Applied Thermal Engineering 185 (2021) 116414

Available online 5 December 20201359-4311/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Coatings utilization to modify the effective properties of high temperature packed bed thermal energy storage

Silvia Trevisan, Wujun Wang *, Björn Laumert Department of Energy Technology, KTH Royal Institute of Technology, Brinellvägen 68, 100 44 Stockholm, Sweden

A R T I C L E I N F O

Keywords: Coatings Thermal emissivity Radiation heat transfer Packed bed thermal energy storage Effective thermal conductivity

A B S T R A C T

High-temperature thermal energy storage is becoming more and more important as a key component in concentrating solar power systems and as an economically viable large-scale energy storage solution. Ceramics and natural rocks based packed beds are one of the attracting solutions. For application temperatures above 600 ◦C, radiation heat transfer becomes the dominant heat transfer phenomenon and it greatly influences the performance of thermal storage systems. Coatings with different thermal properties (mainly thermal emissivity and thermal conductivity) could be exploited to modify the effective thermal properties of packed beds. In this work, we present a methodology to account for the thermal effect of a coating layer applied over the pebbles of a packed bed. The influences on the packed bed effective thermal conductivity of several characteristics of the coating material, packed bed arrangement, and filler material are investigated. The results show that low emissivity coatings could reduce the effective thermal conductivity of a rock based packed bed of about 58%, with respect to a similar uncoated solution, already at 800 ◦C. A low emissivity coating could also limit the increase in the thermal effective conductivity from the cold to the hot zone of the storage. Coatings would have a higher influence when applied in packed beds with large size particles, relatively high thermal conductivity of the substrate and void fraction. The application of different coatings, with various thermo-physical properties, in different parts of the storage could modify the effective thermal conductivity distribution and enable a partial control of the thermocline degradation, increasing the storage thermal efficiency.

1. Introduction

The energy demand has dramatically increased during the past years [1]. To cope with such an increasing demand while respecting the CO2 emission limitation the installed capacity of renewable energies, such as wind and solar, has seen a large increase. Due to the intermittent nature of such energy sources, the power grid and other infrastructure are experiencing challenges such as intermittency, unpredictability, and mismatch between supply and demand sides [2].

Concentrating solar power (CSP) plants with the integration of thermal energy storage (TES) solutions can provide flexible and dis-patchable power, by decoupling the power generation from solar radi-ation [3]. Besides, large-scale CSP with TES has the potential to offer constant baseload power supply [4]. The need for further CSP efficiency enhancement has recently driven the researchers’ attention toward achieving higher working temperatures. To attain this goal, four major pathways have been proposed for the next generation of CSP plants: high temperature molten salts, particle and gas based systems [5], and liquid

metals [6]. Among these solutions, particle and gas driven CSP plants could guarantee working temperatures higher than 1000 ◦C [7], and they could rely on packed bed TES solutions [5]. This storage solution is particularly suitable for gaseous-based receivers, including air receivers and thermochemical reactors [8].

Packed bed TES also represents a viable solution for large scale, relatively inexpensive and site-independent energy storages in the form of Carnot batteries [9]. This system transforms, via an inverse power cycle or via electrical heaters, electricity into heat; stores thermal energy in an inexpensive TES; and then, when required, heat is used for power generation via a traditional power cycle [9] or a thermoelectric gener-ator [10,11].

A traditional packed bed TES employs a bed of natural rocks or ceramic materials as the filler component. Several studies have been devoted to test the suitability and thermo-chemical stability of different rock samples up to 600 ◦C [12–16]. Comprehensive investigations have been conducted on analyzing specific heat [17], thermal conductivity [18] and thermal emissivity [19] of different rocks and mineral samples. Slags, cheap glass-like byproducts of the steelmaking process, have been

* Corresponding author. E-mail address: wujun@kth.se (W. Wang).

Contents lists available at ScienceDirect

Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

https://doi.org/10.1016/j.applthermaleng.2020.116414 Received 19 August 2020; Received in revised form 24 November 2020; Accepted 30 November 2020

mailto:wujun@kth.sewww.sciencedirect.com/science/journal/13594311https://www.elsevier.com/locate/apthermenghttps://doi.org/10.1016/j.applthermaleng.2020.116414https://doi.org/10.1016/j.applthermaleng.2020.116414https://doi.org/10.1016/j.applthermaleng.2020.116414http://crossmark.crossref.org/dialog/?doi=10.1016/j.applthermaleng.2020.116414&domain=pdfhttp://creativecommons.org/licenses/by-nc-nd/4.0/

Applied Thermal Engineering 185 (2021) 116414

2

also proposed as suitable filler candidates for high temperature TES up to 1000 ◦C [20]. Thermo-physical properties such as density, specific heat and thermal conductivity of different slags samples have also been investigated [20–22]. Though similar values have been presented in these works, all authors agree that the thermo-physical properties of slags vary depending on the upstream raw materials. Ceramics mate-rials, mostly alumina (Al2O3) and silica (SiO2), due to their thermo- chemical stability and well documented properties, have been used in most of the pilot and experimental verifications of high temperature packed bed TES solutions [23–25].

Within a packed bed TES, all three heat transfer mechanisms are present: convection between HTF and pebbles, conduction between adjacent particles and within the single pebble and thermal radiation [26]. For many low temperature applications, radiation heat transfer can be negligible [27]. However, when operating the TES unit at high temperature (above 700 ◦C), the thermal radiation is responsible for an important contribution to the overall heat transport process [28]. A proper evaluation of the radiation heat exchange can sensibly enhance the description of both the temperature distribution and pressure losses for high temperature packed beds [29]. Therefore, it is of great impor-tance to study the effect of the particle surface emissivity on the thermal performance of packed bed TES systems, and modify the surface emis-sivity with pigmented or contact coatings [30] for different applications. Particularly, one of the main techno-economic challenges for a packed bed TES is represented by the temperature, or thermocline, degradation [31]. This phenomenon is the flattening of the temperature gradient

