Thermodynamic Analyses of Fuel Production via Solar-driven Non

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Thermodynamic Analyses of Fuel Production via Solardriven Non-stoichiometric Metal Oxide Redox Cycling —Part I: Revisiting Flow and Equilibrium Assumptions Sha Li, Vincent M. Wheeler, Peter B. Kreider, and Wojciech Lipi#ski Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b02081 • Publication Date (Web): 03 Sep 2018 Downloaded from http://pubs.acs.org on September 4, 2018

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Energy & Fuels

Thermodynamic Analyses of Fuel Production via Solar-driven Non-stoichiometric Metal Oxide Redox Cycling—Part I: Revisiting Flow and Equilibrium Assumptions Sha Li, Vincent M. Wheeler, Peter B. Kreider, and Wojciech Lipiński* Research School of Engineering The Australian National University Canberra, ACT 2601, Australia Abstract We present a thermodynamic model describing the operation of solar thermochemical reduction and oxidation chambers utilizing a non-stoichiometric metal oxide. The system under consideration is a generic reactor implementing an ideal counter-current flow (CF) configuration with prescribed inlet conditions of reactant flow rates and thermodynamic states. Conservation of species and mass as well as Gibbs’ criterion are used to determine the maximum and minimum limits of oxygen non-stoichiometry for reduction and oxidation, respectively, under a CF configuration. The methodology presented here is first used to analyze the previous models appearing in literature. It is found that existing efforts to model the CF configuration can violate Gibbs’ criterion. Motivated by this, a revised CF model is formulated and ensures the criterion is met thereby ensuring process spontaneity of the desired reaction for all conditions existing within the reaction chambers. The model identifies the highest reduction (oxidation) extent possible for a given inlet condition of the reduction (oxidation) chamber. This work offers an enhanced understanding of the CF flow configuration that will lead to more realistic estimates of the upper limit on solar-to-fuel efficiency for a reactor system. Key words: solar fuels, thermodynamics, counter-current flow, optimization 1. Introduction Syngas production via solar-driven, two-step, non-stoichiometric metal oxide cycling offers great promise for the efficient, CO2-neutral production of renewable fuels1, 2. A typical two-step non-stoichiometric metal oxide redox cycle consists of an endothermic reduction step, 1 1 1 M x O y −δ ox → M x O y −δ red + O 2 , ∆δ ∆δ 2

(1)

and an exothermic oxidation step for water (Eq. (2)) or carbon dioxide (Eq. (3)) splitting, depending on the desired product: 1 1 M x O y −δ red + H 2 O(g) → M x O y −δ ox + H 2 , ∆δ ∆δ

(2)

1 1 M x O y −δ red + CO 2 → M x O y −δ ox + CO , ∆δ ∆δ

(3)

where M x O y − δ and M x O y − δ denote the reduced and oxidized metal oxides and ∆δ = δ red − δ ox is the difference in non-stoichiometry of the redox material between the reduced and oxidized states. From Eqs. (1) and (2) or (3), it can be seen that the amount of fuel produced is proportional to the nonstoichiometry change of the redox material, red

ox

n&fuel = n&MO (δ red − δ ox ) = n&MO ∆δ .

ACS Paragon Plus Environment

(4)

