Introduction & Context

Constant‑pressure filtration is a unit operation widely encountered in the food, pharmaceutical, water‑treatment and chemical process industries. When selecting the most suitable method, it is helpful to review the vacuum vs. pressure filtration selection guidelines. 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, and it forms the basis of effective filtration cycle optimization.

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

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

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

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