within the storage during consecutive operation cycles and storage pe-riods. A key cause of thermocline degradation is the axial heat con-duction and thermal radiation [32]. Few passive and active methods to control the thermocline have been proposed. Passive methods act on the TES design. In particular, increasing the height-to-diameter ratio of the TES would ensure steeper thermocline fronts, however pumping work and external thermal losses would also increase [33]. Active thermocline-control methods rely on different TES management strate-gies and enlist, among others, flushing [34] (a periodic complete discharge of the TES), segmenting the TES [35] (divide the storage in multiple sections via ports), and siphoning [36] (extract the HTF at the thermocline position while supplying the hot and cold HTF at top and bottom of the storage). Though thermal efficiency improvements have been achieved by adopting these methods, the operational flexibility of the TES is affected and management complexity is added. The thermo-cline spread could be limited by using low emissivity coatings, which would reduce the effective thermal conductivity of the packed bed. Contrarily, in applications where rapid heat transfer is required – such as gas-cooled reactors [37] – high emissivity coatings might enhance the effective thermal conductivity of ceramics packed bed, base materials generally known for their relatively low thermal conductivity. The ap-plications of coatings over natural rocks could also provide known emissivity values for such material, which would otherwise be strongly influenced by the mineral and granular composition of each pebble. Finally yet importantly, by coating with high-temperature stable coat-ings, the pebbles can be protected from the active gas, such as oxygen

Nomenclature

Symbols A cross sectional area, [m2] ai convective heat transfer coefficient on the inside of the

tank, [W/(m2⋅K)] ao convective heat transfer coefficient on the outside of the

tank, [W/(m2⋅K)] as packed bed surface to volume ratio, [m2/m3] cp specific heat, [J/(kg⋅K)] D storage diameter, [m] ΔL effective length between centers of adjacent particles, [m] dp particle diameter, [m] Δrc coating layer thickness, [m] F*E radiation exchange factor G fluid mass flow rate per unit cross section, [kg/(s⋅m2)] H storage height, [m] h convective heat transfer coefficient packed bed, [W/

(m2⋅K)] hp heat transfer coefficient for contact surface, [W/(m2⋅K)] hRS heat transfer coefficient for radiation solid to solid, [W/

(m2⋅K)] hRV heat transfer coefficient for radiation void to void, [W/

(m2⋅K)] K solid to fluid thermal conductivity ratio k thermal conductivity, [W/(m⋅K)] keff effective thermal conductivity, [W/(m⋅K)] kReff effective thermal conductivity due to radiation, [W/(m⋅K)] l effective slab thickness, [m] n number of contact points on a semispherical surface of a

single pebble rp particle diameter, [m] T temperature, [K] t layer thickness, [K] Uw wall to ambient heat transfer coefficient, [W/(m2⋅K)] V volume, [m3]

Greek β* modified empirical parameter δ* heat transfer parameter μ void fraction εR thermal emissivity γ* modified constant parameter ϑ0 angle corresponding to the boundary of the heat flow area

for one contact point ϕ heat transfer parameter ϕ* modified heat transfer parameter ρ density, [kg/m3] σ Stephan-Boltzmann constant, [W/(m2⋅K4)] ν ratio particle volume to overall volume after coating

Subscripts 1 loose packing arrangement 2 close packing arrangement ∞ ambient C coating F fluid ins1 high temperature insulation ins2 low temperature insulation S solid steel 253 MA austenitic steel p particle TOT total V void

Abbreviations BC Boundary Condition CSP Concentrating Solar Power HTF Heat Transfer Fluid RTC Radiative Transfer Coefficient RTE Radiative Transfer Equations TES Thermal Energy Storage

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3

and steam. Throughout the years, many different methods to describe and

simulate radiation in packed beds have been developed. Unit Cell methods are the most commonly used ones. The whole packed bed ge-ometry is idealized employing a specific repeated unit of fixed geometry and predetermined properties. Thermal radiation is modelled via a thermal radiative conductivity for the unit cell, kReff , which is summed to the effective thermal conductivity of the packed bed [38]. The effective radiative conductivity is defined by means of a radiation exchange factor F∗E as kReff = 4F

*EσdpT

3. F∗E depends only on the particle emissivity and bed void fraction [39,40]. Later studies have shown that the solid thermal conductivity has a large influence on the thermal radiative conductivity contribution and they have defined the radiative exchange factor as F*E = f(Λf ,μ, εR), where Λf = ks/(4dpσT3) is a dimensionless solid ther-mal conductivity [41–44]. Kunii and Smith [26] have directly computed the radiation contribution to the overall packed bed thermal conduc-tivity into a single effective thermal conductivity of the bed. All main correlations, for F∗E or keff from the published literature are summarized in Table 1. Integral-Differential Radiative Transfer Equations (RTE) based models, introduced by Modest in [45], consider balances between emitted, absorbed and scattered radiative energy. To solve these equa-tions, a set of optical properties for particles and HTF is required [46], and thus the microstructure of each individual packing has to be analyzed [28]. Radiative Transfer Coefficient (RTC) numerical models, introduced by Lee et al. in [47], consider both the packed bed micro-structure and the medium radiative properties. The RTC for each par-ticle is calculated via Monte Carlo ray-tracing method. Finally, Particle Scale investigations, which are more in-depth analysis of the packing structure, have been firstly introduced by Cheng et al. in [48]. Asakuma et al. [49] analysed keff , considering the thermal radiation heat transfer, by a new homogenization method, which accurately represents the microstructure of the packed bed. In a successive study [50], the same method was applied to investigate the influence of the Biot number over keff . Wu et al. [51] studied the effect of the spatial scale by developing three different models (long-range, short-range, and microscopic model) and comparing their results with the existing correlations. In a tem-perature range lower than 1215 ◦C, the short-range model was found to be in a good agreement with the experimental data and more compu-tationally efficient than the long-range and microscopic model. Thus the same authors integrated the short-range model into a CFD-DEM model of a packed bed [52]. As an outcome of these works, a particle radiation factor has been proposed as an independent non-dimensional measure of the influence of particle-scale radiation in packed beds.

