3. RO Optimization Modeling
Ahmed et al. and Zarzo and Prats 
state that energy consumption in an RO system is between 50 and 60% of the total cost of the process, with this being the key factor for the use of any technology.
Between 1970 and 2022, the energy consumption in an RO system was reduced by 80% due to lower energy consumption in each of the components of the RO plant. Such is the case for the pretreatment system, high-pressure pumps, the material and configuration of the membrane, energy recovery devices, and post-treatment 
. Initially, the energy consumption exceeded 15 kWh/m3
; today, and with the new advances in general, the consumption is in the order of 2.5 kWh/m3 
Ahmed et al. 
point out that an adequate optimization of the system and greater energy reductions can be achieved with an adequate plant configuration, which can be achieved first through simulations before pilot-scale experimental tests. Geise et al. 
state that the configuration of the membrane is fundamental for the reduction of energy consumption, where part of the driving force is the balance between the selectivity and permeability of the membrane.
In terms of energy consumption, several alternatives for the operation of the system were compared. Lin and Elimelech 
compared two-stage reverse RO (SSRO) systems against a single-stage closed-loop (CC-RO). In the first case, the reject stream from the first stage is converted into feed for the second stage, while for the single-stage system, the reject is mixed with the feed stream that passes back through the membrane. The results show that the single-stage system in a closed circuit is less efficient than a two-stage system because it needs more energy to reduce the entropy generated by the mixture of the rejection to the feed flow 
In another investigation, Chong and Krantz 
propose a low-consumption (EERO) system in which they seek to increase the overall water recovery by sending the retentate from one or more two-stage (SSRO) systems as feed to a countercurrent membrane cascade with recycling (CMCR). This consists in one or more low-salt-rejection RO stages (Stage 1) and high-salt-rejection stages (Stage 2). The results show a lower osmotic pressure differential and thus a lower net specific energy consumption 
King and Hong 
propose a partial split single-pass system (SSO-RO) in which permeate from only the back RO elements in a pressure vessel is mixed with the RO feed to dilute the feed. This results in a high-quality permeate with lower energy demand. The modeling carried out shows that the energy efficiency is maximized for the process when the permeate of the last element is mixed with the feed. This modified process is up to 15% more efficient in permeate purity and energy efficiency than a normal two-step system 
An improvement to the RO system which is proposed to achieve a more uniform flow distribution throughout a pressure vessel is to use a hybrid membrane configuration known as internal staged design (ISD). This system involves the use of low-flux membranes in the front and high-flux membranes in the rear elements 
. This system allows a significant reduction in permeate costs by requiring fewer pressure vessels and fewer membranes 
Han et al. 
improved the system using three different types of RO membranes: high rejection, standard, and high flux. In an experimental boron rejection process, they showed that using three standard membranes at the front and four high-flow membranes at the back, energy savings of 0.41 kW/m3
Jeong et al. 
proposes a model based on a finite difference approximation that allows a better numerical optimization of the ISD system in the presence of colloidal fouling. Compared to conventional designs in which the same membrane is incorporated throughout the vessel, the ISD resulted in higher water flow and higher energy efficiency for long-term operation, without compromising the permeate quality (<400 mg/L).
Kotb et al. 
point out that optimization studies require complex or highly non-linear models with many restrictions. They implemented a simple transport model to determine the operating parameters corresponding to the optimal structure of the RO system, that is, one-, two-, and three-stage arrangements with respect to the minimum cost of permeate production for a given permeate flow
rate with the maximum total dissolved solids 
They suggest that the optimal permeate flow rate increases with the number of stages; while a single-stage RO system is suitable for up to 6 m3
/h, three-stage modules are suitable for production up to 20 m3
4. Statistical Modeling
The use of statistical models to analyze the separation process in RO is essential to knowing the behavior of an RO plant and analyzing the input and output data 
Subramanni and Panda 
, in their research carried out on a desalination process by RO, use statistical regression for the analysis of the experimental modeling. Among the most important conclusions, they point out that the statistical model allows a better understanding of the behavior of the plant’s mechanism and the interaction between the input and output variables through the formulation of statistical models. The ANOVA analysis shows that the total dissolved solids of the permeate is affected by the change in the system recycle ratio. They also point out that the proposed statistical model is useful for the planning, monitoring, and analysis of the separation system.
