A Sewer Dynamic Model for Simulating Reaction Rates of Different Compounds in Urban Sewer Pipe

: A sewer dynamic model (SDM), an innovative use of combined models, was established to describe the reactions of compounds in a pilot sewer pipe. The set of ordinary differential equations in the SDM was solved simultaneously using the fourth-order Runge–Kutta algorithm. The SDM was validated by calculating the consistency between the simulation and observation values. After the SDM was validated, the reaction rate was analyzed. For heterotrophs in the water phase and bioﬁlm, their growth rates were greater than the organism decay rate. For ammonia, the supply rate was greater than the consumption rate at the initial time, but the supply rate was smaller than the consumption rate from the 3rd hour. The supply rate was smaller than the consumption rate for the other six compounds. The supply rate of oxygen was smaller than the consumption rate before the 4th hour because of the microorganism activities, and, subsequently, the supply rate was greater than the consumption rate after the 4th hour because of reaeration. The results of this study provide an insight into the reaction rates of different compounds in urban sewer pipes and an urban water network modeling reference for policymaking and regulation.


Introduction and Background
Massive amounts of people live in cities. People in urban areas utilize energy, environmental resources and develop the land to suit their living conditions. This affects the type of land utilization, hydrology, and landscape. Meanwhile, pollutants are discharged into the environment, and, consequently, environmental resource patterns will be changed. For various uses in household, commercial, and industrial sectors of city life, water resources are indispensable. However, uninterrupted household, commercial, and industrial sewage effluents result in serious pollution of urban water environments [1]. Therefore, for the purpose of water resource protection in a city, it is essential to establish an urban sewer pipe (USP) network that can convey urban sewage from households, commercial districts, or industries to sewage treatment plants (STPs) [2,3]. Urban sewage always contains complex compounds, especially organic compounds. The major categories of organic compounds are lipids (including fats and oils), carbohydrates, and proteins, respectively. After undergoing hydrolysis and microorganisms' reactions, these large molecular compounds are transformed into smaller ones including carbon, nitrogen, phosphorus, and even sulfide with different oxidized states [4][5][6][7][8][9][10][11][12][13][14]. Problems of odor and micropollutants in sewage

Objectives
Since the compounds' RRs have been seldom mentioned in previous studies, the objectives of this study are described as follows: (1) To propose a sewer dynamic model (SDM) mainly according to the kinetics of GDM and other models for calculating the RRs of compounds in the water phase and biofilm of USP. Since the kinetics of GDM have only been used in the simulation of activated sludge process for a long time, this paper represents the first report of the innovative use of combined models for the simulation of reaction rates of different compounds in the USP network. (2) To validate the established model by determining the consistency between the simulation values (SV) and observation values (OV) for all types of compounds. The established SDM can be validated when a high consistency is reported. (3) To anatomize the RRs of different compounds in the USP using the established SDM.

Pilot Sewer Pipe
A plastic pilot sewer pipe (PSP) 21 m long and 0.15 m in diameter was used for the experiments, as shown in Figure 1. The synthetic sewage flowrate and sewer pipe slope could be controlled. The synthetic sewage temperature in the PSP was controlled at 28 centigrade. The synthetic sewage in the tail tank could be conveyed to the head tank using a pump between these two water tanks. The DO concentration in the recirculation tank was monitored using a DO meter. All units including the pipe, highly concentrated synthesis sewage (HCSS) tank, head tank, recirculate tank, and tail tank were sealed to ensure the recirculated sewage was not being oxygenated. was monitored using a DO meter. All units including the pipe, highly concentrated synthesis sewage (HCSS) tank, head tank, recirculate tank, and tail tank were sealed to ensure the recirculated sewage was not being oxygenated.

