**2.1 Modeling of the activated sludge basin**

In broad terms, the wastewater treatment plant's activated sludge basin will be modeled as a plug flow reactor, with the key underlying assumption being that while there will be longitudinal variances along the length of the reactor, there will not be any vertical or latitudinal variances (i.e. for a given differential volume, it will be assumed to be completely mixed).

As seen in the folowing figures, the activated sludge basins are modeled as being a plug flow reactor (PFR) with a specified length, l, cross-sectional area, A, and volumetric flow rate, Q.

This process is defined as being a two-phase system, consisting of the solids phase and the aqueous phase. Inherent to this definition is the presumption that mass transfer of the target compounds to/from the gas phase due to deposition/volatilization is negligible based on their chemical properties (primarily pKa and Henry's Law coefficient).

Fig. 2. Schematic of a plug flow reactor (PFR)

372 Mass Transfer - Advanced Aspects

The examination of concurrent sorption and biodegradation in environmental systems is not actually a recent development. The method of volume averaging described by Hassanizadeh and Gray (1979) has frequently been used to examine contaminant transport in porous media. These analyses proceed by defining the groundwater system as containing two phases, the aqueous phase (groundwater) and the solid phase (soils). Of note, in some cases a more elaborate system could be defined with the solid phase actually consisting of multiple phases (i.e. an organic phase along with an inorganic phase), however it is the corollary with the two-phase system that will be used here. In the groundwater systems, the solid phase is stationary with the aqueous phase moving through it, as indicated below in

Fig. 1. Representation of porous media characteristic of groundwater systems

The analysis begins with a microscopic differential volume that is then expanded through volume averaging, with the ultimate result being the series of advection-dispersion-reaction (ADR) equations that are frequently encountered in groundwater research. This underlying methodology will now be applied to an activated sludge basin, with the key difference being that the solid phase will move in conjunction with the aqueous phase, rather than

In broad terms, the wastewater treatment plant's activated sludge basin will be modeled as a plug flow reactor, with the key underlying assumption being that while there will be longitudinal variances along the length of the reactor, there will not be any vertical or latitudinal variances (i.e. for a given differential volume, it will be assumed to be completely

As seen in the folowing figures, the activated sludge basins are modeled as being a plug flow reactor (PFR) with a specified length, l, cross-sectional area, A, and volumetric flow rate, Q. This process is defined as being a two-phase system, consisting of the solids phase and the aqueous phase. Inherent to this definition is the presumption that mass transfer of the target compounds to/from the gas phase due to deposition/volatilization is negligible based on

their chemical properties (primarily pKa and Henry's Law coefficient).

**2. Model development** 

Figure 1.

being stationary.

mixed).

**2.1 Modeling of the activated sludge basin** 

Having defined it as a two-phase system, the initial focus will be on the differential volume indicated by the ΔZ (and multiplied by A) in Figure 3. This differential volume is shown more clearly in the following figure.

Fig. 3. Close-up of the differential volume

For this volume, a mass balance on the target compound in both phases is performed. First, as a word formulation and then as the mathematical representation of that phase:

#### **2.1.1 Compound mass in the aqueous phase in the differential volume**

(Change) = (Flow in with aqueous phase) – (Flow out with aqueous phase) + (flux from aqueous phase to solid phase) – (biodegradation in aqueous phase)

$$\frac{dm}{dt} = \dot{m}\_{in} - \dot{m}\_{out} + m\_{sorb\\_flux} - \dot{m}\_{bio} \tag{1}$$

Because the differential volume is extremely small, it is assumed to be at steady state, and so the overall change is mass is zero.

$$0 = \left. Q\mathbf{C} \right|\_{\mathbf{Z}} - \left. Q\mathbf{C} \right|\_{\mathbf{Z} + \Delta \mathbf{Z}} + V\_{aq} j\_{sl} - V\_{aq} r\_{bio} \tag{2}$$

Microcontaminant Sorption and Biodegradation in Wastewater Modeled as a Two-Phase System 375

*Z Z ls Z Z ls <sup>Z</sup> X S XS A A d XS*

Because the solids concentration varies minimally throughout the reactor (varying by less than 3% from one end to the other), it can effectively be treated as a constant, resulting in:

At this point it can be seen that there are two governing equations for a target compound in

At this point, two additional assertions can be made to simplify the governing equations and provide a unified theory encompassing transport in both phases. First, it should be noted that mass is conserved throughout the inter-phase mass transfer flux. Because of this,

0 *aq*

Equation 11 can then be combined with equation 10 to yield:

*V*

<sup>−</sup> = ⇒=

+Δ +Δ Δ →

*<sup>X</sup> <sup>j</sup> <sup>X</sup> <sup>j</sup> ZQ dz Q*

*ls*

*sl bio*

*ls*

*sl ls sl ls*

*sl*

*bio*

*<sup>j</sup> Xj j Xj <sup>V</sup>*

*dC dS A X r dz dz Q*

The second assertion to be made is in regards to the sorption mechanism occurring. If the assumption is made that sorption occurs very rapidly, is linear, and is essentially at

> *d dS dC <sup>K</sup>*

( ) 1 *<sup>d</sup> bio <sup>d</sup> bio dC dC A dC A KX r KX r dz dz Q dz Q*

equilibrium (as shown in Ottmar et al. 2010), then the following relation holds:

Taking the derivative with respect to movement through the reactor yields:

This can then be substituted into equation 13, giving:

0

the two different phases present in the PFR.

the following equation holds:

Adding equation 12 to equation 5 gives:

lim *ZZZZ Z <sup>Z</sup>*

+Δ +Δ

( )

*dS A X Xj dz Q*= (10)

*dC A A j r dz Q Q* = − (5)

*dS A X Xj dz Q*= (10)

= + ⇒ =− (11)

*dS A <sup>X</sup> <sup>j</sup> dz Q* = − (12)

+ =− (13)

*S KC* = *<sup>d</sup>* (14)

*dz dz* <sup>=</sup> (15)

+ =− ⇒ + =− (16)

<sup>Δ</sup> (9)

Where C is the concentration in the aqueous phase (μg/L), *Vaq* is the aqueous phase volume, *jsl* is the mass flux to the solid phase (μg/L-time) and *rbio* is the biodegradation rate expression (μg/L-time)

At this point, it is important to highlight the differences in the volumes presented so far. *V* is the total differential volume and *Vaq* is the volume of the aqueous phase within that differential volume, and *Vs* is the volume of the solid phase.

$$V = V\_{aq} + V\_s \Longrightarrow A\Delta Z = V\_{aq} + V\_s \tag{3}$$

A further examination of the activated sludge system allows for a simplifying assumption in regards to the aqueous phase volume. The average solids concentration in an activated sludge tank is 3,250 mg/L for completely mixed tanks (Metcalf and Eddy 1991). These solids have a specific gravity of 1.25, meaning that in one liter of activated sludge, the 3,250 mg of solids will have a volume of 2.6 milliliters. Consequently, the aqueous volume (997.4 milliliters) is almost equal to the total volume (1,000 milliliters).

