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Table of Contents
I. FLUID FLOW AND FLUID DYNAMICS.........................................................................................................1
II. INTRODUCTION........................................................................................................................................2
III. ANALYSIS AND DISCUSSION..................................................................................................................2
A. FLUID-FLOW MEASUREMENT...............................................................................................................2
a. Flow meters...................................................................................................................................... 2
b. Device coefficients.......................................................................................................................... 19
c. Reynolds number............................................................................................................................20
d. Mach number..................................................................................................................................20
e. Equation of continuity of mass.......................................................................................................20
f. Head losses in pipes........................................................................................................................20
g. Drag Force.......................................................................................................................................24
h. Aerodynamic lift..............................................................................................................................24
B. EXAMPLE PROBLEMS..........................................................................................................................25
IV. EXPERIMENTS..................................................................................................................................... 27
A. Experiment 1. Flow meters.................................................................................................................27
B. Experiment 2. Open channel flow Measurement using Weirs...........................................................33
C. Experiment 3. Viscous, Turbulence, and Supersonic flow..................................................................35
D. Experiment 4. Pipe Friction Losses.....................................................................................................35
E. Experiment 5. Characteristics of Nozzles and Jets..............................................................................41
F. Experiment 6. Drag coefficient-Determination of the Drag Coefficient of a Sphere...........................44
V. REFERENCES........................................................................................................................................... 54
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I. FLUID FLOW AND FLUID DYNAMICS
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II. INTRODUCTION
Recent developments in both the theory and the application of fluid mechanics have greatly increased the importance of this subject to mechanical engineers. Aerodynamic and jet devices, gas turbines, turbo compressors, fluid couplings and torque converters--- these and many other applications are commanding the engineer’s attention.
Any effort to isolate the applications of fluid mechanics and treat them separately becomes an illustration of the very close relationship with the other fields of mechanics and with thermodynamics.
Fluid mechanics is one of the basic studies for all mechanical engineers. Fortunately, most of the problems involved maybe solved tough the application of a few simple laws, including the principles of statics, Newton’s law of motion, the equation of continuity of mass, and the conservation of energy as expressed I the general energy equation. Many terms will be defined and discussed as required in this report, such as Reynolds number, Mach number and others.
III. ANALYSIS AND DISCUSSION
A. FLUID-FLOW MEASUREMENT
The importance of flow measurement in the industry has grown in the past 50 year, not just because it was widespread use for accounting purposes, such as custody transfer of fluid from supplier to customers, but also because of its application in manufacturing processes. Examples of the industrial involvement in flow measurement includes food and beverage, oil and gas industrial, medical, petrochemical, power generation, and water distribution and etc.
Flow measurement is the determination of the quantity of a fluid, either a liquid, or vapor, that passes through a pipe, duct or open channel. Flow may be expressed as a rate of volumetric flow (such as gallons per minute, cubic meters per minute, cubic feet per minute), mass rate of flow (such as kilograms per hour, pounds per hour), or in terms of a total volume or mass flow (integrated rate of flow for a given period of time).
Fluid flow measurement can be divided into several types; each type requires specific considerations of such factors as accuracy requirements, cost considerations, and use of the flow information to obtain the required end results. Normally the flow meter is measure flow indirectly by measuring a related property such as a differential pressure across a flow restriction or a fluid velocity in a pipe. A number of different fundamental physical principles are used in flow measurement devices.
a. Flow metersA flowmeter is an instrument used to measure linear, nonlinear, mass or volumetric
flow rate of a liquid or a gas.
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1. Types of flowmeters
1.1. Rotameter
The rotameter is a tapered tube and a float. It is the most widely used variable-area flow meter because of its low cost, simplicity, low pressure drop, relatively wide rangeability, and linear output.
1.2. Piston-type flowmeters
Piston-type flowmeters use an annular orifice formed by a piston and a tapered cone. The piston is held in place at the base of the cone (in the "no flow position") by a calibrated spring. Scales are based on specific gravities of 0.84 for oil meters, and 1.0 for water meters. Their simplicity of design and the ease with which they can be equipped to transmit electrical signals has made them an economical alternative to rotameters for flowrate indication and control.
1.3. Mass Gas flowmeter
Thermal-type mass flow meters operate with minor dependence on density, pressure, and fluid viscosity. This style of flowmeter utilizes either a differential pressure transducer and temperature sensor or a heated sensing element and thermodynamic heat conduction principles to determine the true mass flow rate. Many of these mass flowmeters have integral displays and analog outputs for data logging. Popular applications include leak testing and low flow measurements in the milliliters per minute.
1.4. The ultrasonic doppler flow meters
The ultrasonic doppler flow meters are commonly used in dirty applications such as wastewater and other dirty fluids and slurries which ordinarily cause damage to conventional sensors. The basic principle of operation employs the frequency shift (Doppler Effect) of an ultrasonic signal when it is reflected by suspended particles or gas bubbles (discontinuities) in motion.
1.5. Turbine Flow meters
The turbine meter can have an accuracy of 0.5% of the reading. It is a very accurate meter and can be used for clean liquids and
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viscous liquids up to 100 centistokes. A minimum of 10 pipe diameters of straight pipe on the inlet is required. The most common outputs are a sine wave or squarewave frequency but signal conditioners can be mounted on top for analog outputs and explosion proof classifications. The meters consists of a multi-bladed rotor mounted at right angles to the flow and suspended in the fluid stream on a free-running bearing.
1.6. Paddlewheel Sensors
It is one of the most popular cost effective flowmeters for water or water like fluids. Many are offered with flow flittings or insertions styles. These meters like the turbine meter require a minimum of 10 pipe diameters of straight pipe on the inlet and 5 on the outlet. Chemical compatibility should be verified when not using water. Sine wave and Squarewave pulse outputs are typical but transmitters are available for integral or panel mounting. The rotor of the paddlewheel sensor is perpendicular to the flow and contact only a limited cross section of the flow.
1.7. Positive Displacement Flowmeters
These meters are used for water applications when no straight pipe is available and turbine meters and paddlewheel sensor would see too much turbulence. The positive displace ment are also used for viscous liquids.
1.8. Vortex Meters
The main advantages of vortex meters are their low sensitivity to variations in process conditions and low wear relative to orifices or turbine meters. Also, initial and maintenance costs are low. For these reasons, they have been gaining wider acceptance among users. Vortex meters do require sizing, contact our flow engineering.
1.9. The Orifice Plate
An orifice is an opening (usually circular) with a closed perimeter through which fluid flows. It used primarily to measure or to control the flow fluid. The upstream face of the orifice maybe rounded or sharp. An orifice with prolonged side, such as piece of pipe, having a length of two or three times its diameter, is called a short tube. Longer tubes such as culverts under
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embankments are usually treated as orifice although they may also be treated as short pipes.
The orifice meter consists of a flat orifice plate with a circular hole drilled in it. There is a pressure tap upstream from the orifice plate and another just downstream. There are in general three methods of placing the taps. The coefficient of the meter depends upon the position of tap
Advantages
The orifice meter is recommended for clean and dirty liquids and some slurry services.
