Steady-State Methodology for Activated Sludge Model 1 (ASM1) State Variable Calculation in MBR

Abstract

The complexity of Activated Sludge Model No. 1 (ASM1) is one of the main obstacles slowing its widespread use, particularly among wastewater treatment plant (WWTP) professionals. In this paper, a simplification procedure based on steady-state mass balances is proposed for the conventional activated sludge process (ASP) configuration, consisting of an aerated bioreactor and a perfect settler (without particular compounds in the outlet). The results do, in fact, show perfect suitability to a membrane bioreactor process (MBR). Both organic carbon and nitrogen removal were investigated. The proposed approach was applied to ASM1, and simple analytical expressions of the state variables were obtained. These analytical expressions were then validated by comparison to simulations given by the original ASM1 (implemented in GPS-X software). A strong match (less than 4% of error overall) was obtained between both results in the steady-state; consequently, these analytical expressions may be useful as tools for quickly estimating the main state variables, feeding the filtration models, or identifying the interaction between operating parameters. Moreover, this enables a sensitivity analysis, covering relevant factors such as kinetics or operating parameters. For instance, the sludge retention time (SRT) effect is lower on X_BH and X_S at high SRT (˃20 days), while it is more pronounced on X_P and X_I as their variations with SRT are linear.

1. Introduction

The importance of modeling activated sludge processes (ASPs) for domestic wastewater treatment has increased in recent decades, particularly with the advent of the widely accepted activated sludge models (ASMs) [1]. ASMs have shown promising predictive results, but at the expense of complex process models [2]. Even with the use of the continuously stirred tank reactor (CSTR) assumption, a highly nonlinear system of ordinary differential equations is obtained [3]. This complexity is one of the main obstacles holding up the widespread us of ASMs, particularly among WWTP professionals asking for model simplification [2,3]. Since a full model is not always necessary for all users and all applications, various application-motivated ASM simplification studies have been conducted, all concerning only Activated Sludge Model No. 1 (ASM1) [4]. Different strategies for ASM1 simplification have been adopted, such as model dimension reduction [5,6,7], linearization of reaction rates [8,9], or whole mass balance equations [10].

Furthermore, to efficiently control the process, a sensitivity analysis is required to identify the most influential parameters. The most common method consists of making a variation of one parameter at a time in a specific range, generally at +10% and −10% of the parameter’s usual value, and then quantifying the influence of this change for all state variables. However, this method results in a high number of simulations to perform, with, most often, long calculation times [11]. Thus, the International Water Association has launched a task group on membrane bioreactor (MBR) modeling and control, which works on multiobjective performance assessment, to enhance good MBR modeling [12].

In this paper, another simplification procedure based on steady-state mass balances is proposed for ASM1. It provides the analytical expressions of the main state variables once stabilized operation is reached. The objective is to explain, step by step, the methodology used to obtain analytical expressions for a very common ASP configuration used in WWTP (aerated CSTR) in order to provide a rapid estimation of the sludge composition and active biomass concentrations. The impact of each kinetic, stoichiometric, or operating parameter on a state variable is directly given, without a time-consuming sensitivity analysis. These expressions can be used mainly by domestic wastewater professionals as practical but still quite accurate tools for new plant design or for existing WWTP control. In addition, they may be adopted to simply feed the fouling model of MBR units.

2. Materials and Methods

2.1. Biological Pathway and Variables

In this study, ASM1 was adopted. All relationships between state variables are described in the Petersen matrix [4]. This matrix enables the writing of each state variable mass balance depending on the plant configuration. For this research, a basic but very common ASP configuration was implemented. It consists of a single aerated continuously stirred bioreactor, followed by a settling tank with a sludge recycle line from the settler to the bioreactor (Figure 1). Total retention of particulate compounds was assumed in the settler. Hence, the system could be assimilated to an MBR as perfect solid/soluble separation is ensured. Only aerobic processes associated with organic carbon removal and nitrification takes place within the bioreactor. Despite the importance of hydrodynamic phenomena in ASP modeling, hydrodynamics effects were neglected; the bioreactor and the settler were treated as CSTRs [2] (Figure 1).

As was common in previous studies, it was assumed that there were (i) no limitations in the dissolved oxygen (S_O), ammonia (required for biomass growth), or alkalinity, and (ii) no denitrification. These assumptions lead to obtaining biomass concentrations (X_BH, X_BA) in relation to their respective substrate (COD and ammonium) instead of reducing biomass growth due to alkalinity or nitrogen. Subsequently, the settler presented complete retention of the suspended solids and, of course, active biomass (X_BA and X_BH). As it was assumed that all particulate compounds were entirely retained, the equations developed are perfectly adapted to MBR. The Petersen matrix used is the one developed for ASM1. The following figure summarises the biological pathway (Figure 2).

In addition, it was assumed that all particulate compounds were entirely retained by the settler, meaning that these compounds were totally recycled to the bioreactor. For this reason, mass balances of state variables were applied only for the bioreactor. The set of differential equations comes from Henze et al. [4]. Soluble compounds were linked to hydraulic retention time (HRT), whereas particular compounds were linked to sludge retention time (SRT).

2.2. Operating Parameters

Based on ASM1, analytical expressions of state variables in the steady-state operation were developed. To validate the obtained expressions, simulations using full ASM1 were conducted by means of the GPS-X software developed by Hydromantis in order to generate realistic data. Simulation conditions are presented in

Table 1. Simulation operating conditions for expression validation (HRT = 0.33 day).

Condition	SNHi (mgN/L)	SSi (mgCOD/L)	OLR (gCOD/L/day)	COD/N (-)	SRT (day)
1–4	20	200	1.2	11.43	60, 40, 20, 10
5–8	45	500	2.1	11.67	60, 40, 20, 10
9–12	45	200	1.2	6.67	60, 40, 20, 10
13–16	70	800	3	11.76	60, 40, 20, 10
17–20	70	500	2.1	8.24	60, 40, 20, 10
21–24	70	200	1.2	4.71	60, 40, 20, 10

, demonstrating how influent concentration varied in the soluble substrate (S_S,i and S_NH,i). The additional substrates were fixed to X_Si = 200 mgCOD/L, X_NDi = 10 mgN/L, and S_NDi = 5 mgN/L.

