Transient modeling and influence of operating parameters on thermodynamic performance of miniature Joule–Thomson cryocooler

Accepted Manuscript Transient Modeling and Influence of Operating Parameters on Thermodynamic Performance of Miniature Joule–Thomson Cryocooler Jing Cao, Yu Hou, Weibin Wang, Jie Cai, Jiapeng Li, Jun Chen, Liang Chen, Shuangtao Chen PII: DOI: Reference:

S1359-4311(18)31210-9 https://doi.org/10.1016/j.applthermaleng.2018.07.103 ATE 12461

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

24 February 2018 18 July 2018 18 July 2018

Please cite this article as: J. Cao, Y. Hou, W. Wang, J. Cai, J. Li, J. Chen, L. Chen, S. Chen, Transient Modeling and Influence of Operating Parameters on Thermodynamic Performance of Miniature Joule–Thomson Cryocooler, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/j.applthermaleng.2018.07.103

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Transient Modeling and Influence of Operating Parameters on Thermodynamic Performance of Miniature Joule–Thomson Cryocooler Jing Cao1, Yu Hou1, Weibin Wang1, Jie Cai1, Jiapeng Li2, Jun Chen2, Liang Chen1, Shuangtao Chen1* 1

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an Shaanxi 710049, PR China 2

Kunming Institute of Physics， Kunming, Yunnan 650223, PR China *Corresponding author: [email protected]

Abstract: The miniature Joule–Thomson (J–T) cryocooler has been widely used in many applications. Herein, a transient model that considers the real gas properties, the distributed throttling effects and the compressible flow is developed. The density correction term is incorporated in the pressure correction equation to improve the convergence. The mass flow equation of the orifice is coupled with the momentum equations of the high-pressure fluid as the boundary condition. The proposed numerical model is verified against the experimental results, and the effects of distributed throttling, inlet pressure, inlet temperature, load temperature, and pressure vessel discharge are investigated. The developed model and the results can be used for the accurate prediction of the system performance and optimization of the J–T cryocooler. Keywords: miniature Joule–Thomson cryocooler; argon; Hampson-type heat exchanger; thermodynamic performance

Nomenclature ρ——density u——fluid velocity t——time x——spatial coordinate A——cross-sectional area p——fluid pressure τ——stress f——Fanning friction factor h——enthalpy α——convective heat transfer coefficient T——temperature cp——heat capacity k——thermal conductivity l——wet perimeter m——mass flow rate Cd——discharge coefficient γ——specific heat ratio dci——capillary inner diameter

1.

Dhel——helix diameter G——mass flux ' ——correction value Subscripts w——finned capillary wall s——shield m——mandrel h——high-pressure fluid c——low pressure fluid i——inner wall parameter o——outer wall parameter amb——ambient e——right boundary of pressure element P/E——node of pressure element

Introduction

With the advantages of light weight, compact structure, quick start, fast cooling, and high reliability, the Joule–Thomson (J–T) cryocooler has been widely used to provide cooling at liquid nitrogen temperature for infrared detectors, thermal cameras cryosurgery probes and dephlegmators [1-3]. The miniature J–T cryocooler is generally based on an open refrigeration cycle comprising a gas storage tank, heat exchanger, throttling device, and evaporation chamber. High-pressure gas flows from the pressure vessel into the heat exchanger where it could be precooled by the backflow. The isenthalpic process in the throttling device causes the high-pressure gas to change to low pressure, and the low temperature fluid is used to cool the heat load in the evaporation chamber. After the evaporation, the low-pressure and low-temperature vapor flow back through the outer channel of the heat exchanger to cool down the high-pressure fluid flowing in the capillary.

