The standard differential pressure (DP) flow equation is derived directly from Bernoulli's Equation (conservation of energy) and the Continuity Equation (conservation of mass) for an incompressible, frictionless fluid flowing through a horizontal conduit.

1. Bernoulli's Equation:

$$P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$$

Where $P$ is static pressure, $\rho$ is density, and $v$ is velocity at the inlet (1) and throat (2).

2. Continuity Equation:

$$Q = A_1 v_1 = A_2 v_2 \implies v_1 = v_2 \left(\frac{A_2}{A_1}\right) = v_2 \beta^2$$

Where $A$ is the cross-sectional area and $\beta = d/D$. Substituting $v_1$ back into Bernoulli's equation yields:

$$(P_1 - P_2) = \frac{1}{2}\rho \left(v_2^2 - v_2^2 \beta^4\right) = \frac{1}{2}\rho v_2^2 \left(1 - \beta^4\right)$$

Solving for the throat velocity $v_2$:

$$v_2 = \frac{1}{\sqrt{1 - \beta^4}} \sqrt{\frac{2(P_1 - P_2)}{\rho}}$$

Multiplying by the throat area $A_2$ to find volumetric flow $Q$, and introducing an empirical discharge coefficient $C_d$ to account for viscous shear friction and contractions, we get:

$$Q = \frac{C_d \cdot A_2}{\sqrt{1 - \beta^4}} \sqrt{\frac{2 \Delta P}{\rho}}$$

The Reader-Harris/Gallagher (RG) equation is the globally accepted empirical correlation used in ISO 5167-2 to calculate the discharge coefficient ($C_d$) of orifice plates. It replaced older, less accurate correlations like the Stolz equation.

Significance:

• It was developed by fitting a massive database of over 9,000 experimental data points gathered under highly controlled lab conditions.
• It accounts for three specific tap geometries: flange taps, D and D/2 taps, and corner taps through dimensionless location parameters ($L_1$ and $L'_2$).
• It incorporates Reynolds number dependency ($Re$) and diameter ratios ($\beta$) to model boundary layer effects and throat velocity profile deviations.
• It reduces the flow measurement uncertainty of standard orifice installations to less than 0.5% for typical industrial ranges.

For incompressible liquids, density $\rho$ remains constant as the fluid accelerates through the throat. However, for compressible gases and steam, the pressure drop $\Delta P$ causes the fluid to expand, reducing its density locally at the throat.

The Gas Expansion Factor ($Y$) corrects for this density decrease. It is defined as the ratio of mass flow rate of a compressible fluid to that of an incompressible fluid at the same upstream density:

$$Y = \frac{\dot{m}_{\text{compressible}}}{\dot{m}_{\text{incompressible}}}$$

ISO 5167-2 defines the empirical expansion factor for orifice plates as:

$$Y = 1 - (0.3512 + 0.2568\beta^4 + 0.93\beta^8)\cdot\left[1 - \left(\frac{P_2}{P_1}\right)^{1/\kappa}\right]$$

Where $\kappa$ is the isentropic exponent (specific heat ratio $C_p/C_v$) and $P_2/P_1 = 1 - \Delta P / P_1$. For liquids, $Y = 1.0$.

DP meters create differential pressure by restricting flow, converting static pressure into dynamic pressure. Part of this drop is recovered downstream, but friction and eddy turbulence cause a permanent pressure loss (PPL, denoted as $\Delta\varpi$).

Example Calculation: Consider $\Delta P = 100 \text{ kPa}$ and $\beta = 0.5$, $C_d \approx 0.60$:

1. Orifice Plate: High turbulence behind the sharp face leads to maximum loss: $$\Delta\varpi \approx \Delta P \cdot (1 - \beta^2) = 100 \cdot (1 - 0.25) = 75 \text{ kPa lost}$$
2. Flow Nozzle: Guided curve reduces boundary separation: $$\Delta\varpi \approx \Delta P \cdot \frac{1 - \beta^2}{1 + \beta^2} = 100 \cdot \frac{0.75}{1.25} = 60 \text{ kPa lost}$$
3. Venturi Tube: Converging/diverging profile (7-15° angle) allows smooth deceleration, recovering up to 90% of pressure: $$\Delta\varpi \approx \Delta P \cdot 0.12 = 12 \text{ kPa lost}$$

Conclusion: Venturis offer massive energy savings, translating to lower pumping and compression utility costs over the plant's lifetime.

The vena contracta is the point in a fluid stream where the fluid velocity is at its maximum, and the cross-sectional area of the flow stream is at its minimum. This occurs slightly downstream of the physical orifice plate restriction due to fluid momentum preventing the flow lines from making a sharp 90-degree turn.

