A NEW SOLAR FUELS REACTOR USING A LIQUID METAL HEAT TRANSFER FLUID: MODELING AND SENSITIVITY ANALYSIS

A numerical model is developed to analyze a new solar thermochemical reactor using liquid metal as a heat transfer fluid. Reaction kinetics for both reduction and oxidation in two-step redox cycles using a metal oxide are modeled by fitting to experimental data. The transient model includes the heat and mass transfer and reaction kinetics for an analysis of the efficiency and to elucidate limiting factors. The reactor can achieve an


INTRODUCTION
2][3][4][5] This idea has gained increased attention over the last few years as interest has shifted to two-step partial redox cycles. 6-24A well-known example is the production of hydrogen fuel from water through two-step partial redox cycles using metal oxides that undergo the following two reactions: Step 1: Reduction Reaction MO +H (g) δ δ   (2)  In these reactions, the solid phase metal oxide serves as an oxygen storage material (OSM), denoted as 2 MO .In Step 1,   the OSM is heated to a high temperature H T (e.g.1200-1500 o C) and is subject to a low oxygen pressure ( 2 O p ) environment where it endothermically releases oxygen from its lattice.The heat required to break the chemical bonds is supplied by the high temperature solar process heat and the oxygen release is driven by the entropy increase experienced by the O2 molecules upon liberation.This can be seen from the change in Gibbs free energy for the reaction G H T S      where H  represents the strength of the metal oxygen bonds, and at sufficiently high temperatures T S  can become greater than H  prompting the oxygen release.This step is followed by Step 2, where the OSM in a reduced state 2-δ MO and is cooled to a lower temperature L T (e.g.500-800 o C), such that the thermodynamic driving force is reversed.In this second reaction, the OSM consumes the oxygen in H2O to refill its oxygen vacancies, as the metal oxygen bond strength now dominates G  at lower temperatures.This reaction liberates hydrogen, thereby producing fuel.After Step 2, the OSM can be reheated and cycled through these two reaction steps without consuming the OSM.Many metal oxides can be used as an OSM, and they can be divided into two categories, volatile and non-volatile materials. 25Separation of the volatile product to avoid recombination of the products results in the low efficiency of splitting water for volatile OSMs; 26 therefore, more efforts have been devoted to research on non-volatile materials, such as Fe3O4, and CeO2. 7,9any solar reactor concepts have been proposed to produce fuel with this two-step water splitting process.For example, Roeb et al. built a reactor which where the OSM served as the receiver and was coated on honeycomb absorbers. 27In their design, two identical receiver-reactors were constructed such that a quasi-continuous operation of the two-step water splitting cycle is feasible.Kaneko et al. built a rotary-type solar reactor using Ni-Mn ferrites and ceria as the OSM. 28Muhich et al. developed a solar reactor using counter-rotating rings where OSM is coated on the surfaces. 19In this design, the rings rotate between high temperature reduction zone and low temperature oxidation zone to enable better heat recuperation and continuous fuel production.This design allows recuperating the sensible heat in the high temperature part of the reactor via radiation, and simulations indicate a recuperative efficiency > 50%. 15,16Gokon et al. proposed a windowed reactor, using an internally circulating fluidized bed for the thermal reduction of NiFe2O4/m-ZrO2 particles and demonstrated the concept with a laboratory scale reactor. 29Perkins et al. tested an aerosol reactor for the dissociation of ZnO(s) to Zn(g). 26Scheffe et al. also used an aerosol reactor to carry out thermal reduction of ceria. 30Koepf et al. used an inverted conical-shaped cavity where the reactant powder ZnO descends continuously as a moving bed to produce hydrogen. 31espite these various efforts, the efficiency of these reactors is far below 20%, which has been cited as the efficiency regime where such processes can become economically viable at scale. 32The largest efficiency reported in the literature is an average of 1.73% with the peak of 3.53%. 12This is in contrast to thermodynamic analyses that suggest much higher efficiencies of over 50% are possible. 33The key discrepancy between these thermodynamic models and reality is that they do not account for chemical reaction kinetics nor the inherent limitations on heat and mass transport.Several studies 13,16 that have constructed more comprehensive models that consider transport kinetics, have agreed with experiments and shown why the reactors are extremely inefficient -predicting efficiencies on the order of 1%.The conclusions of these models has been that most of the incident sunlight is simply reemitted as thermal radiation after being absorbed at such high temperatures on the walls of the cavity. 13In such designs the cavity walls serve two purposes and facilitate two forms of energy conversion, namely as the receiver which converts the sunlight to heat via optical absorption and as the reactor which converts the thermal energy to energy stored in the chemical bonds of the reaction products.
Recently we have introduced a new concept for the reactor design, which separates the receiver and reactor into two separate devices to reach higher efficiency. 34In our analysis, we considered two efficiencies, one for solar to thermal energy .We proposed a new concept that separates the two conversions steps: solar to thermal, and thermal to chemical.The first energy conversion, solar to thermal, is carried out by a receiver.The second energy conversion, thermal to chemical, is performed by a reactor which can be designed and operated separately.To achieve a high overall efficiency, it is crucial to optimize the reactor.
In this study, we model the reactor and carry out an analysis to optimize the design and operation.Based on the reactor designed, we have constructed a model that has fully incorporated chemical reaction kinetics, heat and mass transfer inside the reactor, which is rare except the work from Keene et al. 13 We carry out a detailed study on the reactor concept to estimate the efficiency of this reactor design and identify its limitations.A sensitivity analysis is performed to optimize the reactor design and operating parameters.Since the main focus of this paper is to show the modelling of a new reactor concept, the reactor and its operation will be introduced only briefly here, but is discussed in more detail elsewhere 34 .

