อัปเดต 2026-09-07

Cooldown / Heat-Up Time

Lumped-capacitance transient — Incropera Ch.5, Biot-gated

Time for a solid body or vessel content at uniform temperature to cool or heat toward an ambient fluid, using the single-node lumped-capacitance exponential model, gated by the Biot-number validity criterion.

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Resolves body geometry (surface area A_s, volume V, characteristic length L_c=V/A_s) for pipe / sphere / slab / vessel presets, forms the thermal time constant τ=ρVc/(hA_s), evaluates T(t) directly or inverts the closed-form logarithm for the time to a target temperature, and reports the Biot number to flag whether the lumped assumption still holds (never throws — returns the number plus a validity verdict).

สมการ

Lumped Thermal Time Constant

τ = \frac{ρ V c}{h A _{s}}

From the energy balance ρVc·dT/dt = −hA_s(T−T∞); Incropera, Fundamentals of Heat and Mass Transfer, §5.1.

Body Temperature at Time t

T (t) = T_{\infty} + (T_{i} - T_{\infty}) e^{- t / τ}

Closed-form exponential decay about the environment; time-to-target inverts as t=−τ·ln[(T_target−T∞)/(Ti−T∞)].

Biot Number (lumped-capacitance validity gate)

B i = \frac{h L _{c}}{k _{so l i d}}

Lumped capacitance is valid for Bi<0.1 (≤~5% internal non-uniformity, Incropera §5.2); Lc=V/A_s.

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