After that, results given by the analytical expressions and simulations were compared. ASM1 default parameters at 20 °C, given in

Table 2. ASM1 default parameters under aerobic conditions at 20 °C [4].

Parameter		Symbol	Unit	Value
Stoichiometric parameters	Heterotrophic yield	YH	gCOD/gCOD	0.67
	Autotrophic yield	YA	gCOD/gN	0.24
	Fraction of biomass yielding particulate products	fP	-	0.08
	Nitrogen content of the biomass	iXB	gN/gCOD	0.086
	Nitrogen content of the inert lysis products	iXP	gN/gCOD	0.06
Kinetic parameters	Heterotrophic maximum specific growth rate	µH	1/day	6
	Heterotrophic decay rate	bH	1/day	0.62
	Autrotrophic maximum specific growth rate	µA	1/day	0.8
	Autrotrophic decay rate	bA	1/day	0.18
	Half-saturation coefficient for heterotrophic biomass	KS	gCOD/m3	20
	Maximum specific hydrolysis rate	kh	gCOD/gCOD/day	3
	Half-saturation coefficient for hydrolysis of XS	KX	gCOD/gCOD	0.03

, were used.

2.3. Methodology

2.3.1. Internal Loop Management

According to the biological pathway (Figure 2), a fraction of heterotrophic biomass is generated from autotrophic cell lysis products. In the same way, a fraction of autotrophic biomass is generated from heterotrophic cell lysis products. To reduce the complexity of mass balances, it is crucial to evaluate the contribution of each population fraction growing on the other population residues to the total concentration—in other words, to estimate the two ratios i_XBH|XBA and i_XBA|XBH given by Equations (1) and (2):

$${\mathrm{i}}_{{\mathrm{X}}_{{\mathrm{BH}|\mathrm{X}}_{\mathrm{BA}}}}=\frac{{\mathrm{X}}_{{\mathrm{BH}|\mathrm{X}}_{\mathrm{BA}}}}{{\mathrm{X}}_{{\mathrm{BH}|\mathrm{COD}}_{\mathrm{i}}}{+ \mathrm{X}}_{{\mathrm{BH}|\mathrm{X}}_{\mathrm{BA}}}}$$

(1)

$${\mathrm{i}}_{{\mathrm{X}}_{{\mathrm{BA}|\mathrm{X}}_{\mathrm{BH}}}}=\frac{{\mathrm{X}}_{{\mathrm{BA}|\mathrm{X}}_{\mathrm{BH}}}}{{\mathrm{X}}_{{\mathrm{BA}|\mathrm{N}}_{\mathrm{i}}}{+ \mathrm{X}}_{{\mathrm{BA}|\mathrm{X}}_{\mathrm{BH}}}}$$

(2)

where X_BH|XBA is the amount of heterotrophic biomass growing on the particulate organic substrate (X_S) generated during autotrophic biomass decay, and X_BH|CODi is the amount of heterotrophic biomass growing on the organic substrate present in the influent (COD_i).

In order to calculate X_BH|XBA, we had to isolate the heterotrophic biomass growing on the debris of the autotrophic biomass from the heterotrophic biomass growing on the organic substrate present in the influent. For this purpose, the same basic ASP system described above (Figure 1) was used with an influent consisting only of nitrogen compounds, i.e., no organic carbon substrate was present.

In steady-state conditions, mass balance on ammonium is given by

$$\mathrm{Q}\left({\mathrm{S}}_{{\mathrm{NH}}_{\mathrm{i}}}{+\mathrm{S}}_{{\mathrm{ND}}_{\mathrm{i}}}{+\mathrm{X}}_{{\mathrm{ND}}_{\mathrm{i}}}\right)+\left({\mathrm{i}}_{\mathrm{XB}}-{\mathrm{f}}_{\mathrm{p}}{\mathrm{i}}_{\mathrm{XP}}\right){\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}{\mathrm{V}=\mathsf{\mu }}_{\mathrm{Am}}\left({\mathrm{i}}_{\mathrm{XB}}+\frac{1}{{\mathrm{Y}}_{\mathrm{A}}}\right)\left(\frac{{\mathrm{S}}_{\mathrm{NH}}}{{\mathrm{S}}_{\mathrm{NH}}{+ \mathrm{K}}_{\mathrm{NH}}}\right){\mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}\mathrm{V}$$

(3)

where Q is the influent flow rate (L/day), and V is the bioreactor volume (L).

The term i_XB-f_Pi_XP represents the amount of particulate organic nitrogen (X_ND) produced during active biomass decay. It should be mentioned that according to Equation (3), ammonium transformation from X_ND was total and not limited by hydrolysis and ammonification rates.

Mass balance on the autotrophic biomass is described as follows:

$${\mathsf{\mu }}_{\mathrm{Am}}\frac{{\mathrm{S}}_{\mathrm{NH}}}{{\mathrm{S}}_{\mathrm{NH}}{+ \mathrm{K}}_{\mathrm{NH}}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}{\mathrm{V} = \mathrm{Q}}_{\mathrm{W}}{ \mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}+{\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}\mathrm{V}$$

(4)

where Q_W is the waste sludge flow rate (L/day).

Equation (4) means that the autotrophic biomass concentration is stabilized when the growth rate is compensated by the wastage and the decay rate.

