3.1 Biological Wastewater Treatment: Activated Sludge Process

Comprehensive practice flashcards on activated sludge wastewater treatment covering mass balances, kinetic derivations, reactor configurations, oxygen demands, and model parameters.

In an activated sludge system with settling and recycle, what is the primary advantage of decoupling the solids residence time ($\theta_x$) from the hydraulic residence time ($\theta_d$)?

It allows operators to maintain a large $\theta_x$ to achieve high substrate removal efficiency while keeping a small hydraulic retention time ($\theta_d = \frac{V}{Q}$), thereby substantially reducing the required aeration basin volume.

What are typical ranges for solids residence time ($\theta_x$) and hydraulic residence time ($\theta_d$) in municipal wastewater treatment?

Typical $\theta_x$ ranges from $4\text{ to }10\text{ d}$, whereas typical $\theta_d$ ranges from $4\text{ to }10\text{ h}$.

What boundary is defined by the red dashed line in this activated sludge schematic, and what key mass assumptions are applied to the secondary clarifier?

The red dashed line represents the overall system boundary for mass balances around the aeration basin and settler; the key assumptions are negligible biomass in the settler and that soluble substrate biodegradation occurs exclusively within the aeration basin.

What is the mathematical definition of solids residence time ($\theta_x$ or $\theta_c$) for an activated sludge system where biomass is concentrated only in the aeration basin?

$\theta_x = \theta_c = \frac{XV}{(Q - Q_w)X_e + Q_w X_w}$ (or $\frac{XV}{(Q - Q_w)X_e + Q_w X_r}$ when wasting from the underflow).

What is the steady-state equation relating effluent substrate concentration ($S_e$) to solids residence time ($\theta_x$) and kinetic coefficients?

$$S_e = \frac{(\theta_x b + 1)K_s}{\theta_x Y \hat{q} - (\theta_x b + 1)}$$

Does steady-state effluent substrate concentration ($S_e$) in a completely mixed activated sludge system depend on hydraulic residence time ($\theta_d$) or aeration tank volume ($V$)?

No, $S_e$ depends exclusively on the solids residence time ($\theta_x$) and kinetic parameters ($Y$, $\hat{q}$, $K_s$, $b$), and does not depend on $\theta_d$ or $V$.

What is the steady-state expression for biomass concentration ($X$) in the aeration basin of a CSTR with cell recycle?

$$X = \frac{\theta_c}{\theta} \frac{Y(S_0 - S_e)}{1 + b\theta_c}$$

What are the two equivalent formulations for the specific substrate utilization rate ($U$)?

$$U = \frac{\hat{q} S}{K_s + S} = \frac{S_0 - S}{X \theta}$$

What is biomass washout, and what formula defines the minimum solids residence time ($\theta_x^{\min}$) required to prevent it?

Washout occurs when $\theta_x$ is too short for cell growth to balance cell loss, resulting in $S_e = S_0$ and $X = 0$. It is defined by $\theta_x^{\min} = \frac{K_s + S_0}{S_0(Y\hat{q} - b) - b K_s}$.

What is the theoretical lower limit of the minimum solids residence time ($[\theta_x^{\min}]_{\lim}$) as influent substrate concentration approaches infinity ($S_0 \rightarrow \infty$)?

$$[\theta_x^{\min}]_{\lim} = \frac{1}{Y\hat{q} - b}$$

What is $S_{\min}$, and what equation gives its value as $\theta_x \rightarrow \infty$?

$S_{\min}$ is the minimum substrate concentration that can sustain a steady-state microbial population; below this concentration, net specific growth is negative and biomass disappears. It is defined by $S_{\min} = \frac{b K_s}{Y\hat{q} - b}$.

How is the microbiological safety factor ($SF$) defined in activated sludge design, and what values are typical?

$SF = \frac{\theta_x}{\theta_x^{\min}}$, with typical values ranging from $5$ to several hundreds ($2\text{--}20$ for High Rate, $20\text{--}70$ for Moderate Rate, and $100+$ for Low Rate).

Under the assumption that effluent biomass ($X_e$) and waste flow ($Q_w$) are negligible, what expression defines the recirculation ratio ($R = \frac{Q_r}{Q}$)?

$$\frac{Q_r}{Q} = \frac{X}{X_r - X}$$

What formula defines the solids loading rate ($SLR$) applied to a secondary settling tank of area $A$?

$$SLR = \frac{(Q + Q_r)X}{A}$$

What is the mass balance relationship between waste sludge production ($Q_w X_r$) and total net solids production ($\frac{XV}{\theta_c}$)?

