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Applications of energy equation

Flow Measurement Flow measurement is the quantification of bulk fluid movement.

Flow can be measured in a variety of ways.

Positive-displacement flow meters accumulate a fixed volume of fluid and then

count the number of times the volume is filled to measure flow.

Other flow measurement methods rely on forces produced by the flowing

stream as it overcomes a known constriction, to indirectly calculate flow.

Flow may be measured by measuring the velocity of fluid over a known area.

Units of measurement of flow:

When gases or liquids are transferred for their energy content, the flow rate may also

be expressed in terms of energy flow, such as J/hour or BTU/day.

The energy flow rate is the volumetric flow rate multiplied by the energy content

per unit volume,

or mass flow rate multiplied by the energy content per unit mass.

In engineering contexts, the volumetric flow rate is usually given the symbol , and

the mass flow rate, the symbol . (m dot).

For a fluid having density , mass and volumetric flow rates may be related by

Types of fluid flowmeters

Orifices, Venturies, Nozzles, Rotameters, Pitot Tubes, Calorimetrics, Turbine, Vortex, Electromagnetic, Doppler, Ultrasonic, Thermal, Coriolis are all flow meters. The most common principals for fluid flow metering are:

Differential Pressure Flowmeters Velocity Flowmeters Positive Displacement Flowmeters Mass Flowmeters Open Channel Flowmeters

Differential Pressure Flowmeters In a differential pressure drop device the flow is calculated by measuring the pressure drop over an obstructions inserted in the flow. The differential pressure flowmeter is based on the Bernoullis Equation, where the pressure drop and the further measured signal is a function of the square flow speed.

The most common types of differential pressure flowmeters are:

1. Orifice Plates 2. Flow Nozzles 3. Venturi Tubes 4. Variable Area - Rotameters

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1. Orifice Plate With an orifice plate, the fluid flow is measured through the difference in pressure from the upstream side to the downstream side of a partially obstructed pipe. The plate obstructing the flow offers a precisely measured obstruction that narrows the pipe and forces the flowing fluid to constrict.

The orifice plates are simple, cheap and can be delivered for almost any application in any material. The TurnDown Rate for orifice plates are less than 5:1.

Example - Turndown Ratio for an Orifice Meter

The turndown ratio - TR - for an orifice meter with maximum flow of 12 kg/s and a minimum flow of 3 kg/s can be calculated as: TR = (12 kg/s) / (3 kg/s) = 4 normally expressed as turndown ratio of 4:1

Their accuracy are poor at low flow rates. A high accuracy depend on an orifice plate in good shape, with a sharp edge to the upstream side. Wear reduces the accuracy.

2. Venturi Tube Due to simplicity and dependability, the Venturi tube flowmeter is often used in applications where it's necessary with higher TurnDown Rates, or lower pressure drops, than the orifice plate can provide. In the Venturi Tube the fluid flowrate is measured by reducing the cross sectional flow area in the flow path, generating a pressure difference. After the constricted area, the fluid is passes through a pressure recovery exit section, where up to 80% of the differential pressure generated at the constricted area, is recovered.

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With proper instrumentation and flow calibrating, the Venturi Tube flowrate can be reduced to about 10% of its full scale range with proper accuracy. This provides a TurnDown Rate 10:1.

3. Flow Nozzles Flow nozzles are often used as measuring elements for air and gas flow in industrial applications.

The flow nozzle is relative simple and cheap, and available for many applications in many materials. The TurnDown Rate and accuracy can be compared with the orifice plate.

The Sonic Nozzle - Critical (Choked) Flow Nozzle When a gas accelerate through a nozzle, the velocity increase and the pressure and the gas density decrease. The maximum velocity is achieved at the throat, the minimum area, where it breaks Mach 1 or sonic. At this point it's not possible to increase the flow by lowering the downstream pressure. The flow is choked. This situation is used in many control systems to maintain fixed, accurate, repeatable gas flow rates unaffected by the downstream pressure.

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Recovery of Pressure Drop in Orifices, Nozzles and Venturi Meters After the pressure difference has been generated in the differential pressure flow meter, the fluid pass through the pressure recovery exit section, where the differential pressure generated at the constricted area is partly recovered

As we can see, the pressure drop in orifice plates are significant higher than in the venturi tubes.

4.Variable Area Flowmeter or Rotameter

The rotameter consists of a vertically oriented glass (or plastic) tube with a larger end at the top, and a metering float which is free to move within the tube. Fluid flow causes the float to rise in the tube as the upward pressure differential and buoyancy of the fluid overcome the effect of gravity.