Several experimental and numerical studies have been performed to investigate the thermodynamic process and flow behavior of J–T cryocoolers. The model proposed by Xue et al. [4] and Ng et al. [5] were used to study the thermodynamic performance at the steady state. Damle and Atrey [6, 7] studied the transient behaviors of the J–T cryocooler with different volumetric storages. The accumulative term of the momentum equation was ignored in the transient model proposed by Chou et al. [8] and Chien et al. [9, 10]. Hong et al. [11, 12] reported the transient thermodynamic behaviors of the cryocooler with a pressurized vessel both experimentally and numerically. Liu et al. [13] developed an optimization model for the coiled and finned tube heat exchanger using the response surface methodology. In addition, other scholars have studied the heat exchanger performance of the J–T cryocooler using mixed refrigerant [14, 15], the wire-type J–T cryocooler [16], the Dewar detector cooler assembly (DDCA) cooled by a J–T cryocooler [17], and the hybrid microcooler precooled a J–T cryocooler [18]. The efficiency of the Hampson-type heat exchanger is crucial to the performance of a J–T cryocooler. Some influence factors on the efficiency of the spiral coiled heat exchanger have been studied, including the tube diameter [19], nonuniform distribution of heat flux [20], Dean number, and pitch size [21]. However, owing to the compact structure and complex flow, few have reported the theoretical and experimental studies of the heat exchanger used in the miniature J–T cryocooler. In the proposed models of previous works, the mass flow rate and pressure distribution were simplified by a given parameter or an empirical correlation. Herein, a miniature J–T cryocooler model is developed, considering the impact of throttle device on the mass flow rate,

the distributed J–T effect in the capillary tube, the variations in thermophysical properties, and the axial heat conduction of the fluid and solid regions. Owing to the large variation in the pressure and temperature of the high-pressure fluid, the Navier–Stokes equations for compressible flow are used to describe the flow and heat transfer in the J–T cryocooler. The density correction term should be considered to solve the coupled velocity and pressure. To better describe the outlet boundary conditions of the high-pressure fluid, the mass flow correlation of the orifice is coupled with the momentum equations through an iterative approach. The transient thermodynamic performance of the J–T cryocooler is studied under different inlet pressures, inlet temperatures, and heat load temperatures. The model and findings herein can provide important guidelines for the optimization of the J–T cryocooler.

2.

Mathematical model

2.1 Geometrical parameters The Hampson-type heat exchanger includes a finned capillary tube, a mandrel, and a shield, which is typically used in the miniature J–T cryocooler. The capillary tube is finned with a ribbon and wound in a helix within the annular space of the co-axial cylinders [22].The design of the spiral fin would strengthen the flow turbulence, and enhance the convective heat transfer of the hot and cold fluid, thus leading to a high effectiveness of the heat exchanger. Table 1 presents the geometric parameters of the J–T cryocooler. 2.2 Governing equations In the transient model, the following assumptions are made: (1) the flow and heat transfer are one dimensional; (2) the radial heat conduction in the fluid and solid is neglected;

(3) the copper fin of the helical capillary tube has a fin efficiency of 100%; (4) the radiation heat transfer between the shield and the surrounding environment is considered, but the mandrel is assumed to be adiabatic. The mass conservation equation in the fluid control volume is     u   0 t x

(1)

The momentum conservation equation in the fluid control volume is A

  u  t

 A

   uu  x

  A

p   wl x

(2)

where the wall shear stress equation is determined by w 

f u 2 2

(3)

The energy conservation equation in the fluid control volume is A

  h t

 A

   uh  x

   l  Tw  T 

(4)

The energy conservation equations for the solid control volume of the finned helical capillary tube, mandrel and shield are as follows: w AwC pw

Tw T    kw Aw w t x  x

    h l pi Th  Tw    c l po Tc  Tw  

(5)

m Am C pm

Tm T    km Am m t x  x

    c l pm Tc  Tm  

(6)

s As C ps

Ts T    ks As s  t x  x

 4 4    c l ps Tc  Ts    amb lso Tamb  Ts  

(7)

2.3 Discretization of governing equations and computational grid The finite volume method is used to discretize the governing equations. The control volumes (CVs) of the high-pressure fluid are set along the spiral direction, which is different from that of the low-pressure fluid in the axial direction. A staggered grid is used to solve the coupled equations of pressure and velocity, as shown in the Figure 1. The outlet velocity is correlated with the outlet pressure at the boundary of the high-pressure fluid. The CV

arrangement for the finned tube is the same as those of the high-pressure fluid. Additionally, the CV division of the mandrel and shield is same as that of the low-pressure fluid. Owing to the helical structure of the high-pressure flow channel, the number of CVs is different from that of the outer annular fluid. 2.4 Pressure and density correction In the pressure correction algorithm, a more accurate value of the density significantly improves the accuracy for pressure correction. The large variation in the pressure and temperature cause a significant change in the density, which affects the change of velocity; hence, a correction term must be included for a better convergence of the continuity equation. The central difference scheme (CDS) and upwind difference scheme (UDS) exhibit some drawbacks in the discretization of the Navier–Stokes equations. However, the deferred correction approach [23], a combination of the CDS and UDS, can effectively overcome the instability and false diffusion problems caused by the discretization scheme, and improve the calculation accuracy. Using the calculation of interface density as an example, we obtain

e    eCDS  1     eUDS

(8)

where ρeCDS is the interface density obtained by the calculation of CDS, ρeUDS is the interface density obtained by the calculation of UDS, and αρ is the blending factor typically assumed as 0.9–1. The equation of state (EOS) of the arbitrary gas could be expressed as a function of density. Therefore, the Taylor formula of EOS at p0 could be written as f  p 

f  p0  0!



f '  p0  1!

 p  p0  

f ''  p0  2!

