Comparing the Energy Content of Batteries, Fuels, and Materials

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Comparing the Energy Content of Batteries, Fuels, and Materials Nitash P. Balsara*,†,‡,§ and John Newman†,§,4 †

Environmental Energy Technologies Division and ‡Materials Sciences Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720, United States § Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California 94720, United States 4 Research Triangle Institute, 3040 East Cornwallis Road, Research Triangle Park, North Carolina 27709-2194, United States ABSTRACT: A methodology for calculating the theoretical and practical specific energies of rechargeable batteries, fuels, and materials is presented. The methodology enables comparison of the energy content of diverse systems such as the lithium-ion battery, hydrocarbons, and ammonia. The methodology is relevant for evaluating the possibility of using batteries and renewable fuels in the transportation and grid sectors where fossil fuels are traditionally used as energy sources. It is also relevant for choosing alternatives to petrochemicals for sustainable production of commodity materials. Instructors may consider introducing the proposed methodology in an introductory chemistry class. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Chemical Engineering, Physical Chemistry, Textbooks/Reference Books, Alcohols, Alkanes/Cycloalkanes, Industrial Chemistry, Electrolytic/Galvanic Cells/Potentials, Thermodynamics

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based on standard Gibbs free energy of formation of the elements and compounds of interest. In most cases, data for the calculations were obtained from the authoritative work of Wagman et al.;1 other sources2−8 are only used when the material of interest was not covered by Wegman et al.1 Data on practical specific energies of some systems is also provided.

here is considerable interest in using rechargeable batteries as energy-storage devices for transportation and electric-grid-related applications. This path is motivated by the finite supply of fossil fuels, the geopolitical complexity of ensuring access to existing reservoirs, forecasts of rising fuel costs, and the environmental impact of burning the fuel to produce energy. One is thus interested in comparing the amount of energy released when a battery is discharged with that obtained when a particular compound undergoes a chemical reaction and produces energy. Whereas the literature contains numerous comparisons of the specific energy of battery technologies and hydrocarbons typically found in fuel, the methodology used to obtain these values is usually not specified. A methodology is provided here for comparing the specific energies of batteries and a wide variety of petroleumbased and renewable fuels. The methodology is also used to evaluate the energy content of materials used in commodity production. The reason for including the latter systems is the recognition that problems underlying future technologies are not related to energy conversion alone. Many high-volume material products (e.g., polyethylene) are also made from nonrenewable raw materials. The objective of this paper is to introduce students and instructors to a simple approach for calculating the theoretical specific energy of batteries, fuels, and other materials. This requires specifying reactions for extracting the energy from the systems of interest. A methodology for specifying these reactions is presented. This is followed by specifying the reactions of selected battery chemistries, which is straightforward as the reactions and products of battery chemistries are generally well-accepted. The calculated specific energies are © XXXX American Chemical Society and Division of Chemical Education, Inc.

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SELECTED MATERIALS AND REACTIONS FOR EXTRACTING ENERGY Table 1 lists a selection of high-free-energy compounds and elements that are used in energy conversion and the production of material products. Reactions listed in Table 1 are used to calculate a consistent measure of the theoretical energy content of each system. The combustion reactions given in Table 1 (A2, A3, and A14) lie at the heart of a wide variety of technologies, including internal combustion engines and turbines for electricity generation. Only normal isomers of the compounds listed in Table 1 are considered (n-alkanes and n-butanol). The energycontaining materials of interest include elements such as lithium and compounds such as methanol. The term “active material” is used to refer to these elements and compounds. Wheat and coal are included due to their historical significance as energy sources for human activities. However, these materials are complex mixtures of compounds, and quantifying the thermodynamic properties of such mixtures is outside the scope of this paper. The major components of wheat are C, H, and O, and the approximate molar ratios C:H:O are 1:2:1.

