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 flowthrough system a steadystate mass balance around the reactor can be expressed:
[Inflow]  [Outflow] +
[Reaction] = [Accumulation]
All components of the mass balance are expressed as mass rate (M_{i} T^{1}). Mass balance can be expressed for any component of interest (M_{i}). In case of clarifiers, the mass balance will be derived for suspended solids. 
(1) 
[Inflow] = X_{a}(Q_{0}+Q_{r}) 
(2) 
[Outflow] = Q_{e}X_{e} + Q_{w}X_{r} + Q_{r}X_{r} 
(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 (nonsteady) conditions accumulation is significant. Depth of sludge blanket can increase or decrease depending on gradient of flux into the clarifier. 

[Inflow]  [Outflow] + [Reaction] = [Accumulation] 

[X_{a}(Q_{0}+Q_{r})]
 [Q_{e}X_{e} + Q_{w}X_{r} +
Q_{r}X_{r}] + [0] = [0]
The expression Q_{e}X_{e} + Q_{w}X_{r} is sum of mass of suspended solids lost from the clarifier with the effluent and mass wasted. If this mass is not wasted, X_{a} would continuously increase and the system would not be in steady state. By other words, at steady state [Q_{e}X_{e} + Q_{w}X_{r}] 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: 

[X_{a}(Q_{0}+Q_{r})]
 [Q_{r}X_{r}] = 0
By rearranging the equation and substituting Q_{r}/Q_{0} = R 
(5) 
X_{a}(1 + R)
= X_{r} R
Concentration of recycle can be calculated 
(6) 
X_{r} = X_{a}(R+1)/R 
(7) 
By decreasing R, Eq. 7 is followed as
long as X_{r} can increase by thickening. Then the
clarifier is at steady state, what also means that the sludge blanket
level is steady.
Finally 

vX_{a} = vR(X_{r}
 X_{a})
where v is the overflow rate Q_{e}/A. See plot of Eq. 8 in Flux theory. Detailed mathematical development leading to Eq. 8 is shown in Appendix. 
(8) 