Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part I: Steady cases

International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx

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

Thermal performance of cascaded thermal storage with phase-change materials (PCMs). Part I: Steady 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 Entropy Entransy Steady case

a b s t r a c t Cascaded latent thermal storage is an efficient technique for balancing the gap between demand and supply in renewable energy utilization. In this work, the steady cascaded thermal storage system with multiple phase-change materials (PCMs) of different phase-change temperatures is theoretically optimized from the viewpoint of thermodynamics. Analytical solutions for optimal temperatures of heat transfer fluid (HTF) and PCMs are obtained based on entropy and entransy concepts. The corresponding qualifications for optimization solutions are also discussed. With an increase in stage number or that in NTU, both thermal and exergy efficiencies increase. Compared with single-stage thermal storage, the cascaded thermal storage with multiple PCMs can not only improve the thermal performance, but also extend the application scope of thermal energy with multi-grade thermal energies provided. The temperature distribution for HTF and PCMs with entransy optimization is linear, while that with entropy optimization is geometric progression. The application situations of entropy and entransy in thermal energy utilization are identified by comparing optimal result based on entropy with that based on entransy. This optimization can not only be used to establish the steady cascaded thermal storage system, but also guide the filtration of PCMs used in such systems. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction As a manner bridging the gap between supply and demand of renewable energies, thermal storage can be used for various applications, such as solar utilization [1], shaving of electric peak [2], waste heat recovery [3], energy-saving in buildings [4], and so on. There are mainly three thermal storage patterns: sensible storage [5], latent storage [6] and thermo-chemical storage [7]. In sensible storage, thermal energy is temporarily stored with rise/fall of temperature in the storage media. In latent storage, considerable latent heat of phase-change material (PCM) is absorbed/desorbed in the solid–liquid phase-change process for storing thermal energy. In thermo-chemical storage, large amount of enthalpy change in endothermic/exothermic chemical reactions is used to store/release thermal energy. Compared with the sensible thermal storage, latent thermal storage with PCM owns the advantages of nearly constant temperature, and much higher energy density per unit volume. As to the comparison of latent thermal storage versus thermo-chemical storage, complexity, safety and harsh ⇑ Corresponding author. E-mail addresses: [email protected] (H.J. Xu), [email protected] (C.Y. Zhao).

operation condition of the system are still challenges confronting the thermo-chemical storage even though with larger energy density. Thus, PCM with high latent heat is now an effective and feasible solution to store thermal energy. Nevertheless, the poor thermal conductivity of PCMs restricts the improvement in thermal efficiency and thermal storage rate. Various heat transfer enhancement techniques for solid–liquid phase-change process of PCM are proposed [8], including extended fins [9], porous media [10], PCM microcapsulation technology [11,12], highly conductive additives [13], heat transfer enhancement assisted with heat pipe [14], and so on. Even so, the thermal performance of the thermal storage system with only one PCM still needs to be promoted, especially for large difference between heat transfer fluid (HTF) and surrounding temperatures. In recent several decades, cascaded thermal storage with multiple PCMs has emerged as a potential and efficient technique for thermal energy utilization. Fig. 1 shows the comparison between systems of the single-stage thermal storage (Fig. 1(a)) and the five-stage thermal storage (Fig. 1(b)) [15]. Compared with the single-stage system, the five-stage system owns much wider operation range in temperature and more uniform temperature difference between HTF and PCM, from which the driving force of phase-change heat transfer can be improved. In addition, the multi-stage system can also

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054 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 I: Steady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054

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Nomenclature A cp E e Ex h k m n NTU Q S s T T0 Te U W X

heat transfer area, m2 1 heat capacity at constant pressure, J
kg
K1 entransy flow, W
K 1 specific entransy, J
K
kg exergy, W convective heat transfer coefficient, W
m2
K1 thermal conductivity, W
m1
K1 mass flow rate, kg
s1 stage number of cascaded thermal storage system number of heat transfer units heat transfer rate, W entropy flow, W
K1 1 specific entropy, J
kg
K1 temperature, K inlet temperature, K environment temperature, K overall heat transfer coefficient, W
m2
K1 work output of the heat engine, W dimensionless position along HTF flow path

Greek symbols a temperature ratio d thickness, m h dimensionless temperature H fixed heat transfer rate, W

(a) Single stage

k

l n

r /

U X

Lagrange multipler, K1 Lagrange multipler, K Lagrange multipler, K Lagrange multipler, K1 entransy dissipation rate, W
K fixed entransy dissipation rate with entransy optimization, W
K fixed entropy generation rate, W
K1

Subscripts d dissipation e environment E entransy Ex exergy g generation HTF heat transfer fluid m melting max maximum min minimum opt optimal PCM phase-change material S entropy tot total w wall

(b) Five stage

Fig. 1. Comparison of single-stage thermal storage and five-stage thermal storage [15].

promote the global thermal performance, and provide multi-grade thermal energies. Up to now, a number of studies have been published on cascaded thermal storage and the relevant researches were mainly emphasized on global performance analysis [16–20], and numerical prediction [21–25]. Farid and Kanzawa [16] conducted a mathematical modeling on cascaded heat storage with air as the HTF. It was found that the heat transfer rate of cascaded heat storage system with different PCMs is larger than that with a single PCM. Wang et al. [17] performed a theoretical study on the cascaded heat storage with composite PCMs by assuming the

phase-change temperature distribution of PCMs as linear. It was reported that the phase-change time of cascaded storage system is decreased by as much as 25–40%. Mosaffa et al. [18] performed thermal analysis of cascaded cold storage system with multiple PCMs for the free cooling application and provided cooling load and COP of the system. Aldoss et al. [19] presented a comparison between thermal storage system with single-PCM and that with multi-PCM, and suggested that the thermal storage system with three-stage PCMs is an economic design. Yang et al. [20] performed a thermal performance analysis of cascaded latent thermal storage unit, and found that the overall performance of thermal storage

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 I: Steady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054