Notwithstanding the large amount of research on modelling of the radiation heat transfer within packed beds, still very little investigation has been done on the possibility of varying the radiative properties of

the filler material by applying coatings for the final purpose of modi-fying the effective thermal conductivity. Thus, in order to fill the iden-tified technical gap in the previous studies, we developed a methodology for modifying the effective thermal conductivity of a packed bed TES by utilizing coatings. In this paper, the methodology will be implemented to investigate the influence of several parameters, such as the thermal properties of the coating material as well as of the main filler material, on the packed bed effective thermal conductivity. Furthermore, this methodology is also included in a complete packed bed thermal energy storage one-dimensional – two-phase model to investigate the effective thermal conductivity distribution of coated and uncoated packed beds during operation.

2. Methodology

In this work, a new methodology to modify the effective thermal properties of a packed bed TES utilizing coatings has been investigated. The model presented improves the approach presented in [26] by including the influence of the coating. Though ideally more precise, particle scale models have not been used in this study, as they show some disadvantages: they require information on the packing structure, which would be challenging to gather for industrial scale applications, they are computationally very expensive and, in previous reported studies, their accuracy have been measured only against previous unit cell approaches [51]. Therefore, the presented study has been limited to unit cell models, which are the most widely used in packed bed studies, and they can provide quite simple and computationally inexpensive solutions.

2.1. Packed bed effective thermal properties

The application of coatings on the particle surfaces, particularly if the coating thermo-physical properties are different from the properties of the main substrate, would affect the effective packed bed properties. The following two sections describe a methodology to account for the coating influence on the packed bed effective heat capacity and thermal conductivity.

2.1.1. Effective heat capacity Generally, the effective heat capacity of packed bed TES is calculated

as the weighted average between the fluid and solid filler heat capacity by considering the void fraction of the bed, as shown in Eq. (1). (ρcp

)

eff = μ(ρcp

)

F +(1 − μ)(ρcp

)

S (1)

where, ρ is the density, cp is the specific heat and μ is the packed bed void fraction. However, when adding a coating on the surfaces of the pebbles,

Table 1 Main correlations for radiation exchange factor and effective thermal conductivity.

Reference Radiation exchange factor or effective thermal conductivity Comments

[39] F*E =εR + Br1 − Br

for μ = 0.4 Br = 0.149909 − 0.24791εR++0.290799ε2R − 0.20081ε3R+

+0.0651042ε4R

[40] F*E =

2Br + εR(1 − Br)2(1 − Br) − εR(1 − Br)

[42]

F*E =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

[1 −

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅(1 − μ)

√ ]μ +

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅(1 − μ)

√

2εR

− 1

B + 1B

1

1 +1

(2εR

− 1)

Λf

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

B = 1.25(

1 − μμ

)10 /9

[44] F*E = a1εRtan− 1

(

a2(Λs)a3

εR

)

+ a4 for μ = 0.476 a1 = 0.5756, a2 = 1.5353, a3 = 0.8011, a4 = 0.1843

[26] keff = μ[kF + hRVΔL] +

(1 − μ)ΔL1

kFlv

+ hp + hRS+

lsks

Parameters defined in paragraph 2.1.2

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4

its influence must be accounted for in the solid component. The pebble covered by the coating is sketched in Fig. 1(a). The volume of the coating can be evaluated considering the coating thickness on the pebble sur-face, Δrc. For sake of simplicity, the thickness will be considered as uniform all over the pebble and for all particles in the bed. The ratio between the original particle volume and overall volume after the coating application is defined as ν = Vp/Vtot = r3p/(rp + Δrc)

3. Thus, the effective heat capacity of the packed bed, once including the coating, can be calculated as: (ρcp

)

eff = μ(ρcp

)

F +(1 − μ)[ν(ρcp

)

S + (1 − ν)(ρcp

)

c

](2)

2.1.2. Effective thermal conductivity In the presented model, the following heat transfer mechanisms are

included in the packed bed effective thermal conductivity:

1. Heat transfer in the fluid region by conduction and by radiation from void to void (considering a non-absorbing gas);

2. Heat transfer through the solid phase: a. Heat transfer through the contact surface between adjacent

particles b. Conduction through the stagnant layer of fluid close to the contact

region c. Radiation between pebbles surfaces void (considering a non-

absorbing gas) d. Conduction through the solid pebbles e. Conduction through the coating layer

These heat transfer mechanisms are schematically shown in Fig. 1(b) and their relationships have been described in the form of associated thermal resistances in Fig. 2. Particularly, with respect to the models presented in literature the coating conduction resistance has been added. The overall heat transfer in the fluid and solid phases are in parallel with each other. Looking at the solid phase, the mechanisms a, b and c are in parallel, while mechanisms d and e are in series with the equivalent resistance provided by a, b and c.

The base Kunii and Smith (K&S) model presented in [26] evaluates the packed bed effective thermal conductivity as:

keff = μ[kF + hRV ΔL] +(1 − μ)ΔL

1kFlv +hp+hRS

+ lsks

(3)

where, kF and ks are the thermal conductivities of the fluid and solid, hRV and hRS are the heat transfer coefficients for the thermal radiation from void to void and between solid surfaces, respectively, ΔL is the effective length between the centers of two adjacent particles along the main direction of the heat flow, lv is the thickness of a film of fluid which

would lead to the same heat resistance as the stationary fluid close to the contact point, ls is the thickness of a slab of solid which would offer the same heat transfer resistance as the pebble and hp is the heat transfer coefficient for the heat transfer through the contact surfaces between adjacent particles. Thermal radiation is accounted by means of the two heat transfer coefficients hRV and hRS, which are defined as in Eq. (4) and (5), respectively.

hRV =[

4σT3 + μ2(1 − μ)

(1 − εR)εR

]

(4)

hRS = 4σT3(

εR2 − εR

)

(5)

where εR is the thermal emissivity of the particle surface, and σ is the Stephan-Boltzmann constant, σ = 5.67 × 10− 8 W/(m2⋅K4).

To account for the coating influence an additional thermal resistance can be added, modifying Eq. (3) as:

keff = μ[kF + hRV ΔL] +(1 − μ)ΔL1

kFlv +hp+hRS

+ lsks +lckc

(6)

where, lc is the thickness of the coating layer that would lead to the same thermal resistance and kc is the thermal conductivity of the coating material.

