# Parametric Investigation and Univariant Optimization of Domestic Stirling Refrigerator

Authors: Muluken Z. Getie; François Lanzetta; Sylvie Bégot; Bimrew T. Admassu; Steve Djetel-Gothe
Journal: International Journal of Energy Research
 (2023)

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## Abstract

As the temperature range of the cooling machine affects the cooling performance, parametric optimization of moderate temperature Stirling refrigerator has valuable concept. In this paper, a nonideal modified simple numerical model was used to investigate the effects of parameters and to propose optimized Stirling refrigerator design for moderate temperature applications. The analysis was conducted through simulating the numerical model using MATLAB code for beta configuration FEMTO-60 prototype operating as a refrigerator. The effects of shuttle heat loss and mass leakage on the cooling performance of domestic Stirling refrigerator have been studied. Moreover, the effects of operating parameters (operating frequency and charging pressure) and design parameters (phase angle, regenerator length and porosity, displacer height, swept volume ratio, and piston diameter to stroke ratio) have been investigated. A cooling power of 625 W and COP of 1.3 have been found at a charging pressure of 22.5 bar and an operating frequency of 6 Hz using the optimized design parameters. The optimized parameters (operating and specified design parameters) could deliver from 48% to 60% more cold production and from 30% to 40% more COP than the existing design with a safe working condition.


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## Body

## 1. Introduction

Refrigeration and air conditioning systems, which use energy to produce cold, are part of day-to-day life in the world. The rise of fuel and other energy prices and global warming push scientists to set the problem of researching alternative and advanced technologies for energy conversion and development of alternative technologies with reasonable efficiency and ecologically safe energy systems. These technologies are based on thermodynamic cycles and used for residential as well as industrial utilities. For instance, the solar energy decentralized systems have been studied for housing sector [1].Despite great scientific strides over the last century, the most widely available refrigeration technologies used today are the same as those used for the last century except for some improvements. The state of the art of alternative refrigerating technologies has been reviewed and identified that the demand of different technologies would be increased in the market [2]. Different researches have been conducted to enhance the thermal performance of cooling. From the working fluid perspective, nanofluids have been studied for cooling performance for different applications [3, 4]. The researches investigate to enhance the heat sink performance. Even if Stirling refrigerators were proposed in the mid of the 1800s [5], still they have very limited market for domestic cooling and air conditioning. The quantities of studies made the on Stirling refrigeration for domestic (near to ambient) temperature applications are very limited as compared to researches conducted about the Stirling cryocooler and Stirling engine. A miniature Stirling cooler machine has been modeled numerically and investigated the effect of parameters on the performance of the cooler [6].A Stirling cycle refrigerator is a cooling device that operates in a reversed regenerative thermodynamic cycle. There are different methods to design and predict the performance of the Stirling machine and classified as zero-order, first-order (approximate design), second-order (decoupled), third-order (nodal design method), and fourth-order (CFD simulation) analyses [7–9].A detailed review was conducted by [10] for high and moderate temperature differential Stirling refrigerator applications. The operating principle, arrangement, and driving mechanisms of the Stirling cycle refrigerator have been presented in the research in addition to different modeling methods followed and their findings. All arrangements are operating with the identical thermodynamic cycle but with different mechanical designs. The different driving mechanisms and arrangements are also presented in [11].An analytical model developed by [12] with different losses included has been used as a basis for large number of second-order numerical models. Low-temperature differential Stirling engines have been studied using numerical investigation [13]. The peak efficiency has been reported at a lower speed which is near to 80 rpm. A parametric investigation of gamma-type low-temperature differential Stirling engine has been conducted using CFD [14]. The researcher reported that phase angle is very significant on engine performance, particularly on efficiency. However, the research did not show the effect of phase angle on machine performance as a function of operating frequency. Theoretical and experimental evaluations of the gamma-type Stirling refrigerator were studied by [15, 16]. Definite regions for the coefficient of performance and thermal efficiency have been identified for duplex Stirling refrigerator and heat engine, respectively [17]. The performance of the duplex refrigerator has been investigated analytically through the assumption of polytropic process to investigate the effect of imperfect regeneration [18]. The effects of parameters on cooling performance have been investigated for a duplex Stirling cooler [19]. The researchers have pointed out that cooling performance has been greatly sensitive to the number of parameters. On the other hand, a trigeneration system that converts thermal energy simultaneously to mechanical, cooling, and heating power has been developed and studied [20]. The study includes theoretical analysis and experimental validation of the combined system.The work losses that include mechanical friction and finite speed losses have been evaluated based on the first law of thermodynamics [21]. Moreover, the effects of irreversibilities (pressure losses and regenerator imperfection) on the Stirling engine power and efficiency have been determined with direct method [22]. Parametric investigations were conducted for an alpha-type Stirling cryocooler [23]. The researcher has stated that conduction heat loss was the major heat loss and the major work loss was due to the mechanical friction loss. The effect of porosity of regenerator, phase angle, pressure, and speed on the cooling performance of the Stirling machine has been investigated [24] for different working fluids. The output parameters, such as the power consumed and the coefficient of performance, were examined under the range of charging pressure and operating speed for a domestic refrigerator [25, 26]. It was reported that the COP varied between 0.1 and 0.9 [25] under different parameters for the V-type integral Stirling refrigerator (VISR). A theoretical model has been developed for a liquid piston Stirling refrigerator which was powered by solar and experimentally tested for residential use [27].Artificial neural network (ANN) has been applied to study the effect of different parameters and operating conditions for the heat-to-cool gamma-type Stirling machine [28]. It has been reported that ANN model could predict the performance of a Stirling machine accurately and within a reasonable time.Beta-type Stirling engine was studied with multiobjective optimization combined with CFD analysis [29]. The researchers conducted sensitivity analysis of seven design parameters of heat exchangers for maximizing power output and efficiency while minimizing pressure drop power loss. A beta-type Stirling cycle refrigeration machine was designed mathematically and tested with experiment [30]. These authors investigated the influence of working fluids, phase angle, swept volume ratio, and the dead volume ratio on the performance of the cooling machine. An optimum cooling performance was stated in a phase angle ranging between 96.1° and 108.3°, and the maximum cooling capacity was obtained when the ratio of volume is almost equal to the ratio of temperature. The effect of swept volume ratio on performance of beta-type Stirling engine and its impact on phase angle and overlap volume have been studied [31]. A numerical investigation and an experimental validation with a focus on evaluating the effects of geometric parameters, such as the expansion space, volume of the dead space, and compression swept were conducted for air-filled beta-type Stirling cycle refrigerator [32]. Djetel-Gothe et al. [33] conducted the design, manufacturing, and testing of a beta Stirling machine for domestic refrigeration applications.Beta-type Stirling refrigeration machine has been studied [34] using the finite physical dimensions thermodynamics (FPDT) method and 0-D modeling using the Schmidt model with imperfect regeneration. The researchers presented the results of both thermodynamic models in comparison with the experimental investigations. An energetic analysis was also reported that allows the development of a system of equations describing the processes that took place at each element of the machine. The performances of refrigerator such as power requirement, cooling load, and coefficient of performance were analyzed [35] for moderate temperature applications using various working fluids (air, nitrogen, helium, and hydrogen). Moreover, loss effect analysis as a function of operating frequency and charging pressure was conducted for such beta-type Stirling refrigerator by [36]. The finding showed that the effect of major power loss (fluid friction) and heat loss (regenerator imperfection) increased as operating frequency and charging pressure increased. That is, the COP of the refrigerating machine showed a decreasing trend at higher operating frequency as the share of the major losses increased. The effect of displacer height on the performance of beta-type Stirling engine has been presented by [37]. However, the researcher did not show the effect of displacer height on different losses.Generally, there are lots of researches on optimization of high temperature difference (engine and cryocooler) applications of the Stirling machine. However, there are very limited researches on the optimal design of moderate temperature Stirling refrigerator. Even though there are few researches on the parametric investigation of the Stirling refrigerators, they are restricted to less number of parameters and did not show the effect of such parameters on different types of losses. Moreover, researches conducted so far on the Stirling refrigerators did not investigate the combined effect of parameters. Therefore, the purpose of this paper was to investigate the effect of parameters on the performance of an air-filled beta-type Stirling cycle refrigerator and to propose optimized parameters for moderate temperature cooling applications. The effects of individual parameters such as displacer length and the effect of combined parameters have been conducted through the univariant optimization technique. A numerical model called modified simple model which was written and validated in the authors’ previous article has been used for the analysis using MATLAB simulation.


