Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants

Available online at www.rajournals.in RA JOURNAL OF APPLIED RESEARCH ISSN: 2394-6709 DOI:10.47191/rajar/v11i12.03 Volume: 11 Issue: 12 December 2025 International Open Access Impact Factor8.553 Page no.- 1107-1116 1107 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants Sarwan S. Sandhu*1, Nil N. Panchal2 1,2Department of chemical and Materials Engineering, University of Dayton, Dayton, OH 45469, USA ARTICLE INFO ABSTRACT Published Online: 12 December 2025 Corresponding Author: Sarwan S. Sandhu An electrochemical system; utilizing the sodium in the molten state, and carbon dioxide and oxygen gas mixture as the reactant species; to produce electrical energy has been formulated to predict the standard-state open-circuit voltage as a function of the cell temperature. The cell open-circuit voltage as a function of temperature can be predicted for the non-standard states of the species taking part in the overall cell reaction. In addition, the developed formulation is capable of predicting the following: The cell operational voltage, electric power density, and efficiency of the cell to deliver electric power density as a function of the cell geometric current density: For the case of the cell operation as a batch electrochemical reactor, the provided formulation can predict the open-circuit cell voltage as a function of the cell temperature, pressure, the overall reaction extent and the initial composition of the cell cathode-side reactant mixture. The rates of the reactant-species consumption and product-species production can also be predicted at a given cell geometric current density. Some examples of the computed data presented in the form of plots are given below. (1) The cell open-circuit voltage decreases from 3.26 to 2.95 volt for the increase in the cell temperature from 100 to 300℃. (2) For the cell with the solid electrolyte (sodium beta-alumina solid electrolyte) of 10𝜇𝑚 thickness, the decrease in the cell operational voltage is of the order of 3𝑚𝑉 at 200℃ for the cell current range of 10 𝑡𝑜 100 𝑚𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 . (3) At the cell temperature of 200℃,𝑃 𝑃0=4, the electric power density increases from about 0.033 to 0.327 𝑊∙ 𝑐𝑚−2 over the geometric current range of 10 to 100 𝑚𝐴∙ 𝑐𝑚𝑔𝑒𝑜𝑚 −2 . (4) The efficiency to deliver electric power ranges from 99.98 to 99.80 over the geometric current density of range of 10 to 100 𝑚𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 for the cell with the solid electrolyte thickness of 10𝜇𝑚 at 200℃ and 𝑃 𝑃0=4. KEYWORDS: Sodium-CO2 fuel cell, Electrochemical energy production, Open-circuit voltage, Power density, Sodium betaalumina electrolyte. 1. INTRODUCTION The analyses and developments of the electrochemical systems have been carried out here at the University of Dayton since 1989 [some typical references: 1-7]. Very recently, we decided to develop a fuel cell, using carbon dioxide gas as one of the cell reactants, to generate electric power. Our current effort is focused on the development of a fuel cell system for applications in the space vehicle systems. Such fuel cell systems can take carbon dioxide from the combustion exhaust gas mixture discharged from the combustor of a space vehicle coupled with the fuel cell. Carbon dioxide or its mixture with oxygen gas would react with the fuel cell anode-side reactant sodium in the molten state to generate electric power. This power coupled with the supersonic power of a space vehicle system is expected to result in the hypersonic speed. On the Earth’s