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Year: 2009 | Volume: 5 | Issue: 1 | Page No.: 69 - 79

Ibrahim H. Mustafa, G. Ibrahim, Ali Elkamel and A.H. Elahwany

**Abstract**

**Problem Statement: **The activated sludge system needs to improve the operational performance and to achieve more effective control. To realize this, a better quantitative understanding of the biofloc characteristics is required. The objectives of this study were to: (i) Study the biofloc characteristics from kinetics-mass transfer interaction point of view by quantification of the weight of the aerobic portion of the activated sludge floc to the total floc weight. (ii) Study the effect of bulk concentrations of oxygen and nitrates, power input and substrates diffusivity on the portion aerobic portion of the floc. **Approach: **An appropriate mathematical model based on heterogeneous modeling is developed for activated sludge flocs. The model was taking into account three growth processes: Carbon oxidation, nitrification and de-nitrification in terms of four components: substrate, nitrate, ammonia, and oxygen. The model accounts for the internal and external mass transfer limitations and relates the external mass transfer resistance with power input. The floc model equations were two- point boundary value differential equations. Therefore a central finite difference method is employed. **Results: **The percentage aerobic portion increased with increasing with oxygen bulk concentrations and power input and decreases when the bulk concentration of ammonia and substrate increases. Both will compete to consume the internal oxygen by autotrophic and heterotrophic bacteria through aerobic growth processes. The biofloc activity through the profiles was either totally active or partially active. The totally active biofloc is either totally aerobic or aerobic and anoxic together. **Conclusions: **The heterogeneous floc model was able to describe the biofloc characteristics and reflects the real phenomena existing in the activated sludge processes.

• | The environmental conditions such as temperature and pH are
constant^{ [13]}. |

• | Uniform density of the biomass and constant density of heterotrophs
and autotrophs in flocs^{[1]}. |

• | External mass transfer resistance due to boundary layer is considered. |

• | Average floc size is assumed to be constant instead of considering
the floc size distribution in the system.^{ [9, 11]} |

• | Liquid phase is assumed to be mixed completely to keep the oxygen concentration DO constant in each zone in the reactor. |

**Derivation of the biofloc model: **The reactions within the floc matrix were assumed to follow the IAWPRC kinetic model of Henze, *et al*.^{ [4]}. A differential shell of a spherical floc is shown in Fig. 1. In figure the diffusional resistance of dissolved oxygen (C), readily biodegradable substrate (S), nitrate nitrogen (Z) and ammonia- nitrogen (H) inside the floc respectively. The differential equations will be considered in the dimensionless form to reduce the number of parameters, to simplify the solution technique and to be able to perform the appropriate comparison between several models and also for the ease of scale-up of the processes.

Fig. 1: | Floc model |

Where:

N_{i} |
= | The flux or mass transfer rate of component (i) per unit time per unit areaatradius (r) and |

(N_{i} +Δ N_{i}) |
= | That flux at radius (r + Δ r). |

r | = | The variable for floc radius. |

R_{p} |
= | The floc radius. |

Δr | = | The thickness of the differential shell. |

i | = | Refers to the substrate S, nitrate Z,ammonia H and oxygen C. |

**Steady state substrate ( s) mass balance: **Applying a component mass balance on the substrate through a differential element Δr gives

(1) |

Where:

N_{s} |
= | The flux or mass of substrate transportedper unit time per unit area at radius (r) |

N_{s} +Δ N_{s } |
= | The flux or mass of substrate transportedper unit time per unit area at radius (r+ Δr). |

A_{r} |
= | The surface area of the floc of a radius (r). |

A_{r+ Δ r} |
= | The surface area of the floc of a radius (r+Δ r). |

Rs | = | The process rate of substrate Substituting |

For A_{r }= 4п r^{2} and A_{r+} _{Δr }= 4п ( r+Δr)^{2} we get:

(2) |

(3) |

Since

(4) |

Then,

(5) |

But Ficks’ law of diffusion is:

(6) |

Or

(7) |

Where D_{S} is the substrate diffusivity coefficient and substituting in Eq. 5 gives:

(8) |

Equation 8 can be reduced to:

(9) |

With the boundary conditions

(9a) |

(9b) |

Kg_{s} is the mass transfer coefficient of substrate, s_{b}
is the bulk concentration and s_{s }is the surface concentration in
mgl^{-1} Transformation of the equation into dimensionless form using
the following two dimensionless variables:

(10) |

Subject to the following B.Cs

(11) |

Where:

(12) |

**Steady state nitrate (z) mass balance: **Applying a component mass balance on the nitrate through a differential element Δr gives

(13) |

Boundary conditions:

(14) |

(15) |

Where

(16) |

**Steady state ammonia (H) mass balance: **Applying a component mass balance on the ammonia through a differential element Δr gives

(17) |

Boundary conditions:

(18) |

Where:

(19) |

**Steady state oxygen (c) mass balance: **Applying a component mass balance on the oxygen through a differential element Δr gives

(20) |

Boundary conditions:

(21) |

Where

(22) |

S, Z, H and C are dimensional concentrations of substrate, nitrate, ammonia
and oxygen respectively.