## 2. Methodology

The design of the Stirling machines require an understanding of the processes that govern their operation. The consistent and efficient operation of the Stirling machine depends upon the dynamic mechanism of the engine (phase angle, drive system, and configuration), operating condition, and performance of all heat exchangers which are interdependent. This is a difficult task to design a system where thermal, fluid, and mechanical design criteria are required to be considered.In this paper, a numerical model called the modified simple model which was written and validated in the authors’ previous article is used to conduct the parametric investigation and optimization of domestic Stirling cycle refrigerator [38]. The model was validated against experimental work.The control volumes for the numerical analysis include two working volumes (compression space (c) and expansion space (e)) as well as three heat exchangers (hot heat exchanger (h), regenerator (r), and chiller (cr)) as demonstrated in Figure 1.Figure 1
Schematic adiabatic model for a reversed ideal Stirling cycle.During the analysis, the ideal adiabatic condition is taken into consideration and modified with inclusion of relevant losses. The summary of such modified ideal adiabatic model is shown in Table1. The additional power and heat losses incorporated in decoupled way are presented in Table 2, and the detail is presented in the authors’ previous work [38]).Table 1
Summary of modified ideal adiabatic model (detail is presented in [38]).
ParametersEquationsPressure (Pa)P=RM−mleakVc/Tc+Vh/Th+Vr/Tr+Vcr/Tcr+Ve/TeDP=−PγDVc/Tch+DVe/Tcre+γRDQshut/CpTch−Tcre/TchTcre−2γRDmleakVc/Tch+γVh/Th+Vr/Tr+Vcr/Tcr+Ve/TereMass (kg)mi=PVi/RTi (where i=c,h,r,cr,e)Change of mass (kg.s-1)Dmi=miDP/P=DP/RVi/Ti (where i=h,r,cr)Dmc=PDVc+VcDP/γ/RTch+DQshut/CpTch+DmleakDme=PDVe+VeDP/γ/RTcre−DQshut/CpTcreWhereDQshut=πs2kgdd/8JLdTc−TeMass flow (kg.s-1)ṁch=−Dmc−DQshut/CpTch−Dmleakṁhr=ṁch−Dmh, ṁrcr=ṁcre+Dmcrṁcre=Dme−DQshut/CpTcreWhereDmleak=dpπP+Pbuffer/4RTgUpJ−J3/6μP−Pbuffer/LpConditional temperature (K)Ifṁch>0, Tch=Tc; if not, Tch=ThIfṁcre>0, Tcre=Tcr; if not, Tcre=TeTemperature variation (K)DTe=TeDP/P+DVe/Ve−Dme/me, DTc=TcDP/P+DVc/Vc−Dmc/mcDQh,mod=VhDPCv/R−CpTchṁch−ThṁhrDQr,mod=VrDPCv/R−CpThṁhr−TcrṁrcrEnergy (J) and power (W)DQcr,mod=VcrDPCv/R−CpTcrṁrcr−TcreṁcreDWmod=DWc+DWe where DWc=−PDVc and DWe=−PDVeTable 2
Summary of additional losses included [38].
LossesEquationsLoss due to regenerator imperfection (W)Q̇rl=ṁcp1−εTc−Te, ε=NTU/NTU+1NTU=NstAwg/A, Nst=0.023Pr−0.6Re−0.2Conduction heat loss in the regenerator wall (J)Qwrl=kAwg/LfTwh−TwcrHeat conduction losses (W)Q̇cond=kA/LΔTPumping loss (W)Q̇p=1−ηṁCpTc−TeWhereṁ=2fddLdJ/RTc−TePa and η=1−4ṁ0.6Cp0.6J/rπkg0.6Ld0.6ddLosses due to pressure drop in heat exchangers (J)Wfr=∫02πΔPdVe/dθdθWhereΔP=ΔPh+ΔPr+ΔPcr andΔPi=2frμViGili/midi2 (where i=h,r,cr)Mechanical friction losses (J)Wmec.fr=2ΔPmec.frVswcΔPmec.fr=0.94+0.045sn105/31−1/3τ1−1/τLosses due to finite speed of piston (J)Wfin−sp=2Δpfin.spVswcAndΔpfin.sp=1/2PaUp/Cc+PaUp/Cea=3γ and C=3RTGas spring hysteresis losses (W)Ẇhys=1/32ωγ3γ−1TwPmeankgVd/2Vt2Awg


## 3. Experimental Validation

The setup and detail experimental results on a particular prototype for moderate cooling application are presented in [33, 38].To assess the validity of the model, the trend of P-V diagram within the machine is investigated. Therefore, Figure2 is drawn to show a comparison of P-V diagram for the experimental and numerical results. Figure 2(a) demonstrates the compression space, and similarly, Figure 2(b) illustrates the expansion space P-V diagram evaluation for numerical simulation results with experimental results. Moreover, the thermodynamic work (W=∫PdV) of compression and expansion spaces has been computed. The thermodynamic work of the compression space for the experiment and simulation is found as 93.4 J and 101.4 J, respectively. This shows that the prediction error is 8.5%. Similarly, the expansion space thermodynamic work for the experiment and simulation is 52.9 J and 55.2 J, respectively, with the prediction error of 4.3%. From the results, we could conclude that there is a good agreement of the simulation work with experimental results.Figure 2
P-V diagram comparison. (a) Compression space and (b) expansion space (P=17.5bar and f=12.1 Hz).
(a)(b)The experimental validation for cooling performance of the Stirling refrigerator under investigation could be shown in Figure3 for the operating frequencies of 7.3 Hz and 9.7 Hz and cold end temperature ranging from 229 K to 273 K. As demonstrated in Figure 3, the comparison between experimental and simulation results of cooling power, COP, and the associated relative errors has been presented. Particularly, Figure 3(a) illustrates the comparison of the cooling load of simulation and experimental results. Figure 3(c) shows the associated relative error evaluated in predicting the COP the numerical model in comparison with experimental value using the prototype Stirling refrigerator. The maximum relative prediction error of -7.7% was found for an operating frequency of 9.7 Hz. It could be confirmed from this figure that the cooling load for simulation result is in close agreement with the experimental results at all temperature ranges and for both operating frequencies (7.3 Hz and 9.7 Hz).Figure 3
Cooling load and COP comparison between simulation and experimental results (P=17.5bar, f=7.3 Hz, and f=9.7 Hz).
(a)(b)(c)(d)Figure3(b) demonstrates the COP comparison for both cases. Figure 3(d) illustrates the associated relative error evaluated in predicting the COP of the numerical model in comparison with experimental value using the prototype Stirling machine. It has been seen that the maximum relative prediction errors were found as 5% and 23.9% for the operating frequencies of 7.3 Hz and 9.7 Hz, respectively.Furthermore, the bibliographic review has been conducted on moderate cooling Stirling refrigerator. The summary of review has been prepared including the optimized parameters, working condition, and performance achievements of different researches and compared with the current optimized study as presented in Table3. It shows the need to perform parametric analysis and optimization on moderate temperature Stirling refrigerator.Table 3
Comparison between the results of the previous and current studies on moderate Stirling refrigerator.
Ref.ConfigurePressure (bar)Frequency (Hz)Cooling (W)Max. COP, fluidsOptimized parameters[30]Beta7-109-171000.9-1 using HeEffects of some design parameters have been investigated. But there is no combined optimization[25]Alpha2.5-143.3-16.671200.1-0.9 using HeNo parametric investigation[32]Beta1.0131.8-3.317-250.66 using airEffects of different design parameters investigated, but no combined optimization[15]Gamma3-94.17-24.172390.32 using HeMultiobjective optimization of cooling load, COP, and pressure drop conducted. Design parameters was not optimized[16]Gamma34.83-5.331120.27 using HeNo parametric investigation[33] (FEMTO 60 prototype)Beta15-205-125500.4-0.9 using nitrogenNo design parametric investigation[38] (FEMTO 60 prototype)Beta15-252-125600.86 using airNo design parameter investigation, but the effect of operating parameters is investigatedCurrent study (FEMTO 60 prototype)Beta15-22.52-126251.3 using airDesign parameters have been optimized


## 4. Result and Discussion

The effects of different operational variables and design (geometric) variables have been investigated. In addition, the effects of losses (shuttle heat losses and mass leakage) that have direct impact on operational parameters (pressure and temperature of working fluid) have been also studied. Unless otherwise specified, the results in this section have been analyzed at hot heat exchanger temperature of 300 K, chiller temperature of 270 K, and using air as a working fluid. The design parameters of FEMTO 60 engine model as presented in [33, 38] are used for the investigation. In this section of the paper, numerical investigation results have been presented showing effects of the abovementioned parameters with appropriate justification.