Fig. 3 schematically shows the equivalent slab thickness for the solid

Fig. 1. (a) Idealized geometry of particle with coating layer (green); (b) unit cell with considered heat transfer mechanisms.

Fig. 2. Equivalent thermal resistance model.

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5

particle, void and coating considered in the unit cell. Eq. (6) can be rewritten as:

keffkF

= μ[

1 +hRVkF

β*dp,TOT]

+(1 − μ)β*

1dp,TOT

lv +dp,TOT

kF (hp+hRS)+ lsdp,TOT

(kFks

)

+ lcdp,TOT

(kFkc

) (7)

where, β* = ΔL/dp,TOT, and its value ranges between 0.895 for close packing of sphere and 1 for loose packing arrangement (as for quantity β introduced in [26]), ls is the height of the cylinder, oriented along the main heat transfer direction, and having the same volume as the spherical particle (ls = 2dp/3, [26]). Similarly, lc can be calculated as the height of the coating layer, on the basis of the previously obtained cylinder, with the same volume of the actual coating, as in Eq. (9).

Vc =43

π[(

rp + Δrc)3

− r3p]= πr2plc (8)

lc =43

1r2p

[Δr3c + 3r

2pΔrc + 3rpΔr

2c

](9)

Eq. (7) can then be rewritten as:

keffkF

= μ[

1 +hRVkF

β*dp,TOT]

+(1 − μ)β*

11

ϕ*+

dp,TOTkF (hp+hRS)

+ γ*(

kFks

)

+ δ*(

kFkc

) (10)

where, the introduced non-dimensional parameters γ*, δ* and ϕ* are defined as in Eqs. (11) to (13), where the overall particle diameter with the applied coating, dp,TOT = dp + 2Δrc, is considered.

γ* =ls

dp,TOT=

2dp/3dp + 2Δrc

(11)

δ* =lc

dp,TOT=

23

(Δr3c + 3r2pΔrc + 3rpΔr2c

)

r2p(rp + Δrc

) (12)

ϕ* =ϕdp

dpdp,TOT

(13)

where, ϕ is evaluated following the same procedure as explained in [26] and summarized in Eqs. (14) and (15).

ϕ = ϕ2 +(ϕ1 − ϕ2)μ − μ2μ1 − μ2

(14)

ϕi =12

(K− 1

K

)2

sin2ϑO,i

ln[K − (K − 1)cosϑO,i

]− K− 1K (1 − cosϑO,i)

−23

1K

(15)

In Eq. (14) the coefficients ϕ1 and ϕ2, corresponding to the ϕ values for the loose and close packing arrangements, respectively, are evalu-ated according to Eq. (15), the void fractions for the two extreme ar-rangements are defined as μ1 = 0.476 and μ2 = 0.26. In Eq. (15) K is the ratio between the solid and fluid thermal conductivity (K = kS/kF), and sin2ϑ0,i = 1/ni, where ni corresponds to the number of contact points on a semispherical surface for each particle and it is equal to 1.5 in the loose packing arrangement and to 3

̅̅̅4

√in the close packing arrangement.

The developed methodology has been validated against the experi-mental data obtained during the SANA project and presented in [37]. The coating material has been considered equal to the main base ma-terial. Some relevant data used for the validation are summarized in Table 2, while the results of the validation are reported in Fig. 4. As pointed out in the IAEA report [37] the calculated results are expected to be lower than the experimental ones. The reason for such difference is that in the experimental facility heat was transported also by gas con-vection in the packed bed, which is not included in the effective thermal conductivity calculation. However, the general trend of the effective thermal conductivity with respect to increasing temperature can be clearly seen and is properly captured the proposed approach.

2.2. Choice of investigated coating characteristics

In order to investigate the potential influence of multiple types of coating materials, the variation of different thermal properties will be considered. Table 3 summarizes the considered coatings properties. Such characteristics have been chosen in order to explore the full range of available coatings. Low, medium and high emissivity coatings, εR, have been idealized. Specifically, the low εR assumed values are repre-sentative of materials such as aluminum oxide [53] or tantalum [54] based coatings and commercial low emissivity paints [55]. Ceramic materials such as silica show medium thermal emissivity [56]; while elevated εR are usually achieved by high emissivity paint (Pyromark 2500 [57]). Additionally, since the thermal emissivity is often depen-dent on the actual temperature of the sample, coatings with increasing and decreasing emissivity with temperature, ε+R and ε−R , have also been simulated. Increasing thermal emissivity is usually observed in metals and some ceramic materials, such as oxidized Inconel [58] and mag-nesium oxide [59]. While decreasing εR have been observed in aluminum oxides and glass [56]. Looking at the thermal conductivity, kc, low values (~10− 2) have been assumed as representative of insu-lating materials based coatings, while intermediate values (~100) have been considered to represent ceramic based coatings. As an example, oxides such as zirconia (ZrO2) and mullite (78% Al2O3 – 22% SiO2) show a thermal conductivity between 1.7 and 2 W/(m⋅K) and between 2 and 6 W/(m⋅K), respectively; while nickel-chromium alloys (i.e. Inconel 600) have thermal conductivity in the range from 10 to 33 W/(m⋅K) [60]. High thermal conductivity values (~102) have been assumed as a model for some metallic based coatings [56]. Since most materials present temperature dependent thermal conductivity, these behaviors have been idealized by a linearly increasing and decreasing function along the

Fig. 3. Idealized schematics of the equivalent slabs for heat exchange within the unit cell.

Table 2 Data considered during the methodology validation procedure, as gathered from [37].

Parameter Value

Density, ρ [kg/m3] 1700 Specific heat, cp [J/(kg⋅K)] 1840 Thermal conductivity, ks [W/(m⋅K)] 24 Particle diameter, dp [m] 0.059 Considered coating thickness, Δrc [m] 0.0005

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6

considered working temperature range, k+c and k−c , respectively. Finally, different coating thicknesses, Δrc, have been investigated and idealized as uniform all over the particle surfaces.

