Process Engineering Tool
Ergun Equation Calculator
Packed Bed Pressure Drop
Calculate pressure drop through packed beds using the Ergun equation, including viscous and inertial contributions, total bed pressure loss, and useful engineering parameters.
- Basic & Advanced modes
- Pressure drop calculation
- Unit conversion
- Reynolds number
- Flow-to-velocity calculation
- Viscous & inertial analysis
- Shareable results
Enter your parameters and calculate to see the pressure drop result here.
Pressure Drop
—kPa/m
Viscous vs. Inertial Contribution
Detailed Calculation Breakdown
Viscous term
150 × μ × (1 − ε)² × v / (ε³ × dp²)
= — Pa/m
Inertial term
1.75 × ρ × (1 − ε) × v² / (ε³ × dp)
= — Pa/m
ΔP/L = Viscous Term + Inertial Term = — Pa/m
Total ΔP = (ΔP/L) × Bed Length = — Pa
Compare a Parameter
See how pressure drop changes as one parameter varies, holding everything else constant. Run a calculation above first.
Calculate a base result, then choose a parameter to compare.
How to Use This Calculator
- 1
Enter the fluid dynamic viscosity
Use the value for your process fluid at operating conditions.
- 2
Enter the fluid density
Use the density at the same operating conditions as the viscosity.
- 3
Enter the packed-bed porosity
The void fraction of the bed, between 0 and 1.
- 4
Enter the particle diameter
Use a representative or equivalent diameter for the packing.
- 5
Enter the superficial velocity
Or, in Advanced Mode, calculate it from a volumetric flow rate and column diameter.
- 6
Enter the packed-bed length
The total length of packing the fluid travels through.
- 7
Click Calculate Pressure Drop
Or press Enter from any field.
- 8
Review the results
Check the pressure drop, viscous and inertial contributions, and the detailed calculation.
What This Tool Offers
- Standard Ergun equation calculation
- Basic and Advanced modes
- Automatic unit conversion
- Total pressure drop & pressure drop per unit length
- Viscous and inertial contribution percentages
- Packed-bed Reynolds number
- Flow-rate to superficial-velocity calculation
- Column cross-sectional area
- Parameter comparison
- Detailed calculation breakdown
- Copy result
- Share result
- Mobile responsive interface
Formula Explanation
ΔP/L = 150 × μ × (1 − ε)² × v / (ε³ × dp²) + 1.75 × ρ × (1 − ε) × v² / (ε³ × dp)
Viscous / Laminar Term
150 × μ × (1 − ε)² × v / (ε³ × dp²)Dominates at low velocity; driven by fluid viscosity and particle size.
Inertial / Turbulent Term
1.75 × ρ × (1 − ε) × v² / (ε³ × dp)Dominates at high velocity; driven by fluid density and velocity squared.
| Symbol | Meaning |
|---|---|
| ΔP/L | Pressure drop per unit packed-bed length |
| μ | Dynamic viscosity of the fluid |
| ε | Void fraction or porosity of the packed bed |
| v | Superficial fluid velocity |
| dp | Characteristic particle diameter |
| ρ | Fluid density |
Particle Reynolds Number
Rep = ρ × v × dp / μ
The particle Reynolds number compares inertial to viscous forces at the scale of a single packing particle. It is a reference value for a packed bed, not a substitute for the full Ergun calculation, and it should not be read using the flow-regime thresholds used for flow inside a straight pipe. Low Rep corresponds to conditions where the viscous term tends to dominate the pressure drop; high Rep corresponds to conditions where the inertial term tends to dominate.
Engineering Reference: Typical Effects on Pressure Drop
These trends follow the Ergun model's assumptions and are not absolute rules for every packing or operating condition.
| Parameter | Typical effect on pressure drop |
|---|---|
| Smaller particle diameter | Strongly increases pressure drop |
| Larger particle diameter | Decreases pressure drop |
| Higher viscosity | Increases pressure drop |
| Higher density | Increases inertial pressure drop |
| Higher velocity | Increases pressure drop |
| Higher porosity | Generally decreases pressure drop |
| Longer bed | Increases total pressure drop |
Bed Porosity Reference
Porosity (ε) is the fraction of the packed-bed volume occupied by void space, where 0 < ε < 1. Common values for randomly packed beds are often in the range of about 0.30–0.50, but the appropriate value depends on the specific packing material and packing method — it should not be assumed the same value applies to every case.
