Solidification behaviors and parametric optimization of finned shell-tube ice storage units

International Journal of Heat and Mass Transfer 146 (2020) 118836

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Solidification behaviors and parametric optimization of finned shell-tube ice storage units Chengbin Zhang a, Qing Sun b, Yongping Chen a,b,c,⇑ a

Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China College of Electrical, Energy and Power Engineering, Yangzhou University, Yangzhou 225127, PR China c Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Environmental Science and Engineering, Suzhou University of Science and Technology, Suzhou 215009, PR China b

a r t i c l e

i n f o

Article history: Received 3 July 2019 Received in revised form 30 August 2019 Accepted 4 October 2019

Keywords: Solidification Natural convection Ice storage Parametric optimization

a b s t r a c t The present paper reports on a numerical study of the ice storage process in finned shell-tube ice storage (STIS) units, with a focus on the special solidification behavior using water as the phase-change material (PCM). The proposed model is experimentally verified using an energy-discharging process in an ice storage unit. The effects of natural convection and buoyancy reversal on the solidification behavior are examined and investigated. Moreover, the Taguchi method is utilized to optimize the fin geometry of STIS units. The results indicate that the natural convection and buoyancy reversal are negatively correlated with the ice storage performance. An increase of the superheat factor leads to the enhancement of the buoyancy reversal intensity, which is not conducive to the acceleration of the solidification rate. In addition, the increases in fin height, fin width, and fin number are positively correlated with ice storage performance. It is demonstrated that the fin height is the dominant factor affecting the overall ice storage performance, and it is independent of the superheat factor. From the perspective of trade-off between ice storage rate and ice storage capacity, the optimal fin parameters for the STIS unit are fin height H = 40 mm, fin number N = 10, fin width D = 3 mm for engineering applications. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction In the utilization of renewable and fossil energy, significant efforts have been made to develop energy-storage technologies, owing to the discrepancy between energy demand and supply [1–3]. Latent heat storage technology based on phase-change materials (PCMs) is the most attractive, because of its inherent advantages including high energy-storage capacity, isothermal energy-storage process, and small volume fluctuation. As a typical latent heat storage technology, shell-tube ice storage (STIS) units have been widely utilized in technical applications such as food preservation [4], peak load shifting [5], building energy conservation [6], and air conditioning [7]. The ice storage in a finned STIS unit is a complex heat-transfer process accompanied by solidification phase change and natural convection (NC). Moreover, NC and water density reversal (although often not properly understood) could significantly affect the overall performance of a finned STIS

⇑ Corresponding author at: Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China. E-mail address: [email protected] (Y. Chen). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118836 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

unit, using water as the PCM. Therefore, it is important to investigate solidification behavior in STIS units for practical engineering applications. Like most PCMs, an important challenge during the ice storage process is determining how to accelerate the solidification rate in shell-tube ice storage units. To offset the efficiency losses arising from the inherent drawback of poor thermal conductivity, comprehensive numerical [8,9] and experimental [10,11] studies have been performed over the past few decades to investigate the icing and melting processes. Several thermal enhancement methods, including the use of extended surfaces [12,13], employing multiple PCMs [14,15], thermal conductivity enhancement [3,16], and the microencapsulation of PCMs [17,18] have been proposed to meet the above challenge. Ismail et al. [9] carried out an experimental and numerical study on the solidification performance in an ice storage unit. They reported that the increase in diameter of spherical shell led to the increase of complete solidification time, while encapsulating materials of high thermal conductivity reduced the complete solidification time. The solidification of stearic acid in a vertical energy storage system was investigated experimentally by Liu et al. [13]. The results indicated that the enhancement factor during solidification was estimated to be as high as 250% and the

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C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

Nomenclature Latin letters ice area (m2) As Ac, Aft cross-section area of chamber and finned tube (m2) C mushy zone constant Cp specific heat capacity (kJ kg
1 K
1) df degree of freedom g gravitational acceleration (m s
2) Fo Fourier number h enthalpy of the material (kJ kg
1) H fin height (mm) i row of orthogonal array j column of orthogonal array K dimensionless permeability K0 an empirical coefficient L characteristic length (m) Lp latent heat of fusion (kJ kg
1) n total row of the orthogonal array ni number of factors nj number of interactions N fin number Nn number of nodes Nr number of levels p pressure (Pa) q ice storage rate (kW) q* dimensionless ice storage rate R inner radius of chamber (mm) Ri inner radius of finned tube (mm) Ro outer radius of finned tube (mm) Sh energy source item (kJ m
3 s
1) Su, Sv momentum source items (kg m
2 s
2) t time (s) tc complete melting time (s) T temperature (K) T0 initial temperature (°C) Ttrans transition temperature of water (°C) Tw cold wall temperature (°C) u, v velocities in  and y directions (m s
1) x, y cartesian coordinates (m) yi value in ith row of orthogonal array

fine fin was recommended for more effective enhancement. Zhang et al [8] inserted the metal foam into PCM for solidification improvement of the ice storage unit. Seeniraj et al. [15] utilized the combination of fins and multiple PCMs to achieve performance enhancement of a solar dynamic LHTS module. Fukai et al. [16] performed an experimental and numerical study on the thermal characteristics of the carbon-fiber brush/PCM composite. They concluded that the effective thermal conductivity of the composite was about three times as large as that of pure PCM. Of these methods, extending surfaces using metal fins has become the most attractive candidate, owing to its low cost, high efficiency, and ease of fabrication. The available studies on finned ice storage units focus primarily on the effect of the fin structure, such as the fin shape [19,20], fin number [21,22], fin height [22], and fin width [9], on the ice storage performance. Sheikholeslami et al. [19] compared the solidification enhancements in the ice storage units with snowflake-shaped fin and nanoparticle dispersion. The results indicated the energy discharging enhancement of the case applying snowflake-shaped fin was higher significantly. Zhai et al. [21] conducted a configuration optimization of the finned ice storage unit and concluded that the optimized parameters for annular fin pitch,

