Water scarcity is becoming a global challenge exacerbated by population growth, socioeconomic development, and climate change. Global water consumption has been increasing by approximately 1% per year since the 1980s, and a similar rate is anticipated to continue until 2050 (World Water Assessment Programme, 2019). It is estimated that more than 2 billion people are experiencing significant water stress, and about 4 billion people face severe water scarcity for at least one month per year (World Water Assessment Programme, 2019).

Desalination offers a viable solution to clean water production from saline and brackish groundwater resources (Burn and Gray, 2016; Kucera, 2019). According to the International Desalination Association (IDA) (https://idadesal.org), as of the end of 2019, there are more than 21,000 desalination plants across the world, and the total global cumulative desalination contracted capacity is 126.57 million m³/day (⁠$1megagallon=3785.4m3$⁠). Commonly used desalination techniques are summarized in Table 1.1. Among them, reverse osmosis (RO) is unquestionably the most adopted industrial desalination technique, followed by multi-stage flash (MSF), multi-effect distillation (MED), nanofiltration (NF), and electrodialysis (ED). The most recent data show that RO accounts for approximately 70% of the desalinated water produced worldwide (Jones et al., 2019).

With regard to source of feed, seawater accounts for 61% of global desalinated water production, followed by brackish water (21%) and river water (8%) (Jones et al., 2019). Seawater desalination has helped severely water-stressed coastal states (e.g., California) to meet their needs in water supply. Designed by IDE Americas and built by Poseidon Water, the Claude “Bud” Lewis Carlsbad desalination plant, shown in Fig. 1.1, is “the largest, most technologically advanced and energy-efficient seawater desalination plant in the nation” (Carlsbad Desalination Plant, n.d.). It is located adjacent to the Encina Power Station in Carlsbad, CA. Commissioned in 2015, the Carlsbad plant produces water at a rate of nearly 50 million gal/day (MGD), accounting for about one-third of all water generated in San Diego County (Carlsbad Desalination Plant, n.d.). More seawater reverse osmosis (SWRO) facilities are proposed in coastal cities in California, including Huntington Beach, Monterey Bay and others (California Water Resources Control Board, n.d.).

The discovery of the phenomenon of osmosis, in 1748, is credited to French physicist Jean-Antoine Nollet for observing, that a solvent could pass selectively through a pig's bladder—a natural semipermeable membrane (Sant, 2019). To explain osmosis, consider two solutions of different salt concentrations and equal hydraulic pressures that are separated by a semipermeable membrane, as shown in Fig. 1.2. Driven by osmotic pressure in the high-salinity compartment, water is extracted from the low-salinity compartment. This process is called direct osmosis. If a sufficiently high pressure is applied to the high-salinity section, however, the direction of water flux will be reversed; this is the process of reverse osmosis (RO).

In this sense, the direction of water flow across the membrane depends on the relative magnitude of transmembrane osmotic and hydraulic pressures. For RO, the water flux J_w can be described by

where L_p is the hydraulic permeability of the membrane, ΔP is the transmembrane pressure, and Δπ is the transmembrane osmotic pressure.

From thermodynamics, the osmotic pressure of salt water π (Pa) can be calculated by (Sandler, 2017)

where $aH2O$ is the activity of water at temperature T (Kelvin), $V¯H2O$ is the partial molal volume of water (m³/mol) at temperature T, and R is the gas constant (8.3145 J/mol/K). In the desalination community, several simplified formulas have been used to estimate the osmotic pressure. For example, the following equation is used by FilmTec (DuPont, 2020):

where π is the osmotic pressure in bar and $∑mi$ is the summation of molalities of all ionic and nonionic constituents in the water in mol/kg. A similar formula is used by Hydranautics, except that the factor 1.12 is replaced by 1.19 (Hydranautics, 2001). For dilute solutions, the van't Hoff equation typically is used (Fritzmann et al., 2007), i.e.,

where c is the concentration of the solutes, i is the dissociation parameter (or the total number of moles of ions and molecules per mole of the solute assuming complete dissolution), and ϕ is the osmotic coefficient that accounts for non-idealities.

It is worth noting that the osmotic pressure of seawater is about 10% lower than that of an NaCl solution of equal total dissolved solids (TDS), due to the fact that there are other species with high molar weight in the former (Crittenden et al., 2012).

A noteworthy breakthrough in the RO membrane history was the asymmetric semipermeable cellulose acetate membrane developed by Sidney Loeb and Srinivasa Sourirajan at UCLA in the late 1950s (Sidney and Srinivasa, 1964; Cohen and Glater, 2010). The anisotropy of the membrane, i.e., a thin skin at the separating surface supported by a thicker spongy sublayer, resulted in water fluxes at high levels viable for commercial application. In 1972, John Cadotte of North Star Research developed the first interfacial polyamide membrane that had better salt rejection and water flux at lower operating pressures than the cellulose acetate membranes. Later, Cadotte invented the aromatic interfacial composite membrane (Cadotte et al., 1981) at FilmTec (now part of DuPont) known as FT30, which set the standard for today's RO membranes.

