Development of a 100-K pneumatically driven split-type cryogenic Stirling cryocooler based on experimental and numerical study

Cryogenics 105 (2020) 102998

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Development of a 100-K pneumatically driven split-type cryogenic Stirling cryocooler based on experimental and numerical study

T

Chin-Hsiang Cheng , Jhen-Syuan Huang ⁎

Department of Aeronautics and Astronautics, National Cheng Kung University, No. 1, University Road, Tainan 70101, Taiwan

ARTICLE INFO

ABSTRACT

Keywords: Split type Stirling cooler Thermodynamic model Experiment

This study is aimed at the development of a 100-K class pneumatically driven split-type cryogenic Stirling cryocooler. The pneumatically driven displacer is floating in the expander and is supported dynamically by the gas enclosed in a bounce space. An efficient theoretical model combining the dynamic and thermodynamic analysis is developed to simulate the transient behavior of the cooler in the starting period by improving the weakness of the existing models. In parallel, a prototype cooler is built to validate the theoretical model. Experimental measurements of cold head temperature, cooling load and coefficient of performance are conducted. A close agreement between the numerical and the experimental data is found. Results show that as the charged pressure is 5 bar and the rotation speed is 2000 rpm, the zero-load temperature of the cold head of the developed cryocooler can reach 103 K in 15 min. Effects of rotation speed on the phase angle, heat absorption rate, displacer stroke and power input are predicted, and the optimal operating ranges of the rotation speed at different charged pressures are presented.

1. Introduction In the past several decades, Stirling cooler has played an important role in cryogenic applications of aeronautics and astronautics, national defense, optical sensor and medicine sectors because of its compact configuration [1]. At a cooling load below 10 W, Stirling cooler gives the best relative Carnot efficiency (~20%) compared with other types of cryocoolers. In addition, the lifetime has been raised up to 10 years owning to the improvement of the compressor [2]. Besides, Stirling cooler also features the flexibility of working gases in use, that can be air, nitrogen, helium, or hydrogen [3]. One of the most important applications of Stirling cooler is in IR system. The IR system has a cooled thermal imager equipped with an infrared detector, for example, Mercury Cadmium Telluride (MCT) and Indium Antimonide (InSb) [4]. For lower thermal noise, the detector is operated at cryogenic temperature ranging from 70 to 80 K. Compared with uncooled ones, the cooled imager leads to better performance in temperature sensitivity, measuring distance, spatial resolution and signal synchronization. As to the theoretical modeling, in a first-order analysis, typically an isothermal hypothesis is utilized to calculate the cyclic variation in pressure and mass. Walker derived a closed-form solution for the cooling load of a Stirling refrigerator with multiple expansions [5]. Finkelstein modified the traditional isothermal analysis by introducing

⁎

a concept of finite heat transfer [6]. Urieli and Berchowitz included the concepts of effectiveness of the regenerator and considered the friction loss of the working fluid [7]. Ataer and Karabulut proposed a thermodynamic analysis of a V-type Stirling cooler [8]. The working space was divided into 14 control volumes while most nodes were placed within the regenerator. Razani et al. took irreversibility in the Stirling refrigerator into consideration [9]. Analytical expressions of exergetic efficiency and normalized cooling capacity were carried out in terms of pressure ratio across regenerator, regenerator effectiveness, and phase angle between pressure and mass flow. A commercial software SAGE [10] was also applied to the development of a Stirling cryogenic cooler by Veprik, et al. [11]. In this study, authors conducted experiments to validate the numerical predictions, and found greater deviation in the higher power regime. More recently, Cheng, Huang and Yang [12] developed a 90-K beta-type Stirling cooler with rhombic-drive mechanism. The authors presented a thermodynamic model in which the pressure drop in the regenerator was taken into consideration. Performance measurements under different heat loading, charged pressure, and operating speed for verifying present model were also conducted. In these above-mentioned theoretical studies, experimental verification was always important. In a split-type Stirling cooler, the compressor and expander chambers are separated, and in between the two chambers there is a

Corresponding author. E-mail address: [email protected] (C.-H. Cheng).

https://doi.org/10.1016/j.cryogenics.2019.102998 Received 21 April 2019; Received in revised form 23 October 2019; Accepted 15 November 2019 Available online 22 November 2019 0011-2275/ © 2019 Elsevier Ltd. All rights reserved.