Returning to equation 2, by dividing both sides by Q and V results in:

$$\frac{\left.\mathbf{C}\right|\_{Z+\Delta Z} - \left.\mathbf{C}\right|\_{Z}}{\Delta Z} = \frac{A}{Q}j\_{sl} - \frac{A}{Q}r\_{bio} \tag{4}$$

Taking the limit as the differential length approaches zero yields:

$$\lim\_{\Delta Z \to 0} \frac{\mathbb{C}|\_{Z + \Delta Z} - \mathbb{C}|\_Z}{\Delta Z} = \frac{A}{Q} j\_{sl} - \frac{A}{Q} \eta\_{bio} \Rightarrow \frac{d\mathbb{C}}{dz} = \frac{A}{Q} j\_{sl} - \frac{A}{Q} \eta\_{bio} \tag{5}$$

Which is the PFR governing equation for the aqueous phase. The process is repeated for the solid phase.

#### **2.1.2 Compound mass in the solid phase in the differential volume**

(Change) = (Flow in with solid phase) – (Flow out with solid phase) + (flux from solid phase to aqueous phase)

$$\frac{dm}{dt} = \dot{m}\_{in} - \dot{m}\_{out} + m\_{sorb\\_flux} \tag{6}$$

Again, because the differential volume is extremely small, it is assumed to be at steady state, and so the overall change in mass is zero.

$$0 = QX\big|\_{\mathbb{Z}} \operatorname{S}|\_{\mathbb{Z}} - QX\big|\_{\mathbb{Z}+\Lambda\mathbb{Z}} \operatorname{S}|\_{\mathbb{Z}+\Lambda\mathbb{Z}} + VX\big|\_{\mathbb{Z}+\Lambda\mathbb{Z}} j\_{\operatorname{ls}} \tag{7}$$

Where *X* is the solids concentration (kg/L), S is the compound concentration in the solid phase (μg/kg solids), and *jls* is the mass flux to the aqueous phase (μg/kg solids-time). Dividing both sides by Q and V results in:

$$\frac{\left.\left.\left[X\right]\_{Z+\Delta Z}S\right|\_{Z+\Delta Z}-\left.X\right|\_{Z}S\right|\_{Z}}{\Delta Z} = \frac{\left.\left.\left.\left.\left.\right|\_{Z}S\right|\_{Z}\right|\_{Z}}{\left.\left.\right|\_{Z+\Delta Z}j\_{ls}\right|}\tag{8}$$

Again, taking the limit as the differential length approaches zero yields:

$$\lim\_{\Delta Z \to 0} \frac{\left. \mathbf{X} \right|\_{Z + \Delta Z} \left. \mathbf{S} \right|\_{Z + \Delta Z} - \left. \mathbf{X} \right|\_{Z} \left. \mathbf{S} \right|\_{Z}}{\Delta Z} = \frac{A}{Q} \mathbf{X} \Big|\_{Z + \Delta Z} j\_{\rm ls} \Rightarrow \frac{d(\mathbf{X} \mathbf{S})}{dz} = \frac{A}{Q} \mathbf{X} \Big|\_{Z + \Delta Z} j\_{\rm ls} \tag{9}$$

Because the solids concentration varies minimally throughout the reactor (varying by less than 3% from one end to the other), it can effectively be treated as a constant, resulting in:

$$X\frac{dS}{dz} = \frac{A}{Q}Xj\_{ls} \tag{10}$$

At this point it can be seen that there are two governing equations for a target compound in the two different phases present in the PFR.

$$\frac{d\mathbf{C}}{dz} = \frac{A}{Q}\mathbf{j}\_{sl} - \frac{A}{Q}\mathbf{r}\_{bio} \tag{5}$$

$$X\frac{dS}{dz} = \frac{A}{Q}Xj\_{ls} \tag{10}$$

At this point, two additional assertions can be made to simplify the governing equations and provide a unified theory encompassing transport in both phases. First, it should be noted that mass is conserved throughout the inter-phase mass transfer flux. Because of this, the following equation holds:

$$0 = \frac{V\_{aq}}{V}j\_{sl} + Xj\_{ls} \Longrightarrow j\_{sl} = -Xj\_{ls} \tag{11}$$

Equation 11 can then be combined with equation 10 to yield:

$$X\frac{dS}{dz} = -\frac{A}{Q}j\_{sl} \tag{12}$$

Adding equation 12 to equation 5 gives:

374 Mass Transfer - Advanced Aspects

Where C is the concentration in the aqueous phase (μg/L), *Vaq* is the aqueous phase volume, *jsl* is the mass flux to the solid phase (μg/L-time) and *rbio* is the biodegradation rate

At this point, it is important to highlight the differences in the volumes presented so far. *V* is the total differential volume and *Vaq* is the volume of the aqueous phase within that

A further examination of the activated sludge system allows for a simplifying assumption in regards to the aqueous phase volume. The average solids concentration in an activated sludge tank is 3,250 mg/L for completely mixed tanks (Metcalf and Eddy 1991). These solids have a specific gravity of 1.25, meaning that in one liter of activated sludge, the 3,250 mg of solids will have a volume of 2.6 milliliters. Consequently, the aqueous volume (997.4

*V V V AZ V V* = *aq* + ⇒Δ= + *s aq <sup>s</sup>* (3)

+Δ <sup>−</sup> = − <sup>Δ</sup> (4)

=− + (6)

0 *QX S QX S VX ZZ ZZZZ ZZ ls j* +Δ +Δ +Δ =− + (7)

*Z Z ls*

<sup>−</sup> <sup>=</sup> <sup>Δ</sup> (8)

+Δ

<sup>Δ</sup> (5)

*sl bio*

differential volume, and *Vs* is the volume of the solid phase.

milliliters) is almost equal to the total volume (1,000 milliliters). Returning to equation 2, by dividing both sides by Q and V results in:

Taking the limit as the differential length approaches zero yields:

**2.1.2 Compound mass in the solid phase in the differential volume** 

*dm*

*dt*

0

Δ →

and so the overall change in mass is zero.

Dividing both sides by Q and V results in:

solid phase.

to aqueous phase)

lim *ZZ Z*

+Δ

*ZZ Z*

*C C A A j r Z QQ*

*sl bio sl bio <sup>Z</sup> C C A A dC A A <sup>j</sup> <sup>r</sup> <sup>j</sup> <sup>r</sup> Z Q Q dz Q Q*

Which is the PFR governing equation for the aqueous phase. The process is repeated for the

(Change) = (Flow in with solid phase) – (Flow out with solid phase) + (flux from solid phase

.. . *in out sorb* \_ *flux*

*mm m*

Again, because the differential volume is extremely small, it is assumed to be at steady state,

Where *X* is the solids concentration (kg/L), S is the compound concentration in the solid phase (μg/kg solids), and *jls* is the mass flux to the aqueous phase (μg/kg solids-time).

