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Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part II: Unsteady cases H.J. Xu a,b, C.Y. Zhao a,⇑ a b

Institute of Engineering Thermophysics, Shanghai Jiao Tong University, Shanghai 200240, China College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao 266580, China

a r t i c l e

i n f o

Article history: Received 1 April 2016 Received in revised form 14 September 2016 Accepted 15 October 2016 Available online xxxx Keywords: Cascaded thermal storage Optimization Exergy Entropy Entransy Unsteady case

a b s t r a c t Cascaded latent thermal storage can find its applications in renewable thermal energies with timedependent temperatures. This paper presents thermal performance of cascaded heat storage with unsteady inlet temperature of heat transfer fluid (HTF). The optimization of temperatures of HTF and phase-change materials (PCMs) is performed based on exergy, entropy, and entransy. Corresponding analytical/numerical solutions with these concepts are obtained. The qualifications for existence of optimization solutions are proposed. The optimization result of unsteady case is compared with that of steady case. The fluctuation of HTF temperature is transferred along the HTF flow path. The HTF temperatures in different stages exhibit similar fluctuating trend, but the fluctuating amplitude is diminished along the HTF flow path. With an increase in stage number, the difference between temperatures of outlet HTF and environment is gradually decreased with obviously improved thermal performance. As NTU increases, thermal performance is gradually increased, but with decreased increasing amplitude. The optimization result with entransy is also compared with that of entropy. The optimal exergy efficiency based on entropy is greater than that based on entransy, while the optimal thermal efficiency based on entransy is superior to that based on entropy. The optimization can be applied to select PCMs for unsteady cascaded thermal storage. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Thermal energy storage can be used for solar thermal utilization [1], electricity peak shaving [2], waste heat recovery [3], LNG cold energy utilization [4], control of thermal environment in buildings [5], geothermal energy utilization [6], thermal management of electronics [7], and so on. Among the three main thermal storage methods (sensible [8], latent [9] and chemical [10] storages), latent thermal storage with phase-change material (PCM) owns the advantages of reliable safety, relatively large energy density, and good technology maturity. Thus, the potential of this kind of storage method for large-scale industrial thermal energy storage can be expected. Recently, the cascaded thermal storage technique using multiple PCMs of different phase-change temperatures has received much attention. Compared to single stage thermal storage, the superiority of cascaded storage can be found in four aspects: (1) the temperature difference in cascaded thermal storage can be homogenized with enhanced driving forced of heat ⇑ Corresponding author. E-mail addresses: [email protected] (H.J. Xu), [email protected] (C.Y. Zhao).

transfer especially in the end storage section; (2) thermal energy in heat transfer fluid (HTF) can be sufficiently used by decreasing the difference between temperatures of outlet HTF and environment; (3) cascaded thermal storage improves the temperature range for practical operation of thermal storage; (4) multi grade thermal energies are supplied in cascaded thermal storage. Recent research studies on cascaded thermal storage [11–18] are summarized in Part I of this paper. Up to now, most publications on system optimization and analysis for cascaded thermal storage are focused on the steady-state working condition. Yet, in practical thermal energy utilization, the temperature of heat source or cold source is fluctuated with time, more or less. Little work has been done for thermal storage process with HTF of unsteady inlet temperature [19,20]. Tao et al. [19] numerically investigated the effect of unsteady inlet temperature on singlestage thermal storage unit. Li et al. [20] performed a thermal analysis for two-stage thermal storage for solar thermal utilization. If the cascaded thermal storage system is optimized with steadystate optimization, thermal performance of the cascaded thermal storage system would be poor than the optimal value in unsteady case. To this end, it is essential to establish the optimization

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.066 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: H.J. Xu, C.Y. Zhao, Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part II: Unsteady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.066

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Nomenclature A cp E Eu e Ex Exu m n NTU Q Qu S Su Sg Sgu s T T0 Te t U X

heat transfer area, m2 1 heat capacity at constant pressure, J kg K1 entransy flow, W K entransy in unsteady case, J K 1 specific entransy, J K kg exergy flow, W exergy in unsteady case, J mass flow rate, kg s1 stage number of cascaded thermal storage system number of heat transfer units heat transfer rate, W heat transfer in unsteady case, J entropy flow, W K1 entropy in unsteady case, J K1 entropy generation rate, W K1 entropy generation in unsteady case, J K1 1 specific entropy, J kg K1 temperature, K inlet HTF temperature, K environment temperature, K time, s overall heat transfer coefficient, W m2 K1 dimensionless position along HTF flow path

Greek symbols h dimensionless temperature H fixed heat transfer rate, W Hu fixed heat transfer in unsteady case, J k; ku Lagrange multiplier (steady, unsteady), K1

procedure of cascaded thermal storage with unsteady thermal energy input. As to optimization of cascaded thermal storage, three concepts in thermodynamics can be employed: exergy, entropy, and entransy. Exergy represents the maximum useful work [21], and entropy stands for the irreversibility of conversion of heat to mechanical work. These two concepts are correlative with each other. Entransy is a new parameter created by Guo et al. [22] for describing the heat transfer irreversibility. In our previous work [23,24], the concepts of exergy and entransy were used to optimize the cascaded thermal storage system, respectively. In the companion part of this paper (Part I), the steady cascaded thermal storage system is optimized with entropy. In this paper (Part II), the optimizations of cascaded thermal storage system with concepts of exergy, entropy, and entransy are conducted for unsteady case. The optimization in unsteady case is compared with that in steady case. Effects of stage number, number of transfer units, and storage time on thermal performance of unsteady cascaded thermal storage system are presented. The optimization results of exergy, entropy and heat transfer combination, and, entransy and heat transfer combination, are compared with each other.

l; lu n; nu r; ru s

/ /u

U

Uu X Xu

Lagrange multiplier (steady, unsteady), K Lagrange multiplier (steady, unsteady), K Lagrange multiplier (steady, unsteady), K1 storage time, s entransy dissipation rate, W K entransy dissipation in unsteady case, J K fixed entransy dissipation rate in entransy optimization, WK fixed entransy dissipation in entransy optimization of unsteady case, J K fixed entropy generation rate in entropy optimization, W K1 fixed entropy generation in entropy optimization of unsteady case, J K1

Subscripts av time-average E entransy e environment HTF heat transfer fluid m melting max maximum min minimum opt optimal PCM phase-change material pr practical S entropy tot total u unsteady

exchanges thermal energy with these PCMs. Taking the solar collector as an example, the temperature of heat transfer fluid for collecting the heat absorbed from solar radiation depends on the varying intensity of solar heat flux. The total storage time is s and the environment temperature is T e . For this unsteady working condition, the fluctuating inlet HTF temperature is known and the varying inlet HTF temperature with time is an important factor affecting the final optimization result. To vividly present the optimization result of unsteady case, the fluctuating inlet HTF temperature as a function of time is set as quadratic in Eq. (1).