Experiment Procedures
The HCSS was synthesized using milk powder, sucrose, acetates, and other reagents. For neutralization, a NaOH solution was adopted to control the pH of HCSS at 7-7.5. Table 1 shows the major HCSS compositions. The HCSS was periodically added into the PSP at certain quantities to be mixed with the synthetic sewage in the PSP.
A certain amount of activated sludge was added into the PSP at a fixed influent flowrate. This demonstrated that the biofilm remained in a stable condition if the effluent concentration from the PSP revealed a constant value, i.e., the biofilm grew stably on the PSP bottom. When the batch experiment was finished, the biofilm was acclimated under a steady condition again. The steady flow velocity and the pH were controlled at 0.6 m s −1 and 6.0-7.2, respectively. In the experiment, synthetic sewage of 500 mL was taken from the sampling port every 1 h and the concentrations of total chemical oxygen demand (COD), ammonia, nitrate, and DO were analyzed.

Establishment of SDM
When establishing the SDM, several kinetic reaction rate equations (RREs) from previous models were incorporated into GDM [5][6][7][11][12][13][14][18][19][20][21]. The GDM kinetics were only used in the simulation of the activated sludge process. In this study, the hydraulic condition was gravity flow in an open channel, neither completely mixing flow nor plug flow in the reactor. The active heterotrophic and autotrophic biomass was separated into two phases, i.e., the water phase and biofilm phase, in the open channel. Furthermore, the reaeration of oxygen in gravity flow was considered. Therefore, this study represents the

Experiment Procedures
The HCSS was synthesized using milk powder, sucrose, acetates, and other reagents. For neutralization, a NaOH solution was adopted to control the pH of HCSS at 7-7.5. Table 1 shows the major HCSS compositions. The HCSS was periodically added into the PSP at certain quantities to be mixed with the synthetic sewage in the PSP.
A certain amount of activated sludge was added into the PSP at a fixed influent flowrate. This demonstrated that the biofilm remained in a stable condition if the effluent concentration from the PSP revealed a constant value, i.e., the biofilm grew stably on the PSP bottom. When the batch experiment was finished, the biofilm was acclimated under a steady condition again. The steady flow velocity and the pH were controlled at 0.6 m s −1 and 6.0-7.2, respectively. In the experiment, synthetic sewage of 500 mL was taken from the sampling port every 1 h and the concentrations of total chemical oxygen demand (COD), ammonia, nitrate, and DO were analyzed.

Establishment of SDM
When establishing the SDM, several kinetic reaction rate equations (RREs) from previous models were incorporated into GDM [5][6][7][11][12][13][14][18][19][20][21]. The GDM kinetics were only used in the simulation of the activated sludge process. In this study, the hydraulic condition was gravity flow in an open channel, neither completely mixing flow nor plug flow in the reactor. The active heterotrophic and autotrophic biomass was separated into two phases, i.e., the water phase and biofilm phase, in the open channel. Furthermore, the reaeration of oxygen in gravity flow was considered. Therefore, this study represents the first report of this innovative use of combined models to simulate the RRs of different compounds in the USP network. The stoichiometric constants and kinetic constants from previous models were also adopted [5][6][7][11][12][13][14][18][19][20][21]. Table 2 shows the definitions of compounds in the established SDM. The RREs are shown in Table 3.  Table 3. Symbols and definitions for reaction rate equations.

No. Reaction Equations
By combining the related constants and RREs in Table 3, the complete SDM for all types of compounds can be listed as follows: Water 2021, 13, 1580 The values of relative constants for the established SDM are shown in Table 4. The flow conditions and reaeration effects are shown in Table 5. The equation set from Equation (1) to Equation (12) formed an ordinary differential equation (ODE) system. The set of ODEs was then solved simultaneously using the subroutine (SB) of the fourth-order Runge-Kutta algorithm (RK4) [22]. The flow chart for the computer program is shown in Figure 2.