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reacted with 1.5 mols of O2 and the required moles of other molecules from the basis set, if necessary, to give 1 mol of CO2 and 2 mol of H2O. The moles of O2, CO2, and H2O chosen occur naturally in the oxidation reaction of methanol (A1). One may view reaction A1 as the primary reaction. The other reactions are constructed so that they are consistent with this reaction; that is, they use 1.5 mol of O2 and produce 1 mol of CO2 and 2 mol of H2O. The methodology for arriving at the other reactions is clarified using two examples. For octane, one can write the reaction as

Table 1. Reactions Used To Compute the Energy Content of Fuels and Other Materials Reaction Number

Reaction

Active Material

A1

methanol

A2

methane

A3

octane

A4

ethylene

A5

ethanol

A6

butanol

A7

hydrogen

A8

carbon monoxide

A9

ammonia

A10

calcium carbide

A11

calcium

A12

lithium

A13

wheat

A14

coal

3 O2 → CO2 + 2H 2O 2 3 3 3 3 CH4 + O2 → CO2 + H 2O 4 2 4 2

CH3OH +

3 3 24 27 C8H18 + O2 → CO2 + H 2O 25 2 25 25

1 3 C2H4 + O2 → CO2 + H 2O 2 2 1 3 3 C2H5OH + O2 → CO2 + H 2O 2 2 2 1 3 5 C4 H 9OH + O2 → CO2 + H 2O 4 2 4

aC8H18 +

3 O2 + bCO2 + c H 2O → CO2 + 2H 2O 2

Coefficients a, b, and c are obtained by balancing this equation, yielding a = 3/25, b = 1/25, and c = 23/25. For ammonia, one has to introduce an additional molecule from the basis set, N2. One can thus write the reaction as

3 O2 → 3H 2O 2 3 3CO + O2 → 3CO2 2 3 2NH3 + O2 → N2 + 3H 2O 2 3 3 3 3 CaC2 + O2 → CO2 + CaCO3 5 2 5 5

3H 2 +

a NH3 + b N2 +

3 O2 + cCO2 + d H 2O → CO2 + 2H 2O 2

Balancing this equation yields a = 2, b = −1, c = 1, and d = −1. The balanced reactions for octane and ammonia are identical to those in Table 1. CO2 appears on the right hand side of all of the reactions in Table 1 except for the cases of calcium and lithium (A11 and A12). Students may wish to test their understanding of the concepts presented in this section by obtaining the other reactions given in Table 1.

3 O2 + 3CO2 → 3CaCO3 2 3 6Li + O2 + 3CO2 → 3Li 2CO3 2 3 3 3 3 CH 2O + O2 → CO2 + H 2O 2 2 2 2 3Ca +

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SELECTED RECHARGEABLE BATTERIES The five rechargeable battery chemistries of interest are leadacid, nickel−metal-hydride, lithium-ion, lithium−sulfur,9 and lithium−oxygen.10 The reactions of interest are given in Table 2. The reactants in reactions B1 through B5 are the energyproducing components of the battery present at the beginning of the discharging step, and the products are the components present at the end of the discharging step, assuming that the reactions are driven to completion. Lithium−sulfur and lithium−oxygen cells are not commercial currently but may be important in the future. The system B2′ described above is an “academic” version of a nickel−metal-hydride battery as it is based on well-defined compounds. Practical nickel−metalhydride batteries are generally described by a reaction wherein one of the reactants is a complex metal alloy, M. The composition of M for a particular battery was determined to be Ni0.56Al0.08Y0.05La0.10Ce0.07Pr0.01Nd0.02Mn0.05Co0.05.11 The mass of 1 mol of this alloy is 73.16 g/mol compared to 106.42 g/mol, the molar mass of Pd used in the academic battery.

3 3 6 3 C2H 2 + O2 → CO2 + H 2O 5 2 5 5

Similarly, the major components of coal are C and H, and the approximate molar ratio C:H is 1:1 (applies to bituminous coal). Formaldehyde and acetylene are used as model compounds that mimic the properties of wheat and coal, respectively, because they have approximately the same average molar compositions as the complex mixtures. In reality, the reactions used to extract energy from wheat and coal are much more complex than those specified above; one may regard eqs A13 and A14 as the simplest possible starting points for quantifying the energy content of these complex mixtures. In our scheme (Table 1), the active materials are reacted with oxygen. In addition, the chemical reactions involve a set of molecules that are defined as the basis set: CO2, H2O, N2, CaCO3, and Li2CO3. The basis set comprises abundantly available low-free-energy materials. The active materials are

Table 2. Reactions Used To Compute the Energy Content of Batteries Reaction Number

Battery Type

Reaction

B1

lead-acid

Pb + PbO2 + 2H 2SO4 → 2PbSO4 + 2H 2O

B2′

academic nickel−palladium-hydride

Pd 2H + NiOOH → 2Pd + Ni(OH)2

B2

nickel−metal-hydride

MH + NiOOH → M + Ni(OH)2 (M is metal alloy)

