Kwara State University Civil Engineering Hydraulics Lecture Notes PDF

KWARA STATE UNIVERSITY, DEPARTMENT OF CIVIL ENGINEERING Civil Engineering Hydraulics (CEE 333) Lecture Notes Dr. A.G. Adeogun 8/24/2017 0 Contents CHAPTER ONE........................................................................................

KWARA STATE UNIVERSITY, DEPARTMENT OF CIVIL ENGINEERING Civil Engineering Hydraulics (CEE 333) Lecture Notes Dr. A.G. Adeogun 8/24/2017 0 Contents CHAPTER ONE................................................................................................................................................... 4 1.1 Introduction to Hydraulics................................................................................................................ 4 1.2 Matter............................................................................................................................................... 4 1.3 The Viscous Flow.............................................................................................................................. 4 1.4 Pressure Drop................................................................................................................................... 5 1.4.1 Causes of Pressure drop in pipes.............................................................................................. 5 1.4.2 Determination of the fluid (liquid or gas) pressure drop along a pipe or pipe component..... 5 1.4.3 Pressure drop by gravity or vertical elevation......................................................................... 6 1.4.4 Pressure drop in gasses and vapor........................................................................................... 6 1.5 A shear stress................................................................................................................................... 6 1.5.1 General shear stress................................................................................................................. 7 1.5.2 Shear Stress (fluid).................................................................................................................... 7 1.5.3 Shear stress in term of velocity gradient.................................................................................. 7 1.6 Dynamic viscosity and Kinematic viscosity....................................................................................... 7 1.7 Compressibility and Bulk modulus................................................................................................... 8 1.8 Reynolds Number............................................................................................................................. 9 1.8.1 Physical significance of Reynolds Number:............................................................................ 10 1.8.2 Use of Reynolds (Re) number in Selection of pipe friction Coefficient:................................. 10 CHAPTER TWO................................................................................................................................................ 11 2.0 Open Channel Flow............................................................................................................................ 11 2.1 Introduction.................................................................................................................................... 11 2.2 Types of flow in Open channel....................................................................................................... 11 2.2.1 Classification based on change of depth with respect with time and space......................... 11 2.2.2 Classification of flow channel based on Reynolds number.................................................... 12 2.2.3 Classification of flow on the basis of the variation of density with the flow......................... 13 2.2.4 Classification of flow on the basis of the ratio of inertial to gravity forces (Froude number) 14 2.3 Differences between pipe flow and open channel flow................................................................ 14 2.4 Properties of open channels.......................................................................................................... 17 2.4.1 Artificial channels................................................................................................................... 17 2.4.2 Natural channels..................................................................................................................... 17 2.5 Geometric properties necessary for analysis................................................................................. 17 2.6 Fundamental equations................................................................................................................. 18 2.6.1 Conservation of matter.......................................................................................................... 18 2.6.2 Conservation of energy.......................................................................................................... 19 1  2.6.3 Conservation of momentum.................................................................................................. 19 2.6.4 The Continuity Equation (conservation of mass)................................................................... 19 2.7 Velocity distribution in open channels........................................................................................... 