The energy efficiency ratio of heat storage in one shell-and-one tube phase change thermal energy storage unit

Applied Energy 138 (2015) 169–182

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The energy efficiency ratio of heat storage in one shell-and-one tube phase change thermal energy storage unit Wei-Wei Wang a, Liang-Bi Wang b,⇑, Ya-Ling He a a b

Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China Key Laboratory of Railway Vehicle Thermal Engineering of MOE, Department of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

h i g h l i g h t s  A parameter to indicate the energy efficiency ratio of PCTES units is defined.  The characteristics of the energy efficiency ratio of PCTES units are reported.  A combined parameter of the physical properties of the working mediums is found.  Some implications of the energy efficiency ratio in design of PCTES units are analyzed.

a r t i c l e

i n f o

Article history: Received 27 June 2014 Received in revised form 20 September 2014 Accepted 24 October 2014

Keywords: Phase change material Energy efficiency ratio Heat storage property Numerically investigate

a b s t r a c t From aspect of energy consuming to pump heat transfer fluid, there is no sound basis on which to create an optimum design of a thermal energy storage unit. Thus, it is necessary to develop a parameter to indicate the energy efficiency of such unit. This paper firstly defines a parameter that indicates the ratio of heat storage of phase change thermal energy storage unit to energy consumed in pumping heat transfer fluid, which is called the energy efficiency ratio, then numerically investigates the characteristics of this parameter. The results show that the energy efficiency ratio can clearly indicate the energy efficiency of a phase change thermal energy storage unit. When the fluid flow of a heat transfer fluid is in a laminar state, the energy efficiency ratio is larger than in a turbulent state. The energy efficiency ratio of a shelland-tube phase change thermal energy storage unit is more sensitive to the outer tube diameter. Under the same working conditions, within the heat transfer fluids studied, the heat storage property of the phase change thermal energy storage unit is best for water as heat transfer fluid. A combined parameter is found to indicate the effects of both the physical properties of phase change material and heat transfer fluid on the energy efficiency ratio. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction With the increasing power consumption of industrial, commercial, and residential activities, the problems of energy shortage and air pollution have become serious. To help relieve this situation, the use of renewable energy, such as wind energy and solar energy [1–5], on a global scale is highly recommended. However, these types of energy have some shortcomings: they are unstable and can be unreliable due to their dependence on the weather, time, and season. Thus, thermal energy storage (TES) units have become a necessary component in applying renewable energy. The main task of the energy storage, then, is to eliminate the mismatch between energy supply and energy demand. ⇑ Corresponding author. E-mail address: [email protected] (L.-B. Wang). http://dx.doi.org/10.1016/j.apenergy.2014.10.064 0306-2619/Ó 2014 Elsevier Ltd. All rights reserved.

TES includes sensible, latent, and thermal–chemical heat storage units. The latent TES system with solid–liquid phase change has gained greater attention due to its advantages. It has high energy storage density and heat charging/discharging at a nearly constant phase change temperature. These characteristics result in a greater flexibility and more compactness of the phase change material (PCM) heat storage system [6]. Therefore, phase change thermal energy storage (PCTES) has been a main topic in research for the last 20 years. The state of the art developments are summarized in many review papers [7–10]. Zalba et al. [7] carried out a review of the history of solid–liquid PCTES with phase change materials and applications. Sharma et al. [8] summarized the analysis of the available thermal energy storage systems for different applications. Agyenim et al. [9] performed a review of the materials, heat transfer, and phase change problem formulation for latent heat thermal

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Nomenclature a discrete equation coefficients c specific heat capacity (J/(kg K)) C constant in Eq. (10) E energy efficiency ratio f fluid flow resistance factor h local convective heat transfer coefficient (W/(m2 K)) H latent heat (J/kg) i the first i axial node j the first j radial node k thermal conductivity (W/(m K)) l length of the tube (m) L dimensionless length of the tube l/Ri n constant in Eq. (10) Nu Nusselt number 2hRi/kf Pr Prandtl number mf/af q heat storage rate (W) Q heat energy stored (J) r radial coordinate (m) R radius (m) or dimensionless radius of the tube r/Ri R1 dimensionless radius of the tube Rw/Ri R2 dimensionless radius of the tube Ro/Ri Ra Rayleigh number Re Reynolds number 2URi/mf St Stanton number h/((qc)fU) Ste Stefan number H/(cpDT) t time (s) t0 time (s) Ri/U T temperature (K) U velocity (m/s) w, e, n, s west, east, north, and south faces of control volumes P, N, S, W, E the center, north, south, west, and east nodes W mechanical energy (J) x axial coordinate (m) X dimensionless axial coordinate x/Ri Greek symbols Dp pressure drop (Pa) DR dimensionless radial space step Dr/Ri DX dimensionless axial space step Dx/Ri a thermal diffusivity (m2/s) b thermal expansion coefficient (/K)

energy storage systems. Al-Abidi et al. [10] completed a review of thermal energy storage for air conditioning systems. In most PCTES systems, as shown in Fig. 1, a shell-and-tube is the core unit of PCTES. There are many reported studies on this unit [11–22]. Refs. [11–15] use experimental methods to investigate the performance characteristics of this unit. Trp [11] experimentally analyzed the transient heat transfer performance during phase change material melting and solidification. Akgun et al. [12,13] studied on melting/solidification characteristics of three kinds of paraffin as PCM. A novel tube-in-shell storage geometry was introduced and the effects of the Reynolds number and Stefan number on the melting and solidification behaviors were examined. Wang et al. [14] used b-aluminum nitride as additive to enhance the thermal conductivity and thermal performance of form-stable composite phase change materials. Mddrano et al. [15] experimentally evaluated the performance of commercial heat exchanger used as PCM thermal storage systems. Numerous experimentally validated mathematical models of the unit have been developed over the years. These models have been used to determine the performance of the unit for design [16–21]. Trp et al.

