Filtration – Theories, Equations & Industrial Filters

Fundamental Concepts of Liquid Filtration

• Filtration = passage of a liquid-solid suspension through a porous barrier that retains the solids.
• Core relationship (generalised):
  • $\text{Rate of filtration} = \frac{\text{Driving force}}{\text{Total resistance}}$
  • Commonly expressed as the volumetric rate $\frac{dV}{dt}$ (e.g., L s$^{-1}$ or m$^{3}$ s$^{-1}$).
• Driving force = pressure differential between the upstream (feed) and downstream (filtrate) sides.
• Resistance is dynamic – it rises continuously as solids deposit and a filter cake builds.
• Because resistance changes, filtration is non-steady-state; highest rate occurs at the very start when cake thickness = 0.
• Once a cake becomes established its outer surface behaves as an auxiliary filter medium, enhancing particle capture but adding hydraulic resistance.

Major Contributors to Hydraulic Resistance

• $\Delta P = P<em>1 - P</em>2$ (difference between upstream and downstream pressures).
• Effective capillary length $L$ (≈ cake thickness once cake forms).
• Radius and tortuosity of flow channels within the cake or depth medium.
• Surface area of the bed and filter medium.
• Viscosity of the filtrate $\eta$ (Pa·s).

Visualising the Cake as a Capillary Bundle

• Powder bed treated as innumerable small capillaries.
• Pressure drop distributed along capillary length – analogous to laminar flow in tubes (Poiseuille flow).

Poiseuille’s Equation (Capillary Model)

• For laminar flow of an incompressible Newtonian fluid through a straight, cylindrical capillary: $V = \frac{\Delta P\, r^{4}}{8\, L\, \eta} \qquad (1)$ where
  • $V$ = volumetric rate (m$^{3}$ s$^{-1}$),
  • $r$ = capillary radius (m),
  • $L$ = capillary length (m) ≈ cake thickness,
  • $\eta$ = viscosity (Pa·s),
  • $\Delta P$ = pressure difference (Pa).
• Highlights extremely strong dependence on pore radius (fourth-power). Small reductions in pore size greatly reduce rate.
• Limitations: assumes uniform, straight capillaries – real filter cakes are tortuous, irregular, compressible.

Darcy’s Law (Empirical Bed Law)

• Henry Darcy generalised Poiseuille’s idea for complex porous beds: $V = \frac{K A \Delta P}{\eta L} \qquad (2)$ where
  • $A$ = total cross-sectional area of bed (m$^{2}$),
  • $K$ = permeability coefficient (m$^{2}$), a lumped parameter capturing porosity, pore size distribution & compressibility,
  • other symbols as before.
• Permeability definition: volume flow rate of a unit-viscosity fluid through unit thickness and unit area of cake under unit pressure gradient.
• Applicable to sands, glass beads, depth filters and conventional cakes.
• If cake compresses, $K$ decreases with pressure – must be treated as variable in design.

Kozeny–Carman Equation (Structure-Aware Model)

• Adds structural parameters (porosity, specific surface): $V = \frac{A\, \Delta P\, \varepsilon^{3}}{K\, L\, \eta\, S^{2}\, (1-\varepsilon)^{2}} \qquad (3)$ where
  • $\varepsilon$ = porosity (dimensionless),
  • $S$ = specific surface area of particles (m$^{2}$ m$^{-3}$),
  • $K$ = Kozeny constant (≈ 5 for many beds).
• Demonstrates that high porosity and low particle surface area enhance flow.
• Useful for predicting how particle size reduction (↑ S) or cake consolidation (↓ ε) slows filtration.

Filter Leaf

Principle & Mechanism

• Surface filtration on a leaf-shaped element; acts as a sieve/strainer.
• Vacuum or positive pressure accelerates flow.

Construction

• Rigid drainage screen or grooved plate encased by a narrow metal frame (any geometry).
• Entire assembly covered with filter cloth.
• Interior connects to a filtrate outlet manifold that can be evacuated or pressurised.

Working Sequence

• Leaf immersed in open slurry tank.
• Vacuum applied → slurry drawn through cloth → solids deposit on cloth forming cake → filtrate passes into drainage channels and out.
• On completion, reverse air/steam pulse detaches cake; leaf can be rinsed externally.

Practical Notes / Applications

• Modular: multiple leaves mounted in vertical leaf filters for edible oil, syrup, pharmaceutical solutions.
• Low capital, easy cake discharge; limited to slurries with moderate–low cake resistance.

Rotary Drum Vacuum Filter (RDVF)

Principle

• Continuous rotation of a perforated, cloth-covered drum through a slurry under vacuum.
• Combines sequential zones for cake formation, drainage, optional compression, cake washing, drying and mechanical discharge.

