# A Comparative Analysis of Statistical Models and Mathematics in Reverse Osmosis Evaluation Processes as a Search Path to Achieve Better Efficiency

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Model Concentration Polarization

#### 1.1.1. Mathematical Modeling

#### 1.1.2. Membrane Modeling Approaches

#### 1.1.3. Mathematical Modeling in Reverse Osmosis

_{v}and J

_{s}= the flux of the solvent and solute, respectively

_{p}= the solvent permeability coefficient (the membrane permeability of water)

_{p}= the transmembrane pressure or system operating pressure

_{π}= osmotic pressure

_{s}= solute transport coefficient

_{δ,1}= solute concentration at the membrane surface (feed side)

_{p}= solute concentration in the permeate

_{s}= solute concentration within the membrane

_{δ1}membrane, generating the concentration by polarization [5,43].

_{p}and J

_{v}is shown in Equation (3):

_{p}= the permeate flow

_{f}, with respect to the solute concentration in the permeate stream, C

_{p}.

_{p}parameter, such that it must be adjusted with the Arrhenius equation [28]:

#### 1.2. RO Optimization Modeling

^{3}; today, and with the new advances in general, the consumption is in the order of 2.5 kWh/m

^{3}[45].

^{3}were achieved.

^{3}/h, three-stage modules are suitable for production up to 20 m

^{3}/h [30].

#### 1.3. Statistical Modeling

^{3}/day they use an energy consumption of only 1.2 to 1.3 kWh/m

^{3}.

## 2. Goals

## 3. Review and Analysis Process

- -
- The type of model applied
- -
- Reverse osmosis system efficiency considerations
- -
- Ease of operation
- -
- Economic criteria
- -
- Energy efficiency
- -
- The efficiency and effectiveness in the results of contaminant rejection and flux production

- -
- Keywords: reverse osmosis, mathematical model, statistical model, heavy metals;
- -
- Databases: Scopus, Google Scholar;
- -
- Main conclusions: physical analysis of the membrane, energy efficiency in the process, main operating variables of the process.

## 4. Results

^{2}h, turned out to be the highest value in this study. For the second case, the optimal solution was found to be a cooling inlet temperature of 13.9 °C, a feed inlet temperature of 59 °C and a feed flow rate of 205 L/h. Under these plant operating conditions, a maximum specific yield index of 188.703 kg/kWh and a specific energy consumption of 5.3 kWh/m

^{3}were experimentally obtained [57].

^{3}/d.

#### Mechanism for the Identification and Application of the Model

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Solution diffusion scheme, source [40].

**Figure 3.**Energy consumption in one-stage, two-stage, and closed-loop RO systems, source [31].

**Figure 4.**Low-consumption RO system proposed by, source [48].

**Figure 6.**ISD system optimization using finite difference approximation source [54].

Model | Typology | Characteristic | Efficiency and Effectiveness in the Results of Contaminant Rejection and Flux Production | ||||
---|---|---|---|---|---|---|---|

Type of Model Applied | Reverse Osmosis System Efficiency Considerations | Ease of Operation | Economic Criteria | Energy Efficiency | |||

Concentration Polarization | Matemática—Kedem model [5,14,19,34,35,41] | Physical behavior of the membrane | Evidence physical/chemical phenomena that govern the process | Determine key operating variables—Shows the behavior of the system interrelating result—widely used in experimental processes [5]. | Energy consumption is the most important factor of the total cost—Electricity price variability generates changes in process conditions such as pressures, concentrations in permeate flows [8,9,20,21]. | The model allows the process to be analyzed at low pressures, identifying the most sensitive operating variables that allow industrial scaling with energy efficiency [10,38]. | Rejection of contaminants greater than 99% |

Physical configuration | Specific energy consumption (SEC) | 1 Stage, 2 Step and CCRO [31,47] | The compensation quantifies the relationship between mean water flow and SEC at module scale with 1-stage, 2-stage, and closed-loop—Operation based on the kinetics [47] | Quantify compensation using the relationship between average water flow and SEC in module-scale reverse osmosis processes with 1-stage, 2-stage, and closed-loop configurations [47] | The reduction of the normalized SEC will have a significantly stronger impact In absolute energy savings, system configuration and operating conditions also have significant impacts on energy or economic costs of pretreatment processes and energy recovery [45]. | A 1-stage RO process usually yields the highest average water flux but also consumes the most energy a 2-stage RO process typically consumes the least energy and yields a moderate average water flux and a CC-RO process yields the lowest average water flux and consumes more energy than a 2-stage RO process but less energy than a 1-stage RO process [31,47] | The optimization of RO operation based on the kinetics-energetics tradeoff should be conducted in the range of operation conditions that would not drastically undermine other aspects of RO performance |