e U

u K

l m G

H

q s Ds

w

energy efficiency ratio related stored heat energy (m3) liquid fraction specific heat capacity ratio qc/(qc)f dynamic viscosity (Pa s) kinematic viscosity (m2/s) thermal diffusion ratio a/af dimensionless temperature density (kg/m3) dimensionless time t/t0 dimensionless time step Dt/t0 related mechanical energy (m3)

Superscripts k iteration k 0 old value Subscripts e effective f heat transfer fluid (HTF) front the melt front surface of PCM hcp PCM relative HTF hcw tube wall relative HTF i internal tube or initial in inlet l liquid phase m melting o external tube out outlet p phase change material P, N, S, W, E the center, north, south, west, and east nodes r radial direction s solid phase t time thermp PCM relative HTF thermw tube wall relative HTF w tube wall w, e, n, s west, east, north, and south faces of control volumes s dimensionless time

[16] presented a mathematical model regarding the conjugated problem of transient forced convection and solid–liquid phase change heat transfer based on the enthalpy formulation. The transient heat transfer phenomenon of the unit was analyzed. Fang et al. [17] investigated the effects of different multiple PCMs on the melted fraction, heat storage capacity and heat transfer fluid (HTF) outlet temperature of the unit. Adine and Qarnia [18] numerically analyzed the thermal behavior of the unit. Tao et al. [19,20] performed the numerical study on thermal energy storage performance of PCM in the unit with enhanced tubes. Wang et al. [21]

Fig. 1. Schematic of the shell-and-tube phase change thermal energy storage unit.

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numerically studied the heat charging and discharging characteristics of such kind unit. A CFD model of the PCM system within a tube-in-tank assembly has been developed and validated [22,23]. In order to optimize the design of the PCTES unit, it is necessary to pay attention to the efficiency of the unit. Tay et al. [24,25] numerically investigated the heat transfer effectiveness of the tube-in-tank PCTES system using the effectiveness-number of transfer unit (NTU) method. Amin et al. [26] demonstrated that a suitable relationship for the effective thermal conductivity was developed as a function of the Rayleigh number. It has been proven experimentally that the effectiveness-NTU method is applicable for PCM encapsulated in spheres in a tank. The heat transfer effectiveness of PCM encapsulated in a sphere system has been experimentally investigated using the effectiveness-NTU method [27]. Gil et al. [28] directly measured the change in effectiveness through the application of square radial fins and showed a 20% increase. The investigation demonstrated that a correlation existed between the effectiveness of heat transfer and the mass flow rate. Above studies pay more attention to the heat transfer processes, but ignore another important aspect that pumping HTF will consume energy. Few studies have addressed the energy consumed to pump HTF. It is believed that optimization design of the PCTES unit should consider the energy consumed in pumping HTF. Motivated by this, in this paper, from aspect of the energy consuming to pump HTF, we focus on developing a dimensionless parameter to indicate the how much energy is consumed in charging or discharging process. Then, we demonstrate the characteristics and possible indications of this parameter in designing and operating of the PCTES unit.

(3) the outer wall of the tube is assumed to be an adiabatic boundary. 2.2. Mathematical model In formulating a mathematical model to represent this physical system, the system is divided into the following three subsections: (1) heat transfer fluid flow in the tube; (2) the tube wall; and (3) the region filled by the phase change material. Based on these simplifications, the thermal storage process of the PCTES unit shown in Fig. 2 can be developed as follows. The model for HTF is,

ðqcÞf pR2i

T f jx¼0 ¼ T f;in ;

The physical model for the PCTES unit, which is shown in Fig. 1, is a shell-and-tube configuration. The inner tube is made of copper. HTF flows through the inner tube. The phase change material fills the annular space. The outer surface of the storage unit is wellinsulated with thick pipe insulation. To develop this model, the following assumptions are made: (1) the viscous dissipation in the fluid is negligible compared to the heat convection in the flow, and the flow of HTF is assumed to be onedimensional fluid flow; (2) the thermophysical properties for HTF, the tube, and PCM are constants as shown in Tables 1 and 2; and

 @T f  ¼0 @x x¼l

ð2Þ

T f jt60 ¼ T pi

ð3Þ

The model for the tube wall is,

ðqcÞw

    @T w @ @T w 1 @ @T w þ kw rkw ¼ @x r @r @t @x @r

ð4Þ

The boundary and initial conditions are,

kw

  @T w  @T w  ¼ hðT  T Þj ; k w w f r¼R i @r r¼Ri @r r¼Rw    @T p  @T w  @T w  ¼ kp ; ¼ ¼0 @r r¼Rw @x x¼0 @x x¼l

ð5Þ

T w jt60 ¼ T pi

ð6Þ

The model for PCM is as follows:

ðqcÞp

    @T p @ @T p 1 @ @T p @u þ  qp H kp rkp ¼ @x r @r @t @x @r @t

ð7Þ

where u is the liquid fraction of PCM, which is estimated as:

8 > < > :

2.1. Physical model

ð1Þ

The boundary and initial conditions are,

2. Physical model and mathematical model To account for the mechanical energy consumed in pumping HTF to complete a charging or discharging process, the charging or discharging time is needed. There are two methods of obtaining it: the experimental and numerical methods. Available results [16– 25] show that the validated numerical method can also predicate the performances of charging or discharging processes reasonable. Thus, in the present study, the numerical method is used. The first step in using the numerical method is to develop a mathematical model of the single shell-and-tube PCTES unit.

