Data Source & Methodology

This calculator provides a determination of mass and volumetric flow rate for fluids and gases based on the principles of differential pressure measurement. The calculations are rigorously based on the international standard:

  • Authoritative Source: International Organization for Standardization (ISO)
  • Reference: ISO 5167-2:2003, "Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full - Part 2: Orifice plates".

All calculations are based strictly on the formulas and data provided by this source. The tool calculates for turbulent flow where the Reynolds number ($Re$) is sufficiently high.

The Formulas Explained

The fundamental equation for calculating mass flow rate ($Q_m$) through an orifice plate is:

$$ Q_m = \frac{C_d}{\sqrt{1-\beta^4}} \cdot Y \cdot \frac{\pi}{4} d^2 \sqrt{2 \cdot \Delta P \cdot \rho_1} $$

Key Components of the Equation

1. Beta Ratio ($\beta$): The ratio of the orifice diameter ($d$) to the pipe diameter ($D$).

$$ \beta = \frac{d}{D} $$

2. Incompressible Flow (Liquids): For liquids, the density is constant, so the Expansibility Factor ($Y$) is equal to 1. The formula simplifies to:

$$ Q_m = \frac{C_d}{\sqrt{1-\beta^4}} \cdot \frac{\pi}{4} d^2 \sqrt{2 \cdot \Delta P \cdot \rho} $$

3. Compressible Flow (Gases): For gases, the fluid expands and its density changes as it passes through the orifice. The Expansibility Factor ($Y$) accounts for this. This calculator uses the empirical formula for $Y$ (Reader-Harris/Stolz equation as per ISO 5167):

$$ Y = 1 - (0.41 + 0.35\beta^4) \frac{x_T}{\kappa} $$

where $x_T = \frac{\Delta P}{P_1}$ is the pressure ratio and $\kappa$ is the specific heat ratio.

4. Volumetric Flow Rate ($Q_v$): This is found by dividing the mass flow rate by the fluid density at upstream conditions ($\rho_1$).

$$ Q_v = \frac{Q_m}{\rho_1} $$

Glossary of Variables

  • $Q_m$ (Mass Flow Rate): The mass of fluid passing through the orifice per unit of time (e.g., kg/s).
  • $Q_v$ (Volumetric Flow Rate): The volume of fluid passing through the orifice per unit of time (e.g., m³/s).
  • $d$ (Orifice Diameter): The internal diameter of the orifice hole, in meters (m).
  • $D$ (Pipe Diameter): The internal diameter of the pipe, in meters (m).
  • $\Delta P$ (Differential Pressure): The pressure drop between the upstream ($P_1$) and downstream ($P_2$) taps, in Pascals (Pa).
  • $\rho_1$ (Upstream Density): The density of the fluid before the orifice plate, in kg/m³.
  • $P_1$ (Upstream Pressure): The absolute static pressure of the fluid before the plate (Required for gases), in Pascals (Pa).
  • $C_d$ (Discharge Coefficient): A unitless value that accounts for friction and non-ideal flow. It is determined empirically and is often ~0.61 for sharp-edged plates.
  • $\beta$ (Beta Ratio): The unitless ratio $d/D$.
  • $Y$ (Expansibility Factor): A unitless factor that accounts for the expansion of compressible fluids (gases). $Y=1$ for liquids.
  • $\kappa$ (Specific Heat Ratio): A unitless property of a gas, also known as the isentropic exponent $\gamma$. (e.g., ~1.4 for air, ~1.33 for steam).

How It Works: A Step-by-Step Example (Liquid)

Let's calculate the flow rate of water through a 100mm pipe with a 50mm orifice plate.