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A high δred achieved from the reduction step (Eq. (1)) or a low δ ox from the oxidation step (Eq. (2) or (3)) will enable a large non-stoichiometry swing and thus high fuel production per mole of working material. Thermodynamically, the reduction step is favored at relatively high temperature and low oxygen partial pressure conditions. To achieve the low oxygen partial pressure, various methods have been studied and compared,3, 4 including vacuum pumping5, 6, inert gas sweeping, the use of a chemical scavenger5, 6, and their possible combination. Inert gas purging has been employed much more frequently than the other methods in laboratory tests7-12 and in theoretical analyses13-16. The efficiency with which solar energy can be used to drive the redox reactions described in Eqs. (1)–(3) critically depends upon the design of the thermochemical reactor built to interface the solar, gas, and solid inputs. Different reactor designs feature different solid–gas flow patterns. Four ideal solid–gas flow patterns have been studied extensively in the literature from the thermodynamics point of view: fixed-bed flow (FF) 9, 17-20, mixed flow (MF)3, 13-15, 21, parallel or co-current flow (PF) 19, and counter-current flow (CF)13, 14, 22. The FF model is a transient description of semi-batch reactor operation using a stationary solid reactant. Its transient equilibrium condition between solid and gas phases has been examined by Venstrom et al.19. A MF reactor is treated to have the same equilibrium composition everywhere within the reactor and at the exit. A zero-dimensional analysis has appeared in prior work3, 14, 15, 21. Both PF and CF assume a steady-state plug flow, which indicates that the flow of reactants through the reactor is orderly, without mixing or overtaking, and the composition of each species varies along a flow path.23 The PF model represents a conservative reactor design which assumes that the solid–gas equilibrium is only established at the reactor exit19, leaving large chemical potential underutilized at the intermediate zone due to the co-current flow arrangement. The CF model is widely recognized to offer an upper limit of the solar-to-fuel efficiency3, 13, 14, 19. However, efficiencies predicted by most prior work are unrealistically high4, 13, 22 and are sometimes accompanied by nonphysical results, such as zero sweep gas supply requirements to the reduction chamber below a certain oxidation temperature. This is mainly due to the unsound assumption that chemical equilibrium is enforced either at both inlet and outlet9 or at each infinitesimal point along the flow path4, 22 of the reduction and oxidation chambers. Hathaway et al.9 further pointed out that efficiencies of over 2% predicted for isothermal operation by prior efforts are caused by neglecting the parasitic work and/or the application of the flawed CF model to either the reduction or oxidation chamber. Many studies, including Krenzke and Davidson24, Brendelberger et al.14 and Hathaway et al.9 point out that the previous CF model is flawed due to the assumption of chemical equilibrium at both inlet and outlet when applied to the reduction chamber. To amend the flawed CF model for the reduction chamber, Brendelberger et al.14 proposed a refined analytical CF model in differential form by considering the coupling characteristics between the oxygen release from the metal oxide and the oxygen uptake by the sweep gas. The minimum sweep gas molar flow rate (here also called “demand” or “supply”) to maintain the low oxygen partial pressures during reduction can then be determined with only one equilibrium condition assumed at the gas inlet. In addition, they developed a numerical CF model to account for the possibility of a final reduction extent below the upper limit determined by the analytical model. Results for the numerical model show a dramatic shrink in the sweep gas demand. The analytical model by Brendelberger et al.14 is pioneering in offering a more accurate sweep gas demand under gas inlet equilibrium. However, that equilibrium state is enforced at gas inlet which is not fully justified and is, therefore, open for discussion. Generally, enforcing equilibrium at a given thermodynamic state does not necessarily guarantee the desired reaction will spontaneously occur at all thermodynamic states within a reaction system. Another constructive effort to address the flawed CF reduction model was made by Ehrhart et al.16 While still maintaining the assumption of equilibrium established at both inlet and outlet, they predicted the solar-to-fuel efficiency by using more realistic sweep gas flow rates in two ways: (i) the existing two-equilibrium model for sweep gas demand is multiplied by a scaling factor ranging from 1 to 1000, and (ii) a threshold value is used if the sweep gas demand predicted by the existing twoequilibrium model is too low. Though their work correctly demonstrates a drop in efficiency at higher


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sweep gas demands, it is still unclear what the exact minimum sweep gas demand is if equilibrium is to be achieved at both inlet and outlet points, if possible at all. A similar CF model applied to the oxidation chamber is widely adopted in this field.4, 13, 16, 22, 25, 26 However, unlike the CF reduction model which yields an obvious non-physical result of negligible sweep gas requirement, the CF oxidation model has not been recognized as flawed, to the best of our knowledge. This is mainly due to its plausible non-zero oxidizer demand to achieve the assumed equilibrium state(s), which may not necessarily guarantee the reaction spontaneity everywhere within the reactor. Bulfin et al.20 stated that the required amount of oxidizer may have been underestimated in some previous work. This might partly explain why the solar-to-fuel efficiencies by Ehrhart et al.16 are still high (above 20%) using a more realistic sweep gas flow rate scaled up by 1000 times, a similar conclusion drawn by Hathaway et al.9. This motivates us to question the base assumptions of the CF oxidation model which adopts the same unsound assumption as the CF reduction model: chemical equilibrium is enforced at both inlet and outlet points. Yet, few efforts have been made to analyze the CF oxidation model in depth. Lack of a rigorous understanding of oxidation chamber operation may lead to predictions of unattainably high solar-to-fuel efficiencies and overestimation of technical and commercial prospects of a technology using a given metal oxide. In the present work, we analyze the CF operation of both the reduction and oxidation steps using statements of conservation of mass and species as well as Gibbs’ criterion. Existing models describing the CF configuration are shown to fit within this framework under limiting—and sometimes unphysical— assumptions. Based on this analysis, we propose a revised model for CF operation in both reduction and oxidation steps to address the flaws of previous CF models and to offer a framework for guiding realistic modeling of metal oxide redox systems. 2. Problem statement The CF configuration to be analyzed is illustrated in Fig. 1. Figure 1(a) shows a reduction chamber, and Fig. 1(b) shows an oxidation chamber using the case of water splitting. An illustration of the PF configuration in a reduction and oxidation chamber is included in Figs. 1(c) and (d), respectively. First, we focus on the reduction chamber. The oxidized metal oxide with a flow rate of n&MO enters the reduction chamber at a non-stoichiometry state of δox . Sweep gas flows from the opposite end with a flow rate of n& N and an oxygen partial pressure of pO ,red,in . After a period of start-up time, the reduction system 2

2

reaches steady-state with a constant distribution of non-stoichiometry and oxygen partial pressure for the streams of metal oxide and sweep gas. The metal oxide leaves the reduction chamber at a reduced nonstoichiometry state of δred while the sweep gas exits with an oxygen partial pressure of pO ,red,out . Next, 2

consider the oxidation chamber in the case of water splitting. The reduced material moves into the oxidation chamber at a non-stoichiometry state of δred . On the opposite end, the water stream with an initial flow rate of n&H2O enters the chamber. The water stream has been heated to the oxidation temperature and is thermolyzed such that an equilibrium oxygen partial pressure of pO

2 ,ox,in

exists at the

inlet. Here, we define a new term—the local conversion ratio of water, X i , as the local flow rate of hydrogen to the inlet water supply rate,

Xi = where n&H

2 ,ox,i

n&H 2 ,ox,i n&H2O

(5)

denotes the flow rate of hydrogen at some thermodynamic state i along the flow path.

