Biot Number Calculator
Enter the convective heat transfer coefficient, characteristic length, and thermal conductivity to compute the dimensionless Biot number. Switch the solve-for mode to find any of the four quantities given the other three. Choose a geometry to auto-compute characteristic length from shape dimensions, or pick a material from the built-in library to fill thermal conductivity. The result panel tells you whether the lumped-capacitance (uniform-temperature) assumption is valid for your object.
What is the Biot number?
The Biot number (Bi) is a dimensionless parameter in heat transfer that compares the external convective thermal resistance at a body's surface to the internal conductive thermal resistance within the body. It is defined as Bi = h * Lc / k, where h is the convective heat transfer coefficient (W/m^2 K), Lc is the characteristic length (m), and k is the thermal conductivity of the solid (W/m K). Named after the French physicist Jean-Baptiste Biot, it was introduced to distinguish situations where temperature varies significantly through the interior of a solid from those where the solid can be treated as thermally uniform.
Characteristic length for common geometries
The characteristic length Lc is always the volume-to-surface-area ratio (V/A) for the lumped-capacitance method. For a sphere of radius r, this gives Lc = r/3. For a long cylinder of radius r the result is Lc = r/2. For a finite cylinder of radius r and height H it becomes Lc = rH / (2(r+H)). For a plane slab of half-thickness L (insulated on one face) it is Lc = L. For a cube of side s it is Lc = s/6. Choosing the correct characteristic length is essential: using the wrong geometry underestimates or overestimates Bi and leads to an incorrect judgment about whether lumped capacitance is valid.
Lumped capacitance and what Bi < 0.1 means
When Bi < 0.1, the internal thermal resistance is negligible compared to convective resistance at the surface. The entire body can be treated as a single node at one uniform temperature, and the transient temperature response follows a simple exponential decay: T(t) = T_inf + (T_0 - T_inf) * exp(-h * A * t / (rho * V * c_p)). This is the lumped-capacitance or lumped-system model. It allows a first-order ordinary differential equation to replace a full partial differential equation, making analysis very straightforward. Most thin metal objects cooled by natural convection satisfy Bi < 0.1 with ease, while poor conductors such as glass, concrete or plastics rarely do even for small sizes.
Reverse-solve: designing for a target Biot number
This calculator lets you solve for any one of the four quantities given the other three. To find the heat transfer coefficient needed to keep Bi below 0.1 (ensuring lumped validity), select "Solve for h", enter your geometry and material, and set the target Bi to 0.1. The result is the maximum allowable h. Conversely, to find the minimum thermal conductivity k required so that a part with known h and Lc stays in the lumped regime, select "Solve for k" and set Bi = 0.1. These reverse-solve modes are especially useful in design: you can set a thermal constraint and back out the required material or cooling specification.
Typical heat transfer coefficients by fluid and mode
The convective coefficient h varies enormously with fluid and flow conditions. Natural convection in air typically gives h = 5-25 W/(m^2 K). Forced air (fans, blowers) raises this to 25-250 W/(m^2 K). Water in natural convection gives roughly 200-1000, and forced water flow 500-10000 W/(m^2 K). Boiling and condensation can reach 5000-100000 W/(m^2 K). Knowing a realistic range for h before computing Bi helps you understand whether lumped capacitance is feasible for your application.
Biot number regimes and analysis methods
| Biot number (Bi) | Regime | Temperature profile | Recommended method |
|---|---|---|---|
| Bi < 0.001 | Perfect conductor | Uniform throughout | Lumped capacitance |
| 0.001 - 0.1 | Good conductor | Nearly uniform | Lumped capacitance (error < 5%) |
| 0.1 - 1 | Transitional | Moderate gradients | One-term series or numerical |
| 1 - 10 | Moderate resistance | Significant gradients | Series solution or FEM |
| 10 - 100 | High resistance | Strong gradients | Full spatial analysis |
| Bi > 100 | Perfect insulator | Steep gradients near surface | FEM / CFD required |
Guidance on which heat transfer analysis method to use based on the Biot number value.
Frequently asked questions
What does Biot number tell you in heat transfer?
The Biot number compares the surface convection resistance to the internal conduction resistance. A low Bi (below 0.1) means the solid conducts heat internally so readily that temperature is nearly uniform throughout. A high Bi means the surface is losing heat faster than the interior can supply it, creating steep temperature gradients inside the body. It determines whether you can use the simple lumped-capacitance approach or must solve the full heat equation.
Why is Bi < 0.1 the threshold for lumped capacitance?
The threshold of 0.1 is a practical engineering rule. When Bi < 0.1, the maximum temperature difference between the centre and the surface of the body is less than about 5% of the overall driving temperature difference. For most engineering purposes this error is acceptable and the single-node lumped model gives reliable results. The exact error depends on geometry, but 0.1 is the universally accepted cut-off.
How is characteristic length calculated for a sphere, cylinder, or slab?
Characteristic length is always Lc = V/A (volume divided by total surface area). For a sphere of radius r: Lc = r/3. For an infinitely long cylinder of radius r: Lc = r/2. For a finite cylinder of radius r and height H: Lc = rH / (2(r+H)). For a plane slab of half-thickness L (insulated on one face and exposed on the other): Lc = L. For a cube of side s: Lc = s/6. This calculator computes these automatically when you select the geometry.
What is the difference between the Biot number and the Nusselt number?
Both are dimensionless ratios involving convection and conduction, but they apply to different regions. The Nusselt number compares convection to conduction within the fluid at the boundary layer and uses the fluid thermal conductivity. The Biot number compares external convection to internal conduction within the solid and uses the solid thermal conductivity. A common mistake is substituting one for the other: always use the solid material thermal conductivity k_solid when computing Bi.
How do I use the reverse-solve mode?
Select "Solve for" the quantity you want to find. If you want to know what heat transfer coefficient will achieve a target Bi, choose "Heat transfer coefficient (h)", enter your material k and characteristic length Lc, and set the desired Bi. The calculator rearranges the formula as h = Bi * k / Lc. Similarly for k or Lc. This is useful in thermal design when you know the cooling limit you must stay within.
What are typical Biot numbers for real applications?
A copper sphere cooled by natural air convection has Bi well below 0.001, so lumped capacitance is perfectly valid. A steel billet quenched in water might have Bi around 0.2-5, requiring a series solution. A concrete wall exposed to convection can easily reach Bi = 10-100. Electronics chips cooled by a heat sink with thermal interface material often have Bi in the range 0.01-0.5, depending on the assembly. The reference table in this calculator maps these regimes to recommended analysis methods.