Why is Bi < 0.1 the usual rule?

It means resistance to conduction inside the body is small compared with convection resistance at the surface, so the internal temperature stays nearly uniform.

What does the time constant τ mean?

After one τ, the temperature difference from ambient has fallen to e^{-1}, about 37% of its initial value. After about 5τ the body is very close to ambient.

Lumped Capacitance Cooling

Q: Why is Bi < 0.1 the usual rule?

It means resistance to conduction inside the body is small compared with convection resistance at the surface, so the internal temperature stays nearly uniform.

Q: What does the time constant τ mean?

After one τ, the temperature difference from ambient has fallen to e^{-1}, about 37% of its initial value. After about 5τ the body is very close to ambient.

The lumped capacitance method treats a solid body as having one uniform internal temperature while it cools or heats by convection. Its validity is controlled by the Biot number Bi = hLc/k, where Lc = V/A. When Bi < 0.1, internal conduction is fast enough that temperature gradients inside the body are small. The transient follows Newton cooling: T(t) = T∞ + (T0 − T∞) exp(−t/τ), with τ = ρVc/(hA). This simulator compares shapes, material properties, convection coefficient, and object size to show how Bi and τ change. It omits radiation, nonuniform h, phase change, contact resistance, and multidimensional conduction for high-Bi cases.

Who it's for: Heat-transfer, thermal engineering, materials processing, cooking/cooling intuition, and electronics thermal introductions.

Key terms

Lumped capacitance
Biot number
Newton cooling
Time constant
Characteristic length

The lumped capacitance method treats the body as internally uniform in temperature. It is most reliable when Bi < 0.1; larger Bi means conduction inside the object creates significant gradients.

Live graphs

How it works

Newton cooling for a lumped body: Biot number, characteristic length, time constant τ = ρVc/(hA), and validity check Bi < 0.1.

Key equations

Bi = h L_c / k, L_c = V/A

T(t) = T∞ + (T0 − T∞) exp(−t/τ), τ = ρ V c / (h A)

Frequently asked questions

Why is Bi < 0.1 the usual rule?: It means resistance to conduction inside the body is small compared with convection resistance at the surface, so the internal temperature stays nearly uniform.
What does the time constant τ mean?: After one τ, the temperature difference from ambient has fallen to e^{-1}, about 37% of its initial value. After about 5τ the body is very close to ambient.

Other simulators in this category — or see all 46.

View category →

NewSchool

Pipe Friction & Moody Chart

Reynolds number, relative roughness, Swamee-Jain/Colebrook-style friction factor, and Darcy-Weisbach head loss.

Launch Simulator

NewSchool

Open-Channel Flow & Hydraulic Jump

Rectangular-channel Froude number, critical depth, conjugate depths, energy loss, and subcritical/supercritical regimes.

Launch Simulator

NewSchool

Multilayer Wall Conduction

Three layers in series: R″ = L/k, q″ and U; T(x) sketch and preset plaster/brick/wool.

Launch Simulator

NewSchool

Adiabatic Cloud Parcel

Lift moist air: dry Γ, LCL, toy moist lapse; RH and cartoon cloud vs height.

Launch Simulator

NewSchool

Engine Cycles Compared (P–V)

One canvas: Carnot, Otto, Diesel, Stirling, Rankine sketch — switch cycles, same formulas as standalone labs.

Launch Simulator

NewSchool

Diesel Cycle (PV)

Air-standard: adiabat compress, isobaric heat, adiabat expand, isochoric out; η(ρ_c, β).

Launch Simulator

Lumped Capacitance Cooling

Live graphs

How it works

Key equations

Frequently asked questions

More from Thermodynamics

Pipe Friction & Moody Chart

Open-Channel Flow & Hydraulic Jump

Multilayer Wall Conduction

Adiabatic Cloud Parcel

Engine Cycles Compared (P–V)

Diesel Cycle (PV)

Lumped Capacitance Cooling

Live graphs

How it works

Key equations

Frequently asked questions