Introduction

Conventional biological nitrogen removal (BNR) consists of two successive steps: autotrophic nitrification and heterotrophic denitrification. Nitrification consists of two steps: ammonia is first oxidized to nitrite by ammonia-oxidizing bacteria (AOB), and then nitrite is oxidized to nitrate by nitrite-oxidizing bacteria (NOB). From a biochemical perspective, AOB utilize ammonium (NH4 +-N) as their electron donor and NOB utilize nitrite (NO2 −) as electron donor. Both AOB and NOB utilize dissolved oxygen (DO) as their electron acceptor, suggesting that a competition for DO between AOB and NOB exists in nitrification systems. Shortcut nitrification/denitrification and shortcut nitrification/anammox are two promising technologies to replace conventional biological nitrogen removal (BNR) (Guo et al. 2009; Van Loosdrecht and Jetten 1998). The benefits of shortcut nitrification processes include lower oxygen and carbon requirements (Beccari et al. 1983; Turk and Mavinic 1987; van Kempen et al. 2001). Enrichment of AOB and washout or inhibition of NOB are important for realizing the partial nitrification process.

When competing for DO, NOB are often at a disadvantage due to their higher dissolved oxygen half-saturation coefficients. Thus, DO control has been adopted by many researchers to achieve shortcut nitrification. However, DO concentrations for achieving stable shortcut nitrification varied in different studies, as shown in Table 1 of the supporting information (SI). As apparent from SI-Table 1, DO values range from 0.16 to 5 mg DO/L. The wide differences in operational DO concentrations resulted directly from the changes in pH, free ammonia (FA), free nitrous acid (FNA), and temperature, as well as the type of system (attached growth or suspended growth). Thus, optimization of DO concentration at any given operational conditions such as pH, FA, FNA, temperature is critical for the design and operation of a successful shortcut nitrification system.

In this study, we developed a mathematical model proposing the concept of the minimum/maximum substrate (MSC) concentrations to include oxygen limitations and the effect of pH, FA and FNA, temperature, and SRT at a given ambient TNN and TAN concentration. The choice of free ammonia as the AOB substrate is rationalized by Hellinga et al. (1999) and Van Hulle et al. (2007), who demonstrated from batch tests that NH3 rather than NH4 + is the actual substrate, which is also supported by the fact that biomass actually can only transport the uncharged NH3 over its membrane (Anthonisen et al. 1976). The model was also validated with data from the literature.

Methodology

General MSC equation

S min is the minimum substrate concentration to support steady-state biomass (Rittmann and McCarty 1980; Rittmann and McCarty 2001). S min and DOmin can be derived from the Monod equation as Eqs. 1 and 2, respectively (Rittmann and McCarty 2001).

$$ {S}_{\min }={K}_{\mathrm{s}}\times \frac{b}{\frac{\mu_{\max}\times \mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}-b} $$
(1)
$$ {\mathrm{DO}}_{\min }={K}_{\mathrm{O}}\times \frac{b}{\frac{\mu_{\max}\times S}{S+{K}_{\mathrm{S}}}-b} $$
(2)

S min refers to the minimum electron donor. K s and K o are the Monod half-saturation concentrations for the electron donor (S) and electron acceptor (DO), respectively; μ max is the maximum growth rate and b is the endogenous decay rate.

MSC equation for AOB

In this study, by adopting free ammonia as substrate, Eqs. 1 and 2 are converted to Eqs. 3 and 4.

$$ {FA}_{\min }={K}_{FA}\times \frac{b}{\frac{\mu_{\max}\times \mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}-b} $$
(3)
$$ {\mathrm{DO}}_{\min }={K}_{\mathrm{O}}\times \frac{b}{\frac{\mu_{\max}\times FA}{FA+{K}_{FA}}-b} $$
(4)

In which K FA is the Monod half-saturation rate concentration for FA.

MSC equation for NOB

In this study, TNN is chosen as the substrate for NOB (Boon and Laudelout 1962; Park and Bae 2009). Eqs. 1 and 2 are converted to Eqs. 5 and 6:

$$ TNN={K}_{TNN}\times \frac{b}{\frac{\mu_{\max}\times \mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}-b} $$
(5)
$$ {\mathrm{DO}}_{\min }={K}_{\mathrm{O}}\times \frac{b}{\frac{\mu_{\max}\times TNN}{TNN+{K}_{TNN}}-b} $$
(6)

In which K TNN is the Monod half-saturation concentration for TNN.

