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Investigation on the pressure drop characteristics of cryocooler regenerators under oscillating flow and pulsating pressure conditions
Cryogenics 44 (2004) 203–210 www.elsevier.com/locate/cryogenics
Investigation on the pressure drop characteristics of cryocooler regenerators under oscillating flow and pulsating pressure conditions Sungryel Choi, Kwanwoo Nam *, Sangkwon Jeong
1
Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejon 305-701, Republic of Korea Received 4 June 2003; received in revised form 31 October 2003; accepted 17 November 2003
Abstract This paper proposes a new oscillating flow model of the pressure drop in oscillating flow through regenerator under pulsating pressure. In this oscillating flow model, pressure drop is represented by the amplitude and the phase angle with respect to the inlet mass flow rate. In order to generalize the oscillating flow model, some non-dimensional parameters, which consist of Reynolds number, Valensi number, gas domain length ratio, oscillating flow friction factor and phase angle of pressure drop, are derived from a capillary tube model of the regenerator. Two correlations in the model are obtained from the experiments for the twill square screen regenerators under various operating frequencies and inlet mass flow rates. It is found that the oscillating flow friction factor is a function of Reynolds number while the phase angle of pressure drop is a function of Valensi number and the gas domain length ratio. Experiment also shows the effect of the mesh weave style on the oscillating flow friction factor and the phase angle. Proposed oscillating flow model can accurately describe the amplitude and the phase angle of the pressure drop through the regenerator. 2003 Elsevier Ltd. All rights reserved. Keywords: Friction factor; Oscillating flow; Phase angle; Pressure drop; Regenerator
1. Introduction Recently, cryocooler studies for Stirling or pulse tube refrigerators have been profoundly done due to great interest of environmental issue and demands for efficient future energy system. As a regenerator separates the warm and cold spaces of such a regenerative cryocooler, the accurate prediction of the pressure drop through the regenerator is very important for the optimal design. Most previous researches [1–3] have adopted correlations of the pressure drop based on the unidirectional steady flow experiments. Miyabe et al.Õs studied the steady flow pressure drop over the wire screen regenerator and presented the following friction factor [3]. fM ¼
33:6 þ 0:337; Rel
fM ¼
DP ; 0:5qu2 n
u ¼ u0 =b
ð1Þ
*
Corresponding author. Fax: +82-42-869-8207. E-mail addresses: [email protected] (K. Nam), skjeong@ kaist.ac.kr (S. Jeong). 1 Fax: +82-42-869-3210. 0011-2275/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2003.11.006
Here, u0 is frontal flow velocity. Since the regenerator in the cryocooler typically experiences bidirectional oscillating flow and large pulsating pressure conditions, it is clear that there is a fundamental discrepancy between the analysis based on the steady flow correlation and the real phenomenon in the combined oscillating flow and pulsating pressure conditions. Therefore, the recent efforts have been made to predict more accurately the pressure drop characteristic of the regenerator in cryocoolers. Some researchers [4–6] performed the pressure drop experiments subjected to oscillating flow in the regenerator and presented the maximum or the cycle-averaged friction factors for low operating frequencies (under 10 Hz). Validities of their results for cryocooler regenerators are still questionable because the oscillating flow was not combined with pulsating pressure and the frequencies were limited to low range. In case of the oscillating flow with pulsating pressure conditions, Helvensteijn et al. [7] presented the friction factor of the regenerator for 30 through 70 Hz operations, but the mass flow rate in the regenerator was not directly measured but calculated indirectly from the
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S. Choi et al. / Cryogenics 44 (2004) 203–210
Nomenclature Aff ADP D dc dh dw ev fosc fF i L l Ma _ M m_ n P p R Re r T t U
free flow area (¼ ev pD2 =4) amplitude of pressure drop diameter of regenerator diameter of capillary tube hydraulic diameter of screen wire diameter of screen porosity oscillating flow friction factor steady pffiffiffiffiffiffiffi flow friction factor 1 length of regenerator mesh distance Mach number amplitude of mass flow rate mass flow rate number of packed screens pressure pitch gas constant Reynolds number (¼ U1 dh =v) radial direction temperature time amplitude of axial velocity component
u Va v z