Greek symbols b liquid fraction d thermal expansion coefficient (K
1) D fin width (mm) Dh latent heat (kJ kg
1) DT superheat factor g calculation constant k thermal conductivity (W m
1 K
1) l dynamic viscosity (kg m
1 s
1) q density (kg m
3) U ice fraction x contribution ratio X the interface between metal fin and PCM Subscripts A, B, C controllable factors AB interaction between factor A and B e error column of orthogonal array f fin ind independent factor int interaction between two factors l liquid phase nf without fin p PCM ref reference state s solid phase total total factors wf with fin Abbreviations ANOVE analysis of variance DoF degree of freedom Exp experiment HTF Heat transfer fluid MS mean square NC natural convection Num numerical PCM phase change material SS sum of squares STIS shell-tube ice storage

the number of rectangular fins and fin thickness were 40 mm, 9 and 1 mm, respectively. Velraj et al. [22] analyzed the solidification enhancement of the ice storage unit with longitudinal fins and discussed the effects of the number of fins and tube radii. However, few efforts have been devoted to determine which of the fin structure parameters is the most important factor affecting ice storage performance. Generally, the effect of NC is usually overlooked in investigations on the discharging process of a latent heat storage unit, because the solidification process is dominated by thermal conduction [8,19,20]. As opposed to conventional PCM, the special nature of water density reversal (i.e., the evolution of water density with temperature reverses when the temperature is larger or less than the transition temperature) leads to the buoyancy reversal during the solidification process. In this context, it is of particular importance to study the solidification behavior of water in the STIS units with consideration of NC. Therefore, several attempts have been devoted to investigate the effect of NC recently [23–25]. Jmal et al. [23] performed a numerical study on the effect of NC of liquid PCM on the solidification performance of STIS units for the air conditioning systems. The results implied the important role of NC in

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

3

solidification heat transfer. The similar conclusion was drawn by Ezan et al. [25] when they investigated the solidification process in a rectangular geometry. Sun et al. [24] conducted a lattice Boltzmann modeling of the dendritic growth of binary alloys in a forced melt convection. It was concluded that the dendritic growth was significantly affected by NC. However, few studies have discussed how NC affects the solidification behaviors in available literature. The relationship between the buoyancy reversal and the NC is still unclear. In addition, the effect of buoyancy reversal on the ice storage performance of STIS units is still waiting to be explored. Therefore, the further study on the solidification performance of water in the STIS units with consideration of NC is of great significance for the practical engineering applications in ice storage. The purpose of the theoretical investigation of ice storage performance is to optimize the geometric structure of ice storage units. Until recently, it has remained a challenge to obtain a comprehensive evaluation of the maximization of discharging performance in a finned shell-tube ice storage unit, because it is costly and time consuming to experimentally or numerically conduct a traversal of the effects of the structural parameters on ice storage performance. Fortunately, it has been proven that the Taguchi method is an efficient method of achieving engineering optimization, owing to its advantage with respect to minimizing the variability around the target when bringing the performance value to the target value. Further, it has been successfully utilized to optimize heat exchangers [26], heat sinks [27], and heat pumps [28]. In this context, the Taguchi method is introduced to perform a quantitative evaluation of all the fin parameters (i.e., fin length, fin number and fin width) affecting the ice storage performance and to determine primary factors related to the optimization of the design of shell-tube ice storage units. In this paper, an unsteady model of the solidification process of water in the STIS unit is developed and numerical solved with a particular focus on the effect of NC. In order to ensure the reliability of the numerical predictions, the present model is verified by a solidification experiment in a finned STIS unit. The solidification performance of water in the STIS unit with NC is analyzed and compared with corresponding case without NC. The coupling effect of buoyancy reversal and the NC on the ice storage process is discussed in depth. Moreover, the optimization of fin geometry is performed through the Taguchi method.

ature gradient of PCM along the axial direction is far smaller than that across the radial direction. Therefore, it is reasonable to conclude that the solidification behavior of the cross section of a finned STIS unit is capable of representing the practical discharging process in a finned STIS unit.

2. Mathematical model

The enthalpy-porosity method [32,33] is used to investigate the solidification process of water in a finned STIS unit. A 2D unsteady model of the ice storage process is established and analyzed. The governing equations, including the continuity, momentum, and

The geometric structure of a finned STIS unit is illustrated in Fig. 1. As shown in the figure, it consists of an ice storage chamber and a finned tube. The finned tube is assembled concentrically in the chamber, and the fins are equally spaced around the tube. The heat transfer fluid (HTF) flowing through the inner tube and the PCM completely fills the interspace between the ice storage chamber and finned tube. The pioneer studies [9,29] have verified the feasibility and the applicability of using two-dimensional model to study the solidification performance of real threedimensional horizontal STIS units due to the fact that the temper-

Fig. 1. Schematic of a finned shell-tube ice storage unit.

2.1. Governing equations To understand the solidification behavior in a horizontal STIS unit considering NC, a two-dimensional (2D) model of the ice storage process in a STIS unit was developed in this study. As illustrated in Fig. 2, the inner radius of the finned tube (Ri) is 20 mm, and the outer radius (Ro) is 25 mm. The inner radius of the ice storage chamber (R) is 85 mm and the thickness of the chamber is ignored. In the investigation, the structural parameters of fins, including the fin height H, fin width D, and fin number N are variable, but the total volume of the STIS unit is identical. In the current study, the water is applied as the PCM (owing to its high thermal energy-storage density), and Aluminum 6061 is selected as the fin material due to its low cost and appropriate thermal conductivity [30]. To prevent heat loss to the ambient, the ice storage unit is wrapped by the heat-insulating layer. Moreover, water with as potential PCM for ice storage applications. Detailed thermophysical properties of the materials utilized in this paper are shown in Table 1. To describe the unsteady discharging process accompanied by a phase change in a finned STIS unit, the following assumptions are made to simplify the mathematical model: 1) The water is the Newtonian and incompressible, and the NC is laminar without viscous dissipation. 2) The thermo-physical properties of water, such as Cp and k, are set to be different constant values for liquid and solid phases, respectively. The density of water is a function of temperature, as determined by Eq. (12). 3) The water has three states during the discharging process: solid, liquid, and mushy zones. 4) The wall boundaries between PCM and fin are treated as noslip conditions. 5) The volume variation and undercooling effect are all ignored [25,31].

Fig. 2. Cross section of a horizontal STIS unit.