The rangeability is 4 to 1 The pressure loss is medium Typical accuracy is 2 to 4% of full scale The required upstream diameter is 10 to 30 The viscosity effect is high The relative cost is low
Equation
Theoretical velocity, υt = √2gh
Actual velocity, υ = C υ√2gh
Theoretical discharge, Qt=A√2gh
Actual discharge, Q=CA √2 gh
H=h+υA 2
2 g+PA
γ−
PB
γ
H=Head Upstream−Head Downstream
1.10. The Venturi Meter
Venturi meter is an instrument used in measuring the discharge through pipes. It is consist of a converging tube which is connected to the main pipe at the inlet and ending I a cylindrical section called the throat and a diverging sectionwhich is connected again to the main pipe at the outlet. The angle of divergence is kept small to reduce the head lost caused by turbulence as the velocity is reduced. In the venturi meter the fluid is accelerated through a converging cone of angle 15-20o and the pressure difference between the upstream side
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of the cone and the throat is measured and provides a signal for the rate of flow.
High pressure and energy recovery makes the venturi meter suitable where only small pressure heads are available.A discharge coefficient cd = 0.975 can be indicated as standard, but the value varies noticeably at low values of the Reynolds number.The pressure recovery is much better for the venturi meter than for the orifice plate.
The venturi tube is suitable for clean, dirty and viscous liquid and some slurry services.
The rangeability is 4 to 1 Pressure loss is low Typical accuracy is 1% of full range Required upstream pipe length 5 to 20 diameters Viscosity effect is high Relative cost is medium
Equation:
υ12
2g+P1
γ+Z1=
υ22
2 g+P2
γ+Z2
Neglecting head lost, the increase in kinetic energy is equal to the decrease in potential energy. This statement is shown as the venture principle.
1.11. The Nozzle
A nozzle is a converting tube installed at the end of a pipe or hose for the purpose of increasing the velocity of the issuing jet. Nozzles used for determining fluid's flow rate through pipes can be in three different types:
The ISA 1932 nozzle - developed in 1932 by the International Organization for Standardization or ISO. The ISA 1932 nozzle is common outside USA.
The long radius nozzle is a variation of the ISA 1932 nozzle. The venturi nozzle is a hybrid having a convergent section similar to the ISA 1932
nozzle and a divergent section similar to a venturi tube flowmeter.
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Advantages
o The flow nozzle is recommended for both clean and dirty liquidso The rangeability is 4 to 1o The relative pressure loss is mediumo Typical accuracy is 1-2% of full rangeo Required upstream pipe length is 10 to 30 diameterso The viscosity effect higho The relative is medium
The discharge through a nozzle can be calculated using the equation
Q=C An√2 gh
1.12. Pitot Tube
Pitot tube is a bent (L-shaped or U-shaped) tubes with both ends open and is used to measure the velocity of fluid flow or velocity of air flow as used in airplane speedometer. When the tube is placed in a moving stream with open end oriented into the direction flow, the liquid enters the opening until the surface in the tube rises a distance above the stream surface. An equilibrium condition is then established, and the quantity of liquid in the tube remains steady. The face of the tube facing the stream is called the stagnation point.
Equation: υ=√2gh
b. Anemometer
An anemometer is a device for measuring wind speed, and is a common weather station instrument. The term is derived from the Greek word anemos, meaning wind. The first known description of an anemometer was given by Leon Battista Alberti in around 1450[1].
Anemometers can be divided into two classes: those that measure the wind's speed, and those that measure the wind's pressure; but as there is a close connection between the pressure and the speed, an anemometer designed for one will give information about both.
An anemometer is an apparatus that is employed to determine the wind's velocity (direction and speed) and pressure. Most anemometers only measure one of these factors. Since wind velocity and wind pressure are closely connected; pressure, direction and speed can be derived mathematically from an anemometer that is only specifically devised to measure wind velocity.
Anemometers can be built from scratch utilizing some materials found at home and a few electrical and electronic parts that can be easily obtained from hardware and electronics stores. A home-made anemometer will usually have a main rotating shaft supported by
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bearings. On top would be at least three horizontal spokes, and at the end of each spoke would be cups to catch the wind. The main rotating shaft is attached to a dynamo which produces electricity when the main shaft rotates. The output in electricity then passes through a connected small circuit board and finally on to a measurement display.
The main component of the circuit board is a microcontroller unit, which is a vastly simplified cousin of the PC's microprocessor. It's available to consumers and electronics hobbyists and is found in various appliances like remote controls, power tools and even toys. The type of microcontroller often used for home-made anemometers is one that can release proportional voltage based on incoming electrical frequency. The rotation of the main shaft affects the electrical frequency put out by the dynamo; the resulting output voltage from the microcontroller is then registered on an attached measurement display, which can be as generic as a multimeter.
These types of home-made anemometers need to be calibrated, meaning a proportional relationship between wind speed and electrical voltage needs to be established. Wind speed is measured in kilometers per hour or meters per second, and although this is evident in the rotation of the main shaft, quantifying this would entail setting up a mechanical device that directly measures the number of rotations the shaft makes in a given time period. Mechanical means of measurement are usually prone to error when dealing with something as turbulent as wind. To test and calibrate your home-made anemometer, you can simulate various wind speeds by driving around in your car with the apparatus attached outside the car.
Velocity anemometers
Cup anemometers
A simple type of anemometer is the cup anemometer, invented (1846) by Dr. John Thomas Romney Robinson, of Armagh Observatory. It consisted of four hemispherical cups each mounted on one end of four horizontal arms, which in turn were mounted at equal angles to each other on a vertical shaft. The air flow past the cups in any horizontal direction turned the cups in a manner that was proportional to the wind speed. Therefore, counting the turns of the cups over a set time period produced the average wind speed for a wide range of speeds. On an anemometer with four cups it is easy to see that since the cups are arranged symmetrically on the end of the arms, the wind always has the hollow of one cup presented to it and is blowing on the back of the cup on the opposite end of the cross.
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Windmill anemometers
The other forms of mechanical velocity anemometer may be described as belonging to the windmill type or propeller anemometer. In the Robinson anemometer the axis of rotation is vertical, but with this subdivision the axis of rotation must be parallel to the direction of the wind and therefore horizontal. Furthermore, since the wind varies in direction and the axis has to follow its changes, a wind vane or some other contrivance to fulfill the same purpose must be employed. An aerovane combines a propeller and a tail on the same axis to obtain accurate and precise wind speed and direction measurements from the same instrument. In cases where the direction of the air motion is always the same, as in the ventilating shafts of mines and buildings for instance, wind vanes, known as air meters are employed, and give most satisfactory results.
Hot-wire anemometers
Hot wire anemometers use a very fine wire (on the order of several micrometers) electrically heated up to some temperature above the ambient. Air flowing past the wire has a cooling effect on the wire. As the electrical resistance of most metals is dependent upon the temperature of the metal (tungsten is a popular choice for hot-wires), a relationship can be obtained between the resistance of the wire and the flow speed. Additionally, PWM (pulse-width modulation) anemometers are also used, wherein the velocity is inferred by the time length of a repeating pulse of current that brings the wire up to a specified resistance and then stops until a threshold "floor" is reached, at which time the pulse is sent again.
Hot-wire anemometers, while extremely delicate, have extremely high frequency-response and fine spatial resolution compared to other measurement methods, and as such are almost universally employed for the detailed study of turbulent flows, or any flow in which rapid velocity fluctuations are of interest.
Laser Doppler anemometers
Laser Doppler anemometers use a beam of light from a laser that is split into two beams, with one propagated out of the anemometer. Particulates (or deliberately introduced seed material) flowing along with air molecules near where
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the beam exits reflect, or backscatter, the light back into a detector, where it is measured relative to the original laser beam. When the particles are in great motion, they produce a Doppler shift for measuring wind speed in the laser light, which is used to calculate the speed of the particles, and therefore the air around the anemometer.