Similarly, the oxidation chamber will achieve steady-state after a dynamic period. The metal oxide leaves


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the reaction chamber at an oxidized non-stoichiometry state of δ ox while the gaseous stream exits with an outlet oxygen partial pressure of pO

2 ,ox,out

and a final conversion ratio of Xout .

(a)

(b)

(c)

(d)

Fig. 1. Schematic of flow configurations and mass and species conservation for (a) CF in the reduction chamber, (b) CF in the oxidation chamber, (c) PF in the reduction chamber, and (d) PF in the oxidation chamber.

Consider a reactor system with known inlet conditions and power input. To evaluate system performance, it is essential to understand: (i) What is the maximum outlet reduction extent of the working material given a fixed sweep gas supply? (ii) Similarly, what is the minimum outlet reduction extent of the working material given a fixed oxidizer supply? We offer answers to these questions using a revised thermodynamic model that will be elaborated in the following section.


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Energy & Fuels

Several assumptions are made to assist the development of the revised thermodynamic CF model: (i) the CF configuration under study is an ideal, steady-state, one-dimensional plug flow; (ii) a single constant flow rate of sweep gas n&N and oxidizer n&H O is supplied to each individual chamber to maintain the 2

2

assumed steady-state operation; (iii) both reduction and oxidation chambers are operated under atmospheric pressure, psys =1 atm; (iv) the reduction and oxidation chambers are operated isothermally at Tred and Tox, respectively; (v) the oxidation chamber is assumed to be a perfect oxygen exchange sponge in which any oxygen released from the water thermolysis reaction is absorbed by the metal oxide, thus creating a constant oxygen partial pressure along the flow path (this assumption will be elaborated further in section 3.2); (vi) reaction kinetics and mass transport are fast enough such that the solid–gas equilibrium is only limited by thermodynamics; (vii) all gas species are treated as ideal gases; (viii) the redox material is assumed to be 100% chemically active in participating in the redox cycling. 3. Methodology The revised CF model proposed in this work is fundamentally based on conservation of mass and species as well as the first and second law of thermodynamics. First, the expressions of sweep gas and oxidizer demand are derived for each reaction process using the principle of conservation of mass and species. Second, Gibbs’ criterion is employed to ensure the process spontaneity at every infinitesimal point along the solid–gas flowing path in both reduction and oxidation reactor systems. 3.1 Conservation of mass and species A schematic of mass and species conservation for the reactor system is illustrated in Fig. 1. Fig. 1(a) and 1(b) represent the CF configuration implemented in the respective reduction and oxidation chambers, while Fig. 1(c) and 1(d) represent PF in the reduction and oxidation chambers, respectively. In an ideal reactor system implementing CF or PF, the composition of both solid and gas streams varies from one thermodynamic state to another along the flowing direction. Consequently, the mass and species rate balance for a reaction component can be made for a differential control volume of the reactor system. Here, we focus on the reduction chamber implementing the CF configuration first. When applied to a differential control volume dVred,i at some thermodynamic state i, the species and mass conservation for oxygen can be simplified to 1 dn&O2 ,red,i = − n&MO dδ i 2

(6)

Molar flow rates can be expressed in terms of partial pressures using the Dalton model and the assumption of ideal gas behavior,

n&O2 ,red,i n&N 2

=

pO2 ,red,i psys − pO2 ,red,i

.

(7)

Note that the pressure of the system is held constant at atmospheric pressure (assumption (iii)). By differentiating Eq. (7) and substituting it into Eq. (6), the sweep gas flow rate can be determined by dδ 1 n& N 2 = − n&MO *i , 2 dpi

(8)

where we have introduced the quantity pi* = pO2 ,red,i ( psys − pO2 ,red,i ) . This expression was previously −1

derived by Brendelberger and coworkers.14


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Next, we consider the oxidation chamber using water splitting as a case study under the CF configuration. The oxidation reaction by steam, Eq. (2), can be modelled by combining a water thermolysis (WT) reaction, 1 H 2 O(g) → H 2 + O 2 , 2

(9)

with a metal oxide oxidation reaction by oxygen,13, 24, 27, 28 1 1 1 M x O y −δ red + O 2 → M x O y −δ ox . ∆δ 2 ∆δ

(10)

A statement of conservation of oxygen for the volume dVox,i appearing in Fig. 1(b) yields 1 1 dn&O 2 ,ox,i = − dn&H 2 O,ox,i − n&MO dδ i . 2 2

(11)

The analogous statement for hydrogen gives dn& H 2O,ox,i = − dn&H 2 ,ox,i .