From Equation (4), the following can be produced:

$${\mathsf{\mu }}_{\mathrm{Am}}\frac{{\mathrm{S}}_{\mathrm{NH}}}{{\mathrm{S}}_{\mathrm{NH}}{+ \mathrm{K}}_{\mathrm{NH}}}{= \mathrm{b}}_{\mathrm{A}}+\frac{1}{\mathrm{SRT}}$$

(5)

According to Equations (3) and (5), autotrophic biomass concentration can be expressed by Equation (6):

$${\mathrm{X}}_{\mathrm{BA}{|\mathrm{N}}_{\mathrm{i}}}= \frac{{\mathrm{Y}}_{\mathrm{A}}\left({\mathrm{S}}_{{\mathrm{NH}}_{\mathrm{i}}}+{\mathrm{S}}_{{\mathrm{ND}}_{\mathrm{i}}}+{\mathrm{X}}_{{\mathrm{ND}}_{\mathrm{i}}}\right) \mathrm{SRT}/\mathrm{HRT}}{\left({1+\mathrm{Y}}_{\mathrm{A}}{\mathrm{i}}_{\mathrm{XB}}\right)\left({1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\right)-{\mathrm{Y}}_{\mathrm{A}}{\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\left({\mathrm{i}}_{\mathrm{XB}}-{\mathrm{f}}_{\mathrm{p}}{\mathrm{i}}_{\mathrm{XP}}\right)}$$

(6)

Meanwhile, mass balance on the heterotrophic biomass is given by Equation (7):

$${\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}{\mathrm{V} = \mathrm{Q}}_{\mathrm{W}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{X}}_{\mathrm{BA}}}{+ \mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{X}}_{\mathrm{BA}}}\mathrm{V}$$

(7)

Consequently, heterotrophic biomass concentration growing on autotrophic biomass debris is determined as follows:

$${\mathrm{X}}_{\mathrm{BH}|{\mathrm{X}}_{\mathrm{BA}}}= \frac{{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}}{{1 + \mathrm{b}}_{\mathrm{H}}\mathrm{SRT}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{N}}_{\mathrm{i}}}$$

(8)

For the calculation of X_BA|XBH, the same approach was used to isolate the autotrophic biomass growing on the debris of the heterotrophic biomass. This time, the influent consisted only of organic carbon substrate. However, it was assumed that there was enough ammonium in the bioreactor for biomass growth. Therefore, mass balance on the organic carbon substrate is shown in Equation (9):

$$\mathrm{Q}\left({\mathrm{S}}_{{\mathrm{S}}_{\mathrm{i}}}{+ \mathrm{X}}_{{\mathrm{S}}_{\mathrm{i}}}\right)+\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}\mathrm{V} = \frac{1}{{\mathrm{Y}}_{\mathrm{H}}}{\mathsf{\mu }}_{\mathrm{Hm}}\frac{{\mathrm{S}}_{\mathrm{S}}}{{\mathrm{S}}_{\mathrm{S}}{+ \mathrm{K}}_{\mathrm{S}}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}\mathrm{V}$$

(9)

Concerning the heterotrophic biomass, the mass balance in steady-state operation is depicted in Equation (10):

$${\mathsf{\mu }}_{\mathrm{Hm}}\frac{{\mathrm{S}}_{\mathrm{S}}}{{\mathrm{S}}_{\mathrm{S}}{+ \mathrm{K}}_{\mathrm{S}}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}{\mathrm{V} = \mathrm{Q}}_{\mathrm{W}}{ \mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}+{\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}\mathrm{V}$$

(10)

This results in Equation (11):

$${\mathsf{\mu }}_{\mathrm{Hm}}\frac{{\mathrm{S}}_{\mathrm{S}}}{{\mathrm{S}}_{\mathrm{S}}{+ \mathrm{K}}_{\mathrm{S}}}{= \mathrm{b}}_{\mathrm{H}}+\frac{1}{\mathrm{SRT}}$$

(11)

Based on Equations (9) and (11), the expression of heterotrophic biomass concentration was obtained:

$${\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}=\frac{{\mathrm{Y}}_{\mathrm{H}}\left({\mathrm{S}}_{{\mathrm{S}}_{\mathrm{i}}}{+ \mathrm{X}}_{{\mathrm{S}}_{\mathrm{i}}}\right) \mathrm{SRT}/\mathrm{HRT}}{{ 1+ \mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}$$

(12)

Mass balance on the autotrophic biomass is represented by Equation (13):

$${\mathrm{Y}}_{\mathrm{A}}{\mathrm{i}}_{\mathrm{XP}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}{\mathrm{V} = \mathrm{Q}}_{\mathrm{W}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{X}}_{\mathrm{BH}}}{+ \mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}|{\mathrm{X}}_{\mathrm{BH}}}$$

(13)

where i_XP(1 − f_P)b_H X_BH|CODi is the particulate organic nitrogen amount (X_ND) released during biomass decay.

Thus, autotrophic biomass concentration growing on heterotrophic biomass debris is given by Equation (14):

$${\mathrm{X}}_{\mathrm{BA}|{\mathrm{X}}_{\mathrm{BH}}}=\frac{{\mathrm{Y}}_{\mathrm{A}}{\mathrm{i}}_{\mathrm{XP}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{{1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}}{\mathrm{X}}_{\mathrm{BH}|{\mathrm{COD}}_{\mathrm{i}}}$$

(14)

At last, i_XBH|XBA and i_XBA|XBH can be calculated. Their expressions are functions of different operating conditions: total COD (COD_i) and nitrogenous compounds (N_i) present in the influent, HRT, and SRT. Numerous calculations were conducted to estimate i_XBH|XBA and i_XBA|XBH for various COD_i and SRT values. We varied COD_i to cover a wide range of COD/N ratios encountered in domestic wastewater treatment. It was found that SRT had a negative effect on i_XBA|XBH, unlike COD_i. Exactly the opposite trend was noticed for i_XBH|XBA, with a positive effect of SRT on i_XBH|XBA and a negative effect of COD_i on i_XBH|XBA. More importantly, for all simulations, i_XBH|XBA did not exceed 2%, whereas i_XBA|XBH exceeded 5% in some cases, notably at values of SRT lower than 10 days. In light of these findings, the fraction of heterotrophic biomass X_BH|XBA growing on the particulate organic substrate (X_S) generated by the decay of the autotrophic species was overlooked for the remainder of this study. This means that the autotrophic biomass does not interfere with heterotrophic biomass activity, reducing the interaction between both populations.

2.3.2. Expressions of State Variable Concentrations in the Bioreactor

For the reasons stated earlier, autotrophic lysis products were not accounted for in mass balances related to heterotrophic activity.

