Measurement of Flow

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Measurement of Flow

• P M V Subbarao
• Professor
• Mechanical Engineering Department

An Essential Requirement in CV Based Industrial
Appliances.
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Mathematics of Flow Rate

• The Scalar Product of two vectors, namely
  velocity and area..

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Important characteristics of the dot product

• The first point to note about the definition is
  that the coordinate system does not enter the
  definition.
• The second point to note is that because
  cosqcos(-q), the order is not important, that
  is, the scalar product is commutative.
• The scalar product is also distributive.
• All these qualities help in development of a
  single instrument to measure the scalar product.

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Types of Flow Measurement Technologies

• Variable Area (rotameters)
• Rotating Vane (paddle turbine)
• Positive Displacement
• Differential Pressure
• Vortex Shedding
• Thermal Dispersion
• Magnetic Magnetic
• Thermal Mass
• Coriolis Mass
• Ultrasonic

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Some Facts About Variable Area Flowmeters

• Called float type float type, rotameter, or
  variable area flowmeters.
• By far the most common specified, purchased, and
  installed flowmeter in the world

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Variable Area Flowmeters

• Fluid flow moves the float upward against
  gravity.
• Float will find equilibrium when area around
  float generates enough drag equal to weight -
  buoyancy.
• Some types have a guide rod to keep float stable.
• Low Cost (pricing usually starts lt 50)
• Simple Reliable Design
• Can Measure Liquid or Gas Flows
• Tolerates Dirty Liquids or Solids in Liquid

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Measuring Principles of Variable Area Flowmeters

• Flow Rate Analysis.
• The forces acting on the bob lead to equilibrium
  between
• the weight of the bob rbgVb acting downwards
• the buoyancy force rgVb and
• the drag force Fd acting upwards.

• Where Vb is the volume and
• rb is the density of the bob,
• r is the density of the fluid, and
• g is the gravitational acceleration

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• The drag force results from the flow field
  surrounding the bob and particularly from the
  wake of the bob.
• In flow analyses based on similarity principles,
  these influences are accounted for by empirical
  coefficient CL or CT in the drag law for

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The volume flow rate through the rotameter is
Where m is the open area ratio, defined as
And D is the tube diameter at the height of the
bob.
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for laminar flow
where the parameter a is defined in terms of a
constant K Vb/D3b characteristic of the shape of
the bob
for turbulent flow
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With either laminar or turbulent flow through the
rotameter, the flow rate is proportional to m.
If the cross-sectional area of the tube is made
to increase linearly with length, i.e.,
then since the cone angle f of the tube is small,
and the flow rate is directly proportional to the
height h of the bob.
h
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Similarity Analysis.

• The basic scaling parameter for flow is the
  Reynolds number, defined as

• where UIN is the velocity at the rotameter inlet,
  and the tube diameter D is represented by its
  value at the inlet, equal to the bob diameter Db.
• Through the Reynolds number regimes of laminar
  or turbulent flow, and particularly important for
  the rotameter flow regimes with strong or weak
  viscosity dependence can be distinguished.
• It has been found to be practical for rotameters
  to use an alternative characteristic number, the
  Ruppel number, defined as

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where mb rbD3b is the mass of the bob. By
combining Equations, the mass flow through the
rotameter can be written as
The relationship between the Ruppel number and
the Reynolds number
The advantage of the Ruppel number is its
independence of the flow rate. Since the Ruppel
number contains only fluid properties and the
mass and the density of the bob, it is a constant
for a particular instrument.
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Design Charts for Laminar Rotameters
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Design Charts for Turbulent Rotameters
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• End fitting flange shown
• flowmeter body
• rotation pickup magnetic, reluctancetype shown
  
• permanent magnet
• pickup cold wound on pole piece
• rotor blade
• rotor hub
• Rotor shaft bearing journal type shown
• rotor shaft
• diffuser support and flow straightener
• diffuser
• (12) flow conditioning plate (dotted) optional
  with some meters.

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Theory

• There are two approaches described in the current
  literature for analyzing axial turbine
  performance.
• The first approach describes the fluid driving
  torque in terms of momentum exchange, while the
  second describes it in terms of aerodynamic lift
  via airfoil theory.
• The former approach has the advantage that it
  readily produces analytical results describing
  basic operation, some of which have not appeared
  via airfoil analysis.
• The latter approach has the advantage that it
  allows more complete descriptions using fewer
  approximations.
• However, it is mathematically intensive and leads
  rapidly into computer-generated solutions.

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Eliminating the time dimension from the
left-hand-side quantity reduces it to the number
of rotor rotations per unit fluid volume, which
is essentially the flowmeter K factor specified
by most manufacturers.
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• In the ideal situation, the meter response is
  perfectly linear and determined only by geometry.
  