Greek letters b opening area ratio c specific heat ratio DP pressure drop e expansion parameter l viscosity m kinematic viscosity (¼ l=qm ) q density /DP phase difference between the pressure drop and the inlet mass flow rate x angular frequency Subscripts 1 warm-end of regenerator 2 cold-end of regenerator avg average value between both sides of the regenerator m mean value in one cycle Superscript dimensionless quantities
pressure data in the reservoir and their computational model. Ju et al. [8] measured the mass flow rate and presented the correlation of the maximum friction factor only at 50 Hz. Relationship of the phase shifts between the velocity and the pressure wave was also presented. Flow patterns in cryocooler regenerators are very complex due to irregular geometry and flow conditions. This complexity makes it impossible to solve full Navier–Stokes equation under exact geometric boundary conditions. Unlike steady and purely oscillating flow, the phase shift and the amplitude attenuation of pressure exist through the regenerator for oscillating flow under pulsating pressure. In this situation, it is hard to define a conventional friction factor because the instantaneous mass flow rate is different along the regenerator. In previous works [7,8], the friction factor was defined as follows. f ¼
ADP dh 2qu2avg L
axial component of velocity Valensi number (¼ dh2 x=v) radial component of velocity axial direction
drop and the mean velocity have often different phase angles. Therefore, a new model is required to describe the pressure drop characteristic more precisely in such a regenerator. In this paper, a new pressure drop model is proposed to represent the pressure drop with two variables, i.e. the amplitude and the phase angle with respect to the oneside mass flow rate. The dimensionless parameters are derived from a capillary tube model of regenerator and experiments are performed to obtain functional forms of the parameters. The experimental data are presented for twill square screen regenerators under various operating frequencies and mass flow rates. Experiments are also carried out to examine the effects of the screen weave style and to ensure the reliability of the experimental measurements. Finally, the pressure drop data from the present oscillating flow model are compared with those from the steady flow friction factor.
ð2Þ
Here, uavg represents the mean amplitude of Darcian velocity between the warm and the cold-ends of the regenerator. This definition is similar to the steady flow friction factor and presumes that the phase angle of the pressure drop is equal to that of the mean Darcian velocity. However, in cryocooler operation, the pressure
2. Theory 2.1. Steady flow model vs. oscillating flow model Although regenerators used in cryocoolers are subjected to oscillating flow and pulsating pressure condi-
S. Choi et al. / Cryogenics 44 (2004) 203–210
tions, most previous researches have used a pseudosteady flow model involving the pressure drop such as a form of Eq. (3). In this paper, the following expression is designated as the steady flow model. m_ avg DP ¼ ADP ð3Þ _ avg M _ avg expðixtÞ. where m_ avg ¼ M As an alternative method to describe the pressure drop, a new model is proposed as the form of Eq. (4). This model, which is called an oscillating flow model in contrast to the steady flow model, represents the pressure drop in two variables; the amplitude and the phase angle with respect to the mass flow rate at the warm-end of the regenerator. DP ¼ ADP expði/DP Þ
m_ 1 _1 M
ð4Þ
_ 1 expðixtÞ. where m_ 1 ¼ M 2.2. Non-dimensionalization of oscillating flow model In order to generalize the proposed oscillating flow model, the process to find out the relevant dimensionless parameters is required prior to the experiments. A useful technique to find the dimensionless parameters is to consider the governing equations of the flow in small capillary tube. The regenerator in this analysis is assumed to be a bundle of numerous small capillary tubes. We used the following equations as the governing equations (continuity and momentum equations) of the flow in each capillary tube [9]. oq 1 oðqvrÞ oðquÞ þ þ ¼0 ot r or oz ou ou ou oP l o ou q þ r þv þu ¼ ot or oz oz r or or
ð5Þ ð6Þ
The variables are normalized by appropriate reference values as follows. q P u v ; ; P ¼ ; u ¼ ; v ¼ qm Pm U1 U1 dc =L r z r ¼ ; z ¼ and t ¼ t x dc L q ¼
These non-dimensionalized variables are substituted into the Eqs. (5) and (6) and rearranged to give the dimensionless form of the governing equations with the significant dimensionless parameters. oq 1 oðq v r Þ oðq u Þ þe þ ¼0 ð7Þ r or oz ot ou ou ou q þe v þu ot or oz e oP 1 1 o ou þ r ¼ ð8Þ c Ma2 oz Va r or or
205
U1 : ratio of gas diswhere, e (expansion parameter) ” xL placement length to tube length, Va (Valensi number) ” dc2 x=v: ratio of tube inner diameter to viscous ffi pffiffiffiffiffiffiffiffiffiffi diffusion length, Ma (Mach number) ” U1 = cRTm . The expansion parameter can be represented in terms of Reynolds number and the gas domain length ratio defined as dc =L.