  • Fluid Type: Liquid (Water)
  • Orifice Diameter ($d$): 50 mm = $0.05 \text{ m}$
  • Pipe Diameter ($D$): 100 mm = $0.1 \text{ m}$
  • Differential Pressure ($\Delta P$): 25 kPa = $25000 \text{ Pa}$
  • Fluid Density ($\rho$): 998 kg/m³ (water at 20°C)
  • Discharge Coefficient ($C_d$): 0.61
  1. Calculate Beta Ratio ($\beta$):
    $\beta = d / D = 0.05 \text{ m} / 0.1 \text{ m} = 0.5$
  2. Calculate Orifice Area ($A_d$):
    $A_d = (\pi/4) \cdot d^2 = (\pi/4) \cdot (0.05)^2 = 0.001963 \text{ m}^2$
  3. Calculate Velocity Term:
    $1 / \sqrt{1 - \beta^4} = 1 / \sqrt{1 - 0.5^4} = 1 / \sqrt{1 - 0.0625} = 1.0328$
  4. Calculate Mass Flow ($Q_m$):
    For liquids, $Y=1$.
    $Q_m = C_d \cdot (1 / \sqrt{1 - \beta^4}) \cdot A_d \cdot \sqrt{2 \cdot \Delta P \cdot \rho}$
    $Q_m = 0.61 \cdot 1.0328 \cdot 0.001963 \cdot \sqrt{2 \cdot 25000 \cdot 998}$
    $Q_m = 0.001235 \cdot \sqrt{49,900,000} \approx 0.001235 \cdot 7064$
    $Q_m \approx 8.72 \text{ kg/s}$
  5. Calculate Volumetric Flow ($Q_v$):
    $Q_v = Q_m / \rho = 8.72 \text{ kg/s} / 998 \text{ kg/m}^3$
    $Q_v \approx 0.00874 \text{ m}^3/\text{s}$

Frequently Asked Questions (FAQ)

What is the Discharge Coefficient ($C_d$) and how do I find it?

The $C_d$ is an efficiency factor. It corrects the theoretical flow rate for real-world energy losses due to friction and turbulence. A value of 0.61 is a common approximation for sharp-edged orifices, but the true value depends on the Reynolds number ($Re$) and Beta Ratio ($\beta$). For high-accuracy work, you must use the Reader-Harris/Stolz equation (defined in ISO 5167) or use the value provided by your orifice plate manufacturer.

Why is the Beta Ratio ($\beta$) important?

The Beta Ratio is critical. ISO 5167 specifies that its calculations are only valid for $0.1 \le \beta \le 0.75$. Outside this range, measurement uncertainty increases significantly. A very small $\beta$ creates a large, permanent pressure loss, while a very large $\beta$ creates a small, difficult-to-measure differential pressure.

What's the difference between mass flow and volumetric flow?

Mass Flow ($Q_m$) is the mass of a substance passing a point per unit time (e.g., kg/s). It is independent of temperature and pressure changes. Volumetric Flow ($Q_v$) is the volume passing per unit time (e.g., m³/s). For gases, this value changes dramatically with pressure and temperature. Therefore, mass flow is the standard for gas measurement.

Why is the Expansibility Factor ($Y$) needed for gases but not liquids?

Liquids are considered incompressible, meaning their density ($\rho$) does not change when pressure changes. Gases are compressible. As a gas approaches the orifice, its pressure drops and it expands, causing its density to decrease. The $Y$ factor (which is always $\le 1$) corrects for this change in density, which is not accounted for in the basic Bernoulli equation.

When should I *not* use this calculator?

This calculator is for single-phase, Newtonian fluids in a full, circular pipe. Do not use it for:

  • Multi-phase flow (e.g., liquid and gas mixed, or liquid with suspended solids).
  • Non-Newtonian fluids (e.g., slurries, gels, polymers).
  • Laminar flow (very low Reynolds number, $Re < 4000$). This tool assumes turbulent flow.
  • Choked flow (when gas velocity reaches the speed of sound, though a basic correction is applied).

Tool developed by Ugo Candido.
Engineering content reviewed by the CalcDomain Editorial Board for ISO 5167 compliance.
Last accuracy review: