Applied hydraulic engineering Lecture notes

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Applied Hydraulic EngineeringCE6403 APPLIED HYDRAULIC ENGINEERING L T P C 3 1 0 4 OBJECTIVES: To introduce the students to various hydraulic engineering problems like open channel flows and hydraulic machines. At the completion of the course, the student should be able to relate the theory and practice of problems in hydraulic engineering. UNIT I UNIFORM FLOW 9 Definition and differences between pipe flow and open channel flow - Types of Flow - Properties of open channel - Fundamental equations - Velocity distribution in open channel - Steady uniform flow: Chezy equation, Manning equation - Best hydraulic sections for uniform flow - Computation in Uniform Flow - Specific energy and specific force - Critical depth and velocity. UNIT II GRADUALLY V ARIED FLOW 9 Dynamic equations of gradually varied and spatially varied flows - Water surface flow profile classifications: Hydraulic Slope, Hydraulic Curve - Profile determination by Numerical method: Direct step method and Standard step method, Graphical method - Applications. UNIT III RAPIDLY VARIED FLOW 9 Application of the energy equation for RVF - Critical depth and velocity - Critical, Sub-critical and Super-critical flow - Application of the momentum equation for RVF - Hydraulic jumps - Types - Energy dissipation - Surges and surge through channel transitions. UNIT IV TURBINES 9 Impact of Jet on vanes - Turbines - Classification - Reaction turbines - Francis turbine, Radial flow turbines, draft tube and cavitation - Propeller and Kaplan turbines - Impulse turbine - Performance of turbine - Specific speed - Runaway speed - Similarity laws. UNIT V PUMPS 9 Centrifugal pumps - Minimum speed to start the pump - NPSH - Cavitations in pumps - Operating characteristics - Multistage pumps - Reciprocating pumps - Negative slip - Flow separation conditions - Air vessels, indicator diagrams and its variations - Savings in work done - Rotary pumps: Gear pump. TOTAL (L :45+T:15) : 60 PERIODS OUTCOMES:  The students will be able to apply their knowledge of fluid mechanics in addressing problems in open channels.  They will possess the skills to solve problems in uniform, gradually and rapidly varied flows in steady state conditions.  They will have knowledge in hydraulic machineries ( p umps and turbines) . TEXT BOOKS: 1. Jain. A.K., "Fluid Mechanics", Khanna Publishers, Delhi, 2010. 2. Modi P.N. and Seth S.M., "Hydraulics and Fluid Mechanics", Standard Book House, New Delhi, 2002. 3. Subramanya K., "Flow in open channels", Tata McGraw Hill, New Delhi, 2000. REFERENCES: 1. Ven Te Chow, "Open Channel Hydraulics", McGraw Hill, New York, 2009. 2. Rajesh Srivastava, "Flow through open channels", Oxford University Press, New Delhi,2008. 3. Bansal, "Fluid Mechanics and Hydraulic Machines", Laxmi Publications, New Delhi, 2008. 4. Mays L. W., "Water Resources Engineering", John Wiley and Sons ( W SE) , New York,2005.Sl.No Contents Page No. UNIT 1 UNIFORM FLOW 1 1.1 Introduction 1.2 2 DifferencesbetweenPipeFlowandOpen ChannelFlow 1.3 2 Types of flow 1.4 3 Propertiesofopenchannels 1.5 5 Fundamentalequations 1.6 8 Velocitydistributionin openchannels 1.7 9 Steady Uniformflow 1.7.1 TheChezyequation 10 11 1.7.2 TheManningequation 1.8 12 BestHydraulicCross-Section 1.9 16 Computationsin UniformFlow 1.10 20 Specific Energy 1.11 23 Critical Flow and Critical Velocity UNIT II GRADUALLYVARIEDFLOW 2.1 Varied Flow 24 2.2 24 Gradually Varied Flow in Open Channel 2.3 25 Transitions between Sub and Super Critical Flow 2.4 Classification of profiles 27 2.5 30 Profile Determination 32 2.5.1 Thedirectstepmethod32 2.5.2 The standardstep method 33 2.5.3 Graphical Integration Method UNIT 3 RAPIDLY VARIED FLOW 3.1 The Application of the Energy equation for Rapidly Varied Flow 37 3.1.1 The energy (Bernoulli) equation 37 3.2 Critical , Sub-critical and super critical flow 38 3.3 Application of the Momentum equation for Rapidly Varied Flow 40 3.4 Hydraulic jump 42 3.4.1 Expression for Hydraulic Jump 42 3.4.2 Loss of Energy due to HydraulicJump 42 3.4.2 Loss of Energy due to HydraulicJump 42 3.4.4 Classification of Hydraulic Jumps 42 UNIT 4 TURBINES 4.1 Introduction 44 4.2 Breaking Jet 44 4.3 Classification of Turbines 44 4.4 Impulse turbines 45 4.5 Reaction turbines 45 4.6 Turbines in action 46 4.7 Kaplan turbine 46 4.7.1Applications 47 4.7.2 Variations 48 4.7.2.1 Propeller Turbines 48 4.8 49 Francis Turbine 4.9 52 Specific speed UNIT 5 PUMPS 5.1 53 Centrifugal Pumps53 5.1.1 Volute type centrifugal pump 5.1.2 Impeller 53 5.1.3 Classification 54 5.1.4 Single and double entry pumps 54 5.1.5 Pressure Developed By The Impeller 