$Q_w X_r = \frac{XV}{\theta_c} - Q X_e$, where total net solids production is $\frac{XV}{\theta_c} = Q_w X_r + Q X_e = Y Q(S_0 - S) - b XV$.

What equation gives the steady-state oxygen required based on a $BOD_L$ mass balance when using $BOD_5$ data with a ratio of $\frac{BOD_5}{BOD_L} = 0.68$?

$$O_2\text{ required} = \frac{Q(S_0 - S_e)}{0.68} - 1.42 \frac{XV}{\theta_c}$$

What is the theoretical oxygen equivalent factor of biological cells ($VSS$) based on complete oxidation ($C_5H_7O_2N + 5O_2 \rightarrow 5CO_2 + NH_3 + 2H_2O$)?

$1.42\text{ mg } O_2/\text{mg } VSS$ (or $1.42\text{ kg } O_2/\text{kg } VSS$).

What is the stoichiometric oxygen requirement for nitrification of ammonia ($NH_4^+ + 2O_2 \rightarrow NO_3^- + 2H^+ + H_2O$)?

$4.57\text{ mg } O_2/\text{mg } N$ (calculated from $\frac{2 \times 32}{14}$).

What activated sludge configuration is depicted in this process flow schematic?

Step-aeration (step feeding) activated sludge, where influent wastewater is introduced at multiple points along the length of the aeration basin to even out organic loading and oxygen demand.

Which activated sludge modification is shown in this schematic diagram?

Contact stabilization process, which uses a contact tank ($15\text{--}60\text{ min}$ retention) to remove easily degradable organics followed by clarification and a stabilization tank to aerate and oxidize settled solids.

What are the primary operational trade-offs of the contact stabilization process?

It requires a smaller total aeration volume due to concentrated biomass ($X$) in the stabilization tank, but is more susceptible to shock loads, has little or no nitrification capability, and requires greater operational skill.

What are the main engineering characteristics and limitations of conventional plug-flow activated sludge?

It features long narrow aeration tanks ($L:W > 5$) with aeration throughout; it provides the highest theoretical substrate removal rate within a given time, but is susceptible to shock and toxic loads due to limited dilution and exhibits steep substrate and oxygen demand gradients.

In this graph showing substrate and dissolved oxygen (DO) uptake along reactor length, what do the solid, small-dash, and large-dash curves represent?

The solid lines represent plug flow ($PF$), the small-dash lines represent step aeration ($SA$), and the large-dash horizontal lines represent continuous-stirred tank ($CSTR$) reactors.

What is tapered aeration in an activated sludge system?

A modification of plug-flow aeration where air supply is reduced progressively along the length of the basin to match the declining substrate concentration and oxygen uptake rate.

What are the primary advantages and operational disadvantages of pure oxygen activated sludge systems?

They support high organic loading rates and higher biomass ($MLSS$) concentrations, but carry high operating costs for oxygen generation, require covered tanks, and risk dropping pH due to $CO_2$ accumulation in the headspace.

In the Metcalf & Eddy design example with total effluent $BOD_5 = 20\text{ mg/L}$, $X_e = 22\text{ mg/L } VSS$ ($65\%$ biodegradable), and $\frac{BOD_5}{BOD_L} = 0.68$, what is the soluble effluent $BOD_5$ ($S_e$)?

$S_e = 6.2\text{ mg/L } BOD_5$. Biodegradable $VSS = 22 \times 0.65 = 14.3\text{ mg/L}$; particulate $BOD_5 = 14.3 \times 1.42 \times 0.68 = 13.8\text{ mg/L}$; and soluble $BOD_5 = 20 - 13.8 = 6.2\text{ mg/L}$.

In this linearized cell mass balance plot of $\frac{1}{\theta_c}$ versus $U$, what parameters do the slope and y-intercept represent?

The slope of the line equals the microbial yield coefficient ($Y$), and the y-intercept equals $-b$ (the negative endogenous decay coefficient).

In this Lineweaver-Burk style plot of $\frac{1}{U}$ versus $\frac{1}{S}$, what kinetic parameters are obtained from the slope, y-intercept, and x-intercept?

