Multi-objective optimization of thermal performance of packed bed latent heat thermal storage system based on response surface method

Journal Pre-proof Multi-objective Optimization of Thermal Performance of Packed Bed Latent Heat Thermal Storage System Based on Response Surface Method

Long Gao, Gegentana, Zhongze Liu, Baizhong Sun, Deyong Che, Shaohua Li PII:

S0960-1481(20)30179-8

DOI:

https://doi.org/10.1016/j.renene.2020.01.157

Reference:

RENE 13020

To appear in:

Renewable Energy

Received Date:

10 June 2019

Accepted Date:

31 January 2020

Please cite this article as: Long Gao, Gegentana, Zhongze Liu, Baizhong Sun, Deyong Che, Shaohua Li, Multi-objective Optimization of Thermal Performance of Packed Bed Latent Heat Thermal Storage System Based on Response Surface Method, Renewable Energy (2020), https://doi.org/10.1016/j.renene.2020.01.157

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Journal Pre-proof Multi-objective Optimization of Thermal Performance of Packed Bed Latent Heat Thermal Storage System Based on Response Surface Method Long Gao, Gegentana, Zhongze Liu, Baizhong Sun*, Deyong Che, Shaohua Li School of Energy and Power Engineering, Northeast Electric Power University, Jilin 132012, China *Corresponding

author. Tel.: +86 18600231496; fax: +86 432 64807281.

E-mail address: [email protected] (B. Sun), [email protected] (L. Gao)

Abstract: In this study, a simplified numerical model was developed to calculate the internal thermal distribution rules and thermal performance index of a packed bed latent heat thermal storage system (PBLHTES). The accuracy of the model was verified by comparing numerical results with experimental data. The response surface method (RSM) was applied to study and optimize the thermal performance of the PBLHTES. The quadratic regression model of three response indices was established based on 130 groups of Box–Behnken design (BBD) model scheme results, and rationality was verified through analysis of variance (ANOVA). The results showed the degree of influence of single factors and interactive factors on each performance index. The combined effect of the main interactive factors was analyzed in detail. The multi-objective optimized results indicated that effective heat storage time, effective heat storage, and exergy efficiency decreased by 30.24%, increased by 39.81%, and improved by 7.50%, respectively, compared with the basic working conditions before optimization. Keywords: packed bed latent heat storage system; thermal performance; response surface method; multi-objective optimization Nomenclature

Greek symbols

A cross-sectional area of storage, m2 cp specific heat, J⋅kg-1⋅K-1 D heat storage tank diameter, m Dc packing ball diameter, mm e response E expectation function Ex exergy Exchar total thermal storage exergy Exdest rate of exergy destruction h heat transfer coefficient, W⋅m-2⋅K-1 Hm latent heat of fusion, kJ⋅kg-1 hv volumetric coefficient of heat transfer, W⋅m-3⋅K-1 k heat conductivity, W⋅m-1⋅K-1 L effective filling height, m m fluid flow rate, L⋅min-1

β γ ε ρ σ φ

1

liquid rate, % regression coefficient void ratio density, kg/m3 statistical error exergy efficiency, %

Abbreviation ANOVA analysis of variance BBD Box–Behnken design CCD central composite design HTF heat transfer fluid HVAC heating, ventilation, air-conditioning, and cooling MS Mean squares PBLHTES packed bed latent heat thermal

Journal Pre-proof n Nu Pr Qeff Re t T T0 teff Tm u x y z Δx

number of variables Nusselt number Prandtl number of the fluid effective thermal storage, MJ Reynolds number time, min temperature, ℃ environment temperature, K effective thermal storage time, ℃ phase change temperature, K flow rate, m⋅s-1 axial coordinate, m system response index independent variable difference in axial distance, m

PCM RSM SS

storage system phase change material response surface method sum of squares

Subscripts f heat transfer fluid i index for cell number in inlet of heat transfer fluid ini initial state l liquid out outlet of heat transfer fluid p phase change material s solid

1 Introduction The development of renewable energy sources is rapidly increasing to alleviate the global energy and environmental crisis and improve the energy structure. However, owing to the intermittent nature and volatility of renewable energy, efficient energy storage technologies have become the key to large-scale and low-cost development of renewable energy [1]. Heat storage technologies have attracted considerable attention owing to their wide applications in solar power generation [2], industrial waste heat utilization [3], HVAC [4], and other energy fields. Among these technologies, a packed bed latent heat thermal storage system (PBLHTES) is considered as an ideal passive heat storage system because of its simple structure, low cost, and good heat transfer performance [5]. However, the design parameters and operating characteristics of the PBLHTES differ according to systems. Hence, it is extremely important to explore the influence of independent parameters and their interactions on the thermal performance and the parameter optimization of the PBLHTES [6]. The main factors that affect the thermal performance of the PBLHTES include operating, geometric, and thermophysical parameters [7]. Among them, operating parameters are mainly determined by the heat source and design conditions of a system. For example, a solar thermal power system [8] and a solar hot water system [9] have different operations and design conditions. Geometric parameters are quite flexible. However, they must meet the structural constraints of design capacity and materials. Thermophysical parameters depend on phase change material (PCM) and heat transfer fluid (HTF) material, and they are typically considered as functions of temperature under isobaric conditions. As the single design experiment of the PBLHTES in different application environments requires considerable time and cost, researchers generally design and optimize such systems using simulations and subsequent experimental verifications [5]. In recent years, research on the thermal performance of the PBLHTES has mainly adopted numerical simulations to analyze the influence of a single factor on a performance index. The calculation of the temperature distribution rules of packed bed systems with time 2

