 48 56 62

Reference:- Book of Pramod R.Bhave P/86


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Quotes from different experts for HW formula(source internet)

The original research used by Hazen and Williams when theypublished their book of hydraulic tables indicates that the researcherstested pipes in the range of about 0.26 to 10.4 feet/sec for iron pipeand calculated C coefficients ranging up to 148.5 (this being for newcast iron pipe, coated and well laid as reported by Williams, Hubbeland Fenkell).

Why Hazen-Williams? Not only is it terribly inconvenient to use for any product other than water, as it does not consider viscosity, but italso fails to differentiate between laminar and turbulent flowconditions. In its unmodified form, it should only be used in thetransition zone and slightly there above, for large diameter pipelineswith cool water.

The HazenWilliams formula is frequently used for the design of large-diameter pipes, without regard for its limited range of applicability. Thispractice can have very detrimental effects on pipe design, and couldpotentially lead to litigation.

Finally, it is important to point out that the indiscriminate application of the HazenWilliams formula either in the design or verification of water-

supply systems is far from a simple academic problem. It may lead toserious practical and conceptual implications in otherwisestraightforward computations.

The HW equation is used because it is an equation that has a singleterm. It was developed exclusively for water in distributions systemsusing THAT distribution system to calibrate the equation. It continuesto be put to use because it works, and complicated networks can beanalyzed through this single equation. It is not good for laminar flowbut the cases of laminar flow in municipality water systems areextremely rare, and typically the when there is laminar flow it is not a

case when the system is interesting (3am) and even then the error is inthe conservative direction.Good technique in using the HW equation includes field monitoring andselected flow tests to see what the current performance of the pipeis. The different qualities of water will change the flow characteristicsof the pipe over time. It is difficult to look at a 50 year old pipe andmeasure the /d roughness for the d-w equations without taking thepipe out of service.HW equation is for water and nothing else. It uses a valid simplificationbecause it uses is bounded by the use in the application. Thatsimplification allows the engineer to evaluate extremely complexnetworks with excellent results.


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The Hazen-Williams formula gives good results for liquids which havekinematic viscosities around 1.2EE-5FT2/sec (which corresponds to 60Deg F water).

At extremely high and low temperatures, the H-W formula can be as

much as 20% in error for water.

The H-W formula should only be for water in turbulent flow.Use of the H-W also requires a knowledge of the coefficient C.


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Modified Hazen-Williams Formula

The Modified Hazen Williams formula has been derived from Darcy-Weisbach and Colebrook-White

equations and obviates the limitations of Hazen-Williams formula. V = 3.83CRd0.6575(gs)0.5525 / v0.105 Where, CR = coefficient of roughness d = pipe diameter g = acceleration due to gravity s = friction slope v = viscosity of liquid For circular conduits, v20 0 c for water =10-6m2 / s and g = 9.81m / s2

The Modified Hazen Williams formula derived as

V = 143.534 CR r0.6575S0.5525h = [L(Q / CR)1.81]/994.62D4.81in which,V = velocity of flow in m/s.CR = pipe roughness coefficient, ( 1 for smooth pipes; < 1 for rough pipes);r = hydraulic radius in m;

s = friction slope;D = internal diameter of pipe in m;h = friction head loss in m;L = length of pipe in m; andQ = flow in pipe in m3 / s

History of Chezy formula:-

A ntoine Chzy was born at Chalon-sur-Marne, France, on September 1,1718, and died on October 4, 1798. He retired in 1790 under conditions of extreme poverty. It was not until 1797, a year before his death, that the effortsof one of his former students, Baron Riche de Prony, finally resulted inChzy's belated appointment as director of the Ecole des Ponts etChausses.In 1749, working in Amsterdam, Cornelius Velsen stated:"The velocity should be proportional to the square root of the slope."In 1757, in Hannover, Germany, Albert Brahms wrote:"The declarative action of the bed in uniform flow was not only equal to theaccelerative action of gravity but also proportional to the square of thevelocity."They were working on the general laws and theories of Torricelli andBernoulli. These are some of the main ideas that Chzy used to develop his


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formula. However, the credit goes to Chzy because his formula is not onlythe first but also the most lasting resistance formula.Chzy was given the task to determine the cross section and the relateddischarge for a proposed canal on the river Yvette, which is close to Paris, butat a higher elevation. Since 1769, he was collecting experimental data from

the canal of Courpalet and from the river Seine. His studies and conclusionsare contained in a report to Mr. Perronet dated October 21, 1775. The originaldocument, written in French, is titled "Thesis on the velocity of the flow in agiven ditch," and it is signed by Mr. Chzy, General Inspector of des Ponts etChausses. It resides in file No. 847, Ms. 1915 of the collection of manuscripts in the library of the Ecole.The Chezy formula can be used to calculate mean flow velocity in conduitsand is expressed asv = c (R S)1/2wherev = mean velocity (m/s, ft/s)c = the Chezy roughness and conduit coefficientR = hydraulic radius of the conduit (m, ft)S = slope of the conduit (m/m, ft/ft)In general the Chezy coefficient - c - is a function of the flow Reynolds Number - Re - and the relative roughness - /R - of the channel. is the characteristic height of the roughness elements on the channelboundary.If velocity is known friction losses in terms of S=slope could be calculated

Usage with Manning coefficientThis formula can also be used with Manning's Roughness Coefficient, insteadof Chzy's coefficient. Manning derived [1] the following relation to C basedupon experiments:

whereC is the Chzy coefficient [m/s],R is the hydraulic radius (~ water depth) [m], andn is Manning's roughness coefficient.

History of Darcy Weisbach equation:Historically this equation arose as a variant on the Prony equation; this variantwas developed by Henry Darcy of France, and further refined into the formused today by Julius Weisbach of Saxony in 1845. Initially, data on thevariation of f with velocity was lacking, so the DarcyWeisbach equation wasoutperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many special-case situations in favor of a variety of empirical equations valid only for certain flow regimes, notably the Hazen -Williams equation or the Manning equation, most of which were significantlyeasier to use in calculations. However, since the advent of the calculator, ease of calculation is no longer a major issue, and so the DarcyWeisbachequation's generality has made it the preferred one.
http://www.engineeringtoolbox.com/hydraulic-equivalent-diameter-d_458.htmlhttp://www.engineeringtoolbox.com/reynolds-number-d_237.htmlhttp://www.engineeringtoolbox.com/reynolds-number-d_237.htmlhttp://www.engineeringtoolbox.com/surface-roughness-ventilation-ducts-d_209.htmlhttp://www.engineeringtoolbox.com/surface-roughness-ventilation-ducts-d_209.htmlhttp://en.wikipedia.org/wiki/Ch%C3%A9zy_formula#cite_note-0http://en.wikipedia.org/wiki/Prony_equationhttp://en.wikipedia.org/wiki/Henry_Darcyhttp://en.wikipedia.org/wiki/Julius_Weisbachhttp://en.wikipedia.org/wiki/Saxonyhttp://en.wikipedia.org/wiki/Empirical_equationhttp://en.wikipedia.org/wiki/Hazen-Williams_equationhttp://en.wikipedia.org/wiki/Hazen-Williams_equationhttp://en.wikipedia.org/wiki/Manning_equationhttp://en.wikipedia.org/wiki/Calculatorhttp://en.wikipedia.org/wiki/Calculatorhttp://en.wikipedia.org/wiki/Manning_equationhttp://en.wikipedia.org/wiki/Hazen-Williams_equationhttp://en.wikipedia.org/wiki/Hazen-Williams_equationhttp://en.wikipedia.org/wiki/Empirical_equationhttp://en.wikipedia.org/wiki/Saxonyhttp://en.wikipedia.org/wiki/Julius_Weisbachhttp://en.wikipedia.org/wiki/Henry_Darcyhttp://en.wikipedia.org/wiki/Prony_equationhttp://en.wikipedia.org/wiki/Ch%C3%A9zy_formula#cite_note-0http://www.engineeringtoolbox.com/surface-roughness-ventilation-ducts-d_209.htmlhttp://www.engineeringtoolbox.com/surface-roughness-ventilation-ducts-d_209.htmlhttp://www.engineeringtoolbox.com/reynolds-number-d_237.htmlhttp://www.engineeringtoolbox.com/reynolds-number-d_237.htmlhttp://www.engineeringtoolbox.com/hydraulic-equivalent-diameter-d_458.html


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Head loss formHead loss can be calculated with

wherehf is the head loss due to friction;L is the length of the pipe;D is the hydraulic diameter of the pipe (for a pipe of circular section, thisequals the internal diameter of the pipe);V is the average velocity of the fluid flow, equal to the volumetric flow rate per unit cross-sectional wetted area ;g is the local acceleration due to gravity ;f is a dimensionless coefficient called the Darcy friction factor. It can be foundfrom a Moody diagram or more precisely by solving Colebrook equation .