*X S XS <sup>A</sup> X j Z Q*

*ZZZZ Z Z*

+Δ +Δ

Again, taking the limit as the differential length approaches zero yields:

<sup>−</sup> = − ⇒= −

expression (μg/L-time)

$$\frac{d\mathbf{C}}{dz} + X\frac{d\mathbf{S}}{dz} = -\frac{A}{Q}r\_{bio} \tag{13}$$

The second assertion to be made is in regards to the sorption mechanism occurring. If the assumption is made that sorption occurs very rapidly, is linear, and is essentially at equilibrium (as shown in Ottmar et al. 2010), then the following relation holds:

$$S = K\_d \mathcal{C} \tag{14}$$

Taking the derivative with respect to movement through the reactor yields:

$$\frac{dS}{dz} = \mathbf{K}\_d \frac{d\mathbf{C}}{dz} \tag{15}$$

This can then be substituted into equation 13, giving:

$$\frac{d\mathbf{C}}{dz} + K\_d X \frac{d\mathbf{C}}{dz} = -\frac{A}{Q} r\_{\text{bio}} \Rightarrow (1 + K\_d X) \frac{d\mathbf{C}}{dz} = -\frac{A}{Q} r\_{\text{bio}} \tag{16}$$

Microcontaminant Sorption and Biodegradation in Wastewater Modeled as a Two-Phase System 377

( ) *<sup>L</sup> dC k X L z dz C z vR K*

At this point, the challenge is in finding the appropriate mathematical expression for how the concentration of substrate (BOD or COD) changes as it moves through the reactor. The simplest model is that of a linear decrease from the concentration entering the reactor, *L0*, to

<sup>1</sup> ( ) <sup>1</sup>

⎛ ⎞ =− + ⎜ ⎟ ⎝ ⎠

( )

*L*

*f*

0 0

*L f Integration*

*L f*

1

0 0

*dC k X K L L L z dz*

⎛ ⎞ ⇒ =− + − − ⎜ ⎟ ⎝ ⎠

( ) ( )

2

Applying the initial condition that at the beginning of the reactor, when *z* = 0, *C* = *C0*, it can be seen that *B* = *C0*, which results in the following governing equation for the aerobic basins:

> ( ) ( ) <sup>1</sup> <sup>2</sup> 0 0 1 2

*k X K Lz L Lz*

*L f*

( ) ( )

*L f*

1 2

<sup>1</sup> <sup>2</sup> 0 0

*k X K Ll L Ll K vR l*

<sup>⎡</sup> <sup>⎤</sup> − +− − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

<sup>1</sup> 0 0

*k Xl KL LL*

<sup>⎡</sup> <sup>⎤</sup> − +− − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

<sup>1</sup> <sup>0</sup>

From this governing equation, the aqueous drug concentration at the end of the reactor, can then be determined. The mass of drug compound in the sorbed phase is calculated from the

*k X KLL*

⎛ ⎞ − ++ ⎜ ⎟ ⎝ ⎠

( ) ( )

*L f*

1 1 2 2

*L f*

1 2

<sup>⎡</sup> <sup>⎤</sup> − +− − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>=</sup> (29)

1 2 0 0

⎡ ⎤ =− + − − + ⎢ ⎥ ⎣ ⎦

*k X C K Lz L Lz Const*

⎝ ⎠

*L L Lz dC k X <sup>l</sup> dz*

1

⎛ ⎞ − − ⎜ ⎟

( )

(25)

(27)

(28)

(30)

0 0 ( ) <sup>1</sup> ( ) *Lz L L Lf <sup>l</sup>* =− − (26)

Rearranging equation 24 and configuring it for integration yields:

1

=− +

( )

( )

<sup>1</sup> ln

*L*

*L*

1

( ) ( ) <sup>1</sup> <sup>2</sup> 0 0

*k X K Lz L Lz K vR l*

⎡ ⎤ − +− − ⎢ ⎥ ⎣ ⎦

*vRK l*

0

0

*C Ce*

*C Ce*

*C Ce*

=

⇒ =

⇒ =

equilibrium sorption condition by means of the following equations:

0

0

1 2

*L f*

*L*

*L*

*K vR <sup>l</sup> C Ce*

From this, the concentration at the end of the PFR can be calculated by setting *z* = l, and:

*L*

*K vR*

*L*

*K R*

θ

*L*

*C z vRK l*

1

*C z vR K*

the concentration leaving the reactor, *Lf*:

Substituting this back into equation 25 gives:

This equation can then be integrated, giving:

*C Be*

⇒ =

Then by defining that (1 + *KdX*) = *R*, where *R* is the retardation factor, equation 16 ultimately reduces to:

$$R\frac{d\mathbf{C}}{dz} = -\frac{A}{Q}r\_{bio} \Rightarrow \frac{d\mathbf{C}}{dz} = -\frac{1}{vR}r\_{bio} \tag{17}$$

Where *v* is the linear velocity along the length of the reactor as defined by *Q*/*A*. At this point, the next step is to further examine *rbio*, the biodegradation rate term. One of the most common approaches is to model biodegradation under a first-order process. This is manifested by:

$$
\eta\_{\text{bio}} = k\_1 \mathbf{C}(z) \tag{18}
$$

Adding equation 18 back into equation 17 gives:

$$\frac{d\mathbf{C}}{dz} = -\frac{k\_1}{vR}\mathbf{C}(z)\tag{19}$$

This expression can then be integrated:

$$\int \frac{d\mathbf{C}}{\mathbf{C}(z)} = -\frac{k\_1}{v\mathcal{R}} \int dz \implies \ln|\mathbf{C}| = -\frac{k\_1}{v\mathcal{R}} z + \text{Const}\_{\text{Integration}} \implies \mathbf{C} = B e^{-\frac{k\_1}{v\mathcal{R}}z} \tag{20}$$

By applying the initial conditions that when *z* = 0, *C* = *C0*, the equation can then be defined:

$$\mathbf{C}\_0 = \mathbf{B} \Longrightarrow \mathbf{C} = \mathbf{C}\_0 e^{-\frac{k\_1}{v\mathcal{R}}z} \tag{21}$$

For the final step, the concentration at the end of the basin can be calculated by setting *z* = l.

$$C = C\_0 e^{-\frac{k\_1}{\upsilon R}l} \Rightarrow C\_0 e^{-\frac{k\_1 \theta}{R}} \tag{22}$$

Where *θ* is the hydraulic retention time, which is equal to the length divided by the linear fluid velocity.

As has been observed experimentally, however, biodegradation sometimes does not quite appear to follow true first-order kinetics, but rather, a sort of substrate-enhanced process. One of the challenges with this is developing a relevant mathematic model that is grounded in physical principles and observations. To this end, the following expression is proposed for the biodegradation rate:

$$r\_{bio} = k\_1 X C(z) \left( 1 + \frac{L(z)}{K\_L} \right) \tag{23}$$

Where *L(z)* is the concentration of substrate (BOD or COD) present and *KL* is the Monod half saturation coefficient. Substituting this equation into equation 17 gives:

$$\frac{d\mathbf{C}}{dz} = -\frac{k\_1 X}{vR} \mathbf{C}(z) \left( 1 + \frac{L(z)}{K\_L} \right) \tag{24}$$

Rearranging equation 24 and configuring it for integration yields:

$$\frac{d\mathbf{C}}{d\mathbf{C}(z)} = -\frac{k\_1 X}{v\mathcal{R}} \left( 1 + \frac{L(z)}{K\_L} \right) dz \tag{25}$$

At this point, the challenge is in finding the appropriate mathematical expression for how the concentration of substrate (BOD or COD) changes as it moves through the reactor. The simplest model is that of a linear decrease from the concentration entering the reactor, *L0*, to the concentration leaving the reactor, *Lf*:

$$L(z) = L\_0 - \frac{1}{l} (L\_0 - L\_f) \tag{26}$$

Substituting this back into equation 25 gives:

376 Mass Transfer - Advanced Aspects

Then by defining that (1 + *KdX*) = *R*, where *R* is the retardation factor, equation 16 ultimately

*dC A dC Rr r dz Q dz vR*

Where *v* is the linear velocity along the length of the reactor as defined by *Q*/*A*. At this point, the next step is to further examine *rbio*, the biodegradation rate term. One of the most common approaches is to model biodegradation under a first-order process. This is

<sup>1</sup> ( ) *dC k C z*

*dC k <sup>k</sup> dz C z Const C Be*

By applying the initial conditions that when *z* = 0, *C* = *C0*, the equation can then be defined:

0 0

*C B C Ce vR*

For the final step, the concentration at the end of the basin can be calculated by setting *z* = l.

