Orifice plate

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An orifice plate is a device which measures the rate of fluid flow. It uses the same principle as a Venturi nozzle, namely Bernoulli's principle which says 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.

Flat-plate, sharp-edge orifice

An orifice plate is basically a thin plate with a hole in the middle. It is usually placed in a pipe in which fluid flows. As fluid flows through the pipe, it has a certain velocity and a certain pressure. When the fluid reaches the orifice plate, with the hole in the middle, 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.

1 Flow of incompressible or compressible fluids through an orifice
2 Flow of gases through an orifice
3 See also
4 External links
5 References

[edit] Flow of incompressible or compressible fluids through an orifice

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

$P_1 + \frac{1}{2}\cdot\rho_1\cdot V_1^2 = P_2 + \frac{1}{2}\cdot\rho_1\cdot V_2^2$

or:

$P_1 - P_2 = \frac{1}{2}\cdot\rho_1\cdot V_2^2 - \frac{1}{2}\cdot\rho_1\cdot V_1^2$

with:

$Q = A\cdot V$ or $V = Q / A$

and $Q 1 = Q 2$ :

$P_1 - P_2 = \frac{1}{2}\cdot\rho_1\cdot \bigg(\frac{Q_1}{A_2}\bigg)^2 - \frac{1}{2}\cdot\rho_1\cdot\bigg(\frac{Q_1}{A_1}\bigg)^2$

Solving for $Q 1$ :

$Q_1 = A_2\;\sqrt{\frac{2\;(P_1-P_2)/\rho_1}{1-(A_2/A_1)^2}}$

and:

$Q_1 = A_2\;\sqrt{\frac{1}{1-(d_2/d_1)^4}}\;\sqrt{2\;(P_1-P_2)/\rho_1}$

and introducing the beta factor $β = d 2 / d 1$ as well as the coefficient of discharge $C d$ :

$Q_1 = C_d\; A_2\;\sqrt{\frac{1}{1-\beta^4}}\;\sqrt{2\;(P_1-P_2)/\rho_1}$

And finally introducing the expansion factor $Y$ to account for the compressibility of gases and 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 (either compressible or incompressible) upstream of the orifice:

$(1)\qquad Q_1 = C\;Y\;A_2\;\sqrt{2\;(P_1-P_2)/\rho_1}$

Multiplying by the density upstream of the orifice to obtain the equation for the mass flow of the fluid (either compressible or incompressible) at any point in the fluid flow:^[1]^[2]^[3]^[4]

$(2)\qquad \dot{m} = \rho_1\;Q_1 = C\;Y\;A_2\;\sqrt{2\;\rho_1\;(P_1-P_2)}$

where:
$Q 1$	= upstream volumetric flow rate, m³/s
$\dot{m}$	= mass flow rate at any point, kg/s
$C d$	= coefficient of discharge, dimensionless
$C$	= orifice flow coefficient, dimensionless (often denoted as $K$ )
$Y$	= expansion factor, 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
$β$	= 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 ·s)
$P 2$	= fluid downstream pressure, Pa with dimensions of kg/(m ·s)
$ρ 1$	= upstream 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.^[2]

The parameter $\sqrt{1-\beta^4}$ is often referred to as the velocity of approach factor ^[1] 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 $β$ 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.

The expansion factor, $Y$ , accounts for the compressibility of gases. It is 1.0 for incompressible fluids and it can be calculated for compressible gases.^[2]

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 as a flowmeter.

[edit] Flow of gases through an orifice

The expansion factor $Y$ , which allows for the change in the density of an ideal gas as it expands adiabatically, is given by:^[2]

$Y =\;\sqrt{r^{2/k}\bigg(\frac{k}{k-1}\bigg)\bigg(\frac{\;1-r^{(k-1)/k\;}}{1-r}\bigg)\bigg(\frac{1-\beta^4}{1-\beta^{4}\;r^{2/k}}\bigg)}$

For values of $β$ less than 0.25, $β 4$ approaches 0 and the last bracketed term in the above equation approaches 1. Thus, for the large majority of orifice plate installations:

$(3)\qquad Y =\;\sqrt{r^{2/k}\bigg(\frac{k}{k-1}\bigg)\bigg(\frac{\;1-r^{(k-1)/k\;}}{1-r}\bigg)}$

where:
$Y$	= Expansion factor, dimensionless
$r$	= $P 2 / P 1$
$k$	= specific heat ratio ( $c p / c v$ ), dimensionless

Substituting equation (3) into the mass flow rate equation (2):

$\dot{m} = C\;A_2\;\sqrt{2\;\rho_1\;\bigg (\frac{k}{k-1}\bigg)\bigg[\frac{(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}}{1-P_2/P_1}\bigg](P_1-P_2)}$

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 the non-choked (i.e., sub-sonic) flow of real gases through an orifice:

$(4)\qquad \dot{m} = C\;A_2\;\sqrt{2\;\rho_1\;P_1\;\bigg (\frac{k}{k-1}\bigg)\bigg[(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}\bigg]}$

Using the ideal gas law and the compressibility factor (which corrects for non-ideal gases), the equivalent equation is obtained for the non-choked flow of real gases through an orifice:^[3]^[4]^[5]

$(5)\qquad \dot{m} = C\;A_2\;P_1\;\sqrt{\frac{2\;M}{Z\;R\;T_1}\bigg(\frac{k}{k-1}\bigg)\bigg[(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}\bigg]}$

where:
$\dot{m}$	= mass flow rate at any point, kg/s
$C$	= orifice flow coefficient, dimensionless (often denoted as $K$ )
$A 2$	= cross-sectional area of the orifice hole, m²
$ρ 1$	= upstream real gas density, kg/m³
$P 1$	= upstream gas pressure, Pa with dimensions of kg/(m ·s)
$P 2$	= downstream pressure in the orifice hole, Pa with dimensions of kg/(m ·s)
$M$	= the gas molecular mass, kg/kmol (also known as the molecular weight)
$R$	= the Universal Gas Law Constant = 8314.5 Pa·m³/(kmol·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.

[edit] See also

[edit] External links

[edit] References

^ ^a ^b Lecture, University of Sydney
^ ^a ^b ^c ^d Perry, Robert H. and Green, Don W. (1984). Perry's Chemical Engineers' Handbook, Sixth Edition, McGraw Hill. ISBN 0-07-049479-7.
^ ^a ^b 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 Procedures
^ ^a ^b "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, page 2.68, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005. PGS2 CPR 14E