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

with multi-stage PCMs is better than that with single-stage PCM. For numerical prediction, Seeniraj and Lakshmi Narasimhan [21] presented a 1-D numerical simulation on the five-stage thermal storage system with the finite difference method and found that the increase in melting rate by using multiple PCMs is appreciable. Tian and Zhao [22] performed a 2-D numerical simulation on metal-foam enhanced cascaded thermal storage with three PCMs and found that the heat transfer rate of metal-foam enhanced cascaded thermal storage system is improved by 2–7 times compared with that of the single-stage thermal storage system without metal foam. Chiu and Martin [23] used COMSOL to numerically simulate the three-stage cold storage device and reported that the thermal performance of cascaded latent cold storage device can be improved by 10–40% compared with the single-stage storage device. Tao et al. [24] numerically studied the two-stage heat storage system for high-temperature applications with the entransy dissipation calculated. Li et al. [25] numerically simulated the three-stage shell-and-tube heat storage unit for solar applications with the finite volume method. In theory, thermal performance of cascaded storage system is superior to that of single-stage storage system, but only a few experimental studies [26–29] were reported for cascaded thermal storage. In 1990, Farid et al. [26] performed an experimental investigation for cascaded thermal storage with several PCMs in the same device, and found that the heat transfer rate is increased by 15% compared with the singlestage thermal storage system. Wang et al. [27] conducted a three-stage thermal storage experiment with three PCMs and found that the thermal storage rate is obviously improved by using multiple PCMs. Michels and Pitz-Paal [28] performed an experimental study of three-stage thermal storage in a shell-and-tube heat exchanger under realistic operation parameters. They indicated that the poor thermal conductivity of PCMs is an obstacle to make full use of the cascaded thermal storage technique. Peiro et al. [29] presented an experimental comparison between the heat storage unit of two stages and that of single stage. The effectiveness of the latent heat storage tank with two-stage PCMs is 19.36% higher than that with single-stage PCM. The major problem for cascaded thermal storage is that the phase-change processes of PCMs can hardly sync due to limited quantity of PCM in each-stage. The phase-change temperature distribution of PCMs has immediate impact on heat transfer rate and thermal efficiency of cascaded thermal storage system. Synchronization for phase-change processes of multiple PCMs can push the cascaded thermal storage technique to the best function. Thus, the thermal optimization of cascaded thermal storage system is necessary for obtaining the optimal thermal performance. In early studies on thermal optimization of cascaded thermal storage, the temperature distribution of multiple PCMs was treated as continuous [30,31]. In Lim et al. [30], the exergy optimization for cascaded thermal storage system was performed for n = 1, 2, and 1 with the continuous PCM temperature assumption, but the global exergy optimization was confused with the single-stage exergy optimization. In Aceves et al. [31], an exergy-based performance optimization for cascaded thermal storage system by assuming the PCM temperature as continuous was also presented, and the implicit optimization solution was solved numerically. Gong and Mujumdar [32] performed a thermal optimization of cascaded thermal storage system with the stage number 1, 2, 3 and 5 by optimizing the exergy efficiency. But the optimization solution is very complex for application. The present authors [33] conducted an exergy-based optimization of cascaded heat/cold storage with multiple PCMs and obtained the unified optimization solution for arbitrary stage number. Since the method of entropy generation (or entropy production) minimization [34] is a powerful tool for modeling and optimization of the practical thermal process with irreversibility, it is essential

to employ this method for optimizing the cascaded thermal storage. Thus, the first aim in Part I of this paper is to explore the optimization solution of cascaded latent thermal storage system with constant HTF inlet temperature using the entropy generation minimization method. Recently, Guo et al. [35] presented a new physical quality, entransy, for describing the irreversibility of heat transfer process, and put forward the extremum principle of entransy dissipation. Even though entransy is a new concept, reasonable optimization results of cascaded thermal storage with entransy can be obtained, such as Tao et al. [24], and Xu and Zhao [36]. Thus, the second aim in Part I of this paper is to compare the optimal result of entropy optimization with that of entransy optimization and identify the application situations of entransy and entropy. The optimal solutions with minimization of entropy generation rate and those with minimization of entransy dissipation rate are also used for guiding the selection of PCMs in cascaded thermal storage system. The optimization of cascaded thermal storage with unsteady inlet HTF temperature will be presented in Part-II of this paper. 2. Physical problem 2.1. Problem description The schematic diagram for a cascaded latent thermal storage system of steady case is shown in Fig. 2. Multiple PCMs with grade phase-change temperatures are arranged along the flow-path of HTF that flows through a channel, and exchanging heat with HTF. The environment temperature is denoted as Te and the inlet temperature of HTF is T0. For the ith-stage unit, the PCM temperature is denoted as Tmi and the outlet HTF temperature is Ti (i = 1, 2, . . ., n). Some assumptions are put forward as: (1) temperature difference in each PCM unit is neglected; (2) thermophysical properties of PCMs and HTF are not affected by temperature; (3) the PCM sensible heat is neglected compared with its large latent heat; (4) undercooling/superheating in thermal storage/release of PCMs is neglected; (5) the HTF temperature variation in the direction normal to flow direction is neglected; (6) the inlet HTF temperature is independent of time with no fluctuation. With the above assumptions, the HTF enthalpy change for the ith-stage thermal storage unit equals to the heat transfer rate from HTF to PCM.

Q i ¼ mcp jT i1  T i j ¼ U i Ai

jðT i1  T mi Þ  ðT i  T mi Þj



ln
TTi1i TTmimi


16i6n ð1Þ

Tm1 T0

Tm2 T1

Tm1

... T2

Tm2

...

...

Tmi Ti-1

Tmi

Ti

Tm,n-1 Tn-2

...

Tm,n-1

Tmn Tn-1

Tn

Tmn

Environment temperature Te Fig. 2. Cascaded thermal storage system with constant inlet HTF temperature.

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 I: Steady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054

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Eq. (1) can be simplified as

  U i Ai ¼ expðNTUÞ ¼ C i ¼ exp mcp

T i1  T mi T i  T mi

16i6n

ð2Þ

The PCM phase-change temperature can be expressed with the HTF temperature.

T mi ¼

C i T i  T i1 Ci  1

16i6n

ð3Þ

The total heat transfer rate for the cascaded thermal storage system is

Q tot ¼

n X

Q i ¼ mcp jT n  T 0 j

ð4Þ

i¼1

On the right hand side of the above equation, the three terms respectively stand for entransy transfer rate by heat transfer, rate of entransy flow by mass transfer, and the entransy accumulation rate in the system. For cascaded thermal storage with assumptions of constant thermophysical properties, steady flow and heat transfer, and constant pressure, the entransy dissipation rate in the ithstage thermal storage unit can be obtained as [36]

/i ¼

1 Ci þ 1 mcp ðT i1  T i Þ2 2 Ci  1

16i6n

ð11Þ

The total entransy dissipation rate is as follows

/tot

" # n n X X 1 Ci þ 1 2 ¼ /i ¼ mcp ðT i1  T i Þ 2 C 1 i¼1 i¼1 i

ð12Þ

2.2. Thermodynamic considerations of thermal storage based on entropy

3. Performance optimizations of steady cascaded thermal storage

Based on the second law of thermodynamics for the open systems [37], the entropy generation rate equation of a simple open system can be expressed as:

In cascaded latent thermal storage system, the inlet HTF temperature is known and the temperature distribution of multi PCMs significantly influences the thermal performance of the whole system, which is very important for the thermal performance.

!

