Scalable, Self‐Contained Sodium Metal Production Plant for a Hydrogen Fuel Clean Energy Cycle

In this chapter, we present a detailed design study of a novel, scalable, self-contained solar powered electrolytic sodium (Na) metal production plant meant to enable a hydrogen (H 2 ) fuel, sustainable, closed clean energy cycle. The hydrogen fuel is gener- ated on demand inside a motor vehicle using an efficient hydrogen generation apparatus that safely implements a controlled chemical reaction between either ordinary salinated (sea) or desalinated (fresh) water and sodium metal. The sodium hydroxide (NaOH) byproduct of the hydrogen generating chemical reaction is stored temporarily within the hydrogen generation apparatus and is recovered during motor vehicle refueling to be reprocessed in the self-contained sodium (Na) metal production plant. The electric power for NaOH electrolysis is produced using photovoltaic (PV) device panels spatially arrayed and electrically interconnected on a tower structure that maxi- mizes the use of scarce land area. Our analysis shows that the scalable, self-contained sodium (Na) metal production plant using solar power is technically and economically viable for meeting the hydrogen fuel clean energy needs of all the motor vehicles in the U.S.A. by constructing approximately 450,000 scalable, self-contained sodium (Na) metal production plant units in the southwestern desert region that includes West Texas, New Mexico, Arizona and Southern California.


Introduction
There is a need in the modern world for sustainable means of producing clean energy economically, on a very large scale. The planet's human population is inexorably increasing toward the 10 billion mark [1][2][3]. The rapid growth in population presents both opportunities for companies seeking to broaden markets for their products, and challenges for governments, as the growing populations demand their share of prosperity. It is well known that prosperity generating economic growth requires energy [4][5][6]. To meet the demands for prosperity, carbon based fossil fuel consumption has increased accordingly, resulting in unacceptable levels of air pollution in major conurbations in both advanced and developing countries [7][8][9][10]. Much of the pollution comes from burning fossil fuels inside internal combustion engines (ICEs) of motor vehicles and ships. Coal burning thermal power plants used for electricity generation also contribute substantially to the rise in air pollution [11,12].
Scientific research has hitherto yielded various solutions to the clean energy challenge using innovative approaches ranging from development of hybrid gasoline-electric motor vehicles (HEVs), plug-in hybrid gasoline-electric motor vehicles (PHEVs), pure battery electric vehicles (BEVs), fuel cell electric vehicles (FCEVs), advanced catalysts for reducing exhaust emissions as well as carbon capture technologies applicable to coal burning thermal power plants [13][14][15][16][17]. These solutions however, work only to mitigate the problem of carbon based fossil fuel emissions and do not address the fundamental problem, namely, how to circumvent carbon based fossil fuels in energy generation and ground transport applications. Although HEV, PHEV, BEV and FCEV technologies offer promise in reducing pollution at least locally, they represent at best, an incomplete remedy to a major problem. HEVs and PHEVs still require combustion of gasoline inside an ICE while BEVs require copious quantities of electrical energy for battery charging, generated by power plants connected to the electric grid. The power plants supplying the electric grid can be hydroelectric or nuclear, but far more often, are fossil fuel burning thermal power plants that are only incrementally more efficient than internal combustion engines in motor vehicles [18]. Since hydroelectric generating capacity in the United States of America (U.S.A.) has already been reached and construction of new nuclear power stations is fraught due to well substantiated fears of radiological leaks, it becomes apparent that the only way to meet the increased demand for electricity from electric vehicle proliferation is by constructing more fossil fuel burning thermal power plants [19,20].
The FCEVs exist at present in small numbers as vehicle prototypes that function primarily as technology demonstrators [21,22]. FCEVs are unique however, because they represent the only motor vehicle technology that uses hydrogen (H 2 ) fuel to generate electric energy to power a motor driving the wheels of the vehicle. Although in existence in various forms since the 1960s, FCEVs have not proliferated for manifold reasons, the principal ones being the absence of means for safely storing hydrogen fuel on board, coupled with a lack of means to economically generate sufficiently pure hydrogen (H 2(g) ) fuel to prevent poisoning sensitive catalysts that might be present in the fuel cells [23,24]. The existing methods of storing hydrogen on board motor vehicles utilize cryogenic storage of liquid hydrogen (H 2(l) ), storage of hydrogen (H 2(g) ) gas at pressures as high as 70 MPa (10,153 psi) in cylinders made from composite material, and storage as a metal hydride (MH X ) in tanks filled with porous metal sponge or powder comprised of light group 1 and 2 metals and/or transition metal elements, namely, Titanium (Ti) or Nickel (Ni) [25][26][27][28][29][30]. Such direct hydrogen storage methods however, are impractical due to the high cost of suitable transition metals Ti and Ni, and moreover, because an infrastructure is needed to supply hydrogen directly in large volume to fill liquid or gas tanks or to saturate or replenish the metal sponge within the storage reservoir inside a motor vehicle, a procedure fraught with all of the well known safety risks associated with handling large volumes of elemental hydrogen [31,32]. Furthermore, the existing industrial method of generating hydrogen (H 2(g) ) gas using steam reforming of natural gas, the latter containing mostly methane (CH 4 ), produces significant quantities of carbon monoxide (CO) even after application of the shift reaction, the latter meant to transform the CO into carbon dioxide (CO 2 ) [33,34]. The presence of even minute quantities of CO on the parts per million (ppm) order of magnitude in H 2(g) fuel, results in rapid poisoning of sensitive platinum (Pt) catalysts present in the latest generation of low operating temperature, proton exchange membrane (PEM) fuel cells [35]. Catalysts based on a mixture of platinum and ruthenium (Pt-Ru) developed to overcome the sensitivity of pure Pt to carbon monoxide poisoning are not cost effective for large scale application in motor vehicle transport applications due to the dearth of ruthenium [36]. Since hydrogen production by conventional steam reforming methods generates significant quantities of CO and CO 2 , it becomes difficult to justify using the approach to generate hydrogen (H 2 ) fuel for FCEVs given that the purpose of advancing such technology is to eliminate carbon based fossil fuel emissions.
Despite challenges, hydrogen (H 2 ) which is stored in near limitless quantity in seawater is the only alternative fuel that is more abundant and environmentally cleaner with the potential of having a lower cost than nonrenewable carbon based fossil fuels. We have shown in previous published work that a novel apparatus and method for safely generating hydrogen fuel at the time and point of use from ordinary salinated (sea) or desalinated (fresh) water (H 2 O) will enable a vehicle range exceeding 300 miles per fueling using direct combustion of the H 2 fuel in appropriately configured internal combustion engines of the Otto or Diesel types, which is comparable to the vehicle ranges presently achieved with gasoline or Diesel fuels, while providing a sustainable, closed clean energy cycle [37]. The novel hydrogen generation apparatus enables hydrogen (H 2(g) ) fuel to be produced on demand in the motor vehicle using a controlled chemical reaction where liquid water (H 2 O (l) ) is made to react with solid sodium (Na (s) ) metal reactant to produce hydrogen (H 2(g) ) gas and sodium hydroxide (NaOH (s) ) byproduct according to Eq. (1). 2Na ðsÞ þ 2H 2 O ðlÞ ! H 2ðgÞ þ 2NaOH ðsÞ (1) The high purity hydrogen (H 2(g) ) fuel produced on demand by the novel hydrogen generation apparatus can be used to safely power FCEVs without contaminating the sensitive Pt catalysts present in PEM fuel cells or any other types of catalysts in fuel cells, because the hydrogen is not derived from carbon based fossil fuels, and therefore does not contain even trace amounts of carbon monoxide or sulfur compounds. The seawater reactant can be concentrated to as much as 252.18 grams of sea salt solute per kilogram of seawater solution to provide a fusion temperature T Eu ¼ -21.2 C (251.95 K), that is equivalent to the eutectic temperature of a 23.18% by weight NaCl in NaCl-H 2 O solution [38,39]. The concentrated sea salt in seawater solution allows the hydrogen generator to operate reliably over a wide ambient temperature range from -21.2 C (251.95 K) to 56.7 C (329.85 K) prevailing in the 48 conterminous states of the U.S.A. [37]. The sodium hydroxide (NaOH) byproduct of the hydrogen generating chemical reaction is stored temporarily within the hydrogen generation apparatus and is recovered during motor vehicle refueling. The NaOH is subsequently reprocessed by electrolysis to recover the sodium (Na) metal for reuse in generating hydrogen fuel.
In this chapter, we describe in detail our company's design approach for constructing a novel, scalable, self-contained electrolytic sodium (Na) metal production plant that uses electric power sourced from the sun. The solar powered electrolytic production plant is meant to form an integral part of a hydrogen fuel, sustainable, closed clean energy cycle in conjunction with the novel, hydrogen generation apparatus, enabling Na metal to be produced cost effectively without negative impact to the environment [37].

