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Granular & Fluidized Bed Simulator

Ergun Equation Simulator — Pressure Drop in Packed Beds

Compute the pressure drop per unit length dP/L through a bed of spherical particles in real time from the Ergun equation dP/L = 150 mu u (1-eps)^2 / (eps^3 dp^2) + 1.75 rho u^2 (1-eps) / (eps^3 dp). From superficial velocity u, particle diameter dp, voidage eps and viscosity mu the tool returns dP/L, particle Reynolds number Re_p, minimum fluidisation velocity u_mf and the laminar share, and visualises the bed and the log-log dP/L vs u curve from the fixed-bed to the fluidisation regime.

Parameters
Superficial velocity u
mm/s
Particle diameter d_p
mm
Voidage eps
Dynamic viscosity mu
x10^-3 Pa.s

Defaults: u = 10 mm/s, dp = 2.0 mm, eps = 0.40, mu = 1.0 x 10^-3 Pa.s (water at 20 C), with fluid density rho = 1000 kg/m^3 fixed. The minimum fluidisation velocity u_mf assumes a particle density rho_p = 2500 kg/m^3 (sand).

Results
—
dP/L pressure gradient
—
Particle Re_p
—
Min. fluidisation u_mf
—
Laminar share (term 1)
Packed-bed side schematic

Side view of a cylindrical packed bed. Many spherical particles (diameter d_p) are packed at voidage eps, and fluid flows upward at superficial velocity u. Particle size and density mirror the current d_p and eps values.

dP/L vs u curve (log-log)

Horizontal axis: u (mm/s, log10). Vertical axis: dP/L (Pa/m, log10). Low velocities follow a slope-1 laminar regime, high velocities a slope-2 turbulent regime. The yellow marker shows the current (u, dP/L) and the dashed lines are the laminar and inertial asymptotes.

Theory & Key Formulas

For a bed of spherical particles, the pressure drop per unit length is the sum of a laminar and an inertial contribution:

$$\frac{\Delta P}{L} = \frac{150\,\mu\,u\,(1-\varepsilon)^2}{\varepsilon^3\,d_p^2} + \frac{1.75\,\rho\,u^2\,(1-\varepsilon)}{\varepsilon^3\,d_p}$$

$u$ is the superficial velocity (m/s), $d_p$ the spherical particle diameter (m), $\varepsilon$ the voidage, $\mu$ the dynamic viscosity (Pa.s) and $\rho$ the fluid density (kg/m^3). The first term is the Carman-Kozeny laminar flow contribution; the second is the Burke-Plummer inertial contribution. The particle Reynolds number is:

$$\mathrm{Re}_p = \frac{\rho\,u\,d_p}{\mu\,(1-\varepsilon)}$$

The laminar-approximation minimum fluidisation velocity (with particle density $\rho_p$) follows from the first term alone:

$$u_{mf} = \frac{\varepsilon^3\,d_p^2\,(\rho_p-\rho)\,g}{150\,\mu\,(1-\varepsilon)}$$

This tool fixes $\rho = 1000$ kg/m^3, $\rho_p = 2500$ kg/m^3 and $g = 9.81$ m/s^2.

What is the Ergun equation simulator?