20 2.8 Laminar and Turbulent flow........................................................................................................... 21 2.9 Best Hydraulics Section.................................................................................................................. 22 CHAPTER THREE............................................................................................................................................. 33 3.0 PIPE FLOW.......................................................................................................................................... 33 3.1 Introduction.................................................................................................................................... 33 3.2 Summary of pipe flow formula....................................................................................................... 33 Pipe flow formulae are as summarised below:.......................................................................................... 33 3.3 Turbulent flow in pipes................................................................................................................... 35 3.4 Determination of Head loss in pipes.............................................................................................. 36 CHAPTER 4...................................................................................................................................................... 44 4.0 Losses in pipe...................................................................................................................................... 44 4.1 Minor losses................................................................................................................................... 44 4.2 Major losses.................................................................................................................................... 45 4.3 Equivalent size of a pipe................................................................................................................. 49 4.3.1 Parallel pipe system................................................................................................................ 51 Chapter 5........................................................................................................................................................ 53 5.0 FLOW MEASUREMENT....................................................................................................................... 53 5.1 Flow Measurement through Pipes................................................................................................ 53 5.1.1 Venturi meter and orifice meter............................................................................................ 53 5.1.2 Pitot tube................................................................................................................................ 54 5.1.3 Flow measurement through orifices and mouthpieces......................................................... 55 5.2 Flow measurement in Open Channels........................................................................................... 58 5.2.1 Rectangular notch.................................................................................................................. 58 6.0 Fluid Flow Concepts and Measurements........................................................................................... 61 2 List of Text Books 1.0 Hand book of hydraulic Engineering Armando Lencastre, Ellis Horwood Series in Civil Engineering Ellis Horwood limited – Publishers – Chichester Halsted press: a division of John Wiley & Sons, New York – Chichester – Brisbane 1987 edition 2.0 Civil Engineering Hydraulics Essential Theory with worked Examples R.E. Feather Stone, C. Nalluri 3.0 Open Channel Hydraulics by H. French 4.0 Understanding Hydraulics by Les Hamill. 2nd Edition Palgrave Macmillan 5.0 Water Resources Engineering Ray K. Linsley, Joseph B. Franzini, David L. Freyberg, George Tchobanoglous. 4th edition. McGRAW-HILL INTERNATIONAL Editions. Civil Engineering Series 6.0 A text book of fluid mechanics and Hydraulic Machines in SI Units by Er. R.K. Rajput, S.Chand & Company ltd. Ram Nagar, New Delhi – 110055 7.0 Fluid Mechanics by J.F.Douglas and J.M Gasiorek 3  CHAPTER ONE 1.1 Introduction to Hydraulics Hydraulics is a topic in applied science and engineering dealing with the mechanical properties of liquids or fluids. At a very basic level, hydraulics is the liquid version of pneumatics. Fluid mechanics provides the theoretical foundation for hydraulics, which focuses on the engineering uses of fluid properties. Engineering hydraulics is concerned broadly with civil engineering problems in which the flow or management of fluids, primarily water, plays a role. Solutions to this wide range of problems require and understanding of the fundamental principles of fluid mechanics in general and hydraulics in particular and these are summarized in this section. 1.2 Matter Matter is recognized in nature as solid, liquid or gas (or vapour). When it exists in liquid or gaseous form, matter is known as a fluid. The common property of all fluids is that they must be bounded by impermeable walls in order to remain in an initial shape. If the restraining walls are removed the fluid flows expands until a new set of impermeable boundaries is encountered. Provided there is enough fluid or it is expandable enough to fill the volume bounded by a set of impermeable walls, it always conforms to the geometrical shape of the boundaries. In other words, a fluid by itself offers no lasting resistance to change of shape. Hence the fluid can be defined as a substance which readily changes its shape under the action of very small forces. 