Kinetics and Structural Optimization of Cobalt-Oxide Honeycomb Structures Based on Thermochemical Heat Storage

Abstract

Thermochemical heat storage is an important solar-heat-storage technology with a high temperature and high energy density, which has attracted increasing attention and research in recent years. The mono-metallic redox pair Co₃O₄/CoO realizes heat storage and exothermic process through a reversible redox reaction. Its basic principle is to store energy by heat absorption through a reduction reaction during high-irradiation hours (high temperature) and then release heat through an exothermic-oxidation reaction during low-irradiation hours (low temperature). This paper presents the design of a cobalt-oxide honeycomb structure, which is extruded from pure Co₃O₄, a porous media with a high heat-storage density and a high conversion rate. Based on the experimental data, a three-dimensional axisymmetric multi-physics numerical model was developed to simulate the flow, heat transfer, mass transfer, and chemical reaction in the thermochemical heat-storage reactor. Unlike the previous treatment approach of equating chemical reactions with surface reactions, the model in this paper considers the consumption and generation of solids and the diffusion and transfer of oxygen in the porous medium during the reaction process, which brings the simulation results closer to the real values. Finally, the influence of the physical parameters of the honeycomb-structured body on the storage and exothermic process is explored in a wide range. The simulation results show that the physical-parameter settings and structural design of the cobalt-oxide honeycomb structure used in this paper are reasonable, and are conducive to improving its charging/discharging performance.

1. Introduction

The development of sustainable energy systems based on renewable energy sources is important for reducing fossil-fuel dependence and reducing carbon-dioxide emissions. Solar energy is a renewable energy source with high levels of energy [1], but it is volatile and intermittent due to weather, geography, human activities, and other factors. Solar thermal-storage systems improve the stability of solar-energy systems by storing heat during the time in the sun (the “charging” phase) and releasing it when there is a shortage of sunlight (the “discharging” phase) [2].

At present, there are three main thermal-energy-storage methods: sensible energy storage (SES), latent energy storage (LES), and thermochemical energy storage (TCES) [3]. Compared with SES and LES, TCES systems have the advantages of high energy-storage temperature, high energy density, and low heat loss in the energy-storage process. The energy-storage density of TCES systems is 5 to 10 times higher than in LES and SES [4]. In addition, TCES has the advantages of long-term energy storage and long-distance transportation, so it has received increasing attention and research in recent years.

The TCES-system types include metal hydrides, metal sulfates, metal carbonates, metal hydroxides, metal oxides, ammonia, organic, etc. [4,5]. Compared with other types of TCES system, metal-oxide systems use open-loop operation and air as the reactant and heat-transfer fluid. The structures of these systems are relatively simpler and do not require gas storage, heat exchangers, or other devices, and the redox-reaction-temperature range is compatible with that of concentrated solar power (CSP) plants [6].

In 2008, General Atomics conducted a comprehensive screening and study of TCES materials, including 16 pure metal oxides, and cobalt oxide was found to be the best choice in terms of reaction completion and re-oxidation kinetics [7].

The mono-metallic redox pair Co₃O₄/CoO has a very high energy density (∆H_298.15 K = 844 kJ/kg), as well as excellent reversibility and recyclability, and is currently recognized as one of the most promising high-temperature redox pairs [8]. Therefore, researchers conducted tests and studies on Co₃O₄/CoO systems in different forms, such as: Co₃O₄ powder, honeycomb support coated with Co₃O₄, and extruded porous structures made entirely of pure Co₃O₄.

Müller et al. [9] combined in situ X-ray diffraction with thermogravimetry and differential scanning calorimetry to study the redox reactions of cobalt-oxide powder under different conditions, and the results showed that the isothermal energy-storage cycling of the cobalt-oxide system is feasible. Muroyama et al. [10] found that the total mass change of cobalt-oxide powder remained stable over 10 cycles, indicating that the cobalt-oxide powder has excellent reversibility.

While earlier studies focused more on storage systems using the pure cobalt-oxide-powder form, more recent studies have focused on cobalt-oxide-storage systems using forms such as pellets and honeycombs. Karagiannakis et al. [11] conducted a study on cobalt-oxide pellets, in which no significant degradation was observed after 10 redox cycles, and the energy-storage density tested was about 525 kJ/kg. In comparison, the cobalt-oxide-storage density in the honeycomb form used by Pagkoura et al. [12] was 549 kJ/kg, which was greater than that of the pellet and powder forms. Similarly, the honeycomb form of Co₃O₄ also has the ability to perform 10 redox cycles. It is worth noting that for larger CoO particles, complete oxidation is not possible due to the formation of a Co₃O₄ layer on the particle surface [7].

