Numerical Analysis on the Effect of Thermal Conductivity and Thermal Contact Conductance on Heat Transfer during Dehydration Reaction in a Fixed Packed Bed Reactor for Thermochemical Heat Storage

Abstract

The generation of electrical power from photovoltaics and wind energies is often mismatched with the demand of energy scheduled by industrial processes. Chemical heat storage technology allows the conversion of heat from electricity surplus, solar heat or industrial waste heat into chemical energy, which can be reutilized on-demand, either for industrial processes requiring heat or for re-conversion into electricity. This energy buffer technology would contribute in stabilizing the mismatch between demand and offer and improve the resilience of energy supply into the decarbonized Society. A main bottleneck for practical utilization of chemical heat storage is related to the poor heat transfer ability in the packed bed reactor. In this work it is tried to clarify numerically the effects of thermal conductivity and thermal contact conductance on the exergy efficiency of heat transfer during the dehydration reaction (heat storage mode). A fixed packed bed reactor with flat geometry based on Ca(OH)₂ dehydration has been considered: chemical heat storage performance indicators like average heat storage rate and exergy efficiency were obtained via numerical simulations by changing the values of thermal conductivity of packed bed’s material, thermal contact conductance between packed bed and reactor’s wall and size of the packed bed. It was found that thermal conductivity enhancement is necessary but not sufficient for achieving the highest average heat storage rate, thermal contact conductance results important for maximizing the benefits of thermal conductivity enhancement, while a decrease of size of the packed bed results beneficial for achieving higher exergy efficiencies.

1. Introduction

Large-sized industrial plants for steel-making or chemical processes are well known heavy users of energy and carbon dioxide emitters, associated either with the process itself or related to the generation of electrical power required for the operation. Aiming at contributing to the decarbonization of Society, many efforts and innovative technologies are required for cutting the carbon dioxide emissions associated to these processes. The demand of electricity or heat from industrial systems can be scheduled, however this is not the case for generation of electricity from power plants fueled by renewable energies, like solar and wind sources. It is therefore required to find cheap and reliable technologies for storing massive amounts of surplus of electricity generation or unutilized heats, and then re-utilize the stored energy when generation is scarce. This can be done by charging and discharging batteries, previous conversion from AC to DC, and vice versa. An alternative way is to convert the surplus electricity generation directly into heat, which can be stored into thermal energy storage (TES) materials.^1,2) This power-to-heat-to-power (P2H2P) concept finds application in Carnot batteries, which include a variety of technologies and methods for storing electricity via conversion into heat and reconvert it into electricity when necessary, like reported in recent review works from Dumont et al.³⁾ or by Novotny et al.⁴⁾. Especially for the industrial applications, chemical heat storage is a candidate TES alternative to sensible heat storage or latent heat storage, characterized by larger energy density, but still difficult to be utilized in practical applications because of the poor heat transfer properties of the reagents utilized, which are mainly hydroxides or carbonates.⁵⁾ One of the most studied chemical reaction for chemical heat storage is the endothermic dehydration of calcium hydroxide (heat storage mode) and the reverse exothermic hydration of calcium oxide (heat output mode). This reaction occurs at temperatures around 500°C and has been extensively investigated for accumulation of solar heat. Many examples of basic study on reagent and its application in prototype reactors can be found in Literature.^{6,7,8,9,10,11,12)} In order to make it possible to utilize these cheap and readily available chemicals, many research efforts are undergoing for enhancing the thermal properties of these materials. Examples are given by vermiculite,¹³⁾ nanoparticles,¹⁴⁾ boron nitride,¹⁵⁾ carbonaceous supports,¹⁶⁾ ceramic honeycomb.¹⁷⁾ Other strategies for improvement of heat transfer into the packed bed reactor consist in the utilization of metallic foams inserted into the packed bed as in Groll et al.,¹⁸⁾ while in Stengler et al. the study focuses on the utilization of metallic fins.¹⁹⁾ Still, the conversion of electricity into heat is sometimes an un-efficient process from the point of view of utilization of exergy. The exergy efficiency of Joule heating is in fact high when the temperature utilization of heat is high, while it becomes gradually lower when approaching ambient temperatures. Exergy analysis of chemical heat storage systems based on Ca(OH)₂/CaO+H₂O reactions are reported in Obermeier et al.²⁰⁾ and Gupta et al.²¹⁾ A methodology for an efficient exergy efficient design of packed bed reactor is proposed by Neveu et al.,²²⁾ aiming at optimizing an elemental volume using an analytic approach and by introducing the concept of “exergy diffusivity”. In practice, the shape of the reactant loaded in the packed bed reactor consists in figures of slabs or pellets, or sometimes compacted powders, which are characterized by a certain value of effective thermal conductivity and thermal contact conductance between packed bed materials and Joule-heated reactor’s wall. The purpose of this work is to provide a more practical analysis of the heat transfer process during chemical reaction, aiming at better understanding the heat transfer process during dehydration reaction: the focus is on this reaction as it is the most critical aspect when converting electricity into heat energy. The numerical analysis will show the merits of thermal conductivity enhancement, the effect of thermal contact conductance and the influence of packed bed’s thickness on the values of heat storage rates. Moreover, the exergy analysis applied to the heat transfer process will enable to observe how the exergy efficiency is affected by these parameters.