@T f @T f ¼ ðqcÞf pR2i U  2hpRi ðT f  T w Þ @t @x

u ¼ 0;

Tp < Tm

0 < u < 1; T p ¼ T m

u ¼ 1;

Tp > Tm

solid mushy

ð8Þ

liquid

and kp is the thermal conductivity of PCM, which can be expressed as follows:

kp ¼ ukpe þ ð1  uÞkps

ð9Þ

In Eq. (9), kpe is the effective thermal conductivity taking into account the effect of natural convection during the melting of PCM. It is expressed by the following correlation [29]:

kpe =kpl ¼ CRan

ð10Þ

The values of C and n in Eq. (10) are 0.099 and 0.24, respectively. Therefore,

Table 1 Thermal properties of HTF. Tf,in (K) Air [32] Liquid NH3 [32] Water [32,33] Ethylene glycol solution 40 v/v% [35] Ethylene glycol solution 60 v/v% [35] Ethylene glycol solution 80 v/v% [35] Heat conduction oil [34]

310 310 310 310 310 310 310

qf (kg/m3) 1.136 579.5 995 1051 1076 1098 955

kf (W/(m K)) 2

2.74  10 0.437 0.62 0.431 0.36 0.304 0.128

cf (J/(kg K))

lf  105 (kg/(m s))

Pr

1005 4943 4178 3535 3171 2770 1910.2

1.899 11.52 76.9 177 284 491 379

0.696 1.303 5.182 14.516 25.012 44.741 56.56

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Table 2 Thermal properties of PCM and tube.

n-Octadecane [31] Paraffin C18 [10] Polyglycol E600 [10] CaCl26H2O [10] Gallium [8] Methyl palmitate [8] Copper [32]

Tm (K)

q (kg/m3)

kpl (W/(m K))

kps (W/(m K))

c (J/(kg K))

H (kJ/kg)

mpl  106 (m2/s)

bp  106 (/K)

300.7 298.15 295.15 303.05 302.91 302.15 300

771 880 1126 1710 5904 852 8930

0.148 0.148 0.189 0.53 27.5 9.17  103

0.358 0.15 0.19 1.09 40.6 1.39  102 398

2222 2900 2050 1400 370 2000 386

243.5 244 127.2 190.8 80.3 205

4.013 5.0 5.151 2947 0.329 4.413

900 760 1300 508 9.0 1430

The boundary and initial conditions are,

Adiabatic boundary

T p Ro

PCM Tube wall

r

HTF

x

   @ Hw  Nu ¼ ðHf  Hw Þ ; 2Pthermw Khcw @R R¼1 R¼1   Pthermp Khcp @ Hp  @ Hw  ¼   @R  P K @R 

h

T f,in

T w Rw

T f Ri

l

Fig. 2. Thermal storage process of the shell-and-tube phase change thermal energy storage unit.

kp ¼ ukpl CRan þ ð1  uÞkps

ð11Þ

ukpl CRan þ ð1  uÞkps k ap ¼ p ¼ ðqcÞp uðqcÞpl þ ð1  uÞðqcÞp s

ð12Þ

3

gbp ðT f;in  T m Þd

 d¼

apl mp

;

rp;front  Rw

for charging

Ro  rp;front

for discharging

;

ð13Þ

k apl ¼ pl ðqcÞpl

 Tp

t60

 @T p  kp @r 

  @ Hw  @ Hw  ¼ ¼ 0; @X X¼0 @X X¼L

    Pthermp @ Hp Pthermp @ Hp @ Hp @ @ 1 þ  ¼ 2 2R @X R@R Ste @s RePr @X RePr @R @u  @s

  @ Hp  @ Hp  ¼ ¼ 0; @X X¼0 @X X¼L

Khcw ¼ ð14Þ

> :

u ¼ 0;

ap ¼

H ¼ ðT  T m Þ=ðT f;in  T m Þ

ð17Þ

Khcp ¼

The boundary and initial conditions are,



 

Hf js60 ¼ T pi  T m = T f;in  T m



ð22Þ

 

ð23Þ 

ð24Þ

2URi

mf

Pthermw ¼

R2 ¼

Hp < 0

;

Pr ¼

mf af

ð25Þ

aw Rw ; R1 ¼ af Ri

ð26Þ

Ro Ri

ð27Þ

solid

Hp > 0

ð28Þ

liquid

uCRan Pthermpl ð1  uÞPthermps Khcpsl ap ¼ þ af u þ ð1  uÞKhcpsl u þ ð1  uÞKhcpsl

ð29Þ

ð30Þ

ðqcÞp ðqcÞpl uþðqcÞps ð1  uÞ ¼ ðqcÞf ðqcÞf

¼ Khcpl u þ Khcps ð1  uÞ

ð31Þ

ð18Þ

Khcpl ¼

The model of the tube wall is,

     @ Hw Pthermw @ @ Hw @ @ Hw þ R ¼2 @X @X R@R @s RePr @R



Hp s60 ¼ T pi  T m = T f;in  T m

uapl CRan ð1  uÞaps K hcpsl þ u þ ð1  uÞKhcpsl u þ ð1  uÞKhcpsl

Pthermp ¼

@ Hf @ Hf Nu ¼ 2 ðHf  Hw Þ RePr @s @X  @ Hf  ¼ 0; @X X¼L

u ¼ 1;

