1. Introduction

Water stress is the ratio of water use relative to water availability (“demand-driven scarcity”) where the dark-red regions are middle east countries. But the demands are high in middle countries of Saudi Arabia, United Arab Emirates (UAE), Southern California, and Singapore[1]. The shortages are expressed as dark-red color (severe) to yellow. Another figure from Google of “water stress by country 2040” shows that water stress regions are increasing very rapidly[2]. For world-wide of water consumption, 70% water goes for agriculture, 11.0% for municipal and 19% for industrial use. More than anything else, a half century ago, the late US President John F. Kennedy[3] said that “Anyone who can solve the problems of water will be worthy of two Nobel prizes-one for peace and one for science”[4]. The current world water problem is much severer than the 1960s' Kennedy days. A scientific literature on water scarcity “Four billion people facing severe water” (out of about 7 billion) published in Science Advances shows a highest citation record of “2563” since its publication in 2016[5].

Securing drinking water is one of the four major issues for sustaining planet earth: CO₂ reduction, particulate matter, and micro-plastics. Lee and Chang et al. [6] cited the fate of planet earth by big-bang theory of physics[7]: the sun expands as the sun gets older and its hydrogen resource for its nuclear fusion is depleted to engulf nearby stars such as Earth. The earth will be too hot for humans to live, and the planet Mar may be a place for human to move for further living. This may be one hundred million years from now, but the current global warming is more urgent enough to keep CO₂ level below 2.7% from the industrial age from of 1850 to the today[8]. Water covers approximately 70.9% of the Earth's surface, mostly in seas and oceans The total volume of water on Earth is estimated at 1.386 billion km³ (333 million cubic miles), with 97.5% being salt water and only 2.5% being the fresh water, only 0.3% is in liquid form on the surface. The average salt concentration of the earth water is 3.5% and 96.5% of water is in oceans[9]. There are four categories of the earth water[10]: briny water (brine pools, 50+ ppt, 50 g/L) where ppt is parts per 1000, saline water (30~50 ppt, 3~5%), brackish water (0.5~30 ppt, 0.05~3.0%), fresh water (0~0.5 ppt, 0~0.05%, 500 ppm), In this figure drinking water is 0.01% (100 ppm, mg/L). Removing 80% of 3.0% salt from seawater may yield osmotic pressure of 0.9% (9000 ppm) equivalent to that of human blood[11]. If the osmotic pressure is higher than that of blood, the solution is called hypertonic, the same isotonic and lower, hypotonic. If the red blood cells are placed in hypotonic solution, the red cells burst and will shrink its volume in hypertonic solution.

In this work 3.0%, 3.5% and 4.5% NaCl concentrations are selected for the calculation of theoretical energies for ideal or non-leaking membrane of sigma (σ) = 1.0, the selectivity. 3.0% is for Korean seawater, 3.5% the earth seawater concentration and 4.5% is for the Middle East countries.

Since Loeb and Sourirajan invented practical reverse osmosis (RO) in 1962[12], reverse osmosis is considered the most important technology for seawater desalination while MSF (multistage flash distillation) or MED (multi-effect distillation) were popular with waste heat utilization, with solar heat[13-14]. Obtaining 1m³ water from seawater or removing the same from the fermentation broth needs thermodynamic energy of 736.2 (= 714.2 / 0.97) kWh by a thermal means, 1.933 kWh or less by RO. The advantage of the thermal method is to recover 100% of water and salts while the water recovery by RO is limited to about 30~50% because of increased osmotic pressure.

Multistage flash evaporation (MSF) lowers this energy to 25 kWh by recycling evaporation energy[13], and RO in practice needs 4 kWh per m³ of water. A smaller thermal desalination system may need more than 25 kWh because of lower recovery efficiency.

There are a variety of methods in desalination: distillation (five), ion exchange, membrane (four), freezing desalination, geothermal, solar (three), methane hydrate, high grade water recycling and wave-powered desalination[15]. The current RO has many applications [16]: ultrapure water in semi-conductor industries in addition to the well-known applications.