Sodium metal production plant characteristics
For its successful implementation, the hydrogen fuel, sustainable, closed clean energy cycle requires a means of producing quantities of sodium (Na) metal cost effectively on a large scale by electrolysis of sodium hydroxide (NaOH), the latter created as a byproduct of hydrogen (H 2(g) ) fuel generation inside motor vehicles according to Eq. (1). The electrolysis is performed either on pure sodium hydroxide (NaOH) or on a mixture of NaOH and sea salt, the latter consisting primarily of sodium chloride (NaCl), according to Eqs. (2) and (3) [40][41][42].
The electrical cost of electrolysis can be estimated from the standard reduction potentials of the oxidation and reduction half reactions that occur at the anode and cathode, respectively of the electrolysis cell when implementing Eqs. (2) and (3) [43].
Cathode ðreductionÞ: Na þ ðaqÞ þ e À ! Na ðsÞ E r ¼ À2:71 V Anode ðoxidationÞ: 2Cl À ðaqÞ ! Cl 2ðgÞ þ 2e From Eqs. (4)-(6), the minimum potentials of E ov ¼ -4.07 V and E ov ¼ -3.11 V are needed to electrolyze NaCl and NaOH, respectively. These voltages are significantly higher than the potential of E ov ¼ -1.23 V needed to electrolyze ordinary H 2 O (l) to produce H 2(g) at the cathode and O 2(g) at the anode, however, the benefit from not having to store volatile H 2(g) in very large industrial quantities and to transport it between the production plants and refueling stations, outweighs the added electrical cost of producing the solid Na (s) metal.
In the United States of America, the only clean renewable source of energy available in sufficient abundance to implement Eqs. (2) and (3) on a large scale is the radiant energy from the sun that illuminates vast tracts of flat, arid, desert land in West Texas, New Mexico, Arizona and Southern California. The weather in the southwestern U.S.A. is mostly warm and arid with high solar irradiance all year and therefore, constitutes the ideal location for constructing scalable, self-contained solar powered electrolytic sodium (Na) metal production plant units by the thousands [44][45][46][47][48]. Each sodium (Na) metal production plant has to be capable of operating autonomously as a self-contained factory, requiring minimal maintenance and resources. The diagram showing all of the material and energy inputs and outputs of the self-contained sodium (Na) metal production plant is presented in Figure 1.
In Figure 1, electric power for the Na metal production plant is produced using photovoltaic (PV) device panels spatially arrayed and electrically interconnected on a vertical tower structure that maximizes the use of scarce real estate or land area. Up to N P ¼ 30,000 PV panels, each having an active area A P ¼ 1 m 2 are mechanically assembled onto the tower, yielding a total PV device panel array active area given as A PA ¼ N P · A P ¼ (30,000 PV panels) · (1 m 2 ) ¼ 30,000 m 2 . Sodium hydroxide (NaOH) or a mixture of NaOH and NaCl recovered from motor vehicle hydrogen generators during refueling, must be supplied to the electrolytic cells to replenish the consumed reactants. When operating the hydrogen generation apparatus in warm tropical climates, NaOH exclusively can be recovered from motor vehicle hydrogen generators during refueling because desalinated (fresh) water (H 2 O (l) ) can safely be used as a reactant without risk for it to freeze. Sodium (Na) metal is produced at the electrolytic cell cathode, and steam (H 2 O (g) ), oxygen (O 2(g) ) gas and some chlorine (Cl 2(g) ) gas are produced at the cell anode, the latter resulting from electrolysis of the NaCl in sea salt. The H 2 O (g) and O 2(g) can be released directly to the atmosphere while the Cl 2(g) must be collected, condensed to a liquid and stored in bottles, for subsequent sale to customers that consume chlorine including the paper and polymer (plastic) manufacturing industries [49]. It is also possible to collect and condense the steam generated at the anode and use the liquid water (H 2 O (l) ) for crop irrigation in arid, desert environments where water resources are limited. The layout of the self-contained sodium (Na) metal production plant is shown in Figure 2.
The self-contained sodium (Na) metal production plant shown in Figure 2, consists of a solar tower that comprises a photovoltaic (PV) device panel array active area given as A PA ¼ 30,000 m 2 . It also consists of a prefabricated Quonset or Q-type metal building having a semicircular cross section, assembled onto a concrete pad foundation that houses electrical switch gear, voltage step down DC-DC converters, the sodium hydroxide (NaOH) electrolytic cells, sodium (Na) metal packaging unit and chlorine (Cl 2 ) gas separation and bottling unit. A control room permits monitoring the operation of the plant. Two above ground storage tanks are shown located outside of the Q-type metal building with one on either side, for storing aqueous sodium hydroxide (NaOH (aq) ) solution. Each above ground storage reservoir has a liquid volume capacity of V NaOH(aq) ¼ 37,850 L (10,000 Gal), and is used to replenish the NaOH in the electrolytic cells after the reactant in the cells has been consumed by electrolysis.  Many factors influence the production yield of sodium (Na) metal during a normal day of plant operation. The two most important factors include the power conversion efficiency of the PV device panels and the magnitude and duration of solar irradiance incident on the PV panels. Other factors affecting the Na metal yield include the power conversion efficiency of the voltage step down DC-DC converter and the efficiency of the electrolytic cell in recovering the Na metal from fused NaOH (l) or from a mixture of fused NaOH (l) and NaCl (l) . The efficiency of the voltage step down DC-DC converter depends mainly on how much power is dissipated or lost in the solid state transistors as a result of high frequency on-off switching. The efficiency of the electrolytic cell in this work is viewed in terms of the number of electrons from the electrolytic cell current I CELL , flowing through an electrolytic cell actually needed to produce an atom of Na metal. An alternate definition of the electrolytic cell efficiency might consider the theoretical electric power required to be supplied to the electrolytic cell to produce one mole of Na metal as calculated from the known Gibbs free energy of NaOH (l) , divided by the actual power required to be supplied to the electrolytic cell to produce one mole of the Na metal, a definition that we do not consider here. Ideally, each electron flowing through the electric circuit comprising the electrolytic cell should produce one atom of Na metal. In the analysis that follows, it will be assumed that the voltage step down DC-DC converters and the electrolytic cells have efficiencies η DC-DC ¼ 100% and η CELL ¼ 100%, respectively.

Scalable photovoltaic tower concept
The solar tower comprising the photovoltaic (PV) device panel array supplies electric power to the sodium (Na) metal producing electrolytic cells. The amount of electric energy supplied by the solar tower is a key determinant of the quantity of Na metal that can be electrochemically separated from the NaOH reactant. The energy conversion efficiency of the photovoltaic (PV) device panels is therefore a critical parameter for determining the amount of Na metal that can be produced by the self-contained sodium (Na) metal production plant. Ideally, the photovoltaic (PV) device panels of the solar tower should have maximum optical to electric energy conversion efficiency approaching the thermodynamic limit η PVmax ¼ 93% [50,51]. According to the Shockley-Queisser theory, such a high conversion efficiency requires PV devices comprising manifold semiconductor junctions [52,50]. Contemporary, commercially available single junction, monocrystalline silicon photovoltaic (PV) devices attain energy conversion efficiencies ranging between η PV ¼ 15-18% for front-illuminated silicon devices and up to η PV ¼ 21.5% for backilluminated silicon devices as summarized in Table 1.
In Table 1, the PV device panels from the Sunpower manufacturing company stand out as having the highest efficiency, due to the use of back-illumination of the silicon device layer. Multijunction devices that offer higher efficiencies remain at a research and development stage and have not reached a level of maturity or cost effectiveness to be ready for release as commercial PV panel products [53]. Our company, AG STERN, LLC is researching the development of advanced, high efficiency PV devices based on novel, very high transmittance, back-illuminated, silicon-on-sapphire semiconductor substrates expected capable of transmitting 93.7% of the total solar irradiance into the semiconductor device layer and therefore capable of achieving an energy conversion efficiency η PV ¼ 90% with proper engineering of the semiconductor photon absorbing layers and PV device structure [54][55][56][57].
The solar tower providing power to the electrolytic cells must be scalable in electric power output, and thus capable of allowing PV device panels to be replaced as higher efficiency ones become available, without affecting the overall operation of the self-contained Na metal production plant in any way, other than increasing the Na metal yield. Since the highest performing PV device panels currently offered commercially are listed in Table 1, it is possible to calculate the expected power output of the solar tower comprising a PV device panel array active area of A PA ¼ 30,000 m 2 , under ASTM direct normal air mass (AM) 1.5D standard terrestrial solar spectral irradiance with a total irradiance Irr AM1.5D ¼ 887 W/m 2 , as a function of the PV device efficiency, as shown in Figure 3.  It is clear in Figure 3, that single junction, monocrystalline silicon PV devices having an efficiency η PV ¼ 21.5% can be arrayed to generate electric power given as P ST ¼ 5.72 MW for electrolysis. Once η PV ¼ 90% efficient PV device panels will become available, the electric power output can be increased to a substantial P ST ¼ 23.9 MW, entailing that just 50 of the self-contained sodium (Na) metal production plants can generate peak power given as P ST-50 ¼ 50 · 23.9 MW ¼ 1195 MW, matching the power generating capacity of a large commercial nuclear power station.