T 0 ðtÞ ¼

1 ðt 21; 600Þ2 þ 1000 2; 332; 800

ð1Þ

Accordingly, the default variation of inlet HTF temperature with time is shown in Fig. 2. This means that the HTF fluctuates from 800 K to 1000 K in the form of quadratic function of time (12 h),

2. Description of unsteady working condition The cascaded thermal storage system in steady case is described in Part I of this paper. In practical thermal energy utilization, the temperature of thermal source is usually fluctuant. The schematic diagram of the cascaded thermal storage system with fluctuating inlet HTF temperature is shown in Fig. 1. HTF with unsteady inlet temperature flows through the cascaded thermal storage unit with multiple PCMs of different phase-change temperatures, and

Fig. 1. Cascaded thermal storage system with fluctuating inlet temperature of HTF.

Please cite this article in press as: H.J. Xu, C.Y. Zhao, Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part II: Unsteady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.066

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Fig. 2. Default variation of HTF inlet temperature with time.

which is the case of solar thermal collection with medium temperature. To mathematically model this problem, the basic assumptions are proposed here. (1) Bulk-phase assumption (lumped parameter method) is used for PCM; (2) thermo-physical properties are considered as constant; (3) the PCM sensible heat is neglected; (4) undercooling/superheating in thermal storage/release of PCMs is neglected; (5) the crosssectional distribution of HTF temperature is uniform; (6) the fluctuation amplitude of inlet HTF temperature is much less than the temperature difference between heat source and environment. Based on the thermal balance equation of the ith-stage storage unit, the recursion equation between HTF and PCM temperatures is obtained as

Ti ¼

1 Ci 1 T i1 þ T mi Ci Ci

16i6n

ð2Þ

The HTF temperature is expressed with PCM temperature, as

T0 T i ¼ Qi

j¼1 C j

þ

i X Cj 1 T mj Qi j¼1 l¼j C l

16i6n

ð3Þ

Thus, the heat transfer rate in the ith-stage is

Q i ¼ mcp ðT i1 T i Þ 0

1

Q u;tot ¼ Q u;HTF

s

C j¼1 j

Rs 0

C l¼j l

T 0 dt

s

i

Te ð7Þ

3. Performance optimizations of unsteady cascaded thermal storage The advantages of cascaded thermal storage can be found in two aspects. Firstly, compared with single-stage thermal storage, the temperature difference of cascaded thermal storage system can be homogenized by adjusting the PCM temperature with increased stage number. Secondly, thermal energy embedded in HTF can be used more sufficiently by reducing the difference between temperatures of HTF and environment. Thus, PCM temperature optimization of multi-stage is a critical issue of cascaded thermal storage, which requires the concepts of thermodynamics, such as exergy, entropy, and entransy. Related optimizations will be given in the following section.

In the cascaded thermal storage system, the exergy flow of the ith-stage unit is

ð4Þ

Te Exi ¼ mcp ðT i1 T i Þ 1 T mi

! i1 X Cj 1 1 Te þ T mj T mi ¼ mcp ðC i 1Þ Qi 1 Qi Ci T mi j¼1 j¼1 C j l¼j C l T0

l¼j

The heat transfer in the ith-stage with storage time from 0 to

s

is

Q u;i ¼

g1st ¼

3.1. Exergy optimization

C B i1 X C B T0 Cj 1 1 C 16i6n ¼ mcp ðC i 1ÞB þ T T mj mi C BY i i Ci Y A @ j¼1 Cj Cl j¼1

! Rs Xi1 C 1 T 0 dt j 1 0 ðC i 1Þ Qi þ Qi T mj C T mi i¼1 j¼1

Xn

ð8Þ

When the inlet HTF temperature varies with time, the ith-stage PCM exergy is calculated as

Z s Q i dt

ð5Þ

! i1 T 0 dt X Cj 1 1 ¼ Exi dt ¼ mcp sðC i 1Þ Qi þ T mj T mi Q Ci 0 s j¼1 C j j¼1 il¼j C l Te ð9Þ 1 T mi

The overall heat transfer in the cascaded thermal storage system can be obtained by summing the heat transfer in each stage.

The total exergy of the cascaded thermal storage system with multi-stages is

0

"R s

T dt 0 0 ¼ mcp ðC i 1Þ Q þs i j¼1 C j

i1 X j¼1

Cj 1 1 T mj T mi Qi C i C l¼j l

16i6n

Q u;tot ¼

Exu;i

n X Q u;i

Exu;tot ¼

i¼1

"R s !# n i1 X X T dt Cj 1 1 0 0 ¼ mcp ðC i 1Þ Qi þs T mj T mi Qi Ci i¼1 j¼1 j¼1 C j l¼j C l

Rs

Z s

!#

0

n n X X Exu;i ¼ mcp s ðC i 1Þ i¼1

ð6Þ

The thermal efficiency based on the first law of thermodynamics can be defined as

i¼1

! i1 T dt X Cj 1 1 Te 0 0 1 þ T mj T mi Qi Qi Ci T mi s j¼1 C j j¼1 l¼j C l Rs

ð10Þ

The exergy efficiency based on the second law of thermodynamics is

Please cite this article in press as: H.J. Xu, C.Y. Zhao, Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part II: Unsteady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.066

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g2nd ¼

Exu;tot Exu;HTF

! Rs Xi1 C 1 T 0 dt j Te 1 0 ðC 1Þ Q þ Q T T 1 i mj mi i i T mi Ci i¼1 j¼1

The entropy generation rate can be expressed with the temperatures of PCMs, as

Xn

s

¼

C j¼1 j

C l¼j l

Rsh 0

i

T 0 T e þT e ln

s

Te T0

Sg;i

dt

2 0 1 6 B C 6 B1 C 1 C T mi 6 i C ¼ mcp 6lnB þ C i1 6 B C Ci X C 1 A T0 j 4 @ i Qi1 þ Qi1 T mj C j¼1 j

ð11Þ The condition for the maximum total exergy is

@Exu;tot @ ¼ mcp s @T mk @T mk ! " Rs # n i1 X T dt X Cj 1 1 Te 0 0 ðC i 1Þ Q þ T T 1 Q mj mi i i Ci T mi i¼1 j¼1 j¼1 C j l¼j C l ¼0 16k6n

8 3 2 > > > > 7 6 > k1 < T 6 R s T dt X C j 1 T mj 17 0 7 e 6 ¼ mcp sðC k 1Þ þ 7 6 0 Qk k > T mk 6sT mk l¼1 C l j¼1 Y T mk C k 7 > > 5 4 > Cl > : l¼j X ) n 1 Te Cj 1 Te þ ¼0 16k6n 1 1 Qj Ck T mk T mj Cl j¼kþ1 l¼k

ð13Þ Eq. (13) cannot be solved analytically and the optimal values of T mk ð1 6 k 6 nÞ can be obtained by numerical method. From Eq. (10), there is no significant influence of inlet HTF temperature fluctuation on the total exergy, which is generally in accordance with Ref. [23]. This is verified by Table 1 of cascaded heat storage (n = 2, 3, 4, 5) for the comparison of exergy optimization between steady and unsteady cases. With exergy for optimization, the numerical result of unsteady case is in coincidence with the result of steady case in Ref. [23]. 3.2. Entropy optimization Based on the entropy balance equation for a simple open system, the entropy generation rate of the heat transfer process in the ith-stage unit of cascaded thermal storage is

Sg;i ¼ mcp ln

Ti mcp ðT i1 T i Þ þ T mi T i1

ð14Þ

Table 1 Exergy optimizations in steady and unsteady cases (C i ¼ 11:78, T 0;av ¼ 933:33 K). Stage number

PCM temperature

Unsteady case (K)

Steady case (K)

n=2

Tm1 Tm2

617.07 413.15

617.07 413.15

n=3

Tm1 Tm2 Tm3

675.64 504.93 377.35

675.62 504.91 377.34

Tm1 Tm2 Tm3 Tm4

709.21 565.60 451.07 359.63

709.01 565.41 450.89 359.57

Tm1 Tm2 Tm3 Tm4 Tm5

729.90 607.02 504.88 419.91 349.25

729.93 607.09 504.91 419.94 349.26

n=4

n=5

T mi

l¼j

i1 X C j 1 T mj þ Qi T mi C j¼1 j¼1 j l¼j C l

T0 Qi

Cl

!

3 7 C i 17 7 7 Ci 7 5

ð15Þ

The entropy generation for the ith-stage thermal storage unit is

ð12Þ

The above equation can be simplified as

@Exu;tot @T mk

þðC i 1Þ

j¼1

Sgu;i ¼ mcp

8 > > > >

1

0

B B 1 Ci 1 ln B BC i þ C i @

0

T mi 0 þ QTi1 j¼1

Ci 1 þ Qi j¼1 C j

Cj

Pi1 j¼1

C 1

Qji1 T mk l¼j

Rs 0

Cl

C C !C Cdt A

9 > > !> > 1 =

i1 X T 0 dt C j 1 T mj þ sðC i 1Þ Qi T mi T mi C i > > j¼1 l¼j C l > > ;

ð16Þ

The total entropy generation can be calculated as 0 1 2 n n 6Z s X X B 1 Ci 1 C T mi Cdt Sgu;tot ¼ Sgu;i ¼ mcp 6 ln B Pi1 C j 1 @C i þ C i A 4 T0 0 Qi1 þ j¼1 Qi1 T mj i¼1 i¼1 C j¼1 j

3 ! Rs Xi1 C j 1 T mj 1 7 C i 1 0 T 0 dt 7 þ sðC i 1Þ þ Qi Qi j¼1 T mi C i 5 T mi C j¼1 j l¼j C l

l¼j

Cl

ð17Þ

The total entropy generation is combined with the total heat transfer for optimization, and two kinds of optimizations are performed: heat transfer optimization with fixed entropy generation, and entropy generation optimization with fixed heat transfer. 3.2.1. Heat transfer optimization with fixed entropy generation For heat transfer optimization with fixed entropy generation, the Lagrange function can be established as

F u;S ðT m1 ; T m2 ; . . . ; T mi ; . . . ; T mn ; nu Þ ¼ Q u;tot þ nu ðSgu;tot Xu Þ "R s !# n i1 X X T dt Cj 1 1 0 0 þs T mj T mi ¼ mcp ðC i 1Þ Qi Qi Ci i¼1 j¼1 j¼1 C j l¼j C l 8 2 1 0 > > > > 6 > C B < n 6Z s X C B 1 Ci 1 T mi 6 Cdt þ nu mcp 6 ln B þ C B i1 > 6 C C X i i 0 > A @ C j 1 i¼1 4 T0 > > þ T Q Q mj > i1 i1 : Cj Cl j¼1

Ci 1 þ Qi j¼1 C j

Rs 0

j¼1

i1 X T 0 dt C j 1 T mj 1 þ sðC i 1Þ Qi T mi T mi C i j¼1 l¼j C l

l¼j

3

9 > > > > > =

!7 7 7 7 Xu > 7 > > 5 > > ; ð18Þ

The Lagrange equations are

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@F u;S @T mk

!# 8 n " Rs i1 X C 1 R s X T 0 dt > C j 1 C i 1 > 1 i 0 > T dt þ ðC 1Þ s T T þ þ Q Q Q i mj i i i > 0 0 C i mi T mi > Cj Cl Cj > j¼1 l¼j j¼1 < i¼1 j¼1 0 1 2 @ ¼ mcp n i1 X X @T mk > > C 1 C 6R s B 1 C i 1 T mi @ > > Qji Pi1 Adt þ sðC i 1Þ 4 0 ln @Ci þ C i C j 1 > mcp nu @T mk T0 > C þ T mj Q Q : i1 i1 l¼j l j¼1 i¼1 j¼1 C j¼1 j

2 n 6Z X

l¼j

Cl

C T mi C Cdt Pi1 C k 1 A 0 0 þ k¼1 QTi1 Qi1 T mk Cj Cl j¼1 l¼k 3 ! Rs i1 X C i 1 0 T 0 dt C j 1 T mj 1 7 7 þ Qi þ sðC i 1Þ 7 Xu ¼ 0 Qi T T mi C i 5 mi j¼1 j¼1 C j l¼j C l

@F u;S 6 ¼ mcp 6 4 @nu i¼1

B1 C 1 B i ln B þ @C i Ci

ð20Þ The variables for PCM temperatures ðT m1 ; T m2 ; . . . ; T mn Þ are implicit in the Lagrange equations, which cannot be analytically obtained. The numerical solution of heat storage optimization with fixed entropy generation can be derived by comprehensively solving the Lagrange equations with the (n + 1) variables. The numerical result of this optimization in unsteady case for cascaded heat storage (n = 2, 3, 4, 5) is shown in Table 2 with comparison between steady and unsteady cases. From Table 2, the numerical value of optimal PCM temperature for unsteady case is higher than that for steady case and the Lagrange multipliers for these two conditions are also different from each other. This means, the heat optimization in unsteady case, different from the steady case, is essential for cascaded thermal storage system with fixed entropy generation. 3.2.2. Entropy generation optimization with fixed heat transfer If the entropy generation is treated as the objective function and total heat transfer is treated as the constraint, the corresponding Lagrange function is established as

Gu;E ðT m1 ; T m2 ; . . . ; T mn ; lu Þ ¼ Sgu;tot þ ru ðQ u;tot Hu Þ 2 0 6Z B n 6 s X B 1 Ci 1 6 ¼ mcp 6 ln B BC i þ C i 6 0 @ i¼1 4 QT 0

i1

C j¼1 j

(

1 C C T mi Cdt C i1 X A C j 1 þ Qi1 T mj j¼1

l¼j

Cl

"R s !# ) n i1 X X T dt Cj 1 1 0 0 ðC i 1Þ Q þ s T T H Q u mj mi i i Ci i¼1 j¼1 j¼1 C j l¼j C l ð21Þ

The corresponding Lagrange equations are

@Gu;S @T mk

! ¼0 16k6n > 7> > C1i 5 > > > ;

ð19Þ

Table 2 Heat optimizations in steady and unsteady cases with fixed entropy generation ðXu ¼ 2160 J K1 ; X ¼ 0:05 W K1 ; T 0;av ¼ 933:33 K. Stage number

PCM temperature

Unsteady case (K)

Steady case (K)

n=2

Tm1 Tm2

709.09 542.94

706.74 539.41

n=3

Tm1 Tm2 Tm3

743.726 605.467 492.911

741.68 602.31 489.13

n=4

Tm1 Tm2 Tm3 Tm4

764.479 647.937 549.163 465.439

762.62 645.08 545.65 461.55

n=5

Tm1 Tm2 Tm3 Tm4 Tm5

778.188 678.305 591.219 515.319 449.161

776.45 675.66 587.96 511.63 445.22

"R s !# n i1 X X T dt @Gu;S Cj 1 1 0 0 ¼ mcp ðC i 1Þ Qi þs T mj T mi Hu ¼ 0 Qi Ci @ ru i¼1 j¼1 j¼1 C j l¼j C l ð23Þ The explicit solutions cannot be derived in the above equations. These equations can be solved with numerical method. The numerical result of the optimization in this section is shown in Table 3 by Table 3 Entropy generation optimizations in steady and unsteady cases with fixed heat (Hu ¼ 8:64 106 J;H ¼ 200 W;Ci ¼ 11:78; T 0;av ¼ 933:33 K).

3 !7 Rs Xi1 C j 1 T mj 1 7 C i 1 0 T 0 dt 7 þ sðC i 1Þ þ Qi Qi j¼1 T mi C i 7 T mi C 5 j¼1 j l¼j C l þ ru mcp

T mj T mi

1

0 s

9 > > > > > > 3=

Stage number

PCM temperature

Unsteady case (K)

Steady case (K)

n=2

Tm1 Tm2

739.45 589.65

739.44 589.63

n=3

Tm1 Tm2 Tm3

790.45 679.72 584.50

790.44 679.71 584.48

n=4

Tm1 Tm2 Tm3 Tm4

814.79 727.59 649.72 580.18

814.79 727.59 649.72 580.18

n=5

Tm1 Tm2 Tm3 Tm4 Tm5

828.73 757.00 691.47 631.61 576.92

828.69 756.95 691.41 631.55 576.87

8 1 2 0 > > > > Rs C B > n 6 X > C 6R s B 1 C 1 Pi1 Cj 1 > T mi > Cdt þ QC ii1 0 T 0 dt þ sðC i 1Þ 6 ln B þ i > > j¼1 Qi T mi Ci C 6 0 BC i i1 < X C C A @ C j 1 @ j¼1 j l¼j l i¼1 4 T0 þ T Q Q ¼ mcp mj i1 i1 Cj Cl @T mk > > > ( "R s j¼1 j¼1 l¼j !# ) > > n i1 X X > T dt > 0 C j 1 H > 1 0 u > þru mcp ðC i 1Þ Qi þs Qi T mj C T mi > i : Cj Cl i¼1

j¼1

j¼1

l¼j

39 > > !7 > > > > 7 T mj 1 7> > > Ci 7 > T mi 5= ¼0 16k6n > > > > > > > > > > ;

ð22Þ

Please cite this article in press as: H.J. Xu, C.Y. Zhao, Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part II: Unsteady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.066

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taking cascaded heat storage (n = 2, 3, 4, 5) as an example. The comparison of entropy generation optimization with fixed heat transfer between steady and unsteady cases is also presented in Table 3. We find that the numerical optimization result of unsteady case is almost equal to that of steady case. The minimal difference is the error of numerical calculation and that of truncation. This proves that there is no need to perform entropy generation optimization with the constraint of fixed heat transfer in unsteady case. 3.3. Entransy optimization Based on the entransy balance equation for a simple open system with flow and heat transfer, the entransy dissipation rate of the ith-stage thermal storage unit is

1 Ci þ 1 /i ¼ Q i T mi þ mðe1 e2 Þ ¼ mcp ðT i1 T i Þ2 2 Ci 1 16i6n

ð24Þ

The entransy dissipation rate can be expressed with the PCM temperatures as i1 X 1 T0 Cj 1 1 /i ¼ mcp ðC 2i 1Þ Qi þ T mj T mi Qi 2 C i j¼1 j¼1 C j l¼j C l

!2

16i6n

all the PCM temperatures are embedded in another kind of variables for optimization, and we can define this kind of new variables as follows

Mi ¼

i1 X Cj 1 1 T mj T mi Qi Ci j¼1 l¼j C l

16i6n

ð29Þ

Thus, the above Lagrange multiplier can be simplified as

F u;E ðM 1 ; M 2 ; . . . ; Mn ; ku Þ ¼ Q u;tot þ ku ð/u;tot Uu Þ Z s n X T0 ¼ mcp ðC i 1Þ Qi

!

þ M i dt 0 i¼1 j¼1 C j 2 3 !2 Z s n X 1 T 0 2 þ ku 4 mcp ðC i 1Þ þ Mi dt Uu 5 Qi 2 0 i¼1 j¼1 C j

Since the phase-change temperatures of PCMs (T mi ) are independent of time, the values of Mi do not vary with time. The Lagrange equation for the Lagrange extremum value is as follows

" !# Rs T dt @F u;E 0 0 ¼0 16i6n ¼ mcp sðC i 1Þ 1 þ ku ðC i þ 1Þ Mi þ Qi @Mi s j¼1 C j

ð31Þ

ð25Þ With a fluctuating inlet HTF temperature, the entransy dissipation of the ith-stage unit is

/u;i ¼

Z s

/i dt

0

¼

1 mcp ðC 2i 1Þ 2

Z s 0

T0 Qi

j¼1 C j

þ

i1 X Cj 1 1 T mj T mi Qi Ci j¼1 l¼j C l

dt ð26Þ

The total entransy dissipation of the cascaded thermal storage system is n X /u;i i¼1

¼

Z s n X @F u;E 1 T0 ¼ mcp ðC 2i 1Þ þ Mi Qi 2 @ku 0 i¼1 j¼1 C j

!2 dt Uu ¼ 0

ð32Þ

The optimal values of these variables ðku ; M1 ; M 2 ; . . . ; M n Þ are deducted as

!2

16i6n

/u;tot ¼

ð30Þ

Z s n i1 X X 1 T0 Cj 1 1 mcp ðC 2i 1Þ þ T mj T mi Qi Qi 2 C i 0 i¼1 j¼1 j¼1 C j l¼j C l

!2 dt ð27Þ

In the following, two kinds of optimizations are presented by combining the total heat transfer with the entransy dissipation: heat transfer optimization with fixed entransy dissipation, and entransy dissipation optimization with fixed heat transfer.

ku;opt

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u P u s nj¼1 CCjj 1 þ1 u ¼ u Rs 2 u u2Uu Pn C 2i 1 R s T 0 T 0 dt dt tmcp 0 2 i¼1 Q 0 s 0

ð33Þ

i

C j¼1 j

Rs T dt 1 0 0 Mi;opt ¼ Q i k ðC s j¼1 C j u;opt i þ 1Þ

16i6n

ð34Þ

where ‘+’ and ‘’ respectively represent cold storage and heat storage. According to the relation between T mi and Mi , the optimal solution of PCM temperature is

Rs T mi;opt ¼

0

T 0 dt

s

þ

! Ci 1 þ C j þ 1 C i þ 1 ku;opt

i1 X Cj 1 j¼1

16i6n

ð35Þ

Based on the recursive relation between T i;opt and T mi;opt , the solution of HTF temperature is

!Rs

3.3.1. Heat transfer optimization with fixed entransy dissipation Taking the total thermal energy as the objective function and the entransy dissipation as the constraint, the Lagrange function can be established as

T i;opt ¼ Qi

F u;E ðT m1 ; T m2 ; . . . ; T mn ; ku Þ ¼ Q u;tot þ ku ð/u;tot Uu Þ Z s n X T0 ¼ mcp ðC i 1Þ Qi

Within storage time, the maximum heat transfer for cascaded thermal storage can be obtained as

i¼1

0

j¼1 C j

þþ

i1 X

!

