Analysis and Prediction of Concentration Polarization in a Pilot Reverse Osmosis Plant with Seawater at Different Concentrations Using Python Software

Abstract

Reverse osmosis (RO) is the most widely used technology in seawater desalination, accounting for around 70% of installations worldwide due to its efficiency and lower energy consumption compared to conventional thermal processes. However, a major challenge for RO is concentration polarization (CP), a phenomenon that reduces permeate flow, increases osmotic pressure, and compromises salt rejection, affecting the useful life of the membranes. In this work, an RO pilot plant was operated with synthetic solutions ranging from 4830 to 39,850 mg L⁻¹ at pressures between 0.69 and 5.79 MPa, to analyze and predict CP behavior. The results obtained showed salt rejection percentages ranging from 98.80% to 99.63%. The adjusted polynomial models presented correlation coefficients close to unity, which supports their high predictive capacity and statistical robustness for estimating the behavior of CP as a function of pressure. These models were implemented in Python software, allowing for the simulation of non-experimental scenarios and the anticipation of critical conditions that could compromise the RO process. Therefore, this work provides a robust predictive simulation tool to optimize RO processes and ensure the sustainable supply of drinking water in regions with water availability problems.

1. Introduction

Water is a vitally important resource that covers around 71% of the Earth’s surface. Most of it is found in seas and oceans with high salt concentrations, while only 3% is available as fresh water [1,2,3]. Of this proportion, 0.3% is found in the main sources of fresh water, rivers and lakes, which facilitate its availability for human consumption. In addition to these sources, access to groundwater bodies is also a potential source, which is why approximately 2.5 billion people depend on this source of fresh water [4]. Water resources have a wide range of applications, from basic daily needs to livestock, agriculture, and industry, which are key to regional, national, and global development [5]. However, the quantity and distribution of water have changed significantly over time due to the growing impact of human activities, inadequate policies, and climate change, causing supply imbalances [6]. This has made the provision of drinking water and water for food production and industry a challenge, particularly in arid areas where surface water bodies are overexploited, as large volumes of water are required to sustain human life and activity [7].

Northern Mexico suffers from severe water shortages, as demonstrated by CONAGUA’s Mexico Drought Monitor in April 2025, which detected extreme drought in the states of Sonora, Chihuahua, Sinaloa, Durango, and Coahuila (Figure 1). Brown indicates exceptional drought, red indicates extreme drought, light brown indicates severe drought, beige indicates moderate drought, yellow indicates abnormally dry, and gray indicates no drought. Most of the state of Sonora is colored red, and much of the southern region is colored brown [8]. Mexico’s water is distributed for various uses: 76% of the water is allocated to agriculture, 14% is used for public supply, 5% is used for thermoelectric power plants, and the other 5% is used for industry [1]. In Mexico, and especially in the southern region of Sonora, most of the water is used for agricultural crops, so it is necessary to find alternative sources of water to avoid water shortages.

An agreement must be reached that allows for sustainable water use, in which water quality, human health, and economic growth are stable and mutually reinforcing. From a social perspective, equitable access to safe water is a key indicator of justice and cohesion. On the other hand, in the economic sphere, its availability and quality are strategic factors for sustainable development [9].

In order to reach a point where water can be used sustainably, and taking into account that it is a finite resource, ways have been devised to recover or reuse it. The main method is found in the water cycle, where water evaporates, condenses, and precipitates in lakes or rivers. Still, it is too slow to be sustainable for current industrial and agricultural use. Therefore, other methods have been devised to obtain drinking water for human consumption. These include multi-stage flash evaporation (MSF), multi-effect distillation (MED), and one of the most important methods today, reverse osmosis (RO) [10]. In the case of MSF, seawater is evaporated in chambers at a pressure lower than the saturation pressure at the existing temperature, which results in sudden evaporation. This process is achieved by introducing preheated seawater through condensers at each stage, thus condensing the salt-free vapor formed by the flash effect. Before entering the first chamber, the seawater receives external heat from an external motive steam stream. When entering the chamber, which is at a lower pressure, the seawater suddenly expands, vaporizing the pure water and leaving the salts at the bottom of the tank [11,12,13]. MSF has several drawbacks, including the high economic cost per m³ of desalinated water, electricity costs, and infrastructure installation costs [12,13]. MED applies the same principle as the MSF process. The main difference lies in the way evaporation is carried out. In MED with thin-film evaporators, evaporation occurs naturally on one side of the tubes of an exchanger, taking advantage of the latent heat released by the condensation of steam on the other side of the exchanger.

In the MED process, seawater, preheated in the condensation stage of the steam generated in the last effect, enters the first effect where its temperature is raised to the boiling point with heating steam. The seawater is sprayed onto the surface of the evaporator tubes, where a thin film immediately forms, which promotes evaporation. The steam produced is collected in this effect and sent to the evaporator tubes of the next effect, which operates at a lower temperature and pressure than the previous one. The brine from the first effect is also sent to the next effect, where it is sprayed, forming a thin film on the surface of the tubes through which the steam circulates, repeating the evaporation process. The vapor from each stage is thus converted into desalinated water when it condenses in the evaporator of the next stage. The process is repeated several times depending on the number of stages in the system [12,13,14], and like MSF, it is inefficient and involves high energy costs [15].

