Industrial Crystallizer (Cooling & Evaporative MSMPR) Yield & CSD Calculator

Industrial crystallization creates a solid crystalline phase from a supersaturated liquid solution. Driving supersaturation is achieved via temperature reduction (cooling crystallization) for systems with steep positive solubility curves (e.g. KCl, KNO₃), solvent removal (evaporative crystallization) for flat solubility systems (e.g. NaCl), or vacuum adiabatic flashing.

1. Austin-Mullin Hydrate Crystal Yield Equation

When a hydrated salt crystallizes (such as $\text{CuSO}_4 \cdot 5\text{H}_2\text{O}$ or $\text{Na}_2\text{SO}_4 \cdot 10\text{H}_2\text{O}$), water molecules are incorporated directly into the solid crystal lattice, reducing the free solvent mass in the mother liquor. The theoretical crystal yield $Y$ is defined by the Austin-Mullin mass balance:

$$Y = \frac{W_0 \left[ C_0 - C_f \left(1 - \frac{E}{W_0}\right) \right]}{1 - C_f \cdot (R_{hyd} - 1)}$$

Where:

• $W_0$ = Initial mass flow of pure solvent (water) entering with the feed ($\text{kg/h}$).
• $C_0, C_f$ = Initial and final equilibrium solubilities ($\text{kg anhydrous solute / kg solvent}$).
• $E$ = Solvent evaporated ($\text{kg/h}$).
• $R_{hyd} = \frac{M_{hydrate}}{M_{anhydrous}}$ = Molecular weight ratio of hydrated crystal to anhydrous solute ($R_{hyd} = 1.0$ for anhydrous salts).

2. MSMPR Population Balance & Crystal Size Distribution

For an ideal continuous Mixed-Suspension Mixed-Product Removal (MSMPR) crystallizer obeying Randolph-Larson population balance theory under McCabe's $\Delta L$ law (size-independent growth rate $G$):

$$\frac{d(G \cdot n)}{dL} + \frac{n}{\tau} = 0 \implies n(L) = n_0 \cdot \exp\left(-\frac{L}{G \cdot \tau}\right)$$

Where $n(L)$ is population density ($\text{number}/\text{m}^3\cdot\mu\text{m}$), $n_0 = B_0 / G$ is nuclei population density, and $\tau = V_{cryst} / Q_{slurry}$ is mean magma residence time. The mass-based crystal size distribution $w(L)$ follows a gamma distribution:

$$w(L) = \frac{1}{6} \cdot \left(\frac{L}{G \tau}\right)^3 \cdot \exp\left(-\frac{L}{G \tau}\right) \cdot \frac{1}{G \tau}$$

The dominant crystal size $L_D$ (the peak of the mass distribution curve where the greatest mass fraction exists) is exactly:

$$L_D = 3 \cdot G \cdot \tau$$

And the volume-weighted mean size is $\bar{L}_{43} = 4 \cdot G \cdot \tau$.

3. Nucleation Rate ($B_0$) & Magma Density ($M_T$)

Magma density $M_T$ is the concentration of solid crystals suspended in the active vessel slurry ($M_T = Y / Q_{slurry}$). The total third moment of the population density relates directly to magma solids content, yielding the fundamental relationship for secondary nucleation rate $B_0$:

$$M_T = 6 \cdot k_v \cdot \rho_c \cdot n_0 \cdot (G \tau)^4$$ $$B_0 = n_0 \cdot G = \frac{M_T \cdot G}{6 \cdot k_v \cdot \rho_c \cdot (G \tau)^4}$$

Where $k_v$ is the volumetric crystal shape factor ($\pi / 6 \approx 0.5236$ for spheres/cubes).