Journal Pre-proof is the basis of thermal performance analysis. A few scholars have established the one-dimensional nonequilibrium energy equation based on porous media for the simplified packed bed physical model, which can significantly improve computational efficiency. Based on this mathematical model, Felix et al. [10] analyzed the effects of HTF inlet temperature, mass flow rate, phase change temperature range, and capsule radii on the thermal performance of the PBLHTES using spherical capsules. The study concluded that the difference in the complete melting time is 31.6% between the cases when the PCM is melted over a temperature range and when it is melted at a fixed temperature. Nallusamy et al. [11] studied a combined sensible and latent heat storage unit in solar hot water system and analyzed the influence of HTF flow rate and HTF inlet temperature on the performance of the storage unit. Experiment results indicated that the charging time is decreased by 14% for the inlet fluid temperature of 66 ℃ when the flow rate is increased from 2 to 6 L/min. Karthikeyan et al. [12] studied the thermal performance parameters of the PBLHTES used in low temperature solar air collectors and analyzed the influence of the size of the PCM ball, HTF inlet temperature and HTF mass flow rate on charging time. It was found that the increase in temperature of the HTF at the inlet from 67 to 80 ℃ increases the temperature difference between the HTF and the PCM, and this reduces the charging time from 565 to 200 min. And it stated that increasing the difference between HTF inlet temperature and the phase change temperature of the PCM has a significant effect on charging time. Wu et al. [13] analyzed the dynamic thermal performance of a molten-salt packed bed thermal energy storage system and showed that increasing the PCT from 360 to 380 ℃ leads to an increase in the effective discharging efficiency from 40.03% to 97.37%; increasing the inlet velocity from 5.55 × 10-4 to 1.295 × 10-2 ms-1 leads to a significant decrease in the effective discharging efficiency from 97.84% to 73.18%; increasing the capsule diameters from 0.02 to 0.1 m the corresponding effective discharging efficiency from 98.33% to 73.78%. Peng et al. [14] studied the behavior of the PBLHTES and analyzed the effects of PCM capsule diameter, HTF inlet velocity, and storage tank height on the temperature profiles of packed bed. The results showed that with the decrease in the particle diameter from 45 mm to 5 mm, the nondimensional charge time decreases from 3.36 to 2.41; when the inlet velocity is increased by 27 times, the nondimensional charge time is reduced by 41%. Pakrouh et al. [15] conducted a numerical simulation study on the PBLHTES using an effective packed bed model, and the experiment results indicated that employing smaller capsules is more effective when the inlet temperature is lower in a way that reducing the diameter from 60 mm to 10 mm for the inlet temperature of 40 ℃ results in 11% improvement in the efficiency while the same reduction in the diameter brings about 21% enhancement when the inlet temperature is 30 ℃. Raul et al. [16] studied the heat storage and release characteristics of spherical capsules and the PBLHTES. The results showed that the maximum efficiency of the storage is found to be 75.69% for charging and discharging inlet HTF temperatures of 180 and 120 ℃ respectively and as the discharging inlet HTF temperature is elevated from 120 to 140 ℃, thermal efficiency of the storage system is declined from 75.69% to 49.88%. In terms of thermal performance optimization, Yang et al. [17] studied the PBLHTES using spherical capsules filled with three kinds of PCM. They proved that the energy efficiency of multiple-type packed bed is nearly 10% higher than that of single-type system 3

Journal Pre-proof and the exergy efficiency of multiple-type system is higher than that of single-type system. Cheng et al. [18] designed a PBLHTES based on composite PCMs for high temperature solar cooling application, optimized the thermophysical parameters of the PCM, and studied the influence of HTF inlet temperature and HTF flow rate on the thermal performance of the unit in the PBLHTES using a newly developed material. The results showed that as the inlet HTF temperature is elevated from 7 °C to 12 °C, the average charging rate, maximal charging capacity and exergy efficiency decline by 73.9%, 44.7% and 10.4%, respectively. The average charging rate increases by 25%, while the exergy efficiency drops by 38.7% as the inlet HTF flow rate increases from 50 L/h to 250 L/h. Li et al. [19,20] studied the thermal properties of a high-temperature PBLHTES using a new PCM. Their results showed that when the inlet temperature of the HTF increases from 425 ℃ to 465 ℃, the efficiency of the system increases from 77.4% to 86.1%. When the mass flow rate of the HTF increases from 180 kg/h to 260 kg/h, heat storage rate increases by 28.4%. On this basis, a new type of double-layer variable-diameter macro package form was proposed to improve the thermal performance of the device. Jan et al. [21] used the multi-objective optimization method to optimize the exergy efficiency and material cost of a packed bed heat storage system with air as the HTF by considering the height of the filling bed, the upper and lower radii, the thickness of the insulating layer, and the diameter of particles as design variables. It can be seen from these studies that the optimization of an industrial-scale storage allowed identifying a design with an exergy efficiency that was only 4.8% below that of the most efficient design, but a cost that was 81.3% lower than the cost of the most efficient design. Compared to brute-force design approaches, the optimization procedure can reduce the computational time by 91-99%. Optimization methods generally include mathematical optimization algorithms and parameter research. At present, the literature mainly focuses on parameter research, but such analyses have certain limitations. Among mathematical optimization methods, the response surface method (RSM) is a well-developed and understood mathematical modeling and statistical technique, which can model and analyze the problem that the response factor of a system is affected by numerous effective factors. This method has been widely used in experimental design, product development, and process optimization to determine the optimal parameters in the least number of iterations. Therefore, the method can be used to efficiently and accurately analyze the influence of single factor or multifactor interactions on each index of the thermal performance of the PBLHTES and to obtain the optimization scheme. The RSM is a very highly reliable and efficient method for analysis and optimization. The RSM is used to evaluate and optimize the thermal performance of the PBLHTES in this study. First, based on the porous media method, a one-dimensional nonequilibrium two-phase mathematical model is established, which is used to calculate the internal temperature distribution and thermal performance index of the PBLHTES. Second, nine parameters, such as void ratio, HTF inlet temperature, and PCM phase change temperature, are selected as the factors that influence thermal performance. Effective heat storage time, effective heat storage, and exergy efficiency are selected as responses and joint optimization indices. The Box–Behnken design (BBD) model is used to generate 130 groups of schemes, and the quadratic regression models of the three response indices are obtained, which proves the accuracy of the regression model and the effectiveness of the prediction. The degree of 4

Journal Pre-proof influence of each factor and its interaction on the response index of thermal performance is determined by analysis of variance (ANOVA). Further, the influence law of the main factors is analyzed. Finally, using the minimum effective heat storage time, maximum effective heat storage, and maximum exergy efficiency as the joint optimization objectives, the parameter combination is optimized using the expectation function method and verified. This study will be helpful for the design and further performance optimization of PBLHTES parameters.

