Thermal energy storage performance of a three-PCM cascade tank in a high-temperature packed bed system

Journal Pre-proof Thermal energy storage performance of a three-PCM cascade tank in a hightemperature packed bed system

Qianjun Mao, Yamei Zhang PII:

S0960-1481(20)30056-2

DOI:

https://doi.org/10.1016/j.renene.2020.01.051

Reference:

RENE 12913

To appear in:

Renewable Energy

Received Date:

12 April 2019

Accepted Date:

11 January 2020

Please cite this article as: Qianjun Mao, Yamei Zhang, Thermal energy storage performance of a three-PCM cascade tank in a high-temperature packed bed system, Renewable Energy (2020), https://doi.org/10.1016/j.renene.2020.01.051

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Journal Pre-proof Thermal energy storage performance of a three-PCM cascade tank in a high-temperature packed bed system Qianjun Mao*, Yamei Zhang School of Urban Construction, Wuhan University of Science and Technology, Wuhan 430081, China * Corresponding author. Tel./fax:+86 27 68893616 E-mail addresses: [email protected]

Abstract: Solar storage tanks are key to ensuring the high efficiency of concentrated solar power plants, and phase change materials are the most important storage energy media influencing system efficiency. Therefore, as energy storage or release mechanisms are a focus of related research. In this study, numerically analysed the thermal performance of a small capsule of three different phase change materials for a packed bed solar energy storage system. Air and molten salt are used as the heat transfer fluid (HTF) and the phase change material (PCM), respectively. A model based on a concentric-dispersion model and the enthalpy method was used to analyse the phase transition of the PCM. The equation was solved using the finite- difference method, and the results were verified using previous experimental data. The influence of particle diameter, porosity, and height-todiameter ratio of the storage tank on the total storage energy, storage capacity ratio, axial temperature curve, and utilization ratio of the PCM were studied. It was found that he storage capacity and utilization rate of 3-PCM energy storage tanks are relatively high. And that increase from 86.07% to 86.65% and 86.07% to 86.67%, respectively, when the porosity is reduced from 0.6 to 0.1. This results in an increase in the total storage energy of 5.2 ×1012 Wh to 1.3 ×1013 Wh. Similarly, when the particle diameter decreases from 0.6 to 0.1, the storage capacity ratio and utilization rate increase from 85.8% to 87.3% and 85.6% to 87.4%, respectively. However, although these increases are larger, the increase in total energy storage is small. Finally, it was found that the shape of the tank has no effect on the storage capacity at a fixed tank volume. The proposed model provides a reference value for energy storage in a concentrating solar thermal power (CSP) system.

Key words: Thermal energy storage; Packed bed; PCM; Utilization ratio 1

Journal Pre-proof Nomenclature A

area of bed (m2)

V

tank volume (m3)

D

diameter of bed (m)

𝑑𝑝

particle diameter (m)

H

storage tank height (m)

𝐸𝑥𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑

total net exergy in (Wh)

𝐸𝑥𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑

total net exergy out (Wh)

𝐸𝑠𝑡𝑜𝑟𝑒𝑡𝑜𝑡𝑎𝑙

total energy stored in PCM particles (Wh)

ℎ𝑝

heat transfer coefficient (W/ m2 K)

ℎ𝑣

volumetric heat transfer coefficient (W/ m2 K)

𝑈𝑤

overall heat transfer coefficient (W/ m2 K)

hi

inner heat transfer coefficient (W/ m2 K)

T

temperature (°C)

Nu

Nusselt number

∆P

pressure drop (Pa)

Re

Reynolds number

Pr

Prandtl number

𝑐𝑝

specific heat (J/kg K)

U

velocity (m/s)

L

latent heat (J/kg)

t

time (s)

Subscripts ave

average

in

inlet

out

outlet

a

air

p

particles

s

solid

n

axial grid steps

r

radial grid steps per sphere

j

index for insulation

e

environment

f

fluid

b

bed

l

liquid

s

solid

ch

charging

dch

discharging

Superscripts i

index for time steps

2

Journal Pre-proof Greek symbols ε

porosity

ρ

density (kg/m3)

α

thermal diffusivity (m2/s)

k

thermal conductivity (W/m k)

η

efficiency

σ

capacity ratio

γ

utilization ratio

μ

dynamic viscosity (kg/m s)