Using fanning friction factor the formula of Darcy-Weisbach becomes asunder

Hf= 4xfx lx vxv/ 2 x g xd

The Darcy Friction factor (which is 4 times greater than the Fanning Frictionfactor) used with Weisbach equation has now become the standard head lossequation for calculating head loss in pipes where the flow is turbulent.

Darcys friction factor f= 64/ Re where Re= Reylonds number

Fannings friction factor f=16/ Re where Re= Reylonds number

Pressure drops seen for fully-developed flow of fluids through pipes can bepredicted using the Moody diagram which plots the DarcyWeisbach frictionfactor f against Reynolds number Re and relative roughness / D. Thediagram clearly shows the laminar, transition, and turbulent flow regimes asReynolds number increases. The nature of pipe flow is strongly dependent onwhether the flow is laminar or turbulent
http://en.wikipedia.org/wiki/Head_losshttp://en.wikipedia.org/wiki/Hydraulic_diameterhttp://en.wikipedia.org/wiki/Volumetric_flow_ratehttp://en.wikipedia.org/wiki/Hydraulic_diameterhttp://en.wikipedia.org/wiki/Earth%27s_gravity#Variations_on_Earthhttp://en.wikipedia.org/wiki/Darcy_friction_factor_formulaehttp://en.wikipedia.org/wiki/Moody_diagramhttp://en.wikipedia.org/wiki/Colebrook_equationhttp://en.wikipedia.org/wiki/Moody_diagramhttp://en.wikipedia.org/wiki/Friction_factorhttp://en.wikipedia.org/wiki/Friction_factorhttp://en.wikipedia.org/wiki/Friction_factorhttp://en.wikipedia.org/wiki/Friction_factorhttp://en.wikipedia.org/wiki/Moody_diagramhttp://en.wikipedia.org/wiki/Colebrook_equationhttp://en.wikipedia.org/wiki/Moody_diagramhttp://en.wikipedia.org/wiki/Darcy_friction_factor_formulaehttp://en.wikipedia.org/wiki/Earth%27s_gravity#Variations_on_Earthhttp://en.wikipedia.org/wiki/Hydraulic_diameterhttp://en.wikipedia.org/wiki/Volumetric_flow_ratehttp://en.wikipedia.org/wiki/Hydraulic_diameterhttp://en.wikipedia.org/wiki/Head_loss


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Manning formula

The Manning formula, known also as the GaucklerManning formula, or GaucklerManningStrickler formula in Europe, is an empirical formula for open channel flow, or free-surface flow driven by gravity . It was first presentedby the French engineer Philippe Gauckler in 1867, [1] and later re-developed

by the Irish engineer Robert Manning in 1890.The GaucklerManning formula states:

where:V is the cross-sectional average velocity (ft/s, m/s)

k is a conversion constant equal to 1.486 for U.S. customary units or 1.0 for SI units n is the GaucklerManning coefficient (independent of units)Rh is the hydraulic radius (ft, m)

S is the slope of the water surface or the linear hydraulic head loss(ft/ft, m/m) (S = hf/L)The discharge formula, Q = A V, can be used to manipulate GaucklerManning's equation by substitution for V. Solving for Q then allows anestimate of the volumetric flow rate (discharge) without knowing the limiting or actual flow velocity.The GaucklerManning formula is used to estimate flow in open channelsituations where it is not practical to construct a weir or flume to measure flow

with greater accuracy. The friction coefficients across weirs and orifices areless subjective than n along a natural (earthen, stone or vegetated) channel
http://en.wikipedia.org/wiki/Empirical_relationshiphttp://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Manning_formula#cite_note-0http://en.wikipedia.org/wiki/Irish_peoplehttp://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Robert_Manning_%28engineer%29http://en.wikipedia.org/wiki/U.S._customary_unitshttp://en.wikipedia.org/wiki/SI_unitshttp://en.wikipedia.org/wiki/Hydraulic_headhttp://en.wikipedia.org/wiki/Discharge_%28hydrology%29http://en.wikipedia.org/wiki/Discharge_%28hydrology%29http://en.wikipedia.org/wiki/Hydraulic_headhttp://en.wikipedia.org/wiki/SI_unitshttp://en.wikipedia.org/wiki/U.S._customary_unitshttp://en.wikipedia.org/wiki/Robert_Manning_%28engineer%29http://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Irish_peoplehttp://en.wikipedia.org/wiki/Manning_formula#cite_note-0http://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Empirical_relationshiphttp://en.wikipedia.org/wiki/File:Moody_diagram.jpg


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reach. Cross sectional area, as well as n', will likely vary along a naturalchannel. Accordingly, more error is expected in predicting flow by assuming aManning's n, than by measuring flow across a constructed weirs, flumes or orifices. The formula can be obtained by use of d imensional analysis. Recently this

formula was derived theoretically using the phenomenological theory of turbulence .

Hydraulic radiusThe hydraulic radius is a measure of a channel flow efficiency. Flow speedalong the channel depends on its cross-sectional shape (among other factors), and the hydraulic radius is a characterization of the channel thatintends to capture such efficiency. Based on the 'constant shear stress at theboundary' assumption, hydraulic radius is defined as the ratio of the channel'scross-sectional area of the flow to its wetted perimeter (the portion of thecross-section's perimeter that is "wet"):

where:Rh is the hydraulic radius (m),

A is the cross sectional area of flow (m2),P is wetted perimeter (m).

The greater the hydraulic radius, the greater the efficiency of the channel andthe less likely the river is to flood. For channels of a given width, the hydraulicradius is greater for the deeper channels.

The hydraulic radius is not half the hydraulic diameter as the name maysuggest. It is a function of the shape of the pipe, channel, or river in which thewater is flowing. In wide rectangular channels, the hydraulic radius isapproximated by the flow depth. The measure of a channel's efficiency (itsability to move water and sediment ) is used by water engineers to assess thechannel's capacity.GaucklerManning coefficient

The GaucklerManning coefficient, often denoted as n, is an empiricallyderived coefficient, which is dependent on many factors, including surface

roughness and sinuosity . When field inspection is not possible, the bestmethod to determine n is to use photographs of river channels where n hasbeen determined using GaucklerManning's formula.

