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Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Finite time thermodynamic analysis of a solar duplex Stirling refrigerator

Dongdong Dai, Zhichun Liu, Fang Yuan, Rui Long, Wei Liu⁎

School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

H I G H L I G H T S

• A solar duplex Stirling refrigerator system is proposed.

• A thermodynamic model considering finite time is developed.

• Polytropic process in expansion/compression is adopted.

• Two constraints including power and time are considered.

A R T I C L E I N F O

Keywords:Solar energyStirling heat engineStirling refrigeratorFinite time thermodynamics

A B S T R A C T

Stirling engine is a machine that has the same theoretical efficiency as that of a Carnot cycle under ideal con-ditions and has an aptitude for utilizing sustainable energy such as solar and biomass energy. In this study, asolar duplex Stirling refrigerator system was proposed to capture solar energy for refrigeration. The system iscomposed of a solar collector for collecting solar thermal energy, a Stirling heat engine for converting thermalenergy into power, and a Stirling refrigerator to utilize power for cooling. The system is modeled based on finitetime thermodynamics. Moreover, two constraints i.e., power and cycle time equality of the engine and re-frigerator are considered in this model. In addition, the linearized heat loss of the solar collector, finite rates ofheat transfer, regenerative heat loss, and conductive thermal bridging loss of the Stirling engine and refrigeratorare considered in this model. Finally, the effects of design parameters on the performance of the system areinvestigated. This may serve as clean technology for solar refrigeration.

1. Introduction

Renewable energy such as solar, wind, tidal, and geothermal energycontributes to a supplement to energy and a reduction in environmentalpollution. Among these renewable energy, solar energy, as one of themost widely distributed energy all over the world, has been paid moreattention owing to the increase in demand for energy and environ-mental protection. The energy consumption of the refrigeration in-dustry has increased during the last few decades. Researches on therefrigerators driven by solar energy have aroused wide attentionespecially in the countries and regions with the availability of a hugeamount of solar energy and long daily sunny hours.

To date, a variety of solar refrigerators have been developed. Thesesolar refrigerators can be classified into solar thermal, solar electric,and other newly emerging technologies [1]. A solar thermal refrigera-tion system uses solar heat to provide energy to drive the refrigerationpart. As a kind of solar thermal refrigeration technologies, thermo-mechanical refrigeration systems use Rankine or Stirling engine to

convert solar heat to mechanical work for driving a vapor compressionrefrigerator. Sorption refrigeration including absorption [2], adsorption[3], and desiccant cooling [4] is the other kind of solar thermal re-frigeration technologies. A solar electric refrigeration system is com-posed mainly of photovoltaic panels and an electrical refrigerationdevice. Technologies like vapor compression, thermoelectric elements,Stirling refrigerator, thermo-acoustic, and magnetic cooling have beenused in combination with photovoltaic panels for solar electric re-frigeration [1].

Besides solar electric and solar thermal refrigeration, some newlyemerging technologies have been developed. A solar-powered electro-chemical refrigeration system was proposed to collect the solar elec-trical energy to drive the electrochemical refrigerator [5]. An ejectorrefrigerator is powered by solar thermal energy and based upon theejector refrigeration [6]. Besides, some combined or hybrid systemshave been developed in solar refrigeration researches [7].

Stirling engines are appealing engines with regard to renewableenergy utilization as they have high theoretical efficiency, low

https://doi.org/10.1016/j.applthermaleng.2019.04.098Received 29 August 2018; Received in revised form 20 April 2019; Accepted 22 April 2019

⁎ Corresponding author.E-mail address: w_liu@hust.edu.cn (W. Liu).

Applied Thermal Engineering 156 (2019) 597–605

Available online 23 April 20191359-4311/ © 2019 Elsevier Ltd. All rights reserved.

