Solidification time and special casting processes

Why it matters

How long a casting takes to freeze decides whether risers can feed it, how fine its grains are and how many parts a die can produce per hour. Chvorinov's rule turns this into a one-line calculation, and it is the basis of riser design. The special casting processes (die, investment, centrifugal, shell, continuous) exist because sand casting cannot give the accuracy, finish, production rate or soundness some parts need.

Key ideas

How a casting solidifies. A pure metal or eutectic freezes at one temperature, so a solid skin grows inward from the mould wall with a plane front. An alloy freezes over a range of temperatures, giving a "mushy zone" of dendrites and liquid; wide-freezing-range alloys are harder to feed and prone to dispersed shrinkage porosity. A typical ingot shows a thin chill zone of fine equiaxed grains at the wall, a columnar zone of grains growing opposite to the heat flow, and a central equiaxed zone.

Total solidification shrinkage has three parts: liquid contraction (pouring temperature to liquidus), solidification shrinkage (liquidus to solidus, usually the largest) and solid contraction (solidus to room temperature). The first two are fed by risers; the third is handled by the pattern's shrinkage allowance.

Chvorinov's rule. Heat leaves through the mould wall, and the mould's resistance controls the rate in sand casting. The total solidification time is proportional to (V/A)ⁿ, where V/A is the casting modulus. With n = 2, doubling the modulus makes freezing take four times as long. The constant B (the mould constant) depends on:

• the metal: density, latent heat, specific heat and superheat (a higher pouring temperature means more heat to remove, so B and t rise);
• the mould: its thermal conductivity, density and specific heat (a metal mould extracts heat much faster than sand, so B is far smaller). B is found experimentally for a given metal–mould pair; take it from data or derive it from a test casting. Chvorinov's rule assumes a thick mould wall, a mould resistance that controls heat flow, and similar shapes; it is less accurate for thin sections and metal moulds.

Shape matters. For a given volume, the sphere has the smallest surface area, so the largest modulus and the longest freezing time. That is why risers are made compact (sphere or H ≈ D cylinder) and why thin fins freeze first.

Skin thickness. In sand moulds the solid skin grows roughly as d = k·√t, so after four times as long the skin is only twice as thick.

Special casting processes.

• Shell moulding – resin-bonded sand cured on a heated metal pattern into a thin shell; better finish and accuracy than green sand, suited to medium-volume production.
• Investment (lost-wax) casting – a wax pattern is dipped in ceramic slurry and stucco, the wax is melted out and the shell fired. No parting line and no draft are needed, giving intricate parts with fine finish (turbine blades, surgical implants, jewellery). Patterns are expendable, so it costs more per part.
• Permanent-mould (gravity die) casting – metal mould filled by gravity; fine grain because of fast cooling, used mainly for aluminium, magnesium and copper alloys.
• Pressure die casting – metal is injected into a steel die under high pressure. Hot-chamber machines (injector submerged in the melt) suit low-melting zinc, tin and lead alloys; cold-chamber machines (metal ladled into a shot sleeve) suit aluminium, magnesium and brass, which would attack an immersed injector. High rate, thin walls, good accuracy; die cost demands large volumes, and trapped air can cause porosity.
• Centrifugal casting – the mould spins. In true centrifugal casting (pipes, cylinder liners) no core is needed; denser metal is thrown outward and lighter slag and gas collect at the bore, which is machined off. Semi-centrifugal casting makes solid symmetric parts (wheels, pulleys); centrifuging spins several small cavities around a central sprue. The spinning speed is set by a G-factor (centrifugal acceleration ÷ g), typically about 60–80 for horizontal true centrifugal casting (take the value from your data book).
• Continuous casting – liquid steel is poured into a water-cooled copper mould and withdrawn continuously as slabs, blooms or billets.
• Squeeze casting and lost-foam casting – pressure applied during solidification for near-zero porosity; a polystyrene pattern that vaporises as metal is poured.

Formulas

t_s = B·(V/A)ⁿ

• t_s = total solidification time (s), V = casting volume (m³ or mm³), A = cooling surface area (m² or mm²), V/A = modulus M (m or mm), n = exponent (usually 2, range about 1.5–2), B = mould constant (s/m² or s/mm² when n = 2). Keep the units of B consistent with those of M.

t₂ / t₁ = (M₂ / M₁)²

• comparison of two castings of the same metal in the same type of mould and pouring conditions (same B).