The correction terms  ' and p ' are

 p  p0   ...  2

f n  p0  n!

 p  p0 

n

 Rn  p 

(9)

 '    0

(10) p '  p  p0

(11) From Eq. (9), the difference expression with first-order precision could be obtained:    0   O  p p p  p0

(12) Based on the gas EOS, the relationship between the correction density and correction pressure is      p'  C p'  p 

'

(13) where Cρ is determined by the type of gas and the process that it undergoes. The interface density and the interface density correction expression could be expressed as

 'e   0.5  e  CP p 'P   0.5  e  CE p 'E (14) The coefficient λe- and λe+ correspond to the coefficient of the UDS and CDS, respectively. 2.5 Boundary conditions The orifice is an important factor that affects the mass flow rate, which is typically used as throttling device of the J–T cryocooler. In addition to the geometry structure of orifice, the temperature and pressure before throttling are also important factors to determine the mass flow. The empirical formula [24] can be written as  1

 2  2 1 m  Cd A  P      1

(15) where Cd is the discharge coefficient of the orifice, and the value is assumed to be 0.62. In the studies of Hong et al. [11, 12] and Damle et al. [6], the mass flow rate was set as the constant value in a single time step, and the relationship between the velocity and pressure at the exit boundary was uncertain. However, herein, the boundary velocity varies with pressure by using Eq. (15). The initial velocities are set as 0, and once t>0, choking flow occurs without delay. The inlet temperature and pressure are known in all conditions. The boundary conditions for all solid regions and fluids are set as [6]

x  0 & t  0 , T  Thin , p  phin , dTw  0 , dTm  0 , dTs  0 , dx

dx

dx

x  L & t  0 , T  Ta,e , p  pcout , dTw  0 , dTm  0 , dTs  0 , dx

dx

dx

where Ta,e is the saturated temperature of the fluid after isenthalpic throttling under the pressure pcin . 2.6 Correlations of heat transfer coefficient and friction factor According to the research of Timmerhaus et al. [25], Chou et al. [8], and Damle et al. [15, 26], the correlations of the Fanning friction factor, and the heat transfer coefficient for the flow inside the capillary finned tube and the outer fluid region are expressed as  d  f h  0.184 1  3.5 ci  Re0.2 Dhel  

(16)

fc  0.184Re0.2

(17) 

 h  0.023c ph Gh Re0.2 Pr 2/3 1  3.5 

dci   Dhel 

(18)

c  0.26c pc Gc Re0.4 Pr 2/3

(19)

2.7 Physical properties The physical properties of argon, which is the working fluid in the present study, are calculated using the commercial software, REFPROP. The finned tube, mandrel, and shield are made of copper, stainless and Monel alloy, respectively. The thermal conductivities of the solid are a function of temperature, and the correlations are shown in Table 2 [27, 28]. 2.8 Solution algorithm For the high-pressure fluid, the inlet pressure and temperature are provided while the outlet velocity is determined by Eq. (15). The SIMPLEC algorithm and staggered grid system are used to solve the discretized equations and obtain the variables (e.g. p, T, u) of the high-pressure fluid. For the low-pressure fluid, the outlet pressure is assumed to be the ambient pressure, while its inlet mass flow rate is calculated from Eq. (15). Further, the inlet temperature could be solved by the inlet pressure and enthalpy. For the solid region, the tri-diagonal matrix algorithm (TDMA) is employed to solve the discretized conduction equations, which improves the convergence and computational efficiency of the model.

3.

