Numerical study of a copper oxide-based thermochemical heat storage system

Numerical study of a copper oxide-based thermochemical heat storage system Zhen Cao, Bas Joris de Leeuw , Tianchao Xie , Abhishek K. Singh * Department of Thermal and Fluid Engineering, University of Twente, Drienerlolaan 5, 7522 NB Enschede, the Netherlands ARTICLE INFO Keywords: Numerical model Thermochemical heat storage Metal oxide High temperature ABSTRACT A multi-physics model is developed to investigate the performance of high-temperature thermochemical heat storage using the redox looping cycle of CuO/Cu 2 O. The model involved flow in free and porous domains, heat transfer in the CuO pellet-packed porous domain (i.e., convection between fluid and pellets, conduction and radiation among pellets), and the endothermic/ exothermic reaction. The reaction rate was estimated using a non-parametric kinetic approach which depends on temperature and extent of the reaction. The model was validated within <10 % error margin by the experimental measurements of the temperature inside the reactor and the molar fraction of O 2 at the reactor outlet. The validated model is used to determine the temperature variation and reaction evolution in the pellet-packed domain. In the end, parameter studies were implemented, including inlet mass flow rate, reduction temperature, and oxidation temperature. It was found that a large inlet mass flow brings about a high output temperature, and the reaction runs faster with the larger inlet mass flow. Similarly, increasing the furnace temperature during the reduction process (reduction temperature) also increases the output temperature and accelerates the reaction. In contrast, increasing the furnace temperature during the oxidation process (oxidation temperature) only slightly affected the reaction in the present case. This model could provide useful insights into reactor design, scale-up, and operating conditions to improve the energy storage system performance. Nomenclature A s specific surface area of the reactive bed, (m −1 ) C p specific heat, (J/kg⋅K) c surf_CuO surface site density of CuO, (mol/m 2 ) c i molar concentration of species i, (mol/m 3 ) c pe, i volumetric site density of species, i (mol/m 3 ) Ddiameter, (m) Girradiation, (W/m 2 ) H 0 standard enthalpy change of the reaction at 298.15 K h fc heat transfer coefficient between the reactor wall and gas, (W/m 2 ⋅K) h nc heat transfer coefficient on the external insulation wall, (W/m 2 ⋅K) Joutgoing radiation, (W/m 2 ) nrefractive index (continued on next page) * Corresponding author. E-mail address: [email protected] (A.K. Singh). Contents lists available at ScienceDirect Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite https://doi.org/10.1016/j.csite.2024.105315 Received 16 August 2024; Received in revised form 7 October 2024; Accepted 20 October 2024 Case Studies in Thermal Engineering 63 (2024) 105315 Available online 21 October 2024 2214-157X/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). (continued) qheat flux (W/m 2 ) R i volumetric reaction rate of species i, (mol/m 3 ⋅s) R pe,i surface reaction rate of species i, (mol/m 2 ⋅s) rdimensionless radius in Eq. (15), and dimensionless fact in Eq. (20) S m mass source term due to the reaction, (kg/m 3 ⋅s) S h heat source term due to the reaction, (W/m 3 ) Ttemperature, (K) Vvolumetric flow rate, (L/min) Greek symbols α extent of reaction α fl interstitial heat transfer coefficient, (W/m 2 ⋅K) βRosseland mean extinction coefficient ε porosity κpermeability λthermal conductivity, (W/m⋅K) μ dynamic viscosity, (Pa⋅s) ρ density, (kg/m 3 ) σ Stepfan-Boltzmann constant (W/m 2 ⋅K 4 ) ψ reaction conversion Subscript f fluid s solid ins insulation ms modified solid pe pellet i species rm reactive material rb reactive bed ib inert bed qw quartz wool 1. Introduction Thermal energy storage (TES) is a key and promising technology to secure the reliable utilization of renewable energy sources such as solar and wind for heat and power applications. The electricity or heat from renewable resources can be stored in the form of heat and be flexibly released according to the demand, effectively overcoming its inherent challenge of intermittency and demand-supply mismatch. Accordingly, national and international initiatives have been initiated to support TES to accelerate renewable energy deployment for decarbonization and carbon footprint reduction, including REPowerEU in the European Union [1], the Inflation Reduction Act (IRA) in the United States [2], and the 14th Five-Year Plan for Renewable Energy in China [3]. Depending on the storage principle, TES can be categorized as sensible heat storage (SHS), latent heat storage (LHS), and thermochemical energy storage (TCES). SHS has been developed on a commercial scale, e.g., molten salts for concentrated solar power (CSP) plants, which stores heat by increasing temperature and has a simple design. However, SHS usually has a low energy storage capacity (~180 MJ/m 3 ) [4] and suffers from salt corrosiveness, high material cost, and large storage volumes. In comparison, by employing phase change materials (PCM), LHS achieves a doubled energy storage capacity of ~360 MJ/m 3 and works at near-constant temperatures [5], but it has risks of leakage, phase segregation, container corrosion, etc. On the contrary, TCES based on endothermic/exothermic reactions, is promising because of several benefits, such as high energy storage capacity (~1800 MJ/m 3 [6]), high thermal stability, and working temperatures (>900 K), and low losses. Thus, TCES is currently being broadly explored and it includes gas-gas reactions (e.g., ammonia synthesis), gas-liquid reactions (e.g., ammonium hydrogen sulfate), and gas-solid reactions (e.g., redox system) [7]. Among these, the gas-solid TCES is preferable to modulate renewable energy input/output, and typical reactive materials contain hydroxides, metal hydrides, carbonates 3 , perovskites, and metal oxides [8]. For example, Han et al. [9] investigated the decomposition performance and reaction kinetics of magnesium hydroxide for thermochemical heat storage and constructed a pilot-scale reactor to explore the heat storage performance [10]. It is