Agrafiotis et al. [13] tested cyclic-reduction-oxidation operations on silicon carbide foams and honeycombs coated with Co3O4 and showed that this support exhibited repeatable, quantitative, cyclic-reduction–oxidation behavior. Schrader et al. [14] investigated the feasibility of Co₃O₄ redox energy-storage systems for large-scale applications and confirmed their feasibility in a thermodynamic analysis. Tescari et al. [15] conducted an experimental study of cordierite honeycomb coated with nearly 90 kg of Co₃O₄. The test results showed that 57% of the material reacts during charging and 51% during discharging, but the overall volumetric energy density of the reactor is lower compared to that of pure cobalt oxide because the majority of the reactor volume is occupied by the cordierite honeycomb. Pagkoura C et al. [16] designed a pure Co₃O₄ porous structure, which had a good reaction performance, but the thin lamellar structure is a great test of mechanical strength and is not convenient for stacking for scale-up applications. Based on this, a Co₃O₄ honeycomb structure was designed in this paper. It is prepared from pure cobalt oxide and has high volumetric energy density, high mechanical strength, and large capacity-expansion flexibility.

Kinetic studies of Co₃O₄/CoO redox energy-storage systems are still limited. Wong et al. [7] found that the reduction and oxidation processes of Co₃O₄/CoO are related to heat-transfer and diffusion mechanisms, respectively, and calculated the activation energy of the reduction process. Muroyama et al. [10] performed kinetic experimental tests on cobalt-oxide powder and obtained the apparent kinetic parameters of the Co₃O₄‘s thermal decomposition process and the CoO’s oxidation process. Singh et al. [17] constructed a 3D model of the reaction kinetics of cordierite honeycomb coated with Co₃O₄. Pagkoura et al. [16] constructed a simple chemical-reaction model for pure Co₃O₄ porous structures. However, these models neglected mass-transfer processes, such as Co₃O₄/CoO solid consumption and generation processes and instead equated the reaction inside the solid with the surface reaction. In this paper, the construction of a three-dimensional simulation model of a Co₃O₄ honeycomb structure is reported, which can better describe the heat-storage-and-release process by considering the mass-transfer process and the porous-media properties of the honeycomb structure. In addition, the influence of the physical parameters of the honeycomb structure on the process of charging/discharging is explored.

2. Experimental Setup

2.1. Honeycomb-Structure-Body Preparation

In this study, cobalt-oxide honeycomb structure was prepared by extrusion and high-temperature calcination process, which was based on technical-grade cobalt-oxide powder (Co₃O₄, CAS: 1308-06-1, 99.0% metal basis, Aladdin Industrial Corporation) and polyvinyl alcohol pellets ((C₂H₄O)_x, CAS: 9002-89-5, Sinopharm Chemical Reagent Co., Ltd., Shanghai, China) as raw materials. The polyvinyl alcohol aqueous solution with a mass fraction of 20% was prepared by adding deionized water to polyvinyl alcohol pellets, which acted as binders at room temperature and underwent a decomposition reaction above 200 °C.

Parts of polyvinyl alcohol aqueous solution were added to the cobalt-oxide powder one by one and stirred thoroughly, after which the stirred mixture was placed in the mold and extruded by hydraulic press to obtain the green bodies with honeycomb structures. The green bodies were soft in texture, and had good mechanical strength and thermal shock resistance after high-temperature drying and calcination under air atmosphere.

The high-temperature drying-and-calcination process is shown in

Table 1. High-temperature drying-and-calcination process for green bodies of cobalt-oxide honeycomb structures.

Serial Number	Temperature/°C	Constant Temperature Holding Time/h
1	200	0.5
2	400	0.5
3	800	8

, and the heating rate between the process steps was 10 °C/min.

The dimensions of the cobalt-oxide honeycomb structure are shown in Figure 1a, with a width and length of 32 mm, a length of 30 mm, a central channel and a sidewall-surface-channel radius of 2.5 mm, and a single volume of approximately 22,959.36 mm³. In this study, a total of four cobalt-oxide honeycomb structures were prepared for the study of heat-storage and release properties, with masses of 69.92 g, 70.43 g, 70.53 g, and 70.92 g, respectively, and the total mass was 281.8 g. The successfully prepared cobalt-oxide honeycomb structures are shown in Figure 1b, and the XRD results of the formed cobalt-oxide honeycomb structure are shown in Figure S1.

2.2. Honeycomb-Structure-Body Charging/Discharging Test Platform

The cobalt oxide (Co₃O₄/CoO) system completes the charging (heat storage) and discharging (heat release) processes through redox reactions. In the high-temperature state, the charging step is completed by the reduction reaction based on cobalt oxide, and, similarly, the discharging step is completed by the oxidation reaction at a low temperature (compared to the reduction reaction temperature). The basic reactions of the charging and discharging processes can be expressed as [15]:

$$\mathrm{C}{\mathrm{o}}_3{\mathrm{O}}_4\leftrightarrow 3\mathrm{C}\mathrm{o}\mathrm{O}+1/2{\mathrm{O}}_2\hspace{1em}∆{\mathrm{H}}_{298.15 \mathrm{K}}=844 \mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}$$

(1)

Reaction equilibrium temperature ${\mathrm{T}}_{\mathrm{e}\mathrm{q}}=890°\mathrm{C}$.

