Baoshan Xie

Nicolas Baudin

Jérôme Soto

Yilin Fan

Lingai Luo
email: lingai.luo@univ-nantes.fr

Experimental and numerical study on the thermocline behavior of packed-bed storage tank with sensible fillers

Keywords: Thermal energy storage, Packed-bed thermocline, Sensible fillers, Flow diffuser, Insulation

Packed-bed thermocline tank with sensible fillers is a cost-effective option for thermal energy storage (TES). In real charging and discharging, the thermocline stability is disturbed and thermocline expansion occurs due to various factors, leading to the decreased global performances of storage tank.

The purpose of this work is to experimentally and numerically study the impact of inlet configuration and insulation on the thermocline expansion in a lab-scale tank under various operating parameters.

Firstly, a DC-3P model considering the wall effect and heat loss is developed and validated by experimental results, capable of precisely predicting the performance of storage tank under the tested conditions. Especially, the temperature difference between the fluid and solid filler center is detected both in experiments and in modeling, indicating the existence of temperature gradient inside large solid particles. Results also show that the injecting flow causes the radial temperature non-uniformity and thermocline expansion at the near-entrance region of the tank, but its effect on the global performance of the storage tank is rather limited. Good insulation of the tank could reduce the thermocline thickness by about 20%, and increase the energy efficiency and the capacity ratio by 5-7% and 3-5%, respectively.

Nomenclature

A

Superficial area (m 2 )

Greek symbols Bi

Biot number, Bi=hꞏ(V/A)/λ α Thermal diffusivity (m 2 s -1 ), α= λ/ (ρꞏC p ) C p Specific heat capacity (J kg -1 K -1 ) λ Thermal conductivity (W m -1 K -

Introduction

The yearly growing in energy demand worldwide and the carbon dioxide emission in 2021 have reached about 4.6% and 5%, respectively [START_REF] Taylor | Guidelines for evaluating and expressing the uncertainty of NIST measurement results[END_REF]. As a promising alternative to traditional fossil sources, renewable energies get great attention to mitigate climate change and environmental pollution concerns.

In 2020, renewable energy represented 22.1 % of energy consumed in European countries and is still significantly expanding to meet the goal of European Green Deal by 2050 [2,3]. Nevertheless, the intermittent and fluctuating nature of these renewable sources (e.g., solar energy) calls for the development and deployment of efficient thermal energy storage (TES) techniques to improve the system's flexibility and dispatchability [4][5][START_REF] Palomba | Application of numerical methods for the design of thermocline thermal energy storage: Literature review and critical analysis[END_REF]. For example, more than 70% of the concentrated solar power (CSP) plants under construction have combined with TES system and this ratio is still increasing for future more powerful plants [START_REF] Pelay | Thermal energy storage systems for concentrated solar power plants[END_REF]. According to the storage mechanism, TES systems can be classified into sensible [START_REF] Gautam | A review on sensible heat based packed bed solar thermal energy storage system for low temperature applications[END_REF], latent [START_REF] Soto | Thermal cycling aging of encapsulated phase change material -Compressed expanded natural graphite composite[END_REF], and thermochemical [START_REF] Pelay | Integration of a thermochemical energy storage system in a Rankine cycle driven by concentrating solar power: Energy and exergy analyses[END_REF] heat categories. Among them, sensible heat storage is currently the most mature and commonly-used technology for various applications.

Thermocline-based TES technology involves a type of single-tank storage system where hot and cold heat transfer fluids (HTF) are both stored in the same tank [START_REF] Lou | A review on the performance indicators and influencing factors for the thermocline thermal energy storage systems[END_REF]. The thermocline or thermal stratification zone with a temperature gradient is formed around the interface between the hot and cold fluids. It periodically moves up/down in the axial direction during the charging/discharging operations.

Compared to the conventional two-tank TES system, the thermocline concept is a more competitive option with about 37% cost reduction, by cutting the needed two tanks into one and by using the cheaper sensible fillers (e.g., rocks or concretes) to substitute the amount of some expansive liquid HTF used as storage media (e.g., oil or molten salt) [START_REF] Rodrigues | Effect of porous material properties on thermal efficiencies of a thermocline storage tank[END_REF]. For all these reasons, the sensible-heat thermocline packedbed (SHTPB) TES system has become a research focus in recent years.

In theory, the thermocline stability and thickness affects the SHTPB system's performance: a higher level of thermal stratification or a thinner thermocline zone means higher energy and exergy efficiencies [START_REF] Roux | Thermocline thermal energy storage optimisation combining exergy and life cycle assessment[END_REF][START_REF] Rosen | The exergy of stratified thermal energy storages[END_REF]. Nevertheless, the thermal stratification tends to become unstable and expanded over dynamic and cycling operations. This thermocline degradation, or decay, can be due to many factors [START_REF] Hoffmann | Experimental and numerical investigation of a thermocline thermal energy storage tank[END_REF][START_REF] Davenne | Stability of packed bed thermoclines[END_REF]. On one hand, the low heat transfer rate between HTF and solid fillers or the heat diffusion, inside the HTF, the solid media, or the wall phases would result in the expanded thermocline thickness in the axial direction [START_REF] Geissbühler | An assessment of thermocline-control methods for packed-bed thermal-energy storage in CSP plants, Part 2: Assessment strategy and results[END_REF]. On the other hand, the flow injection, the heat loss to the environment, and the wall properties would cause the non-uniform radial temperature distribution inside the tank, lowering the thermocline stability and increasing the thickness [START_REF] Cascetta | Experimental and numerical research activity on a packed bed TES system[END_REF].

Uniform radial flow velocity and temperature profiles are usually assumed in one-dimensional (1D) models (e.g., Schumann's model) for modeling SHTPB systems due to the flow homogenizing effect of the porous fillers [START_REF] Nicolas | Flexibility and robustness of a high-temperature air/ceramic thermocline heat storage pilot[END_REF]. The necessity of additional flow diffuser instead of inlet port in packed-bed storage tanks to guarantee the good and stable thermal stratification is controversy in early stages. Study was focused on installing special types of flow diffuser in SHTPB tanks to effectively utilize the bed domain [START_REF] Sanderson | Performance and efficient design of packed bed thermal storage systems. Part 1[END_REF]. Later, Bruch et al. [START_REF] Bruch | Experimental and numerical investigation of a pilot-scale thermal oil packed bed thermal storage system for CSP power plant[END_REF] experimentally observed that there was no significant inhomogeneity of radial temperature in a SHTPB tank without diffuser. The authors stated that the 1D heat transfer model was adequate for the configuration where tank height to diameter aspect ratio H/Dtank =3. Yin et al. [START_REF] Yin | Experimental research on thermal characteristics of a hybrid thermocline heat storage system[END_REF][START_REF] Yin | Thermocline characteristics of molten-salt thermal energy storage in porous packed-bed tank[END_REF] also experimentally tested a near-tube tank (H/D tank =5) and showed that the porous fillers could help to maintain the HTF as an ideal plug flow pattern. However, depending on different filler configurations and heat transfer mechanisms, the impact of injecting flow on the thermal performance is different. For example, Reddy and Pradeep [START_REF] Reddy | Stability analysis of the thermocline thermal energy storage system during high flow rates for solar process heating applications[END_REF] numerically observed that the radial non-uniform temperature distribution in their SHTPB tank (H/D tank =4) is negligible in laminar flow owing to the porous packed bed working as flow distributor, but not in turbulent condition. Wang et al.

[25] simulated a flow annular diffuser (80% annular area) at the inlet of a SHTPB tank (H/D tank 1/3) and found that the non-uniform radial flow has a limited influence on output energy but it could affect the interstitial heat transfer, thus improving the thermal performance. Recently, Vannerem et al. [START_REF] Vannerem | Experimental and numerical investigation of the impact of operating conditions on thermocline storage performance[END_REF] analyzed this problem by comparing three types of baffle-type flow diffusers under different flow rates.

It has been reported that the solid filler could act as a natural distributor and the fluid distribution at the inlet did not impact the global storage performance due to the flow being restricted to a small fraction of the packed bed in their SHTPB tank (H/D tank 2). In brief, more investigations are still needed to

show from what distance to the inlet such radial temperature gradient can be eliminated by the porous bed with different packing properties (e.g., particle size, porosity etc.), and to further clarify the relation between the thermocline expansion due to the injecting fluid flow and the global thermal performance reduction of the SHTPB tank.

The heat losses and associated wall impact are another influencing factor on the radial temperature distribution of SHTPB tank. Early studies [START_REF] Cascetta | Experimental investigation of a packed bed thermal energy storage system[END_REF][START_REF] Cascetta | Comparison between experimental and numerical results of a packed-bed thermal energy storage system in continuous operation[END_REF] reported the existence of radial temperature gradient for about 30-40% of the tank radius due to the heat loss, leading to the higher centerline temperature than that in near-wall region. This radial temperature gradient cannot be taken in account in the 1D model and will cause the smaller amount of energy stored in the SHTPB tank than predicted [START_REF] Cascetta | Experimental and numerical research activity on a packed bed TES system[END_REF][START_REF] Cascetta | A study of a packed-bed thermal energy storage device: Test rig, experimental and numerical results[END_REF].

Hoffmann et al. [START_REF] Hoffmann | Experimental and numerical investigation of a thermocline thermal energy storage tank[END_REF][START_REF] Hoffmann | A thermocline thermal energy storage system with filler materials for concentrated solar power plants: Experimental data and numerical model sensitivity to different experimental tank scales[END_REF] reported that the heat losses would lead to the unstable thermocline, reduced output power, and decreased outlet temperature. Xie et al. [START_REF] Xie | Wall impact on efficiency of packed-bed thermocline thermal energy storage system[END_REF] have developed adapted transient models to systematically explore the wall impact on the dynamic thermocline behavior of the SHTPB tanks by including the wall and insulation heat capacity in the governing equations. The most influencing factors have also been identified to provide useful design guidelines. Nevertheless, the experimental investigations of a SHTPB tank under two extreme conditions, with or without insulation, on the thermocline expansion are still insufficient, especially for tanks operated with storage temperature range less than <100 °C. Table 1 lists main experimental investigations using sensible fillers reported in the literature.

Therefore, this work is devoted to investigating the dynamic thermocline behaviors of the packedbed system under the impacts of inlet configuration and insulation in different operational conditions, including HTF flow rate, inlet temperature, filler particle size, etc.. At first, a numerical dispersionconcentric three-phase phase (DC-3P) model was developed and its applicable range was determined in view of model validation. Then, to assess the influences of radial temperature non-uniformity due to the injecting flow and heat loss, an experimental lab-scale tank of SHTPB was fabricated and tested under controlled conditions. Three inlet configurations (using a structured inlet diffuser) and two insulation configurations were compared. In the end, the effects of heat diffusion in fluid and solid phases and the heat convection between the HTF and solid fillers on the axial thermocline expansion are further discussed.