NEW REACTOR DESIGN
In our new concept, two separate devices are used, a solar receiver which handles the solar-thermal conversion, and a thermal-chemical reactor which converts the thermal energy to fuel energy.Each of these two devices can be designed and optimized separately to achieve a higher efficiency.These two devices are connected through a high temperature liquid metal (LM) which serves as a heat transfer fluid.This concept is schematically illustrated in Figure 1.In this scheme, the overall conversion of solar energy to chemical energy to be stored in hydrogen fuel is divided into two parts: solar to thermal, and thermal to chemical (fuel).The efficiency of the former conversion is Figure 2 New reactor design concept using LM.(a) Thermochemical reactor schematic showing an array of sealed reaction chambers interconnected by a piping network.The chambers are denoted by square boxes and the piping network consists of pipes that do not intersect, but are overlaid on top of each other, so that each acts as a dedicated conduit for LM circulation between a pair of chambers.(b) A schematic of an individual reaction chamber, which consists of an array of pipes carrying an LM through the inner bore.(c) A cross-section of one tube inside of the chamber.
We recently developed a new reactor design as shown in Figure 2, 34 which is discussed briefly here.It is envisioned that the LM would be tin (Sn) and the reactor would be largely constructed out of graphite.Sn melts at 232°C and does not boil until 2602°C at 1 atm, thus it would remain liquid and stable over the entire temperature range of interest.Graphite is inexpensive, easily machinable into complex shapes and is stable up to ~ 3000°C.Most importantly, graphite and Sn are fully chemically compatible at all temperatures, since they do not form any chemical compounds (e.g., no corrosion).Our design employs an array of sealed reaction chambers that are interconnected in a piping network which allows the LM to transfer heat between the chambers (see Figure 2(a)).In our system, all reaction chambers go through the two reaction steps at different times and therefore have different temperatures at any given instant.In the proposed reactor concept, the two reaction steps Eqs. ( 1) and (2), as well as the heat recuperation, occur cyclically and semi-continuously in separate reaction chambers.Each individual reaction chamber consists of an array of pipes (see Figure 2(b)), contained in a hermetically sealed outer housing.Each pipe serves as a containment material for the LM flowing through its inner bore (see Figure 2(b) and (c)).Each pipe also has the OSM coated around its outer diameter.The reaction and heat recuperation strategy for an eightchamber reactor system is shown in Figure 3.While this figure shows only the first stage, one complete cycle of reaction and heat recuperation includes eight stages, where the remaining stages follow a similar sequence.Each stage has three steps, pumping, purging and preheating: Pumping step: The first step shown in Figure 3 is the pumping step, where Chamber 1 with initial temperature H T  , which is lower than H T , is heated by the LM from the solar receiver and undergoes the reduction step (Equation ( 1)).In this step, a vacuum pump is used to lower the total pressure of Chamber 1 to approximately 10 -2 atm.Simultaneously, on the opposite side of the reactor network, Chamber 5 at the initial temperature L T  is cooled by LM at the temperature of L T , and then undergoes the oxidation step (Equation ( 2)).Also simultaneously, the chambers in between 1 and 5 exchange thermal energy with chambers amongst themselves to recuperate heat.
Purging step: The pumping step described above is followed by the purging step, where low pressure steam is introduced to purge the oxygen in the chamber further reducing the oxygen partial pressure Preheating step: Finally in the preheating step, the LM at the highest temperature TH in Chamber 1 exchanges heat with Chamber 2 to prepare Chamber 2 for the reduction step in the next stage, while Chamber 5 exchanges heat with Chamber 6 for the reoxidation reaction.
In the next stage, the entire sequence of the pumping, purging, and preheating steps are repeated while the locations of the corresponding chambers rotate clockwise; i.e.Chamber 2 undergoes the reduction step and Chamber 6 performs the oxidation step.The entire sequence of the pumping, purging and preheating steps are repeated in a cyclic manner without having to physically move any of the chambers.As a result, the cyclic operation is achieved by moving the LM heat transfer fluid instead, which allows for efficient recuperation with minimal high temperature moving parts.