55 5.1.6 Manometric Head 55 5.1.7 Energy Transfer By Impeller 56 5.1.8 Slip and Slip Factor 57 5.1.9 Losses in Centrifugal Pumps 57 5.1.10 Losses in pump 58 5.1.11 Pump Characteristics 58 5.1.12 Characteristics of a centrifugal pump 59 Operation of Pumps in Series and Parallel 5.2 60 60 5.2.1 Pumps in parallel 60 5.2.2 Pumps in series 61 5.2.3 Minimum Speed For Starting The Centrifugal Pump 5.2.4 Net Positive Suction Head (NPSH) 61 5.2.5 Cavitation 61 5.2.6 Multistage Pump 62 5.3 62 Reciprocating Pumps 62 5.3.1 Comparison 5.3.2 Description And Working 62 64 5.3.3 Flow Rate and Power 5.3.4 Slip 64 5.3.5 Coefficient of discharge 65 5.3.6 Indicator Diagram 65 65 5.3.7 Acceleration Head67 5.3.8 Work done by the Pump 68 5.4 Air Vessels Types of positive displacement pump 5.5 68CE6303 APPLIED HYDRAULIC ENGINEERING Unit 1 UNIFORM FLOW Prerequisite 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. 1.1 Introduction 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. 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 2 elevation z of the pipe centre line, the piezometric head y and the velocity head V /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 h . f 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. SCE DEPARTMENT OF CIVIL ENGINEERING Page 1CE6303 APPLIED HYDRAULIC ENGINEERING 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 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 m 3 /s in large rivers or spillways. In each case the flow situation is characterised 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. 1.2 Differences between Pipe Flow and Open Channel Flow 1.3 Types of flow The following classifications are made according to change in flow depth withrespect to time and space. Figure of the types of flow that may occur in open channels SCE DEPARTMENT OF CIVIL ENGINEERING Page 2CE6303 APPLIED HYDRAULIC ENGINEERING Steady and Unsteady: Time is the criterion. Flow is said to be steady if the depth of flow at a particular point does not change or can be Considered constant for the time interval under consideration. The flow is unsteady if depth changes with time. Uniform Flow: Space as the criterion. Open Channel flow is said to be uniform if the depth and velocity of flow are the same at every section of the channel. Hence it follows that uniform flow can only occur in prismatic channels. For steady uniform flow, depth and velocity is constant with both time and distance. This constitutes the fundamental type of flow in an open channel. It occurs when gravity forces are in equilibrium with resistance forces. Steady non-uniform flow. Depth varies with distance but not with time. This type of flow may be either (a) gradually varied or ( b ) rapidly varied. Type ( a ) requires the application of the energy and frictional resistance equations while type ( b ) requires the energy and momentum equations. Unsteady flow The depth varies with both time and space. This is the most common type of flow and requires the solution of the energy momentum and friction equations with time. In many practical cases the flow is sufficiently close to steady flow therefore it can be analysed as gradually varied steady flow. 1.4 Properties of open channels 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 ( a lthough this may change with age – particularly grass lined channels.) Analysis of flow in such well defined channels will give reasonably accurate results. 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. SCE DEPARTMENT OF CIVIL ENGINEERING Page 3CE6303 APPLIED HYDRAULIC ENGINEERING 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. 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 ( D ) – the ratio of area to surface width ( A/B ) m 1.5 Fundamental equations The equations which describe the flow of fluid are derived from three fundamental laws of physics: 1. Conservation of matter ( o r mass) 2. Conservation of energy 3. Conservation of momentum Although first developed for solid bodies they are equally applicable to fluids. Brief descriptions of the concepts are given below. Conservation of matter This says that matter can not 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 reduces to one of conservation of mass. SCE DEPARTMENT OF CIVIL ENGINEERING Page 4CE6303 APPLIED HYDRAULIC ENGINEERING 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 ( p otential 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. 