2 Mathematical model 2.1 Model description The physical model of the PBLHTES is shown in Figure 1. The heat accumulator shell is a vertical cylinder with an external covering of a thermal insulation material and an internal filling of a spherical encapsulated PCM; current equalizers are arranged at the top and bottom. The charge inlet and discharge outlet are arranged on the upper part of the water tank, and the corresponding charge outlet and discharge inlet are arranged on the lower part. In the process of thermal charging, the hot HTF enters the water tank from the charge inlet and flows out evenly through the flow equalizer. The HTF flows from the top to the bottom through the gap around the packing ball, thus transferring the heat to the PCM and finally flowing out from the charge outlet. In the thermal discharging process, the cold HTF enters from the discharge inlet, absorbs the PCM heat from the bottom up, and finally flows out from the discharge outlet. This work mainly studies the thermal charging process of the physical model. Charge inlet

HTF in

Discarge outlet

x Dc

PCM Capsules

L

Charge outlet

Discarge inlet D HTF out

Fig. 1. Schematic diagram of the physical model and numerical calculation area.

The application environment of the PBLHTES is the solar heating system used in a cold area of China (Jilin Province, China). According to the operation characteristics of solar collectors, a solution of propylene glycol is selected as the HTF and myristate acid is used as the PCM [22]. The main geometric, operating, and thermophysical parameters are shown in Table 1. 5

Journal Pre-proof Table 1 Main parameters of PBLHTES Parameter categories

Geometric parameters

Operating parameters

Thermophysical parameters

[22]

Main parameters

Values

Effective filling height Heat storage tank diameter Packing ball diameter Void ratio HTF inlet temperature HTF flow rate PCM phase change temperature PCM phase change latent heat PCM specific heat capacity (l/s) PCM thermal conductivity (l/s) HTF specific heat capacity HTF thermal conductivity HTF viscosity

2300 mm 1200 mm 50 mm 0.4 333–363 K 20–40 L·min -1 325.8 K 186.6 kJ·kg -1 2480(l) /2200(s) J·kg -1 ·K -1 0.221 W·m -1 ·K -1 3530 J·kg -1 ·K -1 0.36 W·m -1 ·K -1 6.6 mm 2 ·s -1

2.2 Governing equation For the above physical model, the effective filling area is selected as the calculation area of the numerical model. Based on the porous media method, a one-dimensional two-temperature nonequilibrium packed bed energy equation is established to calculate the change rule of the temperature distribution of the HTF and PCM with time along the axial direction. The following simplifications and assumptions are considered to improve computational efficiency: (1) the heat loss of the wall facing the surrounding environment is neglected; (2) the thermophysical properties of the HTF and PCM are constant; (3) in the HTF and PCM, only axial heat conduction is considered and radial flow and heat conduction are neglected; (4) the thermal conduction of the PCM packaging shell is neglected; (5) the initial temperature and flow rate are uniform; (6) the effect of radiation heat transfer is neglected. Based on the above assumptions, the governing equation of the heat storage process is established as follows: HTF:

ερs c p, f A x

T f t

 mc p, f

T f x

 x  εk f A x

 2T f x 2

 hv A x(T f  Tp )

(1)

According to the phase change law, the PCM passes through three stages: sensible heat storage in the solid state (I), latent heat storage (II), and sensible heat storage in the liquid state (III). PCM Ⅰ:

(1  ε ) ρs c p,s A x

Tp.s t

 (1  ε )ks A x

 2Tp.s x 2

 hv A x(T f  Tp,s ) (2)

PCM Ⅱ:

(1  ε ) ρs A x

βH m  hv A x(T f  Tm ) t 6

(3)

Journal Pre-proof



0   T T   s sol Tliq  Tsol  1

if Ts  Tsol if Tsol  Ts  Tliq if Ts  Tliq

PCM Ⅲ:

(1  ε ) ρl c p,l A x

Tp.l t

 (1  ε )kl A x

 2Tp.l x 2

 hv A x(T f  Tp,l )

(4)

The relationship between the volumetric heat transfer coefficient, hv, and the surface heat transfer coefficient, h, between the HTF and PCM is as follows [23]:

hv  6h(1  w ) Dc

(5)

The empirical correlation proposed by Wakao and Funazkri [24] is used to calculate the heat transfer coefficient on the surface of the packing ball.

Nu  2  1.1Pr1 3Re0.6

(6)

k f Nu

h

DC

(7)

The finite difference method is utilized to solve the numerical discretization of this equation using MATLAB. The above equations are discretized using the finite difference method with the second-order accurate central finite difference operator in space and the forward difference approximation in time. The initial conditions and boundary conditions are as follows: Initial conditions:

t  0, Tp  Tini , T f  Tini

(8)

Boundary conditions:

x  0, T f  Tin , Tp  Tin ; x  L,

T f x

 0,

Tp x

 0

(9)

2.3 Objective function In this work, effective heat storage time, effective heat storage, and exergy efficiency are selected as the optimization indices of the thermal performance of the PBLHTES. Among these, effective heat storage time and effective heat storage represent the heat storage efficiency and heat storage ability of the PBLHTES, respectively. The exergy efficiency based on the second law of thermodynamics represents the irreversible loss of the heat storage process. Dimensionless time:

τ  7

tu f L

(10)

Journal Pre-proof Effective heat storage time:

teff  tT

(11)

out =Tin

Effective heat storage: t

Qeff (t)   mc f (T f,in  T f,out )dt

(12)

0

Exergy efficiency:

φex  1 

Exdest Exchar  Exin  Exout Exin  Exout

(13)

Here, exergy loss is

Exdest  Exin  Exout  Exchar

(14)

Inlet and outlet exergies are t

Exin   mc f ρ f (Tin T0  T0 ln 0

Tin )dt T0

t

Exout   mc f ρ f (Tout T0  T0 ln 0

(15)

Tout )dt T0

(16)

The total heat storage exergy is the sum of the exergies stored in the PCM and HTF:

Exchar  ExHTF  ExPCM

(17)

2.4 Grid validation 6960

Tini = 32 ℃ , Tin = 70 ℃ ε = 0.5, H = 460 mm, D = 360 mm dc = 55 mm

Charging time/s

6940

6920

6900

6880

6860 0

1000

2000

3000

4000

5000

grid number

Fig. 2. Grid validation.