The slope is $\frac{K_s}{\hat{q}}$, the y-intercept is $\frac{1}{\hat{q}}$, and the negative x-intercept equals $-\frac{1}{K_s}$.

How can oxygen requirements be calculated directly from specific oxygen uptake rate measurements without needing the $BOD_5/BOD_L$ ratio?

Using the expression $O_2\text{ required} = A \cdot Q(S_0 - S) + B \cdot XV$, where $A$ is the oxygen used per unit mass of substrate removed for synthesis, and $B$ is the oxygen used per unit mass of mixed liquor volatile suspended solids ($MLVSS$) per day for endogenous decay.

How do influent conservative (inert) suspended solids ($X_{i,in}$) accumulate inside the aeration basin ($X_i$)?

$X_i = \frac{\theta_c}{\theta} X_{i,in}$, meaning inert solids concentrate directly by the ratio of solids retention time to hydraulic retention time.

In the Choduba and Tucek model for activated sludge solids, what do the parameters $\beta$ and $f_M$ designate?

$\beta$ represents the proportion of organic biomass that is non-biodegradable ($\frac{X_{b,N}}{X_{b,D} + X_{b,N}} = 0.1\text{--}0.2$), and $f_M$ represents the mineral (inorganic) fraction of new cells ($\frac{X_{b,M}}{X_b}$).

How do standard aggregate measures ($VSS$ and $TSS$) relate quantitatively to the actual active cell mass ($X_a$) in activated sludge?

For activated sludge, volatile suspended solids are approximately $VSS = 2.5 X_a$, and total suspended solids are approximately $TSS = 7 X_a$ (in anaerobic systems, $TSS = 10\text{ to }100 X_a$).

What effect does increasing solids residence time ($\theta_c$) have on total sludge production and total oxygen demand?

Increasing $\theta_c$ decreases total sludge production (because endogenous decay dominates at longer SRTs) and increases oxygen demand (because more biomass is consumed and oxidized internally).

What happens to the steady-state effluent substrate concentration ($S_e$) when the solids residence time ($\theta_c$) is increased?

$S_e$ decreases asymptotically toward $S_{\min}$ according to the equation $S_e = \frac{(\theta_c b + 1)K_s}{\theta_c Y \hat{q} - (\theta_c b + 1)}$, because a longer cell residence time allows for more complete biological substrate conversion.

What happens to the total net solids production rate ($Q_w X_r + Q X_e$) when solids residence time ($\theta_c$) is increased?

Solids production decreases because at higher $\theta_c$, endogenous decay ($b X V$) consumes and oxidizes a greater proportion of the synthesized biomass according to $\frac{X V}{\theta_c} = Y Q(S_0 - S_e) - b X V$.

What happens to the total daily oxygen requirement ($\text{O}_2\text{ required}$) when solids residence time ($\theta_c$) is increased?

The oxygen requirement increases according to $\text{O}_2\text{ required} = \frac{Q(S_0 - S_e)}{0.68} - 1.42 \frac{X V}{\theta_c}$, because less organic matter leaves the system unoxidized as waste biomass and more undergoes internal endogenous respiration.

What happens to the minimum solids residence time required to prevent washout ($\theta_x^{\min}$) if the half-velocity constant ($K_s$) increases?

$\theta_x^{\min}$ increases according to $\theta_x^{\min} = \frac{K_s + S_0}{S_0(Y \hat{q} - b) - b K_s}$, because a higher $K_s$ corresponds to a lower substrate affinity, requiring more reaction time to balance growth with biomass loss.

What happens to the required recycle flow rate ($Q_r$) when the return sludge concentration ($X_r$) increases at a fixed aeration basin concentration ($X$)?

$Q_r$ decreases according to the recirculation ratio equation $\frac{Q_r}{Q} = \frac{X}{X_r - X}$, because a thicker underflow stream returns the needed mass of organisms at a smaller volumetric flow rate.

What happens to the required recycle flow rate ($Q_r$) when the target aeration basin biomass concentration ($X$) is increased at fixed $Q$ and $X_r$?

$Q_r$ increases according to $\frac{Q_r}{Q} = \frac{X}{X_r - X}$, because raising $X$ increases the numerator while reducing the denominator $(X_r - X)$, demanding a higher recirculation rate.

What happens to the steady-state aeration basin biomass concentration ($X$) when influent substrate concentration ($S_0$) increases at fixed $\theta_c$ and $\theta_d$?