2. Birth of S-ZERO RO Technology

Since 2005, there have been worldwide efforts of seawater desalination by FO[17,18]. With forward osmosis, a 3.5% of VFAs were enriched to 14% with the saturated NaCl draw agent. However, the separation of NaCl-VFAs from NaCl/NaCl-VFAs mixtures was difficult and not useful for any other uses[19]. Chang had an idea of using NaCl itself as a draw solution and proposed the 2-component seawater desalination RO system consisting of two-components: water and salt, but more exactly speaking, an enriched salt solution that serves the role of a draw solution in forward osmosis. From 2013 to 2019 Δπ= 0 RO, mass balance, and energy-saving S-ZERO were developed. [20-22]. The following five topics of 1~6 will be discussed covering from current reverse osmosis (C-RO) Δπ= 0 RO and finally Singularity ZERO (S-ZERO) RO.

The limitation of C-RO lies in water recovery from feed inputs of 3.0% (600 W), 3.5% (500 W) and 300 W from 4.5%: Their theoretical energies are around 1.00 kWh/m³W.

1 m³W refers to the energy required to obtain 1 m³ of pure water that is different from 1 m³-sol. The 3.0%, 3.5% and 4.5% /m³-solutions contain 30kg-salt (30 S, 970 W), (35 S, 965 W) and (45 S, 955 W), respectively. The 1.00 kWh/m³W of theoretical energy is claimed equivalent to 4.00 kWh/m³W in C-RO system.

(Source: OLI System Inc. OLI Stream. Analyzer 2.0, Morris Plains, NJ, U.S.A (2011) [24]).

Table 1A shows a 3% solution input water (970 W) with fixed salt (30 S). With 30 S for 1% solution we need 2970 W, 2% 1470 W, 3% 970 W. At the saturation of 26.47% we need 83.34 W with 30 S. For 3.0%, 3.5% and 4.5% the saturated water increases from 83.34 W, 97.23 W, and 125 W. The highest water recovery will be 886.66 W for 3.0%, 868.77 W for 3.5% and 830 W for 4.5% solutions. A 3.0% input has 30 S and 970 W and the saturated 26.47%, 30S* / 100 / 36 = 83.33 Ws “3” recurring. The maximum recovery will be 970 W – 83.33 W = 886.66. 6 recurring. Wrec water (recoverable water) decreases with an. increase of salt input% (Table 1B). The saturation concentration 26.47% comes from the solubility of 36 g - salt to 100 g - water. % is given as 36 / 136 = 9 / 34 = 26.47058.

wt% = S / (W + S), salt % per 3%, 1000 kg solution, S = salt (kg), W = water (kg), kWh/m³ 58.44 (molecular weight) of NaCl. OLI (bar). Since osmotic pressure depends molecules of NaCl, it is important to know molecular weights of our interest. OLI is a service company providing osmotic pressures of our interest. In kWh of 3% is 23.74 bar / 36 = 0.6594. In this paper simple W means solution based on m³-NaCl solution while m³W means values based on water recovered. The information about NaCl solution can be available at the Wikipedia[25].

Rather than % solution W/S ratio more convenient. 3% solution W/S ratio = (1.00 – 0.03) / 0.03 = 32.33, at the saturation.

W/S ratio = 25 / 9 = 2.77 (7 recurring). W/S ratio of 3.5% (1 – 0.035 / 0.035) = 27.57, 4.5% (1 – 0.045) / 0.045 = 21.22. The W/S ratios of 3.0%, 3.5% and 4.5% are different at the start but approaches the same 2.77 at the end.

For a 3.5% and 4.5% input similar tables of Table 1A can be made. The above table is based on assumption of ideal membrane selectivity (σ=1, ideal). Sg (abbreviation of sigma), σ = sigma, membrane selectivity, if σ= 1, no salt but only water passing (ideal membrane). but σ= 0 no selectivity against salt. RO membranes have a very high σ= 0.9975 (Toray Korea) for 30 minutes with a 3.2% solution.

Theoretical Energy is defined as the energy required to overcome the osmotic pressure of any solution. For example, a 3% solution has a wide range of concentrations from nearly 0% to its saturation 26.47%. A 1% solution has 7.61 bar (OLI osmotic pressure) and its theoretical energy is 7.61 / 36 = 0.211 kWh/m³-sol. For a 26.47% solution the theoretical energy is 347.71 bar / 36 = 9.658 kWh/m³-sol. A question may arise why “atm” instead of bar is not used? Another question is that osmotic pressure is dependent on volume or not?