feasible to scale such TCES up for large-scale utilization due to their abundant availability, thermal/chemical stability, and potential low cost per unit of stored heat. Hydroxides and metal hydrides work at relatively low temperatures, and hydroxides require extra water storage and steam generation sections while hydrides and carbonates need additional hydrogen storage and CO 2 storage, respectively. Such characteristics make these systems more complex and limit their applications to some extent. In contrast, perovskites and metal oxides-based systems perform charging/discharging by redox reactions in which oxygen is available from ambient air, and no separate oxygen storage is needed. Furthermore, they operate at comparatively higher temperatures >1000 K [6], making it a promising technology to secure renewable energy deployment, e.g., concentrated solar power (CSP) applications. Perovskites are nonstoichiometric redox materials which have continuous/quasi-linear partial reduction over a wide temperature range. Babiniec et al. [11] examined the potential of La-Sr-Co-Mn and La-Sr-Co-Fe-based perovskites as thermochemical heat storage materials and obtained a maximum reaction enthalpy of 250 kJ/kg. Chen et al. [12] investigated Sr-based perovskites for heat storage at medium-to-high temperatures, and SrFeO 3 exhibited excellent redox capacity. In addition, CaMnO 3 is identified as one of the most promising perovskites of twelve Ca-Mn and Sr-Fe-based compositions in terms of heat storage Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 2 performance [13]. A high reaction enthalpy of 390 kJ/kg was achieved by Al/Ti-doped CaMnO 3 [14], while Mastronardo et al. [15] demonstrated favourable thermodynamic properties of Fe-doped CaMnO 3 . Pein et al. [16], for the first time, tested the thermochemical heat storage performance of foams made of CaMnO 3 , a large step to a large-scale demonstration of monolithic perovskite structures. In contrast to the perovskites, the metal oxides-based redox takes place at specific equilibrium temperatures. Wong [17] investigated 74 pure metal oxides by thermodynamic calculations and yielded 16 pure metal oxides for TCES. Cr 5 O 12 , Li 2 O 2, and Mg 2 O were further excluded because of their low equilibrium reaction temperatures. Fig. 1 comprehensively compares the material cost, energy density, and reaction temperature among the screened 13 metal oxides. It shows that PtO 2 , Rh 2 O 3 , and UO 3 are too expensive. which cost two orders of magnitude above the U.S. Department of Energy (DOE) target of $15/kWh. In contrast, BaO, Co 3 O 4 , CuO, Mn 2 O 3 , Mn 3 O 4 , Fe 2 O 3 , and V 2 O 5 have acceptable costs and preferable energy density, which are suitable for TCES of high temperatures (>1000 K). Agrafiotis et al. [18] carried out lab-scale studies of Co 3 O 4 -coated and Mn 2 O 3 -coated foams and honeycombs. They proposed cascaded various metal oxides to overcome thermocline temperature distribution within storage blocks. Tescari et al. [19] conducted the first pilot study of Co 3 O 4 -based TCES, with the storage unit made of cordierite honeycomb coated with 88 kg of cobalt oxide. The performed 22 charging/discharging cycles achieved a high storage efficiency of 86 %. Carrilo et al. [20] revisited the BaO 2 /BaO cycle for TCES by TGA measurements within a temperature range of 773–1173 K, which showed excellent reversibility over 30 cycles and a high energy density of 390 kJ/kg. Karagiannakis et al. [21] experimentally compared the redox performance of Co 3 O 4 /CoO and Mn 2 O 3 /Mn3O 4 . It was concluded that cobalt oxide was superior to manganese oxide concerning energy density and redox kinetics. Furthermore, the small particles of Mn 2 O 3 /Mn 3 O 4 were prone to sintering and degrading their oxidation kinetics because oxygen diffusion was hindered [22] while doping with other metal oxides could modulate surface morphology and boost oxidation kinetics [23]. Hayes et al. [24] presented an experimental demonstration of MgO/MnO mixture-based TCES on a moving-bed reactor which achieved high-temperature (>1273 K) heat storage. Rahmatian et al. [25], for the first time, investigated the TCES performance of MgO/MnO mixtures in a pressure vessel that closely approximates a realistic engineering implementation of a redox storage module, obtaining an average energy storage density of 2428 ±469 MJ/m 3 in working temperatures of 1273–1773 K and working pressure of 0.18–11 bar. CuO/Cu 2 O works at temperatures of around 1300 K which is preferred by emerging CSP plants with high solar power, and it has higher energy storage density and is cheaper than many other metal oxides like BaO 2 /BaO, Mn 2 O 3 /Mn3O 4 , and Co 3 O 4 /CoO [17]. Alonso et al. [26] implemented the first experimental campaign of the solar redox reaction of copper oxides in which they developed a rotary kiln storage reactor heated by a solar furnace. It was found that in comparison with the reduction in Argon atmosphere, the redox cycle in air resulted in strong particle coalescence which inhibited the redox reaction. Deutsch et al. [27] investigated the redox kinetics of CuO/Cu 2 O by TGA and fixed-bed reactor and developed kinetics models using the non-parametric kinetic (NPK) approach. As mentioned before, the sintering of metal oxides degrades the redox performance, and it can be eliminated to some extent by performing the reduction in a non-oxygen atmosphere inside a rotary kiln reactor. Sintering could be a critical issue for the CuO/Cu 2 O because its reduction temperature (~1300 K) is very close to the Cu 2 O melting point (~1500 K). Doping CuO with some sintering-resistant materials is a good way to address this problem. Gigantino et al. [28] prepared yttria-stabilized zirconia (YSZ)-doped CuO granules. The granules exhibited high redox reversibility over 100 cycles and achieved an energy storage density of 470–615 kJ/kg. As indicated by the literature aforementioned, tremendous progress has been made regarding metal oxides for high-temperature TCES, explored mostly by TGA analysis and lab-scale reactors. This is an important step towards the practical application of this technology, but many essential insights into reactors