Effect of pH

The pH can influence nitrification directly by changing the enzymatic reaction mechanism (Park et al. 2007; Van Hulle et al. 2007) and indirectly by changing the concentrations of FA and FNA, which inhibit AOB and NOB (Hellinga et al. 1999; Park and Bae 2009; Van Hulle et al. 2007).

The direct pH effect on the maximum specific substrate utilization rate of AOB or NOB can be captured by the empirical bell-shaped Eq. 7 (Park et al. 2007):

$$ \begin{array}{c}\hfill q=\frac{q_{\max }}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\hfill \\ {}\hfill \left(\left|pH-{pH}_{\mathrm{opt}}\right.\left|<w\right)\right)\hfill \end{array} $$
(7)

q and q max are, respectively, the maximum specific substrate utilization rate at a given pH and at the optimal pH. w is the pH range within which the q is larger than one-half of q max.

Since the biomass yield is a constant, μ is also affected by pH similar to q:

$$ \mu =\frac{\mu_{\max }}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\} $$
(8)

μ is the maximum specific substrate utilization rate at a given pH.

FA and FNA, both of which inhibit AOB and NOB, are influenced by pH. FA and FNA concentrations can be calculated based on pH and TAN or TNN concentration (Anthonisen et al. 1976; Park and Bae 2009).

$$ FA=\frac{17}{14}\times \frac{\mathrm{TAN}\times {10}^{pH}}{ \exp \left(\frac{6344}{273+T}\right)+{10}^{pH}} $$
(9)
$$ FNA=\frac{47}{14}\times \frac{TNN}{ \exp \left[\left(\frac{-2300}{273+T}\right)\times {10}^{pH}\right]+1} $$
(10)

where T is temperature (°C).

Given that FA is the substrate for AOB and FNA is not a substrate for AOB, the inhibition of FA and FNA may potentially be modeled by the following substrate inhibition model (Metcalf&Eddy 2014; Vadivelu et al. 2006) and non-substrate inhibition model (Hellinga et al. 1999), as shown below in Eqs. 11 and 12, respectively.

$$ \mu ={\mu}_{\max}\times \frac{\mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}\times \frac{FA}{K_{FA}+FA+\frac{FA^2}{K_{IFA}}} $$
(11)
$$ \mu ={\mu}_{\max}\times \frac{\mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}\times \frac{FA}{FA+{K}_{FA}}\times \frac{K_{\mathrm{IFNA}}}{K_{\mathrm{IFNA}}+FNA} $$
(12)

Thus, integrating the direct and indirect effects of pH on AOB yields:

$$ {\mu}_{obs}=\frac{\mu_{\max }}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\times \frac{\mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}\times \frac{FA}{K_{FA}+FA+\frac{FA^2}{K_{IFA}}}\times \frac{K_{\mathrm{IFNA}}}{K_{\mathrm{IFNA}}+FNA} $$
(13)

K IFA and K IFNA are the inhibition concentrations for FA and FNA on AOB. μ obs is the observed specific biomass growth rate.

A non-substrate inhibition model was adopted for FA inhibition of NOB as FA is not a substrate for NOB. The model by Boon and Laudelout (1962) was chosen to describe the FNA inhibition of NOB. Thus, integrating the direct and indirect effects of pH on NOB yields the following:

$$ {\mu}_{obs}=\frac{\mu_{\max }}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\times \frac{\mathrm{DO}}{\mathrm{DO}+{K}_{\mathrm{O}}}\times \frac{TNN}{\left({K}_{TNN}+TNN\right)\left(1+\frac{FNA}{K{\acute{\mkern6mu}}_{\mathrm{IFNA}}}\right)}\times \frac{K{\prime}_{IFA}}{K{\prime}_{IFA}+FA} $$
(14)

K′IFA and K′IFNA are the inhibition concentration for FA and FNA on NOB.

