solidification-simulation

name	solidification-simulation
description	Solidification simulation — Stefan problem, enthalpy method, Scheil equation for microsegregation, dendrite arm spacing (PDAS/SDAS), hot tearing criterion (Rappaz-Drezet-Gremaud), shrinkage porosity (Niyama criterion), phase field modeling, CALPHAD thermodynamics, mold filling (filling fraction, gate/runner design), MAGMASOFT/ProCAST/AnyCasting, gating ratio, Chvorinov's rule, and casting defect prediction.
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Solidification Simulation — Complete Skill

Solidification Fundamentals

Heat Transfer During Solidification

Energy equation with latent heat: ρ × ∂H/∂t = ∇·(k∇T) [H = enthalpy; k = thermal conductivity; no source term except at interface]

Latent heat at solidification front: At interface: ρ × L × v_n = [k_s × (∂T/∂n)_s - k_l × (∂T/∂n)_l] [Stefan condition; v_n = interface velocity; L = latent heat; subscripts s,l = solid, liquid]

Solidification time (Chvorinov's rule): t_s = B × (V/A)² [t_s = solidification time [s]; V = volume [m³]; A = surface area [m²]; B = mold constant] B depends on: mold material, thermal diffusivity, superheat, latent heat Sand mold, steel: B ≈ 600–900 s/cm²; permanent mold (die): B ≈ 50–150 s/cm²; investment: B ≈ 200–400 s/cm² V/A = modulus [cm]; equal modulus → equal solidification time → feeding criterion

Temperature profile (1D semi-infinite mold): T_mold(x,t) = T_i + (T_melt - T_i) × erfc(x/(2√(α_m t))) [α_m = mold diffusivity; T_i = initial mold T] Solidification rate: dx_s/dt = √(α_m × π) × ... [from Stefan condition; complex expression]

Enthalpy Method (Numerical)

Implicit Enthalpy Formulation

Used in solidification simulation software: H(T) = ρ × [c_p × T + L × f_s(T)] [enthalpy as function of T; f_s = solid fraction] For pure metal: f_s = 0 (T > T_m), 0.5 at T_m (latent heat), 1 (T < T_m) → step function For alloy: f_s changes continuously over mushy zone T_liq to T_sol

Mushy zone: region between liquidus and solidus; mixture of solid dendrites and liquid Permeability of mushy zone: k_perm = K_0 × f_l³ / ((1-f_l)²) [Carman-Kozeny; K_0 = permeability constant; f_l = liquid fraction]

Numerical method: Finite difference or FEM; implicit time integration for stability Temperature update: T^(n+1) from H^(n+1) using Newton iteration (H-T coupling is nonlinear)

Microsegregation and Scheil Equation

Scheil-Gulliver Equation

Assumptions: complete mixing in liquid; no diffusion in solid; local equilibrium at interface f_S = 1 - (C_L/C_0)^(1/(k-1)) [f_S = solid fraction; C_L = liquid composition; C_0 = initial alloy composition; k = partition coefficient = C_S/C_L at interface]

Composition in solid (inverse Scheil): C_S(f_S) = k × C_0 × (1 - f_S)^(k-1)

Lever rule (alternative — complete back-diffusion in solid): f_S_lever = (C_0 - C_L)/((C_0 - C_L) + (C_S - C_0)) → simpler expression for equilibrium

Effect of k: k < 1: solute rejected into liquid (Cu in Al, Si in Al); final liquid enriched → eutectic at last point to freeze k > 1: solute absorbed into solid; rare (Mn in Fe has k < 1; C in Fe k ≈ 0.3)

Segregation index: CS_ratio = C_S_max / C_S_min; higher ratio → more heterogeneous composition

CALPHAD integration: Use Thermo-Calc or Pandat for multicomponent Scheil simulation Accounts for multiple alloying elements and phase transformations simultaneously

Dendrite Arm Spacing

Primary and Secondary Dendrite Arm Spacing

Primary dendrite arm spacing (PDAS): λ₁ = a₁ × G^(-0.5) × V^(-0.25) [m; G = thermal gradient [K/m]; V = solidification velocity [m/s]; a₁ = alloy-specific constant] For Al-Si alloys: a₁ ≈ 1.25×10⁻³ (SI units) Typical: λ₁ = 100–1,000 μm (investment casting) to 10–50 μm (HPDC)

Secondary dendrite arm spacing (SDAS) — more commonly measured: λ₂ = a₂ × t_f^(1/3) [m; t_f = local solidification time [s] = (T_liq - T_sol) / cooling_rate; a₂ = alloy constant] For Al alloys: a₂ ≈ 5.5×10⁻⁶ (gives λ₂ in meters; t_f in seconds) λ₂ ≈ 20–200 μm → predicts cast microstructure cell size Mechanical property correlation: UTS ∝ λ₂^(-n) [n ≈ 0.3–0.5; finer SDAS → better properties]

Cooling rate from SDAS: dT/dt = (T_liq - T_sol) / t_f = ((T_liq - T_sol) / a₂)³ × λ₂^(-3) [K/s] Measured SDAS → infer cooling rate → compare with simulation

Hot Tearing (Hot Cracking)

Rappaz-Drezet-Gremaud (RDG) Criterion

Hot tearing mechanism: solidification shrinkage in semi-solid; liquid can't feed → pore nucleates → crack Occurs in mushy zone near end of solidification (f_s > 0.85–0.95)

RDG criterion: Δp_max = [∫(ε̇_p - ε̇_s×(1+β)) / (K_0 × f_s³/(1-f_s)²) × μ_L] df [pressure drop in mushy zone] β = volumetric shrinkage during solidification (Al: β ≈ 0.06) ε̇_p = plastic strain rate; ε̇_s = solidification rate; μ_L = liquid viscosity