In natural streams, n values vary greatly along its reach, and will even vary ina given reach of channel with different stages of flow. Most research showsthat n will decrease with stage, at least up to bank-full. Overbank n values for a given reach will vary greatly depending on the time of year and the velocityof flow. Summer vegetation will typically have a significantly higher n valuedue to leaves and seasonal vegetation. Research has shown, however, that nvalues are lower for individual shrubs with leaves than for the shrubs withoutleaves. This is due to the ability of the plant's leaves to streamline and flex asthe flow passes them thus lowering the resistance to flow. High velocity flows
http://en.wikipedia.org/wiki/Weirhttp://en.wikipedia.org/wiki/Flumehttp://en.wikipedia.org/wiki/Orificehttp://en.wikipedia.org/wiki/Dimensional_analysishttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Shear_stress#Shear_stress_in_fluidshttp://en.wikipedia.org/wiki/Wetted_perimeterhttp://en.wikipedia.org/wiki/Wetted_perimeterhttp://en.wikipedia.org/wiki/Hydraulic_diameterhttp://en.wikipedia.org/wiki/Sedimenthttp://en.wikipedia.org/wiki/Sinuosityhttp://en.wikipedia.org/wiki/Sinuosityhttp://en.wikipedia.org/wiki/Sedimenthttp://en.wikipedia.org/wiki/Hydraulic_diameterhttp://en.wikipedia.org/wiki/Wetted_perimeterhttp://en.wikipedia.org/wiki/Wetted_perimeterhttp://en.wikipedia.org/wiki/Shear_stress#Shear_stress_in_fluidshttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Dimensional_analysishttp://en.wikipedia.org/wiki/Orificehttp://en.wikipedia.org/wiki/Flumehttp://en.wikipedia.org/wiki/Weir


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COMPARISION BETWEEN HAZEN-WILLIAMS FORMULA AND DARCYSFORMULA

Friction loss

Diameter VelocityHazenWilliams

DarcyWeisbach

%difference

15 1 180 135.92 -24.4925 1 99.64 81.54 -18.1750 1 44.38 40.77 -8.1380 1 25.65 25.48 -0.66100 1 19.77 20.38 3.09125 1 15.24 16.31 7.02150 1 12.32 13.59 10.31200 1 8.81 10.19 15.66225 1 7.67 9.06 18.12250 1 6.78 8.15 20.21300 1 5.48 6.79 23.91350 1 4.58 5.82 27.07400 1 3.92 5.09 29.85500 1 3.02 4.07 34.77600 1 2.44 3.39 38.93700 1 2.04 2.91 42.65800 1 1.75 2.55 45.71900 1 1.52 2.26 48.681000 1 1.34 2.038 52.09

Diameter VelocityHazenWilliams

DarcyWeisbach

%difference

15 1.5 383.17 305.81 -20.1925 1.5 211.14 183.48 -13.1050 1.5 94.05 91.74 -2.4680 1.5 54.35 57.33 5.48100 1.5 41.89 45.87 9.50125 1.5 32.29 36.69 13.63150 1.5 26.1 30.58 17.16200 1.5 18.66 22.93 22.88225 1.5 16.26 20.39 25.40250 1.5 14.38 18.34 27.54300 1.5 11.63 15.29 31.47350 1.5 9.71 13.11 35.02400 1.5 8.31 11.46 37.91500 1.5 6.41 9.18 43.21600 1.5 5.18 7.65 47.68700 1.5 4.32 6.55 51.62800 1.5 3.7 5.73 54.86900 1.5 3.22 5.09 58.071000 1.5 2.85 4.58 60.70


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COMPARISION OF HAZEN WILLIAMS & DARCY'S FORMULA

Diameter Velocity

Loss asper Hazenwilliams

DarcyWeisbach Diff in %

Diff in0.9 & 1.8velocity

300 0.45 1.29 1.4310.85271

300 0.9 4.5249 5.517121.92756

300 1.8 16.33 22.685 38.91611 16.98855

100 0.8846 15.75 15.951.269841

100 1.9167 65.96 74.8913.53851

12.26867

200 0.92 7.57 8.6514.26684

200 1.8 26.32 33.2526.32979

12.06294

400 0.9 3.24 4.15

28.0864

2

400 1.87 12.62 17.8341.28368

13.19726

500 0.9 2.5 3.31 32.4

500 1.8 9.02 13.2747.11752

14.71752


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CALCULATION OF DISCHARGE OF WATERTHROUGH " V " NOTCH

Q=2.362 X C X H ^2.5

C=0.60

Depth in InchDischarge

lpm1 9.212 48.13 134

3.5 1944 275

4.5 3685 506

5.5 6406 795

6.5 9687 1162

7.5 13758 1607

8.5 1870

9 21559.5 246410 2798

10.5 315611 3420

11.5 436112 4830


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VELOCITY OF FREE FLOW PIPE

V= ( g * X^2/ 2*y)^0.5

V= Velocity in mt / second

X= Horizontal ordinate of free flow

Y= Vertical ordinate of free flow

g=9.81

Discharge inliters per hours

Distance "x" in mts measured frompipe where y=0.30 mt

Diameter of pipe 0.25 0.5 0.75 180 mm 18283 36566 54849 73133100 mm 28567 57135 85702 114270125 mm 44637 89273 133910 178546150 mm 64277 128553 192830 257107


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BERNOULLIS EQUATION

In the 1700s, Daniel Bernoulli investigated the forces present in a movingfluid. This slide shows one of many forms of Bernoulli's equation. Theequation appears in many physics, fluid mechanics, and airplane textbooks.The equation states that the static pressure ps in the flow plus the dynamic

pressure , one half of the density r times the velocity V squared, is equal to aconstant throughout the flow. We call this constant the total pressure pt of theflow.

Applications of Bernoulli's EquationThe fluids problem shown on this slide is low speed flow through a tube withchanging cross-sectional area. For a streamline along the center of the tube,the velocity decreases from station one to two. Bernoulli's equation describesthe relation between velocity, density, and pressure for this flow problem.Since density is a constant for a low speed problem, the equation at thebottom of the slide relates the pressure and velocity at station two to theconditions at station one.
http://www.grc.nasa.gov/WWW/K-12/airplane/bernnew.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/state.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/pressure.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dynpress.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dynpress.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dynpress.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dynpress.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/pressure.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/state.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/bernnew.html


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The Orifice PlateThe orifice meter consists of a flat orifice plate with a circular hole drilled in it.There is a pressure tap upstream from the orifice plate and another justdownstream. There are in general three methods of placing the taps. Thecoefficient of the meter depends upon the position of taps .

Flange location - Tap location 1 inch upstream and 1 inch downstreamfrom face of orifice

"Vena Contracta" location - Tap location 1 pipe diameter (actual inside)upstream and 0.3 to 0.8 pipe diameter downstream from face of orifice

Pipe location - Tap location 2.5 times nominal pipe diameter upstreamand 8 times nominal pipe diameter downstream from face of orifice

The discharge coefficient - cd - varies considerably with changes inarea ratio and the Reynolds number . A discharge coefficient cd = 0.60may be taken as standard, but the value varies noticeably at low valuesof the Reynolds number.

Discharge Coefficient - c d

Diameter Ratio

d = D 2 / D1

Reynolds Number - Re

104

105

106

107

0.2 0.60 0.595 0.594 0.594

0.4 0.61 0.603 0.598 0.598

0.5 0.62 0.608 0.603 0.603

0.6 0.63 0.61 0.608 0.608

0.7 0.64 0.614 0.609 0.609
http://www.engineeringtoolbox.com/reynolds-number-d_237.htmlhttp://www.engineeringtoolbox.com/reynolds-number-d_237.html


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Required upstream pipe length 5 to 20 diameters Viscosity effect is high Relative cost is medium

The NozzleNozzles used for determining fluid's flow rate through pipes can be in threedifferent types:

The ISA 1932 nozzle - developed in 1932 by the InternationalOrganization for Standardization or ISO. The ISA 1932 nozzle iscommon outside USA.

The long radius nozzle is a variation of the ISA 1932 nozzle. The venturi nozzle is a hybrid having a convergent section similar to

the ISA 1932 nozzle and a divergent section similar to a venturi tubeflowmeter.