Solar irradiance conditions
The electric power output of the solar tower comprising photovoltaic (PV) device panels depends on the magnitude and duration of solar irradiance incident on the PV panels, in addition to the PV device energy conversion efficiency described in Section 2.1. It can be assumed that the PV device energy conversion efficiency is constant and might only decrease in value slowly over time [58]. In contrast, the solar irradiance incident on the PV device panels can vary on a daily basis and is determined by two principal factors, namely, the solar geometry and prevailing atmospheric or meteorological conditions.
The sun is effectively a large hydrogen (H 2 ) fusion reactor, spherical in shape with a radius R sun ¼ 6.96 · 10 8 m located at a mean distance from the earth given as r sun-m ¼ 1.496 · 10 11 m [59,60]. The total power output of the sun is given as P sun ¼ 3.8 · 10 26 W that radiates isotropically in all directions, resulting in a surface temperature of the solar black body T sun ¼ 5800 K. Of the total solar power output, earth receives only P earth ¼ 1.7 · 10 17 W which if converted to electric power, vastly exceeds the earth's energy needs [61]. The sun therefore constitutes an excellent potential source of clean radiant energy to be harnessed on earth.
The solar geometry has a key role in determining how much radiant energy from the sun will be incident on the PV devices located on earth. To understand how the solar geometry influences the solar irradiance at the earth's surface, it will be assumed that earth follows a stable elliptical orbit around the sun described by Eq. (7) [59].
Perihelion: r sun ð0 Þ ¼ að1 À eÞ In Eq. (7), the distance r sun , represents the distance from the center of the sun to the center of the earth and the angle θ, represents the angle between the present position of the earth in orbit around the sun, and the perihelion position when it is closest to the sun. The eccentricity e ¼ 0.01673, describes the shape of the elliptical orbit of the earth around the sun and the value a, represents the length of the semi-major axis of the orbit defined as the mean distance from the center of the sun to the center of the earth given as a ¼ {r sun (0 ) þ r sun (180 )} / 2 ¼ r sun-m ¼ 1.496 · 10 11 m. The period of earth's elliptical rotation around the sun is approximately T es ¼ 365.24 days and the period of rotation around its own axis is approximately T ea ¼ 86,400 seconds or 24 hours, the latter known as a mean solar day [62][63][64][65][66]. The earth's angular velocity about its own axis is given as ω ea ¼ 7.292115 · 10 À5 rad/sec, although the rotation is slowing over time [67]. The obliquity or tilt of the earth's axis of rotation with respect to a line perpendicular to the plane of its elliptical orbit around the sun is given as ε ¼ The solar declination angle δ, is the angle made by a ray of the sun (passing through the center of the earth and the center of the sun), and the equatorial plane of the earth. The solar declination angle has a range given as -23.44 ≤ δ ≤ þ23.44 . On the summer solstice day when δ ¼ þ23.44 , the sun shines most directly on the earth's latitude ϕ T-CAN ¼ þ23.44 , known as the Tropic of Cancer. On the winter solstice day when δ ¼ -23.44 , the sun shines most directly on the earth's latitude ϕ T-CAP ¼ -23.44 , known as the Tropic of Capricorn. The longitude at the Greenwich prime meridian λ PM ¼ 0 [68]. The solar declination angle can be calculated using a geocentric reference frame with the sun located on a celestial sphere and earth located at the center of the celestial sphere, according to Eq. (8) [60].
In Eq. (8), λ e represents the ecliptic longitude of the sun, that is the sun's angular position along its apparent orbit in the plane of the ecliptic. The plane of the ecliptic is tilted by the obliquity of the ecliptic angle ε ¼ 23.44 representing the angle between the plane of the ecliptic and the plane of the celestial equator, the latter being earth's equator projected onto the celestial sphere. The ecliptic intersects the equatorial plane at the points corresponding to the spring and fall equinoxes occurring approximately on March 21 and September 21, respectively with λ e ¼ 0 at the March 21 equinox. The Eq. (8) is usually formulated in terms of the day number in a year to account approximately for the elliptic nature of earth's orbit around the sun, as given by Eq. (9) [69].
δ ¼½0:006918 À 0:399912cosðΓÞ þ 0:070257sinðΓÞ À 0:006758cosð2ΓÞ þ 0:000907sinð2ΓÞ À 0:002697cosð3ΓÞ þ 0:00148sinð3ΓÞð180=πÞ In Eq. (9), the solar declination angle δ, is given as a function of the day angle Γ ¼ 2π(N day À 1) / 365, with day number N day , in a year where N day ¼ 1 corresponds to January 1 st . The Eq. (9) provides sufficient accuracy due to the relatively small eccentricity value e in Eq. (7). The Eq. (9) shows that the solar illumination of the earth's surface throughout the year occurs most directly between tropical latitudes, namely, the Tropic of Cancer (δ ≈ þ23.44 ) and the Tropic of Capricorn (δ ≈ À23.44 ). At the time of the vernal and autumnal equinoxes, the earth's equator is located in the plane of the sun and every location on earth receives 12 hours of sunlight.
The PV devices can only generate significant electric power during daylight hours therefore, it is essential to verify that a specific geographic location on earth as defined by the latitude ϕ, and longitude λ, will receive sufficient hours of high incident solar irradiance during the year, before constructing the self-contained sodium (Na) metal production plant. To accurately describe the sun's position in the sky from sunrise to sunset as well as the resulting number of daylight hours, the apparent solar time (AST) is used to express the time of the day. The sun crosses the meridian of the observer at the local solar noon when AST ¼ 12, however, the latter time does not coincide exactly with the 12:00 noon time of the locality of the observer. A conversion between the local standard time (LST) and the AST can be made by using equation of time (ET) and longitude corrections. The equation of time accounts for the length of the day variation as a result of the slight eccentricity of the earth's orbit around the sun as well as the tilt of the earth's axis of rotation with respect to a line perpendicular to the plane of its elliptical orbit around the sun. The ET is given by Eq. (10) and has the dimension of minutes [69,70].
In Eq. (11), the local longitude λ L , represents the exact longitude value for the location relative to the Greenwich prime meridian of λ PM ¼ 0 . The standard longitude λ S , is calculated from λ L as λ S ¼ 15 · (λ L / 15 ) where the quantity in the parentheses is rounded to an integer value. In Eq. (11), if λ L is east of λ S then the longitude correction is positive and if λ L is west of λ S then the longitude correction is negative, while DST ¼ 0 minutes or 60 minutes depending on whether daylight savings time applies. If in effect, DST occurs from March until November.
The hour angle h, represents the angular distance in degrees between the longitude λ, of the observer and the longitude whose plane contains the sun, and is calculated according to Eq. (12) [60].
In Eq. (12), the multiplier of 15 arises because the earth rotates around its own axis 15 in 1 hour, and the AST has a value 0 < AST < 24 hours. When the sun reaches its maximum angle of elevation at the longitude of the observer, it corresponds to the local solar noon time where h ¼ 0 . The solar elevation angle α s , above the observer's horizon and its complement, the solar zenith angle θ sz , can be calculated according to Eq. (13) [60].
The solar azimuth angle γ s , in the horizontal plane of the observer is given by Eq. (14) [60].
The Eq. (14) uses the convention of the solar azimuth angle defined as positive clockwise from north meaning that east corresponds to 90 , south corresponds to 180 and west corresponds to 270 . The solar azimuth angle provided by Eq. (14) should be interpreted as 0 ≤ γ s ≤ 180 when h < 0 , meaning the sun is located east of the observer, and interpreted as 180 ≤ γ s ≤ 360 when h > 0 , meaning the sun is located west of the observer. The sunrise equation allows to calculate the number of hours of sunlight per day that depends on the solar declination angle δ, and on latitude ϕ, as shown in Eq. (15), which is derived from Eq. (13) by setting α s ¼ 0 and solving for h.
In Eq. (15), the hour angle h, has a value À180 < h < 0 at sunrise and 0 < h < þ180 at sunset, and when divided by 15 /hour, yields the number of hours before and after the local solar noon time (AST ¼ 12) when h ¼ 0 , corresponding to sunrise H sr , and sunset H ss , respectively. Using Eqs. (9) and (15), it is possible to calculate for any geographic location on earth the number of daylight hours on any specific day of the year. The maximum angle of solar elevation α s-max , above the observer's horizon that occurs at the local solar noon time is calculated according to Eqs. (16) and (17) that are derived from Eq. (13) by setting h ¼ 0 .
The Eq. (16) is applicable for the northern hemisphere while Eq. (17) is applicable for the southern hemisphere. Formulations for the solar declination, elevation, azimuth and zenith angles which account with greater accuracy for the elliptic nature of earth's orbit around the sun exist in the scientific literature often as part of solar position algorithms however, they can be rather complex [71][72][73][74][75][76]. The air mass can be calculated as a function of the true solar zenith angle θ sz , given by Eq. (13), considering the effect of atmospheric refraction, according to Eq. (18) [77].