This work is fitted to small flow rates (Reynolds numbers, 𝑅𝑒<15), relatively large particle size (sphere Biot number, Bi s >0.1) and low temperature (<100 °C). It is expected to provide useful design and operating guidelines for alleviating thermocline expansion in SHTPB tanks in practical applications.

Experimental setup

Test-rig description

The laboratory-scale SHTPB tank tested has a cylindrical shape with a height of 398 mm and an inner diameter of 194 mm (Fig. 1a), the total useful volume being 11.8 L. It has a transparent polycarbonate wall (3 mm thickness) enclosed by an insulation layer made of black nitrile rubber (25 mm thickness). Glass sphere as the sensible filler is packed inside the tank randomly. Water is used as the HTF. The top and bottom ports are simple tubes (Φ= 11.6 mm) located at the center of each cover of the tank. In charging, hot water first enters from the tank top port and transfers heat to solid fillers inside to store heat. In discharging, cold water enters from tank bottom port to drive the stored hot water from tank top to release heat. The charging or discharging stops when each outlet temperature reaches the corresponding threshold point according to the designed requirement.

In total, 32 thermocouples have been used to measure the local temperature at different locations in the storage tank, including 17 for the fluid and 15 for the solid. They were all attached to a plastic tree which has been installed inside the tank carefully (in Fig. 1b). In axial direction, both fluid and solid temperatures were measured at 7 heights of the centerline (C in Fig. 1b). They are at 49 mm distance one from another and notated as TC-1 to TC-7 from the bottom to the top of the tank. To investigate the radial temperature distribution, the fluid and solid temperatures at the middle (B in Fig. 1b; tank diameter coordinate r * = 0.55) and outer (A in Fig. 1b; r * = 0.95) positions were measured in four radial arms (TC-1; 3; 5; 7). In addition, the inlet and outlet fluid temperatures were also measured at the top and bottom ports, respectively. Thermocouples for glass spheres were installed in the center of sphere and sealed by glue with a similar thermal conductivity as glass (cf. Fig. 1a). The fluid thermocouples have been specially attended to avoid contact with the surface of the solid. The influence of contact is negligible with a condition of 1.3>Bi s >0.1 due to the conduction resistance of the solid being larger than the convection resistance. The glass was chosen as filler in the experiment, mainly because it has adapted thermal conductivity for operating at a larger Bi s number and is easily available in diverse sizes on the market.

Tested cases

Six configurations of SHTPB tank were experimentally investigated to evaluate the impacting factors of inlet configuration and insulation, under different mass flow rates and inlet temperatures, etc., on the thermocline. Detailed parameters of these cases are listed in Table 2 and introduced as below.

Case 1 vs 3: To verify the temperature gradient inside the particle and to explore the influence of particle size on the thermocline behavior, two sizes of glass spheres of 7 and 10 mm were used as fillers.

Case 1 and case 3 were tested under different inlet temperatures and flow rates. for a single-media thermocline TES tank in our previous works [START_REF] Lou | Optimized flow distributor for stabilized thermal stratification in a single-medium thermocline storage tank: A numerical and experimental study[END_REF]. In Case 5, the space between the diffuser (z 1 =10 mm) and the tank top/bottom was filled with one layer of glass spheres to make full use of the tank volume, both the porous bed and the plate diffuser acting as the inlet configuration. In comparative Case 6, no solid fillers were packed in this gap to eliminate the effect of porous fillers, and the height was set to be larger (z 1 =35 mm) to amplify the impact of inflowing jet and mitigating effect of the plate-type diffuser. In this way, the effects of the diffuser on the radial temperature distribution, the impacted height, and and the global thermal efficiency were investigated under different inlet temperatures and flow rates. Case 1 vs Case 3: effect of particle size on the temperature gradient inside the sphere.

Case 1, 3 vs Case 2, 4: effect of insulation on the thermocline dynamics and system's efficiency.

Case 3 vs Case 5, 6: effect of diffuser on the thermocline dynamics and system's efficiency.

Numerical model

A dispersion-concentric three-phase (DC-3P) numerical model has been developed based on the following assumptions:

(1) HTF is incompressible plug flow.

(2) Thermophysical properties of fluid and solid are set as constant (Table 3).

(3) Solid sphere is modeled as dispersion concentric because of the possible non-ignorable temperature gradient inside (Bi s >0.1).

(4) Wall is modeled as a separate phase in 1D (axial direction), taking the wall impact (heat storage/release in the wall) into account [START_REF] Xie | Wall impact on efficiency of packed-bed thermocline thermal energy storage system[END_REF].

(5) Insulation layer is simplified and represented by a thermal resistance based on our earlier study [START_REF] Xie | Wall impact on efficiency of packed-bed thermocline thermal energy storage system[END_REF].

(6) Heat loss to the environment considers both the natural convection and the radiation.

With these assumptions, the governing equations for computation domain (Fig. 4) can be written as follows.

Table 3 Thermophysical properties of different materials used in this study. Ambient (20 °C, 100 kPa) Air 1.17

Material

Governing equations

Fluid [START_REF] Rady | Thermal performance of packed bed thermal energy storage units using multiple granular phase change composites[END_REF]:

𝜀 • 𝜌 • 𝐶 , • 𝑢 • • 𝜆 , • ℎ • • 𝑇 𝑧, 𝑟 𝐷 𝑝 2 𝑇 𝑧 ℎ • • 𝑇 𝑧 𝑇 𝑧 (1a)
Solid filler (2D spherical radius coordinate):

𝜌 • 𝐶 , • , • • 𝑟 • 𝜆 • , (1b)
Wall:

𝜌 • 𝐶 , • • 𝜆 • ℎ • • 𝑇 𝑧 𝑇 𝑧 ℎ • • 𝑇 𝑇 𝑧 (1c)
where 𝑇 temperature, 𝑡 time, 𝑢 interstitial velocity, 𝜀 porosity, 𝐶 specific heat, 𝜌 density, 𝜆 thermal conductivity, ℎ heat transfer coefficient (HTC), 𝐴 superficial area, and 𝑉 volume are presented; 𝑧 is the axial and 𝑟 is the spherical coordinate; subscript of f, s, w, eff, amb represent fluid, solid, wall, effective value, and ambient, respectively; For example, 𝑇 is the solid surface temperature, 𝑇 is the wall temperature calculated at wall middle position of , and the ratios of area to volume are defined as:

• , , (2)
where 𝑅 and 𝑅 are internal and middle radius of tank, and 𝐷 is the particle diameter (cf. Fig. 4).

The effective fluid thermal conductivity (𝜆 , ) in axial direction contributed by stagnant thermal diffusion and turbulent thermal contribution is given as:

𝜆 , 𝜀 • 𝜆 0.5 • 𝑅𝑒 • 𝑃𝑟 • 𝜆 (𝑅𝑒>0.8) (3)
The convection term between solid filler surface and HTF is added in the boundary condition which will be introduced in later text. The solid-fluid HTC (ℎ ) is calculated as:

ℎ • 1.26 • • • • • 𝑅𝑒 • 𝑃𝑟 (𝑅𝑒<74) (4a)
The equivalent HTC of fluid-to-wall center (ℎ ) and the equivalent HTC of wall center-toambient (ℎ ) are defined as:

• ln • • 𝑅 (4b)
• ln

• • ln • • 𝑅 (4c)
where the fluid-to-wall convective HTC (ℎ ) on inner wall surface (Eq. 5a), the insulation-to-ambient air convective HTC (ℎ ) on outer insulation surface based on a correlation of natural convection on vertical standing wall (Eq. 5b), and radiative HTC (ℎ ) based on the Stefan-Boltzmann law (Eq. 5c), are all regarded as constant and calculated based on the tank outer surface temperature when the fluid temperature is equal to (𝑇 ):

ℎ • 0.6 • 𝑅𝑒 • 𝑃𝑟 , (1<𝑅𝑒<40) (5a) ℎ ⎩ ⎪ ⎨ ⎪ ⎧ 0.825 . • . ⎭ ⎪ ⎬ ⎪ ⎫ (Ra air <10 12 ) (5b) ℎ • • (5c)
where all empirical correlations used have been introduced in our previous work [START_REF] Tuttle | A novel dynamic simulation methodology for high temperature packed-bed thermal energy storage with experimental validation[END_REF] and all dimensionless numbers (𝑅𝑒, 𝑅𝑎, 𝑃𝑟) are defined and listed in the nomenclature.

Boundary and initial conditions

In a charging/discharging process, the boundary and initial condition are described in Table 4. It assumes that the initial condition of charging is fully discharged and with a homogenous coldest operating temperature, while the cutoff temperature after charging is the beginning of discharging. The model was solved with the MATLAB function ode45 for the time derivatives after the convergence study, as introduced in Supplementary Material S1 (in Fig. S1). Fig. 5 illustrates a flow chart of numerical algorithm of the DC-3P model and calculation of system's efficiency.

Key performance indicators and associated uncertainty

Several indicators are used to assess the thermocline expansion as well as the overall performance of the SHTPB tank, introduced as follows.

Charging/discharging time

The charging/discharging cutoff time ( 𝑡 ) relies on the threshold or cutoff outlet fluid temperature at the termination condition, defined as Eq. 6 in this study [START_REF] Fasquelle | A temperature threshold evaluation for thermocline energy storage in concentrated solar power plants[END_REF].

𝑇 , 𝑇 20% • 𝑇 𝑇 (6a) 𝑇 , 𝑇 20% • 𝑇 𝑇 (6b)
where the outlet temperature threshold coefficient (𝜃) is equal to 0.2 which is considered useful in generation of superheated steam in CSP plant, usually used in many works [30][51-53]. It means the outlet fluid temperature changes by 20% of the operational temperature difference in charging and discharging. Nevertheless, the coefficient of 𝜃 is determined arbitrarily, discussing between 0.05, 0.1, and 0.3 etc., and its selection is only used to compare performance in different tanks. Therefore, it's necessary to choose the value depending on the experimental requirements or the application of interest.

Normalized parameters are defined for a better comparison under different operational conditions.