3

Journal Pre-proof 1. Introduction Solar thermal energy storage plays an important role in energy services [1-3] such as water heating, air conditioning, and waste heat recovery systems [4-6]. Concentrated solar power plants, which are used worldwide, rely on the heat of the sun to generate electricity [7-9]. Furthermore, because solar energy is inexhaustible and pollution-free, the development of solar thermal energy storage is crucial for ensuring a sustainable energy future [10-12]. Currently, packed bed energy storage is a technical and economical solar energy storage system. The simple structural form of the packed bed and its heat transfer performance which is an improvement on the traditional two-tank system, have led researchers to propose multiple mathematical simulations for packed beds. The packed bed is divided into sensible heat storage and latent heat storage system according to the physical state of the material used[13, 14]. Studies of packed bed can be based on three main things: numerical models, the materials used, and structures. Four models can be used to study packed beds in MATLAB, these are the single-phase, Schumann, concentric dispersion, and continuous solid-phase models. Franklin et al.[15] successfully proposed the first Schumann model, which is now widely used and is a preferred choice for verification for many scholars. Yang, Bellan and Oró et al.[16-18] proposed the development of a continuous-solid model, focusing on the thermodynamic characteristics of thermal energy storage in a packed bed. Cheng et al. [19]used a concentric dispersion model to study packed beds in energy storage applications. The four models were also extensively analysed using different packed bed numerical models to determine how energy storage demand could best be met.[20, 21] The study of materials related to packed beds is also important. Hänchen and Niyas. [22, 23] proposed a sensible heat transfer model for packed beds. Zanganeh et al. [24] proposed a packed bed that combined sensible heat storage and latent heat storage. Other researchers [25-28]conducted a single-phase packed bed and performed parameter analysis. Izquierdo-Barrientos et al.[29]proposed a two-phase packed bed to use to analyse thermodynamic properties. Yang and Peng et al.[30, 31]proposed a three-PCM spherical capsule packed bed heat storage system. Researchers have also proposed studies on the dimensions and other aspects of packed beds. One-dimensional means that only axial heat and mass transfer are considered in the heat transfer process, while two-dimensional means that heat and mass heat transfer are considered both in the 4

Journal Pre-proof radial direction and in the axial direction. Some researchers [32-34] proposed a one-dimensional dimensionless calculation program and used this program to simulate the heat storage and release process of a sensible hot packed bed thermal storage system. Other researchers [35-37] developed a two-dimensional program of transients in a thermal storage system, and obtained a twodimensional transient temperature distribution of a tank. In summary, there have been extensive simulation studies on packed beds. However, the short literature review above shows the following: (1) the temperature field in porous media and the phase transition process in PCM capsules have not been studied in depth, therefore, we chose to study the concentric-dispersion model. (2) The majority of previous research is focused on singlephase or sensible heat storage, therefore, this study, proposes a thermal energy storage system for three different PCMs. The latent heat storage energy in PCMs significantly increases the energy density, which can reduce the storage size and cost. Therefore, the use of three different phase change materials allows for a wider temperature range and is more suitable for application in real energy storage systems. (3) Many studies only consider one dimension, the axial [32-34]. However, considering that there is also a large heat loss in the radial direction, this study considers twodimension. The heat transfer process between the PCM and HTF during charging and discharging is studied in detail, with particular focus on the phase change process. The effects of capsule diameter, porosity, and height-diameter ratio on heat transfer and energy storage are also investigated. Our proposed model can accommodate all sensible heat storage and latent heat storage processes and provide a scheme with which to optimize the basic configuration and operating conditions of the storage system, enabling the selection of more suitable parameters to improve system storage capacity and utilization. Therefore, this study presents a good alternative for thermal energy storage in concentrating solar thermal power (CSP) systems.

2. Numerical analysis 2.1. Model description and governing equations The packed bed system is known for its very simple structure and high storage capacity. Fig. 1 shows the basic structure of the energy storage tank used in this study. Like the traditional twotank form, it uses a cylindrical tank and that is insulated on the outer circumference. H is the height 5

Journal Pre-proof of the tank, D is the diameter of the tank, ε is the porosity, and dp is the diameter of the phase change material capsule. The tank is divided into three main parts: the first is filled with PCM1, the second with PCM2, and the third with PCM3. Table 1 shows the main features and parameters of the packed bed used in this study [38]. The changes in HTF parameters with temperature are as follows [39]: 𝜌𝑎𝑖𝑟 = ―5.75399 × 10 ―16 ∙ 𝑇(℃)5 +3.02846 × 10 ―12 ∙ 𝑇(℃)4 ―6.18352 × 10 ―9 ∙ 𝑇(℃)3 + (1)

6.29927 × 10 ―6 ∙ 𝑇(℃)2 ―3.5422 × 10 ―3 ∙ 𝑇(℃) +1.2507 𝜇𝑎𝑖𝑟 = 6.10504 × 10 ―10 ∙ 𝑇(℃)3 ―2.13036 × 10 ―6 ∙ 𝑇(℃)2 +4.71398 ×