In Eq. (19), Irr 0 ¼ 1367 W/m 2 , represents the solar constant or total irradiance of the AM 0 standard solar spectral irradiance in space prior to attenuation by earth's atmosphere and the factor 0.9751 is included because the spectral interval of 0.3-3.0 μm is used by the detailed SOLTRAN model from which the Parameterization Model C is derived. The factor E 0 , represents the effect of the eccentricity of earth's orbit around the sun on the solar constant Irr 0 , due to the periodically varying distance between the earth and sun, as given by Eq. (20) [69].
In Eqs. (21), (23) and (25), AM a represents the air mass at the actual atmospheric pressure P and is given as AM a ¼ AM · (P / P 0 ) where AM, calculated using Eq. (18), corresponds to standard atmospheric pressure P 0 ¼ 101325 Pa. The actual atmospheric pressure P, can be calculated using the isothermal atmosphere formula in Eq. (26).
The Eq. (26) can be used to calculate the atmospheric pressure P in pascals (Pa) at an elevation h PV , in meters (m) above mean sea level, assuming a constant ambient atmospheric temperature T in Kelvin (K), over the difference in elevations. The molar mass of air M air ¼ 0.028964 kg/mol, earth's gravitational acceleration near the surface g 0 ¼ 9.80665 m/sec 2 , and the universal gas constant R g ¼ 8.3144621 J/K•mol [43,81,82]. In Eq. (22), where l o represents the vertical ozone layer thickness in centimeters (cm) at normal temperature and surface pressure (NTP), and AM is given by Eq. (18). The vertical ozone layer thickness is assumed in the present work to have a mean annual value l o ¼ 0.35 cm (NTP). In Eq. (24), U 1 ¼ w 0 ðP=101325Þ 0:75 ð273=TÞ 0:5 · AM, where w' represents the precipitable water vapor thickness in centimeters (cm) under the actual atmospheric conditions with pressure P in pascals (Pa), and temperature T in Kelvin (K). In Eq. (25), k a represents the aerosol optical thickness and has a range 0 < k a < 1, depending on how clear or aerosol saturated the atmosphere is. The value of k a is generally measured at the optical wavelengths of 380 nm and 500 nm due to low ozone absorption and calculated using the formula, k a ¼ 0:2758 Á k ajλ¼380 nm þ 0:35 Á k ajλ¼500 nm . The aerosol optical thickness can vary daily, however, it is assumed in the present work to have a mean annual value k a ¼ 0.1 reflecting clear days.
In Table 2, geographic and climate characteristics are specified for four prospective locations of the scalable, self-contained solar powered electrolytic sodium (Na) metal production plant including El Paso, Texas; Alice Springs, Australia; Bangkok, Thailand and Mbandaka, Democratic Republic of Congo (DRC).
In Table 2, the geographic coordinates use the sign convention of þ/latitude ϕ, for locations north and south of the equator, respectively, and þ/longitude λ, for locations east and west of the prime meridian (λ PM ¼ 0 ). According to the Köppen-Geiger climate classification system, El Paso has an arid, desert, cold (BWk) climate, Alice Springs has an arid, desert, hot (BWh) climate, Bangkok has a tropical, savanna (Aw) climate and Mbandaka has a tropical, rainforest (Af) climate [88]. The mean annual local solar noon air mass values in Table 2 are calculated using Eqs. (9) and (16) [84][85][86][87]. vapor levels with a long rainy season and sun obscuring rain clouds. Other factors that can significantly affect incident solar irradiance on the PV device panels include the geographic elevation above mean sea level and aerosols comprising solid and liquid sunlight obscuring particulates in the air that can occur naturally due to dust storms and volcanic activity or from human activity such as slash and burn agriculture [89]. According to the accumulated world meteorological data, the southwestern region of the U.S.A. and the central region of Australia, both constitute nearly ideal locations for the self-contained sodium (Na) metal production plants due to the existence of an arid, desert climate, vast tracts of flat open land, sparse human population and many clear days with high solar irradiance throughout the year [47,48,90,91].

Sodium metal production plant architecture
The architecture of the self-contained sodium (Na) metal production plant has to provide an optimal balance between high performance, reliability and cost effective operation. The highest performance can be achieved by constructing the solar tower of the plant shown in Figure 2, in a manner that allows the optical k-vectors from the sun to be normally incident onto the photovoltaic (PV) device panels throughout the entire period of daylight from sunrise to sunset. The Figure 4 shows a solar tower architecture with fixed PV device panels, wherein the panels can rotate and tilt with the solar tower as a single unit to follow the sun's overhead trajectory.
In Figure 4, the solar tower is fabricated using modular sections comprised of high strength, lightweight aluminum alloy that can be fitted end to end and bolted together until the final slant height of the structure S h ¼ 300 m is achieved. Up to N B-L/R ¼ 100 branches that constitute levels, comprised of modular sections each having a final length B l ¼ 50 m, extend out on each side from the central column of the solar tower resulting in a width for the structure of approximately S w ¼ 100 m. The PV device panels are mounted along the top and bottom of the branches projecting from the central column of the solar tower. The projecting branches of the solar tower also enable ground glass, light diffusing panels to be placed between the rows of solar panels that allow sunlight to penetrate and effectively illuminate the land area beneath the solar tower to support crop cultivation including rice paddy fields. The central column of the solar tower is fixed at the base to a static pylon and approximately midway up the height of the column, to a boom for elevating the solar tower to a tower elevation angle π/2 -α s , that is the complement of the solar elevation angle α s , the latter given in Eq. (13). The opposite end of the boom is fixed to the center of a railroad flatcar mounted on a linear train track that is capable of being displaced along the track to elevate and lower the solar tower. An electric motor module mounted midway up the height of the column enables the PV device panels to collectively rotate left and right about the axis of the central column to track the solar azimuth angle γ s , the latter given in Eq. (14), thereby allowing the optical k-vectors from the sun to impinge at normal incidence onto the PV device panels throughout most of the day of operation. When the sun has set at the end of the day or when inclement weather of sufficient severity is expected, the solar tower can be lowered to be parallel and nearly flush with the ground as shown in Figure 4, with metal louvers drawn over the PV device panels to safeguard against damage to the panels from storms, strong winds and flying debris that can occur, albeit rarely, in the southwestern U.S.A. [92,93].

Electrical design of the solar tower
The electrical design of the solar tower comprising PV device panels has to accommodate scalability in the power output level, where it is possible to supplant the existing PV device panels with newer and more efficient ones when they become available, without having to modify other components in the solar tower. It is therefore necessary to optimally dimension the electrical conductors embedded within the branches and central column that transmit the electric power generated by the photovoltaic (PV) device panels to the electrolytic cells, according to the magnitude of the current expected to be transmitted once η PV ¼ 90% efficient PV device panels become available to be installed on the branches of the tower as shown in Figure 4. To develop an accurate design for the aluminum current carrying conductors of the solar tower, it becomes necessary to define the electrical interconnection topology of the photovoltaic (PV) device panels installed on the solar tower. In Figure 5, the equivalent circuit model of the photovoltaic (PV) device panel array installed on the solar tower is shown, with the corresponding straight line approximation of the current versus voltage curve for the PV device panel array.
