Fluid Mechanics - Essay Example

Fluid Mechanics Table of contents S.No Topic Page 2 3 4 5 6 7 8 9 10 11 Introduction Impact of a jet on vanes Pelton Wheel turbine Francis turbine Kaplan turbine Pumps - Introduction Centrifugal pumps Reciprocating pumps Conclusion Glossary Works cited 2 2 6 8 9 11 11 17 19 20 21 Introduction All matter basically exists in one of the 3 states: solids, liquids and gases. The latter two belong to the fluid state. Fluid mechanics is the branch of applied mechanics related to the statics and dynamics of fluids - both liquids and gases. The analysis of the behavior of fluids is built over the fundamental laws of mechanics which relate continuity of mass and energy with force and momentum together with solid mechanics properties. Hydraulic machines are those machines which convert hydraulic energy into mechanical energy (further converted into electrical energy) or mechanical energy into hydraulic energy. The former types of machines are called turbines and the latter, pumps. Impact of a Jet onto a vane A combination of both pump and turbine is used in fluid couplings and torque converters for the transmission of power smoothly through a fluid medium. The analysis of impact of fluid jets on vanes is typically involved in the design of an efficient turbo-machine. When a vane moves away from the jet as shown in the below figure, the mass flow arriving at the vane is considerably reduced because some of the mass leaving the nozzle results in a growing column of fluid between the jet and the nozzle. This is what happens in turbines where the vanes are part of a revolving wheel. We need only consider the simplest case of movement in a straight line in the direction of the jet. Assuming the velocity of the jet is v and the velocity of the vane is u as shown above, the velocity of the fluid arriving would be v - u. This is the relative velocity, that is, relative to the plate. The mass flow rate arriving on the plate is then calculated as M = ρA(v-u) The initial direction of the fluid is the direction of the jet. However, due to the plate movement, the velocity of the fluid as it leaves the edge is not at 90o to the initial direction. The initial forward velocity of the fluid = v The final forward velocity of the fluid = u The change in forward velocity = v-u The force on the plate = mñv = m (v-u) Since m = ñA (v-u) then the force on the plate is F = ñA(v-u)2 Moving curved vane Turbine vanes are normally curved and the fluid joins it at the same angle as the vane as shown in the figure The true velocity of the fluid leaving the nozzle is v1 and velocity of the vane is u. The fluid arrives on the vane with relative velocity v1-u as before, which is shown in the above figure. This is a relative velocity with respect to someone moving with the vane. In the absence of friction, the velocity of the fluid over the surface of the vane will be v1-u at all points. At the tip where the fluid leaves the vane, it will have two velocities. The fluid will be flowing at v1-u over the vane but also at velocity u in the forward direction. The true velocity v2 at the exit as shown above must be the vector sum of these two. The vector diagram is illustrated below: If only the force acting on the vane in the direction of movement is required, then the horizontal component of v2 must be determined. Because this direction is the direction in which the vane is whirling around the centre of the wheel, it is called the velocity of whirl vw2. The velocity v1 is also in the direction of whirling so it implies that v1 = vw1. Vw2 may be found by drawing the above vector diagram to scale or by using trigonometry. In this case it can be shown that vw2 = u + (v1-u) (cosè) The horizontal force on the vane becomes Fh = m (vw1-vw2) = m (v1-vw2) This further simplifies down to Fh = m(v1-u) (1-cosè) This force moves at the same velocity as the vane. The power developed by a force is the product of force and velocity and is called the Diagram Power (D.P.) The diagram power developed by a simple turbine blade is D.P. = mu (v1-u) (1-cosè) This work involving the force on a moving vane is the foundation behind turbine problems and the geometry of the case considered is that of the Pelton Wheel turbine. Fluid friction losses For fluid entry without shock and minimum energy loss, the relative velocity vector must be tangential to the blade tip at inlet. The direction of the fluid jet changes as it flows over the smooth surface of the vane, and at the outlet tip it emerges with a relative velocity w2 inclined at angle b2. If the vane surface is consider very smooth, the relative velocities w1 and w2 may be assumed equal. But in practice, there is always some friction which reduces the magnitude of w2. System efficiency Efficiency of jet is ratio of the work done by jet to the kinetic energy of jet Next, we briefly discuss some of the important types of turbines. Pelton Wheel The Pelton wheel is among the most efficient of water turbines, and is an impulse machine, as it makes use of Newton’s second law to extract energy from a jet of fluid. The water flows along a tangential path to the path of the runner. Streams of water are directed against a series of spoon shaped buckets around the edge of a wheel. The contour of the bucket