2.3. Selection of packed bed properties and filler material

Different pebbles materials have been considered in order to study the potential influence of coating application on them. Since most of the available packed bed TES consists of either ceramic-based particles or natural rocks, the selection have been limited to these two categories. The main thermo-mechanical properties of the studied materials are summarized in Table 4. Since these materials can be packed according to different geometries and the resulting void fraction is affected by the arrangement, the influence of the void fraction, μ, ranging between 0.27 and 0.48 (loose and close packing typical values [28]) have been

analyzed. Finally, particle diameters between 0.005 m and 0.1 m have also been modelled. In order to limit the discussion to the methodology and potential benefit and drawbacks of applying coatings to pebbles for packed bed applications, the thermal expansion of the coating material has not been considered in this study. However, when considering the actual applicability of specific coatings the thermal expansion of the coating envelop and the substrate material have to be matched to avoid coating cracks and damage caused by thermal stresses and thermal shock. The thermal and chemical stability of the coating and base par-ticle material have also to be considered.

2.4. Packed bed thermal energy storage modelling

In order to assess the potential performance of applying coating on particles in packed bed TES units, a traditional packed bed thermody-namic model, based on the model of Schumann [61], has been devel-oped and is summarized by the two equations for the fluid and the solid phases, Eq. (16) and (17), respectively.

∂Tf∂t +

Gρf μ

∂Tf∂x =

keff ,fρf cp,f μ

∂2Tf∂x2 +

hasρf cp,f μ

(Ts − Tf

)+

UwDπρf cp,f Aμ

(T∞ − Tf

)(16)

∂Ts∂t =

keff ,sρscp,sμ

∂2Ts∂x2 +

hasρscp,s(1 − μ)

(Tf − Ts

)(17)

where T is the temperature, ρ is the density, cp is the specific heat, and the subscripts f and s indicate the fluid and the solid. G is the fluid mass flow rate per unit cross section area, h is the convective heat transfer coefficient between the fluid and the filler material (evaluated by the correlation proposed in [62]), as = 6(1 − μ)/dp is the packed bed surface to volume ratio, D and A are the storage diameter and cross sectional area, and Uw is the heat transfer coefficient between the wall and the ambient air, defined as in Eq. (18)

1Uw

=1ai+

D2∑n

j=1

1kj

lndj+1dj

+1ao

doutD

(18)

where, ai and ao are the convective heat transfer coefficients on the in-side and outside of the tank, as from [56,63,64], and kj are the thermal conductivities of the considered insulation and steel layers, as shown in Fig. 5, their values and the thickness of each layer (tins1, tins2, tsteel used to evaluate the diameters dj) are defined in Table 5. The effective thermal conductivity, calculated using the previously proposed method, has been divided between a fluid and a solid component (keff ,f and keff ,s)

Fig. 4. Model validation with respect to experimental data from SANA project presented in [37].

Table 3 Selected thermo-physical properties for the investigated coatings.

Thermal Emissivity εR[–]

Thermal Conductivity kc [W/(m⋅K)]

Coating Thickness Δrc [10− 3

m]

0.1 0.05 0.05 0.5 2 0.5 0.9 10 1 ε+R = (0.6/1400)⋅(T − 273.15) +

0.2 150 2

ε−R =− (0.6/1400)⋅(T − 273.15) +

0.8

k+c = (90/1400)⋅(T − 273.15) +10

k−c =− (90/1400)⋅(T − 273.15) +

100

Table 4 Packed bed TES filler material thermo-mechanical properties [56,61].

Materials Thermal Conductivity ks[W/(m⋅K)]

Specific heat cp,s[J/ (kg⋅K)]

Density ρs[kg/(m3)]

Thermal Emissivity εR[–]

Alumina, Al2O3

19.25 785 3665 0.33

Silica, SiO2 1.4 705 2410 0.65 Natural Rocks -

Basalts 2.65 807.5 2915 0.85 Fig. 5. Packed bed geometry sketch and boundary conditions during charge,

discharge and storage periods.

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according to the method proposed in [65]. The TES model has been implemented and solved in Matlab R2019b® [66] exploiting the avail-able variable step and variable order numerical differentiation solver [67]. More detailed insights on the modelling of the packed bed TES, together with its validation against experimental data, have been pre-sented in a previous work of the authors [68].

The storage cyclic thermal behavior has been simulated as a series of five consecutive cycles of charge, discharge and standstill, after which a constant cyclic behavior was obtained. The overall duration of a cycle has been taken to be 24 h. Alumina pebbles have been assumed. The values selected for the main parameters describing the TES are sum-marized in Table 5. Fig. 5 schematically represents the packed bed TES together with the considered boundary conditions at the inlet and outlet sections during charge (red), discharge (blue) and storage periods (black). The arrows represent the main HTF flow direction during charge (red) and discharge (blue). The cylindrical surface consists of an inner steel layer (grey) and high temperature (yellow) and low temperature (light green) insulation layers, whose combined effects lead to the overall the heat transfer coefficient between wall and ambient air, Uw.

The presented packed bed TES model has been employed to evaluate the temperature and the effective thermal conductivity distributions in a TES unit considering various coating materials and including different coatings in different sections of the storage unit.

3. Results and discussion

In this section, the main results of the application of the exposed method under different conditions are presented and discussed. The influence of some coating properties (thermal emissivity, thermal con-ductivity and thickness) and of some typical packed bed properties (particle diameter, void fraction and particle base material) on the effective thermal conductivity of packed bed are investigated. The effective thermal conductivity variation during operation of a repre-sentative TES unit is studied as well.

3.1. Influence of coating properties

In order to assess the influence of coating layers, an uncoated packed bed made of Al2O3 particles (thermo-physical properties listed in Table 4) with hydraulic diameter dp = 0.01 m and void fraction µ = 0.39 is considered as the base reference case (K&S in Fig. 6 and Fig. 7). The coating properties are modified one at a time while keeping the others properties equal to reference values defined as: emissivity εR = 0.9, coating thermal conductivity kc = 10 W/(m⋅K) and coating thickness Δrc = 5⋅10− 4 m.

