Biot Number Calculator — Heat Transfer Lumped Capacitance
The Biot number (Bi) is the key dimensionless criterion in transient heat transfer. When Bi < 0.1, the solid is thermally thin and the simple lumped capacitance model applies — temperature is nearly uniform throughout. When Bi ≥ 0.1, internal temperature gradients matter and a distributed solution is needed.
W/(m²·K)
m
W/(m·K)
Bi = h · Lc / k (dimensionless)
- 1
Numerator h × Lc
25 × 0.05 = 1.25 W/(m·K) - 2
Biot number Bi = h × Lc ÷ k
1.25 ÷ 50 = 0.0250Bi < 0.1 → lumped capacitance valid (< 5% error).
How does this calculator work?
Bi = h·Lc/k compares surface convection to internal conduction. Bi < 0.1 means the solid is thermally thin — use the lumped capacitance decay T(t) = T∞ + (Ti − T∞)·e^(−t/τ). Above 0.1 gradients matter; above 1, use Heisler charts or numerical methods. Lc = Volume/Surface Area (sphere: r/3, cylinder: r/2, slab: half-thickness).
Formula
How this is calculated
The Biot number compares the internal conduction resistance of a solid (Lc/k) to the external convective resistance at its surface (1/h). A large convective coefficient h or a thick body (large Lc) drives Bi up; a high thermal conductivity k drives it down. When Bi is small, heat moves through the solid so easily that temperature differences inside it are negligible — justifying the lumped capacitance model T(t) = T∞ + (Ti − T∞)·exp(−t/τ), where τ = ρ·c·Lc/h is the time constant. The standard engineering threshold is Bi < 0.1, which limits the lumped model error to roughly 5%.
Above 0.1, internal temperature gradients grow important. Between 0.1 and 1, the lumped model is sometimes still used with explicit acknowledgment of reduced accuracy. Above 1, the surface and core temperatures differ significantly throughout the transient and you need Heisler charts, the one-term Fourier series, or numerical methods (FEM/FDM) appropriate to your geometry.
The characteristic length Lc = V/As is not the same as a geometric dimension: for a sphere Lc = r/3, for a long cylinder Lc = r/2, for a flat plate heated on one side Lc = full thickness, on both sides Lc = half-thickness. Always use the shape-appropriate formula, or compute V/As directly from your geometry.
Frequently asked questions
The standard engineering criterion is Bi < 0.1 (using Lc = V/As). Below this threshold the temperature variation within the solid is less than about 5% of the overall driving temperature difference, making the lumped model acceptably accurate. Some references relax this to Bi < 0.3 with explicit error acknowledgment.
Lc = Volume ÷ Surface Area. For a sphere of radius r: Lc = r/3. For a long cylinder of radius r: Lc = r/2. For a flat slab of thickness L heated on both sides: Lc = L/2. For a cube of side a: Lc = a/6. For complex shapes, compute V and As numerically and divide.
Use the Heisler chart approach or the one-term Fourier series solution appropriate for your geometry (plane wall, long cylinder, or sphere). For complex shapes, use finite-element or finite-difference numerical methods. The Biot number itself (using the relevant characteristic length) still appears as a parameter in those solutions.
Also known as
TG we-Calculate Editorial Team. (2026). Biot Number Calculator — Heat Transfer Lumped Capacitance [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/biot-numer-calculator
TG we-Calculate Editorial Team. "Biot Number Calculator — Heat Transfer Lumped Capacitance." TG we-Calculate. 2026. https://we-calculate.com/calculator/biot-numer-calculator.
TG we-Calculate Editorial Team, "Biot Number Calculator — Heat Transfer Lumped Capacitance," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/biot-numer-calculator
@misc{wecalculate_biot_numer_calculator, title = {Biot Number Calculator — Heat Transfer Lumped Capacitance}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/biot-numer-calculator}}, year = {2026}, note = {TG we-Calculate} }
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