$X$ increases linearly according to $X = \frac{\theta_c}{\theta_d} \frac{Y(S_0 - S_e)}{1 + b \theta_c}$, since a larger mass of incoming organic substrate provides more energy to support and grow a higher microbial population.

What happens to aeration basin biomass concentration ($X$) when reactor volume ($V$) increases at a constant flow rate ($Q$) and constant $\theta_c$?

$X$ decreases because increasing $V$ raises the hydraulic residence time ($\theta_d = \frac{V}{Q}$), lowering the concentration ratio $\frac{\theta_c}{\theta_d}$ in the expression $X = \frac{\theta_c}{\theta_d} \frac{Y(S_0 - S_e)}{1 + b \theta_c}$.

What happens to the concentration of conservative (inert) suspended solids ($X_i$) in the aeration basin when the ratio $\frac{\theta_c}{\theta}$ increases?

$X_i$ increases proportionally according to $X_i = \frac{\theta_c}{\theta} X_{i,in}$, meaning inert particles concentrate significantly within the basin relative to influent levels.

What happens to the minimum sustainable substrate concentration ($S_{\min}$) if the microbial endogenous decay coefficient ($b$) increases?

$S_{\min}$ increases according to $S_{\min} = \frac{b K_s}{Y \hat{q} - b}$, meaning faster internal cell decay requires higher ambient substrate concentrations to prevent negative net growth.

In activated sludge mass balance equations, what parameter does the symbol $X$ represent?

The active biomass concentration in the aeration basin, typically quantified as mixed liquor volatile suspended solids ($MLVSS$) or mixed liquor suspended solids ($MLSS$).

In biological wastewater modeling, what parameter does $S$ (or $S_e$) represent?

The soluble biodegradable organic substrate concentration (typically measured as soluble $BOD_5$ or soluble $COD$) present in the reactor or leaving in the effluent.

In activated sludge operations, what parameter is designated by $\theta_c$ (or $\theta_x$)?

The solids residence time ($SRT$), also called mean cell residence time ($MCRT$) or sludge age, which measures the average time that microbial biomass remains in the treatment system.

In reactor hydraulics, what parameter is represented by $\theta_d$ (or $\theta$)?

The hydraulic residence time ($HRT$), calculated as the aeration tank volume divided by the influent volumetric flow rate ($\theta_d = \frac{V}{Q}$), representing the average time liquid stays within the aeration tank.

In biological reaction kinetics, what does the parameter $K_s$ represent?

The half-velocity constant (or half-saturation constant), defined as the substrate concentration at which the specific substrate utilization rate reaches half of the maximum rate ($\frac{\hat{q}}{2}$).

In Monod kinetic equations, what parameter does $Y$ represent?

The microbial yield coefficient, which represents the mass of active biomass synthesized per unit mass of organic substrate consumed.

In activated sludge cell mass balances, what does the coefficient $b$ (or $k_e$ / $k_d$) represent?

The endogenous decay coefficient, representing the fraction of biomass lost per unit time due to internal maintenance, cell lysis, and endogenous respiration.

In biological substrate kinetics, what does the parameter $\hat{q}$ (or $k$) denote?

The maximum specific substrate utilization rate, defined as the maximum mass of substrate metabolized per unit mass of active biomass per unit time.

In an activated sludge system with settling and recycle, what is the primary advantage of decoupling the solids residence time ($\theta_x$) from the hydraulic residence time ($\theta_d$)?

It allows operators to maintain a large $\theta_x$ to achieve high substrate removal efficiency while keeping a small hydraulic retention time ($\theta_d = \frac{V}{Q}$), thereby substantially reducing the required aeration basin volume.

What are typical ranges for solids residence time ($\theta_x$) and hydraulic residence time ($\theta_d$) in municipal wastewater treatment?

Typical $\theta_x$ ranges from $4\text{ to }10\text{ d}$, whereas typical $\theta_d$ ranges from $4\text{ to }10\text{ h}$.

What boundary is defined by the red dashed line in this activated sludge schematic, and what key mass assumptions are applied to the secondary clarifier?

The red dashed line represents the overall system boundary for mass balances around the aeration basin and settler; the key assumptions are negligible biomass in the settler and that soluble substrate biodegradation occurs exclusively within the aeration basin.

What is the mathematical definition of solids residence time ($\theta_x$ or $\theta_c$) for an activated sludge system where biomass is concentrated only in the aeration basin?

