silo-design

name	silo-design
description	Silo and hopper design — Janssen wall loads, Reimbert, hopper flow modes (mass flow/funnel flow), critical outlet, arching, EN 1991-4, AS 3774, bin activators.
metadata	{"priority":7,"promptSignals":{"phrases":["silo design","hopper design","bulk material storage","bin design","Janssen pressure","mass flow hopper","arching silo"],"minScore":3}}

Silo and Hopper Design — Complete Skill

Bulk Solid Properties (Required Inputs)

• Bulk density ρ_b [kg/m³] — poured and tapped (ratio = compressibility)
• Internal friction angle φ [°] — particle-on-particle
• Wall friction angle φ_w [°] — particle-on-wall material
• Effective angle of internal friction δ [°]
• Unconfined yield strength f_c [kPa] — from Jenike shear cell test
• Flow factor ff (from Jenike charts)

Key instrument: Jenike shear cell (ASTM D6128); measures f_c, δ, φ_w at various consolidation stresses

Janssen Wall Pressure (Vertical Cylindrical Bins)

Vertical pressure: p_v = (ρ_b × g × R) / (K × μ') × [1 - exp(-K × μ' × z / R)]

K = Rankine ratio = (1 - sinφ)/(1 + sinφ) [for loose fill; or use K = 0.4 if uncertain] μ' = tan(φ_w) = wall friction coefficient R = hydraulic radius = A_cross / perimeter [m] z = depth below surface [m]

Asymptotic (deep bin) pressure: p_v,∞ = ρ_b × g × R / (K × μ')

Horizontal (hoop) pressure: p_h = K × p_v

Wall force: F_h = p_h × π × D × dz (per unit height ring)

Important: Eccentric discharge, asymmetric flow → UNSYMMETRIC pressures; EN 1991-4 provides patch load model

Reimbert Method (Alternative Classic)

Used for tall silos: p_v = γ_b × (H/4) × [1 - (H/(H+4C))²] C = (A_cross × tanφ_w) / perimeter

Hopper Pressures (Jenike-Johansson)

Fill (static) pressures: same as bin above down to junction Discharge (dynamic) pressures: switch load from bin-like to hopper-like at some depth above outlet → pressures increase significantly during discharge initiation

Switch pressure factor: p_switch ≈ 1.5–2.0 × static p_v at junction

Flow Mode Classification

Mass Flow

All material moves whenever any is discharged Required if: uniform first-in/first-out, segregation-sensitive, time-sensitive consolidation

Mass flow condition (Jenike): θ_hopper ≤ θ_c (critical hopper half-angle) θ_c from Jenike charts (function of δ and φ_w)

• Conical hopper: θ_c ≈ 20–30° from vertical (steeper than funnel flow)
• Plane flow (wedge): θ_c ≈ 30–40° from vertical

Funnel Flow

Central flow channel, stagnant zones at periphery Acceptable when: material is free-flowing, no segregation concern, time of storage short

Critical Outlet Diameter (No-Arching)

Cohesive arching prevention: d_c = H(θ) × f_c(σ₁) / (ρ_b × g)

H(θ) = hopper flow factor function:

• Conical: H = 2 + sinθ_hopper
• Plane flow: H = 1 + sinθ_hopper

Rat-holing (piping): d_rat-hole = G(φ) × f_c(σ_c) / (ρ_b × g) G(φ) = (1 + sinφ) / cosφ (cylindrical channel)

Design rule: outlet diameter D_outlet > max(d_c, d_rat-hole) × 1.2 safety factor

Wall Load Combinations (EN 1991-4:2006)

EN 1991-4 defines 4 load cases:

• Load case 1: symmetrical filling
• Load case 2: symmetrical discharge
• Load case 3: non-symmetrical filling (eccentric inlet)
• Load case 4: non-symmetrical discharge (eccentric outlet)

Slenderness classification: h_c/d_c > 2 → TALL silo (Janssen fully developed) 0.4 < h_c/d_c < 2 → INTERMEDIATE h_c/d_c < 0.4 → SQUAT (full hydrostatic governs)

Partial safety factors (EN): Favorable: γ_F = 0.9; Unfavorable: γ_F = 1.5 (bulk solids are uncertain)

Structural Design — Wall Thickness (Steel Silo)

Hoop stress: σ_θ = p_h × r / t ≤ F_y / γ_m

Buckling check (thin shells): σ_cr = 0.605 × E × (t/r) [classical; use knockdown factor α ≈ 0.6 for imperfect shells] EN 1993-1-6 provides detailed buckling design method

Minimum wall thickness: t_min = p_h × r / (F_y / 1.1) PLUS corrosion allowance (1.5–3 mm for carbon steel)

Bin Activation and Flow Aid Devices

Vibrating bin activators: mass-flow inducement; vibrate at 10–50 Hz; break cohesion Air cannons: high-pressure air burst (5–10 bar, 0.2–2 L tank) breaks stagnant material Fluidization pads: inject low-pressure air through porous bottom/cone; best for powders Agitators / sweeping arms: rotary; used in hoppers for sticky materials

Discharge Rate (Beverloo Equation)

For free-flowing granular material: Q_mass = C_B × ρ_b × g^0.5 × (D_o - k × d_p)^2.5

C_B ≈ 0.58; k ≈ 1.4; D_o = outlet diameter; d_p = particle diameter

Feeder sizing: Belt: 60–150 m/min; screw: RPM → volumetric output; rotary valve: rev/min → volume per revolution

AS 3774 (Australian Standard — equivalent scope to EN)

Uses similar Janssen approach; defines Type 1 (squat) and Type 2 (tall) bins Requires dynamic load amplification on discharge (1.35–1.5× static)

Output

Provide: p_v_max [kPa], p_h_max [kPa], flow mode (mass/funnel), critical outlet d_c [mm], hopper half-angle θ [°], switch load at junction, wall thickness t [mm] (hoop + buckling), applicable standard (EN 1991-4 or AS 3774), and flow aid recommendation.