Introduction & Context

Constant-pressure filtration is a unit operation widely encountered in the food, pharmaceutical, water-treatment and chemical process industries. During the cycle a fixed pressure difference \( \Delta P \) is maintained across a growing filter cake and the supporting medium; the filtrate volume \( V \) is recorded as a function of time \( t \). The resulting data are correlated with the classical filtration equation to obtain two empirical constants: the cake resistance coefficient \( K_p \) and the medium resistance coefficient \( B \). Once these constants are known for a given slurry/filter pair, the time required to reach any target filtrate volume—or the volume attainable in a fixed cycle time—can be predicted a-priori for any scale of equipment. This capability is essential for cycle-time optimisation, filter-sizing, process scheduling and scale-up from laboratory leaf-tests to industrial filter-presses or rotary drums.

Methodology & Formulas

Unit conversions
Dynamic viscosity: \( \mu \,[\text{Pa·s}] = \mu_{\text{cP}} \times 0.001 \)
Pressure: \( \Delta P \,[\text{Pa}] = \Delta P_{\text{bar}} \times 10^{5} \)
Cake resistance coefficient
\[ K_p = \frac{\mu \, r \, v}{2\,A^{2}\,\Delta P} \quad [\text{s·m}^{-6}] \]
where
- \( r \) = specific cake resistance [m·kg^-1]
- \( v \) = mass of dry cake solids per unit filtrate volume [kg·m^-3]
- \( A \) = filtration area [m²]
Medium resistance coefficient
\[ B = \frac{\mu \, R_f}{A\,\Delta P} \quad [\text{s·m}^{-3}] \]
with \( R_f \) the filter-medium resistance [m^-1].
Filtration time for target volume
The integrated rate equation for constant-pressure filtration is \[ t = \left(K_p\,V + B\right)\,V \] giving the elapsed time \( t \) required to collect a filtrate volume \( V \).

Flow-regime check
A Reynolds number based on estimated cake thickness \( \delta \) and superficial velocity \( u = V/(A\,t) \) is \[ Re = \frac{\rho\,u\,\delta}{\mu} \]

with \( \rho \) the filtrate density. Acceptable limits are:

Regime	Reynolds Range
Laminar (Darcy)	\( Re \leq 1000 \)
Non-Darcy (Forchheimer)	\( Re > 1000 \)

Empirical validity windows
Typical food-industry ranges for the key parameters are:

Parameter	Typical Range
Specific cake resistance \( r \)	\( 10^{10} \)–\( 10^{12} \) m·kg^-1
Pressure difference \( \Delta P \)	0.2–0.8 bar
Solids ratio \( v \)	1–10 kg·m^-3

Plot t/V versus V from the test data; the slope equals μ α c /(2 A² ΔP) and the intercept equals μ R_m/(A ΔP).

Measure filtrate viscosity μ, solids concentration c, and filtration area A.
Calculate specific cake resistance α from the slope and medium resistance R_m from the intercept.
Check linearity (R² ≥ 0.98) to validate the incompressible-cake assumption.

Run a pilot test at 0.5, 1, 2, and 4 bar; plot α versus ΔP.

Choose the highest ΔP where α increases < 10 % per bar—this indicates negligible cake compression.
Ensure downstream pressure remains atmospheric to avoid gas breakout or cake disruption.

Maintain geometric and kinematic similarity.

Keep cake thickness ≤ 10 mm on the drum to avoid cracking; scale cycle time t_cycle ∝ L² where L is characteristic cake thickness.
Use the same ΔP and slurry concentration; adjust drum speed so t_{cake formation} equals lab t at the same V/A.

Non-linear deviation signals cake compression or medium blinding.

Check if slope increases—indicates compressible cake; switch to lower ΔP or add body-feed.
If intercept rises, clean or replace medium; verify pore size distribution hasn’t changed.

Worked Example: Constant-Pressure Leaf Filter Sizing

A specialty-chemical plant needs to clarify an aqueous pigment slurry before packaging. A bench-scale leaf filter with 0.05 m² of filtration area is operated at a constant vacuum of 0.4 bar to collect design data. The lab test targets 5 L of filtrate. Determine the filtration time and the resulting cake thickness.

Knowns

Filtrate viscosity, μ = 1.5 cP (= 0.0015 Pa·s)
Applied pressure difference, ΔP = 0.4 bar (= 40,000 Pa)
Filter area, A = 0.05 m²
Mass of solids per unit volume of filtrate, v = 4 kg·m^-3
Specific cake resistance, r = 1.2×10¹¹ m·kg^-1
Medium resistance, R_f = 1×10¹⁰ m^-1
Target filtrate volume, V = 0.005 m³
Water density (for Re check), ρ = 1000 kg·m^-3

Step-by-step calculation

Convert viscosity to SI units: μ = 1.5 cP × 0.001 = 0.0015 Pa·s.
Convert pressure to SI units: ΔP = 0.4 bar × 100,000 = 40,000 Pa.
Compute the cake-filtration constant, K_p: \[ K_p = \frac{\mu\,v\,r}{2\,A^2\,\Delta P} \] \[ K_p = \frac{0.0015 \times 4 \times 1.2 \times 10^{11}}{2 \times (0.05)^2 \times 40,000} = 3.6 \times 10^6\ \text{s·m}^{-6} \]
Compute the medium-filtration constant, B: \[ B = \frac{\mu\,R_f}{A\,\Delta P} \] \[ B = \frac{0.0015 \times 1 \times 10^{10}}{0.05 \times 40,000} = 7.5 \times 10^3\ \text{s·m}^{-3} \]
Apply the constant-pressure filtration equation to find time, t: \[ t = K_p\,V^2 + B\,V \] \[ t = 3.6 \times 10^6 \times (0.005)^2 + 7.5 \times 10^3 \times 0.005 = 90 + 37.5 = 127.5\ \text{s} \]
Convert time to minutes: t = 127.5 s ÷ 60 = 2.125 min.
Estimate cake thickness, L_c: \[ L_c = \frac{v\,V}{A\,\rho_{\text{cake}}} \approx \frac{v\,V}{A\,\rho_{\text{water}}} \] \[ L_c = \frac{4 \times 0.005}{0.05 \times 1000} = 0.001\ \text{m} = 1\ \text{mm} \]
Check average filtrate velocity and Reynolds number for laminar-flow assumption: \[ u = \frac{V}{A\,t} = \frac{0.005}{0.05 \times 127.5} = 0.00078\ \text{m·s}^{-1} \] \[ Re = \frac{\rho\,u\,L_c}{\mu} = \frac{1000 \times 0.00078 \times 0.001}{0.0015} = 0.52 \ll 2300\ \text{(laminar)} \]

Final Answer

The 5 L filtrate batch requires 127.5 s (≈ 2.1 min) of filtration time and produces a cake approximately 1 mm thick.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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