Mass balances on the soluble organic substrate (S_S) and the heterotrophic biomass (X_BH) gave Equations (15) and (16), similar to Equations (9) and (10), respectively:

$$\mathrm{Q}\left({\mathrm{S}}_{{\mathrm{S}}_{\mathrm{i}}}{+ \mathrm{X}}_{{\mathrm{S}}_{\mathrm{i}}}\right)+\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}\mathrm{V} = \frac{1}{{\mathrm{Y}}_{\mathrm{H}}}{\mathsf{\mu }}_{\mathrm{Hm}}\frac{{\mathrm{S}}_{\mathrm{S}}}{{\mathrm{S}}_{\mathrm{S}}{+ \mathrm{K}}_{\mathrm{S}}}{\mathrm{X}}_{\mathrm{BH}}\mathrm{V}$$

(15)

$${\mathsf{\mu }}_{\mathrm{Hm}}\frac{{\mathrm{S}}_{\mathrm{S}}}{{\mathrm{S}}_{\mathrm{S}}{+ \mathrm{K}}_{\mathrm{S}}}{\mathrm{X}}_{\mathrm{BH}}{\mathrm{V} = \mathrm{Q}}_{\mathrm{W}}{ \mathrm{X}}_{\mathrm{BH}}+{\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}\mathrm{V}$$

(16)

According to what is indicated in Equation (15), it was supposed that there was no limitation of soluble organic substrate production by the hydrolysis step rate. Furthermore, it is noted that Equation (16) is similar to Equation (11).

Based on Equation (16), the concentration of the soluble organic substrate (S_S) could be determined:

$${\mathrm{S}}_{\mathrm{S}}=\frac{{\mathrm{K}}_{\mathrm{S}}\left({1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)}{{\mathsf{\mu }}_{\mathrm{Hm}}\mathrm{SRT}-\left({1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)}$$

(17)

In addition, Equation (16) permitted the substitution of the switching function related to the soluble organic substrate (S_S). Combining Equations (15) and (16), an expression of the heterotrophic biomass concentration in the bioreactor was obtained:

$${\mathrm{X}}_{\mathrm{BH}}=\frac{{\mathrm{Y}}_{\mathrm{H}}\left({\mathrm{S}}_{{\mathrm{S}}_{\mathrm{i}}}{+ \mathrm{X}}_{{\mathrm{S}}_{\mathrm{i}}}\right) \mathrm{SRT}/\mathrm{HRT}}{{ 1+ \mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}$$

(18)

Moving to autotrophic biomass activity, mass balance on ammonium was slightly different from what was indicated in Equation (3) as particulate organic products of the heterotrophic biomass were included:

$$\mathrm{Q}\left({\mathrm{S}}_{{\mathrm{NH}}_{\mathrm{i}}}{+\mathrm{S}}_{{\mathrm{ND}}_{\mathrm{i}}}{+\mathrm{X}}_{{\mathrm{ND}}_{\mathrm{i}}}\right)+\left({\mathrm{i}}_{\mathrm{XB}}-{\mathrm{f}}_{\mathrm{p}}{\mathrm{i}}_{\mathrm{XP}}\right)\left({\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}+{\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}\right){\mathrm{V}=\mathsf{\mu }}_{\mathrm{Am}}\left({\mathrm{i}}_{\mathrm{XB}}+\frac{1}{{\mathrm{Y}}_{\mathrm{A}}}\right)\left(\frac{{\mathrm{S}}_{\mathrm{NH}}}{{\mathrm{S}}_{\mathrm{NH}}{+ \mathrm{K}}_{\mathrm{NH}}}\right){\mathrm{X}}_{\mathrm{BA}}\mathrm{V}$$

(19)

For autotrophic bacteria, mass balance is given by Equation (20):

$${\mathsf{\mu }}_{\mathrm{Am}}\frac{{\mathrm{S}}_{\mathrm{NH}}}{{\mathrm{S}}_{\mathrm{NH}}{+ \mathrm{K}}_{\mathrm{NH}}}{\mathrm{X}}_{\mathrm{BA}}{\mathrm{V} = \mathrm{Q}}_{\mathrm{W}}{ \mathrm{X}}_{\mathrm{BA}}+{\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}\mathrm{V}$$

(20)

From the latter, the same relation in Equation (5) was found. Equation (20) allows the calculation of ammonia concentration in the bioreactor:

$${\mathrm{S}}_{\mathrm{NH}}=\frac{{\mathrm{K}}_{\mathrm{NH}}\left({1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\right)}{{\mathsf{\mu }}_{\mathrm{Am}}\mathrm{SRT}-\left({1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\right)}$$

(21)

Autotrophic biomass concentration was determined by substituting the ammonium-switching function in Equation (19) using Equation (20):

$${\mathrm{X}}_{\mathrm{BA}}{= \mathrm{Y}}_{\mathrm{A}}\frac{\left({\mathrm{S}}_{{\mathrm{NH}}_{\mathrm{i}}}{+\mathrm{S}}_{{\mathrm{ND}}_{\mathrm{i}}}{+\mathrm{X}}_{{\mathrm{ND}}_{\mathrm{i}}}\right)\mathrm{SRT}/\mathrm{HRT}-{\mathrm{X}}_{\mathrm{BH}}\left({\mathrm{i}}_{\mathrm{XB}}{+\mathrm{f}}_{\mathrm{P}}{\mathrm{i}}_{\mathrm{XP}}{\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)}{{1+\mathrm{i}}_{\mathrm{XB}}{\mathrm{Y}}_{\mathrm{A}}{+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\left({1+\mathrm{Y}}_{\mathrm{A}}{\mathrm{f}}_{\mathrm{P}}{\mathrm{i}}_{\mathrm{XP}}\right)}$$

(22)

The term X_BH(i_XB + f_pi_XPb_HSRT) reveals the influence of heterotrophic lysis products on autotrophic biomass concentration.

