Understanding and modelling ammonia partitioning and transport across reverse osmosis membrane

Domain: biology
Published: 2026-03-31
Editorial outcome: minor_revision

Abstract
Reverse osmosis (RO) is increasingly applied for the reclamation of ammonia-rich wastewater. However, the
mechanisms governing NH /NH + transport across RO membranes remain unclear. In this study, we develop an
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ammonia partitioning and transport (APT) model to quantitatively describe NH /NH + retention and transport,
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incorporating pH-dependent NH /NH + partitioning and membrane charge variation. The theoretical model is
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validated through laboratory-scale RO experiments under varying pressures, feed concentrations, and pH conditions.
In contrast to stable water permeability, the permeability coefficient of total ammonia nitrogen (TAN) is sensitive
to pH value. TAN retention is optimal (89.13.2%) near neutral pH, decreasing at higher pH due to increased NH
3
fraction and at lower pH due to reduced membrane charge. By decoupling diffusion, advection, and
electromigration, the APT model shows distinct NH₄⁺ and NH₃ flux contributions, revealing pH-dependent transport
mechanisms governed by speciation and membrane charge. Molecular dynamics simulations further differentiate
the transport pathways of NH₃ and NH₄⁺, showing that NH
4
+ experiences a significantly higher energy barrier.
Validations with both synthetic and real manure streams demonstrate good predictive accuracy for advancing RO
applications in ammonia-rich streams.

Introduction
Ammonia is essential to securing the food supply and supporting global population growth. Synthesized by the
Haber-Bosch process, ammonia has sustained about half of humanity over the past century 1. Currently, the Haber-
Bosch process accounts for 2% of global energy consumption and up to 5% of global greenhouse gas emissions 2-
4. This ammonia is consumed and cycled within the food chain and ends up in the excreta. In the U.S., 60% of
ammonia emissions are attributed to the livestock manure wastewater produced in intensive animal feeding
operations 5. Excess manure nutrients lead to overapplication, increasing nutrient losses and reducing nutrient use
efficiency. The high water content and low nutrient density increase costs to transport manure nutrients to cropland,
with agronomic needs restricting the ability to use manure effectively.

Ammonia in manure can be directly reused through a shortcut resource recovery process 6,7. To achieve clean
water that meets discharge or even drinking standards, anaerobic treatment is applied to livestock manure, followed
by a series of membrane treatment processes 8, to recover energy, nutrients, and clean water out of waste streams.
As the dominant technology for water purification with high energy efficiency, reverse osmosis (RO) is increasingly
used for wastewater reclamation 9, particularly for clean water recovery and ammonia concentration 10,11. Compared
with treating the wastewater by nitrification-denitrification and resynthesizing ammonia from the Haber-Bosch
process, this shortcut was estimated to save energy as high as 16.2-19.7 kWh per kg-N 11, making it a more
sustainable solution 12-14.

In practice, it has been found that total ammonia nitrogen (TAN) is difficult to fully retain in a single-pass RO,
and thus, a two-pass RO system is often employed to minimize the presence of TAN in the final clean water. Recent
studies reported experimental observation of significant variability in ammonia retention efficiency by RO
membranes, ranging from 0% to 98% 10,11,15,16. These results are particularly sensitive to the operating pH 11. It is
commonly believed that this variability results from the NH /NH + speciation under varying pH, and NH and NH +
3 4 3 4
have their respective permeabilities, which are different from each other. RO membrane is effective for only
rejecting the charged ions, so a high permeability of TAN was observed at alkaline pH, where TAN is mainly in the
form of NH 10,17. However, even at acidic pH where all the TAN exists in the form of NH +,
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its permeability is not
simply a constant due to its interaction with the membrane charge 11. Current understandings are limited to
experimental observations and qualitative descriptions. A fundamental understanding of TAN transport through RO
membranes remains limited; thus, a quantitative model is needed to establish the relation between TAN transport
and pH variation.

Researchers have made significant efforts to develop models to describe the transport of water and salts through
RO membranes 18,19 (Supplementary Table 1). Recent experimental and computational studies demonstrated the
presence of advective transport of solutes within the RO membrane 9,20,21. Correspondingly, advanced models
conceptualizing water and solute transport through porous RO membranes were developed, incorporating more
detailed mechanisms, including Donnan steric partitioning and advective solution friction 22,23. They have shown
improved performance in simulating and fitting data for inert salt ions 24,25. By introducing the fast acid–base
reaction assumption, this modeling framework allows the inclusion of amphoteric ions. Purely simulated results of
pH variation in RO showed the possible feasibility of this modeling framework 26. By adjusting multiple fitting
parameters, reasonable agreement with experiments was achieved when only inert ions and pH were considered 27.
Once additional amphoteric ions (e.g., NH₄⁺) are included, the model’s generalizability declines markedly, and its
predictions of permeate pH, amphoteric ions, and bivalent ions become less reliable even under a single feed pH 28.
The highly coupled and complex equations render the transport of amphoteric ions strongly dependent on the local
membrane pH. For instance, NH₃ transport is governed by the coupled transport of NH₄⁺ and H⁺ 28.

In this study, we propose that the transport of amphoteric ions is governed by uncoupled transport equations of
charged and neutral species, with the solution pH subsequently corrected through equilibrium recalculation. We
analyzed and elucidated the separation and transport of TAN in RO membranes, as well as the complex changes
induced by pH. Based on solute transport theory that considers friction between substances, we developed a
theoretical model, named ammonia partitioning and transport (APT) model, to describe NH /NH + transport in RO
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membranes. This model takes into account diffusion, advection, and electro-migration of solutes, while also
considering the interactions between water, solutes, and the membrane matrix. RO experiments were conducted
under different feed concentrations, pH levels, and operating pressures to validate the model. Our theoretical and
experimental observations reveal the changes in TAN removal performance resulting from the pH-dependent shifts
NH /NH + speciation and RO membrane surface charge, as well as the differences in NH /NH + transmembrane
3 4 3 4
transport behavior, and extend the applicability of the solution-friction model in describing TAN transport in RO
membranes.