surface, such a system can be utilized simultaneously to generate electric power and reduce the concentration levels of carbon dioxide (one of the global warming gases) from the ambient atmospheric air. Sodium “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1108 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 supply relative to that of lithium is unlimited [8]. Also, sodium is cheaper than lithium. Furthermore, the cell reaction product sodium carbonate is a valuable product, especially, used in making soaps and chemicals, in cleaning and bleaching, etc. Figure 1 shows the sketch of the fuel cell system intended to be developed. Figure 1: Sodium/carbon dioxide-oxygen gas mixture fuel cell (the sketch not to a scale). The main components of the fuel cell sketched above are: A (anode): Sodium as the fuel cell anode-side reactant in the liquid phase; sodium melting point = 97.8 ℃ and boiling point = 883 ℃. E (electrolyte): Sodium beta-alumina solid electrolyte (NaBASE), Na-𝛽” 𝐴𝑙2𝑂3 (𝑠). C (cathode): Sputtered gold (Au) layer film on a thin microporous nickel foam layer. The cell cathode-side reactant: gas mixture of carbon dioxide and oxygen for example. P (plenum): Cell cathode – side plenum for holding a gaseous reactant or a gaseous reactant mixture. Configured for a reactant gas or gas-mixture flow parallel, or perpendicular to porous interface between the cathode and plenum as well as for the stagnant flow conditions. The fuel cell system sketched in Figure 1, must be experimentally characterized with respect to its electrochemical performance for the initial investigation over the operational conditions of: Temperature range: 25 – 300 ℃; the cell cathode-side reactant gas mixture pressure range: 1 – 5 bar; the reaction stoichiometric and non – stoichiometric compositions of carbon dioxide and oxygen gas mixture; and the cell geometric current density range: 10 – 100 𝑚𝐴∙ 𝑐𝑚𝑔𝑒𝑜𝑚 −2 . 2. FORMULATION The overall cell reaction is: 2 𝑁𝑎(𝑙)+𝐶𝑂2 (𝑔)+1 2𝑂2 (𝑔)⇌𝑁𝑎2𝐶𝑂3 (𝑠) (2-1) Na  =−2, 2( )g CO  =−1, 2( )g O  =−0.5,  𝑁𝑎2𝐶𝑂3(𝑠)=1. Per mole of reaction, Equation (2-1), occurring, the charge involved is: 𝑛𝐹=2𝐹 𝑐𝑜𝑢𝑙𝑜𝑚𝑏𝑠; where 𝐹 (called the Faraday constant) =96487.00 [𝑐𝑜𝑢𝑙𝑜𝑚𝑏𝑠∙(𝑔−𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡)−1]. For the occurrence of the above reaction in the forward direction at the reference temperature of 𝑇0=298.15 𝐾; the standard – state enthalpy and Gibbs – free energy changes are given as follows by Equation (2-2) and (2-3), respectively, ∆𝐻0𝑜=∆𝐻𝑓,𝑁𝑎2𝐶𝑂3 (𝑠),𝑇0 𝑜−(2∆𝐻𝑓,𝑁𝑎(𝑙),𝑇0 𝑜+∆𝐻𝑓,𝐶𝑂2 (𝑔),𝑇0 𝑜+0.5∆𝐻𝑓,𝑂2 (𝑔),𝑇0 𝑜) (2-2) ∆𝐺0𝑜=∆𝐺𝑓,𝑁𝑎2𝐶𝑂3 (𝑠),𝑇0 𝑜−(2∆𝐺𝑓,𝑁𝑎(𝑙),𝑇0 𝑜+∆𝐺𝑓,𝐶𝑂2 (𝑔),𝑇0 𝑜+0.5∆𝐺𝑓,𝑂2 (𝑔),𝑇0 𝑜) (2-3) The Gibbs free energy change for occurrence of reaction (2-1) in the net forward direction with the chemical species in their respective standard states at the cell temperature, 𝑇[𝐾], is given as: ∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇 𝑜=∆𝐻0𝑜−(∆𝐻0𝑜−∆𝐺0𝑜)(𝑇 𝑇0) (2-4) The cell standard open – circuit voltage at temperature, 𝑇[𝐾], is given by: 𝐸𝑇𝑜=(−∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇 𝑜) 𝑛𝐹 =(−∆𝐻0𝑜+(∆𝐻0𝑜−∆𝐺0𝑜)𝑇 𝑇0) 𝑛𝐹 (2-5) “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1109 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 For the cell operation at 𝑇[𝐾] with the chemical species involved in it, not in their respective standard states, the Gibbs free energy change, ∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇, is given by: ∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇=∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇 𝑜+𝑅𝑇𝑙𝑛  [] i i ia   (2-6) where