1.3 The Viscous Flow This is a type of fluid flow in which there is a continuous steady motion of the particles, the motion at a fixed point always remaining constant. The viscosity is that property of a fluid which by virtue of cohesion and interaction between fluid molecules offers resistance to shear deformation. Different fluid deform at different rates under the action of the same shear stress. A liquid's viscosity depends on the size and shape of its particles and the attractions between them. For example, honey has a much higher viscosity than water. A liquid's viscosity depends on the size and shape of its particles and the attractions between them. For example, honey has a much higher viscosity than water. A fluid that has no resistance to shear stress is known as an ideal fluid or inviscid fluid. Zero viscosity is observed only at very low temperatures, in super fluids. Otherwise, all fluids have positive viscosity. If the viscosity is very high, for instance in pitch, the fluid will appear to be a solid in the short term. A liquid whose viscosity is less than that of water 4 is sometimes known as a mobile liquid, while a substance with a viscosity substantially greater than water is called a viscous liquid. 1.4 Pressure Drop Pressure drop is defined as the difference in pressure between two points of a fluid carrying network. Pressure drop occurs when frictional forces, caused by the resistance to flow, act on a fluid as it flows through the tube. The main determinants of resistance to fluid flow are fluid velocity through the pipe and fluid viscosity. Pressure drop increases proportional to the frictional shear forces within the piping network. A piping network containing a high relative roughness rating as well as many pipe fittings and joints, tube convergence, divergence, turns, surface roughness and other physical properties will affect the pressure drop. High flow velocities and / or high fluid viscosities result in a larger pressure drop across a section of pipe or a valve or elbow. Low velocity will result in lower or no pressure drop. 1.4.1 Causes of Pressure drop in pipes  Friction  Vertical pipe difference or elevation  Changes of kinetic energy 1.4.2 Determination of the fluid (liquid or gas) pressure drop along a pipe or pipe component Determine the Pressure drop in circular pipes by using the equation: L = Length of Pipe D = Pipe Diameter = Density; v= Flow Velocity If you have valves, elbows and other elements along your pipe then it is possible to calculate the pressure drop with resistance coefficients specifically for the element. The resistance coefficients are in most cases found through practical tests and through vendor specification documents. If the resistance coefficient is known, then we can calculate the pressure drop for the element. 5  Drop; ρ = density; v=Flow Velocity =Coefficient (determined by test or vendor’s specification) 1.4.3 Pressure drop by gravity or vertical elevation The expression for pressure drop by gravity of elevation is as written as : Where: Drop; = Density; g = Acceleration of Gravity, 1.4.4 Pressure drop in gasses and vapor Compressible fluids expansion caused by pressure drops (friction) lead to the velocity increase. Therefore, the pressure drop along the pipe is not constant. The mathematical expression used to estimate the pressure drop is as written as: where: p1 = Pressure incoming; T1 = Temperature incoming, ρ = density p2 = Pressure leaving; T2 = Temperature leaving We set the pipe friction number as a constant and calculate it with the input-data. The temperature, which is used in the equation, is the average of entrance and exit of pipe. 1.5 A shear stress A shear stress, denoted as (Greek: tau), is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. 6  1.5.1 General shear stress The formula to calculate average shear stress is force per unit area. where: F = the force applied; A=the cross-sectional area of material with area parallel to the applied force vector 1.5.2 Shear Stress (fluid) Shear Stress ( ) is a measure of the force of friction from a fluid acting on a body in the path of that fluid. In the case of open channel flow, it is the force of moving water against the bed of the channel. Shear stress is calculated as: Where: τ = Shear Stress ( ); ɣ = Unit Weight of Water ) D = Average water depth (m); S Sw = Water Surface slope (m/m) 1.5.3 Shear stress in term of velocity gradient The equation relating to the shear stress with velocity gradient is as written as follows: Where: = shear stress ( ); = velocity gradient or rate of deformation = the coefficient of dynamic (or absolute) viscosity ( or kg/ms) 1.6 Dynamic viscosity and Kinematic viscosity Dynamic viscosity is the constant in shear stress in term of velocity gradient, while the ratio of dynamic viscosity to density is known as kinematic viscosity. 7 Where: = Kinematic viscosity ( ); = Density ( ); = Dynamic viscosity ( or kg/ms) 1.7 Compressibility and Bulk modulus Compressibility is measured as the change in volume of a substance due to a change in the pressure applied on it, while Bulk modulus of material is defined as change in pressure per volume strain. Mathematically (a) where p = pressure; V= volume Also density But for a unit mass of fluid, the density or (b) Substituting (b) into (a) gives (c) Equation ( c ) is the compressibility of the fluid and K has the unit of pressure i.e ( ). 