T

emissions, low noise, and multi-fuel capability. Stirling engines drivenby solar energy facilitates the conversion of solar energy into power,which is environmentally friendly owing to its nature. Stirling engine-based units are considered best among the most effective low-powerrange solar thermal conversion units [8]. As mentioned above, Stirlingrefrigerators have also been utilized by numerous engineering compa-nies in refrigeration devices. A duplex Stirling refrigerator is an in-tegrated refrigerator that is composed of a Stirling heat engine and aStirling refrigerator used for cooling. It is a promising device owing toits high efficiency, reliability, and compactness, and low noise, emis-sion, and operation cost [9]. Certain researchers studied the duplexStirling refrigerator for domestic refrigeration applications [10,11]. Asthe Stirling engine is competitive in solar energy systems, the duplexStirling refrigerator can also be considered in solar refrigeration.

In analyzing and designing Stirling engines, researchers proposedseveral models using classical thermodynamics [12–14]. With the de-velopment of finite-time thermodynamics (FTT) [15,16], several Stir-ling engine models have been proposed using FTT. Considering the ir-reversibility in the external heat transfer processes, Blank et al. [17]analyzed an endoreversible Stirling engine with FFT and obtained theefficiency at maximum power. Wu et al. [18,19] investigated the op-timal performance of a Stirling engine with finite-speed effect in theregenerating processes and finite heat transfer in the isothermal pro-cesses. Analyzing the irreversibility of regenerator caused by tempera-ture difference, Dai et al. [20] obtained the theoretical limits of re-generator and Stirling engines via FTT. Using the FTT method, Kaushiket al. [21,22] developed a Stirling heat engine model subjects to a finiteheat capacitance rate of working substance, the heat leak between tworeservoirs, and regenerative losses. Dai et al. [23] constructed a refinedmodel considering the regenerative loss that was supplied by the heatsource and optimized the Stirling engine with a multi-objective opti-mization method based on MOPSOCD. Ahmadi et al. [24–26] optimizedand compared the finite time thermodynamic model of Stirling enginewith other thermal models.

Stirling engines driven by solar energy have also been investigated.Kongtragool et al. [27] presented a theoretical investigation regarding

the optimum absorber temperature of a once-reflecting full conicalsolar concentrator for maximizing the total efficiency of a solar Stirlingengine. Considering the limited heat transfer and regenerative loss,Chen et al. [28] investigated the performance of a combined systemconsisted of a Stirling engine and a solar collector using FTT. Li et al.[29] applied FTT to optimize the maximum power output and thecorresponding thermal efficiency of a solar Stirling engine. Ahmadiet al. [30–33] adopted a multi-objective optimization method in de-signing the solar Stirling engines and obtained considerably realisticresults compared with the single-objective optimization method.

The theory of FTT has also been successfully used in the analysis ofStirling refrigerators. With a given cooling rate, Chen et al. [34] de-veloped an irreversible Stirling refrigerator model that considered thefinite rate heat transfer, regenerative loss, and heat leak. Tyagi et al.[35] proposed an irreversible Stirling cryogenic refrigerator model in-cluding external and internal irreversibilities along with finite heatcapacities of external reservoirs. They also optimized the Stirling re-frigerator model that they proposed with the thermoeconomic criterion[36]. Based on NSGA-II algorithm, Ahmadi et al. [37] optimized theinput power, coefficient of performance, and cooling rate of a Stirlingrefrigerator simultaneously using the method of multi-objective evo-lutionary approaches. Combining the second version of non-dominatedsorting genetic algorithm (NSGA-II), Varun [38] developed a multi-objective optimization for three Stirling models.

Only a few thermodynamic models of the duplex Stirling re-frigerator are found in the literature. Erbay et al. [39] conducted athermodynamic analysis of a duplex Stirling refrigerator considering itsengine and refrigerator separately but under the constraint of the workequality. According to the advantages of solar Stirling engines andduplex Stirling refrigerators, we proposed a solar duplex Stirling re-frigerator system consisting of a solar collector and a duplex Stirlingrefrigerator for converting solar heat energy for cooling indirectly.Unlike the previous thermodynamic analysis under the only constraintof the work equality, we took finite time into consideration and twoconstraints, power and cycle time equality, were considered as theconstraints of the duplex system. The proposed model was analyzed