Effect of temperature

The Monod maximum growth rate (μ max), the half-saturation concentration (K S), and the endogenous decay rate (b) were adjusted for temperature:

$$ {\mu}_{\max }={\mu}_{\max 20}\times {\theta}_{\mu}^{T-20} $$
(15)
$$ {K}_{\mathrm{S}}={K}_{\mathrm{S}20}\times {\theta}_{K_{\mathrm{S}}}^{T-20} $$
(16)
$$ b={b}_{20}\times {\theta}_b^{T-20} $$
(17)

where T is temperature (°C); θ is the temperature coefficient.

SRT effect

The effect of SRT on the MSC equations is shown in Eq. 18:

$$ {\mu}_{\max}\times \frac{\mathrm{DO}}{K_{\mathrm{O}}+\mathrm{DO}}\times \frac{S}{S+{K}_{\mathrm{S}}}-b-\frac{1}{SRT}=0 $$
(18)

Integration of effects

DOmin equations for AOB and NOB are shown in Eqs. 19 and 20, respectively.

$$ {\mathrm{DO}}_{\min }=\frac{\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\times {K}_{\mathrm{o}}}{\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{o}\mathrm{pt}}\right)\right]\right\}}{\left(1+\frac{FNA}{K_{\mathrm{IFNA}}}\right)\times \left(1+\frac{K_{FA20}\times {\theta}_{K_{FA}}^{T-20}}{FA}+\frac{FA}{K_{IFA}}\right)}-{b}_{20}\times {\theta}_b^{T-20}-\frac{1}{SRT}}for\;AOB $$
(19)
$$ {\mathrm{DO}}_{\min }=\frac{\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\times {K}_{\mathrm{o}}}{\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{o}\mathrm{pt}}\right)\right]\right\}}{\left(1+\frac{FNA}{K{\acute{\mkern6mu}}_{\mathrm{IFNA}}}\right)\times \left(1+\frac{FA}{K{\acute{\mkern6mu}}_{IFA}}\right)\times \left(1+\frac{K_{TNN20}\times {\theta}_{K_{TNN}}^{T-20}}{TNN}\right)}-{b}_{20}\times {\theta}_b^{T-20}-\frac{1}{SRT}}\;for\;NOB $$
(20)

FAmin and FAmax for AOB and TNNmin and TNNmax for NOB can be calculated from Eqs. 21–24, respectively, since FA for AOB and TNN for NOB have two limiting values with FA and FNA inhibition. The lower limiting value (FAmin or TNNmin) has the same meaning as the traditional S min, i.e., the minimum substrate concentration to support steady-state biomass while the higher value (FAmax or TNNmax) represents the maximum substrate concentration able to sustain steady-state biomass, above which inhibition by FA or FNA will lead to wash out of AOB or NOB.

For AOB,

$$ {FA}_{\min }=\frac{\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\times \mathrm{DO}}{\left(1+\frac{FNA}{K_{\mathrm{IFNA}}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-{b}_{20}\times {\theta}_b^{T-20}-\frac{1}{SRT}-\sqrt{{\left(\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\times DO}{\left(1+\frac{FNA}{K_{\mathrm{IFNA}}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-{\mathrm{b}}_{20}\times {\theta}_{\mathrm{b}}^{T-20}-\frac{1}{SRT}\right)}^2-4\times \frac{{\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)}^2\times {K}_{FA20}\times {\theta}_{K_{FA}}^{T-20}}{K_{IFA}}}}{2\times \left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\times \frac{1}{K_{IFA}}} $$
(21)
$$ {FA}_{\max }=\frac{\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\times \mathrm{DO}}{\left(1+\frac{FNA}{K_{\mathrm{IFNA}}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-{b}_{20}\times {\theta}_b^{T-20}-\frac{1}{SRT}+\sqrt{{\left(\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}}{2}\times \left\{1+ \cos \left[\frac{\pi }{w}\times \left(pH-{pH}_{\mathrm{opt}}\right)\right]\right\}\times \mathrm{DO}}{\left(1+\frac{FNA}{K_{\mathrm{IFNA}}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-{b}_{20}\times {\theta}_b^{T-20}-\frac{1}{SRT}\right)}^2-4\times \frac{{\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)}^2\times {K}_{FA20}\times {\theta}_{K_{FA}}^{T-20}}{K_{IFA}}}}{2\times \left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\times \frac{1}{K_{IFA}}} $$
(22)

