The Biot number calculator above answers a question that comes before almost every transient heat transfer problem: can you treat this object as having one temperature, or do you have to solve for a temperature that varies through it? That single decision is the difference between a one-line exponential and a partial differential equation, and the Biot number is how you make it defensibly rather than by feel.
Arb Digital builds free physics calculators that own one job properly rather than burying it inside a bigger tool. Several heat tools on this site assume a uniform body temperature without ever saying so. This page is the check that tells you whether that assumption was allowed, and it reports the time constant that follows if it was.
What This Biot Number Calculator Does
The Biot number is Bi = hLc ÷ k, a dimensionless group with no units at all. It compares the internal conduction resistance of a body, which scales as Lc divided by k, with the external convection resistance at its surface, which scales as one over h. When the number is small, heat moves through the inside far more easily than it escapes from the surface, so the interior stays nearly isothermal and the surface film is the bottleneck.
The conventional threshold is 0.1. Below it, the lumped capacitance model, in which the whole body has a single temperature that decays exponentially towards the fluid, is accurate to better than about 5 per cent. Above it, the centre of the body lags its surface enough that you need a spatial solution: series solutions with tabulated coefficients for simple shapes, or numerical work for anything else.
The calculator derives the characteristic length from the shape you pick, computes the Biot number, states the verdict, and then gives the thermal time constant τ = ρcLc ÷ h that the lumped model implies. The grid also reports the Biot number computed on the full dimension rather than the volume-over-area length, because textbooks genuinely disagree about which to use and the two can differ by a factor of three.
How to Use It
- Choose the shape honestly. A plate cooled on both faces has a half-thickness characteristic length; one cooled on a single face with the other insulated has the full thickness. Getting this backwards doubles or halves your answer.
- Enter the solid's conductivity, not the fluid's. The k in the Biot number is always the material being heated or cooled. Using the air's conductivity is a classic slip and gives a wildly wrong result.
- Estimate h as carefully as you can. It is the dominant uncertainty. If you only know it to within a factor of two, your Biot number is only known to within a factor of two, which matters most when the answer lands near 0.1.
- Read the verdict before the number. The tool states plainly whether lumped capacitance is defensible, marginal or unusable for your case.
- Use the time constant if the verdict allows it. One time constant takes the body 63.2 per cent of the way to the fluid temperature, and the grid gives the time to reach 99 per cent.
The Formula: How the Biot Number Is Calculated
The characteristic length is defined as volume divided by surface area, Lc = V ÷ As. For a sphere of radius R that gives (4πR³/3) ÷ (4πR²) = R ÷ 3. For a long cylinder it gives R ÷ 2, and for a plate cooled on both faces it gives the half-thickness. The Biot number is then that length times h, divided by k.
The conduction and convection terms come from the two mechanisms described in OpenStax University Physics Volume 2, section 1.6 on mechanisms of heat transfer, which gives conduction as P = kA(Th − Tc) ÷ d and sets out convection with a surface coefficient. Transient conduction and the lumped-capacitance limit are treated at length in MIT OpenCourseWare 2.51, Intermediate Heat and Mass Transfer.
When the Biot number is small enough, an energy balance on the whole body gives ρVc dT/dt = −hAs(T − T∞), whose solution is a decaying exponential with time constant τ = ρcV ÷ (hAs) = ρcLc ÷ h. That is Newton's law of cooling, and the Newton's law of cooling calculator is the tool that solves it once this page has told you that you are allowed to.
Work the defaults. A sphere of radius 0.05 m gives Lc = 0.05 ÷ 3 = 0.0166667 m. With h = 50 and k = 15, Bi = 50 × 0.0166667 ÷ 15 = 0.8333 ÷ 15 = 0.05556. That is comfortably below 0.1, so lumped capacitance holds. The time constant is ρcLc ÷ h = 8,000 × 500 × 0.0166667 ÷ 50 = 66,666.7 ÷ 50 = 1,333.3 seconds, and 99 per cent settling takes 4.605 time constants, or 6,140 seconds.
Why 0.1 Is the Threshold, and What Happens Either Side
The 0.1 figure is not a law of nature. It comes from comparing the exact series solution for a shape against the single-exponential lumped solution and asking where the difference stops mattering. At Bi = 0.1 the centre-to-surface temperature difference inside the body is roughly 5 per cent of the difference between the body and the fluid, which is smaller than the error in your value of h in almost every real problem.
Below about 0.01 the internal gradient is negligible in any practical sense and the lumped model is essentially exact. Between 0.1 and 1 you are in an awkward middle band: the lumped model is wrong by perhaps 10 to 25 per cent on the centre temperature, which may or may not matter depending on what you are doing with the answer. Above 1 the internal resistance dominates and the surface reaches the fluid temperature long before the core notices, which is why a thick roast keeps cooking after it leaves the oven.
The important asymmetry is which way the error runs. The lumped model always predicts the centre cools faster than it really does, because it ignores the time heat needs to travel out of the middle. If you are sizing a quench for a safety-critical hardening step, or predicting when a core has reached a food-safety temperature, that error is in the unsafe direction, and a Biot number above 0.1 is a reason to do the harder calculation rather than to round it away.