Nomenclature

A collector aperture area, m2

C collector concentrating ratiocv isochoric heat capacity of the working fluid, J/(g·K)c1, c1, c1, c1 constanter regenerator effectiveness of Stirling engineer′ regenerator effectiveness of Stirling refrigeratorh conduction/convection coefficient, W/(m2·K)I direct solar flux intensity, kW/m2

k thermal leak coefficient, W/KM constant rate of regeneration in engine, s/KM′ constant rate of regeneration in refrigeration, s/Kn mole number of working fluid, moleP output power of engine, WP′ power consumption of refrigerator, Wq cooling rate, WQ heat, JR ideal gas constant, J/(mol·K)t time, sT temperature, KV volume, m3

W work, J

Greek symbols

δ Stefan’s constant, W/(m2·K4)

ε emissivity factor of the collectorη0 collector optical efficiencyηe thermal efficiency of Stirling engineηs thermal efficiency of dish collectorλ volume ratio of engineλ′ volume ratio of refrigeratorσR coefficient of performanceσo overall coefficient of performanceτ cyclic period of engine, sτ′ cyclic period of refrigerator, s

Subscripts

0 ambientapp collector aperturee engineh expansion process in engineh′ expansion process in refrigeratorH heat sourceH′ heat sink of refrigeratorl compression process in enginel′ compression process in refrigeratorL heat sinkL′ heat source of refrigeratorr regenerationrec absorberu useful gain of dish solar collector

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using the FTT method, resulting in a considerably realistic investigationof the performance of the system. In addition, we studied the effects ofthe design parameters on the overall performance of the solar duplexStirling refrigerator system for application guidance.

2. Thermodynamic analysis of solar duplex Stirling refrigeratorsystem

The solar duplex Stirling refrigerator is a system for converting solarenergy for indirect cooling. As an integrated system, the solar duplexStirling refrigerator consists of a dish solar collector for collecting solarheat energy, a Stirling cycle engine for converting solar heat energy towork, and a Stirling cycle refrigerator for cooling. The T-S diagram ofthe solar duplex Stirling refrigerator system is depicted in Fig. 1. AStirling cycle consists of two isothermal branches and two isochoricregenerating branches. As finite-time thermodynamics indicates, withina finite time, there are temperature differences between the workingfluid and external heat reservoirs. The working fluid of the Stirlingengine works between the heat source with constant temperature THand heat sink with constant temperature TL. The working fluid of theStirling refrigerator absorbs heat from the heat source with constanttemperature TL′, and rejects heat to the heat sink with constant tem-perature TH′. The power and cycle time are two constraints of the solarduplex refrigerator system; the produced power by the engine is con-sumed by the refrigerator, and the cycle time of the Stirling engine isequivalent to that of the Stirling refrigerator. The mathematical modelof the duplex Stirling refrigerator system is performed considering thethree components of the duplex system separately as the dish solarcollector, the heat engine, and the refrigerator sides on finite-timethermodynamics.

2.1. Analysis of dish solar collector

The dish solar collector employed in this study is a device that cantrack the sun, focus solar energy into a cavity absorber, and convertsolar heat energy to thermal energy for heating expansion space of theStirling engine. According to the previous research [29,33], actualuseful heat gain of the dish solar collector considering conduction,convection, and radiation losses is given by

= − − + −Q IA η A h T T εδ T T[ ( ) ( )]u app rec H H0 04

04 (1)

where I is the direct solar flux intensity, Aapp is the collector aperturearea, η0 is the collector optical efficiency, Arec is the absorber area, h isconduction/convection coefficient, TH is the absorber temperature, T0 isthe ambient temperature, ε is emissivity factor of the collector, δ is theStefan’s constant.

Thermal efficiency ηs of the dish collector is defined as the ratio ofthe useful collected energy to the solar energy input

= = − − + −η QIA

ηIC

h T T εδ T T1 [ ( ) ( )]su

appH H0 0

404

(2)

where C=Aapp/Arec is the collector concentrating ratio. With a given ηs,the absorber temperature TH can be obtained from Eq. (2) with differentdirect solar flux intensities.