Fig. 6(a) reports the behavior of keff over a full range of emissivities, the dashed lines highlight the specific cases summarized in Fig. 6(b) (pink lines for constant εR cases and black lines for ε+R and ε−R ). As it is expected, since thermal radiation becomes increasingly more prominent as the temperature increases and the dominant heat transfer mode at working temperatures higher than 600 ◦C, keff increases with tempera-ture. Higher values of the emissivity of the coating would lead to higher keff due to an enhanced thermal radiation heat exchange within the packed bed. Fig. 6(b) shows the effect of different specific coating emissivities, as listed in Table 3 and reported in Fig. 6(a). Mod K&S stands for the results of the proposed modified Kunii and Smith approach to account for the coating thermal resistance. Table 6 sum-marizes the coefficient of an exponential fitting, in the form keff =a⋅exp(b⋅T), of the curves shown in Fig. 6(b). For the case with a linearly decreasing thermal emissivity of the coating (ε−R ), a linear fitting has been considered since it leads to a higher coefficient of determination, R2. Increasing εR the effective thermal conductivity increases with

Table 5 Values of the main parameters and design variables used for the packed bed TES model.

Parameter Symbol Value Unit

Charge Temperature Tch 900 ◦C Discharge Temperature Tdisch 300 ◦C Ambient Temperature T∞ 20 ◦C Charge Phase Duration tch 6 h Discharge Phase Duration tdisch 10 h Specific HTF Mass Flow Rate Charge Gch 0.669 kg/(m2⋅s) Specific HTF Mass Flow Rate Discharge Gdisch 0.402 kg/(m2⋅s) Packed Bed TES Height H 9.335 m Packed Bed TES Diameter D 7.748 m Void Fraction µ 0.4 – Particle Hydraulic Diameter dp 0.035 m Thermal Conductivity High Temperature

Insulation kins1 0.3 W/(m⋅K)

Thermal Conductivity Low Temperature Insulation

kins2 0.1 W/(m⋅K)

Thermal Conductivity 253 MA ksteel 17 W/(m⋅K) Thickness High Temperature Insulation Layer tins1 0.1 m Thickness Low Temperature Insulation Layer tins2 0.2 m Thickness 253 MA Layer tsteel 0.005 m External Heat Transfer Coefficient Uw 0.41 W/

(m2⋅K)

Fig. 6. (a) Effective thermal conductivity versus temperature for reference case including coatings with different emissivity, εR, and considering Al2O3 particles as base material, dp = 0.01 and μ = 0.39 and a coating with kc = 10 W/(m⋅K) and Δrc = 5⋅10− 4m; (b) Modified effective conductivity behavior with temperature including different coating emissivity and considering Al2O3 particles as base material, dp = 0.01 m, μ = 0.39 and a coating with kc = 10 W/(m⋅K) and Δrc = 5⋅ 10− 4 m.

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temperature at a higher rate, as demonstrated also by the increasing values of coefficient b in Table 6. Indeed, at higher thermal emissivities, the radiation heat exchange and its exponential dependence on the temperature assume a dominant role. A temperature dependent emis-sivity of the coating, such as the simulated representative cases ε+R and ε−R , might be exploited in order to modify the dependence of keff on the temperature. A linearly increasing ε+R would lead to a steeper change of keff (green dashed line); whilst in comparison, a linearly decreasing ε−R would ensure a flatter profile of keff on the investigated temperature range (light blue dashed line). This qualitative view is confirmed by the fitting coefficients reported in Table 6. ε+R leads to the maximum value of the coefficient b, higher than the value achievable for a very high but constant coating thermal emissivity. In contrast, ε−R strongly hampers the influence of the thermal radiation on the packed bed and a linear profile of the effective conductivity can be observed. A coating with a thermal emissivity that increases with temperature could lead to larger variation of the effective thermal conductivity along the axis of the packed bed between the high and low temperature regions. Conse-quently, thicker thermocline regions would be recorded during opera-tion and storage periods, and faster heat exchange would be achievable in the high temperature region. Coatings with decreasing thermal emissivity could be exploited to ensure a more uniform distribution of the effective thermal conductivity along the packed bed, a thinner thermocline region, and limited thermocline degradation during TES operation.

At a reference working temperature of 800 ◦C, a high emissivity coating (εR=0.9) could lead to an increase of the effective thermal conductivity of the packed bed of 84% with respect to a similar uncoated

system. Instead, a low emissivity coating (εR=0.1) would ensure a reduction of the original uncoated keff of about 30%.

Fig. 7(a) summarizes the effect of different coating thermal con-ductivity, kc (as listed in Table 3), on keff . Notwithstanding a high εR, a coating material with a very low thermal conductivity would have a significant impact on the heat flow in the solid particles, leading to a lower effective thermal conductivity for the packed bed (grey solid line). At low temperatures, the reduction is more significant (− 33.1%). At higher temperature, the difference between the coated and uncoated keff (for low kc) decreases to a significantly lower level (− 15.3%). This is due to the increased importance of the radiative heat transfer and the higher εR of the coating with respect to the original particle. As it is expected, a higher kc would be helpful in enhancing the bed effective thermal conductivity. However, very large kc values do not lead to a wide spread of the effective thermal conductivity and similar trends can be observed for all cases including kc higher than 10 W/(m⋅K) (red line). This is explained by the fact that for kc higher than 10 W/(m⋅K) the thermal resistance related to conduction through the coating layer becomes negligible with respect to the other thermal resistances included in the cell unit. Fig. 7(b) shows the effective thermal conductivity for different coating layer thicknesses (as listed in Table 3). Thicker coatings lead to higher keff , which is due both to the larger amount of coating material (which is assumed to have a higher thermal conductivity), and to the larger overall particle diameter (which facilitate the thermal radiative heat exchange). The increase of keff with Δrc is mostly linear and more relevant at small particle diameters.

From the discussion above, it can be concluded that in order to modify keff , the coupled influence of the investigated factors can be exploited. High emissivity coatings with a relatively high thermal con-ductivity (kc≥10 W/(m⋅K)) and applied with a thick layer could maxi-mize keff . While low emissivity coatings with a low thermal conductivity could minimize keff .