Grid independence is analyzed for the case of Nallusamy’s study [11]. A uniform grid is employed along the flow direction for the discretization of the computational domain. Grid sizes of (Nx)46 × (Rx)10, (Nx)80 × (Rx)10, (Nx)100 × (Rx)10, (Nx)200 × (Rx)10, (Nx)230 × (Rx)10, (Nx)400 × (Rx)10, and (Nx)460 × (Rx)10 are applied to examine the grid 8

Journal Pre-proof independence of the solution. The results are shown in Figure 2, which demonstrates that a grid size of (Nx)200 × (Rx)10 provides an accurate solution. 2.5 Model validation The differences between the numerical simulation and experimental results are compared to verify the validity of the numerical model. The experimental results of Nallusamy et al. [11] are used to verify the numerical calculation results. In the experimental system, the volume of the packed bed heat storage tank is 47 L, the PCM is paraffin, the HTF is water, and the diameter of the packing ball is 55 mm. The PCM temperatures in the heat storage process at x/L = 0.25 and 1.00 obtained via numerical simulation and experiments are compared, as shown in Figure 3. Owing to temperature stratification, the phase change occurs at different times and heights. Overall, the numerical simulation results are consistent with the experimental data. The maximum deviation at x/L = 0.25 is 9.16%, and the average deviation is 5.61%. The maximum deviation at x/L = 1.00 is 8.78%, and the average deviation is 5.12%. This validates the proposed model. The main causes of the deviation include the simplifications and assumptions considered in the hypothesis of the mathematical model, in addition to the uncertainty of the experiment.

345 340

Temperature /K

335 330 325 320

experimental data x/L = 1.00 numerical data x/L = 1.00 experimental data x/L = 0.25 numerical data x/L = 0.25

315 310 305 0

1

2

3

4

5

6

Nondimensional time , 

Fig. 3. Comparison between numerical calculation and experimental results.

3 Design and optimization of the response surface method 3.1 Design of the response surface method The main purpose of this study is to evaluate the influence of nine independent parameters on three thermal performance indices and to optimize the thermal performance of the PBLHTES. RSM design is used to obtain the interaction between important influencing factors and response indicators and to realize multi-objective optimization. A multiple-order or higher-order regression equation can be obtained by fitting the function relationship between an independent variable and a response index, and a multivariable problem can be 9

Journal Pre-proof analyzed and optimized using regression equations [25]. This is typically achieved using a second-order polynomial model, as shown below:

y  γ0 

n

γ z a =1

a a



n

n



a =1 b = a +1

γab za zb 

n

γ a =1

z σ

2 aa a

(18)

where y is the system response index, z a and z b are independent variables, n is the number of variables, γ is the regression coefficient of truncation, linearity, quadratic term, and their interaction, and σ is the statistical error. The common design types of the RSM include central composite design (CCD) and BBD. The experimental design principle of BBD is different from that of CCD, and the number of test combinations in BBD is less than that in CCD when the number of factors is the same. In this work, geometrical parameters (packing ball diameter, void ratio), operating parameters (HTF inlet temperature, HTF flow rate), and thermophysical parameters (HTF specific heat capacity, HTF thermal conductivity, PCM phase change temperature, PCM phase change latent heat, PCM thermal conductivity) are selected as the design factors. The response indices are effective heat storage time, heat storage efficiency, effective heat storage based on the first law of thermodynamics, and exergy efficiency based on the second law of thermodynamics. The BBD method is used to generate 130 groups of design schemes, and the corresponding variable levels and ranges are shown in Table 2. Table 2 Factors and responses of BBD model Types Parameters Items Units Low High Packing ball diameter A mm 40 100 Void ratio B 0.35 0.5 HTF inlet temperature C K 333 363 -1 HTF flow rate D L·min 20 40 HTF specific heat E J·kg -1 ·K -1 3000 4000 capacity Factors HTF thermal conductivity F W·m -1 ·K -1 0.1 3 PCM phase change G K 308 328 temperature PCM phase change latent H kJ·kg -1 150 200 heat PCM thermal conductivity I W·m -1 ·K -1 0.1 2 Effective heat storage t min time Response Effective heat storage Q kW Exergy efficiency η 3.2 Multi-objective optimization Desirability expectation functions are used in the RSM to realize multi-objective optimization. The objective function, E (X), which is referred to as the expectation function, reflects the range required for each response (ei). The expected range is from 0 to 1, and the closer that a response is to 1, the more ideal it is. The objective function is the geometric 10

Journal Pre-proof mean of all transform responses. 1r i

E  ( e1r1  e2r2  e3r3  enrn )

n

 (  eiri ) i 1

1r i

(19)

where n is the number of responses. If any response or factor exceeds its expected range, the entire function becomes zero. The objective functions in this work include effective heat storage time, effective heat storage, and exergy efficiency. The expectation of effective heat storage time is the minimum, and the expectations of effective heat storage and exergy efficiency are the maximum. For synchronous optimization, each response must have low and high values to achieve the goal.

4 Results and discussion 4.1 Temperature distribution Figure 4 shows the temperatures of the HTF and PCM calculated at different axial positions for the main parameters of the PBLHTES provided in Table 1. The change in the temperatures of the HTF and PCM is recorded at four different positions (x/L = 0.25, 0.5, 0.75, 1) along the axial height in the heat storage device. The variation trends of the eight curves in the figure are similar. In the process of heat storage, heat is transferred from the HTF to the PCM, and the temperature of the HTF is higher than that of the PCM at each position. When the value of x/L is small, the heating rate of the material is high and the phase change time is short. On the contrary, when the value of x/L is high, the heating rate of the material is low and the phase change time is long. This is mainly because the temperature difference between the fluid and material is large. The heat transfer rate is high when it is close to the inlet section; hence, the heating rate is high. As the distance to the inlet section increases, the temperature difference between the fluid and material decreases. Hence, the heating rate and heat transfer rate decrease. Each group of curves undergoes three stages. The first stage is the solid-state heating stage, in which the heating rate gradually increases to a stable value and then gradually decreases at the end of the stage to smoothly transition to the next stage. In this process, the PCM is heated by solid-state heat conduction and no phase change occurs; this is evidently the heat storage stage. The second phase involves phase change, in which the PCM changes from solid to liquid and temperature remains almost constant. In this process, the PCM absorbs a large amount of heat and stores it in the form of latent heat; this is the stage of latent heat storage. The third stage comprises liquid heating, in which the entire PCM has been melted into the liquid state, and this is the sensible heat storage stage. The heating rate in the third stage is similar to that in the first stage but higher than that in the second stage.