MODELLING METHOD
In this section we describe the mathematical model used in our simulations.The reactor chamber can be divided into four layers: the bulk gas, OSM, OSM support, and LM.Our model consists of mass and energy balance equations for all these layers as well as the reaction kinetics.One of the main assumptions in our model is that the behavior of each pipe can be described by that of a single pipe.Under this assumption our model simulates only one single pipe inside of the reactor, as shown in Figure 4 (a), to estimate the performance of the entire chamber.All extensive variables such as flow rates and energy expenditures are multiplied by the total number of pipes in a reaction chamber.This assumption is valid if the gas and LM are distributed equally to all pipes 36 , which can be realized by carefully designing the distributor at the gas inlet and collector at the gas outlet. 37Flow simulations of the base reactor geometry indicated that with four symmetric inlet and outlet ducts, the flow was quite uniform and only resulted in efficiency changes ~ 1%.

MODELLING ASSUMPTIONS
To further simplify our model, the following assumptions were made for the bulk gas layer:  The ideal gas law is valid everywhere.
 The pressure drop in the axial direction in the chamber is negligible.Our estimation using empirical correlations suggests that the pressure drop is below 10% for the range of flow rates considered in this study, and the influence of the pressure change of such Pumping Purging Preheating a magnitude on the heat and mass balances is negligible.
 The radial distribution of 2 O P within the bulk gas phase, outside the boundary layer, can be neglected.This assumes the gas temperature and concentration are functions of z and t but not R .
 Diffusion and heat conduction in the axial direction of bulk gas flow are neglected, since the gas density and thermal conductivity is low.The volume that the gas occupies around a single OSM pipe is given as free space 36 , which encircles the OSM pipe.The radius of this free space is given as fs R (Figure 4 (a) and (b)).
For the OSM layer, the following assumptions are made:  The OSM layer is a porous medium which consists of OSM particles of the diameter 32 d .
 There is no mass or thermal convective flow inside the pores of the OSM layer, because the pores are sufficiently small. The diffusion rate in the OSM is calculated using Chapman-Enskog theory 38 , while diffusion in the OSM particle is assumed to be significantly faster, such that it presents negligible impedence 9,13 . Mass transfer between the OSM layer and the OSM support layer is neglected; i.e. the OSM support material is non-permeable to H2 and O2, which can be realized by a nonreactive oxide diffusion barrier layer to block the gas penetration.For the LM layer, the following assumptions are made:  The radial temperature distribution of the LM can be ignored; i.e. the LM temperature is a function of z and t but not R .
The last assumption is justified by the high thermal conductivity of the LM, as fully developed laminar flow in a pipe gives

LM Nu 
. 39 This high Nusselt number leads to a heat transfer coefficient of approximately 12,000 W m -2 K -1 .The thermal resistance of LM with such a high conductivity is negligible compared to the conductive resistance of the OSM layer, which has thermal conductivities of only up to 1.0 W m -1 K -1 , and thicknesses of 3 mm respectively.Under these assumptions, the simulation of one single pipe can be modeled in a two dimensional spatial domain as shown in Figure 4(b).