determines the direction of water velocity. Pressure is exerted on the bucket by the water and the water is decelerated as it flows out the other side of the bucket at low velocity. As a result, there is a transfer of the waters momentum to the turbine and this impulse performs work on the turbine. For gaining best power and efficiency, the turbine system is designed in such a way that the water-jet velocity is twice the velocity of the bucket. Because water is almost incompressible, almost all of the available energy is extracted in the first stage of the hydraulic turbine. Pelton wheels are the generally preferred turbines for hydro-power, when the available water source has relatively high hydraulic head at low flow rates. Pelton wheels are manufactured in various sizes. The largest units can be up to 200 megawatts. The smallest Pelton wheels are only a few inches across, and can be used to tap power from mountain streams. Depending on water flow and design, Pelton wheels operate best with heads from 15 metres to 1,800 metres. They are found to exhibit high efficiency in high head applications and hence, more power can be extracted from a water source with high-pressure and low-flow than from a source with low-pressure and high-flow, even though theoretically both the flows contain the same power. They have a high efficiency, generally over 90%. Pelton turbines are suited for high head, low flow applications. They are used in storage power stations with downward gradients up to 2,000 meters and can contain up to 6 nozzles. Francis Turbine The Francis turbine is the most commonly used water turbine, which was developed by James B. Francis. It is an inward flow reaction turbine that combines radial and axial flow and operates in a head range of ten meters to several hundred meters. Their foremost use is in electrical power production. The Francis turbine is a reaction turbine, which means that the working fluid changes pressure as it moves across the turbine, surrendering its energy. The turbine is located between the high pressure water source and the low pressure water exit, usually at the base of a dam. The inlet is spiral shaped and guide vanes direct the water tangentially to the runner. This radial flow exerts pressure on the runner vanes, causing the runner to spin. As the water moves through the runner, it’s spinning radius decreases, further acting on the runner. This property along with the waters pressure helps the turbines to harness water energy. The turbines exit tube is specially shaped to help decelerate the water flow and recover kinetic energy. They are best suited for sites with high flows and low to medium head. These turbines are very expensive to design, manufacture and install, but have a long life, typically for decades. Francis turbine has a wide range of applications and can be used for fall heights of 2–800 meters. Francis type units cover a wide head range, from 20 meters to 700 meters and their output varies from a few kilowatts to close to 1,000 megawatt. Large Francis turbines have found to operate at high efficiencies, typically over 90%. In addition to electrical production, they may also be used for pumped storage. Kaplan Turbine The Kaplan turbine was a direct evolution of the Francis turbine and was developed in 1913 by an Austrian professor Viktor Kaplan Francis turbine The Francis turbine is a type of water turbine [i] that was developed by James B. Francis [i]. ...  . The Kaplan turbine is a propeller-type water turbine that has adjustable blades. It allows efficient power production in low head applications that was not feasible with Francis turbines. Kaplan turbines are now widely used for high-flow, low-head power production. The Kaplan turbine is an inward flow reaction turbine and the design combines radial and axial features. The inlet is a scroll-shaped tube that wraps around the turbines wicket gate. Water is directed tangentially, through the wicket gate, and spirals on to a propeller shaped runner, making it spin. The outlet is in the form of a specially shaped draft tube that helps decelerate the water and release kinetic energy. The turbine does not need to be at the lowest point of water flow, as long as the draft tube remains full of water. A higher turbine location, however, increases the suction that is imparted on the turbine blades by the draft tube. The resulting pressure drop may lead to a phenomenon called cavitation, which is a term used to describe the behavior of voids or bubbles in a liquid. Kaplan turbine efficiencies are typically over 90%, but may be lower in very low head applications. Kaplan turbines are widely used throughout the world for electrical power production. They cover the lowest head hydro sites and are especially suited for high flow conditions. Inexpensive micro turbines are manufactured for individual power production with as little as two feet of head. Large Kaplan turbines are individually designed for each site to operate at the highest possible efficiency, usually over 90%. They are highly expensive to manufacture, but operate for a very long time Kaplan turbines are more suited to situations where there is a low head and a considerable amount of discharge. They have been used for exploiting many hydro sources previously discarded