Cj 1 1 T mj T mi dt Qi C i C j¼1 l¼j l

2 3 !2 Z s n i1 X X 1 T0 Cj 1 1 2 4 þ ku mcp ðC i 1Þ þ T mj T mi dt Uu 5 Qi Qi 2 Ci 0 i¼1 j¼1 j¼1 C j l¼j C l

T0

j¼1 C j

0

j¼1 C j

T 0 dt

s

þ

i X Cj 1 j¼1

1 16i6n C j þ 1 ku;opt ð36Þ

jQ u jmax ¼

n X C j 1 mcp s C þ 1 jku;opt j j¼1 j

ð37Þ

If the inlet HTF temperature is independent of time, the optimization solution is simplified as

ð28Þ

With fluctuating inlet HTF temperature, the variables to be optimized in this kind of optimization are PCM temperatures T mi and Lagrange multiplier ku , respectively. From the above equations,

þ 1 Qi

1

ku;opt

ﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X u umcp s n C j 1 t j¼1 C j þ1 ¼ kopt ¼ 2Uu

ð38Þ

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7

H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx Table 4 Heat optimizations in steady and unsteady cases with fixed entransy dissipation ðUu ¼ 6:048 107 J K, U ¼ 1400 W K, C i ¼ 11:78, T 0;av ¼ 933:33 KÞ.

lu;opt ¼ Pn

C j 1 j¼1 C j þ1

PCM temperature

Unsteady case (K)

Steady case (K)

n=2

Tm1 Tm2

908.80 884.88

884.54 836.96

n=3

Tm1 Tm2 Tm3

913.24 894.85 876.46

893.36 856.78 820.21

n=4

Tm1 Tm2 Tm3 Tm4

915.77 900.98 886.18 871.38

898.40 868.97 839.54 810.11

T mi;opt ¼

Tm1 Tm2 Tm3 Tm4 Tm5

917.41 905.11 892.80 880.50 868.20

901.65 877.18 852.72 828.25 803.78

T0 T i;opt ¼ Qi

T mi;opt

16i6n

j

Rs 0

T 0 dt

s

Pi1 C j 1

i þ C iCþ1 Hu C j 1 mcp s

j¼1 C j þ1

Pn

16i6n

j¼1 C j

þ 1 Qi

!Rs

1

0

j¼1 C j

T 0 dt

s

Pi

C j 1 j¼1 C j þ1

Pn

C j 1 j¼1 C j þ1

/u;min ¼

Xn 1 mcp i¼1 ðC 2i 1Þ 2 0 12 Rs Z s T dt T 1 H 0 0 u 0 @Q A dt Q þ P i mcp s 0 s ij¼1 C j ðC i þ 1Þ nj¼1 CCj 1 j¼1 C j þ1 j

ð48Þ

of

Gu;E ðM 1 ; M2 ; . . . ; Mn ; lu Þ ¼ /u;tot þ lu Q u;tot Hu !2 Z s n X 1 T0 ¼ mcp ðC 2i 1Þ þ M dt Qi i 2 0 i¼1 j¼1 C j " ! # Z s n X T0 þ lu mcp ðC i 1Þ þ Mi dt Hu Qi 0 i¼1 j¼1 C j

T 0 dt þ Mi Q s ij¼1 C j

!

Pi1 C j 1

i þ CiCþ1 H C j 1 mcp

j¼1 C j þ1

T mi;opt ¼ T 0

Pn

16i6n

ð49Þ

j¼1 C j þ1

Pi

C j 1 j¼1 C j þ1

H

C j 1 j¼1 C j þ1

mcp

T i;opt ¼ T 0 Pn

16i6n

ð50Þ

3.4. Solution analysis

ð41Þ

# þ lu ¼ 0 1 6 i 6 n ð42Þ

! Rs n X T dt @Gu;E 0 0 ¼ mcp ðC i 1Þ Qi þ sM i H u ¼ 0 @ lu i¼1 j¼1 C j

From Eqs. (44)–(48), if the time-averaging value of inlet HTF temperature in unsteady case is set as the inlet HTF temperature in steady case, the optimization result in terms of PCM temperatures and minimum entransy dissipation for unsteady case is exactly the same with those for steady case. When the inlet HTF temperature is independent of time, the above solution can be simplified as

This simplified solution is also in coincide with the solution for steady case in Xu and Zhao [24].

The fixed heat transfer ðHu Þ is positive for heat storage, while negative for cold storage. The corresponding Lagrange equations are as follows: 0

16i6n

The minimum value of total entransy dissipation can be calculated as

ð40Þ

3.3.2. Entransy dissipation optimization with fixed heat transfer If the entransy dissipation is the objective function and heat ~ Þ is the constraint, the Lagrange function is in the form transfer ðH

@Gu;E ¼ mcp sðC i 1Þ ðC i þ 1Þ @Mi

Hu

mcp s

ð47Þ

This is strictly accordant with the steady optimization solution in Xu and Zhao [24]. Thus the feasibility of this optimization solution can be validated. For cascaded heat storage (n = 2, 3, 4, 5), the heat optimization result in unsteady case is shown in Table 4 for fixed entransy dissipation. The comparison between the results of steady and unsteady cases is also provided in Table 4. The result of unsteady case is obviously higher than that of steady case for heat optimization with fixed entransy dissipation. This means this kind of unsteady-case optimization is significant for practical application.

Rs

ð46Þ

j¼1 C j þ1

j

"

ð45Þ

The analytical solutions for temperatures of PCMs and HTF are

ð39Þ 16i6n

1 Hu n X mc ps C j 1 ðC i þ 1Þ C þ1 j¼1

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! u i1 X Cj 1 2U Ci u Xn C 1 ¼ T0 t þ 16i6n C þ 1 Ci þ 1 mcp j¼1 Cjj þ1 j¼1 j

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u i X 2U Cj 1 u T i;opt ¼ T 0 t Xn C 1 mcp j¼1 C j þ1 j¼1 C j þ 1

ð44Þ

mcp s

Rs T dt 0 0 þ Mi;opt ¼ Q s ij¼1 C j

Stage number

n=5

Hu

1

ð43Þ

From the above equations, the basic variables ðlu ; M 1 ; M2 ; . . . ; M n Þ can be obtained as

For unsteady/steady cascade thermal storage systems in this paper and previous studies [23,24], the temperature distribution of multi-stage PCMs should not stride over the environment temperature. That is, the last-stage PCM temperature should be greater than the environment temperature for heat storage, while the former should be smaller than the later for cold storage. The qualification for meaningful heat/cold storage can be expressed as

