DEFINITIONS & MASS BALANCE

System definition

The system is defined in Fig. 1. (see Notation). Figure 1: Definition scheme of Activated Sludge Process

Mass Balance

For any flow-through system a steady-state mass balance around the reactor can be expressed:

 [Inflow] - [Outflow] + [Reaction] = [Accumulation] All components of the mass balance are expressed as mass rate (Mi T-1). Mass balance can be expressed for any component of interest (Mi). In case of clarifiers, the mass balance will be derived for suspended solids. (1) [Inflow] = Xa(Q0+Qr) (2) [Outflow] = QeXe + QwXr + QrXr (3) [Reaction] = 0 (there are reactions occurring in clarifier which can decrease or increase mass of suspended solids, e.g. respiration, denitrification etc. but their influence on mass balance is negligible). [Accumulation] = 0 (At steady state, accumulation is zero. Under dynamic (non-steady) conditions accumulation is significant. Depth of sludge blanket can increase or decrease depending on gradient of flux into the clarifier. [Inflow] - [Outflow] + [Reaction] = [Accumulation] [Xa(Q0+Qr)] - [QeXe + QwXr + QrXr] +  =  The expression QeXe + QwXr is sum of mass of suspended solids lost from the clarifier with the effluent and mass wasted. If this mass is not wasted, Xa would continuously increase and the system would not be in steady state. By other words, at steady state [QeXe + QwXr] is equivalent to the increase of mass in the reactor due to the reaction with substrate (growth of biomass). (4) Steady state mass balance equation is: [Xa(Q0+Qr)] - [QrXr] = 0 By rearranging the equation and substituting Qr/Q0 = R (5) Xa(1 + R) = Xr R Concentration of recycle can be calculated (6) Xr = Xa(R+1)/R (7) By decreasing R, Eq. 7 is followed as long as Xr can increase by thickening. Then the clarifier is at steady state, what also means that the sludge blanket level is steady. Finally vXa = vR(Xr - Xa) where v is the overflow rate Qe/A. See plot of Eq. 8 in Flux theory. Detailed mathematical development leading to Eq. 8 is shown in Appendix. (8)  