Two types of dimensionless time for different conditions were used: one is the dimensionless time based on flow rate, 𝑡 * , as a ratio of actual time to the residence time of HTF flowing through the tank in a plug flow pattern. Another is the dimensionless time based on the energy, 𝑡 * , as the ratio of input energy by HTF to maximum stored energy inside the tank used in Supplementary Material.

𝑡 * 𝑡 • (7 a)

𝑡 * 𝑡 • • • , (7b)
𝑧 * and 𝑇 * is the dimensionless height and dimensionless temperature, defined as:

𝑧 * ( 7 c ) 𝑇 * (7d)

Thermocline thickness

The dimensionless thermocline thickness (𝐿 * ) is determined by the length of stratification region in bed [START_REF] Chang | The design and numerical study of a 2 MWh molten salt thermocline tank[END_REF], according to the physical boundary within a certain temperature range before reaching the top or bottom of the tank.

𝐿 * %• %• (8)
In charging, 𝑧 𝑇 𝑇 𝑛% • 𝑇 𝑇 0;

In discharging, 𝑧 𝑇 𝑇 𝑛% • 𝑇 𝑇 𝐻;

where n is a threshold value that is defined as the hottest/coldest temperature differs 𝑛 % to the operational temperature range 𝑇 𝑇 , setting as 5% in this study. In experiment, the thermocline thickness is calculated based on the fluid temperature variation at the axial centerline of the storage tank.

Energy efficiency and capacity ratio

The charging process is discussed in this study. The stored energy (∆𝐸 ) is the amount of thermal energy stored in solid and fluid phases from beginning (t=0) to ending (t=t ch ) in charging.

Charging energy efficiency (𝜂 ) is thus defined as the ratio of the stored energy of ∆𝐸 to the input energy by HTF ∆𝐸 ).

𝜂 ∆ ∆ ∆ ∆ • , • (9a)
where 𝑇 is the inlet temperature of HTF and T o is the reference temperature (20 °C), respectively.

The heat loss ratio of charging (𝜂 , is defined as the ratio of the heat loss (∆𝐸 ) to the total input energy (∆𝐸 in charging.

𝜂 , ∆ ∆ • , • ∆ • , • (9b)
where T out is the outlet temperature of the HTF.

Capacity ratio (𝐶 ) [START_REF] Hänchen | High-temperature thermal storage using a packed bed of rocks -Heat transfer analysis and experimental validation[END_REF][START_REF] Lou | Single-tank thermal energy storage systems for concentrated solar power: Flow distribution optimization for thermocline evolution management[END_REF] in charging is the ratio of the actual stored energy to the theoretical maximum stored energy.

𝐶 ∆ ∆ , ∆ , ∆ , ∆ , , ∆ , , ( 9 c )
In addition, the stored energy in the tank at a certain time for fluid and solid is calculated in modeling:

∆𝐸 𝑡 𝜋 • 𝑅 • 𝜌 • 𝐶 , • 𝜀 • 𝑇 𝑇 𝜌 • 𝐶 , • 1 𝜀 • 𝑇 𝑇 𝑑𝑧 (9d)
where 𝑇 is the average temperature of single solid spheres at the same height. In experiment, the stored energy is calculated using the average temperatures of solid and fluid for a number of small unit volumes.

More precisely, the tank volume is firstly divided into small cells by thermocouple points and the temperature of such single unit volume is considered as homogenous.

Uncertainty analysis

The uncertainties of experimentally measured parameters and calculated parameters are listed in

Assess measurement error:

* 1 The standard uncertainty is defined by standard deviation: 𝑆𝐷 ∑ 𝑥 𝑥̅ , where 𝑥 is measured parameter and 𝑛 is the repeated measurement times [START_REF] Taylor | Guidelines for evaluating and expressing the uncertainty of NIST measurement results[END_REF].

Assess calculation error:

* 2 The components uncertainty is defined as :𝜗 ∑ 𝑆𝐷 , where 𝑦 𝑓 𝑥 , 𝑥 … . . 𝑥 is the in-directly parameter, and m is the number of components [START_REF] Stern | Summary of exerimental uncertainty assessmet methodology[END_REF].

Results and discussion

This work experimentally investigated the thermocline expansion of SHTPB TES system due to various factors, including the inlet diffuser configuration, the insulation, the filler diameter, the HTF flow rate and the working temperatures. The developed DC-3P was firstly validated by comparison with the experimental data and then was used to conduct numerical parametric study.

Comparison between numerical and experimental results

Validation and comparison between numerical (DC-3P) and experimental results are performed to address some interesting but inadequately investigated issues, including the temperature distribution of tank, the temperature gradient inside the solid particles, and the applicable range of the DC-3P model.

Fluid temperature profiles validation

The experimental data of Case 1 (𝐷 =7 mm, Péclet number 𝑃𝑒=12, Particle Reynolds number 𝑅𝑒=3, 𝐵𝑖 =0.76, charging process) was first used for numerical model validation. Fig. 6a shows the comparison of fluid temperature evolution at axial positions (cf. C in Fig. 3b; r=0). At first sight, a good agreement between numerical and experimental results can be found. Fig. 6b further shows the variation of temperature difference ( ∆𝑇 ) vs. charging time. The corresponding standard deviation ( 𝑆𝐷 ) is calculated based on Eq. 10. For most of the thermocouples in the axial centerline of the tank, ∆𝑇< ±2.5 °C and 𝑆𝐷< 0.7 °C can be observed, indicating that the temperature evolution at these positions can be well predicted by the developed DC-3P model. However, for TC-7 near the top inlet, the discrepancy (∆𝑇≈10 °C and 𝑆𝐷≈2 °C ) is still noticeable, implying the possible departure from the assumed plug flow used in this 1D model at this position. This is mainly due to the impact of inflowing HTF jet on the radial velocity distribution and consequently on the thermocline expansion and evolution, which will be further investigated in detail and discussed in later sub-sections.

During charging process, the energy efficiency difference at the cutoff time between numerical and experimental results is around 2%, which is within the experimental uncertainty of 5% (cf. Fig. S4). At a final cutoff fluid outlet temperature of 43.8 °C (𝜃=0.2), the energy efficiency is calculated to be 𝜂 =83.3% by modelling and is 𝜂 =85.5% in experiment, showing a good agreement. In conclusion, the DC-3P model has a good accuracy in simulating the large-size sphere (𝐵𝑖 >0.1) and tank with a large wall volume of SHTPB.

Temperature gradient inside sphere particles

The temperature gradient inside sensible fillers, reflected by the temperature difference between the solid sphere center and the surrounding HTF, has been reported by numerical modelling but inadequately investigated nor validated in experiments. To address this question, Fig. 7 presents the time and volume-averaged temperature gap between solid sphere center and fluid as defined in Eq. 11, obtained both by experimental measurements of all thermocouples and by DC-3P modelling. The solid Biot number 𝐵𝑖 ranges from 0.6-1.1 for 7 mm sphere fillers (Case 1) and from 0.7-1.2 for 10 mm sphere fillers (Case 3) under the tested flow rates, implying that the dispersion-concentric model is necessary to be used [START_REF] Wang | Thermal performance analysis of packed-bed thermal energy storage with radial gradient arrangement for phase change materials[END_REF]. Both numerical and experimental results show that the ∆𝑇 increases with the increasing of flow rate and inlet fluid temperature (𝑇 ). Comparing two sphere sizes, the difference between the numerical and experimental results is smaller than 0.3 °C for 10 mm fillers and about 1 °C for 7 mm fillers, respectively, showing good agreement. Contrary to theory, the temperature gap in experiments is larger for small-size particles, maybe due to heat transfer area difference of the flow thermal front [START_REF] Hoffmann | Experimental and numerical investigation of a thermocline thermal energy storage tank[END_REF] as well as the measurement uncertainties and difficulties. But in general, the ∆𝑇 value up to 1.7 K can be detected in experiments and due to the large 𝐵𝑖 number, the conduction resistance of the solid sphere is much higher than the convection resistance between the fluid and solid.

Thereby, the temperature difference confirms the existence of noticeable temperature gradient inside sphere under certain circumstances which is verified in experiments for the first time.

Experimental study on the influence of inlet configuration and insulation

Evolution of temperature field

Fig. 9 shows the evolution of the fluid temperature cartography inside the SHTPB tank experimentally measured for one fully charging and discharging cycle. Case 3 with porous bed inlet (𝐷 =10 mm) has been tested at two flow rates, i.e., 𝑉 =0.3 (𝑅𝑒=5) and 1.5 L min -1 (𝑅𝑒=15), to highlight the impact of inflowing jet on the thermocline expansion in SHTPB. The colormap is constructed from data interpolation of 17 thermocouples measurements (linear interpolation in the axial direction). The dimensionless time equaling 𝑡 * =1 (cf. Eq. 7a) means that the theoretical plug flow reaches the tank top or bottom. It can be observed that at the beginning of charging (𝑡 * =0), the tank has a quasi-homogeneous temperature of 𝑇 (blue color). But for the beginning of discharging, the temperature distribution inside tank is a bit less homogeneous since the ending of fully charging (𝜃=0.01, for better observation) has been used as the initial condition for discharging.

By examining the temperature color maps, it can be observed that the thermocline region grows thicker over charging or discharging time. Moreover, the shape of thermocline region is not flat, indicating that temperature gradient exists in the radial direction. The impact of penetrating flow inject on the thermocline expansion can be clearly seen, especially with a high flow rate (𝑉 =1.5 L min -1 ) involving a more important inertial force. It seems that the solid fillers alone are not totally capable of buffering the momentum-dominated injecting flow under this tested condition, resulting in the nonuniform fluid distribution which destabilizes the temperature stratification. Note that the shape of thermocline is a little bit different in charging and discharging due to the influence of the gravity and the lowered temperature at the near-wall region (heat loss).

Impact of inlet configuration

The testing results of Case 3 (packed bed), Case 5 (packed bed + diffuser) and Case 6 (diffuser) in Fig. 3 are compared to assess the performance of different inlet configurations to mitigate the impact of penetrating flow injection. Fig. 10 shows the evolution of radial fluid temperature differences over charging time for the center to outer (𝑇 -𝑇 ) and for the center to the middle (𝑇 -𝑇 ) positions at TC-7 (near the top inlet, under the diffuser; z*=0.875) and TC-5 (near middle tank height; z*=0.625) levels.