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

Nomenclature

T Ta V Win Y

Variables A cross section area (m2) COP coefficient of performance cd damping (N s m−1) cp constant-pressure specific heat (J kg−1 K−1) D diameter (m) f friction factor F force (N) g gravitational acceleration (m s−2) G gap between displacer and cold cylinder (m) kf thermal conductivity of gas (W m−1 K−1) L length (m) m mass (kg) Nr number of sections within regenerator mesh number Nrm Nu Nusselt number P pressure (Pa) Pch charged pressure (bar) Pr Prandtl number Q heat (J) Qcond , Qpu , Qsh conduction loss, pumping loss, shuttle loss (W) Qload cooling load (W) r eccentric radius of crank (m) R thermal resistance (K W−1) gas constant (J kg−1 K−1) Rconst Re Reynolds number Sd stroke of displacer (mm)

temperature (K) ambient temperature (K) volume (m3) indicated power input (W) position (m)

Greek symbols α ε φ θ Ω ζ

phase angle (rad) fluctuation of thermal resistance porosity crank angle (rad) rotation speed (rpm) coefficient of spring constant

Subscripts b c d e ey ew h j p r rm t tol

bounce chamber compression chamber displacer expansion chamber expansion cylinder cold head hot chamber index of nodal points within regenerator piston regenerator chamber wire mesh tube chamber total chamber

connecting tube. It is well anticipated that critical values of cooling load, cooling temperature and COP will appear at a certain operating frequency since the vibrating behavior of the displacer is dependent on the pulsating frequency of the driving motor. An experimental study on the phase angle between piston and displacer is performed in [13]. Normalized cooling load reached its maximum in the range of 0.75–0.85 in frequency ratio while the natural frequency of the oscillation of the displacer was set to be 63.0, 67.4 and 74.0 Hz. Wang et al. designed a split cooler which was able to be operated above 100 Hz, and the authors found that optimum frequency for COP under different damping coefficients is around 100 Hz [14]. Li and Grosu studied a Stirling cryocooler by an isothermal model, and they found that the optimum phase angle for COP is predicted to be between 85° and 90° [15]. In this study, a prototype of the split-type Stirling cooler with a pneumatically driven displacer is developed by using numerical and experimental methods. Pressure oscillation is generated in the compressor consisting of a crank mechanism and a rotary motor. Crank driven compressor has merit of precise positioning and low cost of manufacturing. The displacer in the expander is constrained pneumatically by a gas (or say, gas spring) enclosed in a bounce space. Dynamic behavior of the floating displacer depends on the pressure forces induced by the working gas. A number of computation nodes are placed in the regenerator to increase the accuracy of heat transfer between working fluid and solid matrix. The rotation speeds which bring about a lowest cooling temperature at different charged pressures are acquired in both numerical simulations and experiments.

(1) For the thermodynamic model of a Stirling cooler proposed in [12], mass flow rate through the regenerator was evaluated simply based on the velocity of the displacer, and the fluid mass contained in the regenerator chamber is assumed to be constant. This assumption is deviated from the reality. In this study, a differential form of the ideal-gas equation of state is derived. Based on the differential form, mass change in each chamber can be expressed in terms of the changes in pressure and volume of the chamber, so as to remove the above assumption of [12]. (2) In the existing models [5–15], the cold head temperature of the cryocooler is prescribed with a constant value. However, this is not practical since in the starting period, the cold head temperature is varying from ambient temperature toward a stable, lowest temperature and hence, one cannot prescribe the cold head temperature with a constant value in simulation. To improve the weakness of the existing models, the present study is aimed to investigate transient behavior of the cryocooler in the starting period by incorporating a dynamic model with the thermodynamic model. In the model, the cold head temperature is not necessarily to be prescribed as a constant value. (3) Thermal resistances of heat transfer between walls in the expansion and the compression chambers and the working fluids were typically approximated with constant values. However, as the moving parts are moving back-and-forth in the cylinder, the surface areas on the walls for heat transfer change periodically. This implies that the thermal resistances should be varied periodically with time. In this analysis, the effect of variable thermal resistances is taken into consideration.