For NOB,

$$ {TNN}_{\min }=\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}\times \mathrm{DO}}{\left(1+\frac{FA}{K{\acute{\mkern6mu}}_{IFA}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\left(1+{K}_{TNN20}\times {\theta}_{K_{TNN}}^{T-20}\times \frac{f_{FNA}}{K{\acute{\mkern6mu}}_{\mathrm{IFNA}}}\right)-\sqrt{{\left(\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}\times \mathrm{DO}}{\left(1+\frac{FA}{K{\acute{\mkern6mu}}_{IFA}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\left(1+{K}_{TNN20}\times {\theta}_{K_{TNN}}^{T-20}\times \frac{f_{FNA}}{K{\acute{\mkern6mu}}_{\mathrm{IFNA}}}\right)\right)}^2-4\times {\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)}^2\times {K}_{TNN20}\times {\theta}_{K_{TNN}}^{T-20}\times \frac{f_{FNA}}{K{\acute{\mkern6mu}}_{\mathrm{IFNA}}}}}{2\times \left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{SRT}\right)\times \frac{f_{FNA}}{K{\acute{\mkern6mu}}_{\mathrm{IFNA}}}} $$
(23)
$$ {\mathrm{TNN}}_{\max }=\frac{\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}\times \mathrm{DO}}{\left(1+\frac{\mathrm{FA}}{K{\overset{\acute{\mkern6mu}}{\kern0.333em }}_{\mathrm{IFA}}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{\mathrm{SRT}}\right)\left(1+{K}_{\mathrm{TNN}20}\times {\theta}_{K_{\mathrm{TNN}}}^{T-20}\times \frac{f_{\mathrm{FNA}}}{K{\overset{\acute{\mkern6mu}}{\kern0.333em }}_{\mathrm{IFNA}}}\right)+\sqrt{{\left(\frac{\mu_{\max 20}\times {\theta}_{\mu}^{T-20}\times \mathrm{DO}}{\left(1+\frac{\mathrm{FA}}{K{\overset{\acute{\mkern6mu}}{\kern0.333em }}_{\mathrm{IFA}}}\right)\times \left({K}_{\mathrm{O}}+\mathrm{DO}\right)}-\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{\mathrm{SRT}}\right)\left(1+{K}_{\mathrm{TNN}20}\times {\theta}_{K_{\mathrm{TNN}}}^{T-20}\times \frac{f_{\mathrm{FNA}}}{K{\overset{\acute{\mkern6mu}}{\kern0.333em }}_{\mathrm{IFNA}}}\right)\right)}^2-4\times {\left({b}_{20}\times {\theta}_b^{T-20}+\frac{1}{\mathrm{SRT}}\right)}^2\times {K}_{\mathrm{TNN}20}\times {\theta}_{K_{\mathrm{TNN}}}^{T-20}\times \frac{f_{\mathrm{FNA}}}{K{\overset{\acute{\mkern6mu}}{\kern0.333em }}_{\mathrm{IFNA}}}}}{2\times \left({\mathrm{b}}_{20}\times {\theta}_{\mathrm{b}}^{T-20}+\frac{1}{\mathrm{SRT}}\right)\times \frac{f_{\mathrm{FNA}}}{K{\overset{\acute{\mkern6mu}}{\kern0.333em }}_{\mathrm{IFNA}}}} $$
(24)

For AOB, using the relationship between TAN and FA of Eq. 9, TANmin and TANmax can be derived as Eqs. 25 and 26, respectively.

$$ {\mathrm{TAN}}_{\min }={FA}_{\min}\times \frac{14}{17}\times \frac{\left[ \exp \left(\frac{6344}{273+\mathrm{T}}\right)+{10}^{pH}\right]}{10^{pH}} $$
(25)
$$ {\mathrm{TAN}}_{\max }={FA}_{\max}\times \frac{14}{17}\times \frac{\left[ \exp \left(\frac{6344}{273+\mathrm{T}}\right)+{10}^{pH}\right]}{10^{pH}} $$
(26)

Modeling simulations

Simulations were conducted based on Eqs. 19–24 using six cases to evaluate the effect of pH, temperature, and SRT on a CSTR without biomass recycle. The kinetic parameter values used for modeling are summarized in Table 1 while the simulation conditions are presented in Table 2.