11

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335

x/L = 0.25 x/L = 0.50 x/L= 0.75 x/L = 1.00

Temperature /K

330 325

325

320

Dc = 40 mm m = 20 L/min Tini = 303 K Tinlet = 333 K

320 315

315

310 305

310

0.0

0.5

1.0

1.5

PCM temperature

305 0

1

2

3

HTF temperature 4

5

6

7

Nondimensional time , 

Fig. 4. Temperature distribution of PBLHTES.

4.2 ANOVA and model diagnosis The “sum of squares (SS)” of each item in the model is used to evaluate the volatility of experimental data. “Mean squares (MS)” is the estimate of variance terms, which is obtained by dividing the sum of squares of each part by the respective degrees of freedom. The F value is used to compare the mean square of each term with the mean square of the residual error, and it is used to test the accuracy of the model. If the Prob>F value is less than 0.05, then the parameters have a significant impact on the response. “R-squared” is used to check the goodness of fit. “Adjusted R-squared” is a measure of changes around the mean value of the model interpretation, and it is adjusted according to the number of items in the model. “Pred R-squared” is the model’s measure of change in new data. “Adeq Precision” is the signal-to-noise ratio. The comparison of the range of the predicted values of the design point with the average prediction error shows that the model is available when the prediction error is larger than 4. The quadratic regression models of effective heat storage time, effective heat storage, and exergy efficiency are established based on the numerical results of BBD, as shown below.

12

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The ANOVA results of the effective heat storage time, effective heat storage, and exergy efficiency models are shown in Table 3. It can be seen from the F value and Prob>F value that all three models are effective, among which effective heat storage shows the highest accuracy and effective heat storage time shows the lowest accuracy. Based on the values of “R-Squared” and “Adjusted R-squared”, it can be seen that each of the models has a high fit quality. Furthermore, “Adeq Precision” is larger than 4, which indicates that the models can be used for performance prediction and design optimization. Table 3 ANOVA results F value Prob>F R-Squared Adjusted R-squared Adeq Precision Effective heat 62.16 <0.0001 0.9781 0.9624 33.942 storage time Effective heat 342.90 <0.0001 0.9960 0.9931 85.250 storage Exergy efficiency 154.08 <0.0001 0.9911 0.9846 73.185 When the P value is less than 0.05, parameters A, B, C, D, E, F, G, H, AF, CD, CG, DF, DG, C2, D2, F2, and G2 have important effects on effective heat storage time, parameters B, C, D, E, F, G, H, BC, BE, BH, DE, DF, EF, and B2 have important effects on effective heat 13

Journal Pre-proof storage, and parameters A, B, C, E, F, G, H, AF, BG, CF, CG, DE, EF, GH, C2, F2, and G2 have an important influence on exergy efficiency. According to the above results, not only single factors but certain interactive factors, such as AF, CD, and CG, have an important influence on effective heat storage time, effective heat storage, and exergy efficiency. Therefore, it is necessary to further analyze the influence of the interactive factors on the thermal performance of the PBLHTES. The quadratic regression model is used to diagnose and analyze the relationship between the estimated response value and actual response value based on a plot of predicted values vs. actual values. As shown in Figure 5, most design points are closely distributed along the diagonal, which indicates that the relationship is linear and that the regression model can be used to predict the thermal performance index of the PBLHTES.

14

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Fig. 5. RSM prediction model diagnosis. 15

Journal Pre-proof 4.3 Effect of parameters on performance 4.3.1 Effect on effective heat storage time According to ANOVA, the factors that have significant influence on effective heat storage time include A, B, C, D, E, F, G, H, AF, CD, CG, DF, DG, C2, D2, F2, and G2. According to the Prob>F value, B (void ratio) is the least influential factor among the seven important single factors, and the other items are of the same importance. Among the five important interaction factors, AF (diameter of packing ball and thermal conductivity of HTF) has the strongest influence. The AF item is analyzed in detail using a contour plot and a 3D surface plot. Figure 6 shows the effect of AF on effective heat storage time under the condition that the other factors remain unchanged. There is a quadratic parabolic relationship between the thermal conductivity of the HTF and effective heat storage time. Effective heat storage time increases with the diameter of the encapsulated ball, which indicates an interactive effect between these two parameters. When the diameter of the encapsulated ball is small, the heat transfer time from the ball wall to the ball center is short because of the low amount of the PCM encapsulated in the ball. On the contrary, the heat transfer area of the HTF and PCM is increased, and convective heat transfer is enhanced. Therefore, the gradient of effective heat storage time increases with the thermal conductivity coefficient. When the diameter of the packing ball is larger, on one hand, the heat transfer area of the HTF and PCM decreases and convective heat transfer decreases. On the other hand, effective heat storage time is affected by the thermal conductivity of the PCM and the heat transfer process of phase change. Hence, effective heat storage time decreases with change in thermal conductivity because of the large amount of the PCM encapsulated in the ball. In practical application, the thermal conductivity of the HTF can be improved by adding nanoparticles into the HTF. However, it is easier to adjust the diameter of the encapsulated ball compared with regulating HTF thermal conductivity.

Fig. 6. Effect of packing ball diameter and HTF thermal conductivity on effective heat storage time.