2. Theoretical model

Fig. 1 shows the schematic of a split-type Stirling cooler which consists of compression chamber, connecting tube, hot chamber, regenerator chamber and expansion chamber. The compressor and the expander relate to each other by using a tube. At the bottom of the

Compared with the existing studies [5–15], the present theoretical model features the following novelties: 2

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

subject to a specified cooling load (Qload ). Volume variations of the compression, hot and expansion chambers are calculated in terms of the displacements of piston and displacer while no boundary deformation occurs in tube and regenerator chamber. Pressure in the cylinder is firstly calculated by

De

Cold head Expansion chamber Regenerator chamber G

Ld

Vc Tc

Ley

Yp

Dc

θ

Db

+

Vh Th

+

Nr

Vr , j

j=1

+

Tr , j

Ve Te

(3)

(4)

dm = (PdV + VdP ) RT

Lb

The above equation can be used to determine magnitude and direction of the gas velocity on the boundary between chambers. In this study, it is assumed that pressure drops exist only in the axial direction within the tube and the regenerator chambers. Pressure of the hot chamber calculated in Eq. (3) is assumed to be the average value in the cooler, and hence, pressures of other chambers will be modified with the pressure drop. Flow in the tube chamber is treated as fully-developed state, and friction factor is acquired from Moody diagram. Minor losses result from sudden concentration and expansion at the two ends are taken into consideration as well. For regenerator consisting of stainless steel wire mesh, the friction factor correlation proposed by [16] is used herein as

Bounce Compression Tube chamber chamber chamber Fig. 1. Schematic diagram of split-type Stirling cooler.

expander exists an isolated bounce chamber which is functioned as a gas spring. A pneumatically driven displacer is floating in the expander and is supported dynamically by the bounce space. The regenerator installed inside the displacer moves with the regenerator in the expander under the influence of gravity and gas pressures in the expansion, hot, and bounce chambers.

fr , j = 129 Rer , j + 2.91Rer ,0.103 j

2.1. Dynamic analysis

=1

where md and cd are mass and damping of the displacer, and Fd is the total force exerted on the displacer, which is determined with

Pe Ae

md g

(6)

Nrm Drm 4

where Nrm and Drm are mesh number and wire diameter. The empirical correlation of Nusselt number as a function of Reynolds and Prandtl numbers was proposed by [17] for the stacked wire mesh as

(1)

md Y¨d + cd Yd = Fd

(5)

where Rer,j is Reynolds number which is evaluated based on hydraulic diameter of wire mesh, and porosity of the mesh. Porosity of stacked wire mesh is derived from its geometrical pattern as

Note that the oscillation of the piston can be prescribed by the given rotation speed and geometrical parameters of the parts of the cooler, whereas the oscillation of the displacer must be determined by a dynamic analysis based on equation of motion.

Fd = Pb Ab + Ph Ah

+

Vt Tt

By differentiating the above equation, pressure change in a time step, dP, is determined. Once the pressure change is obtained, mass change in each chamber can be estimated by the ideal-gas equation of state in terms of the changes in pressure and volume of the chamber as

Yd

Hot chamber

r

mtol Rconst

P=

0.33 Nur , j = 0.68Rer0.6 , j Pr

(2)

(7)

Prandtl number is assigned to be 0.67 as helium is used as working gas. Convective heat transfer coefficient can then be calculated from the above equation. It is noted that the dependence of dynamic viscosity and thermal conductivity on the pressure and temperature are included in this model based on the equations presented by [18]. Temperature variation of the working gas in each node can be updated with the help of energy equation. As the piston and the displacer are moving, the heat transfer areas as well as the thermal resistances in the control volumes are periodically varied with time. To include this effect, the thermal resistances of heat transfer in the chambers are modeled as function of the crank angle as

The first term in the above equation represents the pneumatic force that results from the gas pressure in the bounce chamber, in which Pb is calculated in terms of Vb with an isothermal process assumption, and ζ is an empirical coefficient used to represent the effect of the gas leakage into/from the bounce chamber. In this study, coefficient of spring 2 constant is fitted as = 1.10 0.0169Pch + 0.00105Pch (Pch in bar), which is obtained based on a comparison between the experimental and numerical data. A gravitation force exists as the displacer is assumed to be placed vertically. 2.2. Thermodynamic analysis In Fig. 2, a nodal diagram showing the control volumes is used to analyze the time-dependent thermodynamic process. It is expected that there exists a large temperature gradient in axial direction in the regenerator and a great amount of heat transfer takes place between working gas and regenerative porous material. The regenerator is divided into 20 sections (Nr = 20), and a computation node is placed in each section of the regenerator. By adding the number of sections and nodes, the accuracy of predictions of heat transfer between working fluid and solid matrix can be increased. Wall temperatures of the compression, tube and hot chambers are fixed at ambient temperature. On the other hand, wall of the expansion chamber is assumed be thermally insulated from the surrounding, whereas the cold head wall is

R c = Rc (1 + sin )

(8)

Rh = Rh (1

(9)

sin( + ))

Compression Tube Vc , Pc mc , Tc

.