• In some flowmeter designs, the rotor blades are
  helically twisted to improve efficiency.
• This is especially true of blades with large
  radius ratios, (R/a).
• If the flow velocity profile is assumed to be
  flat, then the blade angle in this case can be
  described by tan b Constant X r.
• This is sometimes called the ideal helical
  blade.
• In practice, there are instead a number of rotor
  retarding torques of varying relative magnitudes.
  
• Under steady flow, the rotor assumes a speed that
  satisfies the following equilibrium

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• The difference between the actual rotor speed,
  rw, and the ideal rotor speed, rwi , is the rotor
  slip velocity due to the combined effect of all
  the rotor retarding torques , and as a result of
  which the fluid velocity vector is deflected
  through an exit or swirl angle, q.
• Denoting the radius variable by r, and equating
  the total rate of change of angular momentum of
  the fluid passing through the rotor to the
  retarding torque, one obtains

NT is the total retarding torque
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Industrial Correlations for Frictional Losses
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Electromagnetic Flowmeters

• Magnetic flowmeters have been widely used in
  industry for many years.
• Unlike many other types of flowmeters, they offer
  true noninvasive measurements.
• They are easy to install and use to the extent
  that existing pipes in a process can be turned
  into meters simply by adding external electrodes
  and suitable magnets.
• They can measure reverse flows and are
  insensitive to viscosity, density, and flow
  disturbances.
• Electromagnetic flowmeters can rapidly respond to
  flow changes and they are linear devices for a
  wide range of measurements.
• As in the case of many electric devices, the
  underlying principle of the electromagnetic
  flowmeter is Faradays law of electromagnetic
  induction.
• The induced voltages in an electromagnetic
  flowmeter are linearly proportional to the mean
  velocity of liquids or to the volumetric flow
  rates.

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• As is the case in many applications, if the pipe
  walls are made from nonconducting elements, then
  the induced voltage is independent of the
  properties of the fluid.
• The accuracy of these meters can be as low as
  0.25 and, in most applications, an accuracy of
  1 is used.
• At worst, 5 accuracy is obtained in some
  difficult applications where impurities of
  liquids and the contact resistances of the
  electrodes are inferior as in the case of
  low-purity sodium liquid solutions.
• Faradays Law of Induction
• This law states that if a conductor of length l
  (m) is moving with a velocity v (m/s1),
  perpendicular to a magnetic field of flux density
  B (Tesla), then the induced voltage e across the
  ends of conductor can be expressed by

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The velocity of the conductor is proportional to
the mean flow velocity of the liquid. Hence, the
induced voltage becomes
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Ultrasonic Flowmeters

• There are various types of ultrasonic flowmeters
  in use for discharge measurement
• (1) Transit time This is todays
  state-of-the-art technology and most widely used
  type.
• This type of ultrasonic flowmeter makes use of
  the difference in the time for a sonic pulse to
  travel a fixed distance.
• First against the flow and then in the direction
  of flow.
• Transmit time flowmeters are sensitive to
  suspended solids or air bubbles in the fluid.
• (2) Doppler This type is more popular and less
  expensive, but is not considered as accurate as
  the transit time flowmeter.
• It makes use of the Doppler frequency shift
  caused by sound reflected or scattered from
  suspensions in the flow path and is therefore
  more complementary than competitive to transit
  time flowmeters.

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Principle of transit time flowmeters.
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Transit Time Flowmeter

• Principle of Operation
• The acoustic method of discharge measurement is
  based on the fact that the propagation velocity
  of an acoustic wave and the flow velocity are
  summed vectorially.
• This type of flowmeter measures the difference in
  transit times between two ultrasonic pulses
  transmitted upstream t21 and downstream t12
  across the flow.
• If there are no transverse flow components in the
  conduit, these two transmit times of acoustic
  pulses are given by

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Since the transducers are generally used both as
transmitters and receivers, the difference in
travel time can be determined with the same pair
of transducers. Thus, the mean axial velocity
along the path is given by
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Example

• The following example shows the demands on the
  time measurement technique
• Assume a closed conduit with diameter D 150 mm,
  angle f 60, flow velocity 1 m/s, and water
  temperature 20C.
• This results in transmit times of about 116 s and
  a time difference
• Dt t12 t21 on the order of 78 ns.
• To achieve an accuracy of 1 of the corresponding
  full-scale range, Dt has to be measured with a
  resolution of at least 100 ps (1X1010s).
• Standard time measurement techniques are not able
  to meet such requirements so that special
  techniques must be applied.
• Digital timers with the state-of-the art Micro
  computers will make it possible to measure these
  time difference.

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Point Velocity Measurement

• Pitot Probe Anemometry Potential Flow Theory
  Bernoullis Theory .
• Thermal Anemometry Newtons Law of Cooling.
• Laser Anemometry Doppler Theory.