0 0 *<sup>k</sup> <sup>k</sup> <sup>l</sup> C Ce Ce vR <sup>R</sup>*

Where *θ* is the hydraulic retention time, which is equal to the length divided by the linear

As has been observed experimentally, however, biodegradation sometimes does not quite appear to follow true first-order kinetics, but rather, a sort of substrate-enhanced process. One of the challenges with this is developing a relevant mathematic model that is grounded in physical principles and observations. To this end, the following expression is proposed

> 1 ( ) ()1 *bio*

saturation coefficient. Substituting this equation into equation 17 gives:

*L z r k XC z*

= + ⎜ ⎟

<sup>1</sup> ( ) ()1

=− + ⎜ ⎟

Where *L(z)* is the concentration of substrate (BOD or COD) present and *KL* is the Monod half

*dC k X L z C z dz vR K*

1 1 ln

*C z vR vR*

1

=− ⇒ =− (17)

<sup>1</sup> ( ) *bio r kCz* = (18)

*dz vR* = − (19)

=⇒= (21)

= ⇒ (22)

*Integration*

=− ⇒ =− + ⇒ = ∫ ∫ (20)

1

−

1 1

− −

*k z*

θ

*L*

*L*

*K* ⎛ ⎞

⎛ ⎞

⎝ ⎠

⎝ ⎠

1

−

*k z vR*

(23)

(24)

*bio bio*

reduces to:

manifested by:

fluid velocity.

for the biodegradation rate:

Adding equation 18 back into equation 17 gives:

This expression can then be integrated:

( )

$$\begin{split} \frac{d\mathbf{C}}{\mathbf{C}(z)} &= -\frac{k\_1 X}{vR} \Bigg( \mathbf{1} + \frac{\mathbf{L}\_0 - \frac{1}{I} \left( \mathbf{L}\_0 - \mathbf{L}\_f \right) z}{K\_L} \Bigg) dz \\ &\Rightarrow \frac{d\mathbf{C}}{\mathbf{C}(z)} = -\frac{k\_1 X}{vR K\_L} \Bigg( \mathbf{K}\_L + \mathbf{L}\_0 - \frac{1}{I} \left( \mathbf{L}\_0 - \mathbf{L}\_f \right) z \Bigg) dz \end{split} \tag{27}$$

This equation can then be integrated, giving:

$$\begin{split} \ln \left| \mathbf{C} \right| &= -\frac{k\_1 X}{\upsilon R K\_L} \Bigg[ \left( K\_L + L\_0 \right) z - \frac{1}{2l} \left( L\_0 - L\_f \right) z^2 \Bigg] + \text{Const}\_{\text{Integration}} \\ \Longrightarrow \mathbf{C} &= \mathbf{B} e^{-\frac{k\_1 X}{K\_L \upsilon R} \left[ \left( K\_L + L\_0 \right) z - \frac{1}{2l} \left( L\_0 - L\_f \right) z^2 \right]} \end{split} \tag{28}$$

Applying the initial condition that at the beginning of the reactor, when *z* = 0, *C* = *C0*, it can be seen that *B* = *C0*, which results in the following governing equation for the aerobic basins:

$$\mathbf{C} = \mathbf{C}\_0 e^{-\frac{k\_1 X}{K\_L \upsilon R} \left[ (K\_L + L\_0) z - \frac{1}{2l} (L\_0 - L\_f) z^2 \right]} \tag{29}$$

From this, the concentration at the end of the PFR can be calculated by setting *z* = l, and:

$$\begin{aligned} \mathbf{C} &= \mathbf{C}\_{0}e^{-\frac{k\_{1}X}{K\_{L}\upsilon R}\left[ (K\_{L} + l\_{0})l - \frac{1}{2l}(l\_{0} - L\_{f})l^{2} \right]} \\ \implies \mathbf{C} &= \mathbf{C}\_{0}e^{-\frac{k\_{1}Xl}{K\_{L}\upsilon R}\left[ (K\_{L} + l\_{0}) - \frac{1}{2}(l\_{0} - L\_{f}) \right]} \\ \implies \mathbf{C} &= \mathbf{C}\_{0}e^{-\frac{k\_{1}X\theta}{K\_{L}R}\left( K\_{L} + \frac{1}{2}l\_{0} + \frac{1}{2}L\_{f} \right)} \end{aligned} \tag{30}$$

From this governing equation, the aqueous drug concentration at the end of the reactor, can then be determined. The mass of drug compound in the sorbed phase is calculated from the equilibrium sorption condition by means of the following equations:

Microcontaminant Sorption and Biodegradation in Wastewater Modeled as a Two-Phase System 379

solids concentration in each compartment. The right-hand side of the fourth row contains the estimated fraction of each drug that exists in the sorbed phase, as calculated using mass in the aqueous phase and the mass in the sorbed phase. The equilibrium sorbed fraction and computed sorbed fraction are only different from each other when two streams with markedly different solids concentrations mix. For our model, this happens at the beginning of the aerobic basin, where effluent from the primary clarifiers mixes with return activated (RAS) sludge from underneath the secondary clarifiers. The fifth row contains two concentrations calculated from the aforementioned parameters: the biosolids concentration

The first modeled location corresponds to WWTP influent. The aqueous volume for this compartment is set to 1L, but this could be scaled based on actual flow rates. Masses and concentrations for COD and solids are set to match the characteristics of any specific WWTP. The total suspended solids (TSS) in the raw influent and the COD concentration (L) can be based on information from a treatment plant or from various references. Influent total drug masses can be taken from projection calculations by Ottmar et al. (2010b) for a plant with P/Q (service population over daily flow) equal to the target plant. Aqueous-phase and sorbed-phase drug masses can then be calculated assuming equilibrium conditions, using *Kd* values previously determined and the presumed TSS concentration in influent wastewater.

1 1 <sup>1</sup> <sup>1</sup> 11 1

From this, the mass in the sorbed phase is equal to the total mass multiplied by the sorbed fraction. The mass in the aqueous phase is equal to the total mass minus the mass in the

The second modeled location corresponds to the exit of the primary (1°) clarifiers. It was assumed that the overall flow splits into two smaller flows at this location, namely: primary sludge and primary effluent. For an example, it can be said that 60% of TSS and 40% of COD are removed into the primary sludge stream. The remaining TSS and COD flow from the primary clarifier into the activated sludge basin with the primary effluent. Based on these parameters and an assumed solids concentration of 45,000 mg/L for the primary sludge, a mass balance on the solids can be used to calculate the mass of solids (first row, second column) and the aqueous volume (first row, first column) leaving the primary clarifier as effluent or primary sludge. The following two equations are used, with the first being the mass balance equation and the second being the balance equation for a non-compressible

> *solids in solids primaryeff solids primaryslud* \_\_ \_ *ge in in primaryeff primaryeff primarysludge primarysludge*

> > *QQ Q in* = *primaryeff p* + *rimarysludge*

*Mass Mass Mass*

⇒= +

*XQ X Q X Q* = +

*ss sorbed sorbed d ss K K X f X f <sup>f</sup> K X* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup> ⇒ + = ⇒= <sup>−</sup> − −+ ⎝ ⎠

(31)

1 11 *<sup>d</sup> d ss sorbed*

(X) at left and COD concentration (L) at right, both in mg/L.