Results
Core theories governing the TAN transport
Generally, ammonia recovery technologies target TAN, which is distributed in solution between ammonia
molecules and ammonium ions in a specific ratio according to pH (Figure 1a). Their transport across the RO
membrane involves spatial partitioning at the membrane-water interface on both sides (Eq. 1.1 and 1.2), as well as
transport within the membrane (Eq. 2.1 and 2.2), while taking into account the concentration gradient caused by
concentration polarization on the pressurized feed side 18.

c
NH+,m
= c
NH+,s
𝛷
st
𝛷
di
e−z∆𝜑 (1.1)
4 4
c = c 𝛷 (1.2)
CNH3,m NH3,s st
in which, c (mM) is the concentration just inside the membrane at the interface, and c (mM) is the concentration
m s
at the outside surface of the membrane after concentration polarization. For the charged NH +, three effects are
4
considered for partitioning at the membrane-water interface: steric exclusion, 𝛷 , dielectric exclusion, 𝛷 , and
st di
Donnan effect, e−z∆𝜑, in which ∆𝜑 is the gradient of dimensionless potential. For the uncharged NH , only steric
3
exclusion, 𝛷 , is needed (Figure 1c). Note: Equation 1.1 also applies to other ions in the system.
st
The theoretical basis for TAN transport within the membrane is the extended Nernst-Planck equation, where the
charged NH + is controlled by advection, diffusion, and electro-migration 24, while uncharged NH is only controlled
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by advection and diffusion. pH also determines the charge state of the functional groups on the RO membrane.
Specifically, the carboxylic groups (whose dissociation constants are 10-5.23 and 10-8.97, respectively) on the fully
aromatic polyamide membrane will be deprotonated and thus be more negatively charged at higher pH (Figure 1b)
29. This further influences the ion transport via the co-ion partitioning theory 30,31. The driving force for solute
transport across the membrane is generated by the chemical potential gradient, plus the advection by the permeate
flux 24,26,28.
dc NH+ d𝜑
J = K c v −K ε D ( 4 +z c ) (2.1)
NH 4 + a,NH 4 + NH 4 + f d,NH 4 + NH 4 + dx NH 4 + dx
dc
J = K c v −K ε D
NH3
(2.2)
NH3 a,NH3 NH3 f d,NH3 NH3 dx
where J (mol m−2 s−1) is the solute flux, K and K are the advective and diffusional hindrance factors, respectively,
a d
c is the concentration at local point, v (m s−1) is the permeate flow velocity, ε is the effective membrane porosity,
f
D (m2 s−1) is the diffusivity in water, 𝜑 is the dimensionless electrical potential inside the membrane over the
coordinate, x (m), through the membrane. Equation 2.1 also applies to other ions in the system. For the uncharged
ammonia molecules, the charge z = 0, so equation 2.1 becomes 2.2, where, correspondingly, electromigration is not
included 32.
Water transport is established through an equation based on the force balance acting on water molecules within
the membrane 22,24:
dPt
T= RTf (𝜈 −𝜈 )+RT∑ f c (𝜈 −𝜈 ) (3)
dx f−m m f i−f i i f
i
where the first term on the right-hand side refers to the friction between fluid and the membrane, while the
summation of the second term reflects the friction between fluid and all solutes. 𝜈 (m s−1) is the velocity of the
m
membrane, which is 0 in the case of a static membrane. 𝜈 (m s−1) and 𝜈 (m s−1) are the velocities of solute and
i f
permeate fluid, respectively. v can be obtained by dividing solute flux J (mol m−2 s−1) by its concentration, c (mol
i i i
m−3). R (J K−1 mol−1) is the gas constant, T (K) is the absolute temperature, and f are the friction coefficients, where
the friction coefficient between fluid and membrane, f (mol s m−5), is a fitting parameter, and the friction
f−m
coefficient between fluid and solute, f (s m−2), is the inverse of the solute diffusion coefficient in water 24.
i−f
We conducted a series of RO experiments to investigate the effects of operating pressure, feed concentration,
and pH on TAN transport to validate the model. The results of varying operating pressure and feed concentration
are presented in supporting information (Supplementary Discussion 1 and 2, Supplementary Figure 1−3). We focus
on the effect of pH in the main text. The parameters used in the model are detailed in Supplementary Table 2.

Impact of pH value on TAN transport
Although previous studies have observed a significant effect of pH on TAN rejection in RO 11,33, no fundamental
explanation has been provided yet. The APT model enables a quantitative description of TAN transport across the
RO membrane, demonstrating a great fitting performance with an R² of 0.956 for TAN permeability (Figure 2b)
and 0.988 for TAN rejection (Figure 2c). The influence of pH values on TAN transport was examined using a bench-
scale RO at a crossflow velocity of 6.2 mm s–1, an operating hydraulic pressure of 10 bar, a feed TAN concentration
of 10.7 mM, and a temperature of 25 ± 1 °C.
We observed stable water permeability independent of feed pH variations. As illustrated in Figure 2a, both
experimental results (0.539  0.019 L m−2 h−1 bar−1) and model calculations (0.554 L m−2 h−1 bar−1) demonstrate a
stable water transport coefficient across the tested feed pH range (5.5 to 11). This is probably because water
transport primarily depends on the frictional interaction between water molecules and the membrane 24. Although
the water transport is theoretically coupled with ion transport, the ion-associated term in Eq. 3 is negligible in dilute
solutions. Therefore, observable changes in water permeability only occur when the membrane properties are
significantly altered, such as by swelling under extreme pH conditions 34 or compaction under excessively high
pressure 35.
TAN permeability coefficient initially decreases slightly as feed pH increases, followed by a sharp increase
when the pH exceeds 8 (Figure 2b). The model results indicate that the lowest permeability coefficient occurs at
pH 7.3, with a value of 0.656 L m⁻² h⁻¹. Correspondingly, TAN rejection exhibits a parabolic trend (Figure 2c),
with the highest experimental rejection of 89.13.2% observed near pH 7 and a model-predicted maximum of 88.7%
at pH 7.3. Both high and low pH levels in the feed solution led to a reduced TAN rejection efficiency. This arch-
shaped behavior has also been reported in previous studies 11,36. To better understand the arch-shape, we use the
APT model to separate the pH-dependent transport behavior of TAN into individual transport characteristics of
NH₃ and NH₄⁺.
NH₃ permeability remains constant across the feed pH range of 5.5 to 11, with a value of 24.75 L m⁻² h⁻¹ (Figure
2b). As a result, its rejection remains consistently low at 17.2% (Figure 2b). This is because NH₃ is uncharged and
therefore does not interact with the membrane charge. However, the NH₃ transport flux increased from 0.00851
mmol m⁻² h⁻¹ to 44.7 mmol m⁻² h⁻¹ when the feed pH value varied from 5.5 to 10.4 (Figure 2d). This results from
the pH-dependent distribution of NH₃ and NH₄⁺. Given that the permeability of NH₃ is approximately 30 times
higher than that of NH₄⁺, an increase in NH₃ fraction at high feed pH significantly boosts overall TAN permeability
and reduces rejection efficiency.
NH₄⁺ permeability slightly decreases from 1.18 L m⁻² h⁻¹ to 0.21 L m⁻² h⁻¹ as feed pH increases from 5.5 to 11,
respectively. As shown in the inset of Figure 2b, there are two distinct drops for NH + permeability, one at a feed
4
pH value of 5.5 and the other at 8.1. Both drops could be attributed to the increased negative charge of the membrane
at elevated feed pH (Figure 1b), which enhances the electrostatic repulsion of co-ions and reduces their partitioning
into the membrane, thus reducing their transport. The counter-ions (i.e., NH +) have to match the charge neutrality
4
with co-ions in the permeate, thus reducing their transport and increasing NH
4
+ rejection (Figures 2b and 2c) 37. For
example, the flux contributed solely by NH₄⁺ is slightly higher at pH 6.2 (7.24 mmol m⁻² h⁻¹) than at pH 7.0 (6.34
mmol m⁻² h⁻¹), which can be attributed to the lower membrane surface charge at more acidic pH. At pH values
below 8, NH₄⁺ is the dominant TAN species, and thus its transport behavior governs the behavior of TAN. Although
the second drop for NH + permeability is also due to membrane charge, at feed pH values above 8, NH becomes
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the dominant species, and the drop of NH + permeability contributes less significantly to the TAN permeability
4
variation. A similar trend is also observed for Na⁺ (Supplementary Figure 5), supporting that the transport of
charged ions is mainly governed by electrostatic interaction.
In addition to total TAN concentration and the NH₃/NH₄⁺ speciation (Figure 2e), the APT model also accurately
predicts permeate pH values, showing good agreement with experimental results (R2 = 0.973, Figure 2f). Notably,
the permeate pH is constantly higher than the feed pH. For instance, with a feed pH of 8.45, the permeate pH
increases to 9.6 (Figure 2f). This pH elevation results from the re-establishment of equilibrium on the permeate
side after the individual transport of NH₃ and NH₄⁺. Specifically, while NH₃ and NH₄⁺ exist in equilibrium on the
feed side, NH₃ transports across the RO membrane more readily than NH₄⁺. As a result, NH₃ temporarily reaches a
higher concentration on the permeate side before re-equilibration occurs, driving the equilibrium toward NH₄⁺
formation, thereby increasing the permeate pH. Despite being based on different theoretical assumptions, our pH
prediction is consistent with outcomes derived from hydrated proton transport theory 26.