3.2. Influence of packed bed characteristics

The following analysis has been performed in order to assess the influence of different characteristics of the packed bed and under which conditions it might be relevant to consider the utilization of specific coatings. To isolate the effects of the bed parameters the coating prop-erties have been kept constant as: emissivity εR = 0.9, thermal con-ductivity kc = 10 W/(m⋅K) and thickness Δrc = 5⋅10− 4 m.

Fig. 8(a) shows the behavior of the effective thermal conductivity with variable particle diameter and working temperature. It can be seen

Fig. 7. (a) Effective thermal conductivity versus temperature for reference case including coating with different thermal conductivities, kc W/(m⋅K), and considering Al2O3 particles as base material, dp = 0.01, μ = 0.39 and a coating with εR = 0.9 and Δrc = 5⋅10− 4 m; (b) Effective thermal conductivity versus temperature for reference case including different coating layer thicknesses, Δrc[m], and considering Al2O3 particles as base material, dp = 0.01, μ = 0.39 and a coating with εR = 0.9 and kc = 10 W/(m⋅K).

Table 6 Fitting coefficients for keff obtained at different εR. An exponential fitting in the form keff = a⋅exp(b⋅T), with T in K, is considered, except for the ε−R case where a linear fitting is applied. For fitting purposes, the temperature range [673.15–1473.15] K (equivalent to [400–1200] ◦C) is considered.

Coating Emissivity, εR [–] Effective conductivity, keff [W/ (m⋅K)]

R2

a b

0.1 0.29753 1.243 ⋅ 10− 3 0.9998 0.5 0.26629 1.887 ⋅ 10− 3 0.9984 0.9 0.30778 2.051 ⋅ 10− 3 0.9956 ε+R = (0.6/1400)⋅(T − 273.15) + 0.2 0.18608 2.227 ⋅ 10

− 3 0.9996

ε−R = − (0.6/1400)⋅(T − 273.15) + 0.8 0.00268⋅T − 0.88653 0.9989

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that the particle diameter has a dominant influence on the effective thermal conductivity. For a working temperature of 800 ◦C, a keff value of 2.88 W/(m⋅K) can be calculated for a dp of 0.01 m, while increasing dp to 0.05 m and 0.1 m, the keff can reach 9.62 and 16.66 W/(m⋅K), respectively. This behavior is due to the prevailing influence of thermal radiation in the overall heat transfer mechanism for large particle di-ameters compared to for small particles. At such conditions, the packed bed has larger void spaces and radiation encounters less obstructions. In contrast, for small particle size the thermal radiation requires higher temperatures to begin to dominate the overall heat transfer.

Fig. 8(b) represents the effective thermal conductivity behavior with variable packed bed void fraction and working temperature. An almost uniform behavior of keff with respect to μ can be noted. Such results are related to two contrasting phenomena. On one hand, a larger void fraction means larger empty spaces where the thermal radiation takes place, and thus it leads to an increase of keff . On the other hand, a higher void fraction means a lower amount of solid and coating material, and, as a result, decreasing thermal conductivity related to the conduction through these two solid components (considering ks,kc >> kf ). At high void fractions a larger variation of keff with temperature can be observed. At such μ, the reduced solid material amount largely affects the overall properties at low temperature, while increasing the tem-perature the radiation heat transfer begins to take over a prevalent role causing higher values of keff .

Fig. 9 shows the effective thermal conductivity of packed bed made of the materials listed in Table 5 when applying a high (εR=0.9) and a low (εR=0.1) emissivity coatings. In order to simplify the model and limit the focus of this work to the main methodological aspects, the idealized coating material have been assumed to be stable over all base materials. Equal coating thermal conductivity (kc=10 W/(m⋅K)) and thickness (Δrc=5 × 10− 4 m) have been considered for both coatings.

The high emissivity coating application have the largest potential to modify the effective thermal efficiency of an Al2O3 packed bed. An in-crease of 80.5% can be achieved at 800 ◦C. This is justified by two main factors. Firstly, the low original emissivity of Al2O3, which is greatly enhanced by the coating, increasing the radiation heat transfer. Sec-ondly, the high thermal conductivity of Al2O3; the higher ks the larger the enhancement of keff achievable by means of the coating addition. A relatively low increase in the effective thermal conductivity (+11.1%) is recorded for the rock base materials at 800 ◦C. This is mostly due to the assumed original high emissivity of the rocks, which is only slightly increased by the coating. At 800 ◦C, the increase of keff for the SiO2 base material is about 25.1%. Though the original emissivity of SiO2 is relatively low, the improvement of the effective thermal conductivity

keff is hindered by a low ks. In contrast, the application of low thermal emissivity coatings has a

major impact on rock based packed beds, with a keff reduction of about 58.5% at 800 ◦C. This reduction is mainly due to the elevated original thermal emissivity of the rock pebbles. Low effective thermal conduc-tivities are obtained for a SiO2 based packed bed with low emissivity coatings, with keff slightly above 1.0 W/(m⋅K) even at 1200 ◦C. Only a 28.9% reduction in keff is recorded for an Al2O3 based packed bed, however the effective thermal conductivity remains below 2.0 W/(m⋅K) over the temperature range considered.

In summary, it has been observed that the application of a high emissivity coating could enable a significant increase in the packed bed effective thermal efficiency, in particular when the bed is made of large size particles and with a high void fraction. In contrast, in order to limit the thermocline degradation along the packed bed during storing time, low emissivity coatings can be exploited, and their effectiveness would be maximized at large particle sizes and high void fractions. In this way, the effective thermal conductivity of the packed bed could be reduced without largely affecting the thermal conductivity of the base material, and a uniform temperature distribution within each particle could still be assumed.

Fig. 8. (a) Modified effective thermal conductivity behavior with temperature and particle diameter, considering Al2O3 particles as base material, μ = 0.39 and a coating with εR = 0.9, kc = 10 W/(m⋅K), and Δrc = 5 × 10− 4 m; (b) Modified effective thermal conductivity behavior with temperature and void fraction, considering Al2O3 particles as base material, dp = 0.01 and a coating with εR = 0.9, kc = 10 W/(m⋅K), and Δrc = 5⋅10− 4 m.