8 1.8 Reynolds Number Reynolds Number is the ratio of the inertial forces to the viscous forces and it is a dimensionless quantity. In the 19th century, Osborne Reynolds, an English scientist working at the University of Manchester demonstrated through experiment that the flow of a liquid through a pipe is essentially of two types. These are : (a) Laminar flow and (b) Turbulent flow The intermediary flow between (a) and (b) is refers to as Transitional flow. He classified fluid flow into the above categories based on the ratio of inertial force to viscous force acting on particles within such flow. This ratio is known as Reynolds number denoted by Re. where ρ= density; v= flow velocity; L= characteristic length such as the pipe diameter (D) = the coefficient of dynamic (or absolute) viscosity ( or kg/ms) The flow of fluid in a pipe and open channel can then be categories as follows: Type of flows Pipes Open channels Laminar flow Re < 2000 Re < 500 Transitional flow Re 2000 ≤Re≤ 4000 500 ≤Re≤ 2000 Turbulent flow Re > 4000 Re > 2000 Laminar flow is characterized by the gliding of concentric cylindrical layers past one another in orderly fashion. The velocity of the fluid is at its maximum at the pipe axis and decreases sharply to zero at the wall. The pressure drop caused by friction of laminar flow does not depend on the roughness of pipe. Turbulent flow: There is an irregular motion of fluid particles in directions transverse to the direction of the main flow. The velocity distribution of turbulent flow is more uniform across the pipe diameter than in laminar flow. 9  1.8.1 Physical significance of Reynolds Number: 1. It signifies the relative predominance of inertia to viscous forces. 2. It is very useful in determining whether the flow is laminar or turbulent. 3. Require to determine friction coefficient of pipe & roughness coefficient of channel 1.8.2 Use of Reynolds (Re) number in Selection of pipe friction Coefficient: The pipe friction coefficient is a dimensionless number. The friction factor for laminar flow condition is a function of Reynolds number only, for turbulent flow it is also a function of the characteristics of the pipe wall. (a) Determine Pipe friction coefficient at laminar flow: Pipe Friction Coefficient; Re = Reynolds number Note: Perfectly smooth pipes will have a roughness of zero. (b) Determine Pipe friction coefficient at turbulent flow (in the most cases): = Pipe Friction Coefficient; Re = Reynolds Number k = Absolute Roughness; D = Diameter of Pipe Practice Questions 1. Two horizontal plates are placed 1.25 cm apart, the space between them being filled with oil of viscosity 14 poise (1.4 Ns/m2). Compute the shear in the oil if the upper plate is moved with a velocity of 4 m/s. 2. (i) Differentiate between Compressibility and Bulk Modulus of a Fluid (ii) List three (3) Physical significance of Reynolds Number 3. Glycerine of viscosity 0.9Ns/m2 and desisity 1260kg/m3 is pumped along a horizontal pipe 6.5m long of diameter d=0.01m at a flow rate of Q=1.8 litres/min. Determine the flow Reynolds number and verify whether the flow is laminar or turbulent. Calculate the pressure loss in the pipe due to frictional effects. 10  CHAPTER TWO 2.0 Open Channel Flow 2.1 Introduction Open-channel flow, a branch of hydraulics and fluid mechanics, is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways, but differ in one important respect: the free surface. Open-channel flow has a free surface, whereas pipe flow does not. The flow of water in an open channel is a familiar sight, whether in a natural channel like that of a river, or an artificial channel like that of an irrigation ditch. Its movement is a difficult problem when everything is considered, especially with the variability of natural channels, but in many cases the major features can be expressed in terms of only a few variables, whose behavior can be described adequately by a simple theory. The principal forces at work are those of inertia, gravity and viscosity, each of which plays an important role. In this course, we shall consider only water as the fluid, as it is by far the most common one, and for which the most experimental data are available. Other fluids will behave in about the same way, under conditions that are specified in the course. 2.2 Types of flow in Open channel 2.2.1 Classification based on change of depth with respect with time and space. a) Types of flows based on change of depth with respect to time. i. Steady flow: This implies that, depth of flow does nott change with time (dy/dt =0). ii. Unsteady flow: This implies that the depth of flow changes with time (dy/dt≠0). The differentiation between steady and unsteady flow depends on the view point of the observer and is a relative term rather than absolute classification. b) Types of flow based on change of depth with respect to space i. Uniform flow: This implies that the depth of flow does not vary with distance (dy/dx=0) ii. Non-uniform flow: This implies that the depth of flow does vary with distance (dy/dx≠0) c) Types of flow based on the combination of the above conditions i. Steady uniform flow: This implies that the depth of flow does not change with time and distance. ii. Steady non-uniform flow: This implies that the depth of flow does not change with time but changes with distance. iii. Unsteady uniform flow: This implies that the depth of flow changes with time but does not change with distance. iv. Unsteady non-uniform flow: This implies that the depth of flow changes with time and distance. 