Biot number

The Biot number (Bi) is a dimensionless quantity in heat transfer analysis that quantifies the ratio of internal thermal resistance due to conduction within a solid body to the external thermal resistance due to convection at its surface.^[1] It is defined by the formula Bi = \frac{h L}{k}, where h is the convective heat transfer coefficient (in W/m²K), L is the characteristic length of the body (typically volume divided by surface area, in m), and k is the thermal conductivity of the solid (in W/mK).^[1] This parameter arises from balancing Fourier's law of conduction with Newton's law of cooling, helping to assess the temperature distribution during transient heating or cooling processes.^[1] Introduced by French physicist Jean-Baptiste Biot in 1804 during his studies of conduction-convection interactions, the Biot number provides a criterion for simplifying heat transfer models.^[1] When Bi is much less than 0.1, internal conduction is rapid compared to surface convection, allowing the lumped capacitance approximation where the body's temperature is assumed uniform (with errors under 5%).^[2] Conversely, for Bi greater than 0.1, significant temperature gradients develop internally, necessitating solutions to the full conduction equation, such as in finite element or analytical methods for irregular geometries.^[3] The number is also analogous in mass transfer contexts, where it compares diffusive resistance inside a body to convective mass transfer at the surface, using Bi_m = \frac{k_m L}{D_e} (with k_m as the mass transfer coefficient and D_e as effective diffusivity).^[3] In practical applications, the Biot number is essential for designing systems involving transient thermal behavior, such as cooling of electronic components, food processing, battery thermal management, and conjugate heat transfer in ducts or reactors.^[2] For instance, low Bi values confirm uniform temperatures in small or high-conductivity objects exposed to fluids, while high values indicate the need for detailed internal modeling to predict hotspots or efficiency losses.^[2] Its use extends to optimizing industrial processes by balancing material properties, geometry, and flow conditions to minimize thermal stresses or energy consumption.^[3]

Definition and Derivation

Mathematical Expression

The Biot number, denoted as Bi, is defined by the formula

Bi = \frac{h L_c}{k},

where h is the convective heat transfer coefficient, L_c is the characteristic length of the solid body, and k is the thermal conductivity of the solid.^[4] The convective heat transfer coefficient h quantifies the rate of heat transfer by convection at the surface of the solid, with units of W/m²·K. The characteristic length L_c represents a geometric scale of the body, typically taken as the ratio of its volume to surface area (L_c = V/A) for irregular shapes, and has units of meters (m).^[4] The thermal conductivity k measures the solid's ability to conduct heat internally, with units of W/m·K. Dimensional analysis confirms that the Biot number is dimensionless, as the units of h L_c (W/m·K) cancel with those of k (W/m·K). Specific examples of the characteristic length include L_c = r/3 for a sphere of radius r, and L_c = t/2 for a flat plate of thickness t exposed on both sides.^[5]^[6]

Derivation from Governing Equations

The derivation of the Biot number begins with the one-dimensional unsteady heat conduction equation for a solid body, assuming no internal heat generation and constant thermal properties:

\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2},

where T is the temperature, t is time, x is the spatial coordinate, and \alpha = k / (\rho c_p) is the thermal diffusivity, with k denoting thermal conductivity, \rho the density, and c_p the specific heat capacity.^[3]^[7] This equation is subject to an initial condition, typically T(x,0) = T_i (uniform initial temperature), and boundary conditions. At the surface x = L (where L is the characteristic length, such as half-thickness for a slab), the boundary condition equates the conductive heat flux to the convective heat flux via Newton's law of cooling:

-k \frac{\partial T}{\partial x} \bigg|_{x=L} = h (T(L,t) - T_\infty),

with h as the convective heat transfer coefficient and T_\infty the ambient fluid temperature.^[3]^[1] To reveal the dimensionless groups governing the problem, non-dimensional variables are introduced: the dimensionless temperature \theta = (T - T_\infty)/(T_i - T_\infty), the dimensionless position \xi = x/L, and the dimensionless time (Fourier number) Fo = \alpha t / L^2. Substituting these into the heat equation yields the dimensionless form:

\frac{\partial \theta}{\partial Fo} = \frac{\partial^2 \theta}{\partial \xi^2}.