4.3.2 Effect on effective heat storage According to ANOVA, the factors that have an important influence on effective heat storage include B, C, D, E, F, G, H, BC, BE, BH, DE, DF, EF, and B2. According to the Prob>F value, among the seven important single factors, B, C, E, G, and H have the strongest 16

Journal Pre-proof influence and D (HTF flow) has the weakest influence. Among the five important interactive influencing factors, BC (void ratio and HTF inlet temperature) has the strongest influence. BC is analyzed in detail using a contour plot and a 3D surface plot. As shown in Figure 7, as HTF inlet temperature increases and the void ratio decreases, the interaction is enhanced and effective heat storage increases. The increase in HTF inlet temperature can increase the heat transfer temperature difference and improve the sensible heat storage capacity of the HTF and PCM. After the PCM packing balls are piled up in the heat storage tank, the void portion is filled with the HTF. The void ratio reflects the proportion of the HTF and PCM in the internal space of the heat storage tank. When the void ratio decreases, the proportion of the PCM increases because the PCM has latent heat storage ability and the sensible heat storage ability of the PCM is stronger than that of the HTF. Therefore, effective heat storage increases when the void ratio decreases, while the interaction is enhanced as the inlet temperature of the HTF increases. For the design points in the diagram, the void ratio is 0.42, HTF inlet temperature is 348 K, and effective heat storage is 579.015 MJ.

Fig. 7. Interactive effects of void ratio and HTF inlet temperature on effective heat storage.

4.3.3 Effect on exergy efficiency According to the results of ANOVA, the factors that have an important effect on exergy efficiency include A, B, C, E, F, G, H, AF, BG, CF, CG, DE, EF, GH, C2, F2, and G2. According to the Prob>F value, the seven single factors are of equal importance. Among these seven factors, AF (diameter of packing sphere and HTF thermal conductivity) and CG (HTF inlet temperature and PCM phase change temperature) have the strongest influence. The F values of AF and CG are 35.01 and 36.46, respectively. Therefore, the CG item is analyzed in detail using a contour plot and 3D surface plot. As shown in Figure 8, there is a quadratic parabolic relationship between HTF inlet temperature and exergy efficiency. When PCM phase change temperature is high, the gradient of the change in exergy efficiency decreases gradually. When PCM phase change temperature is low, the gradient of the change in exergy efficiency first decreases and then increases. It can be seen from Equation (13) that exergy efficiency is not only related to exergy loss but also to the heat exergy at the inlet and outlet. Thermodynamic analysis shows that the irreversible loss caused by the heat transfer temperature difference is the main cause of exergy loss. At a given ambient temperature, the temperature difference between the system and 17

Journal Pre-proof environment is a key factor that affects heat exergy. It can be seen from Figure 8(a) that when the Tm value is low, exergy efficiency is low because the overall temperature difference between Tm and Tin in the variation range is large. When Tin is 333K, the system temperature difference is the smallest and heat exergy is low. When Tin is gradually increased to 351K, exergy efficiency is mainly reduced owing to the gradual increase in the heat transfer temperature difference between the HTF and PCM, and this temperature difference plays a major role. From 351K to 363K, the heat transfer temperature difference continues to increase, but system heat exergy also increases, which causes exergy efficiency to increase slightly at a lower level. When the inlet temperature of the HTF is low, the gradient of exergy efficiency changes significantly as PCM phase change temperature decreases. When HTF inlet temperature is high, the gradient of exergy efficiency is small. This is because when Tin is large, Tin is considerably different from Tm in the range of variation, and this temperature difference plays a major role in the effect of exergy efficiency. The gradient of exergy efficiency decreases with Tin. PCM phase change temperature is approximately constant in the process of heat storage. Owing to the interaction between the increase in PCM phase change temperature and the decrease in HTF inlet temperature, the temperature difference of heat transfer decreases, the irreversible loss of the heat transfer process decreases, and the exergy efficiency of heat transfer increases.

Fig. 8. Effect of HTF inlet temperature and PCM phase transition temperature on exergy efficiency.

4.4 Multi-objective optimization According to the above analysis, numerous parameters affect the thermal performance of the PBLHTES. In addition, the interactions of the parameters are complex, which may have opposing effects on different evaluation indices. Therefore, to improve the thermal performance of the PBLHTES, it is necessary to find the optimal parameter combination by integrating multiple evaluation indices. In this study, the desirability method in the RSM is adopted to optimize multi-objective thermal performance. The expected value ranges from 0 to 1, and the closer a value is to 1, the closer is a predicted response to the optimal result. All factors in the method are within the scope of the design. Effective heat storage time is set to the minimum value, and effective heat storage and exergy efficiency are set to the maximum values. Table 4 shows the objectives, upper limits, lower limits, weights, and the importance of each factor and response. 18

Journal Pre-proof Table 4 Multi-objective optimization model Name

Goal

Lower Limit

Upper WeightWeightImportance Limit

A: Packing ball diameter B: Void ratio C: HTF inlet temperature D:HTF flow E: HTF specific heat capacity F: HTF thermal conductivity G: PCM phase transition temperature H: PCM phase change latent heat J: PCM thermal conductivity coefficient Effective heat storage time Effective heat storage Exergy efficiency

is in range is in range is in range is in range is in range is in range is in range is in range

40 0.35 333 20 3000 0.1 308 150

100 0.5 363 40 4000 3 328 200

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

3 3 3 3 3 3 3 3

is in range

0.1

2

1

1

3

0.5 1 1

3 3 3

minimize 108.13 478.37 1 maximize 439.021 721.798 0.5 maximize 0.6446 0.9412 0.5

A large number of optimization results of thermal performance are obtained based on the above constraints. The optimum parameters are as follows: packing ball diameter is 56 mm, the void ratio is 0.49, HTF inlet temperature is 363 K, HTF flow rate is 33.11 L/min, HTF specific heat capacity is 3999.75 kJ/℃, HTF thermal conductivity is 2.38 W/m‧K, PCM phase change temperature is 327.90 K, PCM phase change latent heat is 180.26 kJ/kg, PCM thermal conductivity is 1.49, predicted optimal effective heat storage time is 108.12 min, effective heat storage is 721.80 MJ, exergy efficiency is 0.86, and the expected value is 0.95. The background of this work is that the sensible heat storage tank is transformed into the PBLHTES in solar heating systems. Therefore, the external size of the tank is maintained constant. In the optimized geometric parameters, the diameter and void ratio of the packing ball increase by 12% and 22.5%, respectively, compared with the preset value. This provides a design basis for the selection and filling combination of PCM packing ball shell size. Among the optimized operating parameters, HTF inlet temperature is the maximum within the range of change and HTF flow rate is in the middle value within the range of change. Therefore, in the process of charging, inlet temperature should be increased as much as possible and flow rate should be adjusted to fluctuate close to the target value to determine rated operating conditions. Among the optimized thermophysical property parameters, the specific heat capacity and thermal conductivity of the HTF are 13.3% and 5.6 times higher than the preset values, respectively. PCM phase change temperature is almost unchanged, thermal conductivity increases by 5.7 times, and the latent heat of phase change decreases by 3.4%. The optimization results of the thermophysical properties of the materials are consistent with the design rules of the new fluid and composite PCM, which can be realized by adding high thermal conductivity nanometal materials or carbon materials in the HTF and PCM [26,27]. 4.5 Verification of optimal parameters 19