2.2. Analysis of Stirling heat engine

In practice, regenerative heat loss should be considered owing to theimperfect regenerator. The regenerative effectiveness, defined as theratio of heat transfer in real to that in ideal during regenerative pro-cesses, can be determined using the following expression

= = −−

= −−

e QQ

T TT T

T TT Tr

real

ideal

3 2

4 2

4 1

4 2 (3)

Denote the ratio of working fluid temperature in the Stirling engineas

=γ TT

2

4 (4)

Subsequently, the working fluid temperature in state 1, 2, and 3 canbe expressed as

⎧

⎨⎩

= − −== − +

T γ e TT γTT γ e γ T

[1 (1 ) ]

[(1 ) ]

r

r

1 4

2 4

3 4 (5)

It is necessary for the Stirling engine to proceed within finite time,and thus, the heat transfer between the external heat reservoirs andworking substance occurs with a finite temperature difference. FromSection 2.1 we can obtain the absorber with temperature TH, which isused as the heat source of Stirling heat engine. The working fluid ofStirling heat engine expands with variable temperature owing to thefinite time and convection heat transfer between the working fluid andconstant temperature heat source. Instead of an isothermal process, apolytropic process is adopted here to describe the expansive process.Denote the polytropic exponent by nH and the state equation of theworking fluid in the expansive process can be presented as

= = =pV p V p V cn n n3 3 4 4 1H H H (6)

where c1 is a constant. Similarly, the state equation of the working fluidin the compressive process can be stated as

= = =pV p V p V cn n n1 1 2 2 2C C C (7)

where nC and c2 are the polytropic exponent and constant in the com-pressive process, respectively.

Combining the equations above and the ideal gas state equation, wecan obtain the expression of polytropic exponents as

Fig. 1. T-S diagram of the solar duplex Stirling refrigerator system.

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599

=+ −

+ne e γ

λln( (1 ) )

ln1H

r r(8)

and

= +− +( )

nλ

ln

ln1C

γe e γ(1 )r r

(9)

where λ is the ratio of volume during the regenerative processes, i.e.,

= =λ VV

VV

1

2

4

3 (10)

Denoting the variable temperature of working fluid by T, the heatabsorbed from the absorber and released to the heat sink can be de-scribed as

= −δQ α T T dt( )h h H (11)

and

= −δQ α T T dt( )l l L (12)

where αh is high-temperature side convection heat transfer coefficient,αl is the low-temperature side convection heat transfer coefficient, theheat sink temperature is TL.

The energy conservation equations in expansive and compressiveprocesses can be expressed as

= +δQ mc dT pdVh v (13)

and

− = +δQ mc dT pdVl v (14)

where cv is the isochoric heat capacity of the working fluid.From Eqs. (11)–(14) we can obtain the formulas in expansive and

compressive processes as

=− +

−

( )( )

dtc n

α T VdV

(1 ) 1cR H

h Hn c V

mR

1vg

Hg

1

(15)

and

=− +

−

( )( )

dtc n

α T VdV

(1 ) 1cR C

l Ln c V

mR

2vg

Cg

2

(16)

The expansion and compression durations can be derived by in-tegrating Eqs. (15) and (16) as follows

=+ −

−−−

−

−tm R c n

α nmR T c VmR T c V

( (1 ))( 1)

lnhg v H

h H

g Hn

g Hn

1 41

1 31

H

H (17)

and

=+ −

−−−

−

−tm R c n

α nmR T c VmR T c V

( (1 ))( 1)

lnlg v C

l C

g Ln

g Ln

2 21

2 11

C

C (18)

where th indicates the expansion duration and tl indicates the com-pression duration.

As a finite-speed effect exists in the regenerating processes [18], thetime of the regeneration can be assumed to be proportional to thetemperature difference of the working fluid in the two isothermalprocesses, i.e.,

= = −t t M T T( )r r1 2 4 2 (19)

where M is the constant rate of the two regenerating processes. For anon-uniform temperature change, M is the average rate of temperaturechange.