Construction Details

• Drum: 1–3 m diameter, ≈ 3.5 m long → up to 20 m$^{2}$ filtration area.
• Curved surface = perforated plate + coarse mesh support + filter cloth.
• Interior divided radially into sectors; each sector connects via internal pipes to a centre barrel and a rotating valve ported to various vacuum/pressure lines.
• External trough holds slurry; knife blade (doctor blade) scrapes off dried cake.

Operating Cycle (one revolution < 60 s)

1. Pick-up Zone – sector submerges; vacuum draws slurry; cake builds.
2. Drainage Zone – still under vacuum; excess liquor removed; optional compression rollers squeeze cake.
3. Washing Zone – sprays water/solvent; separate vacuum line collects washings.
4. Drying Zone – hot air or inert gas; residual moisture ↓ to < 1 % possible.
5. Cake Removal – positive pressure or blow-back loosens cake; doctor knife scrapes.
6. Cloth passes through rinse/spray before re-entering slurry.

Advantages & Uses

• Continuous, automatic, moderate capital.
• Ideal for large-volume, easily filterable slurries (e.g., mineral concentrates, fertilizer, pharma intermediates).
• Limited by cloth blinding for fine, compressible cakes; vacuum constraints for very viscous filtrates.

Plate and Frame Filter Press

Principle

• Discrete batch surface filtration; slurry pumped under pressure into alternating plates & frames lined with cloth.
• Plates provide drainage channels; frames provide cake cavities.

Construction Elements

• Plates (1 dot symbol): solid, grooved/studded faces, filtrate outlet.
• Frames (2 dots): open cavity, same outer dimensions, variable thickness to suit desired cake thickness.
• Both drilled/eyed to form common feed (slurry) and/or filtrate manifolds when stacked.
• Material: aluminium alloy, lacquered/epoxy-coated; can be steam-sterilised.
• Stack sequence example: Plate–Cloth–Frame–Cloth–Plate (1•2•1•2•1 …) then clamped between a fixed head & follower using a screw or hydraulic ram.

Filtration Step

1. Slurry pumped into feed channel; enters each frame.
2. Liquid passes through cloth to adjacent plates; solids accumulate in frame.
3. Two half-cakes grow from each cloth until meeting centrally; cake thickness ≈ ½ frame thickness per side.
4. Filtrate exits via plate drainage grooves to common outlet manifold.
5. As cake thickens, resistance ↑; when flow becomes uneconomically low, feed is stopped.
6. Press opened; each cake peeled away or dropped under gravity.

Cake Washing (Optional)

• Uses special washing plates (3 dots) and a distinct wash channel.
• Sequence:
  1. Normal filtration until frames full.
  2. Close filtrate valves on washing plates.
  3. Pump wash liquor into wash channel → through washing plates → cloth → cake → opposite cloth → adjacent plates.
  4. Washed liquor exits via outlets of one-dot plates.
• Allows counter-current washing with high efficiency; crucial for pharmaceutical or food products where mother liquor must be displaced.

Design / Operational Considerations

• Frame thickness chosen for optimum cycle time vs. capacity.
• Cloth material selected for chemical compatibility, pore size, ease of cake release.
• Pressures up to 7 bar common; higher pressures available with membrane squeeze plates.
• Batch nature implies labour and downtime for cake discharge; still popular for fine, compressible slurries requiring high clarity filtrate.

Interconnections, Conceptual Importance & Exam Tips

• Poiseuille → theoretical baseline; Darcy → practical engineering law; Kozeny–Carman → links microstructure to Darcy’s $K$.
• In equipment design: apply Darcy-type models to size area $A$ for required throughput, given target pressure drop and viscosity.
• Cake washing efficiency relates to cake porosity (ε) and thickness – same parameters as in Kozeny–Carman.
• Compressible cakes (e.g., pharmaceuticals) necessitate low pressure leaf filters or membrane press technology; incompressible mineral cakes suit RDVF.
• Ethical/pharma context: sterilisation ability (aluminium, steam) and complete product recovery (leaf blow-back, plate-and-frame wash) are vital for patient safety and GMP compliance.

Quick Numerical Reminders for Calculations

• For Newtonian fluid laminar flow in capillary: Re < 2100 (ensures Poiseuille validity).
• Convert viscosity: 1 cP = $1\times10^{-3}$ Pa·s.
• Vacuum typically limited to ≈ 0.08–0.09 MPa gauge (≈ 700 mm Hg) → set upper bound for $\Delta P$ on atmospheric drum/leaf filters.
• In Darcy calculations, remember to convert cake thickness from mm to m.

Real-World Relevance & Further Study

• Water treatment (sand filters) governed by Darcy’s law; design of rapid gravity filters uses similar equations.
• Biomedical devices (dialysers) adapt Darcy-type models for membrane permeance.
• Emerging depth filters (3D-printed lattices, ceramic foams) still characterised by $K$ and Kozeny-Carman correlations.
• Sustainability: optimizing cake washing reduces solvent/water usage; energy footprint tied to $\Delta P$ (pump/vacuum) – a design trade-off.