Sand Casting Line

What is modelled

A sand mould is filled from a bottom-pour ladle. Heat leaves the casting through the mould wall, which behaves like a semi-infinite solid, so the solid skin grows with the square root of time and the whole casting freezes in a time proportional to its modulus squared (Chvorinov). A riser sized by the modulus rule keeps feeding liquid metal while the casting shrinks. Line speed, loop length and the ten flasks are taken from the 3D page, which has no casting physics.

Equations

• Pour dh/dt = -Cd Ag sqrt(2 g h) / Al; fill time (Al / Cd Ag) sqrt(2/g)(sqrt(h0) - sqrt(h1)).
• Mould heat flow q = b (Tm - T0) / sqrt(pi t), skin s = 2 b (Tm - T0) sqrt(t/pi) / (rho dH), dH = L + c (superheat).
• Chvorinov t = B M2, B = (pi/4)(rho dH / b (Tm - T0))2, M = V / A.
• Riser: M_r = f M; a riser of height 1.5 D has M_r = 0.2143 D.
• Line rate 3600 v / pitch; dwell L_line / v.

Checks run in the page

• Heat stored in the mould by implicit finite differences against the semi-infinite solution.
• Ladle drain by RK4 integration against the closed form.
• Riser modulus, freezing-time ratio and feeding volume.
• Conveyor flasks stepped in time against the line rate and the line dwell.

Honest simplifications

Textbook simplifications: the mould is dry sand with a constant heat uptake b, the solid metal offers no resistance, the metal is at its melting point plus a superheat that adds only to the latent load, no mould-metal air gap, one dimensional skin growth, the casting modulus uses all six faces, a fixed discharge coefficient of 0.8, an assumed 14 percent riser feeding efficiency and typical handbook property values. A teaching estimate, not a foundry process design.

• Press Too short a line and read how many seconds the flask is short of being solid.
• Press Chilled mould: a higher heat uptake cuts the solidification time.
• Raise the riser factor and watch the riser freeze later while its volume grows quickly.