In contrast to the thermal desalination systems mentioned above, reverse osmosis (membrane desalination system) has several advantages over technologies that desalinate seawater using heat (phase change), listed below: high-quality water, lower energy consumption, greater removal of contaminants, compact design, automation, and lower maintenance [16,17,18,19,20]. However, as with all technologies, there are advantages and disadvantages. One disadvantage is that 60 to 70% of the cost per m³ of desalinated seawater is due to the energy used [16]. However, this can be mitigated to a large extent by harnessing direct solar energy with photovoltaic cells, as in the case of Dévora-Isiordia et al. (2022), who adapted solar panels to the reverse osmosis process, making it more sustainable and efficient [21].

The other disadvantage is concentration polarization, which consists of the fact that, when water passes through the semipermeable RO membrane, preferably from the feed to the permeate, the salt remains close to the membrane on the feed side unless the salt diffuses back into the main body of the feed solution (Figure 2a). This phenomenon is called concentration polarization (CP) (Figure 2b), which has adverse effects on membrane performance, such as reduced permeate flow, increased permeate concentration, increased operating pressure, and decreased salt rejection, due to the increase in feed water concentration in the boundary layer of the active layer of the RO membrane, which represents an increase in osmotic pressure [22,23,24,25,26,27]. The acceptable limit value for the CP phenomenon is 1.2, as proposed by Kucera (2015) [25]. According to boundary layer theory, concentration polarization is described as follows:

First, the presence of a boundary layer of thickness δ_bl is assumed, such that salt diffusion from the membrane to the main body of the feed stream occurs in the boundary layer (Figure 2b). When considering the mass balance between the plane at a distance y and the membrane wall on the permeate side [22]:

$${-D}_{BA}{\displaystyle \frac{d{C}_B}{dy}}+v{C}_B={vC}_{B3}$$

(1)

where D_BA is the diffusion coefficient (m² s⁻¹) of solute B in solvent A (mainly water) in the boundary layer, C_B is the concentration of the solute, and v is the velocity of the solution.

The performance of reverse osmosis plants is closely related to operating conditions and feed water quality. Therefore, accurate and efficient models are essential for both process operation and design [28,29]. However, developing an accurate mechanical model that takes into account multiple key factors—such as water permeation rate, flow rate, concentration polarization, feed water temperature, and fouling—poses a significant challenge. Although various tools and software programs have been created to support the design of reverse osmosis plants, most of these focus primarily on analyzing the overall performance of the system, leaving the optimization of individual modules in terms of water quality, energy efficiency, and concentration polarization in the background [28]. Al-Mutaz et al. (2017) combined the diffusion transport model of the solution and film theory to obtain an explicit expression of water flow through the reverse osmosis process [30]. This formula helped to calculate concentration polarization using limited data on the properties of water, salt, and the membrane.

The main objective of this research was to evaluate concentration polarization through the operation of an RO desalination plant to obtain compact predictive mathematical models based on operating pressure. With the results obtained for the eight feed concentrations, reliable mathematical models with R² greater than 0.9778 were obtained that predicted concentration polarization with nonlinear behavior. In addition, to provide statistical robustness and present an analysis of errors between different input concentrations, operating pressure variations, and CP response, two tests were performed: constant variance analysis and Durbin–Watson test. The software used is Minitab 17, version 21.1.0. These models were replicated for permeate flow and permeate flux, which exhibit linear behavior. All the models described were programmed in Python software (version 3.13) to simulate the behavior of the main variables in the reverse osmosis process.

2. Materials and Methods

2.1. Reverse Osmosis Pilot Plant Set-Up

A reverse osmosis pilot plant was used (SW-0.38K-125, Pure Aqua, Inc., Santa Ana, CA, USA), which is the same equipment used by Dévora-Isiordia et al., (2023) [31], with a capacity of 1 L min⁻¹ (Figure 3).

Eight synthetic seawater solutions at different concentrations were used to conduct this experiment: 4830; 9950; 15,030; 20,200; 24,860; 29,860; 34,850, and 39,850 mg L⁻¹. To prepare each concentration, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4 kg of Instant Ocean^MR salt were weighed, respectively, and each was diluted in 100 L of distilled water.

2.2. Calibration Curve and Performance of the Desalination Plant

If there is no correct correlation for the YSI-556 multiparameter device (Yellow Springs, OH, USA), it must be calibrated with standard solutions of 10,000, 20,000, and 35,000 mg L⁻¹. Subsequently, to obtain the calibration curve between actual seawater and synthetic seawater, commercial Instant Ocean^® salt (St. Blacksburg, VA, USA) and a YSI 556 MPS multiparameter device were used. Finally, the electrical conductivity reading was adjusted using Equation (2), shown in

Table 1. Equations for the calculation of the performance of a desalination plant.

