Optimization of an air-filled Beta type Stirling refrigerator

Accepted Manuscript Title: Optimization of an air-filled beta type stirling refrigerator Author: Houda Hachem, Ramla Gheith, Fethi Aloui, Sassi Ben Nasrallah PII: DOI: Reference:

S0140-7007(17)30081-6 http://dx.doi.org/doi: 10.1016/j.ijrefrig.2017.02.019 JIJR 3563

To appear in:

International Journal of Refrigeration

Received date: Revised date: Accepted date:

3-11-2016 13-2-2017 17-2-2017

Please cite this article as: Houda Hachem, Ramla Gheith, Fethi Aloui, Sassi Ben Nasrallah, Optimization of an air-filled beta type stirling refrigerator, International Journal of Refrigeration (2017), http://dx.doi.org/doi: 10.1016/j.ijrefrig.2017.02.019. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

International Journal of Refrigeration (2017)

Optimization of an air-filled Beta type Stirling refrigerator Houda HACHEMa*, Ramla GHEITHa, Fethi ALOUI b, Sassi Ben Nasrallah a a

Université de Monastir, École Nationale d’Ingénieurs de Monastir, Laboratoire LESTE, Avenue Ibn El Jazzar 5019 Monastir, Tunisie b University of Lille North of France - University of Valenciennes (UVHC), LAMIH CNRS UMR 8201, Department of Mechanics, Campus Mont Houy, 59313 Valenciennes Cedex 9 - France * Corresponding author. Email:[email protected]; Tel. +216 73500511; Fax. + 21673500514

Highlights 

Parametric theoretical and experimental studies of Stirling refrigerator are proposed.



Refrigerator’s cooling capacity, COP, losses, and cold temperature are discussed.



Maximum refrigeration power is achieved when regenerator porosity is 85%.



Dead volume is a harmful volume to be reduced.

Abstract

The Stirling machine has many successful applications mainly thanks to its high efficiency, fast cool-down, small size, light weight, low power consumption and high reliability (heating and cooling). A Beta type Stirling refrigerator is studied experimentally and numerically. The mathematical model takes into account complex phenomena related to compressible fluid mechanics, thermodynamics and heat transfer losses. A special attention is paid to the effect of geometric parameters such as dead and swept volumes respectively in compression and expansion spaces. Regenerator length, diameter and porosity are also discussed and optimal parameters are proposed. The effect of speed on the cold end temperature and refrigerator’s performances is also investigates. Net cooling capacity, inputted power and COP were estimated at different conditions. Results allow understanding the physical processes occurring in the refrigerator and for predicting its performance.

Keywords

Stirling refrigerator, inputted power, cooling capacity, COP, losses, cold end

temperature.

Nomenclature Variables A

: Gas cross section (m2)

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Awet L D x P V u Qc Qe Qr Qvis Qimp Qcd W m t T h cv CP COP f Re R Pr St ϕ NUT r

: wetted area (m2) : Length (m) : Diameter (m) : Piston position (m) : Pression (Pa) : Volume (m3) : Piston velocity (m.s-1) : Cooling capacity at compression space per cycle (J) : Heat rejection at expansion space per cycle (J) : Heat stored in regenerator per cycle (J) : Energy loss by viscous friction per cycle (J) : Energy loss by imperfection per cycle (J) : Energy loss by conduction per cycle (J) : Total refrigerator work input per cycle (J) : Mass (Kg) : Mass flow rate (kg.s-1) : Time (s) : time step (s) : Temperature (K) : Heat transfer coefficient (W.m-2.K-1) : Specific heat capacity at constant volume(J.kg-1.K-1) : Specific heat capacity at constant pressure (J.kg-1.K-1) : coefficient of performance : Friction factor : dynamic viscosity (Pa. s) : Reynolds number :Perfect gas constant (kJ.kg−1.K−1) : Prandt number : Stantan number : Porosity : Thermal conductivity (W m-1 K-1) : Number of heat transfer unit : Regenerator efficiency : Density (kg m-3) : Phase shift. : Frequency

Indices e c de dc w r g 0 in out min

: Expansion : Compression : Expansion dead volume : Compression dead volume : Regenerator wall : Regenerator : Gas : Initial or starting value : Inlet : Outlet : Minimum

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1. Introduction

Numerous researches focus on developing an adequate refrigeration system at specified temperature with low power input, long lifetime, high reliability and maintenance, free operation with minimum vibration and noise, compactness and light weight. The demand of Stirling refrigerators and cryocoolers has increased due to the ineffectiveness of Rankine cooling systems at lower temperatures [1]. Stirling machine operates in a closed thermodynamic regenerative cycle with the same working fluid repeatedly compressed and expanded at different temperature levels so there is a net conversion of heat to work or vice versa. It can be used as engine, cooling machine or heat pump [2]. The Stirling cycle, invented by Robert Stirling in 1815, has the advantages of using alternative energy sources and avoiding environmentally-harmful refrigerants. In 1834, John Herschel designed the regenerative cooling engine for making ice [3]. Air was used as working gas to achieve cooling effects. In 1946, the Philips Company in Holland ameliorated the machine, by using helium as working fluid (high thermal conductivity, high ratio of specific heats), to liquefy air on the cold tip [4]. The main components of the Stirling refrigerator are piston, compressor space, regenerator and expansion cold space. The piston and regenerator follow a reciprocating motion with a phase difference of 90° that is achieved by crank shaft mechanism coupled to a motor.

Several simulations of Stirling refrigerator were developed. These simulations provided an edge to the developers, as it offers an accurate analysis of the performance of the refrigerator before manufacturing it. It saves time and money. The classical analysis of the ideal Stirling cycle was given by Schmidt (first order approach) [5, 6]. He made assumptions such as steady state procedure with ideal regenerators and no pressure drops during the isothermal processes. The Schmidt analysis assumed that the efficiency of heat exchangers is 100%. In reality, the heat exchangers efficiencies are expected to be less than that. This drop in heat exchanger efficiency will influence the pressure ratio so the refrigeration capacity will be reduced substantially [7].The second order approach was largely developed by Martini. He added realistic losses to the model developed by Schmidt. Realistic losses included pressure drop loss, shuttle loss, pumping loss, static heat loss, regenerator ineffectiveness [8]. He applied decoupled independent correction to consider all losses which have been thoroughly 3

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described in Martini’s 1982 Stirling engine design manual [3]. Wu et al. (1995) [9], evaluated the performance of a real refrigerator (an endoreversible Carnot refrigerator) and calculated the maximum specific cooling load. Chen et al. (1998) [10] studied an air refrigeration cycle with non-isentropic compression and expansion. Results show that it exist a maximum value of COP and that the cooling load has a parabolic dependence on COP, unlike the monotonically decreasing behaviour in the case of an endoreversible air refrigeration cycle. Kaushik et al. (2002) [11] studied internal and external irreversibilities inside Stirling and Ericsson heat pumps cycles. Results show that the effects of internal irreversibilities are more important than external ones. Tyagi et al. (2002, 2004) [12, 13] applied the concept of finitetime thermodynamics to study an irreversible Stirling cryogenic refrigerator cycle for different operating conditions. He studied the effect of fluid temperature, effectiveness, heat rates and irreversibility parameters on the performance of the Stirling and the Ericsson refrigeration cycles. He found that the effect of the internal irreversibility parameter is more pronounced than that of other parameters. Otaka et al. (2002) [14] designed and tested a Beta type Stirling cycle machine of 100 W capacity. The effect of many parameters such as dead volume ratio, working fluids, the ratio of the compression volume to the expansion volume, and the phase difference between power piston and displacer is studied. From their results, they found that the refrigeration produced by nitrogen was 28% less than that produced by helium. Ataer et al. (2005) [15] elaborated a thermodynamic analysis of a V-type Stirlingcycle refrigerator to evaluate work, instantaneous pressure and COP of the Stirling-cycle refrigerator. Based on simulation and experiments, Le’an et al. (2009) [16] investigated a Vtype integral Stirling refrigerator (VISR) when it is applied to the domestic refrigeration. The power consumption and the coefficient of performance (COP) are investigated under various rotating speeds and charged pressures. After comparison between two working fluids, the result showed that the optimum rotational speed for the cooling capacity is different between nitrogen and helium. The COP has a peak value around 600 rpm for nitrogen, and for helium around 900 rpm. So the VISR is applicable to the domestic refrigeration. Chen et al. (2009) [17] described and experimented a pneumatic free piston and free displacer Stirling cryocooler design (FPFD). The effects of pneumatic and thermodynamic parameters including charging pressure, working frequency, damping coefficient of displacement and displacer phase shift are studied experimentally and theoretically. They propose the expressions of three phase shifts of piston and displacer function of working parameters. Based on exergy flow through a Stirling Refrigerator, Razani et al. (2010) [18] proposed a model including irreversibilities of the refrigerator due to external heat transfer with the 4

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reservoirs, heat leak, flow and heat transfer in regenerator, expansion and compression efficiencies. Their model proposes to find the adequate compromise between cooling capacity and efficiency of the Stirling refrigerator. Zhu et al. (2010) [19] introduced a nodal analysis for the simulation of the pulse tube Stirling machine with warm gas-driven displacer which has a displacer rod. Numerical results show that it has high potential to be used as a cryogenic refrigerator, room temperature refrigerator and engine. Tekin et al. (2010) [20] studied numerically the effect of different working fluids on the performance of the VSR using adiabatic regenerator boundary conditions. They concluded that the COP of the VSR is higher when the hydrogen is used as working fluid. He et al. (2014) [21] developed a mathematical model based on thermodynamic theory to evaluate the lifetime of the refrigerator. The results show that when the refrigerator is operated at uniform distribution of the water vapor partial pressure in the regenerator, the cooling capacity is reduced over 10% at about 631 h and the power consumption of compressor is increased over 20% at about 1168 h. Cun-quan et al. (2015) [22] established a theoretical model to predict the dynamic performance of a new type of pneumatically-driven split-Stirling-cycle cryocooler. The model takes into account the working characteristics of the compressor motor and the principal losses of cooling, including regenerator inefficiency loss, solid conduction loss, shuttle loss, pump loss and radiation loss. Results show that the gas pressure has amplitude attenuation and phase delay effects and the gas mass flow rate is similar from the compressor to cold chamber. Dang et al. (2016) [23] investigates numerically effects of geometrical parameters of the compressor on the performances of Stirling-type pulse tube cryocooler. From their study, they found that adjusting the piston diameter is an effective approach to change the optimal operating frequency. In order to verify the theoretical model, Dang et al. (2016) [24] compared their numerical results to experimental ones obtained using a compressor that achieves an electric input capacities ranging from 0 W to 200 W with the mean motor efficiency of 81.5%, and the matched cooler achieves the cooling capacity of 10 W at 80 K with higher than 17% of Carnot efficiency. Results show good agreements between simulated and experimental results. Recently, Pan et al. (2016) [25] investigates numerically and experimentally a Vuilleumier (VM) refrigerator. The simulation results show that the shuttle loss and regenerator loss are the main factors limiting the performance of VM cryocooler. Results show that the optimal phase angle between hot and cold displacer is about 94°, and the optimal length of stainless steel screen is about 40 mm for this crycooler. Wang et al. (2016) [26] studied numerically geometric and operating parameters of Stirling-type pulse tube refrigerators (PTRs). They demonstrate that adopting the mass-spring displacer to 5

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feed the expansion power back into the compression space is an effective way of improving the performance of PTRs. The simulation of the modified PTR with the mass-spring displacer indicates that the expansion power absorbed by the displacer takes up to about 20% of the compression power. Results show that COPs at 77 K can be achieved at a range of 0.13–0.14 with the relative Carnot COPs of around 0.4. Sauer et al. (2017) [27] presents a numerical simulation that directly takes into consideration the interactions between the cylinder wall temperatures and the overall gap loss. Results underline the advantages of third-order models and the need to apply them for advanced design optimization purposes. Erbay et al. (2017) [28] investigated thermodynamically the duplex Stirling by considering the combined effects of diverse constructional and operational parameters which are namely the temperature ratios for heat engine and refrigerator and the compression ratios for both sides. Results show that the work rate increases with the swept volume in the cylinder and the temperature difference between the heat source and sink. However, the heat engine cannot meet the demand of the cooler at the higher temperature ratios when the compression ratio is beyond 10. In previous study, Hachem et al. [29] presents an experimental study of an air-filled Beta Stirling machine used as refrigerator and as heat pump. The effect of thermal insulation of the two heat sources on receptive machine’s energy and exergy efficiencies were studied. Results show that the Stirling refrigerator produces the maximum useful cooling effect output at less electric power consumption at an optimal operating speed equal to 155 rpm. In this paper, parametric theoretical and experimental studies of an air filled Beta type Stirling refrigerator are proposed. A numerical model taking into account thermal losses that occurs in the Stirling refrigerator’s regenerator is developed. Several operating parameters (speed, hot end temperature and water flow rate) are experimented, in order to describe the cooling behavior of the Stirling refrigerator and to validate numerical results. Several geometric parameters (dead and swept volumes of hot chamber and of cold chamber, regenerator length, diameter and porosity) are varied. The influence of every parameter on the Stirling refrigerator cooling capacity, COP, losses (internal conduction loss, imperfection loss and viscous friction loss) and cold end temperature values are presented and discussed.

2. Mathematical model The present model describes the dynamic behaviour of the Stirling refrigerator. In the engine modeling, the temperatures respectively of the hot source and the cold source are imposed as an input data [30]. However, in the refrigerator modeling, theses temperatures are considered as output results calculated at the end of the modeling. Only the expression of 6

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piston displacement which is proportional to the imposed power is introduced in the energy equation in order to calculate the temporal evolution of temperature until the quasi-steady state. In this model the Stirling refrigerator is devised into three parts: compression chamber, expansion chamber and regenerator. Fig. 1 shows the schematic diagram for a Stirling refrigerator constituted of compression and expansion spaces separated with a regenerator of diameter Dr, length Lr and packed with porosity

by woven screen of cooper. The

assumptions made in this model are: - The working fluid is air and it is assumed to be ideal gas. - Losses by leakage of mass are neglected. -The motion of the compression and expansion pistons is considered to be periodic and it is prescribed by following equations: (1)

(2)

is the frequency, expansion cylinders.

and and

and expansion cylinders and

are length of dead spaces respectively in the compression and are length of swept spaces respectively in the compression is the phase shift.

- The wall of the compression and expansion spaces are at a constant temperature so any heat transfer to or from the walls is considered to be with a cold reservoir for expansion chamber and with a hot reservoir for compression chamber.

2.1.Compression and expansion spaces equations Energy balance equations respectively in compression and expansion spaces are written as follow: (3) (4) Where

and

are the wall temperatures,

and

are mass of air respectively of the

expansion and the compression chambers.

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The mass flow rate of the air in compression and expansion spaces is not constant during a cycle.

and

depends on the nature of the flow. The mass flow rate respectively in

expansion and compression chambers is expressed as follow: (5) (6) Where

and

are diameters, ue and uc are velocities of pistons respectively of the

expansion and the compression chambers. According to equations (1) and (2), temporal evolution of ue and uc are expressed as follow: (7) (8) The mass of working fluid in compression and expansion spaces at every time step can be calculated by: (9) (10) Pressures respectively in expansion and compression spaces

and

are calculated when

assuming ideal gas behaviour for the air as follow: (11) (12) The pressure in compression and expanssion spaces at every time step can be calculates as follow: (13) (14)

Where

is the average pressure in working spaces and

is the pressure drop due to

viscous force in regenerator.

The interface temperatures Tin and Tout depends on the flow direction and can be calculated using the definition of the effectiveness of the regenerator as follow:

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if

(15)

if

(16) (17) (18)

The heat transfer coefficients in compression and expansion chambers are calculated as follow: (19)

Where the friction factor for circular tubes is calculated as follows [31]:

If

then

(20)

If

then

(21)

If

(22)

then 2.2.Regenerator equations

The regenerator is constituted of a porous cylinder of length Lr and diameter Dr. Energy balance equation in the regenerator is writen as: (23) Where Tr is the temperature of air inside regenerator, Tw,r is the temperature of the solid matrix in the regenerator,

is the density of air in the regenerator, cp is the specific heat per

unit mass of air at constant pressure and hr is the convective heat transfer coefficient in the regenerator.

is the heat generated by losses in the regenerator due

to axial conduction loss, imperfect heat regeneration and pressure drop due to fluid friction. These losses are calculated in the next paragraph. Heat transfer coefficient in the regenerator is calculated as follow: (24)

Where

is the Stantan number and it is calculated using the following correlation: (25)

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The temperature of the solid matrix in the regenerator Tw is obtained when solving the folowing equation: (26)

Mw is the mass of the regenerative matrix, cw is the specific heat per unit mass of the regenerator matrix. xr is the location along the regenerator from compression chamber to expansion chamber. The velocity of air in the regenerator depends on the pressure difference between compression and expansion chambers. The mass flow rate through the regenerator can be obtained from the following equation [32]: (27)

(28)

where A is the gas cross section, f and

are respectively the friction factor and the

hydraulic diameter and are calculated using the following correlations [31]: (29) (30)

where is the porosity of the regenerator and

is the solid wire diameter.

2.3.Major Stirling refrigerator regenerator losses

The energy loss in the regenerator represents 86% of total losses in a Stirling machine [33, 34]. Three types of thermal losses are estimated in regenerator and considered in this study: internal conduction loss, imperfection loss and viscous loss due to pressure drop. The internal conduction loss due to the important gradient of temperature between regenerator interfaces through regenerator is calculated as follow [35]: (31)

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Where A is the effective cross sectional area normal to the heat flow path (m2), λw is the conductivity of the material through which heat conduction occurs (W m-1 K-1), L is the length of heat conduction path (m), The regenerator thermal conductivity depends on the porosity ϕ, on the thermal conductivity of the working fluid λg and of the material λm. It can be obtained by the Gotting formula [36]: (32) The regenerator role is to store energy from the hot gas and restores to it during its passage in the opposite direction toward the cooler. The amount of heat stored or restored by the regenerator matrix depends on the characteristics of the used material. To estimate this loss, the effectiveness of the regenerator εr is estimated as follow: (33) Where NUT is the number of heat transfer units of the regenerator. It depends on the heat transfer coefficient hr, on the wetted area of the heat exchanger surface Awet, on the mass flow rate through the heat exchanger

(kg s-1) and on the specific heat capacity at constant

pressure Cp (J.kg-1.K-1). Andit is calculated as: (34) Thus, the imperfection loss in regenerator is evaluated as: (35) Energy loss by viscous friction in the regenerator is calculated using the following expression: (36) Where

is the mass flow rate through the regenerator (kg s-1) and ρw is the density of the

solid matrix (kg m-3). 2.4.Refrigerator performances Heat transfer and indicated work are evaluated using the following expressions: (37) (38)

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(39) Thus real COP is: (40)

Differential equations are solved numerically using the method of characteristics. The integral of equation (26) is solved using Euler method. The simulation uses geometrical parameters of the Beta type Stirling refrigerator given in Table 1. The initial pressure and all initial temperatures are taken to be respectively 1 atm, and 300 K. Equation 21 can be reduced to: (41) With

and

Two ordinary differential equations are obtained based on the method of characteristics reduces: (42) (43) 3. Experiments 3.1.Setup The experimental set up is composed of a Beta type Stirling refrigerator using air as working fluid and arranged in a unique cylinder of 300 cm3 in which two pistons moves: the power piston (M) and the displacer (D). Cold is recovered at the expansion space and at the compression spaces calories are lost at the atmosphere. A copper porous structure called regenerator is situated between both working spaces. The regenerator present a thermal barrier between heat sources and store calories during compression then return them to the working fluid during expansion. Fig. 2 (a) presents a sectional view of the prototype which describe the refrigerator working cylinder components (Expansion and compression spaces, regenerator (R), displacer (D) and Working piston (W)) and its drive system. The technical and geometrical characteristics of the Stirling engine are summarized in Table 1. An electric engine is coupled to the Stirling refrigerator by a pulley and belt transmission as shown in Fig. 2 (c). It provides a mechanical work which will be given to cold calories when

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it is driven in the direction of clockwise. In order to measure temperatures, pressure, volume and speed evolutions during time, an acquisition system composed of two CASSY cards is used (Fig. 2 (b)) to convert the analogical signal, given by sensors, into a digital signal which will be treated by a processing unit. The first card records pressure and volume signals acquisitions. The second one records hot and cold spaces temperatures and the machine rpm. The Experimental setup gives as the possibility to see the effect of three operating parameters (speed, hot end temperature and water flow rate) of the Stirling refrigerator. However, numerical investigation allows as investigating the effect of geometric parameters (dead and swept volumes of hot chamber and of cold chamber, regenerator length, diameter and porosity). 3.2.Results Experiments have been realized at ambient temperature and pressure. The heat sink is realized by water circulation around the cylinder. Speed and input work are regulated by an electrical engine. The compression space temperature is adjustable when changing the water flow rate or the water inlet temperature using an external cooler. In order to analyse the effect of functional parameters, the refrigerator is tested under different operating conditions (speeds, water flow rate and different hot end temperature) depicted in table 2. Having regard to experimental data fluctuation (outcome of thermocouples), fitting curves for temperatures are plotted in Fig. 3. From these experimental evolutions, it is clearly seen that the cold end temperature decreases when increasing speed, increasing hot side temperature or decreasing water flow rate. 3.3.Validation In order to compare experimental and numerical (thermodynamic model) results, the geometrical parameters of the Beta Stirling engine were introduced in the model. The most important data usually reported for Stirling refrigerator are the COP and the cooling capacity. Experimental and numerical evolutions of COP and cooling capacity versus crank speed are illustrated respectively in Fig. 4(a) and Fig. 4(b). These evolutions are obtained when supposing as initial temperature T0=300°C and as initial filling pressure Pi=1atm. Unfortunately, the maximum rotation speed obtained with the experimental apparatus is 200 rpm. Both experimental and numerical curves follow a parabolic shape with a maximum. The optimal COP of 65% is obtained at 150 rpm. Optimal cooling capacity about 25W is obtained at 210 rpm. However, they don’t reach the same optimum value of speed.

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The maximum COP can be explained as a compromise between losses that disfavour high heating rate, and losses that disfavour high cooling rates. When the imposed speed is less than 150 rpm, losses disfavour high heating rate. However, when the imposed speed exceeds this optimal value (150 rpm), losses disfavour high cooling rate and that is mean that there is no need to give more power to the machine because the imposed power is no more converted to cold calories.

4. Numerical Results and discussions 4.1.Cooling behavior Initially, the refrigerator was at ambient temperature and once the linear motor is switched on, the temperature of the cold side drops quickly and reaches a steady state low end temperature of 210 K when turning at 10 Hz. The temperature is observed to follow a gradually descending periodic behavior. The cooling curve can be seen in Fig. 5. The simulation was run for 100 seconds. The Stirling refrigerator needs only 1 min and 30 seconds to reach such a low temperature. During the cool-down process, the amplitude of gas pressure in the cold chamber decrease until the quasi-steady state. Thus the area of the PV diagram, that represents the actual work given to the machine, decreases as shown in Fig. 6. During the cool-down process, the temperature of the working gas decreases progressively thus gas density and viscosity increases which lead to an increase of viscous losses due to pressure drop across the Stirling refrigerator’s singularities, which explain the cooling capacity decrease [22]. 4.2.Speed effect The refrigerator limit working temperature is an important feedback parameter in the refrigerator operation [22]. The refrigerator limit cold end temperature depends on the machine’s rotation speed. As shown in Fig. 7, higher is the speed, lower is the limit cold end temperature. For high speed values (>240rpm), the steady state cold end temperature remains stable. This means that the speed increasing leads to the increase of losses which disfavour high cooling rate. The optimal cold end temperature to achieve the maximum refrigerating

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COP is about 219K, whereas the one to achieve the maximum cooling capacity is about 233K (Fig. 8). In the next following results, the simulation was run for 50 seconds for 3 Hz. 4.3. Effect of dead and swept volumes of the hot chamber The effect of compression space swept volume is illustrated in Fig. 9. When increasing the compression swept volume, the inputted power increases thus the refrigerating COP decreases. However, the cooling capacity increases and reaches a maximum at about 160 cm3. It means that there is no need to increase the power because energy input is no more converted into cold rather lost due to internal and external thermal and mechanical losses. The cooling capacity of the Stirling refrigerator decreases when increasing compression swept volume over 160 cm3. Fig. 10 illustrates the evolution of cooling capacity and COP versus compression space dead volume. The COP increases with dead volume however the cooling capacity curve follows a parabolic evolution with a maximum at 100 cm3 of compression space dead volume. When the dead volume of the compression space exceeds 100 cm3, the cooling capacity decreases and the refrigerator COP undergo a little bit of rise. 4.4.Dead and swept Volumes of the cold chamber effect Figures 11 and 12 emphasize respectively the effect of swept and dead volumes of the hot chamber. The curves respectively of cooling capacity, inputted work and COP versus the expansion volumes dead volumes have no optimum values.The produced cooling capacity, the inputted work as well as the refrigerator COP increase with the expansion space swept volume as it is illustrated in Fig. 11. An additional improvement of the refrigerator performances can be achieved by maximising expansion space swept volume. The evolution of Stirling refrigerator performances versus the expansion space dead volume are plotted in Fig. 12. Higher is the expansion space dead volume during the cool-down process, lower is the compression input power, the produced cooling capacity and the refrigerator’s COP.The dead volume of the cold chamber decreases considerably the cooling capacity of a Stirling refrigerator. In order to improve the cooling capacity, it is recommended to increase the swept volume of the cold chamber and to reduce its dead volume (Fig. 13). 4.5. Regenerator proprieties

Regenerator effectiveness is a key point for Stirling refrigerator performances. The main characteristics of such heat exchanger are:

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• Its small size compared to the dimensions of the Stirling machine. • The constituting material is a very good heat conductor. • The fluid injection temperatures are not constant of the two cold and hot sides. • The time variation respectively of pressure, air flow and gas density are important. • The thermos-physical properties of the regenerator component materials can vary greatly during the cool down process. The regenerator is a sensitive organ since it must combine a high exchange surface, high heat capacity while providing the least loss possible load. Several numerical and experimental studies in the literature focus on the regenerator efficiency [37-40] in order to increases the COP of the Stirling refrigerator. The regenerator is the set of 44% of viscous loss, 33% of internal conduction loss and 22% of imperfection loss respectively from the total losses inside a Stirling engine [30].

4.5.1. Regenerator length and diameter

Evolutions of cooling capacity and refrigerator COP versus regenerator length are plotted in Fig. 14. It shows a parabolic evolution with nearly the same optimum value of 60 mm of length. Over these value losses caused namely by the conduction loss, the imperfection loss and the friction loss due to pressure drop increases as shown in Fig. 15. Conduction loss decreases with the rise of regenerator length, imperfection loss keep nearly a constant value about 2.2W. However, viscous loss reaches a minimum value about 8W at 50 mm of regenerator length. Cold end temperature increases with the regenerator length. The evolution of Stirling refrigerator performances versus the regenerator diameter are presented in Fig. 16. Curves reached an optimum value when the diameter is equal to 22 mm. Likewise the optimal diameter correspond to the minimum energy loss in the regenerator. Fig. 17 presents respectively regenerator losses evolution and cold end temperature versus regenerator diameter. The cold end temperature decreases with the regenerator diameter rise. When the regenerator diameter is less than 22 mm, cold end temperature undergoes a little bit of diminution. However, over this value the cold end temperature decreases greatly. Conduction loss increases with the rise of regenerator diameter, imperfection loss keep nearly a constant value about 2.2W. However, friction loss curve follows a parabolic shape with a minimum at about 23 mm of regenerator diameter for a cold end temperature of 219K. Fig. 18 presents the effect of the regenerator form factor on

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the Stirling refrigerator COP. The optimal COP and cooling capacity values are observed when the form factor is around 0.095. 4.5.2. Regenerator porosity Fig. 19 presents respectively the evolution of cooling capacity and refrigerator COP versus the regenerator porosity. Both curves reached an optimum value. Thus, regenerator which porosity is 85% was considered as the most suitable one maximizing the Stirling refrigerator cooling capacity and COP and minimizing heat and friction losses. The refrigerator achieved a maximum coefficient of performance and a maximum cooling capacity respectively of 65% and 25.5W. 5

Conclusion A key issue in designing and optimizing Stirling refrigerators is to build a precise

thermodynamic model to predict its performances. A thermodynamic model is established and parametric study of a Beta Stirling prototype under different operating and geometrical parameters were performed. The model shows a good concordance with the experimental results. Results show that: -The optimal speed corresponding to the maximum COP is different from its corresponding to the maximum cooling capacity. The maximum can be explained as a compromise between losses that disfavour high heating rate, and losses that disfavour high cooling rates. -In order to reach cold temperatures, it is recommended to increase the swept volume of the cold chamber and to reduce its dead volume. -Dead volume of the hot chamber is a harmful volume to be reduced. -To achieve maximum refrigeration power, regenerator porosity must be about 85%. -Several irreversible losses due to thermal conduction, imperfect loss and viscous loss between the two heat reservoirs directly diminish refrigeration power. These losses are function of regenerator length and diameter which optimal values for the Beta type Stirling refrigerator prototype are respectively about 60 mm and 22 mm.

Acknowledgement This work was supported by the laboratory LAMIH CNRS UMR 8201 (University of Valenciennes), the laboratory LESTE (ENIM, Monastir, Tunisia) and the European Commission within the International Research Staff Exchange Scheme (IRSES) in the 7th Framework Programme FP7/2014-2017/ under REA grant agreement n°612230. These supports are gratefully acknowledged.

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References [1] McFarlane, P.K., 2014. Mathematical model and experimental design of an air filled Alpha Stirling refrigerator, thesis, University of Notre Dame, Patrick K. McFarlane. [2] Beale, W., 1971, Stirling cycle type thermal device. US Patent 3552120. [3] Martini, W., 1982. Stirling Engine Design Manual. Second Editions, Martini Engineering, 2303 Harris, Washington, 99352, USA. [4] http://www.stirlingultracold.com/history [5] Schmidt, G., 1871. The theory of lehmans calorimetric machine. ZVereines Deutcher Ingenieure 151. [6] Walker, G., 1983. Cryocoolers, Plenum Press, New York. [7] Hachem, H., Gheith, R., Aloui, F., Ben Nasrallah, S., 2015. Impact of operating parameters on heat exchangers efficiencies inside a Beta type regenerative Stirling machine. In: To Be Presented at ASME-JSME-KSME Joint Fluids Engineering Conference (AJK) 2015. [8] Walker, G., Weiss, M., Fauvel, R., Reader, G., 1989. Microcomputer simulation of Stirling cryocoolers. Cryogenics 29(8), 846–849. [9] Wu, Ch., 1995. Maximum obtainable specific cooling load of a refrigerator. Energy Conversion and Management 36(1), 7-10. [10] Chen, L., Xiaoqin, Z., Sun, F., 1998. Cooling load versus COP characteristics for an irreversible air refrigeration cycle. Energy Conversion and Management 39, 117-125. [11] Kaushik, S.C., Tyagi, S.K., Bose, S.K., Singhal, M.K., 2002. Performance evaluation of irreversible Stirling and Ericsson heat pump cycles. International Journal of Thermal Sciences. 41(2), 193–200. [12] Tyagi, S.K., Kaushik, S.C., Singhal, M.K., 2002. Parametric study of irreversible Stirling and Ericsson cryogenic refrigeration cycles. Energy Conversion and Management. 43(17), 2297–2309. [13] Tyagi, S.K., Lin, G., Kaushik, S.C., Chen, J., 2004. Thermoeconomic optimization of an irreversible Stirling cryogenic refrigerator cycle. International Journal of Refrigeration 27(8), 924–931.