### 4.1. Effect of Mass Leakage and Shuttle Heat Losses

The exact effects of mass leakage and shuttle heat losses could exactly be known with noninclusion and inclusion of these losses in differential equation. To evaluate the effect of mass leakage and shuttle heat losses on the temperature, pressure, and overall performance of the refrigerating machine, the model was run subsequently without mass leakage and shuttle heat losses, only with inclusion of mass leakage, only with inclusion of shuttle heat losses, and with mass leakage and shuttle heat losses included in the differential equation. That is, the difference is in the noninclusion and inclusion of the shuttle heat losses and mass leakage loss in the differential equation.The summary of effects of shuttle heat losses on the average temperature of the working fluid, P-V diagram, cooling power, and COP is illustrated in Table4. From Table 4, it could be seen that the working fluid temperature is largely affected by shuttle heat losses. The shuttle heat losses have the effect of decreasing hot gas temperature and increasing cold side gas temperature as part of the heat energy is transferred with the displacer motion from hot space to cold space. When shuttle heat loss is incorporated in the differential equation, the average hot gas temperature drops by 13 K and the average cold gas temperature raises by 9.2 K at an operating condition of (17.5 bar and 7.5 Hz). On the other hand, since the mass of working gas decreases in the compression space, the gas temperature in the compression space increases when mass leakage is incorporated in the differential equation.Table 4
Summary of effect of shuttle heat and mass leakage losses.
No.Case of shuttle heat and mass leakage lossesEffect on temperature (T) and pressure (P) of gasCooling (W)COP1Without including shuttle heat and mass leakage losses in the differential equations (shuttle heat loss treated separately)—450.60.82When only mass leakage is included in the differential equation (shuttle heat loss treated separately)AverageTc increase by 6 K, average Te remains almost unchanged, ΔT increase by 6 K, and the effect is not too much on P-V diagram446.20.783When only shuttle heat losses are included in the differential equations (mass leakage not included)AverageTc decrease by 13 K, average Te increase by 9.2 K, and ΔT decrease by 22.2 K, having a recognized effect on P-V diagram3470.654When both shuttle heat and mass leakage losses are included directly in the differential equationAverageTc decrease by 7 K, average Te increase by 9.6 K, and total ΔT decrease 16.6 K, having a recognized effect on P-V diagram3430.64Furthermore, the effects of shuttle heat losses on the overall performance of the refrigerating machine have been evaluated as shown in Table4. The cooling power and COP of the refrigerating machine are found, respectively, as 450.6 W and 0.8 when shuttle losses are evaluated separately and subtracted from cooling power. On the other hand, the cooling power and COP are decreased to 347 W and 0.65 when shuttle heat losses are incorporated in the differential equation. Furthermore, when we incorporate the mass leakage loss in the differential equation, the cooling power and COP have been reduced to 343 W and 0.64.Moreover, the effects of shuttle heat losses on the P-V diagram have been investigated. Figure4 shows the P-V diagram at a charging pressure of 17.5 bar and an operating frequency of 7.5 Hz where the analysis is performed first without including shuttle heat losses and then including it in the differential equations. It is noticeable that the P-V loop becomes slenderer when shuttle heat losses are incorporated in the differential equation. This is because when heat is shuttled from compression space to expansion space, the pressure difference between the two spaces decreases. This resulted in a lower area enclosed by the P-V diagram.Figure 4
Effect of shuttle heat losses on the P-V diagram (Tcr=270 K, Th=300 K, P=17.5bar, and f=7.5 Hz).
(a)
P-V in compression and expansion spaces(b)
Net P-V diagram


### 4.2. Effect of Operating Parameters on Cooling Performance

Even though the operating frequency and pressure of the Stirling refrigerator are well known to have a significant influence on machine in terms of its overall performance, the exact effects of these parameters have to be investigated for such moderate cooling application. The variation of P-V diagram as functions of operating frequency and charging pressure has been investigated in addition to effect of such parameters on cooling performance so as to predict the operating condition of the machine.Figures5(a) and 5(b) demonstrate the effect of operating frequency on the P-V diagram at 6 Hz and 12 Hz, respectively. The compression volume pressure remains almost similar, and the expansion volume pressure variations are highly reduced as operating frequency increases from 6 to 12 Hz. These are demonstrated by the thermodynamic work. The compression volume (thermodynamic work of the compression space) varies from 91.0 J to 91.8 J as operating frequency changes from 6 Hz to 12 Hz. The thermodynamic work of the expansion space decreases from 57.5 J to 29.4 J as operating frequency changes from 6 Hz to 12 Hz. The reduction of the P-V diagram area for the expansion space shows that as frequency increases, the amount of heat transfer per cycle to the working fluid decreases as the working fluid spends lesser time in the heat exchangers. However, the P-V diagram in compression spaces remains almost the same. Even though it is expected to decrease as the heat transfer per cycle decreases with operating frequency, the amount of input power requirement increases due to the increase of fluid friction losses, finite speed losses, and mechanical friction losses. The hollow curve in expansion pressure volume curve of Figure 5(b) shows that the higher pressure drop is around crank angle of 40°-60° and is more critical as the operating frequency increases.Figure 5
Effect of operating frequency on the P-V diagram (Tcr=270 K, Th=300 K, and P=17.5bar).
(a)
P-V diagram at 6 Hz(b)
P-V diagram at 12 HzSimilarly, the effect of charging pressure on the P-V diagram has been investigated as shown in Figure6, where P-V curves of the two chambers of the Stirling refrigerator are compared for the charging pressures 15 bar and 25 bar. The lowest and highest pressures inside the compression space were about 11.7 bar and 18.9 bar, and the lowest and highest pressures in expansion spaces were about 13.9 bar and 16.6 bar when the charging pressure is 15 bar. At a charging pressure of 25 bar, the lowest and highest pressures inside the compression space were about 19.6 bar and 31.5 bar, and the lowest and highest pressures in expansion spaces were about 23 bar and 27.7 bar. As shown in the figure, the area corresponding to the volume of the fluid for both P-V diagrams of compression and expansion spaces increases as pressure increased from 15 bar to 25 bar. The compression volume (thermodynamic work of the compression space) increases from 78.3 J and 132.2 J as charging pressure changes from 15 bar to 25 bar. The expansion volume (thermodynamic work of the expansion space) increases from 24.2 J to 44.6 J as charging pressure changes from 15 to 25 bar. The net increase in compression space area is higher than the net increase in the area of expansion space. This confirms that the net power consumed per cycle increases with charging pressure, which leads to decrease in COP.Figure 6
Effect of charging pressure on the P-V diagram (Tcr=270 K, Th=300 K, and f=7.5 Hz).
(a)
P-V diagram at 15 bar(b)
P-V diagram at 25 barThe power demand, as a function of the operating parameters, is studied and presented in Figure7. The analysis is conducted with operating frequency ranging from 1 to 12 Hz and charging pressure vary from 5 to 25 bar. The increase in pressure of the Stirling refrigerator leads to an increase of the work done on the fluid and an increase in the mass of working fluid that would result in higher fluid friction (pressure) losses which require more power to drive the refrigerator. The demand input power initially increases nearly linear and then increases at a higher rate with an increase in operating frequency which is in a good agreement with [23]. This is because the fluid friction loss increases at higher rate with operating frequencies as presented by the authors’ previous work [36]. On the other hand, as operating frequency increases, the number of cycles per time increases which demands higher input power.Figure 7
Trend of power consumed for refrigerating machine with respect to operating frequency at different charging pressure.Figure8 illustrates the effect of operating frequency on the cooling power for a range of charging pressure. As the refrigerator charging pressure increases, the mass of working fluid increases that results in a higher heat removal rate and hence higher cooling power. Furthermore, it could be seen from this figure that cooling power increases linearly with an increase in operating frequency. In other words, increasing the operating frequency of the Stirling refrigerator will result in higher cooling power at the fixed cooling temperature. The choices of operating frequency and charging pressure could be dependent on the cooling temperature requirement and the COP.Figure 8
Trend of cooling power with respect to operating frequency at different charging pressure.In Figure9, the trend of COP is demonstrated as a function of operating frequency at different charging pressures. As COP is the ratio of cooling power and power demand, it can easily be determined from results of Figures 7 and 8. Even though both cooling power and power demand increase with operating frequency, the increase in power demand is higher than the increase in cooling power. This results in a reduction in COP of the refrigerator at higher frequencies. COP shows a peak value with respect to operating frequency, and this was also reported by [32]. The higher the charging pressure, the lower the frequency in which the peak COP existed. This is due to the fact that at higher pressure, fluid friction increases at higher rate with an increase in operating frequency, and hence, the COP will tend to reach its peak value at relatively lower frequency.Figure 9
COP versus operating frequency at different charging pressure.On the other hand, the COP increases with charging pressure; this is because the effect of an increase of mass of working fluid on the heat removal rate is higher than the increase of fluid friction losses.