Table 1 Various kinetic parameters selected for the model simulations of modified Park’s model (at 20 °C)
Full size table
Table 2 Operational conditions for six simulation cases
Full size table

Results and discussion

Impact of pH: cases 1 and 2

Figure 1a, b shows the DO MSC curves for AOB and NOB at different pHs. Generally, if the operating DO is above the DOmin curves for AOB and/or NOB, the conditions support the growth of AOBs, NOBs, or both. FA inhibition increases with pH increase while FNA inhibition increases with pH decrease. For AOB, when the TAN concentration is below TANmin or over TANmax, DOmin approaches infinity, which identifies the washout range (WR) for AOBs. Similar boundaries of TNNmin or over TNNmax also exist for NOB.

Fig. 1
figure 1

Comparison with MSC curves (A–F represent case 1–6, respectively)

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Case 1 simulates the effect of pH on AOB and NOB at a constant TNN concentration of 50 mg N/L and TAN concentration of 0–1000 mg N/L. At pH 7, AOB and NOB curves intersect at 38 mg N/L, implying that below 38 mg N/L, it is impossible to wash out NOB and maintain AOB by DO control as DOmin for NOB is lower than DOmin for AOB. The intersection TAN concentrations at pH 7.5, 8, 8.5, and 9 are 10, 3, 0, and 0 mg N/L, implying that the minimum operational TAN concentration decreases as pH increases. Both TANmin and TANmax concentrations for AOB decrease as pH increases. For example, for pH in the 7 to 9 range, the TANmin concentration for AOB decreases from 24 to 0.3 mg N/L. Similarly, the TANmax values for AOB at pH 7 and 7.5 are over 1000 mg N/L and at 8, 8.5, and 9 are approximately 489, 200, and 90 mg N/L. Due to FA inhibition, the TANmax for NOB also decreases from 126 mg N/L at pH 7 to 0.3 mg N/L at pH 9. In this analysis, both TANmin and TANmax represent the TAN concentration at which DOmin reaches 5 mg /L. When the DO is 2 mg/L and TNN concentration is constant at 50 mg N/L, the TAN concentration range for shortcut nitrification at pH 7, 7.5, 8, 8.5, and 9 are 74–1000, 43–1000, 16–489, 5–200, and 1–90 mg N/L, respectively. Yan and Hu (2009) operated a CSTR at SRTs of 1 to 2 days, DO of 2 mg/L, temperature of 35 °C, pH 8, and achieved nitrite accumulation at a TAN of 150 and TNN of 180 mg N/L for CSTR. The computed DOmin for AOB growth and complete NOB washout based on this model for the conditions of Yan and Hu (2009) are 0.29–0.78 mg/L which are less than 2 mg/L and consistent with the experimental results.

Case 2 simulates the effect of pH on AOB and NOB at a constant TAN concentration of 50 mg N/L and TNN concentration of 0–1000 mg N/L (Fig. 1b). The DOmin for NOB are always higher than those for AOB, implying that shortcut nitrification is achievable at any pH, especially at pH over 8, when DOmin for NOB goes to infinity meaning that NOB are always washed out. The optimal pH is around 8 as the DOmin range for AOB with TNN ranging from 0 to 1000 mg/L are 0.15–0.17 mg/L at pH 8, 0.24–0.26 mg/L at pH 8.5, and 0.66–0.68 mg/L at pH 9. TNN concentrations exert minimal impact on the DOmin for AOB. As apparent from cases 1 and 2, the TAN has a much greater effect on the DOmin of AOB and NOB than the TNN. There is a turning point which corresponds to the minima of the DOmin curves (Fig. 1b as an example) on the TAN-DO curve for AOB and on the TNN-DO curve for NOB, when TNN or TAN is fixed. For example, in Fig. 1a, when TAN is lower than the turning point value, DOmin for AOB will increase significantly with TAN decrease. As discussed above, pH 8 seems to be optimal for AOB at constant TAN values at 50 mg N/L. However, the optimal pH may not be applicable to specific circumstances. For example, if an effluent TAN of 800 mg N/L and TNN of 50 mg N/L is required, only pH 7 or 7.5 can be utilized.