ð16Þ

The model of HTF becomes,

Hf jX¼0 ¼ 1;



 @ Hp  ¼0 @R R¼R2

0 < u < 1; Hp ¼ 0 mushy

To get more general results, the above-mentioned models should be normalized by using following parameters:

s ¼ t=t0 ; t0 ¼ Ri =U;

ðqcÞw ; ðqcÞf

Re ¼

H 1 ; ¼ cp ðT f;in  T m Þ Ste 8 > <

2.3. Normalizing of mathematical model

L ¼ l=Ri ;

ð21Þ

The boundary and initial conditions are,

¼ 0; r¼Ro

ð15Þ

R ¼ r=Ri ;

Hw js60 ¼ ðT pi  T m Þ=ðT f;in  T m Þ

The model of PCM is,

Nu ¼ f ðRe; PrÞ;

¼ T pi

X ¼ x=Ri ;

ð20Þ

R¼R1

In the above equations, the related parameters are defined as:

The boundary and initial conditions are,

  @T w  @T p  kw ¼ k ; p @r r¼Rw @r r¼Rw   @T p  @T p  ¼ ¼0  @x x¼0 @x x¼l

hcw

  Pthermp Khcp @ Hp  @ Hw  ¼   ;  @R R¼R1 Pthermw Khcw @R R¼R1

where,

Ra ¼

thermw

R¼R1

ð19Þ

ðqcÞpl ; ðqcÞf

Pthermpl ¼

Khcps ¼

ðqcÞps ; ðqcÞf

apl aps ; Pthermps ¼ af af

Khcpsl ¼

ðqcÞps ; ðqcÞpl ð32Þ

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3. The energy efficiency ratio of the PCTES unit

4. Numerical method and its validation

To define a parameter that can indicate the energy efficiency ratio of the PCTES unit, the mechanical energy consumed by fluid flow of HTF inside the tube should be considered. For the present case, in the time period of tmax, the energy consumed is as follows:

4.1. Discretization of the model

W tmax ¼

Z

t max

0

pR2i U Dpdt

ð33Þ

where tmax is the time needed to complete phase change process, and

1 Dp ¼ qf U 2 lf =ð2Ri Þ 2

ð34Þ

In this period, the heat energy stored in the unit is,

Z

Q tmax ¼

t max 0

Z

2pRi hðT f  T w Þdxdt

H ð36Þ

Z

Q tmax ¼ 2pR3i St ðqcÞf ðT f;in  T m Þ smax

W tmax 0:5qf U 2

!

For PCM,

Z

ðHf  Hw ÞdXds

ð37Þ

L

¼ pR3i

Z

0:5Lf ds

Usmax ¼ 2St Wsmax

R

smax

RL

ðHf  Hw ÞdXds

smax

0:5Lf ds

smax

1 ukþ1  ukP P Ste Ds

R RL

ðHf  Hw ÞdXds

smax

0:5f ds

ð44Þ

4.2. Solving of the discretization equations Eqs. (42)–(44) are solved using the iteration method. At each time step, the liquid fraction u is updated using the method proposed by Voller [30]:

ukþ1 ¼ ukP þ P

kþ1 ap;P SteHp;P

ð45Þ

a0p;P

where,

ap;E ¼

R P DR 

;

ðdXÞe

2

Pthermp  RePr

ap;W ¼

e



2

R P DR ; ðdXÞw Pthermp  RePr

w

DX  DR  R P Ds

ap;N ¼



Rn DX ; ðdRÞn Pthermp 

2

RePr

n

R s DX 

ðdRÞs

2

Pthermp RePr

ð46Þ

ð47Þ

 s

When u is greater than 1 or smaller than 0, the liquid fraction is corrected by:

ð39Þ



ð40Þ

Eqs. (42)–(44) are closely coupled via the boundary conditions, thus they must be solved using an iteration procedure at each time step until convergence has been achieved.

The final form energy efficiency ratio of the heat storage unit is,

R



ap;S ¼

h Nu ¼ ðqcÞf U RePr

Us max 2Nu ¼ Ws max LRePr

Ds ! k k k k k k k k Pthermp Hp;E  2Hp;P þ Hp;W Hp;E  Hp;W Hp;N  2Hp;P þ Hp;S ¼2 þ þ RePr RP DR 2DX 2 2DR2

ð38Þ

smax

R

Hkp;P

ap;P ¼ a0p;P þ ap;E þ ap;W þ ap;N þ ap;S ; a0p;P ¼

Considering the relationship of St and Re, Pr and Nu,

Es max ¼

Hkw;E  Hkw;W Hkw;N  2Hkw;P þ Hkw;S þ R P DR 2DR2

ð43Þ

kþ1 p;P 

Then, the energy efficiency ratio of the heat storage process can be expressed by:

St ¼

þ

ð35Þ

The second is the ratio of mechanical energy dissipated in the time period of tmax to the dynamic pressure of HTF:

Esmax ¼

k Hkþ1 Pthermw w;P  Hw;P ¼2 Ds RePr Hkw;E  2Hkw;P þ Hkw;W

2 DX 2

The physical meaning is the heat energy stored per unit consumed of mechanical energy in the time period of tmax. It can efficiently present the energy efficiency ratio of the heat storage unit in the whole phase change process. When the phase change process is completed, etmax means the heat energy stored per unit consumed of mechanical energy in phase change process. To make a more general presentation of the energy efficiency ratio of a heat stored unit, etmax should be normalized. For convenience, we introduce two related parameters. One is the ratio of heat stored in the time period of tmax to the heat released by unit volume of HTF with temperature difference Tf,in  Tm:

Ws max ¼

ð42Þ

For the tube wall,

0

Q tmax W tmax

Us max ¼

kþ1 Hf;P  Hkf;P Hkf;E  Hkf;W Nu ¼ 2 ðHk  Hkw;P Þ RePr f;P Ds DX

l

The energy efficiency ratio of the unit can be defined as:

etmax ¼

The finite volume method is used to discretize the governing equations. For easy capturing of the discretization process, the schematic diagram of grid system is shown in Fig. 3. For HTF,

ð41Þ

As indicated by the above-mentioned normalization, the parameter Esmax keeps the physical meaning of etmax, but this is not the exact value of the ratio of heat storage of PCM to pumping energy of HTF because two other parameters, Usmax and wsmax, are introduced.

u ¼ 0; if u < 0 u ¼ 1; if u > 1

ð48Þ

4.3. Convergence criterion For a given time step, the converged results were assumed to have been reached when the following criteria have been satisfied:

       kþ1    kþ1  k   Hkp  6 1:0  108 Hf  Hkf  þ Hkþ1 w  Hw  þ Hp

ð49Þ

Furthermore, in order to verify the numerical solutions, the numerical procedure keeps the overall energy balance. That means that at any time, the stored sensible and latent energy by PCM must be

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(δX )e

(δX )w

N n (δX )e

(δX )w

W

P w

ΔX

ΔR

E

x

e

(a)

φ

W

w

P

RP Rs

(δR)n

s

e

E Rn

(δR)s

S

ΔX

(b)

Fig. 3. Schematic diagram of grid system in the shell-and-tube phase change thermal energy storage unit: (a) the grid system of HTF and (b) the grid system of PCM or tube wall.

Fig. 4. Validation results for grid independency.

equal to the change in HTF internal energy or the energy crossing the surface of the inner tube. 4.4. Selection of grid size In order to validate the solution independency of the computational grid size, six different grid sizes for the same working conditions are tested. These are 70  20, 70  30, 70  40, 100  20, 100  30, and 100  40, respectively. In the charging process, the results of the heat stored using the six different grid sizes are shown in Fig. 4. As shown in the figure, the relative solution deviations with the six different grid sizes are small. The grid size of 70  40 can be regarded as a grid size through which grid-independent results can be obtained. Thus, in the following simulation, the grid size 70  40 is adopted in the charging process. 4.5. Selection of time step To select a suitable time step, we selected three different time steps. These are 0.005, 0.01, and 0.1, respectively. In the charging

Fig. 5. Calculated time-wise variation of total heat storage using different time steps.

process, the heat stored using three different time steps is shown in Fig. 5. This figure shows that the lines of heat storage almost overlap for different Ds. Thus, the time step of 0.01 is adopted in the following simulations. 4.6. Validation of numerical method The available experimental data obtained by Lacroix [31] is used to validate the reliability of the computational model. In Lacroix’s experiment, the inner tube radius of the phase change heat storage unit is 0.00635 m. The dimensionless thickness of the inner tube’s wall is 0.24, and the dimensionless radius of the outer tube is 2.03. The dimensionless length of the unit tube is 157.48. The annular space is filled with PCM (n-Octadecane). The thermal properties of PCM and the tube are given in Table 2. PCM melting temperature is 300.7 K, and the temperature differences between the inlet of HTF and the melting point of PCM are 5, 10, and 20 K, respectively. The thermal properties of HTF are given in Table 1. The mass flow rate of HTF is 0.0315 kg/s, which corresponds to the water velocity 0.25 m/s. In the charging process, the initial

W.-W. Wang et al. / Applied Energy 138 (2015) 169–182

175

Fig. 6. Time-wise variation of PCM temperature: numerical data (solid lines) versus experimental data (points): (a) Tf,in  Tm = 5 K, (b) Tf,in  Tm = 10 K, and (c) Tf,in  Tm = 20 K.

temperature of PCM is 18 K below the melting point of PCM. Two copper-constantan thermocouples are installed inside PCM at locations A (X = 80.31, R = 1.56) and B (X = 149.61, R = 1.40). Correspondingly, the temperatures of the two above-mentioned monitory points are T1 and T2, respectively. Fig. 6a–c show the comparisons between experimental data and numerical data. As shown in these figures, the agreement between the present numerical results and the experimental data is well within the experimental uncertainties. These results indicate that the present computational model can be used for investigating the characteristics of the phase change heat storage unit.

5.1. The characteristics of the energy efficiency ratio From Eqs. (33)–(36), the energy efficiency ratio of the PCTES unit is related with time. There are two definitions. One is the transient energy efficiency ratio at time point t:

Rl

eðtÞ ¼

0

2pRi hðT f  T w Þdxdt

pR2i U Dpdt

ð50Þ

The second is the average energy efficiency ratio over a period from the beginning of the process to the time point mentioned:

Rt Rl 0

0

2pRi hðT f  T w Þdxdt Rt 2 pRi U Dpdt 0

5. The energy efficiency ratio of the PCTES unit

et ¼

Based on the energy efficiency ratio of the PCTES unit defined in Eq. (41), we numerically investigated the characteristics of the energy efficiency ratio in many cases. The results and discussions will be presented in the following sections.