Equation 31, below, can be re-arranged to solve for this:

**2.2.1 Influent** 

sorbed phase.

aqueous fluid:

**2.2.2 Primary clarification** 

$$\begin{aligned} \textit{Mass}\_{drug,total} &= \textit{Mass}\_{drug,aq} + \textit{Mass}\_{drug,sorbed} \\ \textit{Mass}\_{drug,sorbed} &= f\_{sorbed} \textit{Mass}\_{drug,total} \\ \implies \frac{\textit{Mass}\_{drug,sorbed}}{f\_{sorbed}} &= \textit{Mass}\_{drug,aq} + \textit{Mass}\_{drug,sorbed} \\ \implies \textit{Mass}\_{drug,sorbed} &- f\_{sorbed} \textit{Mass}\_{drug,sorbed} = f\_{sorbed} \textit{Mass}\_{drug,aq} \\ \implies \textit{Mass}\_{drug,sorbed} &= \frac{f\_{sorbed} \textit{Mass}\_{drug,aq}}{1 - f\_{sorbed}} \end{aligned}$$

The reduction in COD and the change in solids masses are based on the treatment plants operating characteristics.

#### **2.2 Overall approach for wastewater treatment processes**

Having developed a governing equation for simultaneous sorption and biodegradation, the model could then be applied to a wastewater treatment, as a single entity. Each wastewater treatment plant process is characterized by a set of ten parameters, each of which has been assigned to a specific cell in a table with five rows and two columns, as shown in Figure 4.

Fig. 4. Overall model schematic of a wastewater treatment plant. Note the key for the definition of individual cell values

The first row of each box contains the aqueous volume (V) of each compartment, as scaled to one liter, and the known mass of solids in that compartment. The second row contains the total mass of drug within each compartment (at left) and the mass of drug that exists in the aqueous phase within each compartment (at right). These drug masses are in units of μg. The third row contains the mass of COD substrate within each compartment (at left) and the mass of each drug that exists as sorbed phase (at right). COD mass is in mg, and drug mass is in μg. The total mass of drug compound (second row, first column) will always be equal to the sum of the mass in the aqueous phase (second row, second column) and the mass in the sorbed phase (third row, second column). The left-hand side of the fourth row contains the estimated fraction of each drug that exists in the sorbed phase assuming equilibrium conditions. This is based on the Kd value for each drug, as measured previously, and the solids concentration in each compartment. The right-hand side of the fourth row contains the estimated fraction of each drug that exists in the sorbed phase, as calculated using mass in the aqueous phase and the mass in the sorbed phase. The equilibrium sorbed fraction and computed sorbed fraction are only different from each other when two streams with markedly different solids concentrations mix. For our model, this happens at the beginning of the aerobic basin, where effluent from the primary clarifiers mixes with return activated (RAS) sludge from underneath the secondary clarifiers. The fifth row contains two concentrations calculated from the aforementioned parameters: the biosolids concentration (X) at left and COD concentration (L) at right, both in mg/L.

### **2.2.1 Influent**

378 Mass Transfer - Advanced Aspects

*Mass Mass*

*Mass f Mass f Mass*

*sorbed drug aq*

*sorbed*

The reduction in COD and the change in solids masses are based on the treatment plants

Having developed a governing equation for simultaneous sorption and biodegradation, the model could then be applied to a wastewater treatment, as a single entity. Each wastewater treatment plant process is characterized by a set of ten parameters, each of which has been assigned to a specific cell in a table with five rows and two columns, as shown in Figure 4.

Fig. 4. Overall model schematic of a wastewater treatment plant. Note the key for the

The first row of each box contains the aqueous volume (V) of each compartment, as scaled to one liter, and the known mass of solids in that compartment. The second row contains the total mass of drug within each compartment (at left) and the mass of drug that exists in the aqueous phase within each compartment (at right). These drug masses are in units of

The third row contains the mass of COD substrate within each compartment (at left) and the mass of each drug that exists as sorbed phase (at right). COD mass is in mg, and drug mass

g. The total mass of drug compound (second row, first column) will always be equal to the sum of the mass in the aqueous phase (second row, second column) and the mass in the sorbed phase (third row, second column). The left-hand side of the fourth row contains the estimated fraction of each drug that exists in the sorbed phase assuming equilibrium conditions. This is based on the Kd value for each drug, as measured previously, and the

μg.

, ,

*drug aq drug sorbed*

,,,

*drug sorbed sorbed drug sorbed sorbed drug aq*

, ,,

*drug total drug aq drug sorbed*

1

*f Mass <sup>f</sup>* <sup>=</sup> <sup>−</sup>

, ,

⇒− =

= +

*drug sorbed sorbed drug total*

*Mass Mass Mass*

,

*sorbed*

*f*

*Mass*

*Mass*

operating characteristics.

definition of individual cell values

is in μ *drug sorbed*

*Mass f Mass*

=

⇒ =+

⇒ ,

*rbed*

,

**2.2 Overall approach for wastewater treatment processes** 

*drug so*

The first modeled location corresponds to WWTP influent. The aqueous volume for this compartment is set to 1L, but this could be scaled based on actual flow rates. Masses and concentrations for COD and solids are set to match the characteristics of any specific WWTP. The total suspended solids (TSS) in the raw influent and the COD concentration (L) can be based on information from a treatment plant or from various references. Influent total drug masses can be taken from projection calculations by Ottmar et al. (2010b) for a plant with P/Q (service population over daily flow) equal to the target plant. Aqueous-phase and sorbed-phase drug masses can then be calculated assuming equilibrium conditions, using *Kd* values previously determined and the presumed TSS concentration in influent wastewater. Equation 31, below, can be re-arranged to solve for this:

$$K\_d = \frac{1}{X\_{ss}} \left( \frac{1}{1 - f\_{sorbed}} - 1 \right) \Longrightarrow K\_d X\_{ss} + 1 = \frac{1}{1 - f\_{sorbed}} \Longrightarrow f\_{sorbed} = 1 - \frac{1}{1 + K\_d X\_{ss}} \tag{31}$$

From this, the mass in the sorbed phase is equal to the total mass multiplied by the sorbed fraction. The mass in the aqueous phase is equal to the total mass minus the mass in the sorbed phase.