Decoupling diffusion, advection, and electro-migration
The APT model also provides further insight into the flux contributions of NH + and NH during their transport
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across the membrane. The charged species NH + is transported by advection, diffusion, and electro-migration 24,
4
whereas the transport of uncharged species NH is governed by advection and diffusion. As shown in Equations 2.1
3
and 2.2, advection flux is the product of solute concentration, fluid velocity, and a hindrance factor for transport in
narrow pore, while diffusion flux is driven by concentration gradients and electromigration by potential gradients
along the transport direction. Both diffusion and electromigration are also influenced by the diffusion coefficient,
membrane porosity, and hindrance factor.
For NH +, the highest total transmembrane flux of 13.81 mmol m−2 h−1 occurs at pH 5.3. As pH gradually
4
increases to around 6, the significant increase in membrane charge causes a notable decrease in total flux to 7.727
mmol m−2 h−1. Further increasing pH to 9 results in a second notable decrease in total flux to 1.616 mmol m−2 h−1,
eventually reducing it to nearly zero as pH further increases due to the combined effect of increased membrane
charge and decreased NH + proportion.
4
Although the total NH + flux exhibits a decreasing trend with increasing feed pH value, the individual flux
4
contributions from diffusion, advection, and electromigration show distinct behaviors (Figure 3a). At acidic feed
pH conditions, diffusion of NH + ions is negligible, whereas advection and electromigration fluxes occur in opposite
4
directions. Advection transports NH + towards the permeate side along with water molecules, while electro-
4
migration contributes to a negative flux, thereby acting as a hindrance to NH + transport. For example, at pH 6,
4
where the membrane charge is 68.60 mM, the fluxes from diffusion, advection, and electromigration are 2.03,
481.14, and −475.44 mmol m−2 h−1, respectively, resulting in a total flux of 7.73 mmol m−2 h−1.
As the feed pH value increases to around 8.4, electro-migration flux becomes more negative to −740.8 mmol
m−2 h−1 due to increased membrane charge to −156.2 mM, until the NH + concentration decreases rapidly at pH 9
4
as the dissociation equilibrium shifts towards NH . Similarly, as counter-ions, more NH + ions partition into the
3 4
membrane with the increasing membrane charge at pH larger than 8, resulting in a higher in-membrane NH +
4
concentration, and therefore a high advection flux of 1001 mmol m−2 h−1 at pH 8.8. However, we note that the
increased advection does not lead to a higher total transport flux due to the high electromigration hindrance of
−665.1 mmol m−2 h−1 at pH 8.8. Molecular dynamics (MD) simulation also reveals the strong electrostatic
adsorption when NH + ions transport out of the membrane, as discussed in the next section. Further increasing feed
4
pH to 10 significantly reduces NH + concentration. Combined with the effect of membrane charge and potential
4
changes, it drastically decreases the electromigration flux. Given that electro-migration is more sensitive to NH +
4
concentration, the reduction in electro-migration is more significant than that of advection for the feed pH range of
8 – 10. These substantial changes in both electromigration and advection result in a reverse concentration gradient
within the membrane, driving a reverse diffusion flux that balances the advection flux. The reverse diffusion flux
reaches a maximum of –477.94 mmol m−2 h−1 at pH 9.3, where the electromigration flux decreases to −327.09
mmol m−2 h−1 while the advection flux is still as high as 806.08 mmol m−2 h−1.
For NH , the advection and diffusion fluxes increase from both almost 0 to 0.5645 and 46.46 mmol m−2 h−1,
3
respectively, as the feed pH value increases from 5.3 to 11, while electromigration is absent due to its neutrality.
Therefore, we observe an increasing total NH flux from almost 0 to 47.02 mmol m−2 h−1, of which the diffusion
3
flux is the major contributor, with advection contributing only marginally about 1.2% (Figure 3b). Such a trend is
attributed to the NH speciation, as the sharp increase begins at pH 8, where the NH +/NH equilibrium starts to
3 4 3
shift towards NH formation.
3
Notably, although the total NH₃ flux, of which the highest is 47.02 mmol m−2 h−1, is significantly higher than
that of NH₄⁺ with a highest of 13.81 mmol m−2 h−1, the component fluxes differ greatly in magnitude. For example,
the advective flux of NH + in the range of 256.9−1001 mmol m−2 h−1 is approximately 3−4 orders higher than that
4
of NH in the range of 0−0.5645 mmol m−2 h−1. This is because of their distinct partitioning mechanisms at the
3
membrane-water interface (Figure 1c). The Donnan effect drives cations (NH₄⁺) to partition at higher concentrations
within the membrane to balance its negative charge, resulting in a higher concentration inside the membrane and
thus a larger advection flux for NH₄⁺. However, its total NH₄⁺ flux remains relatively low due to the negative
electromigration flux. In contrast, NH₃ is unaffected by membrane charge or the Donnan effect and is subject only
to steric exclusion at the interface, leading to a lower concentration inside the membrane and thus a smaller
advection flux. However, free from electrostatic interactions, NH₃ maintains a sustained concentration gradient
across the membrane from the feed side to the permeate side, resulting in a high diffusion flux which dominates
NH overall transport.
3

