FLUID THEORY RELATING TO WATER

Transcription

1 FUID TEORY REATING TO WATER DEFINITION OF A FUID Fluids are substances which are capable o lowing and which under suitable temperature conditions, conorm to the shape o containing vessels. When in equilibrium, luids cannot sustain tangential or shearing orces. All luids have some degree o compressibility and oer little resistance to change o orm. Fluids can be roughly divided into liquids and gasses. The main dierences between them are: (i) liquids are practically incompressible whereas gasses are compressible and (ii) liquids occupy deinite volumes and have ree suraces under the action o the Earth's gravity ield whereas a given mass o gas expands until it occupies all portions o any containing vessel. Plastics are dierent rom liquids and gasses in that they possess a yield stress, which must be exceeded beore low commences. Substances such as tomato sauce, butter and many other organic substances are such luids. Blood, which has a very low yield stress, is also regarded as a plastic. INTERNATIONA SYSTEM OF UNITS (SI) The International System o Units (SI) is a coherent set o units, one or each physical quantity. In this system the undamental mechanical dimensions are length (the metre) mass (the kilogram) and time (the second). SI units have been used in Australia since 1970 (superseding the British Engineering system) and unless otherwise indicated, are used throughout these notes. Physical quantity Name Symbol length metre m mass kilogram kg time interval second s electric current ampere A thermodynamic temperature kelvin K amount o substance mole mol luminous intensity candela cd SI derived units with special names (italics) and compound names relevant to hydraulics Physical quantity Name Symbol orce, weight 1 newton N (kg m/s ) pressure, stress pascal Pa (N/m ) energy, work, quantity o heat joule J (Nm) area square metre m volume cubic metre m 3 volumetric low rate cubic metre per second m 3 /s velocity (linear) metre per second m/s density (mass density) kilogram per cubic metre kg/m 3 Table 1 BRITIS ENGINEERING SYSTEM OF UNITS In this system o units (still in wide use around the world) the undamental mechanical dimensions are length (the oot) orce (the pound or pound weight) and time (the second). The unit o mass in this system is the slug and is deined in the ollowing manner using Newton's second law: 1 The mass o an object is the quantity o matter it contains; this is constant and expressed in units related to the kilogram. The mass o an object diers rom its weight, which is the measure o the orce o gravity acting on the object; this is measured in newtons. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 1 004, R.E. Deakin

2 orce (pounds) = mass (slugs) acceleration (t/sec ) Then weight (pounds) = mass (slugs) g (t/sec ) weight W (pounds) or mass M (slugs) = g (t/sec ) By this equation, the units o mass (slugs) in this system are lb-sec /t. Some British Engineering system quantities are set out below Physical quantity Name Symbol length oot t mass slug lb-sec /t time interval second s temperature ahrenheit F orce, weight pound (or pound orce) lb (or lb) pressure, stress pound per square inch psi energy, work oot-pound-orce t-lb area square oot t volume cubic oot t 3 volumetric low rate cubic eet per second t 3 /s velocity (linear) eet per second t/s density (mass density) slug per cubic oot slug/t 3 Table CONERSION FACTORS and USEFU REATIONSIPS The ollowing relationships can be used to determine conversion actors between quantities in both systems Conversion Factors (exact relationships, Mechtly,1973) Useul Relationships Force 1 oot = metres 1 pound = kilograms Using the conversion actors above and the internationally accepted values o the acceleration due to gravity (see National Institute o Standards and Technology Reerence on Constants, Units and Uncertainty, g = m/s = t/s the ollowing relationships or orce (weight), mass and density are 1 lb = N 1 slug = kg 1 slug/t = kg/m 3 3 C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 004, R.E. Deakin

3 Pressure 1 millibar (mb) = 100 pascal (Pa) 1 mm o g (0 C) = Pa 1 in o g (3 F) = Pa 1 mb = mm o g = in o g 1 std. atmosphere = mb (exactly) = mm o g = in o g 1 psi = Pa = KPa Temperature The relationships (exact) between degrees Fahrenheit (ºF) and degrees Celsius (ºC) are ( )( ) C = 5/9 F 3 F = ( 9/5 ) C + 3 DIMENSIONA ANAYSIS Dimensional analysis is the mathematics o dimensions o quantities and is a useul tool or modern luid mechanics. In any equation expressing a physical relationship between quantities, absolute numerical and dimensional equality must exist. In general all such physical relationships can be reduced to the undamental quantities o orce F, length and time T (or mass M, length and time T). Dimensional analysis is useul in converting one system o units to another. The ollowing table shows dimensions o various quantities (a) in terms o orce F, length and time T and (b) in terms o mass M, length and time T. (a) (b) Quantity Symbol F--T M--T Area A in m olume v in m 3 elocity or v Acceleration a or g in m/s A v or v a or g T T 1 T T Force F ( mass acceleration ) in N F F M T Mass M in kg M 1 F T M Density ρ in kg/m 3 ρ 4 F T M 3 Pressure p in N/m or Pa p F M 1 T Absolute viscosity μ in Pa s μ F T M 1 T 1 Rate o discharge Q in m 3 /s Q 3 1 T T 3 Shearing stress τ in N/m or Pa τ F M 1 T Weight W in N W F M T Table 3 C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 3 004, R.E. Deakin