### 4.3. Effect of Design Parameters on Cooling Performance

#### 4.3.1. Effect of Phase Angle

The phase angle determines the dynamics of piston-displacer movements and thus the volume variation during the cycle which is directly related to the work input and output. The effect of phase angle on cooling power and COP at different operating frequencies has been indicated in Figures10 and 11, respectively.Figure 10
Cooling power versus phase angle (P=17.5bar).
(a)
Cooling power using air(b)
Cooling power using heliumFigure 11
COP versus phase angle at different operating frequency (P=17.5bar).
(a)
COP using air(b)
COP using heliumFigure10 shows that there is a single optimum angle (considering cooling power) as a function of operating frequency for a single type of working fluid. This confirms that the phase angle for optimum cooling power is independent of operating frequency. It should be noted that the optimum phase angles have been found near to 90° and 95° for air and helium, respectively. These obtained phase angles closely agree with the results reported for the engine by [39]. It could be stated that the optimum phase angle for gases like air with lower thermal capacity is lower.Figure11 demonstrates the COP of a Stirling refrigerator as a function of the phase angle. It could be seen that the optimum phase angle for the peak COP varies with the operating frequency, and the optimum phase angle for COP is mostly found at lower phase angle than optimum phase angle for cooling power. The higher the operating frequency, the lower the phase angle associated for the peak COP. Generally, the peak COP as a function of phase angle varies with operating frequency. This is because the fluid friction losses depend both on operating frequency and phase angle (as it affects swept volume).


#### 4.3.2. Effect of Length and Porosity of Regenerator

The regenerator is an essential part of the Stirling machines as it recovers a large part of the thermal energy of the working fluid. On the other hand, regenerator is typically the largest source of power losses in the system. Hence, the effects of regenerator parameters should be investigated based on the application of the Stirling machine. Even though many researches have been conducted so far on the effect of regenerator parameters, to set optimized parameters for such particular arrangement and application, the effects of length and porosity of regenerator have been investigated as shown in Figures12–15.Figure 12
Cooling power and COP versus length of regenerator (ϕ=0.64, P=17.5bar, and f=7.5 Hz).Figure 13
Variation of different losses versus length of regenerator (ϕ=0.64, P=17.5bar, and f=7.5 Hz).Figure 14
Cooling power and COP versus regenerator porosity (P=17.5bar and f=7.5 Hz).Figure 15
Variation of fluid friction and regenerator imperfection losses versus regenerator porosity (P=17.5bar and f=7.5 Hz).Figure12 shows the performance of a Stirling refrigerator (regenerator porosity ϕ=0.64, P=17.5bar, and f=7.5 Hz) for a range of regenerator length. As shown in Figure 12, parabolic evolutions of cooling power and COP have been demonstrated as a function of regenerator length with peak values found nearly at the same point (25 mm). It could be stated that both cooling power and COP show a decreasing trend as the regenerator length increases after the peak value. This phenomenon results from decreasing of most of the heat losses and increasing of fluid friction loss with regenerator length as confirmed in Figure 13. From Figure 13, it could be understood that regenerator imperfection loss decreases and fluid friction loss increases at higher rate than other losses. As the length of the regenerator increases, fluid friction loss will increase which will result in lower performance of the Stirling refrigerator.Figure14 shows the impact of regenerator porosity on the cooling performance of the Stirling refrigerator. It could be seen that the cooling power of the refrigerator decreases with an increase in porosity. This is because while the porosity of the regenerator increases, the heat transfer area of the matrix and the amount the heat transfer decrease. The working gas goes into the heater at a higher temperature. On the other hand, due to a higher pressure drop, a low porosity regenerator leads to higher fluid friction losses. The curve shows that the peak COP value is found at a porosity of 0.75 which is similar to the research conducted by [40] on Stirling engine. This result is mainly accompanied from fluid friction and regenerator imperfection losses as shown in Figure 15. The effect of other losses such as conduction heat loss are less significant for such moderate temperature applications. Figure 15 also demonstrates that the decreasing of fluid friction with an increase in regenerator porosity leads to an increase in COP. On the other hand, regenerator imperfection loss increases at higher rate than the decreasing rate of fluid friction loss just after a porosity value of 0.75 that results in an optimum COP. Hence, the optimum design point in case of regenerator porosity could be at a compromise between a higher cooling power and peak COP.


#### 4.3.3. Effect of Displacer Height

Stirling machine designs utilize displacers as a means of transferring the working fluid between the cold and hot regions. The height of the displacer affects the performance of the Stirling machine by changing the compression ratio as the swept volume of compression space varies with displacer height [37]. Figure 16 presented the impact of displacer height on cooling performance (cooling power and COP) using air as a working fluid at a charging pressure of 17.5 bar and an operating frequency of 7.5 Hz. In this analysis, the height of the displacer is allowed to vary, still keeping the total length (height plus length of displacer rod) of the displacer fixed. As displacer height increases, keeping the total length fixed, the compression volume decreases that results in a lower swept volume compression ratio (Vr=Vswc/Vswe).Figure 16
Cooling power and COP versus displacer height (P=17.5bar and f=7.5 Hz).It has been noticed from Figure16 that cooling power decreases monotonously with an increase in displacer height and there exists a peak value of COP. The peak COP value is found as 0.72 at a displacer height of 10 cm with the associated cooling power of 544.4 W and a swept volume compression ratio of 1.58. Hence, with the same operating condition and keeping all other design parameters constant, the peak cold production and COP could be increased by 61.6% and 13.3%, respectively, only through displacer height optimization. The main reason for such improvement in cooling performance is the increase of swept volume compression ratio.


#### 4.3.4. Effect of Swept Volume Ratio

Figure17 demonstrates the effect of the ratio of swept volume on cooling power and COP of the Stirling refrigerator at charging pressure of 17.5 bar and an operating frequency of 7.5 Hz. It can be seen from Figure 17 that the peak values of cooling power and COP are found as 383.6 W and 0.7, respectively. These peak values are found at a swept volume ratio of 1.7. Hence, with the same operating condition and only through swept volume ratio optimization, the peak cold production and COP could be improved by 11.8% and 10.9%, respectively. The analysis is conducted by keeping the total volume constant and still varying the two swept volumes through changing strokes of the displacer and power piston.Figure 17
Cooling power and COP versus swept volume ratio (P=17.5bar and f=7.5 Hz).Both Figures16 and 17 show the effect of volume ratio on the performance of the Stirling refrigerator keeping the overall cylinder volume fixed. However, in Figure 16, due to the variation of displacer height, the swept volume of compression space varies keeping the swept volume of expansion space fixed, whereas in Figure 17, both compression and expansion spaces vary keeping the summation of the two swept volumes still fixed. Figure 18 illustrates the cooling power and COP of the Stirling refrigerator as a function of swept volume ratio for the comparison of these two cases. It is seen in Figure 18 that a better COP could be achieved when the volume ratio variation is due to the variation of compression space swept volume through displacer height variation. From this figure, it could also be concluded that the optimum cooling performance for the domestic Stirling refrigerator can be found for the swept volume ratio ranging from 1.4 to 1.8.Figure 18
COP comparison versus swept volume ratio for different cases of volume ratio (P=17.5bar and f=7.5 Hz).