$\theta_x = \theta_c = \frac{XV}{(Q - Q_w)X_e + Q_w X_w}$ (or $\frac{XV}{(Q - Q_w)X_e + Q_w X_r}$ when wasting from the underflow).

What is the steady-state equation relating effluent substrate concentration ($S_e$) to solids residence time ($\theta_x$) and kinetic coefficients?

$$S_e = \frac{(\theta_x b + 1)K_s}{\theta_x Y \hat{q} - (\theta_x b + 1)}$$

Does steady-state effluent substrate concentration ($S_e$) in a completely mixed activated sludge system depend on hydraulic residence time ($\theta_d$) or aeration tank volume ($V$)?

No, $S_e$ depends exclusively on the solids residence time ($\theta_x$) and kinetic parameters ($Y$, $\hat{q}$, $K_s$, $b$), and does not depend on $\theta_d$ or $V$.

What is the steady-state expression for biomass concentration ($X$) in the aeration basin of a CSTR with cell recycle?

$$X = \frac{\theta_c}{\theta} \frac{Y(S_0 - S_e)}{1 + b\theta_c}$$

What are the two equivalent formulations for the specific substrate utilization rate ($U$)?

$$U = \frac{\hat{q} S}{K_s + S} = \frac{S_0 - S}{X \theta}$$

What is biomass washout, and what formula defines the minimum solids residence time ($\theta_x^{\min}$) required to prevent it?

Washout occurs when $\theta_x$ is too short for cell growth to balance cell loss, resulting in $S_e = S_0$ and $X = 0$. It is defined by $\theta_x^{\min} = \frac{K_s + S_0}{S_0(Y\hat{q} - b) - b K_s}$.

What is the theoretical lower limit of the minimum solids residence time ($[\theta_x^{\min}]_{\lim}$) as influent substrate concentration approaches infinity ($S_0 \rightarrow \infty$)?

$$[\theta_x^{\min}]_{\lim} = \frac{1}{Y\hat{q} - b}$$

What is $S_{\min}$, and what equation gives its value as $\theta_x \rightarrow \infty$?

$S_{\min}$ is the minimum substrate concentration that can sustain a steady-state microbial population; below this concentration, net specific growth is negative and biomass disappears. It is defined by $S_{\min} = \frac{b K_s}{Y\hat{q} - b}$.

How is the microbiological safety factor ($SF$) defined in activated sludge design, and what values are typical?

$SF = \frac{\theta_x}{\theta_x^{\min}}$, with typical values ranging from $5$ to several hundreds ($2\text{--}20$ for High Rate, $20\text{--}70$ for Moderate Rate, and $100+$ for Low Rate).

Under the assumption that effluent biomass ($X_e$) and waste flow ($Q_w$) are negligible, what expression defines the recirculation ratio ($R = \frac{Q_r}{Q}$)?

$$\frac{Q_r}{Q} = \frac{X}{X_r - X}$$

What formula defines the solids loading rate ($SLR$) applied to a secondary settling tank of area $A$?

$$SLR = \frac{(Q + Q_r)X}{A}$$

What is the mass balance relationship between waste sludge production ($Q_w X_r$) and total net solids production ($\frac{XV}{\theta_c}$)?

$Q_w X_r = \frac{XV}{\theta_c} - Q X_e$, where total net solids production is $\frac{XV}{\theta_c} = Q_w X_r + Q X_e = Y Q(S_0 - S) - b XV$.

What equation gives the steady-state oxygen required based on a $BOD_L$ mass balance when using $BOD_5$ data with a ratio of $\frac{BOD_5}{BOD_L} = 0.68$?

$$O_2\text{ required} = \frac{Q(S_0 - S_e)}{0.68} - 1.42 \frac{XV}{\theta_c}$$

What is the theoretical oxygen equivalent factor of biological cells ($VSS$) based on complete oxidation ($C_5H_7O_2N + 5O_2 \rightarrow 5CO_2 + NH_3 + 2H_2O$)?

$1.42\text{ mg } O_2/\text{mg } VSS$ (or $1.42\text{ kg } O_2/\text{kg } VSS$).

What is the stoichiometric oxygen requirement for nitrification of ammonia ($NH_4^+ + 2O_2 \rightarrow NO_3^- + 2H^+ + H_2O$)?