As shown in Fig. 1.3, flat sheet thin film composite (TFC) membrane used for modern commercial RO module construction consists of three layers: a polyester support base, a microporous polysulfone support layer, and an ultra thin active polyamide layer for salt rejection.

Some basic configurations used for constructing membrane modules include hollow fiber, spiral wound, tubular, plate and frame, and capillary (Scott, 1998). Spiral wound is the most common form of packing RO membranes (Bartels et al., 2008) because it maintains a good balance among packing density, resistance to fouling, ease of cleaning, water flux, and fabrication cost (Schwinge et al., 2004). Presently, spiral wound RO membrane separation is being used in a wide variety of industrial applications, including portable water production from brackish groundwater and ocean water (Elimelech and Phillip, 2011; Ruiz-García and Ruiz-Saavedra, 2015), municipal and industrial wastewater treatment (Bodalo-Santoyo et al., 2003; Dialynas and Diamadopoulos, 2009), whey and lactose concentration (Balannec et al., 2005), boiler feed water pretreatment (Čuda et al., 2006), ultra pure water production for use in microelectronics and pharmaceuticals (Ganzi and Parise, 1990; Lee et al., 2016), water reuse in the pulp and paper industry (Pizzichini et al., 2005), solvent recovery from lube oil filtrates (White and Nitsch, 2000), and many others.

A detailed geometry of a spiral wound RO module is shown in Fig. 1.4. It is comprised of membranes, feed spacers (or rententate spacers or concentrate spacers), permeate spacers, and a perforated tube. Stereomicroscope images of feed and permeate spacers are shown in Fig. 1.5.

To construct a membrane module, flat sheet membranes are each folded in half with the dense salt rejection layers facing inward. A feed spacer is inserted between each of the folded membranes, forming a sandwich structure. The feed spacers keep the membrane leaves apart and promote transverse mixing of salt (Li et al., 2016). Then, the permeate spacer is attached to the perforated permeate tube, and the membrane sandwich prepared earlier is inserted between the permeate spacers. The back of each membrane is sealed to the edges of the permeate spacer. The finished membrane layers are rolled around the permeate tube to form the spiral shape. Feed water flows longitudinally into the feed channels. Pure water permeates through the membrane, enters the permeate channels, and travels in the spiral direction before entering the permeate tube. The remainder of the feed, including salts that are blocked by the membrane, exits the feed channels at the end of the RO module.

The most common diameters of an RO membrane element are 2.5 in. (61 mm), 4 in. (99 mm), and 8 in. (201 mm). The typical length is 40 in. (1016 mm), although 14 in. (356 mm) and 21 in. (533 mm) are available for small and compact systems. Some larger diameter RO elements (e.g., 16 in. and 18 in.) have emerged in the market which have the benefits of cost saving and ease of operation.

Several (typically, six to eight) spiral wound RO elements are connected in a series and enclosed in a fiberglass-reinforced plastic pressure vessel (as shown in Fig. 1.6). The inner diameter of the pressure vessel is sized to match the outer diameter of the element brine seal, which prevents the feed water from bypassing the RO element.

The pressure vessels are the building blocks of an RO system. Multiple pressure vessels can be connected in parallel, in series, or both, to form an array. The detailed arrangement depends on feed characteristics, permeate quality requirements, and safe operation of membranes, which will be discussed in the next subsection.

Membrane manufacturers provide general procedures for the design of an RO membrane separation process. For example, the design procedure suggested by FilmTec (DuPont, 2020) is summarized as follows:

1. Collect the feed water analysis data and product quality requirements.

2. Select the flow configuration and number of passes. The standard flow configuration is plug flow, although concentrate recirculation also is used in special applications. One pass design is the most common, although usually a multiple pass configuration is used to attain ultra pure standards of separation. The latter also is referred to as permeate staging, in which the permeate from one pass is used as the feed in the subsequent pass.

3. Select the membrane element type according to feed water salinity, feed water fouling tendency, required rejection, and energy requirements. The standard element size for systems greater than 10 GPM (2.3 m³/h) is 8-in. diameter and 40-in. length.

4. Select the average flux. Calculate the number of elements and number of pressure vessels. The system flux typically is based on pilot studies, previous experience, or membrane manufacturer's design guidelines (details will be provided next). Typically, there are six to eight RO elements connected in a series in a pressure vessel.

5. Select the number of stages and staging ratio in each pass. Multiple stage also is referred to as brine staging, in which the brine from one stage is used as the feed in the subsequent stage. The number of stages depends on the system recovery, the number of membrane elements per vessel, and the feed water quality. A general rule is to use one stage for 40%–60% recovery, two stages for 70%–80% recovery, and three stages for 85%–90% recovery.