1.67555 × 10 ―5 10 ―3 ∙ 𝑇(℃) + (2) 𝑐𝑝,𝑎𝑖𝑟 = 1.28806 × 10 ―13 ∙ 𝑇(℃)4 ―4.46054 × 10 ―10 ∙ 𝑇(℃)3 +4.48772 × 10 ―7 ∙ 𝑇(℃)2 + 1.82754 × 10 ―5 ∙ 𝑇(℃) +1.00651 (3) 𝑘𝑎𝑖𝑟 = ―4.44955 × 10 ―15 ∙ 𝑇(℃)4 +2.41702 × 10 ―11 ∙ 𝑇(℃)3 ―4.09601 × 10 ―8 ∙ 𝑇(℃)2 + (4)

7.91034 × 10 ―5 ∙ 𝑇(℃) +0.242006

Fig. 1. Schematic diagram of the packed bed thermal storage system

6

Journal Pre-proof

Table 1 Main characteristics and design parameters of the packed bed thermal storage system used in this study Parameters

Values

Height of tank(H)

15m

Diameter of tank(D)

30m

Porosity(ε)

0.3

Heat transfer fluid(HTF)

air

Operating temperature(T)

550℃

The storage process of the packed bed is divided into charging and discharging processes. During the charging process, the hot HTF flows from the top of the tank into the gap, fills these and transfers heat using small phase change material capsules to store heat. After the heat is stored, it flows out at a lower temperature. During the opposite process, i.e., the discharging process, the cold HTF flows from the bottom of the tank and out with the higher temperature. the physical properties of a three-PCM packed bed based on other studies in the literature [40], the details of which are as shown in Table 2. Table 2 Properties of the PCM filler material

Properties Average Density ρ(kg/m3) Thermal conductivity K (W/m K) Specific heat 𝑐𝑝(kJ/kg K) Melting point T(℃) Laten heat of fusion L(kJ/kg)

PCM-1(56:44)

PCM-2(38.5:61.5)

Na2CO3-Li2CO3

MgCl2-NaCl

2320 2.09 2360 496 370

2480 0.6 1450 435 350

PCM-3 KOH

2044 0.5 1470 380 150

This study, used a transient concentric-dispersion model to analyse the thermal performance of the energy storage tank effectively, assuming that the capsules are identical and isotropic. To improve the simulation, the following assumptions are made: (1) The heat loss of the tank is ignored and the HTF is considered a medium that exchanges energy with the environment through the tank wall. (2) The spheres are evenly distributed and of equal size, and equally porous. (3) The radiation between the particles and the HTF is ignored. (4) The air flow direction in the tank is axial and incompressible. (5) The physical properties of the air change uniformly with temperature, as in equations (1–4), 7

Journal Pre-proof and are calculated at an average temperature of 𝑇𝑎𝑣𝑒 = (𝑇𝑖𝑛 + 𝑇𝑜𝑢𝑡)/2. (6) Fluid flow and heat transfer are axisymmetric. Based on the above assumptions, the governing energy equations for the fluid phase and the solid phase are defined as follows. For the fluid phase,

(

∂𝑇𝑎

∂𝑇𝑎

∂2𝑇𝑎

)

𝑐𝑝,𝑠𝜌𝑎 𝜀 ∂𝑡 + 𝑈 ∂𝑥 = 𝛼𝑎𝑥𝑐𝑝,𝑎𝜌𝑎𝜀 ∂𝑥2 ―

6(1 ― 𝜀)ℎ𝑝

4𝑈𝑤

𝑑𝑝

𝐷

(𝑇𝑠,𝑅0 ― 𝑇𝑎) ―

(𝑇𝑖𝑛𝑓 ― 𝑇𝑎)

(5)

For the solid phase, ∂𝑇𝑠

∂2𝑇𝑠

2∂𝑇𝑠

(6)

𝜌𝑠𝑐𝑝,𝑠 ∂𝑡 ― 𝑘𝑠𝑟 ∂𝑟 = 𝑘𝑠 ∂𝑟2

The boundary conditions for the initial and solid phases are defined as follows: ∂𝑇𝑆

(7)

= 0,𝑎𝑡 𝑟 = 0

∂𝑟 ∂𝑇𝑠

𝑘𝑠 ∂𝑟 = ℎ𝑝(𝑇𝑎 ― 𝑇𝑠,𝑟 = 𝑅0), 𝑎𝑡 𝑟 = 𝑅0

(8)

Similarly, the boundary conditions for the initial and fluid phases are defined as below: ∂𝑇𝑎 ∂𝑥