dc 1 ð9Þ L Va Consequently, the governing non-dimensional parameters for oscillating flow and pulsating pressure in capillary tube are Reynolds number, Valensi number, Mach number and the gas domain length ratio. To adopt the above non-dimensionalized parameters to wire screen regenerators, the diameter of the capillary tube is set to be the hydraulic diameter of the wire screen defined as follows. ev dc ¼ dh ¼ dw ð10Þ 1 ev
e ¼ Re
Velocity amplitude at the warm-end of the regenerator (U1 ) is calculated as the following formula. _ 1 =ðqAff Þ U1 ¼ M
ð11Þ
Amplitude of the pressure drop (ADP ) can be obtained from the oscillating flow friction factor defined in a similar way to Fanning steady flow friction factor [2]. fosc
ADP dh 2qm U12 L
ð12Þ
In general, the oscillating flow velocity through the cryocooler regenerator is so small that the Mach number is usually very low [9]. Therefore, the oscillating flow friction factor and the phase angle is to have the functional relationships as follows. dh fosc ¼ function Re; Va and ð13Þ L dh /DP ¼ function Re; Va and ð14Þ L From the above functional relationships, the oscillating flow friction factor and the phase angle can be determined by measuring the mass flow rate and the pressure at the ends of the regenerator.
3. Experiments 3.1. Regenerator specimens Stainless steel wire screen is widely used as popular regenerator materials for cryocoolers. It can be classified according to the wire diameter and the weave style. The wire diameter and the porosity determine the hydraulic diameter of the wire screen by Eq. (10). Three types of
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S. Choi et al. / Cryogenics 44 (2004) 203–210
Fig. 1. Microscopic photos of #200 plain and twill screen (left: plain, right: twill).
regenerator (#400 twill, #250 twill and #200 twill screen regenerators) are fabricated for the experiment. Also, they are divided into plain square and twill square screens according to the weave style. The magnified pictures of the two kinds of screens are shown in Fig. 1. Plain square is weaved such that the wires are woven in a simple over and below pattern. On the other hand, twill square is weaved such that each wire passes over two wires and then below two. In order to investigate the effects of the weave style, #200 plain screen regenerator is fabricated and compared with the #200 twill. One more #400 twill screen regenerator is made to confirm the reliability and the repeatability of the experimental results in this study. Table 1 shows the specification of the five regenerators as mentioned above. All of the specimens are designed to have the same porosities within 3% error in order to eliminate the uncertainty caused by different porosity. This porosity value is also reasonable in most regenerative cryocoolers. 3.2. Experimental apparatus and measurements Fig. 2 shows the schematic diagram of the experimental apparatus in which the mass flow rates, the pres-
sures, and the temperatures at both sides of the regenerator are measured. A linear compressor generates oscillating flow and pulsating pressure inside the regenerator specimen. The inner diameter and the length of the regenerator housing are 12.25, 85 mm, respectively. The compressor is controlled by an inverter and a transformer, so that the experimental data are obtained with various inlet mass flow rates and frequencies (45–80 Hz). Two fine hot wires (TSI model: 1260A-T1.5) and two piezoelectric pressure transducers (Kistler model 601A) are used to measure the mass flow rate and the dynamic pressure, respectively. Since the diameter of the hot wire is 3.8 lm and the natural frequency of the piezoelectric pressure sensor is 150 kHz, there is no need to compensate for the time lag of the acquired signal. The detailed information about the experimental apparatus can be found in the authorsÕ other research paper [10]. Although the oscillating flow model assumes that the pressure wave and mass flow rate are first order harmonic functions, the actual waves include higher order components. Therefore, only the first order components of the actual waves are taken by means of curve fitting to get the amplitude and the phase angle. Fig. 3 shows the example that compares the measured data with the first order components.