Sg ¼

dS Q þ mðs2  s1 Þ  dt T source

ð5aÞ

!

where Q is the absorbed heat of the open system from environment and T source is the source temperature. The specific entropy change from state 1 to state 2 is

s2  s1 ¼ cp ln

T2 p  Rg ln 2 T1 p1

ð5bÞ

For the ith-stage unit in cascaded thermal storage system, the entropy generation rate can be obtained with neglected pressure change.

Sg;i ¼ mcp ln

Ti mcp ðT i1  T i Þ þ T mi T i1

ð6Þ

The above equation is simplified as

(

Sg;i ¼ mcp ln

Ti ðC i  1Þ2 T i1 Ci  1 þ  Ci T i1 Ci C i T i  T i1

) ð7Þ

The total entropy generation rate is

Sg;tot ¼ mcp

( ) n X Ti ðC i  1Þ2 T i1 Ci  1 ln þ  Ci T i1 Ci C i T i  T i1 i¼1

ð8Þ

The concept of entransy can be defined as

dE ¼ Q
dT ¼ T
dQ

ð9Þ

From the specific case in Guo et al. [35] with the assumption of constant specific heat capacity of the object, the value of entransy always losses after a heat transfer process with finite temperature, which is called the entransy dissipation. Thus, the equation of entransy dissipation rate for the open system with multiple heat transfer units can be expressed as

X X X Q i T source;i
Dt þ me
Dt  me
Dt  ðEs;tþDt  Es;t Þ i

out

in

ð10aÞ Generally, the equation of entransy dissipation rate for flow and heat transfer in a simple open system can be written as

/ ¼ QT source þ mðe1  e2 Þ 

dE dt

In this part, we applied entropy to optimize the PCM temperature distribution of cascaded thermal storage system by combining the heat transfer rate with the total entropy generation rate. The Lagrange multipler method [38] is used to form the optimization function. Two cases are considered: (1) optimization of heat transfer rate with given entropy generation rate, and (2) optimization of entropy generation rate with fixed heat transfer rate. 3.1.1. Optimization of heat transfer rate with given entropy generation rate For optimizing the total heat transfer rate based on entropy, we established the Lagrange function by treating the total heat transfer rate as the objective function and the given entropy generation rate ðSg;tot ¼ XÞ as the constraint. The optimization function is

F S ðT 1 ; T 2 ; . . . ; T n ; gÞ ¼ Q tot þ nðSg;tot  XÞ ¼ mcp ðT 0  T n Þ ( " # ) n X Ti ðC i  1Þ2 T i1 Ci  1 X ln þ  þ n mcp Ci T i1 Ci C i T i  T i1 i¼1

2.3. Conceptual extension of entransy to thermal storage

Ed ¼

3.1. Entropy optimization

ð10bÞ

ð13Þ

To obtain the optimization solution, all the partial derivatives of Lagrange function are set as zero.

8 h i ðC iþ1 1Þ2 ðC i 1Þ2 > @F S < nmcp  ðC i T i T i1 Þ2 T i1 þ ðC iþ1 T iþ1 T i Þ2 T iþ1 ¼ 0 1 6 i 6 n  1 ¼ n h i o 2 @T i > : mcp n 1  ðC i 1Þ 2 T i1  1 ¼ 0 i¼n T ðC T T Þ i

i i

i1

ð14Þ @F S ¼ Sg;tot  X @n ( ) n X Ti ðC i  1Þ2 T i1 Ci  1 X ¼ mcp ln þ  Ci T i1 Ci C i T i  T i1 i¼1 ¼0

ð15Þ

From the above equations, the following conditions can be obtained.

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 2  2 T iþ1;opt C i  1 C iþ1 T iþ1;opt  T i;opt T m;iþ1;opt ¼ ¼ 16i6n1 C iþ1  1 C i T i;opt  T i1;opt T i1;opt T m;i;opt ð16aÞ nopt ¼

1 1 T i;opt



ðC i 1Þ2

ðC i T i;opt T i1;opt Þ

2

T i1;opt

i¼n

¼X ð17Þ Eqs. (16) and (17) can be solved numerically. For further simplification, Eq. (16a) becomes by setting ai ¼ T

T i;opt i1;opt

; C1 ¼ C2 ¼

. . . ¼ C i ¼ . . . ¼ C n ¼ C, as



T n;opt T0

1n

ð18Þ

Eq. (17) can be simplified as follows

lnðaÞ þ

ðC  1Þ2 1 C1 X ¼0   Ca  1 C C nmcp

ð19Þ

Eq. (19) cannot be analytically solved and a can be derived by mathematical iteration. For cascaded heat storage, the value of a is less than 1, while it is greater than 1 for cascaded cold storage. When the heat transfer is poor (small HTF temperature variation) or the stage number is sufficiently large, the value of a is very close to 1. Approximately, the natural logarithm term in the above equation can be expanded as Taylor series.

ða  1Þ2 ða  1Þ3 ða  1Þk þ . . . þ ð1Þkþ1 2 3 n

lnðaÞ ¼ ða  1Þ  þ Rk ðaÞ

ð20Þ

where Rk is the error term of the Taylor formula. By omitting the terms with orders higher than 1, the original equation becomes

ðC  1Þ2 1 C1 X  ða  1Þ þ ¼0  Ca  1 C C nmcp 

a  2þ

X

ð21Þ

nmcp

ð22Þ

Thus, the two approximate solutions of temperature ratio can be obtained as

aa;b

1 X ¼1þ  2 nmcp

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   X 1 X Cþ1 þ C nmcp 4 nmcp

ð23Þ

These two approximate solutions respectively correspond to cold storage (+) and heat storage () with multi PCMs. The comparison between the approximate solution (Eq. (23)) and the exact solution of Eq. (19) is shown in Fig. 3. From Fig. 3, the difference between exact solution and approximate solution of temperature ratio for cascaded heat/cold storage is decreased with an increase in stage number. From Eq. (18), T n;opt is derived from a and HTF temperature is calculated as

T i;opt ¼ T 0

 i T n;opt n T0

16i6n



ðCT n;opt T n1;opt Þ

2

T 0 an 2

a 1  ðC1Þ ðC a1Þ2

ð25Þ

  i1 i n n T T T 0 Cð n;opt Þ  n;opt T0 T0 C1

16i6n

ð26Þ

Since the objective function is positive for heat storage and negative for cold storage, the optimal solution corresponds to the maximum absolute value of heat transfer rate for cascaded heat/cold storage. The maximum heat transfer rate is as follows

jQ jmax ¼

mcp ðT 0  T n;opt Þ heat storage mcp ðT n;opt  T 0 Þ cold storage

ð27Þ

For cascaded heat storage, the temperatures of PCMs are above the environment temperature and the last-stage PCM temperature should be higher than the environment temperature. For cascaded cold storage, the PCM temperatures are lower than the environment temperature and it makes sense that the last-stage PCM temperature is lower than the environment temperature. This can be expressed as

