Settling Velocity (Newton)
Calculate the settling (terminal) velocity of a particle in turbulent regime by Newton's law, v_t = √(4·g·d·(s − 1) ÷ (3·C_d)), from the particle diameter d (m), the solids relative density s = ρ_s/ρ_w and the drag coefficient C_d (dimensionless, ≈ 0.44 for spheres in turbulent regime). While Stokes' Law holds for SMALL particles (laminar regime, particle Reynolds < 1), Newton's law holds for LARGE, dense particles — gravel, crushed stone, coarse sand — that sink fast, generating TURBULENT flow around them (particle Reynolds > ~1000). In this regime, drag is no longer proportional to velocity (Stokes) but to its SQUARE, and the terminal velocity grows with the SQUARE ROOT of diameter (not the square, as in Stokes) — large particles sink fast, but the size dependence is milder. The drag coefficient C_d ≈ 0.44 is roughly constant in this range (the 'Newton region' of the sphere drag curve). This calculation is fundamental in designing coarse-particle classifiers and separators, sizing settling basins for coarse solids, coarse-sediment transport and hydraulic transport of gravel and granular ore. For the intermediate range between Stokes and Newton, transition correlations are used. Enter the diameter, the relative density and the drag coefficient.
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Velocidade de sedimentação (Newton)
A velocidade de sedimentação (queda terminal) de uma partícula em regime turbulento, pela lei de Newton, é v_t = √(4·g·d·(s − 1) ÷ (3·C_d)), a partir do diâmetro da partícula d, da densidade relativa dos sólidos s = ρ_s/ρ_w e do coeficiente de arrasto C_d (≈ 0,44 para esferas em regime turbulento). Enquanto a Lei de Stokes vale para partículas pequenas (regime laminar, Reynolds de partícula < 1), a lei de Newton vale para partículas grandes e densas — cascalho, brita, areia grossa — que afundam rápido, gerando um escoamento turbulento ao seu redor (Reynolds de partícula > ~1000). Nesse regime, o arrasto deixa de ser proporcional à velocidade (Stokes) e passa a ser proporcional ao seu quadrado, e a velocidade terminal cresce com a raiz do diâmetro (não com o quadrado, como em Stokes) — partículas grandes afundam rápido, mas a dependência com o tamanho é mais suave. O coeficiente de arrasto C_d ≈ 0,44 é aproximadamente constante nessa faixa (a 'região de Newton' da curva de arrasto da esfera). Este cálculo é fundamental no projeto de classificadores e separadores de partículas grossas, no dimensionamento de bacias de sedimentação para sólidos grosseiros, no transporte de sedimentos grossos e no transporte hidráulico de cascalho e minério granulado. Para a faixa intermediária entre Stokes e Newton, usam-se correlações de transição. Informe o diâmetro, a densidade relativa e o coeficiente de arrasto.
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Settling Velocity (Stokes)
Calculate the settling (terminal) velocity of a particle in laminar regime by Stokes' Law, v_s = g·d²·(ρ_s − ρ_w) ÷ (18·μ), from the particle diameter d (m), the solids ρ_s and water ρ_w densities (kg/m³) and the fluid dynamic viscosity μ (Pa·s). The settling velocity is the speed at which an isolated particle SINKS in a still fluid, when weight (minus buoyancy) balances drag. Stokes' Law (1851) holds for the LAMINAR regime (small particles, particle Reynolds < ~1) — fine sand, silt, clay — and has the remarkable property that velocity grows with the SQUARE of diameter: particles twice as large sink four times faster. This calculation is fundamental in many fields: particle settling and separation (settling tanks, thickeners, water and effluent clarifiers), grain-size classification by sedimentation (pipette or hydrometer test), sediment transport in rivers and reservoir deposition, and hydraulic transport (the particle settling velocity sets the critical deposition velocity in the pipe). For large particles (higher Reynolds), Stokes' Law fails and Newton's terminal velocity (turbulent regime) is used. Enter the particle diameter, the densities and the viscosity.
Particle Reynolds Number
Calculate the particle Reynolds number in settling, Re_p = (ρ_w·v_s·d) ÷ μ, from the fluid density ρ_w (kg/m³), the settling velocity v_s (m/s), the particle diameter d (m) and the dynamic viscosity μ (Pa·s). The particle Reynolds number characterizes the flow regime around a particle settling (or being transported) in a fluid, comparing inertial and viscous forces. Its value sets WHICH settling-velocity formula is valid: for Re_p < ~1, the flow around the particle is LAMINAR and Stokes' Law holds (drag proportional to velocity); for Re_p > ~1000, the flow is TURBULENT and Newton's law holds (drag proportional to velocity squared); in the intermediate range, transition correlations are used (such as Allen's or drag-coefficient expressions vs Re_p). So when computing a settling velocity by Stokes' Law, it is ESSENTIAL to verify afterwards that Re_p < 1 — if not, the Stokes result is wrong and the correct regime's formula must be used. The particle Reynolds number is thus the 'checker' that validates the settling calculation, and it is central in designing settling tanks, classifying particles and hydraulic solids transport. Enter the fluid density, the settling velocity, the particle diameter and the viscosity.
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Calculate the maximum acceleration of a cycloidal-motion cam follower, a_max = (2π·h·ω²) ÷ β², from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Cycloidal motion is considered the BEST cam motion law for HIGH SPEEDS, and is standard in precision, high-rpm cams. Its decisive feature is that acceleration is a FULL SINE wave starting at zero, rising to a maximum, passing through zero, going to a minimum and returning to zero — i.e., acceleration is CONTINUOUS and starts and ends smoothly at ZERO at the ends, WITHOUT the discontinuities of SHM and parabolic. This means finite, continuous JERK, eliminating shocks and minimizing vibration excitation — the follower 'glides' smoothly without jolts. The price is a slightly HIGHER maximum acceleration than parabolic (2π ≈ 6.28 vs 4 in the factor) and SHM (π²/2 ≈ 4.93), but the dynamic SMOOTHNESS amply compensates at high speed. The name comes from the cycloid curve describing the displacement. Racing-engine valve cams, fast textile and packaging machines use cycloidal or derived (polynomial) profiles precisely to run at high rpm with low vibration. Enter the lift, the angular velocity and the rise angle.
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