Hollow-fibre modules are immersed in the activated-sludge tank (MLSS). Coarse air bubbles from the diffuser scour the membrane surface while permeate is sucked from inside the fibres.
$$V = \frac{Y_{obs}\,S_0\,Q\,SRT}{X_v},\qquad A_{membrane} = \frac{Q}{J},\qquad L_p = \frac{J}{\Delta P}$$
Reactor volume V, required membrane area A and permeability L_p. Y_obs: observed yield, S_0: influent BOD, X_v: MLVSS (≈0.8·MLSS), J: design flux, ΔP: trans-membrane pressure TMP.
$$Y_{obs} = \frac{Y}{1+k_d\,SRT},\qquad P_{x,vss} = Y_{obs}\,S_0\,Q$$
Apparent yield corrected for SRT and waste-sludge production. We assume Y=0.4 (sludge yield) and k_d=0.05 day⁻¹ (endogenous decay). Longer SRT lowers Y_obs and shrinks the waste-sludge stream.
$$\text{foulingRate} \propto J^{2},\qquad T_{CIP} = \frac{30\ \text{kPa}}{\text{foulingRate}}$$
Pushing the flux up makes fouling progress super-linearly, shortening the time to a +30 kPa TMP rise and therefore the CIP interval. Operating costs change sharply around 15 LMH.