T m;n;opt ¼

CT n;opt  T n1;opt C1



> Te

heat storage

< Te

cold storage

ð28Þ

The above condition is greatly affected by C, which is related to the optimization target. If the overall heat transfer area is restricted in practical application, the heat transfer area is decreased with an increase in stage number, leading to the decreased value of C. In this case, the above qualification is implicit and rather complex. If the heat transfer area of a single-stage thermal storage unit is constant, the stage number has no influence on the value of C. In this case, Eq. (28) can be simplified by setting the point function as:

f S ðnÞ ¼ an 

1 n1 C  1 T e a  C C T0



> 0 heat storage < 0 cold storage

ð29Þ

When n ¼ 1, the above inequality becomes

(

a

> C1 þ C1 C < þ 1 C

Te T0

heat storage

C1 T e C T0

cold storage

ð30Þ

By setting n ¼ 1, Eq. (29) becomes



1 X a þ1¼0 C nmcp

1 T n;opt

¼

ðC1Þ2 T n1;opt

The PCM phase-change temperature can be expressed as

This can be simplifies as 2

1

T m;i;opt ¼

( ) n X T i;opt ðC i  1Þ2 T i1;opt Ci  1 ¼ mcp ln þ  Ci T i1;opt Ci C i T i;opt  T i1;opt i¼1

a1 ¼ a2 ¼ . . . ¼ ai ¼ . . . ¼ an ¼ a ¼

nopt ¼

ð16bÞ

Based on Eq. (15), the following equation can be derived.

Sg;tot

The corresponding Lagrange multipler can be derived as

ð24Þ

8  1 n > > < > TT e 0 a > T e 1n > :< T0

heat storage

ð31Þ

cold storage

The fixed entropy generation rate ðXÞ and stage number ðnÞ is fully or partially embedded in temperature ratio ðaÞ. From Eqs. (19) and (29), the function for heat storage is decreased with an increase in stage number and increased with an increased in X. While, the value of a for cold storage is increased with an increase in stage number or an increased in X. Thus, there always exists a critical stage number for the existence of optimization solution by adjusting X, as



f S ð1Þ < . . . < f S ðncr þ 1Þ 6 0 < f S ðncr Þ < . . . < f S ð1Þ heat storage f S ð1Þ < . . . < f S ðncr Þ < 0 6 f S ðncr þ 1Þ < . . . < f S ð1Þ cold storage ð32Þ

In practical cascaded thermal storage system, the stage number is finite. The fixed entropy generation rate and stage number jointly determine the qualifications of cascaded heat/cold storage.

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 I: Steady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054

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

1.06

Temperature ratio α

Approximate solution αa (Eq.(23)), cold storage 1.04

Exact solution in Eq.(19), cold storage Approximate solution αb (Eq.(23)), heat storage

1.02

Exact solution in Eq.(19), heat storage

1.00

0.98

NTU=0.182 Ω=0.0015 W/K

0.96

0.94 0

50

100

150

200

Stage number n Fig. 3. Comparison of temperature ratio between exact solution and approximate solution.

Due to the absence of the analytical solution for optimization of heat transfer rate with fixed entropy generation rate, the exact form of value ranges for fixed entropy generation rate and stage number cannot be explicitly derived. 3.1.2. Entropy generation rate optimization with fixed heat transfer rate For optimizing the entropy generation rate, we can also regard the total value of entropy generation rate as the objective function and the total heat transfer rate ðQ tot ¼ HÞ as the constraint. In this way, the Lagrange function is established with the Lagrange multipler ðrÞ as

GS ðT 1 ; T 2 ; . . . ; T n ; kÞ ¼ Sg;tot þ rðQ tot  HÞ ( ) n X Ti ðC i  1Þ2 T i1 Ci  1 ¼ mcp ln þ  Ci T i1 Ci C i T i  T i1 i¼1 þ r½mcp ðT 0  T n Þ  H ð33Þ The first-order partial derivatives of the Lagrange function are

8 h i ðC iþ1 1Þ2 ðC i 1Þ2 > <  T þ T mc ¼0 16i6n1 p i1 iþ1 2 2 @GS ðC i T i T i1 Þ ðC iþ1 T iþ1 T i Þ ¼ h i 2 ðC 1Þ @T i > i : mcp 1  T r ¼0 i¼n Ti ðC T T Þ2 i1 i i

i1

ð34Þ @GS ¼ mcp ðT 0  T n Þ  H ¼ 0 @r

ð35Þ

The value of fixed heat storage ðHÞ is positive for heat storage, while negative for cold storage. From Eq. (35), the outlet HTF temperature is expressed with fixed heat transfer rate, as

T n;opt ¼ T 0 

H mcp

ð36Þ

ropt ¼

1 T i;opt



ðC i  1Þ2 T i1;opt ðC i T i;opt  T i1;opt Þ2

i¼n

ð37bÞ

The above equations (Eqs. (36) and (37)) can be solved numerically. These equations can be further simplified by setting ai ¼ T i;opt =T i1;opt ; C 1 ¼ C 2 ¼ . . . ¼ C i ¼ . . . ¼ C n ¼ C. Eq. (37a) becomes

a1 ¼ a2 ¼ . . . ¼ ai ¼ . . . ¼ an ¼ a ¼  ¼

1

1n

H

 1 T n;opt n T0 ð38Þ

mcp T 0

The corresponding optimized HTF temperature is

 T i;opt ¼ T 0 1 

H mcp T 0

ni

16i6n

ð39Þ

From Eq. (34), the Lagrange multipler in Eq. (37b) is

ropt ¼

1



ðC  1Þ2 T n1;opt

ðCT n;opt  T n1;opt Þ2 8 9 > > 1 > > > 2 < = ðC  1Þ ð1  mcHp T 0 Þn > mcp ¼ 1 2 1   > mcp T 0  H > > > n > : ; C 1 H 1 > T n;opt

ð40Þ

mcp T 0

The optimal phase-change temperature distribution of PCMs is expressed as

T m;i;opt ¼

  ni  i1 n T 0 C 1  mcHp T 0  1  mcHp T 0 C1

16i6n

ð41Þ

The minimum entropy generation rate for cascaded thermal storage can be obtained as

2

 2  2 T iþ1;opt C i  1 C iþ1 T iþ1;opt  T i;opt T m;iþ1;opt ¼ ¼ 16i6n1 C iþ1  1 C i T i;opt  T i1;opt T i1;opt T m;i;opt

3   2 61 H ðC  1Þ 1 C  17 7 Sg;min ¼ nmcp 6  1   4n ln 1  mcp T 0 þ C 5 C n C 1  mcHp T 0  1

ð37aÞ

ð42Þ

Based on Eq. (34), the following equation can be derived

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

optimization solution in this section exists unconditionally. In Regions H3 and C3, the given heat transfer rate exceeds the maximum applicable enthalpy embedded in HTF, and the optimization solution does not exist. In Regions H2 and C2, the value of H is in the range