(12)

By substituting Eq. (12) into Eq. (11) and using Eq. (5), we arrive at a relationship between the flow rates for all materials carrying oxygen, 1 1 n&H 2 O dX i − n&MO dδ i . 2 2

dn&O2 ,ox,i =

(13)

Once again, using the Dalton model and the assumptions of ideal gas and an isobaric oxidation system, we can write an expression for n&H2O in terms of quantities related to oxygen,

n&O2 ,ox,i n&H2O

=

pO2 ,ox,i psys − pO2 ,ox,i

.

(14)

Differentiate Eq. (14) and substitute it into Eq. (13) to get an expression for the flow rate of water supplied to the oxidation chamber in terms of quantities at an arbitrary thermodynamic state within the chamber, n&H 2 O =

dδ i 1 , n&MO 1 2 dX i − dpo*x,i 2

(15)

* where we have introduced the quantity pox , i = pO 2 ,ox,i ( psys − pO 2 ,ox,i ) . Relationships between sweep gas

−1

or oxidizer flow rate and relevant thermodynamic properties are thus obtained based on the principles of mass and species conservation for the CF configuration—Eqs. (8) and (15). These expressions determine the number of moles possible for both gases and solids at some thermodynamic state within the reaction chamber. However, these expressions alone are not enough to ensure the spontaneity of the desired reaction within the system or to answer the questions posed at the end of Section 2. For this we require Gibbs’ criterion as discussed in the following subsection. 3.2 Gibbs’ criterion


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Energy & Fuels

According to the second law of thermodynamics and conservation of energy principle, Gibbs’ criterion must be satisfied for any chemical process to spontaneously take place at a specified temperature and pressure, N

dG ]T , p = ∑ µ j d n j ≤ 0 ,

(16)

j =1

with

pj  o  g j + RT ln pref µj =   g o + RT ln a j  j

(ideal gas)

(17)

(ideal solid)

where G ]T , p , µ j , g °j , p j , and a j are the Gibbs function of a system at fixed temperature and pressure, the chemical potential of species j in a multicomponent system, the molar Gibbs function of pure component j at temperature T and 1 bar, the partial pressure of component j if j is an ideal gas, and the activity of component j if j is ideal solid, respectively. First, consider the reduction chamber system. Changes in the amounts of each species are related to the changes in non-stoichiometry state by −

dnMOox,i dnMOred,i dnO2 ,red,i = = = dξ red,i 1 1 1 ∆δ i ∆δ i 2

(18)

where ξ red,i is referred to as the local reaction extent of the reduction reaction. For the reduction chamber system, Eq. (16) takes the form dGred,i  T

, psys

red

= µ MOox ,i dnMO ox ,i + µ MO red ,i dnMO red ,i + µ O 2 ,i dnO 2 ,red,i + µ N 2 ,i dnN 2 ,i ≤ 0 .

(19)

Introducing Eqs. (17) and (18) into Eq. (19), and noting that dnN 2 ,i = 0 because nitrogen is inert, gives

dGred,i T

red , psys

 1  1 1 µMOox ,i + µMOred ,i + µO2 ,i  dξ red,i = − 2 ∆δ i  ∆δ i   °  1 aMOred ,i 1 pO2 ,red,i =  ∆Gred (δ i ) + RTred  ln + ln  ∆δ i aMO ,i 2 pref  ox 

   dξ ≤ 0.   red,i 

(20)

° Here the standard reaction Gibbs function ∆Gred for the reduction reaction is determined by ° o ° (δ i ) = ∆H red (δ i ) − Tred ∆S red (δ i ) , ∆Gred

(21)

o ° where ∆H red , and ∆S red are the standard reaction enthalpy and entropy of reaction Eq. (1). These values have been determined experimentally by authors studying, for example, ceria32 and zirconia-doped ceria.29, 30 The two solids must have identical activities, aMO red ,i = aMO ox ,i , and the occurrence of the

reduction process means d ξ red,i > 0 . Together, these yield


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1

 pO ,red,i  2 ° ∆Gred ( pO 2 ,red,i , δ i ) = ∆Gred (δ i ) + RTred ln  2  ≤0.  pref 

(22)

This expression is written in terms of only two undetermined values for the reactor under study: the partial pressure of oxygen and the reduction non-stoichiometry coefficient at some thermodynamic state on the flow path of the metal oxide and sweep gas stream. It must be satisfied for the reduction reaction to proceed spontaneously. Next, consider the oxidation chamber where two reactions—Eq. (9) and Eq. (10)—occur simultaneously. From Eq. (9), changes in the amounts of each component are related to the stoichiometric coefficients by

−

dnH2O,i 1

=

dnH2 ,i 1

=

dnO2 ,ox,i = dξ WT,i , 1 2

(23)

where ξ WT,i is the local reaction extent of the WT reaction. Similarly, the changes in the amounts of species for reaction Eq. (10) can be expressed as −

dnO2 ,ox,i dn dn = − MOred,i = MOox ,i = dξ ox,O2 ,i , 1 1 1 2 ∆δ i ∆δ i

(24)

where ξ ox ,O 2 ,i is the local reaction extent of metal oxide oxidation. Because the component O2 is involved in both reactions, the total differential change in O2 is given by dnO2 ,ox,i =

1 1 dξ WT,i − dξ ox,O 2 ,i . 2 2

(25)

For the oxidation chamber operated under contant pressure (1 bar) and temperature ( Tox ), Gibbs’ criterion, Eq. (16), becomes

dGox,i T

ox , psys

= µH2 ,i dnH2 ,i + µH2O,i dnH2O,i + µO2 ,i dnO2 ,i + µMOox ,i dnMOox ,i + µMOred ,i dnMOred ,i ≤ 0.