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. The Continuity Equation (c onservation 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. Mass flow entering = mass flow leaving So for the time interval δt : Figure of a small length of channel as a control volume Considering the control volume above which is a short length of open channel of arbitrary cross- Section then, if ρ is the fluid density and Q is the volume flow rate then section then, if mass flow rate is ρ Q and the continuity equation for steady incompressible flow can be written SCE DEPARTMENT OF CIVIL ENGINEERING Page 5CE6303 APPLIED HYDRAULIC ENGINEERING 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 u and the cross-sectional area is A then: 1 Similarly at the downstream face, face 2, where mean velocity is u and the cross- 2 sectional area is A then: 2 Therefore the continuity equation can be written as The Energy equation (c onservation of energy): Consider the forms of energy available for the above control volume. If the fluid moves from the upstream face 1, to the downstream face 2 in time d t over the length L. The work done in moving the fluid through face 1 during this time is Where p is pressure at face 1 1 The mass entering through face 1 is Therefore the kinetic energy of the system is: If z is the height of the centroid of face 1, then the potential energy of the fluid 1 entering the control volume is : The total energy entering the control volume is the sum of the work done, the potential and the kinetic energy: We can write this in terms of energy per unit weight. As the weight of water entering the control volume is ρ A L g then just divide by this to get the total energy per unit 1 1 weight: At the exit to the control volume, face 2, similar considerations deduce If no energy is supplied to the control volume from between the inlet and the outlet then energy leaving = energy entering and if the fluid is incompressible SCE DEPARTMENT OF CIVIL ENGINEERING Page 6CE6303 APPLIED HYDRAULIC ENGINEERING This is the Bernoulli equation. Note: 1. In the derivation of the Bernoulli equation it was assumed that no energy is lost in the control volume - i.e. the fluid is frictionless. To apply to non frictionless situations some energy loss term must be included. 2. The dimensions of each term in equation 1.2 has the dimensions of length ( units of meters) . For this reason each term is often regarded as a "head" and given the names 3. Although above we derived the Bernoulli equation between two sections it should strictly speaking be applied along a stream line as the velocity will differ from the top to the bottom of the section. However in engineering practise it is possible to apply the Bernoulli equation with out reference to the particular streamline The momentum equation ( momentum principle) Again consider the control volume above during the time δt By the continuity principle : = d Q1 = dQ 2 = dQ And by Newton's second law Force = rate of change of momentum It is more convenient to write the force on a control volume in each of the three, x, y and z direction e.g. in the x-direction Integration over a volume gives the total force in the x-direction as As long as velocity V is uniform over the whole cross-section. SCE DEPARTMENT OF CIVIL ENGINEERING Page 7CE6303 APPLIED HYDRAULIC ENGINEERING This is the momentum equation for steady flow for a region of uniform velocity. Energy and Momentum coefficients In deriving the above momentum and energy ( B ernoulli) equations it was noted that the velocity must be constant ( e qual to V) over the whole cross-section or constant along a stream-line. Clearly this will not occur in practice. Fortunately both these equation may still be used even for situations of quite non-uniform velocity distribution over a section. This is possible by the introduction of coefficients of energy and momentum, a and ß respectively. These are defined: where V is the mean velocity. And the Bernoulli equation can be rewritten in terms of this mean velocity: And the momentum equation becomes: The values of α and ß must be derived from the velocity distributions across a cross- section. They will always be greater than 1, but only by a small amount consequently they can often be confidently omitted – but not always and their existence should always be remembered. For turbulent flow in regular channel a does not usually go above 1.15 and ß will normally be below 1.05. We will see an example below where their inclusion is necessary to obtain accurate results. 