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The float rises until the annular area between the float and tube increases sufficiently to allow a state of dynamic equilibrium between the upward differential pressure and buoyancy factors, and downward gravity factors. The height of the float is an indication of the flow rate. The tube can be calibrated and graduated in appropriate flow units. The rotameter meter typically have a TurnDown Ratio up to 12:1. The accuracy may be as good as 1% of full scale rating. Magnetic floats can be used for alarm and signal transmission functions.

Velocity Flowmeters In a velocity flowmeter the flow is calculated by measuring the speed in one or more points in the flow, and integrating the flow speed over the flow area.

1. Pitot Tubes

The pitot tube are one the most used (and cheapest) ways to measure fluid flow, especially in air applications as ventilation and HVAC systems, even used in airplanes for the speed measurent.

The pitot tube measures the fluid flow velocity by converting the kinetic energy of the flow into potential energy. The use of the pitot tube is restricted to point measuring. With the "annubar", or multi-orifice pitot probe, the dynamic pressure can be measured across the velocity profile, and the annubar obtains an averaging effect.

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2. Calorimetric Flowmeter

The calorimetric principle for fluid flow measurement is based on two temperature sensors in close contact with the fluid but thermal insulated from each other.

One of the two sensors is constantly heated and the cooling effect of the flowing fluid is used to monitor the flowrate. In a stationary (no flow) fluid condition there is a constant temperature difference between the two temperature sensors. When the fluid flow increases, heat energy is drawn from the heated sensor and the temperature difference between the sensors are reduced. The reduction is proportional to the flow rate of the fluid. Response times will vary due the thermal conductivity of the fluid. In general lower thermal conductivity require higher velocity for proper measurement. The calorimetric flowmeter can achieve relatively high accuracy at low flow rates. Turbine Flowmeter There is many different manufacturing design of turbine flow meters, but in general they are all based on the same simple principle: If a fluid moves through a pipe and acts on the vanes of a turbine, the turbine will start to spin and rotate. The rate of spin is measured to calculate the flow. The turndown ratios may be more than 100:1 if the turbine meter is calibrated for a single fluid and used at constant conditions. Accuracy may be better than +/-0,1%.

3. Vortex Flow Meter An obstruction in a fluid flow creates vortices in a downstream flow. Every obstruction has a critical fluid flow speed at which vortex shedding occurs. Vortex shedding is the instance where alternating low pressure zones are generated in the downstream.

These alternating low pressure zones cause the obstruction to move towards the low pressure zone. With sensors gauging the vortices the strength of the flow can be measured.

4. Electromagnetic Flowmeter An electromagnetic flowmeter operate on Faraday's law of electromagnetic induction that states that a voltage will be induced when a conductor moves through a

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magnetic field. The liquid serves as the conductor and the magnetic field is created by energized coils outside the flow tube. The voltage produced is directly proportional to the flow rate. Two electrodes mounted in the pipe wall detect the voltage which is measured by a secondary element. Electromagnetic flowmeters can measure difficult and corrosive liquids and slurries, and they can measure flow in both directions with equal accuracy. Electromagnetic flowmeters have a relatively high power consumption and can only be used for electrical conductive fluids as water.

5. Ultrasonic Doppler Flowmeter

The effect of motion of a sound source and its effect on the frequency of the sound was observed and described by Christian Johann Doppler. The frequency of the reflected signal is modified by the velocity and direction of the fluid flow If a fluid is moving towards a transducer, the frequency of the returning signal will increase. As fluid moves away from a transducer, the frequency of the returning signal decrease. The frequency difference is equal to the reflected frequency minus the originating frequency and can be use to calculate the fluid flow speed.

Positive Displacement Flowmeter

The positive displacement flowmeter measures process fluid flow by precision-fitted rotors as flow measuring elements. Known and fixed volumes are displaced between the rotors. The rotation of the rotors are proportional to the volume of the fluid being displaced. The number of rotations of the rotor is counted by an integral electronic pulse transmitter and converted to volume and flow rate. The positive displacement rotor construction can be done in several ways:

Mass Flowmeters

Mass meters measure the mass flow rate directly.

1. Thermal Flowmeter The thermal mass flowmeter operates independent of density, pressure, and viscosity. Thermal meters use a heated sensing element isolated from the fluid flow path where the flow stream conducts heat from the sensing element. The conducted heat is directly proportional to the mass flow rate and the he temperature difference is calculated to mass flow. The accuracy of the thermal mass flow device depends on the calibrations reliability of the actual process and variations in the temperature, pressure, flow rate, heat capacity and viscosity of the fluid.