Journal Pre-proof The optimal parameter combination and optimal index combination of prediction are obtained by utilizing the expectation method. The optimal parameter combination is introduced into the mathematical model of the system to validate the optimization results. The calculation results show that effective heat storage time is 118.82 min, effective heat storage is 713.40 MJ, and exergy efficiency is 0.87 for the optimal parameter combination. After joint optimization, effective heat storage time, effective heat storage, and exergy efficiency decrease by 30.24%, increase by 39.81%, and improve by 7.50%, respectively, compared with the basic working conditions before optimization. The errors between the predicted and actual results are 9.0%, 1.2%, and 1.0% for effective heat storage time, effective heat storage, and exergy efficiency, respectively. Among them, the error in effective heat storage time is relatively high; this is consistent with the accuracy analysis of the previous model. Further optimization of the quadratic regression model is required to reduce the error between predicted and actual results.

5 Conclusions A one-dimensional simplified numerical model was established to calculate the internal temperature distribution and thermal performance index of the PBLHTES. The RSM was used to analyze the influence of the geometric, operational, and thermophysical parameters of the PBLHTES on evaluation indexes and their statistical significance. Different from previous studies, the minimum effective heat storage time, maximum effective heat storage, and maximum exergy efficiency were considered as the joint optimization objectives of multi-objective optimization. The main conclusions of this study are summarized as follows: (1) The internal thermal distribution rule of the PBLHTES is obtained. The variation trend of the PBLHTES along the axial displacement curve is consistent. As the x/L value increases, the heating rate of the solid-state sensible heat storage stage decreases, latent heat storage time increases, and the differences in the liquid-state sensible heat storage stage are minimal. (2) The quadratic regression models for effective heat storage time, effective heat storage, and exergy efficiency are obtained. ANOVA results verify the accuracy of the models and obtain the significant influencing parameters of each index. The linear relationship between predicted and actual values proves the validity of predicting the thermal performance index of the PBLHTES. (3) The interactive influence of important factors on each performance index are analyzed in detail. According to the results of ANOVA, AF, BC, and CG are determined as the main interactive factors of effective heat storage time, effective heat storage, and exergy efficiency, respectively. The interactive influence of each item is analyzed using contour plots and 3D surface plots. (4) The optimal parameter combination with the goal of the minimum effective heat storage time, maximum effective heat storage, and maximum exergy efficiency is obtained. The optimization results can be used to guide HTF and PCM design, PCM packaging and filling design, and operation parameter adjustment. After joint optimization, effective heat storage time, effective heat storage, and exergy efficiency decrease by 30.24%, increase by 39.81%, and improve by 7.50%, respectively, compared with the basic working conditions before optimization. 20

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Authors’ contributions Long Gao: Conceptualization, Methodology, Writing - Review & Editing, Supervision Gegentana: Software, Validation, Formal analysis, Investigation, Data Curation, Writing Original Draft Zhoneze Liu: Software, Validation, Formal analysis, Resources, Data Curation, Writing Review & Editing Baizhong Sun: Supervision, Project administration Deyong Che: Project administration, Funding acquisition Shaohua Li: Project administration

Acknowledgements Funding: This work was supported by National Key Research and Development Program of China (grant No. 2018YFB0905104) and Science and Technology Planning Project of Jilin Province (grant No. 20180201006SF).

References [1] Dominkovic DF, Bacekovic I, Pedersen AS, Krajacic G. The future of transportation in sustainable energy systems: Opportunities and barriers in a clean energy transition. Renewable & Sustainable Energy Reviews 2018; 82:1823–1838. https://doi.org/10.1016/j.rser.2017.06.117. [2] Kabir E, Kumar P, Kumar S, Adelodun AA, Kim KH. Solar energy: Potential and future prospects. Renewable & Sustainable Energy Reviews 2018; 82:894–900. https://doi.org/10.1016/j.rser.2017.09.094. [3] Olivier D, Remi D, Mattia DR, Roy D, Vincent L. Technical and economic optimization of subcritical, wet expansion and transcritical Organic Rankine Cycle (ORC) systems coupled with a biogas power plant. Energy Conversion and Management 2018; 157:294–306. https://doi.org/10.1016/j.enconman.2017.12.022. [4] Atinafu DG, Dong WJ, Huang XB, Gao HY, Wang G. Introduction of organic-organic eutectic PCM in mesoporous N-doped carbons for enhanced thermal conductivity and energy storage capacity. Applied Energy 2018; 211:1203–1215. https://doi.org/10.1016/j.apenergy.2017.12.025. [5] de Gracia A, Cabeza LF. Numerical simulation of a PCM packed bed system:A review. Renewable and Sustainable Energy Reviews 2017; 69:1055-1063. https://doi.org/10.1016/j.rser.2016.09.092. [6] Elfeky KE, Ahmed N, Wang QW. Numerical comparison between single PCM and multi-stage PCM based high temperature thermal energy storage for CSP tower plants. Applied Thermal Engineering 2018; 139:609-622. https://doi.org/10.1016/j.applthermaleng.2018.04.122. [7] Li MJ, Qiu Y, Li MJ. Cyclic thermal performance analysis of a traditional Single-Layered 21