are still missing, and not easy to obtain experimentally, e.g., local temperature, species transport, and reaction advancement in reactors, as well as effects of working conditions on the reactor performance. Therefore, numerical models are considerably required to explore these insights, disclosing flow dynamics, heat/mass transfer, and reaction evolution inside reactors, and then shedding light on the reactor design and optimization. Tescari et al. [29] numerically studied the thermal behaviour of a rotary kiln reactor using Co 3 O 4 as the reactive material, with particular attention to radiation modelling. In addition to the rotary kiln reactor, fixed-bed reactors are extensively investigated numerically. To improve gas/solid heat transfer, cordierite honeycombs and foams were used as the reaction bed which was coated with reactive materials [18]. Numerical simulations have been carried out to reveal the thermochemical behaviour of Co 3 O 4 /CoO in honeycomb-bed reactors [30–32], and Fig. 1. Energy density against material cost (data from Ref. [17]). Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 3 parametric studies, e.g., porosity, were conducted using the validated model. These studies provided detailed information on the temperature and reaction evolution at the reduction/oxidation stage. In addition, a pellet bed is another type of reactor, in which reactive materials are processed as pellets, including fluidized pellet bed and pellet-packed bed [33]. Wang et al. [34] established a heat and mass transfer model to investigate the thermal reduction of iron–manganese oxide particles in a high-temperature packed-bed solar thermochemical reactor, giving a comprehensive description of the transport phenomena in the reactor. Huang et al. [35] developed a high-fidelity 1D model for a moving magnesium-manganese-oxide bed which presented good predictions of local temperatures and oxygen concentrations at the outlet. Korba et al. [36] experimentally and numerically investigated a moving-bed magnesium-manganese-oxide reactor for thermochemical energy storage aiming to achieve high-temperature heat extraction of more than 1273 K. The performance of the reactor was further investigated through two parametric studies (particle flow rates and gas extraction ratios), which demonstrated a strong dependence of reactor efficiency on both operating parameters. Wang et al. [37] proposed a CFD model to reveal the heat transfer mechanism and heat storage performance of calcium hydroxide/oxide in a shell-tube thermochemical energy storage device. Zheng et al. [38] proposed a computational fluid dynamics-discrete element method (CFD-DEM) model to study the TCES performance of fluidized CaCO 3 pellets under direct concentrated solar irradiation. The model not only reveals the thermal behaviour of the reaction bed reliably but also illustrates the moving of pellets. Hamidi et al. [39] experimentally and numerically explored the reduction of iron-manganese oxide particles in a packed-bed reactor, in which a 2D transient model was developed, accounting for heat transfer, mass transfer, and thermochemical reaction kinetics. The model was capable of predicting oxygen concentration and reaction conversion. A reliable numerical model is a must for accurate prediction of storage reactor outputs. As mentioned earlier, CuO is a highly suitable material for CSP applications because of its high volumetric energy density (~2558.7 MJ/m 3 ), low cost (~$30.5/MJ), and high reaction temperature (~1393.15 K). The TCES performance of CuO has been experimentally investigated in the literature, but it still lacks numerical modelling to have more insights into the thermal redox behaviour in a storage reactor. Therefore, in the present study, a numerical model for the CuO pellet-packed bed is developed and validated against experimental measurements. The validated model is used to enhance the understanding of the thermal and reactive behaviour of the bed. Further insights into ways to improve and optimize the reactor are provided by performing a parametric study using the validated model. 2. Numerical modelling A lab-scale CuO pellet-packed bed reactor was designed and experimentally investigated by Gigantino et al. [28] in which the thermochemical redox cycle of CuO/Cu 2 O pair was examined in isobaric and isothermal operational modes. In the isobaric redox cycles, each cycle consisted of a charging phase during which the air inflow temperature was increased to a desired set point above the onset temperature of reduction, followed by a discharging phase during which the air inflow temperature was decreased to a set-point value of 870 ◦C. In the isothermal redox cycles, the cycle was performed by varying the oxygen partial pressure of the inlet gas flow while keeping its temperature constant. The reactor was initially preheated with an airflow of constant temperature reaching a quasi-steady state. Then gas flow was switched to nitrogen to activate the reduction and air to trigger the oxidation. Charging and discharging phases were alternated by switching the gas inflow to nitrogen and air, respectively. Fig. 2 shows the schematics of the experimental apparatus mainly including a pellet-packed reactor, a heating element, insulation, Fig. 2. Schematics of the reaction unit (adapted from Ref. [28]). Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 4 etc. The reactor is made of a quartz tube with an inner diameter of 50 mm. The tube was packed with CuO pellets which were synthesized using CuO powders and Y 2 O 3 -stablized ZrO 2 (YSZ) powders, i.e., 50 wt% CuO-YSZ pellets. The CuO pellets mentioned in the following represent 50 wt% CuO-YSZ pellets. The reactive bed composed of CuO pellets has a height of 40 mm and a porosity of 0.4067. To diminish radiation heat loss from the CuO pellets to the ambient, YSZ beads were packed right above the CuO pellets, i.e., an inert bed of a height of 60 mm and a porosity of 0.3833. Meanwhile, the reactive bed was supported by a porous quartz disk. Gas, flowing through a silicon-infiltrated silicon carbide (SiSiC), was heated by an electric furnace which mimicked concentrated solar heating, and then went into the reactive bed crossing quartz wool which helped to distribute the flow uniformly. The reactor was thermally insulated using multi-layer insulation to decrease heat loss, including glass wool blanket, pyrogenic insulation, and fire bricks. The present simulation was carried out based on the reactor shown in Fig. 2, and Table 1 summarizes important dimensions of the reactor. 