Sonic anemometers
Sonic anemometers, first developed in the 1970s, use ultrasonic sound waves to measure wind velocity. They measure wind speed based on the time of flight of sonic pulses between pairs of transducers. Measurements from pairs of transducers can be combined to yield a measurement of velocity in 1-, 2-, or 3-dimensional flow. The spatial resolution is given by the path length between transducers, which is typically 10 to 20 cm. Sonic anemometers can take measurements with very fine temporal resolution, 20 Hz or better, which make them well suited for turbulence measurements. The lack of moving parts makes them appropriate for long term use in exposed automated weather stations and weather buoys where the accuracy and reliability of traditional cup-and-vane anemometers is adversely affected by salty air or large amounts of dust. Their main disadvantage is the distortion of the flow itself by the structure supporting the transducers, which requires a correction based upon wind tunnel measurements to minimize the effect. An international standard for this process, ISO 16622 Meteorology—Sonic anemometers/thermometers—Acceptance test methods for mean wind measurements is in general circulation.
Ping-pong ball anemometers
A common anemometer for basic use is constructed from a ping-pong ball attached to a string. When the wind blows horizontally, it presses on and moves the ball; because ping-pong balls are very lightweight, they move easily in light winds. Measuring the angle between the string-ball apparatus and the line normal to the ground gives an estimate of the wind speed.
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Plate anemometers
These are the earliest anemometers and are simply a flat plate suspended from the top so that the wind deflects the plate. In 1450, the Italian art architect Leon Battista Alberti invented the first mechanical anemometer; in 1664 it was re-invented by Robert Hooke (who is often mistakenly considered the inventor of the first anemometer). Later versions of this form consisted of a flat plate, either square or circular, which is kept normal to the wind by a wind vane. The pressure of the wind on its face is balanced by a spring. The compression of the spring determines the actual force which the wind is exerting on the plate, and this is either read off on a suitable gauge, or on a recorder. Instruments of this kind do not respond to light winds, are inaccurate for high wind readings, and are slow at responding to variable winds. Plate anemometers have been used to trigger high wind alarms on bridges.
Tube anemometers
James Lind's anemometer of 1775 consisted simply of a glass U tube containing liquid, a manometer, with one end bent in a horizontal direction to face the wind and the other vertical end remains parallel to the wind flow. Though the Lind was not the first it was the most practical and best known anemometer of this type. If the wind blows into the mouth of a tube it causes an increase of pressure on one side of the manometer. The wind over the open end of a vertical tube causes little change in pressure on the other side of the manometer. The resulting liquid change in the U tube is an indication of the wind speed. Small departures from the true direction of the wind cause large variations in the magnitude.
c. Weir
A weir, also known as a lowhead dam, is a small overflow-type dam commonly used to raise the level of a river or stream. Weirs have traditionally been used to create mill ponds in such places. Water flows over the top of a weir, although some weirs have sluice gates which release water at a level below the top of the weir. The crest of an overflow spillway on a large dam is often called a weir.
Weirs are overflow structure which are built across an open channel for the purpose of measuring or controlling the flow of liquids. Weirs have been commonly used to measure the flow water, but are now being adopted to measure the flow of other liquids.
Classification of weirs
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According to shape, weirs may be rectangular, triangular, trapezoidal, circular, parabolic or of any other regular form. The most commonly used shapes are the rectangular, triangular and the trapezoidal shapes. Weirs may be sharp-crested or broad-crested.
The flow over a weir may either be free or submerged. If the water surface downstream from the weir is lower than the crest, the flow is free, but the surface is higher than the crest, the flow is submerged.
Types
There are different types of weir. It may be a simple metal plate with a V-notch cut into it, or it may be a concrete and steel structure across the bed of a river. A weir which causes a large change of water level behind it, compared to the error inherent in the depth measurement method, will give an accurate indication of the flow rate.
Broad-crested weir A broad-crested weir is a flat-crested
structure, with a long crest compared to the flow thickness (Chanson 1999,2004, Henderson 1966, Sturm 2001). When the crest is “broad”, the streamlines become parallel to the crest invert and the pressure distribution above the crest is hydrostatic. The hydraulic characteristics of broad-crested weirs were studied during the 19th and 20th centuries. Practical experience showed that the weir overflow is affected by the upstream flow conditions and the weir geometry.
Sharp crested weir (fayoum weir)
A sharp-crested weir allows the water to fall cleanly away from the weir. Sharp crested weirs are typically 1/4" or thinner metal plates. Sharp crested weirs come in many different shapes such as rectangular, V-notch and Cipolletti weirs.
Combination weir
The sharp crested weirs can be considered into three groups according to the geometry of weir: a) the rectangular weir, b) the V or triangular notch and c) special notches, such as trapezoidal, circular or parabolic weirs. For accurate flow measurement over a wider range of flow rates, a combination weir combines a V-
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notch weir with a rectangular weir. An example is manufactured by Thel-Mar Company and has flow rates engraved along the side of the weir. This is typically used in pipes ranging from 4" to 15" in diameter.
V-notch weir
The V-notch weir is a triangular channel section, used to measure small discharge values. The upper edge of the section is always above the water level, and so the channel is always triangular simplifying calculation of the cross-sectional area. V-notch weirs are preferred for low discharges as the head above the weir crest is more sensitive to changes in flow compared to rectangular weirs.
Equation
For rectangular weir:
Q=23C √2 g L¿
It is a common practice to combine 23√2gh into a single coefficient Cw called
the weir factor. The general formula for a discharge through a rectangular weir considering velocity of approach then becomes,
Q=Cw L¿Where:
Q- dischargeCd- coefficient of dischargeCw- weir factor L- lengthH- total heighth- height
a. Viscous, Turbulent and Supersonic flow
Laminar and turbulent flow
If we were to take a pipe of free flowing water and inject a dye into the middle of the stream, what would we expect to happen?
This
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this
or this
Actually both would happen - but for different flow rates. The top occurs when the fluid is flowing fast and the lower when it is flowing slowly.
The top situation is known as turbulent flow and the lower as laminar flow.
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In laminar flow the motion of the particles of fluid is very orderly with all particles moving in straight lines parallel to the pipe walls. But what is fast or slow? And at what speed does the flow pattern change? And why might we want to know this?
The phenomenon was first investigated in the 1880s by Osbourne Reynolds in an experiment which has become a classic in fluid mechanics.
He used a tank arranged as above with a pipe taking water from the centre into which he injected a dye through a needle. After many experiments he saw that this expression
where r = density, u = mean velocity, d = diameter and m = viscosity
would help predict the change in flow type. If the value is less than about 2000 then flow is laminar, if greater than 4000 then turbulent and in between these then in the transition zone. This value is known as the Reynolds number, Re:
Laminar flow: Re < 2000
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Transitional flow: 2000 < Re < 4000
Turbulent flow: Re > 4000
What are the units of this Reynolds number? We can fill in the equation with SI units:
i.e. it has no units. A quantity that has no units is known as a non-dimensional (or dimensionless) quantity. Thus the Reynolds number, Re, is a non-dimensional number.
We can go through an example to discover at what velocity the flow in a pipe stops being laminar.
If the pipe and the fluid have the following properties:
water density r = 1000 kg/m3
pipe diameter d = 0.5m
(dynamic) viscosity, m = 0.55x103 Ns/m2
We want to know the maximum velocity when the Re is 2000.
If this were a pipe in a house central heating system, where the pipe diameter is typically 0.015m, the limiting velocity for laminar flow would be, 0.0733 m/s.
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Both of these are very slow. In practice it very rarely occurs in a piped water system - the velocities of flow are much greater. Laminar flow does occur in situations with fluids of greater viscosity - e.g. in bearing with oil as the lubricant.
