Orifice plate

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Flat-plate, sharp-edge orifice

ISO 5167 Orifice Plate

An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a restriction plate). Either a volumetric or mass flow rate may be determined, depending on the calculation associated with the orifice plate. It uses the same principle as a Venturi nozzle, namely Bernoulli's principle which states that there is a relationship between the pressure of the fluid and the velocity of the fluid. When the velocity increases, the pressure decreases and vice versa.

Description

An orifice plate is a thin plate with a hole in the middle. It is usually placed in a pipe in which fluid flows. When the fluid reaches the orifice plate, the fluid is forced to converge to go through the small hole; the point of maximum convergence actually occurs shortly downstream of the physical orifice, at the so-called vena contracta point (see drawing to the right). As it does so, the velocity and the pressure changes. Beyond the vena contracta, the fluid expands and the velocity and pressure change once again. By measuring the difference in fluid pressure between the normal pipe section and at the vena contracta, the volumetric and mass flow rates can be obtained from Bernoulli's equation.

Uses

Orifice plates are most commonly used to measure flow rates in pipes, when the fluid is single-phase (rather than being a mixture of gases and liquids, or of liquids and solids) and well-mixed, the flow is continuous rather than pulsating, the fluid occupies the entire pipe (precluding silt or trapped gas), the flow profile is even and well-developed and the fluid and flow rate meet certain other conditions. Under these circumstances and when the orifice plate is constructed and installed according to appropriate standards, the flow rate can easily be determined using published formulae based on substantial research and published in industry, national and international standards. ^[1]

Once the orifice plate is designed and installed, the flow rate can often be indicated with an acceptably low uncertainty simply by taking the square root of the differential pressure across the orifice's pressure tappings and applying an appropriate constant. Even compressible flows of gases that vary in pressure and temperature may be measured with acceptable uncertainty by merely taking the square roots of the absolute pressure and/or temperature, depending on the purpose of the measurement and the costs of ancillary instrumentation.

Orifice plates are also used to reduce pressure or restrict flow, in which case they are often called restriction plates.^[2] ^[3]

Incompressible flow through an orifice

By assuming steady-state, incompressible (constant fluid density), inviscid, laminar flow in a horizontal pipe (no change in elevation) with negligible frictional losses, Bernoulli's equation reduces to an equation relating the conservation of energy between two points on the same streamline:

$P_{1}+{\frac {1}{2}}\cdot \rho \cdot V_{1}^{2}=P_{2}+{\frac {1}{2}}\cdot \rho \cdot V_{2}^{2}$

or:

$P_{1}-P_{2}={\frac {1}{2}}\cdot \rho \cdot V_{2}^{2}-{\frac {1}{2}}\cdot \rho \cdot V_{1}^{2}$

By continuity equation:

$Q=A_{1}\cdot V_{1}=A_{2}\cdot V_{2}$ or $V_{1}=Q/A_{1}$ and $V_{2}=Q/A_{2}$ :

$P_{1}-P_{2}={\frac {1}{2}}\cdot \rho \cdot {\bigg (}{\frac {Q}{A_{2}}}{\bigg )}^{2}-{\frac {1}{2}}\cdot \rho \cdot {\bigg (}{\frac {Q}{A_{1}}}{\bigg )}^{2}$

Solving for $Q_{{}}$ :

$Q=A_{2}\;{\sqrt {{\frac {2\;(P_{1}-P_{2})/\rho }{1-(A_{2}/A_{1})^{2}}}}}$

and:

$Q=A_{2}\;{\sqrt {{\frac {1}{1-(d_{2}/d_{1})^{4}}}}}\;{\sqrt {2\;(P_{1}-P_{2})/\rho }}$

The above expression for $Q$ gives the theoretical volume flow rate. Introducing the beta factor $\beta =d_{2}/d_{1}$ as well as the coefficient of discharge $C_{d}$ :

$Q=C_{d}\;A_{2}\;{\sqrt {{\frac {1}{1-\beta ^{4}}}}}\;{\sqrt {2\;(P_{1}-P_{2})/\rho }}$

And finally introducing the meter coefficient $C$ which is defined as $C={\frac {C_{d}}{{\sqrt {1-\beta ^{4}}}}}$ to obtain the final equation for the volumetric flow of the fluid through the orifice:

$(1)\qquad Q=C\;A_{2}\;{\sqrt {2\;(P_{1}-P_{2})/\rho }}$

Multiplying by the density of the fluid to obtain the equation for the mass flow rate at any section in the pipe:^[4]^[5]^[6]^[7]