R = 8.314 (𝐽−𝑚𝑜𝑙−1−𝐾−1) = the universal – gas constant, 𝑎𝑖 = activity of a chemical species i taking part in the cell reaction, Equation (2-1), and 𝑣𝑖 = stoichiometric number of species i in the cell reaction, Equation (2-1); 𝑣𝑖 is a negative value for a reactant species being consumed and a positive number value for a product species being generated. The term  [] i i ia   represents the product of [(𝑎𝑖𝑣𝑖)] factor over all species (i) taking part in reaction, Equation (2-1). It is well known that: ∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇=[−𝑛𝐹𝐸𝑇] (2-7) Therefore, 𝐸𝑇=(−∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇 𝑛𝐹 ) Inserting the information from Equation (2-6) into the above equation and simplifying, 𝐸𝑇=(−∆𝐺𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛,𝑇 𝑜𝑛𝐹 )−(𝑅𝑇 𝑛𝐹)𝑙𝑛  i i ia      (2-8) Or 𝐸𝑇=𝐸𝑇𝑜−(𝑅𝑇 𝑛𝐹)𝑙𝑛  i i ia      (2-9) where 𝑎𝑖 = activity of a species i to account for its non – ideal thermodynamic behavior. For the chemical species taking part in the overall cell reaction, Equation (2-1), 𝑎𝑁𝑎 (𝑠 𝑜𝑟 𝑙)=1,𝑎𝑁𝑎2𝐶𝑂3 (𝑠)=1 (pure chemical species); for a species i in a gas phase mixture 𝑎𝑖 (𝑔)=(∅ 𝑖𝑝𝑖 𝑃𝑜)=(∅ 𝑖𝑦𝑖𝑃 𝑃𝑜) (2-10) Here 𝑝𝑖, 𝑦𝑖 = partial pressure and mole fraction of a species i, respectively; 𝑃,𝑃𝑜 = gas mixture pressure and standard-state pressure of 𝑃𝑜 (= 1 bar); and ∅ 𝑖 = ∅ 𝑖 (𝑇,𝑦𝑖,𝑃), fugacity coefficient of a gas species i to account for its non-ideal gas behavior. For a gaseous species i in a gas mixture following ideal gas behavior at the mixture temperature and pressure conditions, ∅ 𝑖 = 1. Then 𝑎𝑖 (𝑖𝑑𝑒𝑎𝑙 𝑔𝑎𝑠)=(𝑦𝑖 𝑃 𝑃𝑜) (2-11) Generally, activity of a chemical species i in a solid homogenous phase mixture is given as:   () () i i s is aa  == (2-12) where 𝑥𝑖 = mole fraction of a species i in the solid-phase mixture and 𝛾𝑖(𝑠)=𝛾𝑖(𝑠)(𝑥𝑖,𝑇) = activity coefficient of a species i in a solid phase mixture to account for its non-ideal behavior. For the ideal solid phase mixture behavior, 𝛾𝑖(𝑠) = 1.0. For a pure solidstate species, i, 𝛾𝑖(𝑠)= 1.0. It is also noted that 𝛾𝑖 (𝑠) is a weak function of pressure. Activity of a chemical species i in a liquid phase mixture following non-ideal solution behavior is given as:   () () 0 i i i l il i f aa f   ==   , (2-13) where 𝑥𝑖 = mole fraction of a species i in a liquid solution mixture; ( ) ( ) ( ) [ , ] i l i l i Tx  = = activity coefficient of a species i in a liquid-phase solution mixture to account for its non-ideal energetic interaction behavior in a non-ideal solution mixture; generally, it is a weak function of pressure. (𝑓𝑖 𝑓𝑖0)=(𝑝𝑢𝑟𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑖 𝑓𝑢𝑔𝑎𝑐𝑖𝑡𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑚𝑖𝑥𝑡𝑢𝑟𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑎𝑛𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 𝑝𝑢𝑟𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑖 𝑓𝑢𝑔𝑎𝑐𝑖𝑡𝑦 𝑎𝑡 𝑡ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑 𝑚𝑖𝑥𝑡𝑢𝑟𝑒 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑎𝑛𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑓 (𝑃𝑜=1 𝑏𝑎𝑟)) (2-14) This (𝑓𝑖 𝑓𝑖0) is given by: “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1110 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 () 0 0 P il P V dP RT i i fe f          = (2-15) The exponential in Equation (2-15) is known as a Poynting factor. 