2.1. Model description To reliably simulate heat and mass transfer in the thermochemical heat storage unit as described in Fig. 2, several physical processes need to be considered, e.g., gas flow in free zones, gas flow in porous zones (SiSiC foam, quartz wool, quartz disk, reactive bed, and inert bed), heat transport in the pellet-packed bed, and chemical reactions in the bed. 2.1.1. Fluid flow model Referring to the volumetric flow rate in reference [28], Reynolds number was calculated to determine the flow modes, i.e., laminar or turbulent, as Re= ρ 4V/( π D2 rb)Drb μ (1) Where, the volumetric flow rate V is 10 L/min, and the reactor diameter D rb is 50 mm. Accordingly, the Reynolds number was estimated as 272, using air properties at 25 ◦C. Because the air density decreases and viscosity increases with increasing temperature, the Reynolds number is even lower at the practical working temperature to stimulate the redox reaction. Therefore, laminar flow was assumed for the present free and porous flow. The flow in the free and the porous domain was described by the Navier-Stokes equations and the Brinkman equations, respectively. Thus. •For the free-media flow, the governing equations are as follows: ∂ρ ∂ t+ ∇ • ( ρ U) = 0 (2) ρ∂ U ∂ t+ ρ U• ∇U= − ∇p+ ∇ • [ μ (∇U+ ∇UT)−2 3 μ (∇ • U)I]+ ρ g(3) •For the porous-media flow, the governing equations are as follows: ∂ερ ∂ t+ ∇ • ( ρ U) = Sm(4) Table 1 A few geometric parameters of the TCES unit. Reactive bed (packed 50 wt% CuO-YSZ pellets) The diameter of the CuO-YSZ pellets, D pe 1 mm The porosity of the CuO-YSZ pellets, ε pe 0.63 The weight of the reactive material, m rm 107 g The porosity of the reactive bed, ε rb 0.4067 The initial surface site density of CuO, c surf_CuO_init 21490 mol/m 2 The specific surface area of the reactive bed, A s 3.984 ×10 7 m −1 The diameter of the reactive bed, D rb 50 mm The height of the reactive bed, h rb 40 mm Inert bed (packed YSZ beads) and others The diameter of the YSZ beads, D bead 2.3 mm The porosity of the inert bed, ε ib 0.3833 The height of the inert bed, h ib 60 mm The height of the quartz disk, h qd 2.5 mm The porosity of the quartz disk, ε qd 0.1 The height of the quartz wool, h qw 20 mm The porosity of the quartz wool, ε qw 0.9682 The porosity of the SiSiC foam, ε SiSiC 0.84 The diameter of the gas inlet, D inlet 14 mm Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 5 ρ∂ U ∂ t+ ρ U• ∇U ε = − ε ∇p+ ∇ • [ μ (∇U+ ∇UT)−2 3 μ (∇ • U)I]−( εμ κ+Sm ε )U+ ερ g(5) Where I is the unit tensor. ε is the porosity and κ is the permeability that was estimated by the Kozeny-Carman model. S m is a mass source for species, e.g., oxygen consumption or generation in the reactive porous domain. U is the velocity vector. μ is the fluid viscosity, and ρ is the fluid density. 2.1.2. Heat transfer model In this study, heat transfer was considered in several domains, including thermal insulation, free flow domain, porous reactive domain (CuO pellet-packed domain), and porous non-reactive domains (SiSiC foam, quartz wool, inert bed). •In thermal insulation, there are no convection and heat sources. Heat transfer is determined by thermal conduction, which was expressed as ρ insCp,ins ∂ T ∂ t= ∇ • λins∇T(6) where ρ ins and C p,ins are the density and heat capacity of the thermal insulation material, respectively. λ ins is the thermal conductivity of the thermal insulation material. •In the free flow domain, the rate of energy change depends on the heat flux into the domain by convection and conduction, and the rate of work done. Therefore, the energy equation is given as ρ Cp ∂ T ∂ t+ ρ CpU• ∇T− ∇ • λ∇T=U• ∇p+ ∇ • {[ μ (∇U+ ∇UT)−2 3 μ (∇ • U)I]•U}(7) •In the porous media domain, it was supposed that heat was transported by the thermal convection between the fluid and solid, the thermal conduction, and thermal radiation between pellets, as shown in Fig. 3. Two energy equations were established for the solid Fig. 3. Heat transfer mechanism in the reactive pellet-packed bed. Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 6 and fluid phases, respectively, which were supposed to be in a thermal non-equilibrium state. Accordingly, the energy equation was solved separately for the fluid and solid. ερ Cp ∂ Tf ∂ t+ ρ CpU• ∇Tf− ∇ • ε λ∇Tf=U• ∇p+ ∇ • {[ μ (∇U+ ∇UT)−2 3 μ (∇ • U)I]•U}+ α fsAs(Tf−Ts)+Sh(8) (1− ε ) ρ sCp,s ∂ Ts ∂ t− ∇ • (1− ε )λeff ∇Ts= α fsAs(Ts−Tf)+Sh(9) Where ε is the porosity of the domain. α fs is the interstitial convective heat transfer coefficient. A s is the specific surface area of the porous domain. α fs is the interstitial heat transfer coefficient. S h is the thermochemical heat which only exists in the reactive porous domain while it is excluded from equations (8) and (9) in the other porous domains. It is worth noting that an effective thermal conductivity was used in equation (9). The effective thermal conductivity accounts for the thermal conductivity of the pellet-packed bed and the heat radiation occurring inside the bed. Concerning the pellet-packed bed, its thermal conductivity is not the value of pure materials (pellets) due to its porous characteristics. The thermal conductivity of the bed could be calculated using Krupiczka equation [40], as below λms =λf(λs λf)0.28−0.757 log ε −0.057 log(λs/λf)(10) Meanwhile, the radiative heat transfer cannot be omitted in the porous domain due to high operating temperatures (>1000 K). For the optically thick porous media, the heat radiation can be included by adding a thermal conductivity term in the conduction equation, accounting for the heat radiation. In this study, the Rosseland approximation was used [41], which is formulated as λR=16n2 σ 3 3β(11) Where n is the refractive index which is around 1 for air, and β is the Rosseland extinction coefficient of the solid material. Then the final effective thermal conductivity used in