#### 4.3.5. Effect of Ratio of Diameter to Stroke of Piston (dP/s)

Figure19 illustrates the cooling power and COP versus the ratio of piston diameter to stroke keeping the piston swept volume constant. It can be seen that both cooling power and COP increase in a wide dome-shaped trend with an increase in piston diameter to stroke ratio (dP/s), and similar trend has been presented by [41]. This phenomenon is mainly due to the variation of different power losses with the (dP/s) ratio as can be seen in Figure 20. It can be seen from Figure 20that the total power loss decreases mainly due to the decreasing of mechanical friction and piston finite speed losses as the stroke of piston decreases with an increase in dP/s ratio. On other hand, even though conduction heat loss increases, since shuttle heat losses decreases (see Figure 20(b)), cooling power increases in a wider dome shape as the ratio (dP/s) increases. This is because the shuttle heat loss has more effect than conduction heat loss for such moderate temperature application. As the ratio (dP/s) increases from 1.2 to 2.6, the cooling power and COP enhanced by 17.3% and 32%, respectively.Figure 19
Cooling power and COP variation with ratio between diameter and stroke of piston (P=17.5bar and f=7.5 Hz).Figure 20
Power loss variation with the ratio between diameter and stroke of piston (P=17.5bar and f=7.5 Hz).
(a)
Losses with diameter to stroke ratio(b)
Losses with diameter to stroke ratio


### 4.4. Effect of Combined Optimized Parameters

In the previous sections of this paper, we use univariant analysis in which only one parameter is investigated at a time while all other parameters are retained to prefixed values. This approach is simplistic and allows us to assess the most influential parameters as well as the exact effect of each parameter. However, by using only univariant optimization, it would be difficult to find the best combination of parameters for optimum machine performance. Hence, for the second-order numerical model, after investigating a univariant optimization, a trade-off between multiple conflicting variables is indispensable.Therefore, in this section, we try to investigate parameters with higher impact as presented in Sections4.2 and 4.3 and then combine them to find an optimized combination. Different combinations of parameters which have positive impact on cooling performance have been first selected.Regarding operating parameter optimization, it can be seen from the optimized results presented in Section4.2 that relatively the lower operating frequency (3 to 8 Hz) at higher charging pressure is found to be the better operating parameters. But, due to the restricted operating conditions of the machine for safe working conditions and to minimize manufacturing cost, the average proposed operating frequency and charging pressure are set as 6 Hz and 22.5 bar, respectively.The major design parameters considered for the analysis of combined parameters are regenerator length, porosity, displacer height, swept volume ratio, and diameter to stroke ratio of the piston. The different combinations of these parameters were simulated, and lastly, a combination of parameters with more enhanced cooling performance has been proposed as presented in Table5.Table 5
Specifications of the Stirling refrigerator for optimized analysis.
No.ParametersExisting value [38]Optimized value1Hot sink temperature (K)3053052Cold source temperature (K)2702703Diameter of piston (mm)60624Diameter of displacer (mm)59615Stroke of piston (mm)40376Regenerator diameter (mm)82827Regenerator length (mm)50408Wire diameter (μm)1121129Porosity (%)647210Compression space swept volume (cm3)10314011Expansion space swept volume (cm3)11311312Compression dead volume (cm3)4.244.2413Expansion dead volume (cm3)4.244.2414Working gasAirAir15Frequency (Hz)5-133-816Charging pressure (bar)15-2020-25The results of cooling performance for combined parametric optimization have been compared with the analysis results using existing parameters as demonstrated in Figure21. It could be seen from the figure that the performance of the Stirling refrigerator has been improved significantly with optimized parameters. It is found that all optimized design parameters that are quite influential as individual parameters could enhance the COP of the refrigerator from 7% to 19% from the existing design value. Collectively, the optimized parameters (operating and specified design parameters) could deliver from 48% to 60% more cold production and from 30% to 40% more COP than the existing design in a safe working condition.Figure 21
Comparison of results of existing and optimized cooling performance versus operating frequency.


## 4.1. Effect of Mass Leakage and Shuttle Heat Losses

The exact effects of mass leakage and shuttle heat losses could exactly be known with noninclusion and inclusion of these losses in differential equation. To evaluate the effect of mass leakage and shuttle heat losses on the temperature, pressure, and overall performance of the refrigerating machine, the model was run subsequently without mass leakage and shuttle heat losses, only with inclusion of mass leakage, only with inclusion of shuttle heat losses, and with mass leakage and shuttle heat losses included in the differential equation. That is, the difference is in the noninclusion and inclusion of the shuttle heat losses and mass leakage loss in the differential equation.The summary of effects of shuttle heat losses on the average temperature of the working fluid, P-V diagram, cooling power, and COP is illustrated in Table4. From Table 4, it could be seen that the working fluid temperature is largely affected by shuttle heat losses. The shuttle heat losses have the effect of decreasing hot gas temperature and increasing cold side gas temperature as part of the heat energy is transferred with the displacer motion from hot space to cold space. When shuttle heat loss is incorporated in the differential equation, the average hot gas temperature drops by 13 K and the average cold gas temperature raises by 9.2 K at an operating condition of (17.5 bar and 7.5 Hz). On the other hand, since the mass of working gas decreases in the compression space, the gas temperature in the compression space increases when mass leakage is incorporated in the differential equation.Table 4
Summary of effect of shuttle heat and mass leakage losses.
No.Case of shuttle heat and mass leakage lossesEffect on temperature (T) and pressure (P) of gasCooling (W)COP1Without including shuttle heat and mass leakage losses in the differential equations (shuttle heat loss treated separately)—450.60.82When only mass leakage is included in the differential equation (shuttle heat loss treated separately)AverageTc increase by 6 K, average Te remains almost unchanged, ΔT increase by 6 K, and the effect is not too much on P-V diagram446.20.783When only shuttle heat losses are included in the differential equations (mass leakage not included)AverageTc decrease by 13 K, average Te increase by 9.2 K, and ΔT decrease by 22.2 K, having a recognized effect on P-V diagram3470.654When both shuttle heat and mass leakage losses are included directly in the differential equationAverageTc decrease by 7 K, average Te increase by 9.6 K, and total ΔT decrease 16.6 K, having a recognized effect on P-V diagram3430.64Furthermore, the effects of shuttle heat losses on the overall performance of the refrigerating machine have been evaluated as shown in Table4. The cooling power and COP of the refrigerating machine are found, respectively, as 450.6 W and 0.8 when shuttle losses are evaluated separately and subtracted from cooling power. On the other hand, the cooling power and COP are decreased to 347 W and 0.65 when shuttle heat losses are incorporated in the differential equation. Furthermore, when we incorporate the mass leakage loss in the differential equation, the cooling power and COP have been reduced to 343 W and 0.64.Moreover, the effects of shuttle heat losses on the P-V diagram have been investigated. Figure4 shows the P-V diagram at a charging pressure of 17.5 bar and an operating frequency of 7.5 Hz where the analysis is performed first without including shuttle heat losses and then including it in the differential equations. It is noticeable that the P-V loop becomes slenderer when shuttle heat losses are incorporated in the differential equation. This is because when heat is shuttled from compression space to expansion space, the pressure difference between the two spaces decreases. This resulted in a lower area enclosed by the P-V diagram.Figure 4
Effect of shuttle heat losses on the P-V diagram (Tcr=270 K, Th=300 K, P=17.5bar, and f=7.5 Hz).
(a)
P-V in compression and expansion spaces(b)
Net P-V diagram


## 4.2. Effect of Operating Parameters on Cooling Performance

Even though the operating frequency and pressure of the Stirling refrigerator are well known to have a significant influence on machine in terms of its overall performance, the exact effects of these parameters have to be investigated for such moderate cooling application. The variation of P-V diagram as functions of operating frequency and charging pressure has been investigated in addition to effect of such parameters on cooling performance so as to predict the operating condition of the machine.Figures5(a) and 5(b) demonstrate the effect of operating frequency on the P-V diagram at 6 Hz and 12 Hz, respectively. The compression volume pressure remains almost similar, and the expansion volume pressure variations are highly reduced as operating frequency increases from 6 to 12 Hz. These are demonstrated by the thermodynamic work. The compression volume (thermodynamic work of the compression space) varies from 91.0 J to 91.8 J as operating frequency changes from 6 Hz to 12 Hz. The thermodynamic work of the expansion space decreases from 57.5 J to 29.4 J as operating frequency changes from 6 Hz to 12 Hz. The reduction of the P-V diagram area for the expansion space shows that as frequency increases, the amount of heat transfer per cycle to the working fluid decreases as the working fluid spends lesser time in the heat exchangers. However, the P-V diagram in compression spaces remains almost the same. Even though it is expected to decrease as the heat transfer per cycle decreases with operating frequency, the amount of input power requirement increases due to the increase of fluid friction losses, finite speed losses, and mechanical friction losses. The hollow curve in expansion pressure volume curve of Figure 5(b) shows that the higher pressure drop is around crank angle of 40°-60° and is more critical as the operating frequency increases.Figure 5
Effect of operating frequency on the P-V diagram (Tcr=270 K, Th=300 K, and P=17.5bar).
(a)
P-V diagram at 6 Hz(b)
P-V diagram at 12 HzSimilarly, the effect of charging pressure on the P-V diagram has been investigated as shown in Figure6, where P-V curves of the two chambers of the Stirling refrigerator are compared for the charging pressures 15 bar and 25 bar. The lowest and highest pressures inside the compression space were about 11.7 bar and 18.9 bar, and the lowest and highest pressures in expansion spaces were about 13.9 bar and 16.6 bar when the charging pressure is 15 bar. At a charging pressure of 25 bar, the lowest and highest pressures inside the compression space were about 19.6 bar and 31.5 bar, and the lowest and highest pressures in expansion spaces were about 23 bar and 27.7 bar. As shown in the figure, the area corresponding to the volume of the fluid for both P-V diagrams of compression and expansion spaces increases as pressure increased from 15 bar to 25 bar. The compression volume (thermodynamic work of the compression space) increases from 78.3 J and 132.2 J as charging pressure changes from 15 bar to 25 bar. The expansion volume (thermodynamic work of the expansion space) increases from 24.2 J to 44.6 J as charging pressure changes from 15 to 25 bar. The net increase in compression space area is higher than the net increase in the area of expansion space. This confirms that the net power consumed per cycle increases with charging pressure, which leads to decrease in COP.Figure 6
Effect of charging pressure on the P-V diagram (Tcr=270 K, Th=300 K, and f=7.5 Hz).
(a)
P-V diagram at 15 bar(b)
P-V diagram at 25 barThe power demand, as a function of the operating parameters, is studied and presented in Figure7. The analysis is conducted with operating frequency ranging from 1 to 12 Hz and charging pressure vary from 5 to 25 bar. The increase in pressure of the Stirling refrigerator leads to an increase of the work done on the fluid and an increase in the mass of working fluid that would result in higher fluid friction (pressure) losses which require more power to drive the refrigerator. The demand input power initially increases nearly linear and then increases at a higher rate with an increase in operating frequency which is in a good agreement with [23]. This is because the fluid friction loss increases at higher rate with operating frequencies as presented by the authors’ previous work [36]. On the other hand, as operating frequency increases, the number of cycles per time increases which demands higher input power.Figure 7
Trend of power consumed for refrigerating machine with respect to operating frequency at different charging pressure.Figure8 illustrates the effect of operating frequency on the cooling power for a range of charging pressure. As the refrigerator charging pressure increases, the mass of working fluid increases that results in a higher heat removal rate and hence higher cooling power. Furthermore, it could be seen from this figure that cooling power increases linearly with an increase in operating frequency. In other words, increasing the operating frequency of the Stirling refrigerator will result in higher cooling power at the fixed cooling temperature. The choices of operating frequency and charging pressure could be dependent on the cooling temperature requirement and the COP.Figure 8
Trend of cooling power with respect to operating frequency at different charging pressure.In Figure9, the trend of COP is demonstrated as a function of operating frequency at different charging pressures. As COP is the ratio of cooling power and power demand, it can easily be determined from results of Figures 7 and 8. Even though both cooling power and power demand increase with operating frequency, the increase in power demand is higher than the increase in cooling power. This results in a reduction in COP of the refrigerator at higher frequencies. COP shows a peak value with respect to operating frequency, and this was also reported by [32]. The higher the charging pressure, the lower the frequency in which the peak COP existed. This is due to the fact that at higher pressure, fluid friction increases at higher rate with an increase in operating frequency, and hence, the COP will tend to reach its peak value at relatively lower frequency.Figure 9
COP versus operating frequency at different charging pressure.On the other hand, the COP increases with charging pressure; this is because the effect of an increase of mass of working fluid on the heat removal rate is higher than the increase of fluid friction losses.


## 4.3. Effect of Design Parameters on Cooling Performance

### 4.3.1. Effect of Phase Angle

The phase angle determines the dynamics of piston-displacer movements and thus the volume variation during the cycle which is directly related to the work input and output. The effect of phase angle on cooling power and COP at different operating frequencies has been indicated in Figures10 and 11, respectively.Figure 10
Cooling power versus phase angle (P=17.5bar).
(a)
Cooling power using air(b)
Cooling power using heliumFigure 11
COP versus phase angle at different operating frequency (P=17.5bar).
(a)
COP using air(b)
COP using heliumFigure10 shows that there is a single optimum angle (considering cooling power) as a function of operating frequency for a single type of working fluid. This confirms that the phase angle for optimum cooling power is independent of operating frequency. It should be noted that the optimum phase angles have been found near to 90° and 95° for air and helium, respectively. These obtained phase angles closely agree with the results reported for the engine by [39]. It could be stated that the optimum phase angle for gases like air with lower thermal capacity is lower.Figure11 demonstrates the COP of a Stirling refrigerator as a function of the phase angle. It could be seen that the optimum phase angle for the peak COP varies with the operating frequency, and the optimum phase angle for COP is mostly found at lower phase angle than optimum phase angle for cooling power. The higher the operating frequency, the lower the phase angle associated for the peak COP. Generally, the peak COP as a function of phase angle varies with operating frequency. This is because the fluid friction losses depend both on operating frequency and phase angle (as it affects swept volume).


### 4.3.2. Effect of Length and Porosity of Regenerator

The regenerator is an essential part of the Stirling machines as it recovers a large part of the thermal energy of the working fluid. On the other hand, regenerator is typically the largest source of power losses in the system. Hence, the effects of regenerator parameters should be investigated based on the application of the Stirling machine. Even though many researches have been conducted so far on the effect of regenerator parameters, to set optimized parameters for such particular arrangement and application, the effects of length and porosity of regenerator have been investigated as shown in Figures12–15.Figure 12
Cooling power and COP versus length of regenerator (ϕ=0.64, P=17.5bar, and f=7.5 Hz).Figure 13
Variation of different losses versus length of regenerator (ϕ=0.64, P=17.5bar, and f=7.5 Hz).Figure 14
Cooling power and COP versus regenerator porosity (P=17.5bar and f=7.5 Hz).Figure 15
Variation of fluid friction and regenerator imperfection losses versus regenerator porosity (P=17.5bar and f=7.5 Hz).Figure12 shows the performance of a Stirling refrigerator (regenerator porosity ϕ=0.64, P=17.5bar, and f=7.5 Hz) for a range of regenerator length. As shown in Figure 12, parabolic evolutions of cooling power and COP have been demonstrated as a function of regenerator length with peak values found nearly at the same point (25 mm). It could be stated that both cooling power and COP show a decreasing trend as the regenerator length increases after the peak value. This phenomenon results from decreasing of most of the heat losses and increasing of fluid friction loss with regenerator length as confirmed in Figure 13. From Figure 13, it could be understood that regenerator imperfection loss decreases and fluid friction loss increases at higher rate than other losses. As the length of the regenerator increases, fluid friction loss will increase which will result in lower performance of the Stirling refrigerator.Figure14 shows the impact of regenerator porosity on the cooling performance of the Stirling refrigerator. It could be seen that the cooling power of the refrigerator decreases with an increase in porosity. This is because while the porosity of the regenerator increases, the heat transfer area of the matrix and the amount the heat transfer decrease. The working gas goes into the heater at a higher temperature. On the other hand, due to a higher pressure drop, a low porosity regenerator leads to higher fluid friction losses. The curve shows that the peak COP value is found at a porosity of 0.75 which is similar to the research conducted by [40] on Stirling engine. This result is mainly accompanied from fluid friction and regenerator imperfection losses as shown in Figure 15. The effect of other losses such as conduction heat loss are less significant for such moderate temperature applications. Figure 15 also demonstrates that the decreasing of fluid friction with an increase in regenerator porosity leads to an increase in COP. On the other hand, regenerator imperfection loss increases at higher rate than the decreasing rate of fluid friction loss just after a porosity value of 0.75 that results in an optimum COP. Hence, the optimum design point in case of regenerator porosity could be at a compromise between a higher cooling power and peak COP.