OURBE to Calculate Sensitive Constants and Initial Biomass
In the model, there were many kinetic constants. The oxygen uptake rate batch experiments (OURBE) adopted from our previous studies [23][24][25][26][27] were used to measure the sensitive kinetic constants including maximum specific growth rate for heterotrophs, organism decay rate for heterotrophs, maximum specific growth rate for autotrophs, organism decay rate for autotrophs, and initial biomass of heterotrophs (Z H ) and autotrophs (Z A ).
The OURBE equipment consisted of several tanks of fixed height and volume and magnetic stirrers for agitation and aeration. Highly stable oxygen meters connected to the data processer were adopted to measure DO. Since the OURBE equipment was sealed and airtight, the actual respiration rate of microorganisms at any time did not depend on oxygen input into OURBE. The actual oxygen uptake rate (OUR) values could be measured using the DO values. A certain amount of biofilm was sampled from the PSP and added into the OURBE equipment. Substrates containing glucose, NH 4 SO 4 , and KH 2 PO 4 were added, resulting in a total volume of 1000 mL in the OURBE equipment, in which the pH value was controlled at 7. The OURBE equipment was periodically aerated to measure OUR. Furthermore, the kinetic constants and initial biomass of Z H and Z A could be measured [23][24][25][26][27].
For those constants that are nearly fixed values in domestic sewage, the default values from GDM and previous studies were used, as shown in Tables 4 and 5 [5,6,[11][12][13][14]. Water 2021, 13, x FOR PEER REVIEW 6 of 16

Experimental Results
In the PSP experiment, the volatile suspended solids (VSS) in the water phase were nearly 1.0 mg L −1 , so the Z HW concentration was considered to be 1.0 mg L −1 . The autotrophs' biomass could not be easily measured using OURBE because of their low concentration, so Z AW was considered to be 0.0 mg L −1 . The values of Z HF and Z AF were approximately 400.0 and 0.1 mg L −1 in accordance with OURBE. These values were close to those in our previous in situ survey [28]. The Standard Methods [29]

Experimental Results
In the PSP experiment, the volatile suspended solids (VSS) in the water phase were nearly 1.0 mg L −1 , so the ZHW concentration was considered to be 1.0 mg L −1 . The autotrophs' biomass could not be easily measured using OURBE because of their low concentration, so ZAW was considered to be 0.0 mg L −1 . The values of ZHF and ZAF were approximately 400.0 and 0.1 mg L −1 in accordance with OURBE. These values were close to those in our previous in situ survey [28]. The Standard Methods [29]

Model Validation
To verify the consistency between the SVs and OVs of different compounds in the water phase, the coefficient of determination (R 2 ) and correlation coefficient (R) were used as follows [30]:

Model Validation
To verify the consistency between the SVs and OVs of different compounds in the water phase, the coefficient of determination (R 2 ) and correlation coefficient (R) were used as follows [30]: where R 2 is the coefficient of determination, R is the correlation coefficient, sv is the SV, sv is the mean SV, ov is the OV, ov is the mean OV, and n is the number of concentrations. R 2 is used to explain how differences in OV can be explained by the difference in SV. It is represented as a value between 0.0 and 1.0. If the R 2 value is higher than 0.7, this value is generally considered as having a strong effect [30]. R is used to calculate the strength of the linear relationship between SV and OV. A value of R between 0.7 and 1.0 indicates a highly positive linear relationship. In this study, R 2 and R were used to justify whether SV and OV were consistent or not, providing a measure of how well OVs were replicated by the SDM model. Figure 2 also shows the consistency between the SVs and OVs of selected compounds in the water phase. The R 2 values between SVs and OVs for S BCS , N O3 , N H3 , N SP , and S O2 were 0.83, 0.70, 0.87, 0.91, and 0.73, respectively. The R values for those were 0.97, 0.81, 0.91, 0.97, and 0.93, respectively. These R 2 and R were greater than 0.7 and 0.81, respectively. According to the study proposed by Moore et al. [30], the SVs and OVs of the selected compounds were highly consistent.

