Biot Number (Bi): Conduction–Convection Coupling in Heat Transfer

The Biot number, symbol Bi, is the ratio between internal conduction resistance and external convection resistance. It underpins every decision about whether a body heats uniformly or develops gradients. This guide follows Bi from its nineteenth-century origin to modern ISO 80000-5 notation, demonstrates how to estimate it for irregular geometries, and shows how researchers, product engineers, and energy auditors rely on Bi to interpret transient temperature data. Use it alongside our Reynolds number calculator and the Nusselt number explainer to select correlations that match your flow regime and surface conditions.

Definition and Physical Meaning

ISO 80000-5 presents the Biot number as a dimensionless quantity linking heat conduction and convection. The formal definition is Bi = h·L_c / λ, where h is the convective heat-transfer coefficient in W·m⁻²·K⁻¹, L_c is a characteristic length representing the ratio of volume to surface area, and λ (often written k) is the material thermal conductivity in W·m⁻¹·K⁻¹. Bi therefore compares how easily heat leaves a body through its surface with how quickly it conducts through its interior. When Bi ≪ 1, the temperature is nearly uniform and lumped-capacitance models are defensible; when Bi ≥ 0.1, spatial gradients matter and partial differential equations must track the thermal wave moving inward.

The characteristic length is not a universal constant. For slabs of thickness 2L, textbooks adopt L_c = L; for cylinders of radius r and long length, L_c = r/2; for spheres, L_c = r/3. ISO 13786 and ASTM E1225 emphasise that irregular shapes should use the ratio volume/surface area to preserve energy balance. In computational models, analysts sometimes tune L_c so that Bi reproduces detailed simulations, but the physical meaning—comparing resistances—remains.

Historical Development and Standards

Jean-Baptiste Biot and early heat conduction

Jean-Baptiste Biot introduced the ratio bearing his name while analysing thermal conduction in solids in 1804. Working with Pierre-Simon Laplace, he framed heat flow as a diffusion process. The Biot number allowed him to compare internal conduction to external radiation losses, an important concept before the establishment of the Stefan–Boltzmann law. Joseph Fourier’s analytical solutions published in 1822 expanded upon Biot’s work, embedding the ratio into transient conduction theory. By the late nineteenth century, engineers used Bi to evaluate quenching in metallurgy and the heating of telegraph cables, but notation and characteristic lengths varied widely.

Modern formalisation through ISO and ASME

The rise of international standardisation after the Second World War unified terminology. ISO 80000-5 and ASME PTC 19.1 both codify Bi as the convection-to-conduction resistance ratio. Heat transfer textbooks by Jakob, Rohsenow, and Incropera reinforced the threshold Bi ≤ 0.1 for lumped methods, though practitioners note that Bi ≈ 0.3 can still yield acceptable approximations if transient data are short or measurement uncertainty is large. Today, metrology institutes supply reference materials with known thermal conductivity and volumetric heat capacity, allowing Bi to be calculated with traceable inputs when calibrating transient plane-source devices.

Conceptual Foundations and Analytical Tools

Resistance networks and energy balances

Bi fits naturally into thermal resistance analogies. Internal conduction resistance equals L_c/(λ·A), while external convection resistance equals 1/(h·A). Their ratio simplifies to Bi, making it easy to add contact resistances or multiple convection paths in series–parallel networks. Because Bi is independent of the surface area A, it applies equally to microelectronic dies and large castings provided the characteristic length matches the dominant heat-flow path. Engineers leverage this property to compare different designs on a common scale before running costly finite element analyses.

Lumped-capacitance criterion

The lumped-capacitance model treats an object as having uniform temperature T(t) that changes according to the first- order differential equation ρ·c·V·dT/dt = h·A·(T∞ − T). Solving yields an exponential approach to ambient temperature with time constant τ = ρ·c·V/(h·A). This model implicitly assumes Bi ≪ 1. When Bi approaches 0.1, the surface cools faster than the core, causing errors that can exceed 10 %. Engineers can compensate by using Heisler charts or separation-of-variables solutions that incorporate Bi explicitly. The charts plot dimensionless temperature θ/θ₀ versus Fourier number Fo = α·t/L_c² for various Bi, offering quick estimates without computers.