The initial condition becomes \theta(\xi,0) = 1, and the surface boundary condition transforms to:

\frac{\partial \theta}{\partial \xi} \bigg|_{\xi=1} = -Bi \, \theta(1, Fo),

where Bi = h L / k is the Biot number, emerging as the dimensionless coefficient that balances the convective heat transfer at the surface against the internal conductive resistance within the solid.^[7]^[8] The Biot number was originally introduced by Jean-Baptiste Biot in 1804 during his analysis of the interaction between thermal conduction in a solid and convection at its surface.^[1]

Physical Interpretation

Significance of Bi Values

The Biot number (Bi) represents the ratio of internal conductive resistance within a solid body, given by the characteristic length L divided by its thermal conductivity k, to the surface convective resistance, given by the reciprocal of the heat transfer coefficient h.^[1] This ratio quantifies the balance between internal heat conduction and external convection, determining whether temperature variations within the body are negligible or significant during transient heat transfer processes.^[3] When Bi ≪ 1, typically less than 0.1, the internal conductive resistance is much smaller than the surface convective resistance, leading to negligible temperature gradients within the body and allowing the assumption of a uniform temperature throughout.^[1] In such cases, conduction dominates internally, enabling the use of simplified lumped system analysis with an error of less than 5% in temperature predictions.^[9] For example, small metal objects in low-convection environments often exhibit this behavior, where the entire body responds uniformly to external temperature changes.^[3] At Bi ≈ 1, the internal and surface resistances are comparable, resulting in moderate temperature gradients that require more detailed analytical or numerical solutions to accurately describe the internal temperature distribution.^[1] For Bi ≫ 1, generally greater than 10, the internal conductive resistance dominates over the surface convective resistance, producing steep temperature gradients within the body while the surface temperature closely approximates the ambient fluid temperature, akin to the limit of infinite h.^[3] Here, convection at the surface limits the overall heat transfer rate, and internal conduction governs the temperature profile, necessitating distributed models for analysis.^[9]

Relation to Temperature Profiles

The Biot number governs the nature of spatial temperature distributions in a body during convective heat transfer by quantifying the relative importance of internal conduction versus surface convection resistances. When Bi ≪ 1, the temperature profile remains flat and uniform across the body, exhibiting no appreciable spatial variation because conduction efficiently equalizes temperatures internally relative to the slower surface heat exchange.^[1] For finite Biot numbers, steady-state temperature profiles are linear due to the competing influences of conduction and convection, while transient profiles begin with an initial uniform distribution and progressively evolve toward surface-dominated conditions as heat penetrates the interior.^[10] When Bi ≫ 1, pronounced steep temperature gradients form near the surface, with the core region substantially lagging the surface response and the overall profile resembling a pure conduction solution under fixed-surface-temperature (Dirichlet) boundary conditions.^[1] In slab geometry under steady-state conditions with fixed temperature at one side (\xi=0) and convection at the other (\xi=1), the dimensionless temperature profile takes the linear form \theta(\xi) = 1 - \frac{\mathrm{Bi}}{1 + \mathrm{Bi}} \xi, where \theta = \frac{T - T_\infty}{T_0 - T_\infty} is the normalized temperature, T_0 is the fixed temperature, and \xi = x/L is the normalized position. This illustrates that the profile's slope—and thus the gradient steepness—is given by \frac{\mathrm{Bi}}{1 + \mathrm{Bi}}, which is approximately proportional to Bi for small Bi and approaches 1 for large Bi.^[10] Typical visualizations of \theta versus \xi for Bi = 0.1, 1, and 10 underscore this progression: at Bi = 0.1, the profile is nearly horizontal with minimal gradient (slope ≈ 0.09); at Bi = 1, a moderate slope of 0.5; and at Bi = 10, a steep slope ≈ 0.909 emphasizing the dominance of internal resistance.^[10]

Applications in Heat Transfer

Lumped System Analysis (Bi ≪ 1)

The lumped system analysis, also known as the lumped capacitance method, applies to transient heat conduction problems where internal thermal resistance is negligible compared to surface convection resistance, corresponding to Bi ≪ 1. This approximation is valid when Bi < 0.1, which limits the error in predicted body temperature to less than 5% relative to exact solutions accounting for internal gradients.^[11] The key assumption is that the temperature T within the body remains spatially uniform at any instant during the transient process, varying only with time T(t), which holds for bodies with high thermal conductivity or small characteristic dimensions.^[9] The governing equation arises from an overall energy balance on the body, equating the rate of change of internal thermal energy to the convective heat transfer rate at the surface:

\rho V c_p \frac{dT}{dt} = -h A_s (T - T_\infty)

where \rho is density, V is volume, c_p is specific heat capacity, h is the convective heat transfer coefficient, A_s is surface area, and T_\infty is the ambient fluid temperature.^[9] Introducing the dimensionless temperature \theta = \frac{T - T_\infty}{T_i - T_\infty} (with T_i as initial temperature) and dimensionless time t^* = \frac{\alpha t}{L_c^2} (the Fourier number Fo, where \alpha = k/(\rho c_p) is thermal diffusivity and L_c = V/A_s is characteristic length), the equation simplifies to

\frac{d\theta}{dt^*} = -{\rm Bi} \ \theta.

The analytical solution is an exponential decay:

\theta = \exp(-{\rm Bi} \ t^*),

indicating that the normalized temperature difference diminishes proportionally to the product of Bi and Fo.^[9] This method finds applications in scenarios involving rapid internal conduction relative to surface heat transfer, such as cooling small electronic components where uniform temperature simplifies thermal management predictions.^[12] It is also employed for transient cooling of food products, like packaged dairy or small produce, to estimate chilling times without detailed spatial modeling.^[13] In biomedical engineering, the approach models heat transfer in small biotissue samples or perfused vessels, aiding analysis of thermal therapies where blood flow enhances effective conductivity.^[14] However, the lumped capacitance method becomes inaccurate for larger bodies or low-conductivity materials, where Bi > 0.1 introduces substantial internal temperature gradients, causing prediction errors to exceed 5% and necessitating distributed models.^[11]

Distributed Systems (Finite Bi)

When the Biot number exceeds approximately 0.1, significant temperature gradients develop within the solid due to the comparable magnitudes of internal conduction resistance and surface convection resistance, necessitating distributed parameter models that account for spatial variations in temperature.^[9] In such cases, the transient heat conduction equation, a partial differential equation of the form \frac{\partial T}{\partial t} = \alpha \nabla^2 T, must be solved subject to appropriate initial and boundary conditions, including convective boundaries where -k \nabla T = h (T - T_\infty).^[9] For steady-state conditions in a slab with a fixed temperature face at \xi = 0 and convection at the opposite face \xi = 1, the dimensionless temperature profile is given by \theta(\xi) = 1 - \frac{\text{Bi} \, \xi}{1 + \text{Bi}}, where \theta = \frac{T - T_\infty}{T_s - T_\infty}, \xi = x / L (with T_s the specified temperature at \xi=0), and L is the slab thickness.^[15] This linear profile arises from solving \frac{d^2 \theta}{d \xi^2} = 0 with the boundary conditions \theta(0) = 1 and \frac{d \theta}{d \xi} \big|_{\xi=1} = -\text{Bi} \, \theta(1).^[15] Transient solutions for finite Biot numbers in simple geometries are often obtained using separation of variables, yielding infinite series expansions in terms of eigenvalues that depend on Bi.^[9] However, for practical engineering calculations, Heisler charts provide graphical representations of the dimensionless midplane temperature \theta_0 = \frac{T_0 - T_\infty}{T_i - T_\infty} as a function of the Fourier number \text{Fo} = \alpha t / L^2 for various Bi values, applicable to infinite plates, long cylinders, and spheres.^[16] Supplementary charts in the Heisler method allow determination of temperature profiles at any location and the cumulative heat transfer, facilitating rapid assessment without full series computation.^[16] For irregular geometries or complex boundary conditions where analytical solutions are infeasible, numerical methods such as finite difference or finite element techniques discretize the governing PDE to simulate the temperature evolution, incorporating the Biot number in the convective boundary discretization.^[17] The finite difference approach, for instance, approximates spatial derivatives on a grid and advances time explicitly or implicitly, enabling solutions for Bi in the range of 1 to 10.^[17] In applications like the heat treatment of hot die steels, where Bi ≈ 1–10 due to moderate convection during quenching, distributed models predict internal temperature gradients to avoid cracking or uneven hardening, outperforming lumped approximations.^[18] Similarly, insulation design for pipes or walls with finite Bi uses these methods to optimize thickness by evaluating spatial temperature drops and heat loss rates.^[17]