(9)

= 0,𝑎𝑡 𝑟 = 0

(10)

𝑇𝑎 = 𝑇𝑖𝑛,𝑎𝑡 𝑥 = 𝐻

The equations for the heat transfer coefficient with respect to particle convection are as follows [4] 𝜌𝑎𝑈𝑑𝑝

𝑅𝑒 = 𝑃𝑟 = ℎ𝑝 =

𝑘𝑃𝐶𝑀𝑁𝑢 𝐷𝑝

=

(11)

𝜇𝑎 𝑐𝑝,𝑎𝜇

(12)

𝑘𝑎

[

𝑘𝑃𝐶𝑀 𝐷𝑝

1

(

2 + 1.1 𝑅𝑒0.6𝑃𝑟

)]

3

(13)

The volumetric heat transfer coefficient is determined using the following equation [38] ℎ𝑣 =

6(1 ― 𝜀) 𝐷𝑝 ℎ𝑝

(14)

The total heat transfer coefficient is determined by the heat transfer resistance between the heat transfer fluid and the tank wall as below [41]. 1 𝑈𝑤

1

𝑛

1

= ℎ𝑖 + 𝑟𝑏𝑒𝑑∑𝑗 = 1𝑘𝑗𝑙𝑛

The inner wall heat loss coefficient, ℎ𝑖, is given as follows [39]:

8

( ) 𝑟𝑗 + 1 𝑟𝑗

(15)

Journal Pre-proof 𝑘𝑎

1

[(

ℎ𝑖 = 𝐷𝑝 0.203𝑅𝑒

1 3

𝑃𝑟

) + (0.22𝑅𝑒

3

𝑃𝑟0.4)

0.8

]

(16)

For the packed bed mode, the effective thermal conductivity of the fluid phase is based on the following equation[42]: 𝑘𝑒𝑓𝑓 = 𝑘𝑎 Where 𝜑 = 1 ― 𝜀

𝛽=

[

1 + 2𝛽∅ + (2𝛽3 ― 0.1𝛽)∅2 + ∅30.05𝑒𝑥𝑝(4.5𝛽)

(𝑘𝑃𝐶𝑀 ― 𝑘𝑎)

1 ― 𝛽∅

]

(17)

(𝑘𝑃𝐶𝑀 + 2𝑘𝑎).

The pressure drop equation for the packed bed is determined to be the following [4] ∆𝑝 = 150𝐻

(1 ― 𝜀)2𝜇𝑎𝑈 𝜀

3

𝑑𝑝2

+1.75𝐻

1 ― 𝜀𝜌𝑎𝑈2 𝜀3

𝑑𝑝2

(18)

2.2 PCM particle packing material Equations 5 and 6 can be applied to all processes of sensible heat storage and latent heat storage; however, for the latter, changes in temperature-dependent physical property parameters should be considered to accurately simulate latent heat storage. There are three PCM phases in the transition process: the solid phase, the phase transition itself, and the liquid phase. Similarly, the latent heat storage also contains sensible heat storage. In this study, as change in the density of the PCM is relatively small, the solid and liquid densities are set as equal, and the average values of density, thermal conductivity, and thermal diffusivity are considered during the phase change. (1) When the PCM is in a solid state; i.e., it has not yet reached the melting state, the relevant equation is as follows: 𝑐𝑝 = 𝑐𝑝𝑠,𝜌 = 𝜌𝑠,𝑘𝑝 = 𝑘𝑠

(19)

(2) When the PCM is in the phase change process; i.e., the PCM has reached the melting temperature, the equation is as below [4] 𝑐𝑝 =

𝐶𝑝𝑠 + 𝐶𝑝𝐿 2

𝐿

+ 𝑇𝐿 ― 𝑇𝑠,𝜌 = 𝜌𝑠,𝑙,𝑘𝑝 =

𝑘𝑠 + 𝑘𝑙 2

(20)

(3) When the PCM is in the liquid phase, the melting process has been completed and the new relevant equation is as follows: 𝑐𝑝 = 𝑐𝑝𝑙,𝜌 = 𝜌𝑙,𝑘𝑝 = 𝑘𝑙

(21)

2.3 Performance indicators The amount of energy stored, storage capacity ratio, utilization ratio, and efficiency are all key 9

Journal Pre-proof indicators for a packed bed. Therefore, we calculated the net storage exergy recovered and supplied [43], as follows: 𝑡

[

( )]𝑑𝑡

(22)

[

( )]𝑑𝑡

(23)