4 MASS DENSITY OF A SUBSTANCE ρ (rho) The density o a substance is its mass per unit volume. where dm is an element o mass dv is an element o volume dm ρ = (1) dv The density o water is 1000 kg/m 3 at 4 C (Giles 1976) and may be regarded as constant or practical changes in pressure. In the British Engineering system, the relative density o water is taken as 1.94 slug/t 3 at 40 F When dealing with liquids the variable γ (gamma) is used where γ = ρg () and g is the gravitational acceleration; oten taken as having an average value o 9.81 m/s. In the British Engineering system γ is known as the speciic weight although in the SI system, the use o the term speciic is conined to descriptions o properties per unit mass. The term speciic weight is still widely used in hydraulics although it is not deined in the SI system. REATIE DENSITY OF A BODY The relative density o a body is the dimensionless number denoting the ratio o the mass o the body to the mass o an equal volume o a substance taken as a standard. Relative densities o liquids are reerred to water at 4 C as a standard. mass o the substance relative density o a substance = mass o an equal volume o water density o substance = density o water The relative density o water is 1.00 and o mercury is Relative density is commonly known as speciic gravity but or the reasons mentioned above the term speciic is not used in the SI system. (3) ISCOSITY OF A FUID It is well known that luids such as olive oil or sauce low more slowly through a given tube than would water or petrol. This dierence is ascribed to the presence o an internal luid riction known as viscosity; olive oil being more viscous than water. iscosity o a luid is the amount o resistance to a shearing orce and is due primarily to interaction between luid molecules. dy U d Fixed Plate Moving Plate y F Reerring to Figure 1, consider two large parallel Figure 1 plates at a small distance y apart, the space between the plates illed with a luid. The bottom plate is ixed and the upper plate is moving with a constant velocity U due to a constant orce F. Experiments have shown that whenever a luid lows in a channel the layer o luid at the channel wall actually adheres to it and is stationary, ie, there is no slip. So the luid in contact with the upper plate will adhere to it and will move at a velocity U. The luid in contact with the lower plate will have a velocity o zero. I distance y and velocity U are not to great, the velocity variation (or gradient) will be a straight line. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 4 004, R.E. Deakin

5 It is known (rom experiments) that the orce F varies with the area o the plate, with velocity U, and inversely with distance y. Since, by similar triangles, U y = d dy then AU F d F or y A dy The ratio F A is known as the shear stress and is denoted by the symbol τ (tau). The units o shear stress (τ ) are Newtons per square metre (N/m ) or Pascals (Pa). I a constant o proportionality μ (mu), called the absolute viscosity (or dynamic viscosity) is introduced into the equation above, then The units o absolute viscosity ( μ ) are Pascal-second (Pa s). d τ = μ (4) dy Equation (4) is Newton's law o viscosity, and luids according with this equation are known as Newtonian luids. In the second book o his amous Principia (Philosophiae naturalis principia mathematica The Mathematical Principles o Natural Philosophy published in 1687), Newton considered the circular motion o luids as part o his studies o the planets and wrote hupothesis The resistance arising rom the want o lubricity in the parts o a luid, is, other things being equal, proportional to the velocity with which the parts o a luid are separated rom one another. (Streeter 1971) iscosities o liquids decrease with temperature, but are not appreciably aected by pressure. PRESSURE p A luid under pressure exerts a orce on any surace in contact with it. Fluid pressure is transmitted with equal intensity in all directions and acts normal to any plane. In the same horizontal plane, the pressure intensities in a liquid are equal. Consider an element o area Δ A o a surace on which a orce Δ F is exerted, the pressure p is ΔF df p = lim = A 0 ΔA da For conditions where the orce F is uniormly distributed over an area A then The units o pressure are newtons per square metre (N/m ) or pascals (Pa). F p = (6) A (5) A unit o pressure in the SI system named in honour o Blaise Pascal ( ) a French mathematician, physicist, religious philosopher, and master o French prose. Pascal laid the oundation or the modern theory o probabilities, ormulated what came to be known as Pascal's law o pressure, and propagated a religious doctrine that taught the experience o God through the heart rather than through reason. Pascal tested the theories o Galileo and Evangelista Torricelli (an Italian physicist who discovered the principle o the barometer). To do so, he reproduced and ampliied experiments on atmospheric pressure by constructing mercury barometers and measuring air pressure, both in Paris and on the top o a mountain overlooking Clermont-Ferrand. These tests paved the way or urther studies in hydrodynamics and hydrostatics. Pascal invented the syringe and created the hydraulic press, an instrument based upon the principle that became known as Pascal's aw: pressure applied to a conined liquid is transmitted undiminished through the liquid in all directions regardless o the area to which the pressure is applied. is publications on the problem o the vacuum ( ) added to his reputation (Encyclopaedia Britannica 1999). C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 5 004, R.E. Deakin

6 ARIATION OF PRESSURE WIT DEPT Consider Figure which shows a small (horizontal) element o luid mass dm at rest, submerged within a luid body. surace The weight (orce = mass acceleration) o the element is dw = g dm = gρ dv where dv = dadh is the elemental volume. ence dw = ρg dadh (7) dh h df = p da 1 area da The orces acting on or by the luid element must be in equilibrium, so df1 + dw = df (8) dw df = (p + dp) da Figure Substituting (7) into (8) and using (5) gives p da + ρg dadh = ( p + dp ) da Cancelling terms and rearranging gives dp = ρg dh (9) Assuming that g and ρ are constant, integrating (9) gives p h dp = ρg dh p1 h1 Thus, the dierence in pressure between any two points at dierent levels in a liquid is given by p p = ρg( h h ) (10) 1 1 I point 1 is on the ree surace o the liquid and h is positive downward (10) becomes p = ρgh (11) where p (in Pa) is known as gauge pressure. Gauge pressures are oten expressed in bar where 1 bar = 10 Pa or millibar (mb) where 1 mb = 100 Pa = 1 hpa. In the British Engineering system, the units o pressure are pounds per square inch (psi). 5 ABSOUTE AND GAUGE PRESSURE I pressure is measured above absolute zero, it is known as absolute pressure. I pressure is measured either above or below or below atmospheric pressure as a base, it is called a gauge pressure. This is because practically all pressure gauges read zero when open to the atmosphere and measure only the dierence between the pressure o the luid to which they are connected and that o the surrounding air (Daugherty & Ingersoll 1954). C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 6 004, R.E. Deakin