At small values of Re above 2000 the flow exhibits small instabilities. At values of about 4000 we can say that the flow is truly turbulent. Over the past 100 years since this experiment, numerous more experiments have shown this phenomenon of limits of Re for many different Newtonian fluids - including gasses.
What does this abstract number mean?
We can say that the number has a physical meaning, by doing so it helps to understand some of the reasons for the changes from laminar to turbulent flow.
It can be interpreted that when the inertial forces dominate over the viscous forces (when the fluid is flowing faster and Re is larger) then the flow is turbulent. When the viscous forces are dominant (slow flow, low Re) they are sufficient enough to keep all the fluid particles in line, then the flow is laminar.
In summary:
Laminar flow
Re < 2000 'low' velocity Dye does not mix with water Fluid particles move in straight lines Simple mathematical analysis possible Rare in practice in water systems.
Transitional flow
2000 > Re < 4000 'medium' velocity Dye stream wavers in water - mixes slightly.
Turbulent flow
Re > 4000
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'high' velocity Dye mixes rapidly and completely Particle paths completely irregular Average motion is in the direction of the flow Cannot be seen by the naked eye Changes/fluctuations are very difficult to detect. Must use laser. Mathematical analysis very difficult - so experimental measures are used Most common type of flow.
Laminar FlowWhere the Reynolds number is less than 2300 laminar flow will occur and the
resistance to flow will be independent of the pipe wall roughness.
Turbulent flowTurbulent flow occurs when the Reynolds number exceeds 4000.
ViscousA type of fluid movement in which all particles of the fluid, flow in a straight
line parallel to the axis of a containing pipe or channel with little or no mixing or turbidity. The flow of a fluid through a duct under conditions such that the mean free path is small in comparison with the smallest, transverse section of the duct.
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. For example, high-viscosity magma will create a tall, steep stratovolcano, because it cannot flow far before it cools, while low-viscosity lava will create a wide, shallow-sloped shield volcano. Put simply, the less viscous the fluid is, the greater its ease of movement (fluidity). [1] All real fluids (except superfluids) have some resistance to stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or in viscid fluid.
The study of viscosity isknown as rheology.
Viscosity coefficients can be defined in two ways:
Dynamic viscosity, also absolute viscosity, the more usual one; Kinematic viscosity is the dynamic viscosity divided by the density.
υ =µρ
Turbulent flow
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In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Nobel Laureate Richard Feynman describes turbulence as "the most important unsolved problem of classical physics."[1] Flow that is not turbulent is called laminar flow. While there is no theorem relating Reynolds number to turbulence, flows with high Reynolds numbers usually become turbulent, while those with low Reynolds numbers usually remain laminar. For pipe flow, a Reynolds number above about 4000 will most likely correspond to turbulent flow, while a Reynold's number below 2100 indicates laminar flow. The region in between (2100 < Re < 4000) is called the transition region. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag due to boundary layer skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.
Supersonic flowFluid motion in which the Mach number M, defined as the speed of the fluid
relative to the sonic speed in the same medium, is more than unity. It is, however, common to call the flow transonic when 0.8 < M < 1.4, and hypersonic when M > 5
d. Device coefficientsCoefficient of discharge, Cd
Cd=actualdischarge
theoretical discharge
Coefficient of velocity, Cv
C v=actual velocity
theoretical velocity
Coefficient of contraction, Cc
C c=area of the stream∨ jet
ar eaof opening
e. Reynolds number
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Reynolds number is a dimensionless parameter equal to the ratio of the inertia forces to the friction forces.
Re=VDµk
=VDρµd
=VDɤµd g
Where:V= velocityD= diameterµk=kinematic viscosityµd= dynamic viscosityρ= densityɤ= sp. Weightg= acceleration due to gravity
For non-circular pipes, use D=4R, then he formula becomes;
Re=4VRρµd
=4VRµk
R=cross−sectional area of pipe , Apipe perimeter , P
f. Mach number
Mach number measures the elastic or compressibility forces as compared with inertia forces . For air flow higher than 0.3 Mach, the flow is considered compressible. It means that there is a noticeable change in density.
Machnumber=flow velocity (V )
speed of sound (V a)= V
√kRT
g. Equation of continuity of mass
p1 A1V 1=p2 A2V 2
h. Head losses in pipes
Head losses in pipes may be classified into two; the major head loss, which is caused by pipe friction along straight sections of pipe of uniform diameter and uniform roughness, and minor head loss, which are caused by changes in the velocity or directions of flow, and are commonly expressed in terms of kinetic energy.
1. Major Head Loss1.1.Darcy-Weisbach Formula (pipe friction equation)
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h f=fLV 2
D 2g= fL8Q2
π2g D5 =0.0826 fLQ 2
D5
Where:f= coefficient of friction or friction factorL= length of the pipeD= pipe diameterV= velocityQ= dichargehf= friction loss
Value of f:For laminar flow:
f=64Re
hf=32 µd LV
2
D 2g
For turbulent flow:
In smooth and rough pipes, where Vs=friction velocity
f=8Vs2
V 2
For smooth pipes, Re between 3,000 and 100,000
f=0.316
R e0.25
For smooth pipes with Re up to about 3,000,000
1
√ f=2 log (Re √ f )−0.80
I.2. Manning FormulaThe manning formula is one of the best-known open-channel formulas and is commonly used in pipes.
V=1nR2/3S1/2(SI units )
V=1.486n
R2/3S1/2(Englishunits)
h f=6.35n2 LV 2
D4 /3 (SI units)
Where:n= roughness coefficientR=hydraulic radius
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S= slope of energy grade line=hf/LV= velocity
Table 1.2. Values of n to be used with Manning formula
Nature of surface nmin max
Neat cement surface 0.010 0.013Wood-stave pipe 0.010 0.013Plank fumes, planed 0.010 0.014Vitrified sewer pipe 0.010 0.017Metal fumes, smooth 0.011 0.015Concrete, precast 0.011 0.013Cement mortar surfaces 0.011 0.015Plank fumes, unplaned 0.011 0.015Common-clay drainage tile 0.011 0.017Concrete, monolithic 0.012 0.016Brick with cement mortar 0.012 0.017Cast iron- new 0.013 0.017Cement rubble surfaces 0.017 0.030Riveted steel 0.017 0.020Corrugated metal pipe 0.021 0.025Canals and ditches, smooth earth 0.017 0.025Metal fumes, corrugated 0.022 0.030Canals:Dredged in earth, smooth 0.025 0.033In rock cuts, smooth 0.025 0.035Rough beds and weeds on sides 0.025 0.040Rough cuts, jagged and irregular 0.035 0.045Natural steams:smoothest 0.025 0.033roughest 0.045 0.060Very weedy 0.075 0.150
I.3. Hazen Williams FormulaV=1.318C1R
0.63S0.54 (Englishunits)V=0.849C1R
0.63S0.54(SI units)For circular pipes flowing in full,
Q=0.4322C1D2.63S0.54(Englishunits)
Q=0.278C15 D2.63S0.54(SI units)And,
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h f=10.67 LQ1.85
C11.85 D 4.87
Where:C1=Hazen William coefficientD= pipe diameterR= hydraulic radiusS= slope of the EGL
Table 1.3. value for C1for Hazen Williams Formula
Description of pipe C1
Extremely smooth and straight pipe 140New, smooth cast iron pipes 130Average cast iron pipes 110Vitrified sewer pipe 110cast iron pipes 100cast iron pipes 80New riveted steel 110Smooth wooden or wood stave 120
2. Minor Head Loss
2.1.Sudden Enlargement
hL=(V 1−V 2 )2
2 g,∈m
2.2.Gradual enlargement
hL=k (V 1−V 2 )2
2 g,∈m
2.3.Sudden Contraction
hL=kcV
2
2gV= velocity of the smaller pipeKc= coefficient of sudden contraction
D2/D1 0.0 0.1 0.2 0.3 0.4 0.5 06 .07 .08 0.9 0.1Kc 0.50 0.45 0.42 0.39 0.36 0.33 0.28 0.22 0.15 0.06 0.00
A special case of sudden contraction is the entrance loss for pipes connected to a reservoir; the values of Kc are as follows:
Flush connection……………………………...0.50Projecting connection……………………...0.10Rounded connection………………………..0.05
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Pipe projecting into reservoir…………..0.80Slightly rounded entrance………………..0.25Sharp-cornered entrance………………..0.05
i. Drag Force
The drag coefficient is a well known parameter used to characterize the drag force a body immersed in a fluid experiences due to relative motion between the body and the fluid. Before using data collected from a wind tunnel experiment for complex shapes, the data for the drag force on a sphere should be analyzed and compared to results published in authoritative references. Published results are most often expressed in terms of a plot or mathematical correlation between the drag coefficient and the Reynolds number.
