hot-rolling

name	hot-rolling
description	Hot rolling of metals — rolling force (Sims/Bland-Ford), draft and reduction per pass, roll pressure distribution, forward slip, spread, dynamic recrystallization (DRX) kinetics (Avrami equation), static recrystallization between passes, controlled rolling (TMCP) of HSLA steels, rolling schedule design, finishing temperature, coiling temperature, mill configuration (reversing/tandem), shape defects (camber, edge wave, center buckle), ASTM A6 structural steel, API 5L line pipe plate.
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Hot Rolling — Complete Skill

Rolling Mechanics

Draft and Reduction

Nomenclature: h₀ = entry thickness [mm]; h₁ = exit thickness [mm] Δh = draft = h₀ − h₁ [mm] r = fractional reduction = Δh/h₀ [dimensionless; typical 15–40% per pass hot rolling] w₀ = entry width; w₁ = exit width; L = bite length (arc of contact)

Bite angle α: cosα = 1 − Δh/(2R) → α ≈ √(Δh/R) [small angle; R = roll radius [mm]] Max bite condition: tanα ≤ μ (coefficient of friction) μ_hot rolling ≈ 0.3–0.4 (lubricant); 0.4–0.6 (dry/scale) Max draft: Δh_max = μ² × R [mm; from bite condition]

Arc of contact length: L_d = √(R × Δh) [mm; projected arc length] More precisely: L_d = √(R × Δh − Δh²/4) ≈ √(R × Δh) for Δh << R

Roll Force

Simplified roll force (average pressure method): F = p_avg × L_d × w [N; p_avg = average roll pressure; w = strip width [mm]] p_avg ≈ σ̄_f × Q_F [σ̄_f = mean flow stress; Q_F = force multiplier ≈ 1.15–1.25 for friction]

Mean flow stress for hot rolling: σ̄_f from Zener-Hollomon parameter Z: Z = ε̇ × exp(Q_def/(R×T)) [ε̇ = strain rate [s⁻¹]; Q_def = activation energy [J/mol]; R = 8.314 J/mol·K; T = temperature [K]] For low-carbon steel: Q_def ≈ 310 kJ/mol; A = 1.9×10⁻⁷; n = 4.5 (Sellars-McTegart const.) Z = A × [sinh(ασ̄_f)]^n → solve for σ̄_f [α ≈ 0.012 MPa⁻¹ for steel]

Sims formula (hot rolling): p/k ≈ 1 + (π/4) × (L_d/Δh) [k = shear flow stress = σ_f/√3; simplified for no-tension rolling] More complete: includes correction for front/back tension and roll flattening

Roll flattening (Hertz): R' = R × (1 + C_R × F/(R × w × Δh))^(−1) ≈ R + C_F × F/w [Hitchcock's formula; R' = flattened roll radius; C_R depends on roll material]

Forward Slip

Forward slip (S): S = (v₁ − v_R) / v_R [v₁ = exit strip speed; v_R = roll surface speed] S ≈ (h₀ − h₁)/(2h₁) × [1 − μ × L_d × (2/3) / h_avg] [approximate; depends on friction and geometry] Typical: S = 1–8% for hot strip rolling

Spread: Width increase: Δw = w₁ − w₀ [typically small in hot strip mill; controlled by edger rolls] Spread equation: Δw/Δh ≈ f(R/h_avg, μ, w/h) [various empirical models; width gain = 2–8% of draft for plate]

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Dynamic Recrystallization (DRX)

DRX Kinetics

Critical strain for DRX initiation: ε_c ≈ 0.8–0.9 × ε_p [ε_p = strain at peak stress; DRX starts before peak] ε_p = a × d₀^m × Z^n [d₀ = initial grain size [μm]; constants from literature] For C-Mn steel: ε_p ≈ 5.7 × 10⁻⁴ × d₀^0.3 × Z^0.17

DRX fraction (Avrami equation): X_DRX = 1 − exp(−k_DRX × ((ε − ε_c)/ε_0.5)^n_DRX) Where: ε_0.5 = strain for 50% DRX = A × d₀^q × Z^s; k_DRX ≈ 0.693; n_DRX ≈ 2 (hot rolling) X_DRX approaches 1 when ε >> ε_0.5 → fully dynamically recrystallized austenite

DRX grain size: d_DRX ≈ C × Z^(−m) [fine grain; C and m from calibration; for C-Mn steel: d_DRX ≈ 2.2×10⁷ × Z^(−0.27) μm] DRX produces fine austenite grain → fine final ferrite grain after transformation

Static Recrystallization (SRX) Between Passes

SRX kinetics (inter-pass): t₀.₅_SRX = A_SRX × ε^(−p) × ε̇^(−q) × d₀^² × exp(Q_SRX/(RT)) [t₀.₅ = time for 50% SRX] For C-Mn steel: A_SRX = 2.3×10⁻¹⁶; p = 4; q = 0; Q_SRX = 230 kJ/mol SRX fraction in inter-pass time t_ip: X_SRX = 1 − exp(−0.693 × (t_ip/t₀.₅)²)