Sh_{s}, Sh_{z},, Sh_{h} and Sh_{c} are Sherwood
numbers of substrate, nitrate, ammonia and oxygen respectively.

S_{b}, Z_{b}, H_{b} and C_{b} are bulk concentrations
of substrate, nitrate, ammonia and oxygen respectively.

S_{s}, Z_{s}, H_{s} and C_{s} are surface concentrations
of substrate, nitrate, ammonia and oxygen respectively.

R_{s}, R_{z}, R_{h} and R_{c} reaction rates
associated with floc matrix as defined by Henze, *et al*.^{[4]}
and α is the ratio between K_{CA} and K_{CH }.

**Model parameters: **Table 1, 2 and
3 give the average values of stoichiometric, kinetic, switching
and rheological parameters at neutral pH and 20 ^{0}C for domestic wastewater.
They are based on the IAWPRC task group by Henze *et al*.^{[4]}
as a basic reference in addition to other references shown in the tables. Some
parameter values are dependent on specific factors in the wastewater and on
environmental conditions i.e. the power input is representative of not actual
used plant.

Table 1: | Stoichometric and kinetic parameter values |

**Evaluation of mass transfer coefficients: **It has been found that at high agitation intensities, turbulence is expected to affect mass transfer rate at the biofloc surface. In this case, the concept of local isotropic turbulence may be applied^{[14,15]}. The isotropic turbulence Re-number, Re_{e}, for the floc particle diameter d is given by:

(23) |

Table 2: | Saturation and switching functions |

Table 3: | Rheological properties parameters |

Moo Yoong and Blanch^{ [15]} developed a correlation for rigid surface particle mass transfer in biochemical reactors in terms of the energy input to the system as follows:

(24) |

Where:

The mass transfer coefficient (k_{L}) is seen to be dependent on (P/V)^{ 1/4} which can be expressed by the effect of power input on interfacial area^{[15]}. These relations are used to calculate the mass transfer coefficients of the considered four components as a function of the power input.

**Solution technique: **The floc model equations describing the diffusional limitations inside the flocs are two- point boundary value differential equations. Therefore a central finite difference method was employed. In the floc model a large number of points are used inside this floc to give a better accuracy.

**RESULTS **

The weight of aerobic portion (zone) to the total floc, known as the percentage aerobic ratio, was studied as a function of the bulk liquid concentrations of the components: substrate, ammonia, oxygen, ammonia and nitrate, besides energy input and substrate diffusivities. Figure 2 shows the effect of floc size on the percentage aerobic ratio at the corresponding bulk concentrations. In Fig. 2, the floc is totally aerobic when its size is very small, where the percentage aerobic ratio is constant at 100% as shown in the horizontal line (AB). In the part (BC) the percentage aerobic ratio continuously decreases as the floc size increases.

Fig. 2: | Effect of the floc size on the % |

Fig. 3: | Profiles of nitrates, ammonia, substrate and oxygen along
the floc radius at C_{b}=10, S_{b}=60, H_{b}=0.4,
and Z_{b}=0.1 mg l^{-l} |

Figure 3 gives an example for biofloc profiles of totally active totally aerobic where the bulk concentration of oxygen is very high, so that its internal concentration was not completely consumed through the floc, where it decreases from 85% at the surface to 55% at the center. The nitrate profile shows that the internal nitrate is continuously produced through the floc, where its internal concentration increases from 183% at the surface to 350% at the center.

The substrate profile shows that the percentage internal concentration of substrate reduced from 73% at the surface to a very small value approaching zero at the center. The ammonia profile behaves nearly the same as the substrate profile, where there is no large difference between them.

Figure 4 gives an example for biofloc profiles of totally active but aerobic and anoxic, the aerobic portion represents 16% of the floc and the anoxic represents the rest. It is shown in the figure that the percent of the internal nitrate concentration to its bulk concentration increases from 113% at the floc surface to 119.1% at the limit of the aerobic portion. It is shown from the substrate and oxygen profiles that their internal concentrations decreased. In Fig. 4, it is shown that nitrate, in the anoxic zone, was reduced by the denitrifying bacteria until reaching the floc center where its internal concentration becomes 76% of the bulk concentration.