B3

lithium-ion

2Li 0.5CoO2 + LiC6 → 2LiCoO2 + C6

B4

lithium−sulfur

2Li + S → Li 2S

B5

lithium−oxygen

2Li +

B

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Table 3. Standard Free Energy of Formation of Compounds Discussed in this Paper Materials in Basis Set

ΔGf,i° /(kJ/mol)

Active Materials in Batteries

ΔGf,i° /(kJ/mol)

Energy-Containing Active Materials

ΔGf,i° /(kJ/mol)

CO2 H2O N2, O2 CaCO3 Li2CO3

−394.359 −237.129 0 −1128.79 −1132.06

PbO2 PbSO4 H2SO4 Li2O Pd2H NiOOH Ni(OH)2 LiCoO2 Li0.5CoO2 LiC6 Li2S C6, Pd, M, MH, Pb, S, Li, O2

−217.33 −813.14 −690.003 −561.18 −1.15 −329.4 −458.6 −619.65 −427.49 −13.61 −421.33 0

CH4 C2H2 C2H4 CH2O C8H18 CO CH3OH C2H5OH C4H9OH CaC2 NH3 Ca, Li

−50.72 +209.20 61.42 −102.53 +17.32 −137.168 −166.27 −174.78 −162.5 −64.9 −16.45 0

Table 4. References Used To Obtain Nonzero Standard Free Energy of Formation of Materials Discussed in this Paper Materials in Basis Set CO2 H2O CaCO3 Li2CO3

Reference 1, 1, 1, 1,

p p p p

Active Materials in Batteries

2−83 2−38 2−272 2−296

PbO2 PbSO4 H2SO4 Li2O Pd2H NiOOH Ni(OH)2 LiCoO2 Li0.5CoO2 LiC6 Li2S

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SPECIFIC ENERGY CALCULATIONS A consistent measure of the theoretical specific energy of the systems of interest (A1−A14 and B1−B5) is ΔG°, defined as ΔG° =

∑ reactants, i

siΔG°f , i −

∑ products, i

Reference 1, 1, 1, 1, 5, 3, 3, 8, 7, 7, 2,

p p p p p p p p p p p

2−119 2−120 2−58 2−290 3622 643 643 5 87 94 D-73

Energy-Containing Materials CH4 C2H2 C2H4 C8H18 CO CH3OH C2H5OH C4H9OH CaC2 NH3

Reference 1, 1, 1, 2, 1, 1, 1, 6, 1, 1,

p p p p p p p p p p

2−83 2−92 2−93 D-97 2−83 2−84 2−95 2.520 2−272 2−64

The basis set must be expanded if one is interested in calculating the specific energy of compounds containing elements that we have not yet considered. For example, if compounds containing sodium or chlorine are of interest, then one may consider expanding the basis set to include NaCl. Figure 1A shows ΔG° of the materials of interest on a log scale. The energy content of most organic compounds is 650 ± 50 kJ/(1.5 mol O2 consumed). This reflects the fact that most of the energy of these molecules is stored in the C−H bond. Of the organic molecules considered, methanol has the highest ΔG° value. Most of the high-free-energy systems on this scale are inorganic compounds and elements starting with ammonia at the low end, hydrogen at an intermediate level, and the reactive metals, calcium and lithium, at the high end. In spite of their differences, calcium and lithium have similar ΔG° values. It is instructive to examine the specific energy of active materials on a mass basis (Figure 1B,C). Specific energy can be defined in two ways. The first definition is the specific energy, E1, using the mass of the active material in the corresponding reactions for normalization

siΔG°f , i (1)