Wc

.

Qc

.

Pt mt , Tt

mct

.

Qt

Hot

Regenerator (j=1~Nr) Expansion Cold head

Vh , Ph m.h , Th mth

.

Wh

.

Qh

.

mhr

Pr,j mr,j , Tr,j

. .

mrr,j

.

Qr,j

Qrm,j

mrm,j , Trm,j

Ve , Pe me , Te

.

mre

Fig. 2. Thermodynamic model of the cooler. 3

.

We

. Tew

Qe

.

Qload

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

(10)

R e = Re (1 + sin( + ))

Power input of the cycle (Win ) can be calculated by taking the absolute value of sum of the indicated power by the working gas. When a cooling load (Qload ) is applied to the cold head, coefficient of performance (COP) can be determined to represent the refrigeration performance of the cooler, which is written as

where ε represents the fluctuation of the thermal resistance due to the change in heat transfer area. In this study, values of ε, Rc , Rh and Re are selected to be 0.2, 0.5 K/W, 0.9 K/W and 0.9 K/W based on the experimental comparison. It is noted that thermal resistance in compression chamber is smaller due to the fin heat sink formed on the external surface of the compression chamber. In addition, the cold head placed on the top of the expansion chamber is assumed to be a lump-capacity object so that its temperature can be predicted based on energy balance as

mew d (cew Tew ) =

COP =

The cold head temperature starts from the ambient temperature and gradually descends to a very low temperature after a period of time. The low temperature reached is an index of the performance of the cooler. There are three thermal losses of heat transfer considered in the present analysis: (1) shuttle loss, (2) pumping loss and (3) conduction loss. Shuttle and pumping losses take place in the gap between the displacer and the cold cylinder. Empirical equations are expressed as in [19]:

Qsh =

Qpu =

0.65kf Dd Sd2 (Th

( Dd )0.6Ld (cp

3. Prototype development and experimental setup A prototype of the pneumatically driven split-type cryogenic Stirling cryocooler is built in parallel. A gas pipe is used to connect the compressor and the expander chambers. Piston and compression cylinders are made of aluminum alloy and coated with Teflon in order to reduce friction. Also, the clearance between piston and cylinder is controlled at a few micrometers to avoid the leakage of working gas. It is also noted that in this study the charge pressure is ranged from 1 to 5 bars only in experiments so as to avoid significant gas leakage. Stainless steel with low thermal conductivity is chosen as the material of the expansion cylinder. The wall thickness of the cylinder is decreased to only 0.3 mm, which can reduce the conduction loss in the axial direction to a certain extent. Displacer is made of epoxy resin filled with glass fiber because of its light weight and good strength. As an essential component, regenerator is made of porous mediums that provide enough heat transfer area. Stainless steel wire mesh which has been proven efficient for the

Te ) (12)

5.4GLd

|Ph

Pe |)1.6 (Th

(

Te ) G 2.6

)

Th + Te 1.6 1.6 1.5k f0.6 Rconst 2

(13)

On the other hand, conduction loss takes place within the wall of the cold cylinder in axial direction, and is calculated by

Qcond = key Aey

Ta Tew Ley Lb

(15)

Fig. 3 shows the flow chart of the computation. The typical size of the time step is assigned to be 10−5 s. At each time step, iteration of the coupled physical quantities is continued till the convergence criterion is satisfied. The convergence criterion is defined with the relative error of 10−12. The computation is marched to the next time step consecutively. As the steady operation regime is reached, the computation is terminated.

(11)

Qe + Qcond + Qload

Qload Win

(14)

START

END

Geometrical parameters and initial conditions

YES NO

t i+1 =t i +Δt i+1

Steady operation

i

θ =θ +dθ

Dynamic analysis of displacer, Ydi

1

Heat transfer and work, Q,W YES

Convergence

Volume and pressure variation i 1 c

i 1 h

T

i 1 e

dV , dV , dV , dP

new

T old / T old

10

12

NO

Temperature

Mass flow rate and pressure drops

i 1 c

T , Tt , Thi 1 , Tri, j1 , Trmi ,1j , Tei 1 , Tewi 1 Fig. 3. Flow chart of computation. 4

i 1

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

Table 1 Specifications of the present cryocooler.

Table 2 Operating parameters for the baseline case.

