CHAPTER 20

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=110 BGCOLOR="#666699" TEXT="#080000" bgproperties="fixed" background="04a00840erlcjm.gif"> <div> <FONT SIZE="5" COLOR="#FFFFFF" FACE="Arial" ><B>The World of Rotodynamic PUMPS</B></FONT><B>     |     </B><A HREF="index.htm" TARGET="_top" TITLE="The World of Rotodynamic PUMPS"><U><B>home</B></U></A></div> <div> <A HREF="id17.htm" TARGET="_top" TITLE="INTRODUCTION"><U>INTRODUCTION</U></A>   |   <A HREF="id18.htm" TARGET="_top" TITLE=" CONTENTS"><U>   CONTENTS</U></A>   |   <A HREF="id19.htm" TARGET="_top" TITLE="CHAPTER 1"><U>CHAPTER 1</U></A>   |   <A HREF="id20.htm" TARGET="_top" TITLE="CHAPTER 2"><U>CHAPTER 2</U></A>   |   <A HREF="id21.htm" TARGET="_top" TITLE="CHAPTER 3"><U>CHAPTER 3</U></A>   |   <A HREF="id22.htm" TARGET="_top" TITLE="CHAPTER 4"><U>CHAPTER 4</U></A>   |   <A HREF="id23.htm" TARGET="_top" TITLE="CHAPTER 5"><U>CHAPTER 5</U></A>   |   <A HREF="id24.htm" TARGET="_top" TITLE="CHAPTER 6"><U>CHAPTER 6</U></A>   |   <A HREF="id25.htm" TARGET="_top" TITLE="CHAPTER 7"><U>CHAPTER 7</U></A>   |   <A HREF="id26.htm" TARGET="_top" TITLE="CHAPTER 8"><U>CHAPTER 8</U></A>   |   <A HREF="id27.htm" TARGET="_top" TITLE="CHAPTER 9"><U>CHAPTER 9</U></A>   |   <A HREF="id28.htm" TARGET="_top" TITLE="CHAPTER 10"><U>CHAPTER 10</U></A>   |   <A HREF="id29.htm" TARGET="_top" TITLE="CHAPTER 11"><U>CHAPTER 11</U></A>   |   <A HREF="id30.htm" TARGET="_top" TITLE="CHAPTER 12"><U>CHAPTER 12</U></A>   |   <A HREF="id31.htm" TARGET="_top" TITLE="CHAPTER 13"><U>CHAPTER 13</U></A>   |   <A HREF="id32.htm" TARGET="_top" TITLE="CHAPTER 14"><U>CHAPTER 14</U></A>   |   <A HREF="id33.htm" TARGET="_top" TITLE="CHAPTER 15"><U>CHAPTER 15</U></A>   |   <A HREF="id34.htm" TARGET="_top" TITLE="CHAPTER 16"><U>CHAPTER 16</U></A>   |   <A HREF="id35.htm" TARGET="_top" TITLE="CHAPTER 17"><U>CHAPTER 17</U></A>   |   <A HREF="id36.htm" TARGET="_top" TITLE="CHAPTER 18"><U>CHAPTER 18</U></A>   |   <A HREF="id37.htm" TARGET="_top" TITLE="CHAPTER 19"><U>CHAPTER 19</U></A>   |   <B>CHAPTER 20</B>   |   <A HREF="id39.htm" TARGET="_top" TITLE="CHAPTER 21"><U>CHAPTER 21</U></A>   |   <A HREF="id40.htm" TARGET="_top" TITLE="CHAPTER 22"><U>CHAPTER 22</U></A>   |   <A HREF="id41.htm" TARGET="_top" TITLE="AUTHOR"><U>AUTHOR</U></A></div> <div> </div> </td> </tr><tr> <td valign="top" width=190 BGCOLOR="#CCCCFF" TEXT="#080000" background="04411c00egdysm.gif"> <bR> </td> <td valign="top" height=370 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> CHAPTER 20</div> <div> CHAPTER 20-Fluid Mechanics and pumps </div> <div> <B> </B><FONT SIZE="3" COLOR="#FF0000" FACE="Arial" > INTRODUCTION TO FLUID MECHANICS </FONT></div> <div> <B>Physical science dealing with the action of fluids at rest or in motion, and with applications and devices in engineering using fluids. Fluid mechanics is basic to such diverse fields as aeronautics, chemical, civil, and mechanical engineering meteorology, naval architecture , oceanography . </B></div> <div> <B>Fluid mechanics can be subdivided into two major areas, fluid statics, which deals with fluids at rest, and fluid dynamics, concerned with fluids in motion. </B></div> <bR> <div> <B>The term hydrodynamics is applied to the flow of liquids or to low-velocity gas flows where the gas can be considered as being essentially incompressible. </B></div> <bR> <div> <B>Aerodynamics is concerned with the theory of flight, and compressible fluid flow or gas dynamics with the behaviour of gases under flow conditions, where velocity and pressure changes are sufficiently large to require inclusion of the compressibility effects. </B></div> <bR> <div> <B>Applications of fluid mechanics involve all kinds of flow machinery, including jet propulsion, hydraulics, turbine, compressors and pumps. Hydraulics mainly concerns machines and structures such as hydraulic turbines, dams, and hydraulic pressures, using water or other liquids. </B></div> <div> <B> </B></div> <div> THE