Engineering & Scientific
Lumped-Capacitance Biot Gate (Bi <= 0.1) & Newtonian Thermal Quench Time
Verify lumped-system validity via Characteristic Length L_c = V / A_s and Biot number Bi = h * L_c / k, then calculate thermal time constant tau and exact cooling/heating time.
Always check Biot <= 0.10 before trusting exponential cooling formulas
Newton’s exponential cooling law T(t) = T_inf + (T_i - T_inf) * EXP(-t / tau) assumes internal conduction inside a solid part is much faster than surface convection, keeping internal temperature nearly uniform (< 5% core-to-surface lag). That assumption holds if and only if the dimensionless Biot number Bi = h * L_c / k <= 0.10, where characteristic length L_c = Volume / Surface_Area (R/3 for a sphere, R/2 for a long cylinder, half-thickness L for a two-sided plate).
When Bi <= 0.10, the thermal time constant is tau = (rho * c_p * L_c) / h in seconds, and the time required to cool or heat from T_i to T_target in an ambient bath T_inf is t = -tau * LN((T_target - T_inf) / (T_i - T_inf)).
| Thermal Quantity | Formula | Example Quench Value | Engineering Meaning |
|---|---|---|---|
| Characteristic Length L_c | Volume / Area | 0.0050 m (5 mm) | Volume-to-surface ratio |
| Biot Number Bi | (h * L_c) / k | 0.0100 | Bi <= 0.10 -> LUMPED_VALID |
| Thermal Time Constant tau | (rho * c_p * L_c) / h | 50.00 s | 63.2% temperature decay time |
| Time to Reach 70 °C | -tau * LN((T_tgt - T_inf)/(T_i - T_inf)) | 69.31 s | Exact 75% quench completion |
Biot Number Validity Gate Formula
Compute Bi = (h * (V / A_s)) / k and flag GRADIENT_EFFECTS whenever Bi > 0.10 so engineers switch to 1-term Heisler/Fourier series when internal resistance dominates.
=IF(E3 <= 0.1, "LUMPED_VALID", "GRADIENT_EFFECTS")Exact Time to Target Temperature Formula
Multiply -tau by LN((T_target - T_inf) / (T_i - T_inf)) after verifying T_target lies strictly on the approach side of T_inf.
=ROUND(-E4 * LN((B9 - B8) / (B7 - B8)), 2)Try the example
Use the original values or change them to test your understanding.