Energy-Efficient Reverse Osmosis (EERO) | The energy-efficient reverse osmosis (EERO) desalination process was developed to achieve a highfull water recovery [48] | Feeds retentate from one or more stages of Single Stage Reverse Osmosis (SSRO) in series to a countercurrent membrane cascade with recycle (CMCR) consisting of a reverse osmosis (RO) terminal stage and one or more low salt rejection stages | The process, it develops with an operational strategy that involves increasing the pressure to the low salt rejection stage of the CMCR to compensate for the use of membranes with a higher salt rejection than required | An achieve 75% total water recovery at a lower total cost of water production than conventional SSRO operated with only 50% water recovery [48] | The EERO process can reduce the osmotic pressure differential by 50% relative to conventional SSRO for the same total water recovery | The EERO process significantly reduces the infrastructure costs—The 1-2 and 2-2 configurations of the EERO process can achieve reductions of 3.7% and 6.2%, respectively, in the total cost of water production for operation at 75% total water recovery relative to conventional SSRO operating at only 50% total water recover associated with pre-treatment [48] | |

Hybrid membrane configuration combining SWRO | Combining SWRO elements of different productivity and rejection within the same vessel [50,51] | The hybrid membrane interstage design (HID) is evaluated to improve the SEC efficiency of the reverse osmosis process | The system allows estimating the energy efficiency of the HID under three feeding conditions: high concentration and high temperature (Case 1); low concentration and high temperature (Case 2); and low concentration and low temperature (Case 3) | HID application can save up to 0.41 kWh/m^{3} of SEC. | Configuration of seven elements per vessel, each element of the membrane would produce one seventh (14.3%) of the total flow of permeate [50] | Temperature is a more important design factor than recovery rate for HID application. | |

Internally Staged Design (ISD) | Numerically optimized, based on a finite difference method [54] | It is a method of systematic optimization to find better sequences of membrane elements in a pressure vessel. | A large-scale RO process was numerically modeled to assess the impact of the membrane element configuration on the long-term operation in the presence of colloidal contaminants [54] | The proposed method for optimizing ISD is useful for more economical and efficient design and is a good reference for manufacturers to further improve their RO membranes [54] | The ISD system improves water recovery rate and energy efficiency of SWRO processes during a long-term operation | The optimization ISD shows higher water flow and higher energy efficiency in long-term operation (90 days) compared to conventional designs [30]. | |

Statistics | Reverse Osmosis (RO) Steady State Statistical Models | Constructions of correlations between inputs/outputs [16] | Understanding of the mechanism/behavior of the interaction between input and output variables of the desalination plant by formulating regression models. | Show interaction between input and output variables—They are used to plan and shows the sensitivity of variables against the operation of the system—Characteristic data of the current are used (flow rate, concentration, and pH) over a period | The model developed here is useful for planning, monitoring, and analysis of the current separation system | The model is obtained after multivariate resulting analysis that the P values are smaller than α < 0.052, indicating independently distributed residuals with mean residual values for a confidence level of 95% and 99% that are insignificant | The proposal requires more field as well as experiments to confirm findings based on physical or chemical viewpoints [16] |

New fouling index, β called the “permeation coefficient” | Development of a new fouling index that is more reliable and feasible than the SDI [56] | Definition of a new fouling index β called “permeation coefficient” under natural environmental conditions from a statistical point of view | Regression models allow predicting the rate of return and the specific performance index that takes into account energy consumption based on different variables | The analysis of the performance of the membranes based on their fouling, allows better criteria of its preservation to improve its performance, allowing better energy and economic savings. | It provides new insights into the performance and shortcomings of SDI from a statistical point of view. | adoption of a new fouling index would require further field testing, as well as experiments or theory to confirm the findings based on physical or chemical points of view [56] | |