#### **2.2.2 Primary clarification**

The second modeled location corresponds to the exit of the primary (1°) clarifiers. It was assumed that the overall flow splits into two smaller flows at this location, namely: primary sludge and primary effluent. For an example, it can be said that 60% of TSS and 40% of COD are removed into the primary sludge stream. The remaining TSS and COD flow from the primary clarifier into the activated sludge basin with the primary effluent. Based on these parameters and an assumed solids concentration of 45,000 mg/L for the primary sludge, a mass balance on the solids can be used to calculate the mass of solids (first row, second column) and the aqueous volume (first row, first column) leaving the primary clarifier as effluent or primary sludge. The following two equations are used, with the first being the mass balance equation and the second being the balance equation for a non-compressible aqueous fluid:

$$\text{Mass}\_{\text{solids\\_in}} = \text{Mass}\_{\text{solids\\_primary}\text{eff}} + \text{Mass}\_{\text{solids\\_primary}\text{sludge}}$$

$$\implies \text{X}\_{in}\text{Q}\_{in} = \text{X}\_{primary\text{eff}}\text{Q}\_{primary\text{eff}} + \text{X}\_{primary\text{sludge}}\text{Q}\_{primary\text{sludge}}$$

$$\text{Q}\_{in} = \text{Q}\_{primary\text{eff}} + \text{Q}\_{primary\text{sludge}}$$

Microcontaminant Sorption and Biodegradation in Wastewater Modeled as a Two-Phase System 381

is merged with recycled activated sludge (R.A.S.). Modeling at this location requires a twostep mathematical process. The first step comprises arithmetic addition of the physical

(1 ,1 )

+

= + (3 ,1 ) *rdRow stColumn*

(1 ,2 )

*Mass ndRow ndColumn*

(2 ,2 )

(3 ,2 )

, , . ..

*ier drug aq R A S*

0

0

*drug sorb StartASba drug sorb Leaving Clarifier drug sorb R A S*

= +

, sin , . .. , 1

*COD StartASba COD Leaving Clarifier COD R A S*

, , sin , , . .. ,, 1

, sin , . .. , 1

*Solids StartASba Solids Leaving Clarifier Solids R A S*

= +

0

0

*Mass Mass Mass rdRow ndColumn*

As mentioned previously, the addition of the masses of drug compound in the solids phases and in the aqueous phases produces a condition whereby the equilibrium sorption conditions are not satisfied, owing to the marked increase in solids concentration (a jump from 120 mg/L to 3000 mg/L). This can be seen by re-visiting equation 31 and by the fact that the fraction of drug compound sorbed that is calculated based the masses in the aqueous phase and in the solids phase is not the same as the fraction sorbed calculated based on the solids concentration and the distribution coefficient. Because laboratory batch tests have shown that sorption happens quite rapidly and can be assumed be essentially at equilibrium, a subsequent series of data cells is used to make the conversion to an equilibrium condition. This is done by setting the mass of drug in the sorbed phase (3rd row, 2nd column) of the "Start A.S. EQM" data set equal to the total mass (2nd row, 1st column) in the "Start A.S. Basin" data set multiplied by the fraction sorbed at equilibrium (4th row, 1st column). The mass of drug in the aqueous phase (2nd row, 2nd column) of the "Start A.S. EQM data set is then calculated by subtracting the aforementioned calculated mass in the

The fourth modeled process is the activated sludge basin. This compartment is modeled as a plug flow reactor, which makes it possible to compute extent of pharmaceutical biodegradation and sorption as a function of travel time. A more rigorous, first-principlesbased approach is needed for this process because both sorption and biodegradation are occurring and need to be accounted for simultaneously. The use of a plug-flow model allows for the appropriate formulation of the fate and transport of the compounds and

The fifth modeled location comprises the secondary clarifier. Secondary clarification, like the equilibrium portion of the activated sludge basin, is modeled assuming plug flow conditions with equilibrium sorption. Extent of pharmaceutical biodegradation in this compartment is once again computed as a function of time in the reactor and the biodegradation rate coefficient, in this case, k1/2. A decreased rate constant is used to account for the lack of aeration during secondary clarification and the presumption that the biomass are less actively degrading COD and drugs in the clarifiers relative to the activated sludge basins. After the secondary clarifier, the process stream splits into two streams: the effluent stream and the sludge recycle stream. The volume (first row, first column), the

*Mass Mass Mass stRow ndColumn*

properties from the two feeder streams:

*StartASba leaving clarifier <sup>o</sup> RAS*

= +

*Mass Mass*

sorbed phase from the total mass.

phases (aqueous and solid).

**2.2.5 Secondary clarification** 

**2.2.4 Aerobic activated sludge treatment** 

sin 1 . ..

=

, , sin ,, 1

*drug aq StartASba drug aq Leaving Clarif*

*Mass Mass Mass*

*Q Q Q stRow stColumn*

*Xin* is the solids concentration in the influent, *Qin* is the aqueous flow rate (unit volume), *Xprimaryeff* is the solids concentration in the primary effluent, *Qprimaryeff* is the flow rate leaving the primary clarifier, *Xprimarysludge* is the solids concentration in the primary sludge, and *Qprimarysludge* is the sludge flow rate. Of these six, only *Qprimaryeff* and *Qprimarysludge* are unknown, but with two equations, they can be determined.

Concerning the transport of COD, a similar mass balance approach was used:

$$\begin{aligned} \text{Mass}\_{\text{COD\\_in}} &= \text{Mass}\_{\text{COD\\_primary}} + \text{Mass}\_{\text{COD\\_primary}} \\ \implies \text{L}\_{\text{in}}\text{Q}\_{\text{in}} &= \text{L}\_{\text{primary}\text{eff}}\text{Q}\_{\text{primary}\text{eff}} + \text{L}\_{\text{primary}\text{sludge}}\text{Q}\_{\text{primary}\text{sludge}} \end{aligned}$$

The flow rates have already been determined (previously with the solids mass balance), and the amount of COD leaving the primary clarifier is set as part of this plant's operating characteristics (40% removal). Consequently, the masses of COD (third row, first column) and the concentrations (fifth row, second column) can be calculated.

For evaluation of drug compound transport, each phase is treated as a separate process. Beginning with the aqueous phase, we begin with the familiar mass balance. Here, we assume that substantial biodegradation does not occur in the primary clarifier due to the short hydraulic retention time and anoxic conditions:

$$\begin{aligned} \text{Mass}\_{\text{drag},\text{aq},\text{In}} &= \text{Mass}\_{\text{drag},\text{aq},\text{primary}} + \text{Mass}\_{\text{drag},\text{aq},\text{primary}} \\ \implies \text{C}\_{\text{in}}\text{Q}\_{\text{in}} &= \text{C}\_{\text{primary}\text{eff}}\text{Q}\_{\text{primary}\text{eff}} + \text{C}\_{\text{primary}\text{sludge}\text{ge}}\text{Q}\_{\text{primary}\text{sludge}\text{ge}} \end{aligned}$$

Because the drug compound will be dissolved in the aqueous phase, the concentrations (in *μ*g/L) will not change, so *Cin* will be equal to *Cprimaryeff* and *Cprimarysludge*. Consequently, the transport of drug mass in the aqueous phase will be proportional to the transport of the aqueous phase itself. For example, if the aqueous phase flow rate (first row, first column) leaving the primary clarifier as the primary sludge is equal to 0.4% of the volume in the influent, then the mass of drug compound in the aqueous phase of the primary sludge (second row, second column, Primary Sludge) will be equal to 0.4% of the mass of drug compound in the aqueous phase of the influent (second row, second column, Influent). The transport of drug compound in the sorbed phase will be similarly governed, with the

basis being a mass-balance approach, as outlined in the following equation:

$$\begin{aligned} \text{Mass}\_{\text{drag},sorb,in} &= \text{Mass}\_{\text{drag},sorb,primary} + \text{Mass}\_{\text{drag},sorb,primary} \\ \implies \text{S}\_{\text{int}}X\_{\text{in}}\text{Q}\_{\text{in}} &= \text{S}\_{\text{primary}\#\text{ff}}X\_{\text{primary}\#\text{ff}}\text{Q}\_{\text{prinnary}\#\text{dug}} + \text{S}\_{\text{prinnary}\text{sludge}\#\text{dug}}\text{Q}\_{\text{prinnrary}\text{sludge}\#\text{dug}} \end{aligned}$$

Because the only transport process occurring is a physical separation of the sludge, the sorbed concentration (in *m*g/kg sludge) will not change, so *Sin* will be equal to *Sprimaryeff* and *Sprimarysludge*. For example, if 60% of the solids from the influent (first row, second column, influent) go to the primary sludge (first row, second column, primary sludge), then 60% of the total drug mass in the sorbed phase from the influent (third row, second column, influent) will go with the primary sludge (third row, second column, primary sludge).