4. Results and discussions 4.1. Results and correlations for twill screen regenerator Fig. 4 shows the results for the #400 twill screen regenerator. The oscillating flow friction factor is a function of Reynolds number, but independent of Valensi number. On the other hand, the phase angle is proportional to Valensi number, but independent of Reynolds number. The vertical scattering of the data is
Table 1 Geometric properties of regenerator specimens (regenerator length ¼ 85 mm, inner diameter ¼ 12.25 mm) Type of wire screen
Number of screens
Wire diameter dw [lm]
Porosity ev
Hydraulic diameter dh [lm]
Mesh distance l [lm]
Pitch p [lm]
Opening area ratio b
#400 #250 #200 #200 #400
1261 1039 691 827 1261
30 38 52 52 30
0.679 0.697 0.679 0.680 0.679
63.5 87.4 110.0 110.5 63.5
33.5 63.6 75 75 33.5
63.5 101.6 127.0 127.0 63.5
0.278 0.392 0.349 0.349 0.278
*
twill twill twill plain twill
One more #400 twill mesh was tested to verify the repeatability of the experiment.
ω Linear compressor
T1 m& 1 P1
T2 Regenerator
Orifice
P2 m& 2
Fig. 2. Schematic diagram of experimental apparatus.
Reservoir
S. Choi et al. / Cryogenics 44 (2004) 203–210
207
7
Pressure drop [bar]
1.0
Real data Sine fit
Va = 0.157 Va = 0.175 Va = 0.192 Va = 0.210 Va = 0.227 Va = 0.245 Va = 0.262 Va = 0.280
6
0.5
5 4
0.0
fosc
3
-0.5 2 -1.0 0
1 90
180
270
360 0
Angle [deg]
(a)
0
5
10
15
20
(a)
Inlet mass flow rate [kg/sec]
0.0008
30
35
40
45
50
0.7 0.6
Real data Sine fit
0.0004
0.5
0.0000
φ∆ P
-0.0004
0.4 0.3 0.2 0.1
-0.0008 0 (b)
90
180
270
360
Angle [deg]
not significant considering the error due to the first order curve fitting used in the data processing. Both the #250 twill and #200 twill screen regenerators have the same tendency as the #400 twill screen. The effects of the gas domain length ratio can be recognized from the experimental data comparisons among #400 twill, #250 twill and #200 twill screen regenerators as shown in Fig. 5. It is clear that the gas domain length ratio has a considerable effect on the phase angle, but little effect on the oscillating flow friction factor. Hence, Eq. (13) and (14) are simplified as the following equations. fosc ¼ functionðReÞ dh /DP ¼ function Va and L
ð15Þ ð16Þ
The oscillating flow friction factor of the tested twill screen regenerators is well fitted by the following correlated equation as shown in Fig. 6(a). 39:52 þ 0:01 ¼ Re
0.0 0.00
Re = 7.5 ~ 15 15 ~ 22.5 22.5 ~ 30 30 ~ 37.5 37.5 ~ 45 0.05
0.10
(b)
Fig. 3. First order harmonic component for #400 twill screen at 70 Hz (a) pressure drop, (b) inlet mass flow rate.
fosc
25
Re
ð17Þ
0.15
0.20
0.25
0.30
Va
Fig. 4. Oscillating flow friction factor and phase angle for #400 twill screen (a) friction factor, (b) phase angle.