From Eqs. (17)–(19), the duration of each branch of the Stirlingengine cycle can be obtained, and thus, the cyclic period τ can be ob-tained as

= + + +τ t t t th l r r1 2 (20)

Work produced during expansive and compressive processes can beexpressed as follows

∫= =−

− −W pdVmR T

nγ e

1(1 )(1 )h V

V g

Hr

4

3

4

(21)

and

∫= =−

− −W pdVmR Tn

γ e1

(1 )(1 )l V

V g

Cr

4

1

2

(22)

As no work is produced during regenerative processes, the cyclework of the Stirling engine cycle can be determined as

⎜ ⎟= + = ⎛⎝ −

−−

⎞⎠

− −W W Wn n

mR T γ e11

11

(1 )(1 )h lH C

g r4(23)

In the expansive process, the heat absorbed from the external re-servoir is used to produce work and increase the internal energy of theworking fluid, which can be expressed as

⎜ ⎟= + = − − ⎛⎝ −

+ ⎞⎠

Q W U γ eR

nc mTΔ (1 )(1 )

1h h rg

Hv 4

(24)

For a Stirling heat engine, a conductive thermal leaking loss existsbetween the two external heat reservoirs. The conductive thermalleaking loss can be assumed to be proportional to the cycle time and canbe expressed as

= −Q k T T τ( )h leak h leak H L_ _ (25)

where kh_leak represents the thermal leak coefficient between the ab-sorber and the heat sink.

The heat released from the absorber is absorbed by the workingfluid and consumed via conductive thermal leaking loss. As a result, thenet heat released from the absorber QH and absorbed by the heat sinkQL is given by

= +Q Q QH h h leak_ (26)

Therefore, the power output can be obtained by dividing the workoutput by the cycle period, i.e.,

= =− − −

+ +− −( )

P Wτ

mR T γ e

t t t

(1 )(1 )

2n n g r

h l r

11

11 4H C

(27)

The thermal efficiency of the Stirling heat engine can be expressedas

=+

=− − −

− − + + −

− −

−( )( )

η WQ Q

mR T γ e

γ e c mT k T T τ

(1 )(1 )

(1 )(1 ) ( )

eh h leak

n n g r

rR

n v h leak H L

_

11

11 4

1 4 _

H C

g

H (28)

2.3. Analysis of Stirling refrigerator

Regenerative heat loss should be considered owing to the imperfectregenerator in the Stirling refrigerator. As shown in Fig. 1(b), the re-generative effectiveness of the Stirling refrigerator can be defined as

=′′

= −−

= −−′

′ ′

′ ′

′ ′

′ ′e

QQ

T TT T

T TT Tr

real

ideal

3 2

4 2

4 1

4 2 (29)

The temperature ratio and volume ratio can be denoted as

′ = ′

′γ T

T2

4 (30)

and

′ = =′

′

′

′λ V

VVV

2

1

3

4 (31)

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600

Subsequently, the temperature of working fluid in states 1′, 2′, and3′ can be represented as

⎧

⎨⎩

= − − ′= ′

= − ′ + ′

′ ′ ′

′ ′

′ ′ ′

T e γ TT γ T

T e γ γ T

[1 (1 )]

[ (1 ) ]

r

r

1 4

2 4

3 4 (32)

In a Stirling refrigerator, the working fluid absorbs heat from theheat source at a low temperature TL′, and rejects heat to the heat sink ata high temperature TH′. Owing to the finite time and finite temperaturedifference between the working fluid and external reservoirs, the tem-perature of the working fluid is different from that of the heat sink orheat source. Instead of isothermal processes, polytropic processes areadopted here to describe the expansive and compressive processes.Denoting the polytropic exponents by nH′ and nC′, the state equations ofthe working fluid in expansive and compressive processes can be ex-pressed as

= = =′ ′ ′ ′′ ′ ′pV p V p V cn n n

3 3 4 4 3C C C (33)

and

= = =′ ′ ′ ′′ ′ ′pV p V p V cn n n

1 1 2 2 4H H H (34)

where c3 and c4 are two constants.According to Eqs. (29)–(34), the expressions of polytropic exponents

in expansive and compressive processes can be represented as follows

=′

+′− ′ + ′′nλ

ln

ln1C

e γ γ1

(1 )r

(35)

and

=′

+′

− − ′′

′

nλ

ln

ln1H

e γγ

1 (1 )r

(36)

Denoting the variable temperature of the working fluid by T, theheat absorbed from the absorber and released to the heat sink can bedescribed as

= −′ ′ ′δQ α T T dt( )h h H (37)

and

= −′ ′ ′δQ α T T dt( )l l L (38)

where αh′ is high-temperature side convection heat transfer coefficient,αl′ is the low-temperature side convection heat transfer coefficient, theheat sink temperature is TH′ and the heat source temperature is TL′.