Equation	Equation Number	Description and Reference
$\mathrm{y}=\Omega [{\displaystyle \frac{\sigma }{1+\left({\mathcal{o}}^0+{\mathcal{o}}^1\mathrm{y}\right)\left(\mathrm{t}-25°C\right)}}-\mathsf{\sigma }{y}^{25}{]}^{\alpha }$	(2)	Adjustment interactions between the conductivity of synthetic seawater and seawater [31]
${\mathsf{\mu }}_{\mathrm{a}}=\mathrm{e}\mathrm{x}\mathrm{p}(5.495921\times {10}^5{\mathrm{T}}^{-2}-1.66779\times {10}^3 {\mathrm{T}}^{-1}-7.612821)$	(3)	Viscosity of distilled water [32]
$F_v={\displaystyle \frac{V_p}{A_{me}}}$	(4)	Permeate flux [22,23]
${\mu }_s={\mu }_a[1+\sqrt{z}\left(a_1+a_2{\mathrm{T}}^3\right)+\mathrm{z}\left(a_3{+a}_4{\mathrm{T}}^3\right)+a_5{z}^2{\mathrm{T}}^3]$	(5)	Viscosity of the salt water [32,33]
$\%{\mathrm{R}}_{\mathrm{o}}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{f}}-{\mathrm{C}}_{\mathrm{p}\mathrm{e}}}{{\mathrm{C}}_{\mathrm{f}}}}\times 100\%$	(6)	Observed salt rejection [22,23,31]
$\%{\mathrm{R}}_{\mathrm{i}}={\displaystyle \frac{1}{\mathrm{C}\mathrm{P}\left(\frac{1}{{\mathrm{R}}_{\mathrm{o}}}-1\right)+1}}\times 100\%$	(7)	Intrinsic rejection of salts [31,34]

.

With the data taken by the reverse osmosis pilot plant, different results were obtained such as membrane resistance (R_me), distilled water viscosity (μ_me), permeance (L_p), permeate flux (F_v), nominal operating pressure (P) and flow rate (V_p), percentage of observed rejection (%R_o) and percentage of intrinsic rejection (%R_i). For the experiments, 5 replicates were performed (Table 1).

2.3. Polarization Concentration

To determine the increase in feed water concentration in the active layer of the RO membrane [31,34], the concentration polarization phenomenon was calculated using the following equation.

$$CP={\displaystyle \frac{\mathsf{\Delta }\mathrm{P}-{\mathrm{F}}_{\mathrm{v}}{\mu }_s{\mathrm{R}}_{\mathrm{m}\mathrm{e}}}{({\mathrm{C}}_{\mathrm{f}}-{\mathrm{C}}_{\mathrm{p}\mathrm{e}})}}$$

(8)

2.4. Reverse Osmosis Plant Operation

A reverse osmosis plant with a production capacity of 1.44 m³ d⁻¹ was used. The pilot plant was operated at different concentrations (Section 2.1) and at various flow rates from 27.6 to 278.4 L h⁻¹ (Figure 4), with a total of 53 experiments, each experiment having 5 repetitions for a total of 265 tests.

For each experiment, 30 min were allowed for equipment stabilization, after which readings were taken of the equipment’s operating pressure, permeate flow rate, reject flow rate, salt concentration (both permeate and reject), temperature, and electrical conductivity. The sample was collected using a 2 L beaker with five repetitions at 15 min intervals. This data was recorded in Excel version 16.66.1 spreadsheets containing Equations (2)–(8) to obtain the concentration polarization results.

2.5. Prediction of Polarization Concentration with Python

To develop the prediction program for the Reverse Osmosis Pilot Plant for the parameters of permeate flow, flux, and concentration polarization (CP), we used the free software Python version 3.13 (Creator: Guido Van Rossum, Amsterdam, The Netherlands). Previously, to develop the software script, mathematical models were obtained for each concentration as a function of pressure, using Microsoft Excel version 16.66.1. A linear and polynomial adjustment was performed to select the model that best fits the predictions, considering the value of the coefficient of determination (R²) closest to 1, together with a descriptive statistical analysis (standard deviation and variance). Once the best-fitting model was defined, it was fed into the Python program to design the algorithm.

3. Results and Discussion

3.1. Concentration Polarization Versus Pressure

Table 2. Flux, salt rejection and polarization with brackish water at 4830 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	0.69	26.45	27.6	9.86	99.30	1.01
2	1.52	27.18	73.2	26.14	99.51	1.02
3	1.93	29.16	102.0	36.43	99.50	1.02
4	2.76	27.54	136.8	48.86	99.65	1.04
5	3.10	31.23	168.0	60.00	99.70	1.06
6	3.59	32.46	204.0	72.86	99.64	1.10
7	4.83	29.56	241.2	86.14	99.68	1.22
8	5.38	34.43	278.4	99.43	99.64	1.29

CP: ● normal, ● transition, ● operational risk.