Journal Pre-proof and of a novel Multi-Layered Packed-Bed molten salt Thermocline Tank. Renewable Energy 2018; 118:565-578. https://doi.org/10.1016/j.renene.2017.11.038. [8] Moncef B, Abdessalem BHA, Seif ET, Amenallah G. Optical and thermal evaluations of a medium temperature parabolic trough solar collector used in a cooling installation. Energy Conversion and Management 2014; 86:1134–1146. https://doi.org/10.1016/j.enconman.2014.06.095. [9] Huang P, Vanesa CB, Liu Y, Ma HZ. The governance of urban energy transitions: A comparative study of solar water heating systems in two Chinese cities. Journal of Cleaner Production 2018; 180:222–231. https://doi.org/10.1016/j.jclepro.2018.01.053. [10] Felix Regin A, Solanki SC, Saini JS. An analysis of a packed bed latent heat thermal energy storage system using PCM capsules: Numerical investigation. Renewable Energy 2009; 34(7):1765–1773. https://doi.org/10.1016/j.renene.2008.12.012. [11] Nallusamy N, Sampath S, Velraj R. Experimental investigation on a combined sensible and latent heat storage system integrated with constant/varying(solar) heat sources. Renewable Energy 2007; 32(7):1206–1227. https://doi.org/10.1016/j.renene.2006.04.015. [12] Karthikeyan S, Ravikumar SG, Kumaresan V, Velraj R. Parametric studies on packed bed storage unit filled with PCM encapsulated spherical containers for low temperature solar air heating applications. Energy Conversion and Management 2014; 78:74–80. https://doi.org/10.1016/j.enconman.2013.10.042. [13] Wu M，Xu C，He YL. Dynamic thermal performance analysis of a molten-salt packed-bed thermal energy storage system using PCM capsules. Applied Energy 2014; 121(5):184–195. https://doi.org/10.1016/j.apenergy.2014.01.085. [14] Peng H, Dong HH, Ling X. Thermal investigation of PCM-based high temperature thermal energy storage in packed bed. Energy Conversion and Management 2014; 81:420–427. https://doi.org/10.1016/j.enconman.2014.02.052 [15] Pakrouh R, Hosseini MJ, Ranjbar AA, Bahrampoury R. Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model. Energy 2017; 140:861–878. https://doi.org/10.1016/j.energy.2017.08.055. [16] Raul A, Jain M, Gaikwad S, Saha SK. Modelling and experimental study of latent heat thermal energy storage with encapsulated PCMs for solar thermal applications. Applied Thermal Engineering 2018; 143:415–428. https://doi.org/10.1016/j.applthermaleng.2018.07.123. [17] Yang L, Zhang XS, Xu GY. Thermal performance of a solar storage packed bed using spherical capsules filled with PCM having different melting points. Energy and Buildings 2014; 68:639–646. https://doi.org/10.1016/j.enbuild.2013.09.045. [18] Cheng XW, Zhai XQ, Wang RZ. Thermal performance analysis of a packed bed cold storage unit using composite PCM capsules for high temperature solar cooling application. Applied Thermal Engineering 2016; 100:247–255. https://doi.org/10.1016/j.applthermaleng.2016.02.036. [19] Li MJ, Jin B, Ma Z, Yuan F. Experimental and numerical study on the performance of a new high-temperature packed-bed thermal energy storage system with macroencapsulation of molten salt phase change material. Applied Energy 2018; 221:1–15. https://doi.org/10.1016/j.apenergy.2018.03.156. [20] Li MJ, Jin B, Yan JJ, Ma Z, Li MJ. Numerical and Experimental study on the performance of 22

Journal Pre-proof a new two-layered high-temperature packed-bed thermal energy storage system with changed-diameter macro-encapsulation capsule. Applied Thermal Engineering 2018; 142:830–845. https://doi.org/10.1016/j.applthermaleng.2018.07.026. [21] Marti J, Geissbunhler L, Becattini V, Haselbacher A, Steinfeld A. Constrained multi-objective optimization of thermocline packed-bed thermal-energy storage. Applied Energy 2018; 216:694–708. https://doi.org/10.1016/j.apenergy.2017.12.072. [22] Yuan Y, Zhang N, Tao W, et al. Fatty acids as phase change materials: A review. Renewable and Sustainable Energy Reviews 2014; 29:482-498. [23] Pitam C, Willits DH. Pressure drop and heat transfer characteristics of air-rockbed thermal storage systems. Solar Energy 1981; 27(6):547–553. https://doi.org/10.1016/0038-092X(81)90050-5. [24] Wakao N, Funazkri T. Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed beds: Correlation of nusselt numbers. Chemical Engineering Science 1978; 33:1375–1384. https://doi.org/10.1016/0009-2509(78)85120-3. [25] Tag AT, Duman G, Yanik J. Influences of feedstock type and process variables on hydrochar properties. Bioresource Technology 2018; 250:337–344. https://doi.org/10.1016/j.biortech.2017.11.058. [26] Lachachene F, Haddad Z, Oztop HF, Abu-Nada E. Melting of phase change materials in a trapezoidal cavity: Orientation and nanoparticles effects. Journal of Molecular Liquids 2019; 292:110592. https://doi.org/10.1016/j.molliq.2019.03.051. [27] Gao L, Sun XG, Sun BZ, et al. Preparation and thermal properties of palmitic acid/expanded graphite/carbon fiber composite phase change materials for thermal energy storage. Journal of Thermal Analysis and Calorimetry 2019. https://doi.org/10.1007/s10973-019-08755-y.

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Journal Pre-proof Authors’ contributions Long Gao: Conceptualization, Methodology, Writing - Review & Editing, Supervision Gegentana: Software, Validation, Formal analysis, Investigation, Data Curation, Writing Original Draft Zhoneze Liu: Software, Validation, Formal analysis, Resources, Data Curation, Writing Review & Editing Baizhong Sun: Supervision, Project administration Deyong Che: Project administration, Funding acquisition Shaohua Li: Project administration

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Highlights:     

Developed a simplified numerical model for investigating the thermal performance of PBLHTES Designed the thermal performance optimization scheme of PBLHTES by RSM. Established the quadratic regression model of the performance indices. Analyzed influences of the single factors and the interactive factors on evaluation indices. Obtained the optimal parameter combination of multi-objective optimization.