Fig. 1.

Schematic of the heat transfer process into the packed bed reactor during heat storage mode: heat is transferred to the packed bed for the dehydration of Ca(OH)₂ but the temperature distribution (hence the reaction rate) is controlled by the thermal contact conductance and the evolution of thermal conductivity in the packed bed. (Online version in color.)

2. Numerical Method

2.1. Geometrical Domain and Boundary Conditions

The model presented in a previous work by Zamengo et al.²³⁾ has been modified for considering the planar geometry of this study (1-dimensional problem). The schematic is presented in Fig. 2(a). The packed bed consists alternatively in a packed bed of calcium hydroxide (Ca(OH)₂ pellet) or comprising a composite chemical heat storage material (CHSM) made of Ca(OH)₂ and Silicon-impregnated Silicon Carbide (Si–SiC) foam, as developed by Funayama et al.²⁴⁾ The half-thickness Y/2 [mm] of the packed bed used for the initial parametric study is selected as 25 mm. The planar geometry is considered of infinite length, border effects are neglected. The packed bed is considered to be symmetric, heated from the upper and lower side at a fixed temperature, so it is possible to study only half of the total geometry, as evidenced in Fig. 2(b). The computational domain is divided into n = 50 nodes and finite difference method with implicit scheme is utilized. An appropriate time step for the calculations was preliminary selected in Δt = 1 s and decreased to 0.5 or 0.1 s when thermal conduction in the packed bed was enhanced by both the contributions of high thermal conductivity and high thermal contact conductance. The starting temperature is 300°C on the whole computational domain; then, for time t>0 s, the temperature of the reactor’s wall is raised to a fixed constant temperature T_h = 540°C until the conversion of Ca(OH)₂ in the packed bed, x, is completed by changing from the initial value x = 1 to a final value of x = 0.

Fig. 2.

a) Schematic of the ideal packed bed reactor object of the study and b) computational domain utilized in the calculations; c) by decreasing the porosity of the Si–SiC foam, thermal conductivity can be increased, while the amount of Ca(OH)₂ decreases, causing the energy density of the packed bed to decrease. (Online version in color.)

The governing heat transfer equation on the computational domain is the Fourier equation for conduction in solids with internal heat generation:   

$$\rho c\frac{\partial T}{\partial t}={\lambda }_{bed}{\nabla }^{2}T+s$$(1)

in which s is the contribution of endothermic heat of the chemical reaction. The implicit scheme of the finite difference equation is expressed by the following Eq. (2) valid for the inner regions of the packed bed, while Eqs. (3) and (4) are valid in the contact interface packed bed/wall and at the position y = Y/2 of the packed bed, respectively. The values of Y/2 was selected in among 5 mm, 25 mm and 50 mm.   

$$\begin{array}{l}\frac{{\lambda }_{bed}}{\mathrm{\Delta }y}\left(T_{i+1}-T_i\right)\cdot \mathrm{\Delta }t+\frac{{\lambda }_{bed}}{\mathrm{\Delta }y}\left(T_{i-1}-T_i\right)\cdot \\ \mathrm{\Delta }t=s_i+\rho c\mathrm{\Delta }y\cdot \left(T_i-T_i^{°}\right)\end{array}$$(2)