,

Table 3. Flux, salt rejection and polarization with brackish water at 9950 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	1.38	26.42	45.6	16.29	99.55	1.01
2	1.93	28.64	77.4	27.64	99.56	1.02
3	2.62	27.27	109.2	39.00	99.63	1.05
4	3.03	29.91	142.2	50.79	99.70	1.08
5	3.52	32.01	169.8	60.64	99.68	1.10
6	4.41	31.68	206.4	73.71	99.66	1.22

CP: ● normal, ● transition.

,

Table 4. Flux, salt rejection and polarization with brackish water at 15,030 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	1.86	26.60	42	15.00	99.07	1.02
2	2.48	27.30	77.4	27.21	99.35	1.03
3	3.03	30.15	109.2	39.00	99.33	1.05
4	3.52	29.91	142.2	50.79	99.37	1.10
5	4.41	28.91	169.8	60.64	99.33	1.15
6	4.83	33.25	205.8	73.50	99.34	1.23
7	5.38	35.95	259.8	92.79	99.17	1.29

CP: ● normal, ● transition, ● operational risk.

,

Table 5. Flux, salt rejection and polarization with brackish water at 20,200 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	2.21	27.94	40.80	14.57	98.87	1.03
2	3.17	26.99	78.00	27.86	99.21	1.04
3	3.59	30.21	106.8	38.14	99.23	1.09
4	4.48	28.82	141.0	50.36	99.22	1.19
5	5.38	28.85	179.4	64.07	99.35	1.31

CP: ● normal, ● operational risk.

,

Table 6. Flux, salt rejection and polarization with brackish water at 24,860 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	2.69	27.86	43.20	15.43	98.79	1.04
2	3.03	26.75	59.40	21.21	98.92	1.07
3	3.38	26.60	72.66	25.95	99.32	1.08
4	3.79	29.70	90.60	32.36	99.17	1.10
5	4.00	31.72	107.40	38.36	99.13	1.14
6	4.41	29.45	121.62	43.44	99.26	1.18
7	5.10	27.26	141.00	50.36	99.43	1.31
8	5.38	32.03	154.80	55.29	99.36	1.39

CP: ● normal, ● operational risk.

,

Table 7. Flux, salt rejection and polarization with brackish water at 29,890 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	3.17	27.91	44.40	15.86	98.64	1.05
2	3.59	26.08	61.80	22.07	99.03	1.09
3	4.00	31.09	84.00	30.00	99.05	1.13
4	4.55	28.06	86.40	30.86	99.24	1.16
5	4.69	33.21	108.00	38.57	99.19	1.22
6	4.76	36.29	122.40	43.71	99.14	1.27
7	5.65	32.66	144.00	51.43	99.32	1.41

CP: ● normal, ● transition, ● operational risk.

,

Table 8. Flux, salt rejection and polarization with brackish water at 34,850 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	3.72	27.12	47.06	16.81	98.61	1.12
2	4.41	25.90	62.10	22.18	98.93	1.21
3	4.69	26.26	75.54	26.98	99.11	1.25
4	4.83	30.25	92.21	32.93	99.03	1.27
5	5.38	27.25	97.80	34.92	99.21	1.41
6	5.65	32.53	123.36	44.06	99.01	1.45

CP: ● normal, ● transition, ● operational risk.

and

Table 9. Flux, salt rejection and polarization with brackish water at 39,850 mg L⁻¹.

No.	P (MPa)	T (°C)	Vp (L h−1)	Fv (L m−2 h−1)	Ro %	CP
1	4.14	26.45	45.69	16.32	98.47	1.17
2	4.55	31.76	62.52	22.33	98.53	1.25
3	5.10	26.90	80.11	28.61	99.03	1.34
4	5.45	32.50	95.27	34.03	98.94	1.43
5	5.79	31.72	107.40	38.36	99.05	1.50

CP: ● normal, ● transition, ● operational risk.

are structured as follows: the first column shows the experiment number, operating pressure in MPa, temperature in °C, volumetric flow in L h⁻¹, permeate flux in L m⁻² h⁻¹, observed salt rejection in %, and concentration polarization represented as CP. The following concentrations were used: 4830; 9950; 15,030; 20,200; 24,860; 29,890; 34,850, and 39,850 mg L⁻¹. The thresholds were taken from Kucera’s book (2015) entitled “Reverse Osmosis,” where she establishes that the CP value can be explained using a traffic light system as follows: CP < 1.2 indicates that the reverse osmosis system is within the optimal operating range, represented by green circles [25]. Yellow circles indicate that the system is in a transition phase with values of 1.2 < CP < 1.25, which can cause pressure increase problems, and red circles represent prohibited CP values > 1.25, which pose a critical risk to the operation of the reverse osmosis system.