During the charging/discharging cycle, the reaction is always accompanied by changes in oxygen-volume fraction, and with the premise that the reactor inlet-gas state remains stable, the reactor-outlet-gas oxygen-volume fraction ${\phi }_{{\mathrm{O}}_2}$ can characterize the instantaneous state and the overall process of the reaction in the following way:

$$n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}=n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}-(2\ast {\int }_{t_{ini}}^t{\phi }_{{\mathrm{O}}_2}dt\ast Q_V\ast {\rho }_{{\mathrm{O}}_{2,outlet}})/M_{{\mathrm{O}}_2}$$

(2)

$$n_{\mathrm{C}\mathrm{o}\mathrm{O}}=6\ast {\int }_{t_{ini}}^t{\phi }_{{\mathrm{O}}_2}dt\ast Q_V\ast {\rho }_{{\mathrm{O}}_{2,outlet}}/M_{{\mathrm{O}}_2}$$

(3)

where: $n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$ is Co₃O₄, the instantaneous amount of substance; $\mathrm{m}\mathrm{o}\mathrm{l}$; $n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}$ is Co₃O₄, the initial amount of substance; $\mathrm{m}\mathrm{o}\mathrm{l}$; $t_{ini}$ is the reaction start time, s; $t$ is some time in the reaction process, s; ${\phi }_{{\mathrm{O}}_2}$ is the volume fraction of oxygen in the reactor-outlet gas; $Q_V$ is the reactor-inlet air (20 °C) flow rate, $\mathrm{L}/\mathrm{m}\mathrm{i}\mathrm{n}$; ${\rho }_{{\mathrm{O}}_{2,outlet}}$ is the density of oxygen at the temperature and pressure of the outlet gas, $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$; $M_{{\mathrm{O}}_2}$ is the molar mass of oxygen, 32 $\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$; and $n_{\mathrm{C}\mathrm{o}\mathrm{O}}$ is CoO the instantaneous amount of substance, mol.

Moreover, the oxygen-volume fraction is a convenient physical quantity to collect in this experimental setting. Based on this, a honeycomb-structure-body charging/discharging test platform was designed in this study, and the test platform’s composition is shown in Figure 2.

In the complete test platform, an air compressor powers the air flow, which is measured and controlled by a mass-flow meter (OMEGA, model FMA-2600A). The test sample is placed in a tube furnace (HF-Kejing, model OTF-1200X) corundum tube, which is wrapped with electric heating wires and insulation material. The temperature of the corundum tube wall can be programmed and the cobalt-oxide-honeycomb structure undergoes a reduction/oxidation reaction within a specific temperature range, accompanied by oxygen release/absorption, completing the heat-charging/discharging process. The reacted gas is first cooled to about room temperature by a cooler and then flows through a filter (to prevent system failure due to entrained cobalt oxide particles in the gas) into an oxygen analyzer (Signal, model 8000 M), which records the oxygen content of the outlet gas in real time. Finally, the treated exhaust gas is re-emitted to the atmosphere.

Three K-type thermocoupled temperature-measurement points were arranged on the test bench, one for measuring the temperature of the outer wall of the corundum tube in which the sample section was placed, and the other two for measuring the temperature of the channel of the honeycomb structure closest to the outlet. The temperature-measurement points were arranged in the center channel and the lower right channel, as shown in Figure 1a. In order to minimize the influence of thermocouples on air flow, the thermocouple diameter in the channel of the honeycomb structure was only 0.5 mm.

3. Numerical Model

3.1. Reactor-Geometry Model

Due to the symmetry of the reactor and the honeycomb structure, an axisymmetric 3D model was created. Figure 3 shows the computational domain and symmetry plane of the model geometry. The reactor has two symmetry planes, the xy plane and the xz plane. Air flows from the inlet into the corundum tube, reacts with the honeycomb structure inside the corundum tube, and finally exits through the outlet. The length of the corundum tube is 400 mm and the diameter is 50 mm; the four honeycomb structures are placed in the center of the corundum tube with a spacing S = 5 mm; the diameters of the air-inlet and -outlet pipes are 8 mm. After the mesh-irrelevance check, the computational domain is finally divided into 227,015 grid cells, and the fineness is increased near the interface.

3.2. Flow Model

The continuity equation (Equation (4)) and Navier–Stokes equation (Equation (5)) were used to describe the free flow in the reactor, and the continuity equation (Equation (7)) and Brinkman equation (Equation (8)) were used to describe the flow in the porous region [18].

$$\frac{\partial \rho }{\partial t}+\nabla ·\left(\rho \mathit{u}\right)=0$$

(4)

$$\rho \frac{\partial \mathit{u}}{\partial t}+\nabla ·(\rho \mathit{u}\mathit{u})=-\nabla p+\nabla ·\mathit{\tau }+\mathit{F}$$

(5)

$$\mathit{\tau }=\mu (\nabla \mathit{u}+{\left(\nabla \mathit{u}\right)}^T)-\frac{2}{3}\mu (\nabla ·\mathit{u})\mathit{I}$$

(6)

$$\frac{\partial }{\partial t}\left({\epsilon }_p\rho \right)+\nabla ·\left(\rho \mathit{u}\right)=Q_m$$

(7)

$$\frac{\rho }{{\epsilon }_p}\frac{\partial \mathit{u}}{\partial t}+\frac{\rho }{{{\epsilon }_p}^2}(\mathit{u}·\nabla )\mathit{u}=-\nabla p+\nabla ·\frac{\mathit{\tau }}{{\epsilon }_p}-\left(\mu {\kappa }^{-1}+\frac{Q_m}{{{\epsilon }_p}^2}\right)\mathit{u}+\mathit{F}$$

