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Eqautions_1_Industrial Instrumentation - Flow Measurement Important Equations.pdf
1. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 1 of 17
Industrial Instrumentation
Flow Measurement
Sl. No.
1.
Newtonโs Law of Viscosity: Shear stress is directly proportional to the velocity
gradient.
๐ = ๐
๐ ๐
๐ ๐
๐๐ข
๐๐ฆ
= ๐๐๐ก๐ ๐๐ ๐ โ๐๐๐ ๐๐๐๐๐๐๐๐ก๐๐๐ (๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐๐๐๐๐๐๐ก),
๐ = ๐๐ฆ๐๐๐๐๐ ๐ฃ๐๐ ๐๐๐ ๐๐ก๐ฆ, ๐ = ๐ โ๐๐๐ ๐ ๐ก๐๐๐ ๐ = ๐น ๐ด
โ
Flow
๐ญ = โ๐ธ =
๐ธ๐๐๐๐๐๐๐
๐ป๐๐๐
Volume Flow Rate
๐ธ =
๐ฝ๐๐๐๐๐
๐ป๐๐๐
Flow Velocity
๐ฝ =
๐ญ๐๐๐ ๐น๐๐๐ (๐ธ)
๐จ๐๐๐ (๐จ)
Mass or Weight Flow Rate
๐พ = ๐๐ธ
Hagen Poiseuille Equation
For a Newtonian incompressible fluid, there is a pressure drop in the fluid flow
which is proportional to the fluid viscosity.
(Assumptions: Incompressible Newtonian fluid, laminar flow through pipe of
constant circular cross-section, no acceleration in fluid velocity)
โ๐ =
8๐๐๐
๐(๐ท 2
โ )4
๐ธ =
๐ โ๐ท๐ซ๐
๐๐๐๐๐
๐ = ๐๐๐๐ค ๐๐๐ก๐, โ๐ = ๐๐๐๐ ๐ ๐ข๐๐ ๐๐๐๐๐๐๐๐๐๐, ๐ท = ๐๐๐๐ ๐๐๐๐๐๐ก๐๐,
๐ = ๐ฃ๐๐ ๐๐๐ ๐๐ก๐ฆ, ๐ = ๐๐๐๐ ๐๐๐๐๐กโ
Reynolds Number
A dimensionless number used in fluid mechanics to indicate whether fluid flow
past a body or in a duct is steady or turbulent.
2. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 2 of 17
๐น๐ =
๐๐ ๐
๐
๐ฃ = ๐๐๐๐ค ๐ฃ๐๐๐๐๐๐ก๐ฆ; ๐ = ๐๐๐๐๐๐ก๐๐ ๐๐ ๐๐๐๐; ๐ = ๐๐๐ข๐๐ ๐๐๐๐ ๐๐ก๐ฆ; ๐ = ๐๐๐ข๐๐ ๐ฃ๐๐ ๐๐๐ ๐๐ก๐ฆ,
๐ ๐ < 2000 (๐๐๐๐๐๐๐)
๐ ๐ > 4000 (๐ก๐ข๐๐๐ข๐๐๐๐ก)
๐ ๐2000 <> 4000 ๐๐๐๐๐ ๐๐ก๐๐๐๐๐ ๐น๐๐๐ค
Kingโs Law
The greater the velocity of the gas across the probes, the greater the cooling effect.
The actual mass flow rate is calculated by measuring the variable power required to
maintain this constant temperature difference as the gas flows across the sensor.
๐พ =
๐ฏ
โ๐ป๐ช๐ท
๐ถ๐ = ๐ ๐๐๐๐๐๐๐ โ๐๐๐ก ๐๐๐๐๐๐๐ก๐ฆ, ๐ = ๐๐๐ ๐ ๐๐๐๐ค, ๐ป = โ๐๐๐ก ๐๐๐๐๐ก, โ๐ = ๐๐๐๐๐๐๐๐ก๐ข๐๐ ๐โ๐๐๐๐
Bernoulli's Equation
Bernoulli's principle says that a rise (fall) in pressure in a flowing fluid must always
be accompanied by a decrease (increase) in the speed, and conversely, i.e. an
increase (decrease) in the speed of the fluid results in a decrease (increase) in the
pressure.