7 Gauge pressure Atmospheric pressure Pressure acuum = negative gauge pressure Absolute zero Absolute pressure Absolute pressure Barometric pressure Figure 3 I the pressure is below that o the atmosphere, it is designated as a vacuum and its gauge value is negative. All values o absolute pressure or luids are positive and atmospheric pressure is that measured by a barometer (barometric pressure). Atmospheric pressure varies with altitude, and or a ixed location, varies slightly rom time to time (high- and low-pressure weather systems). For reerence purposes, a standard atmosphere has a value o mb. Figure 3 shows the relationship between dierent types o pressure. Example: At a depth o 60 metres below a ree water surace, what will be the water pressure (gauge pressure)? (use density o water ρ = 1000 kg/m 3 and acceleration due to gravity g = 9.81 m/s ) p = ρgh = ( 1000 )( 9.81)( 60 ) = 588,600 Pa = KPa PRESSURE EAD h Pressure head h represents the height o a column o homogeneous luid o density ρ that will produce a given intensity o pressure. Then p h = (1) ρg Example: What depth o oil (relative density 0.750) will produce a pressure o.75 bar? What depth o water? (use average value o 9810 N/m 3 or ρ g o water) Solution: h oil p = = = 37.4 m ρ g oil 5 h water p = = = 8 m ρ g 9810 water 5 C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 7 004, R.E. Deakin

8 TE FOW OF FUIDS Fluid low is complex and not always subject to exact mathematical analysis. A luid in low experiences, in addition to gravity, pressure orces, viscous and turbulent shear resistances, boundary resistance, and orces due to surace tension and compressibility eects o the luid. The presence o such a complex system o orces in real luid low makes the analysis very complicated. owever, a simpliying approach to problems may be made by assuming the luid to be ideal or perect. Ideal luids are non-viscous (rictionless) and incompressible. Water has a relatively low viscosity and is practically incompressible, and is ound to behave like an ideal luid. Thereore, simpliications can be made in the development o ormulae and the application o those ormulae to practical hydraulic problems. Some deinitions and terminology are given below. aminar and Turbulent Flow Reynolds' Experiment In 1883, the British engineer Osborne Reynolds (see citation below) demonstrated that there were two distinctly dierent types o luid low. e injected a ine, threadlike stream o coloured liquid at the entrance o a large glass tube through which water was lowing rom a tank. A valve at the discharge end permitted him to vary the low. When the velocity in the tube was small, the coloured liquid was visible as a straight line or streamline throughout the length o the glass tube, demonstrating that the particles o water moved in parallel straight lines. As the velocity o the water increased (by opening the valve) there was a point at which the low changed. The line would irst become wavy, and then at a short distance rom the entrance would break into numerous vortices beyond which the colour would be uniormly diused so that no streamlines could be distinguished. With closure o the valve, the process was reversed. The irst type o low (the coloured dye as a streamline) is known as laminar or streamline low. These terms arise rom (i) the luid appears to move by the sliding o laminations o ininitesimal thickness relative to adjacent layers and (ii) individual particles o water appear to move in deinite paths, which combined, orm streamlines. The second type o low (the coloured dye dispersed) is known as turbulent low and is characterised by irregular velocities o individual particles. Reynolds also ound that the nature o low depends on the ratio o the orces o inertia and luid riction due to viscosity. For an incompressible luid o density ρ and viscosity μ lowing through a pipe o diameter D at an average velocity, the ratio o these orces is a dimensionless number known as the Reynolds number NR Reynolds number N R ρd = (13) μ Reynolds established rom the experiment that the transition rom laminar to turbulent low occurs when N 000 (Streeter 1971). R Reynolds, Osborne (b. Aug. 3, 184, Belast, Ire. d. Feb. 1, 191, Watchet, Somerset, Eng.), British engineer, physicist, and educator best known or his work in hydraulics and hydrodynamics. Reynolds was born into a amily o Anglican clerics. e gained early workshop experience by apprenticing with a mechanical engineer, and he graduated at Queens' College, Cambridge, in mathematics in In 1868, he became the irst proessor o engineering at Owens College, Manchester, a position he held until his retirement in e became a ellow o the Royal Society in 1877 and received a Royal Medal in Though his earliest proessional research dealt with such properties as magnetism, electricity, and heavenly bodies, Reynolds soon began to concentrate on luid mechanics. In this area he made a number o signiicant contributions. is studies o condensation and heat transer between solids and luids brought radical revision in boiler and condenser design, while his work on turbine pumps permitted their rapid development. e ormulated the theory o lubrication (1886) and in 1889 developed the standard mathematical ramework used in turbulence work. e also studied wave engineering and tidal motions in rivers and made pioneering contributions to the concept o group velocity. Among his other contributions were the explanation o the radiometer and an early absolute determination o the mechanical equivalent o heat. is paper on the law o resistance in parallel channels (1883) is a classic. The "Reynolds stress" in luids with turbulent motion and the "Reynolds number" used or modelling in luid low experiments are named or him. Copyright Encyclopædia Britannica C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 8 004, R.E. Deakin