ENERGY BALANCE EQUATIONS
The heat transfer between the OSM and LM can be characterized by the heat transfer coefficient LM h (computed from LM Nu ), such that the energy balance equation across the boundary between the LM and OSM support can be written as The energy balance within the OSM support layer is given by: To obtain the temperature of the OSM the heat balance equation within the OSM phase can be written as: Here, we assume that the heat conduction in the OSM phase can be characterized by the overall thermal conductivity, OSM  , which incorporates the effects of the lower thermal conductivity of the gas filling the pore space.The right hand side of this equation is the source term due to the heat of reaction, where i r is the production rate of oxygen or hydrogen and i H  is the corresponding heat of reaction (Equation ( 1) and ( 2)).
The energy balance in bulk gas layer is given by:   The heat transfer coefficient gas h is computed as: 41 gas OSM 1/3 gas gas where: The boundary conditions necessary to solve these set of equations are given in Table 1.
Table 1 Boundary condition to solve energy balance equations.
where inlet T is inlet temperature of LM and purge gas, which depends on the specific step of the cycle.We assume that in the reduction step,

MASS BALANCE EQUATIONS
The O2 or H2 molecules produced in the OSM layer are first transferred from the surface of a ceria particle to the top of the OSM layer via diffusion.For this diffusion, the effective diffusion coefficient is represented by eff D , which is discussed in detail in the supplementary information.The mass balance in the OSM layer is given by: where  is the porosity of OSM, i r is the reaction rate and we used the same reaction kinetics expression in Keene et al. 13 The gas molecules travel across the boundary layer between the bulk gas layer and OSM, where the rate is characterized by the mass transfer coefficient gas k .The mass balance equation is given by:   ,gas ,gas gas gas ,gas gas OSM ,gas ,O OSM SM , , ) ( , ) ( where OSM A is the surface area per unit volume of the OSM layer, given by The gas velocity gas u is obtained from the summation of the mass balance equations for all species:  gas Total Total gas Total gas OSM ,gas ,OSM ( , ) ( The boundary conditions between OSM layer and bulk gas for the mass balance are: This equation is solved with the inlet boundary condition of bulk gas layer, ,gas ,gas, (0, ) The mass transfer coefficients gas k can be computed as

REACTION KINETICS
For the reduction reaction, Eq. ( 1) can be rewritten using Kroger Vink notation as follows: The oxygen production rate can then be described by 13 2 2 The reaction rate constant , where R A is the specific surface area (i.e.surface area of particles per unit volume), and 32 d is the particle diameter of the OSM material.
The equilibrium constant where are enthalpy and entropy of the reduction reaction and are obtained from the experimental data in Panlener 43 .
The partial pressure of oxygen 2 O p is computed from the ideal gas law: The nonstoichiometry ( , , ) R z t  can be computed from the following equation: 13 (1 ) ( , , ) ( , , ) 1 2 ( , , ) 1 0.5 ( , , ) ( , , ) 2 In this reaction rate model, the reaction rate constant In a similar manner to 2 O k  , the reaction rate constant for the .
The change of nonstoichiometry can be given by In this reaction rate model, the reaction rate constant 2 H k  and reaction order m must be obtained from experimental data as described in the following section.

KINETIC PARAMETER ESTIMATION
The equilibrium constant KV K can be modeled by the following equation: 9 where eq  is the equilibrium nonstoichiometry.The experiment reported by Panlener, et al. 43 provides the data necessary to obtain . From the oxygen partial pressure and nonstoichiometry at equilibrium, as shown in Figure 5 one can obtain the plot of lo g ( ) The detailed kinetics of reduction reaction using ceria as an intermediate OSM cannot be found in the literature.Thus here, it was evaluated by fitting the model to the experimental data using a tubular reactor of a packed bed of ceria particles in Chueh and Haile. 8,9In the experiment, the ceria is heated from 800°C to 1500°C.The total gas flow rate is set as 4090 ml min -1 g-OSM -1 with the inlet pressure of O2 of 10 -5 atm.
Assuming the concentration of the gas in the radial direction is uniform, the concentration in the packed bed tube reactor can be described by the following equation:    With the kinetics described by the aforementioned model, the model is complete for reactor design.Nevertheless, it is worth noting that the kinetic model is empirical, and further investigations are needed to quantify the reaction rate more rigorously.

Efficiency Computation
In the proposed design, a general form of the thermal efficiency thermal-chemical  is given by:  34 .
The heat of reaction RXN Q represents the endothermic energy required to liberate oxygen during the reduction step and is given by: The quantity Purge Q accounts for the energy required to preheat the purge gas to L T or H T , and is given by:   gas gas gas gas (1 ) where T  is the ambient temperature, gas  is the efficiency of the gas heat exchanger.