๐ +
1
2
๐๐2
+ ๐๐โ = ๐๐๐๐ ๐ก๐๐๐ก
๐
๐๐
(
๐ฃ2
2
+
๐
๐
+ ๐. โ) = 0
๐ฃ = ๐๐๐๐ค ๐ ๐๐๐๐, ๐ = ๐ ๐ก๐๐ก๐๐ ๐๐๐๐ ๐ ๐ข๐๐, ๐ = ๐๐๐๐ ๐๐ก๐ฆ, ๐ = ๐๐๐๐ฃ๐๐ก๐ฆ, โ = โ๐๐๐โ๐ก
๐๐โ = ๐๐๐๐ฃ๐๐ก๐๐๐ โ๐๐๐,
1
2
๐๐2
= ๐ฃ๐๐๐๐๐๐ก๐ฆ โ๐๐๐ (๐๐ฆ๐๐๐๐๐ ๐๐๐๐ ๐ ๐ข๐๐
๐ฃ2
2
+
๐
๐
+ ๐. โ = ๐๐๐๐ ๐ก๐๐๐ก
๐ฃ2
2๐
+
๐
๐พ
+ โ = ๐๐๐๐ ๐ก๐๐๐ก
๐พ = ๐. ๐
๐๐ฃ2
2
+ ๐ = ๐๐๐๐ ๐ก๐๐๐ก
(๐, โ = ๐๐๐๐ ๐ก๐๐๐ก)
๐๐๐
๐
๐
+ ๐ท๐ =
๐๐๐
๐
๐
+ ๐ท๐ = ๐๐๐๐๐๐๐๐
๐1
๐
โ
๐2
๐
=
1
2
(๐ฃ2
2
โ ๐ฃ1
2)
3. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 3 of 17
(๐ฃ2
2
โ ๐ฃ1
2) = 2
(๐1 โ ๐2)
๐
๐ = ๐ด1๐ฃ1 = ๐ด2๐ฃ2
๐ = ๐ฃ๐๐๐ข๐๐ ๐๐๐๐ค ๐๐๐ก๐, ๐1 = ๐ข๐๐ ๐ก๐๐๐๐ ๐๐๐๐ ๐ ๐ข๐๐, ๐2 = ๐๐๐ค๐๐ ๐ก๐๐๐๐ ๐๐๐๐ ๐ ๐ข๐๐,
๐ฃ1 = ๐ข๐๐ ๐ก๐๐๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ, ๐ฃ2 = ๐๐๐ค๐๐ ๐ก๐๐๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ,
๐ด1 = ๐ข๐๐ ๐ก๐๐๐๐ ๐๐๐๐ ๐ โ ๐ ๐๐๐ก๐๐๐๐๐ ๐๐๐๐, ๐ด2 = ๐๐๐ค๐๐ ๐ก๐๐๐๐ ๐๐๐๐ ๐ โ ๐ ๐๐๐ก๐๐๐๐๐ ๐๐๐๐,
๐ธ =
๐จ๐
โ๐ โ (
๐จ๐
๐จ๐
)
๐
โ
๐(๐ท๐ โ ๐ท๐)
๐
2.