The optimal solution can guarantee the minimization of entropy generation rate. The last-stage PCM temperature for heat storage should be higher than the environment temperature, while that for cold storage should be lower than the environment temperature. This condition should be strictly considered in the optimization, shown as

T m;n;opt ¼

CT n;opt  T n1;opt C1



> Te

heat storage

< Te

cold storage

C1 mcp jT 0  T e j < jHj < mcp jT 0  T e j C

ð43Þ

For a given H in the scope C1 mcp jT 0  T e j < jHj < mcp jT 0  T e j, C there may exist a critical stage number ðncr Þ satisfying the following condition:

Identically, for the constant heat transfer area of a heat/cold storage unit (constant NTU), we set a point function g S ðnÞ from Eq. (43) as follows:  g S ðnÞ ¼ C 1 



H mcp T 0

  1

11n

H mcp T 0

 ðC  1Þ

Te T0





> 0 heat storage < 0 cold storage



mcp T 0

 1  ðC  1Þ

Te T0



ð50Þ

Thus, the qualification of optimization solution can be summarized as

The value of H determines the range of stage number employed for cascaded heat/cold storage. The value of H is positive for heat storage and negative for cold storage. With an increase in H, the value of point function is decreased for heat storage, and increased for cold storage. With an increase in stage number, the point function increases for heat storage, while this value decreases for cold storage. For n ¼ 1, Eq. (44) becomes

H

g S ð1Þ 6 g S ðncr  1Þ 6 0 < g S ðncr Þ < g S ð1Þ heat storage g S ð1Þ < g S ðncr Þ < 0 6 g S ðncr  1Þ 6 g S ð1Þ cold storage

ð44Þ

 g S ð1Þ ¼ C 1 

ð49Þ

8 0 < jHj < C1 mcp jT 0  T e j > <1 6 n < 1 C C1 ncr 6 n < 1 C mcp jT 0  T e j 6 jHj < mcp jT 0  T e j > : inexistence jHj P mcp jT 0  T e j

ð51Þ

Thus, the fixed heat transfer rate and the stage number jointly determine the qualifications of cascaded heat/cold storage.

> 0 heat storage < 0 cold storage ð45Þ

(

The scope of H for n ¼ 1 can be obtained as

0
C1 mcp ðT 0 C

C1 mcp ðT 0 C

 T e Þ heat storage

 T e Þ < H < 0 cold storage

3.1.3. Validation of entropy optimization The entropy optimization results are validated with the exergy optimization result [33], as shown in Fig. 5 in terms of HTF and PCM temperatures. In Fig. 5(a) (cold storage) and Fig. 5(b) (heat storage), the fixed entropy generation rate in Section 3.1.1 is equal to the total entropy generation rate in exergy optimization, while the fixed heat transfer rate in Section 3.1.2 is the heat transfer rate in exergy optimization. As shown, the optimal result in Section 3.1.1, that in Section 3.1.2, and the exergy optimization result are in complete coincide with each other. This means that there is no fundamental difference between the above two optimizations with combination of heat transfer rate and entropy generation rate. There is only one result with exergy optimization for a fixed thermal storage process. The optimal value can be adjusted in the present combination optimization of entropy generation rate and heat transfer rate, including the exergy optimization result. The two entropy optimization solutions, extending the applicable scope of exergy optimization, are easily applied in practical design cascaded thermal storage system.

ð46Þ

For n ¼ 1, Eq. (44) can be deduced as

 g S ð1Þ ¼ ðC  1Þ 1 



H mcp T 0



Te T0



> 0 heat storage < 0 cold storage

ð47Þ

The scope of H for n ¼ 1 is

0 < H < mcp ðT 0  T e Þ heat storage mcp ðT 0  T e Þ < H < 0 cold storage

ð48Þ

Fig. 4 presents the schematic diagram of different regions for value range of fixed heat transfer rate. The number axis is divided into six regions: Regions H1–H3 for cascaded heat storage and Regions C1–C3 for cascaded cold storage. If the value of H is in Region H1 for heat storage or Region C1 for cold storage, the condition in Eq. (43) holds for arbitrary stage number and the

Cascaded cold storage

Region C3

Region C2

Cascaded heat storage

Region C1

Region H1

Region H2

Region H3

Θ 0

Fig. 4. Schematic diagram of different regions for value range of fixed heat transfer rate.

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H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx 1.0

For solution in Section 3.1.1

Ω=Sg,Ex=0.475 W· K-1 For solution in Section 3.1.2

0.9

0.8

n=6 NTU=1.18 T0=10 K

0.7

Te=273.15 K

Θ=QEx=14.86 W

HTF, exergy optimization [33] HTF, entropy, Section 3.1.1 HTF, entropy, Section 3.1.2 PCM, exergy optimization [33] PCM, entropy, Section 3.1.1 PCM, entropy, Section 3.1.2

0.6

0.5

Dimensionless temperature θ

Dimensionless temperature θ

1.0

HTF, exergy optimization [33] HTF, entropy, Section 3.1.1 HTF, entropy, Section 3.1.2 PCM, exergy optimization [33] PCM, entropy, Section 3.1.1 PCM, entropy, Section 3.1.2

0.9 0.8 0.7

n=6 NTU=1.18 T0=1000 K

0.6 0.5 0.4 0.3

For solution in Section 3.1.1

0.2

For solution in Section 3.1.2

0.1

Θ=QEx=368.072 W

Te=273.15 K

Ω=Sg,Ex=0.0995 W· K-1

0.4 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

Dimensionless HTF flow path X

0.4

0.6

0.8

1.0

Dimensionless HTF flow path X

(a) cold storage (T0=10 K)

(b) heat storage (T0=1000 K)

Fig. 5. Comparison of exergy optimization between the present result and that in Ref. [33].