Fd=Cd ɤ APV
2
2g=
Cd AP ρV2
2g
Where:Fd= drag forceCD is a dimensionless drag coefficient,A is the frontal area of the body exposed to the flow (πD2/4 for a sphere),gcis the gravitational constant which allows the left hand side to be expressed in units of force.V= velocity
j. Aerodynamic lift
To assist in determining the properties of the working fluid, air, several proven governing equations can be used, including the ideal gas law, Sutherland’s viscosity correlation, and Bernoulli’s equation. These relationships are valid for steady, incompressible, irrotational flow at nominal temperatures with negligible body forces. The ideal gas law can be used to relate the following
p=ρRT
where p is the pressure of the fluid, R is the universal gas constant (287 J/(kg K)), and T is the temperature of the gas. This expression establishes the relationship between the three properties of air that are of interest for use in this experiment.
Another parameter needed is the viscosity of the working fluid. Sutherland’s viscosity correlation is readily available for the testing conditions and can be expressed as
µ= bT 2
1+ ST
25
where b is equal to 1.458 x 10-6 (kg)/(m s K^(0.5)) and S is 110.4 K. Finally, Bernoulli’equation defines the total stagnation pressure as
po=p+ ρV 2
2
Using the previous governing equations, we can use the Reynolds number. The Reynolds Number is important because it allows the results obtained in this laboratory procedure to be scaled to larger scenarios. The Reynolds number can be expressed as
Re=ρVcµ
where c is a characteristic dimension of the body. For a cylinder, this dimension will be the diameter. As a result, the Reynolds number based on diameter is referenced as ReD.
Aerodynamic Coefficients
Three aerodynamic coefficients are used to explore the lift and drag forces on the test cylinder. First, the pressure coefficient expresses the difference in local pressure, the pressure at one discrete point on the cylinder, over the dynamic pressure.
CP=p−p1
( ρV2
2) 1͚
The theoretical value for Cp can be calculated as
C p=1−4 sin❑2(180−β)
The pressure coefficient can be used in the determination of the 2-D lift coefficient, Cl.
C l=cos (α )∫xc
xc
(Cplower−Cpupper )d ( xc )B. EXAMPLE PROBLEMS
1. Oil having sp. Gr. Of 0.869 and dynamic viscosity of 0.0814 Pa-sec flows through a cast iron pie at a velocity of 1 m/s. the pipe is 50 m long and 150 in diameter. Find the head lost due to friction.
Solution
Re=VDρµd
=(1 ) (0.15 ) (1000 )(0.869)
0.0814=1,601<2000 ,laminar
f=64Re
= 641,601
=0.04
h f=fLV 2
D 2g=
0.04 (50 )(1)2
(0.15 ) (2 )(9.81)=0.68m
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2. What commercial size of new cast iron pipe shall be used to carry 4,490 gpm with a lost of head of 10.56 feet per mile? Assume f= 0.019
Solution Q=4,490 gpm=0.284 m3/sL=1609.76 mhf=10.56ft= 3.22 m
Re=0.0826 fLQ2
D5
3.22=0.0826 (0.019 ) (1609.76 )(0.284)2
D5
D=576 mm
3. The accepted reynold’s for flow in a circular pipe is 2300. For flow through a 5-cm dimeter pipe at what velocity will this occur at 20®C for water flow? Solution µ= .001kg/(m.s)ρ=998kg/m3
Re=ρVdµ
= (998 ) (V )(.05)
.001= 2300 ; V = 0.046
4. During a test on a 2.4m suppressed weir 900 mm high, the head was maintained constant 1t 300 mm. In 38 seconds, 28,800 liters of water were collected. What is the weir factorCw?
Solution
Q=Cw L¿
Q= volumetime
(since t he flow is steady)
¿ 28,80038
=757.9 L/s
Velocity of approach, υa=QA
= 0.75792.4(1.2)
27
υa=0.26316
hυ=υa2
2g=¿¿
Q=Cw (2.4 )¿Cw=1.891
IV. EXPERIMENTS
A. Experiment 1. Flow meters
Calibrating A Pipe Flow Meter
Sketch of Experimental Set-Up
28
Commonly Used Empirical Equation
Photo of Our Set-Up
29
Weight, Time, and Head Experimental Data
Each Team Will Calibrate One Device
30
Photo of the Three Flow Meters
31
Photo: Select a Head Measurement Pair
Calculate Q From Weight & Time
Calculate Delta H & Area
32
Equations For Requested Results
33
B. Experiment 2. Open channel flow Measurement using Weirs
Objective The purpose of this lab is to become familiar with (and verify) equations used for
estimating flow rate over weirs. This will be done by comparing two different methods for estimating the flow rate over a weir in the lab.
Report
For each weir and for each sets of measurements for a different Q, present the data in a tabular form. Show the values measured during the laboratory experiments, namely Q (from the orifices in the open channel), flow depth upstream of the weir, head on the weir, weir height, Cd determined from the laboratory measurements, and Cd determined from the textbook. Also show all intermediate values needed in the computations.
See how well your experimental data fits with the empirical weir curve that was created from other researcher’s data. You need to figure out how best to graph and present this comparison. Do not forget to estimate the uncertainty.Include sample calculations for the first set of measurements for each weir.
Include sample calculations for the first set of measurements for each weir.
34
35
C. Experiment 3. Viscous, Turbulence, and Supersonic flow
Object
This experiment is intended as an introduction to the characteristics of fluid streams, with demonstrations of the effects of stream velocity and calculations of Reynolds number and Mach number.
Apparatus
Two separate pieces of equipment are suggested for this experiment, one using water flow, the other air. The first is a setup similar to that originally used by Osborne Reynolds for demonstrating the critical range between viscous and turbulent flow, shown in Fig. 152. The working section is a clear plastic or glass tube, about 1 ½ in. in diameter, and the dye emerges from a small glass nozzle (Fig. 153) and a compressed-air supply sufficient for steady flow through the nozzle at 50 psig or higher.
The nozzle is equipped with upstream and throat static taps and gages, and discharges to atmosphere. An upstream thermometer is provided, and a thermocouple is used for obtaining impact temperatures in jet.