Discharge Coefficient - c d

Diameter Ratio

d = D 2 / D1

Reynolds Number - Re

10 4 10 5 10 6 10 7

0.2 0.968 0.988 0.994 0.995

0.4 0.957 0.984 0.993 0.995

0.6 0.95 0.981 0.992 0.995

0.8 0.94 0.978 0.991 0.995

The flow nozzle is recommended for both clean and dirty liquids The rangeability is 4 to 1 The relative pressure loss is medium Typical accuracy is 1-2% of full range Required upstream pipe length is 10 to 30 diameters The viscosity effect high The relative is medium


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Pitot tube

A pitot tube is a p ressure measurement instrument used to measure fluid flowvelocity . The pitot tube was invented by the French engineer Henri Pitot in theearly 18th century and was modified to its modern form in the mid-19thcentury by French scientist Henry Darcy . It is widely used to determine theairspeed of an aircraft and to measure air and gas velocities in industrialapplications.

Theory of operationThe basic pitot tube consists of a tube pointing directly into the fluid flow. Asthis tube contains fluid, a pressure can be measured; the moving fluid isbrought to rest (stagnates) as there is no outlet to allow flow to continue. Thispressure is the stagnation pressure of the fluid, also known as the totalpressure or (particularly in aviation) the pitot pressure.The measured stagnation pressure cannot of itself be used to determine thefluid velocity (airspeed in aviation). However, Bernoulli's equation states:

Stagnation pressure = static pressure + dynamic pressure

Which can also be written as

Solving that for velocity we get:
http://en.wikipedia.org/wiki/Pressure_measurementhttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Henri_Pitothttp://en.wikipedia.org/wiki/Henry_Darcyhttp://en.wikipedia.org/wiki/Airspeedhttp://en.wikipedia.org/wiki/Aircrafthttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Bernoulli%27s_equationhttp://en.wikipedia.org/wiki/Bernoulli%27s_equationhttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Aircrafthttp://en.wikipedia.org/wiki/Airspeedhttp://en.wikipedia.org/wiki/Henry_Darcyhttp://en.wikipedia.org/wiki/Henri_Pitothttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Pressure_measurementhttp://en.wikipedia.org/wiki/File:Pitot_tube_types.svg


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where

V is fluid velocity;

pt

is stagnation or total pressure;

p s is static pressure;

and is fluid density.

The dynamic pressure , then, is the difference between the stagnationpressure and the static pressure. The static pressure is generally measuredusing the static ports on the side of the fuselage. The dynamic pressure isthen determined using a diaphragm inside an enclosed container. If the air onone side of the diaphragm is at the static pressure, and the other at thestagnation pressure, then the deflection of the diaphragm is proportional tothe dynamic pressure, which can then be used to determine the indicatedairspeed of the aircraft. The diaphragm arrangement is typically containedwithin the airspeed indicator, which converts the dynamic pressure to anairspeed reading by means of mechanical levers.
http://en.wikipedia.org/wiki/Dynamic_pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Static_porthttp://en.wikipedia.org/wiki/Indicated_airspeedhttp://en.wikipedia.org/wiki/Indicated_airspeedhttp://en.wikipedia.org/wiki/Airspeed_indicatorhttp://en.wikipedia.org/wiki/Airspeed_indicatorhttp://en.wikipedia.org/wiki/Indicated_airspeedhttp://en.wikipedia.org/wiki/Indicated_airspeedhttp://en.wikipedia.org/wiki/Static_porthttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Dynamic_pressure


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Continuity and Conservation of Matter

1. Mass flow rateIf we want to measure the rate at which water is flowing along a pipe. A verysimple way of doing this is to catch all the water coming out of the pipe in abucket over a fixed time period. Measuring the weight of the water in thebucket and dividing this by the time taken to collect this water gives a rate of accumulation of mass. This is know as the mass flow rate.For example an empty bucket weighs 2.0kg. After 7 seconds of collectingwater the bucket weighs 8.0kg, then:

Performing a similar calculation, if we know the mass flow is 1.7kg/s, how longwill it take to fill a container with 8kg of fluid?

2. Volume flow rate - Discharge . More commonly we need to know the volume flow rate - this is morecommonly know as discharge. (It is also commonly, but inaccurately, simplycalled flow rate). The symbol normally used for discharge is Q. The dischargeis the volume of fluid flowing per unit time. Multiplying this by the density of the fluid gives us the mass flow rate. Consequently, if the density of the fluid

in the above example is 850 then:


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An important aside about units should be made here:

As has already been stressed, we must always use a consistent set of unitswhen applying values to equations. It would make sense therefore to alwaysquote the values in this consistent set. This set of units will be the SI units.

Unfortunately, and this is the case above, these actual practical values arevery small or very large (0.001008m3/s is very small). These numbers aredifficult to imagine physically. In these cases it is useful to use derived units,and in the case above the useful derived unit is the litre.

(1 litre = 1.0 10 -3m3 ). So the solution becomes . It is far easier toimagine 1 litre than 1.0 10 -3m3. Units must always be checked, and converted if necessary to a consistent set before using in an equation.

3. Discharge and mean velocity

If we know the size of a pipe, and we know the discharge, we can deduce themean velocity

Discharge in a pipe

If the area of cross section of the pipe at point X is A, and the mean velocity

here is . During a time t, a cylinder of fluid will pass point X with a volume

A t. The volume per unit time (the discharge) will thus be

So if the cross-section area, A, is and the discharge, Q is ,

then the mean velocity, , of the fluid is

Note how carefully we have called this the mean velocity. This is because thevelocity in the pipe is not constant across the cross section. Crossing the


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centreline of the pipe, the velocity is zero at the walls increasing to amaximum at the centre then decreasing symmetrically to the other wall. Thisvariation across the section is known as the velocity profile or distribution. Atypical one is shown in the figure below.

A typical velocity profile across a pipe

This idea, that mean velocity multiplied by the area gives the discharge,

applies to all situations - not just pipe flow.

4. Continuity

Matter cannot be created or destroyed - (it is simply changed in to a differentform of matter). This principle is know as the conservation of mass and weuse it in the analysis of flowing fluids.

The principle is applied to fixed volumes, known as control volumes (or surfaces), like that in the figure below:

An arbitrarily shaped control volume.

For any control volume the principle of conservation of mass says

Mass entering per unit time = Mass leaving per unit time + Increase of massin the control volume per unit time

For steady flow there is no increase in the mass within the control volume, so


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For steady flow

Mass entering per unit time = Mass leaving per unit time

This can be applied to a stream tube such as that shown below. No fluid flowsacross the boundary made by the streamlines so mass only enters and leavesthrough the two ends of this stream tube section.

A stream tube

We can then write

Or for steady flow,

This is the equation of continuity.

The flow of fluid through a real pipe (or any other vessel) will vary due to thepresence of a wall - in this case we can use the mean velocity and write


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When the fluid can be considered incompressible, i.e. the density does not change , 1 = 2 = so (dropping the m subscript)

This is the form of the continuity equation most often used.

This equation is a very powerful tool in fluid mechanics and will be usedrepeatedly throughout the rest of this course.

Some example applications

We can apply the principle of continuity to pipes with cross sections whichchange along their length. Consider the diagram below of a pipe with acontraction:

A liquid is flowing from left to right and the pipe is narrowing in the samedirection. By the continuity principle, the mass flow rate must be the same ateach section - the mass going into the pipe is equal to the mass going out of the pipe. So we can write:

(with the sub-scripts 1 and 2 indicating the values at the two sections)

As we are considering a liquid, usually water, which is not very compressible,

the density changes very little so we can say . This also says thatthe volume flow rate is constant or that


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For example if the area and and the upstream meanvelocity, , then the downstream mean velocity can be calculated by

Notice how the downstream velocity only changes from the upstream by theratio of the two areas of the pipe. As the area of the circular pipe is a function

of the diameter we can reduce the calculation further,

Now try this on a diffuser, a pipe which expands or diverges as in the figurebelow,

If the diameter at section 1 is and at section 2 and the meanvelocity at section 2 is . The velocity entering the diffuser is given by,

Another example of the use of the continuity principle is to determine thevelocities in pipes coming from a junction.


