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Manufacturing · Chapter 2 of 10 · Beginner

Metal Casting

Pour liquid metal into a mould and let it freeze into shape. Simple in principle, but the metal shrinks as it solidifies, and managing that shrinkage is what separates a sound casting from a scrap one.

Readiness check

This chapter uses geometry and a little fluid flow. Tick only what you can do closed-notes.

Compute volume and surface area of simple shapes.
Recall that metals shrink on solidifying.
Use Bernoulli's v = √(2gh) for a falling stream.
Apply continuity Q = Av.
Square and take square roots.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsSkim solidification in Materials Chapter 9 and flow in Fluid Mechanics.

3 or more weak itemsReview Chapter 1 and basic geometry first.

The core idea

A casting solidifies from its surfaces inward at a rate set by how much surface it has per unit volume, and it shrinks as it freezes, so the last metal to solidify must be fed from a reservoir.

t_s = C_m(V/A)²riser solidifies last: (V/A)_riser > (V/A)_castingv_gate = √(2gh)

Heat leaves through the mould walls, so a thick, compact region (high V/A) freezes slowly and a thin one quickly. Chvorinov's rule turns this into a time. Because metal contracts on solidifying, the slowest-freezing spot draws liquid from wherever it can; the founder's job is to make that a sacrificial riser, not a hole in the part. The gating system must also fill the mould fast and smoothly before anything freezes.

The skill works when: you size risers and gates from V/A and flow, so the casting fills cleanly and feeds its own shrinkage.

The skill breaks down when: shrinkage is ignored, leaving porosity, or the mould fills too slowly and freezes short.

The concept. Solidification proceeds inward from the cooled mould walls. A solid shell thickens until the centre, the last liquid, freezes. The more compact the section (higher V/A), the longer that takes.

The skills, taught in order

Casting is solidification managed by geometry and flow. Five skills cover freezing, Chvorinov's rule, risers, gating, and the process choices.

2.1 Solidification and shrinkage

Molten metal cools, nucleates crystals, and grows them into grains as it freezes, releasing latent heat. It contracts in three stages, and each must be allowed for.

Stage	What contracts	Handled by
Liquid contraction	liquid cooling to the freezing point	riser feeding
Solidification shrinkage	the liquid-to-solid change (the largest)	riser feeding
Solid contraction	solid cooling to room temperature	pattern shrinkage allowance

2.2 Chvorinov's rule

Solidification time scales with the square of the volume-to-surface-area ratio: t_s = C_m(V/A)², where C_m is the mould constant. A sphere (lowest surface per volume) freezes slowest; a thin plate fastest. The ratio V/A, the modulus, is the single most useful number in casting design.

2.3 Risers and feeding

A riser is a reservoir of liquid that feeds the casting's shrinkage. For it to work, it must freeze after the casting, so it needs a larger modulus: (V/A)_riser > (V/A)_casting. Good design promotes directional solidification, freezing toward the riser so shrinkage ends up in the sacrificial metal, which is cut off.

2.4 Gating and fluidity

The gating system (pouring basin, sprue, runner, gate) delivers metal to the cavity. The velocity at the gate follows Bernoulli, v = √(2gh) from the sprue height h, and continuity Q = Av sets the fill time. The metal must stay fluid long enough to fill thin sections, a property called fluidity, before it freezes.

2.5 Casting processes

Processes trade detail, finish, and cost against volume.

Process	Finish and tolerance	Volume
Sand casting	rough, loose	one-off to high; any size
Investment casting	excellent, fine detail	low to medium; complex shapes
Die casting	good, smooth	high; non-ferrous, fast
Permanent mould	good	medium; reusable metal mould

Engineering connection: engine blocks, pump housings, turbine blades (investment), and countless brackets begin as castings; the modulus idea also guides where defects hide.

Worked example 1: sizing a riser

A casting has a modulus V/A = 1.5 cm in a sand mould with constant C_m = 2.0 min/cm². Find its solidification time, then size a cylindrical riser (height equal to diameter, all surfaces cooling) that solidifies at least 25% longer.

Figure 1. The riser sits above the casting and must freeze last. A larger modulus (here D = H ≈ 10 cm) buys the extra solidification time so the riser, not the part, supplies the shrinkage.

ProblemFind the casting's solidification time and size the riser in Figure 1.
Given / find(V/A)_cast = 1.5 cm, C_m = 2.0 min/cm², riser t must be ≥ 1.25 t_cast, cylinder H = D. Find t_cast and the riser diameter.
AssumptionsSame mould constant for casting and riser; all riser surfaces lose heat; Chvorinov's rule applies.
ModelChvorinov's rule for the casting time, then invert it for the required riser modulus and convert to a diameter via V/A = D/6 for an H = D cylinder.
Equationst_s = C_m(V/A)² (V/A)_riser = √(t_riser/C_m) V/A = D/6 (H = D cylinder)
Solvet_cast = 2.0(1.5)² = 4.5 min. Required t_riser = 1.25 × 4.5 = 5.625 min, so (V/A)_riser = √(5.625/2.0) = 1.677 cm. With V/A = D/6, D = 6 × 1.677 = 10.1 cm (H ≈ 10 cm).
CheckThe riser modulus (1.68 cm) exceeds the casting's (1.5 cm), as it must to freeze later. A 10 cm riser on a casting of modulus 1.5 cm is bulky, which is why founders use insulating sleeves to shrink risers and improve yield.
ConclusionChvorinov's rule turns "the riser must freeze last" into a diameter. Undersize it and shrinkage porosity moves into the part; oversize it and metal yield falls.

