Microorganisms stay in bioreactor for a significant period of time after they are created. Meanwhile, the microorganisms undergo endogenous respiration that reduces the mass. In addition, dead microorganisms can decay to the debris that can be consumed by other live microorganisms, which leads to mass reduction. Therefore, the observed biosolids yield, Y_{obs}, is always lower than that of the intrinsic yield, Y. The Y_{obs }is the ratio between the total biosolids produced and the total substrate provided and is written as equation (1).
where
Y_{obs } = observed biosolids yield (g MLSS/g COD or BOD)
Q = influent flow rate (L/day)
Q_{x} = mixed liquor removal rate (L/day)
S_{0} = influent COD or BOD (mg/L)
S_{e }= effluent COD or BOD (mg/L)
X_{x} = MLSS or MLVSS of removed mixed liquor (mg/L)
The Y_{obs} can be also estimated from the kinetic parameters. The correlation between Y_{obs} and Y the intrinsic sludge yield, , can be driven assuming biosolids decay is a firstorder reaction.
Rate of biosolids accumulation in system  =  Rate of biosolids production from substrate  –  Rate of biosolids decay  –  Rate of biosolids removal 
———————————–(2)
where
Y = intrinsic sludge yield (g MLSS/g COD or BOD)
k_{d} = firstorder biosolids decay constant (/day)
At steady state, dX/dt equals zero since X in the system is constant. The right side of the equation can be rearranged as equation (3).
By inserting equation (1) and equation (2) into equation (3), the equation below is obtained.
The typical ranges of Y and k_{d} are summarized in Table 1. Four different values exist depending on the unit, but they can be converted each other using the ratio of MLVSS/MLSS and COD/BOD. The COD/BOD ratio is higher than 1.0 in municipal wastewater samples since a portion of substrate converts to microorganisms and does not completely oxidize to CO_{2} during the 5day BOD tests. Nonbiodegradable COD is responsible for the high COD/BOD ratio in some industrial wastewaters. For example, tannery wastewaters have the COD/BOD of approximately 4 (Jenkins, 2004).
Table 1. Typical kinetic parameters for CAS for municipal wastewater
Coefficient  Unit  Range  Typical  Remark 
Y  g MLSS/g COD  0.40.6  0.5  Convertible each other assuming COD/BOD=2 and MLVSS/MLSS=0.8 for municipal wastewater 
g MLVSS/g COD  0.30.5  0.4  
g MLSS/g BOD  0.81.2  1.0  
g MLVSS/g BOD  0.61.0  0.8  
k_{d}  /day  0.0250.075  0.06  Used with MLVSS based Y 
0.020.6  0.05  Used with MLSS based Y  
MLVSS/MLSS  –  0.70.9  0.8  No inorganic coagulant addition was assumed. 
COD/BOD  –  1.252.5^{1)}  2.0 
It is noticeable that the equation (4) is using fixed k_{d} to estimate observed biosolids yields, but k_{d} is variable depending on SRT in practical situations due to the varying degree of nonbiodegradable solids accumulation. As microorganisms are getting aged, they are dying and being reproduced as new microorganisms while the nonrecyclable portions of the biomass such as cell walls are accumulating in mixed liquor. As nonbiodegradable cell debris accumulates, decay rate, k_{d}, should decrease since less portion of biosolids are actually biologically active.
It is noteworthy that the listed Y and k_{d} in Table 1 are estimated based on the experimental results obtained at certain SRT ranges. Therefore, the accuracy of equation (4) diminishes as SRT deviates away from the range of SRT at which k_{d} is obtained. Most known Y and k_{d} in literature are likely obtained from conventional activated sludge process (CAS) and are valid for the SRT common in CAS, e.g. 410 days. Since the common SRT range for MBR is at least 12 days, the accuracy of such equation should not be great unless the Y and k_{d} values measured at a specific MBR condition are used to predict Y_{obs} at slightly different SRT at the same location.
The Y_{obs} of MBR can be more accurately estimated by either empirical equations or activated sludge model (ASM) that takes nonbiodegradable materials into account. Fig. 1 summarizes two empirical curves, one theoretical curve based onASM #1, and one curve based on equation (4). The Y_{obs} predicted by equation (4) agrees well with other Y_{obs} when SRT is less than 10 days, but it starts to deviate significantly from others at higher SRT. ASM#1 predicts higher Y_{obs} than equation (4) because of nonbiodegradable materials accumulated in biosolids. The most direct and accurate correlation might be the empirical curve obtained from well controlled lab experiments (Macomber, 2005). The correlation based on field data (Guildemeister, 2003) results in the highest Y_{obs} in the typical SRT range of 1030 days in MBR. Table 2 summarizes the relation among SRT,Y_{obs} , and F/M.
Fig. 1. Comparison of the correlations predicting biosolids yield.
Table 2. Observed sludge yield and F/M ratio as a function of SRT (Cicek, 2001; Macomber, 2005)
SRT  Y_{obs}  MLVSS/
MLSS 
F/M  
days  g VSS/
g COD 
g MLSS/
g COD 
g COD/
g VSS/d 

2  0.477  0.558  0.855  1.048 
5  0.384  0.461  0.833  0.521 
10  0.329  0.396  0.831  0.304 
20  0.298  0.358  0.832  0.168 
30  0.268  0.328  0.817  0.124 
Effect of inorganic coagulants on biosolids production
If phosphorous discharge limit is tight, inorganic coagulants are often used with or without biological nutrient removal (Lee, 2001). The added aluminium (Al^{3+}) or ferric (Fe^{3+}) ions form insoluble inorganic salts in mixed liquor such as AlPO_{4}, Al(OH)_{3}, AlO(OH), FePO_{4}^{.}2H_{2}O, Fe(OH)_{3}, Ca_{5}PO_{4}(OH)_{2}., etc. Therefore, apparent biosolids production increases and the correlations in Fig. 1 becomes invalid.
© Seong Hoon Yoon