Operating Parameters and Efficiency
We simulated the reduction and oxidation reactions using the mathematical model introduced in Section 3. The two reactions are performed simultaneously in reaction chambers on opposing sides of the circle shown in Figure 2. The reactor design and operating parameters used for this simulation are listed in Table 2.The reactor was designed so that the average fuel output becomes approximately 1.0 kW with maximum efficiency.To minimize heat dissipation from the chamber surface, the surface area must be minimized, and thus a parallel array of tubes with a high surface area per unit volume should be avoided.The length-to-width ratio (L/D) of the reaction chamber is assumed to be 3.0, which is sufficiently small but still may allow us to construct a chamber that would not cause non-uniform distribution of the gas flow.Under such considerations, we used the parameters shown in Table 2.The pipe dimensions ( LM R and Sup R ) were based on low cost thin walled extruded graphite tubes that are commercially available so that the thermal mass is minimized.The number of pipes inside of a chamber is computed based on the size of chamber and size of pipes; no optimization has been done for this parameter and it is treated as a constant.
The operating parameters were also chosen carefully to maximize the efficiency.To save the energy to pump the LM, a small value was chosen for the velocity LM u ; the low velocity would not hinder heat transfer because of the high heat transfer coefficient LM h .Operating parameters for heat recuperation such as preheating time preheat t , initial chamber temperature of reduction step H T  and oxidation step L T  are determined by carrying out cyclic simulation of heat transfer in an eight chamber reactor, and detailed information can be found elsewhere 34 .The duration of the remaining steps shown in Figure 3, where the purging time purge

Reduction reaction
Insulation properties


is 80%.This efficiency is almost one order magnitude higher than the maximum efficiency of a convectional solar thermal chemical reactor design. 12In the next section, with the base case in Table 3, the efficiency bottleneck is identified using a sensitivity analysis.It should finally be noted that there still exist technical and economic uncertainties which must be considered carefully.The proposed reactor concept assumes the hot LM can be circulated between reaction chambers, which requires a thermally stable material to contain the LM.Furthermore, there may be a tradeoff between the overall efficiency and capital cost.In our reactor concept where the area of the OSM in the reactor is independent of the receiver area, the amount of the OSM can be chosen independently from the solar receiver design.To quantify the cost of the OSM, an in depth technoeconomic analysis should be performed, which is beyond the scope of this work specifically.

Heat and Mass Transfer In The Chamber
Achieving fast heating is crucial to generate the vacancies in Step 1 efficiently. 7,9Figure 7 shows the temperature profile of a chamber with the initial temperature of 1410 o C after 40 seconds of heating with the LM at the temperature of 1500 o C. It can be seen that the heat transfer in the radial direction of the OSM is significantly slower than that of the pipe.This is mainly due to the low thermal conductivity of the OSM, which has a porous structure that contains gas internally.Another reason is due to the endothermic reaction which results in a local cooling effect within the OSM.This effect also explains why the temperature in some parts of the OSM region becomes slightly lower than the initial temperature, 1410 °C.However, it is also apparent from Figure 7 that the OSM at bottom left are heated to the desired reaction temperature of 1500 o C in a short amount of time, and the average heating rate of the OSM there can be calculated to be approximately 32 o C/min.Although this heating rate is lower than that of conventional reactor designs, 7 it is sufficient for our reactor concept, because in our eight chamber design, the reaction chamber undergoing reduction only needs to be heated by 90 o C (from TH-=1410 °C to TH=1500 °C) in the preheating step.The heating rate of our reactor design is sufficiently fast, but the rate of oxygen removal becomes the bottleneck to improve the efficiency. 13Figure 8 shows the axial distribution of oxygen partial pressure changes as a function of time in the bulk gas layer.As can be seen in this figure, pO2 increases along the axial domain, since oxygen is accumulated near the outlet of the reactor as the purge gas sweeps through the free volume in the chamber.As the reaction rate decreases with time, the amount of oxygen produced reduces, and this results in a flatter profile of pO2.Nevertheless, the oxygen pressure between z = 0.1 and 0.6 m is still two orders of magnitude larger than the inlet partial pressure, 10 -5 atm, and this high partial pressure of oxygen constrains the oxygen diffusion from OSM layer to the bulk gas layer.This oxygen removal bottleneck is further investigated by analyzing the OSM layer.Figure 9 shows the distribution of P in z indicates that the factor limiting the oxygen removal is not diffusion in the OSM layer but in the bulk gas, as O2 accumulates near the outlet of the reactor (Figure 8).This can be explained by comparing the time scale of diffusion in the radial direction R and convection in the axial direction z.The oxygen diffusion scale in radial direction R is very small for our reactor design due to the high diffusion coefficient at such high temperatures and the low total pressure of the reduction step, which is on the order of Therefore, the removal of oxygen in bulk gas by the convective flow becomes a limiting factor.There are many potential strategies to resolve the bottleneck of oxygen removal.One simple approach to facilitate the oxygen removal is to shorten the reaction chamber length at the expense of lower fuel production rate and potential maldistribution of the gas.Another option is to examine alternative electrochemical based pumping methods, which is out of scope in this study.

