22.1. PRESSURE-BASED FLOWMETERS |
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22.1.9 Equation summary
Volumetric flow rate (Q) full equation:
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ρf |
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Q = N |
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CY A2 |
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P1 − P2 |
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1 − A1 |
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Volumetric flow rate (Q) simplified equation: |
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s |
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Q = k |
P1 − P2 |
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Mass flow rate (W ):
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CY A2 |
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W = N |
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qρf (P1 − P2) |
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Mass flow rate (W ) simplified equation:
q
W = k ρf (P1 − P2)
Where,
Q = Volumetric flow rate (e.g. gallons per minute, flowing cubic feet per second) W = Mass flow rate (e.g. kilograms per second, slugs per minute)
N = Unit conversion factor
C = Discharge coe cient (accounts for energy losses, Reynolds number corrections, pressure tap locations, etc.)
Y = Gas expansion factor (Y = 1 for liquids) A1 = Cross-sectional area of mouth
A2 = Cross-sectional area of throat
ρf = Fluid density at flowing conditions (actual temperature and pressure at the element)
k = Constant of proportionality (determined by experimental measurements of flow rate, pressure, and density)
1650 |
CHAPTER 22. CONTINUOUS FLUID FLOW MEASUREMENT |
The beta ratio (β) of a di erential-producing element is the ratio of throat diameter to mouth diameter (β = Dd ). This is the primary factor determining acceleration as the fluid increases velocity entering the constricted throat of a flow element (venturi tube, orifice plate, wedge, etc.). The following expression is often called the velocity of approach factor (commonly symbolized as Ev ), because it relates the velocity of the fluid through the constriction to the velocity of the fluid as it approaches the flow element:
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Ev = p = Velocity of approach factor 1 − β4
This same velocity approach factor may be expressed in terms of mouth and throat areas (A1 and A2, respectively):
Ev = |
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= Velocity of approach factor |
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Beta ratio has a significant impact on the number of straight-run pipe lengths needed to condition the flow profile upstream and downstream of the flow element. Large beta ratios (where the bore diameter approaches the flowtube’s inside diameter) are more sensitive to piping disturbances, since there is less acceleration of the flowstream through the element, and therefore flow profile asymmetries caused by piping disturbances are significant in comparison to the fluid’s through-bore velocity. Small beta ratio values correspond to larger acceleration factors, where disturbances in the flow profile become “swamped30” by the high throat velocities created by the element’s constriction. A disadvantage of small beta ratio values is that the flow element exhibits a greater permanent pressure loss, which is an operational cost if the flow is provided by a machine such as an engineor motor-driven pump (more energy required to turn the pump, equating to a greater operating cost to run the process).
30“Swamping” is a term commonly used in electrical engineering, where a bad e ect is overshadowed by some other e ect much larger in magnitude, to the point where the undesirable e ect is negligible in comparison.
22.1. PRESSURE-BASED FLOWMETERS |
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When computing the volumetric flow of a gas in standard volume units (e.g. SCFM), the equation becomes much more complex than the simple (flowing) volumetric rate equation. Any equation computing flow in standard units must predict the e ective expansion of the gas if it were to transition from flowing conditions (the actual pressure and temperature it experiences flowing through the pipe) to standard conditions (one atmosphere pressure at 60 degrees Fahrenheit). The compensated gas flow measurement equation published by the American Gas Association (AGA Report #3) in 1992 for orifice plates with flange taps calculates this expansion to standard conditions with a series of factors accounting for flowing and standard (“base”) conditions, in addition to the more common factors such as velocity of approach and gas expansion. Most of these factors are represented in the AGA3 equation by di erent variables beginning with the letter F :
p
Q = Fn(Fc + Fsl)Y FpbFtbFtf Fgr Fpv hW Pf 1
Where,
Q = Volumetric flow rate (standard cubic feet per hour – SCFH)
Fn = Numeric conversion factor (accounts for certain numeric constants, unit-conversion coe cients, and the velocity of approach factor Ev )
Fc = Orifice calculation factor (a polynomial function of the orifice plate’s β ratio and Reynolds number), appropriate for flange taps
Fsl = Slope factor (another polynomial function of the orifice plate’s β ratio and Reynolds number), appropriate for flange taps
Fc + Fsl = Cd = Discharge coe cient, appropriate for flange taps
Y = Gas expansion factor (a function of β, di erential pressure, static pressure, and specific heats)
Fpb |
= Base pressure factor = 14.73 PSI , with pressure in PSIA (absolute) |
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Ftb |
= Base temperature factor = |
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Tb |
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519.67 |
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Ftf |
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519.67 |
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= Flowing temperature factor = q Tf |
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Fgr = Real gas relative density factor = q |
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Fpv = Supercompressibility factor = q |
Zb |
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Zf 1 |
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hW = Di erential pressure produced by orifice plate (inches water column)
Pf 1 = Flowing pressure of gas at the upstream tap (PSI absolute)
1652 |
CHAPTER 22. CONTINUOUS FLUID FLOW MEASUREMENT |
22.2Laminar flowmeters
A unique form of di erential pressure-based flow measurement deserves its own section in this flow measurement chapter, and that is the laminar flowmeter.
Laminar flow is a condition of fluid motion where viscous (internal fluid friction) forces greatly overshadow inertial (kinetic) forces. A flowstream in a state of laminar flow exhibits no turbulence, with each fluid molecule traveling in its own path, with limited mixing and collisions with adjacent molecules. The dominant mechanism for resistance to fluid motion in a laminar flow regime is friction with the pipe or tube walls. Laminar flow is qualitatively predicted by low values of Reynolds number.
This pressure drop created by fluid friction in a laminar flowstream is quantifiable, and is expressed in the Hagen-Poiseuille equation:
Q = k
P D4
µL
Where,
Q = Flow rate
P = Pressure dropped across a length of pipe D = Pipe diameter
µ = Fluid viscosity L = Pipe length
k = Coe cient accounting for units of measurement
Laminar flowmeter elements generally consist of one or more tubes whose length greatly exceeds the inside diameter, arranged in such a way as to produce a slow-moving flow velocity. An example is shown here:
Laminar flowmeter
H L
Tubes
The expanded diameter of the flow element ensures a lower fluid velocity than in the pipes entering and exiting the element. This decreases the Reynolds number to the point where the flow