Once biomass concentrations were determined, concentration expressions of the remaining state variables could be developed. Starting with nitrate and nitrite concentration (S_NO), mass balance on these compounds is shown in Equation (23):

$${\mathrm{r}}_{{\mathrm{S}}_{\mathrm{NO}}}{\mathrm{V} = \mathrm{Q} \mathrm{S}}_{\mathrm{NO}}$$

(23)

The production rate of nitrogen oxides (r_SNO), corresponding to the ammonium oxidation rate, is dependent on the autotrophic biomass aerobic growth rate (r_g,XBA), as indicated in the ASM1 Petersen matrix:

$${\mathrm{r}}_{{\mathrm{S}}_{\mathrm{NO}}}=\frac{1}{{\mathrm{Y}}_{\mathrm{A}}}{\mathrm{r}}_{{\mathrm{g},\mathrm{X}}_{\mathrm{BA}}}$$

(24)

The concentration of nitrogen oxides in Equation (25) was obtained by a combination of Equations (23) and (24):

$${\mathrm{S}}_{\mathrm{NO}}=\frac{{1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}}{\frac{\mathrm{SRT}}{\mathrm{HRT}}{\mathrm{Y}}_{\mathrm{A}}}{\mathrm{X}}_{\mathrm{BA}}$$

(25)

As for lysis products present in the mixed liquor, particulate inert products are generated during biomass decay and extracted with sludge excess, as indicated in Equation (26):

$${\mathrm{f}}_{\mathrm{P}}\left({\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}{+\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}\right){\mathrm{V}= \mathrm{Q}}_{\mathrm{W}}{\mathrm{X}}_{\mathrm{P}}$$

(26)

Consequently, the concentration of these products (X_P) could be expressed as follows:

$${\mathrm{X}}_{\mathrm{P}}{= \mathrm{f}}_{\mathrm{P}}\left({\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}{+\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}\right)\mathrm{SRT}$$

(27)

Since autotrophic lysis products were assumed to be negligible, it became

$${\mathrm{X}}_{\mathrm{P}}{= \mathrm{f}}_{\mathrm{P}}{\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}\mathrm{SRT}$$

(28)

Meanwhile, the concentration of particulate organic substrate (X_S) is ruled by Equation (29):

$$\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\left({\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}{+\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}\right){\mathrm{V}= \mathrm{k}}_{\mathrm{h}}\frac{{\mathrm{X}}_{\mathrm{S}}/{\mathrm{X}}_{\mathrm{BH}}}{{ \mathrm{X}}_{\mathrm{S}}/{\mathrm{X}}_{\mathrm{BH}}{+\mathrm{K}}_{\mathrm{X} }}{\mathrm{X}}_{\mathrm{BH}}{\mathrm{V}+\mathrm{Q}}_{\mathrm{W}}{\mathrm{X}}_{\mathrm{S}}$$

(29)

where the first term is the production rate, and the second term is the hydrolysis rate. The last term, representing the amount extracted by means of waste sludge, is negligible compared to other terms, particularly for operations at high SRT. Consequently, X_S expression is described by Equation (30):

$${\mathrm{X}}_{\mathrm{S}}=\frac{{\mathrm{K}}_{\mathrm{X}}{\mathrm{X}}_{\mathrm{BH}}}{\frac{{\mathrm{k}}_{\mathrm{h}}{\mathrm{X}}_{\mathrm{BH}}}{\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\left({\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}{+\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}\right)} - 1}$$

(30)

In the instance that autotrophic debris was omitted, the expression of X_S became simpler:

$${\mathrm{X}}_{\mathrm{S}}=\frac{\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{{\mathrm{k}}_{\mathrm{h}}-\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{\mathrm{K}}_{\mathrm{X}}{\mathrm{X}}_{\mathrm{BH}}$$

(31)

The concentration of inert particulates could easily be determined by Equation (32):

$${\mathrm{X}}_{\mathrm{I}}=\frac{\mathrm{SRT}}{\mathrm{HRT}}{\mathrm{X}}_{\mathrm{I},\mathrm{i}}{= \mathrm{C}}_{\mathrm{F}}{\mathrm{X}}_{\mathrm{I},\mathrm{i}}$$

(32)

where C_F is the concentration factor defined as the ratio of SRT by HRT.

In the same way as with inert particulates, it was possible to calculate the concentration of the inert colloidal compound (S_Co; [12]). In fact, as S_Co is slightly impacted by antifouling hydraulic turbulence, it thus builds up on the membrane, leading to a microscale fouling layer:

$${\mathrm{S}}_{\mathrm{Co}}=\frac{\mathrm{SRT}}{\mathrm{HRT}+\mathrm{SRT}\left(1-{\mathrm{M}}_{\mathrm{R}}\right)}{\mathrm{S}}_{\mathrm{Co},\mathrm{i}}$$

(33)

where M_R is the membrane inert colloid retention rate.

3. Results

3.1. Equation Validation and Testing

Developed analytical expressions, summarized in

Table 3. Analytical expressions of state variables of an aerated ASP/MBR assimilated to a continuously stirred tank reactor (CSTR).