Biot number in numerical simulation

Computational fluid dynamics (CFD) and finite element solvers incorporate Bi through boundary conditions. Applying a convective boundary equates heat flux q″ = h·(T_s − T∞). If you non-dimensionalise the governing equations, Bi emerges naturally. High-Bi situations require fine meshes near surfaces to resolve steep gradients, while low-Bi cases can use coarser meshes without sacrificing accuracy. Sensitivity studies that vary h, λ, or L_c by ±10 % help quantify uncertainty and maintain alignment with ISO Guide 98-3 (GUM) principles.

Applications Across Industries

Heat treatment and metallurgy

Quenching steel blades or turbine disks demands a low Bi to prevent thermal shock. Metallurgists manipulate Bi by choosing quenchants with specific h values—oil for moderate Bi, polymer solutions for tunable h, and agitated water for high Bi that promotes surface hardening. Monitoring Bi ensures the cooling curve matches phase-transformation targets while avoiding cracks. Data from dilatometry experiments often report Bi alongside cooling rates to document repeatability.

Electronics thermal management

Semiconductor packages strive for Bi close to unity: too low and devices respond sluggishly to dynamic loads; too high and hotspots appear at the junction. Engineers combine high-conductivity materials such as copper heat spreaders with advanced interface materials to tune Bi. Lumped models may suffice for microcontrollers, but power modules with multiple chips require spatially resolved simulations. Linking Bi to the thermal conductivity guide helps teams select substrates and encapsulants that balance conduction and convection resistances.

Built environment and energy auditing

Building scientists use Bi to assess whether walls, floors, or thermal storage tanks can be treated as lumped masses when applying the European standard EN ISO 13790. For thick masonry walls, Bi often exceeds 1, signalling that one-dimensional conduction models are necessary. Conversely, slim interior partitions may exhibit Bi below 0.1, letting auditors approximate the response to heating setbacks with a simple time constant. Pairing Bi with the thermal storage sizing calculator reveals how deeply heat penetrates during night-time charging cycles.

Biot in bioengineering and food processing

Pasteurising beverages, freeze-drying pharmaceuticals, and designing implantable devices all require Bi calculations. In bioprocessing, high Bi indicates that the surface temperature responds rapidly to sterilisation steam, protecting microbial safety, while the core may lag. Food engineers manipulate Bi by adjusting airflow velocity, humidity, and packaging conductivity so that quality parameters such as colour and texture remain within specification. These industries often employ transient thermal probes calibrated to ISO 22007-2 for thermal diffusivity, ensuring Bi estimates trace back to SI units.

Measurement, Estimation, and Uncertainty

Direct measurement of Bi is impractical because it is a ratio. Instead, practitioners determine h via empirical correlations or experiments, measure λ using techniques such as the laser flash method (ISO 18755) or transient plane source, and compute L_c from geometry. To validate Bi, engineers compare predicted temperature histories with thermocouple or infrared data. If deviations exceed uncertainty limits, they refine h estimates or revisit geometric assumptions. Including coverage factors k = 2 for expanded uncertainties ensures reported Bi aligns with metrological best practice.

When experimental determination of h is necessary, the most common approach is a transient cooling test. A specimen is heated uniformly, then exposed to a controlled airflow. By fitting surface temperature data to the lumped-capacitance solution, h is extracted, and Bi is calculated. This method requires Bi < 0.1 to be valid; if the fit residuals are large, iterative inverse methods that solve the heat equation with guessed Bi values provide more accurate results. Modern optimisation routines combine Bayesian inference with CFD to reduce the time required for convergence.

Why the Biot Number Matters

Bi anchors decisions about safety margins, quality control, and energy efficiency. It tells aerospace engineers how quickly composite skins equilibrate during re-entry, guides battery designers when estimating thermal runaway propagation, and informs culinary scientists calibrating sous-vide equipment. Ignoring Bi can lead to thermal cracking, inaccurate sensor readings, or regulatory non-compliance. By explicitly calculating Bi and documenting the supporting data, teams create transparent audit trails that support certifications such as ISO 9001 and AS9100.

The concept also extends beyond heat. Analogous ratios exist for mass transfer (Bi_m) and moisture movement, following the same resistance logic. Multiphysics models often couple thermal Bi with Bi_m to capture simultaneous drying and cooling, ensuring the mathematical treatment remains coherent. Appreciating these parallels strengthens interdisciplinary collaboration across chemical engineering, materials science, and environmental studies.