The energy conservation equations in expansive and compressiveprocesses can be expressed as

− = +′δQ mc dT pdVh v (39)

and

= +′δQ mc dT pdVl v (40)

From Eqs. (37)−(40) we can obtain the formulas in expansive andcompressive processes as

=− +

−

′

′′ ′( )

dtn

αdV

(1 ) 1CcR

hT V

cV

mR

vg

HnC

g3 (41)

and

=− +

−

′

′′ ′( )

dtn

αdV

(1 ) 1HcR

lT V

cV

mR

vg

LnH

g4 (42)

Denoting the compression duration by th′ and expansion duration bytl′, the following expressions can be derived as

=− +

−−−

′′

′ ′

′−

′−

′

′t

m n c Rα n

T mR c VT mR c V

((1 ) )( 1)

lnhC v g

h C

H gn

H gn

3 41

3 31

C

C (43)

=− +

−−−

′′

′ ′

′−

′−

′

′t

m n c Rα n

mR T c VmR T c V

((1 ) )( 1)

lnlH v g

l H

g Cn

g Cn

4 21

4 11

H

H (44)

Since a finite-speed effect exists in the regenerating processes, thetime of the regeneration in the refrigerator cycle can be assumed to beproportional to the temperature difference of the working fluid in thetwo isothermal processes, i.e.,

= = ′ −′ ′ ′ ′t t M T T( )r r1 2 4 2 (45)

where M′ is the constant rate of the two regenerating processes in re-frigeration. For a non-uniform temperature change, M′ is the averagerate of temperature change. Subsequently, we can obtain the cyclicperiod τ′ as

′ = + + +′ ′ ′ ′τ t t t th l r r1 2 (46)

Work produced in expansive and compressive processes can be de-scribed as

∫= = −−′

−

′′

′W pdV λ

nmR T1

1h V

V n

Cg

1

4C

3

4

(47)

∫= = − ′−′

−

′′

′

′ ′W pdV λ

nmR T1

1l V

V n

Hg

1

2H

1

2

(48)

As no work is produced during regenerative processes, the cyclework required in the Stirling refrigerator cycle can be determined asfollows

⎜ ⎟= − = ⎛⎝

− ′−

′ − − ′−

⎞⎠

′ ′

−

′

−

′′

′ ′W W W λ

nγ λ

nmR T1

11

1R l h

n

H

n

Cg

1 1

4H C

(49)

Applying the first law of thermodynamics to the process 1′−2′, theheat absorbed from the external heat source is used to produce workand increase the internal energy of the working fluid, which can beexpressed as

= + = − ′−

− − −′ ′

−

′′ ′ ′

′Q W U λ

nmR T mc e T TΔ 1

1(1 )( )l l

n

Hg l v r

1

4 2H

(50)

A conductive thermal leaking loss also exists between the two ex-ternal heat reservoirs in the Stirling refrigerator cycle. The conductivethermal leaking loss can be assumed to be proportional to the cycle timeand can be expressed as

= − ′′ ′Q k T T τ( )r leak r leak H L_ _ (51)

where kr_leak is the heat leak coefficient between the heat sink and thecooled space.

Because of the conductive thermal leaking loss, the total heat ab-sorbed from the cooled space QL′ in a cycle are given by

= −′ ′Q Q QL l r leak_ (52)

Therefore, the power input, cooling rate and coefficient of perfor-mance of the Stirling refrigerator are obtained as follows

′ =′

=− ′

+ + +

′ ′−−

−− ′

′ ′ ′ ′

− ′

′

′−

′( )P W

τ

γ mR T

t t t tR

λn

λn g

h l r r

11

11 4

1 2

nC

C

nH

H

1 1

(53)

=′ − − − ′ − − ′

+ + +

′−− ′ ′ ′ ′ ′

′ ′ ′ ′

′−

′qmR γ T mc T e γ k T T τ

t t t t

(1 )(1 ) ( )λn g v r r leak H L

h l r r

11 4 4 _

1 2

nH

H

1

(54)

