Hydrodynamic Lubrication in Wire Drawing

Content

Estimate lubricant film thickness and drawing stress for circular wire passing through a pressure-viscous, full-film conical die.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Wire, die, and lubricant inputs

Enter one steady drawing condition. Results update automatically; Calculate provides the same action explicitly.

How to use

  1. This calculator has one mode: steady, circular-wire drawing through a conical reduction and cylindrical bearing under full-film hydrodynamic lubrication.
  2. Enter area reduction in %, entry diameter in mm, reduction half-angle in degrees, entry velocity in m/s, inlet dynamic viscosity in Pa·s, pressure–viscosity coefficient in GPa⁻¹, mean extrusion pressure in MPa, and the dimensionless bearing-length ratio. Use positive magnitudes; α and Lᵦ/d may be zero.
  3. Select Calculate or edit any field to update the result. Select Reset to restore the worked-example defaults.
  4. Read hmin as a screening estimate of lubricant separation and σ as the modeled axial tensile-stress magnitude. Secondary results show exit geometry, continuity velocity, pressure-viscosity ratio, and die-zone lengths. Values are displayed to six significant digits without changing full-precision calculations.
  5. Correct any warning before using a result. Every entered value must be finite; 0<r<100%, 0°<φ<90°, d₁, v₁, η₀, and pₑ must be strictly positive, α and Lᵦ/d must be nonnegative, and αpₑ must not exceed 700.
  6. Do not use this model for starved, boundary, or mixed lubrication; strongly non-isothermal drawing; noncircular sections; transient starts; roughness/die-deflection prediction; material failure; or final die/pass design without independent engineering analysis.

Equations used

The implementation converts all inputs to SI units before calculation. Let r be area reduction as a fraction (entered percentage divided by 100), d₁ and d₂ the entry and exit diameters, A₁ and A₂ their areas, φ the die reduction half-angle in radians, v₁ and v₂ the entry and exit velocities, η₀ the ambient-pressure dynamic viscosity, α the pressure–viscosity coefficient in Pa⁻¹, pₑ the positive mean extrusion pressure, Lᵣ the reduction length, Lᵦ the bearing length, and hmin the minimum film thickness.

A₁ = πd₁²/4
A₂ = (1−r)A₁
d₂ = d₁√(1−r)
v₂ = v₁/(1−r)
η/η₀ = exp(αpₑ)
Lᵣ = (d₁−d₂)/(2 sin φ)
Lᵦ = (Lᵦ/d)d₂
h_min = 3η₀v₁α / [φ(1−exp(−αpₑ))]
α = 0: h_min = 3η₀v₁/(φpₑ)
σ₁ = (4/3)[(Lᵣ+Lᵦ)/d₂]φ[exp(αpₑ)−1]/α
σ₂ = pₑ(A₁−A₂)/A₂ = pₑr/(1−r)
σ = σ₁+σ₂

For α=0, the finite analytic stress limit is σ₁=(4/3)[(Lᵣ+Lᵦ)/d₂]φpₑ. All lengths are positive; pₑ is entered as a positive compressive magnitude while σ is reported as a positive tensile magnitude. The exponential argument αpₑ is dimensionless.

Theory and method

Wire drawing die geometryA circular wire enters a conical die at diameter d one and velocity v one, reduces over length L r and half-angle phi, then exits the bearing at diameter d two and velocity v two with lubricant film h min.d₁, v₁d₂, v₂φReduction LᵣBearing Lᵦh_min

Drawing pulls wire through a converging die. Area continuity sets the smaller exit diameter and higher exit velocity. Relative motion entrains lubricant into the converging clearance; the Reynolds wedge effect generates pressure and a separating film. The Barus relation represents isothermal viscosity growth with pressure. The stress expression combines a pressure-viscous reduction/bearing contribution σ₁ with the direct area-reduction contribution σ₂.

Rheologylab

The model assumes steady incompressible flow, circular geometry, a conical reduction followed by a cylindrical bearing, a representative mean pressure, full-film lubrication, and isothermal Barus behavior. It does not solve coupled plasticity, temperature, starvation, roughness contact, lubricant supply, die elasticity, or wear. Typical bearing lengths are often about 0.3–0.6 exit diameters, while practical die angles and pass reductions depend strongly on material, work hardening, lubricant, temperature, and production constraints.

Worked example

  1. Use r=20%, d₁=2.5 mm, φ=6°, v₁=10 m/s, η₀=0.2 Pa·s, α=12 GPa⁻¹, pₑ=200 MPa, and Lᵦ/d=0.4.
  2. The calculator returns d₂=2.23607 mm, v₂=12.5 m/s, η/η₀=11.0232, Lᵣ=1.26249 mm, Lᵦ=0.894427 mm, hmin=0.756145 µm, and σ=162.497 MPa.
  3. Interpretation: the modeled film is below 1 µm. Compare it with measured combined roughness and verify thermal, material, pressure, and lubricant assumptions before treating the pass as full-film.

Applicability and references

  • Use as a first-pass hydrodynamic screening calculation, not as a substitute for validated drawing-force, thermal, and material models.
  • O. Reynolds (1886), “On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower’s Experiments,” Philosophical Transactions of the Royal Society of London. DOI: 10.1098/rstl.1886.0005.

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