3.2. Entransy optimization For the entransy optimization, the entransy dissipation rate is combined with the heat transfer rate in the cascaded heat/cold storage. In the previous work [36], the entransy optimization of cascaded heat storage was performed with analytical solutions provided. In this part, the optimization solutions of cascaded thermal storage system with entransy will be briefly introduced with the optimization of cold storage proposed. 3.2.1. Optimization of heat transfer rate with given entransy dissipation rate Taking the stored thermal energy as the objective function and the entransy dissipation rate as the constraint, the optimization function can be established as

F E ðT 1 ; T 2 ; . . . ; T i ; . . . ; T n ; kÞ " # n X 1 Ci þ 1 ðT i1  T i Þ2  U ¼ mcp ðT 0  T n Þ þ k mcp
2 C 1 i¼1 i

ð52Þ

This Lagrange function is optimized in Ref. [36]. For the condition C 1 ¼ C 2 ¼ . . . ¼ C n ¼ C, HTF and PCM temperatures, and maximum heat transfer rate are obtained as

T i;opt

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2UðC1Þ > < T 0  i nmc 1 6 i 6 n heat storage p ðCþ1Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : T 0 þ i 2UðC1Þ 1 6 i 6 n cold storage nmcp ðCþ1Þ

T m;i;opt

jQjmax

ð53Þ

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2UðC1Þ > < T 0  iðC1Þþ1 1 6 i 6 n heat storage C1 nmcp ðCþ1Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2UðC1Þ : T 0 þ iðC1Þþ1 1 6 i 6 n cold storage C1 nmcp ðCþ1Þ

ð54Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C1 ¼ 2nUmcp Cþ1

ð55Þ

n X 1 Ci þ 1 mcp ðT i1  T i Þ2 2 C 1 i¼1 i  þ l mcp ðT 0  T n Þ  H

T i;opt ¼ T 0 

i H
n mcp

T m;i;opt ¼ T 0 

/min ¼

16i6n

iðC  1Þ þ 1 H
C1 nmcp

ð57Þ

16i6n

C þ 1 H2
C  1 2nmcp

ð58Þ

ð59Þ

The corresponding qualifications for Sections 3.2.1 and 3.2.2 can be obtained when the last-stage phase-change temperature of PCM is greater than the environment temperature for heat storage and lower than the environment temperature for cold storage. More details can be found in Ref. [36]. The optimization result of cascaded thermal storage system based on entransy will be compared with that based on entropy in the following. 3.3. Parameters and analysis To conveniently express the temperature of cascaded heat/cold storage, the dimensionless temperature is defined by using environment temperature and inlet HTF temperature.

h¼

T  Te T0  Te

ð60Þ

For the ith-stage thermal storage unit, the exergy is calculated as

3.2.2. Optimization of entransy dissipation rate with fixed heat transfer rate The optimization function can be established by treating the total entransy dissipation as the objective function and the heat transfer rate as the constraint condition, as

GE ðT 1 ; T 2 ; . . . ; T i ; . . . ; T n ; lÞ ¼

In the above optimization function, the value of H is positive for heat storage and negative for cold storage. From Ref. [36], for C 1 ¼ C 2 ¼ . . . ¼ C n ¼ C, the optimization solutions of HTF and PCM temperatures, and minimum entransy dissipation rate are

!

  Te Exi ¼ W i;max ¼ mcp ðT i1  T i Þ 1  T m;i " # ðC i  1Þ2 T e T i1 Ci  1 ¼ mcp T i1  T i  þ Te Ci Ci C i T i  T i1

For the cascaded thermal storage system, the total exergy is

Extot ¼

n X i¼1

ð56Þ

ð61aÞ

Exi ¼

n X W i;max i¼1

" # n X ðC i  1Þ2 T e T i1 Ci  1 ¼ mcp T i1  T i  þ Te Ci Ci C i T i  T i1 i¼1

ð61bÞ

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

The thermal efficiency based on the first law of thermodynamics, representing the heating/cooling ability of thermal storage system, is defined as

Q tot mcp jT n  T 0 j ¼ ¼ 1  hn Q HTF mcp jT e  T 0 j

ð62aÞ

The exergy efficiency based on the second law of thermodynamics, standing for the work capability of the stored thermal energy, can be defined as

g2nd

Extot ¼ ¼ ExHTF

n X 2 ½T i1  T i  ðC i 1Þ Ci

T e T i1 C i T i T i1

  T 0 þ T e ln TT0e  1

i¼1

þ C iC1 Te i ð62bÞ

4. Results and discussion

For the thermal storage unit in a certain stage, given that the channel confining HTF is a parallel-plate channel, the overall thermal resistance can be obtained as:

R¼

1 1 d 1 ¼ RPCM þ Rw þ RHTF ¼ þ þ UA hPCM A kw A hHTF A

ð63Þ

Combined with Eq. (2), the parameter NTU is a function of HTF mass flow rate ðmÞ, HTF specific heat ðcp Þ, PCM convective heat transfer coefficient (hPCM , if convection near the wall exists), HTF convective heat transfer coefficient ðhHTF Þ, wall thickness ðdÞ, wall thermal conductivity ðkw Þ, and heat transfer area ðAÞ, expressed as

 NTU ¼ f NTU m; cp ; hPCM ; hHTF ; d; kw ; A

ð64Þ α=1.54

Dimensionless temperature θ

1.0

φtot=2072 W·K

1.0

NTU=1.18 Te=273.15 K Qtot=Θ=72.5 W

0.8

ΔT=17.3 K

0.7

Φ=1399 W·K Sg,tot=0.564 W·K -1

0.6

HTF, entransy, Section 3.2.2 [36] HTF, entropy, Section 3.1.2 PCM, entransy, Section 3.2.2 [36] PCM, entropy, Section 3.1.2

0.5

Fig. 6(a) and (b) present the HTF and PCM temperatures of entransy dissipation rate optimization and entropy generation rate optimization (Section 3.1.1 in present work) and entransy optimization (Section 3.2.1 [36]) respectively for cold storage and heat storage. From Fig. 6, the optimal HTF and PCM temperatures based on entransy exhibit linear distributions, while those based on entropy are geometric. For entropy optimization, the temperature ratio ðaÞ of cascaded cold storage is greater than 1, while that of cascaded heat storage is less than 1. The optimal HTF and PCM temperatures for cold storage are different from those for heat

n=6

Sg,tot=0.452 W·K -1 T0=10 K

0.9

4.1. Optimal temperature of HTF and PCMs

Dimensionless temperature θ

g1st ¼

The parameter NTU stands for the difference between HTF and PCM temperatures. Thus, increasing NTU is benefit for enhancing heat transfer and improving thermal efficiency of thermal storage, which can be achieved by decreasing HTF mass flow rate, HTF specific heat, and wall thickness, and increasing convective heat transfer coefficients of PCM and HTF sides, wall thermal conductivity, and heat transfer area. Since the HTF specific heat represents the ability of carrying thermal energy, decreasing HTF specific heat is usually not feasible. We should adopt other manners to decrease the temperature difference in thermal storage.

0.2

0.4

0.6

0.8

n=6 NTU=1.18 T0=1000 K Te=273.15 K Qtot=Θ=368.07 W

0.6

ΔT=84.55 K Φ=34966 W·K Sg,tot=0.11 W·K -1

0.4

HTF, entransy, Section 3.2.2 HTF, entropy, Section 3.1.2 PCM, entransy, Section 3.2.2 PCM, entropy, Section 3.1.2

0.2

0.0

0.4 0.0

α=0.85 Φ=37648 W·K Sg,tot=Ω=0.0995 W·K -1

0.8

0.0

1.0

0.2

0.4

0.6

0.8

Dimensionless HTF flow path X

Dimensionless HTF flow path X

(a) cold storage (T0=10 K)

(b) heat storage (T0=1000 K)

1.0

Fig. 6. Comparison between optimal temperature based on entropy (Section 3.1.2) and that based on entransy (Section 3.2.2) for fixed heat transfer rate.