Variable Head or Differential Pressure Flow Meter
๐ญ๐๐๐๐ ๐ฝ๐๐๐๐๐๐๐ ๐ฝ = ๐ฌโ๐๐๐
๐
โ
๐ฝ๐๐๐๐๐ ๐ญ๐๐๐ ๐น๐๐๐ ๐ธ = ๐ฌ๐จโ๐๐๐
๐
โ
๐ด๐๐๐ ๐ญ๐๐๐ ๐น๐๐๐ ๐พ = ๐ โ ๐ฌ๐จโ๐๐๐
๐
โ
๐ฝ๐๐๐๐๐๐๐ ๐๐ ๐จ๐๐๐๐๐๐๐ (๐ฌ) =
๐
โ๐ โ (
๐จ๐
๐จ๐
)
๐
๐ = ๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐ ๐๐๐๐ค๐๐๐ ๐๐๐ข๐๐; ๐ = ๐ฃ๐๐๐ข๐๐ ๐๐๐๐ค ๐๐๐ก๐; ๐ = ๐๐๐ ๐ ๐๐๐๐ค ๐๐๐ก๐
๐ด = ๐๐๐๐ ๐ โ ๐ ๐๐๐ก๐๐๐๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐กโ๐๐๐ข๐โ ๐คโ๐๐โ ๐๐๐ข๐๐ ๐๐ ๐๐๐๐ค๐๐๐
โ = ๐๐๐๐๐๐๐๐๐ก๐๐๐ โ๐๐๐ (๐๐๐๐ ๐ ๐ข๐๐) ๐๐๐๐๐ ๐ ๐กโ๐ ๐๐๐ ๐ก๐๐๐๐ก๐๐๐ ๐๐๐๐๐๐๐ก
๐ = ๐๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐ข๐ ๐ก๐ ๐๐๐๐ฃ๐๐ก๐ฆ; ๐ = ๐๐๐๐ ๐๐ก๐ฆ ๐๐ ๐กโ๐ ๐๐๐๐ค๐๐๐ ๐๐๐ข๐๐
๐พ =
๐ถ๐
โ1 โ ๐พ4
๐พ = ๐ ๐๐ก๐๐ ๐น๐๐๐ก๐๐ = ๐๐๐ ๐ก๐๐๐๐ก๐๐ ๐๐๐๐๐๐๐ก ๐๐๐๐๐๐ก๐๐/๐๐๐๐๐ ๐๐๐๐๐๐ก๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ = ๐/๐ท
Coefficient of Discharge (๐ช๐ )
It is the ration of the actual discharge to the theoretical discharge.
๐ช๐๐๐๐๐๐๐๐๐๐ ๐๐ ๐ซ๐๐๐๐๐๐๐๐ ๐ช๐ =
๐ธ๐๐๐๐๐๐
๐ธ๐๐ ๐๐๐
Vena Contracta (ฮฒ)
The Vena Contracta is the downstream point in a fluid stream where the pressure is
the lowest, and the fluid velocity is the highest, and the stream diameter is the least.
4. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 4 of 17
๐ท = ๐ช๐ช. ๐จ๐๐๐๐๐๐๐
Contraction Factor/Coefficient (๐ช๐ช)
The Contraction Coefficient is the ration of the area of the jet at the vena contacta
to the area of the orifice.
๐ช๐ช =
๐จ๐๐๐ ๐๐ ๐ฝ๐๐๐ ๐ช๐๐๐๐๐๐๐๐
๐จ๐๐๐ ๐๐ ๐ถ๐๐๐๐๐๐
=
๐ท
๐จ๐๐๐๐๐๐๐
โ 0.6 โ 0.75
Flow Coefficient (๐ช๐ฝ)
The Flow Coefficient of any device is a relative measure of its efficiency at allowing
fluid flow, and it describes the relationship between the pressure drop (โ๐) across
the orifice/obstruction valve and the corresponding flow rate.
๐ช๐ฝ = ๐ธโ
๐บ๐ฎ
โ๐ท
โ๐ = ๐๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐๐๐ ๐ ๐ข๐๐, ๐๐บ = ๐๐๐ข๐๐ ๐ ๐๐๐๐๐๐๐ ๐๐๐๐ฃ๐๐ก๐ฆ
Rate of Discharge:
๐ = ๐ด1๐1 = ๐ด2๐2
Applying Bernoulliโs equation (ideal flow assumption)
๐1 +
๐๐1
2
2
= ๐2 +
๐๐2
2
2
The differential pressure head โโ is given by:
๐1 โ ๐2
๐๐
= โโ
3.