Fig. 9. Effective thermal conductivity for uncoated and coated (with εR = 0.9 and εR = 0.1) particles of different base materials, considering dp = 0.01, μ =0.39 and a coating with kc = 10 W/(m⋅K) and Δrc = 5⋅10− 4 m.

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3.3. Effective thermal conductivity distribution in a packed bed thermal energy storage

In this section, the influence of a coating application in a represen-tative packed bed TES is investigated. Fig. 10 shows the effective ther-mal conductivity distribution along the x-axis (TES height direction) during a full charge cycle, for the base case with uncoated alumina particles and for the case in which all particles were coated with a low emissivity paint (εR = 0.1). Firstly, it can be seen that the effective thermal conductivity of the packed beds is neither constant nor uniform during operation. During the charge cycle, keff increases together with the packed bed temperature. A distribution of keff develops with low values in the cold bottom part of the TES unit (dimensionless TES height = 1) and high values in the top hot part (dimensionless TES height = 0). This leads to a variation for the base case keff of more than 420% (from about 0.86 W/(m⋅K), in the cold region, to 4.48 W/(m⋅K) in the hot region). The region where keff increases moves along the x-axis of the TES, similarly to the movement of the temperature thermocline. This region enlarges during the charge process, as the thermocline degrada-tion proceeds. Particularly, the slope of the keff increase in the hot part decreases with time (as highlighted by the two black circles and the black arrow in Fig. 10). Applying a low emissivity coating on the particle could limit the increase of keff from the cold to the hot region to less than 220% (from about 0.72 W/(m⋅K), in the cold region, to 2.36 W/(m⋅K) in the hot region). Such a reduction of keff would reflect in reduced axial conduction heat exchange, leading to limited thermocline degradation and self-discharge thermal losses, and enhanced thermal performance. Such an approach could be also seen as a passive thermocline control method.

The potential application of different coatings in different regions of the packed bed TES has been investigated. Fig. 11 shows a schematic representation of the packed bed TES where a high, medium and low thermal emissivity coatings have been applied on the particles of the bottom, mid and top TES layers, respectively. Considering this arrangement the fluid temperature distribution during a charge phase is reported in Fig. 12(a). The corresponding effective thermal conductivity is shown in Fig. 12(b). As expected, sharp changes of keff can be seen at the boundaries between the three regions, and keff increases along with the temperature. The highest values of keff are recorded in the bottom layer of the TES, where the thermal emissivity of the coating is the highest, at the end of the charge. Thanks to the low εR of the coating, in the hot top part of the TES, keff remains low even at high temperature.

The arrangement presented shows that by using different coatings

the heat transfer in the packed bed section could be partially controlled and modified depending on the specific needs. For a packed bed TES, a reduced effective thermal conductivity would be beneficial to limit the thermocline degradation and improve the thermal efficiency. In contrast, for other types of heat exchangers based on packed beds - such as gas-cooled reactors – a higher heat conduction, and therefore higher effective thermal conductivity values, might be favorable.

4. Conclusions

In this work, a methodology to include the thermal effect of a coating layer applied on the pebbles of a packed bed has been presented. The proposed effective thermal conductivity model is based on the commonly used Kunii and Smith model. The heat conduction through the coating layer has been added to the other heat transfer mechanisms and modelled via an effective thickness and thermal resistance. The effects on the effective thermal conductivity of several characteristics of the coating material and of the packed bed have been investigated. From the results discussed, the following conclusions can be drawn:

1. Low emissivity coatings could reduce the effective thermal conduc-tivity of a packed bed, this would limit the thermocline degradation during storage and operation periods. Particularly, the application of low emissivity coatings would reduce the increase of keff from the low to the high temperature regions of an Al2O3 based packed bed by about 200% with respect to the original uncoated case.

2. High emissivity coatings could increase the effective thermal con-ductivity of a packed bed, a keff increase of more than 80% is observed already at 800 ◦C for an Al2O3 based packed bed. This will lead to an enhancement in the heat transfer coefficient and higher heat transfer through the bed.

3. The application of a coating has a particularly large influence when the packed bed is composed of large size particles (dp>0.05 m) with a high void fraction (μ > 0.4) and the filler material has a relatively high thermal conductivity.

4. The application of a coating with a temperature dependent emis-sivity might be exploited to modify the dependence of the effective thermal conductivity on the packed bed temperature. A coating material with a εR linearly decreasing with temperature would inhibit the exponential relationship between temperature and keff .

5. The use of different coatings within the same packed bed unit could enable different distributions of keff and consequently a partial con-trol on the charging process and on the thermocline degradation.

In order to extend the present study, future studies will focus on the thermal stability testing of several coating materials on ceramic pebbles. Besides, the influence of some selected coating materials on a packed

Fig. 10. Effective thermal conductivity distribution along the packed bed TES axis during a whole charge cycle for the base case (uncoated Al2O3 particles) and for the case with applied low emissivity coating (εR = 0.1).

Fig. 11. Packed bed TES schematics with different coating emissivities in different regions.

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bed thermal energy storage will be investigated in a lab prototype.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research has been funded by the Swedish Energy Agency and Azelio AB through the Energy Agency program Electricity from the Sun, project P43284-1.

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Fig. 12. (a) Temperature distribution along the packed bed TES axis during a whole charge cycle; (b) effective thermal conductivity distribution along the packed bed TES axis during a whole charge cycle for the case represented in Fig. 11.

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Coatings utilization to modify the effective properties of high temperature packed bed thermal energy storage1 Introduction2 Methodology2.1 Packed bed effective thermal properties2.1.1 Effective heat capacity2.1.2 Effective thermal conductivity

2.2 Choice of investigated coating characteristics2.3 Selection of packed bed properties and filler material2.4 Packed bed thermal energy storage modelling

3 Results and discussion3.1 Influence of coating properties3.2 Influence of packed bed characteristics3.3 Effective thermal conductivity distribution in a packed bed thermal energy storage

4 ConclusionsDeclaration of Competing InterestAcknowledgementReferences