When solid bodies undergo heating or cooling, temperatures change both with time \(t\) and spatial coordinates. The transient 1D heat equation is:

\[\frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}\]

Lumped Capacitance Method

If conduction resistance within the solid is negligible compared to external convection resistance, temperature inside the solid remains spatially uniform. The criteria for this approximation is a small Biot number:

\[Bi = \frac{h L_c}{k} < 0.1\]

Under this regime, cooling is modeled as: \(\frac{T(t) - T_\infty}{T_i - T_\infty} = e^{-(h A_s / \rho V C_p) t}\).

Analytical Solution (\(Bi > 0.1\))

If conduction resistance is high, separation of variables yields an infinite series solution. For a plane wall, the first-term approximation (Heisler charts) is:

\[\theta^* = C_1 e^{-\zeta_1^2 Fo} \cos(\zeta_1 x^*)\]

Where \(Fo = \alpha t/L^2\) is the Fourier number and \(\zeta_1\) is the first root of the transcendental equation: \(\zeta_1 \tan(\zeta_1) = Bi\).

References

  • Incropera, F. P., & DeWitt, D. P. (2006). Fundamentals of Heat and Mass Transfer. Wiley.