These questions are reasonable questions. For convenience sake, I stick to MKS system to avoid errors. The Van’t Hoff osmotic pressure is given as π= CRT (concentration dependent). But “R” (gas constant) may have many different versions units depending on units of weight kg vs lbs, m vs foot, atmospheric pressure (bar, psi, pascal) and temperatures of F, and C. If Δπ (osmotic pressure difference between the feed and product chamber is 1 bar, the theoretical energy is calculated as 1 bar / m³ × 1 m³ / 36 = 1 kWh / 36 = 0.027 kWh. Thus a 3% solution has a 7.61 / 36 = 0.211 kWh/m³. If you have a tank volume of 5 m³, the energy you need is 0.211 × 5 = 1.056 kWh. In summary theoretical energy is the most important information we can get from osmotic pressure vs solution concentration. There will be further discussion at the osmotic pressure approximation.

Table 1B shows Wsat, Wrec, Energy up to the saturation, and the energy contained in 1 m³ of solution and pure water. Input concentrations selected for 3.0% (Korea), 3.5% (World average) and 4.5% (Middle East).

3. For More Water Recovery Recycling of Enriched Feed Output

Table 2 shows the osmotic pressure difference (Δπ) between the feed and product chambers by recycling a part of the enriched feed, was kept under 70 bar regardless of their osmotic pressures [20]. Conventional RO (C-RO) has a single RO unit while this novel system has an additional π-equalizer that reduces Δπ between the feed and the osmotic equalizer (OE) chambers, replacing the name and function of the C-RO product chamber. Thus, we hypothesized that the RO unit in the Δπ= 0 RO[21] and singularity ZERO (S-ZERO) RO recovers water from a low osmotic pressure solution from the OE-output[22].

At the bottom zero recycle refers to simple RO. The yellow colored errow water recovery beteen 6% and 7%. 100% recycle of enriched the feed output allows Δπ= 0 from 3.0% to 26.47%. In this case the product output is the same as the input. There is no water recovery for 100% recycle. Application of ΔP = 70 bar make differnce between ΔP = 0 bar. The 70 bar will vary feed concentration from 3% to 26.47%, but zero bar applciation the feed concentration will stay as 3.0% without change.

A 90% recycle will allow 100%, 66.60% water recovery up to 26.47%. 50% recycle 21%, and 33.3% recycle up to 12.0%, and 33.30% up to 12.0%, 10% reycle up to 9.0%.

Single point Δπ= 0 exists for every recycle ratio as long as there is a split. Two split streams may have different flow rates, but the concentration between the two are the same and thus Δπ= 0. Later you will see that multi-point Δπ= 0 (singular zero) in the S-ZERO section. Thus theoretical energy needed to overcome osmotic pressure differences between the feed and OE chambers. Making very shorter residence time in the OE chamber allows higher concentration in the OE chamber higher than the feed chamber, in this case Δ π< 0 forward osmsosis.

“1/2 W and 2-fold E” is the abbreviation of successive removal of water to make the feed concentration doubling. From a 3% 1000 kg (30 S, 970 W) a successive withdrawal of water from the feed solution (500 W, 250 W, 125 W, 11.66 W), the feed solution will be enriched to 3%, 6%, 12%, 24%, 26.47%. The amount of the water becomes (500 + 250) + 25 = 875 W, 11.67 W, 83.34 W to make 970 W. “11.67 W” 83.34 W means free water and 83.34 W means saturated water, respectively.

Applying these to a 3.5% solution: 500 W, 250 W, 117.77 W, 97.23 W), 3.5%, 7.0%, 14%, 26.47%) For a 4.5% solution (500 W, 250W, 125W), 4.5%, 9.0%, 18%, 26.47%.

At this point I like to introduce how Singularity ZERO (S-ZERO) RO differs from RO, Fig. 1 includes feed intake, pretreatment, NF (nanofiltration) enters S-ZERO module and enriched feed output is split into two streams: one is discharged to the outside ant the other stream is recycled through OE (osmotic equalizer) chamber where OE input is diluted by feed permeate to the OE chamber. Thus it is expected that OE-output flows counter-currently to the feed input side (FOS-CC). FOS-CC is the abbreviation of Feed Output Split Counter-Current system. There is one more split point: OE-output (diluted) where one stream is enriched to the feed concentration that joins the feed input, and the other RO output for water recovery. One is for enriched feed and the other is for diluted OE-output.

