❄️ Stefan Moving Boundary Solidification Front

Solve two-phase moving boundary Stefan phase change problems: solid front position s(t), total freezing time, solid Stefan number (Stes), and instantaneous cooling heat flux.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

📊 Solver Telemetry ● ACTIVE

👁️ Consultations 43

⚡ Calculs faits 35

💾 Téléchargements 401 📦 Code Fortran 3.4 KB

📅 Mise en service Jun 2026

⏱️ Latence < 1 ms

❄️ Two-Phase Moving Solidification Front Interface s(t)

Real-time visual simulation: Frozen solid crystal layer advancing dynamically into liquid domain

📝 Configuration & Presets

🧊 Ice Rink Freezing Plate 🔋 Paraffin Wax PCM Battery 🥩 Industrial Food Freezing 🏭 Continuous Steel Casting

📐 Geometry & Phase Change Temperatures

Target Thickness L [mm]:

Phase Change Temp T_m [°C]:

Cold Wall Temp T₀ [°C]:

Initial Liquid Temp T_i [°C]:

🧪 Latent Heat & Solid Properties

Latent Heat L_f [kJ/kg]:

Solid Conductivity k_s [W/(m·K)]:

Solid Density ρ_s [kg/m³]:

Solid Specific Heat c_p,s [J/(kg·K)]:

Stefan Problem Formulation:
• Solid Stefan Number: Ste_s = c_p,s (T_m − T₀) / L_f
• Front Constant: λ · e^λ² · erf(λ) = Ste_s / √π
• Front Position: s(t) = 2 λ √(α_s · t) [mm]
• Freezing Time: t_freeze = L² / (4 λ² α_s) [hours]

📊 Stefan Solidification Results

📊 Output Summary
💾 Fortran Source

Total Freezing / Solidification Time

t = 0.21 hours (739 s)

Front Growth Constant: λ = 0.7133 | 1-hr Thickness: 220.8 mm

Ste_s = 1.444

Solid Stefan Number (Ste_s) 1.444 Sensible / Latent heat

Solid Thermal Diffusivity (α_s) 6.65e-6 m²/s k_s / (ρ_s · c_p,s)

1-Hour Solidified Layer 220.8 mm 2 λ √(α_s · 3600)

Liquid Superheat (Ste_l) 0.148 c_p,l (T_i − T_m) / L_f

📈 Solidification Front Position s(t) [mm] vs Time (hours)

📉 Wall Heat Flux q″ [kW/m²] vs Time (hours)

=================================================================
 THERMOFLUIDCALC — STEFAN MOVING BOUNDARY PHASE CHANGE REPORT
=================================================================
Case Title                 : Continuous Casting Steel Shell Solidification Front
Target Thickness (L)       : 100.0 mm
Phase Change Temperatures  : Tm = 1500.0 C, Wall T0 = 900.0 C, Liquid Ti = 1550.0 C
Latent Heat & Solid Props  : Lf = 270.0 kJ/kg, ks = 32.00 W/m.K, rhos = 7400.0 kg/m3 (alpha_s = 6.653e-6 m2/s)
-----------------------------------------------------------------
SOLID STEFAN NUMBER (Ste_s): 1.4444
LIQUID STEFAN NUMBER (Ste_l: 0.1481
FRONT GROWTH CONSTANT (lam): 0.7133
TOTAL SOLIDIFICATION TIME  : 0.205 hours (738.6 seconds)
Layer Thickness after 1 hr : 220.78 mm
=================================================================

📘 Calculation Methodology & Stefan Problem Standards

Neumann Transcendental Solution

Solves the exact non-linear energy balance at the moving phase change interface $x = s(t)$:

                    λ · eλ² · erf(λ) = Stes / √π
                

Parabolic Growth Law

The solid front advances with the square root of time: $s(t) = 2\lambda \sqrt{\alpha_s t}$. Freezing rate decelerates as solid thermal resistance grows.

Key Engineering Assumptions

1D semi-infinite or planar slab geometry with sharp phase interface.
Constant solid thermophysical properties ($\rho_s, k_s, c_{p,s}$).
Applicable to ice makers, PCM thermal storage, casting, and food freezing.

ThermoFluidCalc Engineering Report