Simplified hot tearing susceptibility: HTS = ∫[df_s / permeability] × [solidification rate + thermal contraction] High HTS → hot tearing likely; compare HTS against critical threshold

Hot tearing index: HCS (Hot Cracking Susceptibility) = T_liq - T_sol) / T_sol^(1/2) (empirical Clyne-Davies criterion) Lower is better; for Al: HCS < 3 generally resistant

Shrinkage Porosity — Niyama Criterion

Niyama Criterion

Empirical criterion from thermal analysis: Ny = G / √(dT/dt) [K^(1/2)·s^(1/2)/m; G = temperature gradient; dT/dt = cooling rate]

Porosity prediction: Ny < Ny_critical → porosity likely; Ny_critical ≈ 1–3 K^(1/2)·s^(1/2)/m (material-dependent) For steel: Ny_critical ≈ 1.0; for aluminum: Ny_critical ≈ 0.5 (aluminum less critical)

Physical meaning: High G, high cooling rate → short mushy zone, fast solidification → less time for shrinkage accumulation Low G, slow cooling (far from chill) → long mushy zone → porosity accumulates

Improvement strategies: Add chill at hot spot → increases G; increases local Ny → reduces porosity Increase riser: feeds shrinkage with liquid; reduce last-to-freeze distance Internal chill: metallic insert that melts and helps feed from inside

Mold Filling

Gate and Runner Design

Metal velocity at gate: v_gate = Q_fill / A_gate [m/s; Q_fill = volume flow rate; A_gate = gate area] Target: v_gate = 0.5–1.0 m/s for iron; 0.3–0.7 m/s for Al (avoid turbulence; air entrainment at high velocity)

Gating ratio (pressurized vs. unpressurized): Pressurized: A_sprue : A_runner : A_gate = 1 : 0.75 : 0.5 (runner bottleneck controls; mold full most of fill) Unpressurized: A_sprue : A_runner : A_gate = 1 : 2 : 4 (gate is bottleneck; quieter flow; less oxide) Unpressurized preferred for aluminum (oxide film entrapment reduction)

Bernoulli energy equation (filling): P₁/(ρg) + v₁²/(2g) + z₁ = P₂/(ρg) + v₂²/(2g) + z₂ + losses For sprue of height h: v_gate = √(2gh × η_gate) [η_gate = gate coefficient ≈ 0.75–0.85]

Mold filling time: t_fill = V_mold / Q_fill [ensure t_fill < solidification time of runners]

Simulation Software Overview

MAGMASOFT (MAGMA GmbH): Industry standard; CALPHAD-linked thermodynamics; automatic optimization; all casting processes Porosity, hot tearing, distortion, residual stress analysis

ProCAST (ESI Group): FEM-based; coupled thermal-mechanical; best for investment casting and structural analysis Phase field module for microstructure; CAFE (cellular automaton-finite element) for grain structure

AnyCasting (AnyTech): Specialized for die casting (HPDC); fast fill simulation; spray/cooling optimization

FLOW-3D Cast: Volume-of-fluid (VOF) method; excellent for free surface mold filling; turbulence and entrainment tracking

Thermocouple model validation: Compare T(t) simulation vs. thermocouple data at multiple locations; calibrate mold interface HTC h_interface (sand): 500–1,500 W/m²·K; (die): 2,000–10,000 W/m²·K (pressure-dependent for HPDC)

Casting Defect Correlation

Defect	Criterion	Simulation Parameter
Shrinkage porosity	Niyama Ny < 1	Ny contour map
Gas porosity	Poor degassing; dissolved H₂	H₂ level [cc/100g Al]
Hot tearing	RDG; HCS > threshold	High dε/dt + low Ny near solidus
Cold shut	Insufficient superheat; slow fill	T_liquid vs. T_mold contact
Misrun	T drops below T_liq during fill	Fill fraction animation
Macrosegregation	Flow during solidification	C_L distribution at last-to-freeze

Standards and References

Standard	Scope
AFS Casting Design Manual	Casting design for soundness
ASTM A802	Visual examination of steel castings
ISO 11971	Steel castings — visual examination
Campbell "Castings" (2nd Ed.)	Comprehensive casting metallurgy reference
Rappaz et al. "Modeling in Materials Science"	Solidification modeling reference

Output

Provide: casting description (alloy: composition; T_liq [°C]; T_sol [°C]; L [kJ/kg]; shrinkage β [%]; casting geometry; process: sand/investment/HPDC/gravity die), mold filling (Q_fill [cm³/s]; t_fill [s]; gate velocity v_gate [m/s]; gating ratio; turbulence/oxide risk), Chvorinov modulus (V/A [cm] at critical section; solidification time t_s [s]; comparison between sections), SDAS prediction (local cooling rate dT/dt [K/s]; t_f [s]; λ₂ [μm]; UTS correlation [MPa]), Scheil microsegregation (C_S_max/C_S_min; eutectic fraction at last point; HCS index; CALPHAD analysis recommended?), Niyama criterion map (Ny_min in casting; locations below 1.0; porosity risk assessment), hot tearing susceptibility (RDG or HCS index; alloy sensitivity; locations at risk; geometry modification recommendation), riser/chill design (riser modulus M_r = 1.2 × M_casting; chill location; modulus correction; feeding distance estimate), simulation software recommendation (MAGMASOFT for foundry optimization; ProCAST for structural coupling; FLOW-3D for fill), and applicable standard (AFS Manual; Campbell "Castings" principles).