Heat Transfer · Learn

Transient Conduction

How a body cools over time — and the single number that tells you whether you can treat it as one uniform lump or must track the gradient inside.

▶ Play the lab 📖 Learn the theory

1The unsteady heat equation

When temperatures change in time, the energy stored in the material matters. An energy balance on a control volume gives the heat equation:

$$ \frac{\partial T}{\partial t} = \alpha\,\nabla^2 T, \qquad \alpha = \frac{k}{\rho c} $$

The thermal diffusivity $\alpha$ sets how fast a thermal disturbance spreads: it balances how readily the material conducts ($k$) against how much heat it must store to change temperature ($\rho c$). Metals have large $\alpha$ (respond fast); brick and glass small $\alpha$ (sluggish).

2Lumped capacitance

If a body conducts heat internally far faster than it exchanges heat with its surroundings, its temperature stays essentially uniform — just a single number falling in time. An energy balance, with convection to a fluid at $T_\infty$, becomes one ODE:

$$ \rho c V \frac{dT}{dt} = -hA\,(T - T_\infty) $$

which solves to a pure exponential decay toward the ambient:

$$ T(t) = T_\infty + (T_i - T_\infty)\,e^{-t/\tau}, \qquad \tau = \frac{\rho c V}{hA} $$

The time constant $\tau$ is how long to fall to $1/e$ ($\approx 37\%$) of the initial gap. Bigger body or lower $h$ → slower cooling.

3The Biot number — is lumped valid?

The lumped assumption holds only when internal conduction resistance is small compared to surface convection resistance. That ratio is the Biot number:

$$ \mathrm{Bi} = \frac{hL_c}{k}, \qquad L_c = \frac{V}{A} $$

The rule of thumb: if $\mathrm{Bi} < 0.1$, internal gradients are under a few percent and lumped capacitance is accurate. When $\mathrm{Bi} \gtrsim 1$, the surface sheds heat faster than the core can keep up, a real gradient opens, and you must solve the full PDE. (This is exactly the regime of quench hardening: cool the surface of steel fast while the core stays hot.)

▶ In the lab

Slide $h$ up or switch to glass and watch $\mathrm{Bi}$ cross 0.1 — the lumped (uniform) curve peels away from the true profile, with the error reported live. Open the lab →

4Dimensionless time: the Fourier number

Transient problems share a natural clock, the Fourier number:

$$ \mathrm{Fo} = \frac{\alpha t}{L^2} $$

It's elapsed time measured in units of the diffusion time across the body. When $\mathrm{Fo} \approx 1$, the thermal wave has crossed the object and it's well on its way to equilibrium. Charts and one-term series solutions for slabs, cylinders, and spheres are all written in terms of $\mathrm{Bi}$ and $\mathrm{Fo}$.

5Solving it numerically

For real shapes we step the PDE forward in time on a grid. The simplest scheme is explicit finite differences, but it's only stable if the time step is small enough — for a 1-D surface node with convection,

$$ \mathrm{Fo}\,(1 + \mathrm{Bi}_{\text{cell}}) \le \tfrac12 $$

which is exactly the limit the lab respects so the march never blows up.

Key takeaways

Unsteady conduction: $\partial T/\partial t = \alpha\nabla^2T$, with $\alpha=k/\rho c$ setting the response speed.
If the body stays uniform, it decays exponentially with $\tau=\rho cV/hA$.
$\mathrm{Bi}=hL_c/k<0.1$ → lumped capacitance valid; $\mathrm{Bi}\gtrsim1$ → real internal gradients.
$\mathrm{Fo}=\alpha t/L^2$ is dimensionless time; $\mathrm{Fo}\approx1$ means the wave has crossed.

▶ Play

Open the transient lab

2D Steady Conduction

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