Transient Conduction 1D
Analyze transient heat conduction using the one-term approximation (Heisler charts) for plane walls, long cylinders, and spheres.
Calculation Domain Inputs
Specify geometry parameters, material properties, and boundary conditions to calculate the transient thermal history of the 1D domain using Heisler series approximation:
- Geometry: Choose between a Plane Wall, Long Cylinder, or Sphere.
- Properties ($k, \rho, c_p$): Thermal conductivity, density, and specific heat.
- Convection ($h, T_\infty$): Convection coefficient and ambient temperature of the fluid.
- Point of Interest ($x/L$ or $r/r_0$, $t$): Position coordinate and time elapsed.
Configuration
Wall: $\theta^* = C_1 e^{-\zeta_1^2 Fo} \cos(\zeta_1 x/L)$
Cyl: $\theta^* = C_1 e^{-\zeta_1^2 Fo} J_0(\zeta_1 r/r_0)$
Sphere: $\theta^* = C_1 e^{-\zeta_1^2 Fo} \frac{\sin(\zeta_1 r/r_0)}{\zeta_1 r/r_0}$
• $Bi = hL/k$, $Fo = \alpha t/L^2$
• $\theta^* = (T-T_\infty)/(T_i-T_\infty)$
Results & Visualization
Results and visualizations will appear here after calculation.
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
📘 Calculation Methodology
Mathematical Model & Theory
For transient conduction in 1D shapes (plane walls, long cylinders, spheres) where the Biot number is greater than 0.1, internal temperature gradients are non-negligible. The one-term Heisler approximation is valid for Fourier numbers ($Fo > 0.2$):
Where $\zeta_1$ and $C_1$ are eigenvalues determined by boundary conditions (transcendental equations involving $Bi$).
Assumptions & Boundary Conditions:
- One-dimensional transient heat conduction in the spatial coordinate ($x$ for wall, $r$ for cylinder/sphere).
- Uniform initial temperature ($T_i$) throughout the solid at time $t = 0$.
- Constant material properties (conductivity $k$, density $\rho$, specific heat $C_p$).
- Symmetric temperature profile about the centerline ($x = 0$ or $r = 0$).
- Uniform and constant convection heat transfer coefficient $h$ and temperature $T_\infty$ at the outer surface boundary.
- No internal heat generation.
Academic References:
- Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer.
- Çengel, Y. A. (2015). Heat and Mass Transfer: Fundamentals and Applications.
Worked Engineering Example
A steel plate ($2L = 40$ mm thick, $k = 45$ W/m·K, $\alpha = 1.25 \times 10^{-5}$ m²/s) initially at 400°C is exposed to convection at 20°C with $h = 250$ W/m²·K. Find the temperature at the center after 2 minutes.
Step-by-step Solution:
1. Calculate Biot number:
$$Bi = \frac{h L}{k} = \frac{250 \times 0.02}{45} = 0.1111$$ 2. Calculate Fourier number after $t = 120$ s:
$$Fo = \frac{\alpha t}{L^2} = \frac{1.25 \times 10^{-5} \times 120}{0.02^2} = 3.75 > 0.2 \quad \text{(One-term approx valid)}$$ 3. Find eigenvalues for $Bi = 0.1111$ (Table/Calculated):
$$\zeta_1 \approx 0.32 \text{ rad}, \quad C_1 \approx 1.018$$ 4. Calculate center temperature ratio ($x^* = 0$):
$$\theta_0^* = C_1 e^{-\zeta_1^2 Fo} = 1.018 \times e^{-0.32^2 \times 3.75} = 1.018 \times 0.6811 = 0.6934$$ $$T(0, 120) = T_{\infty} + \theta_0^* (T_i - T_{\infty}) = 20 + 0.6934 \times (400 - 20) = 283.48 \text{°C}$$
Final Result:
The plate center temperature is 283.5°C.