MD simulation reveals the different transport mechanism
The potential of mean force (PMF) of moving one NH or NH + across the polyamide RO membranes is
3 4
evaluated to determine the energetic barriers. The PMFs determine the intrinsic energy barriers at low driving force
(concentration gradient and/or pressure) and low concentration, where molecular transport occurs through a quasi-
static process (no irreversibility). The commercial polyamide membrane is modelled as a crosslinked membrane
composed of m-phenylenediamine (MPD) and trimesoyl chloride (TMC) with a 90% degree of crosslinking 38. The
experimental charge densities of the membrane associated with the pH of 5.5 (−48 mM) and 10 (−402 mM) are
enforced (Eq. 10), which produce a surface charge on the polyamide membrane (Supplementary Figure 10). A
representative initial configuration of the system is shown in Figure 4a. The PMFs determined from umbrella
sampling simulations, as shown in Figure 4b, highlight the significant differences in NH and NH + transport. NH +
3 4 4
demonstrates an adsorption phenomenon (PMF < 0) close to the transition region of the membrane (𝑧 ∈
[−5.5,−4.5]∪[4.5,5.5] ), followed by large energy barriers within the dense region of the membrane (𝑧 ∈
[−4.5,4.5]). In contrast, NH lacks the adsorption behavior (PMF > 0) and only has small energy barriers (~1.5
3
𝑘 𝑇). The effective energy barriers (Δ𝐺‡) of the PMF profiles (Figure 4c) are evaluated based on the generalized
𝐵
De Donder equation (based on transition state theory), Eq. 4, where 𝑊(𝑧) is the PMF 39.
Δ𝐺‡ = 𝑘 𝑇ln∫exp(𝑊/𝑘 𝑇)d𝑧 (4)
B B
The Δ𝐺‡ of NH + is much higher than that of NH , which is consistent with the experimental rejection
4 3
measurements. While Δ𝐺‡ increases with pH for both NH and NH +, the increase is more pronounced for NH +.
3 4 4
This is again consistent with experimental measurements where rejection of NH remains constant (does not
3
increase significantly) while a small increase in rejection is observed for NH +.
4
The slope of the PMF in 𝑧 ∈
(−7,−6)∪(6,7) is used as a metric to estimate the propensity of adsorption (Figure 4d). This demonstrates that
adsorption of NH + is mainly dictated by electrostatic attraction with the polyamide membrane, while NH is not.
4 3
This observation is also corroborated by the number of negatively charged oxygen atoms around NH and NH +
3 4
(Supplementary Figure 7; Figure 4f). Therefore, the large energy barriers with NH
4
+ occur due to its separation
from negatively charged oxygen atoms, while small energy barriers with NH occur due to their small size and
3
neutral charge. The effect of hydrogen bonding on the transport was also studied. While NH + forms more hydrogen
4
bonds than NH
3
in bulk water (Supplementary Figure 8), the maximum number of hydrogen bonds lost during the
transport process is comparable (Figure 4e). That is, hydrogen bonding is not a significant factor in explaining the
difference in the rejection of NH and NH +. While steric effects play a role (e.g. in NH ), their contribution is much
3 4 3
smaller than electrostatic effects (Supplementary Figure 9).
The MD simulations provide a good qualitative comparison with the experimental results and validate our
numerical models. However, several limitations arise in capturing the experimental conditions. As experimental
rejection of molecules approaches 100%, standard non-equilibrium simulations become ineffective at identifying
the rejection mechanisms. In such cases, enhanced sampling simulations under equilibrium conditions 40, as used
in this work, are used to determine intrinsic energy barriers for molecular transport. Such equilibrium simulations
cannot capture the inherent non-equilibrium effects 41, such as concentration or pressure gradients, concentration
polarization, membrane compaction, and energy dissipation (e.g., friction). Therefore, the simulated energy barriers
shown in this work describe a quasi-static transport process under low driving force (pressure or concentration).

Model validation using complex feed matrices
The APT model effectively describes the transport of water and NH /NH + across the RO membrane and
3 4
accurately captures the impacts of operating parameters, including pressure, feed concentration, and feed pH.
Notably, all these model curves are generated using a single set of physicochemical parameters, with fitting
achieved only through the solute partitioning coefficient, 𝛷 , and the water-membrane friction coefficient, 𝑓 .
i f−m
The consistency between model predictions and experimental results highlights the reliability and applicability of
the APT model.
To further validate its robustness, we evaluated its performance with solutions containing more competing ions.
When NaCl was added (Supplementary Figure 6), the model prediction remained stable and accurate in the presence
of 10.7 and 21.4 mM additional Na+ ions. For the real samples collected from the manure treatment plant, the 1st
pass RO permeate was relatively clean, containing moderate levels of TAN (176.7 mg L−1), and small amounts of
Na+, K+, and SO 2− (7.3, 78.6, and 38.8 mg L−1, respectively) (Supplementary Table 3). Since the TAN concentration
4
exceeds the discharge standard, a second RO pass is typically required. The simple composition and absence of
organic matter in it allow us to validate the model directly using this real wastewater. Experimental results with this
real wastewater are consistent with our observations using synthetic wastewater. Similarly, the RO membrane
exhibits optimal TAN rejection around pH 7, with rejection decreasing at both higher and lower pH values (Figure
5b), consistent with our previous observations and discussion. The predicted optimal point is at pH 7.6 with the
highest TAN rejection of 88.3%. The presence of K+ and SO 2− at relatively low concentration levels does not
4
significantly affect TAN transport. Similarly, the transport flux from NH + gradually decreases with increasing pH,
4
while NH flux increases sharply above pH 8, leading to an increase in TAN flux (Figure 5c). The flux from NH +
3 4
at pH 6.2 was 9.01 mmol m−2 h−1 while that from NH was 0.0503 mmol m−2 h−1. These flux contributions are
3
reversed at pH 9.7 as that from NH + decreases to 0.427 mmol m−2 h−1 while NH increases to 41.7 mmol m−2 h−1.
4 3
Model predictions generally matched the experimental results, with only a slight underestimation of TAN rejection
under mildly acidic conditions, which is likely due to minor unmeasured cations present in authentic wastewater
that were not included in the model, leading to a slight overestimation of Cl⁻ partitioning (Supplementary Figure
12). This discrepancy diminishes quickly once NH₃ becomes the dominant species at higher pH.
The ultrafiltration (UF) effluent exhibits higher complexity with a chemical oxygen demand (COD) of 3250 mg
L−1. To avoid the interference from organic matters and unknown components, we prepared a synthetic UF effluent
based on the measured cation composition of the real UF permeate. Results confirmed that high concentrations of
monovalent and divalent cations do not impact the model’s ability to predict TAN transport across different pH
conditions. Notably, at high ionic concentrations, co-ion transport was enhanced due to the increased co-ion
partitioning into the membrane, driven by both the Donnan effect and the partitioning coefficient. When the Donnan
contribution is removed, the intrinsic partitioning coefficient decreases (Supplementary Figure 12). Consequently,
the increased counter-ion fluxes result in slightly lower TAN rejection than in pure NH Cl solutions. The maximum
4
TAN rejection for UF-based synthetic wastewater was 86.0% at pH 7.1 (Figure 5d). TAN fluxes (Figure 5e) also
increase significantly due to the combined effect of higher TAN and competing-ion concentrations, ranging from
35.7 to 53.8 mmol m⁻² h⁻¹, consistent with experimental observations. Because high Mg²⁺ concentrations may
induce precipitation and scaling, validation could not be extended to higher pH values; practically, alkaline pH is
rarely used in real operations. Overall, the results demonstrate the excellent applicability and robustness of the APT
model.