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Total mass flow into the junction = Total mass flow out of the junction 1Q1 = 2Q2 + 3Q3

When the flow is incompressible (e.g. if it is water) 1 = 2 =

If pipe 1 diameter = 50mm, mean velocity 2m/s, pipe 2 diameter 40mm takes30% of total discharge and pipe 3 diameter 60mm. What are the values of discharge and mean velocity in each pipe ?


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DESIGN OF THRUST BLOCK

Pipe bends and thrust blocks forces on anchors due to fluid velocity andinternal pressure

The resulting force on a thrust block or anchor depends on the fluid mass flowand flow velocity and the pressure in the bend.

Resulting force due to Mass flow and Flow VelocityThe resulting force in x-direction due to mass flow and flow velocity can be expressed as:R x = m v (1 - cos) (1)

= A v 2 (1 - cos) (1b) = (d / 2) 2 v 2 (1 - cos) (1c)

whereR x = resulting force in x-direction (N) m = mass flow (kg/s)v = flow velocity (m/s)

= turning bend angle (degrees) = fluid density (kg/m 3 )

d = internal pipe or bend diameter (m) = 3.14... The resulting force in y-direction due to mass flow and flow velocity can be expressed as:R y = m v sin (2)

= A v 2 sin (2b) = (d / 2) 2 v 2 sin (2c)

R y = resulting force in y direction (N) The resulting force on the bend due to force in x- and y-direction can be expressed as:R = (R x

2 + R y 2 )1/2 (3)

whereR = resulting force on the bend (N)

Example - Resulting force on a bend due to mass flow and flow velocityThe resulting force on a 45 o bend with


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diameter 114 mm = 0.114 m water with density 1000 kg/m 3 flow velocity 20 m/s

can be calculated by asResulting force in x-direction:R x = (1000 kg/m 3 ) ((0.114 m) / 2) 2 (20 m/s) 2 (1 - cos45)

= 1196 (N) Resulting force in y-direction:R y = (1000 kg/m

3 ) ((0.114 m) / 2) 2 (20 m/s) 2 sin45 = 2887 (N)

Resulting force on the bendR = ((1196 N) 2 + (2887 N) 2 )1/2

= 3125 (N) Note - if is 90 o the resulting forces in x- and y-directions are the same.

Resulting force due to Static PressureThe pressure and the end surfaces of the bend creates resulting forces in x- and y-directions.

The resulting force in x-direction can be expressed asR px = p A (1- cos ) (4)

= p (d / 2) 2 (1- cos ) (4b) whereR px = resulting force due to pressure in x-direction (N)

p = gauge pressure inside pipe (Pa, N/m 2 ) The resulting force in y-direction can be expressed asR py = p (d / 2)

2 sin (5) whereR py = resulting force due to pressure in y-direction (N) The resulting force on the bend due to force in x- and y-direction can be expressed as:R p = (R px

2 + R py 2 )1/2 (6)

where

R p = resulting force on the bend due to static pressure (N)

Example - Resulting force on a bend due to pressureThe resulting force on a 45 o bend with

diameter 114 mm = 0.114 m pressure 100 kPa

can be calculated by asResulting force in x-direction:R x = (100 kPa) ((0.114 m) / 2)

2 (1 - cos45)= 299 (N)

Resulting force in y-direction:R y = (100 kPa) ((0.114 m) / 2)

2 sin45 = 722 (N)

Resulting force on the bendR = ((1196 N) 2 + (2887 N) 2 )1/2

= 781 (N)


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MINOR HEAD LOSS:-

DUE TO ENLARGEMENT

H= Kx V 2 - V2^2 / 2 x g

SUDDEN ENLARGEMENT

D:D2 K1:1.2 0.11-0.081:1.4 0.26-0.21:1.6 0.4-0.321:1.8 0.51-0.41:2 0.6-0.471:1.25 0.74-0.58

1:3 0.83-0.651:4 0.92-0.721:5 0.96-0.75


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GRADUAL ENLARGEMENT

Degree K2 0.0334 0.0396 0.0468 0.055

10 0.07812 0.115 0.1620 0.3130 0.4940 0.650 0.6760 0.7275 0.7290 0.67


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SUDDEN CONTRACTION

D:D2 K1.2:1 0.07-0.111.4:1 0.17-0.21.6:1 0.26-0.241.8:1 0.34-0.2702:01 0.38-0.292.5:1 0.42-0.3103:01 0.44-0.3304:01 0.47-0.3405:01 0.48-0.35


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LOSS AT BENDS AND ELBOWS=K x V^2 / 2 x g

DESCRIPTIONVALUE OFK

Elbow 45%

Flanged regular 0.2-0.3Flanged long radius 0.18-0.2Screwed regular 0.3-0.42

Elbow 90%

Flanged regular 0.21-0.3Flanged long radius 0.18-0.2

Interconnection of two cylinder pipeswelded but not

rounded 1.25-1.8Screwed shortradius 0.9Screwed mediumradius 0.75Screwed longradius 0.6

BENDS


Angle of bend5% 0.116-0.024

10% 0.034-0.04415% 0.042-0.062

22.50% 0.066-0.15430% 0.13-0.16545% 0.236-0.3260% 0.68590% 1.265

RETURN BEND ( 2 NOS 90%)


Flanged regular 0.38Flanged long radius 0.25Screwed 2.2


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TEES


Standard bifurcating 1.5-1.8Standard 90% turn 1.8

Standard run of tee 0.6Reducing run of tee( In terms of velocityat smaller end)2:1 0.94:1 0.75

OBSTRUCTIONS PIPE AREA :FLOW AREA

VALUE OFK

1.1 0.211.4 1.151.6 2.42 5.553 154 27.35 326 577 72.5

10 121

VENTUREMETER Throat to Inlet diaratio Long tube Short tube

1:3 1-1.2 2.431:2 0.44-0.52 0.722:3 1.25-0.3 0.323:4 0.2-0.23 0.24

ORIFICE METER Orifice to pipe diaratio

VALUE OFK

1:4 4.81:3 2.51:2 12:3 0.43:4 0.24


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GATE VALVE

Description Value of KFully open 0.19Three forth open 1.15Half open 5.6One quarter open 24

BUTTERFLY VALVE

Closure angle

Angle in degree Value of K0 0.3

10 0.4620 1.3830 3.640 1050 3160 94

DIAPHRAM VALVEDescription Value of KFully open 2.3Three forth open 2.6Half open 4.3One quarter open 21


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STOP VALVE

Description Value of KFully open 4Three forth open 4.6Half open 6.4One quarter open 780

CHECK VALVE

Description Value of KSwing check (Fullyopen) 2.5Ball type ( Fully open) 2.5-3.5Horizontal lift type 8-12

FOOT VALVE WITH STRAINER k= 2.5

PRESSURE REDUCING VALVE k= 10

EXIT k= 1( When discharged into air or still water)


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ANALYSIS OF RISING MAIN DESIGN BASED ONFORMULA D=1.2 X Q^0.5

For economic design of rising main with in 1.0 Km the formula for calculationof diameter of pipe line is D= 1.2 x q^0.5 where as diameter worked out is inmm and q= discharge of water in cubic meter per second.

Reason for analysis:- The above formula could be used for vide range of discharge and static head difference, hence detailed analysis is made for therange in which formula could be used.

The analysis is made for range upto 5.0 kms and static head difference fromrange of 5 mt to 60 mts. The analysis is also made for different type of pipessuch as PVC, DI and Mild steel pipes.

The readers of this topic can further explore the analysis for other variety of pipes and range beyond 5.0 kms and static difference beyond 60 mts.

The analysis is required for instant estimation of diameter of rising main.