Results and discussion

3.1 Model validation The experimental data and calculated values of the outlet temperature and mass flow rate at the steady state are compared, as shown in Table 3. The error of the mass flow rate between the experimental and calculated results is less than 7%, and that of the outlet temperature is less than 3.5K. Transient experiments were performed on a J–T cryocooler of similar structure parameters with

that in the experiments by Ng et al.[4, 5]. In addition, the helix diameter and the heat exchanger length were 10 mm and 23 mm, respectively. The schematic diagram and the picture of the experimental apparatus are shown in Figure 2. The GFD-type platinum resistance thermometer element is used to measure the cooling temperature with a maximum absolute error of ±0.1K. The CS200 mass flowmeter produced by SEVENSTAR electronic Co. Ltd is used to measure the mass flowrate with an accuracy of ±0.01g/s. With the inlet pressure of 30 MPa, the comparison between the experimental data and the predictions is shown in Figure 3 in which the nitrogen with an inlet pressure of 30 MPa was used as the working fluid. We observed that the predictions of the transient mass flow rate and temperature are in good agreement with the experimental data during the cool-down process. 3.2 Influence of the distributed Joule–Thomson effect The cooling capacity would be caused by the distributed J–T effect along the flow direction of the capillary tube. It shows that the distributed J–T effect leads to a larger pressure drop of the high-pressure fluid, as shown in Figure 4. The temperatures of the high-pressure fluid and low-pressure fluid are lower with the distributed J–T effect than without it. This is because the contribution of the distributed J–T effect in the capillary tube causes a larger temperature drop in the high-pressure fluid, thus facilitating decrease in the consumption of cooling capacity from the returning cold fluid. 3.3 Influence of inlet pressure The change in mass flow rate with time under the five inlet pressures is shown in Figure 5. As the inlet pressure increases from 14 MPa to 18 MPa, the mass flow rate is increased by 25.88%. Under the same operating condition, the mass flow rate rises rapidly at first, and

subsequently reaches a steady state after 20 s. The comparison of the mass flow rates under different inlet pressures shows that a larger inlet pressure leads to a faster growth rate of the mass flow rate, and that the cryocooler system requires a shorter time to reach stability. The governing equation of the throttling orifice can be used to explain the influence of the inlet pressure on the mass flow rate. Figure 6 shows the variation in pressure and density before throttling. It indicates that the fluid pressure and density before throttling increase with the increase in the inlet pressure. Eq. (15) shows that two variable factors affect the mass flow rate. With the increase in inlet pressure, the pressure p and density ρ before throttling also increase. The empirical correlation of the orifice implies that

p

has a linear relation with the mass flow rate m.

After throttling, the temperature is reduced while the state is changed from single phase to two phases. Figure 7 shows the variation in vapor quality after throttling under different conditions. As shown, the fluid is of the single phase after throttling in the initial period. As time progresses, the vapor quality decreases and the liquid phase appears. The mass fraction of the liquid phase increases until the cryocooler reaches a steady state. The comparison of the vapor quality under different inlet pressures indicates that the fluid can remain at a single phase for a longer time, and the final vapor quality can be a higher value when the inlet pressure is lower. However, the liquid phase after throttling can appear sooner when the inlet pressure is higher, and it requires a shorter time from the start to the stable operation. The mass flow rate and cooling capacity increase with the increase in the inlet pressure, as shown in Figure 8. The relationship between cooling capacity and mass flow rate is approximately linear. The enthalpy of the high-pressure fluid increases with the increase in

the inlet pressure. Additionally, the enthalpy of the low-pressure fluid also increases owing to the relatively constant backflow pressure and the increase in the returning fluid temperature, as shown in Figure 9. According to the results of Figure 8, the enthalpy difference between the high-pressure fluid and low-pressure fluid remains relatively constant. This indicates that the enthalpy difference has little dependence on the inlet pressure when the pressure rises from 14 MPa to 18 MPa. The cooling capacity is primarily determined by the mass flow rate with different inlet pressures. Figure 9 shows the variations in temperature before and after throttling with different inlet pressures. The temperature before throttling is increased by nearly 9 K as the inlet pressure increases from 14 MPa to 18 MPa. Meanwhile, the temperature after throttling (cooling temperature) also increases with the inlet pressure. The cooling temperature is determined by the pressure after throttling because the fluid after throttling is in the two-phase state when the cryocooler reaches a stable state. The increase in the mass flow rate leads to the pressure larger drop in the backflow. Consequently, the cooling temperature increases with the inlet pressure, but the magnitude of change is relatively small. As the inlet pressure changes from 14 MPa to 18 MPa, the cooling temperature is only increased by 3 K. 3.4 Influence of inlet temperature The variation in the cooling capacity and vapor quality with the increase in the inlet temperature is shown in Figure 10. The increase in the inlet pressure will lead to the temperature increase in the high-pressure gas at the outlet, thus causing the temperature of the returning gas to increase in the recuperative heat exchanger[29]. Consequently, the insufficient heat exchange leads to the increase in the vapor quality after throttling. Meanwhile, the

