**1. Introduction**

Due to global population growth and the spread of pollution it is becoming more challenging to provide clean drinking. A sustainable method for obtaining clean water is through seawater desalination by reverse osmosis. A reverse osmosis membrane system is composed of high-pressure pumps, one or more reverse osmosis membranes, and energy recovery devices which are designed to meet purity requirements while also minimizing energy consumption. The pressure of the seawater supplied by the high-pressure pump varies depending on the salt concentration of the seawater. The standard criteria in the seawater desalination process is the concentration of TDS (total dissolved solids) and boron in fresh water. The WHO (World Health Organization) states that the palatability of water with TDS lower than 600 mg/L is considered good and they specify guidelines for 2.4 mg/L of boron [1], although lower values are generally preferred. These WHO criteria have an impact on the design of the seawater reverse osmosis (SWRO) processes such that both salt rejection and boron rejection must be considered.

The removal of boron through reverse osmosis is complicated by the fact that boron exists in seawater mainly as boric acid and borate ions (with negligible concentrations of other boron compounds) [2]. This is a problem because reverse osmosis membranes are known to easily permeate only the negatively charged borate ions while having more difficulty removing the neutrally charged boric acid [3]. For this reason the pH should typically be increased to give higher fractions of borate ions [2]. This has been demonstrated using FilmTec membranes which are shown to give very high boron rejection at high pH values [4]. Additionally, Koseoglu et al. tested FilmTec and Toray membranes and found that around 85–90% rejection is possible at pH 8.2 while pH 11 allows for 98% or higher rejection using both membranes [5]. More recently Ali et al. have developed a membrane material which is able to achieve 99% boron rejection at pH 10 [6]. At a lower pH of 8 Li et al.

**Citation:** Binns, M. Analytical Models for Seawater and Boron Removal through Reverse Osmosis. *Sustainability* **2021**, *13*, 8999. https:// doi.org/10.3390/su13168999

Academic Editors: Julian Scott Yeomans and Mariia Kozlova

Received: 14 July 2021 Accepted: 10 August 2021 Published: 11 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

have developed a membrane modification process which embeds 4-nitrobenzenesulfonyl chloride into an existing membrane and is able to increase the boron rejection from 82.12% up to 93.1% [7]. In addition to the development of new materials, the design of feed spacers inside the membrane modules is also important. A review of the impact of feed spacer design by Haidari et al. discusses their effects on pressure drop, flux through the membrane, and fouling [8]. Additionally, it is suggested by Ruiz-Garcia and Nuez based on experimental and modelling results that feed spacers should be chosen based on the designed operating conditions to reduce energy consumption and enhance the quality of the permeate [9].

In order to meet boron drinking water criteria, a multistage design of reverse osmosis modules is typically required [2]. For example, Tu et al. state that in practice a first stage with natural pH might be used to reduce TDS and a second stage with elevated pH might be used to remove boron [10]. In addition to reverse osmosis, Najid et al. consider and discuss alternative technologies for boron removal including electrocoagulation, adsorption, ion exchange, and various other membrane processes such as forward osmosis and membrane distillation [2]. Their comparison showed that reverse osmosis has the potential to remove boron but can be uneconomical due to high energy requirements and the requirement to alter pH, and hence they suggest that a hybrid process combining different technologies could be the best solution [2]. To reduce the costs of two-stage processes Ban et al. also consider a hybrid process with one stage of forward osmosis followed by a second stage with reverse osmosis [11]. They compare this against a two-stage reverse osmosis design and show that the costs associated with chemically altering the pH can be eliminated by using the hybrid forward osmosis plus reverse osmosis process, but this comes at the expense of higher capital costs [11]. Instead of chemical modification Jung et al. have suggested electrochemical modification using a layer of carbon nanotubes on the membrane surface as a cathode to increase the pH. While this does increase boron rejection over 90% it also causes some scaling [12]. In another recent study a hybrid system is suggested combining electro dialysis as a pretreatment with a nanofiltration reverse osmosis to enhance the overall boron removal [13].

Despite this progress in membrane materials and potential hybrid systems there is still the need for modelling and optimization of such systems. This would allow, for example, the prediction of salt and boron rejection for wide ranges of possible conditions to identify low energy and low cost designs. For example, Ruiz-Garcia et al. use modelling to compare the performance of two different Toray membranes (TM820L-440 and TM820S-400) for the purpose of boron removal over a range of conditions and show that the TM820L-440 generally gives lower boron concentrations of under 1ppm [14].