Moduli (all faces cooling): cube of side a, M = a/6; sphere of radius r, M = r/3; cylinder of diameter D and height H, M = D·H / (2D + 4H); long plate of thickness t, M ≈ t/2.

d = k·√t

• d = skin thickness (m), t = time from pouring (s), k = solidification constant (m/√s) from experiment.

G = ω²·R / g, N = (60 / 2π)·√(G·g / R)

• G = G-factor (dimensionless), ω = angular speed (rad/s), R = mould inner radius (m), g = 9.81 m/s², N = speed (rpm).

Worked examples

Example 1 – solidification time of a cylinder (standard). A steel cylinder, D = 100 mm and H = 200 mm, is cast in sand with B = 2.0 s/mm² and n = 2. All faces lose heat.

1. V = (π/4)·D²·H = (π/4) × 100² × 200 = 1.571 × 10⁶ mm³.
2. A = 2 × (π/4)·D² + π·D·H = 15 708 + 62 832 = 78 540 mm².
3. M = V/A = 1.571 × 10⁶ / 78 540 = 20.0 mm (check: D·H/(2D + 4H) = 20 000/1 000 = 20 mm).
4. t_s = B·M² = 2.0 × 20.0² = 800 s. t_s = 800 s ≈ 13.3 min

Example 2 – cube versus sphere of equal volume (GATE level). A 100 mm sand-cast cube solidifies in 600 s. A sphere of the same volume is cast from the same metal in the same sand. Take n = 2. Find its solidification time.

1. Cube modulus M_c = a/6 = 100/6 = 16.67 mm, so B = t/M² = 600/16.67² = 2.16 s/mm².
2. Equal volume: (4/3)πr³ = a³ → r = a·(3/(4π))^(1/3) = 100 × 0.6204 = 62.04 mm.
3. Sphere modulus M_s = r/3 = 62.04/3 = 20.68 mm.
4. t_s = B·M_s² = 2.16 × 20.68² = 2.16 × 427.6 = 923.6 s. t_sphere ≈ 924 s (1.54 times the cube's time, because the sphere has less surface for the same volume).

Example 3 – speed for true centrifugal casting. A cast-iron pipe with outside diameter 300 mm is cast in a horizontal mould with G = 60. Find the mould speed.

1. R = 0.15 m. Formula: ω = √(G·g/R).
2. ω = √(60 × 9.81 / 0.15) = √3 924 = 62.64 rad/s.
3. N = 60ω/(2π) = 60 × 62.64 / 6.283 = 598 rpm. N ≈ 598 rpm

Common mistakes

• Using C(V/A)ⁿ with mixed units: a B quoted in s/mm² must be used with V/A in mm.
• Including faces that do not lose heat (e.g. the face joined to a riser or casting) in A.
• Assuming a bigger casting with the same shape freezes in proportion to size; time goes with the square of the modulus.
• Saying pouring temperature has no effect: superheat raises B and lengthens freezing.
• Mixing up hot-chamber and cold-chamber die casting; aluminium is cold-chamber.
• Thinking slag goes to the outside in centrifugal casting; it collects at the inner bore.

For GATE ME

Typical items: solidification-time ratio between two shapes (cube, sphere, cylinder, plate) using Chvorinov's rule; riser sizing so the riser freezes after the casting; finding B from one casting and using it for another; skin-thickness growth with √t; G-factor and rpm in centrifugal casting; and matching special processes to parts, alloys and features. Practise the moduli of common shapes until they are automatic.

Quick check

1. If the modulus of a casting doubles, by what factor does its solidification time change (n = 2)?
2. Which shape has the longest solidification time for a given volume?
3. Which die-casting machine is used for aluminium alloys?
4. What is the modulus of a 120 mm cube?
5. In true centrifugal casting, where do slag and inclusions collect?

Answers: 1. Four times. 2. Sphere. 3. Cold-chamber. 4. 20 mm. 5. At the inner (bore) surface, which is machined off.