A non-uniformity factor ( 𝑓 ) defined as the maximum temperature difference in radial direction to the operating temperature range (Eq. 12) is introduced to indicate the (non)-uniformity of radial temperature distribution, as presented at the upper-right corner of Fig. 10. When the value of 𝑓 approaches 0, there is almost no radial temperature gradient, indicating that the shape of thermocline is flattened as assumed in 1D numerical model. From Fig. 10a, it can be seen that the radial temperature difference (∆𝑇) at TC-7 level decreases after adding the plate-type diffuser. The corresponding value of 𝑓 is 0.83 for basic Case 3 (z 1 =0, only the packed bed), 0.66 for Case 6 (z 1 =35 mm, only the diffuser), and 0.30 for Case 5

(z 1 =10 mm, the combined diffuser and packed bed), respectively. As expected, the combined effect of diffuser and packed bed largely alleviates the influence of penetrating HTF jet on the thermocline stability in the entrance, better than that with only the diffuser or only the packed bed.

At TC-5 level near the middle tank height, the impact of penetrating HTF jet on the radial temperature distribution uniformity is shown to be rather limited even at high flow rate, reflected by the small radial temperature differences and low 𝑓 values (<0.2) for all tested inlet configurations (Fig. 10b). The outer radial temperature of Case 6 (no packed bed in the entrance region) is a little higher than the center temperature because of the double-hump shape of thermocline caused by higher velocity in near-wall region [START_REF] Lou | Single-tank thermal energy storage systems for concentrated solar power: Flow distribution optimization for thermocline evolution management[END_REF]. Without packed bed in the entrance, the double-hump phenomenon is more obvious so that the 𝑓 value for Case 6 is higher than other configurations.

The influence of radial thermocline expansion or instability due to the penetrating jet on the global performance of the SHTPB tank needs further discussion. To this end, Fig. 11a shows the energy efficiency ( 𝜂 ) and capacity ratio or utilization (C ratio ) at the cutoff time for different inlet configurations. For all studied inlet configurations, when 𝑇 increases, the 𝜂 slightly decreases mainly due to the higher heat loss, while the C ratio slightly increases because of the larger temperature difference, which will be discussed in detail in the next subsection.

Case 5 with the most uniform radial temperature distribution at the inlet region has the largest capacity ratio and the longest charging time. The near plug flow pattern and the flattened thermocline shape indicate the better utilization of the storage capacity (around 3%) of the SHTPB tank. However, it presents the lowest energy efficiency (Fig. 11a) mainly due to the longest charging time before reaching the cutoff fluid outlet temperature thus the additional amount of heat loss as shown in Fig. 11b.

In reality, the 𝜂 curves for Case 3 with serious radial thermocline expansion and Case 5 are very close (<2%) during the charging operation (Fig. 11b). This implies that the radial temperature non-uniformity and the thermocline expansion near inlet region caused by the penetrating HTF inject may have negligible influence on the energy efficiency of the SHTPB tank. It actually depends largely on the cutoff temperature and the corresponding cutoff time. The energy efficiency and capacity ratio of Case 6 with full water HTF in the entrance (thereby larger volumetric heat capacity than that of glass) are very close to those for Case 3. The near-entrance region of the tank is less utilized due to the strong mixing of hot and cold fluids, such as in single-medium thermocline tanks [START_REF] Lou | Single-tank thermal energy storage systems for concentrated solar power: Flow distribution optimization for thermocline evolution management[END_REF].

Impact of insulation

The impact of heat loss on the performance of the SHTPB tank has been evaluated by comparing the experimental results of Case 3 (with insulation) and Case 4 (without insulation) with the same testing condition. From Fig. 13a, it is found that due to the heat loss, the outlet fluid temperature of no-insulation tank is 5 °C lower than that of the tank with insulation at the ending moment of charging (40 min, t*=2.5). Longer charging time of around 5% is also needed to reach the cutoff temperature (𝜃=0.2) when the tank is not insulated due to the serious tailing effect. Fig. 13b shows the variation of the normalized thermocline thickness over charging time for the two cases. It may be observed that the dimensionless thermocline thickness 𝐿 * values are almost the same before t*=0.34 (11 minutes) for the two cases. That is because, at the beginning of charging, the thermocline expansion at axial centerline is caused by fluid mixing and diffusion primarily. The near-wall region of the SHTPB tank is still occupied by low-temperature HTF (cf. Fig. 9a) thus the influence of heat loss on the thermocline expansion is negligible. But at the second half of the charging, there is an obvious discrepancy between the two 𝐿 * curves, implying the noticeable influence of heat loss on the thermocline expansion.

Comparing the insulation tank and no-insulation tank, the 𝐿 * is calculated to be 0.59 and 0.76 at t*=1.38 (thermocline zone reaching the tank bottom). Fig. 13c shows the variation of input energy (∆𝐸 ), stored energy (∆𝐸 ), and outlet energy (∆𝐸 ) as a function of charging time for the SHTPB tank with/without the insulation. It is observed that both 𝐸 curves increase almost linearly until reaching a plateau. At the cutoff moment of charging, the 𝐸 value of no-insulation tank is 4-7% lower than that of the tank with insulation, mainly due to the heat loss. Fig. 13d shows the energy efficiency (𝜂 ) curve and heat loss ratio (𝜂 , ) curve, not surprisingly both indicating that the SHTPB tank performs better when well-insulated. At the cutoff temperature set as 𝜃=0.2, the SHTPB tank with insulation can improve the energy efficiency by about 5-7%, increase total stored energy by 4-7%, and augment the capacity ratio by about 3-5%.

At last, the impacts of flow rates and inlet temperature with particle sizes on the thermocline evolution in axial direction further investigated experimentally are presented in Supplementary Material.

When the HTF flow rate increases, the fluid temperature at the same height becomes higher because of the insufficient heat exchange (in Section S4, in Fig. S5). The thermocline thickness also increases faster at high HTF low rate (high 𝑅𝑒) (in Section S4, in Fig. S6). Numerically, the smaller filler particle size is more advantageous and an optimal inlet flow rate (about 𝑅𝑒=3) in charging can be identified for this SHTPB under low-temperature operation (in Section S5, in Fig. S7).

Conclusions

In this work, the impacts of inlet configuration and insulation on thermocline behavior and global performance of the SHTPB storage tank were investigated. The temperature evolution of both fluid and solid phases at different axial and radial positions of a lab-scale storage tank was experimentally measured. In parallel, a DC-3P numerical model has been developed and compared with the experimental data under various operating conditions. Main conclusions are summarized as follows.

(1) The developed DC-3P model considering the wall heat capacity and heat loss is proved to be able to predict the global performance precisely. Compared to the experimental data, the temperature standard deviation and energy efficiency difference can be kept below 2.5 °C and within 3%, respectively, under the applicable range of 𝑅𝑒<15 and 𝐵𝑖 <1.3.

(2) Temperature gradient inside spherical particle is shown to be existing due to the conduction resistance of the solid particles. A temperature difference up to 1.7 °C between the particle center and fluid has been observed both numerically and experimentally under our tested conditions.

(3) Injecting fluid in packed bed causes the non-uniform radial temperature distribution and thermocline expansion near the inlet region, especially at higher flow rate (higher Re). The combined effect of diffuser and packed bed as inlet configuration can better maintain the radial temperature uniformity as well as the thermocline stability. Nevertheless, the impact of injecting fluid on the global thermal performance of the tested SHTPB tank is rather limited.

(4) The heat loss without insulation layer could result in a noticeable thermocline expansion (>20%), lowered charging energy efficiency (5-7%), increased cutoff time (4-5%), and decreased capacity ratio (3-5%), even for low-temperature applications of the SHTPB storage tank.

This work can contribute to exploring the influencing factors based on a packing configuration by sensible fillers, enabling the inclusion of thermocline expansion into the design and optimization of lowtemperature thermocline packed-bed system. Our on-going work is focused on the understanding and optimization of multi-layered packing configuration, using diverse phase change materials (PCMs) or mixtures of PCM-sensible material as fillers for different applications

S1. Solving

The equations presented above were discretized by the explicit Runge Kutta (4.5) method with an adaptive time step. The model was solved with the MATLAB function ode45 for the time derivatives. ) from 5 to 20. The particle node has a minor influence on the convergence of cutoff temperature. Thus, N=1000 and M=1000 are used for height and time node differential mesh, respectively, and F=20 is chosen to better capture the temperature gradient inside the particle.

S2. Calibrating experiment measurement

The Curves are fitted linearly for a better comparison using the direct proportion formula, 𝐿 * = 𝑘 • 𝑡 𝐸 * , and the slope value k for each curve is indicated aside. The thermocline expands over time due to various factors discussed before and its thickness can go up to 70% of the total tank height. From Fig. S6 it can be observed that the thermocline thickness increases faster at high HTF rate (high 𝑅𝑒). Whether there is an optimum flow rate value at very low Re condition will be discussed later. In addition, the slope value becomes a bit smaller with increasing 𝑇 𝐻 under the same flow rate. This is because the higher inlet fluid temperature augments the fluid-solid heat transfer rate, thereby slowing the thermocline expansion.

Fig. 1 .

 1 Fig. 1. Laboratory-scale SHTPB tested in LTEN: (a) geometry and dimensions; (b) axial and radial

Fig. 2 Fig. 2 .

 22 Fig.2shows the schematic and the photo view of the experimental setup used to study the SHTPB

Case 1 ,

 1 3 vs Case 2, 4: To explore the impact of heat loss on thermocline expansion, two cases (2, 4) without insulation layer were tested and compared to others with insulation (1, 3). Case 3 vs Case 5, 6: To analyze the buffering effect of porous bed on the allevaiation of inflowing HTF jet, three configurations (Case 3, 5, 6) were tested for comparison, as shown in Fig. 3. Case 3 is the fully packed tank without diffuser, while Cases 5 and 6 introduced the top and bottom diffusers. The plate-type diffuser (6 mm in thickness) having three ring-shaped grooves has been optimized and tested

Fig. 3 .

 3 Fig. 3. Different inlet diffuser configurations experimentally tested in this study: (a) only porous bed;

Fig. 4 .

 4 Fig. 4. Schematic of numerical resolution domain for the developed DC-3P model.

Fig. 5 .

 5 Fig. 5. Flow chart: numerical algorithm of the DC-3P model and calculation of system's efficiency.

Fig. 6 .

 6 Fig. 6. Comparison between numerical results and experimental data: (a) Evolution of fluid

Fig. 7 . 5 . 1 . 3 .

 7513 Fig. 7. Average temperature difference between sphere center and fluid in charging process:

Fig. 8 .

 8 Fig. 8. Comparison of numerical and experimental results on temperature standard deviation (SD) and

Fig. 9 .

 9 Fig. 9. Evolution of fluid temperature distribution experimentally measured for one chargingdischarging cycle (Case 3: 𝑇 /𝑇 =20/50 °C, 𝐷 =10 mm): (a) 𝑉 =0.3 L min -1 ; (b) 𝑉 =1.5 L min -1 .