Parameter Sensitivity Analysis
A sensitivity analysis is performed to determine the influence of the some materials and system parameters on the reactor efficiency.To check the validity of the efficiency, the sensitivity of one of the most uncertain parameters, the reaction kinetic constant for the reduction step , should be optimized.From Figure 11, we can determine the optimal duration of the pumping step, where all other operation parameters are fixed.In this figure, it can be seen that the efficiency reaches its maximum value at around 8.0 minutes, and decreases slightly thereafter.This is due to the trade-off between (1) increased fuel production, and (2) the preheating energy requirement for the purging gas and heat leakage through the insulated chamber surface.From this analysis, the purging time purging t was determined to be 8.0 minutes.Another critical parameter that determines the efficiency is the reduction temperature of reactor which determines the extent of reduction reaction of ceria at a given pressure of oxygen.Figure 12

Q
. This is because of the efficient heat recuperation enabled by the proposed reactor concept.However, employing a higher temperature becomes unrealistic when one considers the constraints on concentrating the sunlight 44,45 , as well as materials constraints associated with the pumps, valves and piping for the fluids.Another important parameter that affects the efficiency is the gas velocity of the purging gas, which determines how fast oxygen can be removed.As discussed previously, in our reactor design, the oxygen removal occurs in two directions: one is the radial direction, which is mainly determined by the oxygen diffusion in the OSM layer, and the other in the axial direction, which is determined by the bulk gas convection and is influenced significantly by the purging gas velocity.Figure 13 shows how the reactor efficiency and the average nonstoichiometry over a stage, as the purging gas velocity of bulk layer ( gas u ) increases.Here, the average nonstoichiometry  is defined as: In this figure,  always increases for a larger value of the bulk gas velocity, which confirms that removal of oxygen from the bulk gas is a limiting factor.However, this is not the case for the reactor efficiency, thermal-chemical  which remains nearly constant when gas u is over 20 m/s.This is because a larger consumption rate of the purge gas requires more preheating energy for the purge gas.In our reactor design, by employing the LM as a heat transfer fluid, the heat of reaction is supplied efficiently while carrying out sensible heat recuperation simultaneously.However, removal of produced oxygen becomes the bottleneck.A sensitivity analysis is carried out to analyze the effect of some parameters.The most effective way to increase the efficiency is to increase the operating temperature.However, a higher temperatures pose other implementation challenges.The tradeoff between the efficiency and capital cost should be investigated carefully.Nevertheless, to our knowledge the reactor concept presented herein is the first two-step metal oxide reactor to indicate that efficiencies as high as 20% are obtainable by using a full transient model including the reaction kinetics.Thus, the reactor concept presented herein offers an interesting pathway to high efficiency solar fuels reactors.

(
solar-thermal  ) and another for thermal to chemical energy ( thermal-chemical  ), thus expressing the overall efficiency solar-chemical  as the product of these two efficiencies, via solar-chemical solar-thermal thermal-chemical    

Figure 1
Figure 1 New receiver-reactor design concept with two separate devices.

Figure 3 A
Figure 3 A schematic diagram of the reaction and heat recuperation strategy for an eight-chamber reactor system.Red lines denote which pipe channels are open and circulate the LM to exchange heat and bring different reaction chambers into equilibrium.Only the first stage is shown in this figure, while the remaining seven stages follow a similar sequence.Each stage includes three steps, pumping, purging and preheating.

2 OP
. During this step, the heat recuperation and oxidation reaction in Chamber 5 still continue in the same way as in the pumping step.
a) four layers of one tube inside of the reactor chamber.(b) Two-dimensional cross-sectional view of single pipe.