2. Coriolis Flowmeter Direct mass measurement sets Coriolis flowmeters apart from other technologies. Mass measurement is not sensitive to changes in pressure, temperature, viscosity

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and density. With the ability to measure liquids, slurries and gases, Coriolis flowmeters are universal meters. Coriolis Mass Flowmeter uses the Coriolis effect to measure the amount of mass moving through the element. The fluid to be measured runs through a U-shaped tube that is caused to vibrate in an angular harmonic oscillation. Due to the Coriolis forces, the tubes will deform and an additional vibration component will be added to the oscillation. This additional component causes a phase shift on some places of the tubes which can be measured with sensors. The Coriolis flow meters are in general very accurate, better than +/-0,1% with an turndown rate more than 100:1. The Coriolis meter can also be used to measure the fluids density.

Open Channel Flowmeters

A common method of measuring flow through an open channel is to measure the height of the liquid as it passes over an obstruction as a flume or weir in the channel.

Common used is the Sharp-Crested Weir, the V-Notch Weir, the Cipolletti weir, the Rectangular-Notch Weir, the Parshall Flume or Venturi Flume.

The Pitot Static Tube.

A Pitot-static tube can measure the fluid flow velocity by converting the kinetic energy in the fluid flow into potential energy.

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Pitot-Static tubes, which are also called Prandtl tubes, are used on aircraft as speedometers. The actual tube on the aircraft is around 10 inches (25 centimeters) long with a 1/2 inch (1 centimeter) diameter. Several small holes are drilled around the outside of the tube and a center hole is drilled down the axis of the tube. The outside holes are connected to one side of a device called a pressure transducer. The center hole in the tube is kept separate from the outside holes and is connected to the other side of the transducer. The transducer measures the difference in pressure in the two groups of tubes by measuring the strain in a thin element using an electronic strain gauge. The pitot-static tube is mounted on the aircraft, or in a wind tunnel , so that the center tube is always pointed in the direction of the flow and the outside holes are perpendicular to the center tube. On some airplanes the pitot-static tube is put on a longer boom sticking out of the nose of the plane or the wing.

Difference in Static and Total Pressure

Since the outside holes are perpendicular to the direction of flow, these tubes are pressurized by the local random component of the air velocity. The pressure in these tubes is the static pressure (ps) discussed in Bernoulli's equation. The center tube, however, is pointed in the direction of travel and is pressurized by both the random and the ordered air velocity. The pressure in this tube is the total pressure (pt) discussed in Bernoulli's equation. The pressure transducer measures the difference in total and static pressure which is the dynamic pressure q.

measurement = q = pt - ps

Solve for Velocity

With the difference in pressures measured and knowing the local value of air density from pressure and temperature measurements, we can use Bernoulli's equation to give us the velocity.

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Second Course Fluid Mechanics2nd year Refrigeration and AC department Kirkuk Technical College

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Moody Chart Moody’s Diagram (Reynold’s Number vs. Relative Roughness)

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Pressure losses due to Friction in Pipes and Darcy weisbach Equation

The Darcy-Weisbach equation with the Moody diagram are considered to be the most accurate model for estimating frictional head loss in steady pipe flow. Since the approach requires a not so efficient trial and error iteration an alternative empirical head loss calculation like the Hazen-Williams equation may be preferred:

f = 0.2083 (100 / c)1.852 q1.852 / dh4.8655 (1)

where

f = friction head loss in feet of water per 100 feet of pipe (fth20/100 ft pipe)

c = Hazen-Williams roughness constant From tables

q = volume flow (gal/min)

dh = inside hydraulic diameter (inches)

The hydraulic diameter is also used to calculate the pressure loss in a ducts or pipe. The hydraulic diameter is not the same as the geometrical diameter in a non-circular duct or pipe and can be calculated with the generic equation dh = 4 A / p where dh = hydraulic diameter (m, ft) A = area section of the duct (m2, ft2) p = wetted perimeter of the duct (m, ft)

In fluid dynamics, the Darcy–Weisbach equation is a phenomenological equation,

which relates the head loss — or pressure loss — due to friction along a given length

of pipe to the average velocity of the fluid flow. The equation is named after Henry

Darcy and Julius Weisbach.

The Darcy–Weisbach equation contains a dimensionless friction factor, known as

the Darcy friction factor. This is also called the Darcy–Weisbach friction

factor or Moody friction factor. The Darcy friction factor is four times the Fanning

friction factor, with which it should not be confused

Head loss can be calculated with

where

hf is the head loss due to friction (SI units: m);

L is the length of the pipe (m);

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D is the hydraulic diameter of the pipe (for a pipe of circular section, this

equals the internal diameter of the pipe) (m);

V is the average velocity of the fluid flow, equal to the volumetric flow rate per

unit cross-sectional wetted area (m/s);

g is the local acceleration due to gravity (m/s2);

fD is a dimensionless coefficient called the Darcy friction factor. It can be found

from a Moody diagram (page 13), or more precisely by solving the Modified

Colebrook equation. Do not confuse this with the Fanning Friction factor, f.