Modelling can also be used to simulate and compare different configurations of separators to further enhance energy efficiency. For example, Al-Obaidi et al. evaluated the performance of a multistage reverse osmosis system with varying operating parameters through a modelling approach [15]. In other work Al-Obaidi et al. also utilized modelling to compare a number of different recycling options in a multistage reverse osmosis membrane process [16]. More recently Alsarayreh et al. also used a modelling approach to investigate different retentate recycle ratios [17].

The review of Alsarayreh et al. shows that a larger number of models have been developed for the prediction of performance for spiral wound reverse osmosis membrane modules [18]. These models can be divided into two categories: numerical models which discretize the length of the module and formulate model equations using finite difference type methods, and analytical models which use integration of model equations to obtain expressions for directly calculating the outlet conditions. While the majority of models developed have been for steady state solutions, model equations can also be solved dynamically as shown by Joseph and Damodaran [19]. Regarding steady state modelling, Ben Boudinar et al. have developed a numerical model solved through finite differences and they show that their model fits well for desalination of brackish water but is less accurate for seawater desalination [20]. They suggest that this is due to an inaccuracy in

the mass transfer coefficient [20]. This has been addressed by Senthilmurugan et al. who also fitted parameters for mass transfer correlations as well as solving the model equations using finite differences [21]. In other studies, such as the work of Geraldes et al., mass transfer correlations for "typical spiral wound modules" are assumed to be valid [22]. While Geraldes et al. do not provide validation for their model they have fitted water and salt transport coefficients which are then used to optimize the configuration and operating conditions of a two-stage desalination system [22].

Following these works numerical models have also been developed to predict the removal of boron. For example, the study of Mane et al. developed a finite elements model to predict the removal of boron [23]. In that study the parameters and correlations for boron transport coefficients developed by Hyung and Kim based on experimental results are used to account for the effects of pH and temperature [24]. Another study of Ruiz-Garcia et al. proposed a function for boron permeability in terms of feed pressure, temperature, and operating time based on plant data which might also be used in modelling studies [25]. Alternatively the model developed by Sassi et al. [26] used a finite-difference type numerical model which accounts for boron permeation using data and correlations from the experimental study of Taniguchi et al. [27]. More recently the study of Du et al. [28] also considered boron removal through numerical models based on a combination of the equations from the studies of Geraldes et al. [22] and Hyung and Kim [24] which they use to optimize a superstructure of different configurations.

A number of studies have also developed analytical models where the model equations are integrated to give analytical expressions. For example, Avlonitis et al. developed equations for calculating the variation of concentration, pressure, and flow rates along the length of the module, although they assume that the mass transfer coefficient is constant along the length [29]. More recently Sundaramoorthy et al. suggested an analytical model which includes the variation of the mass transfer coefficient across the length [30]. They have demonstrated the validity of their approach through the removal of chlorophenol [31] and dimethylphenol [32] from waste water where they show how model parameters and parameters for mass transfer coefficients can be estimated through linear fitting of experimentally measured values. Following these earlier studies an analytical model was developed by Fraidenraich et al. for the desalination of brackish water which they showed to be accurate for the conditions tested [33]. Additionally, Al-Obaidi et al. have published numerous models including the development of analytical expressions from integration [34] and using average pressure and salt concentrations to simplify calculations which can be used to evaluate and test different configurations of modules [35].

While great progress has been made simulating desalination membrane modules using finite-difference type numerical models, these models generally involve large numbers of equations (due to the discretization) which can be solved simultaneously or possibly sequentially using numerical algorithms. Meanwhile, analytical models will have a relatively small number of equations which can be solved using less computational time and simpler algorithms; for example, in a spreadsheet program. Hence, analytical models should be more suitable for the design and optimization of multistage configurations which require the simulation of individual module performance a large number of times, provided they are shown to give reasonable accuracy.

However, to the best of our knowledge, there has not been any analytical model (based on integration of model equations) which has been developed to predict the removal of both salt and boron simultaneously. Therefore, for this reason, a combined salt and boron removal analytical model is developed here. Additionally, in many cases the fitting of parameters for spiral wound reverse osmosis models are often proposed based on nonlinear optimization using least-squares methods. In this study the methods of Sundaramoorthy et al. [30] and Avlonitis et al. [29] are extended such that all the model parameters can be estimated though simpler linear optimization in a new sequential parameter estimation procedure.
