This is the 2D side-view cross-section of the same cooling batch crystallizer as the 3D version — a diametral slice through the vessel, so you're looking at crystals suspended between the two vertical walls instead of orbiting a 3D tank. The driving force is the supersaturation ratio:
S(t) = C(t) / C*(T(t))
C*(T) = 0.10 + 0.0072 · T (linear solubility, g solute / g solvent)
Two competing rate laws consume that supersaturation — this is the classic nucleation/growth trade-off PAT systems are built to manage:
Growth: G(S) = kg · max(S−1, 0)^1.6 (µm/min per crystal)
Nucleation: B(S) = kb · max(S−1, 0)^3.5 (new crystals/min)
Every crystal growing consumes solute mass proportional to its surface area × radial growth (dV = 4πr²·dr), which feeds back into C(t) — so growth alone pulls S back down with no controller at all. The risk is nucleation: because the nucleation exponent (3.5) is much steeper than growth's (1.6), letting S spike creates a burst of tiny new crystals (a wide, hard-to-filter size distribution) instead of growing the seeds you already have.
The inline probe (an FBRM/Raman-style PAT sensor in a real plant, drawn here as the cyan beam sweeping in from the right wall) reports S(t) with a little measurement noise. With PAT feedback ON, a proportional controller throttles the cooling ramp itself:
coolRate_eff = coolRate_base · clamp(1 − Kp·(S − Starget), 0, 1.6)
Cooling pauses when S drifts above the setpoint (letting existing crystals grow the supersaturation back down before more nucleation happens) and speeds back up when S falls below it. With feedback OFF the cooling ramp runs open-loop at a constant rate regardless of what the probe reads — watch how much faster the crystal count climbs and how much smaller the mean size ends up for the same batch.
- Base cooling rate — the open-loop ramp speed; also the ceiling the controller throttles against in closed loop.
- Starget — the supersaturation band the PAT controller tries to hold.
- Seed loading — starting crystal count; more seed surface area means less driving force is "wasted" on nucleation.
- Drag / scroll — drag the vessel to pan, scroll (or pinch) to zoom in on the crystal population.