Methods
Model development
The commercial RO membranes typically use the thin-film composite polyamide membranes produced through
interfacial polymerization 42. The model in this study describes the membrane layer as a polymer network with
continuous, water-filled sub-nanometer channels through which clusters of water molecules and solutes can move
24. The support layer also has a similar polymer network, but with much larger voids, and thus, its effect on water
and solute transport can be neglected during model development, allowing us to focus only on the active layer of
the membrane. In the polymer matrix of the membrane, water exists in a liquid phase, while solutes are dispersed
in the water and transported across the membrane through the aqueous phase.
Previous modeling strategies for the transmembrane transport of amphoteric ions were commonly based on the
assumption of infinitely fast acid-base equilibrations 43, treating amphoteric ions as a group and establishing their
relationship through dissociation equilibrium at every local point inside the membrane 28. However, such method
cannot describe the transport mechanism of NH as its concentration was calculated based on the transport of NH +
3 4
and local pH instead of its own governing equation. To address this, we established individual transport equations
for both NH + and NH according to their respective transport mechanisms (Eqs 1.1-2.2). For simplification, we
4 3
did not set up transport equations for the proton or hydroxyl ion since both of them are in very low concentrations
within the pH range we care about 43, otherwise their involvement would introduce more equations than unknowns
in the coupled system, leading to conflicts of solving. The permeate pH is determined by the re-established
dissociation equilibrium between the transported NH and NH +. Their distribution in solution is determined by the
3 4
following equation 44
c 1
NH3
= (5)
c
NH3
+c
NH
4
+
1+10(pKa-pH)
In studies modeling ion exchange membranes, membrane thickness (100 m) was commonly discretized into
tens of grids 43,45, and theoretically, equilibrium can be reestablished after each grid to improve accuracy. However,
the active layer of RO membrane is already extremely thin (100 – 150 nm) 37, much thinner than a single
discretization layer used in ion-exchange membrane models. So, we reasonably assume the membrane active layer
is sufficiently thin and that solute transport is infinitely fast, to reduce the computational complexity.
Due to the limited pore size of the membrane matrix network (average 0.76 nm) 46 and its inherent charge,
solutes experience steric hindrance and electrochemical potential effects. During transmembrane transport, solutes
must first undergo concentration partitioning at the feed-side membrane-water interface before entering the
membrane and then partition again at the permeate-side interface before entering the permeate (Equation 1.1 and
1.2). The physical meanings and specific explanation for these partitioning coefficients are provided in
Supplementary Method 1. It is noteworthy that steric and dielectric effects are both related to pore size and ionic
Stokes radius. Wang, et al. 24 suggested that some partitioning mechanisms might be interrelated; therefore, 𝛷
st
and 𝛷 can be combined into a single partitioning coefficient, Φ, which is related to the size of the solute molecules or
di
ions. Given the challenges in obtaining detailed information on solute dehydration and post-dehydration size 47, Φ
is used as a fitting parameter in this study, similar to strategies adopted by other researchers 24,48. A recent study
found that the partitioning coefficients of ions vary with feed ionic concentrations and that the partitioning behavior
of co-ions plays a dominant role in controlling the transport 37. In this study, we developed an empirical expression
for the Cl⁻ partitioning coefficient (the main co-ion dominants the transport) as a function of the total cationic
charge concentration in the feed (Supplementary Figure 12).
It is important to note that c (mM) in Eq. (1.1 and 1.2) differs from the solute concentration in the bulk feeding
s
solution due to concentration polarization at the feed-side membrane surface. The relationship between these
concentrations can be calculated using the boundary layer film model 49
𝜈
f
c = c +(c −c )exp( ) (6)
s p f p k
in which c (mM) and c (mM) are the concentrations of solute in the bulk feed solution and in the permeate
f p
solution, respectively; 𝜈 (m s−1, also 103 L m−2 s−1) is the permeate water flux, and k (m s−1) is the mass transfer
f
coefficient of the concentration polarization boundary film which depends on the hydrodynamic conditions over
the membrane surface. The detailed process to calculate this coefficient can be found in Supplementary Method 2.
The extended Nernst-Planck equation is derived from the Maxwell-Stefan theory, which relates the driving force
acting on a molecule/ion and the frictions it encounters, including the friction between solute and membrane, and
between solute and fluid. The derivation process is provided in Supplementary Method 3.
With a given feed solution composition, permeate composition can be solved by combining equations in main
text and charge neutrality (Eq. 7 and 8) as well as steady state condition (Eq. 9).
∑ z J = 0 (7)
i i
i
∑ z c +𝜔X = 0 (8)
i i
i
J = c v (9)
i i,p f
where X (mM) is the volumetric charge density of the membrane and ω is the sign of the membrane charge, here
it is −1 for the negatively charged RO membrane 50.
The membrane charge ωX is a function of local pH due to the dissociation of the charged functional group on
RO membrane (i.e., amine groups and carboxylic groups) which can be expressed as 29
K [H+] [H+]
𝜔X = X /(1+ )−X /(1+ )−X /(1+ ) (10)
NH2 [H+] COOH1 K COOH2 K
COOH1 COOH2
where K and K are the dissociation equilibrium constants of amine and carboxylic groups, respectively,
NH2 COOH
X and X are the fixed number of amine and carboxylic groups, respectively. For the ionized amine groups,
NH2 COOH
they can be described by one dissociation constant, while two values are required for the carboxylic groups, which
is why they are denoted as COOH and COOH 22. Supplementary Method 4 provides more details about the
1 2
explanation of membrane charge.
For inert ions that do not undergo acid-base reactions (e.g., Na+ and Cl−), c represents the final concentration
p
of the ion in the permeate as determined by the model. However, for NH and NH +,
3 4
which have different transport
mechanisms and rates, an acid-base equilibrium must be re-established on the permeate side. The c we get is virtual.
p
These virtual concentrations are used in Eqs. (11) and (12) to calculate the rejection and permeability coefficient.
After that, by reestablishing the equilibrium in the permeate via Eq. 5, we can solve the final concentration of NH
3
and NH +,
4
as well as the permeate pH.
For both the simulated and experimental result, rejection rate of an any solute, or ion, or group of the TAN by
the membrane is defined as
c
p
R = 1− (11)
j c
f
in which c and c are the concentration in permeate and in feed, respectively.
p f
The solute permeability coefficient from the modeling result, B (L m−2 h−1), is calculated by
c J
p w
B = (12)
c −c
s p
where J (L m−2 h−1) is the water flux, which can be calculated from the fluid flow velocity, 𝜈 (m s−1), by
w f
converting the unit. c and c have been defined in Eq. (6) and notably c is the concentration of solute in feed at
s p s
the membrane surface after considering concentration polarization. This permeability coefficient, B, is a
phenomenal parameter introduced by the traditional RO model to describe the transport of solute, obtained by
normalizing the solute flux by its concentration gradient across the membrane and having the same unit as water
flux.
Water molecules within membrane pores remain in the liquid phase and move in clusters 9. The relationship
between pressure and fluid flow velocity can be established through a force balance acting on a volume element of
water. This relationship is also derived from the Maxwell-Stefan theory 22. Under steady-state conditions, water
clusters are driven by a pressure gradient, balanced by frictional forces between the water and the membrane, as
well as between the water and solutes (Eq. 3) 51.
The total pressure, P (Pa), in Eq. (3) is given by
t
P = P −𝛱 (13)
t h
where P (Pa) is the applied hydraulic pressure, and Π ( Pa) is osmotic pressure, which is given by 52
h
𝛱 = RT∑ c (14)
i
i
The water permeability coefficient, A (L m –2 h –1 bar –1), is obtained by normalize the water flux by its driving
force (the applied hydraulic pressure minus the osmosis pressure at membrane surface). The unit of pressure in
denominator is converted to bar before calculation.
J
w
A = (15)
P −∑ RT(c −c )
h i i,s i,p