6. Balance the permeate flow rate. The goal is to reduce variation in fluxes at different locations of the system primarily caused by variations in hydraulic and osmotic pressures. For example, an interstage booster pump may be used in a two-stage seawater RO system.

7. Analyze, refine, and optimize the system. Membrane manufacturers offer software for fine-tuning the system design.

In the following paragraphs, some of these items will be discussed in detail.

The most important factor that should be taken into consideration in the design of RO systems is the fouling tendency of the feed water (DuPont, 2020). There are various types of foulants: colloidal (clays, flocs), biological (bacteria, fungi), organic (oils, polyelectrolytes, humics), and scaling (mineral precipitates) (Baker, 2012). During RO operation, these fine foulants tend to be concentrated at the membrane surface, causing decline in flux and water quality and requiring frequent cleaning. The silt density index (SDI) is a measure of the fouling potential of suspended particles and colloidal materials in the feed water. Turbidity is another term commonly used in the RO community to describe the clarity or cloudiness of water due to the presence of suspended particles and colloidal materials. Even though there is no direct correlation between these two terms, generally it is acknowledged that SDI < 5 in the feed water is equivalent to turbidity < 1 NTU with regard to membrane fouling.

In addition to the SDI, operating conditions such as permeate flux and element recovery also affect the concentration of the foulants at the membrane surface. To minimize membrane fouling and mechanical damage, limits are recommended by membrane manufacturers on the maximum recovery, maximum permeate flow rate, minimum concentrate flow rate, maximum feed flow rate, and maximum pressure drop per RO element in system design (DuPont, 2020). At the system level, arecommended maximum pressure drop and a range of average flux also are recommended (DuPont, 2020). The latter provides a quick estimate of the total number of RO elements required for an RO plant.

Some of the general design guidelines for 8-in. FilmTec RO elements in water treatment applications are listed in Table 1.2.

The maximum pressure drop recommended by FilmTec is 1 bar across a single element or 3.5 bar across multiple elements in a pressure vessel (DuPont, 2020). For design purposes, a maximum of 0.8 bar for any element in a system is used. A pressure drop that is too great may produce the so-called “telescoping effect,” which damages the membrane.

Single-stage RO: The simplest configuration is single-stage RO. As shown in Fig. 1.7, water is pumped to a high pressure before entering the RO unit, which is comprised of multiple pressure vessels connected in parallel. The pump speed and opening of a concentrate valve located at the exit of the unit are manipulated to control the intake flow rate and recovery.

Because the concentrate stream at the outlet of the RO has a high pressure, an energy recovery device (ERD) may be installed to pressurize the feed. One such example is the Pressure Exchanger (PX) developed by Energy Recovery Inc., shown in Fig. 1.8(a). The working principle of the PX is based on a maintenance-free ceramic rotor that transfers pressure from the high-pressure brine to the low-pressure feed (Stover, 2007). The rotor is fit into a sleeve between two end covers while the water in the clearances between the rotor and the sleeve serves as a lubricated hydrodynamic bearing. The ERDs are common in SWRO plants. Figure 1.8(b) shows the PX-type ERDs installed in the Carlsbad SWRO plant; here, 144 devices are used, which reduces the overall energy consumption of the RO by 46%, equivalent to an annual savings of 146 million kW h (Carlsbad Desalination Plant, n.d.).

Concentrate recirculation: The configuration of concentrate recycling is shown in Fig. 1.9. Part of the concentrate stream is recycled and blended with the raw feed water. The purpose of this configuration is to increase the overall system recovery rate and the flow rate in the membrane. For example, if the fresh feed is 100 and the recycle stream is 50, the permeate rate is 75 based on an RO recovery of 50%. The system recovery is 75/100 = 75%. Because concentrate recirculation leads to an increase in the feed salinity, energy consumption could increase for the system. A detailed analysis will be discussed in Chap. 4, Sec. 4.4.2 and Chap. 5, Sect. 5.5.2.4.

Concentrate staging: The concentrate stream is used as feed to a subsequent RO stage for additional water recovery, as shown in Fig. 1.10. The interstage booster pump may or may not be used. The pressure vessels are connected in parallel in each stage. The number of pressure vessels used in each stage, however, is not the same; rather, they are selected in order to maintain a similar flow rate per vessel in each stage. For example, in brackish water RO, a typical recovery is 75% and the staging ratios are usually close to 2:1 for six-element vessels. That is, the first stage recovers 50% of the feed and the second stage recovers 50% of the brine from the first stage, resulting in a total recovery of 50% + 50% × (1 − 50%) = 75%. Both the flow rate and the area in the second stage are one half of those in the first stage. As a result, both stages have comparable cross velocities.

Permeate staging: For some applications, a single-pass RO system may not meet the requirement of permeate quality. To further reduce the salinity of the permeate, it is filtered again in a second RO system, as shown in Fig. 1.11. Because the salinity of the concentrate from the second pass is lower than that of the raw feed, often it is beneficial to blend raw feed with the second-pass concentrate. Examples of multiple pass design include removing pharmaceutical residues from municipal sewage (Heberer and Feldmann, 2008) and using boiler feed water (BFW) for high-pressure steam generation (Schoenberger, 2004). Some seawater RO plants employ a two-pass design in which the second pass is actually brackish water RO (Ayyash et al., 1994).

Split partial second pass: The design of split partial second pass (SPSP) is based on the fact that the front elements in the RO pressure vessel always produce permeate with less TDS than the back elements in a pressure vessel (Rybar et al., 2010). As shown in Fig. 1.12, permeate flows out from both sides of the pressure vessel. Part of the back permeate in the first pass is sent to the second pass for further treatment. The permeate streams in both passes are blended as the final product. The SPSP design is very flexible given that the ratio between the front and back permeate streams can be adjusted to meet the final product quality requirements. By treating only a portion of the permeate stream in the first pass, the SPSP allows reductions in the cost of operation as well as capital investment.

Most membrane manufacturers provide software packages to facilitate the system design and analysis of RO systems. Examples include IMSDesign from Hydranautics, WAVE from DuPont (formerly ROSA), and TorayDS/DS2 from Toray.

In this section, an example of a brackish groundwater desalination facility, the Chino I Desalter, will be used to illustrate how a desalination plant is designed and operated. The figures and narrative are primarily based on the plant operating manual (Inland Empire Utilities Agency, 2005).

The Chino I Desalter is a 14 MGD desalination facility owned and operated by the Chino Basin Desalter Authority (CDA), Chino, CA (see Fig. 1.13). The RO system design and construction have been administered by Separation Processes, Inc.

Raw water is drawn from 14 wells about 500 ft below ground level in the vicinity of the plant. The wellhead facilities convey the raw groundwater to the treatment plant at approximately 40 psig (or 2.7 barg) of pressure. The groundwater has an average TDS concentration of 950 mg/l and nitrate concentration of 170 mg/l. After pretreatment, the feed goes through three parallel processing units—the ion exchanger (5 MGD), RO (7 MGD), and volatile organic compound (VOC) air stripping (2 MGD). The ion exchange system reduces the nitrate levels in raw water to a level of less than 20 mg/l. The RO reduces the TDS in raw water to 35 mg/l. The VOC air stripping unit removes VOCs from the bypass wells. After post-treatment and disinfection, the final water delivered to neighboring cities has the product quality of TDS < 350 mg/l, nitrate < 25 mg/l.

The annual production is 5.1 billion gal of drinking water. Typical power consumption for the entire facility is 1.7 MW, of which 0.5 MW is for the operation of the RO membrane separation. Unquestionably, RO is the most energy intensive unit.

Raw groundwater is routed from the wellhead facilities for pretreatment at the plant, which includes both chemical treatment and physical filtration. The chemical pretreatment consists of sulfuric acid and threshold inhibitor addition.

Sulfuric acid system. As water passes across the RO membrane, the remainder of the feed becomes increasingly concentrated along the membrane channel. For an 80% recovery, the RO brine will be approximately five times more concentrated than the RO feed flow. Some constituents in the water, such as calcium carbonate (CaCO₃), calcium sulfate (CaSO₄), barium sulfate (BaSO₄), and silica (SiO₂), may exceed their solubility limits and precipitate on the RO membranes. The sulfuric acid system (as shown in Fig. 1.14) located upstream of the cartridge filters and RO units supplies sulfuric acid to the raw water to lower its pH to 6.5. This increases the solubility of calcium carbonate, a key precipitate in the CDA, and reduces its scaling potential enough to be within the capability of the threshold inhibitor.

The Langelier saturation index (LSI) is a measure of the scaling potential of calcium carbonate. It is calculated as follows:

where pH_c represents the pH of the RO concentrate stream and pH_cs is the pH of the RO concentrate stream at saturation in calcium carbonate. The latter is a function of the calcium concentration in the feed as calcium carbonate, the concentration of TDS in the feed, the alkalinity in the feed as calcium carbonate, the temperature, and the RO system recovery. No antiscalant is required if the LSI of the RO concentrate stream is negative. If sodium hexametaphosphate (⁠$(NaPO3)6$⁠) is used as the antiscalant, a concentration of 20 mg/l in the concentrate stream can prevent the formation of calcium carbonate scale when LSI < 1. If polymeric organic scale inhibitors are used, scaling is inhibited even when the concentrate has LSI > 1 (DuPont, 2020).

In the Chino I Desalter, the concentrate scaling potential is effectively controlled by acidifying the raw water to achieve a concentrate LSI of 1.9 to 2.0, in conjunction with the use of a proprietary threshold inhibitor. If too much sulfuric acid is added to bring the concentrate LSI considerably lower than 2.0, the sulfate ion concentration may increase, leading to possible scaling of calcium sulfate and barium sulfate. Moreover, it will increase the operating cost of chemicals used for pH adjustment in post-treatment.

It is important to note that variations in feed water chemistry may have an impact on the requirements for sulfuric acid and threshold inhibitor dosage rates. Moreover, an increase in RO system recovery will lead to a higher concentrate LSI, which in turn requires additional acidification of the raw feed so that the concentrate LSI is within the capability of the threshold inhibitor.

The sulfuric acid system also provides sulfuric acid to the clean-in-place (CIP) system which will be discussed in a subsequent section.

Threshold and silica inhibitor system. The threshold and silica inhibitor system (shown in Fig. 1.15) is located between the sulfuric acid system and the cartridge filtration unit, which serves to inhibit the precipitation of sparingly soluble salts blocked by the RO membranes and carried away in the concentrate discharge. Each scale inhibitor has its unique capability for inhibiting carbonate, sulfate, or silica precipitation. There are two feed systems in the Chino I Desalter to facilitate the addition of separate inhibition chemicals. An approach also used in other plants is blending two or more ingredients into a single solution.

The requirements for the inhabitation chemical dosage depend on the concentrations of inorganic constituents in the feed water and the temperature. Variations in the concentration of individual ions, especially calcium, bicarbonate, iron, sulfate, and silica, may require a change in the selection of the inhibitor or the inhibitor dosage. Inhibitor suppliers and membrane manufacturers provide software that calculates the concentrations and solubility limits of various compounds. The input for these calculations includes a complete feed water analysis and operating parameters for the RO system.

It is worth noting that the threshold inhibitors only slow the process of precipitation by interfering with the formation of crystals; they do not stop it. During normal operation, the constituents are carried away in the concentrate stream. Upon shutdown, a flush should be performed to prevent a super-saturated concentrate stream. Figure 1.16 shows precipitation in the RO brine line after 13 years in service.

It should be noted that the inhibitor chemicals, if overdosed, may precipitate on the RO membranes.

Pretreatment filtration system. The pretreatment filtration system (shown in Fig. 1.17) is equipped with five cartridge filters (four in operation and one in standby mode). It removes suspended solids that may be present in the feed water, thereby preventing damage to the RO pumps and preventing premature fouling of the RO membrane elements. Typically, the total suspended solids (TSS) must be reduced to be less than 5 mg/l.

The cartridge filters are designed for unexpected upsets and are not intended for heavy-duty filtration service. A single differential pressure transmitter reads the common pressure drop (dP) across all five cartridge filter housings.

The back of the filter housing includes evenly spaced holes into which the individual filter cartridges fit. A removable plate with the same pattern of holes is placed on the other side of the individual filters prior to closing the filter housing door. This allows each filter to remain in place during operation. Water flows through the cartridge filters and solids are retained. These solids build up over time, causing an increase in dP across the filters. Once the pressure drop reaches a threshold value (between 10 and 12 psi), the cartridge filters are replaced.

The cartridge filters are made of strings of yarn wrapped around a central core. It is a type of depth filter that is capable of trapping a large amount of particles (1 μm). These types of cartridge filters are less susceptible to being “blinded off,” whereby a limited amount of solids plug the outside of the filter, requiring its replacement.

The filtered feed water upstream of the RO high-pressure feed pumps is monitored by a pH analyzer, one conductivity meter, and one turbidity meter (see Fig. 1.18).

The RO is a pressure-driven process in which the pressurized feed water flows tangentially across a semipermeable membrane surface. Water is allowed to permeate through the membrane while more than 99.5% of dissolved salts are blocked, leaving behind a concentrated stream. In the Chino I Desalter, 80% of the feed water is recovered as permeate product with TDS < 35 mg/l.

The remaining 20% with the rejected salts is carried to the brine discharge line. The brine is discharged to the Santa Ana Regional Interceptor (SARI) and treated by the Orange County Sanitation District.

The RO system in the Chino I Desalter has four parallel process trains (shown in Fig. 1.19). Each train is equipped with the following main components:

• RO membrane feed pump with variable frequency drive (VFD);

• RO membranes and pressure vessels;

• Pressure vessel racks and manifolding;

• Isolation, permeate dump, and concentrate control valves;

• Field-mounted pressure, flow, and conductivity instrumentation.

Each train has a dedicated feed pump (Johnston 12GHC nine-stage vertical pump) which has a designed total dynamic head of 625 ft and a designed capacity of 1450 gal/min (GPM). Its characteristic map is shown in Fig. 1.20. The RO feed pumps are equipped with VFDs to adjust the pump speed.

The Codeline pressure vessels in each RO train are configured in a two-stage array with a staging ratio of 2:1 (shown in Fig. 1.21). The first stage has 28 vessels connected in parallel. The second stage has 14 vessels. Each pressure vessel houses seven 40-in. FilmTec BW30-400 spiral wound RO elements each having a membrane area of 400 ft² (or 37 m²). With a designed flux average flux of 15 gpd/ft² (or 25.4 l/m²/h), the RO system produces 7 MGD water.