for economic or environmental reasons, and also found use as wind turbines. Pumps - Introduction Hydraulic machines which convert mechanical energy into hydraulic energy are called pumps. A pump is a device which gives energy to the fluid it transports, which maybe a liquid or a gas, thereby increasing its pressure head, kinetic energy or both. They are widely employed in water supply, agricultural engineering, sewage treatment, chemical and mechanical engineering etc. Centrifugal pumps If the mechanical energy is converted into pressure energy by means of centrifugal force acting on the fluid, the hydraulic machine is called centrifugal pump. All points above or below BEP (Best efficiency point, explained below) have a lower efficiency, and the impeller is subject to vibration, heat, and cavitation. This leads to premature bearing and mechanical seal failures due to shaft deflection and the heat will result in seizure of close tolerance parts and cavitation. The energy created by the pump is kinetic energy according to the Bernoulli Equation. The energy transferred to the liquid corresponds to the velocity at the edge or vane tip of the impeller. The faster the impeller revolves or the bigger the impeller is, the higher will be the velocity of the liquid energy transferred to the liquid. This is explained by the Affinity Laws. The centrifugal pump acts as a reverse of an inward radial flow reaction turbine. That is, the flow in centrifugal pumps is in the radial outward directions. The centrifugal pump works on the principle of forced vortex flow which implies that when a certain mass of liquid is rotated by an external torque, the rise in pressure head at any point of the rotating liquid takes place. The rise in pressure head at any point of the rotating liquid is proportional to the square of tangential velocity of the liquid at that point. Thus at the outlet of the impeller where radius is more, the rise in pressure head will be higher and the liquid will be discharged from the outlet with a relatively high pressure head, due to which, the liquid can be lifted to a high level. System characteristics In a pumping system, the objective generally, is either to transfer a liquid from a source to a required destination, like filling a high reservoir, or to circulate liquid around a system, e.g. as a means of heat transfer in heat exchanger. A pressure is needed to make the liquid flow at the required rate and this must prevail against head losses in the system. These losses are of two types: static and friction head. Static head is simply the difference in height of the supply and destination reservoirs, as in Figure (i). In this illustration, flow velocity in the pipe is assumed to be very small. An example of a system with only static head is pumping into a pressurized vessel with short pipe runs. Static head is independent of flow and graphically, it would be shown as in Figure (ii). (i) (ii) Friction head (sometimes called dynamic head loss) is the friction loss, on the liquid being moved, in pipes, valves and equipment in the system. The friction losses are proportional to the square of the flow rate. A closed loop circulating system without a surface open to atmospheric pressure, would exhibit only friction losses and would have a system friction head loss vs. flow curve as shown below. The performance of a pump can be expressed graphically as head against flow rate. The centrifugal pump has a curve where the head falls gradually with increasing flow. This is called the pump characteristic curve (Head - Flow curve) as shown below. Power and Efficiency Brake Horsepower (BHP) is the actual horsepower delivered to the pump shaft, defined as follows: BHP = Q x Hr x Sp. Gr. / 3960 x Eff. Q = Capacity in gallons per minute, Hr = Total Differential Head in absolute feet, Sp. Gr. = Specific Gravity of the liquid. Eff. = Pump efficiency as a percentage To get the value in S.I units (watts), the BHP is multiplied by 745.7 Water Horsepower (WHP) is the hydraulic horsepower delivered by the pump, defined as follows: WHP = Q x Hr x Sp.Gr./ 3960 Q = Capacity in gallons per minute Hr = Total Differential Head in absolute feet Sp. Gr. = Specific Gravity of the liquid The constant (3960) is the number of foot-pounds in one horsepower (33,000) divided by the weight of one gallon of water (8.33 pounds). To get the value in S.I units (watts), the WHP is multiplied by 745.7 Brake horsepower is always greater than hydraulic horsepower due to the friction in the pump. Pump efficiency is the ratio of these two values. Pump Efficiency = WHP / BHP Hydraulic power, pump shaft power and electrical input power Hydraulic power Ph = Q (m3/s) x Total head, hd - hs (m) x ρ (kg/m3) x g (m/s2) / 1000 Where hd – discharge head, hs – suction head, ρ – density of the fluid, g – acceleration due to gravity Pump shaft power Ps = Hydraulic power, Ph / pump efficiency, ηPump Electrical input power = Pump shaft power Ps / ηMotor Efficiencies: a) Manometric efficiency: The ratio of manometric head to the head imparted by the impeller to the water is known as manometric efficiency. b) Mechanical efficiency: The ratio of the power available at the impeller to the power at the shaft is known as the mechanical efficiency. c) Overall efficiency: It is