Impact of temperature: cases 3 and 4

The effect of temperature in the 10 to 35 °C range on the DO MSC curves for AOB and NOB is shown in Fig. 1c, d. Case 3 is similar to case 1 at a constant TNN concentration of 50 mg N/L and TAN concentration of 0–1000 mg N/L, while case 4 is similar to case 2 at a constant TAN concentration of 50 mg N/L and TNN concentration of 0–1000 mg N/L.

In case 3, both TANmin and TANmax values for AOB decreased as temperature increased. The TANmin and TANmax values for AOB decreased from 11 mg N/L and over 1000 mg N/L at 10 °C to 1.5 and 489 mg N/L at 35 °C. The TANmax for NOB also decreased from 65 mg N/L at 10 °C to 24 mg N/L at 35 °C. The intersection TAN concentration of AOB and NOB curves decreased from 16 mg N/L at 10 °C to 3 mg N/L at 35 °C. From the aforementioned point of intersection to TANmax for AOB, it is possible to maintain AOB and wash out NOB by adopting specific DO control. For example, from TANmax for NOB to TANmax for AOB, nitrite accumulation can always be achieved with DO higher than DOmin for AOB. In case 4, when TAN is fixed at 50 mg N/L, the DOmin for NOB are always higher than that for AOB, meaning that shortcut nitrification is achievable at any temperature in the 10 to 35 °C range.

Impact of SRT: cases 5 and 6

The effect of SRT on DO MSC curves for AOB and NOB is shown in Fig. 1e, f by changing SRT from 5 days to infinity while maintaining pH and temperature constant at 8 and 35 °C, respectively.

Case 5 simulates the effect of SRT on AOB and NOB at a constant TNN concentration of 50 mg N/L and TAN concentration of 0–1000 mg N/L. The TANmin concentrations for AOB at any SRT are lower than 3 mg N/L. Both TANmax values for AOB and NOB increased with SRT. The TANmax concentration for AOB increases from 263 mg N/L at an SRT of 5 days to 489 mg N/L at an infinite SRT. The TANmax for NOB increased from 13 mg N/L at an SRT of 5 days to 24 mg N/L at an infinite SRT. The intersection point of the DO curves for AOB and NOB corresponds to the TAN concentration at which both species survive. The common TAN concentration for both AOB and NOB decreased as SRT increased from 5 days to infinity but always fell within the range of 3 to 4 mg N/L.

Case 6 simulates the effect of SRT on AOB at a constant TAN concentration of 50 mg N/L and TNN concentration of 0–1000 mg N/L. At all SRTs, NOB will be washed out, ending up with nitrite accumulation when DO is higher than DOmin for AOB. From Fig. 1f, DOmin for AOB decreased as SRT increased at a given TNN concentration.

Special applications of the DOMSC curves

Recently, the feasibility and performance of nitrification-anammox system at low nitrogen concentrations (20 to 60 mg NH4-N/L) and low temperatures (5–25 °C) (De Clippeleir et al. 2013; Hu et al. 2013; Lotti et al. 2014; Persson et al. 2014) has elicited the attention of both researchers and practitioners. The MSC curves for various pHs for the special case of shortcut nitrification-anammox process at 10 °C are shown in Fig. 2. In this simulation, SRT were set at 10, 20, and 30 days, temperature was set as 10, 15, and 20 °C, and pH was set at 8. Stoichiometrically, anammox needs an influent that has almost the same concentrations of ammonium- and nitrite-nitrogen (Strous et al. 1997). Thus, the TAN/TNN was set at 1:1.32 with TN as the sum of TAN and TNN, i.e., completely ignoring nitrates. As presented in Fig. 2, the DOmin for both AOB and NOB decreased when SRT or temperature increased. The minimum SRT decreased with temperature increase. At a given condition, for example, a TN of 60 mg N/L, pH 8, the minimum SRT was 13 days at 10 °C, decreasing to 7 days at 15 °C, and further to 3.6 days at 35 °C.