Fig. 2A shows conventional Reverse Osmosis system (C-RO) is a single chamber system of feed and product sub-chambers where “sub-chamber” is used to avoid two chamber systems of the following Δπ= 0 and S-ZERO (Fig. 2B and 2C). Fig. 2A has much smaller product chamber than its feed chamber (0.2 mm vs 1 mm). In nearly all chemical engineering systems of continuous flow, the residence times of feed and outputs should be equal in order to maintain a steady state operation. In contrast to Figs. 2B and 2C there is recycle from feed output (enriched). Fig. 2B has feed and osmotic pressure equalizer sub- chamber (OE) of equal sizes. In this case the role of OE chamber is to reduce “Δπ” a difference in osmotic pressures or concentrations between feed and OE chambers. In order to recover water Fig. 2B needs reverse osmosis chamber (RO).

Fig. 2C shows the energy saving S-ZERO system. Please pay attention to the geometrical difference between Fig. 2B and Fig. 2C. In system B the residence times of the feed chamber and product (OE) chamber are the same meaning RRT = 1.0 while Fig. 2C has a smaller OE chamber than the feed chamber (RRT ≪ 1.0). The strategy of S-ZERO system is to keep FOS (feed output split-stream (enriched) concentration as longer as possible. Shorter, the residence time in the OE chamber (RRT ≪ 1.0), less diluted OE than the OE –input. In this way we can make Δπ equal or less than ZERO (forward osmosis). Fig. 2C has an OE-res- ervoir that Figs. 2A, 2B does not have. S-ZERO principle will be further explained in S-ZERO section.

4. Mass Balance in Δπ=0 Reverse Osmosis System

Δπ= 0 RO system consists of π-equalizing (Δπ reducing = 0) and water recovery chamber (RO).

FOS-CC consists of two split points: the first FOS point is at the end of enriched feed output, and the other is diluted OE-output between feed input and OE output. These two points have roles where the FOS point manages the enriched stream into two parts: discharge to the outside the system, and recycle through OE while the OE split point manages the diluted stream into two parts: recovery of water, and enriched recycle up to 3.0% OE stream to the feed stream. Recovering water from highly diluted stream from OE output by reverse osmosis (RO) makes OE output 1.59% to 3.0%.

Basis: Net input in the system point 1000 SW, 30 S, 970 W where SW is a NaCl solution ⑤ 3.0% 30 S, 970 W. “1/2” will be considered afterwards. At point ④ 500 SW, 500 W is removed to become 6.0% of 500 SW, 470 W; ③ 250 SW or 250 W ➔ 12%, 250 SW, 220 W ② 125 SW, 125 W ➔ 24%, 125 SW, 95 W ① 113.34 Sw or 11.66 W ➔ 26.47%. Conclusion of single point Δπ= 0: 50% recycle of saturated solution yields 50% (15 S) salt removal of the total salt (30 S) but removes water only 4.29% (41.67 Ws) of water resulting in 95.7% water recovery (970 W – 41.67 W = 928.33 W. 928.33 W is broken into two parts: 15 S, 485 W for recycle as 3% solution as 500 SW to feed, and the 443.33 W is recovered by reverse osmosis.

15 S, 485 W (feed) the net output 15 S, 83.34 Wsat × 1 / 2 at the first split point (41.47 Wsat); The second split point = 485 W - 41.67 W = 443.33 W is recovered. Thus, the balance between total input and total output is completed.

Fig. 3B shows how to find a second Δπ= 0 in addition to the FOS point (first Δπ= 0) at point ① One assumption here is that the amount of feed to permeate is proportional the residence times in feed and OE chambers introducing so-called, “relative residence time RRT concept”. In order to a steady state operation in C-RO and Δπ= 0, residence RRT = 1.0. Because of this if RRT = 0.5, see what happens. The water in the feed chamber between ② and ① is 11.86 W. If the residence time of OE is a half of the feed chamber, the permeate will. be a half of 11.86 W, 5.93 W. The total water in OE at ② will be 41.666 W + 5.833 W ≒ 47.5 W. the OE and feed salt concentrations are 24%, like the osmotic pressure difference is zero (Δπ= 0). These Δπ= 0 continue from ①~④. But at OE ⑤ the strict mass balance needs to be ensured. The OE concentrations are 15 S and 928.33 W and the fate was discussed in Fig. 3A.