T mn;opt

> Te

heat storage

< Te

cold storage

ð51Þ

Since the last-stage thermal storage unit owns the least thermal energy quality, the above relation can guarantee that all thermal energies stored in different stages are applicable. Simultaneously, for the fluctuating inlet HTF temperature, the fluctuation of inlet HTF temperature has critical effect on the global optimization solution, and the purpose of optimization is to obtain the global optimization solution within the storage time. Thus, the outlet temperature of the first-stage thermal storage unit should

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H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

be lower than the inlet temperature in cascaded heat storage, while it is greater than the inlet temperature in cascaded cold stor-

T 1;opt

The qualification of fluctuating inlet HTF temperature for Section 3.3.1 can be obtained as

8 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R s 2 u > Rs T dt > u2Uu Pn C 2 1 > > umcp i¼1 i 2 0 T 0 0 s 0 dt > > u Q R i > s T dt > u Cj > 0 j¼1 > T C 1 t 0 > < T0 þ 1 C11 0 s þ C11 þ1 Pn C 1 > > < C1 s j¼1 C j þ1 j ¼ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R s 2 u > > Rs T 0 dt u2Uu Pn C 2 1 > > i u T0 0 s dt > mc 2 p > i¼1 Q 0 u > R i > s T dt u > Cj > 0 j¼1 T C 1 > 1 0 0 > > T0 C11 þ1 t Pn Cj 1 > s : C1 þ 1 C1 s C þ1

cold storage ð55Þ

heat storage

j¼1 j

age. The qualification for modeling the fluctuating inlet HTF temperature can be expressed as

T 1;opt

< T0

heat storage

> T0

cold storage

ð52Þ

If the fluctuation is relatively small as assumed, the global optimization solution exists conditionally. The above two equations defines the conditions of the optimizations in unsteady cascaded thermal storage system from to 3.13.3, which makes the optimization solution physically meaningful. For the exergy optimization and entropy optimization in to 3.13.2, the qualifications in Eqs. (51), (52) also hold. Since explicit

This can be simplified as follows:

8 !2 Rs n < C þ 12 T 0 dt X mcp Ci 1 1 Uu > T0 0 max s : C1 C þ1 2 s i¼1 i 9 ! Rs 2 Z s n X T 0 dt C 2i 1 = T0 0 dt þ 2 Qi s 0 i¼1 ð CÞ ; j¼1

ð56Þ

j

Rs where the term T 0 0 T 0 dt=s is the departure of instantaneous inlet HTF temperature from the time average value of inlet HTF temperature. The value range of fixed entransy dissipation for optimization in Section 3.3.1 can be obtained as

9 8 " !#2 Z !2 Rs Rs < X n n s X T dt T dt mcp Ci 1 C1 þ 1 C 2i 1 = 0 0 0 0 < Uu T0 þ T0 dt max s Qi 2 : i¼1 C i þ 1 C1 2 s s 0 i¼1 ð CÞ ; 2

0

6 B R s T 0 dt n 6 0 X Te mcp 6 Ci 1 B B < 6s B n1 s 2 6 i¼1 C i þ 1 BX C j 1 4 @ þ Cn j¼1

C j þ1

C n þ1

3

12

T mn;opt ¼

0

T 0 dt

s

!

n1 X Cj 1

þ

j¼1

Cn 1 þ C j þ 1 C n þ 1 ku;opt

> T e heat storage

12

Rs T 0 dt n 0 Te C mcp 6 X Ci 1 B 0 < Uu < 4s @Pn1sCj 1 A C 2 þ 1 i þ Cn i¼1 j¼1 C j þ1

n X C 2i 1 þ Qi 2 i¼1 ð j¼1 C j Þ

Z s 0

Rs T0

0

C n þ1

!2 3 T 0 dt dt 5

s

The above equation defines the fixed entransy dissipation, total stage number, storage time, and the temperature fluctuation for optimization solution of cascaded heat/cold storage. For the optimization in Section 3.3.2, based on the qualification for thermal storage, the value range of fixed heat transfer in cascaded thermal storage can be derived with:

Rs T mn;opt ¼

0

T 0 dt

s

Pn1 C j 1

Cn j¼1 C j þ1 þ C n þ1

Pn

C j 1 j¼1 C j þ1

> T e heat storage mcp s < T e cold storage

Hu

ð58Þ

This equation can be simplified as

8 Pn Cj 1 R s > T 0 dt > j¼1 C þ1 > heat storage > 0 < Hu < Pn1 Cj 1j Cn mcp s 0 s T e > < þ j¼1 C j þ1 C n þ1 Pn Cj 1 R s > > T 0 dt j¼1 C j þ1 > 0 > < Hu < 0 cold storage mc s T P > p e C 1 s : n1 j þ Cn

The above equation can be further simplified as

0

ð57Þ

< T e cold storage ð53Þ

2

j

C !2 7 Rs Z s n 7 C X T dt C 2i 1 7 C 0 0 T0 dt 7 C þ Qi 2 7 C s 0 ð i¼1 C Þ 5 A j¼1 j

solutions of exergy and entropy optimizations are absent for cascaded thermal storage system with unsteady inlet HTF temperature, the value ranges of related parameters cannot be analytically provided. Yet, the analytical qualifications for entransy optimizations in Section 3.3 can be obtained in the following. For Section 3.3.1, according to the qualification of thermal storage, the value range of the fixed entransy dissipation for the cascaded heat/cold storage system is obtained with:

Rs

j¼1

ð59Þ

j¼1 C j þ1 C n þ1

ð54Þ

The qualification for fluctuating inlet HTF temperature can be derived for Section 3.3.2 as

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H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

T 1;opt ¼

Rs C 1 1 T dt T0 1 Hu < T 0 heat storage C þ1 0 0 þ 1 Pn 1 C 1 j C1 C1 s mcp s > T 0 cold storage j¼1 C j þ1

ð60Þ This can be simplified as follows:

Rs 8 n X T 0 dt C j 1 > 0 > Hu > C 1Cþ1 mc heat storage > p s max T 0 þ1 s C > 1 j < j¼1 Rs n X > T 0 dt > C j 1 > 0 > Hu < C 1Cþ1 cold storage mc p s min T 0 : þ1 s C 1 j

4. Results and discussion Since the release is similar to the storage for thermal energy, the release of heat/cold can be obtained by drawing an analogy between release and storage of heat/cold. In the following, the optimizations of the cascaded heat storage in unsteady case are performed for presenting the results of temperature distribution, thermal performance, and PCM selection. 4.1. Temperature distribution

j¼1

ð61Þ The value range of fixed heat transfer in Section 3.3.2 can be summarized as

Fig. 3 presents the local HTF temperature in different stages (i = 1,2,. . ., 6) for entropy optimization (Fig. 3(a)) and entransy optimization (Fig. 3(b)). For unsteady inlet HTF temperature, the varying trend of HTF temperature versus time is similar to that

8 Pn Cj 1 Rs R s > T 0 dt T 0 dt > j¼1 C þ1 C 1 þ1 Pn C j 1 0 > < Hu < Pn1 Cj 1j Cn mcp s 0 s T e heat storage > C 1 j¼1 C j þ1 mcp s max T 0 s > < þ j¼1 C j þ1 C n þ1 Pn Cj 1 Rs R s > > T 0 dt T 0 dt j¼1 C j þ1 > C 1 þ1 Pn C j 1 0 0 > < cold storage mc s T H < mc s min T P > p e u p 0 C 1 n1 j¼1 C j þ1 s s C1 : j Cn

ð62Þ

þ

j¼1 C j þ1 C n þ1

HTF temperature T (K)

1000

n=6 Ω=0.1 W·K-1 NTU=0.15

900

unsteady steady

800

inlet

i=1 i=2 i=3 i=4 i=5 i=6

700

of inlet HTF temperature, but with less fluctuation range. Along the HTF flow path, the fluctuation range of HTF temperature is 1000

n=6

Φ=30000 W·K

950

HTF temperature T (K)

The above analysis means that the fluctuation of HTF temperature reduces the possibility of optimization solutions.