equation (9) is calculated as λeff =λms +λR(12) 2.1.3. Species transport model In this study, heat storage and release were achieved by a reversible redox reaction of CuO/Cu 2 O. The CuO was prepared as pellets that were packed in a reaction bed. In the heat storage (charging) process, N 2 was preheated by an oven at a temperature of 950 ◦C when flowing through the SiSiC foam and then enters the pellet bed to activate the reduction. CuO was reduced to Cu 2 O and O 2 was released. Correspondingly, in the oxidation (discharging) process, air preheated by the oven passes through the pellet bed and reacts with Cu 2 O to release heat. The chemical reaction formula is shown below 4CuO+ΔHr⇌2Cu2O+O2(13) In the simulation, CuO and Cu 2 O were regarded as surface species on pellets. They can be produced and consumed in the reaction. Oxygen was transported by airflow, which could be governed by a species transport equation. The rate of change of oxygen concentration results from the convective transport, species diffusion, and production/consumption due to the chemical reaction. The species transport equation is given as ε∂ ∂ t(ci) + U• ∇ci+ ∇ • Ji=Ri(14) Where c i is the molar concentration of bulk species (mol/m 3 ). J i is the tensor that accounts for diffusion and dispersion, expressed as −(Dd,i+De,i)∇ci, and R i is the volumetric reaction rate (mol/(m 3 ⋅s)) that accounts for the reaction. ε is the porosity of the pellet bed. In addition to oxygen which is bulk species, surface species, i.e., CuO and Cu 2 O, were also considered. The surface density change of the surface species could be expressed by the species governing equation inside the pellet. A few assumptions such as the surface species being immobile and the pellet being spherical, were made to formulate the surface species governing equation. The surface density of the surface species only changes along the radial direction while no density variation in the space-angle direction. Then, the change rate of the surface density depends on the diffusion of the gas into the pellet to advance the reaction, and reaction rate. The equation describing the surface species is written as ε pe ∂ ∂ t(cpe,i)+1 r2 per2 ∂ ∂ r(−r2Dpe,i cpe,i ∂ r)=ApeRs pe,i(15) Where c pe,i is the volumetric site density of the surface species (mol/m 3 ). r pe is the radius of the spherical pellet. A pe is the specific surface area of the pellet (m −1 ). Rs pe,i is the surface reaction rate (mol/(m 2 ⋅s)). ε pe is the porosity of the pellet. 2.1.4. Reaction kinetics model The reaction kinetics of the CuO/Cu 2 O pair has to be correctly modelled so that the reaction rate can be calculated to close the Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 7 governing equations above. The present study used the kinetic model developed by Deutsch et al. [27] who experimentally investigated the reaction kinetics of the CuO/Cu 2 O reaction cycle using a simultaneous thermal analysis and a lab-scale pellet-packed-bed reactor. A non-parametric kinetic (NPK) approach was used to develop the kinetics model. The approach relies on the singular value decomposition for the experimental data processing, and establishes the kinetics model, getting rid of empirical coefficients and the Arrhenius law. Normally, the conversion rate of species in the redox reaction depends on the conversion ( ψ ) and temperature (T), which is expressed as d ψ dt =f( ψ )k(T)(16) The conversion ψ is defined as the ratio of generated oxygen to the theoretical amount of oxygen that can be generated, as discussed in Ref. [27]. The measurement could provide a data set including ψ , T, and d ψ /dt which can be written as a matrix A=⎡ ⎣ f( ψ 1)k(T1)⋯f( ψ 1)k(Tm) ⋮ ⋱ ⋮ f( ψ n)k(T1)⋯f(n)k(Tm) ⎤ ⎦(17) Then the matrix A could be processed by the singular value decomposition (SVD) to reduce the data dimensions and extract the most important information, as A=SWVT=∑ min(m,n) 1 S(i)W(i)VT(i)(18) Where i represents the column number of the matrix. The matrix W is a diagonal matrix and the element value decreases along the diagonal. equation (16) indicates that only the first entry of W is significantly larger than zero. Thus, the matrix A can be written as A=S(1)W(1)VT(1) = swvT(19) Where s and v T are the first column of the matrix S and V T , respectively, while w is the first entry of the matrix W. Deutsch et al. [27] reported the ψ -dependent vector s and T-dependent vector v T for the CuO/Cu 2 O reaction in a pellet-packed bed. Then, a kinetic model was established in this study by extracting data there. A 2D-grid file was generated and uploaded in COMSOL to estimate the conversion rate. Fig. 4 shows the relationship between the conversion rate and the temperature as well as the conversion. In principle, the heat source of the reaction is dependent on the enthalpy of formation of species and the reaction rate of the creation of species. In this study, it is defined as Sh= − d ψ dt csurf CuO init 4•rH0(20) H0 is the standard enthalpy change of the reaction at 298.15 K (283 kJ/mol) which is multiplied by a factor r to account for the enhanced enthalpy change at the equilibrium temperature of the reaction. csurf CuO init is the initial surface concentration density of CuO which is 21490 mol/m 2 . 2.2. Simulation domain and mesh In this study, a 2D numerical model is developed to investigate the thermochemical heat storage performance of CuO/Cu 2 O. Fig. 2 demonstrates the reactor configuration which is the object of interest. Considering the symmetric characteristics of the reactor, half of the reactor was chosen as the simulation domain which can fully present the physical processes but reduce the computational load, as shown in Fig. 5. The simulation domain is composed of a flow channel, a heater, and thermal insulation. The flow channel has an inlet/ outlet free-flow section and porous media zones including SiSiC foam, quartz wool, quartz disk, reactive bed, and inert bed. When the gas passed through the SiSiC foam, it was heated by the heater. The heated gas then reached the reactive bed (CuO pellet-packed bed), having the redox reaction. To reduce heat losses, the reactor was wrapped with three layers of thermal insulation marked by the solid Fig. 4. Reaction kinetics of the CuO/Cu 2 O redox reaction: the dependence of the conversion rate on temperature and conversion. Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 8 green/blue/grey colour. Table 2 summarizes the thermal properties of the materials involved in this study, i.e., CuO/Cu 2 O [42], YSZ [43], SiSiC [44], quartz wool [45], glass wool blanket [46], pyrogenic insulation [47], and fire brick [48]. Quadrilateral meshes were generated separately in the flow domain and non-flow domain. In the flow domain, structured quadrilateral mesh was generated by adding mapped nodes, with fined boundary layer meshes. In the domain of insulation and the cavity with heaters, the unstructured quadrilateral mesh was generated by adding free quad nodes. The maximum mesh size was controlled as 3.0 mm, while the minimum mesh size was controlled as 0.12 mm. The maximum mesh growth rate was 1.3. Fig. 5 also shows the mesh in different domains, with an average element quality of around 0.989, indicating a high quality. 2.3. Boundary conditions To solve the governing equations above, boundary conditions have to be given. Fig. 5 also demonstrates the main boundary conditions including symmetry axis, gas inlet, reactor inlet, gas outlet, reactor outlet, and walls, which are summarized in Table 3. 2.3.1. Inlet and outlet boundary conditions The current model involved multi-physical processes including fluid flow, heat transfer, and species transport (reaction). To solve the continuous equation and momentum equation, a mass flow rate was given at the gas inlet, and a gauge pressure of zero was set at the gas outlet The mass flow rate was calculated based on the volume flow rate of the measurement [28]. To solve the species transport equation, an oxygen concentration was defined at the reactor inlet, and a diffusive transport flux of zero was given at the reactor outlet. The current simulation was for an isokinetic redox cycle, meaning that the gas temperature on the reactor inlet was kept constant but the oxygen concentration (partial pressure) at the reactor inlet was varied. The whole process was divided into three phases, i.e., preheating phase (100 min), reduction phase (120 min), and oxidation phase (30 min) which were the same as those in the measurement. In the preheating phase, the reactor was heated by an airflow, and a nitrogen flow was used in the reduction phase while an air flow was used in the oxidation phase. Accordingly, the oxygen concentration at the reactor inlet was 0 and 8.6 mol/m 3 (0.21 atm) at the reduction and oxidation phases, respectively. In terms of the thermal boundary condition, a constant temperature of 293.15 K was set at the gas inlet, while a temperature gradient of zero was given at the gas outlet. 2.3.2. Wall boundary conditions There were inside walls, external walls, and heater walls which were all non-slippery. The external wall included the side wall, top wall, and top wall on which a convective heat flux boundary condition was applied, accounting for the heat loss, expressed as q= hnc(Tw−Tambient). h nc is the convective heat transfer coefficient. On the side wall, h nc was defined as that of external natural convection on cylinder side surfaces while concerning h nc on the top wall and bottom wall, it was the external natural convective heat transfer coefficient on the top surface and bottom surface of hot plates, respectively. The heater wall had a constant temperature of 950 ◦C to mimic the set-point furnace temperature in the measurement. Since the Fig. 5. Simulation domain and mesh. Table 2 Thermal properties of the materials. Material ρ (kg/m 3 )C p (J/kg⋅K) λ (W/m⋅K) CuO 6315 10 −7 T 3 -0.005T 2 +0.6904T +374.18 33 Cu 2 O 6000 2 ×10 −7 T 3 -0.005T 2 +0.562T +309.28 6.28 YSZ (Y 2 O 3 -stablized ZrO 2 ) 6100 −0.0002T 2 +0.5331T +332.96 2 Reactive pellets 0.5((1α ) ρ _CuO + αρ _Cu2O )+0.5 ρ _YSZ 0.5((1α )C p_CuO + α C p_Cu2O )+0.5 C p_YSZ λ f ×(λ s /λ f ) (0.28-0.757log10( ε rb)−0.057log10(λs/ λf)) λ s =0.5((1α ) λ _CuO + α λ _Cu2O )+0.5 λ _YSZ λ f =0.07821 SiSiC foam 480 700 6 Quartz wool/disk 2210 730 0.038/1.26 Glass wool blanket 128 1130 3 ×10 −7 T 2 -0.0002T +0.0457 Pyrogenic insulation 220 1080 4 ×10 −8 T 2 -0.00005T +0.0299 Fire brick 390 1050 2 ×10 −7 T 2 -0.0002T +0.1452 Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 9 region shrinks. Correspondingly, the oxidation reaction occurs and completes the earliest in the side region near the reactor wall. The reaction front fades inwardly from the side to the middle of the reactor and the oxidation rate near the wall is higher than that near the center. 4) The reduction/oxidation performance is affected by the mass flow rate of gas, the reduction temperature, and the oxidation temperature. Large mass flow rates and high reduction temperatures promote the reaction performance. For example, the large mass flow rate and high reduction temperatures correspond to high output temperature and fast reduction reaction because the large flow rate and high reduction temperature both increase the heating rate and the reactive material temperature. In comparison, the output temperature also increases with increasing oxidation temperature, and the oxidation reaction rate is slightly improved by decreasing the oxidation temperature in the concerned case. Fig. 15. Comparison of the reactive material temperature in conditions of various reduction temperatures. Fig. 16. The effect of the reduction temperature on the output temperature. Fig. 17. The effect of the set-point furnace temperature during the oxidation (oxidation temperature) on the reactor performance: (a) the extent of the reaction, (b) the output temperature. Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 16 CRediT authorship contribution statement Zhen Cao: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis. Bas Joris de Leeuw: Validation, Methodology, Investigation, Formal analysis, Conceptualization. Tianchao Xie: Visualization, Software, Data curation. Abhishek K. Singh: Writing – review & editing, Supervision, Resources, Project administration, Investigation, Funding acquisition, Conceptualization. Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Abhishek Kumar Singh reports financial support was provided by the European Commission, through the HORIZON EUROPE, RIA – Research and InnovationActions programmes HORIZON-CL5-2021-D3-03. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to thank the financial support by the European Commission, through the HORIZON EUROPE, RIA – Research and Innovation Actions programmes HORIZON-CL5-2021-D3-03 under grant agreement No. 101084569 (project ABraytCSPfuture) and HORIZON-CL5-2022-D4-01-05 under grant agreement No. 101104182 (project HERCULES). Data availability Data will be made available on request. References [1] R. Vezzoni, Green growth for whom, how and why? The REPowerEU Plan and the inconsistencies of European Union energy policy, Energy Res. Soc. Sci. 101 (2023) 103134, https://doi.org/10.1016/j.erss.2023.103134. [2] J. Bistline, G. Blanford, M. Brown, D. Burtraw, M. Domeshek, J. Farbes, et al., Emissions and energy impacts of the inflation reduction Act, Science 380 (6652) (2023) 1324–1327, https://doi.org/10.1126/science.adg3781. [3] C. Hepburn, Y. Qi, N. Stern, B. Ward, C. Xie, D. Zenghelis, Towards carbon neutrality and China’s 14th Five-Year Plan: clean energy transition, sustainable urban development, and investment priorities, Environ. Sci. Ecotechnology. 8 (2021) 100130, https://doi.org/10.1016/j.ese.2021.100130. [4] F. Raganati, P. Ammendola, Review of carbonate-based systems for thermochemical energy storage for concentrating solar power applications: state-of-the-art and outlook, Energy & Fuels 37 (3) (2023) 1777–1808. https://orcid.org/0000-0002-2005-3877. [5] S. Bellan, T. Kodama, N. Gokon, K. Matsubara, A review on high-temperature thermochemical heat storage: particle reactors and materials based on solid–gas reactions, WIREs Energy and Environment 11 (5) (2022), https://doi.org/10.1002/wene.440. [6] J. Zhao, D. Korba, A. Mishra, J. Klausner, K. Randhir, N. AuYeung, et al., Particle-based high-temperature thermochemical energy storage reactors, Prog. Energy Combust. Sci. 102 (2024) 101143, https://doi.org/10.1016/j.pecs.2024.101143. [7] Y. Yan, K. Wang, P.T. Clough, E.J. Anthony, Developments in calcium/chemical looping and metal oxide redox cycles for high-temperature thermochemical energy storage: a review, Fuel Process. Technol. 199 (2020) 106280, https://doi.org/10.1016/j.fuproc.2019.106280. [8] S. Wu, C. Zhou, E. Doroodchi, R. Nellore, B. Moghtaderi, A review on high-temperature thermochemical energy storage based on metal oxides redox cycle, Energy Convers. Manag. 168 (2018) 421–453, https://doi.org/10.1016/j.enconman.2018.05.017. [9] X.C. Han, H.J. Xu, W.S. Hua, Decomposition performance and kinetics analysis of magnesium hydroxide regulated with C/N/Ti/Si additives for thermochemical heat storage, Appl. Energy 344 (2023) 121322, https://doi.org/10.1016/j.apenergy.2023.121322. [10] X.C. Han, H. Xu, Y. Li, Experimental investigation on thermochemical reaction with gradient-porosity reactor for medium temperature heat storage applications, J. Energy Storage 78 (2024) 110021, https://doi.org/10.1016/j.est.2023.110021. [11] S.M. Babiniec, E.N. Coker, J.E. Miller, A. Ambrosini, Investigation of La Sr1−Co M1−O3−(M =Mn, Fe) perovskite materials as thermochemical energy storage media, Sol. Energy 118 (2015) 451–459, https://doi.org/10.1016/j.solener.2015.05.040. [12] X. Chen, M. Kubota, S. Yamashita, H. Kita, Investigation of Sr-based perovskites for redox-type thermochemical energy storage media at medium-high temperature, J. Energy Storage 38 (2021) 102501, https://doi.org/10.1016/j.est.2021.102501. [13] M. Pein, C. Agrafiotis, J. Vieten, D. Giasafaki, S. Brendelberger, M. Roeb, et al., Redox thermochemistry of Ca-Mn-based perovskites for oxygen atmosphere control in solar-thermochemical processes, Sol. Energy 198 (2020) 612–622, https://doi.org/10.1016/j.solener.2020.01.088. [14] S.M. Babiniec, E.N. Coker, J.E. Miller, A. Ambrosini, Doped calcium manganites for advanced high-temperature thermochemical energy storage, Int. J. Energy Res. 40 (2) (2016) 280–284, https://doi.org/10.1002/er.3467. [15] E. Mastronardo, X. Qian, J.M. Coronado, S.M. Haile, The favourable thermodynamic properties of Fe-doped CaMnO3 for thermochemical heat storage, J. Mater. Chem. A. 8 (17) (2020) 8503–8517, https://doi.org/10.1039/D0TA02031A. [16] M. Pein, L. Matzel, L. de Oliveira, G. Alkan, A. Francke, P. Mechnich, et al., Reticulated porous perovskite structures for thermochemical solar energy storage, Adv. Energy Mater. 12 (10) (2022), https://doi.org/10.1002/aenm.202102882. [17] B. Wong, Thermochemical Heat Storage for Concentrated Solar Power, Final Report for the US Department of Energy, 2011, https://doi.org/10.2172/1039304. [18] C. Agrafiotis, A. Becker, M. Roeb, C. Sattler, Exploitation of thermochemical cycles based on solid oxide redox systems for thermochemical storage of solar heat. Part 5: testing of porous ceramic honeycomb and foam cascades based on cobalt and manganese oxides for hybrid sensible/thermochemical heat storage, Sol. Energy 139 (2016) 676–694, https://doi.org/10.1016/j.solener.2016.09.013. [19] S. Tescari, A. Singh, C. Agrafiotis, L. de Oliveira, S. Breuer, B. Schl¨ ogl-Knothe, et al., Experimental evaluation of a pilot-scale thermochemical storage system for a concentrated solar power plant, Appl. Energy 189 (2017) 66–75, https://doi.org/10.1016/j.apenergy.2016.12.032. [20] A.J. Carrillo, D. Sastre, D.P. Serrano, P. Pizarro, J.M. Coronado, Revisiting the BaO2/BaO redox cycle for solar thermochemical energy storage, Phys. Chem. Chem. Phys. 18 (11) (2016) 8039–8048, https://doi.org/10.1039/C5CP07777J. [21] G. Karagiannakis, C. Pagkoura, A. Zygogianni, S. Lorentzou, A.G. Konstandopoulos, Monolithic ceramic redox materials for thermochemical heat storage applications in CSP plants, Energy Proc. 49 (2014) 820–829, https://doi.org/10.1016/j.egypro.2014.03.089. [22] A.J. Carrillo, D.P. Serrano, P. Pizarro, J.M. Coronado, Thermochemical heat storage based on the Mn2O3/Mn3O4redox couple: influence of the initial particle size on the morphological evolution and cyclability, J. Mater. Chem. A. 2 (45) (2014) 19435, https://doi.org/10.1039/C4TA03409K, 1943. Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 17 [23] T. Block, M. Schmücker, Metal oxides for thermochemical energy storage: a comparison of several metal oxide systems, Sol. Energy 126 (2016) 195–207, https://doi.org/10.1016/j.solener.2015.12.032. [24] M. Hayes, D. Korba, P. Schimmels, J. Klausner, J. Petrasch, N. AuYeung, et al., Experimental demonstration of high-temperature (>1000 ◦C) heat extraction from a moving-bed oxidation reactor for thermochemical energy storage, Appl. Energy 349 (2023) 121625, https://doi.org/10.1016/j.apenergy.2023.121625. [25] N. Rahmatian, A. Bo, K. Randhir, J.F. Klausner, J. Petrasch, Bench-scale demonstration of thermochemical energy storage using the Magnesium-ManganeseOxide redox system, J. Energy Storage 45 (2022) 103682, https://doi.org/10.1016/j.est.2021.103682. [26] E. Alonso, C. P´ erez-R´ abago, J. Licurgo, E. Fuentealba, C.A. Estrada, First experimental studies of solar redox reactions of copper oxides for thermochemical energy storage, Sol. Energy 115 (2015) 297–305, https://doi.org/10.1016/j.solener.2015.03.005. [27] M. Deutsch, F. Horvath, C. Knoll, D. Lager, C. Gierl-Mayer, P. Weinberger, et al., High-temperature energy storage: kinetic investigations of the CuO/Cu2O reaction cycle, Energy & Fuels 31 (3) (2017) 2324–2334, https://doi.org/10.1021/acs.energyfuels.6b02343. [28] M. Gigantino, S. Sas Brunser, A. Steinfeld, High-temperature thermochemical heat storage via the CuO/Cu2O redox cycle: from material synthesis to packed-bed reactor engineering and cyclic operation, Energy & Fuels 34 (12) (2020) 16772–16782, https://doi.org/10.1021/acs.energyfuels.0c02572. [29] S. Tescari, M. Neises, L. de Oliveira, M. Roeb, C. Sattler, P. Neveu, Thermal model for the optimization of a solar rotary kiln to be used as high temperature thermochemical reactor, Sol. Energy 95 (2013) 279–289, https://doi.org/10.1016/j.solener.2013.06.021. [30] A. Singh, S. Tescari, G. Lantin, C. Agrafiotis, M. Roeb, C. Sattler, Solar thermochemical heat storage via the Co3O4/CoO looping cycle: storage reactor modelling and experimental validation, Sol. Energy 144 (2017) 453–465, https://doi.org/10.1016/j.solener.2017.01.052. [31] X. Zhou, M. Mahmood, J. Chen, T. Yang, G. Xiao, M.L. Ferrari, Validated model of thermochemical energy storage based on cobalt oxides, Appl. Therm. Eng. 159 (2019) 113965, https://doi.org/10.1016/j.applthermaleng.2019.113965. [32] M. Ebadi, M. Mehrpooya, A.H. Kani, Sensitivity analysis and optimization of geometric and operational parameters in a thermochemical heat storage redox reactor used for concentrated solar power plants, J. Therm. Anal. Calorim. 147 (11) (2021) 6415–6435, https://doi.org/10.1007/s10973-021-10980-3. [33] J.A. Almendros-Ib´ a˜ nez, M. Fern´ andez-Torrijos, M. Díaz-Heras, J.F. Belmonte, C. Sobrino, A review of solar thermal energy storage in beds of particles: packed and fluidized beds, Sol. Energy 192 (2019) 193–237, https://doi.org/10.1016/j.solener.2018.05.047. [34] B. Wang, L. Li, F. Sch¨ afer, J.J. Pottas, A. Kumar, V.M. Wheeler, et al., Thermal reduction of iron–manganese oxide particles in a high-temperature packed-bed solar thermochemical reactor, Chem. Eng. J. 412 (2021) 128255, https://doi.org/10.1016/j.cej.2020.128255. [35] W. Huang, D. Korba, K. Randhir, J. Petrasch, J. Klausner, N. AuYeung, et al., Thermochemical reduction modeling in a high-temperature moving-bed reactor for energy storage: 1D model, Appl. Energy 306 (2022), https://doi.org/10.1016/j.apenergy.2021.118009. [36] D. Korba, M. Hayes, P. Schimmels, K. Randhir, J. Klausner, N. AuYeung, et al., Continuum modeling of high-temperature (>1000 ◦C) heat extraction from a moving-bed oxidation reactor for thermochemical energy storage, J. Energy Storage 82 (2024) 110579, https://doi.org/10.1016/j.est.2024.110579. [37] W. Wang, Y. Shuai, J. Yang, B.G. Lougou, Y. Huang, Heat transfer and heat storage characteristics of calcium hydroxide/oxide based on shell-tube thermochemical energy storage device, Renew. Energy 218 (2023) 119364, https://doi.org/10.1016/j.renene.2023.119364. [38] H. Zheng, X. Liu, Y. Xuan, C. Song, D. Liu, Q. Zhu, et al., Thermochemical heat storage performances of fluidized black CaCO3 pellets under direct concentrated solar irradiation, Renew. Energy 178 (2021) 1353–1369, https://doi.org/10.1016/j.renene.2021.07.026. [39] M. Hamidi, V.M. Wheeler, X. Gao, J. Pye, K. Catchpole, A.W. Weimer, Reduction of iron–manganese oxide particles in a lab-scale packed-bed reactor for thermochemical energy storage, Chem. Eng. Sci. 221 (2020), https://doi.org/10.1016/j.ces.2020.115700. [40] M. Kaviany, Principles of Heat Transfer in Porous Media, second ed., Springer, 2012. [41] M. Kaviany, B.P. Singh, Radiative heat transfer in porous media, Adv. Heat Tran. 23 (1993) 133–186, https://doi.org/10.1016/S0065-2717(08)70006-6. [42] M. Liu, M.C. Lin, C. Wang, Enhancements of thermal conductivities with Cu, CuO, and carbon nanotube nanofluids and application of MWNT/water nanofluid on a water chiller system, Nanoscale Res. Lett. 6 (2011), https://doi.org/10.1186/1556-276X-6-297. [43] Yttria Stabilized Zirconia Powder. http://www.advancedmaterials.us/4039OR-8601.htm. [44] M. Sans, V. Schick, G. Parent, O. Farges, Experimental characterization of the coupled conductive and radiative heat transfer in ceramic foams with a flash method at high temperature, Int. J. Heat Mass Tran. 148 (2020) 119077, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119077. [45] Fused Quartz Wool. https://www.americanelements.com/fused-quartz-wool-14808-60-7. [46] DALFRATHERM Bulk wool. https://www.promat.com/en/industry/products-solutions/high-temperature-insulation/fibres-textiles/dalfratherm-bulk-wool/. [47] https://www.promat.com/siteassets/industry/downloads/technical-data-sheets-tds/microporous/eng/promatmicrotherm-overstitched-product-data-sheet. pdf?v=49cc93/Download. [48] https://www.morganthermalceramics.com/media/7699/morgan-advanced-materials_thermal-ceramics-productdata-book-e-version_2.pdf. [49] X-l. Xia, X. Chen, C. Sun, Z-h. Li, B. Liu, Experiment on the convective heat transfer from airflow to skeleton in open-cell porous foams, Int. J. Heat Mass Transf. 106 (2017) 83–90, https://doi.org/10.1016/j.ijheatmasstransfer.2016.10.053. Z. Cao et al. Case Studies in Thermal Engineering 63 (2024) 105315 18