$(2)\qquad {\dot {m}}=\rho \;Q=C\;A_{2}\;{\sqrt {2\;\rho \;(P_{1}-P_{2})}}$

where:
$Q_{{}}$	= volumetric flow rate (at any cross-section), m³/s
${\dot {m}}$	= mass flow rate (at any cross-section), kg/s
$C_{d}$	= coefficient of discharge, dimensionless
$C$	= orifice flow coefficient, dimensionless
$A_{1}$	= cross-sectional area of the pipe, m²
$A_{2}$	= cross-sectional area of the orifice hole, m²
$d_{1}$	= diameter of the pipe, m
$d_{2}$	= diameter of the orifice hole, m
$\beta$	= ratio of orifice hole diameter to pipe diameter, dimensionless
$V_{1}$	= upstream fluid velocity, m/s
$V_{2}$	= fluid velocity through the orifice hole, m/s
$P_{1}$	= fluid upstream pressure, Pa with dimensions of kg/(m·s² )
$P_{2}$	= fluid downstream pressure, Pa with dimensions of kg/(m·s² )
$\rho$	= fluid density, kg/m³

Deriving the above equations used the cross-section of the orifice opening and is not as realistic as using the minimum cross-section at the vena contracta. In addition, frictional losses may not be negligible and viscosity and turbulence effects may be present. For that reason, the coefficient of discharge $C_{d}$ is introduced. Methods exist for determining the coefficient of discharge as a function of the Reynolds number.^[5]

The parameter ${\sqrt {1-\beta ^{4}}}$ is often referred to as the velocity of approach factor^[4] and dividing the coefficient of discharge by that parameter (as was done above) produces the flow coefficient $C$ . Methods also exist for determining the flow coefficient as a function of the beta function $\beta$ and the location of the downstream pressure sensing tap. For rough approximations, the flow coefficient may be assumed to be between 0.60 and 0.75. For a first approximation, a flow coefficient of 0.62 can be used as this approximates to fully developed flow.

An orifice only works well when supplied with a fully developed flow profile. This is achieved by a long upstream length (20 to 40 pipe diameters, depending on Reynolds number) or the use of a flow conditioner. Orifice plates are small and inexpensive but do not recover the pressure drop as well as a venturi nozzle does. If space permits, a venturi meter is more efficient than an orifice plate.

Flow of gases through an orifice

In general, equation (2) is applicable only for incompressible flows. It can be modified by introducing the expansion factor $Y$ to account for the compressibility of gases.

$(3)\qquad {\dot {m}}=\rho _{1}\;Q=C\;Y\;A_{2}\;{\sqrt {2\;\rho _{1}\;(P_{1}-P_{2})}}$

$Y$ is 1.0 for incompressible fluids and it can be calculated for compressible gases.^[5]

Calculation of expansion factor

For flow measurement purposes, the expansion factor $Y$ , which allows for the change in the density of an ideal gas as it expands isentropically, is given by the empirical formula:^[5]

$(4)\qquad Y=\;1-{\bigg (}{\frac {1-r}{k}}{\bigg )}{\bigg (}0.41+0.35\beta ^{4}{\bigg )}$

where:
$Y$	= Expansion factor, dimensionless
$r$	= $P_{2}/P_{1}$
$k$	= specific heat ratio ( $c_{p}/c_{v}$ ), dimensionless

and the pressures are measured at orifice plate tappings (such as flange, corner or D+D/2), β is in the range 0.2 to 0.75 and certain other conditions are met.

For smaller values of β (such as restriction plates and discharge from tanks but not generally in flow measurement), Y may be calculated from first principles and the effect of β neglected, giving:

${\dot {m}}=C\;A_{2}\;{\sqrt {2\;\rho _{1}\;{\bigg (}{\frac {k}{k-1}}{\bigg )}{\bigg [}{\frac {(P_{2}/P_{1})^{{2/k}}-(P_{2}/P_{1})^{{(k+1)/k}}}{1-P_{2}/P_{1}}}{\bigg ]}(P_{1}-P_{2})}}$

and:

${\dot {m}}=C\;A_{2}\;{\sqrt {2\;\rho _{1}\;{\bigg (}{\frac {k}{k-1}}{\bigg )}{\bigg [}{\frac {(P_{2}/P_{1})^{{2/k}}-(P_{2}/P_{1})^{{(k+1)/k}}}{(P_{1}-P_{2})/P_{1}}}{\bigg ]}(P_{1}-P_{2})}}$

and thus, the final equation for approximating the non-choked (i.e., sub-sonic) flow of ideal gases through an orifice for values of β less than 0.25:

$(5)\qquad {\dot {m}}=C\;A_{2}\;{\sqrt {2\;\rho _{1}\;P_{1}\;{\bigg (}{\frac {k}{k-1}}{\bigg )}{\bigg [}(P_{2}/P_{1})^{{2/k}}-(P_{2}/P_{1})^{{(k+1)/k}}{\bigg ]}}}$