$4.57\text{ mg } O_2/\text{mg } N$ (calculated from $\frac{2 \times 32}{14}$).

What activated sludge configuration is depicted in this process flow schematic?

Step-aeration (step feeding) activated sludge, where influent wastewater is introduced at multiple points along the length of the aeration basin to even out organic loading and oxygen demand.

Which activated sludge modification is shown in this schematic diagram?

Contact stabilization process, which uses a contact tank ($15\text{--}60\text{ min}$ retention) to remove easily degradable organics followed by clarification and a stabilization tank to aerate and oxidize settled solids.

What are the primary operational trade-offs of the contact stabilization process?

It requires a smaller total aeration volume due to concentrated biomass ($X$) in the stabilization tank, but is more susceptible to shock loads, has little or no nitrification capability, and requires greater operational skill.

What are the main engineering characteristics and limitations of conventional plug-flow activated sludge?

It features long narrow aeration tanks ($L:W > 5$) with aeration throughout; it provides the highest theoretical substrate removal rate within a given time, but is susceptible to shock and toxic loads due to limited dilution and exhibits steep substrate and oxygen demand gradients.

In this graph showing substrate and dissolved oxygen (DO) uptake along reactor length, what do the solid, small-dash, and large-dash curves represent?

The solid lines represent plug flow ($PF$), the small-dash lines represent step aeration ($SA$), and the large-dash horizontal lines represent continuous-stirred tank ($CSTR$) reactors.

What is tapered aeration in an activated sludge system?

A modification of plug-flow aeration where air supply is reduced progressively along the length of the basin to match the declining substrate concentration and oxygen uptake rate.

What are the primary advantages and operational disadvantages of pure oxygen activated sludge systems?

They support high organic loading rates and higher biomass ($MLSS$) concentrations, but carry high operating costs for oxygen generation, require covered tanks, and risk dropping pH due to $CO_2$ accumulation in the headspace.

In the Metcalf & Eddy design example with total effluent $BOD_5 = 20\text{ mg/L}$, $X_e = 22\text{ mg/L } VSS$ ($65\%$ biodegradable), and $\frac{BOD_5}{BOD_L} = 0.68$, what is the soluble effluent $BOD_5$ ($S_e$)?

$S_e = 6.2\text{ mg/L } BOD_5$. Biodegradable $VSS = 22 \times 0.65 = 14.3\text{ mg/L}$; particulate $BOD_5 = 14.3 \times 1.42 \times 0.68 = 13.8\text{ mg/L}$; and soluble $BOD_5 = 20 - 13.8 = 6.2\text{ mg/L}$.

In this linearized cell mass balance plot of $\frac{1}{\theta_c}$ versus $U$, what parameters do the slope and y-intercept represent?

The slope of the line equals the microbial yield coefficient ($Y$), and the y-intercept equals $-b$ (the negative endogenous decay coefficient).

In this Lineweaver-Burk style plot of $\frac{1}{U}$ versus $\frac{1}{S}$, what kinetic parameters are obtained from the slope, y-intercept, and x-intercept?

The slope is $\frac{K_s}{\hat{q}}$, the y-intercept is $\frac{1}{\hat{q}}$, and the negative x-intercept equals $-\frac{1}{K_s}$.

How can oxygen requirements be calculated directly from specific oxygen uptake rate measurements without needing the $BOD_5/BOD_L$ ratio?

Using the expression $O_2\text{ required} = A \cdot Q(S_0 - S) + B \cdot XV$, where $A$ is the oxygen used per unit mass of substrate removed for synthesis, and $B$ is the oxygen used per unit mass of mixed liquor volatile suspended solids ($MLVSS$) per day for endogenous decay.

How do influent conservative (inert) suspended solids ($X_{i,in}$) accumulate inside the aeration basin ($X_i$)?

$X_i = \frac{\theta_c}{\theta} X_{i,in}$, meaning inert solids concentrate directly by the ratio of solids retention time to hydraulic retention time.

In the Choduba and Tucek model for activated sludge solids, what do the parameters $\beta$ and $f_M$ designate?

$\beta$ represents the proportion of organic biomass that is non-biodegradable ($\frac{X_{b,N}}{X_{b,D} + X_{b,N}} = 0.1\text{--}0.2$), and $f_M$ represents the mineral (inorganic) fraction of new cells ($\frac{X_{b,M}}{X_b}$).

