The Orifice Flow Equation

An orifice plate is a thin plate with a precisely machined hole that creates a pressure drop when fluid flows through it. This pressure drop is used to measure flow rate or to intentionally restrict flow. The fundamental orifice equation is:

Q = Cd * A * E * Y * sqrt(2 * dP / rho)

• Q: Volumetric flow rate (m3/s or ft3/s)
• Cd: Discharge coefficient (dimensionless, typically 0.59-0.65)
• A: Orifice area = pi * d2 / 4
• E: Velocity of approach factor = 1 / sqrt(1 - beta4)
• Y: Expansion factor (1.0 for incompressible liquids)
• dP: Differential pressure across the orifice
• rho: Fluid density

Discharge Coefficient (Cd)

The discharge coefficient accounts for the actual flow being less than ideal due to the formation of the vena contracta (the point of minimum flow area downstream of the orifice). Cd depends on:

• Beta ratio: Higher beta = higher Cd (less restriction)
• Reynolds number: Affects Cd at low flows
• Orifice edge geometry: Sharp, square, or rounded edges
• Pressure tap location: Corner, flange, or D-D/2 taps

For sharp-edge orifices per ISO 5167, the Reader-Harris/Gallagher equation provides Cd values with approximately 0.5% uncertainty for beta ratios between 0.2 and 0.75.

Beta Ratio (d/D)

The beta ratio is the ratio of orifice diameter (d) to pipe inner diameter (D). It is a critical design parameter:

beta = d / D

• 0.20 - 0.30: Low beta - high pressure drop, good turndown ratio
• 0.30 - 0.50: Common range for flow measurement
• 0.50 - 0.65: Moderate beta - balance of accuracy and pressure loss
• 0.65 - 0.75: High beta - low pressure drop, reduced accuracy

ISO 5167 Limits: Beta must be between 0.20 and 0.75 for standard correlations. Outside this range, accuracy degrades significantly.

Pressure Tap Locations

Different tap configurations affect the measured differential pressure and the applicable Cd correlation:

• Corner Taps: Located at the orifice plate faces (L1=L2=0). Best for small pipes.
• D and D/2 Taps: Upstream tap at 1D, downstream at D/2. Measures maximum pressure differential.
• Flange Taps: Located 1 inch (25.4mm) from orifice faces. Most common in USA.

Orifice Edge Types

• Sharp-Edge: Standard square edge with sharp upstream corner (ISO 5167). Cd approximately 0.60-0.65. Most common for flow measurement.
• Square-Edge: Thick plate with square edges. Higher Cd (0.65-0.70). Used for restriction orifices.
• Rounded/Beveled: Radiused or chamfered inlet. Highest Cd (0.70-0.98). Lower permanent pressure loss.

Permanent Pressure Loss

Unlike the measured differential pressure, the permanent pressure loss is the unrecoverable energy lost:

dP_permanent = dP * (1 - beta^2) * sqrt(1 - beta^4)

For a beta of 0.5, approximately 73% of the measured dP is permanently lost. This represents pumping energy cost.