9 Steady and unsteady low Flow parameters such as velocity, pressure p and density ρ o a luid low are independent o time in a steady low. For unsteady low,, p and ρ, all or singly, vary with time. For example d dt d dt = 0 or steady low 0 or unsteady low In reality, these parameters are generally time dependent but oten remain constant, on average, over a reasonable time period. In the solution o many practical hydraulic problems, luid low is assumed to be steady. In steady low the streamline has a ixed direction at every point; and is thereore ixed in space. A particle always moves tangentially to the streamline, hence in steady low, the path o a particle is a streamline. Uniorm and non-uniorm low Flow is uniorm i its characteristics at any given instant remain the same at dierent points in the direction o low; otherwise the low is non-uniorm. I s is a distance measured along the low, uniorm and non-uniorm low is expressed mathematically as d = 0 or uniorm low ds d 0 or non-uniorm low ds The low o water through a long uniorm pipe at a constant rate is steady uniorm low. I the low rate is changing then it is unsteady uniorm low. The low o water through a non-uniorm pipe (eg, changing diameter) at a constant rate is steady non-uniorm. I the low rate varies, then the low is unsteady non-uniorm low. Streamlines and Streamtubes Streamlines are imaginary curves drawn through a luid to indicate the direction o motion o individual particles. A tangent to a streamline represents the instantaneous velocity o the luid particles at that point. Streamrubes represent elementary portions o a lowing luid bounded by a group o streamlines which conine the low. I the cross-sectional area o the streamtube is suiciently small, the velocity o the midpoint may be taken as the mean velocity or the section as a whole. The streamtube is used to derive the equation o continuity. EQUATION OF CONTINUITY The equation o continuity results rom the principle o conservation o mass. For steady low, the mass o luid passing all sections in a stream o luid (a streamtube), per unit mass o time is the same. A da 1 da 1 A 1 Figure 4 C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 9 004, R.E. Deakin

10 Considering Figure 4, which shows a streamtube, we may write mass entering stream tube mass leaving stream tube at section 1 = at section second second Since density = mass / volume, volume = area length and velocity = length / time then the equation o continuity can be written as ρ11 A1 = ρ A = constant (14) For incompressible luids and where the density is constant ( ρ1 = ρ), then or all practical purposes the equation o continuity (14) can be expressed as where Q is the discharge in cubic metres per second m 3 /s A is the cross-sectional area in m is the mean velocity o the section in m/s Q = A11 = A = constant (15) ENERGY EQUATION The energy equation results rom application o the principle o conservation o energy to luid low. In scientiic usage, a body is said to possess energy i it is capable o doing work and the amount o work it can do is a measure o its energy. The energy possessed by a lowing liquid consists o internal energy and energies due to pressure, velocity and position. In the direction o low, the energy principle is summarised in the ollowing equation Energy at Energy Energy Energy Energy at Section 1 + = Added ost Exctacted Section Energy Added to luid low is generally rom mechanical devices such as pumps, Energy ost is usually due to riction orces and Energy Extracted rom the low is usually extracted by mechanical devices such as turbines. For steady low o incompressible luids in which the change in internal energy is negligible, this equation simpliies to Bernoulli's theorem A E p p + + z + = + + z ρg g ρg g (16) In Bernoulli's theorem, all the terms are in units o length and are expressed as either "heads" or heights above a p datum. The term is the pressure head and the term is the velocity head. is known as lost head. ρg g A proo o Bernoulli's theorem is set out below, and ollows the derivation given by Giles (1976). DERIATION OF BERNOUI'S TEOREM Consider Figure 5(a) showing, as a ree body, an elementary mass o luid dm, contained within a cylinder o length dl and end-area da. The luid is lowing in the pipe at a steady rate and the direction o motion is the x- axis. Figure 5(b) shows a sectional view o the luid element; the x-axis in the plane o the paper and inclined to the horizontal at an angle θ. The orces acting in the x-direction are due to (i) the pressure acting on the end areas, (ii) the component o the weight and (iii) the shearing orces df S exerted by the adjacent luid. Forces normal to the direction o motion have not been shown acting on the ree body dm. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

11 dm x dl θ s df s df dw = ρ g da dl 1 dz df = p da df = (p + dp) da Figure 5(a) Figure 5(b) Considering orces in the x-direction, Newton's second law ( orce = mass acceleration ) can be written as F x = dm a x, and with the acceleration in the x-direction ax = d dt we obtain d ( df1 df dw sinθ dfs ) = dm (17) dt Remembering that pressure p is orce divided by area, df1 and df equal pda and ( p + dp) da respectively. The weight (orce = mass acceleration) o the element is dw = g dm and dm = ρ dv = ρ dadl, hence dw = ρg dadl. Making these substitutions, (17) becomes d ( pda ( p+ dp) da ρgdadlsinθ dfs ) = ρdadl (18) dt Dividing (18) by ρ g da, cancelling terms, then replacing dl dt with velocity and dl sinθ with dz gives dp dfs d dz ρg ρgda = g (19) df The term S ρg da the area ( perimeter length ) da = dp dl represents the resistance to low in length dl. The shear stress τ = orce area = dfs da and is, hence dfs = τ dp dl and the term dfs τ dp dl τ dl = = (0) ρg da ρg da ρg R The term R in (0) is known as the hydraulic radius and is deined as the cross-sectional area divided by the wetted surace perimeter. In this case, R = da dp. The sum o all the shearing orces is the measure o energy lost due to the low, remembering that work is orce by distance and a body is said to possess energy i it is capable o doing work. This energy loss is called lost head h. Shearing orces are oten also called riction orces, and the lost head h is also known as riction head h lost head dh τ dl = (1) ρg R The units o head loss (or riction loss) are metres i other quantities are in SI units. This can be veriied by dimensional analysis dh 1 ( M T ) ( ) ( )( )( ) = τ dl metres ρgr = 3 M T = = C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