INPUT DATA FOR PVC PIPES:-

P IP ED A T A

Outer dia DIA RATE

METER TYPE CLASS "HWC" Rs/m

180.00 166.90 PVC 10 Kg/scm 140.00 592.00

200.00 185.50 PVC 10 Kg/scm 140.00 725.00

225.00 208.80 PVC 10 Kg/scm 140.00 918.00

250.00 231.90 PVC 10 Kg/scm 140.00 1147.00

280.00 259.80 PVC 10 Kg/scm 140.00 1450.00

315.00 292.30 PVC 10 Kg/scm 140.00 1834.00

Present 1


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demand

Ultimatedemand 1.5

Length in KMStatic

difference1 2 3 4 5

5 167 167 167 167 167

10 167 167 167 167 167

15 167 167 167 167 167

20 167 167 167 167 167

25 167 167 167 167 167

30 167 167 167 167 167

35 167 167 167 167 167

40 167 167 167 167 167

45 167 167 167 167 167

50 167 167 167 167 167

55 167 167 167 167 167

60 167 167 167 167 167

Dia arrived Dia as per 1.2xq^0.5

167 165.1445648

Presentdemand 1.5

Ultimatedemand 2.25

Length in KMStaticdifference 1 2 3 4 5

5 209 209 209 209 209

10 209 209 209 209 209

15 209 209 209 209 209

20 209 209 209 209 209

25 209 209 209 209 20930 209 209 209 209 209

35 209 209 209 209 209

40 209 209 209 209 209

45 209 209 209 209 209

50 209 209 209 209 209

55 209 209 209 209 209

60 209 209 209 209 209


209 202.25996


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Presentdemand 2

Ultimatedemand 3


5 209 232 232 232 232

10 209 232 232 232 232

15 209 232 232 232 232

20 209 232 232 232 232

25 209 232 232 232 232

30 209 232 232 232 23235 209 232 232 232 232

40 209 232 232 232 232

45 209 232 232 232 232

50 209 232 232 232 232

55 209 232 232 232 232

60 209 232 232 232 232Diaarrived Dia as per 1.2xq^0.5

232 233.54968

Presentdemand

2

Ultimatedemand 3

Length in KM

Staticdifference 1 2 3 4 5

5 250 250 250 250 250

10 250 250 250 250 250

15 250 250 250 250 25020 250 250 250 250 250


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25 250 250 250 250 250

30 250 250 250 250 250

35 250 250 250 250 250

40 250 250 250 250 250

45 250 250 250 250 25050 250 250 250 250 250

55 250 250 250 250 250

60 250 250 250 250 250

Diaarrived 250 233.55

Dia as per 1.2xq^0.5

INPUT DATA FOR DI PIPES:-

PI PED AT A

Outer dia DIA RATE


200.00 DI K-7 130.00 1183.00

250.00 DI K-7 130.00 1544.00

300.00 DI K-7 130.00 1925.00

350.00 DI K-7 130.00 2652.00

400.00 DI K-7 130.00 2844.00

450.00 DI K-7 130.00 3717.00


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Presentdemand 3

Ultimatedemand 4.5

Length in KM


5 300 300 300 300 300

10 300 300 300 300 300

15 300 300 300 300 30020 300 300 300 300 300

25 300 300 300 300 300

30 300 300 300 300 300

35 300 300 300 300 300

40 300 300 300 300 300

45 300 300 300 300 300

50 300 300 300 300 300

55 300 300 300 300 300

60 300 300 300 300 300Dia arrived Dia as per 1.2xq^0.5

300 286.03878

Presentdemand 4

Ultimatedemand 6

Length in KM


5 300 300 300 300 300

10 300 300 300 300 300

15 300 300 300 300 300

20 300 300 300 300 300

25 300 300 300 300 300

30 300 300 300 300 300

35 300 300 300 300 300

40 300 300 300 300 300


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Presentdemand 5

Ultimatedemand 7.5

Length in KM


5 400 400 400 400 400

10 400 400 400 400 400

15 400 400 400 400 400

20 400 400 400 400 400

25 400 400 400 400 400

30 400 400 400 400 400

35 400 400 400 400 400

40 400 400 400 400 400

45 400 400 400 400 400

50 400 400 400 400 400

55 400 400 400 400 400


400 369.27447

Presentdemand 6

Ultimatedemand 9

Length in KM


5 400 400 400 400 400

10 400 400 400 400 40015 400 400 400 400 400

45 300 300 300 300 300

50 300 300 300 300 300

55 300 300 300 300 300


300 330.28913


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20 400 400 400 400 400

25 400 400 400 400 400

30 400 400 400 400 400

35 400 400 400 400 400

40 400 400 400 400 40045 400 400 400 400 400

50 400 400 400 400 400

55 400 400 400 400 400


400 404.51

P I P E D A T A

Outer dia DIA RATE


457 417.60 MS 4,7MM 130.00 3653.00

508 468.00 MS 5MM 130.00 4411.00

610 568.00 MS 6,3MM 130.00 6018.00

610 568.00 MS 6,3MM 130.00 6018.00

711 667.00 MS 7MM 130.00 7750.00

813 767.00 MS 8MM 130.00 9764.00

Presentdemand 7

Ultimate

demand10.5


5 418 418 418 418 418

10 418 418 418 418 418

15 418 418 418 418 418

20 418 418 418 418 418

25 418 418 418 418 418

30 418 418 418 418 418

35 418 418 418 418 41840 418 418 418 418 418


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45 418 418 418 418 418

50 418 418 418 418 418

55 418 418 418 418 418

60 418 418 418 418 418

Dia arrived Dia as per 1.2xq^0.5418 356.753034

Presentdemand 10

Ultimatedemand 15


5 468 468 468 468 46810 468 468 468 468 468

15 468 468 468 468 468

20 468 468 468 468 468

25 468 468 468 468 468

30 468 468 468 468 468

35 468 468 468 468 468

40 468 468 468 468 468

45 468 468 468 468 468

50 468 468 468 468 468

55 468 468 468 468 468

60 468 468 468 468 468


468 426.40143

Presentdemand 12.5

Ultimatedemand 18.75


5 468 468 468 468 468

10 468 468 468 468 468

15 468 468 468 468 468

20 468 468 468 468 468

25 468 468 468 468 468

30 468 468 468 468 468

35 468 468 468 468 468

40 468 468 468 468 468


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45 468 468 468 468 468

50 468 468 468 468 468

55 468 468 468 468 468


468 476.73129Presentdemand 15

Ultimatedemand 22.5


5 568 568 568 568 568

10 568 568 568 568 568

15 568 568 568 568 568

20 568 568 568 568 568

25 568 568 568 568 568

30 568 568 568 568 568

35 568 568 568 568 568

40 568 568 568 568 568

45 568 568 568 568 568

50 568 568 568 568 568

55 568 568 568 568 568

60 568 568 568 568 568


568 522.23297


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Conclusion for PVC pipes: For range of length upto 5kms, range of static head from 5 mt to 60 mts andrange of discharge upto 2.0 mld the formula for calculation of diameter d=1.2 x q^0.5 where q=ultimate discharge in cubic meter/second

Conclusion for DI pipes: For range of length upto 5kms, range of static head from 5 mt to 60 mts andrange of discharge from 3.0 upto 6.0 mld (present) theformula for calculation of diameter d=1.2 x q^0.5where q= ultimate discharge in cubic meter/second

Conclusion for MS pipes: For range of length upto 5kms, range of static head from 5 mt to 60 mts andrange of discharge from 7 upto 15.0 mld the formulafor calculation of diameter d=1.2 x q^0.5 where q=present discharge in cubic meter/second

The intention beyond giving such long analysis is thatthe readers shall be guided for further exploring therange of formula for economic design of rising main.


