Burr Grinder

What is modelled

A burr grinder crushes beans between two toothed discs. The gap sets the largest fragment that can leave, so the typical particle size scales with the gap, while a spread of sizes is always produced. The sizes are described by a Rosin-Rammler distribution, a standard model for crushed solids. Smaller particles have far more surface per gram, which matters for anything that happens at particle surfaces.

Equations

• Cumulative undersize F(d) = 1 - exp(-(d/d63)n); characteristic size d63 = c g, c = 0.8 (illustrative), gap g.
• Surface-mean (Sauter) size 1/d32 = ∫ (1/d) dF = Γ(1 - 1/n) / d63 for n > 1.
• Specific surface Sm = 6 / (ρ d32), ρ = 1100 kg/m³ for ground bean.
• Flow and motor: from the 3D model file, flow = 16 g / 12 s · (g / 0.38 mm)0.2 · rpm/rpmnom, motor speed approaching 1081 rpm with time constant 0.36 s.

Checks run in the page

• The dose after 12 s at the nominal gap from the 3D model file against 16 g.
• The Rosin-Rammler cumulative curve from the numerical integral of its density.
• The Sauter size from numerical integration against the gamma-function closed form.
• The motor spin-up e-folding time against the 3D time constant.

Honest simplifications

The proportionality d₆₃ = 0.8 g and the spread n are illustrative, not measured for any grinder or bean, and real grinds are often bimodal. The flow law and the motor come from the 3D page's simplified model, and heat, retention and static are ignored. No taste or extraction claim is made.

• Press Fine espresso grind: the gap closes, the particles shrink and the surface area per gram rises sharply while the dose takes longer.
• Press Coarse brew grind and see the opposite.
• Raise the spread n and watch the distribution narrow: fewer fines and fewer boulders for the same mean.