Table 2 and Table 3 show that at concentrations of 4830 and 9950 mg L⁻¹, since the feed water has low salinity, there is no CP because no salt agglomeration forms in the active layer of the membrane at pressures below 3.59 MPa. However, at pressures close to or greater than 4.83 MPa, exceeded values occur that compromise the operation of the reverse osmosis system. Table 4, Table 5, Table 6 and Table 7 show that there is a transition in the CP variable result at a range of pressure (4.83–5.65 MPa) because at lower pressures, there is salt accumulation in the active layer of the membrane, which causes an increase in osmotic pressure, suggesting a rigorous chemical pretreatment system with the addition of antiscalants to regulate the CP phenomenon.

Table 8 and Table 9 show high CP values at pressures greater than 4.41 MPa. This indicates that at feed water concentrations equal to or greater than seawater, the phenomenon is enhanced in the active layer of the membrane. These results of increased internal and external CP in the membrane reported by Ma et al. (2023) also confirm that increased feed water salinity in a direct osmosis system is a factor that affects flux reduction (Figure 5) and salt rejection due to salt accumulation in the active layer of the membrane [35]. Studies of the polarization factor presented by Wei et al. (2024), using the Lattice–Boltzmann method, found that CP decreases at higher Reynolds numbers due to the promotion of turbulence in the active layer of the membrane [36]. Furthermore, it mentions that the variation in permeate flow promotes the absence of scaling or salt accumulation. On the other hand, Prakash et al. (2024) agree that turbulent flow and velocity fluctuations lead to a decrease in CP [37].

3.2. Salt Rejection, Recovery and Permeate Flux

Results indicate that as salt concentration increases, the salt removal capacity of the pilot plant declines. As shown in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, average salt rejection rates of 99.58%, 99.63%, 99.28%, 99.18%, 99.17%, 99.09%, 98.98%, and 98.80%, respectively. Dévora-Isiordia et al. (2023) reported similar results using a reverse osmosis pilot plant [24]. Specifically, salt rejection for the 5000 mg L⁻¹ solution was 93.68%, whereas that for the 10,000 mg L⁻¹ solution was 89.8% at 23 °C.

In another study published by Dévora-Isiordia et al., (2023) with a solution of 13,535 mg L⁻¹, the pilot plant had an average salt rejection of 99.71%, and with the solution of 35,522 mg L⁻¹ it was 99.59% [31]. The long-term risks of fouling, scaling, and membrane degradation increase as the feed water concentration increases. This suggests more rigorous physical and chemical pretreatment systems. It is important to note that this study was conducted at the pilot level, where dosing concentrations are lower due to scalability. However, the number of filters, concentration, and dosing frequency will depend on the size of the industrial plant. It should be noted that a pilot plant study prevents some of the start-up problems associated with an industrial plant, allowing for variations and studies at the pilot level, which would not be possible in an industrial context. The reverse osmosis membrane module has more difficulty desalinating as the salt concentration in the solutions and the operating time increase. Considering the operating time, authors such as Bai et al. (2023) [38] mention that the process of concentration boundary layer formation occurs in the first few minutes of the desalination process, which is why the technical data sheet provides specifications or limits of use [39]. High pressures and concentration polarization cause more mass transfer of ions to the permeate [22]. With the average salt rejections of the pilot plant and a permeate water concentration value of 478 mg L⁻¹ of total dissolved solids, drinking water suitable for human consumption can be obtained in accordance with Mexican Official Standard NOM-127-SSA1-2021 [40].

Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 and Figure 5 also show that the permeate flux decreases as the salt concentration increases. It is understood that the capacity of the membrane module is affected by the high salt concentration and the effect of concentration polarization [24,39]. For example, at a concentration of 4830 L h⁻¹ (Table 2) and a pressure of 5.38 MPa, the permeate flow and flux were 278.4 L h⁻¹ and 99.43 L m⁻² h⁻¹, respectively, but a concentration of 15,030 mg L⁻¹ at the same pressure (Table 3), 259.8 L h⁻¹ and 92.79 L m⁻² h⁻¹, respectively, there was a decrease in permeate flow and flux of 6.68%. When compared to a concentration of L h⁻¹ (Table 6) and a pressure of 4 MPa, the permeate flow and flux were 107.40 L h⁻¹ and 38.36 L m⁻² h⁻¹, respectively, for the concentration of 29,890 mg L⁻¹ (Table 7) and the same pressure, the results were 84 L h⁻¹ and 30 L m⁻² h⁻¹, respectively, with a decrease in permeate flow and rejection of 21.78%. Giacobbo et al. 2018 determined in their study that concentration polarization causes a reduction in membrane performance [41].

One of the key variables in a reverse osmosis process is conversion, which is defined as the ratio of the permeate flow obtained to the feed flow. Based on Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, it is confirmed according to Kucera (2015) [25] that conversion decreases as the concentration of the feed solution increases: 28.22, 27.6, 26.29, 18.83, 18.19, 17.08, 15.29, and 14.47% average conversion in respective order for each concentration. This coincides with the findings reported by Medina-Collana et al. (2024), whose study shows that increasing the pressure results in better recovery [42].

3.3. Mathematical Models for Prediction of Flux and Concentration Polarization

Table 10. Linear mathematical models of permeate flux as a function of pressure at different salt concentrations.