When the charging process is completed, it presents the overall energy efficiency ratio of the PCTES unit. The time-wise variations of the PCTES unit energy efficiency ratio during the charging process are shown in Fig. 7a and b. With

ð51Þ

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Fig. 7. Time-wise variations of e(t) (a) and et (b).

increasing time, e(t) quickly increases and reaches its maximum value. Then it decreases gradually, finally equaling zero. It is the same as the time-wise variations of the heat storage rate [21], and it goes through three distinct stages. In the first stage, because the PCM temperature is lower than phase change temperature, the heat is stored as sensible heat, the heat storage rate rapidly increases when phase change begins, and the transient mechanical energy consumed by fluid flow of HTF inside the tube is constant. Thus, e(t) rapidly increases and reaches its maximum value, which is about 3  106. The time period of the first stage is about 0.3 min. In the second stage, more of PCM changes into liquid, the heat is stored as latent heat, the heat transfer resistance increases gradually, and the heat storage rate begins to decrease [21]. During this stage, the heat storage rate decreases gradually, the transient mechanical energy consumed by fluid flow of HTF inside the tube is also constant, and e(t) decreases gradually. This time period is about 38 min, and e(t) decreases to about 0.12  106. A larger decrease of e(t) indicates the beginning of the third stage. In the third stage, when the liquid fraction of PCM is equal to 1, the phase change process is complete, the heat is stored as sensible heat, and e(t) decreases slowly, finally reaches its minimum value. With increasing time, et quickly increases and reaches its maximum value. Then it decreases gradually. When the liquid fraction of PCM equals 1, the phase change process is complete and et decreases slowly. After the phase change process is complete, the heat is only stored as PCM liquid sensible heat. The energy storage in this process is very small, but this process consumes too much mechanical energy. In the present study, the heat storage and mechanical energy consumption are not considered after the phase change process is complete. At the time point of tmax, the average energy efficiency ratio in the time period of 0 to tmax is the average energy efficiency ratio of the PCTES unit. In the same way, from Eqs. (37)–(41), that the energy efficiency ratio of the heat storage unit relates with the dimensionless time, there are also two definitions. One is the transient energy efficiency ratio of the heat storage unit at the dimensionless time point s:

EðsÞ ¼

2Nu LRePr

R

L

ðHf  Hw ÞdXds 0:5f ds

ð52Þ

The second is the average energy efficiency ratio of the heat storage unit averaged for the dimensionless time from the beginning of process to the time point mentioned.

Es ¼

R R 2Nu s L ðHf  Hw ÞdXds R LRePr s 0:5f ds

ð53Þ

Fig. 8 shows the dimensionless time-wise variations of E(s) and Es. With the increase of s, E(s) quickly increases and reaches its maximum value. Then it decreases gradually, until finally it equals zero, as in Fig. 8a. It is the same as shown in Fig. 7a, and it also goes through three distinct stages. In the first stage, the maximum value of E(s) is about 0.144. In the second stage, E(s) decreases to about 0.006. The first stage ends within 160, and the second stage ends within 23  103. At the third stage, up to which the liquid fraction of PCM is equal to 1, the phase change process is complete. E(s) decreases slowly and finally equals zero. As Fig. 8b shows, Es has the same time-dependent characteristics as shown in Fig. 7b. As s increases, Es quickly increases and reaches its maximum value, and then decreases gradually. When the liquid fraction of PCM is equal to 1, Es decreases slowly.

5.2. The effects of working conditions on the energy efficiency ratio The effect of HTF initial inlet temperature on Esmax is shown in Fig. 9. The results show that as initial inlet temperature difference increases from 5 K to 25 K, the HTF initial inlet temperature has little effect on Esmax. This is because the parameter Usmax is the heat stored per unit temperature difference and per unit volumetric specific heat capacity of HTF in the time period of smax. The other parameter, wsmax, is the mechanical energy dissipated per unit dynamic pressure of HTF in the time period of smax. This shows that Esmax has more generality. The effect of Re on Esmax is shown in Fig. 10. With the increase of Re, Esmax first increases and then decreases. When Re equals about 2100, the maximum value of Esmax is about 0.025. Re is augmented as the inlet velocity of HTF increases, which leads to the enhancement of the heat transfer coefficient of the tube wall, and Nu increases, but Esmax is not monotonous with increasing Re. This indicates that the PCTES unit has its optimal working condition.

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Fig. 8. Dimensionless time-wise variations of E(s) (a) and Es (b).

Fig. 9. Effect of HTF initial inlet temperature on Esmax.