Parameter

Value

Operating parameter

Value

r (m) Dc (m) De (m) Db (m) Ley (m) Ld (m) Lb (m) G (m) Vt (m3) Vr (m3) Porosity of regenerator, φ Compression ratio, Г

0.005 0.03 0.012 0.006 0.1185 0.07 0.033 0.0002 3.927 × 10−7 3.713 × 10−6 0.675 2.135

Charged pressure, Pch (bar) Rotation speed, Ω (rpm) Ambient temperature, Ta (K) Mass of the displacer, md (kg) Initial position of displacer, Yd0 (m) Damping, cd (N s m−1)

1 1000 300 0.03 0.113 5.5 0.

Cooling load, Qload (W)

for zero-load tests. As a cooling load is applied, the input heat transfer rate is generated by a polyimide film electric heater which is regulated by a DC power supply. Temperature of the cold head is measured by Ttype thermocouple. While the Stirling cryocooler is operating, data of cold head temperature, torque exerted on cooler, rotation speed and cooling load are monitored and recorded for further analysis with a computer-based data-acquisition unit. Based on the data of torque and rotation speed, one can calculate input power for the cooler.

applications whose cooling temperatures are above 30 K. Therefore, this material is utilized to make the regenerator in the present study. Concentricity between the displacer and the expansion cylinder is maintained by a linear bushing. Major geometrical dimensions of the present cooler are shown in Table 1. For measuring performance of the prototype, an experimental apparatus is built, which is displayed in Fig. 4. A DC brushless motor serves as the power source to drive the Stirling cooler. Both the brushless motor and the cryocooler are fastened at the platform. The shafts of the brushless motor and the cooler are precisely aligned. Besides, a rotary transducer is installed to measure the torque exerted on the cooler and the rotation speed as well. Helium is used as the working gas due to its high thermal conductivity and specific heat. The working chambers in the cooler are highly vacuumed, and then pure helium is charged into the cooler to a required operating pressure. A polyimide film electric heater is applied to generate cooling load to the cold head of the cooler for testing. The cold head cylinder is thermally insulated

A parametric study is performed to investigate the effects of the influential parameters on the performance of the cooler. A baseline case is specified as a reference case for comparison in the parametric study. Operating parameters for the baseline case are listed in Table 2. In the baseline case, the cold head is thermally insulated from the surrounding to acquire a zero-load test (Qload =0). Fig. 5 shows the numerical predictions of transient variations in cyclic-average cold head temperature (Tew), position of displacer (Y¯d ), indicated power input (Win ) and heat absorption rate by working gas (Qe ) with the baseline case. As Stirling cooler is started from the ambient temperature, thermodynamic cycle is initiated immediately due to the movement of piston and displacer. For this case, the indicated power input and heat absorption rate are 0.407 W and 0.149 W, respectively. It is seen in Fig. 4 that for this particular case, it takes approximately 1000 s for the cold head to reach a steady cyclic-average cold head temperature of 223 K. The equilibrium position of the displacer is slightly moved from 0.1117 to 0.1122 m. Fig. 6 shows variations in the positions of the piston and the displacer (Yp and Yd) with the baseline case. The stroke of the piston is assigned to be 10 mm, whereas that of the displacer is predicted by the equation of motion, Eq. (1). It is found in this figure that the stroke of the displacer reaches 4.17 mm. Both the displacer and the piston are

Polyimide film Thermocouple

Insulation

Data acquisition Computer

Torque sensor

DC motor

Yd (m)

Controller

Vacuum pump

_

Tew (K)

0.113

300

0.112

250

. Win . Qe

Yd Tew

Helium tank

1

0.8

(a) Schematic of experimental system.

0.6 0.111

Win , Qe (W)

Power supply

4. Results and discussion

.

200

0.4 . 0.11

150

0.109

100

0.2

0

500

1000

1500

2000

2500

0 3000

Time (s) Fig. 5. Numerical predictions of transient variations in cyclic-average cold head temperature, position of displacer, indicated power input and heat absorption rate with the baseline case.

(b) Photos of prototype cooler (left) and experimental system (right).

