Reference ID: MET-D8C4 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Specific cake resistance, denoted α, quantifies how much a filter cake opposes flow per unit mass of solids deposited. It is the key parameter in the constant-pressure filtration equation and is indispensable for sizing batch Nutsche filters, plate-and-frame presses, rotary drums, and any equipment where solids accumulate on a porous medium. A reliable α value allows engineers to predict filtration time, filter area, cake thickness, and washing or drying cycles without resorting to costly pilot trials.
The laboratory determination is normally performed in a Buchner funnel or dead-end filtration cell at constant pressure. By recording cumulative filtrate volume V versus time t and plotting t/V against V, the slope of the resulting straight line gives the information needed to compute α under the assumption of an incompressible cake and laminar flow through the pores.
Extract slope from t/V vs V plot
The linearised filtration law for incompressible cakes is
\[ \frac{t}{V} = \frac{\mu \alpha c}{2 A^2 \Delta P} V + \frac{\mu R_{\text{m}}}{A \Delta P} \]
Hence the slope m (units s m-6) is taken directly from the straight-line region of the experimental data.
Compute specific cake resistance
Rearranging the slope term gives
\[ \alpha = \frac{2 A^2 \Delta P}{\mu c} \cdot m \]
Assumption
Validity Criterion
Symbol / Formula
Typical Threshold
Incompressible cake
Pressure drop across cake
\( \Delta P \leq 1 \) bar
\( \Delta P \leq 100\,000 \) Pa
Laminar pore flow
Reynolds number through cake
\( Re = \dfrac{\rho v d_{\text{p}}}{\mu (1 - \varepsilon)} \)
\( Re < 10 \)
If either criterion is violated, the linear t/V relationship breaks down; the cake may compact or the flow may become turbulent, invalidating the simple formula above.
Specific cake resistance (α) quantifies how much a unit mass of solids in the cake resists filtrate flow. A higher α means you need more filter area, longer cycle time, or higher pressure to hit target throughput. Accurate α values prevent under-sized equipment that bottlenecks production or over-sized units that waste capital.
Run a variable-pressure or constant-pressure leaf-filter test with a stirred 47 mm or 76 mm cell. Record filtrate volume vs. time at 2–4 pressure levels spanning your expected range. Plot t/V vs. V; the slope gives α directly when you know cake solids concentration. For highly compressible cakes, repeat at least three pressures to build an α vs. ΔP curve so you can extrapolate to field conditions.
Temperature: Higher temperature lowers filtrate viscosity, reducing α(apparent) by 2–8 % per °C; correct to reference viscosity before scaling.
Particle size: Finer particles pack into tighter pores, raising α roughly with the inverse square of surface-mean diameter; a 50 % size reduction can triple α.
Always test at plant temperature and with representative milled or precipitated solids.
Only for first-pass estimates. Published data are usually for rigid, low-pressure cakes. Real slurries contain salts, surfactants, or flocculants that change porosity and compressibility by an order of magnitude. Treat literature α as ±50 % accuracy; confirm with bench tests before committing to detailed design.
Measure cake solids mass per unit filtrate volume (c) from the lab test.
Insert α, c, μ (viscosity), ΔP, and target filtrate flow Q into the Ruth filtration equation: A² = (μ αc t Q) / (2 ΔP t – μ αc Q t).
Solve for filter area A; add 15 % safety margin for cloth blinding and cake non-uniformity.
If cake is compressible, use the α value that corresponds to the average ΔP on the full-scale unit, not the lab ΔP.
Worked Example: Determining Specific Cake Resistance in a Plate-and-Frame Filter
A specialty-chemical plant needs to polish 200 L of an aqueous pigment slurry. A lab-scale pressure-leaf filter with 100 cm² effective area is tested at 0.5 bar to obtain design data for the full-scale unit. After 4 min 10 s of constant-pressure filtration, the collected filtrate volume is 200 mL. Using the recorded t/V vs. V data, the slope of the filtration plot is 125,000 s·m⁻⁶. Estimate the average specific cake resistance α for the pigment under these conditions.
Knowns
Filtration pressure, ΔP = 500 mbar = 50,000 Pa
Filter area, A = 100 cm² = 0.01 m²
Filtrate viscosity, μ = 1.1 cP = 0.0011 Pa·s
Slurry solids concentration, c = 4.5 kg·m⁻³
Slope of t/V vs. V plot, m = 125,000 s·m⁻⁶
Step-by-step calculation
Convert the slope to the Ruth filtration coefficient:
\[ \frac{\mu c \alpha}{2 A^2 \Delta P} = m \]
Re-arrange to solve for α:
\[ \alpha = \frac{2 A^2 \Delta P}{\mu c} \cdot m \]
Insert the known values:
\[ \alpha = \frac{2 \cdot (0.01)^2 \cdot 50,000}{0.0011 \cdot 4.5} \cdot 125,000 \]