#### **2.2.3 Preliminary activated sludge treatment**

Entrance into secondary (2°) treatment, "Start A.S. Basin," marks the third modeled location, in particular inlet to the activated sludge basins. Here, the effluent from the primary clarifier is merged with recycled activated sludge (R.A.S.). Modeling at this location requires a twostep mathematical process. The first step comprises arithmetic addition of the physical properties from the two feeder streams:

0 0 sin 1 . .. , sin , . .. , 1 , , sin ,, 1 (1 ,1 ) (1 ,2 ) *StartASba leaving clarifier <sup>o</sup> RAS Solids StartASba Solids Leaving Clarifier Solids R A S drug aq StartASba drug aq Leaving Clarif Q Q Q stRow stColumn Mass Mass Mass stRow ndColumn Mass Mass* = + = + = 0 0 , , . .. , , sin , , . .. ,, 1 , sin , . .. , 1 (2 ,2 ) (3 ,2 ) *ier drug aq R A S drug sorb StartASba drug sorb Leaving Clarifier drug sorb R A S COD StartASba COD Leaving Clarifier COD R A S Mass ndRow ndColumn Mass Mass Mass rdRow ndColumn Mass Mass Mass* + = + = + (3 ,1 ) *rdRow stColumn*

As mentioned previously, the addition of the masses of drug compound in the solids phases and in the aqueous phases produces a condition whereby the equilibrium sorption conditions are not satisfied, owing to the marked increase in solids concentration (a jump from 120 mg/L to 3000 mg/L). This can be seen by re-visiting equation 31 and by the fact that the fraction of drug compound sorbed that is calculated based the masses in the aqueous phase and in the solids phase is not the same as the fraction sorbed calculated based on the solids concentration and the distribution coefficient. Because laboratory batch tests have shown that sorption happens quite rapidly and can be assumed be essentially at equilibrium, a subsequent series of data cells is used to make the conversion to an equilibrium condition. This is done by setting the mass of drug in the sorbed phase (3rd row, 2nd column) of the "Start A.S. EQM" data set equal to the total mass (2nd row, 1st column) in the "Start A.S. Basin" data set multiplied by the fraction sorbed at equilibrium (4th row, 1st column). The mass of drug in the aqueous phase (2nd row, 2nd column) of the "Start A.S. EQM data set is then calculated by subtracting the aforementioned calculated mass in the sorbed phase from the total mass.

### **2.2.4 Aerobic activated sludge treatment**

The fourth modeled process is the activated sludge basin. This compartment is modeled as a plug flow reactor, which makes it possible to compute extent of pharmaceutical biodegradation and sorption as a function of travel time. A more rigorous, first-principlesbased approach is needed for this process because both sorption and biodegradation are occurring and need to be accounted for simultaneously. The use of a plug-flow model allows for the appropriate formulation of the fate and transport of the compounds and phases (aqueous and solid).

#### **2.2.5 Secondary clarification**

380 Mass Transfer - Advanced Aspects

*Xin* is the solids concentration in the influent, *Qin* is the aqueous flow rate (unit volume), *Xprimaryeff* is the solids concentration in the primary effluent, *Qprimaryeff* is the flow rate leaving the primary clarifier, *Xprimarysludge* is the solids concentration in the primary sludge, and *Qprimarysludge* is the sludge flow rate. Of these six, only *Qprimaryeff* and *Qprimarysludge* are unknown,

> *COD in COD* \_\_ \_ *primaryeff COD primarysludge in in primaryeff primaryeff primarysludge primarysludge*

The flow rates have already been determined (previously with the solids mass balance), and the amount of COD leaving the primary clarifier is set as part of this plant's operating characteristics (40% removal). Consequently, the masses of COD (third row, first column)

For evaluation of drug compound transport, each phase is treated as a separate process. Beginning with the aqueous phase, we begin with the familiar mass balance. Here, we assume that substantial biodegradation does not occur in the primary clarifier due to the

> *drug aq In drug aq primaryeff drug aq primaryslud* , , , , , , *ge in in primaryeff primaryeff primarysludge primarysludge*

Because the drug compound will be dissolved in the aqueous phase, the concentrations (in *μ*g/L) will not change, so *Cin* will be equal to *Cprimaryeff* and *Cprimarysludge*. Consequently, the transport of drug mass in the aqueous phase will be proportional to the transport of the aqueous phase itself. For example, if the aqueous phase flow rate (first row, first column) leaving the primary clarifier as the primary sludge is equal to 0.4% of the volume in the influent, then the mass of drug compound in the aqueous phase of the primary sludge (second row, second column, Primary Sludge) will be equal to 0.4% of the mass of drug compound in the aqueous phase of the influent (second row, second column, Influent). The transport of drug compound in the sorbed phase will be similarly governed, with the

Concerning the transport of COD, a similar mass balance approach was used:

*Mass Mass Mass*

*Mass Mass Mass*

basis being a mass-balance approach, as outlined in the following equation:

*Mass Mass Mass*

⇒ = +

**2.2.3 Preliminary activated sludge treatment** 

= +

*drug sorb in drug sorb primaryeff drug sorb primarysludge* , , , , , ,

*SXQ S X Q S X Q*

*in in in primaryeff primaryeff primaryeff primarysludge primarysludge primarysludge*

Because the only transport process occurring is a physical separation of the sludge, the sorbed concentration (in *m*g/kg sludge) will not change, so *Sin* will be equal to *Sprimaryeff* and *Sprimarysludge*. For example, if 60% of the solids from the influent (first row, second column, influent) go to the primary sludge (first row, second column, primary sludge), then 60% of the total drug mass in the sorbed phase from the influent (third row, second column, influent) will go with the primary sludge (third row, second column, primary sludge).

Entrance into secondary (2°) treatment, "Start A.S. Basin," marks the third modeled location, in particular inlet to the activated sludge basins. Here, the effluent from the primary clarifier

*CQ C Q C Q* = +

⇒= +

and the concentrations (fifth row, second column) can be calculated.

⇒= +

short hydraulic retention time and anoxic conditions:

*LQ L Q L Q* = +

but with two equations, they can be determined.

