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 directly relates to the total resistance in cake filtration, making it 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:
\[ \alpha = \frac{2 \cdot (0.01)^2 \cdot 50,000}{0.0011 \cdot 4.5} \cdot 125,000 \]