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601

=

=′ − − − ′ − − ′

− ′

′

′

′

−− ′ ′ ′ ′ ′

−−

−− ′

′−

′− ′

′

′−

′( )

σ QW

mR γ T mc T e γ k T T τ

γ mR T

(1 )(1 ) ( )

RL

R

λn g v r r leak H L

λn

λn g

11 4 4 _

11

11 4

nH

HnC

C

nH

H

1

1 1

(55)

2.4. Analysis of entire system

In the solar duplex Stirling refrigerator system, the work generatedin the Stirling heat engine provides input work for the Stirling re-frigerator simultaneously. As power can be calculated by dividing workby cycle time, the power and cycle time are considered as the twoconstraints of the solar duplex refrigerator system; the produced powerby the engine is consumed by the refrigerator, and the cycle time of theStirling engine is equivalent to that of the Stirling refrigerator. Here weconsider the maximum power of the Stirling heat engine as the inputpower of the Stirling refrigerator, thus the following constraints areobtained

= ′P P (56)

= ′τ τ (57)

With certain given parameters, we can obtain the temperature of theworking fluid in state 2′ and 4′ according to the equations above.

To evaluate the performance of the entire system, the overallcoefficient of performance of the system is adopted and defined as

=σ η η σO s e R (58)

3. Results and discussion

The solar duplex Stirling refrigerator, consisting of a dish solarcollector, a Stirling heat engine, and a Stirling refrigerator, is an en-vironment-friendly system that absorbs solar heat energy for cooling.Certain criteria are applied in this study for evaluation: the outputpower is applied to evaluate the performance of the Stirling engine; thecoefficient of performance is applied to evaluate the performance of theStirling refrigerator; the cooling rate and overall coefficient of perfor-mance are applied to evaluate the performance of the entire system.

To evaluate the effects of the design parameters on the performanceof the solar duplex Stirling refrigerator system, the following

parameters remain constant unless specifically stated to vary [29,36]:n=1mol, αh= αl=200WK−1, λ= λ ′=2, R=8.314 Jmol−1 K−1,cv=3.214 J g−1 K−1, ε=0.9, kh_leak= kr_leak=2.5WK−1, C=1300,αh′= αl′=700WK−1, η0= 0.9, TL=290 K, TL′=200 K,TH′= T0= 288 K, h=20Wm−2 K, δ=5.67× 10−8Wm−2 K−4,M=M′=1×10−5 s K−1.

To observe the effect of the collector optical efficiency on the per-formance of the system, we calculate the power P, cooling rate q,coefficient of performance σR, and overall coefficient of performance σofor different optical efficiencies and the result is shown in Fig. 2. Asobserved from Fig. 2, with the variation of the collector optical effi-ciency from 88% to 94%, the power of the engine and the overallcoefficient of performance of the system increase; the cooling rate in-creases from 1.32 kW reaching a peak at η0 with 91%, and then de-creases to 1.39 kW; the coefficient of performance of the refrigeratorincreases from 0.495 to 0.522 with η0 increases from 0.88 to 0.91, andthen decreases to 0.508 with η0 increasing to 0.94. Therefore, a largecollector optical efficiency is favorable for engine power and overallcoefficient of performance of the system, but an appropriate value ofcollector optical efficiency is required to maximize the cooling rate andthe coefficient of performance of the Stirling refrigerator.

The effects of low-temperature side convection heat transfer coef-ficient of Stirling engine αl on the power of Stirling engine P, coolingrate q, coefficient of performance σR, and overall coefficient of perfor-mance σo are plotted in Fig. 3. As can be observed, increasing the lowtemperature side convection heat transfer coefficient of Stirling enginefrom 200WK−1 to 1200WK−1 increases the power of the Stirlingengine from 7985W to 11,002W with a high speed, and then increasesto 12,897W slowly; the overall coefficient of performance increasesfrom 15.20% to 15.63% rapidly with αl increasing from 200WK−1 to500WK−1, and then increases to 15.81% with a low speed for αl largerthan 500WK−1; the cooling rate remains constant at 1576W; thecoefficient of performance of the refrigerator remains constant at0.5215. Thus, the low-temperature side convection heat transfer coef-ficient of Stirling engine has no effect on the cooling rate and thecoefficient of performance of the refrigerator, but for a satisfactoryperformance with regard to the power and overall coefficient of per-formance, a large low-temperature side convection heat transfer coef-ficient of Stirling engine is required.

