Pressure tappings

There are three standard positions for pressure tappings (also called taps), commonly named as follows:

• Corner taps placed immediately upstream and downstream of the plate; convenient when the plate is provided with an orifice carrier incorporating tappings
• D and D/2 taps or radius taps placed one pipe diameter upstream and half a pipe diameter downstream of the plate; these can be installed by welding bosses to the pipe
• Flange taps placed 25.4 mm (1 inch) upstream and downstream of the plate, normally within specialised pipe flanges.

These types are covered by ISO 5167 and other major standards. Other types include

• 2½D and 8D taps or recovery taps placed 2.5 pipe diameters upstream and 8 diameters downstream, at which point the measured differential is equal to the unrecoverable pressure loss caused by the orifice
• Vena contracta tappings placed one pipe diameter upstream and at a position 0.3 to 0.9 diameters downstream, depending on the orifice type and size relative to the pipe, in the plane of minimum fluid pressure.

The measured differential pressure differs for each combination and so the coefficient of discharge used in flow calculations depends partly on the tapping positions.

The simplest installations use single tappings upstream and downstream, but in some circumstances these may be unreliable; they might be blocked by solids or gas-bubbles, or the flow profile might be uneven so that the pressures at the tappings are higher or lower than the average in those planes. In these situations multiple tappings can be used, arranged circumferentially around the pipe and joined by a piezometer ring, or (in the case of corner taps) annular slots running completely round the internal circumference of the orifice carrier.

Plate

Standards and handbooks are mainly concerned with sharp-edged thin plates. In these, the leading edge is sharp and free of burrs and the cylindrical section of the orifice is short, either because the entire plate is thin or because the downstream edge of the plate is bevelled. Exceptions include the quarter-circle or quadrant-edge orifice, which has a fully rounded leading edge and no cylindrical section, and the conical inlet or conical entrance plate which has a bevelled leading edge and a very short cylindrical section. The orifices are normally concentric with the pipe (the eccentric orifice is a specific exception) and circular (except in the specific case of the segmental or chord orifice, in which the plate obstructs just a segment of the pipe). Standards and handbooks stipulate that the upstream surface of the plate is particularly flat and smooth. Sometimes a small drain or vent hole is drilled through the plate where it meets the pipe, to allow condensate or gas bubbles to pass along the pipe.

Pipe

Standards and handbooks stipulate a well-developed flow profile; velocities will be lower at the pipe wall than in the centre but not eccentric or jetting. Similarly the flow downstream of the plate must be unobstructed, otherwise the downstream pressure will be affected. To achieve this, the pipe must be acceptably circular, smooth and straight for stipulated distances. Sometimes when it is impossible to provide enough straight pipe, flow conditioners such as tube bundles or plates with multiple holes are inserted into the pipe to straighten and develop the flow profile, but even these require a further length of straight pipe before the orifice itself. Some standards and handbooks also provide for flows from or into large spaces rather than pipes, stipulating that the region before or after the plate is free of obstruction and abnormalities in the flow.

Incompressible flow

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 (which expresses the conservation of energy of an incompressible fluid parcel as it moves between two points on the same streamline) can be rewritten without the gravitational potential energy term and reduced to:

${\displaystyle p_{1}+{\frac {1}{2}}\cdot \rho \cdot V_{1}^{'^{2}}=p_{2}+{\frac {1}{2}}\cdot \rho \cdot V_{2}^{'^{2}}}$

or:

${\displaystyle p_{1}-p_{2}={\frac {1}{2}}\cdot \rho \cdot V_{2}^{'^{2}}-{\frac {1}{2}}\cdot \rho \cdot V_{1}^{'^{2}}}$

By continuity equation:

${\displaystyle q_{v}^{'}=A_{1}\cdot V_{1}^{'}=A_{2}\cdot V_{2}^{'}}$   or   ${\displaystyle V_{1}^{'}=q_{v}^{'}/A_{1}}$ and ${\displaystyle V_{2}^{'}=q_{v}^{'}/A_{2}}$ :

${\displaystyle p_{1}-p_{2}={\frac {1}{2}}\cdot \rho \cdot {\bigg (}{\frac {q_{v}^{'}}{A_{2}}}{\bigg )}^{2}-{\frac {1}{2}}\cdot \rho \cdot {\bigg (}{\frac {q_{v}^{'}}{A_{1}}}{\bigg )}^{2}}$

Solving for ${\displaystyle q_{v}^{'}}$:

${\displaystyle q_{v}^{'}=A_{2}\;{\sqrt {\frac {2\;(p_{1}-p_{2})/\rho }{1-(A_{2}/A_{1})^{2}}}}}$

and:

${\displaystyle q_{v}^{'}=A_{2}\;{\sqrt {\frac {1}{1-(d/D)^{4}}}}\;{\sqrt {2\;(p_{1}-p_{2})/\rho }}}$

The above expression for ${\displaystyle q_{v}^{'}}$ gives the theoretical volume flow rate. Introducing the beta factor ${\displaystyle \beta =d/D}$ as well as the discharge coefficient ${\displaystyle C_{d}}$:

${\displaystyle q_{v}=C_{d}\;A_{2}\;{\sqrt {\frac {1}{1-\beta ^{4}}}}\;{\sqrt {2\;(p_{1}-p_{2})/\rho }}}$

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

${\displaystyle (1)\qquad q_{v}=C\;A_{2}\;{\sqrt {2\;(p_{1}-p_{2})/\rho }}}$

Multiplying by the density of the fluid to obtain the equation for the mass flow rate at any section in the pipe:^{[8][9][10][11]}

${\displaystyle (2)\qquad q_{m}=\rho \;q_{v}=C\;A_{2}\;{\sqrt {2\;\rho \;(p_{1}-p_{2})}}}$

where:
${\displaystyle q_{v}{}}$	= volumetric flow rate (at any cross-section), m³/s
${\displaystyle q_{v}^{'}{}}$	= theoretical volumetric flow rate (at any cross-section), m³/s
${\displaystyle q_{m}}$	= mass flow rate (at any cross-section), kg/s
${\displaystyle q_{m}^{'}}$	= theoretical mass flow rate (at any cross-section), kg/s
${\displaystyle C_{d}}$	= coefficient of discharge, dimensionless
${\displaystyle C}$	= orifice flow coefficient, dimensionless
${\displaystyle A_{1}}$	= cross-sectional area of the pipe, m²
${\displaystyle A_{2}}$	= cross-sectional area of the orifice hole, m²
${\displaystyle D}$	= diameter of the pipe, m
${\displaystyle d}$	= diameter of the orifice hole, m
${\displaystyle \beta }$	= ratio of orifice hole diameter to pipe diameter, dimensionless
${\displaystyle V_{1}^{'}}$	= theoretical upstream fluid velocity, m/s
${\displaystyle V_{2}^{'}}$	= theoretical fluid velocity through the orifice hole, m/s
${\displaystyle p_{1}}$	= fluid upstream pressure, Pa   with dimensions of kg/(m·s² )
${\displaystyle p_{2}}$	= fluid downstream pressure, Pa   with dimensions of kg/(m·s² )
${\displaystyle \rho }$	= 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 ${\displaystyle C_{d}}$ is introduced. Methods exist for determining the coefficient of discharge as a function of the Reynolds number.^[9]

The parameter ${\displaystyle {\frac {1}{\sqrt {1-\beta ^{4}}}}}$ is often referred to as the velocity of approach factor^[8] and multiplying the coefficient of discharge by that parameter (as was done above) produces the flow coefficient ${\displaystyle C}$. Methods also exist for determining the flow coefficient as a function of the beta function ${\displaystyle \beta }$ 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, or venturi-nozzle does. Venturis also require much less straight pipe upstream. A venturi meter is more efficient, but usually more expensive and less accurate (unless calibrated in a laboratory) than an orifice plate.

Compressible flow

In general, equation (2) is applicable only for incompressible flows. It can be modified by introducing the expansibility factor, (also called the expansion factor) ${\displaystyle \epsilon }$ to account for the compressibility of gasses.

${\displaystyle q_{m}=\rho _{1}\;q_{v,1}=C\;\epsilon \;A_{2}\;{\sqrt {2\;\rho _{1}\;(p_{1}-p_{2})}}}$

${\displaystyle \epsilon }$ is 1.0 for incompressible fluids and it can be calculated for compressible gases^[9] using empirically determined formulae as shown below in computation.

For smaller values of β (such as restriction plates with β less than 0.25 and discharge from tanks), if the fluid is compressible, the rate of flow depends on whether the flow has become choked. If it is, then the flow may be calculated as shown at choked flow (although the flow of real gases through thin-plate orifices never becomes fully choked^{[lower-alpha 1][13]} By using a mechanical energy balance, compressible fluid flow in un-choked conditions may be calculated as: ^{[10] [11] [14]}

${\displaystyle q_{m}=C\;A_{2}\;{\sqrt {2\;\rho _{1}\;p_{1}\;{\bigg (}{\frac {\gamma }{\gamma -1}}{\bigg )}{\bigg [}(p_{2}/p_{1})^{2/\gamma }-(p_{2}/p_{1})^{(\gamma +1)/\gamma }{\bigg ]}}}}$

or

${\displaystyle q_{v}=C\;A_{2}\;{\sqrt {2\;{\frac {p_{1}}{\rho _{1}}}\;{\bigg (}{\frac {\gamma }{\gamma -1}}{\bigg )}{\bigg [}(p_{2}/p_{1})^{2/\gamma }-(p_{2}/p_{1})^{(\gamma +1)/\gamma }{\bigg ]}}}}$

Under choked flow conditions, the fluid flow rate becomes: ^[10]