where si are the positive stoichiometric coefficients in the reactions and ΔG°f,i are the standard Gibbs free energies of formation of the compounds in the reactions. The values of ΔGf,i° of the systems of interest are listed in Table 3. In most standard chemistry textbooks, Gibbs free energies of reactions are defined as the Gibbs free energy of the products minus that of the reactants. The opposite sign convention was chosen for eq 1 because this paper deals with the extraction of energy from active materials of interest. In all of the reactions, the Gibbs free energy of the reactants is greater than that of the products, that is, ΔG° as defined in eq 1 is positive. Values of ΔG°f,i of most of the materials in Table 3 were obtained from literature.1−6,8 In the case of Li0.5CoO2 and LiC6, half-cell potential data as a function of state of charge reported by Thomas7 were integrated to obtain the reported values, and ΔG°f,i of LiCoO2 was taken from Yokokawa et al.8 The sources from which the data in Table 3 were obtained are specified in Table 4. Octane is used as an example to demonstrate our approach. In this case from eq A3, the theoretical specific energy of 637 kJ/(1.5 mol of O2 consumed) is obtained (ΔG° = (3/25) × 17.32 + (24/ 25) × 394.359 + (27/25) × 237.129). The computed value of ΔG° depends on two factors: the particular basis set of lowfree-energy compounds chosen in this paper, and our use of methanol oxidation as the primary reaction. These factors are not unique. For example, different values of ΔG° would be obtained if LiOH were placed in the basis set instead of Li2CO3.

E1 =

ΔG° s 0M 0

(2)

where the subscript 0 signifies the active material and M0 is the molar mass of the active material. For example, E1 of octane is 637/13.707 = 46.46 kJ/g. The W h is used as a measure of energy instead of kJ due to the widespread use of this measure in the energy sector (1 W h = 3.6 kJ). Figure 1B shows E1 of the materials of interest on a log scale. The second definition is the specific energy, E2, using the mass of the reactants (or products) for normalization C

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Figure 1. Theoretical specific energy of active materials (A) ΔG° on a log scale, (B) E1 on a log scale, and (C) E2 on a linear scale, defined by eqs 1, 2, and 3, respectively. All of the values reported in the figures are calculated using the specific free energies of a set of earth-abundant low-free-energy elements and compounds, CO2, H2O, N2, CaCO3, and Li2CO3, as the basis. This set of elements and compounds, which forms the basis set, is shown explicitly as the baseline in panel C only. The same baseline applies to panels A and B.

E2 =

ΔG° s0M 0 + ∑reactants, i ≠ 0 siMi

arise due to our use of Gibbs free energy differences for calculating E1 and E2, in contrast to enthalpy differences, and our use of model compounds in the calculations. The theoretical specific energy of closed battery systems is clearly given by E2. There are, however, oxygen-based battery systems wherein oxygen gas is not “carried” within the battery. The secondary lithium−oxygen battery (B5) is an example of such a system. In this battery, the specific energy E 2 corresponds to the fully discharged state when the battery is at its heaviest, and E1 is the value in the fully charged state when the battery is at its lightest. For consistency, E2 is used as a measure of the specific energy of this battery. The ordinate in Figure 2 is the theoretical specific energy of selected systems: E2 values of the five battery chemistries (B1− B5) calculated using eq 3. With the exception of the nickel− metal-hydride battery, all of the systems involve standard compounds with well-established methods for obtaining ΔG°f,i.

(3)

where Mi is the molar mass of reactant i. Figure 1C shows E2 of the materials of interest on a linear scale. As ΔG° is positive in all cases, it follows that E1 and E2 are also positive. Figures 1B,C shows that hydrogen has the highest specific energy and carbon monoxide has the lowest specific energy, regardless of whether E1 or E2 is used. Of the organic compounds studied, octane has the highest specific energy. E1 and E2 values of conventional fuels, CH4, the primary component in natural gas, coal, and octane are remarkably similar. The value of E1 of coal (13,177 W h/kg) is greater than the heating value of coal (11,500 W h/kg of carbon in bituminous coal).12 Similarly, E1 of wheat (4898 W h/kg) is greater than the energy content listed on commercial wheatbased bread packages (about 3000 W h/kg). These differences D