Fig. 4: | Profiles of nitrate, ammonia, substrate and oxygen concentrations
along the floc radius at C_{b}=5, H_{b}=3, S_{b}=75,
Z_{b}=2.5mg l^{-l} |

Fig. 5: | Profiles of nitrate, ammonia, substrate, and oxygen concentrations
at along the floc radius S_{b}=45, Z_{b}=1.5, H_{b}=1.5,
and C_{b}=5 mg l^{-l} |

In the ammonia profile the internal concentration decreases from 80% at the surface to 64 % at the limit of the aerobic portion then becomes nearly constant through the anoxic zone until the floc center. The internal substrate concentration reduced through both aerobic and anoxic zones from 76.3% at the surface to 1% at the center. Figures 5 and 6 represent examples for the partially active biofloc. In Fig. 5, the active portion of the floc represents 80% of the floc through the anoxic zone then nearly remains constant in the inactive portion.

Fig. 6: | Profiles of nitrate, ammonia, substrate, and oxygen at concentrations
at along the floc radius S_{b}=600, Z_{b}=0.1, H_{b}=0.15,
and C_{b}=2 mg l^{-l} |

Fig. 7: | Effect of C |

It is shown that the percentage internal concentration of ammonia decreases from 72% at the surface to 31% at the limit of the aerobic zone, then it remains constant in the inactive portion. Figure 6 shows that the internal oxygen and nitrate were completely consumed before reaching the floc center. Ammonia concentration decreased from 75.2% at the surface to 56.7% at the center of the floc then remains constant.

Fig. 8: | Effect of C_{b }on aerobic ratio at different H_{b
}values |

Fig. 9: | Effect of C |

Figure 7, 8 and 9 show
the effect of the change of bulk concentrations of oxygen on the percent of
the aerobic one of the total floc (known as the percentage aerobic ratio). Figure
7 shows that the percentage aerobic ratio increases as the bulk concentration
of oxygen increases but it decreases as the bulk concentration of substrate
increases from 60 to 1200 mg l^{-l} because a substrate works as an
electron donor. Figure 8 shows the effect of the change of
bulk concentration of oxygen at different ammonia bulk concentrations.

Fig. 10: | Effect of Z_{b} on aerobic ratio at different C_{b}
values |

Fig. 11: | Effect of power input on aerobic ratio at different Z_{b
}values |

Figure 9 shows the effect of C_{b} at different Z_{b}.
It is shown that the percentage aerobic ratio will not change as Z_{b}
increases from 1 to 35 mg l^{-l}.Figure 10 shows
the response of the aerobic zone due to change in Z_{b}. As expected,
the percentage aerobic ratio has not changed effectively due to the change of
(Z_{b}). Figure 11 shows the effect of the power
input (energy) on the percentage aerobic ratio at different oxygen bulk concentrations.

Fig. 12: | Effect of substrate diffusivity on aerobic ratio at different
C_{b }values |

It is clear that the percentage aerobic ratio increases sharply when the power input increases more than 200 W m-^{3}_{. }Then as the power input is larger than 200 W/m^{3}, the percentage aerobic increases very slightly, and then it becomes stable. Furthermore, Fig. 11 shows that the percentage aerobic ratio increases when oxygen bulk concentrations increases from 1 to 10 mg ^{-l}. Figure 12 shows the effect of substrate diffusivity on the percentage aerobic ratio at different bulk concentrations of oxygen (C_{b}). At C_{b}=5 and 6.75 mg ^{-l}, the percentage aerobic ratio seems to be large like 63.8 and 79% respectively for very small values of (Ds), then it decreases as (Ds) increases.

**DISCUSSION**

The percentage aerobic portion decreases when the floc size increases due to
exhibition of aerobically inactive zone in the floc (Fig. 2).
The horizontal line (AB) which is close to the center of the floc and represents
100% aerobic ratio confirms the assumption that the concentration of dissolved
oxygen is identical within the bacterial colonies whereas the central bacteria
in the colony may be subjected to even higher dissolved oxygen deficits due
to the diffusional resistances within the bacterial colony. The decrease of
the percentage aerobic ratio in the part (BC) as the increase of the floc size
is due to exhibition of aerobically inactive zones in the floc. The part (BC)
of the figure predicts a hyperbolic decrease in the percentage aerobic ratio
due to the dissolved oxygen deficits as one moves away from the center of the
floc. When the bulk concentrations change, the curve had the same shape but
the part (AB) may be shifted to the right or the left according to the available
bulk conditions. This figure is similar to that obtained by Smith P. G. and
Coakley^{[16]} who studied the predicted oxygen deficits in a 40 μm
diameter floc. In the case of BOD removal, it is well known that the concentration
gradient in the flocs permits more efficient substrate utilization. A similar
phenomenon appears to operate for ammonia. In Fig. 3, 4,
5 and 6, oxygen works as the electron acceptor
in the aerobic portion and substrates and ammonia work as an electron donor,
however in the anoxic portion, the denitrification reactions occurs where the
DO concentration is at low levels, so that nitrate nitrogen plays the role of
an electron acceptor and ammonia works as an electron donor. In Fig.
3, the DO at the center of the floc represents 50% of that at the surface,
so that the floc appears as totally active aerobic. However; the DO in Figure
is consumed through the aerobic portion existing in the range 0.8-1 of the floc
interior. The percent of the internal nitrate concentration increases due to
oxidation of ammonia by autotrophs, where the ammonia bulk concentration exists
in great amounts. Then the anoxic or denitrification portion initiates where
nitrate works as a terminal electron acceptor instead of oxygen producing nitrogenous
compounds and nitrous compounds