𝐸𝑥𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑 = ∫𝑡𝑓𝑖𝑛𝑎𝑙,𝐷𝑐ℎ 𝑚𝑎𝑐𝑝,𝑎 𝑇𝑎,𝑜𝑢𝑡(𝐾) ― 𝑇𝑎,𝑖𝑛(𝐾) ― 𝑇0𝑙𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙,𝐷𝑐ℎ

𝑡

𝐸𝑥𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 = ∫𝑡𝑓𝑖𝑛𝑎𝑙,𝑐ℎ 𝑚𝑎𝑐𝑝,𝑎 𝑇𝑎,𝑖𝑛(𝐾) ― 𝑇𝑎,𝑜𝑢𝑡(𝐾) ― 𝑇0𝑙𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙,𝑐ℎ

𝑇𝑎,𝑜𝑢𝑡 𝑇𝑎,𝑖𝑛

𝑇𝑎,𝑖𝑛

𝑇𝑎,𝑜𝑢𝑡

To calculate the maximum theoretical storage capacity, as defined in Cheng et al [22], we use the following: 𝐸𝑠𝑡𝑜𝑟𝑒𝑑

𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 = 𝜎 = 𝐸𝑚𝑎𝑥 ,𝑠𝑡𝑜𝑟𝑒𝑑 𝐸𝑚𝑎𝑥,

𝑠𝑡𝑜𝑟𝑒𝑑

= 𝑚𝑝𝑐𝑚𝑐𝑝,𝑠(𝑇𝑎𝑖𝑛 ― 𝑇𝑏𝑖𝑛) + 𝑚𝑝𝑐𝑚𝐿

(24) (25)

In contrast, the utilization reflects the energy obtained for the maximum possible energy storage and the maximum energy released during the discharge process [26]. 𝐸𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑑

𝑈𝑡𝑖𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 = 𝛾 = 𝐸𝑚𝑎𝑥,𝑠𝑡𝑜𝑟𝑒𝑑

(26)

However the overall efficiency of the system is also important for evaluating the packed bed system [39], and for that the following equations are required.

𝐸𝑠𝑡𝑜𝑟𝑒𝑡𝑜𝑡𝑎𝑙

𝜂𝑐ℎ = 𝐸𝑥

(27)

𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑

𝐸𝑥𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑑

(28)

𝜂𝐷𝑐ℎ = 𝐸𝑠𝑡𝑜𝑟𝑒𝑡𝑜𝑡𝑎𝑙

𝜂𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝜂𝑐ℎ ∙ 𝜂𝐷𝑐ℎ =

𝐸𝑥𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑 𝐸𝑥𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑

(29)

2.4 Numerical approach Using the mathematical model developed in Section 2.1, the governing equations were solved in Matlab using direct finite difference approximation under the fully implicit scheme, with n nodes in the axial direction and each small sphere having r nodes. The detailed numerical method is shown in Fig. 2, where i is the time step, k is the total charging time, m is the number of cycles, and the numerical analysis was performed by the time variable, mainly through the following steps: Step 1: Enter the packed bed parameters, HTF parameters, and PCM parameters into MATLAB. Step 2: Set the threshold temperature during the charge and discharge time, as shown in equations (30) and (31).The normalized temperature of the charging threshold is 0.39 and the discharging 10

Journal Pre-proof threshold value is 0.74 [44]. 𝑇𝑚𝑎𝑥 = 𝑇𝑏𝑖𝑛 +0.39(𝑇𝑎𝑖𝑛 ― 𝑇𝑏𝑖𝑛)

(30)

𝑇𝑚𝑖𝑛 = 𝑇𝑏𝑖𝑛 +0.74(𝑇𝑎𝑖𝑛 ― 𝑇𝑏𝑖𝑛)

(31)

Step 3: Discretize equations 5 and 6, calculate the HTF temperature of each node during charging and discharging with a time step of 1 s, and use this value to calculate the PCM temperature.

(

𝑐𝑝.𝑎𝜌𝑎 𝜀

𝑇𝑎𝑖𝑛+ 1 ― 𝑇𝑎𝑖𝑛 𝛥𝑡

∂𝑇𝑎

)

+ 𝑈 ∂𝑋 = 𝛼𝑎𝑥𝑐𝑝.𝑎𝜌𝑎𝜀

𝑇𝑎𝑖𝑛++11 ― 2𝑇𝑎𝑖𝑛+ 1 + 𝑇𝑎𝑖𝑛+―11 ∆𝑋

2

―

6(1 ― 𝜀)ℎ𝑝 𝑑𝑝

(𝑇𝑠.𝑅 𝑖𝑛 ― 𝑇𝑎𝑖𝑛+ 1) 0

4𝑈𝑤

―

(𝑇𝑒 ― 𝑇𝑎𝑖𝑛+ 1)

