Calculator Inputs
Choose what you want to solve. Enter known values, select units, then calculate.
Formula Used
The Euler number compares pressure forces to inertial forces in a flow:
Rearranged forms are used to solve for Δp, ρ, or v when needed.
How to Use This Calculator
- Select what you want to solve for (Eu, Δp, ρ, or v).
- Choose the convention that matches your reference.
- Enter the known values and select appropriate units.
- Click Calculate to see results above the form.
- Use Download CSV or Download PDF to export.
Example Data Table
| Case | Δp (Pa) | ρ (kg/m³) | v (m/s) | Convention | Eu (dimensionless) |
|---|---|---|---|---|---|
| Water, moderate drop | 2500 | 998 | 3.0 | k = 1 | 0.27833 |
| Air, higher speed | 400 | 1.20 | 25 | k = 1 | 0.53333 |
| Same as Case 1, dynamic pressure | 2500 | 998 | 3.0 | k = 0.5 | 0.55667 |
These examples illustrate how the selected convention changes Eu by a factor of two.
Engineering Notes
- Large Eu often indicates pressure effects dominate relative to inertia.
- For similarity studies, matching Eu helps replicate pressure behavior across scales.
- Use consistent reference points when defining Δp (inlet-outlet, stagnation-static, etc.).
Euler Number in Fluid Systems: Practical Guide
1) What the Euler Number Represents
The Euler number (Eu) is a dimensionless ratio that compares a characteristic pressure difference to inertial effects in a moving fluid. It is commonly written as Eu = Δp/(ρv²) or Eu = Δp/(0.5ρv²), depending on whether you reference full or dynamic pressure. Because Eu is dimensionless, it supports direct comparison between different fluids, sizes, and operating conditions when the same reference definitions are used.
2) Choosing Δp for Meaningful Results
Δp should represent the pressure change that matters for your engineering question: inlet–outlet drop across a valve, loss through a bend, stagnation–static difference, or pump head converted to pressure. A clear choice of reference points reduces ambiguity and makes Eu values transferable between designs and test reports.
3) Typical Density and Velocity Scales
Density varies widely: air near room conditions is about 1.2 kg/m³, water is near 998–1000 kg/m³, and many oils lie around 800–900 kg/m³. Velocity often ranges from below 1 m/s in large channels to 20–40 m/s in compact ducts. Because Eu scales with 1/v², doubling velocity reduces Eu by roughly a factor of four when Δp is unchanged.
4) Interpreting Large and Small Eu
Large Eu generally indicates that pressure effects dominate over inertia for the selected reference scale, which is common in strong restrictions or high-loss components. Small Eu suggests inertia dominates for the same Δp reference, often associated with smoother passages or smaller pressure gradients. Interpretation is only valid when Δp, ρ, and v represent the same physical location and scale.
5) Relation to Pressure Coefficient and Loss Terms
With consistent definitions, Eu is closely related to commonly used pressure or loss coefficients that scale with Δp/(0.5ρv²). If your field uses the dynamic-pressure form, select the 0.5ρv² convention in this calculator for quick alignment with typical component-loss reporting and many lab measurement formats.
6) Similarity and Scaling Between Models
In similarity studies, matching Eu helps reproduce pressure behavior across different sizes or fluids. For example, a water test rig and an air prototype can be compared if Eu is matched using consistent reference velocity and pressure points. Eu is often considered alongside other dimensionless numbers that capture viscosity or compressibility effects when those are important.
7) Measurement Tips and Data Quality
Use calibrated pressure transducers for small Δp values and verify tubing or tap placement to avoid local disturbances. For density, prefer measured temperature and composition over generic constants when precision matters. For velocity, a volumetric flow rate divided by cross-sectional area often provides a stable mean value.
8) Worked Data Insight
Consider water with Δp = 2500 Pa, ρ = 998 kg/m³, and v = 3 m/s. Using Eu = Δp/(ρv²) gives Eu ≈ 0.278. Using the dynamic-pressure convention Eu = Δp/(0.5ρv²) gives Eu ≈ 0.556. The factor-of-two change highlights why documenting the chosen convention is essential for correct comparison.
FAQs
1) Is the Euler number always dimensionless?
Yes. Eu is a ratio of pressure to inertial terms. When Δp is in pascals, ρ in kg/m³, and v in m/s, the units cancel, leaving a dimensionless quantity.
2) Which convention should I choose: ρv² or 0.5ρv²?
Use the convention that matches your reference or reporting standard. Many loss coefficients use 0.5ρv² (dynamic pressure). Some texts use ρv². The calculator supports both for consistency.
3) What velocity should I use in pipes or ducts?
Typically use the mean velocity: v = Q/A, where Q is volumetric flow rate and A is cross-sectional area. If your Δp is local, consider a local characteristic velocity that matches the measurement location.
4) Can I use gauge pressure for Δp?
Yes, as long as Δp is a pressure difference between two points using the same reference. The absolute offset cancels. Just ensure both measurements use the same datum and the same sign convention.
5) Does compressibility affect Eu calculations?
Eu can still be computed, but interpretation may change if density varies significantly. For high-speed gas flows, use density appropriate to the measurement conditions and consider compressibility metrics in parallel.
6) Why does Eu change a lot when velocity changes slightly?
Because Eu scales with 1/v². A small percentage increase in velocity produces roughly twice that percentage decrease in Eu, squared. This sensitivity makes accurate velocity estimation important for reliable Eu values.
7) What is a common use of Eu in equipment selection?
Eu helps compare pressure behavior of components such as valves, orifices, and restrictions under different operating conditions. When Eu is consistent, pressure-loss performance is often comparable across geometrically similar setups.