Using the ideal gas law and the compressibility factor (which corrects for non-ideal gases), a practical equation is obtained for the non-choked flow of real gases through an orifice for values of β less than 0.25:^[6]^[7]^[8]

$(6)\qquad {\dot {m}}=C\;A_{2}\;P_{1}\;{\sqrt {{\frac {2\;M}{Z\;R\;T_{1}}}{\bigg (}{\frac {k}{k-1}}{\bigg )}{\bigg [}(P_{2}/P_{1})^{{2/k}}-(P_{2}/P_{1})^{{(k+1)/k}}{\bigg ]}}}$

Remembering that $Q_{1}={\frac {{\dot {m}}}{\rho _{1}}}$ and $\rho _{1}=M\;{\frac {P_{1}}{Z\;R\;T_{1}}}$ (ideal gas law and the compressibility factor)

$(8)\qquad Q_{1}=C\;A_{2}\;{\sqrt {2\;{\frac {Z\;R\;T_{1}}{M}}{\bigg (}{\frac {k}{k-1}}{\bigg )}{\bigg [}(P_{2}/P_{1})^{{2/k}}-(P_{2}/P_{1})^{{(k+1)/k}}{\bigg ]}}}$

where:
$k$	= specific heat ratio ( $c_{p}/c_{v}$ ), dimensionless
${\dot {m}}$	= mass flow rate at any section, kg/s
$Q_{1}$	= upstream real gas flow rate, m³/s
$C$	= orifice flow coefficient, dimensionless
$A_{2}$	= cross-sectional area of the orifice hole, m²
$\rho _{1}$	= upstream real gas density, kg/m³
$P_{1}$	= upstream gas pressure, Pa with dimensions of kg/(m·s²)
$P_{2}$	= downstream pressure, Pa with dimensions of kg/(m·s²)
$M$	= the gas molar mass, kg/mol
$R$	= the Universal Gas Law Constant = 8.3145 J/(mol·K)
$T_{1}$	= absolute upstream gas temperature, K
$Z$	= the gas compressibility factor at $P_{1}$ and $T_{1}$ , dimensionless

A detailed explanation of choked and non-choked flow of gases, as well as the equation for the choked flow of gases through restriction orifices, is available at Choked flow.

The flow of real gases through thin-plate orifices never becomes fully choked. Cunningham (1951) first drew attention to the fact that choked flow will not occur across a standard, thin, square-edged orifice.^[9] The mass flow rate through the orifice continues to increase as the downstream pressure is lowered to a perfect vacuum, though the mass flow rate increases slowly as the downstream pressure is reduced below the critical pressure.^[10]

Permanent pressure drop for incompressible fluids

For a square-edge orifice plate with flange taps:^[11]

${\frac {\Delta P_{p}}{\Delta P_{i}}}=1-0.24\beta -0.52\beta ^{2}-0.16\beta ^{3}$

where:

$\Delta P_{p}$ = permanent pressure drop

$\Delta P_{i}$ = indicated pressure drop at the flange taps

$\beta =d_{2}/d_{1}$

And rearranging the formula near the top of this article:

$\Delta P_{i}=P_{1}-P_{2}={\frac {Q^{2}~\rho ~(1-\beta ^{4})}{2~C_{d}^{2}~A_{2}^{2}}}={\frac {Q^{2}~\rho ~(1-\beta ^{4})}{2~C_{d}^{2}~A_{1}^{2}~\beta ^{4}}}$

References

↑ Miller, Richard W (1996). Flow Measurement Engineering Handbook. New York: McGraw-Hill. ISBN 0-07-042366-0.
↑ "Orifice Plates for Flow Measurement & Flow Restriction". Retrieved 1 February 2014.
↑ Flow of Fluids Through Valves, Fittings and Pipe. Ipswich: Crane. 1988. p. 2-14.
↑ 4.0 4.1 Lecture, University of Sydney
↑ 5.0 5.1 5.2 5.3 Perry, Robert H. and Green, Don W. (1984). Perry's Chemical Engineers' Handbook (Sixth Edition ed.). McGraw Hill. ISBN 0-07-049479-7.
↑ 6.0 6.1 Handbook of Chemical Hazard Analysis Procedures, Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989. Handbook of Chemical Hazard Analysis, Appendix B Click on PDF icon, wait and then scroll down to page 391 of 520 PDF pages.
↑ 7.0 7.1 Risk Management Program Guidance For Offsite Consequence Analysis, U.S. EPA publication EPA-550-B-99-009, April 1999. Guidance for Offsite Consequence Analysis
↑ Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases), PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. PGS2 CPR 14E
↑ Cunningham, R.G., "Orifice Meters with Supercritical Compressible Flow", Trans. ASME, Vol. 73, pp. 625-638, 1951
↑ Section 3 -- Choked Flow
↑ Catalog section by AVCO

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