  

$$\frac{{\lambda }_{bed}}{\mathrm{\Delta }y}\left(T_2-T_1\right)\cdot \mathrm{\Delta }t+{\gamma }_h\left(T_h-T_1\right)=s_i+\rho c\frac{\mathrm{\Delta }y}{2}\cdot \left(T_1-T_1^{°}\right)$$(3)

  

$$\frac{{\lambda }_{bed}}{\mathrm{\Delta }\text{y}}\left(T_{n-1}-T_n\right)\cdot \mathrm{\Delta }t=s_i+\rho c\frac{\mathrm{\Delta }y}{2}\cdot \left(T_n-T_n^{°}\right)$$(4)

2.2. Thermophysical Properties of Materials

The thermophysical properties, such as the volumetric heat capacity, are considered to be changing in function of the reacted fraction x from Ca(OH)₂ to CaO accordingly to the following relationship:   

$${\rho }_i{c}_i\left(x_i\right)=x_i\cdot {\rho }_{Ca{\left(OH\right)}_2}{c}_{Ca{\left(OH\right)}_2}+\left(1-x_i\right)\cdot {\rho }_{CaO}{c}_{CaO}$$(5)

in which x = 1 correspond to the hydrated state (100% Ca(OH)₂) and x = 0 represents the full conversion of Ca(OH)₂ into CaO. Concerning the composite material, considering the porosity φ of the Si–SiC matrix, the relationships expressing the volumetric heat capacity for the composite is given by:   

$$\begin{array}{l}{\rho }_i{c}_i\left(x_i\right)=\left[x_i\cdot {\rho }_{Ca{\left(OH\right)}_2}{c}_{Ca{\left(OH\right)}_2}+\left(1-x_i\right)\cdot {\rho }_{CaO}{c}_{CaO}\right]\cdot \\ \phi +{\rho }_{Si-SiC}{c}_{Si-SiC}\cdot \left(1-\phi \right)\end{array}$$(6)

Lacking of measured values, the dependency of effective thermal conductivity on the volume ratio between Ca(OH)₂ and Si–SiC has been estimated via the Braun model²⁵⁾ assuming that the foam pores have spherical shape:   

$$\frac{{\lambda }_{bed}-{\lambda }_{Ca{\left(OH\right)}_2}}{{\lambda }_{Si-SiC}-{\lambda }_{Ca{\left(OH\right)}_2}}\cdot {\left(\frac{{\lambda }_{Si-SiC}}{{\lambda }_{bed}}\right)}^{-\frac{1}{3}}=1-\phi$$(7)

It is assumed that the thermal conductivity for each node in both of the packed beds decreases 50% when conversion of Ca(OH)₂ into CaO is completed, accordingly to the following equation:   

$${\lambda }_i\left(x_i\right)=x_i{\lambda }_{bed}+\left(1-x_i\right)\frac{{\lambda }_{bed}}{2}$$(8)

This relationship was observed in a work from Zamengo et al.²⁶⁾ for Mg(OH)₂ and expanded graphite composite, and it has been considered that it can be a valid assumption also in this case. Lacking of precision on this information, the temperature dependency of thermal properties has been neglected in order to avoid further assumptions. Table 1 summarizes the properties of materials utilized in this work, while Table 2 collects the value of thermal conductivity estimated for the CHSM obtained by utilization of Eq. (7). It is indeed of critical importance for the correct application of the method to obtain the precise measured values of the properties.

Table 1. Summary of thermophysical properties assumed for the materials in the packed bed reactor.

Material	Density ρ [kg·m−3] x = 1 → x = 0	Spec. heat capacity, c [J·kg−1·K−1] x = 1 → x = 0	Thermal conductivity, λ [W·m−1·K−1] x = 1 → x = 0
Ca(OH)2 pellet	1200 → 908	1181 → 710	0.20 → 0.10
Si–SiC	2700	890	180

Table 2. Summary of the thermal conductivity estimated for the CHSM using Eq. (7).