Response surface methodology (RS) | Applied for modeling and optimization of the air gap membrane distillation process used in desalination [58,59,60,65] | Regression models are used to predict the rate of return and the specific performance index that takes into account energy consumption based on different variables | The developed models have been statistically validated by analysis of variance | Two optimal operating conditions were found by solving two different problems. optimization cases: (i) maximization of the performance index and (ii) maximization of the specific performance index. This allows design with greater energy and economic savings in the systems. | From the RS models, the optimal AGMD conditions were determined using the multi-stage Monte Carlo simulation technique. | For the performance index, the optimal solution was a cooling inlet temperature of 13.9 °C, a feed inlet temperature of 71 °C and a feed flow rate of 183 L/h [58,59] | |

Model Predictive Control (MPC) | It is an advanced control algorithm widely used in the process industries, as reverse osmosis plant [62] | In this paper, the QDMC controller is used to control a simulated reverse-osmosis (RO) water desalination system with spiral wound element (SWM). A cascaded control system was designed with the QDMC controller and a PID controller for the desalination process, where the QDMC controller optimizes the set point of the PID controller and directly controls one output. | Support software is available to help engineers adjust QDMC controller parameters. Since the QDMC controller is implemented in an embedded system, the system cost is reduced, which is helpful for RO desalination system application | Dado que todo el controlador QDMC está implementado a través de un chip FPGA, el costo es muy bajo, lo que es útil para una amplia aplicación en plantas prácticas | QDMC control system can also handle the system constraints and is very effective in controlling the complex coupled process of a RO plant. | The model compared the results of the proposed QDMC cascade control system with the traditional two-PID control strategy used in the industry. The model considers three different scenarios, with set point control and disturbance rejections. Based on the simulation results [62] | |

Statistical data processing, applied to research studies related to costs | Report of a mathematical model based on statistical processing related to production costs. The analysis of the cost of one m3 of desalinated water by reverse osmosis (RO) is carried out [63] | A mathematical model is proposed based on expressions related to costs based on production capacity. | All cost data are plotted on bar charts and box-and-whisker plots. The study of atypical values was carried out as well as that of Kolmogorov–Smirnov and Shapiro–Wilk tests were performed based on Hubera’s M, Tukey’s biweight, Hampel’s M and Andrew’s wave estimators. Subsequently, factorial analysis was performed using the Bartlett and Kaiser-Meyer-Olkin tests; Possible mathematical models were analyzed | the model shows that desalination costs can be up to 1.5%, more efficient in the production line compared to the rest of the observed lines | The model provides an innovative aspect in cost analysis because the study focused exclusively in the search for technologically more efficient and lower cost production lines impact on the plant. | The proposed equation corresponds to the mathematical model based on the statistical data adjusted to 98% of the real cost for small desalination plants [63] |

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Villena-Martínez, E.M.; Alvizuri-Tintaya, P.A.; Lora-Garcia, J.; Torregrosa-López, J.I.; Lo-Iacono-Ferreira, V.G.
A Comparative Analysis of Statistical Models and Mathematics in Reverse Osmosis Evaluation Processes as a Search Path to Achieve Better Efficiency. *Water* **2022**, *14*, 2485.
https://doi.org/10.3390/w14162485

**AMA Style**

Villena-Martínez EM, Alvizuri-Tintaya PA, Lora-Garcia J, Torregrosa-López JI, Lo-Iacono-Ferreira VG.
A Comparative Analysis of Statistical Models and Mathematics in Reverse Osmosis Evaluation Processes as a Search Path to Achieve Better Efficiency. *Water*. 2022; 14(16):2485.
https://doi.org/10.3390/w14162485

**Chicago/Turabian Style**

Villena-Martínez, Esteban Manuel, Paola Andrea Alvizuri-Tintaya, Jaime Lora-Garcia, Juan Ignacio Torregrosa-López, and Vanesa Gladys Lo-Iacono-Ferreira.
2022. "A Comparative Analysis of Statistical Models and Mathematics in Reverse Osmosis Evaluation Processes as a Search Path to Achieve Better Efficiency" *Water* 14, no. 16: 2485.
https://doi.org/10.3390/w14162485