11 Moreover, non-uniform flow is termed varied flow. Varied flow is also divided into two. Rapidly varied and gradually varied flows. Rapidly varied flow: This implies that the depth of flow changes rapidly over a relatively short distance. e.g hydraulic jump. Gradually varied flow: This implies that the depth of flow changes rather slowly with distance. e.g. reservoir behind a dam. 2.2.2 Classification of flow channel based on Reynolds number Recall from elementary fluid mechanics that depending on the magnitude of the ratio of the inertia forces to the viscous forces, a flow may be classified as laminar, transitional or turbulent. The basis for this classification is a dimensionless parameter known as Reynolds number. Mathematically: (1) Where U = characteristic velocity of flow, often taken as the average velocity of flow L = characteristics length V = kinematic viscosity a) Laminar flow: This implies that the viscous forces are so large relative to the inertia forces i.e the flow is dominated by the viscous forces. In such a flow, the fluid particles move along definite, smooth earth in a coherent fashion. Re≦500 12  b) Turbulent flow: This implies that the inertia forces are large relative to the viscous forces i.e the flow is dominated by the inertia forces. In such a flow, the fluid particle moves in an incoherent fashion. 2000≦ Re c) Transitional flow: This implies that the flow cannot be classified as Turbulent or Laminar flow. 500 ≤ Re ≤2000 In open channel flow, the characteristic length commonly used in calculating Reynolds number is the hydraulic radius which is the ratio of the flow area A to the wetted perimeter P. 2.2.3 Classification of flow on the basis of the variation of density with the flow a) Homogeneous flow: This implies that in all the spatial dimensions the density of the flow is constant. b) Stratified flow: This implies that the density of the flow varies in any direction. The importance of density stratification is that when stable density stratification exists, i.e density increases with depth or lighter fluid overlies heavier fluid, the effectiveness of the turbulence as a mixing mechanism is reduced. In two-dimensional flow of the type normally encountered in open channels, a commonly accepted measurement of the strength of the density stratification is the gradient Richardson number. (2) Where, When is small relative to , is large, and the stratification is stable. When is large relative to , is small and as , the flow system approaches a homogeneous or neutral condition. 13  2.2.4 Classification of flow on the basis of the ratio of inertial to gravity forces (Froude number) Depending on the magnitude of the ratio of inertial to gravity forces, a flow is classified as subcritical, critical or supercritical. The parameter on which this classification is based is known as the Froude number. (3) Where In an open channel, the characteristic length in estimating Froude number is taken to be the hydraulic depth, which by definition is the flow area A divided by the width of the free surface T. Mathematically: (4) a) Subcritical flow If Froude number is less than 1 (F1), the flow is supercritical and the inertial forces are dominant. 2.3 Differences between pipe flow and open channel flow The flow of water in a conduit may be either open channel flow or pipe flow. The two kinds of flow are similar in many ways but differ in one important respect. Open-channel flow must have a free surface, whereas pipe flow has none. A free surface is subject to atmospheric pressure. In Pipe flow there exist no direct atmospheric flow but hydraulic pressure only. 14  Figure of pipe and open channel flow The two kinds of flow are compared in the figure above. On the left is pipe flow. Two piezometers are placed in the pipe at sections 1 and 2. The water levels in the pipes are maintained by the pressure in the pipe at elevations represented by the hydraulics grade line or hydraulic gradient. The pressure exerted by the water in each section of the pipe is shown in the tube by the height y of a column of water above the centre line of the pipe. The total energy of the flow of the section (with reference to a datum) is the sum of the elevation z of the pipe centre line, the piezometric head y and the velocity head V2/2g , where V is the mean velocity. The energy is represented in the figure by what is known as the energy grade line or the energy gradient. The loss of energy that results when water flows from section 1 to section 2 is represented by hf. A similar diagram for open channel flow is shown to the right. This is simplified by assuming parallel flow with a uniform velocity distribution and that the slope of the channel is small. In this case the hydraulic gradient is the water surface as the depth of water corresponds to the piezometric height. Despite the similarity between the two kinds of flow, it is much more difficult to solve problems of flow in open channels than in pipes. Flow conditions in open channels are complicated by the position of the free surface which will change with time and space. And also by the fact that depth of flow, the discharge, and the slopes of the channel bottom and of the free surface are all inter dependent. Physical conditions in open-channels vary much