As shown in Fig. 5(b), the phase angle (/DP ) cannot be correlated by Valensi number only. Modified phase angle as a new dimensionless parameter with the following definition [11] is introduced to get the proper correlation. /DP /DP ðdh =LÞ
2:62
ð18Þ
Fig. 6(b) shows that the modified phase angle is linearly varied with the Valensi number. Therefore, the phase angle of the twill screen regenerator is well fitted by the following correlated equation as shown in Fig. 6(b). 2:62 dh /DP ¼ 1:32 108 Va ð19Þ L 4.2. Comparison between twill screen and plain screen regenerators As all properties of both #200 twill and #200 plain screen regenerators are identical except for the weave style as shown in Table 1, the difference of their results
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S. Choi et al. / Cryogenics 44 (2004) 203–210 7
7
d h /L=0.00075 (#400 twill mesh) d h /L=0.00103 (#250 twill mesh) d h /L=0.00129 (#200 twill mesh)
6 5
5 4
4
f osc
d h/L=0.00075 (#400 twill mesh) d h/L=0.00103 (#250 twill mesh) d h/L=0.00129 (#200 twill mesh) Correlation curve
6
fosc
3
3 2 2 1 1 0
0 0
20
40
(a)
60
80
0
100
Re 1.4x10 1.0
0.8 0.7
1.0x10 -8
φ
*
∆P
0.6
φ ∆P
Re -8
1.2x10 -8
d h/L=0.00075 (#400 twill mesh) d h/L=0.00103 (#250 twill mesh) d h/L=0.00129 (#200 twill mesh)
0.9
10 20 30 40 50 60 70 80 90 100 110 120
(a)
d h/L=0.00075 (#400 twill mesh) d h/L=0.00103 (#250 twill mesh) d h/L=0.00129 (#200 twill mesh)
8.0x10 -9 6.0x10 -9
0.5 0.4
4.0x10 -9
0.3 2.0x10 -9
0.2 0.1 0.0 0.0
(b)
0.2 0.2
0.4
0.6
0.8
1.0
Va
(b)
0.4
0.6
0.8
Va
Fig. 6. Correlation curve of oscillating flow friction factor and phase angle (a) friction factor, (b) phase angle.
Fig. 5. Comparison of oscillating flow friction factor and phase angle for different mesh number (a) friction factor, (b) phase angle.
4.4. Superiority of the oscillating flow model over the steady flow model implies the effects of weave style. Fig. 7 shows the comparison between their experimental results. The weave style seems to have little effect on the phase angle, but a relatively larger effect appears on the oscillating flow friction factor. The oscillating flow friction factor of the plain screen regenerator is in general larger than that of the twill screen. Since the oscillating flow friction factor has no information about the weave style, an additional non-dimensional parameter may be required to account for the effect of the weave style. 4.3. Reliability and repeatability test In order to ensure the reliability of the experimental results, repeatability test should be performed for two specimens whose all properties are identical. Fig. 8 shows that two identical #400 twill screen regenerators have consistent results. Accordingly, uncertainty due to fabrication procedure is negligible and the experimental results in this study are consistent.