(26)

Substitution of Eqs. (17) and (23)–(25) into Eq. (26) and using aMO red ,i = aMO ox ,i gives

dGox,i  T

ox

, psys

1    2   p X  ° O 2 ,ox,i   i =  ∆GWT + RTox ln      dξ WT,i 1 − X i  psys        1   2   p O 2 ,ox,i  °  +  ∆Gox,O2 (δ i ) − RTox ln  dξ ,  p   ox,O2 ,i sys     

(27)

° ° where ∆GWT and ∆Gox,O are the standard reaction Gibbs function for WT reaction represented by Eq. (9) 2

and oxidation reaction by oxygen as given by Eq. (10), with the latter determined by ° (δ i ) = ∆H oxo (δ i ) − Tox ∆Sox° (δ i ) . ∆Gox,O 2


(28)

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Energy & Fuels

o The quantities ∆H ox and ∆Sox° are the standard reaction enthalpy and entropy, respectively, for the oxidation reaction by oxygen as expressed by Eq. (10) in terms of the local non-stoichiometry coefficient. o o ° Data for ∆H ox and ∆Sox° are negative values of the respective ∆H red and ∆S red since these two reactions have the same expression but proceed in opposite directions.

To determine the differential change of the Gibbs function for the oxidation chamber system, we need to consider the reaction extents of both reactions representing reduced metal oxide oxidation by steam. Since the two reactions are coupled via Eq. (25), dξ WT,i and d ξ ox,O ,i can vary in three possible ways: (i) 2

dξ WT ,i >dξ ox,O 2 ,i , (ii) dξ WT ,i 0 , so 2

∆Gox,H2O ( X i , δ i ) = ∆GW° T + ∆Go°x,O2 (δ i ) + RTox ln

Xi ≤0. 1− Xi

(31)

The inequalities (22) and (30) ensure that the desired reduction and oxidation reactions proceed spontaneously in the reduction and oxidation chambers, respectively. Note that these relationships hold for both CF and PF configurations.

4. Analyses on prior CF models The methodology elaborated above enables us to test the viability of previous CF models applied either to the reduction or oxidation chamber with the second law of thermodynamics. Table 1 lists the models appearing in the literature, each derived making varying assumptions. The model presented in this work is also included in Table 1 for comparison. For each, the sweep gas and oxidizer molar flow rates determined by different CF models can be derived from Eq. (8) and Eq. (15). The N multi appearing in Eq. (33) is the ad-hoc multiplicative factor of the minimum sweep gas flow rate employed by Ehrhart et al.16 The expressions in the table do not appear directly in the referenced papers but can be derived under the assumptions employed in them. The non-stoichiometry coefficients and outlet water conversion that


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appear as a function of temperature and oxygen partial pressure with the subscript “eq” denote quantities that can be evaluated from experimentally-determined equilibrium thermodynamic data. These expressions only guarantee that mass is conserved in the reduction or oxidation process. They cannot, however, guarantee that the reaction will take place within the chamber.

Table 1. Summary of previous CF models and the model used in this work showing the calculation for sweep gas or oxidizer demand. Note that the equations of this work do not enforce any particular equilibrium location, rather they * * ensure that all thermodynamic state points i satisfy the Gibbs criterion. See the text for definitions of pred,i and pox,i .

Expression Sweep gas supply to reduction chamber δ ox,eq (Tred , pO2 ,red,out ) − δ red,eq (Tred , pO2 ,red,in ) 1 n&N2 = − n&MO * 2 pre* d,out − pred ,in

References

(32)

Bader et al.13 and Krenzke and Davidson24

δ ox,eq (Tred , pO2 ,red,out ) − δ red,eq (Tred , pO2 ,red,in ) 1 n&N2 = − n&MO N multi * 2 − pre* d,in ) ( pred,out

(33)

Ehrhart et al.16

dδ eq (Tred , pO2 ,red,in ) 1 n&N2 = − n&MO * 2 dpred,in

(34)

Brendelberger et al.14

(35)

This work

(36)

Bader et al.13, Krenzke and Davidson24, and Ehrhart et al.4, 16, 22.