more than in pipes – the cross-section of pipes is usually round – but for open channel it can be any shape. Treatment of roughness also poses a greater problem in open channels than in pipes. Although there may be a great range of roughness in a pipe from polished metal to highly corroded iron open channels may be of polished metal to natural channels with long grass and roughness that may also depend on depth of flow. Open channel flows are found in large and small scale. For example the flow depth can 15 vary between a few cm in water treatment plants and over 10m in large rivers. The mean velocity of flow may range from less than 0.01 m/s in tranquil waters to above 50 m/s in high-head spillways. The range of total discharges may extend from 0.001 l/s in chemical plants to greater than 10000 m3/s in large rivers or spillways. In each case the flow situation is characterized by the fact that there is a free surface whose position is NOT known beforehand – it is determined by applying momentum and continuity principles. Open channel flow is driven by gravity rather than by pressure work as in pipes. In summary, the comparison between open and pipe flow are as detailed in the table below. Comparison between Open Channel and Pipe Flow 16 2.4 Properties of open channels 2.4.1 Artificial channels These are channels made by man. They include irrigation canals, navigation canals, spillways, sewers, culverts and drainage ditches. They are usually constructed in a regular cross-section shape throughout – and are thus prismatic channels (they don’t widen or get narrower along the channel. In the field they are commonly constructed of concrete, steel or earth and have the surface roughness’ reasonably well defined (although this may change with age – particularly grass lined channels.) Analysis of flow in such well defined channels will give reasonably accurate results. 2.4.2 Natural channels Natural channels can be very different. They are not regular nor prismatic and their materials of construction can vary widely (although they are mainly of earth this can possess many different properties.) The surface roughness will often change with time distance and even elevation. Consequently it becomes more difficult to accurately analyse and obtain satisfactory results for natural channels than is does for man made ones. The situation may be further complicated if the boundary is not fixed i.e. erosion and deposition of sediments. 2.5 Geometric properties necessary for analysis For analysis various geometric properties of the channel cross-sections are required. For artificial channels these can usually be defined using simple algebraic equations given y the depth of flow. 17 The commonly needed geometric properties are shown in the figure below and defined as: Depth (y) – the vertical distance from the lowest point of the channel section to the free surface. Stage (z) – the vertical distance from the free surface to an arbitrary datum Area (A) – the cross-sectional area of flow, normal to the direction of flow Wetted perimeter (P) – the length of the wetted surface measured normal to the direction of flow. Surface width (B) – width of the channel section at the free surface Hydraulic radius (R) – the ratio of area to wetted perimeter (A/P) Hydraulic mean depth (Dm) – the ratio of area to surface width (A/B) Table of equations for rectangular trapezoidal and circular channels. 2.6 Fundamental equations The equations which describe the flow of fluid are derived from three fundamental laws of physics: 1. Conservation of matter (or mass) 2. Conservation of energy 3. Conservation of momentum Although first developed for solid bodies they are equally applicable to fluids. A brief description of the concepts are given below. 2.6.1 Conservation of matter This says that matter cannot be created nor destroyed, but it may be converted (e.g. by a chemical process.) In fluid mechanics we do not consider chemical activity so the law 18 reduces to one of conservation of mass. 2.6.2 Conservation of energy This says that energy can not be created nor destroyed, but may be converted form one type to another (e.g. potential may be converted to kinetic energy). When engineers talk about energy "losses" they are referring to energy converted from mechanical (potential or kinetic) to some other form such as heat. A friction loss, for example, is a conversion of mechanical energy to heat. The basic equations can be obtained from the First Law of Thermodynamics but a simplified derivation will be given below. 2.6.3 Conservation of momentum The law of conservation of momentum says that a moving body cannot gain or lose momentum unless acted upon by an external force. This is a statement of Newton's Second Law of Motion: Force = rate of change of momentum In solid mechanics these laws may be applied to an object which is has a fixed shape and is clearly defined. In fluid mechanics the object is not clearly defined and as it may change shape constantly. To get over this we use the idea of control volumes. These are imaginary volumes of fluid within the body of the fluid. To derive the basic equation the above conservation laws are applied by considering the forces applied to the edges of a control volume within the fluid. 