Feed water is pressurized by the RO feed pump and enters the first stage. The concentrate streams from the first stage are combined and sent to the second stage as feed for additional water recovery. There is no booster pump between the two stages. Permeate produced by both stages is collected and piped to the decarbonators.

The pump speed and concentrate valve position are manipulated in real time by the control system to adjust the feed flow rate and feed pressure, which in turn maintain the desired permeate flow and recovery (80%). The pressure gauges for measuring pump suction and discharge sections are shown in Fig. 1.22; the concentrate control valve is shown in Fig. 1.23.

The RO membrane elements are subject to fouling/scaling by suspended solids and sparingly soluble salts during normal operation. Fouling/scaling is progressive and is accompanied by an increase in pressure drop and a reduction in flux. The RO CIP system provides in situ chemical cleaning of membrane modules while they are enclosed within the RO units. The CIP process involves recirculating chemical cleaning solutions through each stage of the RO at high velocities (to scour loose particles from the surface of the membrane) and elevated temperatures (to enhance solubility of foulants and/or scalants). Chemicals used in the RO CIP system include citric acid, hydrochloric acid, sodium dodecylbenzene sulfonate, and sodium tripolyphosphate. Flow direction in the CIP mode is the same as that in the normal RO operation. Note that this is different from the backwash procedure.

Scaling primarily occurs in the tail end elements of the RO system. For this problem, normal cleaning is desirable, and the cleaning solution will flow in the same direction as the feed water in RO operation. Even so, certain types of foulants, such as biological foulants, particulates, and colloidal matter, usually concentrate in the lead end elements; cleaning with direct flow may carry these foulants through the entire pressure vessel. For this issue, a reverse cleaning method, in which the cleaning solution runs from the brine side to the feed side of the pressure vessel, allows the foulants to leave the system by taking the shortest path. When the reverse cleaning is used, it is critical to limit the cleaning flow rate to prevent telescoping of the RO elements given that the thrust ring is only installed on the downstream end of the pressure vessel.

Backwash of RO by feeding a salted water solution into the feed channel such that direct osmosis occurs (i.e., water flows from permeate to feed side of the membrane) may lift foulants and/or scalants from the surface of the membrane. Manufacturers also may claim that this process could reduce or remove the CIP requirements.

Post-treatment prevents metallic corrosion (thus extending the life of the distribution system), minimizes taste and odor complications, and lowers the risk of bacteria infection to the public. The post-treatment decarbonation is located downstream of the RO system, which removes dissolved carbon dioxide in the RO permeate to raise the pH. Recall that acidification in the pretreatment of the raw feed has converted bicarbonate alkalinity (⁠$HCO3−$⁠) to CO₂, which cannot be removed by the RO unit. The post-treatment decarbonation system (shown in Fig. 1.24) consists of two counter-current forced draft or blower-type decarbonators.

The removal of carbon dioxide from the RO permeate also decreases the amount of caustic soda that serves to stabilize the water product. Other post-treatment chemicals include sodium hypochlorite for disinfection and aqueous ammonia for minimizing trihalomethanes (THMs) which are byproducts of disinfection. The decarbonated effluent enters into the treated water pump station clearwell.

The clearwell also receives flows from the Ion Exchanger and VOC units (see Fig. 1.13). Water discharges from the clearwell via the treated water pump station (shown in Fig. 1.25), which conveys water to the nominal end users in the neighboring cities.

In the Carlsbad desalination plant, 100 MGD seawater is transported to the plant via a 72-in. pipe. The intake goes through multimedia filtration that employs anthracite, sand, and gravel to eliminate algae and organic materials and microfiltration to remove microscopic impurities. After these pretreatment steps, the feed is sent to the RO unit to remove dissolved salts and minerals. The RO unit has more than 2000 pressure vessels, which house more than 16,000 membrane elements. About 50% of the feed is recovered as permeate, which undergoes post-treatment (remineralization, disinfection, and fluoridation) before distribution to end users. Here, 144 ERDs are used to subsidize the pump energy consumption (see Fig. 1.8). The brine stream carrying the salts and minerals is returned to the ocean. More details about the plant design and operation can be found on the Carlsbad desalination plant website (Carlsbad Desalination Plant, n.d.).

The past few decades have witnessed a significant advancement in computing power and a drastic increase in the availability of numerical and engineering solvers, e.g., MATLAB®, ANSYS Fluent®, COMSOL Multiphysics®, ASPEN PLUS™, and PRO/II™. Systematic computer-based methods (Biegler et al., 1997) play an important role in the analysis, design, monitoring, operation, control, and optimization of chemical, physical, and biological processes. With the aid of mathematical models, process systems engineers are able to enhance the performance of existing processes, and to facilitate the scale-up of emerging technologies.