The fifth modeled location comprises the secondary clarifier. Secondary clarification, like the equilibrium portion of the activated sludge basin, is modeled assuming plug flow conditions with equilibrium sorption. Extent of pharmaceutical biodegradation in this compartment is once again computed as a function of time in the reactor and the biodegradation rate coefficient, in this case, k1/2. A decreased rate constant is used to account for the lack of aeration during secondary clarification and the presumption that the biomass are less actively degrading COD and drugs in the clarifiers relative to the activated sludge basins. After the secondary clarifier, the process stream splits into two streams: the effluent stream and the sludge recycle stream. The volume (first row, first column), the

Microcontaminant Sorption and Biodegradation in Wastewater Modeled as a Two-Phase System 383

calculated from the Recycle stream. This value then would be fed into the calculation for Start A.S. Basin, resulting in a circular calculation error. This error is resolved by essentially creating two separate entries in the model for the same process. The first, RAS, provided the values that feed into the Start A.S. Basin process. The second, 95% Calculated, is calculated from the Recycle stream, as described previously. Initially, two arbitrary values are inputted for the mass of drug compound in the aqueous phase (second row, second column) and the in the sorbed phase (third row, second column) for the RAS process. An optimization routine is then executed to minimize the sum of the squared residuals between the drug masses in the RAS line that feed into the activated sludge basin and the RAS line that is calculated as being 95% of the recycle line. By minimizing the difference between the two processes, the recycle loop is effectively closed, allowing for a complete modeling of the

Modeling the simultaneous sorption and biodegradation in wastewater systems has proven to be a challenging problem for researchers. Because the two processes are intrinsically linked, a novel approach was needed to develop a comprehensive mathematical expression to be used in modelling analyses. To that end, the volume averaging methodology commonly employed in groundwater systems was used with one key difference: rather than the having the solid phase be stationary, it was mobile. This paradigm shift allowed for fate and transport modelling throughout a wastewater treatment plant. This new model is sufficiently robust that it can have applications with many different types of compounds in

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Glassmeyer, S., Furlong, E., Kolpin, D., Cahill, J., Zaugg, S., Werner, S., Meyer, M. & Kryak,

Hassanizadeh, M. & Gray, W. (1979). General conservation equations for multi-phase

Joss, A., Zabczynski, S., Gobel, A., Hoffmann, B., Loffler, D., McArdell, C., Ternes, T.,

of illicit and therapeutic pharmaceuticals in wastewater effluent and surface waters in Nebraska. *Environmental Pollution*, Vol. 157, No. 8, (August 2009), pp. 786-791,

environment: Agents of subtle change? *Environmental Health Perspectives*, Vol. 23,

posed by residues in the environment. *Environmental Toxicology and Chemistry*, Vol.

D. (2005). Transport of chemical and microbial compounds from known wastewater discharges: A potential for use as indicators of human fecal contamination. *Environmental Science and Technology*, Vol. 39, No. 3, (November

systems: 1. Averaging procedure. Advances in Water Resources, Vol. 2, No. 3,

Thomsen, A. & Siegrist, H. (2006). Biological degradation of pharmaceuticals in

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wastewater treatment process.

**3. Conclusions** 

**4. References** 

ISSN

solids mass (first row, second column), and the COD mass (third row, first column) in the effluent are set to match the plant operating characteristics (and also effluent regulatory requirements, 5 mg/L solids and 3 mg/L COD). The volume, solids mass, and the COD mass for the sludge recycle are calculated by subtracting the effluent values from the secondary clarifier values:

$$\begin{aligned} \text{Q}\_{\text{Rec-cycle}} &= \text{Q}\_{\text{Leaving2}^0 \text{Clarifier}} - \text{Q}\_{\text{Effluent}} \text{(1stRow, 1st Column)} \\ \text{Mass}\_{\text{Solids,Recçcle}} &= \text{Mass}\_{\text{Solds,Lewing2}^0 \text{Clarifier}} - \text{Mass}\_{\text{Solds,Eifflent.}} \text{(1stRow, 2ndColumn)} \\ \text{Mass}\_{\text{COD,Satur}} &= \text{Mass}\_{\text{COD,Lewing2}^0 \text{Clarifier}} - \text{Mass}\_{\text{COD,Eifflent}} \text{(3rdRow, 1stColumn)} \end{aligned}$$

As with separation after the primary clarifier, the transport of drug compound is modeled to mirror the transport of the phase containing the drug compound (i.e. inter-phase flux, or *j*, is assumed to not be significant owing to equilibrium conditions). If 66% of the aqueous phase goes into the effluent stream (first row, first column, Effluent), then 66% of the drug mass present in the aqueous phase leaving the secondary clarifier will go into the effluent stream (second row, second column, Effluent). The drug masses in the sludge recycle stream are the calculated by subtracting the effluent masses from the masses leaving the secondary clarifier.

### **2.2.6 Sludge recycle**

After separation following the secondary clarifier, the next modeled process is the sludge recycle stream. As mentioned above, the values for the aqueous volume (first row, first column), the solids mass (first row, second column), the mass of COD (third row, first column), the aqueous drug mass (second row, second column), and the sorbed drug mass (third row, second column) for the recycle stream are calculated by subtracting the effluent values from the values leaving the secondary clarifier. The recycle stream is then split into two separate streams, the return activated sludge (RAS), which is pumped back to the beginning of the activated sludge basins, and the waste activated sludge (WAS), which is merged with the primary sludge stream and pumped to the anaerobic digesters (not modeled here). In this case, the treatment plant's operating characteristics define the separation between these two streams, specifically, the RAS is 95% of the recycle stream, whereas the WAS is 5% of the stream. Both the aqueous phase (first row, first column) and the solids phase (first row, second column) are split proportionately with 95% moving to the RAS and 5% moving to the WAS. Additionally, the COD mass dissolved in the aqueous phase (third row, first column) and the drug mass in the aqueous phase (second row, second column) are split proportionately to the phase (95% to RAS, 5% to WAS), as is the mass of drug in the sorbed phase (third row, second column).

The final component of the model is the iterative step that is part of the RAS stream. This two-step process was necessary to eliminate circular calculation errors that arise due to the recycle stream which otherwise would have produced an indeterminate system. This error can be highlighted by looking at just the aqueous phase drug mass. The mass at Start A.S. Basin is calculated from the mass leaving the primary clarifier and the mass in the RAS stream. The mass at Start A.S. EQM is calculated from the mass at Start A.S. Basin. The mass at Leaving A.S. Basin is calculated from the mass at Start A.S. Basin. The mass Leaving 2° Clarifier is calculated from the mass at Leaving A.S. Basin. The mass at Recycle is calculated from the Effluent and the mass at Leaving 2° Clarifier. Finally, the mass in the RAS stream is calculated from the Recycle stream. This value then would be fed into the calculation for Start A.S. Basin, resulting in a circular calculation error. This error is resolved by essentially creating two separate entries in the model for the same process. The first, RAS, provided the values that feed into the Start A.S. Basin process. The second, 95% Calculated, is calculated from the Recycle stream, as described previously. Initially, two arbitrary values are inputted for the mass of drug compound in the aqueous phase (second row, second column) and the in the sorbed phase (third row, second column) for the RAS process. An optimization routine is then executed to minimize the sum of the squared residuals between the drug masses in the RAS line that feed into the activated sludge basin and the RAS line that is calculated as being 95% of the recycle line. By minimizing the difference between the two processes, the recycle loop is effectively closed, allowing for a complete modeling of the wastewater treatment process.