(8)

$${\epsilon }_p=1-\frac{n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}\ast M_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}}{V\ast {\rho }_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}}-\frac{3\ast (n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}-n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4})\ast M_{\mathrm{C}\mathrm{o}\mathrm{O}}}{V\ast {\rho }_{\mathrm{C}\mathrm{o}\mathrm{O}}}$$

(9)

where: $\rho$ is the fluid density, $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$; u is the fluid velocity vector, $\mathrm{m}/\mathrm{s}$; $Q_m$ is the mass source or sink, $\mathrm{k}\mathrm{g}/({\mathrm{m}}^3·\mathrm{s})$; $p$ is the fluid pressure, $\mathrm{P}\mathrm{a}$; τ is the viscous stress tensor, $\mathrm{P}\mathrm{a}$; $\mathsf{\mu }$ is the fluid dynamic viscosity, $\mathrm{P}\mathrm{a}·\mathrm{s}$; I is the unit tensor; ${\mathsf{\epsilon }}_{\mathrm{p}}$ is the solid-phase porosity of porous media (i.e., excluding the central channel and sidewall channel); $\mathsf{\kappa }$ is the permeability of porous media, ${\mathrm{m}}^2$; F is gravity and other bulk forces, $\mathrm{k}\mathrm{g}/({\mathrm{m}}^2·{\mathrm{s}}^2)$; $M_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$ is Co₃O₄ molar mass, 240.8 $\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$; V is the total solid-phase volume of the honeycomb structure, ${\mathrm{m}}^3$; ${\rho }_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$ is Co₃O₄ density, $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$; $M_{\mathrm{C}\mathrm{o}\mathrm{O}}$ is the CoO molar mass, 74.93 $\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$; and ${\mathsf{\rho }}_{\mathrm{C}\mathrm{o}\mathrm{O}}$ is CoO density, $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$.

The Brinkman equation describes a fluid in a porous medium, and the model is an extension of Darcy’s law, with the inclusion of a term considering viscous transport in the momentum balance, and the physical interface is well adapted to the transition between the slow flow in a porous medium described by Darcy’s law and the fast flow in a channel described by the Navier–Stokes equation. This treatment was used by Le Bars et al. [19].

The initial velocity field and gauge pressure of the flow model are 0. The inlet boundary condition is the flow rate, $Q_V=10 \mathrm{L}/\mathrm{m}\mathrm{i}\mathrm{n}$, and the outlet boundary condition is the pressure; the gauge pressure $p=0$.

3.3. Heat-Transfer Model

The equations of the heat-transfer model [20,21] are as follows:

$$\rho C_p\left(\frac{\partial T}{\partial t}+\mathit{u}·\nabla T\right)+\nabla ·\mathit{q}=Q+Q_{vd}$$

(10)

where $C_p$ is the specific heat capacity at constant pressure of the fluid, $\mathrm{J}/(\mathrm{k}\mathrm{g}·\mathrm{K})$; $T$ is the thermodynamic temperature, $\mathrm{K}$; q is the thermal-conductivity heat flux, $\mathrm{W}/{\mathrm{m}}^2$; $Q$ is the heat source other than viscous dissipation, $\mathrm{W}/{\mathrm{m}}^3$; and $Q_{vd}$ is the viscous dissipation in the fluid, $\mathrm{W}/{\mathrm{m}}^3$.

The material properties of Co₃O₄, CoO and air are shown in Table S1.

The heat-transfer equation in porous media [18] is as follows:

$${\left(\rho C_p\right)}_{eff}\frac{\partial T}{\partial t}+\rho C_p\mathit{u}·\nabla T+\nabla ·\mathit{q}=Q$$

(11)

$$\mathit{q}=-{\lambda }_{eff}\nabla T$$

(12)

where ${\left(\rho C_p\right)}_{eff}$ is the effective volumetric heat capacity at constant pressure, $\mathrm{J}/({\mathrm{m}}^3·\mathrm{K})$. When no immobile fluid is present in the pore space, the effective volumetric heat capacity at constant pressure is defined as:

$${\left(\rho C_p\right)}_{eff}={\theta }_s{\rho }_s{C}_{p,s}+{\epsilon }_p{\rho }_f{C}_{p,f}$$

(13)

$${\theta }_s=1-{\epsilon }_p$$

(14)

where θ_s is the volume fraction of solids matrix in porous media and ${\lambda }_{eff}$ is the effective thermal conductivity, $\mathrm{W}/(\mathrm{m}·\mathrm{K})$.

$${\lambda }_{eff}={\theta }_s{\lambda }_s+{\epsilon }_p{\lambda }_f$$

(15)

The initial temperature of the heat-transfer model is 293.15 $\mathrm{K}$. The inlet boundary condition is the temperature, $\mathrm{T}=293.15 \mathrm{K}$. The outer wall of the corundum tube is set up with the temperature boundary condition according to the test temperature.