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C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

Table 1 Thermo-physical properties of the materials [25,37]. Materials

Density q/(kg m
3)

Specific heat capacity Cp/(kJ kg
1 K
1)

Thermal conductivity k/(W m
1 K
1)

Latent heat of fusion Lp /(kJ kg
1)

Phase change point Tp / (°C)

Aluminum 6061 Water Ice

2700

0.963

180

/

/

Eq. (12) 800

4.182 2.217

0.61 2.22

334 334

0 0

energy equations are respectively listed in the cylindrical coordinate system (r, h). The continuity equation is

@ q 1 @ðrqV r Þ 1 @ðqV h Þ þ ¼0 þ @r r @h @t r

ð1Þ

The momentum equations are

! @V r V2 1 @p 2 @V h V r þ tðr2 V r
2 þ ðr  V ÞV r
h ¼ f r

2Þ r @h @t r q @r r þ Sr

ð2Þ

! @V h VrVh 1 @p 2 @V r V h þ tðr2 V h þ 2 þ ðr  V ÞV h þ ¼ fh

2 Þ þ Sh r @h @t r qr @h r ð3Þ The energy equation is

!
 ! @ðqhÞ 1 @ @T 1 @2T þ 2 þ Sh þ r  ðq V hÞ ¼ k r @t r @r @r r @h2

ð4Þ


where T is the temperature, q is the density, t is time, V ¼ ðV r ; V h Þ is the velocity vector, p is the pressure, m is the viscosity, h is the enthalpy of the material, k is the thermal conductivity, fr and fh are the gravity items, and Sr, Sh, and Sh are the source items of the momentum and energy equations. The gravity items in the momentum equations are given by

f r ¼
gdðT
T p Þcosh f h ¼ gdðT
T p Þsinh

ð5Þ ð6Þ

where Tp is the phase-change temperature, d is the thermal expansion coefficient, and g is the gravitational acceleration. In this paper, the phase change point is regarded as the reference state. As stated in [33], the source term Sh in the energy equation is derived from the enthalpy formulation of the convectiondiffusion phase change, and the expression can be written as follows:

Sh ¼

 !  @ðqDhÞ þ r  q V Dh @t

ð7Þ

where Dh is the latent heat content, which is defined as a function of temperature, i.e., Dh = f(T). In the case of an isothermal phase ! change, the term div(q V Dh) vanishes. The enthalpy-porosity method does not track the phase interface explicitly. Instead, a quantity called the local liquid fraction, b, is defined to characterize the mushy zone, with the assumption that the mushy zone is modeled as a ‘‘pseudo” porous medium in which the porosity decreases from 1 to 0 as the material solidifies. In this context, the latent heat content is given by

Dh ¼ bLp

ð8Þ

where Lp is the latent heat of fusion, and b is the local liquid fraction. In the melting point model [32,34] for pure PCMs (such as water), the local liquid fraction of the isothermal solidification process can be written as follows:

b¼

8 0 > > > > < > > > > :

T 6 Tp


T
ðT p
eÞ 2e

e

Tp
e < T < Tp þ e

ð9Þ

T P Tp þ e

1

where 2e is a small temperature range of the mushy zone, and the mushy zone is expected to be narrow, which is usually set to be 0.5 °C [25,35]. The enthalpy of the material is the sum of the sensible heat and the latent heat, and can be expressed as follows:

h ¼ c p T þ Dh

ð10Þ

where cp is the specific heat at constant pressure. Darcy’s law and Kozeny-Carman equation [36] are used to quantitatively characterize the flow and permeability in the mushy zone. The permeability in the mushy zone can be calculated by

K ¼ K0

b3

! ð11Þ

ð1
bÞ2

where K0 is an empirical coefficient, and K is the dimensionless permeability in the mushy zone. The source terms Sr and Sh are used to modify the momentum equations in the mushy zone. As the local liquid fraction increases, the superficial velocity and permeability also increase up to a maximum value of 1 when the PCM completely melts. In a numerical model, this feature can be considered by defining

Sr ¼


Sh ¼


1 Cð1
bÞ2

q b3 þ g 1 Cð1
bÞ2

q b3 þ g

Vr

ð12Þ

Vh

ð13Þ

where g is the calculation constant, which is a small value that is introduced to prevent the denominator from being zero. C is the mushy zone constant, and is set to 107. The NC of water is taken into account during the discharging process. Because the density of water achieves the largest value at 4 °C, the change of water density with temperature is nonlinear. According to Ref. [37], the density of water in the buoyancy term can by calculated by



ql ¼ ql;max 1
c  jT
T trans j1:89



ð14Þ

where ql is the density of the liquid phase, and Ttrans is the transition temperature when the density of water achieves the largest value, ql,max. In this paper, ql,max = 999.97 kg m
3, c = 9.3  10
6, and Ttrans = 4 °C. 2.2. Initial and boundary conditions The whole domain of a finned STIS unit is in the thermal equilibrium at the initial time, so the temperature of the whole domain is identical, as given by

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

T ðr; h; 0Þ ¼ T 0

ð15Þ

where T0 is the initial temperature, which is constant for the whole computational domain. The range of the initial temperature selected here is from 4 to 12 °C, with increments of 4 °C. As illustrated in Fig. 2, the finned STIS unit is wrapped by a heatinsulating layer. Therefore, the wall of the ice storage chamber is regarded as an adiabatic boundary

 @T  ¼0 @r r¼R

ð16Þ

The finned tube is cooled by heat transfer fluid with a constant temperature. Considering that the temperature gradient in the radial direction is much larger than that in the axial direction, the inner wall of the finned tube is defined as an isothermal boundary

Tjr¼Ri ¼ T w

ð17Þ

where Tw is the cold-wall temperature with a constant value of Tw =
10 °C. In addition, the interface between the metal fin and water is a coupled boundary at which the temperature and heat flux are all identical, and it is written by

  T f X ¼ T p X
kf

ð18Þ

  @T f  @T p  ¼
k p @r X @r X

ð19Þ

where X represents the interface between the metal fin and water, and the subscripts f and p represent the fin and PCM, respectively. 2.3. Numerical method In this study, the finite-volume method with a double-precision solver was utilized to solve the 2D, laminar, unsteady solidification process of a finned STIS unit based on the commercial CFD software. The SIMPLE algorithm [38] was used to simulate the coupling of pressure and velocity. The momentum and energy equations were discretized by the QUICK scheme, and the gradient option was processed using the Green-Gauss Cell Based method. To achieve a good convergence, the under relaxation factor was used to deal with the following parameters in the governing equations: volume rate (0.2), pressure (0.2), density (0.5), volume-force source (0.5), and energy source (0.7). The residual convergence criterions in the computation for the continuity, momentum and energy equations were 10
5, 10
5 and 10
6, respectively. A structured grid of quadrilateral meshes shown in Fig. 3 was used to mesh the computational domain of a finned STIS unit. A finned STIS unit with fin height H = 40 mm, fin number N = 8 and fin width D = 5 mm was selected as the test case. Four different grid systems were used to perform the grid independence test.