Material	Porosity of Si–SiC matrix φ [−]	Estimated thermal conductivity [Wm−1K−1]	λ/λref
Ca(OH)2 pellet	n.a.	0.2	1.0
CHSM-98	0.98	0.8	4.0
CHSM-96	0.96	1.8	8.5
CHSM-94	0.94	2.9	14.5
CHSM-92	0.92	4.4	22.0
CHSM-90	0.90	6.0	30.0
CHSM-88	0.88	7.8	39.0
CHSM-86	0.86	9.7	48.5
CHSM-84	0.84	11.8	59.0
CHSM-82	0.82	14.0	70.0
CHSM-80	0.80	16.4	82.0
CHSM-78	0.78	18.8	94.0
CHSM-76	0.76	21.4	107.0
CHSM-74	0.74	24.1	120.5

In order to analyze the effect of thermal conductivity, a certain porosity of the Si–SiC foam was selected in the even values from 98% to 74% (which means φ = 0.98 and φ = 0.74). The corresponding naming of the composites will be respectively CHSM-98, CHSM-96, CHSM-94 and so on until CHSM-74, for a total of 13 different CHSM. The thermal contact conductance coefficient γ_h, has been assumed among the values 50, 150 and 300 Wm⁻²K⁻¹. These values are assumed as good estimations after considering that a value of 60 Wm⁻²K⁻¹ was successfully utilized in a previous work by Zamengo et al.²³⁾ for the numerical analysis of an experimental work on a packed bed reactor of Mg(OH)₂ pellet, which is also the configuration utilized for Ca(OH)₂ pellet.⁹⁾ The contact condition of pellets with reactor wall is actually not optimal, as many gaps of random size are existing. The values of 150 Wm⁻²K⁻¹ and 300 Wm⁻²K⁻¹ are considered as a possible improvement: such values are in fact common for consolidated blocks of solid sorption materials used also in other experimental works, which are reported to be as 180 Wm⁻²K⁻¹ for a consolidated composite of Zeolite 4A-Cu foam²⁷⁾ or up to 400–800 Wm⁻²K⁻¹ of a MnCl₂-carbon composite (IMPEX).¹⁹⁾ In this work, the effect of heat removal from water vapor and the eventual effects of gas diffusion on heat transfer in the packed bed reactor are neglected.

2.3. Reaction Kinetics

The thermodynamics of Ca(OH)₂ dehydration (equilibrium equation) and the temperature dependency of kinetic coefficients, k, are obtained, respectively from Samms et al.²⁸⁾ and Schaube et al.²⁹⁾ in function of temperature and for water vapor condensation temperature of 25°C. For implementation into the finite difference calculations scheme, the temperature dependent reaction rate equation has been linearized as follows, by approximating the dehydration reaction as a first order chemical reaction:   

$$-\frac{dx}{dt}=kx$$(9)

The reaction rate is computed for each computational node i in function of temperature. The Eq. (9), after integration, can be rewritten as:   

$$\text{ln}{x}_i=-k_i\cdot t+ln{x}_0$$(10)

where:   

$$k_i=A_0\cdot e^{-E_a/R{T}_i}$$(11)

At time t + Δt the reacted fraction x′_i is expressed as   

$$\text{ln}{x}_i^{\prime }=-k_i\cdot \left(t+\mathrm{\Delta }t\right)+ln{x}_0$$(12)

By taking the difference between Eqs. (12) and (10), it is obtained:   

$$x_i^{\prime }=x_i{e}^{-k_i\mathrm{\Delta }t}$$(13)

The amount of heat stored at each time step in each calculation node, which needs to be included in the heat transfer Eqs. (2), (3) and (4) can be finally obtained using the following equation:   

$$s_i=-\frac{\mathrm{\Delta }H}{{\varrho }_i{c}_i}\left(x_i^{\prime }-x_i\right)$$(14)

where the enthalpy change of dehydration reaction is assumed as ΔH = −1.404·ρ_bed kJ·m⁻³. From here, the heat storage rate w_chem can be simply obtained as the summation, for each time step, of the s_i for each node i as follows:   

$$w_{chem}={\mathrm{\Sigma }}_i{\rho }_i{c}_i{s}_i\mathrm{\Delta }y/\mathrm{\Delta }t$$(15)

It is reminded that the value of w_chem is expressed in kWm⁻² of reactor wall surface on a computational domain of packed bed having a certain thickness Y/2 (half of the total thickness). From the calculations, it is also extracted the value of ending time of reaction, t_end [min], corresponding to the time at which the conversion x is x = 0 on the whole computational domain. Another important parameter for the evaluation of the chemical heat storage performance is the average heat storage rate, w_chem,avg, obtained as the average value of w_chem between t = 0 and t_end. This parameter is important for understanding the amount of heat that can be transferred to the packed bed during the heat storage operation. The value of total energy stored, q_chem [MJm⁻²] is obtained by integration of the values of w_chem:   

$$q_{chem}={\mathrm{\Sigma }}_t{\mathrm{\Sigma }}_i{w}_{chem,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }i}\mathrm{\Delta }t$$(16)