Venturi Meter
๐1 โ ๐2 =
๐
2
(๐ฃ2
2
โ ๐ฃ1
2)
๐1
๐ค1
+ ๐1 +
๐ฃ1
2
2๐
=
๐2
๐ค2
+ ๐2 +
๐ฃ2
2
2๐
๐1 & ๐2 = ๐๐๐๐ ๐ ๐ข๐๐ ๐๐ก ๐๐๐๐๐ก ๐๐๐ ๐กโ๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
๐ฃ1 & ๐ฃ2 = ๐๐ฃ. ๐ฃ๐๐๐๐๐๐ก๐๐๐ ๐๐ก ๐๐๐๐๐ก ๐๐๐ ๐กโ๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
๐ค1 & ๐ค2 = ๐ ๐๐๐๐๐๐๐ ๐ค๐๐๐โ๐ก ๐๐ ๐๐๐ข๐๐ ๐๐ก ๐๐๐๐๐ก ๐๐๐ ๐กโ๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
๐ด1 & ๐ด2 = ๐๐๐๐ ๐ ๐ ๐๐๐ก๐๐๐๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐๐ก ๐๐๐ ๐กโ๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
๐1 & ๐2 = ๐๐๐๐ฃ๐๐ก๐๐๐ ๐๐ ๐๐๐๐๐ก ๐๐๐ ๐กโ๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
๐, ๐1 & ๐2 = ๐๐๐๐ ๐๐ก๐ฆ, ๐๐๐๐ ๐๐ก๐๐๐ ๐๐ ๐๐๐ข๐๐ ๐๐ก ๐๐๐๐๐ก ๐๐๐ ๐กโ๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
5. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 5 of 17
Considering the venture meter being held horizontal and fluid at inlet & throat of
same density
๐1 = ๐2; ๐1 = ๐2; ๐ = ๐1๐ด1๐ฃ1 = ๐2๐ด2๐ฃ2
๐ฃ2
2
โ ๐ฃ1
2
2๐
=
๐1 โ ๐2
๐ค
By equation of continuity
๐ด1๐ฃ1 = ๐ด2๐ฃ2
๐ฃ1 = (
๐ด2
๐ด1
) ๐ฃ2
๐ฃ2 =
1
โ1 โ (
๐ด2
๐ด1
)
2
โ โ
2๐
๐ค
(๐1 โ ๐2) = ๐ธโ
2๐
๐ค
(๐1 โ ๐2)
Considering few losses, ๐ฃ2 is multiplied with a factor ๐ถ๐ฃ called the coefficient of
velocity.
๐ฃ2(๐๐๐ก๐ข๐๐) = ๐ถ๐ฃ๐ธโ
2๐
๐ค
(๐1 โ ๐2)
Discharge (volume flow rate)
๐ = ๐ด2๐ฃ2 = ๐ถ๐ฃ๐ด2๐ธโ
2๐
๐ค
(๐1 โ ๐2)
Considering contraction factor ๐ถ๐
๐๐๐๐ก๐ข๐๐ = ๐ถ๐๐ถ๐ฃ๐ด2๐ธโ
2๐
๐ค
(๐1 โ ๐2)
๐ธ๐๐๐๐๐๐ = ๐ช๐ ๐จ๐๐ฌ๐ถโ
๐๐
๐
(๐ท๐ โ ๐ท๐) = ๐ช๐ ๐จ๐๐ฌ๐ถโ
๐๐
๐
โ๐ท
๐ท๐๐ ๐โ๐๐๐๐ ๐ถ๐๐๐๐๐๐๐๐๐๐ก ๐ถ๐ = ๐ถ๐๐ถ๐ฃ; ๐ผ = ๐ ๐๐๐๐ ๐ก๐๐๐ก ๐๐๐๐ก๐๐ ๐๐๐ ๐ก๐๐๐๐๐๐๐ก๐ข๐๐,
โ๐ = ๐๐๐๐ ๐ ๐ข๐๐ ๐๐๐๐๐๐๐๐๐๐
๐ค = ๐ ๐๐๐๐๐๐๐ ๐ค๐๐๐โ๐ก ๐๐ ๐๐๐ข๐๐, ๐ธ = ๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐ ๐ด๐๐๐๐๐๐โ, ๐ด2 = ๐๐๐ค๐๐ ๐ก๐๐๐๐ ๐๐๐๐.