NTU=0.15

900 850

unsteady steady inlet

800

i=1 i=2 i=3 i=4 i=5 i=6

750 700

600 0

1x104

2x104

Time t (s)

3x104

4x104

0

1x104

(a) Entropy

2x104

Time t (s)

3x104

4x104

(b) Entransy

Fig. 3. HTF temperature distributions of steady and unsteady cases for n = 6.

Outlet HTF temperature Tn (K)

850

800

750

700

NTU=0.15 Ω=0.1 W·K-1

650

n=1 n=2 n=3 n=4 n=5 n=6

900

Outlet HTF temperature Tn (K)

n=1 n=2 n=3 n=4 n=5 n=6

900

850

800

750

NTU=0.15 Φ=30000 W·K

700

650

0

1x104

2x104

3x104

4x104

0

1x104

Time t (s)

(a) Entropy

2x104

3x104

4x104

Time t (s)

(b) Entransy

Fig. 4. The last-stage HTF temperature in unsteady case.

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H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

diminished gradually. This is due to the fact that even though the HTF temperature is fluctuant, the PCMs with phase-change temperatures independent of time can effectively absorb the temperature fluctuation induced by HTF. Since the thermal energy storage is indirectly reflected in the outlet HTF temperature, the outlet HTF temperature is a critical parameter for evaluating thermal storage. Fig. 4 presents the outlet HTF temperature distribution for different stage numbers in unsteady case with entropy (Fig. 4(a)) and entransy (Fig. 4(b)), respectively. For the entropy and entransy optimizations, with the same inlet HTF temperature variation, the outlet HTF temperature decreases with an increase in stage number. This means that the stored thermal energy can be obviously increased by increasing stage number. In addition, with an increase in stage number, the fluctuation range of outlet HTF temperature is also decreased. Fig. 5 shows the effect of stage number on the outlet HTF temperature for steady case and the time-average value of that for unsteady case respectively with entropy optimization (Fig. 5(a)) and entransy optimization (Fig. 5(b)). The outlet HTF temperature in steady case and the time-average value of that in unsteady case both decrease with the increased stage number, which means higher thermal efficiency with the increased stage number. By comparing steady case with unsteady case, the outlet HTF temperature for steady case is less than the time-average value of that for

unsteady case for both entropy and entransy optimizations. For entropy optimization, the difference between the outlet HTF temperature in steady case and the time-average outlet HTF temperature in unsteady case is almost unchanged with the increasing stage number. However, this difference is increased with an increase in stage number for entransy optimization. The optimal PCM temperature distributions of cascaded heat storage system (n = 4) in steady and unsteady cases are shown respectively in Fig. 6(a) (entropy) and Fig. 6(b) (entransy). The PCM temperature distribution of unsteady case is similar to that of steady case. But the PCM temperature of unsteady case is higher than that of steady case. This means that, for unsteady cascaded heat storage, the optimal PCM temperature in unsteady case rather than that in steady case should be treated as a reference for PCM selection. In addition, along the HTF flow path, the optimal PCM temperature difference between steady and unsteady cases based on entropy is almost unchanged, while that based on entransy is gradually increased. 4.2. Thermal performance The effects storage time on thermal and exergy efficiencies are respectively shown in Fig. 7(a) (entropy) and Fig. 7(b) (entransy). It is undoubtedly that the increase in storage time will increase

650

Outlet HTF temperature Tn (K)

Outlet HTF temperature Tn (K)

550

Tn,av, unsteady

500

Tn, steady

450

Ω=0.1 W·K-1 Te=273.15 K

T0,av=933.3 K

400

350

Tn,av, unsteady

600

Tn, steady 550

Φ=25000 W·K Te=273.15 K

500

T0,av=933.3 K 450

400

350 1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

Stage number n

Stage number n

(a) Entropy

(b) Entransy

8

9

10

Fig. 5. Effect of stage number on outlet HTF temperature for steady and unsteady cases.

700

unsteady T0

unsteady T0

steady T0

PCM temperature Tm (K)

PCM temperature Tm (K)

560

540

520

500

n=4 Te=273.15 K

480

Ω=0.1 W·K-1 NTU=0.2

steady T0 650

n=4 Te=273.15 K

600

Φ=25000 W·K NTU=0.2

550

460 0.0

0.2

0.4

0.6

0.8

Dimensionless HTF flow path X

(a) Entropy

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Dimensionless HTF flow path X

(b) Entransy

Fig. 6. PCM temperature distribution with entropy and entropy optimizations (n = 6).

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11

H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx 0.90

0.90

η1st

0.88

η2nd

0.86

Efficiency

Efficiency

η2nd

η1st

0.85

0.80

0.75

Te=273.15 K

Φ=25000 W·K

1x104

0

Te=273.15 K

0.82

Ω=0.1 W·K-1 NTU=3 n=3

0.80

NTU=3 n=3

0.70

0.84

2x104

3x104

0.78

4x104

1x104

0

2x104

3x104

Storage time τ (s)

Storage time τ (s)

(a) Entropy

(b) Entransy

4x104

Fig. 7. Effect of storage time on thermal and exergy efficiencies for entropy and entransy optimizations.

0.90 0.8

0.85 0.7

Efficiency

Efficiency

0.80

Unsteady Steady

0.75

η1st

0.70

η2nd

0.65

Unsteady Steady

η1st η2nd

0.6

Φ=25000 W·K Te=273.15 K

0.5

Ω=0.1τ J·K-1

Te=273.15 K

0.60

0.4

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

Stage number n

Stage number n

(a) Entropy

(b) Entransy

8

9

10

Fig. 8. Effect of stage number on thermal and exergy efficiencies for steady and unsteady cases with entropy and entransy optimizations.