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SAMPLE DESIGN OF THRUST BLOCK

TOTAL FLOW 150 MLD

PEAKING FACTOR 1.1

PEAKING FLOW 165 MLD

DIAMETER OF PIPE 1100 MM

AREA OF PIPE 0.95 SQMT

VELOCITY IN PIPE 2.01 MT/SECOND

FLOW IN CUMSECS 1.91 CUMECS

HMG OF LINE 82 MT

LOWEST LEVEL ON GROUND 30 MT

MAXIMUM STATIC PRESSURE 52 MT

I.E 5.2 Kg/sqcmi.e 0.52 MpaI.e 520 Kpa

DEGREE OF BEND 45

COS 45/ SIN 45 0.7071

FORMULA

p1xa1 - Fx - p2xa2cos 45 = gQ ( v2cos 45-v1)

Units

p1,p2 pascals A1,A2 sqmtv1,v2 mt/secondg= 1000Q cumecs

540X1000x0,95- Fx- 540X1000x0,95*0,7071=1000X1,91 (2,01*0,7071-2,01)

513000-Fx -362000=(-) 1125

Fx=362000-513000-1125

Fx=152125 newton


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Fx=152125 newton

F=(Fx*Fx +fyxFy)^0,5

F= 215137.2 Newton

Force in Kg 21513.72

Density of concrete 2400 kg/cum

Volume of concrete required 8.964052 cum

Keeping depth as 2 mt

Area required 4.482026 sqmt

Size required 2.11708 mt

There fore provide size 2,5x2,5x2,0

Check for net volume after pipe diameter

Total volume 12.5

Less pipe volume'3*,785*1,1*1,1 2.84955 cum

Net volume 9.65045 cum

Volume required 8.964052 cum

HENCE OK


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CASE STUDY

Purpose of case study:- It is observed that isolation

valves installed on the trunk mains are of reduced size, due to which excess pressure on the body of valve isexerted which had caused some failure of valves. Thefailure mostly occurred in body cracking by excessivehoop stress. The calculation of stress is worked outas under which is the actual case study.

CALCULATION OF TOTAL PRESSURE ON BODY OF VALVES ONREDUCED SECTION OF PIPE LINE

CASE STUDY OF NARMADA PIPE LINE NC-10

DIAMETER OF PIPE LINE 1800 MM 180 cm

DIAMETER OF BUTTERFLY VALVE 1200 MM 120 cm

CARRYING CAPACITY OF LINE 200 MLD

PEAKING FACTOR 1.2

HENCE HOURS OF FLOW 20 HRS

VELOCITY CALCULATIONS

V=200*1000/(20*60*60*0.785*1.8*1.8)1.09mt/second

Velocity at reduced section

V=200*1000/(20*60*60*0.785*1.2*1.2)

2.46

mt/second

Pressure on line P1 10 Kg/sqcm

Pressure on reduced section P2

Considering on same ground level hence Z is nullified from equation

P2= P1 - ( V2^2- V1^2)/( 2*9.81)

P2= 10 - ( 2.46^2- 1.09^2)/( 2*9.81) 9.752115189

Say9.75kg/sqcm


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EQUATING FORCE ( PIPE LINE IN RUNNING CONDITION)

F=P1*A1-P2*A2F=10*(0.785*180*180)-9.75*(0.785*120*120) 144126F= 144126 Kg

Force intensity

f= F/area

f=144126/ 0.785*120*120 12.75

Summing of p+f 9.75+12.75 22.5 kg/sqcm

Which is greater than 20 kg/sqcm body pressure

Hence many butterfly valves on the line failed from the body

EQUATING FORCE ( PIPE LINE IN STATIC CONDITION)

In static condition P1=P2 as velocity is zero

F=P1*A1-P2*A2

F=10*(0.785*180*180)-10*(0.785*120*120) 141300

F= 141300 Kg

Force intensity

f= F/area

f=141300/ 0.785*120*120 12.5

Summing of p+f 10+12.5 22.5 kg/sqcm

Which is greater than 20 kg/sqcm body pressure


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CALCULATION OF TOTAL PRESSURE ON BODY OF VALVES ONREDUCED SECTION OF PIPE LINE

CASE STUDY OF FAILURE OF NRV AT SAMAKHIYALI

DIAMETER OF PIPE LINE 529 MM I.E 559 MM OD MS LINE

DIAMETER OF NRV 350 MM

DISCHARGE FROM LINE540CUM/HR

VELOCITY CALCULATIONS

V=540/(60*60*0.785*0.529*0.529) 0.683 mt /second

Velocity at reduced section

V=540/(60*60*0.785*0.35*0.35)1.56mt/second

Pressure on line P1 7.5 Kg/sqcm

Pressure on reduced section P2

Considering on same ground level hence Z is nullified from equation

P2= P1 - ( V2^2- V1^2)/( 2*9.81)

P2= 7.5 - ( 1.56^2- 0.683^2)/(2*9.81) 7.4Say 7.4 kg/sqcm

EQUATING FORCE ( PIPE LINE IN RUNNING CONDITION)

F=P1*A1-P2*A2

F=7.5*(0.785*529*529)-7.4*(0.785*350*350) 9359.61

F= 9359.61 Kg

Force intensity

f= F/area

f=9359.61/ 0.785*35*35 9.733118419

Summing of p+f 7.4+9.73 17.13 kg/sqcm


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EQUATING FORCE ( PIPE LINE IN STATIC CONDITION)

In static condition P1=P2 as velocity is zero

F=P1*A1-P2*A2

F=7.5*(0.785*52.9*52.9)-7.5*(0.785*35*35) 9263.45

F= 9263.45 Kg

Force intensity

f= F/area

f=9263.45/ 0.785*35*35 9.633121019

Summing of p+f 7.5+9.63 17.13 kg/sq.cm.


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CASE STUDY SHOWING IMPORTANCE OF MINORLOSSES

Purpose of case study:- 31 mld allocation of water was sanctioned from Varsamedi had works and for which size of connection 250 mm diameter wasdesigned by some private consultant. Using theformulas of minor losses which were neglected byprivate consultant , it is established that minior lossesin some cases play the major role. Theoritically it wasestablished that 31 mld could not be conveyedthrough the line due to minor losses and for sufficientdischarge the size of connections to be enhanced to400 mm diameter

CALCULATION OF MINOR LOSSES DUE TO 250 MM DIA SLUICE VALVE ON 864MM DIA MS LINE FROM RAMBAGH TO VARSAMEDI

Water supply in mld 31 mld DIA 0.25

1) LOSS DUE TO TEEConsidering standard birfucation

H=k V x V / 2 x g

k=1.5 to 1.8 for standard birfucation

V for X mld=X x1000/ 24x60x60x0.785x0.25x0.25

V= 7.313

There fore loss =1.8xVxV/ 2x9.81

Loss= 4.91 mts 4.91

2) LOSS DUE TO SLUICE VALVEConsidering fully open condition

H=k V x V / 2 x g

k=0.19 for fully open condition



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Loss= 0.52 mts 0.52

3) LOSS DUE TO GRADUAL EXPANSION

Gradual expansion from 250 mm to 600 mm dia

Velocity V1= 7.313 mt/second

V2 for 40 mld= X x1000/ 24x60x60x0.785x0.6x0.6

V2= 1.27 mt/second

For 30% reducer K=0.49

Loss= V2xV2 - V1xV1 / 2 x g 2.64 2.64

4) Loss due to 250 mm dia pipe 2 mt lengthFor 2 mt length=

Loss= 0.05 mt 0.05

5) Loss in 610 mm dia ms pipe 15 mt 0.042

Total loss= 8.16

Loss due to friction in 864 mm dia mt 3.59

Head of sterrling chamber 3.5

Terminal head required VxV/2x9.81 0.02V= 0.663471

Total head 15.28

Available head 10

The head required for conveying 31 mld through 250 mm

diameter connection is 15.28 mt against the available head of 10 mt , hence it is not possible to convey 31 mld from thepipe line