Fig. 4. Experimental system. 5

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

oscillating at the same frequency; however, a phase angle (α) is formed between the oscillations of the displacer and the piston. The displacer leads the piston by a phase angle of 84°. It is noted that for this case the curves are plotted with 0.2 s as time exceeds 3000 s. Fig. 7 displays temperature distribution within the regenerator at different crank angles. A temperature difference of roughly 65 K between the two ends of the regenerator is established during a complete cycle. It is found that the temperature profile within the regenerator is obviously different from the linear temperature distribution assumed in [12]. In addition, the temperature profile is actually periodically changed in a cycle owning to the mass and energy transfers with the connected chambers. When the operating condition is altered, the behavior of the cooler may be rather different from the baseline case. Experimental and numerical results of the cold head temperature under different charged pressures and rotation speeds are displayed in Fig. 8. It is observed that at 1 bar and 1200 rpm, the steady cold head temperature is measured to be 240 K in experiments, whereas with numerical prediction it is 225 K. At 3 bar and 1600 rpm, the lowest cold head temperature is numerically predicted to be 150 K, and the experimental temperature reading is 132 K. As the charged pressure and rotation speed are increased to 5 bar and 2000 rpm, the cold head temperature of the prototype can reach 103 K in merely 15 min. Basically, the comparison between the numerical and experimental data shows a close agreement. In theory, natural frequency of the displacer increases with the spring constant, which is also proportional to the charged pressure. Therefore, the rotation speed of the split-type Stirling cryocooler should be adjusted to an optimal value, which is compatible with the natural frequency at a specific charged pressure, in order to establish a better phase angle. Fig. 9 provides the comparison between the experimental and the numerical data in the effects of the rotation speed on the steady coldhead temperature, at 1-bar, 3-bar and 5-bar charged pressures. In the experiments, the rotation speed is varied within 800 and 2000 rpm. On the other hand, the numerical predictions are fitted with third-order polynomial functions. A discrepancy of approximately 15 K exists between the two set of data. Nonetheless, the trends of the curves are rather similar. The rotation speed exhibits a subtle influence on the cold head temperature, and one can see that at a specific rotation speed the cold head reaches a minimum temperature. For the case at 1-bar charged pressure, the rotation speed leading to the minimum temperatures is 1200 rpm in experiment, and 1100 rpm in numerical prediction. For 3-bar charged pressure, the minimum temperature occurs at 1600 rpm in experiments and 1800 rpm in numerical prediction. Even though the discrepancy between the experimental and the numerical data is not remarkable, further investigation of improving the accuracy of the model is still necessary. Fig. 10 shows the numerical predictions of the effects of the rotation speed on the behavior of the cryocooler at different charged pressures. Fig. 10(a) displays the dependence of heat absorption rate on the rotation speed. It is noted that there exist optimal rotation speeds leading to the maximum heat absorption rates. The optimal rotation speed is predicted to be 1100, 1800, 2100, 2500 and 3300 rpm for 1, 3, 5, 7 and 9 bar, respectively. The split-type Stirling cooler may be analogous to a forced-vibration system, and hence, dynamic behavior of the displacer must be influenced by the rotation speed caused by motor. Once the rotation speed is changed, the thermodynamic process will be also brought into a new equilibrium state, resulting in different heat absorption rate. Fig. 10(b) illustrates the effect of the rotation speed on the phase angle of the displacement. Typically, the phase angle of Stirling cryocooler is assumed to be 90° in a first-order analysis. However, it is found that the phase angle corresponding to the optimal rotation speed decreases from 76° at 1 bar to 52° at 9 bar in this figure. Due to the existence of pressure drop, finite heat transfer and irreversible heat losses, the phase angle for maximum heat absorption is greatly altered by the charged pressure. Representing the dynamic behavior of the cryocooler, the oscillation

0.14

Yd

0.13

Yp

Yd , Yp (m)

0.12 0.11 0.1 0.09 0.08 0.07 3000

3000.05

3000.1

3000.15

Time (s)

3000.2

Fig. 6. Positions of displacer and piston with the baseline case. The curves are plotted within 0.2 s as time exceeds 3000 s.

310 300 290

Tr, j (K)

280 270 260

=0

250

= /2 = = 3 /2

240 230 220

0

5

10

15

20

j Fig. 7. Temperature distribution within regenerator at different crank angles.

500 450

Operating conditions 1 bar,1200 rpm

400

3 bar,1600 rpm

350

Tew (K)

Exp.Num.

5 bar,2000 rpm

300 250 200 150 100 50

0

10

20

30

Time(min.)

40

50

Fig. 8. Experimental and numerical results for the cold head. temperature for different operating conditions. 6

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

350

350

1 Exp.

Pch

Exp.