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practical specific energy of Li−S cells produced by Sion Power (350 W h/kg) and the rightmost point on the bar represents projections of designs that are currently being developed (600 W h/kg). It should be clear from this example that practical specific energies of other battery systems are also subject to similar uncertainties. For example, current Li−S cells have extremely limited cycle life (25 to 80 cycles) compared to current lithium-ion cells (about 500 cycles). If one were interested in making a lithium-ion cell that works for only 50 cycles, one could envision designs with much higher practical specific energies than those indicated in Figure 2. Thus, in reality, there is a horizontal bar of considerable length associated with each of the systems shown in Figure 2. The lithium−oxygen cell is entirely speculative at this point as it is not clear that this chemistry can be effectively incorporated in a rechargeable battery. An optimistic value for the practical specific energy of this cell is obtained by assuming the same practical-to-theoretical ratio as that of the current lithium-ion cell. Also shown in Figure 2 is a dashed line through the origin with unit slope. Technological improvements in a given system will result in a horizontal shift in the data as has been demonstrated with the Li−S battery. The dashed line in Figure 2 represents the maximum horizontal shift possible for any system. It is clear from Figure 2 that the specific energy of existing rechargeable battery chemistries, lead-acid, nickel− metal-hydride, and lithium-ion, cannot be improved to obtain specific energies typical of traditional fossil fuels. There are, however, some battery chemistries such as lithium−oxygen and Li−S that can, in principle, approach specific energies typical of traditional fossil fuels. Numerical listings of the data in Figures 1 and 2 are given in Tables 5 and 6. Students may wish to test their understanding

Figure 2. Theoretical specific energies, E1, of active materials (open squares) based on eq 2 and E2 of battery systems (filled squares) based on eq 3 versus the practical specific energy. In the case of octane, both E1 and E2 are shown (open and filled square, respectively). The straight dashed line with unit slope represents the upper bound for the practical specific energy of each system. The practical specific energy of all systems really has a range of values, and technological improvements will shift the data horizontally to the right. This is shown explicitly for the Li−S system only. The dashed line represents the maximum horizontal shift possible for a given system.

In the case of the academic nickel−palladium-hydride battery (B2′), ΔG° is dominated by the ΔGf,i° difference between NiOOH and Ni(OH)2; that is, the magnitude of ΔG°f,i of Pd2H is negligible compared to that of NiOOH and Ni(OH)2. The value of ΔG° of the practical nickel−metal-hydride battery was calculated using reaction B2 assuming that ΔGf,i° of the alloy M (with effective molecular weight of 73.16 g/mol) and the metal hydride MH are identically zero. The value of E2 of the academic nickel−metal-hydride battery calculated using reaction B2′ is 116 Wh/kg. This battery is not included in Figure 2. Also included in Figure 2 are E1 values of four compounds (methanol, octane, wheat, and coal) calculated using eq 2. Some may argue that it is more appropriate to compare specific energies of active materials and batteries on the same basis; the value of E2 of octane is included in Figure 2 for this reason. The abscissa in Figure 2 is the practical specific energy of the systems of interest. It is assumed that fuels are oxidized in an engine with 43% efficiency, typical of electrical powergenerating stations. Internal combustion engines typically have somewhat lower efficiencies (about 35%). For clarity of presentation, the practical specific energies of fuels are obtained using a single value of the efficiency (43%). Values of 35, 70, and 150 W h/kg are used for the practical energy densities of lead-acid, nickel−metal-hydride, and lithium-ion cells, based on currently available batteries. These values are based on the mass of active materials, electrolyte, separator, and current collectors in the battery but they do not include other system-related components such as casing, electronic circuits for managing the battery, cooling systems, and so forth. In the case of the Li−S cell, the practical specific energy is depicted using a horizontal bar. The leftmost point on the bar represents the current

Table 5. Data Used in Figure 1 Active Material

ΔG°/(kJ/1.5 mol O2)

E1/(W h/kg)

E2/(W h/kg)

H2 CO CH3OH C2H4 CH4 C2H5OH C4H9OH C8H18 NH3 Ca CaC2 Li wheat (CH2O) coal (C2H2)

711 772 702 662 613 663 650 637 678 2203 875 2213 793 741

32675 2550 6089 13115 14163 7991 9746 12905 5532 5090 6319 14764 4893 13177

3656 1623 2437 2966 2838 2591 2715 2866 2296 2038 2811 2773 2369 3235

of the concepts presented in this paper by calculating the specific energies of systems other than octane and ammonia, given in Tables 5 and 6.