12 Substituting (0) and (1) into (19) and re-arranging gives the equation o motion or steady low o any luid dp d dz dh 0 ρ g + g + + = () This dierential equation or steady low is a undamental luid low equation. When applied to an ideal luid (lost head = 0) it is known as Euler's equation. For incompressible luids (and water is practically incompressible) the integration o () is as ollows dp d dz dh ρ g + g + + = 1 p z p1 1 z1 The evaluation o the last integral yields the total head lost 0. Integrating and substituting the limits gives p p ( z z ) = ρg ρg g g 1 0 (3) Re-arranging (3) gives the customary orm o Bernouilli's theorem (when no energies are added or extracted) p p + + z z ρg g = + + ρg g (4) Note that in equations (3) and (4) all terms are in units o length, a act that can be veriied by dimensional analysis, eg 1 3 ( M )( T ) 1 ( T ) p M T z ρg + g + = + T + = + + For an ideal luid ( = 0 ) and when no other energies are added or extracted Bernoulli's theorem can be expressed as p gz constant ρ + + = (5) In this orm, Bernoulli's theorem can explain some seemingly paradoxical practical results o luid low (air and water). The ollowing extract rom Encyclopaedia Britannica ( ) is illuminating. Bernoulli's theorem indicates that, i an ideal luid is lowing along a pipe o varying cross section, then the pressure is relatively low at constrictions where the velocity is high and relatively high where the pipe opens out and the luid stagnates. Many people ind this situation paradoxical when they irst encounter it. Surely, they say, a constriction should increase the local pressure rather than diminish it? The paradox evaporates as one learns to think o the pressure changes along the pipe as cause and the velocity changes as eect, instead o the other way around; it is only because the pressure alls at a constriction that the pressure gradient upstream o the constriction has the right sign to make the luid accelerate. Paradoxical or not, predictions based on Bernoulli's theorem are well-veriied by experiment. Try holding two sheets o paper so that they hang vertically two centimetres or so apart and blow downward so that there is a current o air between them. The sheets will be drawn together by the reduction in pressure associated with this current. Ships are drawn together or much the same reason i they are moving through the water in the same direction at the same speed with a small distance between them. In this case, the current results rom the displacement o water by each ship's bow, which has to low backward to ill the space created as the stern moves orward, and the current between the ships, to which they both contribute, is stronger than the current moving past their outer sides. As another simple experiment, listen to the hissing sound made by a tap that is almost, but not quite, turned o. What happens in this case is that the low is so constricted and the velocity within the constriction so high that the pressure in the constriction is actually negative. Assisted by the dissolved gases that are normally present, the water cavitates as it passes through, and the noise that is heard is the sound o tiny bubbles collapsing as the water slows down and the pressure rises again on the other side. Copyright Encyclopædia Britannica C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 1 004, R.E. Deakin

13 AN APPICATION OF BERNOUI'S TEOREM Figure 6 shows a pipe with water lowing under pressure, at a mean velocity equal to. A piezometer and a Pitot tube are attached to the pipe. The piezometer is a simple device or measuring luid pressure. It is a thin tube inserted into the pipe at right angles and water will rise in the tube to a height equal to the static pressure head. The Pitot tube is a bent tube with one end pointed directly into the stream low. The water rises in the tube until all its kinetic energy is converted into potential energy. At A, the luid streamlines are parallel but divide as they approach the blunt end o the Pitot tube at B. At this point, there is complete stagnation (zero velocity), since the luid at this point is not moving up, nor down, nor neither to the right or let. The water in the Pitot tube will rise to a height equal to the stagnation pressure head. The dierence h between the static pressure head and stagnation pressure head is the velocity head. Total ead or Energy Gradient piezometer h static h vel Pitot tube h stag C A B datum Figure 6 Applying Bernoulli's theorem rom the pipe cross section through A in the undisturbed luid low to the section through B, considering the centre-line o the pipe as the datum yields pa no loss B 0 p ρg g = (assumed) ρg The static pressure head is h them is the velocity head static pa =, the stagnation pressure head is ρg, hence or an ideal "rictionless" luid h vel h stag pb = and the dierence between ρg h vel = (6) g Note, that in the case o water lowing in an open channel or stream, the static pressure head is zero (since the local pressure is zero gauge). The height to which water will rise in a Pitot tube, in this case, measures the velocity head. In stream gauging measurements using weirs, ormulae are derived based on an assumption that the velocity o low v through a thin strip is a unction o the head h o water above the strip. This assumption can be deduced rom a rearrangement o (6); v = gh APPICATION OF TE BERNOUI TEOREM IN YDRAUIC DESIGN PROBEMS In hydraulic design problems, the Bernoulli theorem should be applied in the ollowing manner (1) Draw a sketch o the system, choosing and labelling all cross sections o the stream (or pipe) under consideration. () Apply the Bernoulli equation in the direction o low. That is, section numbers increase in the direction o low. Select a datum line such that all quantities (heads and elevations) are positive. (3) Evaluate the energy upstream at Section 1 expressing pressure head and velocity head in metres o luid. I required, pressures should be expressed in gauge units. All velocities can be assumed to be average or mean velocities o low. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