Miyamoto et al. 
statistically examined the performance of the sediment density index (SDI) and defined a new fouling index, defined as the “coefficient of permeation”. The research was carried out in a desalination process under normal environmental conditions, where they statistically analyzed the relationships between the amount of filtered water, the elapsed time, and the environmental factors in order to obtain new knowledge about the performance and deficiencies of the use of SDI from a statistical point of view.
Khajet and Cojocaru 
performed the modeling and optimization of the air-gap membrane distillation process using the response surface methodology. The optimization of a solar-powered desalination plant was evaluated through the response surface methodology 
Khajet et al. 
investigated the optimization of an RO plant using solar-powered energy through statistical response surface modeling. They applied the orthogonal type methodology 
for the design of the experiments and a minimum number of experimental runs—as proposed by Taguchi et al. 
—with simultaneous temperature variations, which allowed them to develop a predictive model of salt rejection, permeate flux, and the RO specific performance index. The results obtained from the analysis of variance (ANOVA) confirm that the response surface models developed are statistically validated in order to simulate the OI process. They establish, for example, that for a drinking water production of 0.2 m3
/day they use an energy consumption of only 1.2 to 1.3 kWh/m3
In another investigation, Mohsen and Salen 
evaluated the performance of an RO plant in Wadi Ma’in, Zara, and Mujib to present the state of the art of its operation and maintenance. They collected detailed information on plant design and engineering, water quality, plant personnel, and the cost of operation and maintenance since the plant was commissioned. They analyzed for 150 days the performance of the RO desalination process in terms of recovery, permeate flux, normalized permeate flux during the erratic period, normalized differential pressure throughout the RO system, and salt rejection, and obtained the state of the system operation and maintenance.
Khajet and Cojocaru 
used the response surface methodology for the statistical design of the experiment, which allowed them to evaluate an air gap membrane desalination process. They developed and statistically validated two regression models, one for the performance index and one for the specific performance index that considers energy consumption. The temperature is the one that has the greatest positive effect on the performance index and the feed flow rate for the specific performance index.
De-wei et al. 
, meanwhile, used quadratic dynamic matrix control (QMDC), which is a model-based predictive control (MPC) strategy to evaluate and control an RO desalination system. For the QMDC controller, they installed a field-programmable gate array (FPGA) chip and operator using software developed for this purpose. The results showed that the proposed system performs better than the traditional proportional, integral and derivative (PID) controller systems.
Feo et al. 
analyzed the production capacities and costs in production lines in small RO plants in the Canary Islands. For this purpose, they developed a mathematical model based on expressions related to costs based on production capacity. They collected and processed statistical data. They plotted all of the cost data on bar charts and box-and-whisker plots. They performed the study of outliers, as well as Kolmogorov-Smirnov and Shapiro-Wilk tests based on the Hubera M-wave, Tukey biweight, Hampel M, and Andrew’s estimators. Subsequently, factorial analysis was performed using the Bartlett and Kaiser-Meyer-Olkin tests; they then analyzed the possible mathematical models.
The response surface model was proposed by Box and Wilson 
, and is very useful for the modeling and analysis of the results obtained in applications where the response of interest is influenced by different variables, and where the objective is to optimize said response. Its main advantage, compared to other models that relate a variable at three levels, is that it provides the minimum number of experimental runs. An economic design does so from the point of view of reagents, sample quantification, the payment of external checks, and energy, among others 
In order to form the MSR, it is necessary to consider several phases or steps, one of them being the exploration of the optimal response region, through 2k full factorial experiments or the option of 2k−p fractional factorial designs. This is necessary in order to determine a smaller number of experiments 
Taguchi contributed to the dissemination of the design of experiments because this, unlike what happens with classical design, does not require being an expert in the method in order to be able to apply it 
Taguchi’s method manages to solve design of experiments problems in a practical way. The use of the classic design generally implies the allocation of more resources to experimentation (a greater number of experimental units, more personnel involved, more time, a special place may even be required for experimentation because there are more replicas, and efforts to maintain a homogeneous the experimental conditions, among others). In some cases, the Taguchi method constitutes the only possible way for companies to carry out experimentation 
Likewise, it should be pointed out that Taguchi’s method would not be viable if its conclusions were not certain. The reliability of the results obtained through the design of experiments proposed by Taguchi 
is given by the power of the AO, that is, the power of the ANOVA (ANalysis Of Variance).