5

Moreover, the comparisons of ice front morphologies and dynamic ice volume under different grid systems and time steps were illustrated in Fig. 4. It can be seen from the figure that the morphologies of the ice front at a typical time as well as the ice volume are in better agreement as the number of nodes increases. In particular, the predicted results of the ice front and the ice volume are almost the same for the cases where the number of nodes equals 6265 and 8533. Similarly, the dynamic ice volume variation curves almost coincide when the time step is less than or equal to 0.1 s. Therefore, considering the results accuracy and time consumption, the number of nodes of 6265 and the time step of 0.1 s are selected in this paper. 2.4. Model verification To verify the present model, a visual experiment of the ice storage process in a finned STIS unit is conducted. The schematic of the experimental setup is illustrated in Fig. 5(a). As shown in the figure, the experimental setup is mainly composed of a finned STIS unit, a temperature control system, and a data acquisition system. The components and dimensions of the finned shell and tube IES unit are illustrated in Fig. 5(b). The finned tube is made of aluminum 6061, and the material of the ice storage chamber is a special glass allowing the transmission of infrared rays. A finned tube with fin height H = 40 mm, fin number N = 8, and fin width D = 5 mm is selected here. The heat-transfer fluid of ethylene glycol cycling in the finned tube is pumped by a thermostatic water bath (XODC-2030-II) and provides a constant cold source for the solidification of water in the ice storage charmer. To prevent heat loss to ambient, the external surfaces of the finned STIS unit are wrapped by polyurethane. In the experiment, the STIS unit is in thermal equilibrium with an initial temperature of T0 = 8 °C. Once the solidification starts, the HTF with a temperature of
10 °C is provided in the inner tube of the STIS unit. During the energy discharging process, the temperature variations of PCM are measured by T-type thermocouples (OMEGA), and the acquisition of ice front morphology is conducted through a CCD camera (DFK 23U274). Moreover, the dynamic ice fraction in the STIS unit is measured by the image processing technology based on the Matlab software. Fig. 6 compares the ice fraction and the morphologies of the ice front between the experiment and simulation for the discharging process in a finned STIS unit. In the experiment, the initial temperature of the whole domain is T0 = 8 °C, and the cold wall temperature is Tw =
10 °C. It is evident that initially, the ice front is slightly over predicted by the numerical model relative to the experimental results. The discrepancy is more obvious in the ice fraction at the start time between experimental and numerical results. The reason is that in a practical experiment, it is difficult to cool the inner wall of the finned tube to the desired steadystate temperature within a limited time. Despite this deficiency, the evolution of the ice front and the dynamic variations of the ice fraction during the solidification process predicted by the numerical model agree with the experimental results. Consequently, the numerical model proposed in this paper is reasonable, and can predict the ice storage performance of a finned STIS unit. 2.5. Evaluation criteria and characteristic parameter

Fig. 3. Grid of computational domain.

Ice storage is a complex heat-transfer process that is accompanied by the solidification phase change and NC. To evaluate the performance of a STIS unit quantitatively, a series of evaluation criteria are introduced in this paper, including the ice fraction, dimensionless ice storage rate, enhancement ratio, and contribution ratio. Of these parameters, the ice fraction, dimensionless ice storage rate, and enhancement ratio are independent criteria that only describe the influence of a single factor, while the contribution

6

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

(a)

t = 3000s

t = 8000s

(b)

grid independence test

time step independence test

Fig. 4. Grid system check (T0 = 8 °C): (a) ice front, (b) ice volume variation.

ratio can comprehensively determine the weight of each factor on the thermal performance of a STIS unit. The ice fraction, U, is defined as the ratio of the ice volume to the volume of the interface of the ice storage chamber as follows:

U¼

As Ac
Aft

ð20Þ

where As is the ice area, and Ac and Aft are the cross-section areas of the ice storage chamber and finned tube, respectively. The dimensionless ice storage rate, q*, is defined as the ice storage rate of the finned STIS unit relative to the corresponding case without fins, and is given as

qwf qnf

q ¼

ð21Þ

where the subscripts wf and nf indicate the values of the STIS units with and without fins at the same time, respectively. The Fourier number and superheat factor are utilized to characterize the conditions of the ice storage process. The Fourier number, Fo, is a dimension time characterizing the solidification process in an STIS unit, and it is calculated by

Fo ¼

k



t

qcp L2

ð22Þ

where L is the characteristic length (here L = R0), and the density q is a constant of 999.97 kg/m3. The superheat factor [39,40], DT, is a dimensionless parameter to quantitatively describe the initial temperature difference between HTF and PCM, and determines the driving force of the solidification process. The parameter is defined as

DT ¼

T0
Tw Tp
Tw

ð23Þ

where T0 is the initial temperature, Tp is the phase change point of water, and Tw is the cold-wall temperature. 3. Results and discussion 3.1. Solidification process analysis As a typical PCM, there is a transition point in the relationship between the water density and temperature. Fig. 7 illustrates the variation of the water density with temperature. It is seen that the density of water increases gradually with temperature when it is less than 4 °C, while a further increase in temperature leads to a gradual decrease of the water density. In this context, the buoyancy force of water arising from the density variation is also reversed. By performing a comprehensive investigation of the solidification behaviors in a STIS unit during the discharging process, it is seen that the density reversal of water plays a significant role in the ice storage performance inside a STIS unit. The negative effect of NC and the buoyancy reversal are observed during the discharging process in a finned STIS unit. 3.1.1. Negative effect of NC When an ice storage unit is imposed on by a cooling load, it goes through an unsteady discharging process with temperature variation before approaching final thermal equilibrium. Because the density of water is sensitive to variations in temperature, the buoyancy force arising from the variation of density contributes to the NC of water, which is a very important factor that affects ice storage performance. However, it remains unclear how NC affects the ice storage process, and it is very important to understand whether it is a positive or negative factor in terms of improving the STIS unit performance.

