Solidification, Chvorinov's rule and riser design

Why it matters

Almost every casting metal shrinks as it freezes, and if the last region to freeze is inside the casting, that shrinkage appears as a hidden cavity. Chvorinov's rule tells you which section freezes last; riser design uses it to put the last-freezing metal in a riser that is cut off and remelted, instead of in the part you sell.

Key ideas

How a pure metal and an alloy freeze. A pure metal (or eutectic) freezes at one temperature: a solid skin forms at the mould wall and a plane front grows inward, leaving a pipe-shaped shrinkage at the last point to freeze. An alloy freezes over a range between liquidus and solidus; a mushy zone of dendrites and liquid forms, and the shrinkage is spread out as fine interdendritic (micro) porosity, which is harder to feed.

Grain structure. Next to the cold mould wall a thin chill zone of fine equiaxed grains forms; then columnar grains grow inward along the heat flow; in the centre, where the remaining liquid is nearly uniform in temperature, equiaxed grains form. Faster cooling gives finer grains and higher strength; inoculants promote equiaxed grains.

Three contractions. Liquid contraction (pouring temperature down to liquidus) and solidification shrinkage (liquid to solid; roughly 3% for steel, 4–7% for aluminium alloys, take values from your data book) must be fed by risers. Solid contraction after freezing is handled by the pattern's shrinkage allowance. Grey cast iron shrinks very little on solidification because graphite precipitates and expands — it often needs small or no risers.

Chvorinov's rule. Heat leaves the casting through its surface, so the time to freeze depends on how much heat is stored (proportional to volume V) against the area A through which it escapes. The ratio M = V/A is the casting modulus. Chvorinov's rule says the total solidification time t = B·(V/A)ⁿ, with n = 2 for most sand-casting problems. The mould constant B depends on the metal (latent heat, specific heat, density), the mould (thermal conductivity, density, specific heat), the pouring temperature (superheat) and the initial mould temperature; it is found experimentally. For a given metal–mould combination B is the same for the casting and its riser, so comparisons depend only on modulus.

Shape effect. For a fixed volume, a sphere has the smallest surface area and so the longest freezing time; a thin plate freezes fastest. Moduli of common shapes:

• Sphere of radius r: V/A = r/3 (= D/6).
• Cube of side a: V/A = a/6.
• Cylinder with H = D, all surfaces cooling: V/A = D/6.
• Cylinder with H = D, base in contact with the casting (top riser, base not cooling): V/A = D/5.
• Long plate of thickness t: V/A ≈ t/2.

Riser design. A riser must (1) freeze after the section it feeds — usually designed so the riser modulus is at least 1.2 times the casting modulus (so the riser freezes about 1.44 times later); (2) hold enough liquid to supply the shrinkage volume — only a fraction of an open riser's volume can actually feed (data-book riser efficiency); and (3) be connected so a liquid path stays open (directional solidification toward the riser). A cylinder with H/D ≈ 1 is the best practical shape. Caine's method plots freezing ratio X = (A/V)casting / (A/V)riser against volume ratio Y = V_riser / V_casting on a hyperbola X = a/(Y − b) + c, with constants for each metal from data books. Chills (metal blocks in the mould wall) speed up freezing of thick sections; exothermic sleeves and insulating toppings slow the riser down.

Formulas

t = B × (V/A)ⁿ

• t = solidification time (s), V = volume (mm³ or m³), A = surface area through which heat is lost (mm² or m²), B = mould constant (s/mm² if V/A is in mm and n = 2), n = 2 unless the question gives another value. Typical B for sand moulds is of the order of 2–4 s/mm² (take from the question).

t_riser / t_casting = (M_riser / M_casting)² — the B cancels when the riser and casting are in the same mould.

M_riser ≥ 1.2 × M_casting — modulus (Wlodawer) criterion for a riser.

X = (A/V)_c / (A/V)_r, Y = V_r / V_c, X = a / (Y − b) + c — Caine's method; a, b, c are empirical constants for the metal.

Worked examples

Example 1 (standard). A steel plate casting is 200 × 100 × 20 mm. For this sand mould B = 2.0 s/mm² and n = 2. Find the solidification time.

1. V = 200 × 100 × 20 = 4.0 × 10⁵ mm³.
2. A = 2 × (200 × 100 + 200 × 20 + 100 × 20) = 2 × 26 000 = 52 000 mm².
3. V/A = 4.0 × 10⁵ / 52 000 = 7.69 mm.
4. t = 2.0 × 7.69² = 2.0 × 59.2 = 118 s.

Answer: t ≈ 118 s (about 2 min).

Example 2 (GATE level — riser size). A 150 mm cube is to be fed by a cylindrical side riser with H = D. All riser surfaces lose heat. The riser must take 25% longer to solidify than the casting. Find D.

1. Casting modulus: M_c = a/6 = 150/6 = 25 mm.
2. Riser modulus with H = D: M_r = D/6.
3. Time condition: (M_r/M_c)² = 1.25 → M_r = 25 × √1.25 = 27.95 mm.
4. D = 6 × 27.95 = 167.7 mm.

Answer: D = H ≈ 168 mm. (With the 1.2-modulus rule instead, M_r = 30 mm and D = 180 mm.)

Example 3 (shape effect). A sphere and a cube have the same volume and are cast in the same mould. Find t_sphere / t_cube.

1. Take r = 1: V = 4.189; sphere M = r/3 = 0.3333.
2. Cube side a = 4.189^(1/3) = 1.612; M = a/6 = 0.2687.
3. t_s/t_c = (0.3333/0.2687)² = 1.54.

Answer: the sphere takes about 1.54 times as long.

Common mistakes

• Writing Chvorinov's rule with B in the denominator. Time is proportional to B.
• Using the riser's full surface area when its base sits on the casting. The contact area does not lose heat — for a top riser with H = D the modulus is D/5, not D/6.
• Forgetting to square the modulus ratio when comparing times.
• Mixing units: if B is in s/mm², V/A must be in mm.
• Designing the riser only for freezing time and not checking that it holds enough feed metal (or the reverse).
• Thinking the pattern shrinkage allowance removes the need for risers; it only covers solid contraction.

For GATE PI

Chvorinov's rule is one of the most-asked casting topics. Expect numericals on solidification time, ratio of times for two shapes, riser diameter for a cube or plate with H = D (with or without a base in contact), and Caine's-method problems where constants are given. One-mark questions test which shape freezes slowest, what the mould constant depends on, and the role of chills and exothermic sleeves. Practise moduli of the standard shapes until you can write them without thinking.

Quick check

1. Casting A has twice the modulus of casting B in the same mould. Ratio of freezing times?
2. Modulus of a cylinder with H = D when its base is not cooling?
3. Why does grey cast iron need smaller risers than steel?
4. Which shape of a given volume freezes slowest?
5. What does a chill do to the section it is placed against?

Answers: 1. 4 : 1. 2. D/5. 3. Graphite expansion during solidification offsets most of the shrinkage. 4. A sphere. 5. Speeds up its solidification, promoting directional solidification toward the riser.