Figure 5
Figure 5 Equilibrium nonstoichiometry plotted against partialpressure of O2 at different temperature in 750-1500 °C, the close to the line represents corresponding temperature for the unit of °C .43

i r z t are 2 Ok 2 Ok
given in Eq.(17) and Eq.(21), respectively.The parameter is obtained using the parameter estimation in gPROMS, which minimizes the difference between the model prediction and the experimentally observed values in the O2 generation rate obtained at the exit of the tubular reactor.From this minimization, is estimated to be 499 kmol m -3 s -1 .

Figure 6 ( 2 -
Figure 6 (a) shows the comparison of O2 generation rate at the exit of reactor from model with

Figure 6 ( 2 -.
Figure 6 (a) Comparison of outlet O2 generation rate in the reduction reaction at the exit of reactor from the model and experimental data with the estimated parameter 2 purge gas used for oxidation and reduction reactions, respectively, and G

Figure 7
Figure 7 The temperature distribution of LM, OSM support, OSM and gas in the reduction reaction, after 40s of heating with LM at the initial temperature of 1410 o C.

Figure 8
Figure 8 Distribution of oxygen partial pressure in the bulk gas along the z direction at different times during the reduction reaction.

2 OP 2 OP 2 OP 2 OP
, temperature OSM T and nonstoichiometry  in the OSM layer at 3 min and 8 min after the reduction step begins.At 3 min (Figure 9(a)), a large gradient in oxygen partial pressure is observed in the axial direction z, while the temperature has a steep gradient in the radial direction R .Due to these spatial gradients, the nonstoichiometry  is the lowest at the top right corner, and the highest at the bottom left corner.At 8 min (Figure 9(b)), the still has a gradient in z while the temperature profile is almost uniform within the range of 4°C.Thus the nonstoichiometry (  ) distribution is almost entirely determined by , which is almost uniform in the radial direction but still has a large gradient in the axial direction.This large gradient of 2 O

.
This is significantly smaller than the characteristic time scale of convection in bulk gas which is on the order of Convection gas

function of the kinetic constant 2 OkFigure 9
Figure 9 Distribution of partial pressure of oxygen, temperature and nonstoichiometry in OSM layer at time (a) 3 min and (b) 8 min in the reduction reaction.The influence of the overall reaction time, which is determined by the pumping time pumping t

Figure 10 Figure 11
Figure 10 Sensitivity of efficiency against the kinetic constant for reduction reaction, 2 O k .
shows the influence of the reduction temperature H T on the efficiency th e rm a l-c h e m ic a l  for a fixed value of the oxidation temperature L T .As we can see from this figure, with the increase of reduction temperature, the efficiency of reactor increases almost linearly in this temperature range, despite the significant increase in the sensible heat R eh eat

Figure 12
Figure 12 Reactor efficiency versus operating temperature in reduction step.

Figure 13
Figure 13 Reactor efficiency and average nonstoichiometry versus bulk gas purging velocity.CONCLUSION A mathematical model is developed to describe a new solar thermal chemical reactor using an LM as a heat transfer fluid.Reaction kinetic models for both reduction and oxidation are obtained by fitting to experimental data.The transient model includes the heat, mass transfer and reaction kinetics which allows us to analyze the efficiency of the new reactor concept and to investigate limiting factors.The reactor can achieve the thermal to chemical efficiency of approximately 20%, and the overall efficiency of approximately 16% under some assumptions, which is nearly one order of magnitude higher than reported values in the literature.In our reactor design, by employing the LM as a heat transfer fluid, the heat of reaction is supplied efficiently while carrying out sensible heat recuperation simultaneously.However, removal of produced oxygen becomes the bottleneck.A sensitivity analysis is carried out to analyze the effect of some parameters.The most effective way to increase the efficiency is to increase the operating temperature.However, a higher temperatures pose other implementation challenges.The tradeoff between the efficiency and capital cost should be investigated carefully.Nevertheless, to our knowledge the reactor concept presented herein is the first two-step metal oxide reactor to indicate that efficiencies as high as 20% are obtainable by using a full transient model including the reaction kinetics.Thus, the reactor concept presented herein offers an interesting pathway to high efficiency solar fuels reactors.

nN
Total number of moles of gas produced in a stage by a chamber pipe Number

Table 3 .
The thermal to chemical efficiency thermal-chemical