14 (4) Add, in metres o the luid, any energy contributed by mechanical devices such as pumps. (5) Subtract, in metres o the luid, any energy lost during low (riction losses). (6) Subtract, in metres o the luid, any energy extracted by mechanical devices, such as turbines. (7) Equate this summation o energy to the sum o the pressure head, velocity head and elevation head in Section. (8) I the two velocity heads are unknown, they can be related by the equation o continuity. ENERGY GRADIENT The energy gradient (or energy line or energy grade line) is a graphical representation o the energy at each section. With respect to a datum, the total energy (in metres o luid) can be plotted at each section and the line obtained (passing through these points) is a valuable tool in many low problems. The energy gradient will slope downwards in the direction o low, unless energy is added by mechanical means, or unless no losses are assumed. In the case o no losses assumed, the energy gradient will be a horizontal line. YDRAUIC GRADIENT The hydraulic gradient (or hydraulic grade line) lies vertically below the energy gradient by an amount equal to the velocity head at each section. The two lines (energy gradient and hydraulic gradient) are parallel or all sections o equal cross sectional area. The vertical distance between the centre o the stream and the hydraulic gradient is the pressure head at the particular section. A B' B M p ρ g g C N D C Energy gradient z ydraulic gradient F Datum E Figure 7 Consider Figure 7 which shows two bodies o luid connected by a pipe o constant cross sectional area. I a piezometer tube were erected at B the luid would rise in it to BB' equal to the pressure head p ρ g existing at that section. I the end o the pipe at E were closed so that no low would occur, the height o the column would be BM. The drop rom M to B' when low occurs, is due to two actors, (i) a portion o the pressure head has been converted into velocity head g which the luid has at B, and (ii) there has been a loss o head due to luid riction between A and B. I a series o piezometers were erected along the pipe, the luid would rise in them to various levels. The line drawn through the summits o these imaginary series o luid columns is the hydraulic gradient. The hydraulic gradient represents the pressure along the pipe, as at any point the vertical distance rom the pipe centre line to the hydraulic gradient is the pressure head at that point, assuming the proile has been drawn to scale. At C, this distance is zero, indicating that the pressure is atmospheric at both locations. At D, the pipe is above the hydraulic gradient, indicating that the pressure head is DN, or a vacuum o DN. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

15 I the proile o a pipeline is drawn to scale, the hydraulic gradient enables the pressure head to be determined at any section by measurement on the diagram. In addition, the hydraulic gradient shows the variation o pressure in the entire length o the pipe. The hydraulic gradient is a straight line only i the pipe is straight and o constant cross sectional area. I pipe sizes change, then there will be abrupt jumps in the hydraulic gradient. At A, there is slight curvature in the hydraulic gradient and energy gradient due to streamlines converging as they enter the pipe. For long pipelines, the hydraulic gradient can be drawn rom water surace to water surace without any appreciable loss in accuracy. FRICTION OSSES IN PIPES Bernoulli's theorem (the energy equation) is used in the solution o practical pipe low problems in various branches o engineering. Flow o real luids is more complex than low o ideal luids; shear orces between luid particles and the boundary walls and between luids themselves, result rom the luid's viscosity. Equations that might account or the low (Euler's dierential equation) have no general solution and recourse must be made to experimentation and semi-empirical methods to solve low problems in pipes. An important problem in determining the low o luids in pipes (using Bernoulli's theorem) is determining head loss due to the riction o luids lowing in pipes. A ormula or determining head loss is the Darcy- Weisbach Formula pipe length = riction actor velocity head pipe diameter D g = D g (7) where is a pipe riction actor is the length o the pipe in metres D is the diameter o the pipe is the mean velocity o the low g is the acceleration due to gravity The riction actor in the Darcy-Weisbach ormula can be evaluated using a ormula developed by Colebrook (1939) known as the Colebrook-White equation which is regarded as reliable or all pipes (rough or smooth) (Giles 1976, Featherstone & Nalluri 1988) 1.51 Colebrook-White equation = log ε D NR where ε D is the relative roughness o the pipe is the Reynolds number N R (8) In the Colebrook-White equation, ε D is the ratio o the height o protuberances ε o small particles attached to the inside surace o the pipe to its diameter D. Studies o low in pipes, whether laminar or turbulent (see Reynolds' experiment), have shown that there is a very thin layer o luid attached to the sides o the pipe known as the boundary layer o thickness δ (generally δ < 0.0 mm ). In the early part o the 0th century, the German engineers Nikuradse and Prandtl (see citation below) experimented with pipes o varying diameter by coating them with sand grains o uniorm diameter ε and then subjecting them to lows o varying velocity. Their experiments showed that i the coating thickness ε was greater than the thickness o the boundary layer δ, the surace was regarded as hydraulically rough. And i the coating thickness ε, was less than the thickness o the boundary layer δ, the surace was regarded as hydraulically smooth. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