RR of Z HW
Since the established model was validated, the RRs of different compounds at different flowing times could be calculated. Figure 4 shows the calculated RRs within 6 h.
According to Equation (1), Z HW would grow because of r 1 (aerobic growth of Z HW in the water phase) and r 3 (anoxic growth of Z HW in the water phase), and decay because of r 7 (organism decay of Z HW in the water phase). In the simulation proposed by Tanaka and Hvitved-Jacobsen [31], Z HW would grow from 32.0 to 35.0 mg L −1 within the initial 4 h, revealing positive RRs.
In Figure 4a, the RR for Z HW was 0.155 mg L −1 h −1 at the initial time, began to increase before the 3rd hour, and reached its highest value of 0.202 mg L −1 h −1 at the 3rd-4th hour, afterward the RR decreased gradually to a value of 0.176 mg L −1 h −1 at the 6th hour.
Before the 3rd hour, the r 1 (aerobic growth of Z HW in the water phase) and r 3 (anoxic growth of Z HW in the water phase) with positive signs were high because of high S BCS as shown in Figure 3. Therefore, the sum for the growth rate of Z HW (r 1 +r 3 ) was greater than r 7 (organism decay of Z HW in the water phase) and the RRs began to increase before the 3rd hour. After the 4th hour, the sum for the growth rate of Z HW (r 1 +r 3 ) was less than r 7 because the sum for the growth rate of Z HW became lower. Therefore, the RRs revealed a decreasing trend after the 4th hour.

RR of Z HF
According to Equation (2), Z HF would grow because of r 2 (aerobic growth of Z HF in the biofilm) and r 4 (anoxic growth of Z HF in the biofilm), and decay because of r 8 (organism decay of Z HF in the biofilm). According to a study by Nielsen et al. [32], the biofilms grew linearly at an average rate of 0.2 mm day −1 thickness during a series of experiments, revealing positive RRs.
In Figure 4b, the rates of Z HF were 31.2 mg L −1 h −1 at the initial time, began to increase before the 2nd hour, and reached the highest value of 32.3 mg L −1 h −1 at the 2nd hour, afterward, the RR decreased gradually to a value of 19.9 mg L −1 h −1 at the 6th hour.  Before the 2nd hour, the r2 (aerobic growth of ZHF in the biofilm) and r4 (anoxic growth of ZHF in the biofilm) with positive signs were high because of high SBCS as shown in Figure  3. Therefore, the sum for the growth rate of ZHF (r2 + r4) was greater than r8 (organism decay of ZHF in the biofilm) and the RRs began to increase before the 2nd hour. After the 2nd hour, the sum for the growth rate of ZHF (r2 + r4) was less than r8 because the sum for the growth rate of ZHF became lower. Thus, the RRs revealed a decreasing trend after the 2nd hour.  Before the 2nd hour, the r 2 (aerobic growth of Z HF in the biofilm) and r 4 (anoxic growth of Z HF in the biofilm) with positive signs were high because of high S BCS as shown in Figure 3. Therefore, the sum for the growth rate of Z HF (r 2 + r 4 ) was greater than r 8 (organism decay of Z HF in the biofilm) and the RRs began to increase before the 2nd hour. After the 2nd hour, the sum for the growth rate of Z HF (r 2 + r 4 ) was less than r 8 because the sum for the growth rate of Z HF became lower. Thus, the RRs revealed a decreasing trend after the 2nd hour.

RRs of Z AW and Z AF
According to Equation (3), Z AW would grow because of r 5 (aerobic growth of Z AW in the water phase) and decay because of r 9 (organism decay of Z AW in the water phase). According to Equation (4), Z AF would grow because of r 6 (aerobic growth of Z AF in the biofilm) and decay because of r 10 . In the experiment, the organism biomass of the autotrophs in the water phase was ignored. The value of Z AF was about 0.1 mg L −1 according to OURBE. The calculation indicated that the RRs of Z AW and Z AF were almost zero.

RR of Z E
According to Equation (5), Z E would increase because of r 7 (organism decay of Z HW in the water phase), r 8 (organism decay of Z HF in the biofilm), r 9 (organism decay of Z AW in the water phase), and r 10 (organism decay of Z AF in the biofilm).
In Figure 4c, the RR of Z E was 1.2 mg L −1 h −1 at the initial time, began to increase, and reached the highest value of 1.7 mg L −1 h −1 at the 5th hour, afterward, the RR remained fixed at the value 1.7 mg L −1 h −1 at the 6th hour.
Before the 5th hour, the biomass of Z HW , Z HF , Z AW , and Z AF in r 7 (organism decay of Z HW in the water phase), r 8 (organism decay of Z HF in the biofilm), r 9 (organism decay of Z AW in the water phase), and r 10 (organism decay of Z AF in the biofilm) increased continuously. Thus, the RR of Z E continuously increased before the 5th hour. After the 5th hour, the biomass for the organisms was more stable, so the RR remained fixed at the value of 1.7 mg L −1 h −1 at the 6th hour.