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DISCUSSION Current industries are rapidly converting naturally occurring high-free-energy materials such as hydrocarbons into low-freeenergy compounds in the basis set. This may be viewed as, perhaps, an acceleration of processes that have occurred spontaneously throughout the earth’s history. The question of how we shall produce energy and materials in a world devoid of naturally occurring high-free-energy compounds is an interestE

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ethanol in water. Butanol spills will be easier to contain due to its limited solubility in water (about 8 wt %), but it is more toxic. Gases such as ammonia, hydrogen, and carbon monoxide look promising on paper but are not likely to be used for largescale energy storage due to practical issues such as availability, storage, flammability, and toxicity. The large specific energy of H2 is not necessarily an advantage as there are no large reserves of hydrogen on earth. It implies that any process used to create hydrogen from the compounds in the basis set will require considerable energy input. Reactive solids such as lithium are also not practical for energy storage because activation barriers for releasing the energy are too small. It is, however, worth noting that calcium carbide (A10) was used to provide light in mines but was abandoned in the early 1900s as electric power took root. Many of the systems listed in Figure 1 are used to produce materials that possess lower free energy (e.g., polyethylene from ethylene and fertilizers from ammonia). As humans convert the naturally occurring supply of active materials into materials in the basis set, it is obvious that the only energy source that can be used to pump energy back into the molecules on scales relevant to society is the sun. We shall thus have two choices: either we use the solar energy to produce high-free-energy starting materials and use existing infrastructure to make and transport products or create entirely new solar-energy-based processes to convert low-free-energy compounds directly into active materials and products.

Table 6. Data Used in Figure 2 Specific Energy/(W h/kg) System

Theoretical

Practical

Pb-acid NiMH (B2) Li-ion Li−S Li-oxygen octane wheat methanol octane

218 (E2) 216 (E2) 385 (E2) 2548 (E2) 5217 (E2) 2866 (E2) 4893 (E1) 6089 (E1) 12905 (E1)

35 70 150 350−600 2033 1233 2104 2618 5549

ing one. Batteries may play a significant role in energy storage and distribution, but we shall need to develop chemistries that can safely deliver several thousand cycles before large-scale implementation takes hold. Increasing the specific energy of batteries so that they are comparable to those of fuels is an obvious goal that has always received considerable attention (see Figure 2). The main accomplishment of this paper is establishing the basis for making such comparisons. If rechargeable batteries are to be used to store and deliver renewable energy on a large scale, it is important to compare the energy delivered during its lifetime with that required to make the battery. The data in Figure 2 can be used for making these comparisons. For example, the minimum (theoretical) energy required to create 0.33 kg of Li metal is 4921 W h, assuming Li2CO3 (a member of the basis set) is used as the raw material (4921 W h/kg = E1/3 of Li; see reaction A14 and E1 of Li in Table 5). A Li−S cell with a practical specific energy of 350 W h/kg must thus be cycled about 12 times to recover the theoretical energy required to produce the lithium metal, assuming that the masses of the two electrodes and the electrolyte are roughly equal (4291/350 ≈ 12). If the processes used to make the lithium foil have an energy efficiency of 50%, then 24 cycles are required to recover the practical energy that went into producing the anode of the Li−S cell. It should be evident that the energy extracted from current rechargeable battery chemistries is not negligible when compared to the total energy extracted from the batteries during their lifetime. The appropriateness of a particular definition of theoretical specific energy of active materials (ΔG°, E1, E2, or some other method of normalization) depends on the technology of interest. When a fuel such as octane is burned to create energy using current technology, E2 is appropriate because the engines take O2 from the atmosphere and reject CO2 and H2O to the atmosphere. Different normalization schemes might be needed if the cost of producing energy is the quantity of interest. If, for example, there is a monetary cost for emitting CO2, then a normalizing scheme that includes the cost of reactant acquisition and product disposal can be used for comparing the efficacy of different active materials. The present framework can readily be extended to include such normalization schemes. As we run out of cheap fossil fuel, it is appropriate to ask what kind of molecule is best suited for storing energy. Liquid fuels such as ethanol or butanol, which can be produced by biological processes that use sunlight as the energy source, appear to be logical choices. Ethanol is attractive because it is the least toxic of the active materials considered here. If a large fuel spill were to occur, natural dissipation and dilution of ethanol may prove to be an advantage, in spite of the fact that it may be difficult to contain the spill due to the high solubility of

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This paper was motivated by work done by the PIs in the Batteries for Advanced Transportation Technologies (BATT) Program, supported by the U.S. Department of Energy, Vehicle Technologies Program under Contract No. DE-AC0205CH11231. We thank Evren Oscam, Anna Javier, Adriana Rojas, and Douglas Greer for their help with the manuscript.

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REFERENCES

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(12) Hougen, O. A.; Watson, K. M. Chemical Process Principles, Part One: Material and Energy Balance; John Wiley & Sons: London, U.K., 1947.

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