In the circuit models shown in Figure 5, I PV represents the PV device current, V OC represents the PV device open circuit voltage, R P represents the parallel resistance of the PV device that ideally  should be infinite, R S represents the series resistance of the PV device that ideally should be zero in value, I ST represents the output current of the solar tower and V ST represents the output voltage of the solar tower. The circuit models shown in Figure 5 are applicable for a single PV cell as well as for an array of PV device panels comprised of PV cells connected in series and/or in parallel [94]. The expressions for the Thevenin equivalent circuit voltage V TH and resistance R TH , are given both for the linear equivalent current source and voltage source circuits. The electrical interconnection topology of the PV device panels on the solar tower must achieve an optimal balance in operating parameters including the solar tower supply system maximum voltage  Figure 5. The solar tower must be capable of transmitting P ST ¼ 23.9 MW of electric power as indicated in Figure 3, from the photovoltaic (PV) device panels to the electrolytic cells of the scalable, sodium (Na) metal production plant while maintaining reasonable design values for V ST-MAX and I ST-MAX that will not escalate the cost of the system. To achieve an optimal balance between the V ST-MAX and I ST-MAX parameters of the solar tower supply system, it is necessary to identify the most reliable and cost effective approach for maintaining the PV devices at or near their maximum power point (MPP) during operation. Many methods exist for providing a variable or dynamic load to PV devices and performing maximum power point tracking (MPPT), however, for the present application it is necessary to consider that since there are up to N P ¼ 30,000 PV device panels installed on the solar tower, it becomes uneconomical to use any approach that requires dedicated MPPT hardware for so many panels individually or even for small groups of panels and therefore, a collective solution is required for the entire PV In practice, the MPP output voltage of a single junction, monocrystalline silicon PV cell is given as V MPP-C ≈ 0.4 V which is too low for direct input to a PWM DC-DC converter [95]. Commercial single junction, front-illuminated, monocrystalline silicon PV panels of the type shown in Table 1 [98]. A low voltage DC supply system entails a minor risk of electric arcing through the air and an exemption from specialized protection equipment needed for high voltage. If 15 of the 96 cell module, back-illuminated SPR-X21-345 Sunpower PV panels are connected together in series, then the solar tower supply system maximum voltage V ST-MAX ≈ 900 VDC, which exceeds by 150 V the nominal 750 VDC IEC standard of a low voltage supply. Once η PV ¼ 90% efficient PV device panels will become available, it is expected that they will each have an area A P ¼ 1 m 2 and the number of PV cell modules connected together in series will yield a maximum power point voltage for the panel given as V MPP-P ≈ 30 V that is similar in value to most of the commercial single junction, front-illuminated, monocrystalline silicon PV panels listed in Table 1, that contain 60 PV cell modules connected in series. Under an ASTM AM 1.5G standard terrestrial solar spectral irradiance with a total irradiance Irr AM1.5G ¼ 1000 W/m 2 , the corresponding maximum power point current of the η PV ¼ 90% efficient PV panel can be calculated as I MPP-P ¼ (η PV · Irr AM1.5G ) / V MPP-P ¼ 30.0 A, which is a substantially larger current than the I MPP-P current values listed in Table 1, as might be expected for the more efficient PV panel unit. Therefore, it is necessary to design the current conductors within the scalable solar tower to be capable of transmitting the substantially larger current of more efficient PV panels once they will become available for installation onto the solar tower. The solar tower supply system maximum voltage can be set to V ST-MAX ¼ 450 V, achieved by connecting in series 15 PV device panels with an efficiency η PV ¼ 90% and V MPP-P ≈ 30 V, mounted on a single branch section of the solar tower apparatus as shown in Figure 6.
In Figure 6, each branch section supports the installation of 2 groups of 15 series connected PV panels having an efficiency η PV ¼ 90% for a total N P-Bsec ¼ 30 PV panels per branch section. A total of 5 branch sections should be fastened together end to end, to yield N P-B ¼ 5 · N P-Bsec ¼ 150 PV panels per branch with 5 groups of 15 series connected PV panels installed along the top of the branch and another 5 groups of 15 series connected PV panels installed along the bottom of the branch. Each branch section has a length Bsec l ¼ 10 m, and contains embedded within the aluminum current conductors, yielding a branch length B l ¼ 50 m as shown in Figure 4. The groups of 15 series connected PV panels are electrically connected to a pair of aluminum conductors inside the branch, resulting in parallel electrical interconnection for the groups of 15 series connected PV panels that increases the electric current delivered by the solar tower. Each PV panel has a width P w ¼ 0.66 m and a height P l ¼ 1.5 m for a total PV panel area given as A P ¼ P w · P l ¼ 1 m 2 . Since the height of each PV panel is given as P l ¼ 1.5 m, then each branch has an overall height given as B h ¼ 2 · P l ¼ 3 m. The solar tower that has a height S h ¼ 300 m as shown in Figure 4, can therefore accommodate up to 100 branches extending out from the left and right sides of the central column, yielding a total PV panel array area on the solar tower given as The electric current from each branch section flows into the electrical conductors installed inside the central column of the solar tower as shown in Figure 6. Therefore, electric current that flows from the outermost branch section toward the central column increases as each branch section contributes additional current generated by the 30 PV panels mounted on it. If an ASTM direct normal AM 1.5D standard terrestrial solar spectral irradiance with a total irradiance Irr AM1.5D ¼ 887 W/m 2 , is incident on the PV panels having an efficiency η PV ¼ 90% and V MPP-P ¼ 30 V, then the maximum power point output current can be calculated as I MPP-P ¼ (η PV · Irr AM1.5D ) / (V MPP-P ) ¼ 26.6 A. The result for I MPP-P ¼ 26.6 A, entails that each branch section will contribute I Bsec ¼ 2 · I MPP-P ¼ 53.2 A. Therefore, the current in each branch increases by I Bsec ¼ 53.   bottom most modular section of the central column have to be dimensioned to transmit the maximum current. The diameters of aluminum electrical conductors within the central column sections scale linearly from a diameter D S1 ¼ 2 cm in stage 1 at the top to D S100 ¼ 41.6 cm for stage 100 at the bottom of the solar tower, the latter that transmits the maximum current I ST-MAX ¼ 26,600 A. Figure 6, shows the calculated voltage drop across the aluminum electrical conductors in all 100 modular sections of the central column of the solar tower to be given as V ST-DROP ¼ 3.86 V, corresponding to a low loss considering that the solar tower supply system maximum voltage V ST-MAX ¼ 450 V.
The electrical design of the solar tower described provides manifold advantages including a mostly parallel electrical interconnection architecture for the PV device panels that allows the MPP of the PV device panels to be controlled collectively by controlling the output voltage V ST of the PV device panel array using a voltage step down PWM DC-DC converter designed to have a constant output voltage V OUT , and controllable input voltage V IN , the latter supplied from the PV device panel array and equal to V ST , shown in Figure 5. Other advantages include a low voltage DC solar tower supply system with optimal balance between the maximum voltage V ST-MAX ≈ 450 V and maximum current I ST-MAX ¼ 26,600 A, wherein two independent and electrically isolated power supply feeds are provided from the left side branches and right side branches of the solar tower apparatus, respectively. Yet another important advantage of the design includes a low center of gravity for the solar tower apparatus due to the modular sections comprising the central column being heavier near the base and lighter near the top as a result of larger diameter aluminum electrical conductors placed near the base of the tower.

Electrical design of the sodium hydroxide electrolysis plant
The solar tower apparatus described in Section 3.1 implements a mostly parallel electrical interconnection architecture for the PV device panels that yields two identical, independent and electrically isolated low voltage DC power supplies, each having a maximum voltage V ST-MAX ≈ 450 V and maximum current I ST-MAX ¼ 26,600 A, wherein the two power supply feeds emanate from the left side branches and right side branches of the solar tower apparatus, respectively as shown in Figure 6. The sodium hydroxide (NaOH) electrolysis cell however, requires a substantially lower voltage for operation. The quantity of sodium (Na) metal produced by the NaOH electrolytic cell depends fundamentally on the magnitude of the DC current flowing between the anode and cathode terminals of the electrolysis cell. The Eqs. (4)-(6) provide the standard reduction potentials of the oxidation and reduction half reactions that occur at the anode and cathode, respectively of the electrolytic cell. In practice, the electrolytic cell operating voltage has to be set at approximately V CELL ¼ 4 V for electrolysis of pure NaOH or V CELL ¼ 5 V for electrolysis of a mixture of NaOH and NaCl, the latter derived from sea salt. The higher voltage accounts for the overvoltage effects [99]. The maximum current that can be supplied to an electrolytic cell operating at approximately V CELL ≈ 5 V will be I CELL ≈ 100,000 A and is limited in large measure by the material cost as well as by the mass and physical dimensions of the electrical conductors required to transmit such magnitude of the current.
In a fused or molten state, NaOH (l) is highly corrosive and therefore the only conventional materials capable of withstanding prolonged exposure to its caustic effects at an elevated temperature include graphite, iron and nickel. Graphite however, cannot be used as an anode electrode because it will react with the oxygen (O 2 ) generated to produce carbon dioxide (CO 2 ) and become consumed in the process. Iron can withstand corrosion from fused NaOH (l) , and consequently could be used to fabricate the electrolytic vessel however, as an anode electrode, the reaction with steam (H 2 O (g) ) and O 2 will quickly oxidize and erode iron. The only material suitable for fabricating both the anode and cathode electrodes remains nickel (Ni) which is significantly more expensive than both graphite and iron. The cost of the Ni electrodes therefore becomes an important factor in limiting the maximum current in the electrolytic cell. The other factor limiting the current in the electrolytic cell becomes the PWM DC-DC converter that must supply the large currents at the correct output voltage to the electrolytic cell, safely and reliably. The design of voltage step down PWM DC-DC converters that have a fixed output voltage V OUT and control the input voltage V IN that is variable, has been an active topic of research in the context of photovoltaic power systems applications [94,95]. The requirements of the present application however, entail a specialized type of large scale direct photovoltaic power conversion for chemical electrolysis not hitherto contemplated or addressed in the scientific/ industrial literature. The design for the voltage step down PWM DC-DC converter with a fixed output voltage V OUT and variable input voltage V IN meant to supply the NaOH electrolytic cells is shown in Figure 7 to consist of a multiphase converter topology, with synchronous voltage step down converter circuits connected in parallel.
