1. Introduction

Airflow calculations through a nozzle are crucial for understanding and optimizing pneumatic systems, industrial tools, and flow machines. The goal is to estimate the amount of air flowing through a nozzle at a given pressure, which requires not only the use of formulas but, above all, knowledge of fluid dynamics and thermodynamics.

2. Basics of Fluid Flow

Flow through a nozzle is a complex interaction of pressure, velocity, and volume. The key principles are the conservation of mass and energy, described by the continuity equation and Bernoulli's principle.

The continuity equation is expressed by the formula:

$$\dot{m}=\rho \cdot A\cdot v$$

where:

• $\dot{m}$ — mass flow (kg/s)
• $\rho$ — air density (kg/m³)
• $A$ — cross-sectional area (m²)
• $v$ — flow velocity (m/s)

3. The Role of Compressibility

Air, as a gas, is compressible — its density changes significantly with pressure and temperature. For subsonic flows, isothermal or adiabatic simplifications can be assumed. An example formula for density:

$$\rho =\frac{P}{RT}$$

• $P$ — absolute temperature (K)
• $R$ — air gas constant (≈287 J/kg·K)
• $T$ — absolute temperature (K)

4. Cross-Sectional Area

The cross-sectional area of the nozzle is calculated from the diameter:

$$A=\pi {\left(\frac{d}{2}\right)}^2$$

For diameter in millimeters, the area is obtained in mm², which is often used in engineering with empirical coefficients.

5. Flow Coefficient

To account for turbulence, resistance, and shape, a flow coefficient is used $C$, expressed in l/s/mm²:

$$Q=C\cdot A$$

A typical value is 0.98 l/s/mm² for air under standard conditions.

6. Unit Conversion

• From l/s to l/min:
  $$Q_{l/min}=Q_{l/s}\cdot 60$$
• From l/min to m³/min:
  $$Q_{m^3/min}=\frac{Q_{l/min}}{1000}$$
• From l/min to CFM (cubic feet per minute):
  $$Q_{CFM}=\frac{Q_{l/min}}{28.3168}$$

7. Pressure Conversion

In Anglo-Saxon countries, the PSI unit is used. Conversion from bar is as follows:

$$P_{PSI}=P_{bar}\cdot 14.5038$$

8. From Theory to Practice

Although the above formulas are precise, in practice engineers use a simplified algorithm:

• Nozzle diameter and pressure as input data
• Constant density or flow coefficient
• Flow coefficient $C$
• Conversions to l/min, m³/min, CFM

This approach is practical where advanced CFD analysis is not applied.

9. Practical Aspects

In reality, factors such as:

• Temperature: affects air density
• Altitude: affects atmospheric pressure
• Nozzle geometry: sharp or rounded edges
• Choked flow: when it reaches the speed of sound

10. Summary

Understanding nozzle flow is based on physics — continuity equations, compressibility considerations, and the application of e.g. empirical data.