The OE chamber concentration at point ② was 21.94% while the feed concentration is 24%. We have seen that 50% of FOS point removes 50% of 30 S, 15 S, but removed only 41.67 W and recovered 95.7% water. This imbalance is a key to how to enriching salt in the feed and dilute the OE solution. We recover water from the most diluted OE-output and discharge the most enriched salt from the FOS point.

5. Singularity ZERO (S-ZERO) RO σ = 1.0, σ < 1.0, W/S Ratio

FOS-CC refers to feed output split counter-current system. Feed output is highly enriched, and OE output is highly diluted. The system we are dealing with both ideal membrane (σ = 1.0) and non-ideal (salt leaking) real membranes (σ < 1.0). Singularity ZERO (S-ZERO) system where Feed and OE chambers have Δπ≤ 0 (Δπ= 0 or forward osmosis state).

Another concept W/S ratio is used rather than salt %. W/S ratio: % = S / (S + W) = 1 / (1 + W / S).

If the W/S ratio is same in the feed and OE chamber, Δπ will be zero. Since the W between 24% and 26.47% is 970 - (500 + 250 + 125 = 875 W) - 83.34 W = 11.66 W (free water). If the 11.66 W is permeated from the feed to the OE, the concentration in the OE = 15 / (15 + 41.67 + 11.66) = 21.95%. Thus only a single Δπ= 0 is possible. However, in a S-ZERO system 3% feed (30 S, 970 W) multiple Δπ= 0 is possible. If OE concentrations has the same W/S ratio.

Table 4A shows that any feed concentratrion can be matched with a small amount of OE salt concentration provided their W/S ratio are the same. From the other point for water recovery the water recovery will be higher if OE has less salts.

The selectivity of RE 2521-SHN 3.2% NaCl solution under 55 MPa is 0.9975. The study of selectivity and flux of the above membrane under Δπ= 0 condition was studied under the project by Korea Research Foundation (see: acknowledgement section). The second manuscript on the above subject is under preparation. The contents can be found in the recent Korean patent [21] and WIPO citation. Using the flux and selectivity of above study Table 4B was prepared. For example, σ= 1.0 membrane 500 W removal would lead to 6%, but here σ< 1.0 membrane, you see 5.69%. It is general that theoretical energy consumption would be 1.050 kWh/m³W that is higher than with σ= 1.0 membrane. 0.970 kWh/m³W. The salt leakage was 1.903 g out of 30 g.

For 3.0% input, 3.5% input and 4.5% input water recovery is possible up to 5.69% (500 W). 6.52% (500 W), and 6.22% (300 W). With σ= 1.0, 500 W could have been obtained at 6.0% for 3.0 input, at 7.00% for 3.5% input, and 300 W at 6.49%. More water can be obtained at lower salt concentrations.

The ratio of C-RO, Δπ= 0 to S-ZERO is 6.96-X for 3.0% input and, 5.5-X, for 3.5%, 9.36-X; For higher water recovery 5.5-fold for 3.0%, 6.45-X for 3.4%, 5.82X for 4.5%. Table 5B, MA-total: 10.88 for 3%, 500 W, 4.11 for 750 W for Δπ= O, and 2.84 for S-ZERO. The scale-up experimental result with a 0.1m² membrane area showed the flux of S-ZERO system was 1.8 times that of Δπ= 0 RO[23]. Based on this result, the comparison of C-RO, Δπ= 0 RO, and S-ZERO was made (Table 5B). S-ZERO uses less membrane area, about 1/3 of C-RO. Another reason for this phenomena is that S-ZERO produces at much lower NaCl % than C-RO.

6. Osmotic Pressure Approximation

As an energy unit in RO, kWh is commonly used. For instance, Δπ = 23.74 bar of 3% NaCl solution the specific energy per m³ solution is given as 23.74 / 36 = 0.659 kWh/m³ (solution). 1 kWh = 3600 kJ and 1 bar m³ is equivalent to 100 kJ. In the real RO system, this small value cannot be obtained because upon saturation, the specific energy goes up to 9.5 kWh/m³ (solution) where 1 kWh is 36 bar⋅m³.