### 4.3.3. Effect of Displacer Height

Stirling machine designs utilize displacers as a means of transferring the working fluid between the cold and hot regions. The height of the displacer affects the performance of the Stirling machine by changing the compression ratio as the swept volume of compression space varies with displacer height [37]. Figure 16 presented the impact of displacer height on cooling performance (cooling power and COP) using air as a working fluid at a charging pressure of 17.5 bar and an operating frequency of 7.5 Hz. In this analysis, the height of the displacer is allowed to vary, still keeping the total length (height plus length of displacer rod) of the displacer fixed. As displacer height increases, keeping the total length fixed, the compression volume decreases that results in a lower swept volume compression ratio (Vr=Vswc/Vswe).Figure 16
Cooling power and COP versus displacer height (P=17.5bar and f=7.5 Hz).It has been noticed from Figure16 that cooling power decreases monotonously with an increase in displacer height and there exists a peak value of COP. The peak COP value is found as 0.72 at a displacer height of 10 cm with the associated cooling power of 544.4 W and a swept volume compression ratio of 1.58. Hence, with the same operating condition and keeping all other design parameters constant, the peak cold production and COP could be increased by 61.6% and 13.3%, respectively, only through displacer height optimization. The main reason for such improvement in cooling performance is the increase of swept volume compression ratio.


### 4.3.4. Effect of Swept Volume Ratio

Figure17 demonstrates the effect of the ratio of swept volume on cooling power and COP of the Stirling refrigerator at charging pressure of 17.5 bar and an operating frequency of 7.5 Hz. It can be seen from Figure 17 that the peak values of cooling power and COP are found as 383.6 W and 0.7, respectively. These peak values are found at a swept volume ratio of 1.7. Hence, with the same operating condition and only through swept volume ratio optimization, the peak cold production and COP could be improved by 11.8% and 10.9%, respectively. The analysis is conducted by keeping the total volume constant and still varying the two swept volumes through changing strokes of the displacer and power piston.Figure 17
Cooling power and COP versus swept volume ratio (P=17.5bar and f=7.5 Hz).Both Figures16 and 17 show the effect of volume ratio on the performance of the Stirling refrigerator keeping the overall cylinder volume fixed. However, in Figure 16, due to the variation of displacer height, the swept volume of compression space varies keeping the swept volume of expansion space fixed, whereas in Figure 17, both compression and expansion spaces vary keeping the summation of the two swept volumes still fixed. Figure 18 illustrates the cooling power and COP of the Stirling refrigerator as a function of swept volume ratio for the comparison of these two cases. It is seen in Figure 18 that a better COP could be achieved when the volume ratio variation is due to the variation of compression space swept volume through displacer height variation. From this figure, it could also be concluded that the optimum cooling performance for the domestic Stirling refrigerator can be found for the swept volume ratio ranging from 1.4 to 1.8.Figure 18
COP comparison versus swept volume ratio for different cases of volume ratio (P=17.5bar and f=7.5 Hz).


### 4.3.5. Effect of Ratio of Diameter to Stroke of Piston (dP/s)

Figure19 illustrates the cooling power and COP versus the ratio of piston diameter to stroke keeping the piston swept volume constant. It can be seen that both cooling power and COP increase in a wide dome-shaped trend with an increase in piston diameter to stroke ratio (dP/s), and similar trend has been presented by [41]. This phenomenon is mainly due to the variation of different power losses with the (dP/s) ratio as can be seen in Figure 20. It can be seen from Figure 20that the total power loss decreases mainly due to the decreasing of mechanical friction and piston finite speed losses as the stroke of piston decreases with an increase in dP/s ratio. On other hand, even though conduction heat loss increases, since shuttle heat losses decreases (see Figure 20(b)), cooling power increases in a wider dome shape as the ratio (dP/s) increases. This is because the shuttle heat loss has more effect than conduction heat loss for such moderate temperature application. As the ratio (dP/s) increases from 1.2 to 2.6, the cooling power and COP enhanced by 17.3% and 32%, respectively.Figure 19
Cooling power and COP variation with ratio between diameter and stroke of piston (P=17.5bar and f=7.5 Hz).Figure 20
Power loss variation with the ratio between diameter and stroke of piston (P=17.5bar and f=7.5 Hz).
(a)
Losses with diameter to stroke ratio(b)
Losses with diameter to stroke ratio


## 4.3.1. Effect of Phase Angle

The phase angle determines the dynamics of piston-displacer movements and thus the volume variation during the cycle which is directly related to the work input and output. The effect of phase angle on cooling power and COP at different operating frequencies has been indicated in Figures10 and 11, respectively.Figure 10
Cooling power versus phase angle (P=17.5bar).
(a)
Cooling power using air(b)
Cooling power using heliumFigure 11
COP versus phase angle at different operating frequency (P=17.5bar).
(a)
COP using air(b)
COP using heliumFigure10 shows that there is a single optimum angle (considering cooling power) as a function of operating frequency for a single type of working fluid. This confirms that the phase angle for optimum cooling power is independent of operating frequency. It should be noted that the optimum phase angles have been found near to 90° and 95° for air and helium, respectively. These obtained phase angles closely agree with the results reported for the engine by [39]. It could be stated that the optimum phase angle for gases like air with lower thermal capacity is lower.Figure11 demonstrates the COP of a Stirling refrigerator as a function of the phase angle. It could be seen that the optimum phase angle for the peak COP varies with the operating frequency, and the optimum phase angle for COP is mostly found at lower phase angle than optimum phase angle for cooling power. The higher the operating frequency, the lower the phase angle associated for the peak COP. Generally, the peak COP as a function of phase angle varies with operating frequency. This is because the fluid friction losses depend both on operating frequency and phase angle (as it affects swept volume).


## 4.3.2. Effect of Length and Porosity of Regenerator

The regenerator is an essential part of the Stirling machines as it recovers a large part of the thermal energy of the working fluid. On the other hand, regenerator is typically the largest source of power losses in the system. Hence, the effects of regenerator parameters should be investigated based on the application of the Stirling machine. Even though many researches have been conducted so far on the effect of regenerator parameters, to set optimized parameters for such particular arrangement and application, the effects of length and porosity of regenerator have been investigated as shown in Figures12–15.Figure 12
Cooling power and COP versus length of regenerator (ϕ=0.64, P=17.5bar, and f=7.5 Hz).Figure 13
Variation of different losses versus length of regenerator (ϕ=0.64, P=17.5bar, and f=7.5 Hz).Figure 14
Cooling power and COP versus regenerator porosity (P=17.5bar and f=7.5 Hz).Figure 15
Variation of fluid friction and regenerator imperfection losses versus regenerator porosity (P=17.5bar and f=7.5 Hz).Figure12 shows the performance of a Stirling refrigerator (regenerator porosity ϕ=0.64, P=17.5bar, and f=7.5 Hz) for a range of regenerator length. As shown in Figure 12, parabolic evolutions of cooling power and COP have been demonstrated as a function of regenerator length with peak values found nearly at the same point (25 mm). It could be stated that both cooling power and COP show a decreasing trend as the regenerator length increases after the peak value. This phenomenon results from decreasing of most of the heat losses and increasing of fluid friction loss with regenerator length as confirmed in Figure 13. From Figure 13, it could be understood that regenerator imperfection loss decreases and fluid friction loss increases at higher rate than other losses. As the length of the regenerator increases, fluid friction loss will increase which will result in lower performance of the Stirling refrigerator.Figure14 shows the impact of regenerator porosity on the cooling performance of the Stirling refrigerator. It could be seen that the cooling power of the refrigerator decreases with an increase in porosity. This is because while the porosity of the regenerator increases, the heat transfer area of the matrix and the amount the heat transfer decrease. The working gas goes into the heater at a higher temperature. On the other hand, due to a higher pressure drop, a low porosity regenerator leads to higher fluid friction losses. The curve shows that the peak COP value is found at a porosity of 0.75 which is similar to the research conducted by [40] on Stirling engine. This result is mainly accompanied from fluid friction and regenerator imperfection losses as shown in Figure 15. The effect of other losses such as conduction heat loss are less significant for such moderate temperature applications. Figure 15 also demonstrates that the decreasing of fluid friction with an increase in regenerator porosity leads to an increase in COP. On the other hand, regenerator imperfection loss increases at higher rate than the decreasing rate of fluid friction loss just after a porosity value of 0.75 that results in an optimum COP. Hence, the optimum design point in case of regenerator porosity could be at a compromise between a higher cooling power and peak COP.