4.
Orifice
Vena-contracta is a point where the liquid jet issued from the orifice has the least
diameter, minimum pressure and maximum velocity. It is located at as distance
๐ท1 2
โ from the orifice plate approximately.
Actual velocity at vena-contracta is
6. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 6 of 17
๐๐(๐๐๐๐๐๐) =
๐ช๐
โ๐ โ (
๐จ๐
๐จ๐
)
๐
โ
๐๐
๐
(๐ท๐ โ ๐ท๐) = ๐ช๐๐ฌโ
๐๐
๐
โ๐ท
The jet of liquid coming out of the orifice plate contracts to a minimum area ๐ด0 at
the vena-contracta.
Area of the vena-contracta is ๐จ๐ = ๐ช๐๐จ๐๐๐๐๐๐๐
โด ๐ฃ2(๐๐๐ก๐ข๐๐) =
๐ถ๐ฃ
โ1 โ (
๐ถ๐๐ด0
๐ด1
)
2
โ
2๐
๐ค
(๐1 โ ๐2)
๐ท๐๐ ๐โ๐๐๐๐ ๐๐๐๐ก๐ข๐๐ = ๐ด2๐ฃ2 = ๐ถ๐๐ฃ2(๐๐๐ก๐ข๐๐)
โด ๐ธ๐๐๐๐๐๐ = ๐ช๐๐ช๐
๐จ๐
โ๐ โ (
๐ช๐๐จ๐
๐จ๐
)
๐
โ
๐๐
๐
(๐ท๐ โ ๐ท๐)
Taking into account the effect of temperature (๐ผ)
๐ธ๐๐๐๐๐๐ = ๐ช๐ ๐จ๐๐ฌ๐ถโ
๐๐
๐
(๐ท๐ โ ๐ท๐)
Let ๐พ = ๐ถ๐๐ธ
โด ๐ธ๐๐๐๐๐๐ = ๐ฒ๐ถ๐จ๐โ
๐๐
๐
(๐ท๐ โ ๐ท๐)
Mass Flow across an Orifice Plate
๐ธ๐ =
๐ช๐
โ๐ โ ๐ท๐
๐
๐
๐
๐ ๐
โ๐โ๐ท โ ๐๐
๐๐ = ๐๐๐๐๐๐๐ ๐๐๐๐ค ๐๐๐ก๐, ๐ถ๐ = ๐๐๐ ๐โ๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐ก, ๐ฝ = ๐๐๐๐๐๐ก๐๐ ๐๐๐ก๐๐,
๐ = ๐๐ฅ๐๐๐๐ ๐๐๐๐๐๐ก๐ฆ ๐๐๐๐ก๐๐, ๐ = ๐๐๐๐๐ ๐๐๐๐๐๐ก๐๐,
โ๐ = ๐๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐๐๐ ๐ ๐ข๐๐, ๐ = ๐๐๐ข๐๐ ๐๐๐๐ ๐๐ก๐ฆ
5.