𝐷 𝑇𝑆𝑖𝑟 + 1 ― 𝑇𝑆𝑖𝑟

𝜌𝑠𝑐𝑝,𝑠

𝛥𝑡

𝑖+1 𝑖+1 2𝑇𝑠𝑟 ― 1 ― 𝑇𝑠𝑟 + 1 2𝛥𝑟

― 𝑘𝑠𝑟

(32) 𝑇𝑠𝑖𝑟 ++ 11 ― 2𝑇𝑠𝑖𝑟 + 1 + 𝑇𝑠𝑖𝑟 +― 11

= 𝑘𝑠

𝛥𝑟2

(33)

Step 4: Calculate the average energy stored and heat lost during charging and discharging based on the result of step 3, then repeat steps 3–4 until the threshold temperature is reached. Step 5: Calculate the total storage of the packed bed based on the total energy storage equation [35]. 𝐸𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑑 = 𝐸𝑠𝑡𝑜𝑟𝑒𝑑 𝑎𝑓𝑡𝑒𝑟 𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔 ― 𝐸𝑠𝑡𝑜𝑟𝑒𝑑 𝑎𝑓𝑡𝑒𝑟 𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔

(34)

Step 6: Based on the energy calculated, calculate the storage capacity ratio, utilization ratio, and efficiency. Step 7: Cycle through the entire process again until the last set number of cycles is reached.

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Fig. 2. Detailed numerical method

3. Results and discussion 3.1 Model validation To verify the correctness of the numerical model, we first performed deviation analysis on existing numerical prediction results and experimental results, and used a single-phase transformation model to verify our model. The results of the simulation were verified using the experimental data in Bellan et al.[36]. Air is used as the heat transfer fluid, the inlet temperature is 326 °C, the average equivalent sphere diameter of the PCM particles is 0.02563 m, and the porosity is 0.328. During the charging and discharging process, the temperature curve of each layer and the deviations are compared over time. Fig. 3 and Fig. 4 show the second and eighth rows of temperature profiles during the charging process and discharging process, respectively. A comparison of simulation and experimental results shows that, the average deviations in the second and eighth rows are approximately 8.67% and 7.36%, respectively. The two sets of graphs indicate good agreement between the experimental results and those from the numerical simulation. 12

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330 Charging process Temperature(℃ )

320 310 300

2nd row-Num 2nd row-Exp 8th row-Num 8th row-Exp

290 280

0

50

100

150

200

Time(min) Fig. 3. Comparison between the numerical simulation results of this study and experimental data of the charging process temperature profile

330 Discharging process

Temperature(℃ )

320

2nd row-Num 2nd row-Exp 8th row-Num 8th row-Exp

310 300 290 280

0

50

100

150

200

Time(min) Fig. 4. Comparison between the numerical simulation results of this study and experimental data of the discharging process temperature profile

3.2 Temperature profiles in the packed bed Fig. 5 and Fig. 6 show the instantaneous changes in HTF temperature at various axial positions during the charging and discharging modes, respectively, which can be divided into four regions 13

Journal Pre-proof during the charging and discharging process. The first region represents a sensible heat process (Fig. 5), where the sensible heat below the phase change temperature, and heat is transferred from the HTF to be stored in the packed bed. The second area represents the melting process of the PCM capsule; i.e., the phase change process. This area starts from 𝑇𝑚𝑒𝑙𝑡3 to 𝑇𝑚𝑒𝑙𝑡2. The third area also represents the phase transition process, starting from 𝑇𝑚𝑒𝑙𝑡2 and ending at 𝑇𝑚𝑒𝑙𝑡1. The temperature gradient clearly decreases during these two stages, the smaller the slope, the longer it takes to decrease. This is due to the endothermic process occurring in the PCM capsule. The fourth region also represents a sensible heat storage process, in which the sensible heat is above the phase change temperature. The relatively high temperature during this stage means the heat transfer rate between the PCM capsule and the HTF is very high, and that efficiency is high, heat transfer time is reduced, heat transfer between packed bed and environment is reduced and the heat energy loss is reduced. Moreover, the heat of the HTF can be efficiently transferred to the PCM capsule until the end of the charging process. The reverse process is shown in Fig. 6. In this case, in all processes except the second and the third, the PCM capsule is exothermic and the slope is not as pronounced as the melt slope change because there is no phase change in the PCM until the end of the discharging process.