The Characteristic Length Argument Nobody Warns You About
Two conventions are in circulation and both appear in respectable textbooks. The volume-over-area definition, used by this calculator, gives R/3 for a sphere and R/2 for a long cylinder. The other convention uses the full radius or half-thickness for every shape, on the grounds that it is the actual conduction path length. For a sphere the two differ by a factor of three, which moves a Biot number of 0.05 to 0.15 — from safely lumped to not lumped at all.
There is a reason both survive. The volume-over-area length is the one that makes the lumped time constant come out right, so it is the correct choice when you are asking whether the lumped model applies and then using it. The full-dimension length is the one that appears in the tabulated series solutions for spheres, cylinders and plates, so it is the correct choice when you have already decided to solve the spatial problem and are looking up coefficients.
The grid reports both so you can see the gap and cite the right one. If your Biot number on the volume-over-area basis is under 0.1 but the full-dimension version is over it, you are near the boundary, and the honest thing to say is that the lumped result carries a real error rather than that it passed a test.
Getting the Convective Coefficient Without Guessing
Everything on this page rests on h, and h is not a material property. It depends on the fluid, the flow speed, the geometry, the orientation and whether the flow is free or forced, and it varies over the surface of the same object. The usual route is a Nusselt number correlation for your configuration, which returns Nu and hence h = Nu kfluid ÷ L, and the correlation you choose depends on whether the flow is laminar or turbulent.
That is where the Reynolds number calculator comes in: it is the dimensionless group that decides which correlation is legitimate, in the same way the Biot number decides which conduction model is legitimate. The two are often confused because both are dimensionless and both involve a length, but the Reynolds number is about the fluid and the Biot number is about the solid, and they never substitute for each other.
A useful sanity check is the order of magnitude. Free convection in air rarely exceeds about 25 W/m²·K, forced air runs from tens to a few hundred, forced water reaches thousands, and phase change reaches tens of thousands. If your assumed h sits outside the band for your situation, the Biot number that follows is not worth arguing about.
Where This Sits Next to the Other Heat Tools
This page answers one narrow question and hands off. The heat transfer calculator computes the heat moved for a given temperature change, the thermal conductivity calculator works with the conduction rate through a material, and the specific heat calculator handles the energy stored in a mass for a given temperature rise. None of them tests whether a body is thermally thin, which is what this page exists for.
Once the verdict is favourable, the Newton's law of cooling calculator is the natural next step, since it solves the exponential the lumped model produces. For steady-state building problems rather than transients, the heat loss calculator and the insulation calculator work in R-values and U-values instead, where the same conduction and convection resistances appear in series.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the fluid's conductivity — the k in the Biot number is always the solid being heated or cooled. The fluid's conductivity belongs inside h, not beside it.
- Mixing the two characteristic length conventions — volume over area and full dimension differ by a factor of three for a sphere, which can flip the verdict entirely.
- Treating 0.1 as a hard edge — it is a 5 per cent error criterion. At 0.11 nothing breaks; at 0.5 the lumped model is meaningfully optimistic about how fast the centre responds.
- Forgetting that the error is one-sided — lumped capacitance always says the core cools or heats faster than it really does, so a marginal Biot number is unsafe in exactly the applications that care.
- Confusing it with the Fourier number — the Biot number decides which model to use, while the Fourier number is dimensionless time inside whichever model you chose.
Related Free Tools From Arb Digital
Once this page clears the lumped assumption, solve the transient with the Newton's law of cooling calculator. For the underlying quantities, use the thermal conductivity calculator, the specific heat calculator and the heat transfer calculator. The Reynolds number calculator is the companion dimensionless group on the fluid side and is what you need before choosing a correlation for the convective coefficient. For steady-state envelope work, the heat loss calculator and the insulation calculator take over. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It compares the resistance to conduction inside a body with the resistance to convection at its surface. A small value means the interior stays nearly uniform in temperature while the surface film controls the rate. A large value means the inside lags the surface and you must solve for temperature as a function of position.
Because at that value the temperature difference between the centre and the surface of the body is roughly 5 per cent of the difference between the body and the surrounding fluid. That is smaller than the uncertainty in a typical convective coefficient, so the simpler model costs nothing you can measure.
Both conventions exist. Volume divided by surface area gives radius over three for a sphere and is the one that makes the lumped time constant correct. The full radius is the one used in the tabulated series solutions for spatial problems. This tool reports both so you can cite whichever your reference uses.
No. It is dimensionless by construction: a coefficient in watts per square metre per kelvin, times a length in metres, divided by a conductivity in watts per metre per kelvin, leaves nothing behind. If your answer has units, one of the three inputs is in the wrong system.
They look identical but use different conductivities. The Biot number divides by the conductivity of the solid, and asks about internal gradients. The Nusselt number divides by the conductivity of the fluid, and describes how much better convection moves heat than conduction through a still fluid layer would.
Usually from a Nusselt number correlation chosen for your geometry and flow regime, which needs the Reynolds number first. As a sanity check, free convection in air is single digits to about 25, forced air is tens to hundreds, forced water is thousands and boiling is higher still.
It is density times specific heat times the characteristic length, divided by the convective coefficient. The body closes 63.2 per cent of the gap to the fluid temperature in one time constant, about 95 per cent in three and 99 per cent in 4.6. It is only meaningful when the Biot number allows the lumped model.
This tool is provided for educational and study use. It evaluates a dimensionless ratio from the values you supply and does not validate them, so treat its verdict as a modelling guide rather than a design confirmation. Thermal design for equipment, process safety or food safety should be carried out by a qualified engineer against the applicable standard.
