Multidimensional Transient

Product Solution Principle

For transient heat conduction in multi-dimensional geometries, analytical solutions can be formulated as products of corresponding 1D solutions. This is valid for bodies initially at a uniform temperature $T_i$ subjected to convective boundary conditions with a constant ambient temperature $T_\infty$.

The dimensionless temperature ratio is defined as: $$\theta^*(x_1, x_2, \dots, t) = \frac{T(x_1, x_2, \dots, t) - T_\infty}{T_i - T_\infty}$$ Applying the product rule, we decompose this into: $$\theta^*(x_1, x_2, \dots, t) = \prod_{i=1}^n \theta_i^*(x_i, t)$$ where each $\theta_i^*$ is the 1D solution (one-term approximation) of the Heisler chart equations.

Supported Decomposition Equations:

• Short Cylinder (L₁ × r₀): $$\theta^*_{2D}(x, r, t) = \theta^*_{wall}(x, t) \times \theta^*_{cyl}(r, t)$$
• Rectangular Bar (L₁ × L₂): $$\theta^*_{2D}(x_1, x_2, t) = \theta^*_{wall, 1}(x_1, t) \times \theta^*_{wall, 2}(x_2, t)$$
• Rectangular Block (L₁ × L₂ × L₃): $$\theta^*_{3D}(x_1, x_2, x_3, t) = \theta^*_{wall, 1}(x_1, t) \times \theta^*_{wall, 2}(x_2, t) \times \theta^*_{wall, 3}(x_3, t)$$

Academic References:

1. Incropera, F. P., DeWitt, D. P., Bergman, T. L., & Lavine, A. S. (2011). Fundamentals of Heat and Mass Transfer (7th Edition). John Wiley & Sons. Chapter 5.8 (Multidimensional Effects).
2. Langston, L. S. (1982). Heat Transfer from Multidimensional Objects using Product Solutions. International Journal of Heat Transfer, 25(1), 149-150.
3. Çengel, Y. A., & Ghajar, A. J. (2015). Heat and Mass Transfer: Fundamentals and Applications (5th Edition). McGraw-Hill. Chapter 4.4 (Transient Heat Conduction in Multidimensional Systems).

Worked Engineering Example