RR of S ENM
According to Equation (6), S ENM would be supplied because of r 7 (organism decay of Z HW in the water phase), r 8 (organism decay of Z HF in the biofilm), r 9 (organism decay of Z AW in the water phase), and r 10 (organism decay of Z AF in the biofilm), and consumed because of r 12 (hydrolysis of S ENM ). According to the simulation carried out by Vollertsen et al. [33], in which the maximum specific hydrolysis rate for S ENM was 4.0 day −1 , the S ENM was 40 mg L −1 at the 0th km and 28 mg L −1 at the 20th km, revealing significant biodegradation and negative RRs.
In Figure 4d, the RR of S ENM was −16.2 mg L −1 h −1 at the initial time and increased gradually to a value of −12.1 mg L −1 h −1 at the 6th hour. During the whole experimental time of 6 h, the r 7 (organism decay of Z HW in the water phase), r 8 (organism decay of Z HF in the biofilm), r 9 (organism decay of Z AW in the water phase), and r 10 (organism decay of Z AF in the biofilm) with positive signs were low, but r 12 (hydrolysis of S ENM ) with a negative sign was relatively high. However, r 12 became lower because of decreasing S ENM , so the RR of S ENM increased gradually. It indicated that the supply rate of S ENM was lower than the consumption rate in the experiment.

RR of S BCS
According to Equation (7), S BCS would be consumed because of r 1 (aerobic growth of Z HW in the water phase), r 2 (aerobic growth of Z HF in the biofilm), r 3 (anoxic growth of Z HW in the water phase), and r 4 (anoxic growth of Z HF in the biofilm), and supplied because of r 12 (hydrolysis of S ENM ). According to previous studies reported by Raunkjaer et al. [34], the removal rate of S BCS (acetate) was 4.13 g m −2 h −1 and 1.98 g m −2 h −1 under low-loaded and high-loaded conditions, respectively, revealing significant biodegradation. In the simulation proposed by Vollertsen et al. [33], the S BCS was 30 mg L −1 at the 0th km and 3 mg L −1 at the 20th km, revealing significant biodegradation and negative RRs.
In Figure 4e, the RR of S BCS was −39.2 mg L −1 h −1 at the initial time, began to decrease before the 3rd hour and reached the lowest value of −43.9 mg L −1 h −1 at the 3rd hour, afterward the RR increased gradually to a value of −30.2 mg L −1 h −1 at the 6th hour.
Before the 3rd hour, the r 1 (aerobic growth of Z HW in the water phase), r 2 (aerobic growth of Z HF in the biofilm), r 3 (anoxic growth of Z HW in the water phase), and r 4 (anoxic growth of Z HF in the biofilm) with negative signs were high because of high S BCS , but r 12 (hydrolysis of S ENM ) with a positive sign was relatively low. After the 3rd hour, the sum for consumption rate of S BCS (r 1 + r 2 + r 3 + r 4 ) became lower because of low S BCS . So the RRs revealed an increasing trend after the 3rd hour. The RR of S BCS was highly negative at the initial time and final time. It indicated that the supply rate of S BCS was lower than the consumption rate in the experiment.