State Variable	Analytical Expression
Heterotrophic biomass concentration (gCOD/L)	${\mathrm{X}}_{\mathrm{BH}}=\frac{{\mathrm{Y}}_{\mathrm{H}}\left({\mathrm{S}}_{{\mathrm{S}}_{\mathrm{i}}}{+\mathrm{X}}_{{\mathrm{S}}_{\mathrm{i}}}\right) \mathrm{SRT}/\mathrm{HRT}}{{ 1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{ \mathrm{f}}_{\mathrm{P}}\right)\right)}$
Concentration of Soluble organic compounds (gCOD/L)	${\mathrm{S}}_{\mathrm{S}}=\frac{{\mathrm{K}}_{\mathrm{S}}\left({1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)}{{\mathsf{\mu }}_{\mathrm{Hm}}\mathrm{SRT}-\left({1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)}$
Concentration of particulate organic compounds (gCOD/L)	${\mathrm{X}}_{\mathrm{S}}=\frac{\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{{\mathrm{k}}_{\mathrm{h}}-\left(1-{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{\mathrm{K}}_{\mathrm{X}}{\mathrm{X}}_{\mathrm{BH}}$
Autotrophic biomass concentration (gCOD/L)	${\mathrm{X}}_{\mathrm{BA}}{=\mathrm{Y}}_{\mathrm{A}}\frac{\left({\mathrm{S}}_{{\mathrm{NH}}_{\mathrm{i}}}{+\mathrm{S}}_{{\mathrm{ND}}_{\mathrm{i}}}{+\mathrm{X}}_{{\mathrm{ND}}_{\mathrm{i}}}\right)\mathrm{SRT}/\mathrm{HRT}-{\mathrm{X}}_{\mathrm{BH}}\left({\mathrm{i}}_{\mathrm{XB}}{+\mathrm{f}}_{\mathrm{P}}{\mathrm{i}}_{\mathrm{XP}}{\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)}{{1+\mathrm{i}}_{\mathrm{XB}}{\mathrm{Y}}_{\mathrm{A}}{+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\left({1+\mathrm{Y}}_{\mathrm{A}}{\mathrm{f}}_{\mathrm{P}}{\mathrm{i}}_{\mathrm{XP}}\right)}$
Concentration of inert organic lysis products (gCOD/L)	${\mathrm{X}}_{\mathrm{P}}{=\mathrm{f}}_{\mathrm{P}}\left({\mathrm{b}}_{\mathrm{H}}{\mathrm{X}}_{\mathrm{BH}}{+\mathrm{b}}_{\mathrm{A}}{\mathrm{X}}_{\mathrm{BA}}\right)\mathrm{SRT}$
Soluble ammonia concentration (gN/L)	${\mathrm{S}}_{\mathrm{NH}}=\frac{{\mathrm{K}}_{\mathrm{NH}}\left({1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\right)}{{\mathsf{\mu }}_{\mathrm{Am}}\mathrm{SRT}-\left({1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}\right)}$
Nitrite and nitrate concentration (gN/L)	${\mathrm{S}}_{\mathrm{NO}}=\frac{{1+\mathrm{b}}_{\mathrm{A}}\mathrm{SRT}}{\frac{\mathrm{SRT}}{\mathrm{HRT}}{\mathrm{Y}}_{\mathrm{A}}}{\mathrm{X}}_{\mathrm{BA}}$
Concentration of inert particulates (g/L)	${\mathrm{X}}_{\mathrm{I}}=\frac{\mathrm{SRT}}{\mathrm{HRT}}{\mathrm{X}}_{\mathrm{I},\mathrm{i}}$
Concentration of inert colloidal compounds (gCOD/L)	${\mathrm{S}}_{\mathrm{Co}}=\frac{\mathrm{SRT}}{\mathrm{HRT}+\mathrm{SRT}\left(1-{\mathrm{M}}_{\mathrm{R}}\right)}{\mathrm{S}}_{\mathrm{Co},\mathrm{i}}$

, need to be validated using a data set.

For this purpose, simulations by the GPS-X software were conducted in order to generate realistic data, as explained in Section 2. A number of pseudoexperiments, with widely varying operation conditions, were considered, as stated in Table 1. To be as realistic as possible, the COD/N ratio in the influent, corresponding to the total amount of organic substrates relative to the total amount of nitrogenous compounds present in the influent, must be among the usual range for domestic wastewater, with variable ratio values of less than 12.

For all state variables and reaction rates, simulation results using previously developed analytical expressions were compared to results given by GPS-X simulations using full ASM1 when steady-state was reached. The latter could be identified when all state variables and reaction rates became constant over time. Steady-state is generally obtained after a function period that exceeds three times the SRT value.

An acceptable match was obtained for all state variables, as shown in Figure 3, Figure 4 and Figure 5. The SRT had a positive effect on particular compounds, which increased with rising SRT. The remaining soluble substrates (S_S and S_NH) decreased when SRT increased. This was not the case, however, for S_NO, which increased with SRT because the uptake of ammonia into the biomass decreased with high SRT. Unlike with SRT, increasing HRT had a negative influence on the sludge matrix composition: X_BH, X_BA, X_S, X_P, and X_I were diluted in the influent flow. If there was more substrate (S_S,i), there was a greater rise of particulate organic compound concentrations (X_BH, X_S, and X_P). In fact, the concentration of X_BH, X_S, and X_P are linear with S_S,i.

3.2. Sensitivity Analysis

Following this, a sensitivity analysis of four operating parameters—S_S,i, X_I,i, SRT, and HRT—was performed with the aim of identifying if the developed steady-state expressions had the same sensitivity to these parameters as the full ASM1. For this reason, the normalized sensitivity coefficient, S_ij, giving the relative sensitivity of output i (y_i) respective to an operating parameter, j (x_j), was calculated according to Petersen et al. [13].

$${\mathrm{S}}_{\mathrm{ij}}=\left|\frac{{\mathrm{x}}_{\mathrm{j}}}{{\mathrm{y}}_{\mathrm{i}}}\frac{{\mathrm{dy}}_{\mathrm{i}}}{{\mathrm{dx}}_{\mathrm{j}}}\right|$$

(34)

Depending on the S_ij value, the parameter influence on the outputs could be classified, as shown in

Table 4. Classification of parameter influence according to S_ii value.

Sij Value Range	Influence Classification
Sij ≥ 2	Extremely influential
1 ≤ Sij < 2	Very influential
0.25 ≤ Sij < 1	Influential
Sij < 0.25	Not influential

. The derivatives dy_i and dx_j were determined numerically in a specific range if no analytical expressions were available [14]. Generally, 10% increase (+10%) and 10% decrease (−10%) of one parameter were applied to the calculation of S_ij. In this study, the derivatives were calculated both numerically and analytically thanks to the developed steady-state equations. In the following, the steps of the analytical sensitivity analysis were detailed only for organic compounds in order to minimize the dimension of the tables. It should be mentioned that these analysis steps are still applicable for nitrogen treatment compounds (X_BA, S_NH, and S_NO). Obtained expressions for different derivatives are gathered in

Table 5. Analytical expressions of derivatives of organic compounds based on steady-state equations.

