This is a batch cooling crystallizer. A solution starts saturated at a hot temperature T. Solubility Csat(T) falls as the jacket cools it, so the dissolved concentration C becomes supersaturated:
S = C / C_sat(T) (supersaturation ratio)
σ = S − 1 (relative supersaturation)
Below S ≈ 1 the solution is undersaturated and any crystals redissolve. Between S = 1 and the metastable limit no crystals appear spontaneously — this is the metastable zone. Past that limit, classical nucleation theory gives the primary nucleation rate:
J = A · exp( −B / (ln S)² )
where the exponential term collapses toward zero for small ln S, so nucleation switches on sharply once S is high enough to overcome the surface-energy penalty of forming a new crystal face. Existing crystals also spawn new nuclei by contact/attrition — secondary nucleation — at a rate that grows with crystal count, agitation and σ.
Every crystal then grows by McCabe's ΔL law — a size-independent linear growth rate proportional to supersaturation:
dL/dt = k_g · σ (grow if σ > 0, dissolve if σ < 0)
Growth consumes dissolved solute, which lowers C and self-limits σ — the classic desupersaturation curve. This is exactly the tension a real crystallizer operator manages: cool too fast and uncontrolled primary nucleation floods the vessel with countless tiny crystals; add seed crystals and cool gently and growth dominates, producing a narrow, coarse crystal-size distribution — the basis of pharmaceutical and industrial (sugar, salt, fine-chemical) batch crystallization.
- Cooling rate — how fast T falls; faster cooling pushes S past the nucleation threshold sooner.
- Seed crystals — crystals present at t = 0; more seeds mean more surface area to grow onto, so σ is consumed by growth instead of feeding new nucleation.
- kg — growth-rate constant; scales how fast existing crystals absorb supersaturation.
- Agitation — mixing intensity; raises secondary nucleation from crystal–crystal/wall contact and stirs the suspension visually.