In 1982, Chang solved a simple nonlinear equation of immobilized enzymes of high Thiele moduli by combining analytical and numerical approaches[24]. The Van't Hoff and Lewis equations were known to fit well at low and high solution concentrations, respectively[ 25]. An approximation of OLI osmotic pressures for specific applications was necessary, but a simple polynomial was not enough. The combination of the Van't Hoff and numerical approach worked well within a 0.1% accuracy[24-26]. S-ZERO RO technology would find its applications in desalination and biofuel ethanol enriching, and non-aqueous solutions, for instance, replacing thermal distillation with non-organic polymer membrane distillation[27].

Note that the viscosity and density of NaCl solutions are important. The viscosity varies in the ranges of 0.75~1.80 cp of 0.5~5 M solutions and at temperatures of 20~30°C. It is relatively insensitive to temperature and salt concentration as compared to that of viscous organic polymers. The density at 25°C varies from 1.00409 for 1% and 1.199443 for 26.47%, but the density in the S-ZERO system is assumed to be the same in both chambers. Thus, it is no need to worry about flow rate changes because of the density difference.

7. Higher Recovery, Less Energy and Less Membrane Area

7.1. Advances in reverse osmosis

Kessler and Moody[33,34] made drinking water by forward osmosis from seawater using nutrients as draw agents. McCutcheon et al.[35] of Yale University proposed a novel ammonia—carbon dioxide forward (direct) osmosis desalination process, which requires thermal regeneration. These two desalination by forward osmosis introduces the third component “draw agent”. Forward Osmosis has various applications in areas such as emergency drinks by two-component forward osmosis, make-up water for evaporative cooling tower, brine concentration and landfill leachate treatment, etc.[36]. A hot topic these days is how to realize osmotic power plants invented by Loeb in 1975[37, 32].

In addition to the desalination and future fuel ethanol, there will be increasing demands on the production of volatile fatty acids (VFAs) during the anaerobic digestion of biodegradable organic wastes [12-16]. These works were intended to store VFAs for later use rather than the on-site biogas production. For the removal of the aqueous solution with 0.3% to 3.5% of VFAs to its saturation or salts, RO or FO has been considered. These systems also showed high osmotic pressures that are similar to NaCl solutions Chang and Jung has been working on the enriching of VFAs-Na 3.5% to 14% using forward osmosis with saturated NaCl as the draw solution. As expected, the separation of NaCl and NaCl-VFAs appeared to be economically infeasible for any products[13,14]. Instead, Chang had an idea of using the NaCl solution as a draw solution in seawater desalination. In 2013 simple experiment was carried out to prove osmotic pressure-free RO of water and NaCl two-component desalination[16]. The enriched (saturated) feed stream output can be as high as 26.47% in concentration. Recycling a part of the feed output in the π-equalizer. The Δπ, osmotic pressure difference between the feed and π-equalizer, from 343.70 to zero at the split point of the FOS-CC (feed output split) counter-current system. This keeps the Δπ under 70 bars such that RO operation may be carried out without osmotic pressure limitation, and is named as Δπ= 0 RO[17]. The FOS-CC (feed out split counter-current) consists of two chamber systems outputs)[18] C-RO originated by applying hydraulic pressure (ΔP) to input solution over salt selective membrane (σ). The water flux Jv, in terms of LMH (liter/m²-membrane area. hour) is given as Jw = A (ΔP - σΔπ). A typical ΔP is 70 bar and δ = 1.0 (ideal, salt non-passing).

A membrane manufacturer claims σ= 0.9975 of a 3.2% NaCl solution. A lower concentration (1%) of these compounds exerts low osmotic pressure of about 10 bars. But when saturated (26.47%), at higher con centrations this number goes to 347.7 bars. The highest osmotic pressure of 99.50% fuel ethanol is about 5,000 bars.

Introducing some of the current applications: drinking water purification, water and wastewater purification, food industry, maple syrup production, low-alcohol beer, hydrogen production, aquariums, window cleaning, and ultrapure, etc.[49].

The theoretical energy analysis is important for seeking the insight for energy efficient RO system[21,22]. As an energy unit kWh is commonly used. For instance, Δπ= 23.74 bar of 3% NaCl solution the specific energy per m³ solution is given as 23.74 / 36 = 0.659 kWh/m³-sol'n. 1 kWh = 3600 kJ and 1bar⋅m³ is equivalent to 100 kJ. In the real RO system this small value cannot be obtained because upon saturation, the specific energy goes up to 9.5 kWh/m³ - SW where 1 kWh is 36 bar⋅m³.