(37)

This work

(

)

dδ 1 n&N2 = − n&MO * i 2 dpred,i s.t. ∆Gred ( p

* red,i

, δ i ) ≤ 0 ∀ (p

* red,i

, δ i ) ∈{p

* red,i

≥p

* red,in

, δ i ≥ δ ox }

Oxidizer supply to oxidation chamber

δ red,eq (Tox , pO2 ,ox,out ) − δ ox,eq (Tox , pO2 ,ox,in ) 1 n&H2O = n&MO 1 2 * X out,eq (Tox , pO2 ,ox,out ) − pox,o ut 2 n&H2O = n&MO

dδ i dX i

s.t. ∆Gox,H2O ( X i , δ i ) ≤ 0 ∀ (X i , δ i ) ∈{X i ≥ X in , δ i ≤ δ red }

The models appearing in Table 1 are used herein to calculate the change in Gibbs function for any partial pressure of oxygen between the inlet and outlet partial pressures. The derivation for these calculations is not detailed here but can be found in the Supporting Information. The results of these calculations for ∆Gred and ∆ Gox,H O as a function of oxygen partial pressure are shown in Fig. 2(a) and 2(b), respectively. 2

The CF reduction models corresponding to Eqs. (32) and (33) clearly show that the Gibbs criteria is not satisfied for all points between the inlet and outlet where thermodynamic equilibrium has been imposed— a condition that must be satisfied for the effective operation of a reactor; see Fig. 2(a). We believe this is the source of the unphysical behavior observed for these models as previously recognized by Brendelberger et al.14 and Hathaway et al.9 Note that even the case using Eq. (33) with N multi =1000 does not work for all points within the reactor, because there exists a range of positive values of Gibbs function


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at low oxygen partial pressures between two equilibrium points representing the region of reactor operation. Similar conclusions are reached by considering the ∆Gox,H O in the oxidation chamber resulting from the use of Eq. (36); see Fig. 2(b). The analytical CF model by Brendelberger et al.14 corresponding to Eq. (34) does display a non-positive ∆Gred for the entire range of partial pressures considered here. This is evidence that imposing a single equilibrium point in the reactor system may lead to reasonable estimates of the required sweep gas that also guarantee the spontaneity of the desired reaction. The choice of imposing equilibrium at a thermodynamic state other than the gas inlet is a possibility that will be analyzed in the next section. 2

(a)

(b)

Fig. 2. Changes in the Gibbs function for previous CF models versus local oxygen partial pressure based on the equilibrium assumptions enforced in their work (see Table 1). This figure couples the reduction chamber (a) and the oxidation chamber (b) via the non-stoichiometry swing of δred=0.099 and δox=0.014. The inlet conditions used in this figure, as detailed below, are chosen to showcase the respective models but are not unique. The reactor system is operated with prescribed inlet conditions of (a) Tred=1773 K, pO2 ,red,in = 10−6 bar, δox=0.014 and n& N 2 n& MO =2.57 (Eq. (32), references13, 24), or n& N 2 n& MO =25.7, 257 and 2570 (Eq. (33) with Nmulti=10, 100 and 1000, respectively, reference16), or n& N 2 n& MO =8198 (Eq. (34), reference14) for the reduction chamber, and (b) Tox=1673 K,

pO2 ,ox,in = 3.85 × 10−4 bar, δred=0.099 and n&H2O n&MO =1.2 (Eq. (36), references4, 13, 16, 22, 24) for the oxidation chamber.

5. A revised model of the CF configuration and analysis We propose a revised treatment of the CF configuration based on the methodology detailed in Section 3. We believe the result to be an answer to the questions posed in Section 2 which we restate here using more precisely: (i) For the reduction chamber: given n&MO , n& N , δ ox , and pO 2

2 ,red,in

, what is the maximum metal oxide

reduction extent possible, max{δ red } ? (ii)

Similarly, for the oxidation chamber: given n&MO , n&H O , δ red , and X in , what is the minimum 2

metal oxide reduction extent possible, min{δ ox } ?


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First, we will find a convenient form for the conservation of mass statements for the reduction and oxidation chambers given by Eqs. (8) and (29), respectively. By integrating Eq. (8) from gas inlet point ( pin* , δ red ) to an arbitrary thermodynamic state point ( pi* , δ i ) we get

δ red ( pi* , δ i ) = δ i +

2n& N 2 n&MO

(p

* i

− pin* ) .

(38)

Thus, the thermodynamic states of a reduction system that conserves mass must follow a line segment with a slope that is proportional to the ratio of the nitrogen and metal oxide flow rates (known quantities for our system) where the inlet and outlet conditions correspond to the endpoints. We perform a similar analysis for the oxidation chamber, Eq. (29). Integrating this expression from gas inlet point ( Xin , δ ox ) to an arbitrary thermodynamic state point ( X i , δ i ), we again obtain a linear relationship between the local non-stoichiometry state and a local thermodynamic variable—here the oxidizer conversion ratio,

δ ox ( X i , δ i ) = δ i −

n&H 2O n&MO

( X i − X in ) .

(39)

Consequently, the answers to the proposed questions are formulated. First, we state them in words: (i)

The maximum reduction extent corresponds to the maximum possible ordinate of the gas inlet point on the line segment representing conservation of mass where all points on the line segment satisfy Gibbs’ criterion—Eq. (40).

(ii) The minimum reduction (maximum oxidation) extent corresponds to the minimum possible ordinate of the gas inlet point on the line segment representing conservation of mass where all points on the line segment satisfy Gibbs’ criterion—Eq. (41). These statements can be written more formally as optimization problems that we will demonstrate are able to yield quantitative predictions. For the reduction chamber, we have

max δ red ( pi* , δ i )

s.t.