HTF Temperature[℃ ]

550

Charging process: dp=0.04m; D=30m; H=15m; =0.3

region4

500 region3

Tmelt1

450 400 350

region2

Tmelt2

Tmelt3

region1 0

50

100

Y=0.95

Y=0.90

Y=0.85

Y=0.80

Y=0.75

Y=0.70

Y=0.65

Y=0.60

Y=0.55

Y=0.50

Y=0.45

Y=0.40

Y=0.35

Y=0.30

Y=0.25

Y=0.20

Y=0.15

Y=0.10

Y=0.05

150

200

250

300

Time[min] Fig. 5 Temperature of HTF at different axial positions during the charge process 14

Journal Pre-proof Discharging process:dp=0.04m; D=30m; H=15m; =0.3

HTF Temperature[℃ ]

550

region4

500 450

region3

400

region2

Tmelt2 Tmelt3

350 300

Tmelt1

Y=0.05 Y=0.30 Y=0.55 Y=0.80

0

50

Y=0.10 Y=0.35 Y=0.60 Y=0.85

100

Y=0.15 Y=0.40 Y=0.65 Y=0.90

Y=0.20 Y=0.45 Y=0.70 Y=0.95

150

Y=0.25 Y=0.50 Y=0.75

200

region1 250

300

Time[min] Fig. 6 Temperature of HTF at different axial positions during the discharge process

3.3 Effect of bed porosity Fig. 7 shows the axial temperature curve at a porosity of 0.1-0.6 at 4h. It shows that, at the same axial position, a higher porosity results in a higher HTF temperature. This is because a higher porosity increases the positions of gaps in space, allowing more HTF to be carried through, and weakening the heat transfer. The ability of heat to be transferred from HTF to PCM is then reduced, and the ratio of temperature decrease is lower, therefore, the higher porosity, the higher HTF temperature.

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Fig. 7 Temperature profiles of HTF at t=4 h for different bed porosities

Fig. 8 shows a comparison of the total storage energy at different porosities. The lower the porosity, the higher the proportional increase in energy storage, shown by the initial steep slope. When the phase transition process is reached, the storage speed decreases due to the convective heat transfer that occurs between the particles during the phase transition. At different porosities, the basic states of these energy increases are similar. The Fig. 8 shows that the stored energy decreases as porosity increases, as does the time taken to complete energy storage. For example, for two porosity curves with slopes of 0.3 and 0.6, the maximum energy storage is reduced from 9.8×1012 Wh to 5.6×1012 Wh as porosity increases, therefore, charging time is also reduced from 6 h to 3 h.

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Fig. 8 Total heat storage for different bed porosities

Fig. 9 shows a comparison of the capacity ratio at different porosities, when the steady state is not reached, the porosity and capacity ratio are higher, because the local flow rate increases as porosity initially increases. Then, the capacity ratio also increases due to the large flow between HTF and PCM and an increased heat transfer rate. With time going by, as the porosity increases, when stabilization is achieved, the higher the porosity allows the PCM temperature to rise faster, the heat loss between the packed bed and the environment is higher, Therefore, the higher the final porosity, the smaller the capacity ratio.

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Fig. 9 Capacity ratio for different bed porosities

Fig. 10 shows a comparison between the utilization ratio at different porosities. When heat transfer reaches steady state, a higher porosity leads to a lower utilization ratio, because the corresponding higher temperature results in HTF flowing more quickly. However, HTF cannot move to the thermocline area for a specified period of time at slow speeds, therefore, it cannot reach the bottom of the tank. This results in less energy being wasted and higher utilization taking place during the charge and discharge processes.

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Fig. 10 Utilization ratio for different bed porosities

3.4 Effect of PCM capsule diameter Fig. 11 shows the axial temperature profile of the HTF at the different particle sizes dp =0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 at t=3 h. The smaller the particle size, the steeper the slope of HTF temperature. According to the heat transfer coefficient formula, the heat transfer coefficient of the HTF and the PCM capsules increases as the particle diameter decreases. Therefore, the convective heat transfer between the HTF and the PCM capsule is improved, as observed at Y = 0.15, 0.42, and 0.72, where the respective phase transition melting temperatures are reached and the result is a thinner thermocline in these regions. Therefore, when the phase change process occurs in all three phase change materials, the temperature drop is more than 45% of the dimensionless height.

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Fig. 11 Temperature profiles of HTF at t=3 h for different capsule diameters

Fig. 12 shows that, during the initial stage of the charging process, energy levels increases with increasing charge time, the smaller the particles, the greater this trend (Fig. 10). In the initial stage of the charge process, the temperature gradient is steeper. However, the energy storage then increases slowly during the melting process until the heat storage energy reaches a maximum during the final charging process. Because the Reynolds number of a particle involves the its diameter, the former increases as particle diameter increases. This means that the speed of the HTF increases, leading to uneven local flow and uneven heat, and that this a temperature difference, results in a small difference in maximum energy storage.