Fig. 10 .

 10 Fig. 10. Radial temperature difference inside the SHTPB tank with different tested inlet configurations: (a) TC-7 position; (b) TC-5 position (Case 3, 5 & 6: 𝐷 =10 mm; 𝑉 =1.5 L min -1 ;

Fig. 11 .

 11 Fig. 11. Effect of inlet configuration on the global performance of the SHTPB tank: (a) energy

Fig. 12 .

 12 Fig. 12. Experimental thermal infrared images of charging and discharging process (𝑇 /𝑇 =20/50 °C, 𝐷 =10 mm, 1.5 L min -1 , diffuser 𝑧 =10 mm, no insulation, 𝜖=0.90, 𝑇 =20.7 °C).

Fig. 13 .

 13 Fig. 13. Experimental results comparing the performance of SHTPB tank with/without insulation. (a) inlet and outlet temperature; (b) dimensionless thermocline thickness 𝐿 * ; (c) inlet/outlet and stored energy; (d) energy efficiency and heat loss ratio (Case 3 & 4: T C /T H =20/70 °C, 𝐷 =10 mm, 𝑉 =0.3 L min -1 , 𝜃=0.2).

Fig. S1

 S1 Fig.S1shows the results of height mesh size and time step sensibility test with a variation of the particle

Fig. S1 .

 S1 Fig. S1. Height, time, and particle radius node number independence tests in differential mesh.

Fig. S2 .Fig. S3 .

 S2S3 Fig. S2. (a) Differential scanning calorimetry curves of glass sphere in range of 0-100 °C; (b)

Fig. S4 .

 S4 Fig. S4. Energy efficiency comparison for numerical and experimental in charging process (𝑇 𝐶 /𝑇 𝐻 =20/50 °C, 𝐷 𝑝 =7 mm, 𝑉 ̇𝑓 =0.5 L min -1 ).

Fig. S5 .

 S5 Fig. S5. Experimental fluid temperature profile at the axial centerline of the SHTPB tank for various

Fig. S6 shows

  Fig.S6shows the variation of normalized thermocline thickness as a function of 𝑡 𝐸 * in charging.

Fig. S6 .

 S6 Fig. S6. Evolution of thermocline thickness for various flow rates, particles sizes, and working

Table 1

 1 Summary of experimental investigations on SHTPB TES systems.

	Studies	Years	Solid fillers	HTFs	𝑻 𝑪 /𝑻 𝑯 (°C)	𝜺	𝑯/ 𝑫 𝒕𝒂𝒏𝒌 (m)	Number of thermocouples & arrangement	Diffuser	Insula tion	Num. /Exp.	Tested in Exp. parameters	Main findings
	Faas et al. [32]	1986	Granite Rock and sand	Caloria ® HT 43	179.2/2 95.5	0.22	12/1 8.2	--	--	Yes	Exp.	--	 Low Rankine-cycle efficiency of 21% due to limited upper temperature of TES tank.
	Meier et al. [33]	1991	Rock	Air	150/55 0	0.36	1.2/0 .15	--(axial)	--	Yes	Exp.	--	 Considerable heat losses through the wall.  Smaller measured pressure drop than predicted due to low flow resistance near wall.
	Pacheco et al. [34]	2002	Quartzite rock & sand	Molten salt	290/39 0	0.22	6/3	--(axial and radial)	Yes	Yes	Num. /Exp.	--	 Thermocline thickness well predicted by the model.  higher heat loss than predicted due to the lack of insulation on the top cover of the tank.
	Yang et al. [35]	2012	Rock	Molten salt	300/50 0	0.2	2/1	12 (axial)	No	Yes	Num. /Exp.	--	 Higher thermal storage capacity and efficiency by using fillers with higher density and specific heat, but also higher entropy generation (Num.).
	Yin et al. [22,23]	2014/ 2017	Zirconiu m & silicon carbide	Molten salt	290/39 0	--	0.6/0 .12	6 (axial)	Yes	Yes (heatin g strap)	Exp.	 Inlet HTF temperature  Flow rate  Mixed fillers	 Heat storage efficiency is smaller than 80% due to the thermocline expansion.  Thermocline evolution influenced by HTF velocity.  Piston flow pattern achieved by buffering effect of porous fillers.
	Anderson et al. [36]	2014	Alumina	Air	20/120	0.4	10/9. 56	--(5 for axial outer tank surface)	No	Yes	Num. /Exp.	--	 Vessel heat loss is acceptable due to only loss 12% of supplied energy.
													 No significant inhomogeneity of radial temperature
	Bruch et al. [21]	2014	Silica gravel & silica sand	Oil	≤ 300	0.27	3/1	250 (for fillers and HTF, axial and radial circle)	Yes	Yes	Num. /Exp.	 Cycling number	distribution.  Differed thermocline behavior between multiple cycle and single cycle operation.  Impact of metal tank wall on the thermocline moving in
													charging.
								19 (axial equally
								spaced),
								5 (radial,
	Cascetta et al. [18,29]	2015/ 2016	Alumina	Air	25/300	0.385 -0.395	1.8/0 .58	decreasing distance), 10 (axial outer	Yes	Yes	Num. /Exp.	 Cycling number	 40% radial temperature profile affected by metal wall heat, which cannot be predicted by 1D model.  60% reduction of stored energy after 4 cycles.
								surface)
								5 (circumferential
								outer surface)
													 Higher heat losses and heat diffusion due to low HTF flow
	Hoffmann et al. [15,30]	2016/ 2017	Quartzite rock	Rapese ed oil	160/21 0	0.4	1.8/0 .4	32 (axial and radial equally spaced)	No	Yes	Num. /Exp.	 Flow rate  Particle size	rate.  Stronger forced convection due to high HTF flow rate.  Maximum discharging efficiency of 75% at an optimal flow
													rate of 0.3 kg s -1 .
	Al-Azawii et al. [37,38]	2018/ 2019	Alumina	Air	21.5/15 0	0.375	1/0.1 247	--	Yes (10 cm distance)	Yes	Num. /Exp.	 Flow rate	 Higher heat losses due to low HTF flow rate.  Increased exergy efficiency from 35.7% to 55.4% with increasing flow rate from 0.002 to 0.006 m 3 s -1 .

 Solid filler acts as a natural distributor.  Distributor does not influence storage behavior.  Radial homogeneity is then improved at low fluid velocity.

 Correlations proposed to predict the Nusselt number.

 Flow rate  Inlet HTF temperature  Cold energy storage system is able to successful store cold at night and compensate performance loss of cooler in daytime.  Solid filler  Important role of heat capacity of solid fillers.  Very low heat loss owing to the well-insulated tank walls.  Flow rate  Thermocline stability disturbed by the radial plate-type diffuser.  Model developed based on thermal diffusion to predict the thermocline evolution in axial direction, with an average error of about 13.9%.

 Particle size  Flow rate  Air flow increases, the cycle efficiency increases.  Permeability decreases, superficial velocity increases.

Table 2

 2 Tested SHTPB configurations with different operational conditions. 1

	Geometry parameters
	Configuration

Table 4

 4 Initial and boundary condition in axial 𝑧-direction and spherical radius 𝑟 -direction.

	Initial condition (𝒕=0)
	All domains	𝑇	𝑇	𝑇	𝑇 , where	𝑧 ∈ 0, 𝐻	𝑟 ∈ 0,
		𝑇	𝑇 , (Ch)
		𝑇	𝑇	, , (Dis)
	Boundary condition (𝑡 0)
	Surface of solid particle	𝜆 •	,		ℎ • 𝑇	𝑇	𝑧 ∈ 0, 𝐻	𝑟	𝐷 2
	Center of solid particle	,	0			𝑧 ∈ 0, 𝐻	𝑟	0
	Tank top & bottom	,		,	0		𝑧 0, & 𝑧 𝐻	𝑟 ∈ 0,
	Inlet HTF	𝑇 𝑧 𝐻, 𝑡	𝑇 , (Ch)		--	--
		𝑇 𝑧 0, 𝑡	𝑇 , (Dis)
	Outlet HTF		,	0, (Ch)		--	--
			,	0, (Dis)

Table 5 .

 5 Based on the calibrating measurement results in Supplementary Material S2, details of uncertainty analysis are discussed in S3.

	Table 5The uncertainties of measured and calculated parameters.
	Measured parameters		Uncertainties
	Temperature	𝑇	± 0.37 K
	Volumetric flow rate	𝑉	± 1%
	Particle diameter* 1	𝐷	± 0.1 mm
	Solid density* 1	𝜌	± 1.6% (40 kg m -3 )
	Thermal conductivity* 1	𝜆	± 0.001 W m -1 K -1
	Specific heat capacity* 1	𝐶	± 8 J kg -1 K -1
	Calculated parameters* 2
	Porosity	𝜀	± 0.004
	Energy efficiency	𝜂	± 5%
	Capacity ratio	𝐶	± 2%
	Thermocline thickness	𝐿 *	± 5% (20 mm)

Acknowledgements

This work is supported by the French ANR within the project OPTICLINE (ANR-17-CE06-0013) and by the Chinese Scholarship Council (CSC) with the scholarship for Ms. Baoshan XIE (No. 201908430177). Technical supports from Mr. Julien AUBRIL, Mr. Arnaud ARRIVE and Mr. Jorge FERNANDEZ-PACHECO ARMENTEROS during the experiments are gratefully appreciated.

Experimental and numerical study on the thermocline behavior of packed-bed storage tank with sensible fillers

Baoshan XIE a , Nicolas BAUDIN a , Jérôme SOTO a,b , Yilin FAN a , Lingai LUO a,1 a Nantes Université, CNRS, Laboratoire de Thermique et énergie de 1 Corresponding author E-mail address: lingai.luo@univ-nantes.fr (L. LUO)

S3. Uncertainty analysis

The measurement uncertainties were obtained by evaluating the standard deviation in repeated testing [START_REF] Taylor | Guidelines for evaluating and expressing the uncertainty of NIST measurement results[END_REF]. The analytical uncertainties for in-direct parameters, such as energy efficiency, are determined by the components uncertainties, 𝜗, that are contributed by the measurement uncertainties based on error propagation [2]. In experiment, the average temperature of small unit volume is used to calculate energy. Both measurement uncertainty and analytic interpolation integral uncertainty of the average temperature can influence the energy calculation (average temperature is actually a type of middle point integration method). Firstly, the temperature error (systematic error) is estimated to be ±0.37 K, the measurement uncertainty on average temperature is calculated to be ±0.15 °C, and the numerical integration uncertainty is below that, as calculated in Fig. S3a. Especially due to the different radial thermocouples number, the uncertainty is different for radial thermocouples. Thus, among those, system error of ±0.37 K is used to calculate the in-direct parameters uncertainty. The corresponding uncertainty of stored energy, input energy, and efficiency of the validation case are exhibited in Fig. S3 b-d.

