🧊 2D Transient Conduction & Heisler Multi-D

Calculate 2D multidimensional transient heat conduction in rectangular billets: center core temperature (T0), corner surface temperature, Biot and Fourier numbers, and heat removed (Q/Qmax).

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

📊 Solver Telemetry ● ACTIVE

👁️ Consultations 22

⚡ Calculs faits 21

💾 Téléchargements 321 📦 Code Fortran 3.4 KB

📅 Mise en service Jun 2026

⏱️ Latence < 1 ms

🧊 2D Rectangular Billet Quenching & Isothermal Contour Field

Real-time visual simulation of internal heat diffusion migrating from center core to convective boundaries

📝 Configuration & Presets

🔩 Steel Billet Water Quench 🪟 Aluminum Extrusion Air Cool 🧱 Ceramic Furnace Lining 💻 Silicon CPU Die Surge

📐 Geometry (Half-Widths)

Half-Width L_x [mm]: Total 2Lx = 150 mm

Half-Height L_y [mm]: Total 2Ly = 240 mm

🧪 Material Physical Properties

Thermal Conductivity k [W/(m·K)]:

Density ρ [kg/m³]:

Specific Heat c_p [J/(kg·K)]:

🌡️ Thermal Boundary Conditions

Initial Temp T_i [°C]:

Ambient / Quench T_∞ [°C]:

Convection Coeff h [W/(m²·K)]:

Quench Time t [seconds]:

2D Heisler Product Formulation:
• Dimensionless Temp: θ(x,y,t) = θ_x(x,t) · θ_y(y,t)
• Center Solution: θ₀ = [ C_x e^{−ζ_x² Fo_x} ] · [ C_y e^{−ζ_y² Fo_y} ]
• Transcendental Roots: ζ · tan(ζ) = Bi
• Heat Removed: Q / Q_max = 1 − θ₀ · (sin ζ_x sin ζ_y) / (ζ_x ζ_y)

📊 Transient Conduction Results

📊 Output Summary
💾 Fortran Source

Center Core Temperature (T₀)

T₀ = 194.2 °C (θ = 0.8387)

Corner Temp: 1070.8 °C | Energy Removed: 57.9 %

t = 1800.0 s

Biot Numbers (Bi_x / Bi_y) 6.000 / 9.600 Spatial gradients active

Fourier Numbers (Fo_x / Fo_y) 0.218 / 0.085 Dimensionless times

Thermal Diffusivity (α) 6.82e-7 m²/s k / (ρ · c_p)

Corner Surface Temperature 1070.8 °C Fastest cooled point

📈 Center & Corner Temperature (°C) vs Time (s)

📉 Fraction of Thermal Energy Transferred Q/Q_max (%)

=================================================================
 THERMOFLUIDCALC — 2D TRANSIENT CONDUCTION REPORT
=================================================================
Case Title                 : Refractory Ceramic Furnace Lining Transient Heat-Up
Billet Dimensions          : 2Lx = 150.0 mm, 2Ly = 240.0 mm
Thermophysical Properties  : k = 1.50 W/m.K, rho = 2200.0 kg/m3, cp = 1000.0 J/kg.K (alpha = 6.818e-7 m2/s)
Thermal Conditions         : Ti = 20.0 C, Tinf = 1100.0 C, h = 120.0 W/m2.K, t = 1800.0 s
-----------------------------------------------------------------
Biot Numbers (Bix / Biy)   : 6.0000 / 9.6000
Fourier Numbers (Fox / Foy): 0.2182 / 0.0852
CENTER CORE TEMPERATURE T0 : 194.22 deg C (theta = 0.8387)
CORNER TEMPERATURE T_corn  : 1070.84 deg C (theta = 0.0270)
Energy Transferred (Q/Qmax): 57.87 %
=================================================================

📘 Calculation Methodology & Multidimensional Heisler Standards

Heisler Product Theorem

For multidimensional geometric solids (infinite rectangular bar), the dimensionless temperature is the direct product of 1D infinite slab analytical solutions:

                    θ(x,y,t) = θslab,x(x,t) × θslab,y(y,t)
                

Spatial Temperature Distribution

The corner cools first due to simultaneous two-sided convective exposure ($\theta_{corner} = \theta_{surf,x} \cdot \theta_{surf,y}$), while the core retains thermal energy.

Key Engineering Assumptions

Homogeneous, isotropic solid with constant thermal diffusivity $\alpha$.
Uniform initial temperature $T_i$ and uniform ambient quench convection $h$.
Fourier numbers $Fo > 0.2$ for high-accuracy single-term series representation.

ThermoFluidCalc Engineering Report