Lumped thermal relaxation and Biot criterion
An approximately uniform-temperature body can have an exponential thermal response.
Key knowledge
For a fixed body with approximately uniform temperature, constant heat capacity mc and only convective exchange with a constant-temperature fluid, mc dT/dt=−hA(T−T_∞). With constant h and area A, T(t)=T_∞+(T_i−T_∞)exp(−t/τ), where τ=mc/(hA). A small Biot number hL_c/k_body supports the uniform-temperature approximation by making internal conduction resistance small relative to the surface transfer resistance.
Values and equations
Bi=hL_c/k_body; mc dT/dt=−hA(T−T_∞); τ=mc/(hA); T(t)=T_∞+(T_i−T_∞)exp(−t/τ)
- Bi
- Ratio of characteristic internal conduction resistance to external convective resistance 1
- h
- Convective heat-transfer coefficient W m⁻² K⁻¹
- L_c
- Characteristic internal conduction length chosen for the body m
- k_body
- Body thermal conductivity W m⁻¹ K⁻¹
- m
- Fixed body mass kg
- c
- Effective specific heat for the model process J kg⁻¹ K⁻¹
- A
- Heat-transfer surface area m²
- T,T_i,T_∞
- Body, initial body and constant fluid temperatures K
- t,τ
- Elapsed time and thermal time constant s
Conditions and scope
- Require m,c,h,A,k_body>0, t≥0 and a justified characteristic conduction length L_c>0; use Bi≪1 for the lumped approximation.
- The stated model neglects phase change, internal heat generation, mechanical work and other significant heat-transfer paths.
- Treat c as the effective specific heat appropriate to the constrained process; equality of c_p and c_v is not assumed as an exact law.
Sources · 2
References
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Evidence and review
Small-Biot lumped thermal storage, boundary convection and exponential relaxation toward a constant ambient temperature.
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Foundational review
2026-10-10 · Independent AI editorial inspection of this new compact reference entry, its equations, applicability, sources and knowledge connections.
Sources from other places
Agreement within this scope
With small internal temperature gradients and constant convection conditions, a lumped body approaches ambient temperature exponentially with τ=mc/(hA).
The Biot criterion is a screening approximation whose usefulness depends on geometry and required accuracy.
Indian Institute of Science · NPTEL
- Heat and Mass Transfer · Module5 IndiaEnglish
5.2 equation5.5; 5.3 equations5.8,5.11–5.12, pp2–4
- Heat and Mass Transfer · Module5
Entry details
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- foundation-physics-lumped-thermal-response
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- foundations-audit.1
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- Foundational core completion