S4. Impact of flow rates and inlet HTF temperature

The influence of HTF flow rate and inlet temperature on the thermocline evolution in axial direction is further investigated experimentally and reported in this sub-section. Fig. S5 shows the fluid temperature profile along the tank height (centerline) for various flow rates, particles size, and operating temperatures at charging time of 𝑡 𝐸 * =0.5 (input energy by HTF is equal to the half of ideal maximum stored energy of tank ). When HTF flow rate increases from 0.3 to 1.5 L min -1 and the theoretical input energy by HTF is the same, the fluid temperature at the same z* becomes higher. That's because the shorter residence time at higher flow rate causes insufficient heat exchange between solid and HTF.

S5. Optimal flow velocity

Additional simulations using the DC-3P model have been performed and compared with the experimental data in order to exhibit the existence (or not) of an optimal HTF flow rate (𝑅𝑒) for SHTPB TES tanks. Fig. S7 shows the numerical results of the energy efficiency and remained thermocline thickness at cutoff time with HTF flow rate ranging from 0.12 to 2.5 L min -1 (𝑅𝑒=0. [START_REF] Gautam | A review on sensible heat based packed bed solar thermal energy storage system for low temperature applications[END_REF][START_REF] Soto | Thermal cycling aging of encapsulated phase change material -Compressed expanded natural graphite composite[END_REF][START_REF] Pelay | Integration of a thermochemical energy storage system in a Rankine cycle driven by concentrating solar power: Energy and exergy analyses[END_REF][START_REF] Lou | A review on the performance indicators and influencing factors for the thermocline thermal energy storage systems[END_REF][START_REF] Rodrigues | Effect of porous material properties on thermal efficiencies of a thermocline storage tank[END_REF][START_REF] Roux | Thermocline thermal energy storage optimisation combining exergy and life cycle assessment[END_REF][START_REF] Rosen | The exergy of stratified thermal energy storages[END_REF][START_REF] Hoffmann | Experimental and numerical investigation of a thermocline thermal energy storage tank[END_REF][START_REF] Davenne | Stability of packed bed thermoclines[END_REF][START_REF] Geissbühler | An assessment of thermocline-control methods for packed-bed thermal-energy storage in CSP plants, Part 2: Assessment strategy and results[END_REF][START_REF] Cascetta | Experimental and numerical research activity on a packed bed TES system[END_REF][START_REF] Nicolas | Flexibility and robustness of a high-temperature air/ceramic thermocline heat storage pilot[END_REF][START_REF] Sanderson | Performance and efficient design of packed bed thermal storage systems. Part 1[END_REF][START_REF] Bruch | Experimental and numerical investigation of a pilot-scale thermal oil packed bed thermal storage system for CSP power plant[END_REF][START_REF] Yin | Experimental research on thermal characteristics of a hybrid thermocline heat storage system[END_REF][START_REF] Yin | Thermocline characteristics of molten-salt thermal energy storage in porous packed-bed tank[END_REF][START_REF] Reddy | Stability analysis of the thermocline thermal energy storage system during high flow rates for solar process heating applications[END_REF][START_REF] Wang | Influence of flow distribution on the thermal performance of dual-media thermocline energy storage systems[END_REF]. The inlet fluid temperature has negligible effect on the energy efficiency and the remained thermocline thickness.

Numerically, the smaller filler particle size is more advantageous and an optimal inlet flow rate (about 𝑅𝑒=3) in charging can be identified for this SHTPB under low temperature operation. This finding is in line with other researches for high-or middle-temperature applications


O Sima

Anne De

M.S Vismes Ott

P Dias

Laurent Dryak

D Ferreux

Koskinas

M S Dias

P Dryak

L Ferreux

D Gurau

S Hurtado

P Jodlowski

K Karfopoulos

M F Koskinas

M Laubenstein

Y K Lee

M C Lépy

A Luca

M O Menezes

D S Moreira

J Nikolič

V Peyres

P Saganowski

M I Savva

R Semmler

J Solc

T T Thanh

K Tyminska

Z Tyminski

T Vidmar

I Vukanac

H Yucel

Consistency test of coincidence-summing calculation methods for extended sources

Keywords: Gamma-ray spectrometry, HPGe detector, Coincidence-summing corrections, Self-consistency test

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Coincidence-summing effects represent one of the factors affecting the calibration of the full energy peak (FEP) efficiency for high efficiency HPGe detectors. These effects are enhanced in present day measurements, due to the desire and possibility to achieve high efficiency measurement conditions.

In view of the importance of the coincidence-summing corrections and of the complex procedures applied for evaluating them, the Gamma-Ray Spectrometry Working Group (GSWG) of the International Committee for Radionuclide Metrology (ICRM) carried out a series of actions intended to test the quality of these procedures [START_REF] Lépy | Intercomparison of methods for coincidence summing corrections in gamma-ray spectrometry[END_REF]Lépy et al., 2012;[START_REF] Vidmar | Equivalence of computer codes for calculation of coincidence summing correction factors[END_REF][START_REF] Vidmar | Equivalence of computer codes for calculation of coincidence summing correction factors -PartII[END_REF]. As a continuation, this work presents the results of an action proposed in the meeting of the GSWG organized during the ICRM Conference in Buenos Aires (ICRM GSWG webpage, 2018). It has the specific task of testing the internal self-consistency of the methods applied for the computation of coincidence-summing corrections for volume sources. The proposed test is based on exact relations that should be fulfilled in certain ideal measurement configurations. More precisely, the results obtained using any computation method for one such configuration should be related by exact equations to the results given by the same method for other configurations. Thus, this test does not require experimental data (avoiding the problem of experimental uncertainties and the conformity of the detector model used in the computations with the actual one) or comparisons of a method with other methods (avoiding the debate concerning the selection of a particular reference method).

Theoretical Background

From the point of view of activity measurement by gamma-ray spectrometry, the occurrence of coincidencesumming has two types of effects: coincidence-losses from existing peaks and coincidence summing-in effects [START_REF] Debertin | Gamma-and X-Ray spectrometry with semiconductor detectors[END_REF][START_REF] Gilmore | Practical Gamma-ray Spectrometry[END_REF]. The latter add counts to existing peaks or produces peaks (pure sum peaks) which cannot be assigned to any photon emitted by the decay of the nuclide.

In the absence of coincidence-summing effects, the net count rate in every peak (energy E i ) is proportional with the nuclide activity A and the photon emission probability p i =I  (E i ):

(1)

In a given measurement configuration, the FEP efficiency (E i ) depends only on the energy of the photon. In the presence of coincidence-summing effects, the probability of recording a count in the peak of energy E i depends also on the probability that other photons (energy E j ) interact with the detector simultaneously. The count rate equation can be written similarly to Eq. 1, by introducing the apparent efficiency  app (E i ):

(2)

In this equation F C (E i ,X) represents the coincidence-summing correction factor for the peak of energy E i of the nuclide X. In the case of a point source and neglecting the angular correlation effects, it can be written as [START_REF] Sima | Accurate computation of coincidence summing corrections in low level gammaray spectrometry[END_REF]:

(3)

The first sum describes the coincidence losses from the peak of energy E i due to the simultaneous detection of photon E j (pair coincidences), summed over all photons emitted in cascades with the photon of energy E i ; the second sum describes triple coincidences and so on. The next sums from the equation (which are the only terms contributing in the case of pure sum peaks) describe the summing-in contribution, when relevant; the first term corresponds to complete energy deposition of the photons of energies E p and E q , with E p +E q =E i , summed over all possible combinations, whereas the second term takes into account the events when some counts of this type are lost from the peak due to coincidence losses produced by the detection of another photon E r . Note that the magnitude C of the coincidence effects is given by the sum terms from Equation (3), i.e. the term C=1-F C in the case of usual peaks (irrespective to the fact that only coincidence losses or both losses and summing in contribute) and C=F C in the case of pure sum peaks. In Eq. 3,  represents the FEP efficiency, the total efficiency, and p ij and p ijk , ... are the probabilities of simultaneous emission of the groups of (i,j) and (i,j,k), ... photons. In this equation the angular correlation of cascading photons was neglected.

In the case of volume sources, the terms become more involved. For example, (E j ) from equation ( 3) should be replaced by the quantity (4) where the integration is carried out over the source volume, dv being the volume element around . In the presence of angular correlations, the dependence of the efficiency on the direction of emission of the photon should be introduced, the angular correlation function should be included in the integrand and integrations should be carried out over the directions of emission of the photons (the same procedure is valid also in the case of point sources). The expression in equation (4) can be viewed as an effective total efficiency [START_REF] Arnold | Total versus effective total efficiency in the computation of coincidence summing corrections in gamma-ray spectrometry of volume sources[END_REF] for photon E j , depending also on the energy E i . The terms involving two efficiencies in equation (3) should be replaced by similar integrals of products of three efficiencies, and so on. Note that for the evaluation of F C , the variation of the efficiencies as a function of the emission point within the volume source is needed. Thus, contrary to the case of point sources, the efficiency related quantities required for the computation of F C do not have a directly measurable correspondent. Furthermore, due to the dependence of the effective total efficiency on E j and E i , not only on E j , the number of quantities that should be evaluated is higher. Consequently, it is clear that compared to the case of point sources, the computation of F C for volume sources is more difficult and can take much longer time. The angular correlation effects further complicate the evaluation of F C .

In view of the complexity of the evaluation of F C in the case of volume sources, in some codes explicit or hidden approximations are applied. The uncertainties associated with some of these approximations can be partly revealed by the test described below.

Self-consistency test

Consider three sources, S1, S2 (identical with S1) and S3, with negligible container walls and filled with air (or vacuum). The dimensions of the sources satisfy R 1 =R 2 =R 3 =R, H 2 =H 1 , H 3 =H 1 +H 2 . The sources are measured in the configurations a, b and c presented in Fig. 1. Note that configuration c corresponds to mounting the first two sources one upon the other on the detector. Each configuration is placed in vacuum (or in air) and there are no materials in the vicinity of the detector and sources. In these conditions the interactions within the sources, in the container walls or outside the sources and detector, are negligible. Then the properties of the S3 source measured in configuration c should be identical with the properties of the composite source, obtained by placing S2 above S1.