𝑉𝑖 (𝑙) = the liquid-phase molar volume of species i; it is a very weak function of pressure at temperatures well below the critical state temperature, 𝑇𝑐,𝑖, of a species i. With 𝑉𝑖(𝑙) at 𝑃𝑜 = 1 bar and mixture temperature, Eq. (2-15), leads to: (𝑓𝑖 𝑓𝑖0)=𝑒(𝑉𝑖(𝑙)(𝑃−𝑃𝑜) 𝑅𝑇 ) (2-16) From Equations (2-13) and (2-16), the following equation is obtained: 𝑎𝑖=𝛾𝑖(𝑙)𝑥𝑖𝑒(𝑉𝑖(𝑙)(𝑃−𝑃𝑜) 𝑅𝑇 ) (2-17) Except for high pressures, the exponential term is close to unity. Then, Equation (2-17) becomes:   () () i i l i l i a a x  == (2-18) If the liquid-phase mixture composition is in terms of molar concentrations; then, the activity of a species i is given by: 𝑎𝑖 (𝑙)=[𝛾𝑖,𝑐 ∙ 𝑐𝑖 𝑐(𝑙) 0] (2-19) where 𝑐𝑖 (𝑙) = molar concentration of species i in a liquid-phase mixture and 𝑐(𝑙) 0 = molar concentration = [1 𝑚𝑜𝑙− (𝑙𝑖𝑡𝑒𝑟)−1] and 𝛾𝑖,𝑐 = activity co-efficient of a species i in a non-ideal solution mixture = 𝛾𝑖,𝑐 (𝑇,𝑐𝑖 (𝑙)). 𝛾𝑖,𝑐 is a very weak function of pressure. For the ideal solution behavior, 𝛾𝑖,𝑐=1, and Equation (2-19) leads to: 𝑎𝑖 (𝑙)=𝑐𝑖 (𝑙) 𝑐(𝑙) 0 (2-20) Application of Equation (2-9) to the overall cell reaction, Equation (2-1) leads to: 𝐸𝑇=𝐸𝑇0−(𝑅𝑇 2𝐹)𝑙𝑛[𝑎𝑁𝑎 −2 × (𝑎𝐶𝑂2 (𝑔))−1×(𝑎𝑂2 (𝑔))−0.5×(𝑎𝑁𝑎2𝐶𝑂3 (𝑠))1] Here, 𝑎𝑁𝑎=1,𝑎𝑁𝑎2𝐶𝑂3 (𝑠) =1; then, the above equation becomes: 𝐸𝑇=𝐸𝑇0−(𝑅𝑇 2𝐹)𝑙𝑛[1 𝑎𝐶𝑂2 (𝑔)∙𝑎𝑂2 (𝑔) 0.5 ], (2-21) where 𝑎𝐶𝑂2 (𝑔)=(∅ 𝐶𝑂2 (𝑔)∙𝑦𝐶𝑂2 (𝑔)∙𝑃 𝑃0)=[∅ 𝐶𝑂2 (𝑔)∙𝑦𝐶𝑂2 (𝑔)∙(𝑃 𝑃0)], 𝑎𝑂2 (𝑔) =[∅ 𝑂2 (𝑔)∙𝑦𝑂2 (𝑔)∙𝑃 𝑃0]. Inserting this information into Equation (2-21) and simplifying, the following expression is obtained. 𝐸𝑇=𝐸𝑇0−(𝑅𝑇 2𝐹)𝑙𝑛[1 (∅ 𝐶𝑂2 (𝑔)∙√∅ 𝑂2 (𝑔))(𝑦𝐶𝑂2 (𝑔)∙√𝑦𝑂2 (𝑔))(𝑃 𝑃0)1.5] (2-22) For the gas mixture of 𝐶𝑂2 (𝑔) and 𝑂2 (𝑔) for the cell cathode-side pressure of 5 bar or less, it is assumed that ∅ 𝐶𝑂2 (𝑔) =∅ 𝑂2 (𝑔) =1. Equation (2-22), then, leads to: 𝐸𝑇=𝐸𝑇0−(𝑅𝑇 2𝐹)𝑙𝑛[1 𝑦𝐶𝑂2 (𝑔)∙𝑦𝑂2 (𝑔) 0.5 ∙(𝑃 𝑃0)1.5] (2-23) Further simplification of Equation (2-23) leads to: 𝐸𝑇=𝐸𝑇0+1.5(𝑅𝑇 2𝐹)𝑙𝑛(𝑃 𝑃0)+(𝑅𝑇 2𝐹)[𝑙𝑛𝑦𝐶𝑂2+0.5𝑙𝑛𝑦𝑂2 (𝑔)],(𝑣𝑜𝑙𝑡) (2-24) where 𝐹=96487 𝑐𝑜𝑢𝑙𝑜𝑚𝑏𝑠 𝑝𝑒𝑟 𝑔− 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡. For the case of the cell operation as a batch reactor with respect to the cathode side reactant mixture of 𝐶𝑂2 (𝑔) and 𝑂2 (𝑔) initially fed to the cell plenum in batch-form; it is interesting to relate the changes in the moles of species taking part in the overall cell “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1111 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 reaction, Equation (2-1), and also, relate the species mole fractions to a simple variable, 𝜉 (called the reaction extent). The reaction extent, 𝜉, of the reaction, Equation (2-1), is a measure of progress of its occurrence towards right. A definition of 𝑑𝜉 is given by the following equation for the reaction, Equation (2-1): 𝑑𝑛𝑁𝑎 Na  =𝑑𝑛𝐶𝑂2 (𝑔) 2( )g CO  =𝑑𝑛𝑂2 (𝑔) 2( )g O  =𝑑𝑛𝑁𝑎2𝐶𝑂3 (𝑠) 2 3( )s Na CO  =⋯=𝑑𝑛𝑖 i  =⋯=𝑑𝜉 (2-25) For any species i taking part in the overall cell reaction; 𝑑𝑛𝑖= i  ∙𝑑𝜉 (i = 𝑁𝑎(𝑠 𝑜𝑟 𝑙),𝐶𝑂2 (𝑔),𝑂2 (𝑔) 𝑎𝑛𝑑 𝑁𝑎2𝐶𝑂3 (𝑠).). (2-26) Corresponding to the initial amounts, ′𝜉′=0, and ′𝜉′=𝜉 corresponding to an arbitrary extent of the reaction, Equation (2.1). Integrating, (2-26) ,0 0 i i n ii n dn d   =  (2-27) (𝑛𝑖−𝑛𝑖,0)= i  𝜉 (2-28) Therefore, 𝑛𝑖=(𝑛𝑖,0+ i  𝜉),(𝑚𝑜𝑙𝑒𝑠) of species i. (2-29) For 𝑁𝑎(𝑠) 𝑜𝑟 (𝑙) and 𝑁𝑎2𝐶𝑂3 (𝑠), 𝑛𝑁𝑎(𝑠) 𝑜𝑟 (𝑙) =(𝑛𝑁𝑎(𝑠),0−2𝜉),(𝑚𝑜𝑙𝑒𝑠), (2-30) 𝑛𝑁𝑎2𝐶𝑂3 (𝑠) =𝑛𝑁𝑎2𝐶𝑂3 (𝑠),0+𝜉=0+𝜉=𝜉,(𝑚𝑜𝑙𝑒𝑠), (2-31) 𝑛𝐶𝑂2 (𝑔) =𝑛𝐶𝑂2 (𝑔),0−𝜉,(𝑚𝑜𝑙𝑒𝑠), and (2-32) 𝑛𝑂2 (𝑔) =𝑛𝑂2 (𝑔),0−0.5𝜉,(𝑚𝑜𝑙𝑒𝑠). (2-33) At the reaction extent level of 𝜉, total gas mixture moles in the plenum, 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔𝑎𝑠)=(𝑛𝐶𝑂2(𝑔)+𝑛𝑂2(𝑔))=(𝑛𝐶𝑂2(𝑔),0+𝑛𝑂2(𝑔),0−1.5𝜉),(𝑚𝑜𝑙𝑒𝑠) (2-34) Or, 𝑛𝑡𝑜𝑡𝑎𝑙(𝑔) =(𝑛𝑡𝑜𝑡𝑎𝑙(𝑔),0−1.5𝜉),(𝑚𝑜𝑙𝑒𝑠) (2-35) where 𝑛𝑡𝑜𝑡𝑎𝑙(𝑔),0 =(𝑛𝐶𝑂2(𝑔),0+𝑛𝑂2(𝑔),0)=(𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑎𝑠𝑒𝑜𝑢𝑠 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑓𝑒𝑒𝑑 𝑚𝑖𝑥𝑡𝑢𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑒𝑙𝑙 