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Ring & Rotor Spinning, Textile Physics & Technical Cord Mechanics

Yarn Twist Factor (Ktex), Diameter & Helix Angle Formula

Calculate Metric Twist Factor K_tex = TPM * SQRT(Tex), Equivalent Yarn Cylinder Diameter d (mm), Surface Fiber Helix Angle theta (deg), and Axial Obliquity cos^2(theta) Factor.

Why K_tex = TPM * SQRT(Tex) preserves identical surface fiber helix angle—and why over-twisting drops strength via cos^2(theta)!

Ring and open-end rotor spinners scale Turns Per Meter (TPM) across coarse and fine yarns of linear density Tex (g/km) using Koechlin’s Metric Twist Factor K_tex = TPM * SQRT(Tex)! Geometrically, a cylindrical yarn of bulk packing density rho_yarn (g/cm³, ~0.85–0.95 g/cm³ for ring-spun cotton/polyester) has diameter d_mm = SQRT((4 * Tex) / (1,000 * PI() * rho_yarn)), so unrolling one twist turn reveals Surface Fiber Helix Angle tan(theta) = PI() * (d_mm / 1,000) * TPM—which depends ONLY on K_tex! Moderate twist locks staple fibers together by radial friction, yet excessive crepe/voile twist (>30° helix angle) tilts fibers away from the load axis so Hearle’s Fiber Obliquity Efficiency cos^2(theta) drops yarn tenacity!

Verify on a 25.0 Tex Ring-Spun Weaving Yarn—Linear_Density = 25.0 Tex (B2), Twist_Insertion = 750.0 TPM (B3), and Yarn_Packing_Density rho_y = 0.90 g/cm³ (B4): Metric Twist Factor K_tex hits 750 * SQRT(25) = 3,750.00 turns/m·tex^0.5 (E2), Effective Yarn Diameter equals 0.1881 mm (E3), Surface Helix Angle theta equals ATAN(PI() * 0.00018806 * 750) = 23.90 deg (E4), and Hearle Obliquity Factor cos^2(theta) equals 0.836 (83.6% axial alignment, E5)!

Spinning Twist KPI (25 Tex, 750 TPM, Yarn Packing rho = 0.90 g/cm³)FormulaVerified Output
Metric Twist Factor K_texTurns_Per_Meter * SQRT(Tex)3,750.00 (5,500.00 @ 1,100 TPM)
Effective Yarn Diameter d (mm)SQRT((4 * Tex) / (1000 * PI() * rho_y))0.1881 mm
Surface Helix Angle theta (deg)ATAN(PI() * (d_mm/1000) * TPM)*180/PI()23.90° (33.02° @ 1,100 TPM)
Axial Obliquity Factor cos²(theta)COS(theta_rad)^20.836 (0.703 @ 1,100 TPM!)

Koechlin Metric Twist Factor & Surface Fiber Helix Angle Formula

Validate Tex > 0, Turns_Per_Meter > 0, and Yarn_Packing_Density rho_y in [0.40, 1.50] g/cm³!

=LET(tx_y, B2, tp_m, B3, rh_y, B4, IF(OR(tx_y <= 0, tp_m <= 0, rh_y < 0.4, rh_y > 1.5), "Invalid Spinning Input", LET(dm_m, SQRT((4 * tx_y) / (1000 * PI() * rh_y)), ROUND(ATAN(PI() * (dm_m / 1000) * tp_m) * 180 / PI(), 2))))

High-Twist Crepe / Voile Obliquity Strength Drop Check (Set Twist = 1,100 TPM)

Crank Twist_Insertion B3 from 750 TPM up to 1,100 TPM (high-twist crepe yarn): K_tex E2 jumps to 5,500.00, Surface Helix Angle E4 steepens to 33.02 deg, and Axial Obliquity Factor cos^2(theta) E5 drops from 0.836 down to 0.703—explaining over-twist strength rolloff!

=IF(ISTEXT(E4), E4, LET(dm_m, SQRT((4 * B2) / (1000 * PI() * B4)), th_r, ATAN(PI() * (dm_m / 1000) * B3), ROUND((COS(th_r) ^ 2), 3)))

Try the example

Use the original values or change them to test your understanding.

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