3.4. Reaction Kinetic Model

The kinetic equation describing the reaction in the homogeneous phase is as follows [22]:

$$\frac{dc}{dt}=k(T)f(c)$$

(16)

where: $c$ is the reactant concentration, $\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^3$; $k\left(T\right)$ is the temperature dependence of rate constants; $f\left(c\right)$ is the reaction-mechanism function, which is generally expressed in the form of reaction-order in homogeneous reaction, $f\left(c\right)={(1-c)}^x$; and $x$ is the reaction-order.

When studying the solid-state reactions with the basic concepts of kinetics, since the concentration concept in homogeneous systems is no longer applicable in solid-state reactions, the conversion rate (α) was used instead. The conversion rate is the percentage conversion of reactants to products, which indicates the degree of progress of a reaction in a nonhomogeneous system. Accordingly, the kinetic mode function f(α) is used to replace the reaction-mechanism function in the homogeneous system [23].

The rate constant $k$ in the kinetic equation is very closely related to the temperature. The rate constant–temperature relation which Arrhenius proposed by simulating the equilibrium constant–temperature relation is more commonly used:

$$k=Aexp(-\frac{E}{RT})$$

(17)

where: $A$ is the prefactor, ${\mathrm{s}}^{-1}$; $E$ is the activation energy, $\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}$; and $R$ is the molar gas constant, 8.314472 $\mathrm{J}/(\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K})$.

Therefore, the kinetic equation of the gas–solid reaction studied in this paper can be expressed as:

$$\frac{d\alpha }{dt}=Aexp(-\frac{E}{RT})f(\alpha )$$

(18)

The purpose of the kinetic study is to obtain the “kinetic triplet” (E, A and f(α)) [24] in the above equation, which describes the cobalt-oxide-charging and -discharging reactions.

Although there is a large number of mathematical models for kinetic mode functions f(α), these models can be grouped into three main types: accelerated, decelerated, and s-type (sometimes called autocatalytic) [25]. As can be seen in Section 4.1, the charging and discharging processes of the cobalt-oxide honeycomb structures were all consistent with the decelerating model, which represents a process in which the reaction rate reaches a maximum at the beginning of the process and decreases as the degree of conversion increases. The most common example is the reaction-order model:

f(α) = (1 − α) ^x

(19)

The oxygen-reaction rate is low relative to the oxygen-flow rate in the flowing gas. Therefore, the effect of the oxygen concentration on the reaction rate can be neglected in the reaction kinetic model [16].

In summary, the kinetic equation for the cobalt-oxide reaction can be expressed as:

$$r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}=k_{ox}\left(T\right)\ast f\left({\alpha }_{ox}\right)\ast n_{\mathrm{C}\mathrm{o}\mathrm{O}}-k_{re}\left(T\right)\ast f\left({\alpha }_{re}\right)\ast n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$$

(20)

$$r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}=A_{ox}exp\left(-\frac{E_{ox}}{RT}\right)\ast {\left(1-{\alpha }_{ox}\right)}^y\ast n_{\mathrm{C}\mathrm{o}\mathrm{O}}-A_{re}exp\left(-\frac{E_{re}}{RT}\right)\ast {\left(1-{\alpha }_{re}\right)}^x\ast n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$$

(21)

$${\alpha }_{re}=1-n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}/n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}$$

(22)

$${\alpha }_{ox}=1-n_{\mathrm{C}\mathrm{o}\mathrm{O}}/n_{\mathrm{C}\mathrm{o}\mathrm{O},ini}$$

(23)

where:$r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$ is the reaction rate of cobalt oxide, mol/s, in which positive value represents the consumption of ${\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4$ and negative value represents the formation of ${\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4$; α_re is the conversion rate in the charging process, hereafter referred to as the conversion rate; α_ox is the conversion rate in the discharge process; and $n_{\mathrm{C}\mathrm{o}\mathrm{O},ini}$ is the initial amount of CoO during the discharge process, mol.

The initial state of the honeycomb structure is pure Co₃O₄, and all the CoO is reduced by the Co₃O₄. Consequently,

$$n_{\mathrm{C}\mathrm{o}\mathrm{O}}=3\ast (n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}-n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4})$$

(24)

$$n_{\mathrm{C}\mathrm{o}\mathrm{O},ini}=3\ast n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}$$

(25)

Therefore,

$${\alpha }_{ox}=1-3\ast \left(n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}-n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}\right)/(3\ast n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini})=1-{\alpha }_{re}$$

(26)

$$r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}=({3\ast A}_{ox}exp\left(-\frac{E_{ox}}{RT}\right)\ast {{\alpha }_{re}}^{y+1}-A_{re}exp\left(-\frac{E_{re}}{RT}\right)\ast {\left(1-{\alpha }_{re}\right)}^{x+1})\ast n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}$$

(27)

$${r_{\mathrm{C}\mathrm{o}\mathrm{O}}=-3\ast r}_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$$

(28)

$$r_{{\mathrm{O}}_2}=-0.5\ast r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}$$

(29)

The heat source other than viscous dissipation can be expressed as:

$$Q=r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}\ast ∆H\ast M_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}/V$$

(30)

where a positive value of Q means heat release and a negative value means heat storage.

The initial oxygen-volume fraction of the reaction kinetic model and the inlet boundary condition are the oxygen-volume fraction in the atmosphere.