The heart of systematic methods is a mathematical model, which can be empirical or based on first principles. The hydrodynamics and mass transfer in an RO are governed by coupled partial differential equations (PDEs) (Gu et al., 2020; Luo et al., 2020). In this book, the transport phenomena in a small fraction of an RO feed channel is simulated using COMSOL Multiphysics® (COMSOL AB, 2012). A finite element-based numerical solver, COMSOL Multiphysics® is capable of solving coupled processes involving two or more simultaneously occurring physical fields. Despite the fact that its computational fluid dynamics (CFD) simulation provides a comprehensive and in-depth fundamental understanding of the local transport phenomena, the high computational demand prohibits its direct application in model-based optimization on the process level. The knowledge obtained from CFD simulations may be used for the development of simplified, yet reliable, system-level models. The system-level model also may be obtained via correlation of experimental data.

System models of membrane processes are usually in the form of nonlinear algebraic equations (NAEs) and original differential equations (ODEs). In the following paragraphs, a brief description of the algorithms for solving NAEs and ODEs will be given.

Newton's method for NAEs: Newton's method, also known as the Newton–Raphson method, is a commonly used algorithm for solving coupled NAEs simultaneously. Consider an n nonlinear equation,

or, in vector form,

where f = [f₁, f₂,…, f_n]^T, x = [x₁, x₂,…, x_n]^T, and 0 = [0, 0,…, 0]^T.

The Newton's method is based on the Taylor series expansion with truncation after the first derivatives, i.e.,

where k stands for the iteration counter. Therefore, Eq. (1.6) becomes

or, in a matrix form,

or

where Δx^k+1 = x^k+1 − x^k and J is called the Jacobian matrix,

The procedure is summarized as follows:

1. Make an initial guess $x0=[x10,x20,…,xn0]T$⁠. k = 0.

2. Evaluate the Jacobian matrix using Eq. (1.12).

3. Compute Δx^k+1 = x^k+1 − x^k using Eq. (1.10) or Eq. (1.11).

4. Determine x^k+1 = Δx^k+1 + x^k.

5. If the norm |Δx^k+1| is below the user-specified tolerance, the solution is converged. If not, repeat steps 2–5.

A quasi-Newton method is available in which the derivatives in the Jacobian matrix are based on linear approximation. This is useful when the derivatives are too expensive to compute. For singular or near-singular problems, the Levenberg-Marquardt algorithm may be used by adding a small identity matrix to the matrix J^TJ in Eq. (1.11).

Runge–Kutta method for ODEs: The Runge–Kutta method may be used to solve coupled nonlinear ODEs. Consider the following ODEs with initial conditions:

or

where x = [x₁, x₂,…, x_n]^T and f = [f₁, f₂,…, f_n]^T. According to the 4th order Runge–Kutta (or RK4) algorithm, the approximate solution to x after an integration step h = t_n+1 − t_n is

where

which can be interpreted as the present value plus weighted average of four increments, each corresponding to a different slope.

Once a system-level model is developed, it can be used to analyze and optimize the performance of a membrane process. The optimization aims to find the best design and/or operating conditions to maximize the performance of the membrane. Some commonly used objective functions in the optimization model include cost, energy consumption, and water flux. The design parameters could be membrane layout and area. The operating conditions usually involve pressure applied to the membrane unit.

There are two methods to solve an optimization model: the direct search method (e.g., Fibonacci, golden-section, and Nelder–Mead searches) and the optimization method (e.g., Newton's method). The former attempts to improve a feasible point by searching among neighboring points, generally using function values only. The latter, however, solves an optimization problem, often using gradient information. A description of several commonly used algorithms is provided next.

Nelder–Mead simplex search method: The Nelder–Mead algorithm is a simplex-based direct search algorithm for unconstrained multivariable optimization without derivatives (Lagarias et al., 1998). To minimize f(x) that has n variables, the method begins with a construction of a convex hull of n + 1 vertices and an evaluation of the function at each vertex. At each step in the iteration, the algorithm discards the worst vertex (corresponding to the largest function value) and accepts another point into the simplex via a series of transformations. This process is repeated until the working simplex is sufficiently small or the standard deviation of the function values of the simplex is below a specified tolerance. The Nelder–Mead procedure for min f(x) is summarized as follows:

1. Create an initial simplex S, which is formed by vertices x₁, x₂,…, x_n+1. Evaluate f(x) at each of the vertices.

2. Identify the indices h, g, and l corresponding, respectively, to the maximum, second maximum, and minimum values of the function f(x): $f(xh)=maxif(xi)$⁠, $fg=maxi≠hf(xi)$⁠, and $fl=mini≠hf(xi)$⁠, respectively.

3. Calculate the centroid of excluding the worst vertex x_h, or $xo=1n∑i≠hxi$⁠.

4. Conduct a transformation of the simplex.

   • Perform the reflection x_r = x_o + α(x_o − x_h), where α > 0.

   • If f(x_l) ≤ f(x_r) < f(x_g), accept x_r. Go to step 5.