1.0

T0=10 K Te=273.15 K

0.95

Ω=0.4 W·K -1 n=1: α=2.95 NTU=0.711 n=2: α=1.74 NTU=0.355 n=6: α=1.20 NTU=0.118

0.90

0.85

n=1

0.80

n=2

n=6

HTF PCM 0.2

0.4

0.6

0.8

Dimensionless HTF flow path X

(a) cold storage (T0=10 K)

1.0

T0=1000 K Te=273.15 K

Ω=0.15 W·K -1 n=1: α=0.529 NTU=7.4 n=2: α=0.637 NTU=3.7 n=6: α=0.821 NTU=1.2

0.8

0.6

0.4

0.2

0.0

0.75 0.0

Dimensionless temperature θ

Dimensionless temperature θ

1.00

n=1

n=2

n=6

HTF PCM 0.0

0.2

0.4

0.6

0.8

1.0

Dimensionless HTF flow path X

(b) heat storage (T0=1000 K)

Fig. 7. Temperature distribution for heat transfer optimization with fixed entropy generation rate for different stage numbers.

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H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx 1.0

NTU=0.59

NTU=0.59

0.5

0.8

Ω=0.05 W/K Ω=0.1 W/K Ω=0.2 W/K

0.4 0.3

Efficiency

Last-stage dimensionless temperature θmn

0.6

0.2

ncr=52 0.1

ncr=12

α=0.972

0.0 0

ncr=12

0.4

α=0.98 10

α=0.972

0.2

20

30

40

0

50

α=0.986

α=0.98

α=0.986

ncr=25

ncr=52

ncr=25

0.6

10

η1st

η2nd

Ω=0.05 W/K Ω=0.1 W/K Ω=0.2 W/K

20

30

40

50

Stage number n

Stage number n

(a) θ m,n

(b) efficiency

Fig. 8. Optimization results of heat transfer rate with given entropy generation rate.

1.0

0.8

NTU=0.59

NTU=0.59

ncr=1 10

-1

Θ=300 W Θ=600 W Θ=650 W

ncr=6

-2

10

Efficiency

Last-stage PCM dimensionless temp. θmn

100

20

0.4

η1st

η2nd

Θ=300 W Θ=600 W Θ=650 W

ncr=10

0.2

ncr=10 0

ncr=6

0.6

0.0 40

60

80

100

Stage number n

(a) θ m,n

0

ncr=1

20

40

60

80

100

Stage number n

(b) efficiency

Fig. 9. Optimization result of entropy dissipation rate for given heat transfer rate.

storage. In cold storage, the optimal PCM temperature band based on entropy is broader than that based on entransy. With heat transfer rate optimization with fixed entropy generation rate (n = 1, 2, 6), the HTF and PCM temperatures of cold storage are shown in Fig. 7(a), and those of heat storage are shown in Fig. 7(b). The bending direction of HTF temperature curve of cold storage is different from that of heat storage. With the increased stage number, the difference between temperatures of outlet HTF and environment becomes smaller, and thermal energy carried by HTF can be used more thoroughly. From Fig. 7, compared with the single-PCM system, the multiple-PCM system provides multigrade thermal energies, and extends the temperature range of thermal storage.

4.2. Thermal performance Fig. 8(a) and (b) show the optimization result for heat transfer rate with fixed entropy generation rate respectively for the laststage dimensionless PCM temperature and efficiencies (g1st and g2nd ). From Fig. 8, the increased stage number can lead to the decreased hm;n , and the increased thermal and the exergy efficiencies. The existence of critical stage number ðncr Þ for certain values of fixed entropy generation rate, which restricts the stage number in the range 1 6 n 6 ncr , can also be verified by Fig. 8. In

addition, the increased X leads to the decreased critical stage number. This means that the increased value of fixed entropy generation rate, representing the irreversibility of the thermodynamic process, reduces the stage number range in cascaded thermal storage. In Fig. 9(a) and (b), the effects of stage number on the last-stage PCM dimensionless temperature and efficiencies are respectively presented with fixed heat transfer rate. From Fig. 9(a), the laststage dimensionless PCM temperature increases with an increase in stage number. For fixed heat transfer rate, leading to a fixed thermal efficiency, the exergy efficiency is increased with an increase in stage number. From Fig. 9, the critical stage number may also exist for certain values of fixed heat transfer rate, which restricts the stage number in the range n P ncr . The increased H will result in the increased critical stage number. This is in accordance with the qualification for fixed heat transfer rate. This means it requires more thermal storage units (stage number) for maintaining higher heat transfer rate.

4.3. Selection of PCMs Taking the heat storage with optimizations of Sections 3.1.2 and 3.2.2 as an example, we performe PCM selection in this part. Schematically, the double-pipe heat reservoir is employed for the

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3.1

3.9

5.1

7.6

14.6

B

11

cascaded heat storage device. Na–K (22.2–77.8 wt%) [39] is used as the HTF for carrying heat. Thermo-physical properties (specific heat, density and thermal conductivity) of Na-K are respectively

2.9 86.7 86.3 775.3 677.6 579.9 482.2 384.5 5

740.9 628.8 533.8 453.0 384.5

K2CO3–Li2CO3 (65–35 wt%)/778.2 [41] LiCl–LiF–Li2SO4–Li2MoO4 (51.5–16.2–16.2–16.2 wt%)/675.2 [41] NaNO3/579.2 [42] LiNO3–NaCl (87–13 wt%)/481.2 [43] RT110/385.2 [40]

CaCl2–NaCl–KCl (50–42.75–7.25 wt%)/738.2 [41] MnCl2–KCl–NaCl (45–28.7–26.3 wt%)/623.2 [41] LiCl–LiOH (37–63 wt%)/535.2 [43] LiCl–LiNO3–NaNO3 (1.4–47.9–50.7 wt%)/453.2 [43] RT110/385.2 [40]

3.7 84.9 84.4 751.0 628.9 506.7 384.6 4

711.3 579.5 472.1 384.6

Al/Si/Sb (86.4–9.4–4.2 wt%)/744.2 [41] MnCl2–KCl–NaCl (45–28.7–26.3 wt%)/623.2 [41] KCl–ZnCl2 (68.1–31.9 wt%)/508.2 [40] RT110/385.2 [40]

LiF–KF–NaF–BaF2 (45.7–41.2–11.3–1.8 wt%)/711.2 [41] NaNO3/579.2 [42] Sn–Zn (91–9 wt%)/472.2 [44] RT110/385.2 [40]