The synchronous parallel multiphase voltage step down PWM DC-DC converter power supply shown in Figure 7 offers the inherent advantage of allowing the large load current I OUT ¼ I CELL ¼ 96,500 A to be split among the phases of the converter to match the current carrying capacity of the individual electronic switches, that in practice would consist of power insulated gate bipolar transistors (IGBTs) such as the Model 1MBI3600U4D-120, manufactured by the Fuji Electric Company with a maximum rated collector-emitter voltage and collector-emitter current given as V CE ¼ 1,200 V (at T C ¼ 25 C) and I CE ¼ 3,600 A (at T C ¼ 80 C), respectively. It is possible to use 64 such IGBT devices in up to N φ ¼ 32 phases of the multiphase PWM DC-DC converter with 2 IGBT devices per phase as shown in Figure 7, to deliver the required current I OUT ¼ I CELL ¼ 96,500 A to the 25 series connected NaOH electrolytic cells indicated as having resistances R C1 , R C2 , …, R C25 , without exceeding the electrical ratings of the solid state switches. Using a multiphase converter topology with synchronous voltage step down converter circuits connected in parallel, also allows the current I CELL ¼ 96,500 A to be split among the 32 inductors L 1 -L 32 , present in each of the N φ ¼ 32 phases, thereby not requiring one single large inductor to transmit the full current I OUT , supplied by the PWM DC-DC converter to the load comprising the electrolytic cells. Current sharing occurs between the synchronous voltage step down converter circuits connected in parallel according to Eq. (30).
Assuming the ideal case that current sharing between synchronous parallel voltage step down converter circuits is equal, then  Figure 7, that also corresponds to the output voltage V ST , of the solar tower PV device array has to be controlled reliably to ensure that the PV device panel array operates at or near its maximum power point (MPP).
The synchronous parallel multiphase voltage step down PWM DC-DC converter power supply in Figure 7 with N φ ¼ 32 phases, constitutes a complicated nonlinear dynamical system that can be challenging to model, control and analyze accurately to ensure stable operation under wide ranging conditions [100]. The closed loop control system using voltage control and a single feedback loop that can be applied to the present converter is shown in Figure 8.
The control system for the multiphase voltage step down PWM DC-DC converter shown in Figure 8 consists of a sensor that senses the process parameter to be controlled, namely, the input voltage V IN to the PWM DC-DC converter that is also the output voltage V ST , of the solar tower PV device panel array. The error amplifier compares the scaled process parameter V IN to the set point value V SET and computes the difference as an error signal V ERROR . A proportional, integrator, differentiator (PID) circuit processes the error signal V ERROR , and accordingly generates a control signal V CONTROL , that is supplied to a pulse width modulation (PWM) signal generation circuit to produce the control signals that have the correct frequency, duty cycle and phase shift. The PWM signal to the IGBT switches has to be replicated N φ ¼ 32 times and phase  shifted by φ Shift ¼ 360 / N φ ¼ 11.25 using a driver circuit to control all the synchronous voltage step down converters connected in parallel in Figure 7. In Figure 9, the electronic circuit schematics for some of the different control blocks in Figure 8 are shown.
The electronic circuits shown in Figure 9 are representative of the functional blocks of the PWM DC-DC converter control system. The sensor circuit can consist of a resistive voltage divider with a unity gain op-amp buffer that senses the voltage V IN , at the input of the PWM DC-DC converter according to Eq. (31).
The error amplifier subtracts the input voltage V SEN , scaled by the resistive voltage divider of the sensor circuit, from the set point reference voltage V SET , to calculate an error voltage V ERROR , according to Eq. (32).
The PID unit receives the error voltage V ERROR , as an input and produces a control voltage V CONTROL , given as Eq. (33).
In Eq. (33), the terms G P , G I and G D represent the gains of the proportional, integrator and differentiator circuits, respectively that can be tuned as needed to achieve optimal control characteristics for the PWM DC-DC converter. There exist myriad other ways of controlling the PWM DC-DC converter using only proportional (P) and integrator (I) control functions without the differentiator (D) circuit for example, however, the control system shown in Figures 8 and 9, represents a robust and reliable type of control of the PWM DC-DC converter. The PWM unit is shown to consist of a fixed frequency square wave voltage oscillator with an op-amp integrator circuit that converts the square wave into a triangular wave ramp signal which is then compared to the control voltage V CONTROL , from the PID circuit using a comparator, to generate the PWM signal for the IGBT electronic switches of the PWM DC-DC converter. The driver circuit that generates the N φ ¼ 32 copies of the PWM signal from Figure 8, with each signal and its complement phase shifted by φ Shift ¼ 11.25 for all 64 IGBT switches in the converter is not shown in Figure 9. It could be implemented however, using all digital logic. The control system allows the PWM DC-DC converter to be effectively controlled by setting just one parameter, namely, the reference voltage V SET , to a value that corresponds with the maximum power point (MPP) output of the solar tower PV device panel array to transmit the maximum power available from the PV devices to the NaOH electrolytic cells.
It is possible to gain insight into the operation of the synchronous parallel multiphase voltage step down PWM DC-DC converter with attached solar tower PV device panel array, from the most common modeling approach using small signal analysis based on state space averaging [101]. The open loop transfer function G V (s), of the voltage step down PWM DC-DC converter that yields the small signal response of the input voltage process parameter V IN ¼ V ST to the duty cycle control variable d, can be calculated straightforwardly by making the simplifying assumption, namely, that the power supply has a single phase (N φ ¼ 1) rather than N φ ¼ 32 phases in parallel as depicted in Figure 7. The voltage step down PWM DC-DC converter from Figure 7 with only a single phase φ 1 , can be characterized by two pairs of differential equations that describe the current I O1 , flowing through the inductor L 1 , and the voltage V IN , across the input capacitor C 1 , during the two distinct states of the converter when the switch S H1 is closed and open. The pair of differential equations corresponding to the switch S H1 being closed are given by Eqs. (34) and (35).
The pair of differential equations corresponding to the switch S H1 being open are given by Eqs. (36) and (37).
The state space form of the above four differential equations describing the PWM DC-DC converter having just a single phase φ 1 , is given as Eqs. (38) and (39) The Eq. (38), has the form _ The averaged state space equation can be expressed in a similar form _ x ¼ Ax þ Bu, where the A and B matrices are calculated based on the duty cycle d, of the switch S H1 , the matrices A 1 , B 1 , A 2 , B 2 and are given as The vector x, in the averaged state space equation contains the average values of the state variables <I O1 > and <V IN > and the vector u, contains the input variables V OUT and V TH that are considered as DC values.
When all transients in the PWM DC-DC converter have stabilized and steady state operation is achieved, then _ x ¼ 0, thus the averaged state space equation can be used to express the DC steady state equations given as Eqs. (40) and (41).
Calculating out Eq. (41) yields the DC value results given in Eqs. (42) and (43) for I O1 and V IN .
The result in Eq. .
In Eq. (44), the vector x ¼ X þx and vector u ¼ U þû, where each vector comprises the DC value and a small signal perturbation. The Eq. (44) can be expanded as shown in Eq. (45).
Taking the Laplace transform of the averaged state space equation which has the form _ x ¼ Ax þ Bu, yields the result in Eq. (47).
The open loop transfer function given as G V ðsÞ ¼xðsÞ=d 0 ðsÞ of the PWM DC-DC converter can be obtained by calculating out Eq. (50) to yield the result in Eq. (51).
The analysis to yield the open loop transfer function for the multiphase voltage step down PWM DC-DC converter with N φ ¼ 32 phases requires creating a map that contains all the state space equations describing all possible states of the 64 IGBT switches during a period of duration T P , corresponding to a cycle of operation of the PWM DC-DC converter. For N φ ¼ 32 phases, there will be in effect 32 switching events occurring per cycle of operation with a phase shift φ Shift ¼ 11.25 . Thus, there will exist 32 state space equations of the form given in Eq. (52).
In Eq. (52), the matrices A i and B i describe the PWM DC-DC converter in the subinterval of duration t i between switchings of the parallel phases where P Nφ i¼1 t i ¼ T P . The solutions from Eq. (52) can be stacked to yield a discrete time equation valid over an entire period T P that effectively represents a map for the multiphase PWM DC-DC converter [100]. The state space averaging method used to obtain the open loop transfer function of the PWM DC-DC converter with a single phase given in Eq. (51), can also be applied to calculate the open loop transfer function for the PWM DC-DC converter with N φ ¼ 32 phases in parallel shown in Figure 7 [101]. An analysis of the present N φ ¼ 32 phase, closed loop multiphase PWM DC-DC converter including stability analysis, is highly complex especially when considering parametric variations among the parallel voltage step down converter circuits, and will be described in future work. It is also possible, to operate the multiphase PWM DC-DC converter in an open loop control mode, where a plant operator monitors and controls the electrical resistance of the electrolytic cells, and manually adjusts the duty cycle of the PWM DC-DC converter to deliver maximum power to the electrolytic cells. The plant operator balances the currents I O1 -I O32 , flowing in each phase of the PWM DC-DC converter by manually adjusting the amplitudes of the PWM control signals supplied to the gate terminals of the IGBT switches.