For example, the contribution to the count rate of the decays from each of the two parts of the composite source is equal to the contribution to the count rate of the sources 1 and 2 measured independently in geometries a and b. This is true both in the case of the peak count rate (either in the absence or in the presence of coincidence-summing effects) and in the case of the total count rate. Therefore the following relations should hold (ICRM GSWG webpage, 2018):

Denote by  c , c , F Cc and the respective quantities computed directly for source S 3 , measured in geometry c. If the procedure used to compute these quantities is self-consistent, then their values should be equal, within the range of their uncertainties, with  comp , comp , F Ccomp and .

The participants to the test were requested to compute the coincidence-summing correction factors F C with a statistical uncertainty better than 1%, for several peaks of 60 Co (1173.23, 1332.49, 2505.75 keV), 134 Cs (569.33, 604.72, 795.86, 1400.59 keV), 133 Ba (276.40, 356.01 keV) and 152 Eu (121.78, 1408.01 keV), for the three configurations a, b, c. The parameters of the sources were R=2cm, H 1 =H 2 =2.5 cm, H 3 =5 cm. A simplified model of an n-type HPGe detector [START_REF] Vidmar | Equivalence of computer codes for calculation of coincidence summing correction factors[END_REF]ICRM GSWG webpage, 2018) with radius RD=3 cm, length LD=6 cm, cylindrical inner hole (RH=0.5 cm, LH=4 cm) and negligible dead layer thickness was assumed. The parameters of the Al endcap were: thickness 1 mm, radius 4 cm, distance from the crystal to the endcap 0.5 cm.

The participants were asked to report also the values of the full energy peak efficiency and to provide information on the method applied. The use of nuclide data recommended by the ICRM, based on DDEP (DDEP, 2018), was suggested.

Participants

A number of 21 individual or group participants took part in this action. Several participants have sent more than one set of results, obtained using different computation methods. Thus, 1 participant has sent 6 sets, 2 participants -3 sets, 3 participants -2 sets, resulting in a total number of 33 sets of results being received.

The main characteristics of the procedures applied, as well as the codes of the results received, are presented in Table 1. The procedures differ in the method applied for the computations of the efficiency (including volume source effect), of the required decay data parameters and in the way in which the efficiencies and the decay data are combined [START_REF] Sima | Efficiency calculation of gamma detectors by Monte Carlo methods[END_REF].

General purpose programs such as MCNP [START_REF] Briesmeister | MCNP-A General Monte Carlo N-Particle Transport Code, Version 4C[END_REF][START_REF] Pelowitz | MCNP6 TM user's manual[END_REF][START_REF] Goorley | Initial MCNP6 Release Overview -MCNP6 version 1[END_REF], PENELOPE [START_REF] Salvat | PENELOPE-2014: A code System for Monte Carlo Simulation of Electron and Photon Transport[END_REF] or GEANT 4 [START_REF] Agostinelli | GEANT4-a simulation toolkit[END_REF][START_REF] Allison | Geant4 developments and applications[END_REF] have been used by several participants for the computation of coincidence-summing corrections [START_REF] Thanh | Validation of an advanced analytical procedure applied to the measurement of environmental radioactivity[END_REF][START_REF] García-Toraño | Simulation of decay processes and radiation transport times in radioactivity measurements[END_REF][START_REF] Hurtado | GEANT4 code for simulation of a germanium gamma-ray detector and its application to efficiency calibration[END_REF]. In all the cases the value of F C is obtained by running the code to obtain the peak efficiencies from the simulated spectra both in the presence and in the absence of coincidence emission. MCNP-CP [START_REF] Berlizov | A correlated particle source extension of a general purpose Monte Carlo N-particle transport code[END_REF] and MCNPX, PENELOPE with PENNUC [START_REF] García-Toraño | Simulation of decay processes and radiation transport times in radioactivity measurements[END_REF] and GEANT 4 with the Radioactive Decay Module [START_REF] Hauf | Radioactive Decays in Geant4[END_REF] were used, the decay being fully simulated using ENSDF (MCNP-CP and GEANT 4) or DDEP (PENNUC) decay data.

Another generalist code, developed primarily for applications in reactor physics, (TRIPOLI-4 webpage), was applied for the computation of F C values for the two gamma photons of 60 Co.

MCNP and PENELOPE were also used for providing input values of the efficiencies for other software applied for the computation of F C .

Thus, MCNP was used in combination with ETNA [START_REF] Lépy | ETNA Software for efficiency transfer and coincidence summing correction in gamma-ray spectroscopy, Laboratoire National Henri Becquerel[END_REF]Lépy et al., 2012) by two participants. The complete version of ETNA was used for the evaluation of the efficiencies dependence on the position of the emission point in the volume source, using the efficiency transfer from a reference geometry, for which the efficiency was computed by MCNP. One participant defined as the reference geometries the configurations a, b and c, which were considered also the target geometries. Using the efficiencies dependence on position, ETNA performs a numerical integration for obtaining F C .

In the application of the software developed at CMI [START_REF] Dryak | Correction of the TRUE summations in volume sources[END_REF] (2 participants), the volumes of the sources were decomposed in a number of small domains. One participant evaluated the spectra in the presence/absence of coincidences for each domain and then merged them together. F C was obtained from the ratio of the peak count rates in the combined spectra. The other participant used MCNP to compute , and F C for each domain and then averaged the values. The joint emission probabilities were evaluated using a MATLAB script or numerically. SUMCOR [START_REF] Dias | SUMCOR: Cascade summing correction for volumetric sources applying MCNP6[END_REF] coupled with MCNP code was used in 3 sets of results. In the first, the efficiencies were computed for a number of 200 (geometries a and b) or 400 (geometry c) randomly sampled points and a Semkow type procedure [START_REF] Semkow | Coincidence summing in gamma-ray spectroscopy[END_REF] was applied to obtain individual F C values, that were finally averaged. In the second set, the same point source efficiencies were used for the evaluation of the LS curve, that was further applied for F C computation via a Blaauw and Gelsema (2003) type formalism. The quasi-point source approximation using the efficiencies computed by MCNP for the complete sources was also applied in SUMCOR.

PENELOPE was used for providing input values of the efficiencies for 7 sets of F C results obtained with other software, not with PENNUC. SUMCOR was applied with input from PENELOPE in the same way as with input from MCNP (3 sets of results). The complete version of ETNA for volume sources was also applied with input efficiencies for the reference point source evaluated by PENELOPE; in this case ETNA provided the values of elementary efficiencies for emission points within the volume sources and then performed a numerical integration for obtaining F C . In addition, the efficiencies for the complete volume sources were computed using the transfer method from the point source reference (with efficiencies evaluated by PENELOPE) and the FAST version of ETNA (quasi-point source approximation) was applied for obtaining F C . PENELOPE was also used by 2 participants for the computation of the efficiencies required in the input of the TrueCoinc software [START_REF] Sudar | TrueCoinc'', a software utility for calculation of the true coincidence correction[END_REF].

The standard version of GESPECOR [START_REF] Sima | GESPECORa versatile tool in gamma-ray spectrometry[END_REF][START_REF] Sima | Self-attenuation and coincidence-summing corrections calculated by Monte Carlo simulations for gamma-spectrometric measurements with well-type germanium detectors[END_REF][START_REF] Arnold | Application of GESPECOR software for the calculation of coincidence summing effects in special cases[END_REF][START_REF] Arnold | Calculation of coincidence summing corrections for X-ray peaks and for sum peaks with X-ray contributions[END_REF] was applied by 7 participants. In this code the joint emission probabilities of groups of photons, evaluated analytically [START_REF] Sima | A tool for processing decay scheme data that encompasses coincidence summing calculations[END_REF], are combined with the corresponding integrals of products of efficiencies, obtained by Monte Carlo simulation. The GESPECOR version including the angular correlations was also applied (1 set). Finally, one set of results obtained with GESPECOR deliberately applied the quasi-point source approximation.

2 participants reported results obtained using the EFFTRAN software [START_REF] Vidmar | EFFTRAN-A Monte Carlo efficiency transfer code for gamma-ray spectrometry[END_REF][START_REF] Vidmar | Calculation of total efficiencies of extended samples for HPGe detectors[END_REF][START_REF] Vidmar | Calculation of "LS-curves" for coincidence summing corrections in gamma ray spectrometry[END_REF][START_REF] Vidmar | Calculation of true coincidence summing corrections for extended sources with EFFTRAN[END_REF]. In EFFTRAN the effective solid angle method is used for obtaining the total efficiency for the sample as a whole, in combination with an approximate way of accounting for the contribution from the scattering within the sample itself [START_REF] Vidmar | Calculation of total efficiencies of extended samples for HPGe detectors[END_REF]. The spatial dependence of the efficiency within the volume sources is treated with the LS-curve formalism introduced by Blaauw and Gelsema ( 2003), but contrary to his original approach, the LS factors are obtained by pure calculation, rather than based on a measurement [START_REF] Vidmar | Calculation of "LS-curves" for coincidence summing corrections in gamma ray spectrometry[END_REF][START_REF] Vidmar | Intrinsic error of the point-source and the LS-curve approximations in treatment of true coincidence summing[END_REF]. To arrive at the full-energy-peak efficiencies the effective solid angle values are multiplied with a peak-to-total ratio of a point source obtained from a deterministic model based on [START_REF] Vidmar | A physically founded model of the efficiency curve in gamma-ray spectrometry[END_REF]. Finally, the Andreev recursive formulae [START_REF] Andreev | Consideration of cascade transitions in determining the absolute yield of gamma rays[END_REF] are applied, using the peak and total efficiencies multiplied by the value read off of the LS curve at a corresponding gamma-ray energy.

The ISOCS procedure included in the CANBERRA software was used by one participant. The coincidence summing corrections including up to 3 photons are evaluated using decay data from an internal library [START_REF] Zhu | X-ray true coincidence summing correction in Genie 2000[END_REF]. In the case of extended sources, the effects are evaluated for points located inside of the volume source and then are averaged [START_REF] Kolotov | Estimation of true coincidence corrections for voluminous sources[END_REF].

In most of the cases, DDEP data were used (DDEP, 2018). Exceptions were the applications of MCNP-CP, GEANT 4, TrueCoinc and ISOCS.

In several calculations, only pair coincidences were included, in others all possible coincidences with  photons were considered. There were also differences in the procedures applied to deal with X-Rays. Several participants included the contribution of beta rays and of bremsstrahlung radiation to coincidence summing, but the effect is very small and most results reported did not include these effects. With one exception, angular correlations were neglected.