𝑝𝑙𝑒𝑛𝑢𝑚 ),(𝑚𝑜𝑙𝑒𝑠) (2-36) Carbon dioxide and oxygen mole fractions, at the reaction extent of 𝜉, are given as: 𝑦𝐶𝑂2(𝑔) =𝑛𝐶𝑂2(𝑔) 𝑛𝑡𝑜𝑡𝑎𝑙(𝑔) =𝑛𝐶𝑂2(𝑔),0−𝜉 𝑛𝑡𝑜𝑡𝑎𝑙(𝑔),0−1.5𝜉 , (2-37) and 𝑦𝑂2 (𝑔) =𝑛𝑂2(𝑔) 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔)=𝑛𝑂2(𝑔),0−0.5𝜉 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0−1.5𝜉 (2-38) Equations (2-37) and (2-38) can also be expressed as: 𝑦𝐶𝑂2(𝑔) =[𝑦𝐶𝑂2(𝑔),0−( 𝜉 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0) 1−1.5( 𝜉 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0)] (2-39) 𝑦𝑂2(𝑔) =[𝑦𝑂2(𝑔),0−0.5( 𝜉 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0) 1−1.5( 𝜉 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0)] (2-40) Now, a modified reaction extent, 𝜉𝑚𝑜𝑑, is defined as: 𝜉𝑚𝑜𝑑=( 𝜉 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0) (2-41) “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1112 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 Equations (2-39) and (2-40) reduce to the following forms, 𝑦𝐶𝑂2(𝑔) =𝑦𝐶𝑂2(𝑔),0−𝜉𝑚𝑜𝑑 1−1.5𝜉𝑚𝑜𝑑 , (2-42) 𝑦𝑂2(𝑔) =𝑦𝑂2(𝑔),0−0.5𝜉𝑚𝑜𝑑 1−1.5𝜉𝑚𝑜𝑑 . (2-43) Putting, for 𝑛𝑡𝑜𝑡𝑎𝑙 (𝑔),0=1 𝑚𝑜𝑙𝑒 𝑜𝑓 (𝐶𝑂2(𝑔) 𝑎𝑛𝑑 𝑂2(𝑔)) in initial mixture, 𝜉𝑚𝑜𝑑=𝜉. Inserting the information for 𝑦𝐶𝑂2(𝑔) and 𝑦𝑂2(𝑔) from Equations (2-42) and (2-43), respectively into Equation (2-24) and simplifying leads to: 𝐸𝑇=𝐸𝑇0+1.5(𝑅𝑇 2𝐹)𝑙𝑛(𝑃 𝑃0)+(𝑅𝑇 2𝐹)𝑙𝑛[(𝑦𝐶𝑂2(𝑔),0−𝜉𝑚𝑜𝑑)(𝑦𝑂2(𝑔),0−0.5𝜉𝑚𝑜𝑑)0.5 (1−1.5𝜉𝑚𝑜𝑑)1.5 ],(𝑣𝑜𝑙𝑡) (2-44) It is here noted that the expression within the square brackets, [ ], is a function only of 𝜉𝑚𝑜𝑑. For the application of Equation (2-44) at any cell temperature and pressure conditions, the modified reaction extent, 𝜉𝑚𝑜𝑑, must be such that it is less than 𝑦𝐶𝑂2(𝑔),0; as well as (0.5𝜉𝑚𝑜𝑑) less than 𝑦𝑂2(𝑔),0 and (1.5𝜉𝑚𝑜𝑑) less than 1. Sodium ionic conductivity in the solid electrolyte, sodium beta-alumina solid electrolyte (Na-BASE), 𝑁𝑎−𝛽"𝐴𝑙2𝑂3(𝑠), at a temperature, 𝑇[𝐾], is given by: 𝜎𝑁𝑎+,𝑇=𝜎𝑁𝑎+,𝑇0∙𝑒𝑥𝑝[−𝐸 𝑅(1 𝑇−1 𝑇0)] (2-45) where 𝜎𝑁𝑎+,𝑇0 = sodium ionic conductivity at a reference temperature 𝑇0=298.15 𝐾,( 1 𝑜ℎ𝑚−𝑐𝑚=𝑆 𝑐𝑚);𝐸= activation energy for the ionic transport through the electrolyte, (𝐽∙𝑚𝑜𝑙−1); and 𝑅= the universal gas constant =8.314,(𝐽∙𝑚𝑜𝑙−1∙𝐾−1). Using the supplemental information from Figure S2 [8], 𝜎𝑁𝑎+,𝑇0 and (𝐸 𝑅) were determined. Their values are 𝜎𝑁𝑎+,𝑇0= 3.540×10−3(𝑜ℎ𝑚−1∙𝑐𝑚−1)=3.540 𝑚𝑆∙𝑐𝑚−1 and (𝐸 𝑅)=1248.88𝐾. The cell-voltage loss due to the resistance to the sodium-ion transport through the solid electrolyte sodium-beta alumina (𝑁𝑎−𝛽"𝐴𝑙2𝑂3(𝑠)) separator between the cell electrodes, during the cell discharge period, is given as: (∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑠𝑒𝑝=(𝑖𝑔𝑒𝑜𝑚×𝑙𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑠𝑒𝑝 𝜎𝑁𝑎+,𝑇 ),(𝑣𝑜𝑙𝑡) (2-46) where thickness of the solid electrolyte electr sep l−= . It has been stated that the cell electrode activation overpotential (i.e., the cell electrode polarization voltage loss is negligibly small (i.e., less than 0.050 volt depending on the cell operational temperature and geometric current density [8]). For the analytic work being reported in this paper, it is further assumed that the porous cell cathode is thin enough that the resistance to the mass transfer of chemical species carbon dioxide/oxygen gas from the cathode-side plenum to the highly active reaction sites for the overall cell reaction, Equation (2-1), is negligible. It has been mentioned [9] that a practical successful electrode should be free of species diffusional mass control. Also, it is assumed that sodium ion transport in the cathode and its reaction with the gaseous species oxygen and carbon dioxide