FOUNDERS OF FLUID DYNAMICS OR HYDRODYNAMICS </div> <div> <B>This branch of fluid mechanics deals with the laws of fluids in motion; these laws are considerably more complex and, in spite of the greater practical importance of fluid dynamics, only a few basic ideas can be discussed here. </B></div> <div> <B>Interest in fluid dynamics dates from the earliest engineering application of fluid machines. Archimedes made an early contribution by his invention of the screw pump, the pushing action of which is similar to that of a cork-screw device in a meat grinder. Other hydraulic machines and devices were developed by the Romans, who not only used Archimedes' screw for irrigation and mine pumping but also built extensive aqueduct systems, some of which are still in use. The Roman architect and engineer Vitruvius invented the horizontal waterwheel during the 1st century BC, which revolutionised corn milling. </B></div> <bR> <div> <B>Despite the early practical applications of fluid dynamics, little or no understanding of the basic theory existed, and development lagged accordingly. After Archimedes, more than 1800 years elapsed before the next significant scientific advance was made by the Italian mathematician and physicist Evangelista Torricelli, who invented the barometer in 1643, and formulated Torricelli's law, which related the efflux velocity of a liquid through an orifice in a vessel to the liquid height above it. The major spurt in the development of fluid mechanics had to await the formulation of Newton's laws of motion by the English mathematician and physicist Isaac Newton. These laws were applied to fluids first by the Swiss mathematician Leonhard Euler, who derived the basic equations for a frictionless, or inviscid, fluid. </B></div> <div> <B> </B></div> <div> <B>Euler</B><FONT SIZE="4" COLOR="#000000" FACE="Arial" ><B> </B></FONT><FONT SIZE="2" COLOR="#FF0000" FACE="Arial" ><B>first recognised that dynamical laws for fluids can only be expressed in a relatively simple form if the fluid is assumed incompressible and ideal, that is, if the effects of friction or viscosity can be neglected. Because, however, this is never the case for real fluids in motion, the results of such an analysis can only serve as an estimate for those flows where viscous effects are small. </B></FONT></div> <div> <B> </B></div> <div> Principles applied to Incompressible and Inviscid [Frictionless] Flows </div> <div> <B>These flows follow Bernoulli's principle, named after the Swiss mathematician and scientist Daniel Bernoulli. The principle states that the total mechanical energy of an incompressible and inviscid flow is constant along a streamline. Streamlines are imaginary flow lines that are always parallel to the local direction of the flow, and that for steady flow are also the lines followed by individual fluid particles. Bernoulli's principle leads to an interrelationship between pressure effects, velocity effects, and gravity effects, and indicates that the velocity increases as the pressure decreases. This principle is important in nozzle design and in flow measurements, and it can also be used to predict the lift of a wing in flight. </B></div> <div> <B> </B></div> <div>  The theoretical developments of Viscous Flow, Laminar and Turbulent Motion </div> <div> <B>The first carefully documented friction experiments in low-speed pipe flow were carried out independently in 1839 by the French physiologist Jean Leonard Marie Poiseuille, who was interested in the characteristics of blood flow, and in 1840 by the German hydraulic engineer Gotthilf Heinrich Ludwig Hagen. An attempt to include the effects of viscosity into the mathematical equations was made first in 1827 by the French engineer Claude Louis Marie Navier, and independently by the British mathematician Sir George Gabriel Stokes, who in 1845 perfected the basic equations for viscous incompressible fluids. These are now known as the