Fig. 4 shows the effect of high-temperature side convection heattransfer coefficient of Stirling engine αh on the power P, cooling rate q,coefficient of performance σR, and overall coefficient of performance σo.As shown in Fig. 4, αh has a similar effect to αl: increasing αh from

Fig. 2. Optimum power, cooling rate, coefficient of performance of Stirling refrigerator, and overall coefficient of performance versus collector optical efficiency.

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200WK−1, to 1200WK−1 increases the power of the Stirling enginerapidly from 7986W to 9297W, and then slowly to 9932W; the overallcoefficient of performance increases rapidly from 15.20% to 15.42% atfirst, and then to 15.50% gradually; the cooling rate and the coefficientof the refrigerator remain at 1576W and 0.5215, respectively. There-fore, for a satisfactory performance of the present system, a largecoefficient αh is required to improve the power of the Stirling engineand the overall coefficient of performance.

The effects of thermal conductance between the heat external re-servoirs and working fluid of the refrigerator on the performance of thesolar duplex Stirling refrigerator system are plotted in Figs. 5 and 6. Itcan be observed that the thermal conductance αh′ and αl′ have no effecton the optimum power of Stirling engine. The cooling rate, coefficientof performance of Stirling refrigerator, and overall coefficient of per-formance increase rapidly from 1544W to 1923W, 0.5174 to 0.5204,and 15.65% to 15.74%, respectively, with the thermal conductance αh′increasing from 200WK−1 to 600WK−1, and then slowly to 2121W,0.5212, and 15.76%, respectively, with αh′ increases to 1200WK−1.The cooling rate, coefficient of performance of Stirling refrigerator, and

overall coefficient of performance increase rapidly from 1544W to2480W, 0.5174 to 0.5307, and 15.65% to 16.05%, respectively, withthe thermal conductance αl′ increasing from 200WK−1 to 400WK−1,and then slowly to 4329W, 0.5411, and 16.36%, respectively, with αl′increasing to 1200WK−1. Therefore, for a satisfactory performance ofthe present system, large values of thermal conduction between theheat reservoirs and working fluid of the refrigerator can improve theperformance of the solar duplex Stirling refrigerator system.

4. Conclusions

In this paper, a solar duplex Stirling refrigerator system for con-verting solar heat energy for cooling was proposed. To develop athermodynamic model of the system, the dish solar collector, Stirlingheat engine, and Stirling refrigerator were analyzed using finite timethermodynamics. The linearized heat loss of the solar collector, finiterates of heat transfer, regenerative heat loss, and conductive thermalbridging loss of the Stirling engine and refrigerator were considered inthis model. Different from previous papers that considered only one

Fig. 3. Optimum power, cooling rate, coefficient of performance of Stirling refrigerator, and overall coefficient of performance versus low-temperature side con-vection heat transfer coefficient of Stirling engine.

Fig. 4. Optimum power, cooling rate, coefficient of performance of Stirling refrigerator, and overall coefficient of performance versus high-temperature side con-vection heat transfer coefficient of Stirling engine.

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constraint of work equality for the duplex Stirling refrigerator, weconsidered finite time and obtained two constraints, power and cycletime equalities, resulting in a considerably reasonable system model.

Based on the proposed model, the effects of the collector opticalefficiency, high and low temperature side convection heat transfercoefficients of Stirling engine, and thermal conductance between theexternal heat reservoirs and working fluid of refrigerator on the per-formance of the solar duplex Stirling refrigerator system were in-vestigated. Criteria such as the optimum power, cooling rate, coefficientof performance of Stirling refrigerator, and overall coefficient of per-formance of the system were adopted to evaluate the performance ofthe system. The results obtained in this study may provide certaintheoretical basis for the heat management of a real solar duplex Stirlingrefrigerator system.

Acknowledgments

The work was supported by the National Natural Science

Foundation of China (Grant No. 51736004, 51706076).

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