is such that the solid reaction product is formed on the external surface of the cathode facing towards the cathode-side plenum, P. The reactant species molar consumption rates, corresponding to a fixed value of the geometric current density, 𝑖𝑔𝑒𝑜𝑚,(𝑎𝑚𝑝∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 ); according to the occurrence of reaction, Equation (2-1), are given as follows: 𝑛󰇗𝑁𝑎=(𝑖𝑔𝑒𝑜𝑚 𝐹),(𝑚𝑜𝑙∙𝑠−1∙𝑐𝑚−2) (2-47) 𝑛󰇗𝐶𝑂2(𝑔) =(𝑖𝑔𝑒𝑜𝑚 2𝐹 ),(𝑚𝑜𝑙∙𝑠−1∙𝑐𝑚−2) (2-48) 𝑛󰇗𝑂2(𝑔) =(𝑖𝑔𝑒𝑜𝑚 4𝐹 ),(𝑚𝑜𝑙∙𝑠−1∙𝑐𝑚−2) (2-49) Also, the generation rate of solid product, 𝑁𝑎2𝐶𝑂3(𝑠), from the cell reaction, is given by: 𝑛󰇗𝑁𝑎2𝐶𝑂3(𝑠) =(𝑖𝑔𝑒𝑜𝑚 2𝐹 ),(𝑚𝑜𝑙∙𝑠−1∙𝑐𝑚−2) (2-50) For the case of relatively thin cathode electrode (e.g., 10𝜇𝑚 thickness), the cell voltage drop associated with the sodium ion transport to the external surface of the cathode is expected to be negligibly small; of the order of 10𝑚𝑉 or less depending on the cell operational temperature and the magnitude of the cell geometric current density. For this situation, the cell operational voltage available at a fixed geometric current density, 𝑖𝑔𝑒𝑜𝑚, is: 𝐸𝑇,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙=𝐸𝑇−[|(∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑠𝑒𝑝|+|(∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑐𝑎𝑡ℎ|] (2-51) “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1113 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 where |(∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑐𝑎𝑡ℎ| = (voltage loss for sodium ion transport from the electrolyte separator to the exterior surface of the cell cathode facing the cathode-side plenum. Here, for the assumption of a thin, porous cathode electrode, |(∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑐𝑎𝑡ℎ| is being set equal to zero. Consequently, 𝐸𝑇,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙=𝐸𝑇−[|(∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑠𝑒𝑝|] (2-52) The cell electric power, 𝑃󰇗, is given as 𝑃󰇗=(𝐸𝑇,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙)×𝑖𝑔𝑒𝑜𝑚,[𝑊∙𝑐𝑚−2] (2-53) The cell electric efficiency, 𝜂𝑐𝑒𝑙𝑙, defined as: 𝜂𝑐𝑒𝑙𝑙=𝐸𝑇,𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐸𝑇=𝐸𝑇−∑(|∆𝑉|𝑑𝑟𝑜𝑝,𝑗) 𝑗𝐸𝑇 (2-54) Or, 𝜂𝑐𝑒𝑙𝑙=1−(∑(|∆𝑉|𝑑𝑟𝑜𝑝,𝑗) 𝑗𝐸𝑇) (2-55) where the (Σ𝑗) = (sum of the voltage losses, during the cell operation to deliver electric power to an electric circuit external to the cell, due to the cell electrode reaction polarization, ohmic resistance associated with the electron and ion transport in the interior cell components as well as with the electron transport in the cell anode and cathode current collectors. For the work presented in this paper, 𝜂𝑐𝑒𝑙𝑙 is given by 𝜂𝑐𝑒𝑙𝑙=1−(|(∆𝑉)𝑑𝑟𝑜𝑝−𝑒𝑙𝑒𝑐𝑡𝑟𝑜−𝑠𝑒𝑝| 𝐸𝑇) (2-56) We focused our analysis on the overall cell reaction, Equation (2-1); where the mixture of carbon dioxide and oxygen was assumed to be dry. For the situation of this mixture being moistened with water vapor; there is likelihood for the occurrence of the reaction: 𝑁𝑎+1 2𝑂2(𝑔)+1 2𝐻2𝑂(𝑣)⇄𝑁𝑎𝑂𝐻 (2-57) The standard-state Gibbs free energy change of this reaction, ∆𝐺0𝑜=(−265.208 𝑘𝐽∙𝑚𝑜𝑙−1) whereas that for the reaction, Equation (2-1), is (−650.081 𝑘𝐽∙𝑚𝑜𝑙−1) at 𝑇0=298.15𝐾. ∆𝐺0𝑜 for the reaction, Equation (2-1), is lower (i.e., more negative) than that for the reaction, Equation (2-57), by the factor of 2.45. So, it is envisioned that the reaction, Equation (2-1), occurrence would predominate over the reaction, Equation (2-57). 