Navier-Stokes equations, and they are so complex that they can be applied only to simple flows. One such flow is that of a real fluid through a straight pipe. Here Bernoulli's principle is not applicable because part of the total mechanical energy is dissipated as a result of viscous friction, resulting in a pressure drop along the pipe. The equations suggest that this pressure drop for a given pipe and a given fluid should be linear with the flow velocity. Experiments first conducted near the middle of the 19th century showed that this was only true for low velocities; at higher velocities, the pressure drop was more nearly proportional to the square of the velocity. This problem was not resolved until 1883 when the British engineer Osborne Reynolds showed the existence of two types of viscous flows in pipes. At low velocities the fluid particles follow the streamlines (laminar flow) and results match the analytical prediction. </B></div> <bR> <div> <B>At higher velocities the flow breaks up into a fluctuating velocity pattern or eddies (turbulent flow) in a form that cannot be fully predicted even today. Reynolds also established that the transition from laminar to turbulent flow was a function of a single parameter that has since become known as the Reynolds number. If the Reynolds number, which is the product of velocity, fluid density, and pipe diameter, divided by the fluid viscosity, is less than 2100, the pipe flow will always be laminar; at higher values it will normally be turbulent. The concept of a Reynolds number is basic to much of modern fluid mechanics. </B></div> <bR> <div> <B>Turbulent flows cannot be evaluated solely from computed predictions and depend on a mixture of experimental data and mathematical models for their analysis, with much of modern fluid-mechanics research still being devoted to better formulations of turbulence. </B></div> <bR> <div> <B>The transitional nature from laminar to turbulent flows and the complexity of the turbulent flow can be observed as cigarette smoke rises into very still air. At first it rises in a laminar streamline motion but after some distance it becomes unstable and breaks up into an intertwining eddy pattern. </B></div> <div> <B> </B></div> <div> <B>The developments in theories of Boundary Layer Flow . </B></div> <div> <B>Before about 1860 the engineering interest in fluid mechanics was limited almost entirely to water flows. The development of the chemical industry during the latter part of the 19th century directed attention to other liquids and to gases. Interest in aerodynamics began with the studies of the German aeronautical engineer Otto Lilienthal in the last decade of the 19th century and saw major advances following the first successful powered flight by the American inventors Orville and Wilbur Wright in 1903. </B></div> <bR> <div> <B>The complexity of viscous flows, especially turbulent flows, severely restricted progress in fluid dynamics until the German engineer Ludwig Prandtl recognized in 1904 that many flows could be divided into two principal regions. </B></div> <bR> <div> <B>The region close to the surface consists of a thin boundary layer where the viscous effects are concentrated and where the mathematical model can be greatly simplified. Outside the boundary layer viscous effects can be disregarded and the simpler mathematical equations for inviscid flows can be used. </B></div> <bR> <div> <B>The boundary-layer theory has made possible much of the development of modern aircraft wings and the design of gas turbines and compressors. The boundary-layer model not only permitted a much simplified formulation of the Navier-Stokes equations in the region close to the body surface but also led to further developments of the flow of inviscid fluids that can be applied outside the boundary layer. </B></div> <bR> <div> <B>Much of the modern development of fluid mechanics was made possible by the boundary-layer concept and it