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[14] Otaka, T., Ota, M., Murakami, K., 2002. Study of performance characteristics of a small Stirling refrigerator. Heat Transfer Asian Research.31 (5) . [15] Ataer, O.E., Karabulut, H., 2005. Thermodynamic analysis of the V-type Stirling-cycle refrigerator. International Journal of Refrigeration. 28(2) , 183–189. [16] Le’an, S., Yuanyang, Z., Li, L., Pengcheng, S., 2009. Performance of a prototype Stirling domestic refrigerator. Applied Thermal Engineering 29, 210–215. [17] Chen, X., Wu, Y.N., Zhang, H., Chen, N., 2009. Study on the phase shift characteristic of the pneumatic Stirling cryocooler. Cryogenics 49, 120–132. [18] Razani, A., Dodson, C., Roberts, T., 2010. A model for exergy analysis and thermodynamic bounds of Stirling refrigerators. Cryogenics 50, 231–238. [19] Zhu, Sh., Nogawa, M., 2010. Pulse tube Stirling machine with warm gas-driven displacer. Cryogenics 50(5), 320–330. [20] Tekin, Y., Ataer, O.E., 2010. Performance of V-type Stirling-cycle refrigerator for different working fluids. International Journal of Refrigeration 33 (1) , 12–18. [21] He, Y.-L., Zhang, D.-W., Yang, W.-W., Gao, F., 2014. Numerical analysis on performance and contaminated failures of the miniature split Stirling cryocooler. Cryogenics 59, 12–22. [22] Cun-quan, Z., Cheng, Z., 2015. Experimental study of a gas clearance phase regulation mechanism for a pneumatically-driven split-Stirling-cycle cryocooler. Cryogenics 66, 24– 33. [23] Dang, H., Zhang, L., Tan J., 2016. Dynamic and thermodynamic characteristics of the moving-coil linear compressor for the pulse tube cryocooler. Part A: Theoretical analyses and modeling. International journal of refrigeration 69, 480–496. [24] Dang, H., Tan, J., Zhang, L., 2016. Theoretical and experimental investigations on the optimal match between compressor and cold finger of the Stirling-type pulse tube cryocooler. Cryogenics 76, 33–46. [25] Pan, Ch., Zhang, T., Wang, J., Zhou Y., 2016. Numerical study of a one-stage VM cryocooler operating below 10K. Applied Thermal Engineering 101, 422-431. [26] Wang, K., Dubey, S., Choo, F. H., Duan F., 2016. Modelling of pulse tube refrigerators with inertance tube and mass-spring feedback mechanism. Applied Energy 171, 172–183. [27] Sauer, J., Kuehl, H-D., 2017. Numerical model for Stirling cycle machines including a differential simulation of the appendix gap. Applied Thermal Engineering 111, 819–833. [28] Erbay, L. B., Ozturk, M. M., Dogan, B., 2017. Overall performance of the duplex Stirling refrigerator. Energy Conversion and Management 133, 196–203. 19

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[29] Hachem, H., Gheith, R., Aloui, F., Ben Nasrallah, S., Dincer, I., 2015. Energetic and exergetic performance evaluation of an experimental Stirling machine. Book Springer: Progress in Clean Energy, Chapter 55, Vol. 2. [30] Hachem, H., Gheith, R., Aloui, F., Ben Nasrallah, S., 2015. Global numerical characterization of a γ-Stirling engine considering losses and interaction between functioning parameters. Energy Conversion and Management 96, 532–543. [31] Ureili I., Berchowitz D.M., 1984. Stirling Cycle Analysis. Published by Adam Hilger Ltd. Bristol. [32] Kays, W.M., London, A.L., 1964. Compact heat exchangers. 2nd ed. New York, USA: McGraw-Hill. [33] Tlili, I., 2012. Finite time thermodynamic evaluation of endoreversible Stirling heat engine at maximum power conditions. Renewable and Sustainable Energy Reviews 16, 2234–2241. [34] Tlili, I., Timoumi, Y., Ben Nasrallah, S., 2008. Analysis and design consideration of mean temperature differential Stirling engine for solar application. Renewable Energy 33, 1911–1921. [35] Reader, G.T., Hooper, C., 1983. Stirling engines. London & New York: E&FN Spon. [36] Stouffs, P., 2000. Dimensionnement Optimal des Volumes de Compression et de Détente des Moteurs Stirling. In: To Be Presented at the French Thermal Congress SFT, Vol. 8, pp. 851–856. [37] Tao, Y.B., Liu, Y.W., Gao, F., Chen, X.Y., He, Y.L., 2009. Numerical analysis on pressure drop and heat transfer performance of mesh regenerators used in cryocoolers. Cryogenics 49, 497–503. [38] Dietrich, M., Yang, L.W., Thummes, G., 2007. High-power Stirling-type pulse tube cryocooler: Observation and reduction of regenerator temperature-inhomogeneities. Cryogenics 47, 306–314. [39] Clearman, W.M., Cha, J.S., Ghiaasiaan, S.M., Kirkconnell, C.S., 2008. Anisotropic steady-flow hydrodynamic parameters of microporous media applied to pulse tube and Stirling cryocooler regenerators. Cryogenics 48, 112–121. [40] Huang, P.Ch., Yang, Ch.F., 2008. Analysis of pulsating convection from two heat sources mounted with porous blocks. International Journal of Heat and Mass Transfer 51, 6294–6311.

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Dr

xe xc Figure 1. Schematic presentation of a Stirling machine

0

Figure 2. Experimental setup : Beta type Stirling refrigerator

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20 Cold end temperature (°C)

(a)

111 Log.rpm (tf-18) 200 Log.rpm (tf-27) 149 Log.rpm (tf-32) 126 rpm Log. (tf-35)

10 0 0

2

4

6

8

10

12

-20 -30 -40

Time (min)

20 Cold end Temperature (°C)

(b)

293K 303K 286K

10 0 00:00 -10

02:24

04:48

07:12

12:00

14:24

-30 -40

0 -500:00

Time (min) 02:24

04:48

07:12

-10

09:36

12:00

14:24

16:48

19:12

3.75(Q4) ml/s Log. 1.98(Q2) ml/s Log. 1.02(Q1) ml/s Log.