Equation solving and parameter fitting procedure
The model is solved analytically following a structured two-loop procedure. The outer loop iterates over
experimental conditions investigated in this study, including feed pH, solute concentration, or permeate flux. Under
each fixed condition, an inner loop determines whether the concentration polarization condition (Eq. 6) is satisfied.
At the beginning of each iteration, the solute concentration at the membrane surface is assumed equal to that in the
feed (i.e., c = c , no polarization).
s f
At the feed-side membrane-water interface, the Donnan-steric partition equations (Eqs. 1.1 and 1.2), together
with the electroneutrality condition inside the membrane (Eqs. 8 and 10), form a solvable system that yields the
solute concentrations and the dimensionless potential difference (∆𝜑 ) at the interface. Within the membrane,
transport equations (Eqs. 2.1 and 2.2) are constructed, which, together with Eqs. (7) (8) and (10), form another
solvable system. This system yields the concentrations and potential difference at the permeate-side interior of the
membrane. At the permeate-side membrane-water interface, the Donnan-steric equilibrium conditions (Eqs. 1.1 and
1.2), coupled with Eqs. (8) (9) and (10), provide analytical solutions for the permeate concentrations of all solutes.
The resulting permeate concentrations (c ) are then substituted back into Eq. (6) to compute the membrane
p
surface concentrations that satisfy the concentration polarization condition. If the updated values differ from the
initial guess, the membrane-surface concentrations are updated by averaging the new and old values; the process
iterates until concentrations at the feed, membrane surface, and permeate simultaneously satisfy Eq. (6). At
convergence, the solution set satisfies all coupled relationships, including concentration polarization, Donnan and
steric partitioning, and Nernst–Planck transport. Because NH₃ and NH₄⁺ exhibit different permeabilities, they must
reestablish chemical equilibrium via Eq. (5), which determines the permeate pH. After obtaining solute
concentrations and fluxes, the pressure drop across the membrane is determined through Eqs. (3), (13), and (14).
Parameter fitting was performed by minimizing the discrepancies between measured and calculated values. The
partitioning coefficients were fitted by minimizing the sum of squared errors between measured and predicted TAN
permeability coefficient and the rejection, with a weighing factor to adjust their magnitude to be the same level,
while the fluid–membrane friction parameter was fitted using water permeability measurements. Optimization was
conducted in MATLAB using the ‘fminsearch’ function (Nelder–Mead algorithm). To enhance convergence and
reduce the likelihood of local minima, the partitioning coefficients were constrained to the literature-based range
of 0–0.2 24,28, and the friction parameter was searched within a logarithmic range of 1013–1017. A multi-start strategy
(n = 4) was applied to further reduce the risk of local minima. Minor manual adjustments were applied to ensure a
consistent parameter set across multiple operating conditions (e.g., varying feed pH, operating pressures, and feed
concentrations; Supplementary Figures 1–2). A complete list of parameter values and their origins (literature-based,
experimentally determined, or fitted) is provided in Supplementary Table 2.

Chemicals and Materials
Lab-scale RO experiments were conducted using both synthetic NH Cl solution and real anaerobic effluent from
4
a dairy farm. The synthetic NH Cl solution was prepared by dissolving reagent grade ammonium chloride
4
(Laboratory Grade, Ward’s Science, Rochester, NY) in deionized water. A 1 M NaOH solution and a 1 M HCl
solution were used to adjust the pH of the RO feed. The real wastewater was collected from a dairy farm located in
northern Wisconsin (4439’37.6”N, 8735’15.1”W), where the manure wastewater had been anaerobically digested
and processed by a series of membrane treatments. The wastewater was from the effluent of UF and 1st pass RO,
respectively, which served as the feed for the following 1st pass and 2nd pass RO, respectively. Detailed chemical
information about the water is provided in Supplementary Table 3. The membrane used in the RO setup was a
commercial polyamide thin-film composite RO membrane (BW3E0, Dow Filmtec), purchased from Sterlitech
Corporation (Auburn, WA), designed for brackish water desalination with low pressure requirements. The received
membranes were stored in the dark at 4°C.

RO Setup and Experiments
A bench-scale crossflow RO system (Innovator CF016, Sterlitech Corporation, Auburn, WA) was employed to
carry out the RO experiment. The system featured a rectangular plate-and-frame feed flow channel with dimensions
of 5.3 cm in length, 3.9 cm in width, and 0.23 cm in depth, giving it an active membrane area of 20.6 cm2. A porous
stainless-steel frit provided mechanical support on the permeate side.
All experiments were conducted at 25 ± 1°C, with a crossflow velocity of 6.2 mm s⁻¹ at the membrane surface.
The retentate was recirculated back to the feed tank, and experiments were conducted across a range of operating
pressures from 5 to 20 bar, feed TAN concentrations from 50 to 1270 mg N/L, and feed pH values from 5.5 to 10.5
to determine TAN rejection rates and permeate flux. The feed volume for each experimental condition was 2 L.
Newly installed membranes in the RO system were first equilibrated with deionized water for 5 hours at the
experimental operating pressure. Subsequently, the membranes were equilibrated with the feed solution under
experimental conditions for 30 minutes before permeate samples were collected. The permeate reservoir was placed
on an electronic balance (Scout SPX2201, OHAUS, Pine Brook, NJ) to continuously monitor permeate flux by
tracking weight changes. The volume of permeate collected during the experiment did not exceed 2% of the total
feed volume, allowing us to assume that the composition of the solution in the feed tank remained unchanged. The
TAN, chemical oxygen demand, and total nitrogen were measured using the Nitrogen-Ammonia Reagent Set HR,
the COD Digestion Vials HR, and the Total Nitrogen Reagent Set HR (Hach, Loveland, CO), respectively, with a
spectrophotometer (DR900, Hach, Loveland, CO). The individual concentrations of NH and NH + were calculated
3 4
using Eq. (1). The cation composition of the real wastewater was determined by inductively coupled plasma optical
emission spectrometry (5800 VDV ICP-OES, Agilent, Santa Clara, CA), while the anion composition was
measured using ion chromatography (IC, Thermo Dionex 2100, Thermo Fisher Scientific, Waltham, MA). Repeated
experiments were conducted with different membrane samples cut from the same sheet under identical experimental
conditions to minimize errors due to membrane variability.