has been carried out by such key contributors as the Hungarian-born American aeronautical engineer Theodore von Kármán, and the German mathematician Richard von Mises, by the British physicist and meteorologist Sir Geoffrey Ingram Taylor. </B></div> <div> <B> </B></div> <div> <B> The development of Compressible Flow . </B></div> <div> <B>Interest in compressible flows started with the development of steam turbines by the British inventor Charles Algernon Parsons, and the Swedish engineer Carl Gustaf Patrik de Laval during the 1880s. Here high-speed flow of steam within flow passages was first encountered and the need for efficient turbine design led to improved compressible flow analyses. </B></div> <bR> <div> <B>Modern advances, however, had to wait for the stimulus of successful gas turbine and jet engine development in the 1930s. The early interest in high-speed flows over surfaces arose in the study of ballistics, for which an understanding of the motion of projectiles was needed. </B></div> <bR> <div> <B>Major developments started near the end of the 19th century, involving Prandtl and his students, among others, and increased after the introduction of high-speed aircraft and rockets . </B></div> <bR> <div> <B>One of the basic principles of compressible flows is that the density of a gas changes when the gas is subjected to large velocity and pressure changes. At the same time its temperature also changes, leading to more complex means of analysis. The flow behaviour of a compressible gas depends on whether the flow velocity is smaller or greater than the velocity of sound. The velocity of sound is the name given to the propagation velocity of a very small disturbance, or pressure wave, within the fluid. For a gas it is proportional to the square root of the absolute temperature. For instance, air at 20° C, or 293° on the Kelvin, or absolute, scale (68° F), has a sound velocity of 344.65 m per sec (1130 ft per sec). </B></div> <bR> <div> <B>If the flow velocity is less than the sound velocity (subsonic flow), pressure waves can be transmitted throughout the whole fluid to adjust the flow that rushes toward an object. Thus the subsonic flow approaching a wing will adjust itself some distance upstream to flow smoothly over the surface. In supersonic flow, pressure waves cannot travel upstream to readjust the flow. As a result, the air rushing toward a wing in supersonic flight will not be prepared for the impending disturbance the wing will cause. Instead, it has to redirect very suddenly in the proximity of the wing, where a sharp compression or shock is coupled with the redirection. The noise associated with this sudden shock causes the sonic boom of aircraft flying at supersonic speeds. Compressible flows are often identified by the Mach number, which is the ratio of the flow velocity divided by the sound velocity. Supersonic flows therefore have a Mach number greater than 1. </B></div> <div> <B> </B></div> <div> THE ASSUMPTIONS USED IN CFD FOR INCOMPRESSIBILITY </div> <div> <B>All materials, whether gas, liquid or solid exhibit some change in volume when subjected to a compressive stress. The degree of compressibility is measured by a bulk modulus of elasticity, E, defined as either </B></div> <div> <B>E=dp/(dr /r ), or E=dp/(-dV/V </B><FONT SIZE="2" COLOR="#000000" FACE="Arial" ><B>), </B></FONT></div> <div> <B>where dp is a change in pressure and dr or dV is the corresponding change in density or specific volume. Since dp/d r =c2, where c is the adiabatic speed of sound, another expression for E is,E =rc2. </B></div> <div> <B>In liquids and solids E is typically a large number so that density and volume changes are generally very small unless exceptionally large pressures are applied. </B></div> <div> <B>If an incompressible assumption is made in which densities are assumed to remain constant, it is important to know under what conditions that assumption is likely to be valid. There