4.9 81.7 81.0 710.3 547.5 384.6 3

664.4 505.5 384.6

LiF–KF–NaF–BaF2 (45.7–41.2–11.3–1.8 wt%)/711.2 [41] LiCl–LiOH (34.5–65.5 wt%)/547.2 [43] RT110/385.2 [40]

KCl–MnCl2–NaCl (45.5–34.5–20 wt%)/663.2 [41] KCl–ZnCl2 (68.1–31.9 wt%)/508.2 [40] RT110/385.2 [40]

7.3 74.9 628.9 384.6 2

579.5 384.6

MnCl2–KCl–NaCl (45–28.7–26.3 wt%)/623.2 [41] RT110/385.2 [40]

74.0

14.6

NaNO3/579.2 [42] RT110/385.2 [40]

A

50.1

50.1

A

RT110/385.2 [40] RT110/385.2 [40]

A B

384.6 384.6

A

1

Selected PCMs/Tm,pr (K)

B

B

1

Tm,opt (K) n

Table 1 PCM selection with entropy and entransy optimizations for fixed heat transfer rate (H = 6  104 W, g1st = 81.4%, A for entransy and B for entropy).

g2nd (%)

/tot  106 (W
K)

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

870 J
kg
K1 , 719 kg
m3 , and 26:2W
m1
K1 . HTF flows inside the inner pipe and multiple PCMs are embedded separately in the annular space of the double-pipe heat reservoir. The environment temperature is set as 273.15 K. The radius of the inner pipe is 0:05 m and velocity of HTF is 0:1 m
s1 . The HTF inlet temperature is set as 873:15 K and the overall heat transfer coefficient is estimated as 4500 W
m1
K1 . Based on Eqs. (41) and (58) respectively for entropy and entransy optimizations, the optimal phase-change temperature of PCMs with the fixed heat transfer rate can be obtained, which can be applied for filtering PCMs in cascaded thermal storage. Table 1 shows the PCM selection with optimizations of entropy and entransy for stage number from 1 to 5 (H = 6  104 W, g1st = 81.4%). With an increase in stage number, the exergy efficiency is increased, but the entransy dissipation rate is decreased. The optimal result based on entropy is equal to that based on entransy for single-stage heat storage ðn ¼ 1Þ. For n > 1, the optimal exergy efficiency based on entropy is superior to that based on entransy. While, the optimal entransy dissipation rate based on entropy is greater than that based on entransy for n > 2. Practically, since there are many optional PCMs with phase-change temperatures near the optimal value, some other aspects should be well examined, such as energy density, flammability, toxicity, stability, corrosivity, cost, and so on. 4.4. Discussion on entropy and entransy for optimizing cascaded thermal storage For PCM selection, the optimization method should be chosen from entransy and entropy. Fig. 10 shows the performance comparison between entropy and entransy optimizations for fixed heat transfer rate. From Fig. 10(a), the optimal exergy efficiency based on entransy and that based on entropy are increased with the increased stage number. The optimal exergy efficiency based on entropy is greater than that based on entransy. From Fig. 10(b), the optimal values of entransy dissipation rate and entropy generation rate decrease for the increased stage number, which manifests that the irreversibility of thermal storage process is decreased. The optimal entransy dissipation rate based on entransy is smaller than that based on entropy, while the optimal entropy generation rate based on entropy is less than that based on entransy. Fig. 11(a) and (b) respectively show optimal thermal and exergy efficiencies based on entransy and entropy. In Fig. 11 (a), the fixed entransy dissipation rate, which is set as that for entransy optimization, is used in the entropy optimization. With an increase in NTU, the thermal efficiency is increased. The optimal thermal efficiency based on entropy is lower than that based on entransy. From Fig. 11(b), with an increase in NTU, the optimal exergy efficiency is increased, and the optimal exergy efficiency based on entropy is slightly higher than that based on entransy. When NTU is very large, optimal thermal and exergy efficiencies is nearly unchanged due to sufficient heat transfer. Overall, the entropy generation stands for the capability of heat-work conversion, while the entransy dissipation is a variable for heat transfer irreversibility. Storing exergy and storing heating/cooling ability are two main purposes of thermal storage. Thus, entropy is more appropriate for optimizing cascaded storage system which exports useful work, while entransy is more suitable for cascaded storage system directly providing heat/cold.

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 I: Steady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054

12

H.J. Xu, C.Y. Zhao / International Journal of Heat and Mass Transfer xxx (2016) xxx–xxx 105

Θ=650 W

Entransy dissipation rate φ(W K)

Exergy efficiency η2nd

0.95

NTU=0.59

0.90

0.85

Entransy [36] Entropy

0.80

Entransy [36] Entropy

φ

Sg

10-1

ncr=10

104

ncr=15

ncr=10

Entropy generation rate Sg(W/K)

η1st=0.92

ncr=15

Θ=650 W

NTU=0.59

0.75 101

Stage number n

(a) exergy efficiency

102

101

Stage number n

102

(b) rates of entropy dissipation or entropy generation

Fig. 10. Performance comparison between entransy and entropy optimizations.

0.15

Entransy dissipation rate via entropy optimization 0.10

0.30

140

120

100

n=1 n=3 n=5

80

0.05 60

0.00 10-1

100

Number of transfer units NTU

40 101

Θ=10 W

Exergy efficiency η2nd

Entransy [36] Entropy (Ω=0.15 W/K) n=1 n=3 n=5

Entransy dissipation rate φ(W K)

Thermal efficiency η1st

0.20

0.25

Te=273.15 K

n=2 0.20

0.15

Entransy [36] Entropy

0.10

0.05 10-1

100

101

Number of transfer units NTU

(a) thermal efficiency

(b) exergy efficiency

Fig. 11. Effect of NTU on thermal and exergy efficiencies.

5. Conclusions

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Optimizations of steady cascaded heat/cold storage are performed based on entropy and entransy. Simplified analytical solution for optimal temperatures of HTF and PCMs based on entropy is derived, which is compared with that based on entransy. Qualifications for optimization solutions are also discussed. The conclusions are drawn below. (1) The optimal HTF and PCM temperatures are linear with entransy optimization, while those with entropy optimization are geometric. (2) For optimizing heat transfer rate with fixed entropy generation rate, a critical stage number may exist, leading to n 6 ncr . For optimizing entropy generation rate with fixed heat transfer rate, ncr may also exist with n P ncr . (3) Thermal and exergy efficiencies increase with increased stage number or the increased NTU. Thermal and exergy efficiencies are nearly constant for sufficiently large stage number or NTU. (4) In steady case, entropy should be adopted in cascaded storage system for heatwork conversion, while entransy should be employed in cascaded storage optimization for heating/cooling.

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Acknowledgments 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.

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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 I: Steady cases, Int. J. Heat Mass Transfer (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.054