16 Their equations or determining the riction actor, improved and modiied by others are quoted in the literature as the Kármán-Prandtl equations (Giles, 1976, Featherstone & Nalluri, 1988) smooth pipe surace ( δ 1.7ε ) rough pipe surace ( δ 0.08ε ) 1 > 10 ( NR ) 1 < 10 = log 0.8 (9) D = log 1.14 ε + (30) The Colebrook-White equation (8) the addition o equations (9) and (30) has been ound to accord with observations o low in commercial pipes over a wide range o laminar and turbulent lows and is regarded as suicient or determining the riction actor in pipes (Featherstone & Nalluri 1988). The derivation o the Darcy-Weisbach ormula is an interesting study in the useulness o dimensional analysis and is given below. Prandtl, udwig (b. Feb. 4, 1875, Freising, Ger. d. Aug. 15, 1953, Göttingen), German physicist who is considered to be the ather o aerodynamics. In 1901 Prandtl became proessor o mechanics at the University o annover, where he continued his earlier eorts to provide a sound theoretical basis or luid mechanics. e served as proessor o applied mechanics at the University o Göttingen rom 1904 to 1953 and there established a school o aerodynamics and hydrodynamics that achieved world renown. In 195 he became director o the Kaiser Wilhelm (later the Max Planck) Institute or Fluid Mechanics. is discovery (1904) o the boundary layer, which adjoins the surace o a body moving in air or water, led to an understanding o skin riction drag and o the way in which streamlining reduces the drag o airplane wings and other moving bodies. is work on wing theory, which ollowed similar work by a British physicist, Frederick W. anchester, but was carried out independently, elucidated the process o airlow over airplane wings o inite span. That body o work is known as the anchester-prandtl wing theory. Prandtl made decisive advances in boundary-layer and wing theories, and his work became the undamental material o aerodynamics. e was an early pioneer in streamlining dirigibles, and his advocacy o monoplanes greatly advanced heavier-thanair aviation. e contributed the Prandtl-Glaubert rule or subsonic airlow to describe the compressibility eects o air at high speeds. In addition to his important advances in the theories o supersonic low and turbulence, he made notable innovations in the design o wind tunnels and other aerodynamic equipment. e also devised a soap-ilm analogy or analysing the torsion orces o structures with non-circular cross sections. Copyright Encyclopædia Britannica C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

17 DERIATION OF TE DARCY-WEISBAC FORMUA In the development o Bernoulli's theorem, the lost head or the elemental distance dl was shown as equation (5) dl dh = τ and or the total length o pipe the lost head ρgr can be expressed = τ (31) ρg R where τ is shear stress is the length o the pipe in metres R is the hydraulic radius in metres ρ is the density in kg/m 3 g is the acceleration due to gravity The shear stress τ is known to be a unction o low velocity, pipe diameter D, density ρ, viscosity μ and a coeicient K, the relative roughness o the pipe. K = ε D where ε is the size o the protuberances on the inside o the pipe. Thus, we may write τ = (, D, ρ, μ, K ) (3) The exact orm o (3) is unknown but can be expressed in a general orm a b c d e τ = C D ρ μ K (33) where C is a dimensionless coeicient and the unknown powers o the variables are expressed as the indices a, b, c, d and e. Expressing (33) in terms o the dimensions o each o the variables gives (ignoring C since it is dimensionless) a b c d ( ) ( ) ( ) ( ) ( ) e M T = T M M T a a b c 3c d d d = T M M T Equating the indices o the dimensions M, and T gives the ollowing three equations 1 = c + d 1= a + b 3c d = a d Solving in terms o d, which will allow, D, ρ and μ to be combined, gives a = d c = 1 d b = d Substituting these values into (33) and simpliying gives ρd The term μ relative roughness o the pipe is the Reynolds number K d d 1 d d τ = C D ρ μ K N R ρd = C μ d e ρk e. Combining the Reynolds number, the coeicient C and the = ε D together in a coeicient C gives τ = C ρ (34) C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

18 Substituting (34) into (31) gives = C (35) gr e d where C = CK N. R The hydraulic radius R is the cross-sectional area divided by the wetted surace perimeter ydraulic Radius For a circular pipe lowing ull, the cross-sectional area ( π ) cross-sectional area A R = = (36) wetted surace perimeter P A = D 4, the wetted surace perimeter is the pipe circumerence P = π D and the hydraulic radius is R = D 4. Substituting this expression into (35) gives = C R g 4 = C D g Again, gathering terms into another coeicient the riction actor gives the Darcy-Weisbach ormula = D g (37) The riction actor is a unction o the Reynolds Number the original coeicient C in (33) NR, the relative roughness o the pipe K = ε D and e d = 4C = 8CK N (38) The riction actor is given by the empirically derived Colebrook-White equation (8). R MINOR OSSES In addition to head loss due to luid riction, other head losses occur at entrances and exits o pipes, bends in pipes, changes in pipe size, at valves and other pipe ittings. These minor losses (or local losses), due to changes in either direction or magnitude (or both) o the low velocity can be expressed in the general orm minor head loss = Ke g where K e is the kinetic energy correction actor, is the velocity in the downstream section o diameter D ; D 1 is the diameter upstream. Featherstone & Nalluri (1988) have the ollowing table o representative values o K e or changes in pipe diameters D D K 1 e Table 4 Note that the value K e = 0.5 type are called entry losses. relates to the abrupt entry rom a reservoir into a circular pipe. Minor losses o this C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

19 Other values or minor losses cited in Featherstone & Nalluri (1988) are ead loss at abrupt enlargement ead loss 90º elbow ead loss at 90º smooth bend Discharge (exit) loss A = 1 g A = g = g = g ead loss at a valve = K g where K depends upon the type o valve and percentage o closure SOUTION OF PIPE-FOW PROBEMS In the solution o pipe low problems using Bernoulli's theorem, three basic cases arise Case Given To ind I Q or,, D, μ, ε II,, D, μ, ε Q or III, Q or,, μ, ε D Examples 1, and 3 below describe the solutions to these three basic cases. In these cases the ollowing equations, in various combinations, are used, numbered as in the text Reynolds Number N R ρd = (13) μ Continuity equation Q = A11 = A = constant (15) Darcy-Weisbach equation = D g (7) and (37) 1 Colebrook-White equation log ε.51 = D NR To solve problems o type I, the Reynolds Number, Darcy-Weisbach and Colebrook-White equations can be combined to give an explicit equation or the low velocity (8).51μ = gd log D ε ρd gd (39) C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc , R.E. Deakin