are, in fact, two conditions that must be satisfied before compressibility effects can be ignored. </B></div> <div> <B>Let us define "incompressibility" as a good approximation when the ratio d r /r is much smaller than unity. To determine the conditions for this approximation we must estimate the magnitude of changes in density. In steady flow, the maximum change in pressure can be estimated from Bernoulli's relation to be dp=r u2. Combining this with the above relations for the bulk modulus, we see that the corresponding change in density is dr /r = u2/c2. (1] </B></div> <div> <B>Thus, the assumption of incompressibility requires that fluid speed be small compared to the speed of sound, </B></div> <bR> <div> <B>Condition One: u </B></div> <div> <B>In unsteady flow another condition must also be satisfied. If a significant change in velocity, u, occurs over a time interval t and distance l, then momentum considerations (for an inviscid fluid) require a corresponding pressure change of order dp = r ul/t . Since changes in density are related to changes in pressure through the square of the sound speed, dp=c2dr , this relation becomes </B></div> <div> <B>dr /r = (u/c)l/(ct ). (2) </B></div> <div> <B>Comparing with expression (1), we see that the factor multiplying (u/c) must also be much less than one, </B></div> <bR> <div> <B>Condition Two: l </B></div> <div> <B>Physically, this condition says that the distance traveled by a sound wave in the time interval t must be much larger than the distance l, so that the propagation of pressure signals in the fluid can be considered nearly instantaneous compared to the time interval over which the flow changes significantly. An example of why both conditions are required can be found in the collapse of a vapor bubble. During the collapse process the surrounding liquid can be treated as an incompressible fluid because the collapse velocity is much less than the speed of sound. However, at the instant the bubble vanishes, all the fluid momentum rushing toward the point of collapse must be stopped. If this really happened instantaneously, the collapse pressure would be enormous, i.e., much larger than what is actually observed. </B></div> <div> <B>Since a sound signal requires time to travel out from the collapse point to signal incoming fluid that it must stop, Condition Two is violated (i.e., l ct ). An accurate numerical model of the collapse process, one capable of predicting the correct pressure transients, requires the addition of a bulk compressibility in the liquid. </B></div> <bR> <div> <B>VISCOSITY </B></div> <div> <B>Fluids handling and their behaviour forms a very important part of pump selection and processing. </B></div> <bR> <div> Newtonian and non-Newtonian fluids </div> <div> <B>The following is a list of Newtonian and non-Newtonian fluids. It is important to categorise the type of fluid in order to use the proper friction calculation method and select the proper pump. I would appreciate any comments on the list, and if you have any other fluids that can be added (as long as you know the source of the information). </B></div> <bR> <div> <B>NEWTONIAN </B></div> <div> <B>- water - high viscosity fuel - some motor oils - most mineral oils - gasoline - kerosene - most salt solutions in water - light suspensions of dye stuffs - kaolin (clay slurry) </B></div> <bR> <div> <B>NON-NEWTONIAN </B></div> <div> <B>- oils containing polymeric thickeners, viscosity index improvers and waxy or soot particles </B></div> <bR> <div> <B>BINGHAM PLASTIC, YIELD PSEUDOPLASTIC, YIELD DILATANT </B></div> <div> <B>- thermoplastic polymer solutions - sewage sludge's - digested sewage - clay - mud - ketchup - chewing gum - tar - high concentrations of asbestine in oil </B></div> <bR> <div> <B>PSEUDOPLASTIC </B></div> <div> <B>- GRS latex solutions - sewage sludge's - grease - molasses - paint - starch - soap - most emulsions - printer's ink - paper pulp Viscosity