Sodium metal production plant operating characteristics
The solar cycle described in Section 2.2, the electrical design of the solar tower described in Section 3.1, and the electrical design of the sodium hydroxide (NaOH) electrolysis plant described in Section 3.2, entail that the voltage step down pulse width modulated (PWM) DC-DC converter supplying electricity from the solar tower PV device panel array directly to the NaOH electrolytic cells has to be operated according to a precise protocol.
Prior to sunrise occurring, the sodium hydroxide (NaOH) electrolytic cells are replenished to capacity with concentrated aqueous NaOH (aq) solution from the storage tanks shown in Figure 2, that are located adjacent to the Q-type metal building that houses electrical switch gear, voltage step down (PWM) DC-DC power converter units, the sodium hydroxide (NaOH) electrolytic cells, sodium (Na) metal packaging unit and chlorine (Cl 2 ) gas separation and bottling unit. As sunrise commences, the solar tower photovoltaic (PV) device panel array begins generating electric power. The voltage step down PWM DC-DC converter supplies the electric power from the solar tower PV device panel array to the electrolytic cells at a fixed voltage given as V BAT ¼ V OUT ¼ 124 V. Before NaOH electrolysis and Na metal production can begin, the liquid aqueous NaOH (aq) has to be heated to evaporate all of the water (H 2 O (l) ) content and fuse or melt the remaining anhydrous solid NaOH (s) . The presence of water or moisture in the NaOH being electrolyzed can significantly reduce Na metal yield due to side reactions occurring between the Na metal and residual H 2 O. The sodium hydroxide melts into a liquid state at T f ¼ 594 AE 2 K [102]. Therefore, electric power delivered by the solar tower initially at sunrise, has to be transmitted to electric heating elements for evaporating the water (H 2 O (l) ) from the aqueous NaOH (aq) and to raise the temperature of the fused anhydrous NaOH (l) to the proper temperature for electrolysis. Once the proper temperature of the fused anhydrous NaOH (l) is reached, current from the solar tower can be transmitted to the cathode and anode electrodes of the 25 electrolytic cells electrically connected in series, to begin production of Na metal according to Eqs. (2) and (3).
The electric current is supplied to the 25 electrolytic cells electrically connected in series, by the synchronous parallel multiphase voltage step down PWM DC-DC converter at a fixed output voltage set by the utility scale battery given as V BAT ¼ V OUT ¼ 124 V. The current I OUT ¼ I CELL , supplied by the PWM DC-DC converter from the solar tower to the electrolytic cells is not constant throughout the day, increasing gradually from sunrise until the sun reaches its zenith, and subsequently decreasing gradually as the sun tracks west and eventually sets. To use the electric power generated by the solar tower PV device panel array most efficiently for Na metal production, it is necessary to configure the electrolytic cells as a variable electric load, wherein their electrical resistance can decrease gradually as the sun arcs toward zenith allowing I CELL to increase as more electric power is generated by the solar tower for electrolysis and subsequently, the electrical resistance of the electrolytic cells can begin to increase gradually as the sun passes beyond the zenith toward sunset, when I CELL must decrease as less electric power is generated by the solar tower for electrolysis.
The sodium (Na) metal production plant can effectively be controlled using only two adjustable parameters, including the set point reference voltage V SET , that controls the input voltage V IN , of the PWM DC-DC converter that is equal to the output voltage of the solar tower V IN ¼ V ST , and the electrical resistance of the NaOH electrolytic cells. The set point reference voltage V SET can function either as a coarse or fine adjustment for the current supplied by the solar tower PV device panel array to the electrolytic cells at the fixed voltage The electrical resistance of the electrolytic cells naturally increases slowly over time as the fused NaOH (l) reactant is decomposed by the current flowing between the electrodes of the cell according to Eqs. (2) and (3). The electrolytic cells can be designed for example, with a controllable electrical resistance that can remain relatively constant (or increase or decrease as needed) even as the fused NaOH (l) reactant is consumed, by varying the spacing between the anode and cathode electrodes, where the electrodes can be moved closer together to compensate the loss of reactant volume as it is consumed in the cell.
It is possible to calculate the quantity of Na metal produced throughout the year by the selfcontained sodium (Na) metal production plant sited in the different geographic locations given in Table 2, based on the hours of daylight and the prevailing air mass conditions. It is not necessary to specify a detailed design for the NaOH electrolytic cell to generate an accurate daily estimate of Na metal production yield throughout the year, if it is assumed that maximum electric power available from the solar tower PV device panel array can always be supplied to the electrolytic cells by appropriately controlling the electrical resistance of the NaOH electrolytic cells together with the set point reference voltage V SET of the PWM DC-DC converter. It is assumed that electrolysis of NaOH, and thereby Na metal production can only occur if the available current supplied by the solar tower PV device panel array has a minimum threshold value of I ST ¼ 3,000 A. In Figure 10, the production yield of Na metal is calculated for each day of the hypothetical year 2015, for electrolysis of pure NaOH according to Eq. (2), for the four geographic locations listed in Table 2, using the solar position algorithm (SPA) described in Solar position algorithm for solar radiation applications written by Reda & Andreas in 2004, that is a refined algorithm based on the book, The Astronomical Algorithms written by Meeus in 1998, and is presently regarded as the most accurate [73,74]. The SPA allows the solar zenith θ sz , and azimuth γ s , angles to be calculated with uncertainties of AE0.0003 in the range between -2000 to 6000 years.
The calculation in Figure 10 provides the expected daily sodium (Na) metal production yield under the assumption that the energy conversion efficiency η PV ¼ 90% for the solar tower PV device panel array and furthermore, current is only transmitted to the electrolytic cells when the solar tower PV device panel array receives sufficient solar irradiance to produce the minimum threshold value of current I ST ¼ 3,000 A. The calculation in Figure 10, uses the SPA algorithm to determine the solar zenith angle θ sz , throughout the day from sunrise to sunset for each day of the hypothetical year 2015, and calculates the air mass (AM) using Eq. (18). The direct normal total (spectrally integrated) solar irradiance incident onto the solar tower PV device panel array is calculated in turn, using Eq. (19) in a real time manner, as the solar position and air mass change throughout the day for each day of the year. The results from Figure 10, show that consistent quantities of Na metal are produced throughout the year when the self-contained sodium (Na) metal production plant is located as near as possible to the equator where the length of the day is the most uniform. Further away from the equator, the variability in the length of the day increases however, even at a latitude ϕ ¼ þ31.8 , in El Paso, Texas, the variability in the length of the day is not so significant as to render Na metal production uneconomical during the winter months when the daylight interval becomes reduced. The mean daily sodium (Na) metal production yields for the hypothetical year 2015 are calculated as 41,998 kg/day for El Paso, 41,884 kg/day for Alice Springs, 40,281 kg/day for Bangkok and 40,947 kg/day for Mbandaka, assuming a mean annual aerosol optical thickness value k a ¼ 0.1 reflecting clear days and uninterrupted Na production for all 365 days of the year. Notwithstanding the variability in daily Na metal production throughout the year, El Paso, Texas has the highest elevation above mean sea level when compared with Alice Springs, Bangkok and Mbandaka as shown in Table 2, resulting in an increased solar irradiance incident onto the PV device panel array and thus, the highest mean daily Na metal production. A conservative estimate for the true mean daily sodium (Na) metal production yield for the year might be m Na ¼ 30,000 kg/day of Na metal, as a consequence of nonproductive days due to inclement weather and required plant maintenance. The results in Figure 10, clearly demonstrate that the scalable, self-contained solar powered electrolytic sodium (Na) metal production plant can be constructed almost anywhere on earth and especially in the southwestern region of the U.S.A., to achieve a hydrogen (H 2(g) ) fuel, sustainable, closed clean energy cycle.

Logistics of sodium hydroxide and sodium metal
For the full benefits of the hydrogen (H 2(g) ) fuel based sustainable clean energy economy to be realized, it is essential to overcome the logistical problems inherent with H 2(g) fuel. The H 2(g) fuel possesses the unique characteristic that it can be combusted directly inside an internal combustion engine (ICE) to produce useful work without emission of carbon dioxide (CO 2 ) or sulfur oxides (SO X ) and with minimal emission of nitrogen oxides (NO X ). It can also be converted to electricity directly in a fuel cell to produce useful work with substantially higher efficiency than in an ICE. Regardless of how the versatile H 2(g) fuel is applied to produce useful work, it is essential to provide safe, reliable and economical logistics for its use. The major drawback of H 2(g) fuel remains the difficulty of direct storage. Fortunately, the element sodium (Na) positioned two rows below hydrogen in Group I of the periodic table of elements, is sufficiently electropositive to be capable of chemically releasing the H 2(g) fuel stored in either ordinary salinated (sea) or desalinated (fresh) water (H 2 O) over a wide temperature range [37]. Sodium (Na) is also sufficiently abundant in nature in the form of sodium chloride (NaCl) in seawater, to make its use economical for H 2(g) fuel generation [103]. Therefore, sodium (Na) metal and the sodium hydroxide (NaOH) byproduct resulting from the H 2(g) fuel producing chemical reaction in Eq. (1), constitute the ideal intermediate materials needed to render H 2(g) into a practical and usable fuel by storing the sun's radiated energy as Na metal.
