Biot & Fourier Number Calculator

Calculate Biot and Fourier numbers for lumped capacitance analysis of transient heat conduction.

Natural conv: 5-25, Forced: 25-250
Sphere: r/3, Cylinder: r/2, Slab: L/2
Steel=50, Al=200, Concrete=1.4
Steel=1.4e-5, Al=8.4e-5
Please check your inputs and try again.

Biot and Fourier Numbers Separate Internal and External Thermal Effects

The Biot number compares resistance to heat conduction inside a solid with resistance to convection at its surface, while the Fourier number measures how far thermal diffusion has progressed in time. Bi=hLc/k. When Bi is small, temperature gradients inside the object are weak compared with the surface-to-fluid temperature difference, which is why lumped-capacitance analysis can be accurate. A common engineering guideline is Bi<0.1.

Fo=αt/Lc2, where thermal diffusivity α=k/(ρcp). Large Fo means heat has had more time to diffuse through the characteristic length. The two dimensionless groups answer different questions: Bi tests whether the body can be treated as nearly uniform in temperature, while Fo describes transient penetration and equilibration.

Bi=hLc/k,   Fo=αt/Lc2,   α=k/(ρcp)
SymbolMeaningWhy it appears / units
hConvective heat-transfer coefficientW/(m²·K); controls surface exchange.
LcCharacteristic lengthm; often V/As for lumped analysis.
kThermal conductivityW/(m·K); larger k reduces internal temperature gradients.
αThermal diffusivitym²/s; rate at which temperature disturbances spread.

A small Biot number supports a nearly uniform body temperature, but the transient still takes time; Fourier number helps describe that time evolution. Always use the characteristic length specified by the correlation or geometry, because different conventions can change both dimensionless numbers.

Bi and Fo answer different questions. Bi=hLc/k compares internal conduction resistance with surface convection resistance; Fo=αt/Lc2 measures elapsed diffusion time. Changing time should alter Fo but not Bi when material, geometry, and convection conditions remain fixed.

Worked Examples

Example 1: Steel sphere r=1cm cooling in air: h=25, k=50
Bi=25×(0.01/3)/50=0.0017
Result: Bi<<0.1 — lumped capacitance valid
Uniform temperature in steel — use simple τ=ρcV/hA
Example 2: Concrete wall L=0.1m: h=10, k=1.4
Bi=10×0.1/1.4=0.71
Result: Bi>>0.1 — spatial gradients important
Need Heisler chart or numerical method
Example 3: Small metal sphere approximation
h=20W/m²K, Lc=0.005m, k=50W/mK
Result: Bi=0.002
Internal conduction is much easier than surface convection, supporting a lumped-temperature model.
Example 4: Thermal diffusion time
α=1×10−5m²/s, t=100s, L=0.02m
Result: Fo=2.5
Thermal diffusion has acted over a timescale several times L²/α.

Common Mistakes

⚠️
Using the full thickness when the correlation expects a characteristic length

For lumped analysis Lc=V/As. Slabs, cylinders, and spheres may use different length definitions in transient charts or correlations.

⚠️
Assuming Bi<0.1 means equilibrium has already been reached

Small Bi supports uniform internal temperature, but the object can still be far from the surrounding-fluid temperature.

⚠️
Confusing conductivity with diffusivity

Conductivity k measures heat-flow response to a gradient, while diffusivity α=k/(ρcp) measures how rapidly temperature changes spread.

Frequently Asked Questions

Lumped capacitance method?
Valid when Bi<0.1. Assumes entire body is at one temperature (like a well-stirred pot). Temperature decays as T(t)=T_∞+(T₀-T_∞)×exp(-t/τ) where τ=ρcV/hA.
Fourier number physical meaning?
Fo = αt/L² = (heat conducted rate)/(heat stored rate). Fo>0.2 means enough time for heat to diffuse through the material — quasi-steady temperature profiles exist.
Why is Biot number dimensionless?
The numerator hLc and conductivity k have the same units of W/(m·K), so their ratio has no units. It compares two thermal resistances rather than representing a new physical property.
What does a large Biot number mean?
It means internal conduction resistance is important relative to surface convection, so significant temperature gradients can exist inside the body and a single lumped temperature may be inaccurate.
What does Fourier number physically represent?
Fo compares elapsed time with the characteristic diffusion time L²/α. Values much less than one indicate limited thermal penetration, while larger values indicate more extensive diffusion across the body.
Can Biot and Fourier numbers be used together?
Yes. Transient conduction solutions and charts commonly depend on both. Bi describes the boundary-to-interior resistance balance, while Fo tracks the dimensionless time evolution.
Which characteristic length belongs in the Biot number?
For lumped-capacitance analysis, the common characteristic length is Lc=V/As, where As is the surface area participating in convection. Other transient-conduction correlations may define characteristic length differently. State the geometry and definition before comparing Bi with the usual 0.1 lumped-capacitance guideline.

Formula Explorer connections

Interpretation: This relationship connects pressure, velocity, density, viscosity, geometry or transport in a fluid system. Assumption: Check whether flow is steady, incompressible, laminar, fully developed or one-dimensional. Reynolds and Mach regimes determine whether simplified formulas are valid.

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