	j	SRT	HRT	SS,i	XI,i
i
XBH		$\frac{{\mathrm{dX}}_{\mathrm{BH}}}{\mathrm{dSRT}}=\frac{\frac{{\mathrm{Y}}_{\mathrm{H}}}{\mathrm{HRT}}\left({\mathrm{S}}_{\mathrm{S},\mathrm{i}}{+\mathrm{X}}_{\mathrm{S},\mathrm{i}}\right)}{ {\left[{1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)\right]}^2}$	$\frac{{\mathrm{dX}}_{\mathrm{BH}}}{\mathrm{dHRT}}=\frac{-{ \mathrm{Y}}_{\mathrm{H}}\left({\mathrm{S}}_{\mathrm{S},\mathrm{i}}{+\mathrm{X}}_{\mathrm{S},\mathrm{i}}\right)\frac{\mathrm{SRT}}{{\mathrm{HRT}}^2}}{{1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}$	$\frac{{\mathrm{dX}}_{\mathrm{BH}}}{{\mathrm{dS}}_{\mathrm{S},\mathrm{i}}}=\frac{{\mathrm{Y}}_{\mathrm{H} }\frac{\mathrm{SRT}}{\mathrm{HRT}}}{{1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}$	$\frac{{\mathrm{dX}}_{\mathrm{BH}}}{{\mathrm{dX}}_{\mathrm{I},\mathrm{i}}}=0$
XS		$\frac{{\mathrm{dX}}_{\mathrm{S}}}{\mathrm{dSRT}}=\frac{\frac{{\mathrm{K}}_{\mathrm{X}}\left(1 -{ \mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{{ \mathrm{k}}_{\mathrm{h}}-\left(1 -{ \mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}\frac{{\mathrm{Y}}_{\mathrm{H}}}{\mathrm{HRT}}\left({\mathrm{S}}_{\mathrm{S},\mathrm{i}}{+\mathrm{X}}_{\mathrm{S},\mathrm{i}}\right)}{ {\left[{1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)\right]}^2}$	$\frac{{\mathrm{dX}}_{\mathrm{S}}}{\mathrm{dHRT}}=\frac{\frac{-{ \mathrm{K}}_{\mathrm{X}}\left(1 -{ \mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{{ \mathrm{k}}_{\mathrm{h}}-\left(1 -{ \mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{\mathrm{Y}}_{\mathrm{H}}\left({\mathrm{S}}_{\mathrm{S},\mathrm{i}}{+\mathrm{X}}_{\mathrm{S},\mathrm{i}}\right)\frac{\mathrm{SRT}}{{\mathrm{HRT}}^2}}{{1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}$	$\frac{{\mathrm{dX}}_{\mathrm{S}}}{{\mathrm{dS}}_{\mathrm{S},\mathrm{i}}}=\frac{\frac{{\mathrm{K}}_{\mathrm{X}}\left(1 -{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{{ \mathrm{k}}_{\mathrm{h}}-\left(1 -{\mathrm{f}}_{\mathrm{P}}\right){\mathrm{b}}_{\mathrm{H}}}{\mathrm{Y}}_{\mathrm{H}}\frac{\mathrm{SRT}}{\mathrm{HRT}}}{{1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}$	$\frac{{\mathrm{dX}}_{\mathrm{S}}}{{\mathrm{dX}}_{\mathrm{I},\mathrm{i}}}=0$
SS		$\frac{{\mathrm{dS}}_{\mathrm{S}}}{\mathrm{dSRT}}=\frac{-{ \mathsf{\mu }}_{\mathrm{H}}{\mathrm{K}}_{\mathrm{S}}}{{\left[\left({\mathsf{\mu }}_{\mathrm{H}}-{\mathrm{b}}_{\mathrm{H}}\right)\mathrm{SRT}-1\right]}^2}$	$\frac{{\mathrm{dS}}_{\mathrm{S}}}{\mathrm{dHRT}}=0$	$\frac{{\mathrm{dS}}_{\mathrm{S}}}{{\mathrm{dS}}_{\mathrm{S},\mathrm{i}}}=0$	$\frac{{\mathrm{dS}}_{\mathrm{S}}}{{\mathrm{dX}}_{\mathrm{I},\mathrm{i}}}=0$
XI		$\frac{{\mathrm{dX}}_{\mathrm{I}}}{\mathrm{dSRT}}=\frac{1}{\mathrm{HRT}}{\mathrm{X}}_{\mathrm{I},\mathrm{i}}$	$\frac{{\mathrm{dX}}_{\mathrm{I}}}{\mathrm{dHRT}}=-\frac{\mathrm{SRT}}{{\mathrm{HRT}}^2}{\mathrm{X}}_{\mathrm{I},\mathrm{i}}$	$\frac{{\mathrm{dX}}_{\mathrm{I}}}{{\mathrm{dS}}_{\mathrm{S},\mathrm{i}}}=0$	$\frac{{\mathrm{dX}}_{\mathrm{I}}}{{\mathrm{dX}}_{\mathrm{I},\mathrm{i}}}=\frac{\mathrm{SRT}}{\mathrm{HRT}}$

. After that, expressions of sensitivity coefficients of state variables, depending on operating parameters, were also determined, as shown in Table 5. According to the steady-state equation, the sensitivity analysis can also be easily performed on stoichiometric and kinetic parameters [15].

The sensitivity analysis are calculated

Table 6. Sensitivity coefficients calculated using developed steady-state equations.

	j	SRT	HRT	SS,i	XI,i
i
XBH		${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{BH}},\mathrm{SRT}}=\frac{1}{{ 1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}=\frac{1}{1+0.24 \mathrm{SRT}}$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{BH}},\mathrm{HRT}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{BH}}{,\mathrm{S}}_{\mathrm{S},\mathrm{i}}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{BH}}{,\mathrm{X}}_{\mathrm{I},\mathrm{i}}}=0$
XS		${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{S}},\mathrm{SRT}}=\frac{1}{{ 1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}=\frac{1}{1+0.24 \mathrm{SRT}}$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{S}},\mathrm{HRT}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{S}}{,\mathrm{S}}_{\mathrm{S},\mathrm{i}}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{S}}{,\mathrm{X}}_{\mathrm{I},\mathrm{i}}}=0$
SS		${\mathrm{S}}_{{\mathrm{S}}_{\mathrm{S}},\mathrm{SRT}}=\frac{-{ \mathsf{\mu }}_{\mathrm{H}}\mathrm{SRT}}{ \left({1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)\left[{\mathsf{\mu }}_{\mathrm{H}}\mathrm{SRT} -\left({1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\right)\right]}$	${\mathrm{S}}_{{\mathrm{S}}_{\mathrm{S}},\mathrm{HRT}}=0$	${\mathrm{S}}_{{\mathrm{S}}_{\mathrm{S}}{,\mathrm{S}}_{\mathrm{S},\mathrm{i}}}=0$	${\mathrm{S}}_{{\mathrm{S}}_{\mathrm{S}}{,\mathrm{X}}_{\mathrm{I},\mathrm{i}}}=0$
XP		${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{P}},\mathrm{SRT}}=\frac{{ 2+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}{{ 1+\mathrm{b}}_{\mathrm{H}}\mathrm{SRT}\left(1-{\mathrm{Y}}_{\mathrm{H}}\left(1-{\mathrm{f}}_{\mathrm{P}}\right)\right)}=1+\frac{1}{1+0.24 \mathrm{SRT}}$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{I}},\mathrm{HRT}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{P}}{,\mathrm{S}}_{\mathrm{S},\mathrm{i}}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{P}}{,\mathrm{X}}_{\mathrm{I},\mathrm{i}}}=0$
XI		${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{I}},\mathrm{SRT}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{I}},\mathrm{HRT}}=1$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{I}}{,\mathrm{S}}_{\mathrm{S},\mathrm{i}}}=0$	${\mathrm{S}}_{{\mathrm{X}}_{\mathrm{I}}{,\mathrm{X}}_{\mathrm{I},\mathrm{i}}}=1$