During the charging process, solid Co₃O₄ decomposes to form solid CoO and gaseous oxygen, and during the discharging process, solid CoO and gaseous oxygen combine to reform Co₃O₄ solid. For reasons such as the fact that simulation tools do not support reactions that consume solid materials, previous studies [12,17] modeled the redox reaction as a surface reaction by assuming that all the cobalt oxide is deposited on the surface of the porous medium. Obviously, this modeling approach is not applicable to cobalt-oxide honeycomb structures. In this study, the process of solid consumption and oxygen diffusion is considered to improve the reaction kinetic model, based on which the porosity of the porous medium is also no longer a constant, but changes in real time as the reaction proceeds.

4. Results and Discussion

4.1. Experimental Results

The charging/discharging performance of the cobalt-oxide honeycomb structure was investigated on the aforementioned test platform at temperature windows of 1000–800 °C, 990–800 °C, and 970–800 °C, respectively. The honeycomb structure of the cobalt oxide did not show obvious appearance changes after 10 cycles. The inlet-air-flow rate of the reactor was $Q_V=10 \mathrm{L}/\mathrm{m}\mathrm{i}\mathrm{n}$ (20 °C), the tube-furnace-heating/cooling rate was 10 °C/min, and constant temperatures were maintained for 90 min in each of the high- and low-temperature sections of the test’s temperature window. The tube-furnace-temperature settings for the 1000–800 °C temperature window are shown in

Table 2. 1000–800 °C temperature-window tube-furnace-temperature settings.

Serial Number	Time t/min	Temperature/K
1	0–98	293.15 + 10 × t
2	98–188	1273.15
3	188–208	1273.15 − 10 × (t − 188)
4	208–298	1073.15

, and the other temperature windows settings are similar.

The tube-furnace set temperature, cobalt-oxide-honeycomb-structure channel temperature, and reactor-outlet-gas oxygen-volume-fraction curves at different temperature windows are shown in Figure 4. The outlet-gas oxygen-volume fraction changed greatly during the temperature-rise-and-fall cycle, indicating that the honeycomb structure underwent an obvious redox reaction. The temperatures of the two measurement points in the channel of the honeycomb structure were close to each other, and the channel temperature in Figure 4 is the average of the temperatures of the two measurement points. The appearance and volume of the cobalt-oxide honeycomb structure did not change significantly after the charging/discharging cycle.

Integrating the variation curve of the oxygen-volume fraction of the reactor -outlet gas gives the total mass of oxygen released by the reduction process and absorbed by the oxidation process, which in turn gives the mass of Co₃O₄ consumed in the reduction reaction and generated in the oxidation reaction. Finally, the reduction-conversion rate and -re-oxidation rate (the ratio of the oxidation -conversion rate to the reduction -conversion rate) of the complete reaction process was obtained. The results of the charging and discharging at different temperature windows are shown in

Table 3. Reaction processes within different temperature windows.

Temperature Window/°C	Reduction Start Channel Temperature/K	Conversion Rate	Oxidation Start Channel Temperature/K	Re-Oxidation Rate	Volumetric Energy Density kWh/m3
1000–800	1143.9	71.65%	1229.3	96.08%	340.71
990–800	1145.8	56.59%	1226.2	96.63%	270.46
970–800	1141.9	55.40%	1207.9	98.31%	264.74

. The temperature was the driving force of the reduction reaction, and the charging process started at a temperature range of roughly 1141.9–1145.8 K. As the temperature increased, the conversion rate and volumetric energy density of the honeycomb structure increased significantly. The conversion rate was 71.65% and the volumetric energy density was 340.71 kWh/m³ within the temperature window of 1000–800 °C. The re-oxidation rate of the honeycomb structure was maintained above 96% within different temperature windows, indicating that it has good reversibility.

It is worth noting that when the volumetric energy density in Table 3 was calculated, the volume included the solid-phase volume of the cobalt-oxide honeycomb structure, the volume of the channels, and the volume of the arrangement spacing between the honeycomb structures, and the energy was considered only for the thermochemical energy storage, on which the volumetric energy density calculations below are based. If only the honeycomb solid-phase volume is considered, the volume energy density is 512.86 kWh/m³ within the 1000–800 °C temperature window.

The conversion rate used by Tescari S et al. [9] within the temperature window of 1030–680 °C (the most favorable temperature range for the reaction) was about 57%, the re-oxidation rate was about 89.47%, and the volumetric energy density was about 150.56 kWh/m³. This volumetric energy density included both the thermochemical thermal storage and the sensible thermal storage. In comparison, the pure cobalt-oxide honeycomb structure had a stronger charging/discharging performance.

4.2. Comparison of Simulation Results with Experimental Results

In this paper, a three-dimensional axisymmetric multi-physics field-coupled numerical model was established in COMSOL Multiphysics was used to simulate the flow, heat transfer, mass transfer, and chemical reactions in the thermochemical heat-storage reactor. Based on the charging/discharging test data on the cobalt-oxide honeycomb structure at 970–800 °C, 990–800 °C, and 1000–800 °C, the “kinetic triplet” in Equation (21) was solved, and the results are shown in

Table 4. Results of kinetic triplet solution.