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"Pressure drop through a packed bed" — I read about it in class but it still feels abstract. What is a real example?
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Think of a coffee dripper or a sand filter. Pour water onto a bed of grains and the water needs pressure to make its way through the gaps. Industrial examples are everywhere: catalyst beds in refineries, adsorber columns, drying towers in chemical plants, and gravel filters in civil engineering. The Ergun equation is the standard semi-empirical formula that estimates the pressure gradient dP/L from the superficial velocity u, particle diameter d_p, voidage eps, fluid viscosity mu and density rho. With the defaults (water at 20 C, d_p = 2 mm, eps = 0.40, u = 10 mm/s), the tool reports dP/L of about 2.93 kPa/m — about 0.03 atm per metre of bed.
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There are two terms in the equation. What is each one doing?
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The first term is the laminar (Carman-Kozeny) contribution: at low velocity the pore channels behave like a network of Hagen-Poiseuille tubes and dP/L scales linearly with u. The second term is the inertial (Burke-Plummer) contribution: at high velocity the fluid jets between particles and loses momentum on every impact, so dP/L scales with u^2. Their balance is set by Re_p = rho u d_p / (mu (1-eps)). With the defaults Re_p is 33.3 — a transition regime where laminar contributes about 72 percent and inertia about 28 percent. Drop u to 1 mm/s and the laminar share exceeds 95 percent; push it to 100 mm/s and inertia takes over.
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In the log-log plot the slope changes between low and high velocity. Why?
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On log-log axes the first term has slope 1 (dP/L proportional to u), the second has slope 2 (dP/L proportional to u^2). At low velocity the curve hugs the green dashed laminar asymptote; at high velocity it hugs the red dashed inertial asymptote. In design the first step is to check where the target Re_p sits — laminar regime calls for viscosity-driven corrections, turbulent regime for inertial-impact corrections. In a catalytic reactor running at high velocity, dP/L blows up quadratically and pump power follows, so this is not a regime to enter lightly.
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What is the minimum fluidisation velocity u_mf, and is it linked to fluidised beds?
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Yes — u_mf is the gateway to fluidisation. It is the superficial velocity at which the upward drag on the bed exactly balances its apparent weight (gravity minus buoyancy). Past u_mf the bed lifts and "fluidises". In the laminar limit the first Ergun term gives u_mf = eps^3 d_p^2 (rho_p - rho) g / (150 mu (1-eps)) with rho_p = 2500 kg/m^3 (sand) here. The defaults give u_mf of about 41.9 mm/s, so the current u = 10 mm/s is a fixed bed and u = 50 mm/s already starts fluidisation. At high Re_p you need non-laminar corrections such as the Wen-Yu or Grace correlations.
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Why does dP/L move so much when I change the voidage eps?
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eps^3 sits in the denominator. Just dropping eps from 0.40 to 0.30 takes eps^3 from 0.064 to 0.027 (about 2.4 times smaller), so dP/L jumps by two to three times. Conversely eps = 0.50 cuts dP/L roughly in half. In practice low sphericity, fines, or segregation drop the local eps and cause unexpected pressure rises. Designers usually carry an eps uncertainty of about plus or minus 0.03. Sweep the slider from 0.30 to 0.60 to feel the sensitivity before choosing your safety factor.

FAQ

The Ergun equation is a semi-empirical formula for the pressure drop per unit length dP/L of a fluid through a bed of spherical particles, written dP/L = 150 mu u (1-eps)^2 / (eps^3 dp^2) + 1.75 rho u^2 (1-eps) / (eps^3 dp). The first term is the Carman-Kozeny laminar contribution and the second is the Burke-Plummer inertial one. With the defaults the tool reports dP/L about 2.93 kPa/m, Re_p about 33.3 and a laminar share of about 72.0 percent.
For a packed bed Re_p = rho u dp / (mu (1-eps)). Re_p below 10 is laminar (first term dominates), Re_p above 1000 is fully turbulent (second term dominates), and in between is a transition where both contribute. With the defaults Re_p is about 33.3 and the laminar share is 72 percent. Lowering u to 1 mm/s drops Re_p to about 3.3; raising it to 100 mm/s pushes Re_p to about 333.
u_mf is the superficial velocity at which the upward drag on the bed balances its apparent weight; above u_mf the bed fluidises. In the laminar approximation u_mf = eps^3 dp^2 (rho_p - rho) g / (150 mu (1-eps)) with rho_p = 2500 kg/m^3. With the defaults the tool reports u_mf about 41.9 mm/s, so u = 10 mm/s is firmly in the fixed-bed regime. At high Re_p the Wen-Yu or Grace correlations are used.
At low velocity the first Ergun term dominates and dP/L is proportional to u (log-log slope 1). This is the Hagen-Poiseuille viscous-flow limit inside the pore network — the essence of Carman-Kozeny. At high velocity the second term dominates and dP/L is proportional to u^2 (slope 2); inertial losses and jet impacts produce the Burke-Plummer behaviour. The lower-right chart shows this transition on log-log axes with a yellow marker for the current u.

