Biot Number Calculator

Find the Biot number (Bi = h · Lc / k) for a plate, cylinder, or sphere and check whether the lumped-capacitance heat transfer approximation is valid.

Quick Facts

Formula
Bi = h · Lc / k
Lc = L (half-thickness) for a plate, r/2 for a cylinder, r/3 for a sphere.
Rule of thumb
Bi < 0.1
Below this, the lumped-capacitance (uniform-temperature) approximation is considered valid.

Your Results

Calculated
Biot number (Bi)
-
Conduction resistance / convection resistance
Characteristic length (Lc)
-
Derived from shape and dimension
Internal conduction resistance
-
Lc / k, in m²·K/W
Lumped-capacitance check
-
Bi < 0.1 required for validity

Ready

Enter h, k, shape, and the characteristic dimension, then press Calculate.

About the Biot Number

The Biot number (Bi) is a dimensionless quantity used in transient heat transfer to compare two competing resistances: the resistance to heat conduction inside a solid and the resistance to heat convection across its surface into the surrounding fluid. It answers a practical question whenever an object is heated or cooled — does the whole object change temperature together, or does the surface change much faster than the core?

The formula

The standard definition is:

Bi = h × Lc / k

  • h — the convective heat transfer coefficient at the surface, in W/m²·K (how effectively the surrounding fluid carries heat away from or into the surface)
  • Lc — the characteristic length of the solid, in meters, defined generally as volume divided by surface area (V/As)
  • k — the thermal conductivity of the solid material, in W/m·K (how effectively heat moves through the solid itself)

Characteristic length by shape

For the common shapes used in textbook and design problems, V/As simplifies to a single dimension of the body:

  • Flat plate / slab (cooled on both faces): Lc = L, the half-thickness
  • Long cylinder (cooled around its circumference): Lc = r/2, half the radius
  • Sphere: Lc = r/3, one-third the radius

This calculator applies whichever formula matches the shape you select, using the dimension you enter as L (plate) or r (cylinder and sphere).

Interpreting the result

Bi is the ratio of internal conduction resistance (Lc/k) to external convection resistance (1/h). When Bi < 0.1, conduction inside the solid is so much faster than convection at the surface that the whole object can be treated as having one uniform temperature at any instant — the basis of the lumped capacitance method used in simple cooling/heating time calculations. As Bi grows past roughly 0.1, temperature gradients inside the solid become significant and the lumped assumption breaks down; problems in that range are normally solved with Heisler charts or a full transient conduction (Fourier series) solution instead. At very large Bi (commonly cited as above 40–100), the surface essentially snaps to the fluid temperature immediately while the interior lags well behind.

Working with units

Keep inputs in consistent SI units: h in W/m²·K, k in W/m·K, and the dimension in meters. Because Bi is dimensionless, using consistent units throughout is what makes the ratio meaningful — mixing unit systems (e.g., inches with SI conductivity) will silently corrupt the result.

Frequently Asked Questions

What is the Biot number and why does it matter?
The Biot number (Bi) is a dimensionless ratio that compares the resistance to heat conduction inside a solid to the resistance to heat convection at its surface: Bi = h × Lc / k, where h is the convective heat transfer coefficient, Lc is the characteristic length, and k is the solid's thermal conductivity. It tells you whether an object heats or cools with a roughly uniform internal temperature or develops significant temperature gradients.
What does Bi less than 0.1 mean?
When the Biot number is below about 0.1, internal conduction resistance is small compared to surface convection resistance, so the temperature inside the solid stays nearly uniform at any instant. This is the condition under which the lumped capacitance method, which treats the whole object as one temperature, gives accurate results.
How is the characteristic length Lc calculated?
Characteristic length is generally defined as Lc = V / As, the solid's volume divided by the surface area exposed to convection. For common shapes this simplifies to the half-thickness for a large flat plate cooled on both faces, radius divided by 2 for a long cylinder, and radius divided by 3 for a sphere.
What happens when the Biot number is large?
As Bi rises well above 1 (commonly cited thresholds are 40 to 100), internal conduction becomes the rate-limiting step: the surface reaches the surrounding fluid temperature almost immediately while the center lags far behind, producing steep internal gradients. In that regime the lumped approximation fails and transient conduction solutions, such as Heisler charts or numerical methods, are needed instead.

Common Tip Percentages

ServiceTypical Tip RangeNotes
Restaurant (sit-down)15–20%20%+ for exceptional service
Food delivery10–20%Consider extra for long distances
Taxi / rideshare10–20%More for assistance with luggage
Hotel housekeeping$2–5/nightLeave daily, not just at checkout
Barber / hair salon15–20%On pre-discount total
Pizza delivery10–15% or $3–5 minimumHigher in bad weather

General Household Budget Ratios (50/30/20 Rule)

Category% of Net IncomeExamples
Needs50%Rent, groceries, utilities, insurance
Wants30%Dining out, entertainment, hobbies
Savings / Debt20%Emergency fund, retirement, extra payments

Common Measurement Conversions for Everyday Use

QuantityUS / ImperialMetric Equivalent
1 cup8 fl oz237 mL
1 pint16 fl oz473 mL
1 quart32 fl oz946 mL
1 gallon128 fl oz3.785 L
1 lb16 oz453.6 g
1 ft12 in30.48 cm

Related calculators