decrease in the mass flow rate also reduces the cooling capacity. As the inlet temperature increases from 250 K to 300 K, the cooling capacities are reduced by 68.2% and 100.3% under the inlet pressures of 14 MPa and 18 MPa, respectively. 3.5 Influence of load temperature In addition to the inlet pressure and inlet temperature, the cooling capacity is also affected by the heat load. The results of Figure 11 shows that the cooling capacity and mass flow rate decrease as the heat load temperature increases. Under a higher temperature of the heat load, the system heat load becomes smaller, the cryocooler consumes less working fluid, and a smaller cooling capacity is required to balance the heat leakage of the system. Moreover, the vapor quality after throttling increases as the heat load temperature increases, and also reduces the mass flow rate as well as the cooling capacity. Figure 11 also shows the mass flow rate and cooling capacity under the inlet pressure of 18 MPa is significantly higher than that under the inlet pressure of 14 MPa; this is because the higher inlet pressure leads to the higher mass flow rate and cooling capacity. However, these results are different from the conclusions by Chua et al. [30, 31] because the mass flow rate is constant under the same condition in the study by Chua et al. In the present work, the mass flow rate depends on the pressure and temperature before throttling. As the load temperature increases, the temperature of the returning gas increases, and will lead to the increase in the temperature before the throttling. Consequently, we observed that the vapor quality increases with the increase in the load temperature. 3.6 Transient operation with pressure vessel In a practical operation of the open-cycle J–T cryocooler, the vessel volume is provided, and the inlet pressure decreases during the entire operation period. Figure 12 shows the

transient variations of the reservoir pressure and cooling temperature for the reservoir capacity of 500 cm3. A significant pressure drop is shown during the cooling-down periods that require approximately 0 s to 15 s. Because of the large mass flowrate at a large inlet pressure (see Figure 5), the reservoir pressure decreases more quickly when the initial vessel pressure is higher. After the cool-down processes, the cooling temperature is maintained at a relatively stable value with a slight temperature drop of approximately 0.5K from 15 s to 200 s. According to Figure 5, the decrease in the reservoir pressure decelerates owing to the decrease in the mass flow rate as the inlet pressure decreases. After 200 s, the cooling temperature increases quickly when the reservoir pressure decreases below 4.5 MPa under which no net cooling can be achieved. According to the results shown in Figure 12, we found that the cooling-down time is shorter at a higher initial charge pressure; however, the effective working time is 190 s, which has less dependence on the initial charge pressure. This suggests that the proper regulation of the mass flowrate is necessary to extend the effective working time for the J–T cryocooler.

4.

Conclusions

In this work, a transient model of a miniature J–T cryocooler was developed to study its thermodynamic characteristics. The numerical model could provide good predictions for the transient mass flow and cooling temperature during the cool-down process. The simulation results showed that a higher inlet pressure and a lower temperature could provide more cooling capacity, while a lower inlet pressure and inlet temperature consumed lower gas flow rate. For a J–T cryocooler with a pressure vessel, the cooling temperature and operating time had little difference when the initial charge pressure varied from 14 MPa to 18 MPa, but the cooling

temperature rose quickly when the vessel pressure was lower than 4.5 MPa. The maximum working time could be achieved by minimizing the mass flowrate while maintaining the required cooling capacity.

Acknowledgements This project was supported by the National Nature Science Foundation of China (51706169), the National key basic research program (613322), and the Natural Science Foundation (BK20160388) of Jiangsu Province, China.

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Figure 1 The schematic of control volumes of J–T cryocooler

1

2

3 8 4

9

7 5 6

Figure 2 The schematic diagram of the experimental apparatus 1-gas source; 2-pressure gage; 3-gas filter; 4-flow collector; 5-J-T cryocooler; 6-platinum resistance thermometer; 7-flowmeter; 8-data acquisition system; 9-dynamic vacuum system.