1.0

0.8

Efficiency

the total thermal energy and total exergy that is stored in multi-PCMs. Yet, the thermal energy and exergy of inlet HTF also vary with time. This leads to the complex varying trends of thermal and exergy efficiencies shown in Fig. 7. The thermal efficiencies of entropy and entransy optimizations first decrease and then increase with an increase in storage time. As the storage time changes, the exergy efficiency of entropy optimization varies with twice changes in its varying curve and two extremum values. While, the exergy efficiency of entransy optimization first increases and then decreases with the increased storage time. This is determined by the variation of inlet HTF temperature. Fig. 8(a) and (b) respectively show the effect of stage number on thermal/exergy efficiencies of entropy optimization and that of entransy optimization. The optimal result of steady case is greater than that of unsteady case in terms of thermal and exergy efficiencies. This manifests that the temperature fluctuation of HTF should be kept as small as possible to maintain the better thermal performance. With an increase in stage number, thermal and exergy efficiencies increase but the corresponding increasing amplitudes are decreased. For large stage number, the increase in stage number has little contribution on improving thermal and exergy efficiencies. This means the stage number should not be very large since

0.6

n =2

0.4

Entransy

Ω =6480 J·K-1 Φ =φtot,Sg Te =273.15 K

0.2

Entropy

η1st η2nd

0.0 0

2

4

6

8

10

Number of transfer units NTU Fig. 9. Comparisons of thermal and exergy efficiencies between entransy and entropy optimizations with unsteady HTF inlet temperature.

the increased stage number will lead to a more complex and cost-plus cascaded thermal storage system.

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H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

Table 5 PCM selection for cascaded heat storage of entropy/entransy optimizations in steady and unsteady cases ðn ¼ 2; NTU ¼ 5; U ¼ /tot;Sg ¼ 5:04 104 W KÞ. Optimization Type

Tm (K)

Selected PCMs/Tm,pr (K)

Sg;tot (WK1)

g1st (%)

g2nd (%)

Steady, entropy

585.8 369.2

RbNO3/585.2 [25] Bi–Pb–Sn (52-30–18 mol%)/369.2 [26]

0.150

85.2

74.2

Unsteady, entropy

586.9 370.5

CuCl2–NaCl (73–27 mol%)/587.2 [27] Glutaric acid/370.7 [28]

0.150

85.0

74.1

Steady, entransy

640.6 349.8

CdBr2–NaBr (47–53 mol%)/6340.2 [27] Hexatriacontane/349.4 [29]

0.182

88.1

71.0

Unsteady, entransy

643.6 355.8

CdCl2–PbCl2 (37–63 mol%)/643.2 [27] Acetamide/355.2 [27]

0.177

87.2

71.5

One of the tasks in this work is to analyze the between entransy and entropy in optimizing the cascaded thermal storage. By setting the fixed entransy dissipation in entransy optimization as that obtained from entropy optimization with fixed entropy optimization, the optimal result based on entransy is compared with that based on entropy in terms of thermal and exergy efficiencies, as shown in Fig. 9. From Fig. 9, for both entransy and entropy optimizations, with an increase in NTU, thermal and exergy efficiencies are increased gradually but the corresponding increasing amplitudes are gradually decreased. Simultaneously, for smaller NTU, the increasing amplitudes of thermal and exergy efficiencies are very large, while for larger NTU, the corresponding increasing amplitudes are mild. This means, when heat transfer of thermal storage is very sufficient, the contribution of increasing NTU for improving thermal/exergy efficiencies is negligible. The optimal thermal efficiency based on entransy is obviously higher than that based on entropy, while the optimal exergy efficiency based on entropy is greater than that based on entransy. Thermal efficiency stands for the potential of thermal energy utilization via heat transfer in thermal storage, and exergy represents the conversion ability from thermal energy to mechanical work. Thus, from the viewpoint of unsteady cascaded thermal storage, entransy can be used for optimizing thermal storage with heat transfer purpose and entropy can be applied to optimize thermal storage to be converted to mechanical work. In addition, for smaller NTU, the difference between entropy and entransy optimizations is negligible in terms of thermal and exergy efficiencies. While, this difference is notable for larger NTU. This means, when the heat transfer of thermal storage is very sufficient, the optimal result based on entransy obviously differs from that based on entropy. In this condition, the thermal storage purposes should be especially distinguished for the practical cascaded thermal storage.

4.3. Selection of PCMs in unsteady case The optimization solutions for unsteady cascaded thermal storage can be used for the corresponding PCM selection. The comparison between steady and unsteady cases and that between entropy and entransy optimizations are shown in Table 5. The unsteady inlet HTF temperature varies with time as Eq. (1) shows and the inlet HTF temperature in steady case is the corresponding timeaverage value for unsteady case (933.3 K). The stage number is n = 2 and the number of transfer units is NTU = 5. The fixed entropy generation rate for entropy optimization in steady/unsteady cases is 0.15 WK1. The fixed entransy dissipation rate in entransy optimization for steady/unsteady cases is set as the average value of total entransy dissipation rate in unsteady entropy optimization (5.04 104 WK). Based on the optimization solution, the practical PCMs selected for cascaded heat storage can also be found in Table 5. The optimization result for steady case and that for unsteady case is obviously different. For cascaded heat storage,

the optimal PCM temperature in unsteady case is higher than that in steady case for both entropy and entransy optimizations. As stated above, for thermal storage process, entropy is appropriate for heat-work conversion, while entransy can be applied to estimate heat transfer. From Table 5, the optimal exergy efficiency in steady case is superior to that in unsteady case for entropy optimization, while the optimal thermal efficiency in steady case is higher than that in unsteady case for entransy optimization. This is attributed to that the optimal PCM temperature of unsteady cascaded thermal storage is a global optimal solution rather than a local optimal solution. In practical, it makes little sense for too large stage number, which causes the increased system complexity and operation risk. Simultaneously, it is not much useful when PCM temperature is very close to the environment temperature, since the thermal energy quality is very low. 5. Conclusions Thermal optimizations for unsteady cascaded heat storage are performed based on entransy, entropy and exergy. The equations for entropy and exergy optimizations are numerically solved, and analytical optimization solutions based on entransy are obtained. The qualifications for existence of optimal solutions are proposed for all the optimizations with analytical qualifications obtained for entransy optimizations. The conclusions are as follows. (1) The HTF temperature in each stage exhibits similar fluctuating trend as the inlet HTF temperature does, but the corresponding fluctuating amplitude diminishes along the HTF flow path. (2) With the increased stage number or NTU, the optimal thermal performance is gradually increased, but with decreased increasing amplitude. (3) Optimal thermal performance in steady case is obviously superior to that in unsteady case for entransy and entropy optimizations. (4) In unsteady case, entropy is more suitable for optimizing cascaded thermal storage for heat-work conversion, and entransy is more appropriate for optimizing that for heat transfer. Acknowledgements Financial supports from the National Key Basic Research Program of China (973 project, 2013CB228303), and the National Natural Science Foundation of China (51406238) are acknowledged. References [1] Y. Tian, C.Y. Zhao, A review of solar collectors and thermal energy storage in solar thermal applications, Appl. Energy 104 (2013) 538–553. [2] E. Halawa, W. Saman, Thermal performance analysis of a phase change thermal storage unit for space heating, Renewable Energy 36 (1) (2011) 259–264. [3] A.I. Fernández, C. Barreneche, L. Miró, S. Brückner, L.F. Cabeza, 19 – Thermal energy storage (TES) systems using heat from waste, in: L.F. Cabeza (Ed.), Advances in Thermal Energy Storage Systems, Woodhead Publishing, 2015, pp. 479–492.

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