1.6 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. SCE DEPARTMENT OF CIVIL ENGINEERING Page 8CE6303 APPLIED HYDRAULIC ENGINEERING Figure of velocity distributions Determination of energy and momentum coefficients To determine the values of a and ß the velocity distribution must have been measured ( o r be known in some way) . In irregular channels where the flow may be divided into distinct regions a may exceed 2 and should be included in the Bernoulli equation. The figure below is a typical example of this situation. The channel may be of this shape when a river is in flood – this is known as a compound channel . Figure of a compound channel with three regions of flow If the channel is divided as shown into three regions and making the assumption that α = 1 for each then where 1.7 Steady Uniform flow When uniform flow occurs gravitational forces exactly balance the frictional resistance forces which apply as a shear force along the boundary (channel bed and walls) . SCE DEPARTMENT OF CIVIL ENGINEERING Page 9CE6303 APPLIED HYDRAULIC ENGINEERING Figure of forces on a channel length in uniform flow Considering the above diagram, the gravity force resolved in the direction of flow is and the boundary shear force resolved in the direction of flow is In uniform flow these balance Considering a channel of small slope, (a s channel slopes for unifor and gradually varied flow seldom exceed about 1 in 50) then So 1.7.1 The Chezy equation If an estimate of τ o can be made then we can make use of Equation. If we assume the state of rough turbulent flow then we can also make the assumption the shear force is proportional to the flow velocity squared i.e. Substituting this into equation gives Or grouping the constants together as one equal to C SCE DEPARTMENT OF CIVIL ENGINEERING Page 10CE6303 APPLIED HYDRAULIC ENGINEERING This is the Chezy equation and the C the “Chezy C” Because the K is not constant the C is not constant but depends on Reynolds number and boundary roughness ( s ee discussion in previous section) . The relationship between C and is easily seen be substituting equation 1.9 into the Darcy- Wiesbach equation written for open channels and is 1.7.2 The Manning equation A very many studies have been made of the evaluation of C for different natural and manmade channels. These have resulted in today most practising engineers use some form of this relationship to give C: This is known as Manning’s formula, and the n as Manning’s n . Substituting equation 1.9 in to 1.10 gives velocity of uniform flow: Or in terms of discharge Note: Several other names have been associated with the derivation of this formula – or ones similar and consequently in some countries the same equation is named after one of these people. Some of these names are; Strickler, Gauckler, Kutter, Gauguillet and Hagen. The Manning’s n is also numerically identical to the Kutter n . The Manning equation has the great benefits that it is simple, accurate and now due to it long extensive practical use, there exists a wealth of publicly available values of n for a very wide range of channels. Below is a table of a few typical values of Manning’s n SCE DEPARTMENT OF CIVIL ENGINEERING Page 11CE6303 APPLIED HYDRAULIC ENGINEERING Conveyance Channel conveyance, K , is a measure of the carrying capacity of a channel. The K is really an agglomeration of several terms in the Chezy or Manning's equation: So Use of conveyance may be made when calculating discharge and stage in compound channels and also calculating the energy and momentum coefficients in this situation. 1.8 Best Hydraulic Cross- Section We often want to know the the minimum area A for a given flow Q, slope S and 0 roughness coef- ficient n. This is known as the best hydraulic cross section 2/3 The quantity AR in Mannings’ equation is called the section factor h Writing the Manning equation with R = A/P, we get h Rearranged we get SCE DEPARTMENT OF CIVIL ENGINEERING Page 12CE6303 APPLIED HYDRAULIC ENGINEERING  ( inside ) is a constant; Channel with minimum A is also minimum P  Minimum excavation area A also has minimum P  Best possible is semicircular channel, but construction costs are high Let’s find out what the best hydraulic cross section is for a rectan- gular channel Example: Water flows uniformly in a rectangular channel of width b and depth y. Determine the aspectratiob/yforthe besthydraulic cross section. SCE DEPARTMENT OF CIVIL ENGINEERING Page 13CE6303 APPLIED HYDRAULIC ENGINEERING  Thus best hydraulic cross- section for a rectangular channel occurs when the depth is one- half the width of the channel  Note for 1 b/y 4; Q ˜ .96 Q max Must include freeboard f in design between 5 to 30% of y n Table gives Optimum properties of Open Channel Sections .For trapezoid, half- hexagon .For circular section, half- circle .For triangular section, half- square Design of Erodible Channels Design velocity V small enough not to cause erosion Find maximum permissible velocity based on channel material ( R oberson, Table 4- 3) SCE DEPARTMENT OF CIVIL ENGINEERING Page 14

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