1.0 Experimetal data Simulated data

250

0.9

200

0.8

150

0.7

100

0.6

50

0

40

80

120

Mass flowrate (g/s)

Cooling temperature (K)

300

0.5 160

Time (s)

Figure 3 The comparison of the experimental data and the predictions during the cool-down process

300

15

250

13

200

11

150

ph ph(no μJT) Th

9

100

Th(no μJT)

Temperature (K)

Pressure (MPa)

17

Tc Tc(no μJT)

7

0

10

20

30

40

50 50

Length (mm)

Figure 4 The variations in pressure and temperature along the heat exchanger length with and without J–T effect

0.40

Mass flow rate (g/s)

0.35 0.30 0.25

14MPa 15MPa 16MPa 17MPa 18MPa

0.20 0.15

0

10

20

30

40

50

60

Time (s)

Figure 5 The variation in mass flow rate with time under the five inlet pressures

15.0

600

13.0

500

11.0

400

9.0

300

7.0 14.0

15.0

16.0

17.0

200 18.0

Inlet pressure (MPa)

Figure 6 The variation in pressure and density before throttling

Density (kg/m3)

Pressure (MPa)

Pressure before throttling Density before throttling

1.00 14MPa 15MPa 16MPa 17MPa 18MPa

Vapor quality

0.95

0.90

0.85

0.80

0

10

20

30

40

50

60

Time (s)

Figure 7 The variations in vapor quality after throttling under different conditions

0.38

15.0

0.33

12.5

0.28

10.0

0.23

7.5

0.18 14

15

16

17

Cooling Capacity (W)

Mass flow rate (g/s)

Mass flow rate Cooling Capacity

5.0 18

Inlet pressure (MPa)

Figure 8 The variations in mass flow rate and cooling capacity under different conditions

180

Temperature (K)

170

Temperature before throttling Temperature after throttling

160 100 90 80 14

15

16

17

18

Inlet pressure (MPa)

Figure 9 The variations in temperature before and after throttling with different inlet pressures

20

0.95 14MPa 18MPa 14MPa 18MPa

0.90

12

0.85

8

0.80

4

0.75

0 240

260

280

300

320

Vapor quality

Cooling capacity (W)

16

0.70 340

Inlet temperature (K)

Figure 10 The variation in the cooling capacity and vapor quality with the increase in the inlet temperature

0.45 14MPa 18MPa 14MPa 18MPa

9.5

0.40

9.0

0.35

5.5

0.30

5.0

90

100

110

120

Mass flow rate (g/s)

Cooling capacity (W)

10.0

0.25

130

Load temperature (K)

Figure 11 The variations in the cooling capacity and mass flow rate with different load temperatures

Figure 12 The transient variations of reservoir pressure and cooling temperature for the reservoir capacity of 500 cm3

Table 1 Geometric parameters of the J–T cryocooler [7] Dimension

Heat exchanger

Dimension

(mm)

geometry

(mm)

Cooler geometry

Outside diameter of Outside diameter of shield

4.8

0.5 capillary tube

Inside diameter of Inside diameter of shield

4.5

0.3 capillary tube

Outside diameter of 2.5

Fin height

0.25

Inside diameter of mandrel

2.3

Fin thickness

0.1

Mandrel length

50

Fin pitch

0.3

Helix diameter

3.5

Diameter of orifice

0.1

mandrel

Table 1 Thermal conductivities of metal materials Relative Materials

Correlation errors

Copper(finned tube)

Monel(shield)

2  0.2413T  47.775T  2848,  60 K  T  100 K  k  2  0.0028T  1.525T  608, 100 K  T  200 K 

<1.5%

k  6.5169ln T  14.76,  40 K  T  400K 

<1.0%

k  5.0353ln T  13.797,  40 K  T  400 K 

<1.0%

Stainless steel(mandrel)

Table 2 The comparison of the experimental data and the calculated data Phin

Thin

Mass flow

Mass flow

Experimeta

Error

Tcout

Tcout

simulated

Experimental

simulated

l data[7]

data

data[7]

data

Error

MPa

K

g/s

g/s

%

K

K

K

14.047

291.94

0.28098

0.28632

1.90

284.98

281.80

-3.18

14.966

292.14

0.30378

0.30434

0.18

284.90

282.19

-2.71

16.010

292.25

0.33078

0.32425

-1.90

284.77

282.53

-2.24

16.986

291.04

0.36288

0.34324

-5.41

283.72

281.95

-1.77

17.912

291.49

0.38573

0.36043

-6.56

282.57

282.21

-0.36

Research Highlights 1. A transient model is developed and validated for cryocooler. 2. Distributed throttling effects and compressible flow are included. 3. Coupling between heat exchanger and throttling orifice is solved iteratively. 4. The effects of operating parameters and pressure vessel discharge are analyzed.