2.6.4 The Continuity Equation (conservation of mass) For any control volume during the small time interval δt the principle of conservation of mass implies that the mass of flow entering the control volume minus the mass of flow leaving the control volume equals the change of mass within the control volume. If the flow is steady and the fluid incompressible the mass entering is equal to the mass leaving, so there is no change of mass within the control volume. So for the time interval δt: 19 As, Q, the volume flow rate is the product of the area and the mean velocity then at the upstream face (face 1) where the mean velocity is u1 and the cross-sectional area is A1 then: Similarly at the downstream face, face 2, where mean velocity is u2 and the cross- sectional area is A2 then: Qleaving =u2 A2 Therefore the continuity equation can be written as u1A1 =u2 A2 2.7 Velocity distribution in open channels The measured velocity in an open channel will always vary across the channel section because of friction along the boundary. Neither is this velocity distribution usually axisymmetric (as it is in pipe flow) due to the existence of the free surface. It might be expected to find the maximum velocity at the free surface where the shear force is zero but this is not the case. The maximum velocity is usually found just below the surface. The explanation for this is the presence of secondary currents which are circulating from the boundaries towards the section centre and resistance at the air/water interface. These have been found in both laboratory measurements and 3d numerical simulation of turbulence. The figure below shows some typical velocity distributions across some channel cross sections. The number indicates percentage of maximum velocity. Figure of velocity distributions 20 2.8 Laminar and Turbulent flow As in pipes , and all flow, the flow in an open channel may be either laminar or turbulent. The criterion for determining the type of flow is the Reynolds Number, Re. So for an open channel the limits for each type of flow become Laminar: Re channel < 500 Turbulent: Re channel > 1000 In practice the limit for turbulent flow is not so well defined in channel as it is in pipes and so 2000 is often taken as the threshold for turbulent flow. We can use the ideas seen for pipe flow analysis to look at the effect of friction. Taking the Darcy-Wiesbach formula for head loss due to friction in a pipe in turbulent flow 21 and make the substitution for hydraulic radius R = D/4 and if we put the bed slope So = L/hf then and The Colebrook-White equation gives the f - Re relationship for pipes, putting in R=D/4 the equivalent equation for open channel is where ks is the effective roughness height A chart of the X - Re relationship for open channels can be drawn using this equation but its practical application is not clear. In pipes this relationship is useful but due to the more complex flow pattern and the extra variable (R varies with depth and channel shape) then it is difficult to apply to a particular channel. In practice flow in open channels is usually in the rough turbulent zone and consequently simpler friction formulae may be applied to relate frictional losses to velocity and channel shape 2.9 Best Hydraulics Section A best hydraulics section is that which has minimum wetted perimeter and can convey maximum discharge for a given slope, roughness coefficient and cross-sectional area. It has been determined that Semi-circular section has minimum wetted perimeter and this is the best hydraulic section. Semi-circular section is not commonly used because of difficulties in construction. However, the proportion of the best hydraulic section of a specified geometry shape can be derived as follows: 22 (a) Best Hydraulic Rectangular Section y b Recall A = yb ………..(i) b= P = b + 2y = + 2y A = 2y2 From (i) b= = = 2y R= = = 23 (b) Best Hydraulic Trapezoidal Section 1 y N b Recall A = ( b + Ny)y………………. (i) P = b + 2y ………… (ii) From (i) b = – Ny………………….. (iii) Put (iii) and (ii) P = – N + 2y ……....................................(iv) – –N+2 =0 +N=2 A + Ny2 = 2y2 A = 2y2 – Ny2 = y2 [2 ] From (iii) 24 b = – Ny = b= – = y {2 – = y {2 – = 2y { – However, it can be shown that best hydraulic rectangular section is a special case of best hydraulic trapezoidal section with N = 0 Check Area. A = y2 {2 – N=0 A = y2 {2 = 2y2 bed width b = 2y { – N=0 b = 2y From (iv) 25 Determine the expression for P P = – Ny + 2y = y[2 – = 2y[ – = 4y[ – = 2y [ – ] R= = = Rectangular P = + 2y = + 2y = 2y + 2y = 4y R= = = Example 1 A trapezoidal channel of the most efficient section has side slope 1:1, It is required to carry a discharge of 25m2/s with a slope of 1m in 1500m. Design the section if n = 0.0135. Solution. S = 1m in 1500m = 26  Side slope 1:N =1:1 Q = 25m2/s n = 0.0135 Condition: design the most efficient section From Manning’s equation Q= ………………(i) For the best section trapezoidal A=[ – ] b= – ] R= = P= – ] From (i) = …………… (ii) N=1 A=[ – ] = 1.8285 R= b = 2y[ – ] = 0.8285y 27 From (ii) = 1.8285 x ) 13.0713 = 13.0713 x = 1.8285 = = 11.35 y=( = 2.4866m = 2.5m b = 0.8285 x (2.5) = 2.0713m = 2.1m Example 2 A rectangular section is to be built of rough timber. If given a drop of 2m in a km. What will be the width and depth for the most efficient section if it has to carry a discharge of 1.10m2/s and n = 0.011? Solution Given data 2m S = 2m in a km = 0.002 1km = 1000m Q = 1.10 m2/s, n = 0.001 28 Condition: Design for most efficient section. From Manning’s equation Q= ………………(i) For the best hydraulic or most efficient section for rectangular section. A= , b = 2y, R = From (i) = = 0.2706 = = x( = x 0.2706 x = x = 2[ ] =2 y=[ = 0.5617m b = 2y = 2 x 0.5617 = 1.1234m 29 R= = Example 3 