Derive the lumped capacitance temperature response and explain time constant.

For Bi < 0.1, energy balance gives: ρVc(dT/dt) = -hA(T-T∞). Solving: (T-T∞)/(Ti-T∞) = exp(-t/τ), where τ = ρVc/(hA) is the time constant. Physical interpretation: τ is the time to reach 63.2% of temperature change; 5τ gives 99% approach to equilibrium. For faster response (smaller τ): increase h (forced convection), increase A/V ratio (smaller objects, fins), decrease ρc (lower thermal mass). Example: quenching steel ball (D=20mm) in oil (h=1000 W/m²-K): τ = 7800×(0.01/3)×500/(1000) = 13 seconds. This analysis applies to thermocouple response, small electronics, and heat treatment processes.