decreases with rate of shear </B></div> <bR> <div> <B>DILATANT </B></div> <div> <B>- starch in water - beach sand - quicksand - feldspar - mica - clay - candy compounds - peanut butter Viscosity increases with the rate of shear </B></div> <bR> <div> <B>THIXOTROPIC - RHEOPECTIC </B></div> <div> <B>- most paints (thixo.) - silica gel - greases - inks - milk - mayonnaise - carboxymethyl cellulose - bentonite (rheop.) - gypsum in water (rheop.) - asphalt - glues - molasses - starch - lard - fruit juice concentrates Thix.: decrease viscosity with time Rheop.: increase viscosity with time </B></div> <div> <B>Essentially, the viscosity of a fluid is its resistance to shear strain. Viscous stesses occur when one element layer of a fluid slides over [moves] an adjacent contacting layer under the influence of a shear force. In the case of Newtonian fluids the shear stress is defined as follows </B></div> <div> <B>fs =m.du/dv where m is the modulus of shear or co-efficient of dynamic viscosity, du is the velocity between elemental layers and dy is the distance between these layers. </B></div> <bR> <div> <B>The kinematic viscosity n = m / r </B></div> <div> <B>The stress required to maintain a velocity difference between adjacent planes in a Newtonian fluid is proportional to the velocity gradient from plane to plane. </B></div> <div> <B>However, not all fluids behave in this manner. PAINTS, ADHESIVES, DOPES, etc. Such fluids behave with a non linear velocity gradient expressed as a function as viscous stress. </B></div> <div> <B>There are some industrial substances where the viscosity depends upon both shear rate and time. These are known as thixotropic. When a thixotropic fluid is sheared its viscosity decreases ,while at rest it exhibits high viscosity values. </B></div> <div> <B>This peculiarity of the relationship is that thixotropic fluids often give large pressure drops due to their viscosity in flowing along a pipe-lines, but behave almost like water when passing through restrictions such control valves. </B></div> <div> <B>With technology ever changing and complex materials [fluids] evolving, the understanding and measurement of the mechanical behaviour is critical. The rotodynamic pump does not lend itself to pumping non Newtonian fluids. </B></div> <div> <B> </B></div> <div>  FRICTION IN VERY SMOOTH AND ROUGH PIPES </div> <div> <B>Equating forces due to shear stress at the pipe wall and pressure gradient for horizontal pipe we can deduce the following equation </B></div> <div> <B>[4 t / [rvv/2]] = -[D/ [rvv/2]] [dp/ds] = l = f[Re, K/D] </B></div> <bR> <div> <B>i.e. p1 - p2 = l[L/D] [r][v^2]/2] </B></div> <div> <B>l can also be expressed as 4f where f = t / r[v^2]/2] </B></div> <div> <B> </B></div> <div> <B>LAMINAR FLOW </B><FONT SIZE="2" COLOR="#0000FF" FACE="Arial" ><B>[ p1 - p2 ]= Dp = 32 mLv / [DD] </B></FONT></div> <div> <B><IMG BORDER="0" SRC="Symb075c00b700c800000000.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">                      note for a horizontal pipe </B><FONT SIZE="2" COLOR="#FF00FF" FACE="Arial" ><B>OR [p1 - p2] / w = [h1 - h2] </B></FONT></div> <div> <B><IMG BORDER="0" SRC="Symb075c00b700c8ff00ff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">                                                                   = 32 mLv / w[D^2]-pipe inclined </B></div> <bR> <div> <B>HAGEN-POISEUILLE </B><FONT SIZE="2" COLOR="#FF00FF" FACE="Arial" ><B>LAW FOR LAMINAR VISCOUS FLOW </B></FONT></div> <div> <B>noting that v is the mean velocity ,hence </B></div> <bR> <div> <B>Dp = l[L/D] [rv^2]/2] </B></div> <bR> <div> <B>from which , l =64/Re and f = 16/Re that is dependent upon Re and not </B></div> <div> <B>K/D </B></div> <div> <B> </B></div> <div>  TURBULENT FLOW IN SMOOTH PIPES </div> <div> <B>Experimental work and analytical evaluation by BLASIUS </B></div> <div> <B>l =0.316/ Re ^0.25 </B></div> <bR> <div> <B>MORE REFINED STUDIES gave </B></div> <bR> <div> <B>[1/l ] =[ [2 log[base10]Re