3. COMPUTED DATA AND DISCUSSION The data calculated from the formulation given in section 2 is presented in this section in the form of plots. Figures 2 and 3 shows the standard-state, open-circuit cell voltage calculated using Equation (2-5) as a function of the cell temperature. The open-circuit cell voltage decreases from 3.26 to 2.95 volt for the cell temperature increase from 373.15 to 573.15𝐾. One observes a linear relation between the cell voltage decrease and the cell temperature increase. Figures 4 and 5 show the effect of the cell Cathode-side reactant gas mixture composition on the open-circuit cell voltage, 𝐸𝑇, as a function of temperature at 𝑃 𝑃0=1,2 𝑎𝑛𝑑 3. “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1114 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 In figure 4, a decrease in 𝐸𝑇 with an increase in the cell temperature is observed at a (𝑃 𝑃0) value at the cell Cathodeside stochiometric reactant gas mixture of carbon dioxide and oxygen (carbon dioxide mole function = 2 3 and oxygen mole fraction =1 3). Also, at a cell temperature; 𝐸𝑇 increases with an increase in (𝑃 𝑃0) value; however, the increase in 𝐸𝑇 is small; for example, the increase of ≅0.03 𝑣𝑜𝑙𝑡 for the (𝑃 𝑃0) increase from 1 to 4 at 100℃ (≡373.15𝐾). Figure 5 shows the similar relation between 𝐸𝑇 and (𝑇 𝑎𝑛𝑑 𝑃 𝑃0) for the cell cathode-side nonstochiometric reactant gas mixture (e.g.,carbon dioxide mole fraction 𝑦𝐶𝑂2(𝑔) = 0.1053,oxygen mole fraction 𝑦𝑂2(𝑔) = 0.0526 and nitrogen mole fraction 𝑦𝑁2(𝑔) =1− (𝑦𝐶𝑂2(𝑔)+𝑦𝑂2(𝑔))). Comparison of the data presented in Figure 4 and 5 informs us that the open circuit cell voltage is slightly lower for the cathode-side non-stoichiometric reactant gas mixture at any (temperature,𝑃 𝑃0) pair; for example, lower by about 0.05 𝑣𝑜𝑙𝑡 at 𝑇=200℃ and 𝑃 𝑃0=2. Figures 6 and 7 show the ohmic voltage loss associated with the sodium ion transport through the cell solid electrolyte of thickness 10 and 20𝜇𝑚, respectively, as a function the cell geometric current density ranging from 10 to 100 𝑚𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 at three different cell temperatures: 100,200 and 300℃ At each cell temperature and solid electrolyte thickness, an increase in the ohmic cell voltage loss with an increase in the cell geometric current density is observed in the plots in Figures 6 and 7 as expressed by Equation (2-46). Also, the ohmic voltage loss through the solid electrolyte decreases with an increase in the cell temperature at each cell-geometric current density. This is because of an increase in the sodiumion ionic conductivity in the cell solid electrolyte as expressed by Equation (2-45). Comparison current density of 50 𝑚𝐴∙ 𝑐𝑚𝑔𝑒𝑜𝑚 −2 shows the cell voltage loss of 2.8 and 5.6 𝑣𝑜𝑙𝑡, respectively, for the solid electrolyte thickness of 10 and 20𝜇𝑚. Figures 8 and 9 show the cell operational voltage, 𝐸𝑇−𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙, and electric power density, 𝑃󰇗, as a function of the cell geometric current density, 𝑖𝑔𝑒𝑜𝑚, for the cell solid electrolyte thickness of 10 and 20𝜇𝑚, respectively, with the cell at 𝑃 𝑃0=4 and three different temperatures of 100,200, and 300℃. At any cell geometric current density, a slight decrease in the cell operational voltage is observed with an increase in the cell temperature. For example, at the cell geometric current density of 0.05 𝐴∙ 𝑐𝑚𝑔𝑒𝑜𝑚 −2 (50 𝑚𝐴∙ 𝑐𝑚𝑔𝑒𝑜𝑚 −2 ), the arithmetic-average operational cell voltage decrease is approximately 0.15 𝑣𝑜𝑙𝑡 as seen in Figure 8 over the cell temperature ranging from 200 to 300℃ for the cell solid electrolyte thickness of 10𝜇𝑚. “Fuel Cell for Production of Electric Energy with Carbon Dioxide as One of the Cell Reactants” 1115 Sarwan S. Sandhu1, RAJAR Volume 11 Issue 12 December 2025 Also, at a cell