No.	Concentration (mg L−1)	Linear Mathematical Models of Permeate Flux (L m−2 h−1)	R2
1	4830	y = 18.909x − 1.2749	0.9888
2	9950	y = 19.049x − 9.1121	0.9952
3	15,030	y = 20.691x − 24.089	0.9804
4	20,200	y = 15.752x − 20.316	0.9951
5	24,860	y = 14.698x − 23.087	0.9898
6	29,890	y = 14.762x − 30.007	0.9787
7	34,850	y = 14.667x − 39.767	0.9544
8	39,850	y = 13.225x − 38.271	0.9980

Where y represents CP, and x represents the pressure operation.

shows the linear mathematical models of permeate flux as a function of the operating pressure of the pilot plant. The R² values are very close to 1, indicating that there is considerable linearity, as demonstrated by James et al. (2023) [43]. Various authors, such as Dévora-Isiordia et al. (2023) [24,31], found that pressure is directly proportional to permeate flow; in other words, the higher the pressure, the greater the permeate flow obtained, and they also observed linear behavior very similar to Figure 5. Mohammadi (2009) treated an aqueous effluent containing 1–20 mg L⁻¹ of chromium, and the behavior of the permeate flow against the operating pressure was linear [44,45]. Kucera (2015) cites a study on seawater treated with a FilmTec FT-30 membrane for seawater with a sodium chloride solution of 35,000 mg L⁻¹ and an osmotic pressure of 2.5 MPa [25]. He comments that when the osmotic pressure is overcome, the increase in permeate flow is linear with the increase in the operating pressure of the system. Similarly, Dencheva-Zarco mentions in her work that working with high pressures increases the permeate flow [46].

Figure 6 shows that as the concentration of the feed solution increases, a higher pumping operating pressure is required to overcome the osmotic pressure. This is why, at low concentrations, the pressure range is greater. For example, for a concentration of 4830 mg L⁻¹ (0.69–5.38 MPa), in contrast to the higher feed concentration of 39,850 mg L⁻¹ (4.14–5.79 MPa), the operating pressure range decreases. In other words, the operating pressure range is inversely proportional to the increase in salt concentration [38].

In addition, eight polynomial models of concentration polarization (CP) were obtained, of which five are fourth-order, two are third-order, and one is second-order. All models have an R² very close to 1, which guarantees that the model will predict the actual or experimental value of CP (

Table 11. Polynomial mathematical models for predicting concentration polarization as a function of pressure at different salt concentrations.

No.	Concentration (mg L−1)	Mathematical Models of Concentration Polarization	R2
1	4830	y = −0.0012x4 + 0.0147x3 − 0.0446x2 + 0.0537x + 0.9911	0.9990
2	9950	y = 0.0104x4 − 0.1119x3 + 0.4449x2 − 0.7237x + 1.421	0.9996
3	15,030	y = 0.003x4 − 0.0043x3 + 0.037x2 − 0.0892x + 1.0758	0.9905
4	20,200	y = 0.0026x2 − 0.1045x + 1.1255	0.9947
5	24,860	y = −0.0004x4 + 0.0196x3 − 0.1542x2 + 0.4886x + 0.4832	0.9977
6	29,890	y = −0.1011x4 + 1.75x3 − 11.154x2 + 31.153x − 31.162	0.9978
7	34,850	y = −0.025x3 + 0.3904x2 − 1.816x + 3.7605	0.9953
8	39,850	y = 0.0127x3 − 0.1683x2 + 0.9226x − 0.6666	0.9976

Where y represents CP, and x represents the pressure operation.

).

It can be seen in Figure 7 that the concentration polarization (CP) increases as the operating pressure and the flux obtained (Figure 5) in the pilot plant increase, using nonlinear polynomial models. This contrasts with the findings of Álvarez-Sánchez et al. (2024), who obtained pressure versus concentration polarization graphs, but with linear behavior in cross-flow equipment [47]. According to the results obtained by Dévora-Isiordia et al. (2023), the behavior in their results on concentration polarization has not been linear; instead, they had to use polynomial models to predict CP with greater accuracy [31].

Table 12. Concentration polarization results: theoretical and experimental.