In Fig. 5, the floc is totally active which is aerobic and anoxic where nitrate is produced as a result of oxidizing of the internal ammonia by autotrophic bacteria and anoxic where nitrate is working as electron acceptor. However, the biofloc in Fig. 6 is partially active where both of the oxygen and nitrates are consumed completely before reaching the center. Our resuts agree with that obtained experimentally by Suwa *et al*.^{[23]} who showed that the denitrification reactions occur when the DO concentration is at low levels in the interior portion of the flocs.

The percentage aerobic ratio increases with increasing of oxygen bulk concentration
and it decreases with increasing substrate and ammonia bulk concentrations (Fig.
7, 8 and 9). This is because oxygen
works as an electron acceptor and both substrates and ammonia as electron donors.
The results in Fig. 7, 8 and 9
agree with the results obtained by Muller *et al*.^{[20]} who showed
that the oxygen transfer rate became low at high substrate and ammonium concentration.
Further more, this compatible with the results obtained by Smith^{[21]}
who showed that the stabilization rate of substrates is proportional to the
dissolved oxygen concentration. Furthermore our results agree with Baillod and
Boyle (1970)^{[22]} who showed that substrate uptake decreased through
a dissolved oxygen concentration range between 4-5 mg.l^{-1} for the
flocculated sludge, yet the critical oxygen level dropped to 1-2 mgl^{-1}
for the blended sludge. Ammonia behavior in Fig. 8 is similar
to the behavior of substrate shown in Fig. 6, but autotrophic
bacteria consumed ammonia (Fig. 8) and heterotrophic bacteria
consumed the substrate (Fig. 6).

In Fig. 9 and 10, the nitrates bulk concentration has no effect on the substrates or ammonia or the oxygen concentration inside the aerobic zone. This is because the bioflocs are enriched with the bulk concentration of oxygen. Its effect exists in the anoxic zone only as an electron acceptor. These results are in agreement with M. Kornaros and G Lyberatos^{[17] }who showed that nitrate and nitrite concentrations remained almost constant throughout the aerobic growth and the cell growth rate did not seem to increase significantly (aerobic lag phase), while nitrate and nitrite reducing activity ceased immediately after the exposure to high dissolved oxygen concentrations.

The percentage aerobic ratio in Fig. 11 increases because the increase of the power input leads to increase of the mass transfer coefficient of oxygen rather than other components. Then when the transfer rate of oxygen is saturated the percentage aerobic increases very slightly and becomes stable. However, the percentage aerobic ratio in Fig. 12 decreases when substrate diffusivity increases because more oxygen is consumed inside the aerobic shell and when the oxygen bulk concentration is not enough the percentage aerobic ratio will be very low and seems to be unchanged.

**CONCLUSION**

In this study, a mathematical model was developed for activated sludge floc. The model takes into account three growth processes: {carbonaceous oxidation, nitrification and denitrification} and four components {substrate, ammonia, nitrate and oxygen}. The model accounts for the internal and external mass transfer limitations and relates the external mass transfer resistance with power input. The model is able to describe the effect of liquid bulk concentrations on biofloc characteristics in terms of aerobic weight to the total floc defined as the percentage aerobic ratio. The model was used to study the effect of diffusivity of different substrates was studied. It was found that the percentage aerobic portion increases with increasing with oxygen bulk concentration and power input.

The percentage aerobic portion decreases when the bulk concentration of ammonia and substrate increases. Both will compete to consume the internal oxygen by autotrophic and heterotrophic bacteria through aerobic growth processes.

The percentage aerobic portion was found to be unchanged due to an increase in nitrate bulk concentration. This is compatible with the experimental study by Kornaros *et al*.^{[17]}; Azimi and Horan^{[18]}. It was found also that the percentage aerobic ratio increases with increasing substrate diffusivity. The biofloc activity was studied through the profiles of the above four components. It might be totally active or partially active. The totally active biofloc is either totally aerobic or aerobic and anoxic together. In aerobic activity of the biofloc, the nitrate production was initiated due to the autotrophic reactions of ammonia but in the anoxic activity, the nitrate was consumed due to the denitrification reactions.

Finally, the aerobic portion was found to be more sensitive to changes of bulk concentrations of oxygen, substrate and ammonia in addition to the power input and substrate diffusivity and less sensitive to changes in nitrate bulk concentration.

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