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C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

Cold bath Valve

Valve Temperature signals

Rotameter

T Valve Data logger Cold light Sealing flange

PC

CCD camera

Finned-tube y

x

z

Transparent ice storage chamber (polymethyl methacrylate)

Cold light

(a) Schematic diagram

(ii) Transparent ice storage chamber

(i) Detailed geometries

(iii) Finned-tube

(b) Dimensions and components Fig. 5. Schematic and images of the experimental setup.

Fig. 8. plots the dynamic variations of the ice fraction and dimensionless ice storage rate in finned STIS units with and without NC. As illustrated in the figure, there is a rapid increase in the

ice fraction for the case with NC during the early discharging process, and the increased rate is in agreement with the corresponding case without NC. However, as the discharging process continues,

8

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

(a)

Num

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

Exp

(b)

Fig. 6. Comparison between experiment and simulation: (a) ice front morphology; (b) ice fraction.

Fig. 7. Effect of temperature on water density.

the dynamic ice fraction for the NC case is lower than that without NC, indicating that the overall performance of a finned STIS unit with NC is less efficient than that without NC during the discharging process. Moreover, the evolution of the dimensionless ice storage rate for the NC case is faster than that with NC. It is interesting that there is a rapid increase in the dimensionless ice storage rate during the initial discharging process (i.e., the ice storage performance is enhanced) owing to the expansion of the heat-transfer

area. Then, with the development of NC, the dimensionless ice storage rate drops sharply. Finally, the variation of the dimensionless ice storage rate tends to be gentle owing to the attenuation of NC. The above phenomenon implies that NC is a negative factor for ice storage, and weakens the potential of the ice storage rate of a finned STIS unit. Therefore, in the practical engineering applications of ice storage, it is important to limit the NC to improve the efficiency of a finned STIS unit. To show the negative effect of NC visually, Fig. 9 illustrates the ice front evolution in finned STIS units considering the cases with and without NC. It is seen that NC plays an important role in the evolution of the ice front morphology. The solidification rate in the NC case is clearly slower than the corresponding case without NC during the discharging process. This is because in the STIS unit with NC, the cold energy is partly taken away from the ice front; hence, the solidification rate slows down accordingly. Moreover, as opposed to the ice front morphology in the NC case, the upward NC leads to the lower solidification rate for the top half of water, so the solid-liquid interface in the STIS unit with NC is no longer centrally symmetrical, but is rather an asymmetrical curve. In order to provide a further insight into the effect of NC on ice storage process, Fig. 10 compares the dynamic temperature distribution and the streamline evolution in a finned STIS unit between the NC case and no NC case. It is evident that the temperature distribution of a finned STIS unit with NC is quite different from the

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C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

(a) ice fraction

(b) dimensionless ice storage rate

Fig. 8. Effect of NC on ice storage performance (H = 30 mm, N = 6, D = 3 mm, DT = 1.4).

Fo=0.07

Fo=0.54

Fo=1.79

Fo=7.14

NC

Without NC

Fig. 9. Effect of NC on the ice front evolution (H = 30 mm, N = 6, D = 3 mm, DT = 1.4).

corresponding case without NC. As illustrated in the figure, the cold energy evenly diffuses along the radial direction of the finned STIS unit because of the absence of NC. Therefore, the temperature distribution of water is completely symmetrical. However, because of the effect of NC, the heat transfer of water beyond the liquidsolid interface is enhanced. In this context, the temperature gradient of water is reduced and the temperature distribution in water is more uniform. Moreover, it is interesting to find that the vortex size and vortex number decrease with the continuous discharging process, which leads to a decrease in the intensity of the NC. Fig. 11 quantitatively describes the dynamic temperature variation of PCM at typical positions (1#, 2#, 3#, 4#, see Fig. 11(b)) in finned ice storage units with and without NC. In this study, water is used as the PCM. As shown in the figure, the PCM temperatures at different positions agree well during the discharging process for the case without NC owing to the dominance of thermal conduction and the symmetrical structure of the STIS unit. However, for the case with NC, there is a large difference in the PCM temperature at various positions in the finned STIS unit. For the case where

the superheat factor DT = 1.4 (i.e., the initial temperature T0 = 4 °C), the dynamic temperature variation of PCM in a finned STIS unit with NC is faster than that without NC during the early discharging process. This indicates that NC enhances the heat transfer of water away from the ice front, and owing to its upward trend, the temperature variation of the upper water is significantly more rapid than that of the lower half. As the solidification process progresses, it is evident that the duration of the solidification process for the case with NC is longer than that without NC. This is believed to be because a large amount of cold energy is transferred from the ice front to water far away from ice by NC, and the temperature gradient of water beyond the solid-liquid interface is decreased; hence, the solidification of water nearing the ice front slows down. Moreover, it is interesting to note that the temperature variations of the PCM at different positions tend to be consistent during the later discharging process as a result of the dominance of the thermal conduction of PCM. In summary, the presence of NC is not conducive to the performance enhancement of the discharging process.

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C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

Fo=0.07

Fo=0.54

Fo=1.79

Fo=7.14

NC

Without NC

Fig. 10. Effect of NC on temperature distribution and velocity field (H = 30 mm, N = 6, D = 3 mm, DT = 1.4).

(a) dynamic temperature variation

(b) typical positions

Fig. 11. Effect of NC on temperature evolution (H = 30 mm, N = 6, D = 3 mm, DT = 1.4).