36
Instructions
With the Reynolds demonstration apparatus, establish laminar flow; then increase the rate of flow until a definite disturbance of the dye stream is produced. Make a test at this flow rate and compute the Reynolds number. Establish pure turbulent flow, and then reduce the rate of flow slowly until the steady dye stream appears. Make a test at this flow rate and compute the Reynolds number. Repeat both this procedures at least twice and discuss the procedures.
Conclusion
Laminar flow can be determine if the value of the Reynolds number is less than 2000, and turbulent flow can be determine if the value of the Reynolds number is higher than 2000.
D. Experiment 4. Pipe Friction LossesObjective
To observe the head loss that occurs in a pipe due to frictional resistance, hydraulic gradient, and the relationship between head loss and the Reynold’s number.
Experimental Procedure
1. Level the apparatus on the bench so the manometer stand is vertical. (Assume it is level when we start the lab.)
2. Check to see what manometer is turned on (mercury or water?). The water or mercury manometer is introduced into the circuit by directing the lever on the tap towards the relevant connecting pipe. Select the water manometer on first.
3. Turn on the flow rate in your tub. There is a small knob on the apparatus used to adjust the flow rate. Open this supply valve to allow water to enter the apparatus.
4. Turn the supply valve off. The levels in the two limbs of the inverted U-tube should settle on the same value. If not, check that the flow from the tub has not stopped, or that there are no air bubbles in the system. If this does not work, open the bleed valves slightly to release pressure (please see your TA before you do this).
37
5. Fully open the needle valve to obtain the maximum differential head (approximately 400 mm).
6. Find the flow rate using the volume-time method by timing the collection of a suitable amount of water in a graduated cylinder.
7. Record the temperature of the water filling up the graduated cylinder.
8. Record the pressure heads.
9. Repeat steps 5, 6, and 8 while decreasing the difference in manometer readings by 50 mm down to 300 mm, then by 40 mm down to 180 mm, then by 20 mm down to 100 mm, and then by 10 mm down to 0 mm.
10. Switch to the mercury manometer. Increase the flow until a pressure difference of 10 mm is obtained. Again, measure flow rate and temperature.
11. Repeat step 10 (except only measure temperature for the first trial) increasing the difference by 10 mm up to 60 mm, then by 20 mm up to 200 mm, and then 30 mm up to the maximum difference possible.
CALCULATIONS
GIVEN:• 1 m3 = 1*106 ml• Pipe is made of brass• Length of pipe between piezometer tappings, dl = 524 mm• Nominal Diameter of the pipe, D = 0.003 m• Cross-sectional area of the pipe, A = 7.07 . 10-6 m2
Water Manometer:1. Calculate the Hydraulic Gradient, iH2O, for the water manometer
The hydraulic gradient is equal to the change in hydraulic head per unit length, and is usually a negative number as hydraulic head decreases in the direction of the flow. The hydraulic head is the sum of elevation and pressure that is measured by thee manometer tube, or in other words, the driving force of the fluid flow. The hydraulic gradient can be written as:
i=d hdl
=h1−h2.524m
2. Calculate the Hydraulic Gradient, iHg, for the mercury manometer
38
Mercury’s density is 13.6 times that of water, which must be taken into account when finding the hydraulic gradient.
i=d hdl
=Δh(13.6−1)
.524m
3. Solve for the flow rate, Q, found by using the volume-time method where:
Q=VT
V = Volume of water filled in the graduated cylinder, andT = The time it takes to fill the graduated cylinder to the volume
4. Using the continuity equation, calculate the velocity of the water through the apparatus.Q=AV
a. LAMINAR FLOW CALCULATIONS:In laminar flow, the fluid particles move in straight lines.
1. Calculate the coefficient of absolute viscosity, , from Poiseuille’s equation, using each value of i in the laminar region as indicated in your graph.
µ= i ρ gD 232V
where: i = hydraulic gradientρ= density (dependent on temperature so use the tables in the back of the book to find this)g= gravity = 9.81 m/s2D = inside diameter = 0.003 mV = velocity (from step 4)
2. Average the absolute viscosity values for the laminar region, avg
3. Calculate a Reynold’s number for each flow rate in the laminar region.
R= ρVDµ
4. Make sure that all Reynold’s numbers calculated are less than 2000 as this is the definition for laminar flow. If some of your flow rates are greater than 2000, then they are turbulent and your transition point is incorrectly place. Move your transition point, move the points that were greater than 2000 to your turbulent calculations, and recalculate avg and R for your laminar flow points that are remaining.
39
5. Knowing that the flow is laminar under pressure in a circular pipe, the friction factor can be solved for using the following equation:
f=64R
b. TURBULENT FLOW CALCULATIONS:In turbulent flow, the fluid particles follow random paths.
1. Determine the absolute viscosity, µ, of the turbulent flow region by interpolation using the values from Table 1.
Table 1: Absolute Viscosity ChartTemperature (deg C) µ*10-4 (Ns/m2)
0 17.9010 13.1020 10.1030 8.0040 6.56
2. Calculate a Reynold’s number for each flow rate in the turbulent region.
R=ρVDµ
3. Make sure that all Reynold’s numbers calculated are greater than 2000, as this is the definition for turbulent flow. If some of your Reynold’s numbers are less than 2000, then they are laminar and your transition point is incorrectly placed. Move your transition point, and move the points that had R < 2000 to your laminar calculations.
4. Use Darcy’s equation to calculate the friction factor at each flow rate in the turbulent flow region.
f= i g D2
V 2COMPARE AND CONTRAST:Compare and contrast experimental with theoretical values.
a. Theoretical and experimental slopes- Theoretical Laminar = 1.0- Theoretical Turbulent = 1.85
b. Reynold’s Number, R, at the transition point- Experimental versus 2000
c. Friction factor, f, at the transition point- Experimental versus Moody Diagram at R = 2000
d. Absolute viscosity, - Experimental versus lab handout interpolation
40
EXAMPLE TABLES:
Table 1: Water Friction Loss Data SheetVolume
(ml)Time
(s)Temp. h1 h2 dh Q V I R f u
110170220240300350400450500500550600
22.8322.0722.4321.3623.0824.8323.9923.5523.9922.1322.4023.02
242424242424242424242424
0.1920.1990.2100.2200.2300.2400.2620.2800.3000.3250.3430.358
0.1830.1780.1700.1600.1500.1420.1230.1080.0920.0700.0520.040
0.1130.260.5040.7561.0081.2351.7512.1672.6213.2133.6674.007
4.82E067.70E069.81E061.12E051.30E051.41E051.67E051.91E052.08E052.26E052.46E052.61E05
0.6821.0891.3871.5891.8391.9942.3582.7032.9483.1963.4733.687
0.2160.5050.9621.4431.9242.3563.3424.1365.0026.1326.9977.647
2252.353600.774585.035252.416076.236589.317794.328932.449742.9010561.7811477.9212184.13
0.0280.0180.0140.0120.0110.0100.0080.0070.0070.0060.0060.005
9.05E049.05E049.05E049.05E049.05E049.05E049.05E049.05E049.05E049.05E049.05E049.05E04
Uavg of Laminar Flow 9.34E-04 (Ns/m2)
Table 2: Mercury Friction Loss Data Sheet
Volume(ml)
Time(s)
Temp. h1 (m)
h2(m)
dh(m)
Q(m3/s)
V(m/s)
I R f u(Ns/m2)
41
200 200 150 1501501001001004030
21.1324.46 22.62 24.9728.0921.8626.4435.6823.6525.37
24242424242424242424
0.5080.4250.3900.3780.3680.3500.3450.3330.3200.315
0.050
0.045
0.190
0.205
0.220
0.233
0.250
0.263
0.278
0.285
0.4580.3800.2000.1730.1480.1170.0950.0700.0420.030
9.47E06
8.18E06
6.63E06
6.01E06
5.34E06
4.57E06
3.78E06
2.80E06
1.69E06
1.18E06
1.3391.1570.9380.8500.7550.6470.5350.3960.2390.167
0.8740.7250.3820.3300.2820.2230.1810.1340.0800.057
4424.653822.283099.902808.162496.252071.291712.491269.01765.81535.42
0.0140.0170.0210.0230.0260.0310.0370.0500.0840.120
9.05E049.05E049.05E049.05E049.05E049.49E049.32E049.27E049.22E049.42E04
Table 3: Final Results Table
42
Property Experimental Theoritical % DifferenceR 2071.29 2000 3.56f 0.031 0.030 3.33uavg 9.34E-04 9.05E-04 3.20
EXAMPLE GRAPH:
E. Experiment 5. Characteristics of Nozzles and Jets
Objective
To demonstrate the effect of compressibility on the flow equations for a convergent flow.