Grain growth during SRX: d_SRX² − d_DRX² = C_GG × exp(−Q_GG/(RT)) × t [grain growth if inter-pass time long]

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Controlled Rolling (TMCP — Thermo-Mechanical Controlled Processing)

TMCP Strategy for HSLA Steel

Three-stage TMCP schedule:

1. Reheating: austenitize above T_RX_stop (≥ 1100–1200°C); dissolve carbides, nitrides (NbC, V(C,N)) T_dissolve NbC ≈ 1200–1250°C; ensure complete dissolution for microalloying effect

2. Rough rolling above T_nr (no-recrystallization temperature): T_nr = 887 + 464%C + (6445%Nb − 644√%Nb) + (732%V − 230√%V) + 890%Ti + 363%Al − 357%Si [°C] T_nr typically 900–1050°C for Nb-containing steels Above T_nr: full DRX → refined austenite grain; pancaked austenite not required yet

3. Finish rolling in non-recrystallization region (T_nr to Ar₃): Deform austenite below T_nr → austenite does not recrystallize → builds up sub-structure → pancaked elongated austenite grains Deformation bands, grain boundaries → multiple nucleation sites for ferrite → very fine ferrite grain d_ferrite ≈ 2–6 μm Finishing temperature T_F: typically Ar₃ + 50–100°C [avoid rolling into two-phase α+γ region]

Accelerated cooling (ACC): After finish rolling: ACC from T_F to T_coil at 10–30°C/s Coiling temperature T_coil: 500–650°C for ferrite-pearlite; 450–550°C for acicular ferrite/bainite Lower T_coil → finer precipitate (NbC) → precipitation hardening + grain refinement

TMCP mechanical property benefits: Yield strength improvement: +80–150 MPa vs. conventional rolling Impact transition temperature: −60 to −80°C (Charpy CVN) achievable for offshore/Arctic steels

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Mill Configuration

Mill Types

Reversing mill (plate mill): Single mill stand; strip reversed for multiple passes; used for plate (t = 5–100 mm) Pass schedule designed (Hitchcock iteration): target draft each pass keeping roll force within mill capacity

Tandem hot strip mill (HSM): 5–7 continuous stands; strip passes through all without reversal; high production F1–F7 stands → reducing thickness from ~25 mm to 2–20 mm finished strip Speed cascade: each stand speeds up to maintain constant mass flow: h₀×v₀ = h₁×v₁ = constant

Steckel mill: reversing + heated coiler furnaces; used for stainless, Ti, Ni alloys; maintains high temperature

Rolling Schedule Calculation

Inter-pass time (tandem mill): t_ip = L_table / v_between_stands [table length ÷ strip speed between stands; ms to seconds] Short t_ip → limited SRX → builds sub-structure → finer product grain

Temperature drop per pass: ΔT_deform = σ̄_f × ε / (ρ_s × c_p) [adiabatic heating from deformation; ρ_s × c_p ≈ 4.5 MJ/m³·K for steel] ΔT_radiation ≈ εσ × T_s⁴ × t_ip / (ρ_s × c_p × h) [radiation loss between passes]

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Shape Defects

Camber: asymmetric draft across width → curvature in horizontal plane → fix by leveling roll crown Edge waves: strip edges thicker than center → compressive stress at edges → buckle → fix: positive roll bending or work roll crown Center buckle: strip center thicker → compressive stress at center → buckle → fix: negative roll bending Alligatoring: center splits open at exit → severe over-reduction (>50%) or large T gradient → reduce draft

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Standards

Standard	Scope
ASTM A6/A6M	General requirements for hot-rolled structural shapes, plates
ASTM A36/A572	Structural carbon/HSLA steel plate (rolled)
API 5L	Line pipe steel (hot-rolled plate + seamless)
ASTM A588	HSLA weathering steel
ISO 630	Structural steels (European equivalent)
EN 10025	Hot-rolled structural steel products
ASTM A1008	Cold-rolled reference (for comparison)

Output

Provide: material (grade; composition relevant alloying: %C/%Mn/%Nb/%V/%Ti; T_nr [°C]; Ar₃ [°C]), reheating (T_reheat [°C]; soak time [min]; NbC dissolution confirmed?), roll schedule (pass | h₀ [mm] | h₁ [mm] | Δh | r [%] | T_pass [°C] | ε̇ [s⁻¹] | Z; DRX? X_DRX [%]), roll force estimate (σ̄_f from Z model [MPa]; L_d = √(RΔh) [mm]; F = σ̄_f×Q_F×L_d×w [MN]; within mill capacity [MN]?), recrystallization status (above/below T_nr for each pass; SRX in inter-pass time: X_SRX [%]; austenite grain size after each pass [μm]), TMCP finish (T_F [°C] ≥ Ar₃+50°C?; ACC rate [°C/s]; T_coil [°C]; expected microstructure: ferrite-pearlite/acicular ferrite/bainite), expected properties (σ_y [MPa]; σ_UTS [MPa]; CVN at T_test [J]; Charpy transition temperature [°C]), shape quality (roll crown setting; bending force; edge/center flatness check), and applicable standard (ASTM A6 + grade spec; API 5L if pipeline; confirm chemical composition within specification).