Reaction Process	$\mathbf{A}$$/{\mathbf{s}}^{-1}$	$\mathbf{E}$$/(\mathbf{J}/\mathbf{m}\mathbf{o}\mathbf{l})$	$\mathbf{f}\left(\mathsf{\alpha }\right)$
Reduction	$2·{10}^{30}$	781,560.368	${\left(1-{\alpha }_{re}\right)}^{6.4}$
Oxidation	$8/3·{10}^{-42}$	−814,818.256	${\left(1-{\alpha }_{ox}\right)}^4$

.

By substituting the solution results into Equation (27), the reaction kinetics equation of the cobalt-oxide honeycomb structure is obtained as follows:

$$r_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4}=(8\ast {10}^{-42}\ast exp\left(\frac{98000}{T}\right)\ast {{\alpha }_{re}}^5-2\ast {10}^{30}\ast exp\left(-\frac{94000}{T}\right)\ast {\left(1-{\alpha }_{re}\right)}^{7.4})\ast n_{{\mathrm{C}\mathrm{o}}_3{\mathrm{O}}_4,ini}$$

(31)

The simulation model takes into account the solid consumption and generation of the honeycomb structure during the charging/discharging process, which is accompanied by the change in porosity of the porous medium. Therefore, the porosity changes from a constant to a variable and is calculated by Equation (9).

The volume fraction of the oxygen in the reactor-outlet gas can characterize the transient state and overall process of the charging/discharging reaction. Temperature is an important factor affecting the charging/discharging reaction, but the temperature distribution within the reactor and honeycomb structure is not uniform. This paper uses the reactor-outlet oxygen-volume fraction as the accuracy-verification variable of the simulation model, while the reactor-outlet-gas oxygen-volume fraction is the inevitable result of the charging/discharging reaction, which is easy and reliable to measure.

The simulation and experimental results of the charging/discharging process of the cobalt-oxide honeycomb-structured body within three sets of temperature windows are shown in Figure 5. It can be seen that the simulation results agreed well with the experimental values, and the reaction process was accompanied by a change in porosity.

The root mean square error (RMSE) and coefficient of determination (R²) were used to characterize the degree of fit between the simulation results and the experimental results. Within the three sets of temperature windows, the RMSE of the simulation model simulating the oxygen-volume fraction at the outlet of the reactor in the charging/discharging process of the cobalt-oxide honeycomb structure did not exceed 0.279%, and the R² exceeded 0.9673, which indicated that the simulation model is accurate and can better describe the charging/discharging dynamic process of cobalt-oxide honeycomb. In particular, the simulation accuracy of the oxidation process was slightly lower than that of the reduction process. In addition, the simulation model had the highest accuracy in the temperature window of 970–800 °C, with RMSE = 0.081% and R² = 0.9913 for the complete cycle process.

4.3. Sensitivity Analysis

The reactor has a better performance within the temperature window of 1000–800 °C, but this places a greater burden on the solar heat absorber and involves a higher temperature resistance for the pipe materials, as well as a better thermal insulation performance from the reactor, which is often difficult to achieve in practice. Therefore, this paper explores the effects of the physical parameters, such as the initial porosity, central-channel and sidewall-channel diameter (referred to as bore diameter, D), arrangement spacing (S), and length (L), on the charging/discharging processes in a wide range within the temperature window of 970–800 °C, in order to obtain a better reaction performance.

The initial porosity of the cobalt-oxide honeycomb structure during the test was 49.78%, the bore diameter was 5 mm, the length was 30 mm, and the arrangement spacing was 5 mm.

The initial porosity is determined by the initial total mass and the total solid-phase volume of the honeycomb structure. In this study, the initial total mass was controlled to ensure it was constant, so the initial porosity was determined by the total solid-phase volume of the honeycomb structure. The length of the honeycomb structure was one of the studied variables and was expected to be consistent with the experimental value. Therefore, the volume of a single honeycomb structure was kept constant, and the total volume of the solid phase of the honeycomb structure was adjusted by increasing or decreasing the number of honeycomb structures, which in turn changed the initial porosity of the honeycomb structure. The number of honeycomb structures was 3, 4, 6, 8, and 10, corresponding to initial porosities of 33.04%, 49.78%, 66.52%, 74.89%, and 79.91%, respectively. The simulation results are shown in Figure 6. Overall, the re-oxidation rate is less affected by the physical parameters mentioned above and can always be maintained at a high level. The trends of the conversion rate and volumetric energy density of the cobalt-oxide honeycomb-structure body were opposite to each other: increasing the initial porosity, the bore diameter, and the arrangement spacing or decreasing the length increased the conversion rate, but the volumetric energy density consequently decreased. This pattern was especially obvious when the initial porosity changed: when the initial porosity increased from 33.04% to 79.91%, the conversion rate increased from 48.4% to 68.36% (growth rate, 41.24%), while the volumetric energy density decreased from 312.546 kWh to 127.843 kWh (decrease rate, 59.1%). In particular, the volumetric energy density decreased rapidly when the initial porosity exceeded 49.78%, as shown in Figure 6A.