2.4. Calculation of Exergy Efficiency

The exergy efficiency of the heat transfer process between heated reactor’s wall and packed bed relates to conversion of electricity into heat and the corresponding heat transfer process with a finite temperature difference.³⁰⁾ This was observed to be the main bottleneck of the process in term of exergy conversion, as previously explained in a study for Carnot-batteries using chemical heat storage.³¹⁾ Considering that the amount of heat exchanged, q_h [Jm⁻²], at each time step is:   

$$q_h={\gamma }_h\cdot \left(T_h-T_1\right)\cdot \mathrm{\Delta }t$$(17)

The amount of loss of exergy (or irreversibility) is calculated as:   

$$\text{irr}=q_h\frac{T_a}{T_{bed,avg}}$$(18)

where T_bed,avg is the average temperature of the packed bed reactor and T_a is the reference temperature (298 K). Finally, the exergy efficiency, ψ, which refers only to the overall exergy transfer from the heated wall to the packed bed reactor having a finite temperature difference, is obtained for each time step using the following relationship:   

$$\psi =1-\frac{{{\displaystyle \sum }}_{t=0}^{t=t_{end}}irr}{{{\displaystyle \sum }}_{t=0}^{t=t_{end}}{q}_h}$$(19)

3. Results and Discussion

The numerical model allows to calculate for each computational node, the local temperature T_i and the local reacted fraction x_i. As representative example of results, Figs. 3 and 4 show, respectively, the local temperature and the local reacted fraction for the packed bed of Ca(OH)₂ pellet and the packed bed comprising CHSM-94. It is assumed that Y/2 = 25 mm and γ_h = 150 Wm⁻²K⁻¹ It looks clear that enhancement of thermal conductivity in the packed bed allows to achieve a faster temperature increase in the point farthest from the heated wall (T_c); consequently, the advancement of dehydration reaction, expressed by the reacted fraction x, is more rapid even in the inner regions of the packed bed. It also remarked that the temperature distribution in the packed bed of Ca(OH)₂ pellet is quite broad, while in the packed bed of CHSM-94 the values of T_c and T_m are more similar, therefore dehydration advances with similar rates in the whole bed in the latter case. This happens because, when reaction is completed in the regions in direct contact with the heated reactor’s wall, a layer of CaO is formed and it works as a thermal insulator, impeding heat to be transferred swiftly into the inner regions of the packed bed. This is not the case of CHSM, where the Si–SiC foam still can provide a highly thermal conductive path to the inner region of the packed bed.

Fig. 3.

Temperature change observed at position y = 25 mm (T_c), position y = 12.5 mm (T_m) and position y =0 (T_w) for the packed bed of Ca(OH)₂ pellet and CHSM-94. The thickness of the computational domain is Y/2 = 25 mm. It is considered a) γ_h = 50 Wm⁻²K⁻¹, b) γ_h = 150 Wm⁻²K⁻¹ and c) γ_h = 300 Wm⁻²K⁻¹. (Online version in color.)

Fig. 4.

The local reacted fractions observed at position y = 25 mm (x_c), position y = 12.5 mm (x_m) and position y = 0 (x_w) for the packed bed of Ca(OH)₂ pellet and CHSM-94. The thickness of the computational domain is Y/2 = 25 mm. a) γ_h = 50 Wm⁻²K⁻¹, b) γ_h = 150 Wm⁻²K⁻¹ and c) γ_h = 300 Wm⁻²K⁻¹. (Online version in color.)

An increase of values of thermal conductivity is beneficial for achieving high heat storage rates w_chem, making it possible to store heat in less time, as presented in Fig. 5: it shows a comparison of four different representative CHSM versus Ca(OH)₂ pellet. The heat storage rate of the CHSM results higher than the one of Ca(OH)₂, and it can be observed that a peak value is found for each packed bed at the beginning of the heat storage process. It is remarked that the maximum value of the w_chem peaks tends to approach the same value of 24 kWm⁻² when gradually augmenting the amount of Si–SiC in the CHSM. This is because, despite the reaction is accelerated, the amount of Ca(OH)₂ in the packed bed becomes gradually less.