5.3. The effects of the unit configurations on the energy efficiency ratio Fig. 11 shows the effect of L on Esmax. The PCM quantity in the tube is augmented with the increase of L, which leads to the increase of thermal storage capacity, but Esmax reduces. As L increases from 50 to 300, Esmax decreases from 0.026 to 0.020. The decrease is about 23.1%. However, with the increase of L, the reduction gradient of Esmax weakens gradually. It can be deduced that when the length of the tube reaches a certain value, it will have little effect on Esmax. Fig. 12 shows the effect of R2 on Esmax. As R2 increases, the quantity of PCM in the tube augments, the thermal storage capacity increases, and Esmax is reduced. When R2 increases from 1.3 to 5.0, Esmax is reduced from 0.061 to 0.008. R2 is increased by 3.7 times; Esmax is reduced by about 86.9%. However, with further increase of R2, the effect of R2 on Esmax is weakened. These results

Fig. 10. Effect of Re on Esmax.

imply that when the radius of the outer tube reaches a certain value, it will have little effect on the energy efficiency ratio of the PCTES unit. The above results show that when L increases by about five times, Esmax is reduced by about 23.1%; when R2 increases by about three times, Esmax reduces by about 86.9%. Therefore, the effect of R2 on Esmax is stronger than that of L. These results suggest that more attention should be paid to the role of the tube radius in optimizing the design of the single shell-and-tube phase change heat storage unit. 5.4. The effects of the material properties of HTF and PCM on the energy efficiency ratio Under the same inlet temperature, the effect of HTF physical properties on Esmax is shown in Fig. 13. From this figure, it can be

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Fig. 13. Effect of Pr on Esmax.

Fig. 11. Effect of L on Esmax.

PCM is Methyl palmitate, the value of Esmax is smallest, at about 0.004. When PCM is n-Octadecane, the value of Esmax is about 0.022. Fig. 14 shows the effect of Ste on Esmax. Among these six phase change materials, Ste of Polyglycol E600 is largest, and Ste of Gallium is smallest. The values are about 0.161 and 0.046, respectively. Fig. 15 shows the effect of Pthermp on Esmax. Pthermp of Gallium is largest, and Pthermp of Methyl palmitate is smallest. The values are about 21.808 and 0.109, respectively. Fig. 16 shows the effect of Khcp on Esmax. Among these six phase change materials, Khcp of Paraffin C18 is largest, and Khcp of Methyl palmitate is smallest. It is about 0.614 and 0.41, respectively. From these figures, it can be seen that Esmax is negatively correlated with these three characteristic numbers. Thus, we cannot use the three characteristic numbers as the PCM property factors affecting Esmax. Then, we get a characteristic formula by combining these three characteristic numbers. The effect of StePthermp/Khcp on Esmax is shown in Fig. 17. As StePthermp/Khcp increases, Esmax increases. Therefore, the combined parameter StePthermp/Khcp is PCM property parameter that affects Esmax of the single shell-and-tube phase change heat storage unit. Fig. 12. Effect of R2 on Esmax.

seen that with increasing Pr, Esmax quickly decreases. Then, it slowly decreases after Pr becomes greater than 10. Finally, it reaches a point where it hardly changes. At Pr = 0.696, HTF is air, and Esmax is at its largest, at about 0.183. At Pr = 5.182, HTF is water, and Esmax is about 0.023. When Pr = 56.56, HTF is heat conduction oil, and Esmax is lowest, at about 0.007. When air is used as HTF, although the PCTES unit energy efficiency ratio can reach a high value, it takes a very long time to complete the phase change process, such as the melting time is about 680 min. When HTF is water, the melting time is about 38 min. Therefore, using only Esmax as the parameter to carry out an optimum design of the single shell-and-tube PCTES unit is not sufficient. The heat storage rate of the PCTES unit should be considered at the same time. Under the same working conditions, the effects of PCM physical properties on Esmax are shown in Figs. 14–17, respectively. When PCM is Gallium, the value of Esmax is largest, at about 0.053. When

6. The heat storage rate of the PCTES unit The above results indicate that using the energy efficiency ratio of a PCTES unit as the only parameter to evaluate the heat storage performance of the PCTES unit is not sufficient. It is important that the heat is stored by PCM per unit time in the phase change process. Based on heat energy stored in the unit defined in Eq. (35), we defined the heat storage rate of the PCTES unit in the time period of tmax. It follows that,

qtmax ¼

Q tmax ¼ t max

R tmax R l 0

0

2pRi hðT f  T w Þdxdt t max

ð54Þ

The physical meaning is the heat energy stored by PCM per unit time in the time period of tmax. It presents the heat storage rate of the PCTES unit through the whole phase change process. We numerically investigated the characteristics of qtmax under various parameters. The results and discussions will be presented in the following sections.

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Fig. 14. Effect of Ste on Esmax. Fig. 16. Effect of Khcp on Esmax.

Fig. 15. Effect of Pthermp on Esmax. Fig. 17. Effect of StePthermp/Khcp on Esmax.

6.1. The effects of working conditions on the heat storage rate The effect of HTF initial inlet temperature on qtmax is shown in Fig. 18. The increase of the inlet temperature leads to a decrease in the melting time, but the heat stored by PCM is same. qtmax increases from 18 W to 90 W as the initial inlet temperature difference of HTF increases from 5 K to 25 K. However, as shown in Fig. 9, the initial inlet temperature difference of HTF has little effect on Esmax. These results indicate that when the initial inlet temperature of HTF is larger, the heat storage property of the PCTES unit is better. The effect of Re on qtmax is shown in Fig. 19. With increasing Re, qtmax increases. When Re changes from 200 to 4200, qtmax increases from 24 W to 52 W. Increasing Re leads to the enhancement of the heat transfer coefficient of the tube wall. Then, the melting time decreases and qtmax of the PCTES unit increases. When Re equals

about 2100, Esmax of the PCTES unit reaches its maximum value. When Re is reduced from 2100 to 200, Esmax is reduced from 0.025 to 0.015, thus being reduced by about 40.0%. Also, qtmax is reduced from 40 W to 24 W; thus, it is reduced by about 65.0%. When Re increases from 2100 to 4200, Esmax is reduced from 0.025 to 0.013 (about 48.0%), but qtmax increases from 40 W to 52 W (an increase of about 30.0%). These results suggest that when the HTF flow is laminar, the heat storage property of the PCTES unit is better. 6.2. The effects of the unit configurations on the heat storage rate Fig. 20 shows the effect of L on qtmax. The PCM quantity in the tube is augmented with the increase of L, which increases the thermal storage capacity. The melting time increases. However, the