3.1.2. Buoyancy reversal As stated previously, the presence of NC is negative related to ice storage performance. To explain this effect, it is necessary to analyze the unique phenomenon of buoyancy reversal during the ice storage process. The density singularity of water corresponds to a transition temperature (Ttrans = 4 °C). When the water is initially superheated above the transition temperature, the water will expand accordingly once the cooling load is imposed. However, when water with an initial temperature less than the transition temperature is cooled, it goes on a reverse expansion, i.e., the higher the initial temperature, the higher the density. In other

words, for the case where the superheat factor is greater than 1.4 (i.e., the initial temperature T0 = 4 °C), the NC initially develops downward during the solidification process, while the buoyancy reverses upward when the water is cooled to less than the transition temperature. To explain the buoyancy reversal effect on ice storage performance in a finned STIS unit more effectively, Fig. 12 shows a plot with variations of the dynamic responses of the ice fraction and dimensionless ice storage rate with the different superheat factor. It can be seen that the superheat factor plays an important role in the ice fraction and dimensionless ice storage rate. As the superheat factor increases, the ice fraction of a finned

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

STIS unit decreases accordingly. The lower the superheat factor, the higher the dimensionless ice storage rate, i.e., the ice storage performance is better for a lower superheat factor in a finned STIS unit. These results indicate that the buoyancy reversal is negatively related to ice storage performance. Moreover, as the superheat factor increases, the performance loss arising from the buoyancy reversal also increases. It should be noted that although the overall effect of the buoyancy reversal is negative, at the beginning of the inversion, the downward cold energy in the early stage and the upward cold energy after the inversion converge in the center of the finned STIS unit. This results in a temporary improvement in ice storage performance, as shown in the inset. To understand the effect of buoyancy reversal on the ice storage performance more fully, Fig. 13 depicts the evolution of the temperature distribution and velocity field in finned STIS units with different superheat factors. As expected, the evolution of the temperature and velocity fields has a significant reversal phenomenon with the variation of the superheat factor. For the case where the superheat factor is 1.4, the NC develops upward during the solidification process. However, for this case where the superheat factor is larger than 1.4, the NC shifts from an early downward trend (i.e., vortexes are concentrated in the lower half) to an upward one, and the temperature distribution also changes from ‘‘upper half is hot and lower half is cold” to ‘‘upper half is cold and lower half is hot.” Moreover, with the increase of the superheat factor, the intensity of the NC is stronger, and the temperature distribution is more uniform, which is negatively related to ice storage performance. Fig. 14 plots the temperature variation of water during the discharging process at symmetrical positions for STIS units with different superheat factors. It can be seen that there is a fluctuation in temperature owing to the buoyancy reversal for the cases with superheat factors of 1.8 and 2.2. Then, the temperature of the monitoring points at symmetrical positions also reverses. The above analysis shows that the superheat factor has a negative correlation with the ice storage performance of a finned STIS unit. For better discharging efficiency, a higher superheat factor should be avoided in practical engineering applications.

3.2. Parametric optimization using Taguchi method The Taguchi method is a quality engineering method with advantages of low cost and high efficiency, and it has been widely utilized in product design, quality management, and technological revolution. In the engineering optimization of a process or product, it can be classified into three categories: system design, parameter design, and tolerance design. Of these categories, the parameter design is utilized for the parametric optimization of products. As

(a) ice fraction

11

the most important approach in the Taguchi method, it is based on the concept that the design parameters of products can be imitatively manipulated to achieve less sensitivity to variations in the configurations. In the present study, the specifically constructed table known as ‘‘Orthogonal Arrays” is designated to example layouts. Moreover, a statistical analysis of variance (ANOVA) was also conducted to determine which design parameter is statistically significant. 3.2.1. Performance characteristics It has been documented that fin geometry plays an important role in the ice storage performance of finned STIS units. The unreasonable design of fins not only weakens the efficiency of the discharging processes, but also increases the design cost. To obtain an optimal fin geometry design, the performance characteristics of a finned STIS unit are selected as baseline evaluation criteria. By performing a comprehensive review of the previous studies, it is evident that the influence of fin geometry on the ice storage performance also depends on the initial conditions and the physical model. Therefore, a dimensionless parameter was defined to evaluate the ice storage performance using the Taguchi method. In the present work, the discharging characteristic of a finned STIS unit is described using the fraction. If the primary purpose is to maximize the dynamic ice fraction, the ‘‘higher is better” scenario is suitable for the present study. 3.2.2. Taguchi method 3.2.2.1. Controllable factors and levels. To satisfy the optimization goal of this study, three main controllable factors that affect ice storage performance are selected: fin height (H), fin number (N), and fin width (D). The levels of each factor are illustrated in Table 2, and are selected based on the interior space and volume fraction of finned STIS units. 3.2.2.2. Orthogonal array. Considering the trade-off between the available budget and time consumption, the design of the orthogonal array depends on the total number of degrees of freedom (DoFs) of the numerical simulation experiment. The DoF, df, is defined as the number of comparisons between design parameters. For an independent factor A, the DoF is calculated by

df A ¼ Nr
1

ð24Þ

where Nr is the number of levels related to a factor. Moreover, the DoFs associated with the interaction between two factors are given by the product of the DoFs for each factor. For example, the DoFs of an interaction between A and B can be written as

df AB ¼ df A  df B

(b) dimensionless ice storage rate

Fig. 12. Effect of superheat factor on the ice storage performance (H = 30 mm, N = 6, D = 3 mm).

ð25Þ

12

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

T/K

Fo = 0.07

Fo = 0.21

Fo = 0.54

Fo = 1.79

(a) ΔT = 1.4

(b) ΔT = 1.8

(c) ΔT = 2.2

Fig. 13. Effect of superheat factor on temperature distribution and velocity field (N = 6, D = 3 mm, H = 30 mm).

