How do I calculate firing shrinkage and the green size for a pressed alumina part?

If the mass of ceramic stays the same, volume is inversely proportional to density, and for uniform shrinkage each linear dimension scales with the cube root of the volume ratio: L_fired / L_green = (ρ_green / ρ_fired)^(1/3).

With 55 and 98 percent of theoretical density: (0.55 / 0.98)^(1/3) = 0.561^(1/3) = 0.825. The linear shrinkage is therefore 17.5 percent and the volume shrinkage 44 percent. A 20.0 mm fired diameter needs a green diameter of 20.0 / 0.825 = 24.2 mm. Use the green density of the ceramic alone, excluding the binder, since the binder burns out before densification.

The formula also shows how sensitive the result is to green density. If one pressing comes out at 53 percent instead of 55, the ratio becomes (0.53 / 0.98)^(1/3) = 0.815, and the same 24.2 mm green part fires to about 19.7 mm instead of 20.0 mm. A two-point change in green density moves the fired size by more than 1 percent.

This is why fired dimensions are controlled by controlling the pressing: consistent powder fill weight, granule moisture and binder content, and pressing to a set pressure or a set density. It is also why the die is sized from measured shrinkage of trial parts in each direction, not from the calculation alone.

The hourglass shape comes from density gradients in the green part. In uniaxial die pressing, friction between the powder and the die wall means the pressure falls with distance from the moving punch. With pressing from both ends, the compact is densest near the two faces and least dense at mid-height. Every region sinters to nearly the same final density, so the low-density middle has to shrink more, and the diameter there ends up smaller.

Single-ended pressing gives a taper instead, with the end far from the punch shrinking most. The effect grows with the height-to-diameter ratio, so tall compacts are worse than thin discs.

Ways to reduce the gradient: keep the height-to-diameter ratio low, press from both ends or use a floating die, use free-flowing spray-dried granules with a suitable binder and lubricant, and lubricate the die wall. For tall or complex shapes, cold isostatic pressing applies pressure from all sides and gives much more uniform density, at the cost of a less precise green shape that is then machined in the green state.

During firing, friction against the setter can hold back the bottom face and distort thin parts, so parts are often fired on a bed of coarse alumina grain or on a sacrificial setter of the same material that shrinks along with them.

Be realistic about as-fired tolerances. With shrinkage near 17 percent, holding the fired size within about ±1 percent of the dimension is a normal capability for pressed and sintered parts, which is ±0.2 mm on your 20 mm disc. Anything much tighter is usually achieved by diamond grinding after firing, which is slow and expensive because fully dense alumina is so hard.

The economical route is to do as much shaping as possible before firing, when the compact machines easily, and to leave grinding stock only on the surfaces that really need a close tolerance or a fine finish.