Result. t_cast = 4.5 min; riser diameter and height about 10 cm (modulus 1.68 cm).

Worked example 2: gating and fill time

Metal is poured from a sprue 0.20 m tall into a 1200 cm³ mould cavity through a gate of 400 mm² area. Find the velocity at the gate and the time to fill the cavity.

Figure 2. The sprue height drives the gate velocity through Bernoulli, and continuity sets how fast the cavity fills. Faster filling beats early freezing, but too fast causes turbulence.

ProblemFind the gate velocity and fill time for the mould in Figure 2.
Given / findh = 0.20 m, gate A = 400 mm² = 4.0×10⁻⁴ m², cavity V = 1200 cm³ = 1.2×10⁻³ m³. Find v and the fill time.
AssumptionsFrictionless Bernoulli flow from the sprue top, steady filling, gate runs full.
ModelVelocity from Bernoulli, flow rate from continuity, fill time from cavity volume over flow rate.
Equationsv = √(2gh) Q = Av t_fill = V/Q
Solvev = √(2 × 9.81 × 0.20) = √3.92 = 1.98 m/s. Q = 4.0×10⁻⁴ × 1.98 = 7.92×10⁻⁴ m³/s. t_fill = 1.2×10⁻³/7.92×10⁻⁴ = 1.5 s.
CheckAbout 1.5 s to fill 1.2 litres is plausible for a small casting. A taller sprue would raise v and shorten the fill but risk turbulence and air entrapment, the trade-off behind gating design.
ConclusionGating is the fluid mechanics of casting: enough head to fill before freezing, but smooth enough to avoid defects. Bernoulli and continuity are all it takes to size a gate.

Result. Gate velocity 1.98 m/s; cavity fills in about 1.5 s.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Ignoring shrinkage	Porosity or sink in the thick section	"Where does the last metal freeze?"	Feed the slowest-freezing region with a riser of larger modulus.
Undersized riser	Shrinkage cavity ends up in the part	"Does the riser freeze after the casting?"	Make (V/A)_riser > (V/A)_casting.
Time scales with size, not V/A	Solidification time mis-estimated	"Did I use (V/A)², not volume?"	Chvorinov's rule depends on the modulus squared.
Pattern equals part size	Casting comes out undersized	"Did I add the shrinkage allowance?"	Oversize the pattern for solid contraction.

Practice ladder

Level 1 · Direct skill

A cube of side 8 cm solidifies in a mould with C_m = 2.5 min/cm². Find its solidification time.

Show answer

V/A = (8³)/(6 × 8²) = 512/384 = 1.333 cm. t_s = 2.5 × 1.333² = 4.4 min. The modulus, not the side length, sets the time.

Level 2 · Mixed concept

A sphere and a cube have the same volume. Which solidifies more slowly, and why?

Show answer

The sphere: it has the least surface area for a given volume, so the highest V/A and the longest Chvorinov time. This is why hot-spots and shrinkage concentrate in compact, rounded sections.

Level 3 · Independent problem

A casting needs a final length of 300 mm in an alloy with 1.6% total solid shrinkage. Find the pattern length, ignoring machining allowance.

Show answer

Pattern length = 300 × (1 + 0.016) = 304.8 mm. The pattern is made oversize so the part shrinks to size on cooling, the role of a shrink rule.

Level 4 · Transfer to real engineering

Find a real cast part (an engine block, a manhole cover, a faucet). Identify likely hot spots from the geometry and where you would place risers and gates.

What good work looks like

The thickest, most compact sections flagged as high-modulus hot spots, risers placed to freeze last there, and gates sized to fill before freezing.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my riser modulus exceeds the casting's."

"Give me five castings; I will point to the likely hot spot from the geometry."

"Size the riser for me." Inverting Chvorinov's rule yourself is the skill.

"Will this casting have porosity?" Reasoning from V/A and feeding is the point.

Portfolio task

Analyse one casting: compute its modulus, size a riser with Chvorinov's rule, and estimate the gate velocity and fill time from a chosen sprue height.

Must include: V/A for casting and riser, a Chvorinov time, and a Bernoulli-plus-continuity fill estimate.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. State Chvorinov's rule.

t_s = C_m(V/A)²: solidification time scales with the square of the modulus.

2. What condition must a riser satisfy?

It must freeze after the casting, so (V/A)_riser > (V/A)_casting.

3. Name the three shrinkage stages.

Liquid contraction, solidification shrinkage, and solid contraction.

4. How do you find gate velocity and fill time?

v = √(2gh) (Bernoulli) and t_fill = V/(Av) (continuity).

5. Which casting process gives the finest detail?

Investment casting; sand casting is the roughest but cheapest.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the riser size from a blank page.

+3 daysOne Chvorinov and one gating calculation.

+7 daysMove to bulk deformation, Chapter 3.

+30 daysCompare casting cost with forming in Chapter 10.

Textbook mapping

Item	Mapping
Primary source	Kalpakjian and Schmid, Manufacturing Engineering and Technology, Chapters 10 to 12 (Metal Casting)
Cross-reference	Groover, Ch. 10 and 11 · DeGarmo, casting chapters
Core topics	2.1 Solidification and shrinkage · 2.2 Chvorinov's rule · 2.3 Risers · 2.4 Gating · 2.5 Casting processes
Engineering connection	Engine blocks, housings, and investment-cast turbine blades.
Read next	Chapter 3: Bulk Deformation.