Particle Diameter Reference
Particle diameter (dp) represents the characteristic diameter of the particles forming the packed bed. For non-spherical particles, an equivalent particle diameter may be required depending on the chosen engineering method. This calculator uses the standard Ergun equation exactly as published; it does not apply a sphericity correction unless you account for one in the particle diameter you enter.
Assumptions
- Flow is through a packed bed of particles.
- The packing is reasonably homogeneous or has been appropriately characterized.
- A representative particle diameter is available.
- Fluid properties are appropriately represented by the entered values.
- Superficial velocity, not interstitial velocity, is used in the equation.
- The standard Ergun coefficients (150 and 1.75) are applied.
- The calculation estimates pressure drop and does not replace detailed equipment design.
Important Engineering Notes
Actual packed-bed pressure drop can differ from this theoretical calculation due to factors including:
References
| Reference | Information used | Source |
|---|---|---|
| Ergun, S. (1952) | Original correlation for pressure drop through packed columns; basis of the standard equation and coefficients (150, 1.75) used in this calculator. | Chemical Engineering Progress, 48(2), 89–94 |
| McCabe, W. L., Smith, J. C., & Harriott, P. | Standard chemical engineering treatment of flow through packed beds and the Ergun equation. | Unit Operations of Chemical Engineering (textbook) |
| Perry, R. H., & Green, D. W. (Eds.) | Reference data and design guidance for packed-bed flow and pressure drop. | Perry's Chemical Engineers' Handbook |
Calculating Pressure Drop…
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| dP/L | Pressure drop per unit bed length | Pa/m |
| mu | Fluid viscosity | Pa.s |
| e | Bed void fraction | |
| v | Superficial velocity | m/s |
| dp | Particle diameter | m |
| rho | Fluid density | kg/m3 |
What it means
The Ergun equation is used to calculate the pressure drop per unit length in a packed bed of particles. It is a combination of the Kozeny‑Carman equation for laminar flow (viscous term) and the Burke‑Plummer equation for turbulent flow (inertial term). The equation is: dP/L = 150 μ (1−ε)² v / (ε³ d_p²) + 1.75 ρ (1−ε) v² / (ε³ d_p), where μ is fluid viscosity, ρ is fluid density, v is superficial velocity, d_p is particle diameter, and ε is void fraction. The first term dominates at low Reynolds numbers (creeping flow), while the second dominates at high Re. This equation is essential for designing fixed‑bed reactors, catalytic converters, and packed columns in absorption and distillation. It is also used in filtration and in the design of sand filters. Accurate prediction of pressure drop is critical for sizing pumps or compressors and for ensuring proper fluid distribution. The Ergun equation is widely used in chemical engineering and is a standard reference for packed bed hydrodynamics.
Worked example
Ergun Equation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| μ | 0.001 Pa·s |
| ε | 0.4 |
| v | 0.02 m/s |
| dp | 0.005 m |
| ρ | 1000 kg/m³ |
| Parameter | Value |
|---|---|
| μ | 2×10⁻⁵ |
| ε | 0.4 |
| v | 0.5 |
| dp | 0.01 |
| ρ | 1.2 |
Common mistakes
- Void fraction ε: The fraction of bed volume that is voids (porosity); typical values 0.35‑0.45 for random packing.
- Particle diameter d_p: For non‑spherical particles, use the equivalent sphere diameter.
- Viscosity μ and density ρ: Fluid properties at the bed temperature.
- Superficial velocity v: Flow velocity based on the empty tube cross‑section (not the interstitial velocity).
- Units: Ensure consistent SI units – the pressure drop will be in Pa/m.
Applications
The Ergun equation predicts the pressure drop through a packed bed of particles, combining viscous (Darcy) and inertial (Forchheimer) contributions. It is used extensively in the design of fixed‑bed catalytic reactors, adsorption columns, and packed distillation towers. The equation accounts for void fraction, particle diameter, fluid properties, and velocity. Engineers use it to calculate the required pump or compressor power, to assess the likelihood of fluidisation, and to design distributors. Accurate pressure drop estimation prevents over‑sizing of equipment and ensures uniform flow distribution. The Ergun equation is also applied in filtration, soil mechanics, and other porous media flow problems, making it a versatile tool in chemical and environmental engineering.
- Design of fixed‑bed catalytic reactors (e.g., steam methane reforming)
- Sizing of adsorption and ion‑exchange columns
- Design of packed distillation and absorption columns
- Assessment of fluidisation and entrainment risks
- Porous media flow in filtration and soil mechanics
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