Because velocity is highest at the vena contracta, Bernoulli's principle dictates that static pressure is lowest at this exact point. Sizing tools locate pressure taps relative to this phenomenon:

• Flange Taps: Standardized at 1 inch (25.4 mm) upstream and 1 inch downstream of the orifice plate faces. Extremely common due to manufacturing ease.
• D and D/2 Taps: Locate the upstream tap at 1 Pipe Diameter ($D$) and downstream at $D/2$ (the nominal vena contracta position). Offers maximum differential pressure sensitivity.
• Corner Taps: Taps are located directly in the corner of the flange faces, immediately upstream and downstream of the plate. Favored for small pipes ($D < 50\text{ mm}$).

To achieve the standard uncertainty levels specified in ISO 5167, the flow profile upstream of the primary element must be a fully developed turbulent profile without swirl or asymmetric distortion.

Straight Pipe Runs:

• Upstream straight pipe run requirements vary from 10D to over 44D, depending on the severity of the upstream disturbance (e.g., single elbow vs. two out-of-plane elbows, or control valves).
• Downstream straight run is typically 4D to 8D.
• If straight pipe runs are limited by space, a flow conditioner or straightener (like a Zanker or tube bundle) must be installed to decouple and dissolve swirl patterns before the fluid reaches the plate.

A differential pressure transmitter measures the physical pressure drop ($\Delta P$) across the primary element, which maps to flow rate through a square-root relationship: $Q \propto \sqrt{\Delta P}$. Because of this, the transmitter or control system must extract the square root of the measured $\Delta P$ to linearize the flow signal.

Calibration Procedure (5-Point Calibration):

1. Apply 0%, 25%, 50%, 75%, and 100% of the maximum calibrated $\Delta P$ using a pressure calibrator.
2. Observe the corresponding transmitter current loop output (4 to 20 mA).
3. Square Root Extraction mapping:

   Input DP (%)	Linear Output (mA)	Square Root Output (mA)	Flow Rate (%)
   0%	4.00 mA	4.00 mA	0%
   25%	8.00 mA	12.00 mA	50%
   50%	12.00 mA	15.31 mA	70.7%
   75%	16.00 mA	17.86 mA	86.6%
   100%	20.00 mA	20.00 mA	100%

Modern plants extract the square root in the DCS (Distributed Control System) rather than the transmitter to prevent signal noise amplification at low flow rates (below 10% flow).

In steam flow lines, high-temperature steam will condense inside the transmitter impulse lines, forming liquid water columns. If one impulse line condenses faster than the other, it creates an artificial hydro-static pressure imbalance, causing a zero-shift error in the DP transmitter.

To prevent this, condensate pots (wet legs) are installed at the tap points. They are pre-filled with water so that both impulse lines are kept constantly full of water at equal heights. The hydro-static heads on the positive and negative sides of the DP transmitter cancel each other out:

$$HP_{\text{side}} = P_1 + \rho_{water} \cdot g \cdot H$$

$$LP_{\text{side}} = P_2 + \rho_{water} \cdot g \cdot H$$

$$\Delta P_{\text{measured}} = HP - LP = (P_1 - P_2) = \Delta P_{\text{flow}}$$

This wet-leg arrangement ensures that the hydro-static column $H$ does not corrupt the true dynamic pressure drop across the restriction.

Choosing the ideal Beta ratio ($\beta = d/D$) is a critical balancing act in instrumentation design:

• Low Beta ($\beta < 0.3$): Offers higher differential pressure ($\Delta P$) signal, making it highly sensitive and less prone to measurement noise. However, it creates high permanent pressure loss and risks choking flow in vapor lines.
• High Beta ($\beta > 0.7$): Results in low pressure drop and minimal permanent pressure loss. However, the $\Delta P$ signal is weak, and small errors in the bore edge shape or upstream piping profile will cause huge measurement uncertainties.
• Cd Stability: ISO 5167 recommended ranges ($0.2 \le \beta \le 0.75$) provide a region where the discharge coefficient is stable and least sensitive to Reynolds number fluctuations.

For gases and steam, operating temperature ($T$) and pressure ($P$) are continuously changing. Since gas density $\rho \propto P/T$, an uncompensated flow measurement will have significant errors if the process drifts from the original sizing design conditions.

Correction Formula:

$$Q_{\text{corrected}} = Q_{\text{indicated}} \cdot \sqrt{ \frac{P_{\text{actual}}}{P_{\text{design}}} \cdot \frac{T_{\text{design}}}{T_{\text{actual}}} \cdot \frac{Z_{\text{design}}}{Z_{\text{actual}}} }$$

Flow computers or smart multivariable transmitters continuously read absolute pressure and temperature inputs, recalculating density in real-time to solve this correction factor, ensuring custody-transfer levels of accuracy.