Industrial Filtration & Solid-Liquid Separation Designer Documentation

1. Executive Summary & Objective

The Industrial Filtration & Solid-Liquid Separation Designer provides a comprehensive platform for chemical and pharmaceutical process engineers to translate laboratory Buchner funnel test data into commercial-scale solid-liquid separation equipment.

It computes Specific Cake Resistance ($\alpha$), Filter Medium Resistance ($R_m$), and Cake Compressibility Index ($s$) using the classical Ruth / Darcy filtration model, and performs rigorous sizing for:

1. Agitated Nutsche Filter Dryers (ANFD) (filter plate area, cake thickness, displacement & re-slurry washing, N₂ deliquoring, and contact vacuum drying).
2. Industrial Centrifuges (Vertical Basket, Horizontal Peeler, Inverting Filter, and Pusher centrifuges based on $G$-force and centrifugal dewatering kinetics).
3. Multi-Technology Comparison (OEB containment, particle attrition, and solvent hazard compatibility).

2. Governing Equations & Filtration Kinetics

2.1 Ruth / Darcy Constant-Pressure Filtration Equation

The instantaneous differential filtration rate is governed by Darcy's law for flow through porous media: $$\frac{dt}{dV} = \frac{\mu \cdot (R_c + R_m)}{A \cdot \Delta P} = \frac{\mu \cdot \left( \alpha \cdot \frac{c \cdot V}{A} + R_m \right)}{A \cdot \Delta P}$$

Integrating for constant pressure drop ($\Delta P = \text{constant}$): $$\frac{t}{V} = \left( \frac{\mu \cdot \alpha \cdot c}{2 \cdot A^2 \cdot \Delta P} \right) \cdot V + \frac{\mu \cdot R_m}{A \cdot \Delta P}$$

Linearized as: $$\frac{t}{V} = K_p \cdot V + B$$

Where:

• $V$: Cumulative filtrate volume $[\text{m}^3]$
• $t$: Filtration time $[\text{s}]$
• $\mu$: Liquid dynamic viscosity $[\text{Pa}\cdot\text{s}]$
• $\alpha$: Specific cake resistance $[\text{m/kg}]$
• $c$: Mass of dry cake solids per unit filtrate volume $[\text{kg/m}^3]$
• $A$: Active filtration area $[\text{m}^2]$
• $\Delta P$: Applied differential pressure $[\text{Pa}]$
• $K_p$: Ruth slope constant $=\frac{\mu \alpha c}{2 A^2 \Delta P} \quad [\text{s/m}^6]$
• $B$: Ruth intercept constant $=\frac{\mu R_m}{A \Delta P} \quad [\text{s/m}^3]$
• $R_m$: Filter medium / cloth resistance $[\text{m}^{-1}]$

2.2 Cake Compressibility Correction

For compressible crystals and amorphous precipitates, specific cake resistance increases with filtration pressure: $$\alpha_{plant} = \alpha_0 \cdot \left( \frac{\Delta P_{plant}}{\Delta P_{lab}} \right)^s$$

Where:

• $s = 0$: Incompressible rigid crystals (e.g. coarse NaCl, large API prisms)
• $s = 0.1 - 0.4$: Moderately compressible (standard pharmaceutical API crystal needles/plates)
• $s = 0.5 - 1.0$: Highly compressible (gelatinous, amorphous, or flocculated biomass)

3. Commercial Equipment Sizing

3.1 Agitated Nutsche Filter Dryer (ANFD) Sizing

Filter Area Required for Target Filtration Time ($t_{target}$):

$$A_{ANFD} = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$ Where:

• $a = \Delta P_{ANFD} \cdot t_{target}$
• $b = -\mu \cdot R_m \cdot V_{filtrate}$
• $c = -\frac{1}{2} \mu \cdot \alpha_{plant} \cdot c \cdot V_{filtrate}^2$

Wet Cake Bed Thickness ($h_{cake}$):

$$h_{cake} = \frac{V_{cake_wet}}{A_{ANFD}} = \frac{M_{dry} / \rho_{bulk_wet}}{A_{ANFD}} \quad [\text{mm}]$$

[!WARNING] In pharmaceutical ANFD design, cake bed thickness should typically be kept between $100\text{ mm}$ and $250\text{ mm}$ (max $300\text{ mm}$). Excessively thick cakes lead to crack formation, channeling bypass during washing, and drastically prolonged vacuum drying times.

Displacement Washing Volume and Duration:

• Displacement Wash Volume ($V_{wash}$): $$V_{wash} = n_{wash} \cdot \varepsilon \cdot h_{cake} \cdot A_{ANFD} \quad [\text{L}]$$ (where $n_{wash} \approx 2.0 - 3.0$ cake void volumes, and $\varepsilon$ is porosity)
• Displacement Wash Time ($t_{wash}$): $$t_{wash} = \frac{\mu_{wash} \cdot \alpha_{plant} \cdot (M_{dry} / A_{ANFD}) \cdot V_{wash}}{A_{ANFD} \cdot \Delta P_{wash}} \quad [\text{s}]$$

3.2 Industrial Centrifuge Sizing

Centrifugal Acceleration & $G$-Force:

$$G = \frac{\omega^2 \cdot r_{basket}}{g} = \frac{D_{basket} \cdot (\pi N / 30)^2}{2 \cdot 9.80665} \approx \frac{D_{basket} \cdot N^2}{1790}$$

Centrifugal Driving Pressure:

$$\Delta P_{centrifugal} = \frac{1}{2} \rho_L \cdot \omega^2 \cdot (r_{basket}^2 - r_{pool_inner}^2) \quad [\text{Pa}]$$

Number of Cycles per Total Batch:

$$N_{cycles} = \left\lceil \frac{V_{cake_total}}{V_{basket_holding}} \right\rceil$$

4. Specific Cake Resistance ($\alpha$) Classification

5. Reference Standards & Literature

1. Perry's Chemical Engineers' Handbook: Section 18 Liquid-Solid Operations and Equipment.
2. Svarovsky, L. (2000): Solid-Liquid Separation, 4th Edition, Butterworth-Heinemann.
3. Coulson & Richardson’s Chemical Engineering: Volume 2 (Particle Technology and Separation Processes).
4. ISPE Good Practice Guide: Applied Risk Management for Commissioning and Qualification.
5. FDA Guidance for Industry: Q7 Good Manufacturing Practice Guidance for Active Pharmaceutical Ingredients.