Molecular dynamics simulation
Membrane preparation. Reverse-osmosis polyamide membrane with dimensions of 4.7×4.7×9 nm3 was
prepared with MPD and TMC as described in our previous work 53. Monomers MPD and TMC were placed
randomly inside a box, maintaining a target density of 1.3 g cm–3 and MPD:TMC ratio of 1.5. The initial system
was energy-minimized using the steepest-descent method, followed by crosslinking reactions at 300 K using a
canonical ensemble. During crosslinking, reactive sites within a cutoff distance of 0.45 nm were pulled towards
each other using virtual elastic springs. As the distance between reactive sites decreased below 0.15 nm, the
topologies were updated by adding amide bonds and removing appropriate H and Cl atoms. The crosslinking steps
were stopped after the degree of crosslinking reached 90%. The unreacted monomers were removed and TMC
residues were deprotonated randomly 54 to achieve a charge density of –48 mM (8 TMCs) and –402 mM (66 TMCs)
for pH 5.5 and 10, respectively. It was ensured that an equal number of TMCs were charged on either side of the
membrane. Additionally, TMCs with one bound MPD were deprotonated before deprotonating TMCs with two
bound MPDs.
Potential of mean force. The simulations were performed following below protocol 54. The polyamide
membrane was placed at the center of a simulation box of 4.7×4.7×18 nm3, and NH or NH + was placed at (2.35,
3 4
2.35, 0) nm. Thereafter, the system was filled with TIP3P water, and sodium ions were added to neutralize the
system. The SwissParam forcefield 55 was applied to the polyamide membrane, whereas water, ions, NH , and NH +
3 4
were modelled using the CHARMM36 force field. The accuracy of SwissParam forcefield has been validated for
neutral MPD-TMC membranes with TIP3P water in previous work 38. The system was equilibrated sequentially
using steepest-descent, canonical ensemble (NVT) for 50 ps, and isothermal-isobaric ensemble (NPT) for 20 ns.
The temperature was maintained at 300 K using stochastic velocity rescaling 56. The pressure was maintained at 1
bar using the Berendsen barostat 57 with compressibility of 4.5×10–5 bar–1 along the z-axis. The box dimensions
along the x and y directions were kept fixed. The initial configurations for umbrella sampling were generated by
pulling NH or NH + across the membrane using a force constant of 100 kJ mol–1 nm–2 and a pulling rate of 10−3
3 4
nm ps–1. The displacement vector between the center of mass of the membrane and NH or NH + along the z-axis
3 4
was chosen as the reaction coordinate. A total of 141 umbrella sampling windows were chosen for the reaction
coordinate, ranging between ±7 nm. Each umbrella sampling simulation underwent simulated annealing (20 ns),
NPT equilibration (5 ns), and NPT production of 80 ns. The production simulations were performed with an
umbrella force constant of 200 kJ mol–1 nm–2, Nose-Hoover thermostat 58, and Parrinello-Rahman 59 barostat. The
PMF profiles were calculated with the weighted histogram analysis method using the periodic (along z-axis)
assumption 59. All simulations were performed using GROMACS 2022 60.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript described an ammonia nitrogen partitioning and transport (ANPT) model to predict NH3/NH4+ transport in RO membranes. This model takes into account diffusion, advection, and electro-migration of solutes, while also considering the interactions between water, solutes, and the membrane matrix. This model seems to be useful framework for predicting the transport and rejection of NH3/NH4+ in RO membrane, which is important study for resource recovery. However, the operating conditions examined in this paper appear to be at a gap with regard to their application towards the intended ammonia recovery and reuse. There are some questions remain regarding the points indicated below.\n1. This paper mentioned that it aims to understand the mechanisms of the transport of NH3/NH4+ across RO membranes especially in ammonia-rich streams. However, TAN concentrations have only been investigated within the range of a few tens of mM (such as Figure S2). This is because the present study focuses on manure stream under conditions where ammonia concentration and other ions are dilute. If this model is to be considered universally applicable for ammonia recovery from the various wastewater, it is thought that tests should be conducted at higher pressures (up to 70 bar) and with higher concentrations of ammonia. Is this model unable to deal with the 2nd RO shown in Fig. 5a? For example, other research paper presented a high NH4+ concentration of the industrial wastewater by osmotically assisted RO process over 40,000 ppm of total N using cellulose triacetate membranes (Sep. Purif. Technol, 352, 128159, 2025). Therefore, it is necessary to demonstrate the maximum concentration levels this model can predict. As it stands, the model is only applicable under limited conditions.\n2. Similarly to Question 1, what is the effect of other types and concentrations of coexisting ions on this model? In the authentic wastewater shown in Table S2, all other ions are either close to zero or, at most, in the tens of mg/L order of magnitude. Consequently, these other ions appear to exert negligible influence on the model, likely contributing to the good fitting results observed in Fig. 5. What effects might be expected on this model under conditions where other ion concentrations are high? Such investigations are thought to yield good predictions for actual wastewater.\n3. In the MD simulation of Fig.4, the authors mentioned that “adsorption of NH4+ is mainly dictated by electrostatic attraction with the polyamide membrane, while NH3 is not.” Please compare the extent to which electrostatic attraction, hydrogen bonding, and van der Waals forces influence the interaction between NH₄⁺ or NH₃ with polyamide membranes.", "reviewer_2": "This manuscript develops an Ammonia Nitrogen Partitioning and Transport (ANPT) model for RO that couples steric, Donnan, dielectric partitioning at the interfaces with extended Nernst-Planck/Maxwell-Stefan transport inside the membrane, and explicitly treats NH3 and NH4+ as separate species with re‑equilibration in the permeate. Bench‑scale RO experiments across pressure, TAN concentration, and pH validate the model. MD simulations further show higher free‑energy barriers for NH4+ than NH3, indirectly corroborating the observed phenomena. The model also performs well on an authentic manure RO effluent. The topic is timely and the integration of modeling with experiments and MD is attractive, but several major issues require clarification and improvement. Below are the specific comments.\n1. Literature survey on modeling attempts is insufficient. The Introduction cites foundational RO transport studies and some recent advances, but it lacks a cohesive, critical survey of modeling frameworks directly comparable to ANPT such as other Maxwell-Stefan variants, pH‑coupled or amphoteric‑ion transport models. Please add a section that (i) maps the assumptions, governing equations, and capabilities of prior models to those of ANPT; (ii) identifies what is novel here; and (iii) explains when ANPT reduces to or diverges from earlier models. A compact comparison table would help readers position your contribution. The current manuscript gives only hints at these ideas but does not synthesize the overarching modeling framework.\n2. Lines 167-172 and 209-213: These sections appear contradictory. The first states that counter ion transport is reduced at high pH, whereas the second says NH4+ partition increases at high pH. Please revise these passages for clarity, as they describe key findings of the paper.