Therefore, the total DoFs of the numerical simulation experiment here is given by

df total ¼

ni X 1

df ind þ

nj X

df int

ð26Þ

1

where ni is the number of factors, nj is the number of interactions between structural parameters, the subscripts ‘‘ind” and ‘‘int” represent the DoFs of the independent factor and that of the interaction between two factors, respectively. In the present study, three independent factors are considered, including the fin height, fin number, and fin width. In addition, there are also three interactions between these structural parameters. Thus, the total DoFs of this simulation equals to [3  (3
1)] + [3  (3
1)  (3
1)] = 18. It should be

noted that the noise factor is negligible in this work, because the results are obtained from numerical simulations. It is recommended that the DoFs of an orthogonal array should not be less than the sum of those of all controllable factors. Therefore, a standard orthogonal array of L27 (311) is utilized here to cope with three-level factors. The example layout for the three controllable factors using the L27 orthogonal array is illustrated in Table 3. 3.2.2.3. Analysis of variance. Analysis of variance (ANOVA) is a very practical and effective statistical analysis method in the Taguchi method, and can be used to evaluate the percentage contribution of each factor. Here, the ANOVA and standard methods are carried out to analyze the results obtained from the numerical simulation experiment in order to estimate the importance of structural

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C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

(a) dynamic temperature variation

(b) typical positions

Fig. 14. Effect of superheat factor on the dynamic temperature response (H = 30 mm, N = 6, D = 3 mm).

where SStotal is the total sum of squares, which reflects the total difference between the simulated results. The larger the SStotal, the greater is the difference between the simulated results. n is the total number of rows of the orthogonal array, yi is the corresponding value of the ith row in each column. Moreover, the sum of squares for each factor can be calculated by

Table 2 Structural parameters and corresponding levels. Symbol

Structural parameters

A B C

Fin height, H/(mm) Fin number, N Fin width, D/(mm)

Level 1

2

3

20 6 1

30 8 3

40 10 5

r SSj ¼ n

parameters on the ice storage performance. The steps of the analysis are as follows: Step 1. Calculate the sum of squares (SS) The total sum of squares is given by

SStotal ¼

n X i¼1

n 1 X yi
y n i¼1 i

! ki

i¼1

2

n 1 X
y n i¼1 i

!2 ð28Þ

where SSj is the sum of squares for the independent factor, j is the number of columns in the orthogonal table in which this factor is located, ki is the sum of corresponding results when the level number is i in column j. In addition, the sum of the squares of error, SSe, is the sum of squares corresponding to all the empty columns, and can be expressed as

!2 ð27Þ

2

r X

Table 3 Case layout for a L27 standard orthogonal array. Case

A

B

(A  B)1

(A  B)2

C

(A  C)1

(A  C)2

(B  C)1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 1 1 2 2 2 3 3 3 2 2 2 3 3 3 1 1 1 3 3 3 1 1 1 2 2 2

1 1 1 2 2 2 3 3 3 3 3 3 1 1 1 2 2 2 2 2 2 3 3 3 1 1 1

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3 2 3 1 2 3 1 2 3 1 3 1 2 3 1 2 3 1 2

1 2 3 1 2 3 1 2 3 3 1 2 3 1 2 3 1 2 2 3 1 2 3 1 2 3 1

1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2

(B  C)2 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 1 2 3 3 1 2 1 2 3 2 3 1

1 2 3 2 3 1 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1 3 1 2 1 2 3

1 2 3 3 1 2 2 3 1 1 2 3 3 1 2 2 3 1 1 2 3 3 1 2 2 3 1

14

SSe ¼

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

X

ð29Þ

SSempty

Step 2. Calculation of the degrees of freedom (DoF) The DoF of the total sum of squares and error are respectively calculated by

df total ¼ n
1 df e ¼ df total


ð30Þ X

df j

ð31Þ

where dftotal and dfe represent the DoF of the total sum of squares and error, respectively, and dfj is the DoF of the j column in the orthogonal array. Step 3. Calculate the mean square (MS) The mean square is equal to the ratio of the sum of squares to the DoF. The statistical parameter can be written as

MS ¼

SS df

ð32Þ

Step 4. Calculate the contribution ratio The contribution ratio is introduced to quantitatively characterize the percentage influence of each factor on the target characteristic of the orthogonal test. Taking factor A as an example, the contribution ratio of factor A is given by

-A ¼

SSA - ðMSe  df A Þ SStotal

ð33Þ

3.2.3. ANOVA results The structural parameters of fins play an important role in the ice storage performance of a finned STIS unit. It is generally accepted that the higher the number of fins, the greater is the num-

Fig. 15. Contribution ratio of fin parameters: (a) DT = 1.4; (b) DT = 1.8; (c) DT = 2.2.

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

ber of fins, and the thicker the fins, the better will be the discharging performance of the finned STIS unit. However, owing to the typical thermo-physical property (i.e., density reversal) of water, the contribution percentage of the structural parameters of fins varies during the discharging process. Fig. 15 depicts the contribution ratio of the fin parameters as a function of the Fourier number at different superheat factors. The contribution ratio indicates the effect of fin parameters on the ice fraction of a finned STIS unit. As expected, the influence of the structural parameters on the ice storage performance varies during the discharging process. The fin height is a less important factor compared with the other two factors, namely the fin number and fin width, during the early discharging process (i.e., Fo = 0.18). This is because the NC is dominant during the early discharging process, and the increase in fin height leads to the expansion of the heat transfer area, so more cold energy is removed from the ice front by NC, thus decreasing the ice storage performance. However, the increase in the fin number also expands the heat transfer area, but it restricts the growth of NC, and the increase in the fin width effectively improves the fin efficiency, which is beneficial for the discharging process. For the case in which the superheat factor is 1.4, the contribution ratio of the fin number or fin width is almost twice that of the fin height when Fo is 0.18. As the solidification process continues, the contribution ratio of the fin height increases rapidly, while that of the fin number or fin width decreases gradually. The explanation is that the thermal conduction becomes a more important factor, and the increase in the fin height not only expands the heat transfer area, but also successfully constructs a fast heat flow path from the inner tube to the borderline, which accelerates the ice storage rate. However, the increase in the fin width only enhances the thermal conduction of the fins, and plays a lesser role in the heat transfer between the fins and water. Similarly, the increase in the fin number enhances the heat transfer between the PCM and fins, while failing to improve the heat transfer in the radial direction of the STIS unit. Therefore, once the ice front gradually approaches the fin end, the thermal resistance increases further, so the discharge rate is restricted owing to the poor thermal conduction in the radial direction. Therefore, during the subsequent discharging process (i.e., Fo = 5.36), the most important factor is the fin height, with a contribution ratio of 83.6%, while the contribution ratios of the fin number and fin width are 11.6% and 2.9%, respectively. A closer look at Fig. 15(a), (b), and (c) reveals that the superheat factor plays an important role in the contribution ratios of factors