Pressure (MPa)	$\mathbf{C}\mathbf{P}$ Experimental	$\mathbf{C}\mathbf{P}$ Theoretical	Standard Deviation	Variance	R2 Linear	R2 Polynomial
Brackish water calculations at 4830 mg L−1
0.69	1.01	1.01	0.001	1.1 × 10−6	0.8980	0.9990
1.52	1.02	1.01	0.004	1.3 × 10−5
1.93	1.02	1.02	0.002	2.8 × 10−6
2.76	1.04	1.04	0.001	5.7 × 10−7
3.10	1.06	1.06	0.003	7.0 × 10−6
3.59	1.10	1.09	0.007	5.4 × 10−5
4.83	1.22	1.21	0.005	2.4 × 10−5
5.38	1.29	1.27	0.012	1.5 × 10−4
Brackish water calculations at 9950 mg L−1
1.38	1.01	1.01	0.002	5.2 × 10−6	0.8551	0.9996
1.93	1.02	1.02	0.001	9.2 × 10−7
2.62	1.05	1.06	0.005	2.1 × 10−5
3.03	1.08	1.08	0.002	5.3 × 10−6
3.52	1.10	1.10	0.001	2.2 × 10−6
4.41	1.22	1.22	0.001	3.2 × 10−7
Saline water calculations at 15,030 mg L−1
1.86	1.02	1.01	0.004	1.7 × 10−5	0.9398	0.9905
2.48	1.03	1.03	0.001	9.6 × 10−7
3.03	1.05	1.05	0.001	2.1 × 10−6
3.52	1.10	1.08	0.014	2.0 × 10−4
4.41	1.15	1.15	0.001	5.3 × 10−7
4.83	1.23	1.19	0.029	8.4 × 10−4
5.38	1.29	1.25	0.027	7.5 × 10−4
Saline water calculations at 20,200 mg L−1
2.21	1.03	1.02	0.006	3.6 × 10−5	0.9230	0.9947
3.17	1.04	1.06	0.011	1.2 × 10−4
3.59	1.09	1.09	0.003	1.2 × 10−5
4.48	1.19	1.18	0.007	5.5 × 10−5
5.38	1.31	1.32	0.004	1.6 × 10−5
Saline water calculations at 24,860 mg L−1
2.69	1.04	1.04	0.002	2.6 × 10−6	0.9441	0.9977
3.03	1.07	1.06	0.007	5.5 × 10−5
3.38	1.08	1.08	0.002	2.7 × 10−6
3.79	1.10	1.10	0.003	1.0 × 10−5
4.00	1.14	1.12	0.012	1.5 × 10−4
4.41	1.18	1.17	0.008	6.3 × 10−5
5.10	1.31	1.29	0.012	1.3 × 10−4
5.38	1.39	1.37	0.017	3.0 × 10−4
Saline water calculations at 29,890 mg L−1
3.17	1.05	1.04	0.004	1.4 × 10−5	0.9291	0.9778
3.59	1.09	1.10	0.007	4.9 × 10−5
4.00	1.13	1.10	0.018	3.3 × 10−4
4.55	1.16	1.18	0.015	2.3 × 10−4
4.69	1.22	1.22	0.001	4.9 × 10−7
4.76	1.27	1.24	0.021	4.6 × 10−4
5.65	1.41	1.40	0.009	8.4 × 10−5
Sea water calculations at 34,850 mg L−1
3.72	1.12	1.12	0.000	1.4 × 10−7	0.9740	0.9953
4.41	1.21	1.20	0.007	4.7 × 10−5
4.69	1.25	1.25	0.001	1.4 × 10−6
4.83	1.27	1.28	0.007	4.9 × 10−5
5.38	1.41	1.40	0.009	8.1 × 10−5
5.65	1.45	1.45	0.003	6.4 × 10−6
Sea water calculations at 39,850 mg L−1
4.14	1.17	1.17	0.000	1.1 × 10−7	0.9950	0.9976
4.55	1.25	1.24	0.005	2.2 × 10−5
5.10	1.34	1.35	0.004	1.7 × 10−5
5.45	1.43	1.42	0.008	6.6 × 10−5
5.79	1.50	1.50	0.001	1.5 × 10−6

presents the different ways in which the CP can be obtained experimentally and theoretically (calculated). Descriptive statistics show standard deviation values, which, according to Alvarado and Tenezaca (2023), are a measure of data variability and an indicator of the dispersion of the values of a variable [48]. It is known that an error of less than 10% with respect to the experimental data is permissible and reliable. It is of great importance in classical inference, especially in relation to the study of normal distribution as one of the parameters that determines the distribution, in addition to the population mean [49], but its interest is reduced in traditional inference in finite populations with values less than 10%, which is permitted and acceptable.

The standard deviation value has a minimum and maximum value of 0 and 0.029, respectively, indicating the low dispersion of the experimental data (CP experimental) and calculated data (CP theoretical) in the polynomial concentration polarization models. Similarly, the variance value shows the same trend with values close to zero, with a minimum value of 1.1 × 10⁻⁷ and a maximum value of 8.4 × 10⁻⁴, indicating the reliability of the models.

The data shown in Table 12 present the R² values with a comparison between linear and polynomial models, which shows that the polynomial models fit more closely to the value of 1 than the linear models, suggesting that they are models that predict behavior closer to the experimental data. With the above and the models evaluated, it is possible to obtain the CP value for any operating pressure.

3.4. Prediction of Concentration Polarization Using Python Software

The Python software script (Figure 8) is based on the mathematical models in Table 10 and Table 11. The program code was developed to predict the parameters of permeate flow, flux, and concentration polarization at various pressures, which could not be performed during the experiment.

CP values reported by other authors (

Table 13. Comparison of other models for predicting CP with different authors.