How do standard aggregate measures ($VSS$ and $TSS$) relate quantitatively to the actual active cell mass ($X_a$) in activated sludge?

For activated sludge, volatile suspended solids are approximately $VSS = 2.5 X_a$, and total suspended solids are approximately $TSS = 7 X_a$ (in anaerobic systems, $TSS = 10\text{ to }100 X_a$).

What effect does increasing solids residence time ($\theta_c$) have on total sludge production and total oxygen demand?

Increasing $\theta_c$ decreases total sludge production (because endogenous decay dominates at longer SRTs) and increases oxygen demand (because more biomass is consumed and oxidized internally).

What happens to the steady-state effluent substrate concentration ($S_e$) when the solids residence time ($\theta_c$) is increased?

$S_e$ decreases asymptotically toward $S_{\min}$ according to the equation $S_e = \frac{(\theta_c b + 1)K_s}{\theta_c Y \hat{q} - (\theta_c b + 1)}$, because a longer cell residence time allows for more complete biological substrate conversion.

What happens to the total net solids production rate ($Q_w X_r + Q X_e$) when solids residence time ($\theta_c$) is increased?

Solids production decreases because at higher $\theta_c$, endogenous decay ($b X V$) consumes and oxidizes a greater proportion of the synthesized biomass according to $\frac{X V}{\theta_c} = Y Q(S_0 - S_e) - b X V$.

What happens to the total daily oxygen requirement ($\text{O}_2\text{ required}$) when solids residence time ($\theta_c$) is increased?

The oxygen requirement increases according to $\text{O}_2\text{ required} = \frac{Q(S_0 - S_e)}{0.68} - 1.42 \frac{X V}{\theta_c}$, because less organic matter leaves the system unoxidized as waste biomass and more undergoes internal endogenous respiration.

What happens to the minimum solids residence time required to prevent washout ($\theta_x^{\min}$) if the half-velocity constant ($K_s$) increases?

$\theta_x^{\min}$ increases according to $\theta_x^{\min} = \frac{K_s + S_0}{S_0(Y \hat{q} - b) - b K_s}$, because a higher $K_s$ corresponds to a lower substrate affinity, requiring more reaction time to balance growth with biomass loss.

What happens to the required recycle flow rate ($Q_r$) when the return sludge concentration ($X_r$) increases at a fixed aeration basin concentration ($X$)?

$Q_r$ decreases according to the recirculation ratio equation $\frac{Q_r}{Q} = \frac{X}{X_r - X}$, because a thicker underflow stream returns the needed mass of organisms at a smaller volumetric flow rate.

What happens to the required recycle flow rate ($Q_r$) when the target aeration basin biomass concentration ($X$) is increased at fixed $Q$ and $X_r$?

$Q_r$ increases according to $\frac{Q_r}{Q} = \frac{X}{X_r - X}$, because raising $X$ increases the numerator while reducing the denominator $(X_r - X)$, demanding a higher recirculation rate.

What happens to the steady-state aeration basin biomass concentration ($X$) when influent substrate concentration ($S_0$) increases at fixed $\theta_c$ and $\theta_d$?

$X$ increases linearly according to $X = \frac{\theta_c}{\theta_d} \frac{Y(S_0 - S_e)}{1 + b \theta_c}$, since a larger mass of incoming organic substrate provides more energy to support and grow a higher microbial population.

What happens to aeration basin biomass concentration ($X$) when reactor volume ($V$) increases at a constant flow rate ($Q$) and constant $\theta_c$?

$X$ decreases because increasing $V$ raises the hydraulic residence time ($\theta_d = \frac{V}{Q}$), lowering the concentration ratio $\frac{\theta_c}{\theta_d}$ in the expression $X = \frac{\theta_c}{\theta_d} \frac{Y(S_0 - S_e)}{1 + b \theta_c}$.

What happens to the concentration of conservative (inert) suspended solids ($X_i$) in the aeration basin when the ratio $\frac{\theta_c}{\theta}$ increases?

$X_i$ increases proportionally according to $X_i = \frac{\theta_c}{\theta} X_{i,in}$, meaning inert particles concentrate significantly within the basin relative to influent levels.

What happens to the minimum sustainable substrate concentration ($S_{\min}$) if the microbial endogenous decay coefficient ($b$) increases?

$S_{\min}$ increases according to $S_{\min} = \frac{b K_s}{Y \hat{q} - b}$, meaning faster internal cell decay requires