Fig. 2
figure 2

DOmin curves by SRT(10–30 days) and temperature(10–20 °C) for shortcut nitrification to provide an input to the ANAMMOX process

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Analysis of literature results with the MSC model

Table 3 summarizes the comparison of literature operational conditions and performance data with the computed DOmin values based on this model. Columns 1 to 8 are from the experimental data; columns 10 to 11 are AOB and NOB DOmin values calculated using the experimental values in this model. It can be seen that almost all experimental DO values are higher than DOmin for AOB and lower than DOmin for NOB, suggesting that this model may become a predictive tool for successful shortcut nitrification system.

Table 3 Determination of the minimum DO concentration in shortcut nitrification systems
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When analyzing the effect of SRT in this work, SRT ranged from 5 to 30 days. However, the SHARON process usually has an SRT less than 2 days (Galí et al. 2007a; van Dongen et al. 2001; Hellinga et al. 1998). As apparent from Table 3, this model was unable to predict the SHARON process (Galí et al. 2007b; van Dongen et al. 2001) as the DOmin for AOB by the model suggested that the AOB would be washed out under the Sharon conditions. The reason for the aforementioned discrepancy is different kinetics. The maximum observed growth rates of AOB in the SBR and SHARON were 1.0 and 2.0, and 1.3 and 2.4 day−1 in the study of Galí et al. (2007a) and Galí et al. (2007b), respectively. One set of SHARON kinetic parameters for AOB determined by Van Hulle et al. (2007) at pH 7 and a temperature of 35 °C is as follows: b = 0.045 day−1, K NH3 = 0.75 mgNH3-N L−1, K INH3 was very high and K IHNO2 = 2.04 mgHNO2-N L−1 and Ko = 0.94 mg/L. By using the aforementioned kinetic parameters and μ max of 2.4 day−1, the predicted DOmin for AOB was 1.09 to 1.36 mg/L in the case of van Dongen et al. (2001). The model predictions for several typical Sharon processes are shown in Table 4. Although in some studies the actual DO was not mentioned, the model suggested that the AOB could survive with DO higher than DOmin while NOB would be washed at any operational DO. This indicates that the model could predict the SHARON process based on its specific kinetic parameters. In this study, b of 0.17 day−1 at 20 °C and θ b of 1.04 were used, resulting in b value of 0.31 day−1 at 35 °C, which is actually higher than the b value of 0.23 day−1 at 35 °C reported by Magri et al. (2007). In fact, the reported b values at 35 °C range from 0.045 to 0.31 day−1(Henze et al. 1987; Metcalf&Eddy 2003; Van Hulle et al. 2007; Liu et al. 2016). In addition, the FA and FNA inhibition threshold concentrations range from 5.0 to 27.3 mg FA/L, and 0.09 to 0.97 mg FNA/L (Park and Bae 2009).

Table 4 Model prediction of Sharon reactors based on specific kinetic values
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Model use for bioreactor design

This model not only suggests a DO range in which nitrite accumulation can be successfully achieved, but also provides bioreactor design information such as SRT.

For example, if a CSTR is to be designed to treat wastewater with Q in, influent TAN of 220 mg N/L to achieve an effluent concentration of TAN 20 mg N/L and TNN 200 mg N/L at a pH of 7.5 and a temperature of 35 °C, the SRT ranges from 2.4 days to infinity. Furthermore, the operational DO ranges from 0.31 to 1.19 mg/L at infinite SRT to over 4.5 mg/L at an SRT of 2.4 days.

Conclusions

A model for successful shortcut nitrification conditions determination was derived based on MSC values. In addition, the effect of temperature, pH, and SRT was analyzed. Specific application of this model for shortcut nitrification coupled with anaerobic ammonium oxidation (ANAMMOX), in which the effluent concentrations of nitrite and ammonium from shortcut nitrification were equal, was discussed. Comparison of the model predicted DOmin with experimental data suggested that this model can be a useful and practical tool for shortcut nitrification systems design and operation.