${\displaystyle q_{m}=C\;A_{2}\;{\sqrt {\gamma \;\rho _{1}\;p_{1}\;{\bigg (}{\frac {2}{\gamma +1}}{\bigg )}^{\frac {\gamma +1}{\gamma -1}}}}}$

or

${\displaystyle q_{v}=C\;A_{2}\;{\sqrt {\gamma \;{\frac {p_{1}}{\rho _{1}}}\;{\bigg (}{\frac {2}{\gamma +1}}{\bigg )}^{\frac {\gamma +1}{\gamma -1}}}}}$

where:
${\displaystyle \gamma }$	= heat capacity ratio (${\displaystyle c_{p}/c_{v}}$), dimensionless (${\displaystyle \gamma \approx 1.4}$ for air)
${\displaystyle q_{m}}$, ${\displaystyle q_{v}}$	= mass and volumetric flow rate, respectively, kg/s and m³/s
${\displaystyle \rho _{1}}$	= real gas density under upstream conditions, kg/m³
	and other symbols are defined as above

Coefficient of discharge

Coefficient of discharge for sharp-edged orifice plates with corner, flange or D and D/2 tappings and no drain or vent hole (Reader-Harris/Gallagher equation):

${\displaystyle C=0.5961+0.0261\beta ^{2}-0.216\beta ^{8}+0.000521{\bigg (}{\frac {10^{6}\beta }{Re_{D}}}{\bigg )}^{0.7}+(0.0188+0.0063A)\beta ^{3.5}{\bigg (}{\frac {10^{6}}{Re_{D}}}{\bigg )}^{0.3}+(0.043+0.080\exp(-10{L_{1}})-0.123\exp(-7{L_{1}}))(1-0.11A){\frac {\beta ^{4}}{1-\beta ^{4}}}-0.031(M'_{2}-0.8{M'_{2}}^{1.1})\beta ^{1.3}}$

and if D < 71.2mm in which case this further term is added to C:

${\displaystyle +0.011(0.75-\beta ){\bigg (}2.8-{\frac {D}{0.0254}}{\bigg )}}$ ^{[16][lower-alpha 2]}

In the equation for C,

${\displaystyle A={\bigg (}{\frac {19000\beta }{Re_{D}}}{\bigg )}^{0.8}}$

${\displaystyle M'_{2}={\frac {2L'_{2}}{1-\beta }}}$

and only the three following pairs of values for L₁ and L'₂ are valid:

corner tappings: ${\displaystyle L_{1}=L'_{2}=0}$

flange tappings: ${\displaystyle L_{1}=L'_{2}={\frac {0.0254}{D}}}$^{[lower-alpha 2]}

D and D/2 tappings:

${\displaystyle L_{1}=1}$

${\displaystyle L'_{2}=0.47}$

Expansibility factor

Expansibility factor, also called expansion factor, for sharp-edged orifice plates with corner, flange or D and D/2 tappings:

if ${\displaystyle p_{2}/p_{1}>0.75}$^{[16]: 5.3.2.2} (at least - standards vary)^{[lower-alpha 3]}

${\displaystyle \epsilon =1-(0.351+0.256\beta ^{4}+0.93\beta ^{8}){\bigg [}1-{\bigg (}{\frac {p_{2}}{p_{1}}}{\bigg )}^{\frac {1}{\kappa }}{\bigg ]}}$ ^[16]

but for incompressible fluids, including most liquids

${\displaystyle \epsilon =1}$

where:
${\displaystyle C}$	= coefficient of discharge, dimensionless
${\displaystyle d}$	= internal orifice diameter under operating conditions, m
${\displaystyle D}$	= internal pipe diameter under operating conditions, m
${\displaystyle p_{1}}$	= fluid absolute static pressure in plane of upstream tapping, Pa
${\displaystyle p_{2}}$	= fluid absolute static pressure in plane of downstream tapping, Pa
${\displaystyle q_{m}}$	= mass flow rate, kg/s
${\displaystyle q_{v}}$	= volume flow rate, m3/s
${\displaystyle Re_{D}}$	= pipe Reynolds number, ${\displaystyle {\frac {4q_{m}}{\pi \mu D}}}$, dimensionless
${\displaystyle \beta }$	= diameter ratio of orifice diameter to pipe diameter, ${\displaystyle {\frac {d}{D}}}$, dimensionless
${\displaystyle \Delta p}$	= differential pressure, Pa
${\displaystyle \epsilon }$	= expansibility factor, also called expansion factor, dimensionless
${\displaystyle \kappa }$	= isentropic exponent, often approximated by specific heat ratio, dimensionless
${\displaystyle \mu }$	= dynamic viscosity of the fluid, Pa.s
${\displaystyle \rho _{1}}$	= fluid density in plane of upstream tapping, kg/m³

Overall pressure loss

The overall pressure loss caused by an orifice plate is less than the differential pressure measured across tappings near the plate. For sharp-edged plates such as corner, flange or D and D/2 tappings, it can be approximated by the equation

${\displaystyle {\frac {\Delta {\bar {\omega }}}{\Delta p}}=1-\beta ^{1.9}}$^[16]: 13

or

${\displaystyle {\frac {\Delta {\bar {\omega }}}{\Delta p}}={\frac {{\sqrt {1-\beta ^{4}(1-C^{2})}}-C\beta ^{2}}{{\sqrt {1-\beta ^{4}(1-C^{2})}}+C\beta ^{2}}}}$ ^[16]: 13

where
${\displaystyle \Delta {\bar {\omega }}}$	= overall pressure loss, Pa
	and other symbols are as above

Notes

Citations

Sources

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