1 bar

250

3 bar

Tew (K)

200 150

0.8

COP

250

5 bar

Tew (K)

Tew

300

Num.

0.6

200

COP

300

Num.

150

0.4

100 100

0.2

50 50 0

0 0

1000

2000

3000

4000

Fig. 11. Cold head temperature and COP versus cooling load at charged pressure of 3 bar and rotation speed of 1600 rpm.

heat absorption is reduced by the shuttle loss at large displacer stroke. In Fig. 10(c), the displacer stroke at the optimal rotation speed is changed from 3.7 to 7.3 mm as the charged pressure is elevated from 1 to 9 bar. Based on the parametric analysis, one can figure out that only a certain range of the operating rotation speed is suitable for split-type

stroke of the displacer at different rotation speeds are plotted in Fig. 10(c). Larger stroke causes higher variation in volume of the expansion chamber to increase the cooling load. However, Eq. (12) indicates that the shuttle loss is proportional to the square of stroke. The

3 bar 5 bar 7 bar 9 bar

1 bar 3 bar 5 bar 7 bar 9 bar

14 12

Sd (mm)

Qe (W) .

16

1 bar

0.4

0

1

.

Qload (W)

Fig. 9. Experimental and numerical data of cold head temperature versus rotation speed at various pressures.

0.5

0.5

5000

(rpm)

0.6

0

0.3

10 8 6

0.2

4 0.1 500

1000

1500

2000

2500

3000

3500

4000

2 500

4500

2000

2500

3000

3500

4000

(a) Heat absorption rate versus rotation speed.

(c) Stroke of displacer versus rotation speed.

110 100

1 bar 3 bar 5 bar 7 bar 9 bar

70 60 50

Win (W)

90

4500

80

1 bar 3 bar 5 bar 7 bar 9 bar

120

(degree)

1500

(rpm)

130

80

.

70 60

40 30 20

50

10

40 30 500

1000

(rpm)

0 1000

1500

2000

2500

3000

3500

4000

4500

500

(rpm)

1000

1500

2000

2500

3000

3500

4000

4500

(rpm)

(b) Phase angle versus rotation speed.

(d) Indicated power input versus rotation speed.

Fig. 10. Numerical predictions of effects of rotation speed at different charged pressures. 7

Cryogenics 105 (2020) 102998

C.-H. Cheng and J.-S. Huang

Stirling cryocooler and the range is dependent on the charged pressure. Fig. 10(d) shows the indicated power input versus the rotation speed. As the charged pressure and the speed are increased, the power input required for operation grows rapidly due to the larger pressure oscillation in the cylinder and higher frequency. As a result, the cold head temperature is lowered while the required indicated power is increased significantly. As the cooling load is applied to the cold head, temperature of the cold head is affected, as already described in Eq. (11). Fig. 11 conveys the cold head temperature and COP versus cooling load at 3 bar and 1600 rpm. Cooling load of 0.1 W, 0.5 W and 1 W are applied on the cold head by using the polyimide film electric heater in the experiment. It is observed that the numerically predicted relation between the cooling load and the cold head temperature is roughly linear. At a lower cold head temperature, the agreement between experimental and numerical data is relatively close. For example, at 0.5-W cooling load, the cold head temperature is found to be 172 K in experiment and 169 K in numerical prediction. That is, a deviation of only 1.7% between the two sets of data is noted. However, as the cooling load is increased, the deviation becomes larger. Furthermore, the present model also leads to a larger deviation in COP between the experimental and numerical data at higher cooling loads. The larger deviations observed at high cooling loads were also observed in Ref. [11] using SAGE, that were explained by irreversible compression losses and insufficient heat rejection from the compression chamber, etc., by Veprik, et al. [11]. Besides, the concept of mechanism effectiveness factor proposed by [20] may be employed to modify the theoretical model. The mechanism effectiveness can be adjusted such that the numerical predictions can agree with the experimental data closely. However, in this report authors simply present the original numerical data for readers’ reference without introducing this concept.