A hydrogen (H 2(g) ) fuel clean energy economy based on a sustainable, closed clean energy cycle that uses sodium (Na) metal recovered by electrolysis from sodium hydroxide (NaOH) as a means of storing the sun's radiant energy collected during daytime hours, provides numerous benefits including safe, reliable and economical logistics. The scalable, selfcontained sodium (Na) metal production plant that stores the sun's radiant energy in sodium (Na) metal, can be constructed in almost any geographic location on earth benefitting from ample solar irradiance and clear weather all year. In the U.S.A., the arid, southwestern desert region offers the requisite conditions, including sufficient undeveloped land area to construct scalable, self-contained solar powered electrolytic sodium (Na) metal production plants by the thousands. Using the southwestern desert region that includes West Texas, New Mexico, Arizona and Southern California as a hub for solar powered sodium (Na) metal production by electrolysis of sodium hydroxide (NaOH), it is possible to develop sufficient Na metal production capacity based on the scalable, self-contained sodium (Na) metal production plant described, to meet the U.S.A.'s energy needs for motor vehicle transport and for broader clean electric power applications.
The physical and chemical properties of sodium (Na) metal and sodium hydroxide (NaOH) render these materials ideal from an operational logistical standpoint. The sodium (Na) metal is a solid at room temperature and therefore has negligible vapor pressure. As a result, Na metal can be stored almost indefinitely in hermetically sealed packaging that can be opened much as a sardine can, only when the Na metal must be loaded into a hydrogen generation apparatus to react with either salinated (sea) or desalinated (fresh) water (H 2 O) according to Eq. (1), to produce hydrogen (H 2(g) ) fuel on demand [37]. The sodium hydroxide (NaOH) byproduct of the hydrogen producing chemical reaction in Eq. (1), is also a solid at room temperature in its pure form and has negligible vapor pressure. The NaOH is miscible with water in all proportions, enabling the aqueous NaOH (aq) solution to be readily transferred by pumping into and out of sealed tanks for transport by truck, rail car or pipeline to the remotely located self-contained sodium (Na) metal production plant units for reprocessing by electrolysis according to Eqs. (2) and (3), to recover the Na metal for reuse in generating H 2(g) fuel. The NaOH (aq) transportation/storage tanks of the type shown in Figure 2, can be fiberglass or metal with a corrosion resistant internal rubber liner, and must seal hermetically to exclude ambient air that contains carbon dioxide (CO 2 ) which slowly degrades the NaOH (aq) , albeit not irreversibly.
To obtain a sense for the magnitude of the logistical effort needed to produce and distribute sufficient sodium (Na) metal to fuel all of the motor vehicles in the U.S.A. while recovering the sodium hydroxide (NaOH) byproduct for reprocessing by electrolysis, it is necessary to consider the total number of vehicles in circulation. According to the Bureau of Transportation Statistics at the United States Department of Transportation (DOT), the total number of registered vehicles in the year 2013 in the U.S.A. numbered 255,876,822 [104]. The figure includes passenger cars, motorcycles, light duty vehicles, other 2-axle/4-tire vehicles, trucks with 2-axles/6-tires or more and buses. If it is further assumed that each motor vehicle on average consumes the energy equivalent of 16.2 gallons of 100 octane gasoline (2,2,4-Trimethylpentane) per week, then the corresponding quantity of H 2(g) fuel having an equivalent heating value is given as 15.8 kg. The generation of 15.8 kg of H 2(g) fuel according to Eq. (1), requires that 361.6 kg of Na metal react with approximately 300 kg of water (H 2 O) [37]. Therefore, the total quantity of Na metal consumed per week in the U.S.A. can be calculated as 255,876,822 vehicles · 361.6 kg/week ¼ 92,525,058,835 kg/week. If each solar tower produces a mass m Na ¼ 30,000 kg/day of Na metal, then in one week the Na metal yield per solar tower will be given as 30,000 kg · 7 days ¼ 210,000 kg/week. The number of solar towers required to meet demand for Na metal will be given as N ST ¼ 92,525,058,835 kg/ week / 210,000 kg/week ¼ 440,596 solar towers or approximately N ST ≈ 450,000 solar towers. While the number of solar towers required might seem very large and the task of constructing them onerous, it is in fact possible to construct the sufficient number of towers to provide Na metal for all the motor vehicles in the U.S.A. The self-contained sodium (Na) metal production plants can be constructed at a density of approximately ρ plant ¼ 30 plant units per square mile to prevent mutual shading when the towers are elevated and rotated to track the sun. The solar tower density and layout necessitate a land area given as A land ¼ N ST / ρ plant ¼ 450,000 / 30 ¼ 15,000 square miles, to meet the Na metal demand for all the motor vehicles in the U.S.A. using PV device panels with an efficiency η PV ¼ 90%, and a land area A land ¼ 75,000 square miles using PV device panels with an efficiency η PV ¼ 18% that currently exist commercially. The land area required can be placed into perspective when considering that the area of the state of New Mexico is approximately 121,000 square miles and thus, there exists more than sufficient desert land area for constructing the scalable, selfcontained sodium (Na) metal production plants in the U.S.A.
Our company believes that hydrogen (H 2(g) ) fuel will earn an important role in motor vehicle transport applications for powering smaller 1-5 kW class secondary power fuel cells for onboard continuous recharging of battery electric vehicles (BEVs), a concept implemented successfully in the 1960s using H 2(g) fuel stored in high pressure cylinders [23]. The concept of a smaller hydrogen fuel cell operating at a fixed power output level to continuously recharge an electric storage battery can be extended not only to motor vehicle propulsion systems but also for a broad range of clean electric power applications, including general ground transport that includes commercial trucks, trains, maritime transport as well as powering single family homes, commercial establishments and industrial enterprises. Such an approach will ultimately enable mankind to dispense with use of carbon based fossil fuels for motor vehicle transport applications and most other types of ground based electric power generation.

Conclusion
The technical and economic viability of a novel, scalable, self-contained solar powered electrolytic sodium (Na) metal production plant has been demonstrated for meeting the hydrogen (H 2(g) ) fuel clean energy needs of the U.S.A. The scalable, self-contained sodium (Na) metal production plant uses a solar tower PV device panel array to collect and convert the sun's vast radiant energy emission produced by hydrogen fusion, into electric power that is used to recover sodium (Na) metal from sodium hydroxide (NaOH) or from a mixture of NaOH and NaCl by electrolysis. The Na metal can subsequently be reused to generate H 2(g) fuel and NaOH byproduct by reacting with either ordinary salinated (sea) or desalinated (fresh) water (H 2 O). The scalable, self-contained sodium (Na) metal production plant operation is enabled by a specially designed voltage step down PWM DC-DC converter consisting of a multiphase converter topology with up to 32 synchronous voltage step down converter circuits connected in parallel. The PWM DC-DC converter has a fixed output voltage V OUT ≈ 124 V and variable input voltage V IN ¼ V ST , that corresponds to the output voltage of the solar tower PV device panel array and can be controlled to maintain the PV device panel array operating near the maximum power point (MPP). Each scalable, self-contained sodium (Na) metal production plant consists of two voltage step down PWM DC-DC converters, wherein each unit supplies 25 NaOH electrolytic cells, electrically connected in series, with a current I CELL ¼ 96,500 A, corresponding to approximately one mole of electrons per second. The scalable electrical design of the solar tower allows the PV device panel array to be upgraded with newer and more efficient PV device panels as they become available as a result of progress in scientific research and development. Once PV device panels with an efficiency η PV ¼ 90% will become available, the power output of the solar tower PV device panel array can reach P ST ¼ 23.9 MW that is sufficient for producing a mass quantity of approximately m Na ¼ 30,000 kg of Na metal per day from the electrolysis of NaOH. It therefore becomes possible to meet the hydrogen (H 2(g) ) fuel clean energy needs of all the motor vehicles in the U.S.A. by constructing approximately 450,000 scalable, self-contained sodium (Na) metal production plant units of the type described, in the southwestern desert region of the U.S.A. that includes West Texas, New Mexico, Arizona and Southern California. If the land area needed for the scalable, self-contained sodium (Na) metal production plant units becomes scarce, then purpose built ships equipped with the Na metal production plants can be dispatched into the vast ocean expanses near the equator where high solar irradiance occurs, to convert aqueous NaOH (aq) stored onboard into sodium (Na) metal before returning to port.

Nomenclature a
Length of the semi-major axis of earth's elliptical orbit around the sun (m)