\n3. You state that partitioning coefficients and the fluid-membrane friction (f_f-m) were fitted, with other parameters set to “typical values.” Please (i) describe the fitting protocol (objective function, optimizer, constraints); (ii) report confidence intervals; and (iii) discuss the possibility of multiple optimal solutions. In addition, a global sensitivity analysis on Φ_i, K_a,i/K_d,i, ε, L_m, and k would quantify robustness of conclusions. Table S1 lists many fixed quantities; please indicate which were literature-based (with sources) and which were fitted.\n4. The feed‑side mass‑transfer coefficient k is computed from a Sherwood correlation using an average diffusion coefficient. Please (i) justify the single‑D approach for a multi‑solute, pH‑dependent system, (why not individual k values coupled with the electroneutrality?); (ii) show sensitivity of predictions to k; and (iii) report the film Péclet numbers used. Because CP directly affects surface concentrations, a quantitative CP analysis is important for the pressure and feed concentration sweeps.\n5. The model treats NH3 and NH4+ separately and re‑establishes equilibrium only in the permeate, omitting explicit H+/OH- transport under a “thin membrane/fast transport” assumption in Methods. Please justify this choice and if possible, test sensitivity under finite rate NH3 <--> NH4+ scenarios inside the membrane to assess its effect on predicted permeate pH.\n6. The system of equations is nonlinear and implicit. Please describe the numerical schemes used and boundary conditions especially for Donnan potentials and concentration polarization. Also clarify whether activity coefficients were assumed to be unity and discuss implications.\n7. The MD results showing higher energy barriers for NH4+ than NH3 are valuable. To strengthen the link to ANPT, discuss how these barriers relate to effective partitioning/permeability in the continuum model. Also comment on sensitivity to degree of crosslinking and charge density (the authors simulate 90% crosslinking and two charge states) and then quantify uncertainty in effective energy barrier.\n8. The real permeate used for the second RO pass has moderate conductivity and relatively simple ion composition. Please clarify (i) variability across samples; (ii) potential fouling or compaction effects; and (iii) how ANPT could extend to higher TDS or multivalent rich feeds. A brief discussion of limits of applicability would help practitioners.\nMinor comments:\n- State how Figure 1a was obtained via equilibrium calculations or from literature data.\n- Define all velocity terms (v_f, v_m, v_i).\n- Ensure every result statement refers to the correct figure (e.g., when discussing fitting performance near Line 140, reference the corresponding figure).\n- Standardize units (L based permeability while mmol based fluxes). Using volumetric units may be confusing. Consider expressing results in molar or mass units.\n- Indicate n for replicates in all figure captions and include consistent error bars.\n- Adding an identity line (y=x) would be helpful in Figure 2f.\n- Use TAN consistently (rather than alternating with “AN”)\n- Consider including comparative results from previous studies for flux components (advection, diffusion, electromigration).\n- In Figure 4a, include a labeled coordinate system.\n- Present the key model equations together in the main text for clarity, rather than splitting them across the main text and SI.\n- The opening section on global food demand and ammonia’s role is informative but lengthy. It could be shortened to focus more on the modeling novelty and knowledge gap.\n- Specify spacer geometry used to justify the Sherwood correlation and list physical properties (Re, Sc) used.\n- Describe other experimental conditions (UF pretreatment, volumetric flow, etc.) where relevant.", "reviewer_3": "The manuscript addresses modeling approaches to simulate the transport of ammonia nitrogen species through RO membranes. The topic is interesting and relevant. However, I identified several major issues throughout the paper that must be addressed before further consideration.\nAt its current stage, the manuscript and its results do not convincingly support the proposed results and conclusions, leaving substantial doubts about the robustness and originality of the work.\nMajor Comments\n1. The \"Introduction\" part in Main reads very similarly to previously published papers, particularly in structure and flow, with only minor changes in content. Please clarify the novelty and clearly distinguish this work from earlier studies.\n2. The terminology alternates between TAN and AN throughout the manuscript, which is confusing. Please standardize the notation consistently.\n3. How are concentration gradients at the boundary layers considered in Eqs. (1.1) and (1.2)? Which term(s) represent concentration polarization?\n4. Please include units for all parameters when they are first introduced. For example, indicate the unit of concentration (e.g., mmol/L) in Eqs. (1.1)–(3).\n5. Line 119: Is the coordinate, x, dimensionless or dimensional? If nondimensionalized, please clarify and show the scaling.\n6. Lines 153–154 discuss permeability but use the unit of flux (L/m2/h), particularly for TAN. Please check for consistency, including Figure 2b.\n7. Why was the model validation (Figure 2) performed only within the pH range of 6–10? Since many RO systems operate under slightly acidic conditions, expanding this range would strengthen the applicability of the model.\n8. How were NH3 and NH4+ concentrations in the permeate determined separately? If these were calculated rather than directly measured (as suggested in the Methods), please specify this in the figure captions. Also clarify how pH was measured—especially for the permeate.\n9. Figure 3a differentiates fluxes by mechanism. Are there experimental results that support this separation?\n10. Lines 190–235 describe possible removal mechanisms qualitatively, but without quantitative analysis. Since the paper claims to provide mechanistic insight into TAN removal, more rigorous quantification is needed.\n11. Figure 4 and its corresponding discussion are unclear. Do the simulations account for factors such as concentration gradients, surface charge effects, or other transport parameters? Please specify what variables were included, especially for Figures 4a and 4b.\n12. The validation with real wastewater appears to involve only a single sample (Table S2). If so, it may not be sufficient to generalize the model’s applicability or predictability.\n13. Long-term operation could lead to fouling and changes in surface charge, affecting TAN transport and rejection. Please discuss how these potential effects may influence the model and its applicability."}, "author_response": "We appreciate the reviewer’s constructive suggestions, and we thank the reviewer for the time and effort spent in helping us improve the quality of this work. In response to the reviewers’ concerns regarding model applicability, we have added additional experimental data and corresponding model results. Detailed point-by-point responses and revisions to all other specific comments are provided below. [Full rebuttal text omitted for brevity as per instructions to include verbatim text]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors properly addressed the questions in the revised manuscript. I believe the revised manuscript is now suitable for publication.", "reviewer_2": "The authors have successfully addressed my questions. I am happy with the revised manuscript and SI.", "reviewer_3": "The authors have carefully addressed all of my comments and concerns. The revised manuscript is significantly improved, and I have no further substantive comments."}, "author_response": ""}]

Decision letter