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Fig. 12 Total heat storage for different capsule diameters

Fig. 13 shows the change in the capacity ratio at different particle diameters. During the initial stage of the charging process, the capacity ratio increases with time and smaller particle diameters have a higher capacity ratio. This shows that heat transfer between the HTF and the PCM particles is related to the size of the latter, because a smaller particle diameter can reduce the entropy of the system and form a narrow temperature zone. Therefore, the smaller the particle diameter, the larger the capacity ratio.

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Fig. 13 Capacity ratio for different capsule diameters

Fig. 14 shows the change curves of the utilization ratio for different particle sizes, larger particles size correlates to a lower utilization ratio because their Reynolds number is larger. Therefore, larger particles (with their large Reynolds number) requires a longer distance to exchange the same amount of energy between the HTF and the PCM. This is equivalent to extending the thermocline region, which could potentially result in wasted thermal energy and lower utilization.

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Fig. 14 Utilization ratio for different capsule diameters

3.5 Effect of tank height-to-diameter ratio For packed bed energy storage tanks, we can change the height-to-diameter ratio of the energy storage tanks for a fixed volume, e.g., a large height would result in a smaller diameter, and vice versa. When the ratio is constant, the energy storage capacity of the tank is the same, and the axial temperature curve of the energy storage tank changes to the same form. As shown in Fig. 15, the axial temperature nodes are evenly divided according to height. The stratification of the energy storage tank is also evenly divided. At the dimensionless height, the temperature curves are highly coincident. Therefore, changing the height to diameter ratio has no effect on the curve of axial temperature change (Fig. 16). Similarly, changing the height to diameter ratio produces highly consistent capacity ratios for the energy storage tanks, which confirms that the energy stored is constant for a fixed volume, and has highly consistent utilization rates. Therefore, the aspect ratio of the tank of a fixed volume three-layer packed bed energy storage system has little effect on the energy storage capacity and no effect on the utilization ratio or capacity ratio. Therefore, the influence of aspect ratio of cylindrical tank can be ignored during experiments.

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Fig. 15 Temperature profiles of HTF at t=4 h for different tank height-to-diameter ratios

Fig. 16 Capacity ratio for different tank height-to-diameter ratios

4. Conclusion In this study, we used concentric dispersion models to investigate the thermal properties of latent heat storage in 3-PCM capsules. We considered radial heat transfer and wall heat loss and both the charging and discharging processes. We also considered the effects of porosity, diameter of the PCM capsule, and tank height-to-diameter ratio on the temperature distribution, energy storage, 24

Journal Pre-proof capacity ratio, and utilization ratio of a packed bed storage tanks. The main conclusions of this study are as follows. (1) The thermal behaviour of the 3-PCM packed bed, including sensible heat storage and latent heat storage, can be predicted successfully using a numerical concentric-dispersion model that considers radial heat transfer and wall heat loss. In this study, only simulated latent heat storage, however, if the phase change material was changed to rock, the sensible heat storage could also be obtained. (2) Porosity significantly influences the overall heat transfer performance of the tank. The smaller the porosity, the slower the HTF temperature drops and the higher the effective energy storage, which ranges from 5.6×1012 Wh to 1.35×1013 Wh in this study. The effect of porosity on the capacity ratio and utilization rate is not significant. (3) The particle diameter of the particles have a large effect on the heat transfer between air and the PCM capsules. As the diameter of the particle decreases, the thermocline region becomes thinner and the energy storage capacity increases. Similarly, the capacity ratio and utilization rate also increase significantly when the diameter of the particle increases, in this study these values went from 85.8% to 87.3% and 85.6% to 87.4%, respectively, for a particle diameter ranging from 0.6 to 0.1. (4) For a fixed volume energy storage tank, changing the height to diameter ratio has no effect on the heat transfer behaviour of the system, therefore, the aspect ratio of the tank has no effect on the energy storage of the system. (5) The limitation of this study is that we only studied the effects of three parameters. Future work, which will aim to make a significant contribution to improving the energy storage efficiency of a concentrated solar power system, will focus on other parameters that could help improve energy storage efficiency

Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 51876147 and 51406033).

References 25

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Author Contributions Section Qianjun Mao provides the idea of this study, and Yamei Zhang write the code of the manuscript. All authors read the manuscript and all authors contribute equally.

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Declaration of Interest Statement We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work entitled “Thermal energy storage performance of a three-PCM cascade tank in a high-temperature packed bed system”.

Journal Pre-proof Highlights 1. A three-PCM cascade system using the CSP plant has been proposed. 2. The computer code for this model has been written. 3. The entire process of charging and discharging has been investigated.