Several participants did not report the uncertainties of all the computed quantities.

Concerning the completeness of the results sent, it should be mentioned that not every set received included all the requested results. In fact, only the F C values for 1173 and 1332 keV were reported in every set. From the 33 sets of results received, only 19, respectively 21, included results for the sum peaks with energy 1400 keV ( 134 Cs) and 2505 keV ( 60 Co); apparently in the standard versions of MCNP, ETNA, EFFTRAN, ISOCS and TrueCoinc it is not possible to make calculations for these pure sum peaks, but a trick applied by several participants allowed obtaining results for pure sum peaks using these codes (see Table 1).

Results

According to the principle of the proposed self-consistency test, the value of the quantity x (F C or the efficiency) computed directly for the geometry c, x c , should be equal to the value of the same quantity computed considering the geometry c as composed from a and b, x comp . For testing purposes, an index of self-consistency is defined by: (9)

The value of ISC can be interpreted as follows. If the procedure applied for the computation of the quantity x would be absolutely correct in the case of geometries a and b, then the relative bias of the quantity x calculated with that procedure for geometry c is equal to the value of ISC. On the other hand, if also the values x a and x b computed with that procedure are biased, then the value of ISC is no longer equal to the relative bias of x c , the relation between the two depending on the specific case and in fact the relative bias of x c can be larger than ISC.

In the following, the values of the self-consistency index for coincidence summing effects, ISC(NP) for usual peaks (x=1-F C ) and ISC(SP) for pure sum peaks (x=F C ), as well as the index ISC(AE) for the apparent efficiency (x= app ), will be presented.

In Figs. 2345, the distribution of the self-consistency index is displayed for selected peaks of each nuclide considered. Most values of the self-consistency index are around 0, but there are several values around -0.15, clearly violating the self-consistency. According to Equations ( 7) and ( 9), ISC(NP) and ISC(SP) depend both on the values of F C and on the values of the peak efficiencies. However, the evaluation of the self-consistency index for the peak efficiencies demonstrates that with the exception of the set 30, for which the index is around 0.034 in the energy range from 121 to 1408 keV, in all the other cases the values of the peak efficiencies are self-consistent to a high degree. Thus, if ISC(NP) and ISC(SP) are significantly different from zero, this is due to the method applied for the evaluation of F C , and not of the efficiencies. It should be mentioned that the sets with code numbers 9, 19, 24, 26, 27 and 28 were obtained using the quasipoint source approximation, either deliberately, or because the software requires simply the efficiency values for the complete volume source. In Table 2 the results obtained when the quasi-point source approximation was deliberately applied (specific versions of GESPECOR, ETNA and SUMCOR) are compared with the results obtained with the same software using the averaging over the volume of the sources. As expected, the relative difference between the values calculated using the two procedures is highest in the case of the source with larger dimensions (geometry c) and smallest in the case of the source in geometry b (same volume as in geometry a, but a weaker relative dependence of the efficiencies on the position of the emission point due to longer distance to the detector). In the case of the sets of results obtained using the quasi-point source approximation, the self-consistency index is around -0.15, but the bias (with respect to volume integration) of the results for geometry c is around -0.23; thus, the fact that also the results for geometries a and b are biased, results in a bias higher than ISC in geometry c. The conclusion that the true bias of the results in geometry c is larger than ISC is valid, even if the reference values from Table 2 might also have a small bias.

In a correct approach to the computation of F C the self-consistency index should be zero. If this condition is not met, certainly there should be some approximations and inconsistencies in the computational method. Therefore it is interesting to correlate the computed magnitude of the coincidence-summing effects (1-F C for usual peaks and F C for pure sum peaks) for the case of configuration (a) with the self-consistency index. Several examples are given in the figures 6-8. It can be observed that generally many points belong to one of two groups, the first around ISC=0 (self-consistency fulfilled) and the other around ISC=-0.15 (quasi-point source approximation). In the group around ISC=0, the magnitude of coincidence-summing effects for a given energy is roughly the same, whereas in the group around ISC=-0.15, the values are spread over a wider range. In addition, there are also points that do not belong to any of these two groups, as evident in Fig. 7. In these cases the decay schemes ( 133 Ba and 152 Eu) are more complex and X-rays contributions to the coincidence effects are significant. The differences between specific computation codes in the description of the X-rays contributions, as well as the possible differences in the decay data libraries, might be responsible for these points being located apart from the two groups.

From the point of view of activity calculation, the most important quantity characterizing the measurement is the apparent full energy peak efficiency  app . Generally, for usual peaks the index ISC(AE) is not higher than 4 to 5%; in most cases it is close to 0. This is due to the fact that  app for normal peaks is equal to the product F C . Because F C is relatively small with respect to 1, even if the relative bias of 1-F C is about 15%, as in the case of quasi-point source approximation, the effect on F C is much smaller and consequently, the relative bias introduced in the value of  app is also small. Thus F C and  app for usual peaks are weakly sensitive to the approximations applied in the computation. This is in contrast with the case of pure sum peaks, for which the value of ISC(AE) is about the same as ISC(NP), around -0.15, when the quasi-point source approximation is applied in the computation. In Figure 9 the self-consistency index of the apparent efficiencies is represented as a function of energy, but keeping only the results of the sets for which at least one absolute value of ISC(AE) is higher than 0.05.

Concerning the uncertainties represented in the figures, they correspond to one standard deviation. The values were computed on the basis of the data reported by the participants using uncertainty propagation formula. In this context, the uncertainties of ISC(NP) and ISC(SP) should be interpreted with some caution, because they were computed assuming that the values of F C and of  are not correlated. This is true in the case of some sets of reported results, but not in all cases; however, as the covariance between F C and  was not provided by the participants, it was always disregarded.

The proposed self-consistency test may reveal the existence of some problems in the method for the evaluation of the coincidence-summing effects. However, it may be insensitive to some features of the computation method, even if they imply specific approximations and led to biased results. A trivial example is the following: if all efficiencies are multiplied by the same scale factor, the value of ISC for the peak efficiency remains unchanged. Also, ISC(NP), ISC(SP) and ISC(AE) are insensitive to a scale factor applied to all efficiencies if only pair coincidences contribute to summing effects. A more complex example is provided by the angular correlation effects. These effects were included only in one set of results. This set passed the self-consistency test, as well as many sets in which angular correlations were neglected. In Table 3 the magnitude of the effect of angular correlations on F C is presented. Of course, the discrepancies with respect to isotropy are higher in the case of geometry b, corresponding to the smallest solid angle. In the case of pure sum peaks, it is higher than 7% in geometry b, but it is not negligible also in the other geometries. The effect on F C for usual peaks is much lower, but the effect on 1-F C for these peaks is almost the same as in the case of pure sum peaks. The small difference between the effect on 1-F C for 1173 and 1332 keV and on F C for 2505 keV, all subjected to the same angular correlations, is due to the integration over a larger solid angle in the first case (higher effective solid angle for any interaction than for complete energy deposition in the detector).

Summary and conclusions

A self-consistency test to check some features of several procedures applied for the evaluation of coincidence-summing corrections in the case of extended sources was proposed. The test is based on exact relations between the values of F C and of the peak efficiencies calculated for three ideal measurement conditions. These relations should be valid in the case of correct procedures for the evaluation of F C . The

FiguresFigure

  Figures

  Figure 1. Measurement configurations.

Figure 2 .

 2 Figure 2. Self-consistency index for 60 Co peaks. The values of ISC(NP) for the 1173.23 keV and 1332.49 keV peaks, and of ISC(SP) for the 2505.75 keV peak are displayed in function of the code of the participants. The uncertainty bars correspond to 1 .

Figure 3 .

 3 Figure 3. Self-consistency index for 133 Ba peaks of energy 276.40 keV and 356.01 keV. The values of ISC(NP) are displayed in function of the code of the participants.

Figure 4 .

 4 Figure 4. Self-consistency index for several peaks of 134 Cs. The values of ISC(NP) for the 569.33 keV and 604.72 keV peaks, and of ISC(SP) for the 1400.59 keV peak are displayed in function of the code of the participants.

Figure 5 .

 5 Figure 5. Self-consistency index for 152 Eu peaks of energy 121.78 keV and 1408.01 keV. The values of ISC(NP) are displayed in function of the code of the participants.

Figure 6 .

 6 Figure 6. The distribution of the self-coincidence index as a function of the magnitude C of coincidence summing effects in configuration (a) for the 1173.23 keV peak of 60 Co.

Figure 7 .

 7 Figure 7. The distribution of the self-coincidence index as a function of the magnitude C of coincidence summing effects in configuration (a) for the 356.01 keV peak of 133 Ba.

Figure 8 .

 8 Figure 8. The distribution of the self-coincidence index as a function of the magnitude C of coincidence summing effects in configuration (a) for the pure sum peak of 60 Co at 2505.75 keV.

Figure 9 .

 9 Figure 9. The self-consistency index for the apparent peak efficiency for the sets of results in which at least one absolute value of ISC(AE)>0.05. The uncertainty bars were removed for avoiding a too clumsy figure.

test parameter proposed is the index of self-consistency ISC, equal to 0 if the relations are valid. A value of ISC significantly different from 0 suggests that there are some problems in the computation of the coincidence-summing effects and indicates qualitatively the magnitude of the bias of the results.

The main purpose of this GSWG action was to provide to each user of gamma-ray spectrometry a simple and convincing test of the quality of the procedure applied in his laboratory for the computation of coincidence-summing corrections. The test is particularly useful for emphasizing the limitations of the quasi-point source approximation in the calculation of coincidence-summing corrections for extended sources. In the test, vacuum (or air) sources are considered, whereas actual matrices of the sources present non-negligible photon interaction probabilities; in such measurements the expected bias is higher, because efficiency dependence on the position of the emission point should be stronger than in the case of vacuum sources [START_REF] Arnold | Total versus effective total efficiency in the computation of coincidence summing corrections in gamma-ray spectrometry of volume sources[END_REF].

As with any test, the hypothesis of correctness of the procedure can be ruled out if ISC significantly differs from 0, but the fact that ISC is close to 0 is not a proof of the correctness of the procedure. Therefore the proposed test should be complemented by other tests, including experimental validation of the results.

Table 1

Procedures applied by the participants for the calculation of coincidence-summing corrections. In the Details column, only specific information is listed, standard features (all coincidence orders included, pure sum peaks evaluated, K X-rays included) are not mentioned. In the last column the name of the participant and the code of the set of results (in parentheses) are listed.

Efficiencies computation