a charged pressure within 1 to 5 bar and rotation speed within 800–2000 rpm. A discrepancy of approximately 15 K in the lowest temperature is observed between the two set of data. Nonetheless, the trends of the curves are rather close. Eventually, a prototype cryocooler is successfully built which can reach a low temperature of 103 K in just 15 min at 5-bar charged pressure and 2000-rpm rotation speed under zero-load condition. Acknowledgement Financial support from Ministry of Science and Technology, Taiwan, under Grant MOST 106-2622-E-006 -032 -CC2, is greatly appreciated. References [1] Timmerhaus KD, Reed R. Cryogenic engineering: fifty years of progress. Springer Science & Business Media; 2007. [2] ter Brake HJ, Wiegerinck G. Low-power cryocooler survey. Cryogenics 2002;42(11):705–18. [3] Hachem H, Gheith R, Aloui F, Nasrallah SB. Optimization of an air-filled Beta type Stirling refrigerator. Int J Refrig 2017;76:296–312. [4] Veprik A, Vilenchik H, Riabzev S, Pundak N. Microminiature linear split Stirling cryogenic cooler for portable infrared imagers. Infrared Technology and Applications XXXIII, vol. 6542, pp. 65422F. International Society for Optics and Photonics; 2007. [5] Walker G. Cycle analysis for Stirling refrigerator with multiple expansion stages, perfect regeneration and isothermal processes. Int J Refrig 1990;13(1):13–9. [6] Finkelstein T. Generalized thermodynamic analysis of Stirling engines. SAE Technical Paper, 0148-7191; 1960. [7] Urieli I, Berchowitz DM. Stirling cycle engine analysis. Taylor & Francis; 1984. [8] Ataer ÖE, Karabulut H. Thermodynamic analysis of the V-type Stirling-cycle refrigerator. Int J Refrig 2005;28(2):183–9. [9] Razani A, Dodson C, Roberts T. A model for exergy analysis and thermodynamic bounds of Stirling refrigerators. Cryogenics 2010;50(4):231–8. [10] SAGE User’s Guide v11 Edition, provided on website of SAGE: http://sageofathens. com/Documents/documents.php (available on October 9, 2019). [11] Veprik A, Zechtzer S, Pundak N, Kirkconnell C, Freeman J, Riabzev S. Adaptation of the low-cost and low-power tactical split Stirling cryogenic cooler for aerospace applications. Infrared Technology and Applications XXXVII, vol. 8012, pp. 80122I, 2011. International Society for Optics and Photonics. [12] Cheng CH, Huang CY, Yang HS. Development of a 90-K beta type Stirling cooler with rhombic drive mechanism. Int J Refrig 2019;98:388–98. [13] Park S, Hong Y, Kim H, Lee K. An experimental study on the phase shift between piston and displacer in the Stirling cryocooler. Curr Appl Phys 2003;3(5):449–55. [14] Wang L, Yuan Y, Ge Z, Song Y, Gan Z, Wu Y. Study on a pneumatically driven split stirling cryocooler operating above 100 Hz. In: AIP Conference Proceedings, vol. 1573, pp. 1633–1637, 2014, no. 1. AIP. [15] Li R, Grosu L. Parameter effect analysis for a stirling cryocooler. Int J Refrig 2017. [16] Gedeon D, Wood J. Oscillating-flow regenerator test rig: hardware and theory with derived correlations for screens and felts. NASA Contractor Report, 198442; 1996. [17] Ackermann RA. Cryogenic regenerative heat exchangers. Springer Science & Business Media; 1997. [18] Hans-Detlev K. Numerically efficient modelling of non-ideal gases and their transport properties in Stirling cycle simulation. In: The 17th International Stirling Engine Conference and Exhibition, UK; 2016. p. 572–579. [19] White R. Vuilleumier Cycle Cryogenic Refrigeration. AIR FORCE FLIGHT DYNAMICS LAB WRIGHT-PATTERSON AFB OH; 1976. [20] Senft JR. Optimum Stirling engine geometry. Int J Energy Res 2002;26(12):1087–101.

5. Conclusions In this study, a 100-K class pneumatically driven split-type cryogenic Stirling cryocooler is developed. To achieve this target, an efficient theoretical model of a split-type Stirling cryocooler is attempted by modifying the existing models. The cooler is equipped with a floating displacer which is pneumatically by gravity and gas pressures. Effects of the rotation speed at different charged pressures are studied extensively. Optimal rotation speeds that lead to the lowest cooling temperature are found. As the charged pressure is 1 bar, the optimal rotation speed is found to be 1200 rpm by experiment and 1100 rpm by numerical prediction. As the pressure is increased to 3 bar, the optimal rotation speed is found to be 1600 rpm by experiment and 1800 rpm by numerical prediction. For the split-type Stirling cryocooler, the working frequency should be chosen in accordance with the specifications of the cryocooler and the operating conditions. The present study makes the comparison between the experimental and the numerical data in the steady cold-head temperature typically at

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