   • If f(x_r) < f(x_l), perform the expansion x_e = x_o + γ(x_r − x_o), where γ > 1 and γ > α.

     • If f(x_e) < f(x_r), accept x_e. Go to step 5.

     • Otherwise, accept x_r. Go to step 5.

   • If f(x_r) ≥ f(x_h), perform the inside contraction x_c = x_o + β(x_h − x_o), where 0 < β < 1.

     • If f(x_c) < f(x_h), accept x_c. Go to step 5.

     • Otherwise, perform a shrink transformation.

   • If f(x_g) < f(x_r) < f(x_h), perform the outside contraction x_e = x_o + β(x_r − x_o).

     • If f(x_c) < f(x_r), accept x_c. Go to step 5.

     • Otherwise, perform a shrink transformation.

   • For a shrink transformation, compute n new vertices x_j = x_l + δ(x_j − x_l) for all j ≠ l.

5. If convergence criterion is satisfied, terminate. If not, go to step 2.

Standard values used in the algorithms are α = 1, β = 0.5, γ = 2, and δ = 0.5.

Newton's method for unconstrained optimization: This method uses the optimization

where f is a convex and twice continuously differentiable function. The necessary condition of optimality is

where $∇f=[∂f∂x1,∂f∂x2,…,∂f∂xn]T$⁠. Using the Newton's method presented previously, the iteration formula for Eq. (1.18) is

where H is called the Hessian matrix,

In the damped Newton's method, the gradient term in Eq. (1.19) often is multiplied by a factor between 0 and 1. Other algorithms, such as quasi-Newton and Broyden–Fletcher–Goldfarb–Shanno (BFGS), are available, in which the Hessian matrix is updated by analyzing successive gradient vectors.

Karush–Kuhn–Tucker Theorem for nonlinear programming: Many engineering optimization problems can be formulated as the minimization of an objective function subject to various equality and inequality constraints, e.g.,

where x = [x₁, x₂,…, x_n]^T are design and operating parameters to be optimized.

The Lagrangian of the constrained problem described by Eq. (1.21) is constructed as follows:

According to the Karush–Kuhn–Tucker (KKT) Theorem, also known as the Kuhn–Tucker Theorem, the necessary conditions for the local minimum $x∗$ are

If $gj(x∗)<0$ (inactive), μ_j = 0. If μ_j > 0, $gj(x∗)=0$ (active). The second order sufficient conditions for local minimum is

where y is a nonzero vector that satisfies $[∇gi(x∗),∇hj(x∗)]Ty=0$⁠. Only those $gi(x)$ corresponding to $μi>0$ are applied.

Interior-point method for constrained nonlinear optimization: The interior-point approach to constrained minimization is to solve a sequence of approximate minimization problems (Byrd et al., 1999). By introducing a barrier parameter μ > 0 and l slack variables s_i > 0, the original problem in Eq. (1.21) is approximated by

where s = [s₁, s₂,…, s_l]^T.

As μ decreases to zero, the sequence of approximate solutions to the minimum of f_μ approaches the minimum of f. This method does not require feasibility of the inequality constraints but only forces positive slack variables (Byrd et al., 1999).

Its Lagrangian is constructed as follows:

The optimality conditions of the barrier problem are given by

where $Ah=[∇h1,∇h2,…,∇hm]$⁠, $Ag=[∇h1,∇h2,…,∇hl]$⁠, e = [1, 1,…, 1]^T, and S = diag(s₁, s₂,…, s_l). This equation can be solved iteratively by the Newton's method. Details about how to implement the algorithm can be found in the literature (Byrd et al., 1999).

MATLAB® is used for system-level modeling and optimization in this book (The Mathworks Inc., 2017). Specifically, NAEs are solved by fsolve and ODEs are solved by ode45, if not specifically stated otherwise. Identification of parameters in the system model is done using fminsearch, which is based on the Nelder–Mead algorithm. Optimization of membrane processes is solved by fmincon, which is a gradient-based numerical method designed to handle problems in which the objective and constraint functions are both continuous and have continuous first derivatives.

This book is organized as follows:

• Chapter 1 provides an introduction to desalination, spiral wound membrane, RO design and operation, and systematic methods.

• Chapter 2 focuses on a 1D system-level model developed from 3D CFD simulations.

• Chapter 3 focuses on a quasi-2D module model derived from experimental data. This model accounts for pressure drop in the permeate channel.

• Chapter 4 discusses energy issues in seawater RO desalination and strategies to reduce the specific energy consumption (SEC).

• Chapter 5 presents system-level modeling and optimization of plant operation brackish water reverse osmosis (BWRO).

• Chapter 6 covers systematic analysis and optimization of pressure retarded osmosis (PRO) for power generation from salinity gradient.

• Chapter 7 presents a hybrid RO-PRO design for energy-efficient seawater desalination.

• Chapter 8 discusses dynamic operation of batch RO and batch PRO that can potentially enhance process efficiency.