defined as the ratio of the power output of the pump to the power input to the pump. It is the product of manometric efficiency and mechanical efficiency. Best Efficiency Point (BEP) is the capacity at maximum impeller size at which the efficiency is the highest. The standard convention for centrifugal pump is to plot the pump performance curves showing flow on the horizontal axis and Head generated on the vertical axis. Efficiency, Power & NPSH required, are plotted on the vertical axis, against flow, as illustrated in the below figure. The value, by which the pressure in the pump suction exceeds the liquid vapor pressure, is expressed as a head of liquid and referred to as Net Positive Suction Head Available (NPSHA). The value of NPSH needed at the pump suction to prevent the pump from cavitation is known as NPSH Required (NPSHR). Applications Centrifugal pumps are commonly used in plumbing systems. It is also extensively found in heavy industries such as mining, petroleum refining, paper production, and manufacturing in general. Reciprocating pumps If the mechanical energy is converted into pressure energy by sucking the liquid into a cylinder in which a piston is reciprocating (moving backwards and forwards), which exerts the thrust on the liquid and increases its hydraulic energy, the pump is called reciprocating pump. It is a positive displacement pump, which means it causes a fluid to move by trapping a fixed amount of it, then forcing or displacing that trapped volume into the discharge pipe. The movement of the piston is brought about by connecting the piston rod to crank by means of a connecting rod. The crank is rotated with the aid of an electric motor. Suction and delivery pipes with suction valve and delivery valve are connected to the cylinder. These valves allow water to flow in one direction only (one way valves). Suction valve allows water from suction pipe to the cylinder while delivery valve allows water from cylinder to delivery pipe only. These pumps can be single-stage or multistaged. Multistaged reciprocating pumps have multiple cylinders in series Reciprocating pumps move water or other liquids by a plunger or piston that travels back and forth inside a cylinder. It is used widely for pumping foaming liquids and high viscosity liquids. It can control flow by controlling the speed of the drive with no head loss by throttling as in a centrifugal pump. They are generally used often at high or very high pressures and also as metering pumps because of the constant flow rate. The flow rate can be easily altered by adjusting the RPM (revolutions per minute) of the driver. Applications Reciprocating pumps are capable of working with extremely low flow rates, making them suitable for many chemical injection applications. Reciprocating pumps are also suitable used for filling buffer tanks and gas cylinders and generally used for industrial type uses, including concrete and oil. They have also been used as dosing and metering pumps. Conclusion Fluid mechanics is an important branch of science with significant roles in diverse fields such as aeronautics, chemical, civil, and mechanical engineering, meteorology, naval architecture and oceanography. Fluid mechanics can sometimes be mathematically complex, which then employs numerical methods using computers. The discipline devoted to this approach of problem solving is called Computational Fluid Dynamics (CFD). Glossary Affinity Laws Laws applied to pumps, fans, and hydraulic turbines to express the relationship between several variables involved in pump or fan performance (such as speed, power etc) Bernoulli’s Principle The principle states that, when the speed of a moving fluid increases, the pressure within that fluid decreases. Brake horsepower The work applied to a pump shaft. Cavitation Formation of air cavities or bubbles within fluids when subjected to pressure changes. Centrifugal pump A pump that uses the centrifugal force of the included liquid to impart pressure to that liquid. Fluid mechanics Branch of applied mechanics related to the statics and dynamics of fluids - both liquids and gases. Positive displacement A pump that pushes a liquid by means of a pump of a continuously moving cavity. Pump A hydraulic machine which converts mechanical energy into hydraulic energy. Reciprocating pump A positive displacement pump which has a piston moving back and forth to perform the pumping. Turbine A hydraulic machine which converts hydraulic energy into mechanical energy. Water horsepower The useful power transmitted to a water system. Works Cited “Centrifugal Pumps” 13 Nov 2008. “Fluid mechanics” 15 Nov 2008. “Fluid Mechanics and Machinery” 12 Nov 2008. “Francis Turbine” 14 Nov 2008. “Francis Turbine” 30 Nov 2008. “Hydraulic Machines” 11 Nov 2008. “Kaplan Turbine” 14 Nov 2008. “ Operating Principles Of Centrifugal And Reciprocating Pumps” 14 Nov 2008. Bachus, Larry & Custodio, Angel. (2003), Know and Understand Centrifugal Pumps, Elsevier Advanced Technology, Oxford OX5 lGB, UK “Pelton Turbine” 30 Nov 2008. “Pumps“.12 Nov 2008. “Pumps and pumping systems” 12 Nov 2008. “Pumps, Fans and Turbines - Horsepower” 30 Nov 2008. “Reciprocating Pumps” 12 Nov 2008. “The Engineering toolbox” 13 Nov 2008. “Types of Reciprocating Pumps” 14 Nov 2008. Read More