-15

TC (°C)

09:36

-20

-50 (c)

14

-10

-20 -25

-30 -35 -40 -45

-50 Time (min) Figure 3. Cool down curves(a) at different speeds (a) at different hot end temperature (b) at different water flow rate

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0.8 0.75

30 Experiments Simulated results

Experiments Simulated results 25 Cooling capacity (W)

0.7

COP

0.65 0.6 0.55 0.5 Regime where losses 0.45 disfavor high heating rate

0.4

100

Regime where losses disfavor high cooling rate

150 Speed (rpm)

20

15

10

5

200

100

150 Speed (rpm)

200

(a) (b) Figure 4. Comparison between simulation and experiments (a) COP versus speed (b) Cooling capacity versus speed

Working conditions: f=10Hz Pi=1atm

300 temperature OF space 1 AND 2

Cold end temperature (K)

320

280

Zoom 260

240

220

200

0

10

20

30

40

50 time

60

70

80

90

100

Time (s)

Figure 5. Cool down curve (numerical result)

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3

Cycle 1 Cycle 2 2.5

Cycle n Pressure (bar)

2

1.5

1

0.5 20

40

60

80

100

120

140

160

180

Expansion volume (cm 3)

Figure 6. Cold chamber PV diagram variation during cooling process

Regime where losses disfavor high heating rate

240

Regime where losses disfavor high cooling rate

230

220 210

200 80

100

120

140

160

180 Speed (rpm)

200

220

240

260

280

Figure 7. Cold end temperature versus speed

30

70

25

60

20

50

15 40

10

30

5

24 210

COP (%)

Cooling capacity (W)

Cold end Temprature (K)

250

215

220

225 230 Cold end Temprature (K)

235

240

20 245

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Figure 8. Stirling refrigerator performances versus cold end temperature

Qe COP Qe W 40 50

COP (%)

Cooling capacity and inputted power(W)

100

20

0 20

40

60

80 100 120 Compression space swept volume(cm3)

140

160

0 180

30

80

20

60

10 80

COP (%)

Cooling capacity (W)

Figure 9. Stirling refrigerator performances versus the compression space swept volume

100

120 140 160 3 Compression space dead volume (cm )

180

40 200

Figure 10. Stirling refrigerator performances versus compression space dead volume

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80

100 COP W Qe

80

40

60

20

40

0 40

60

80

100

120 140 160 180 3 Expansion space swept volume(cm )

200

COP (%)

60

20 240

220

Figure 11. Stirling refrigerator performances versus expansion space swept volume 100

100 W Qe COP

50

0 80

50

100

120

140 160 180 Expansion space dead volume(cm3)

200

220

COP (%)

Cooling capacity and inputted power(W)

Cooling capacity and inputted power(W)

International Journal of Refrigeration (2017)

0 240

Figure 12. Stirling refrigerator performances versus expansion space dead volume

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250 245

Cold end temperature(K)

240 235 230 225 220 215 210 205 200 40

60

80

100

120

140

160

180

200

220

240

Expansion space swept volume(cm3)

26

65

24

60

22

55

20

50

18

45

16

40

14

35

12

30

10 20

COP (%)

Cooling capacity (W)

Figure 13. Cold end temperature versus expansion space swept volume

40

60

80

100 120 Regenerator length (mm)

140

160

180

25 200

Figure 14. Cooling capacity and COP evolution versus regenerator length

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30 Tcold Qvis Qimp Qcd

250 Cold end temperature(K)

25

300

Losses (W)

20

200 15

150 10

100

5

20

40

60

80

100 120 Regenerator length (mm)

140

160

180

50 200

26

65

24

60

22

55

20

50

18

45

16

40

14

35

12 18

COP (%)

Cooling capacity (W)

Figure 15. Cold end temperature and losses evolution versus regenerator length

19

20

21 22 Regenerator diameter (mm)

23

24

30 25

Figure 16. Stirling refrigerator performances versus regenerator diameter

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240 Qvis Qimp 230 Qcd Tcold 220

24 22 20

Cold end temperature(K)

Losses (W)

18 16

210

14

200

12 10

190

8

180 170 18

19

20

21 22 Regenerator diameter (mm)

23

24

160 25

Figure 17. Cold end temperature and losses evolution versus Regenerator diameter 1

20

COP(%)

Cooling capacity (W)

30

0.5 10

0 0.05

0.06

0.07

0.08 0.09 Regenerator form factor

0.1

0.11

0 0.12

Figure 18. Effect of the regenerator form factor on Stirling refrigerator performances

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26

70

24

65

22

60

20

55

18

50

16 0.5

COP (%)

Cooling capacity (W)

International Journal of Refrigeration (2017)

0.55

0.6

0.65

0.7 0.75 0.8 Regenerator porosity

0.85

0.9

0.95

45 1

Figure 19. Stirling refrigerator performances versus regenerator porosity

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Table 1. Geometrical properties of the experimental setup Hot space diameter [mm]

60

Hot space height [mm]

49.2

Displacer con rod length [mm]

100

Displacer stroke [mm]

48

Cold space diameter [mm]

60

Cold space height [mm]

35.7

Power piston con rod length [mm]

197

Power piston stroke [mm]

48

Regenerator inner diameter [mm]

22

Regenerator outer diameter [mm]

60

Regenerator height [mm]

59

Regenerator porosity

0.79

Maximum machine volume [cm3]

300

Minimum machine volume [cm3]

160

Table 2. Experimental range of functional parameters Functional parameters

Range

Temperature of the hot side [K]

280 -305

Water flow rate [ml.s-1] Speed [rpm]

1- 4 110 - 200

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