Fig. 5.

The heat storage rates w_chem calculated for the packed beds of Ca(OH)₂ pellet, CHSM-94, CHSM-84 and CHSM-74. In the calculations it is assumed Y/2 = 25 mm and γ_h = 150 Wm⁻²K⁻¹. (Online version in color.)

It is observed that the maximum value of w_chem is achieved by the CHSM-84, which reaches and attains a higher value of w_chem respect CHSM-94 and results always higher than the maximum value achieved by CHSM-74. It is noted that for Ca(OH)₂ pellet and CHSM-94, after an initial peak of w_chem, its value gradually become smaller, indicating that reaction is advancing slowly. On the other hand, when considering CHSM-84 and CHSM-74, they show a quasi-constant value of w_chem and a final sudden drop: this indicates that reaction is advancing in the whole reactor with similar rate, as the distribution of temperature in the packed bed is more uniform. It is understood that the height of the w_chem peak is related to the volume ratio of Ca(OH)₂ to Si–SiC foam and the thermal conductivity of the packed bed: by decreasing the porosity of the Si–SiC foam, the energy density decreases as less Ca(OH)₂ is comprised in the composite. Accordingly, the area under the w_chem curve becomes gradually smaller. On the other hand, a decrease of Si–SiC foam porosity increases the thermal conductivity and allows the reaction to be completed in a shorter time (43 min, 26 min, and 22 min for CHSM-94, CHSM-84, and CHSM-74, respectively).

However, these peak values of w_chem and, more in general, the average heat storage rate, w_chem,avg, achievable by each bed during the dehydration reaction are also dependent on the value of thermal contact conductance γ_h and the size of the packed bed. The results of the numerical analysis are reported in Fig. 6) for Ca(OH)₂ and thirteen different CHSM. On the horizontal axis, different value of enhancement factor λ_bed/λ_ref are reported, where λ_ref is the reference value of thermal conductivity of Ca(OH)₂ pellet. It is observed that by increasing the value of γ_h, the difference becomes larger and larger: in case of CHSMs, it is found that for γ_h = 50 Wm⁻²K⁻¹ a maximum w_chem,avg of 7.4 kWm⁻² is achieved by CHSM-90, for γ_h = 150 Wm⁻²K⁻¹ a maximum w_chem,avg of 17.7 kWm⁻² is achieved by CHSM-84, while for γ_h = 300 Wm⁻²K^-1 a maximum w_chem,avg of 29.0 kWm⁻² is achieved by CHSM-78. In other words, to achieve the highest values of w_chem,avg, a thermal conductivity enhancement of 20x factor could be enough for achieving the highest w_chem,avg if γ_h = 50 Wm⁻²K⁻¹, while it is necessary a 70x factor if γ_h = 150 Wm⁻²K⁻¹ or 94x if γ_h = 300 Wm⁻²K⁻¹.

Fig. 6.

Comparative plot of heat storage rate w_chem,avg in function of thermal contact conductance γ_h for the packed beds. The results refer to the computational domain Y/2 = 25 mm. Depending on the combination between thermal conductivity λ_bed and thermal contact conductance γ_h, different optimal materials can be identified. (Online version in color.)

Still, by increasing the thermal conductivity enhancement factor, less energy is available in the packed bed reactor, therefore it is necessary to determine the more appropriate trade-off between energy storage density q_chem (hence thermal conductivity of packed bed material) and the desired rate of heat storage w_chem,avg. By considering all the combinations of thermal conductivity, thermal contact conductance γ_h, and three different size of the packed bed reactor, the plot in Fig. 7 can be finally obtained. It shows the values of total chemical heat storage q_chem and the corresponding values of average heat storage rate, w_chem,avg divided by the reference results q_chem,ref and w_chem,avg,ref obtained for the packed bed of Ca(OH)₂ pellet having Y/2 = 5 mm and γ_h = 50 Wm⁻²K⁻¹. Moreover, each point expresses by color gradation the value of exergy efficiency ψ obtained from each configuration. It is observed that the largest values of q_chem/q_chem,ref are characterized by low values of exergy efficiency and correspond to the packed bed having larger size Y/2 = 50 mm. The utilization of higher thermal conductive CHSM in the packed beds makes q_chem/q_chem,ref to become lower respect of the Ca(OH)₂ pellet packed bed, but it is possible to achieve the highest heat storage rates and exergy efficiencies, especially by increasing simultaneously the value of thermal contact conductance γ_h. By varying γ_h in fact, a wide range of w_chem,avg/w_chem,avg,ref can be achieved, up to a 7x factor. A high value of thermal contact conductance γ_h is useful for maximizing the benefits achievable by utilization of thermal conductivity enhancer in the packed bed reactor. It also appears evident that, for a fixed size of the packed bed, the CHSM with higher w_chem,avg/w_chem,avg,ref is not always the same. The highest exergy efficiencies could be achieved by decreasing the size of the packed bed.