Flow Nozzle
The discharge through a flow nozzle is
๐ธ๐๐๐๐๐๐ = ๐ฒ๐ฌ๐จ๐๐๐๐๐๐โ
๐
๐
(๐ท๐ โ ๐ท๐)
10. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 10 of 17
๐2 = ๐๐๐๐๐ก ๐๐๐๐๐๐ก๐๐ ๐๐ ๐ฃ๐๐๐ก๐ข๐๐ ๐๐๐ข๐๐
The maximum value of flow in a venture flume occurs when โ2 = (
2
3
) โ
Accuracy of flumes are higher that of weirs.
The free-flow discharge rate (Q) in a Parshall flume is given as
๐ธ = ๐. ๐๐๐๐ฏ๐.๐๐๐
๐ โ ๐๐๐๐ ๐๐๐ ๐ ๐๐๐๐๐๐
๐ธ = ๐. ๐๐๐ฏ๐.๐๐
๐ โ ๐๐๐๐ ๐๐๐ ๐ ๐๐๐๐๐๐
๐ธ = ๐. ๐๐๐ฏ๐.๐๐
๐ โ ๐๐๐๐ ๐๐๐ ๐ ๐๐๐๐๐๐
๐ธ = (๐. ๐๐๐๐๐ณ = ๐. ๐)๐ฏ๐.๐๐
๐๐ โ ๐๐ ๐๐๐๐ ๐๐๐ ๐ ๐๐๐๐๐๐
๐ = ๐น๐๐๐ค ๐๐๐ก๐, ๐ฟ = ๐ค๐๐๐กโ ๐๐ ๐๐๐ข๐๐ ๐กโ๐๐๐๐ก, ๐ป = ๐ป๐๐๐ (๐๐๐๐ก)
13.
Variable-Area Flow Meter
Drag Force
The Drag Force ๐ญ๐ ๐๐๐ is a force acting opposite to the relative motion of the objects
which is a function of the fluid velocity. Drag force is proportional to velocity for a
laminar flow and proportional to the velocity squared for a turbulent flow.
The numerical expression for ๐ญ๐ ๐๐๐
๐ญ๐ ๐๐๐ =
โ๐ท
โ๐ธ
= ๐๐จ
๐๐
๐๐
The Drag Force ๐ญ๐ ๐๐๐ is also expressed as,
๐ญ๐ ๐๐๐ =
๐
๐
๐๐๐
๐ช๐ซ๐จ
๐ = ๐ฃ๐๐ ๐๐๐ ๐๐ก๐ฆ, ๐ = ๐๐๐๐ ๐๐ก๐ฆ, ๐ด = ๐๐๐๐, ๐ช๐ซ = ๐ท๐๐๐ ๐ถ๐๐๐๐๐๐๐๐๐๐ก,
โ๐ = ๐๐๐๐ ๐ ๐ข๐๐ ๐๐๐๐๐๐๐๐๐๐, โ๐ = ๐๐๐๐ค ๐๐๐๐๐๐๐, ๐ฃ = ๐๐๐ข๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ,
Force Balance Equation of Variable Area Flow Meter
๐ญ๐ ๐๐๐ + ๐ญ๐๐๐๐๐๐๐๐ = ๐ญ๐๐๐๐๐๐
๐จ๐(๐ท๐ โ ๐ท๐) + ๐๐๐๐๐ฝ๐ = ๐๐๐๐ฝ๐
(๐ท๐ โ ๐ท๐) =
๐ฝ๐
๐จ๐
๐(๐๐ โ ๐๐๐)
๐๐, ๐๐๐ = ๐๐๐๐ ๐๐ก๐๐๐ ๐๐ ๐๐๐๐๐ก & ๐๐๐ข๐๐ ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ; ๐๐ = ๐ฃ๐๐๐ข๐๐ ๐๐ ๐๐๐๐๐ก.
๐๐, ๐๐ข = ๐๐๐๐ ๐ ๐ข๐๐ ๐๐ก ๐๐๐ค๐๐ค๐๐๐ & ๐ข๐๐ค๐๐๐ ๐๐๐๐๐ ๐๐ ๐๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ.
