📉 Transient Conduction 1D — Heisler Charts
Analyze transient heat conduction using the one-term approximation for plane walls, long cylinders, and spheres.
📝 Configuration
One-Term Approximation (Fo > 0.2):
Wall: θ* = C₁·e-ζ₁²Fo·cos(ζ₁·x/L)
Cyl: θ* = C₁·e-ζ₁²Fo·J₀(ζ₁·r/r₀)
Sphere: θ* = C₁·e-ζ₁²Fo·sin(ζ₁·r/r₀)/(ζ₁·r/r₀)
• Bi = hL/k · Fo = αt/L²
• θ* = (T-T∞)/(Tᵢ-T∞)
Wall: θ* = C₁·e-ζ₁²Fo·cos(ζ₁·x/L)
Cyl: θ* = C₁·e-ζ₁²Fo·J₀(ζ₁·r/r₀)
Sphere: θ* = C₁·e-ζ₁²Fo·sin(ζ₁·r/r₀)/(ζ₁·r/r₀)
• Bi = hL/k · Fo = αt/L²
• θ* = (T-T∞)/(Tᵢ-T∞)
📊 Results & Visualization
Results and visualizations will appear here after calculation.
ℹ️ About the One-Term Approximation
For Fourier number Fo > 0.2, the infinite series solution converges to a single dominant term, giving accurate results (< 2% error).
Eigenvalue equations:
• Wall: ζₙ·tan(ζₙ) = Bi
• Cylinder: ζₙ·J₁(ζₙ)/J₀(ζₙ) = Bi
• Sphere: 1 - ζₙ·cot(ζₙ) = Bi
Applications:
• Heat treatment of metals
• Food sterilization processes
• Thermal protection systems
• Concrete curing analysis
For Fourier number Fo > 0.2, the infinite series solution converges to a single dominant term, giving accurate results (< 2% error).
Eigenvalue equations:
• Wall: ζₙ·tan(ζₙ) = Bi
• Cylinder: ζₙ·J₁(ζₙ)/J₀(ζₙ) = Bi
• Sphere: 1 - ζₙ·cot(ζₙ) = Bi
Applications:
• Heat treatment of metals
• Food sterilization processes
• Thermal protection systems
• Concrete curing analysis