Grey background show the non-connected variables.

and the results are summarized in

Table 7. Sensitivity analysis of state variables from equation (Equ.) or software (GPS-X) according to operating parameter values.

Operating Parameter j	SRT (day)						HRT (h)		SS,i (mgCOD/L)		XI,i (mg/L)
Nominal Value	10		30		50		8		600		100
Output i	Equ.	GPS-X	Equ.	GPS-X	Equ.	GPS-X	Equ.	GPS-X	Equ.	GPS-X	Equ.	GPS-X
XBH	0.30	0.30	0.12	0.13	0.08	0.08	1.00	1.01	1.00	1.00	0.00	0.00
XS	0.30	0.30	0.12	0.12	0.08	0.08	1.00	1.01	1.00	1.03	0.00	0.00
SS	0.16	0.16	0.06	0.07	0.04	0.05	0.00	0.01	0.00	0.05	0.00	0.00
XP	1.30	1.30	1.12	1.12	1.08	1.08	1.00	1.01	1.00	1.00	0.00	0.00
XI	1.00	1.00	1.00	1.00	1.00	1.01	1.00	1.02	0.00	0.04	1.00	1.00

Grey background are used when sensitivity analysis between the parameter is not influential.

. First of all, it can be noticed that the sensitivity analysis based on the steady-state expressions gave almost the same numerically calculated sensitivity coefficient values for all compounds. S_S,i and HRT did not have any influence on the soluble organic substrate concentration S_S.

It was found that SRT was very influential on X_P concentration only and had a low influence on S_S. Concerning SRT influence on S_S, steady-state expressions gave less accurate S_ij values, but, to some extent, these values are still reliable to quantify the low sludge-age influence on S_S. For the remaining organic state variables (X_BH and X_S), it was revealed that unlike with S_S,i and HRT, the sensitivity of these concentrations to SRT strongly depended on the study value range: SRT was influential at lower values (around 10 days) and was not influential at higher values (higher than 30 days). This influence reduction was expected because of the Monod-type kinetics ruling the active biomass growth rate. However, the range-influence of SRT was clearly revealed in the analytical expression of the normalized sensitivity coefficients S_XBH,SRT and S_XS,SRT, thanks to this approach.

Therefore, steady-state expressions of state variables showed high accuracy for concentration predictions and were able to successfully track the process state sensitivity to operating parameters. Consequently, the developed analytical expressions were validated; hence, they can be used for identification of optimum operating conditions. For this reason, rapid simulations were performed using the validated expressions, with the aim of investigating the impact of each operating parameter on system performance. Nominal values of operating parameters (SRT = 30 day, HRT = 8 h, S_S,i = 600 mgCOD/L, X_I,i = 100 mg/L) were considered for these simulations.

4. Discussion

Analyzing the sensitivity of operating, stoichiometric, and kinetic parameters can be challenging and time-consuming. The steady-state equation provides a better understanding of the meaning of the different process variables and main parameters. It also allows the simplest method of plotting the current interactions between the process and the stoichiometric and kinetic parameters. Due to the complexity of the biological processes and membrane filtration phenomena taking place in MBR-based systems, the steady-state equation presents the optimal way to operate an MBR-based system by providing the composition of the suspended solids impacting the macroscale fouling as the value of colloidal and soluble fractions that feed the microscale fouling layer [16].

It is noteworthy to mention that the analytical expressions developed in this study have a wide applicability range, thanks to the low number of assumptions. This makes it possible to fix the initial condition before starting simulation or to explain the impact of an operating parameter on effluent quality (EQ) or on mixed-liquor suspended solids composition. Moreover, the steady-state equation allows the identification of the most sensitive parameters and provides pointers for initiating the elaboration of procedures for model parameter determination. It could also allow WWTP managers to understand which operating variables/parameters should be assessed for solving a determined problem [17,18]. Subsequently, state variables and sensitivity coefficients of organic compounds of an aerated ASP assimilated to a CSTR, with complete retention of particulate compounds, or an aerated MBR assimilated to a CSTR have the analytical expressions summarized in Table 3 and Table 6.

5. Conclusions

An approach based on steady-state ASM1 mass balances is presented for ASM1 simplification. The advantage of the adopted approach is that it uses relatively few assumptions to ensure an accurate model. After development, nine analytical expressions were obtained, describing the concentrations of the main state variables present in the sludge matrix: X_BH, X_S, S_S, X_BA, S_NH, S_NO, X_P, X_I, and S_Co. Validation was then performed by means of data generation, using software simulations of the full ASM1. The steady-state expressions showed satisfying predictions of the state variable concentrations and were able to give a quite-accurate sensitivity analysis of state variables to the operating parameters. Therefore, these relatively simple expressions may be useful to understand the impact of operating parameters, which is of great importance in this field, providing researchers and practitioners with a standardized set of state variable values. Moreover, the sensitivity analysis based on these equations shows the complex interaction between all parameters. It provides a significantly better understanding of activated sludge and MBR applications, such as optimization, control, or design.

In addition, the obtained analytical expressions of this study are still applicable for the design and control of each ASP process that can be assimilated to a CSTR in the steady-state conditions, which is a very common configuration in WWTP. If a different process configuration is set, mass balance equations should be slightly modified to include the configuration impact, and then new analytical expressions for state variables and sensitivity coefficients suited for the new configuration can be generated.