## 4.3.3. Effect of Displacer Height

Stirling machine designs utilize displacers as a means of transferring the working fluid between the cold and hot regions. The height of the displacer affects the performance of the Stirling machine by changing the compression ratio as the swept volume of compression space varies with displacer height [37]. Figure 16 presented the impact of displacer height on cooling performance (cooling power and COP) using air as a working fluid at a charging pressure of 17.5 bar and an operating frequency of 7.5 Hz. In this analysis, the height of the displacer is allowed to vary, still keeping the total length (height plus length of displacer rod) of the displacer fixed. As displacer height increases, keeping the total length fixed, the compression volume decreases that results in a lower swept volume compression ratio (Vr=Vswc/Vswe).Figure 16
Cooling power and COP versus displacer height (P=17.5bar and f=7.5 Hz).It has been noticed from Figure16 that cooling power decreases monotonously with an increase in displacer height and there exists a peak value of COP. The peak COP value is found as 0.72 at a displacer height of 10 cm with the associated cooling power of 544.4 W and a swept volume compression ratio of 1.58. Hence, with the same operating condition and keeping all other design parameters constant, the peak cold production and COP could be increased by 61.6% and 13.3%, respectively, only through displacer height optimization. The main reason for such improvement in cooling performance is the increase of swept volume compression ratio.


## 4.3.4. Effect of Swept Volume Ratio

Figure17 demonstrates the effect of the ratio of swept volume on cooling power and COP of the Stirling refrigerator at charging pressure of 17.5 bar and an operating frequency of 7.5 Hz. It can be seen from Figure 17 that the peak values of cooling power and COP are found as 383.6 W and 0.7, respectively. These peak values are found at a swept volume ratio of 1.7. Hence, with the same operating condition and only through swept volume ratio optimization, the peak cold production and COP could be improved by 11.8% and 10.9%, respectively. The analysis is conducted by keeping the total volume constant and still varying the two swept volumes through changing strokes of the displacer and power piston.Figure 17
Cooling power and COP versus swept volume ratio (P=17.5bar and f=7.5 Hz).Both Figures16 and 17 show the effect of volume ratio on the performance of the Stirling refrigerator keeping the overall cylinder volume fixed. However, in Figure 16, due to the variation of displacer height, the swept volume of compression space varies keeping the swept volume of expansion space fixed, whereas in Figure 17, both compression and expansion spaces vary keeping the summation of the two swept volumes still fixed. Figure 18 illustrates the cooling power and COP of the Stirling refrigerator as a function of swept volume ratio for the comparison of these two cases. It is seen in Figure 18 that a better COP could be achieved when the volume ratio variation is due to the variation of compression space swept volume through displacer height variation. From this figure, it could also be concluded that the optimum cooling performance for the domestic Stirling refrigerator can be found for the swept volume ratio ranging from 1.4 to 1.8.Figure 18
COP comparison versus swept volume ratio for different cases of volume ratio (P=17.5bar and f=7.5 Hz).


## 4.3.5. Effect of Ratio of Diameter to Stroke of Piston (dP/s)

Figure19 illustrates the cooling power and COP versus the ratio of piston diameter to stroke keeping the piston swept volume constant. It can be seen that both cooling power and COP increase in a wide dome-shaped trend with an increase in piston diameter to stroke ratio (dP/s), and similar trend has been presented by [41]. This phenomenon is mainly due to the variation of different power losses with the (dP/s) ratio as can be seen in Figure 20. It can be seen from Figure 20that the total power loss decreases mainly due to the decreasing of mechanical friction and piston finite speed losses as the stroke of piston decreases with an increase in dP/s ratio. On other hand, even though conduction heat loss increases, since shuttle heat losses decreases (see Figure 20(b)), cooling power increases in a wider dome shape as the ratio (dP/s) increases. This is because the shuttle heat loss has more effect than conduction heat loss for such moderate temperature application. As the ratio (dP/s) increases from 1.2 to 2.6, the cooling power and COP enhanced by 17.3% and 32%, respectively.Figure 19
Cooling power and COP variation with ratio between diameter and stroke of piston (P=17.5bar and f=7.5 Hz).Figure 20
Power loss variation with the ratio between diameter and stroke of piston (P=17.5bar and f=7.5 Hz).
(a)
Losses with diameter to stroke ratio(b)
Losses with diameter to stroke ratio


## 4.4. Effect of Combined Optimized Parameters

In the previous sections of this paper, we use univariant analysis in which only one parameter is investigated at a time while all other parameters are retained to prefixed values. This approach is simplistic and allows us to assess the most influential parameters as well as the exact effect of each parameter. However, by using only univariant optimization, it would be difficult to find the best combination of parameters for optimum machine performance. Hence, for the second-order numerical model, after investigating a univariant optimization, a trade-off between multiple conflicting variables is indispensable.Therefore, in this section, we try to investigate parameters with higher impact as presented in Sections4.2 and 4.3 and then combine them to find an optimized combination. Different combinations of parameters which have positive impact on cooling performance have been first selected.Regarding operating parameter optimization, it can be seen from the optimized results presented in Section4.2 that relatively the lower operating frequency (3 to 8 Hz) at higher charging pressure is found to be the better operating parameters. But, due to the restricted operating conditions of the machine for safe working conditions and to minimize manufacturing cost, the average proposed operating frequency and charging pressure are set as 6 Hz and 22.5 bar, respectively.The major design parameters considered for the analysis of combined parameters are regenerator length, porosity, displacer height, swept volume ratio, and diameter to stroke ratio of the piston. The different combinations of these parameters were simulated, and lastly, a combination of parameters with more enhanced cooling performance has been proposed as presented in Table5.Table 5
Specifications of the Stirling refrigerator for optimized analysis.
No.ParametersExisting value [38]Optimized value1Hot sink temperature (K)3053052Cold source temperature (K)2702703Diameter of piston (mm)60624Diameter of displacer (mm)59615Stroke of piston (mm)40376Regenerator diameter (mm)82827Regenerator length (mm)50408Wire diameter (μm)1121129Porosity (%)647210Compression space swept volume (cm3)10314011Expansion space swept volume (cm3)11311312Compression dead volume (cm3)4.244.2413Expansion dead volume (cm3)4.244.2414Working gasAirAir15Frequency (Hz)5-133-816Charging pressure (bar)15-2020-25The results of cooling performance for combined parametric optimization have been compared with the analysis results using existing parameters as demonstrated in Figure21. It could be seen from the figure that the performance of the Stirling refrigerator has been improved significantly with optimized parameters. It is found that all optimized design parameters that are quite influential as individual parameters could enhance the COP of the refrigerator from 7% to 19% from the existing design value. Collectively, the optimized parameters (operating and specified design parameters) could deliver from 48% to 60% more cold production and from 30% to 40% more COP than the existing design in a safe working condition.Figure 21
Comparison of results of existing and optimized cooling performance versus operating frequency.


## 5. Conclusion

The numerical model developed by the authors’ previous work [38] could predict the performance of domestic Stirling refrigerator with a very reasonable accuracy. However, a higher performance Stirling refrigerator could be designed using the developed model if parametric optimization is conducted further. This paper is aimed at analyzing the effects of different operational variables, design (geometric) variables, shuttle heat losses, and mass leakage losses to the buffer space might have on the cooling performance of the Stirling refrigerator. For this purpose, the numerical model called modified simple model which has been developed so far is used to investigate the effects of individual parameters keeping other parameters fixed at design condition. Finally, through combining different optimized parameters, new combinations of parameters have been proposed which could produce enhanced cooling performances of the Stirling refrigerator. Some of the main findings of this study are summarized below.
(i)
Shuttle heat losses have major effect on the cooling performance through affecting the operating condition of the Stirling refrigerator. Hence, to design the Stirling machines, losses that a have direct effect on working condition of working fluid shall be directly incorporated with the differential equations(ii)
The effects of each parameter on cooling performance and on different types of losses have been investigated by using univariant optimization technique(iii)
The cold production and power demand increase with the operating frequency. The COP has a peak value as a function of operating frequency. On the other hand, COP increases with charging pressure. Furthermore, the higher the pressure, the lower the operating frequency in which the peak COP is found(iv)
Cooling power has peak value at a single phase angle irrespective of operating frequencies, whereas the phase angle for the peak COP increases slightly with a decrease in an operating frequency(v)
Shorter displacers with relatively higher compression ratios show better performance for low-temperature differential domestic Stirling refrigerators(vi)
Using combined parametric optimization, a higher performance Stirling refrigerator could be designed. In this research, optimized design parameters that could enhance the overall performance of the existing machine from 30% to 60% are proposed(vii)
The optimized design for such domestic Stirling refrigeration could have a COP of 1.3 and cooling power of 625 W at a cooling temperature of -4°C, charging pressure of 22.5 bar, and operating frequency of 6 Hz



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