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CALCULATION OF MINOR LOSSES DUE TO 400 MM DIA SLUICE VALVE ON 864MM DIA MS LINE FROM RAMBAGH TO VARSAMEDI

Water supply in mld 31 mld DIA 0.4

1) LOSS DUE TO TEEConsidering standard birfucation

H=k V x V / 2 x g

k=1.5 to 1.8 for standard birfucation

V for X mld=X x1000/ 24x60x60x0.785x0.4x0.4

V= 2.857


Loss= 0.75 mts 0.75

2) LOSS DUE TO SLUICE VALVEConsidering fully open condition

H=k V x V / 2 x g

k=0.19 for fully open condition


Loss= 0.08 mts 0.08

3) LOSS DUE TO GRADUAL EXPANSION

Gradual expansion from 400 mm to 600 mm dia

Velocity V1= 2.857 mt/second

V2 for 40 mld= X x1000/ 24x60x60x0.785x0.6x0.6

V2= 1.27 mt/second

For 30% reducer K=0.49

Loss= V2xV2 - V1xV1 / 2 x g 0.33 0.33

4) Loss due to 400 mm dia pipe 2 mt lengthFor 2 mt length=

Loss= 0.05 mt 0.05


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The Affinity Laws

The examples given in the note are not of IS units but

the calculations of IS units could be made in the sameway as the formula have direct relations hence for anytype of unit could be used.

The affinity laws are mathematic relationships that allow for the estimation of changes in pump performance as a result of a change in one of the basicpump parameters (variables). As such, it is important to understand theconcept of variables.

Why change pump variables?

Pump variables are changed to change pump performance. For example, itwould make little sense to completely replace a given pump in order to simplyreduce the head in a system. It may be more efficient to simply turndown theimpeller, i.e., change the diameter variable, to produce the desired result. Inso doing, we would want to know how the flow rate may be affected with thisnew impeller diameter. With the advent of the variable speed drive, a pumpsrotational speed is easily and conveniently adjusted over a broad range. Thisis an excellent method of controlling the flow rate of material streams inprocesses that require variability. What changes in head and flow might occur with the manipulation of a pumps speed using a variable drive? In another example, it may become necessary to estimate the performance of acentrifugal pump whose impeller diameter or speed may not be indicated on astandard performance curve. The approximate curves for a new impeller diameter or new speed could be determined by means of the affinity laws. Inshort, the affinity laws can come to the rescue of an enterprising Engineer.

There are two sets of affinity laws: (1) That set that is based on the impeller diameter variable assigned as a constant(D = C) and (2) that set that is based on the speed variable assigned as aconstant (N = C). Lets take a look at each set separately.The premise of the first set of affinity laws is:For a given pump with a fixed diameter impeller, the capacity will be directlyproportional to thespeed, the head will be directly proportional to the square of the speed, andthe required power willbe directly proportional to the cube of the speed.

Volume Capacity The volume capacity of a centrifugal pump can be expressed like

q1 / q2 = (n1 / n2)(d1 / d2) (1) where


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Example - Pump Affinity Laws - Changing Pump Speed

The pump speed is changed when the impeller size is constant. The initialflow is 100 gpm, the initial head is 100 ft, the initial power is 5 bhp, theinitial speed is 1750 rpm and the final speed 3500 rpm.

The final flow capacity can be calculated with (1a):

q2 = q1 n2 / n1

= (100 gpm) (3500 rpm) / (1750 rpm)

= 200 gpm

The final head can be calculated with (2a):

dp2 = dp1 (n2 / n1)2

= (100 ft) ((3500 rpm) / (1750 rpm))2

= 400 ft

The final power consumption can be calculated with (3a):

P2 = P1 (n2 / n1)3

= (5 bhp) ((3500 rpm) / (1750 rpm))3

= 40 bph


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Example - Pump Affinity Laws - Changing Impeller Diameter

The diameter of the pump impeller is reduced when the pump speed isconstant. The diameter is changed from 8 to 6 inches.

The final flow capacity can be calculated with (1b):

q2 = q1 (d2 / d1)

= ( 100 gpm) ((6 in) / (8 in))

= 75 gpm

The final head can be calculated with (2b):

dp2 = dp1 (d2 / d1)2= (100 ft) ((6 in) / ( 8 in))2

= 56.3 ft

The final power consumption can be calculated with (3b):

P2 = P1 (d2 / d1)3

= (5 bhp) ((6 in) / ( 8 in))3

= 2.1 bph

EXAMPLE 1: (Set 1 Affinity Laws)Problem: A centrifugal pump equipped with a variable frequency (speed) driverunning at 3500 rpm is discharging 240 gallons per minute corresponding witha head of 287 feet. The horsepower is 35.5. If the pumps speed is reduced to2900 rpm, what will be the revised flow rate, head, and power required.Solution:(1) Since there is no reference to the impeller diameter variable, it is safe toassume that D = constant and that theSet 1 Affinity laws are applicable and can be applied.(2) Given, in order, are: N1 = 3500 rpm, Q1 = 240 gpm, H1 = 287 feet, andBHP1 = 35.5 hp.be made to one of the basic pump parameters.

Motor Frequency and Speed (RPM)The speed of the AC motor is determined primarily by the frequency of the ACsupply and the number of poles in the stator winding, according to therelation: RPM = 2 * F * 60/p where RPM = (Synchronous) Revolutions per minute F = AC power frequency p = Number of poles, usually an even number but always a multiple of the number of phases.


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1725T = 315,120

1725 T = 182.7 lb-ft

Calculating Work:Work is applying a force over a distance. Force is any cause that changes theposition, motion, direction, or shape of an object. Work is done when a forceovercomes a resistance. Resistance is any force that tends to hinder themovement of an object. If an applied force does not cause motion the no workis produced.To calculate the amount of work produced, apply this formula:

W = F x D

W = work (in lb-ft)F = force (in lb)D = distance (in ft)

Example : How much work is required to carry a 25 lb bag of groceriesvertically from street level to the 4th floor of a building 30' above street level?

W = F x D W = 25 x 30 W = 750 -lb

Calculating Torque:

Torque is the force that produces rotation. It causes an object to rotate.Torque consist of a force acting on distance. Torque, like work, is measured ispound-feet (lb-ft). However, torque, unlike work, may exist even though nomovement occurs .

To calculate torque, apply this formula:

T = F x D

T = torque (in lb-ft)F = force (in lb)D = distance (in ft)

Example: What is the torque produced by a 60 lb force pushing on a 3'lever arm?

T = F x DT = 60 x 3

T = 180 lb ft


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Calculating Full-load Torque:

Full-load torque is the torque to produce the rated power at full speed of themotor. The amount of torque a motor produces at rated power and full speed

can be found by using a horsepower-to-torque conversion chart. When usingthe conversion chart , place a straight edge along the two known quantitiesand read the unknown quantity on the third line.

To calculate motor full-load torque, apply this formula:

T = HP x 5252rpm

T = torque (in lb-ft)HP = horsepower5252 = constantrpm = revolutions per minute

Example: What is the FLT (Full-load torque) of a 30HP motor operating at1725 rpm?

T = HP x 5252rpm

T = 30 x 52521725

T = 157,5601725

T = 91.34 lb-ft

Calculating Horsepower:

Electrical power is rated in horsepower or watts. A horsepower is a unit of power equal to 746 watts or 33,0000 lb-ft per minute (550 lb-ft per second). Awatt is a unit of measure equal to the power produced by a current of 1 ampacross the potential difference of 1 volt. It is 1/746 of 1 horsepower. The watt

is the base unit of electrical power. Motor power is rated in horsepower andwatts.Horsepower is used to measure the energy produced by an electric motor while doing work.

To calculate the horsepower of a motor when current and efficiency, andvoltage are known, apply thi

note on basics of hydraulics

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