The discharge over a broad triangular notch can be written as : Q = ce ( tan ( ) (a) If an error 1% in measuring as introduced, determine the corresponding error in the computed discharge. (b) A right angle triangular notch is used for gauging the flow of an open channel. If the effective coefficient C, C is 0.593 and an error of 2mins is suspected in observing the head, find the percentage error in computing an estimated discharge of 20l/s. Solution (a) The equation can be written as Q=K …………………. (i) where K = Ce ( tan ( ) = K = K and = = 30  = ……………………….. (ii) Therefore, if is 1%, the error in the discharge, from equation (ii) is = (1%) = 2.5% (b) From equation (i) Q = CDCv ( tan ( ) Where Ce = CDCv = 0.593 Q = 20l/s = 0.02m3/s = 90o therefore, Hence 0.02 = 0.593 x = 0.593 x x 1.9809 x = 0.7518 =( = 0.2344 =234.4mm From equation (ii) 31  = = = 2.13% Exercise 4 The discharge over a sharp triangular and rectangular notches can be expressed as Q = Ce ( tan ( ) and Q =Ce ( respectively. (a) If an error of 1% in measuring is introduced, determine the corresponding error in the computed discharge. (b) If the effective discharge coefficient is 0.593 and an error of 2mm is suspected in observing the head, find the percentage error in computing an estimated discharge of 60l/s. Using the two types of weirs mentioned above. The triangular notch is 60o and the width of the approach channel is 2.0cm. 32  CHAPTER THREE 3.0 PIPE FLOW 3.1 Introduction The term pipe flows refer to flow in a circular closed conduct entirely with fluid. Pipe flow can also be categorized into laminar flow and turbulent flow. In laminar flow, flow particles move in parallel layer in one direction. The parabolic velocity distribution in laminar flow creates a shearing stress where is the rate of change of velocity with depth and μ is the dynamic viscosity. As the shear stress increases, the viscous forces become unable to damp out disturbances and turbulent flow results. The region of change depends on the fluids velocity, density and the size of the conduit, 3.2 Summary of pipe flow formula Pipe flow formulae are as summarised below: (i) V= = 1 Where Q=Flow rate (m3/s) and V=Velocity (m/s) (ii) R= = = R=Hydraulic radius 2 D=Pipe diameter Reynolds number (iii) Re = or Re = 3 V=Fluid Velocity D=Pipe diameter ʋ=Dynamic viscosity 33 Note: Laminar Flow: Re ≤2000 Turbulent flow: Re≥4000 Transition Flow 4000≤Re≤2000 Between the two boundaries there is a transition zone, but the range depends on the flow history. Expression for velocity, rate of flow and pressure loss in circular pipes when laminar flow exists may be derived. Then this is done by considering section of pipe L dislocate apart Average pressure at each section are represented by P1 and P2 (iv) The flow through the pipe can be derived to be: Q= 4 Where ∆P =Change in pressure, µ =Viscosity, r= radius This equation is refer to as Hagen Poiseuille’s flow equation Or in term of D Q= Recall that =V → V= Sin A= If Vm is the value of the mean velocity then VM = = * = 34 3.3 Turbulent flow in pipes Recall that the total energy at a point in a pipe flow can be represented as: H= This is the vertical ordinate at any point above the centre line of the pipe. The energy loss due to the fluid motion from point 1 and 2 can be represented by the Bernoulli’s equation, such that +Z = + +Z +Loss 5 The energy loss is as a result of friction in the pipe flow. This loss has been shown from the basis of experiment to be given as: hf =hL = 6 Where hf=hL =Head loss ʎ =Friction factor d=Diameter of the pipe, L=length of the pipe V=Velocity of flow In terms of flow rate Q Hf =hl = 7 35 Equation 7 is known as the pipe friction equation and is referred to as Darcy-Weisbach equation. The friction factor (f) is dimensionless and it is also a function of Reynolds number F= = = → 8  Re= = Note that hf =L hf=v2 hf=1/d hf depends on surface roughness of pipes hf depends on fluid density and viscosity hf is independent by pressure 3.4 Determination of Head loss in pipes The loss of head in pipes can be determined by the following formula in addition to Equation 6 (a) Manning Formula Usually used in determining the head loss in the open channel. It is equally applicable to turbulent flows in pressure pipe. The formula is written as: HL= 9 Where m=Manning rugosity coefficient L= Length of pipe (m) 36  R= Hydraulic mean depth V= Velocity of flow (m/s) If D is the diameter g pipe, the value of R R= = = Substituting in equation 9 Therefore hL = 10 (B) Hazen-William’s formula This formula is written as V=0.85Cn. R0.63. S0.54 11 Cn =Hazen-Williams coefficient of hydraulic capacity S = Slope of the energy line (m/m) V= Velocity of flow R= Hydraulic radius of the pipe (C) Chezy’s equation of head loss due to friction It has been derived that mean velocity is given by =C Where C= Chezy’s constant Ym =The hydraulic mean depth or hydraulic radius R Ym =R= I = the ratio (The loss of head loss per unit length of the pipe) 37 Worked Examples Example 1 Glycerin of viscosity 0.9Nsm-2 and density 1260kgm-3 is pumped along a horizontal pipe 6.5m long of diameter d=0.001m at a flow rate of Q=1.8 litters min -1. Determine the flow Reynolds number and verify whether the flow is laminar or turbulent. Calculate the pressure loss in the pipe due to frictional effect and calculate the maximum flow rate for laminar flow condition to prevail Solution Mean velocity.u = = { / X 10-3ms-1 =0.382ms-1 Re = =1260 X 0.382 X = 0.535 Therefore the flow is laminar as Re