Hydrodynamic Lubrication in Sheet Rolling

Content

Estimate the lubricant film entrained between a work roll and sheet during hydrodynamic rolling.

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

Rolling and lubricant inputs

Use consistent operating values; Calculate updates the screening result.

Minimum entrained film, hmin

— µm
VRφLt₁t₂

How to use

  1. This calculator has one sheet-rolling mode. Select the reduction factor r=t₂/t₁, where t₁ and t₂ are positive entry and exit thicknesses.
  2. Enter t₁ in mm, work-roll radius R in m, and positive incoming sheet velocity V in m/s.
  3. Enter dynamic viscosity η₀ in Pa·s at inlet temperature, pressure–viscosity coefficient α in GPa⁻¹, and positive mean extrusion pressure pₑ in MPa. Use α=0 only for the constant-viscosity limit.
  4. Select Calculate to update the film thickness and secondary quantities. Inputs also recalculate as they change. Reset restores the worked-example defaults.
  5. Read hmin as the ideal minimum entrained lubricant-film estimate. The secondary results report entry angle, projected contact length, pressure viscosity ratio, and compressive roll load per unit strip width.
  6. Correct any warning before using a result. Do not use this model for thermal, starved, rough, elastically flattened, slipping, emulsion-controlled, mixed-lubrication, or safety-critical rolling design without a process-specific model and validation.

Equations used

All equations are evaluated in SI units. The reduction factor is r = t₂/t₁; t₁ is the incoming sheet thickness, t₂ is the outgoing thickness, R is the work-roll radius, V is the positive incoming sheet velocity, η₀ is dynamic viscosity at inlet temperature and ambient pressure, α is the pressure–viscosity coefficient in the nonnegative Barus model, and pₑ is the positive mean extrusion pressure magnitude.

φ = arccos[1 − (t₁/R)(1 − r)/2]L = RφF′ = pₑLη/η₀ = exp(αpₑ)hmin = 6η₀Vα / {φ[1 − exp(−αpₑ)]}, for α > 0hmin = 6η₀V / (φpₑ), for α = 0

Here φ is the positive entry half-angle in radians, L is projected contact length in m, F′ is positive compressive roll load per unit width in N/m, η/η₀ is dimensionless, and hmin is the minimum entrained film thickness in m. Inputs are converted from mm, GPa⁻¹, and MPa before evaluation. Public values preserve the method’s display rules: φ to whole mrad, L and η/η₀ to 0.01, F′ to whole kN/m, and hmin to 0.01 µm; machine-readable results retain full precision. The accompanying expanded values use up to six significant digits.

Validation limits: Every entered value must be finite. The dimensional inputs t₁, R, V, η₀, and pₑ must be strictly positive; in particular, η₀ must be strictly positive. The coefficient α must be nonnegative. The selected ratio must satisfy 0 < r < 1, and the roll-bite geometry must satisfy −1 ≤ 1 − t₁(1−r)/(2R) < 1. Evaluation is rejected when αpₑ > 700 to prevent a non-finite exponential result.

Worked example

  1. Use r=0.80, t₁=0.5 mm, R=0.25 m, V=10 m/s, η₀=0.1 Pa·s, α=18 GPa⁻¹, and pₑ=200 MPa.
  2. Select Calculate. Expected displayed results are φ=20 mrad, L=5.00 mm, η/η₀=36.60, F′=1000 kN/m, and hmin=5.55 µm.
  3. Interpretation: under the ideal isothermal fully flooded assumptions, the minimum entrained film is about 5.55 µm. Compare this with combined surface roughness, thermal predictions, and measured process behavior before judging separation.

Theory and method

The geometry represents a rigid circular work roll approaching a sheet whose thickness decreases from t₁ to t₂. The exact circular relation gives φ; multiplying by R gives the projected bite length. Uniform pₑ then supplies a screening roll load per unit strip width. The lubricant follows a Barus pressure–viscosity law, η=η₀exp(αp), and the closed-form film equation integrates that response through the idealized inlet. Its α=0 expression is the analytic constant-viscosity limit, avoiding division by zero.

The model assumes steady, isothermal, fully flooded hydrodynamic entrainment; a Newtonian lubricant; rigid roll/sheet geometry; uniform mean pressure; and positive entry speed. It does not resolve thermal thinning, elastic roll flattening, inlet starvation, surface roughness, shear thinning, slip, emulsion concentration, friction, neutral-point position, or mixed/boundary contact. Treat the result as an early screening estimate rather than a rolling-load, product-quality, or lubricant-qualification calculation.

Continue in TriboSolver

Use TriboSolver to refine lubricant-film behavior with process-specific geometry, properties, loading, and independently checked boundary conditions.

References

  • W. R. D. Wilson and L. E. Murch, “A Refined Model for the Hydrodynamic Lubrication of Strip Rolling,” Journal of Lubrication Technology, 98(3), 426–431 (1976), doi:10.1115/1.3452877. This is the primary strip-rolling hydrodynamic-lubrication context; the equations implemented above are explicitly stated so their scope can be checked.