∆Gred ( pi* , δ i ) ≤ 0 ∀ (pi* , δ i ) ∈{(pi* , δ i ):l red (pi* , δ i ) = 0, pi* ≥ pin* , δ i ≥ δ ox },

(40)

and for the oxidation chamber,

min δ ox ( X i , δ i )

s.t.

∆Gox,H2 O ( X i , δ i ) ≤ 0 ∀ (X i , δ i ) ∈{(X i , δ i ):l ox (X i , δ i ) = 0, X i ≥ X in , δ i ≤ δ red }.

(41)

We have defined l red = δ i − δ red + 2( n& N 2 / n&MO )( pi* − pin* ) and l ox = δ ox ( X i , δ i ) − δ i + (n&H2 O / n&MO )( X i − X in ) for convenience. Although these optimization problems are not necessarily mathematically transparent, we believe their meaning to be intuitive. Consider the reduction chamber. The maximization problem formulated in Eq. (40) states that we would like to find the highest possible reduction extent at the gas inlet that both conserves mass and satisfies the Gibbs’ criterion, Eq. (22), for all partial pressures within the chamber. Two additional constraints are also enforced: the reduction extent must always be greater than the most oxidized state and the partial pressure of oxygen must be higher than the inlet partial pressure of oxygen. These are practical conditions that we do not think need further justification. The maximization problem—similarly, the minimization problem—is perhaps most clearly illustrated using a figure, so we have presented this problem graphically in Fig. 3(a), where δ i has been plotted against pi* for a reactor utilizing ceria as a working metal oxide. The gray region represents the area


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satisfying the constraints appearing in Eq. (40): ∆Gred ≤ 0 , pi* ≥ pin* , and δ i ≥ δ ox . It represents the meaningful operation region where reduction will proceed spontaneously. The line segments represent the * possible operation of the reactor such that mass is conserved, where the inlet ( pin* , δ red ) and outlet ( pout , δ ox ) are represented by the endpoints. Different line segments represent different levels of oxygen exchange between ceria and inert gas flow streams. It can be seen that the maximum outlet nonstoichiometry state occurs when the mass conservation line just touches the equilibrium curve—at the point ( peq* , δ eq )—yet never crosses it and stays fully within the gray area. In this example, the equilibrium state is established at some intermediate thermodynamic state. Figure 3(b) shows the analogous graphical representation of the oxidation chamber for the case of oxidation by steam, Eq. (2). The oxidation chamber analysis differs from the reduction chamber only in the fundamental thermodynamic variable, X i instead of pi* , and the equilibrium curve (∆G=0) whose shape depends upon the material-dependent thermodynamic behavior determined experimentally or otherwise. The above analysis and graphical descriptions clarify the currently used flow models in the field and further justify the validity of the model developed here. The prior CF model13, 24 which assumes equilibrium at both inlet and outlet is also illustrated to further demonstrate what was shown in Fig. 2; besides the inlet and outlet points, the whole mass conservation line lies outside the constrained gray area, signifying the desired reduction reaction will not occur. This is true both for the reduction chamber, Fig 3(c), and for the oxidation chamber, Fig. 3(d). The refined analytical CF model by Brendelberger et al.14 is also represented in Fig. 3(c). Their model properly conserves mass and satisfies the Gibbs criterion. However, because of their imposition that the equilibrium condition be reached at the gas inlet, the sweep gas demand is far higher than that predicted by the present analysis under CF configuration. A comparison between the numerical incomplete counter-flow (IC) model by Brendelberger et al.14 and our revised CF model is presented in the Supporting Information for interested readers. The PF model can be analyzed using the same tools—conservation of mass and Gibbs’ criterion—and arguments as were developed above. We do not give the full analysis here, but the interested readers can consult the Supporting Information for more details. If we consider the same circumstances as were considered for the CF model for reduction ( n&MO , n& N , δ ox , and pO ,red,in are known), the maximum 2

2

reduction extent can be calculated without the need for an optimization problem. Unlike the CF model whose mass conservation line is not uniquely determined since only the slope is known, the PF model has a fixed mass conservation line since both the slope and one point on the line ( pin* , δ ox ) are known. This is a consequence of the gas and solid flowing in the same direction. The maximum outlet non-stoichiometry will be achieved when the mass conservation line intersects with the equilibrium curve because the whole mass conservation line must stay within the constrained gray area, as illustrated in Fig. 3(c). Therefore, the mathematical expression of the maximum outlet non-stoichiometry of a reduction system with prescribed inlet conditions under PF configuration can be simply calculated by

δ PF,red,max = δ ox +

2n& N 2 n&MO

* * ( pPF,out ). − pin* ) = δ eq (Tred , pPF,out

(42)

The oxidation chamber follows the same analysis, but using X i as the fundamental thermodynamic variable:

δ PF,ox,min = δ red −

n&H 2O n&MO

( X PF,out − X in ) = δ eq (Tox , X PF,out ) .


(43)

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The resulting line segment representing the operating conditions of the reactor is shown in Fig. 3(d). As can be seen for the case shown in Fig. 3(c), the CF model achieves a higher δ red,max than the PF model for a reduction system with the same inlet conditions. This is in line with the claim that the CF model offers an upper limit on the reactor performance as discussed in the introduction. This claim should hold true as long as the concave shape of the ceria equilibrium curves shown in Fig. 3 is typical of other metal oxides.