Flow rate ๐ธ๐๐๐๐๐๐ is given as
11. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 11 of 17
๐ธ๐๐๐๐๐๐ = ๐ฒ(๐จ๐ โ ๐จ๐)
๐พ = ๐๐๐ก๐๐๐๐ก๐๐ ๐๐๐๐ ๐ก๐๐๐ก, ๐ถ๐ = ๐๐๐ ๐โ๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐ก,
๐ด๐ก = ๐ก๐ข๐๐ โ ๐๐๐๐ ๐๐ก ๐๐๐๐๐ก ๐๐๐ฃ๐๐, ๐ด๐ = ๐๐๐๐๐ก ๐๐๐๐,
(๐ด๐ก โ ๐ด๐) = ๐๐๐๐๐๐ข๐ ๐๐๐๐ข๐๐๐ ๐๐๐๐ ๐๐๐ก๐ค๐๐๐ ๐ก๐ข๐๐ ๐๐๐ ๐๐๐๐๐ก,
๐ธ๐๐๐๐๐๐ = ๐ช๐
๐จ๐๐จ๐
โ๐จ๐
๐
โ ๐จ๐
๐
โ๐๐โโ๐ =
๐ช๐ (๐จ๐ โ ๐จ๐)
โ๐ โ (๐จ๐ โ ๐จ๐)
๐
/๐จ๐
๐
โ๐๐โ
๐ฝ๐
๐จ๐
(๐๐ โ ๐๐๐)
๐๐๐
If the angle of taper is ฮธ (which is very small), then
๐จ๐ =
๐
๐
(๐ซ๐ + ๐๐๐๐๐ฝ)๐
=
๐
๐
๐ซ๐
๐
+
๐
๐
๐๐ซ๐๐๐๐๐ฝ
๐ฆ = ๐๐๐๐๐ก ๐๐๐ ๐๐ก๐๐๐ ๐ค. ๐. ๐ก. ๐๐๐๐๐ก; ๐ท๐ = ๐๐๐๐๐ก ๐๐๐๐๐๐ก๐๐
๐ธ๐๐๐๐๐๐ = ๐ฒ
๐
๐
๐ซ๐๐๐๐๐๐ฝ + ๐ฒ (
๐
๐
๐ซ๐
๐
โ ๐จ๐) = ๐ฒ๐๐ + ๐ฒ๐
14.
Rotameter
By Bernoulliโs theorem and assuming the rotameter to be perfectly vertically
aligned, the energy equation is written as
๐2
๐ค
+
๐ฃ๐2
2
2๐
=
๐1
๐ค
+
๐ฃ๐1
2
2๐
๐๐ ๐ฃ๐2
2
โ ๐ฃ๐1
2
=
2๐
๐ค
(๐1 โ ๐2)
๐ = ๐ ๐ก๐๐ก๐๐ ๐๐๐๐ ๐ ๐ข๐๐; ๐ฃ๐ = ๐๐๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ; ๐ค = ๐ ๐๐๐๐๐๐๐ ๐ค๐๐๐โ๐ก
For static equilibrium of the float at any position
๐ด๐ (๐1 +
๐ฃ๐1
2
2๐
๐ค) + ๐ฃ๐๐ค = ๐ด๐๐2 + ๐ฃ๐๐ค๐
๐๐ & ๐ค๐ ๐๐๐ ๐ฃ๐๐๐ข๐๐ ๐๐๐ ๐ ๐๐๐๐๐๐๐ ๐ค๐๐๐โ๐ก ๐๐ ๐๐๐๐๐ก ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ
By the continuity equation, we have
๐ = ๐
๐๐ด1 = ๐ถ๐๐ฃ๐2
๐ด2
๐ด1 = ๐๐๐๐ ๐๐ ๐๐๐๐๐ก ๐๐ ๐ก๐๐๐๐๐๐ ๐ก๐ข๐๐; ๐ด2 = ๐๐๐๐ ๐๐๐ก๐ค๐๐๐ ๐๐๐๐๐ก & ๐ก๐ข๐๐;