Mass Transfer Analogue

Definition and Formula

The mass transfer Biot number, often denoted as \text{Bi}_m, is a dimensionless parameter used in chemical engineering to characterize the relative resistance to mass diffusion within a solid or fluid phase compared to the resistance due to convective mass transfer at the phase boundary. It serves as an analogue to the thermal Biot number in heat transfer problems, adapting the concept to scenarios involving species diffusion and concentration gradients. This number helps determine whether internal diffusion limits the overall mass transfer rate or if external convection dominates. The formula for the mass transfer Biot number is given by

\text{Bi}_m = \frac{k_m L_c}{D},

where k_m is the mass transfer coefficient with units of m/s, L_c is the characteristic length scale of the system (such as the radius or half-thickness of a particle, in m), and D is the molecular diffusivity of the species (in m²/s). This expression arises from nondimensionalizing the governing mass transfer equations, ensuring \text{Bi}_m is unitless, as the dimensions of k_m L_c (m²/s) match those of D. The parameter is sometimes referred to as the "Biot number for mass transfer" or simply the transfer Biot number in the literature.^[19] Physically, \text{Bi}_m represents the ratio of the internal diffusion resistance (L_c / D) to the external convective mass transfer resistance ($1 / k_m). A low \text{Bi}_m (much less than 1) indicates that internal diffusion is rapid relative to external transfer, allowing for uniform concentration profiles within the phase, while a high \text{Bi}_m (much greater than 1) signifies significant internal resistance, leading to steep concentration gradients. This interpretation directly parallels the thermal case but substitutes diffusivity for conductivity and mass transfer coefficient for heat transfer coefficient. The concept was introduced in mid-20th century chemical engineering texts as an extension of the original thermal Biot number, with seminal development appearing in works like Transport Phenomena by Bird, Stewart, and Lightfoot (1960), which formalized dimensionless groups for multicomponent transport processes.^[19]

Applications in Mass Diffusion

In mass diffusion processes, the mass transfer Biot number, Bi_m = \frac{k_m L}{D}, where k_m is the mass transfer coefficient, L is the characteristic length, and D is the diffusion coefficient, determines the appropriate modeling approach for transient species transport under convective boundary conditions. When Bi_m \ll 1, typically Bi_m < 0.1, internal diffusion resistance is negligible compared to external convection, allowing the lumped concentration approximation. This assumes a uniform species concentration C throughout the domain, reducing the governing equation to the ordinary differential equation

\frac{dC}{dt} = -\frac{k_m A}{V} (C - C_\infty),

where A is the surface area, V is the volume, and C_\infty is the ambient concentration. The solution exhibits exponential decay toward equilibrium, simplifying predictions of overall mass transfer rates. This regime is valid for small particles or high k_m, ensuring rapid internal equilibration.^[3]^[20] For finite Bi_m > 0.1, internal diffusion gradients become significant, requiring solution of the full diffusion equation

\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}

with the Robin boundary condition at the surface,

-D \frac{\partial C}{\partial x} \bigg|_{x=L} = k_m (C(L, t) - C_\infty).

Analytical solutions involve infinite series expansions via separation of variables, with eigenvalues depending on Bi_m, analogous to thermal conduction cases. Dimensionless charts, adapted from heat transfer Heisler diagrams, facilitate practical evaluation of concentration profiles and average diffusion times as functions of Fourier number Fo = Dt/L^2 and Bi_m. High Bi_m \gg 1 approximates a Dirichlet boundary (fixed surface concentration), emphasizing internal diffusion control. The physical interpretation of Bi_m parallels the thermal Biot number, quantifying the ratio of internal diffusive resistance to external convective resistance.^[3]^[20] These concepts apply to various engineering and environmental processes. In the drying of porous solids, such as agricultural grains, low Bi_m enables lumped models to predict uniform moisture reduction during the falling rate period, while finite Bi_m (e.g., around 10–200) requires distributed solutions for accurate drying curves at elevated temperatures like 80°C. Drug release from polymeric capsules or implants in biomedical applications relies on Bi_m to assess membrane permeability versus internal diffusion; for instance, low Bi_m (e.g., Bi_m \to 0) yields quasi-steady release controlled by external transport in vitreous humor environments. Similarly, in contaminant leaching from soils or sediments, finite Bi_m models describe pollutant desorption from porous matrices into groundwater, with lumped approximations valid for small soil aggregates where external convection dominates, aiding remediation design.^[20]^[21]^[22]