RR of N H3
According to Equation (8), N H3 would be consumed because of r 1 (aerobic growth of Z HW in the water phase), r 2 (aerobic growth of Z HF in the biofilm), r 3 (anoxic growth of Z HW in the water phase), r 4 (anoxic growth of Z HF in the biofilm), r 5 (aerobic growth of Z AW in the water phase), and r 6 (aerobic growth of Z AF in the biofilm), and supplied because of r 11 (ammonification of N BS ). In a study by Marjaka et al. [35], the removal of dissolved organic carbon and nitrogen in a USP with a fabricated porous ceramic bed was investigated. When N H3 was used as the nitrogen source, and air was supplied through the aerator, 56% of the initial N H3 was removed within 180 min. On the other hand, the value was about 38% when aeration was not applied. In their nitrification test, the RR of N H3 was negative. But in our study, the RR of N H3 was positive because of the hydrolysis of organic nitrogen.
In Figure 4f, the RR of N H3 was 15.5 mg L −1 h −1 at the initial time, began to decrease before the 5th hour and reached the lowest value of −1.6 mg L −1 h −1 at the 5th hour, afterward, the RR increased slightly to a value of −1.2 mg L −1 h −1 at the 6th hour.
Before the 5th hour, the r 1 (aerobic growth of Z HW in the water phase), r 2 (aerobic growth of Z HF in the biofilm), r 3 (anoxic growth of Z HW in the water phase), r 4 (anoxic growth of Z HF in the biofilm), r 5 (aerobic growth of Z AW in the water phase), and r 6 (aerobic growth of Z AF in the biofilm) with negative signs decreased highly because of high N H3 , but r 11 (ammonification of N BS ) with a positive sign increased slightly. Thus, N H3 decreased before the 5th hour.
It indicated that the supply rate of N H3 was greater than the consumption rate at the initial time, but the supply rate was smaller than the consumption rate from 3rd hour in the experiment. The RR of N H3 could be also regarded as the nitrification rate. Since the supply rate of N H3 was greater than the consumption rate, the nitrification rate was not obvious. This is consistent with the results proposed by Shoji et al. [36], Liang et al. [37], and our previous in-situ study [28].

RR of N O3
According to Equation (9), N O3 would be consumed because of r 3 (anoxic growth of Z HW in the water phase) and r 4 (anoxic growth of Z HF in the biofilm), and supplied because of r 5 (aerobic growth of Z AW in the water phase) and r 6 (aerobic growth of Z AH in the biofilm). In the study proposed by Marjaka et al. [35], nitrite and nitrate concentrations in the nitrification test were low, indicating that the conversion of nitrite and nitrate to nitrogen gas by denitrification might undergo; therefore, the RR of N O3 was negative. In the experiments carried out by AEsøy et al. [38], the denitrification rate of 3.5-4.3 g NO 3 -N m −2 day −1 was observed in the PSP. This indicated the RR of N O3 was negative in the PSP.
In Figure 4g, the RR of N O3 was −0.2 mg L −1 h −1 at the initial time, began to decrease before the 2nd hour, reached the lowest value of −0.7 mg L −1 h −1 at the 2nd hour, afterward the RR increased gradually to a value of 0.0 mg L −1 h −1 at the 6th hour.
Before the 2nd hour, the r 3 (anoxic growth of Z HW in the water phase) and r 4 (anoxic growth of Z HF in the biofilm) with negative signs decreased highly, but r 5 (aerobic growth of Z AW in the water phase) and r 6 (aerobic growth of Z AH in the biofilm) with positive signs increased slightly. Thus, the RR of N O3 decreased before the 2nd hour. After the 2nd hour, r 3 and r 4 became lower because of low N O3 . Therefore, the RRs revealed an increasing trend after the 2nd hour.
This indicated that the supply rate was smaller than the consumption rate in the experiment. The RR of N O3 could be also be regarded as the denitrification rate. Denitrification did not occur obviously because of the low initial concentration and poor denitrification activity. This is consistent with the results proposed by Shoji et al. [36].

RR of N BP
According to Equation (10), N BP would be supplied because of r 7 (organism decay of Z HW in the water phase), r 8 (organism decay of Z HF in the biofilm), r 9 (organism decay of Z AW in the water phase), and r 10 (organism decay of Z AF in the biofilm), and consumed because of r 13 (hydrolysis of N BP ).
In Figure 4h, the RR of N BP was −0.4 mg L −1 h −1 at the initial time and decreased gradually to a value of −0.9 mg L −1 h −1 at the 6th hour.
During the whole experimental time of 6 h, the r 7 (organism decay of Z HW in the water phase), r 8 (organism decay of Z HF in the biofilm), r 9 (organism decay of Z AW in the water phase), and r 10 (organism decay of Z AF in the biofilm) with positive signs were low, but r 13 (hydrolysis of N BP ) with a negative sign was relatively high. Therefore, the RR of N BP decreased gradually to a value of −0.9 mg L −1 h −1 at the 6th hour. This indicated that the supply rate of N BP was smaller than the consumption rate in the experiment.