The bore-diameter change was also achieved by changing the number of honeycomb structures. When 3, 4, 5, 6, and 8 honeycomb structures were used, these numbers correspond to pore sizes of 0.12 mm, 5.00 mm, 6.32 mm, 7.07 mm, and 7.90 mm, respectively. The changes in pore size had a significant effect on the honeycomb-storage/exhaust-heat performance. When increasing the pore size, the oxygen release/absorption increased significantly, the conversion rate and the re-oxidation rate increased, and the volumetric energy density decreased, as shown in Figure 6B. When the pore size was increased from 0.12 mm to 7.90 mm, the conversion rate of the honeycomb structure increased from 44.66% to 63.23% (growth rate, 41.58%), the re-oxidation rate increased from 93.30% to 98.56% (growth rate, 5.64%), and the volumetric energy density decreased from 288.125 kWh to 148.339 kWh (decrease rate, 48.52%). When the bore diameter was larger than 5mm, the volumetric energy density decreased rapidly with the increasing pore size.

The arrangement spacing is the distance between the adjacent honeycomb structures along the corundum tube’s axial direction. The cobalt-oxide honeycomb structure does not achieve the best performance at the spacing distance S = 0. The conversion rate and volumetric energy density corresponding to S = 1 mm are higher than S = 0. When the spacing distance is 1 mm, the volumetric energy density is 264.777 kWh, as shown in Figure 6C. When the arrangement spacing is greater than 10 mm, the conversion rate increases slowly in line with the increase in the arrangement spacing, but the volumetric energy density decreases rapidly, which does not help to improve the overall system performance. The volumetric energy density at S = 5 mm is close to the maximum value at S = 1 mm, but the conversion rate is 3.65% higher than that at S = 1 mm.

When studying the influence of the length (L) on the charging and discharging process of the honeycomb structure, the total mass and solid volume were controlled unchanged, and the length of the honeycomb structure was adjusted by increasing or decreasing the number of honeycomb structures. When the number of honeycomb structures was 1, 2, 3, 4, 6, and 12, the initial lengths were 120 mm, 60 mm, 40 mm, 30 mm, 20 mm, and 10 mm, respectively. Reducing the length of the honeycomb structure can improve the conversion rate, but this increases the difficulty of arranging the honeycomb structure inside the reactor, especially for large-capacity reactors. When the length of the honeycomb structure’s body is greater than 30 mm, the volumetric energy density tends to stabilize as the length continues to increase, but the conversion rate gradually decreases, which adversely affects the reaction process, as shown in Figure 6D.

Taking the physical parameters and reaction results of the cobalt-oxide honeycomb structure prepared in this paper as the benchmark, the conversion rate, volumetric energy density, and re-oxidation rate responded to the sizes of the physical-parameter variations, as shown in Figure 7. The conversion rate is sensitive to the changes in the initial porosity and bore size, especially the initial porosity, and is less sensitive to the length and arrangement spacing in comparison. In addition, the conversion rate is more sensitive to length and alignment spacing becoming smaller than length and alignment spacing becoming larger. Similar to the conversion rate, the volumetric energy density is sensitive to the initial porosity and bore-size changes and responds more significantly to parameter changes. However, the volumetric energy density is less sensitive when the length increases or the arrangement spacing decreases. In contrast, the re-oxidation rate is less sensitive to variations in the physical parameters.

5. Conclusions

In this study, cobalt-oxide honeycomb structures were prepared by extrusion and high-temperature calcination processes. Within the temperature windows of 970–800 °C, 990–800 °C and 1000–800 °C, the volume fraction of the oxygen in the reactor-outlet gas changed greatly, indicating that the honeycomb structures underwent a significant redox reaction, completing the charging (heat storage) and discharging (heat release) processes.

Temperature is the driving force of the charging process, and the charging process starts at a temperature range of roughly 1141.9–1145.8 K. As the temperature increased, the conversion rate and volumetric energy density of the honeycomb structure increased significantly, with a conversion rate of 71.65% and a volumetric energy density of 340.71 kWh/m³ within the 1000–800 °C temperature window. The re-oxidation rate of the honeycomb structure was maintained above 96% within different temperature windows, indicating its excellent reversibility.

Based on the experimental data, a three-dimensional axisymmetric multi-physics field-coupled numerical model was established to simulate the charging and discharging process of the cobalt-oxide honeycomb structure, and the simulation model can describe the charging and discharging process more accurately by considering the change in porosity caused by the mass-transfer process. The RMSE did not exceed 0.279% and the R² exceeded 0.9673 in all three temperature windows, with the highest accuracy in the temperature window of 970–800 °C, where RMSE = 0.081% and R² = 0.9913 during the complete cycle.

Finally, a sensitivity analysis of the cobalt-oxide honeycomb structure was performed, and the results showed that the conversion rate and volumetric energy density were sensitive to variations in the initial porosity and bore diameter, and were less sensitive to the length and arrangement spacing in comparison. The re-oxidation rate always remained above 92% and was less sensitive to changes in the physical parameters. Increasing the initial porosity, the bore diameter, and the arrangement spacing or decreasing the length increased the conversion rate, but the volumetric energy density consequently decreased. Furthermore, the simulation results showed that the physical-parameter settings (${\epsilon }_p$= 49.78%, D = 5 mm, L = 30 mm, and S = 5 mm) and structural design of the cobalt-oxide honeycomb structure used in this paper are reasonable, and are conducive to improving its charging/discharging performance.