Reference ID: MET-6005 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In solid–liquid separation processes such as pressure or vacuum filtration, the resistance offered by the filter cake increases with the applied pressure drop. The cake compressibility correction quantifies this rise through a power-law exponent \(s\), the compressibility coefficient. Accurate prediction of the corrected specific cake resistance \(r\) is essential for sizing filters, estimating cycle times, and scaling from laboratory leaf tests to industrial units.
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Reference State
The reference specific resistance \(r_0\) is measured at a reference pressure drop \(\Delta P_1\).
Power-Law Correction
For any new pressure drop \(\Delta P_2\) within the empirical range, the corrected resistance is
\[
r_2 = r_0 \left(\frac{\Delta P_2}{\Delta P_1}\right)^s
\]
where both pressures must share the same unit (bar in the code).
Empirical Bounds
Parameter
Lower Limit
Upper Limit
\(\Delta P\) (bar)
0.1
15
\(s\) (dimensionless)
0
1
Operation outside these limits triggers a warning because the correlation is no longer validated.
Numerical Safeguard
To prevent division by zero or negative arguments, the code replaces any non-positive pressure with a small positive value \(\epsilon = 10^{-9}\) bar.
Cake compressibility is the tendency of the solids layer formed during filtration to shrink under pressure, reducing porosity and permeability. Correction is required because the declining permeability invalidates the linear form of Darcy’s law used in constant-pressure filtration models; without it, cycle times and cake moisture predictions can be off by 20–40%.
Introduce the empirical specific-cake-resistance function α = α₀(ΔP)ⁿ where:
α₀ is the resistance at unit pressure
ΔP is the applied pressure differential
n is the compressibility exponent (0 = incompressible, 1 = highly compressible)
Replace the constant α in the basic filtration equation with this function and integrate numerically or use the corrected parabolic law t/V = (μ·α₀(ΔP)ⁿ·c/2A²)·V + μ·Rm/A·ΔP.
Run a series of constant-pressure leaf tests at 3–5 different pressures covering your expected plant range. Plot log α versus log ΔP; the slope of the best-fit line equals n. Typical values:
Inorganic salts and crystalline products: 0.1–0.3
Biological sludge: 0.4–0.7
Metal hydroxides: 0.6–0.9
Ensure cake thickness is kept below 15 mm to side-step wall-friction effects that can bias n high.
Yes. Because compressibility is pressure-dependent, scale-up must preserve the same ΔP profile across the cake. On a larger area, maintaining identical ΔP usually requires:
Thicker cakes (higher form time) to keep pressure drop per unit length similar
Lower air or liquid flow rates per unit area to avoid over-compaction
Neglecting this leads to under-estimated cycle times and over-estimated capacity; apply the corrected α function in the scale-up model and validate with a 10–20% safety factor on pilot data.
Worked Example – Estimating the Compressibility-Corrected Specific Cake Resistance
A small confectionery plant is switching from batch to continuous filtration of a starch-based slurry. The filter vendor supplied the incompressible resistance \(r_0\) measured at a reference pressure of 1 bar. Because the new unit will operate at 2 bar, the process engineer must correct for cake compressibility to predict cycle time and cloth area.
Knowns
Reference specific cake resistance, \(r_0\) = 5.00 × 1011 m kg-1
Multiply by the reference resistance:
\[
r_2 = 5.00 \times 10^{11} \times 1.516 = 7.58 \times 10^{11} \text{ m kg}^{-1}
\]
Final Answer
The compressibility-corrected specific cake resistance at 2 bar is 7.58 × 1011 m kg-1. Use this value for sizing the continuous filter and estimating cycle time.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
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