(a) Completed solidification time

15

during the early discharging process (i.e., Fo  0.36). However, the contribution ratio of each factor is unaffected by the superheat factor as the solidification process further progresses (i.e., Fo  1.07). It is interesting that when Fo  0.36, the superheat factor is negatively correlated with the contribution ratio of the fin height, while it is a positive factor of the contribution ratio of the fin width. In other words, an increase in the superheat factor contributes to an increase in the contribution ratio of the fin width, but it simultaneously decreases the contribution ratio of the fin height. Considering as an example the superheat factor within a range of 1.4 to 2.2, the contribution ratio of the fin height experiences a significant drop of 80%, while the contribution ratio of the fin width is greatly increased by 76% at the typical time of Fo = 0.18. This is attributed to the fact that with the increase of the superheat factor, the NC of water beyond the ice front is enhanced owing to the buoyancy reversal. In this context, the increase in the fin height further decreases the ice storage rate, so the corresponding contribution ratio decreases. However, the NC is limited to the increase in the fin number, which means that the contribution ratio of the fin number is not sensitive to the superheat factor. In addition, the increase of the fin efficiency arising from the increase of the fin width enhances the heat transfer between the fin and PCM, which finally improves the ice storage performance during the early discharging process. Nevertheless, the thermal conduction is dominant as the solidification process continues. In this context, the superheat factor plays a less important role in the contribution ratio of all structural factors. By performing the above ANOVA simulation, the influences of the fin geometry on the ice storage performance are clarified as follows: (1) the fin width and fin number play important roles in the early discharging process (i.e., Fo  0.18); (2) as the solidification process proceeds, the fin height is more important to the ice storage performance, and it finally becomes the most important factor in the subsequent discharging process (i.e., Fo = 5.36); (3) for Fo  0.36, as the superheat factor increases, the influence of the fin height decreases, while the influence of the fin width increases; however, the effect of the fin geometry on the ice storage performance is almost unrelated to the superheat factor as the solidification process proceeds (i.e., Fo  1.07). 3.2.4. Fin geometry optimization The ANOVA results indicate that the fin height is the most important factor of the ice storage performance of STIS units and almost independent of the superheat factor. In view of this, the

(b) Ice storage capacity of PCM

Fig. 16. Effects of fin number and fin width on the ice storage performance (DT = 2.2, H = 40 mm).

16

C. Zhang et al. / International Journal of Heat and Mass Transfer 146 (2020) 118836

optimal fin height is H = 40 mm in present work. In order to determine the optimal fin width and fin number, the complete solidification time and the ice storage capacity of PCM for the STIS units with different configurations are depicted in the Fig. 16. The superheat factors for all cases are identical and set to be DT = 2.2. As expected, the fin number plays a more important role in the ice storage performance than that of fin width. The complete solidification time has the largest decrease of 33.2% for the STIS unit with D = 1 mm when the fin number changes from 6 to 10, and the largest decrease in complete solidification time arising from the increase of fin width is only 29.9% for the STIS unit with 6 fins. It is seen that when the fin number is larger than 8 or the fin width is larger than 3 mm, the enhancement of the ice storage performance decreases gradually. As depicted in the Fig. 16(b), although the ice storage performance of the STIS unit is improved with the increase of fin number and fin width, the ice storage capacity of PCM decreases as well. Obviously, the increase of fin width results in a larger decrease in the ice storage capacity of PCM. In view of this, considering the trade-off between ice storage rate and ice storage capacity, the fin width D = 3 mm is recommended in the engineering applications. The decrease in the ice storage capacity of PCM is only 1.2%, while the increase in complete solidification time is 12.2%, so the optimal fin number is 10 in this paper. In the context, with a comprehensive consideration of ice storage rate and ice storage capacity, the optimal fin parameters for the STIS unit are fin height H = 40 mm, fin number N = 10, fin width D = 3 mm. 4. Conclusions A numerical study was carried out to analyze the ice storage performance of a finned shell-tube ice storage (STIS) unit. A thermal energy discharging experiment in a finned STIS unit was also performed to verify the proposed model. By performing a comprehensive analysis of the solidification behaviors, the negative effect of the natural convection and the buoyancy reversal were examined in a finned STIS unit. Furthermore, the Taguchi method was introduced to optimize the fin geometry of STIS units to obtain the best discharging performance, where the geometric parameters considered were the fin width, fin height, and number of fins. The primary conclusions can be summarized as follows: (1) The natural convection is negatively related to the ice storage performance of a finned STIS unit. With the development of natural convection, some cold energy is taken away from the ice front to the periphery, which decreases the temperature gradient of the water beyond the ice front and slows down the solidification rate of water close to the ice front. In addition, the ice front morphology in a STIS unit with natural convection is no longer centrally symmetrical, but an asymmetrical curve. Therefore, it is very important to limit the development of the natural convection in order to improve the efficiency of a finned STIS unit. (2) Buoyancy reversal plays an important role in the energy discharging process of a finned STIS unit. At the beginning of the inversion, the cold energy is congested around the inner tube, resulting in a temporary improvement in ice storage performance. However, the intensity of natural convection is enhanced and the temperature distribution is more uniform because of the buoyancy reversal; hence, the overall discharging rate is weakened. Moreover, with the increase of the superheat factor, the performance loss arising from the buoyancy reversal is increasingly increased. For a better discharging efficiency, a higher superheat factor is unsuitable in practical engineering applications.

(3) Using the Taguchi method, the influences of the fin geometry on the ice storage performance are clarified. The fin width and fin number play less important roles in the discharging process, while the fin height is the most important factor affecting the overall ice storage performance. Moreover, in terms of the final contribution ratio to the ice fraction, the effect of the fin height is independent of the superheat factor. Consequently, increasing fin height is an ideal method for improving ice storage performance. From the perspective of trade-off between ice storage rate and ice storage capacity, the optimal fin parameters for the STIS unit are fin height H = 40 mm, fin number N = 10, fin width D = 3 mm for engineering applications.

Declaration of Competing Interest None. Acknowledgement This work was supported by National Natural Science Foundation of China (Grant Nos. 51725602 and U1737104), Natural Science Foundation of Jiangsu Province (Grant No. BK20170082), ’six talent peaks’ project of Jiangsu Province (Grant No. XNY042), and ’Zhishan Young Scholar’ Program of Southeast University.

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