Apparatus
Armfield Compressible Flow Bench, convergent-divergent duct, two inclined tube manometers, mercury manometer.
43
Figure 1 : The compressible Flow Bench
Procedure
1. An inclined tube manometer to read P0-P1 using the 12.7mm range is connected.
2. Another inclined tube manometer to read P0-P2 using the 25.4mm range is connected.
3. The flow to give approximately equal increments of (P0-P1) is adjusted.
44
4. The readings of both manometers are read for each flow rate.
5. Using the 50.8mm range of an inclined tube manometer and with mercury manometer to measure P0-P1, the steps were repeated.
Results and Calculation
At 1000rpm,
Conversion of unit : mmH2O to kPa
Theoretical value:P0-P2 = (A1/A2)2 (P0-P1)
= (40.733) (0.196kPa)= 7.984 kPa
Vin = √ (2(P0-P1)/ P0) ; P0 = 101.325kPa= √ (2(0.196)/ 101.325)
= 0.0622 m/sVout = √ (2(P0-P2)/ P0)
= √ (2(0)/ 101.325)= 0 m/s
45
(P0-P1)12.7mm
kPa(P0-P2)25.4mm
Theoreticalvalue
(P0-P2)kPa
Vin(m/s)
Vout(m/s)
mmH2O kPa
20 0.196 0.000 7.984 0.0623 0.0000
40 0.392 0.010 15.967 0.0879 0.0140
60 0.588 0.020 23.951 0.1077 0.0199
80 0.784 0.025 31.935 0.1244 0.0222
100 0.980 0.040 39.919 0.1391 0.0281
20 1.176 0.045 47.902 0.1524 0.0298
140 1.372 0.050 55.886 0.1646 0.0314
160 1.568 0.060 63.869 0.1759 0.0344
180 1.764 0.070 71.854 0.1866 0.0372
200 1.960 0.080 79.837 0.1967 0.0397
1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
(P0-P2)(P0-P1)
po-p1
po-p
2
Graph 1
Dicussion
The experiment used inclined manometer test set to measure pressure. From the graph 1, we can see as the different in pressure at convergent duct increase, the pressure of air flow also increase. This is due to the increasing of velocity after passing through the throat although the flow area increases rapidly in the region. When the fluid density decrease, the velocity passing the throat also increase.
A plot of pressure distribution along the nozzle provides a good way of summarizing its behavior. To understand how the pressure behaves there are a few basic rules to remember :
-When the flow accelerates (sub or supersonically) the pressure drops-The pressure rises instantaneously across a shock
-The pressure throughout the jet is always the same as the ambient (i.e. the back pressure) unless the jet is supersonic and there are shocks or expansion waves in the jet to produce pressure differences.-The pressure falls across an expansion wave. Often viscous effects are not important in compressible flows, since the boundary layers are very thin. Flows in the nozzle are easily controlled by varying the backpressure.
From the graph we can see that it is different from the theoretical values that were calculated. This might be due to some misconduct or the condition of the instrument. Apart from that it can also be due to the readings that we have done (parallax) or even the connections between the pipes are not well connected.
46
Conclusion
From this experiment, we were able to demonstrate the effect of compressibility on the flow equations for a convergent flow and how it varied at different points for example from the plot of pressure difference, we could understand how the pressure behaves in a nozzle.
F. Experiment 6. Drag coefficient-Determination of the Drag Coefficient of a Sphere
Abstract
The drag force on a sphere in an air stream was measured at various free stream velocities below 100 ft/sec. This was done in a low speed wind tunnel using an integral balance system to measure the drag force and a pitot tube and venturi meter to measure the velocity.
The raw data were processed according to classical equations of fluid mechanics which define the Reynolds number and drag coefficient. An expression for the drag coefficient in terms of the Reynolds number was developed using a least squares curve fit to the experimental data.
The experimental results are compared to published results over the range tested.
Description of Experimental Setup
Figure1 Low speed wind tunnel
A manually controlled variable speed wind tunnel similar to that shown in Figure1 Low speed wind tunnel was used in this experiment. The wind tunnel was equipped with an integral force balance which measured both drag and lift forces and a multistation manometer tube bank to measure the velocity of the air stream. A separate pitot tube was used to verify the calibration of the built-in manometer. A mercury barometer was used to measure the atmospheric pressure and a thermometer was used to measure the air temperature
47
List of Equipment Used
1. Flotek 250 wind tunnel located in the Mechanical Engineering Laboratory (S/N FT250-2784)2. 2.5-inch diameter smooth calibration sphere wind tunnel accessory3. Pitot tube and differential manometer (Property tag BSW365-22984)4. Mercury barometer fixed to the wall near the wind tunnel.
5. Mercury thermometer (Sargent brand, no tag or serial number)
Procedure
Step 1: The atmospheric pressure and temperature were recorded at the beginning of data collection.
Step 2: The venturi meter velocity gage built in to the wind tunnel was calibrated by inserting the pitot tube at the center of the wind tunnel test section and varying the fan speed so as to produce 0.05-inch changes in the differential manometer attached to the pitot tube. At low speeds (less than 25 miles per hour) the built-in venturi meter was below the first scale reading, thus the pitot tube was used to adjust the fan speed while taking drag measurements. At air speeds greater than 25 miles per hour, the venturi meter was very reliable, so it was used to adjust the fan speed setting during drag force measurements. A calibration table was made that correlated venturi air speed readings with desired pitot tube differential pressure values.
Step 3: The fan speed was set to produce a pitot differential pressure of 0.025 inches.
Step 4: The drag force of the specimen mounting stand was measured using the force balance and recorded on the data sheet. The 2.5-inch spherical test specimen was then mounted and the drag force was measured and recorded on the data sheet.
Step 5: The test specimen was removed from the test stand and the fan was shut off and the drag force indicator was checked to make sure it read zero.
Step 6: The fan was restarted and its speed was adjusted so as to produce a pitot differential pressure of 0.05 inches, then steps 4 and 5 were repeated. This process was continued, increasing the pitot differential pressure by 0.05 inches each run until the differential pressure reached 0.35 inches. At this point, the fan speed was adjusted by referring to the air speed calibration table that was made earlier. Measurements were made up to the maximum free stream air speed capability of the wind tunnel, which was 52 miles per hour (1.50 inches of pitot tube differential pressure).
Step 7: The entire data collection process was repeated in reverse, i.e., starting with the fan running at maximum speed, and lowering the speed by to match those used previously for each drag force reading.