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(a)

(b)

(c)

(d)

Fig. 3. A graphical representation for determination of the optimal outlet non-stoichiometry parameters of the working material given a prescribed set of inlet conditions, as detailed below. The operating conditions and values for this figure are not unique; however, they are chosen to ensure that the results reflect all models under consideration and are easy to visualize. (a) The revised CF model for the reduction system at Tred=1873 K with inlet conditions of pin* = 10 −6 , δ ox =0.04, and n&N2 n&MO =50, (b) the revised CF model for the oxidation system at Tox=1773 K with inlet conditions of X in = 1.52 ×10−3 , δ red = 0.10 , and n&H2O / n&MO =2.0, (c) comparison of CF and PF

models for the reduction system at Tred=1873 K with inlet conditions of pin* = 10 −6 , δ ox =0.04, and n&N2 / n&MO =3.0 for CF revised model, CF prior models13, 24 and PF model, or n&N2 / n&MO =7000 for CF model by Brendelberger et al.14, and (d) comparison of CF and PF models for the oxidation system at Tox=1773 K with inlet conditions of X in = 1.52 ×10−3 , δ red = 0.10 , and n&H2O / n&MO =1.8 for all models under consideration. The subscript “prior” refers to previous work13, 24, and the subscript “revised” refers to this work.


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6. Summary and conclusions A revisited thermodynamic model for a reactor system undergoing metal oxide based two-step nonstoichiometric redox reactions under a counter-current flow configuration has been developed based on the imposition of conservation of mass and satisfaction of Gibbs’ criterion. The model allows for the determination of ideal outlet non-stoichiometry states for both the reduction chamber and the oxidation chamber via the construction of relevant optimization problems. The revisited counter-flow model proposed here fills a gap in the thermodynamic understanding of counter-current non-stoichiometric metal oxide reduction and oxidation reactors. The analysis allowed elucidation of the assumptions (and sometimes shortcomings) of the presently-used counter-current flow models and the parallel flow model. While the analysis approach and the resulting revisited model are themselves interesting, perhaps more interesting will be the incorporation of the model into a full system level model where the overall solarto-chemical efficiency will be determined. The efficiency is highly sensitive to the predicted molar flow rates of sweep gas and oxidizer. We present such an analysis in a parallel work.31

Associated Content Supporting Information Discussion of equilibrium assumptions made in the CF model by Ehrhart et al.16, derivation of changes in Gibbs function versus local oxygen partial pressure for prior CF models as listed in Table 1, mass and species conservation analysis for parallel flow configurations, selected results of optimal outlet nonstoichiometry states for both CF and PF models under a wide range of gas supply rates, and comparison between the numerical ‘incomplete counter-flow model’ by Brendelberger et al.14 and the present CF model. This document is available free of charge via the Internet at http://pubs.acs.org. Author Information Corresponding Author *Email: [email protected]. Notes The authors declare no competing financial interest. Acknowledgements

We gratefully acknowledge the financial support of the China Scholarship Council (Sha Li, grant no. [2015]3022, 201506020092) and the Australian Research Council (Wojciech Lipiński, Future Fellowship, award no. FT140101213). We thank Dr. Roman Bader for the fruitful discussions during the initial development of the revisited thermodynamic model. Nomenclature

aj=activity of component j for ideal solid dV=a differential control volume, m3 g °j = molar Gibbs function of pure component j at temperature T and 1 bar, J mol−1

G ]T , p =Gibbs function of a system at fixed temperature and pressure, J n& =molar flow rate, mol s-1

Nmulti= multiplicative factor appearing in Eq. (33) p=pressure, bar p* =dimensionless pressure as defined by p i * = p i ( p sys − p i )

−1

R = universal molar gas constant, 8.314 J mol−1 K−1


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Energy & Fuels

T=temperature, K X=chemical conversion Greek Symbols ° ∆GWT = standard molar Gibbs free energy for WT reaction as given by Eq. (9), J mol−1

∆H ox° = standard molar enthalpy for reaction (10), J mol -1O ° ∆Hred = standard molar enthalpy for reaction (1), J mol -1O o ∆H WT =standard molar enthalpy for WT reaction as given by Eq. (9), J mol-1

∆Sox° = standard molar entropy for reaction (10), J K -1 mol O-1 ° ∆Sred = standard molar entropy for reaction (1), J K -1 mol O-1

∆δ =non-stoichiometry swing

∆T =temperature swing, K δ =non-stoichiometry

µ j = chemical potential of species j, J mol-1

ξ =reaction extent, mol Subscripts eq=equilibrium condition i=thermodynamic state in=inlet max=maximum min=minimum MO=metal oxide out=outlet ox=oxidation chamber ox,H2O=oxidation reaction with steam as given by Eq.(2) ox,O2=oxidation reaction with oxygen as given by Eq. (10) red=reduction chamber sys=system Superscripts ° =standard condition at T and patm Abbreviations CF=countercurrent flow FF=fixed-bed flow MF=mixed flow MO=metal oxide PF=parallel flow WT=water thermolysis


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Thermodynamic Analyses of Fuel Production via Solar-driven Non

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