Four-Stand Hot Strip Mill

What is modelled

A glowing slab is squeezed by a tandem of four stands. Because the volume of steel passing each stand per second is the same, the strip must travel faster as it gets thinner. Each pair of rolls can only pull the strip in if the draft is small enough for friction to win, and the roll force and drive power follow from the contact length and the strength of hot steel.

Equations

• Bite: the rolls grip while tan α ≤ μ with cos α = 1 − Δh / 2R, so Δhmax = 2R (1 − 1/√(1 + μ²)) (about μ² R for small angles).
• Contact length L = √(R Δh − Δh²/4); mean thickness in the gap hout + Δh/3.
• Roll force F = w L (2/√3) σ Q with friction factor Q = 1 + μ L / (2 hmean); torque arm 0.5 L per roll.
• Continuity w h v = const; forward slip s ≈ (R/hout) φn² with φn = (α/2)(1 − α/(2μ)).

Checks run in the page

• The bite limit by bisection on the roll geometry.
• The contact length and mean gap thickness from the circle geometry.
• Schedule bookkeeping and volume flow through all four stands.
• Drive power against the ideal plastic work.

Honest simplifications

A simplified Ekelund-style model with constant flow stress and Coulomb friction. There is no strip temperature drop, strain-rate or strain hardening, roll flattening, tension between stands, looper control or hydraulic gauge control, and thickness is drawn exaggerated against the rolls. The figures are teaching-scale, not those of any real mill.

• Press Slippery rolls (will not bite) and see which stands cannot grip the slab.
• Press Front-loaded to move the work to the first stands and compare the roll forces.
• Press Thin, fast finish and watch the strip speed climb stand by stand.