Real-world applications

Pressure-drop design of fixed-bed catalytic reactors: hydrodesulphurisation (HDS) in refineries, syngas reforming and automotive catalytic converters all run fluid through fixed catalyst beds, and pump or compressor power is set directly by the bed pressure drop. For a d_p = 3 mm pellet at eps = 0.40 and u = 50 mm/s the tool gives dP/L of about 8 kPa/m, or about 40 kPa (0.4 atm) for a 5 m bed — a major compressor load. Increasing d_p from 3 to 5 mm cuts dP/L by about a factor of 2.8 but worsens internal mass transfer, so the design is always a trade-off between pressure drop and reactivity.

Adsorber and dryer columns and the breakthrough curve: air drying with molecular sieves or silica gel, and VOC removal with activated carbon, typically run at u of 100 to 300 mm/s through packed beds. With d_p = 3 mm, eps = 0.38 and u = 200 mm/s the tool returns dP/L of about 30 kPa/m and Re_p of about 600 — fully turbulent. Real designs balance breakthrough time and mass-transfer-zone length against the pressure drop and choose the column diameter and particle size accordingly. Fines from regeneration cycles lower the local eps and spike the pressure drop, so particle attrition is a key part of lifetime design.

Start-up of fluidised-bed reactors and u_mf: fluid catalytic cracking (FCC), gasifiers and fluidised-bed boilers all operate above u_mf. With d_p = 0.5 mm, eps = 0.45 and mu = 2 x 10^-5 Pa.s (hot air) u_mf lands at a few cm/s, while real operation is 3 to 10 times higher. The laminar-approximation u_mf from Ergun is a starting point; real designs use Geldart particle classification (A, B, C, D) and correlations such as Wen-Yu, Grace and Saxena-Vogel. This tool is ideal for feeling out the fixed-bed-to-fluidisation transition by varying particle size and voidage.

Gravel filters and groundwater treatment: rapid sand filters in waterworks (d_p of 0.5 to 1 mm), gravel filters in groundwater plants and crushed-stone layers under stormwater infiltration tanks all rely on the Ergun (or Forchheimer) equation. With d_p = 0.8 mm, eps = 0.40 and u = 2 mm/s (a typical filtration speed) the tool reports dP/L of about 1 to 2 kPa/m and Re_p of about 4 — laminar. Backwashing pushes u above 50 mm/s so the bed fluidises and the trapped material is washed out. Comparing the design velocity with u_mf in this tool lets you size the backwash pump.

Common misconceptions and pitfalls

The most common mistake is assuming "the Ergun equation works for any packed bed". The Ergun equation was derived for near-spherical particles, narrow size distributions, voidage in the range 0.35 to 0.55 and wall effects that are small. Needle-like or flake catalysts, broad-size sand, and small bed-diameter-to-particle-diameter ratios (D_t/d_p below 10) all push you outside its domain. With non-spherical particles you should use the Sauter mean diameter d_32, add a shape correction (d_p replaced by phi_s d_p) or switch to a Forchheimer or generalised Carman-Kozeny formulation. Real beds typically carry 20 to 30 percent uncertainty on top of the Ergun prediction.

The second pitfall is treating "voidage eps as a zero-order correction". Because eps^3 sits in the denominator, an eps uncertainty of plus or minus 0.02 around 0.40 translates into roughly plus or minus 15 to 20 percent in dP/L. In real columns the packing operation (drop-loading, vibration, steam venting) can move eps by more than 0.05, and initial and operating voidage are not the same. Sweep eps between 0.38, 0.40 and 0.42 here to feel the sensitivity before sizing the safety factor.

The last pitfall is assuming "the laminar approximation of u_mf is always conservative". The formula u_mf = eps^3 dp^2 (rho_p - rho) g / (150 mu (1-eps)) is only valid for Re_p,mf below about 10. For d_p above 1 mm (Geldart B and D particles) Re_p,mf rises above 10 and the laminar u_mf overestimates the real one (the true u_mf is obtained from the laminar plus inertial Ergun crossover). The Wen-Yu correlation Re_p,mf = sqrt(33.7^2 + 0.0408 Ar) - 33.7 (with Ar the Archimedes number) is recommended in that regime. Treat the u_mf value in this tool as a laminar starting point, especially for Geldart D particles where the correction is large.

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