Newton Cooling: Time Constant τ
Lumped object, known T₀ and T_∞. Measure T at several times and recover τ from a fit of ln((T−T_∞)/(T₀−T_∞)) versus t.
Goal
Determine τ from T(t) = T_∞ + (T₀−T_∞)e^{−t/τ}. The slope of the log plot versus t is −1/τ.
Equipment
- Lumped specimen (Bi < 0.1)
- Known T₀, T_∞
- Thermometer
- Clock
Experiment
Theory
When Bi < 0.1 the object is spatially isothermal and Newton cooling applies. The bench hides τ and live T. T₀ and T_∞ are known. You only change the clock t. Bi is known to be valid and is not graded.
Procedure
- Initial and ambient temperatures are fixed and known. τ is hidden. You only change elapsed time t.
- Record. A thermometer logs T with small noise. There is no live T or τ.
- The notebook computes ln((T−T_∞)/(T₀−T_∞)). Repeat for at least 6 times from about 30 s to 420 s.
- Fit the log column versus t; τ = −1/slope. Compare with the reference.
Conclusion
The fitted time constant agrees with the hidden τ. Main uncertainties: thermometer noise and the lumped-capacitance idealization.