temperature, there is a slight decrease in the cell operational voltage with an increase in the cell geometric current density from 0.01 to 0.10 𝐴 ∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 . For example, at 200℃, the decrease in the operational cell voltage is approximately equal to 3𝑚𝑉 over current range of 0.01 to 0.10 𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 . At each cell temperature, the electric power density, 𝑃󰇗, increases with an increase in the cell geometric current density. The cell, with its solid electrolyte thickness of 20𝜇𝑚, shows the similar behavior with regard to the cell operational voltage and electric power density as presented in figure 9. Figure 10 shows the cell efficiency, 𝜂𝑐𝑒𝑙𝑙=𝐸𝑇−𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐸𝑇, to deliver electric power to an external electrical circuit as a function of the cell geometric current density, 𝑖𝑔𝑒𝑜𝑚, at 200℃ and (𝑃 𝑃0)=4 for the cell with its solid electrolyte thickness of 10𝜇𝑚. It is quite obvious that the electric power delivery efficiency ranges from 99.98 to 99.80% over the cell geometric current density ranges from 0.01 to 0.10 𝐴 ∙ 𝑐𝑚𝑔𝑒𝑜𝑚 −2 . 4. CONCLUDING REMARKS IN BRIEF In accordance with our intent to utilize carbon dioxide gas (a global warming gas) as one of the fuel cell reactants to generate electric power; the formulation based on the overall cell reaction, 2𝑁𝑎+𝐶𝑂2(𝑔)+1 2𝑂2(𝑔)=𝑁𝑎2𝐶𝑂3(𝑠), has been developed and presented in Section 2. The formulation can be used, for example, to predict the following: (a) The standard-state, open-circuit cell voltage as a function of the cell temperature. (b) The open-circuit cell voltage as a function of the cell temperature with the cell cathode-electrode side reactive gas mixture of carbon dioxide and oxygen; the gas mixture may be in the ideal or non-ideal thermodynamic’ state. (c) The cell operational voltage and electric power density as a function of the cell geometric current density. (d) Electric power delivery efficiency of the cell as a function of the cell geometric current density. Some computed data examples are given below: (A) The standard-state, open-circuit cell voltage decreases from 3.26 to 2.95 𝑣𝑜𝑙𝑡 for the cell temperature increase from 100 to 300℃. (B) For the cell cathode-side stochiometric reactant gas mixture of carbon dioxide and oxygen at 𝑃 𝑃0=2, the open-circuit cell voltage decreases from 3.26 to 2.95 𝑣𝑜𝑙𝑡 with increase in the cell temperature from 100 to 300℃. Similar open-circuit cell voltage behavior is observed for a non-stochiometric reactant gas mixture. (C) At a given cell temperature and pressure, e.g., 200℃ and 𝑃 𝑃0=4 for the cell having the solid electrolyte, sodium beta-alumina solid electrolyte (𝑁𝑎− 𝐵𝐴𝑆𝐸;𝑁𝑎−𝛽"−𝐴𝑙2𝑂3) of 10𝜇𝑚 thickness, the operational cell voltage change is small, i.e., 3 𝑚𝑉 for the cell geometric current density range of 10 to 100 𝑚𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 . (D) For the cell, with the solid electrolyte thickness of 10𝜇𝑚, at 200℃ and 𝑃 𝑃0=4, the electric power density increases from 3.12×10−2 to 3.11× 10−1 𝑊∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 with an increase in the geometric current density from 10 to 100 𝑚𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 . (E) For the cell, having the solid electrolyte of 10𝜇𝑚 thickness, at 𝑃 𝑃0=4 and 200℃; the electric power delivery efficiency ranges from 99.98 to 99.80% over the cell geometric current density increase from 10 to 100 𝑚𝐴∙𝑐𝑚𝑔𝑒𝑜𝑚 −2 . REFERENCES 1. S.S. Sandhu. A Thermodynamic Simulation of A Hydrogen/Oxygen Fuel Cell Involving Nonideal