๐ถ๐ = ๐๐๐๐๐๐๐๐๐๐๐ก ๐๐ ๐๐๐๐ก๐๐๐๐ก๐๐๐
12. FLOW MEASUREMENT โ SUMMARY OF IMPORTANT EQUATIONS
Er. Faruk Bin, Dept. of AEIE, UIT, BU Page 12 of 17
Thus we have,
๐ธ = ๐ช๐๐จ๐โ
๐๐๐๐
๐จ๐
(
๐๐
๐
โ ๐) = ๐ช๐๐จ๐โ
๐๐๐๐
๐จ๐
(
๐๐
๐
โ ๐)
๐ = ๐๐๐๐ ๐๐ก๐ฆ ๐๐ ๐๐๐ข๐๐; ๐๐ = ๐๐๐๐ ๐๐ก๐ฆ ๐๐ ๐๐๐๐๐ก
๐ธ โ ๐; ๐ฅ = (๐๐๐๐๐ก ๐๐๐ ๐๐๐๐๐๐๐๐๐ก)
Rotameter Flow Rate is also obtained using the equation,
๐ธ๐๐๐๐๐๐ = ๐ฒ
๐
๐
๐ซ๐๐๐๐๐๐ฝ + ๐ฒ (
๐
๐
๐ซ๐
๐
โ ๐จ๐) = ๐ฒ๐๐ + ๐ฒ๐
๐น๐๐๐๐๐๐๐๐ ๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐
15.
Electromagnetic Flow Meter
๐ฌ = ๐ฉ. ๐. ๐
๐ธ = ๐ฃ๐๐๐ก๐๐๐ ๐๐๐๐๐๐๐ก๐๐; ๐ต = ๐๐๐ข๐ฅ ๐๐๐๐ ๐๐ก๐ฆ; ๐ = ๐๐๐๐๐กโ ๐๐ ๐๐๐๐๐ข๐๐ก๐๐;
๐ฃ = ๐ฃ๐๐๐๐๐๐ก๐ฆ ๐๐ ๐๐๐๐๐ข๐๐ก๐๐
The volume flow rate for a circular pipe of diameter (D) is given as,
๐ธ = ๐จ โ ๐ = ๐ โ (
๐ซ
๐
)
๐
โ ๐ =
๐ ๐ซ๐
๐
๐
=
๐ ๐ซ๐
๐
๐ฌ
๐ฉ. ๐ซ
=
๐ ๐ซ๐ฌ
๐๐ฉ
16.
Turbine Flow Meter
๐ญ๐๐๐ ๐น๐๐๐ ๐ธ = ๐ โ ๐
โKโ factor of the turbine element (e.g. pulses per gallon); = ๐๐. ๐๐ ๐๐๐ก๐๐ก๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐๐
๐ฌ = โ
๐ ๐
๐ ๐
๐ธ = ๐ด๐ถ ๐๐๐๐ก๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ ๐ข๐ ๐๐๐๐; ๐ = ๐๐๐ก๐๐ก๐๐๐ ๐๐๐๐๐๐ก๐๐ ๐๐๐๐๐;
17.
Target Flow Meter
๐ญ๐ =
๐
๐
๐ช๐ ๐๐๐ฝ๐
๐จ
๐น๐ = ๐ท๐๐๐ ๐น๐๐๐๐, ๐ถ๐=overall drag coefficient; ๐ = ๐๐๐ข๐๐ ๐๐๐๐ ๐๐ก๐ฆ,
๐ด = ๐ก๐๐๐๐๐ก ๐๐๐๐, ๐ = ๐๐๐ข๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ
18.
Thermal Flow Meter
For Hot Wire Thermal Flow Meter
๐๐ = โ๐ป [๐ฒ + ๐(๐๐ช๐๐๐ ๐ ๐ฝ๐๐๐)
๐
๐
โ
]