Software	Model	CP	Feed Water (mg L−1)	References
Results of this research
Excel, 16.66.1. Phyton 3.13	$CP={\displaystyle \frac{\mathsf{\Delta }\mathrm{P}-{\mathrm{F}}_{\mathrm{v}}{\mu }_s{\mathrm{R}}_{\mathrm{m}\mathrm{e}}}{\left({\mathrm{C}}_{\mathrm{f}}-{\mathrm{C}}_{\mathrm{p}\mathrm{e}}\right)}}$ Film theory	1.01–1.5	4830–39,850	[31,34]
Results from other research studies
Excel 16.66.1. Matlab R2023a	$CP=\mathit{exp}{\displaystyle \frac{J_w}{k_{mt}}}$ Film theory	1.094 a 1.106	5000	Dévora-Isiordia et al., 2023 [24]
Excel, 16.66.1. Matlab R2023a	$CP=\mathit{exp}{\displaystyle \frac{J_w}{k_{mt}}}$ Exponential with activity correction	1.092 a 1.106	10,000	Dévora-Isiordia et al., 2023 [24]
Excel 16.66.1.	$\mathrm{C}\mathrm{P}={\displaystyle \frac{C_m-C_p}{C_f- C_p}}=\mathit{exp}{\displaystyle \frac{J_w}{k_{mt}}}$ Film theory	1.007 a 1.022	10,000	Dévora-Isiordia et al., 2022 [21]
Matlab, R2023a Comsol, 6.2 Aspen V11.0	$\mathrm{C}\mathrm{P}={\displaystyle \frac{C_w}{C_b}}=[1-{Ri+ R}_{i }{\mathit{exp}\left(-{\displaystyle \frac{V_w}{k}}\right)]}^{-1}$ Film theory	1.25	2922 (NaCl)	Kim and Hoek (2005) [50]

) indicate that several model assumptions have been analyzed, such as film theory, exponential with activity correction, and Spiegler Kedem film, with results similar to those reported in this study. CP values from other authors range from 1.007 to 1.25, while those from this study range from 1.01 to 1.50. This indicates that the models analyzed coincide despite being different analysis assumptions.

This programming will allow an environment to determine whether the CP (Concentration Polarization) is greater or less than the maximum allowed value of 1.2 recommended by Kucera (2015) [25]. On the other hand, if the value is greater than 1.25, a warning will be issued indicating that the membranes are polarizing. At this value, the equipment may be damaged due to increased inlet pressure to the membrane module.

4. Conclusions

Eight linear models and eight polynomial models were determined to determine concentration polarization using Excel software 16.66.1, which served as input for creating the code in Python software. For this purpose, eight feed concentrations were used: 4.830, 9.950, 15.030, 20.200, 24.860, 29.860, 34.850, and 39.850 mg L⁻¹ in a reverse osmosis pilot plant. Equations were used that employed variables such as permeate flow, flux, active membrane area, viscosity, salt rejection, constants, and temperature. The results were analyzed using linear and polynomial mathematical models, with the latter being the most accurate (fourth- and third-order) for calculating permeate flow, flux, and concentration polarization, with R² values close to 1 and a better fit than the linear models.

Salt rejection ranged from 98.80% for a concentration of 39,850 mg L⁻¹ to 99.63% for a concentration of 4830 mg L⁻¹, which is sufficient to produce drinking water. This indicates that as the concentration of salts fed into the RO process increases, salt rejection gradually decreases. In a reverse osmosis process, it is very important to take into account the concentration polarization because the membranes can suffer significant damage. When RO processes are on an industrial scale, if a good preventive maintenance program is not promoted in the process, there could be damage to personnel due to the high pressures that are handled or that can be generated with a polarized membrane module. The improvement of a reverse osmosis system in terms of polarization concentration is achieved by using the variables resulting from simulations, which allow the necessary technical adjustments to be made to reduce this phenomenon and increase the efficiency of the system. It is important to perform maintenance after each test in the pilot plant to prevent the membrane from polarizing or to perform continuous maintenance with descaling substances. With regard to maintenance requirements, a 0.5% hydrochloric acid (HCl) solution was applied to the membrane system and the system was left to recirculate for 60 min with changes in flow rate to promote turbulent Reynolds flow, thereby reducing the deposition of calcium carbonate, calcium sulfate, barium sulfate, strontium sulfate, and silica salts. In this sense, chemical and physical pretreatment are very important before membranes, but to protect the life of the membrane modules, acid and base washes are key to better reverse osmosis system performance.

This demonstrates that the models obtained enhance decision-making to predict CP without having to implement or halt the operation of a reverse osmosis plant. In this context, process control and savings in time due to downtime or damage will be avoided. On the other hand, if the model predicts high CP values, adjustments can be made to the antiscalants dosage, application frequency, and operating pressure. These examples of the model’s application will be the results and contributions to the scientific community for its implementation and prevention, to ensure the supply of drinking water to cities, industries, and the agro-industrial sector.