Sparkler Combustion: Metal Particle Sparks (3D)

What is modelled

A hand-held sparkler in a dark garden. Metal powder in the coating burns at the front. Tiny burning particles are thrown off in every direction, fly through the air under gravity and drag, shrink as they burn, glow with the colour of a hot body and, for iron with carbon, burst into fragments. Each streak you see is one simulated particle; the tip light and the garden lighting are driven by the total light the sparks emit.

Equations

• Burning: d²(t) = d0² − K t, so a particle lives tb = d0² / K. K depends on the metal.
• Motion: dv/dt = g − k (v − w) with k = (3/4)(ρair/ρp) Cd |v − w| / d and Cd = 24/Re (1 + 0.15 Re0.687), where w is the wind.
• Colour and brightness: Planck’s law B(λ,T) = 2hc²/λ&sup5; / (ehc/λkT − 1). Wien’s law λmax = 2.898×10−³ m K / T puts the peak in the infrared; only the visible tail is seen. Visible light from a spark is the integral of B over 400-700 nm times its surface area.
• Cooling after burnout: m c dT/dt = −A [h (T − Tair) + ε σ (T⁴ − Tair⁴)] with h = 2 kair / d.
• Bursts: an iron drop that contains carbon breaks into 3 to 5 fragments with random mass fractions that add up to the parent mass. Gas formed inside the drop is the real cause; the page uses a random burst time inside the burn.

Checks run in the page

• Terminal speed of falling particles from the page’s time stepper against the Stokes formula (5 µm) and the root of the force balance (100 and 300 µm).
• The d-squared law against the stepped burn time, and mass conservation in 200 random bursts.
• Planck’s law against Wien’s displacement law and against the Stefan-Boltzmann integral at three temperatures.

Honest simplifications

K, temperatures and size distribution are representative values chosen to look and scale like real sparklers; they are not measured data for a particular product. Particles are treated as isolated spheres that burn down to nothing, so oxide residue (cinders) is ignored. Drag uses a standard sphere correlation without turbulence. Brightness is relative, tone-mapped, and a colour temperature is shown as a blackbody; real sparks are not perfect blackbodies. The garden is lit by a single point light at the tip. Never hold a real sparkler near children, hair or clothing: the burning front is well above 1000 K.

What to try

• Use the slow-motion bursts preset and watch a drop split into several brighter, shorter-lived fragments.
• Switch the metal to titanium or aluminum: hotter, whiter, and no bursts. The spectrum graph shifts left.
• Increase wind and see small particles bend downwind while big ones fly straight.
• Choose the coarse powder: the d-squared law makes bigger sparks last far longer.
• Switch on long exposure to get the familiar light-painting streaks.