Fig. 7.

Comparative plot of average heat storage rate w_chem,avg and total chemical heat storage q_chem in function of thermal contact conductance γ_h and size Y/2 of the packed bed reactor. The color gradation of markers refers to the respective values of exergy efficiency ψ. (Online version in color.)

4. Conclusion

The analysis of dehydration reaction on a packed bed for chemical heat storage showed the sensitivity of the heat storage performance to thermal conductivity of the packed bed material, the thermal contact conductance and the size of the packed bed reactor. It has been found that CHSM having the optimal chemical heat storage rate w_chem,avg require an increase of thermal conductivity larger than 10x factor in comparison with Ca(OH)₂ pellet, but thermal conductivity of the packed bed is not itself able to improve the performances of chemical heat storage for packed bed reactors. Noticeable contributions to the heat storage performance are also related to the combinations of contact conductance γ_h and reduction of size, which play an important role for achieving a high exergy efficiency of the heat storage process. Under the conditions investigated and the assumptions of the numerical model, the average heat storage rate w_chem,avg varied from 7.7 kWm⁻² for CHSM-86 (γ_h = 50 Wm⁻²K⁻¹, Y/2 = 50 mm) to 29.1 kWm⁻² for CHSM-74 (γ_h = 300 Wm⁻²K⁻¹, Y/2 = 50 mm), while the exergy efficiency could range from 0.514 of Ca(OH)₂ pellets (γ_h = 50 Wm⁻²K⁻¹, Y/2 = 50 mm) to 0.575 CHSM-74 (γ_h = 300 Wm⁻²K⁻¹, Y/2 = 5 mm). Further research efforts are required for characterization of both the values of thermal conductivity of CHSM and the value of thermal contact conductance γ_h for improving the calculations. It is expected that this work will contribute as a guidance for development and optimization of packed bed reactors so that chemical heat storage could be soon utilized for contributing in the development of a sustainable industry and cut the related carbon dioxide emissions.

Nomenclature

Δy: distance between temperature nodes [mm]

Δt: time step used in the calculations, [s]

φ: porosity of the ceramic foam enhancer, [−]

γ_h: thermal contact conductance, [Wm⁻²K⁻¹]

λ_bed: effective thermal conductivity of material in the PBR, [Wm⁻¹K⁻¹]

λ_ref: reference value of thermal conductivity of Ca(OH)₂ pellet packed bed, [Wm⁻¹K⁻¹]

ρ: density of material in the PBR, [kgm⁻³]

ψ: exergy efficiency [−]

E_a: activation energy [Jmol⁻¹K⁻¹]

Irr: irreversibility [Jm⁻²]

k: kinetic coefficient of dehydration reaction

CHSM: Chemical heat storage material

c: specific heat capacity of material in the PBR, [Jkg⁻¹K⁻¹]

PBR: packed bed reactor

q_chem: chemical heat storage density per unit surface of PBR, [MJm⁻²]

q_h: heat power supplied by the Joule heater, [Jm⁻²]

T: temperature at time t+Δt, [°C]

T_a: reference ambient temperature, [K]

T^o: temperature at time t, [°C]

T_h: temperature of heater on the PBR’s wall.

T_c: Temperature at the center of the packed bed, at distance Y/2 from the wall, [°C]

T_m: Temperature at middle-way distance between center and wall or the packed bed, [°C]

T_w: Temperature at the contact point with of the packed bed reactor’s wall, [°C]

t: time, [s]

t_end: time of end of reaction

TES: Thermal energy storage

w_chem: chemical heat storage rate, [kWm⁻²]

w_chem,avg: average chemical heat storage rate, [kWm⁻²]

Y/2: half thickness of PBR [mm]

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