RR of N BS
According to Equation (11), N BS would be consumed because of r 11 (ammonification of N BS ) and supplied because of r 13 (hydrolysis of N BP ).
In Figure 4i, the RR of N BS was −18.3 mg L −1 h −1 at the initial time and increased gradually to a value of −0.1 mg L −1 h −1 at the 6th hour. During the whole experimental time of 6 h, the r 11 (ammonification of N BS ) with a negative sign was greater than r 13 (hydrolysis of N BP ) with a positive sign. Thus, the RR of N BS increased gradually. It indicated that the supply rate was smaller than the consumption rate in the experiment.

RR of S O2
According to Equation (12), S O2 would be consumed because of r 1 (aerobic growth of Z HW in the water phase), r 2 (aerobic growth of Z HF in the biofilm), r 5 (aerobic growth of Z AW in the water phase), and r 6 (aerobic growth of Z AF in the biofilm), and supplied because of r 14 (reaeration of oxygen) [39]. In a study by Marjaka et al. [35], the S O2 value was decreased to 2.0 mg L −1 within the first 15 min, and then increased gradually to 6.0 mg L −1 when organic carbon was almost completely consumed. When no aeration was applied, the S O2 concentration decreased to 0.0 mg L −1 within 30.0 min and an anaerobic condition was maintained until 120.0 min. Then S O2 concentration gradually increased to 4.0 mg L −1 during the rest period.
In Figure 4j, the RR of S O2 was −14.0 mg L −1 h −1 at the initial time, 0.0 mg L −1 h −1 at the 4th hour, and 1.0 mg L −1 h −1 at the 6th hour, respectively. According to Figure 3, S O2 was 6.23 mg L −1 at the initial time, began to decrease to the lowest value of 1.98 mg L −1 at the 2nd hour, and then increased gradually to 3.76 mg L −1 at the 6th hour. Therefore, the r 1 (aerobic growth of Z HW in the water phase), r 2 (aerobic growth of Z HF in the biofilm), r 5 (aerobic growth of Z AW in the water phase), and r 6 (aerobic growth of Z AF in the biofilm) with negative signs were greater than r 14 (reaeration of oxygen) with a positive sign before the 2nd hour. After the 2nd hour, r 14 (reaeration of oxygen) was greater than the sum of r 1 , r 2 , r 5 , and r 6 . Thus, the RR of S O2 increased gradually to a value of 1.0 mg L −1 h −1 at the 6th hour.
This indicated that the supply rate was smaller than the consumption rate before the 4th hour because of significant organism activity, and, subsequently, the supply rate was greater than the consumption rate after the 4th hour because of reaeration.

Conclusions
The SDM, an innovative use of combined models, was established to describe the reaction of several compounds in the PSP. The established model was validated by calculating the consistency between the SVs and OVs of different compounds. R 2 and R were greater than 0.7 and 0.81, respectively, revealing that the SVs and OVs of selected compounds were highly consistent. Therefore, the RRs of compounds in the PSP were calculated using the SDM.
For Z HW and Z HF , their growth rates were greater than the organism decay rates in the experiment. For Z E , S ENM , S BCS , N O3 , N BP , and N BS , their supply rate was smaller than the consumption rate in the experiment. For N H3 , the supply rate was greater than the consumption rate at the initial time, but the supply rate was smaller than the consumption rate from 3rd hour in the experiment. The supply rate of S O2 was smaller than the consumption rate before the 4th hour because of significant microorganism activity, and, subsequently, the supply rate was greater than the consumption rate after the 4th hour because of reaeration. The kinetic constants and RREs in the SDM could be applied to predict the real-world situation.

Recommendations for Future Research
The results of this study not only provide an insight into the reaction rates of different compounds in the USP and an urban water network modeling reference for policymaking and regulation but also valuable data to help designers create more reasonable USP planning. However, in situ experiments in the USP are suggested to further explore the SDM in the future.  Data Availability Statement: Some data and models related to this study are available from the corresponding author upon reasonable request.