Öl] - 0.8] and again dependent only on Re </B></div> <div> <B> </B></div> <div> <B>FOR TURBULENT FLOW IN ROUGH PIPES </B></div> <div> <B>Laminar flow will normally break down if the Re number is greater than 2000 approximately. For turbulent flow in smooth pipes for Re up to 10^5, the observed pressure drop is such that as to imply a value of skin friction coefficient </B></div> <div> <B>cf =0.79 Re ^-0.25 </B></div> <bR> <div> <B>Experiment and analysis yield </B></div> <div> <B>[1/l ] =[2 log[base 10][D/K]] +1.14 </B></div> <div> <B>for fully developed turbulence and is now dependent upon [K/D] the roughness factor. </B></div> <bR> <div> <B>NIKURADSE AND COLEBROKE AND WHITE </B><FONT SIZE="2" COLOR="#000000" FACE="Arial" ><B>give; </B></FONT></div> <div> <B>[1/l ] = - [2 log [base10][[2.51/ Re Öl] +[K/3.7D]] </B></div> <bR> <div> <B>AND THESE FORM THE BASIS OF THE UNIVERSAL PIPE FRICTION CHARTS </B></div> <div> <B> </B></div> <div> <B>If we write p1 - p2 =l [L/D] [rvv/2]] in more general form to allow for change in elevation </B></div> <div> <B>p/w becomes [p/w] + z = h </B></div> <bR> <div> <B>h1 - h2 = hf =l [L/D] [r/w]v^2/2] =l[Lv^2]/2gD </B></div> <bR> <div> <B>D`ARCY WEISBACH EQUATION </B><FONT SIZE="2" COLOR="#000000" FACE="Arial" ><B>and more often written in the form of </B></FONT></div> <div> <B>hf =4flv^2/ 2gD </B></div> <div> <B>WHERE l =4f [[NOTE that in the USA SYMBOL f is used for l] </B></div> <div> <B>Pipe manufacturers produce loss of head versus flow charts and nomograms for the various pipe-work materials. </B></div> <bR> <div> <B>HAZEN WILLIAMS </B><FONT SIZE="2" COLOR="#000000" FACE="Arial" ><B>is a popular and reliable basis in the water engineering field. </B></FONT></div> <div> <B>v = 1.318 CR [^0.63][S^0.54] </B></div> <div> <B>where C = CONSTANT DEPENDING ON PIPE MATERIAL AND ITS CONDITION [ IE 100 FOR A NEW CAST IRON PIPE </B></div> <div> <B>R = MEAN HYDRAULIC RADIUS OR HYDRAULIC MEAN DEPTH = AREA /WETTED PERIMETER for circular pipe = d/4 </B></div> <div> <B>S = hf/l =head loss due to friction/length = hydraulic gradient </B></div> <bR> <bR> <bR> <bR> <div> <B>ALTERNATIVELY THE FORMULA MAY BE EXPRESSED as </B></div> <div> <B>v = 0.457[10^-2]D^0.63][S^0.54] </B></div> <div> <B>OR Q = 0.359[10^-5][D^2.63][i^0.54] </B></div> <div> <B>where v = mean velocity in m/s </B></div> <div> <B>Q = quantity in l/s </B></div> <div> <B>D = internal diameter in [mm] </B></div> <div> <B>I = hydraulic gradient [dimensionless] </B></div> <div> <B>C = HAZEN WILLIAMS FRICTION COEFFICIENT </B></div> <div> <B> </B></div> <div> <B>COMPUTER PACKAGES ARE NOW IN USE IN PROBLEM SOLVING </B></div> <bR> <div> THE NAVIER-STOKES EQUATIONS [stresses in a viscous fluid ] </div> <div> <B>Vector form </B></div> <div> <B>r[Du/Dt] = [rho]Fx - [dp/dx] + [mu][nabla^2]u +[1/3][mu][d/dx][div u] </B></div> <bR> <div> <B>and similarly for the y and z directions </B></div> <bR> <div> <B>Where, F= VECTOR [ BODY FORCE with components Fx,Fy,Fz] acting on elemental mass , </B></div> <div> <B>Du/ Dt =acceleration of elemental mass of fluid, </B></div> <div> <B>r[rho] = fluid density , </B></div> <div> <B>p = pressure , </B></div> <div> <B>tau = mu[du/dy] , </B></div> <div> <B>n[kinematic viscosity]= mu /rho </B></div> <div> <B> </B></div> <div> <B>The 3 component equations may be combined into one vectorial equation notably </B></div> <bR> <div> <B>[rho][Du/Dt]=[rho]F - [grad p] + mu[nabla^2]u + [mu][1/3]grad div u </B></div> <bR> <div> <B>and for </B><FONT SIZE="2" COLOR="#0000FF" FACE="Arial" ><B>incompressible flow </B></FONT><B>[from the continuity equation then </B></div> <bR> <div> <B>[div u =0] hence, </B></div> <bR> <div> <B>[rho][Du/Dt] = [rho]F - grad p + mu[nabla^2]u </B></div> <bR> <div> <B>or </B><FONT SIZE="2" COLOR="#FF0000" FACE="Arial" ><B>[Du/Dt] = F - [1/rho] [grad p] + n[nabla^2]u </B></FONT></div> <div> <B> </B></div> <div> <B> </B></div> <div> <B> </B></div> <bR> </td> </tr></table></body>