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Fig. 18. Effect of HTF initial inlet temperature on qtmax. Fig. 20. Effect of L on qtmax.

PCM has larger effect than the axial thermal resistance of PCM on the heat storage property of the PCTES unit. 6.3. The effects of the material properties of HTF and PCM on the heat storage rate

Fig. 19. Effect of Re on qtmax.

increases of the thermal storage capacity and the melting time are different. When L increases from 50 to 300, qtmax of the PCTES unit increases from 13 W to 60 W (about 361.5%). But Esmax of the PCTES unit is reduced from 0.026 to 0.020 (about 23.1%). It appears that when the tube is longer, the heat storage property of the PCTES unit is better. Fig. 21 shows the effect of R2 on qtmax. As R2 increases, the quantity of PCM in the tube is augmented. The thermal storage capacity increases. The PCM melting time increases inherently. As R2 changes from 1.3 to 5.0, qtmax decreases from 99 W to 13 W. This shows that increasing R2 of the tube can reduce qtmax. However, with further increase of R2, the decrease rate of qtmax is reduced. The above results show that when L of the tube increases by about five times, qtmax of the PCTES unit increases by about 361.5%. When R2 of the tube increases by about three times, qtmax decreases about 661.5%. Therefore, qtmax is less affected by L than by R2. These results show that the radial thermal resistance of

Under the same inlet temperature, the effect of HTF physical properties on qtmax is shown in Fig. 22. When HTF is an Ethylene glycol solution of 80 v/v%, qtmax of the PCTES unit is the largest, at about 38 W. When HTF is air, qtmax of the PCTES unit is smallest, at about 2 W. When air is used as HTF, although the energy efficiency ratio can reach a high value, it takes a very long time to complete the heat charging process. When HTF is water, qtmax of the PCTES unit is large, at about 36 W. Therefore, combined with the economic cost and the heat storage rate, water should be used as HTF. Under the same working conditions, the effect of different PCM physical properties on qtmax is shown in Fig. 23. As StePthermp/Khcp increases, qtmax of the PCTES unit increases. In all of the studied cases, when PCM is Gallium, Methyl palmitate, or n-Octadecane, qtmax is about 85 W, 7 W, and 36 W, respectively. Therefore, when the parameter StePthermp/Khcp of PCM is larger, the heat storage property of the PCTES unit is better. It can be deduced that the heat storage rate and the energy efficiency ratio of the PCTES unit are affected by many thermal properties of PCM and HTF, such as latent heat, specific heat capacity, thermal conductivity, and density. The above-mentioned results indicate that the energy efficiency ratio and the heat storage rate of the PCTES unit are two important parameters to carry out an optimum design of the single shell-andtube PCTES unit. 7. Conclusion In the present paper, we have focused on developing a parameter that can be used to indicate the energy efficiency ratio of the single shell-and-tube phase change heat storage unit. Based on the energy efficiency ratio defined, the sensitivity of the energy efficiency ratio on HTF initial inlet temperature, HTF working conditions, the unit structure size, and the material properties of HTF and PCM are numerically investigated, respectively. The following conclusions can be derived:

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Fig. 21. Effect of R2 on qtmax.

181

Fig. 23. Effect of StePthermp/Khcp on qtmax.

energy efficiency ratio and the heat storage rate are reduced. Thus, more attention should be paid to the outer tube’s radius in optimizing design. (4) Under the same working conditions, the heat storage rate is 2 W, 36 W, and 38 W for air, water, and an Ethylene glycol solution of 80 v/v%, respectively, but the energy efficiency ratio of the PCTES unit is 0.183, 0.023, and 0.005, correspondingly. These indicate that water should be used as HTF. (5) The parameter StePthermp/Khcp can indicate the major effects of the physical properties of PCM and HTF on the energy efficiency ratio of the PCTES unit.

Acknowledgments This work was supported by the China National Hi-Tech R&D (863 Plan) Project (2013AA050502), the National Key Basic Research Program of China (973 Program) (2013CB228304), and the National Natural Science Foundation of China (No. 51176155). References Fig. 22. Effect of Pr on qtmax.

(1) Although two other dimensionless parameters are introduced in the definition of the energy efficiency ratio of the PCTES unit, it captures successfully the physical meaning of the heat energy stored per unit of mechanical energy consumed in the whole phase change process. (2) Increasing HTF inlet velocity can enhance Re and Nu, but the two numbers restrict each other for the energy efficiency ratio and the heat storage rate of the PCTES unit. Therefore, when the fluid flow of HTF is laminar, the energy efficiency ratio of the PCTES unit is larger. (3) As the length of the tube increases, the energy efficiency ratio is reduced, but the heat storage rate increases. As the radius of the PCTES unit’s outer tube increases, both the

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