20 The ollowing table or the size o protuberances ε (in mm's) or pipes is useul. The values in the table have been taken rom Diagram A1 in the Appendix o Giles et al (1994) where they were originally given in eet. Kind o Pipe alues o ε in mm or ining (new) Range Design alue Brass Copper Concrete Cast Iron uncoated asphalt dipped cement lined bituminous lined centriugally spun Galvanised Iron Wrought Iron Commercial & Welded Steel Riveted Steel Table 5 (Taken rom Diagram A1, Giles et al (1994), original values in eet) Example 1 (Case I: given Q or,, D, μ, ε to ind ) Determine the lost head in 300 metres o new uncoated 300 mm inside diameter cast iron pipe when water at 15ºC lows at 1.5 m/s. Use: water at 15ºC has density Solution ε = 0.50 mm (Table 5) g = 9.81 m s 3 ρ = 1000 kg m and viscosity 1. Compute the Reynolds Number rom (13) N R 3 μ = Pa s ρd ( 1000 ) ( 0.3) ( 1.5) = = = 398, 31 μ Compute the riction actor rom the Colebrook-White equation (8) 1 log ε.51 = D NR Note that must be computed by iteration since it appears on the et-and Side (S) and Right- and Side (RS) o (8). A table o iterations is shown below Iteration n n RS n Adopt = C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 0 004, R.E. Deakin

21 3. Compute the lost head rom the Darcy-Weisbach equation (7) = = =.5 m D g Example (Case II: given,, D, μ, ε to ind ) Water at 15ºC lows through a 300 mm (inside diameter) steel pipe with a head loss length 300 metres. Calculate the low. Use: water at 15ºC has density Solutions ε = 0.06 mm (Table 5) g = 9.81 m s Method 1 (iterative) 3 ρ = 1000 kg m and viscosity 1. Assume a riction actor. Calculate rom the Darcy-Weisbach equation (7) 3 μ = Pa s o 6 metres in a pipe o = D g giving D = g 3. Compute the Reynolds Number rom (13) N R ρd = μ 4. Compute a "new" riction actor by iteration using the Colebrook-White equation (8) 1.51 = log ε D NR 5. Repeat steps to 4 until there is no change in. Method (direct) Use equation (39), which is a combination o the Reynolds number, Darcy-Weisbach and Colebrook- White equations giving an explicit equation in ε.51μ = log + =.81 m s ρd gd gd D C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 1 004, R.E. Deakin

22 Example 3 (Case III: given, Q or,, μ, ε to ind D ) What size o new uncoated cast iron pipe, 500 metres long will deliver 1 m 3 /s o water at 15ºC with a drop in the hydraulic grade line o 65 metres? Use: water at 15ºC has density Solution ε = 0.50 mm (Table 5) g = 9.81 m s 3 ρ = 1000 kg m and viscosity 3 μ = Pa s In this case, with D unknown, there are three unknowns in the Darcy-Weisbach equation (, and D), three unknowns in the Reynolds number ( NR, and D) and the relative roughness o the pipe ε D is also unknown. Using the Continuity equation (15) to eliminate the velocity in the Darcy-Weisbach equation (7) and in the Reynolds number will simpliy the solution. Since Q = A11 = A = constant (Continuity equation) then Q = A and area A = π D then 4 Q 4Q rearranging gives = A = π D and 16Q = 4 π D. The expressions or and can be substituted into the Darcy-Weisbach and Reynolds number equations to give = 8Q 5 π g D (40) and N R 4ρ Q = (41) π μ D The solution, using equations (40) and (41) and the Colebrook-White equation, is iterative Note: Normally, only one or two iterations are required since standard pipe sizes are usually selected. The next larger pipe-size diameter rom that given by the computation is taken. Iteration A A1. Assume a riction actor say = 0.0 A. Calculate D rom a rearrangement o (40) using the assumed riction actor 8Q 8( 500 )( 1) D = m π g = π ( 9.81)( 65) = (4) A3. Compute the Reynolds Number rom (41) N R 4ρ Q 4 ( 1000 ) ( 1) = = ( 3 = 1, 955, πμd π )( ) A4. Compute the riction actor rom the Colebrook-White equation (8) 1 log ε.51 = D NR Note that must be computed by iteration since it appears on the et-and Side (S) and Right- and Side (RS) o (8). A table o iterations is shown below C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 004, R.E. Deakin

23 Iteration n n RS n Adopt = Repeat the above steps until the value o the riction actor does not change. When this occurs, all equations are satisied. Iteration B B1. Adopt the "new" riction actor rom Iteration A = B. Calculate D rom (4) using the new riction actor D 8Q = 5 π g = B3. Compute the Reynolds Number rom (41) N R m 4ρ Q = =, 03, πμd B4. Compute the riction actor by iteration using the Colebrook-White equation (8). Iteration n n RS n Adopt = Iteration C C1. Adopt the "new" riction actor rom Iteration B = C. Calculate D rom (4) using the new riction actor D 8Q = 5 π g = C3. Compute the Reynolds Number rom (41) N R m 4ρ Q = =, 09, πμd C4. Compute the riction actor by iteration using the Colebrook-White equation (8). Iteration n n RS n Adopt = The riction actor has changed by between Iterations A and B, and only between Iterations B and C. It may be assumed that urther iterations would produce no change. Thus we may adopt the computed pipe diameter rom the last iteration D = m. C:\Projects\Geospatial\Eng Surv 1\Fluid Theory\Fluid.doc 3 004, R.E. Deakin