Theoretical energy of any NaCl % solution can be calculated using thermodynamically[32].

But the easiest thing is, once you know the osmotic pressure any % NaCl solution (either OLI or estimated value, the theoretical energy can be estimated. For instance, 3.0%, 3.5% and 4.5% have osmotic pressures of 23.74, 28.01, 36.75 bars, respectively. Their theoretical energies are 0.6594, 0.7780, 1,021 kWh/m³ - solution.

W/S ratio decrease as more water is removed from the feed to the OE chamber. W/S ratio of 3.0%, 3.5% and 4.5% are (1 - 3%) / 3% = 32.33, 27.57, and 21.22. The saturated solutions have (1 - 0.2647) / 0.2647 = 2.777 - 7 recurring.

7.2. Osmotic pressures, viscosity and density

Molecular weights of small molecules such as NaCl, volatile fatty acids of acetic and butyric acids and their Na salts are 58.5,60,84, 91.5 and 107, respectively. Osmotic pressure depends on their “colligative property” than chemical properties. It is dependent on the ratio between the total number of solute particles (in the solution) to the total number of solvent particles (water) of a solution, such as molarity, normality, and molality. The four colligative properties that can be exhibited by a solution are: Boiling point elevation, Freezing point depression, relative lowering of vapor pressure Osmotic pressure.

Nonlinear equations of immobilized enzymes with high Thiele moduli were solved by combining analytical and numerical approaches[24]. The Van't Hoff and Lewis equations were known to fit well at low and high solution concentrations, respectively[24]. An approximation of OLI osmotic pressures for specific applications was necessary, but a simple polynomial was not enough. The combination of the Van't Hoff and numerical approach worked well within a 1.0% accuracy[ 24-26]. S-ZERO technology would find its applications in desalination and biofuel ethanol enriching, and non-aqueous solutions, for instance, replacing thermal distillation with non-organic polymer membrane distillation[27].

The viscosity and density of NaCl solutions are important and given in Materials and Methods. The viscosity varies in the ranges of 0.75~1.80 cp of 0.5~5 M solutions and at temperatures (T) of 20~30°C. It is relatively insensitive to temperature and salt concentration as compared to that of viscous organic polymers. The density at 25°C varies from 1.00409 for 1% and 1.199443 for 26.47%, but the density in the S-ZERO system are assumed to be the same in the both chambers.

7.3. Volatile fatty acids as future raw materials for fuels and chemicals

The unlimited amounts of resources on the earth (unit tons): air 5.1 × 10¹⁵ (33), Water 1.31 × 10¹⁸ or (333 miles³) [9], Biomass 1,841 billion tons (existing), 172.5 billion tons/year (6), Wastes (google image): million tons/day: 1900 (0.5), 1950 (1.0), 2020 (4.0), 2050(8.0), 2100 (12). Among these renewable resources biomass, but the other resources cannot be said to be depleted. One thing for sure is wastes are increasing very rapidly. Fatty acids can be made from biodegradable organic wastes. If they are not treated properly, the anaerobic digestion will result in global warming CO₂ and CH₄ production. 1 Kg of VFAs may cost less than 0.1$ if the logistics and bioprocessing are properly managed. A fatty acid is a carboxylic acid with an aliphatic chain, which is either saturated or unsaturated. Depending on the carbon number of an aliphatic chain it is classified as short, medium, long, very long: less than 5, “short”, 6~12, “medium” and “13~18” long, greater than 19 “very long”. This carbon number can be as large as 35. However, most naturally occurring fatty acids have an unbranched chain of an even number of carbon atoms, from 4 to 28[31].

In the Chang's laboratory VFAs derived from Korean food wastes were used as electron donor replacing methanol in removing P, N in waste water treatment plant[32].

There are about 10 literatures on microbial production from volatile fatty acids derived from waste organic biomass such as foodwastes including Dr. Fei's KAIST thesis[32~37, 39~43]. Also, one valuable reference on biodegradable plastics is Dr. Doi of RIKEN, Japan[5]. These plastics are made with VFAs derived from organic wastes. Two recent review article on reverse osmosis desalination[44] received 470 citations. Hong and his associates suggested a low energy reverse osmosis based on their theoretical works[45].