48
Step 8: The atmospheric pressure and temperature were recorded at the conclusion of the last measurements.
Data
Temperature at start of experiment: 77° F
Barometric pressure at start of experiment: 29.80 inches of mercury
Pitot Tube Differential Pressure-Δh
(inches)
Venturi Meter
Reading
(miles/hour)
Going Up Coming Down
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
0.025 N/A 0.00 0.010 0.00 0.007
0.05 N/A 0.002 0.010 0.002 0.010
0.10 N/A 0.003 0.015 0.003 0.017
0.15 N/A 0.004 0.020 0.004 0.020
0.20 N/A 0.005 0.020 0.005 0.030
0.25 N/A 0.007 0.030 0.007 0.037
0.30 N/A 0.009 0.035 0.009 0.040
0.35 25.0 0.011 0.045 0.011 0.045
0.40 26.8 0.013 0.050 0.013 0.050
0.45 29.0 0.015 0.060 0.015 0.060
0.50 30.6 0.018 0.070 0.018 0.070
0.55 31.2 0.021 0.075 0.021 0.075
0.60 33.6 0.024 0.080 0.024 0.080
0.65 34.8 0.027 0.080 0.027 0.085
0.70 35.2 0.030 0.085 0.030 0.085
0.75 36.5 0.033 0.090 0.033 0.090
0.80 37.5 0.036 0.095 0.036 0.100
0.90 40.0 0.039 0.105 0.039 0.105
1.00 43.0 0.042 0.120 0.042 0.120
1.10 45.2 0.045 0.130 0.045 0.135
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Pitot Tube Differential Pressure-Δh
(inches)
Venturi Meter
Reading
(miles/hour)
Going Up Coming Down
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
1.20 47.8 0.050 0.145 0.050 0.145
1.30 49.0 0.055 0.150 0.055 0.150
1.40 50.6 0.060 0.150 0.060 0.160
1.50 52.5 0.065 0.165 0.065 0.160
Temperature at end of experiment: 77° F
Barometric pressure at end of experiment: 29.80 inches of mercury
Analysis of Data
1. List of Variables:FD - drag force in lbf
CD - drag coefficientRe - Reynolds numberD - diameter of sphere in inchesρ - density of air in lbm/ft3
u∞ - velocity of air stream in ft/sec2
P - atmospheric pressure in lbf/ft2
Δp - pressure difference in manometer in lbf/ft2
Δh - difference in heights of liquid in manometer in inchesT - atmospheric temperature in °Rμ - viscosity of air in lbm/ft-hrρo - density of oil in manometer in lbm/ft3
2. Calculated ResultsThe following values were used to compute the values in the data reduction equations:
T=77 ° F=537 °R, μ=0.0444
lbm
hr⋅ft , P=29 . 802 in⋅Hg=2105
lbf
ft2. Applying equation Error: Reference
source not found, the density is calculated as 0 . 07349
lbm
ft3. The calculated values of free stream air
velocity (u∞), Reynolds number (Re), and drag coefficient (CD) are given in the table below.
50
Pitot Tube Differential Pressure-Δh
(inches)
Free Stream Air Velocity- u∞
(ft/sec)
Reynolds Number-Re
Drag Coefficient-CD
0.025 9.70 12,302 2.2230.050 13.72 17,398 1.0460.100 19.40 24,604 0.85010.150 23.76 30,134 0.69750.200 27.43 34,795 0.65390.250 30.67 38,902 0.69320.300 33.60 42,616 0.62120.350 36.29 46,030 0.63520.400 38.80 49,209 0.60490.450 41.15 52,193 0.65390.500 43.38 55,017 0.68010.550 45.49 57,702 0.64210.600 47.52 60,268 0.61030.650 49.46 62,729 0.55840.700 51.32 65,097 0.51380.750 53.13 67,382 0.49700.800 54.87 69,592 0.50270.900 58.20 73,813 0.47961.000 61.34 77,806 0.51011.100 64.34 81,603 0.52021.200 67.20 85,232 0.51771.300 69.94 88,712 0.47791.400 72.58 92,061 0.44371.500 75.13 95,292 0.4251
The calculated values for drag coefficient versus Reynolds number are plotted on the next page. A least-squares best fit logarithmic equation for the experimental data was found to be
CD=407 .87 Re−0.5978 .
3. Uncertainty AnalysisThe uncertainty associated with each of the measured variables is given in the table below. These
values were chosen based on the stated accuracy of the instrument, if available. Otherwise, they are reasonable estimates based on values typically reported.
Uncertainty Description Symbol Numerical Value
Pitot tube differential pressure
U Δh 0.05 inch
Drag force U FD0.005 lbf
Sphere diameter U D 0.01 inch
Atmospheric pressure U P 0.005 inch Hg = 0.353 lbf/in2
51
Temperature UT 1°F
Manometer oil density U ρo
3.12 lbm/ft3
Viscosity of air U μ 6 .4×10−5lbm/hr-ft
The detailed calculations of the uncertainty associated with the calculated variables (in accordance with Reference 1) are given in the Appendix. The table below summarizes the results of these calculations.
Uncertainty Description Symbol Numerical Value
Air density U ρ 1 .883×10−4 lbm/ft3
Manometer pressure difference
U ΔP 1 .774×10−3 psi
Free stream air velocity Uu∞2.52 ft/sec
Reynolds number URe 3181
Drag coefficient UC D0.131
10000 1000000.1
1
10
Drag Coefficient vs. Reynolds Number for a Sphere
Experimental Data
Reynolds Number (Re)
Dra
g C
oe
ffic
ien
t (C
D)
Figure 2 Plot of Drag Coefficient versus Reynolds Number
52
A. Discussion of Results
The results of this experiment are best depicted in Figure 2. Over the air speed range tested, the drag coefficient generally decreases as the Reynolds number increases. Similar results are reported in Reference 2 (shown in Figure 3). Owing to the limitations of the low speed wind tunnel used in this
experiment, comparison of results is possible over only a single decade (105≤Re≤106 ). The results calculated using the present experimental data compare very favorably at the upper end of this region, where both curves show Re≈0 . 4 . At the lower end of the region, the present experimental data yielded values of CD near unity, while those reported in Reference 2 remained considerably less than unity.
An examination of the uncertainty in the values of the drag coefficient provides some interesting information. At the lowest velocity the uncertainty calculation yields a probable error of 2.625, which exceeds the calculated value. At the highest velocity the probable error is 0.044, or only approximately 10%. This is so because the uncertainty in the velocity changes relatively little even though the velocity itself increases greatly. The principal factor contributing to this, and therefore to the greater reliability of the values of drag coefficient at higher velocities was the uncertainty in the reading of the manometer tube, which did not vary with velocity. Therefore, the uncertainty due to this factor represented a greater percentage of the velocity and thus the drag coefficient at low velocities that at the higher velocities. For the same reasons the uncertainties in the higher values of the Reynolds number would be less on a percentage basis that those at lower values.
0.01
0.1
1
10
100
1000
Drag Coefficient vs. Reynolds Number for a Sphere
Reynolds Number (Re)
Dra
g C
oe
ffic
ien
t (C
D)
Figure 3 Experimental Values of Drag Coefficient vs. Reynolds Number for a Sphere (from Reference 2)
53
Conclusions
The results of this experiment show that the drag coefficient for a sphere can be calculated reasonably accurately using a low speed wind tunnel. The results obtained agree with other published results at the higher range of velocities used. More accurate results at lower velocities would probably require a more sensitive force balance and air speed indicator.
54
REFERENCES
55
56
1
2