Fig. 9 shows the typical pressure drop and the mean mass flow rate data under oscillating flow and pulsating pressure. The mean mass flow rate in this plot is the averaged value between the mass flow rates at two ends of the regenerator as indicated in Eq. (3). It is clearly observed from Fig. 9 that the phase of the pressure drop is different from that of the mean mass flow rate. The steady flow model expressed by Eq. (3) cannot describe this fundamental phase shifting phenomenon because it is assumed that the pressure drop phase is always equal to the phase of the mean mass flow rate. In the oscillating flow model, the phase of the pressure drop is correlated with respect to the phase of the mass flow rate at the warm-end. Hence, it is not necessary to confine the pressure drop phase equal to the phase of the mean mass flow rate. Fig. 9 clearly shows that the oscillating flow model shall predict the pressure drop better than the steady flow model. We now compare the amplitude of the pressure drop data with the pressure drop which is predicted by the
S. Choi et al. / Cryogenics 44 (2004) 203–210
209
7
5
#200 twill mesh #200 plain mesh
4
6
#400 twill * #400 twill
5 3
f osc
4
f osc
3
2 2 1
0
1 0 0
10
20
30
40
(a)
50
60
70
80
90
100 110
Re
10
15
20
25
30
35
40
45
Re 0.7 0.6
#200 twill mesh #200 plain mesh
0.5
0.4
φ
5
(a)
0.6 0.5
0
φ∆ P
0.3
∆P
0.4 0.3 0.2
0.2
#400 twill * #400 twill
0.1 0.1
0.2
0.4
0.6
0.8
1.0
Va
Fig. 7. Effect of weave style on oscillating flow friction factor and phase angle (a) friction factor, (b) phase angle.
steady flow friction factor. Pressure drop amplitude is readily calculated by the MiyabeÕs steady flow friction factor as indicated in Eq. (1). Here, the flow rate is set to be the mean peak value between the warm and the coldends as mentioned in Eq. (2). Fig. 10 shows the pressure drop calculation from the steady flow friction factor. It is evident that when the Reynolds number increases, difference between the calculated pressure drop and the experimental data becomes large. From these experimental results, we can say that the conventional steady flow friction factor cannot accurately simulate both the amplitude and the phase angle of the pressure drop. Accordingly, the oscillating flow model proposed in this paper is undoubtedly very useful to simulate the pressure drop through the regenerator under actual operating conditions of cryocoolers. 5. Conclusions A new model is presented for accurate description of oscillating flow characteristic through regenerator under
0.1
(b)
0.2
0.3
Va
Fig. 8. Reliability and repeatability test result (a) friction factor, (b) phase angle.
0.0010
1.0
0.0008
0.8
0.0006
0.6
0.0004
0.4
0.0002
0.2
0.0000
0.0
-0.0002
-0.2
-0.0004
-0.4
-0.0006
-0.6
-0.0008
-0.8
-0.0010
0
50
100
150
200
250
300
Pressure drop (bar)
(b)
0.0 0.0
Mean mass flow rate (kg/s)
0.0 0.0
-1.0 350
Phase angle (degree) Fig. 9. Experimental data of the mean mass flow rate and the pressure drop for #200 plain screen regenerator at 75 Hz.
pulsating pressure conditions. The non-dimensionalized correlations of the model are derived from the experimental results of twill screen regenerators (#400, #250, #200).
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S. Choi et al. / Cryogenics 44 (2004) 203–210
50 Hz 60 Hz
1.50
A∆P (bar)
Acknowledgements
#200 twill mesh
1.75
The authors are grateful to Ministry of Science and Technology for the support of this work through the Dual Use Technology Program (grant no. 99-DU-04-A02). This work is also supported by LG Electronics Inc. and the Combustion Engineering Research Center of KAIST.
70 Hz Miyabe's correlation Oscillating flow model
1.25 1.00 0.75 0.50
References
0.25 0.00 10
20
30
40
50
60
70
80
90
Reavg Fig. 10. Comparison between the amplitude of the experimental pressure drop and the calculated value from the steady flow friction factor.
Oscillating flow friction factor (fosc ) and phase difference between pressure drop and mass flow rate at the warm-end (/DP ) can be expressed as follows by Reynolds number (Re), Valensi number (Va) and the gas domain length ratio (dh =L). fosc ¼
39:52 þ 0:01 Re
/DP ¼1:32 10
8
ð5 6 Re 6 100Þ
dh L
2:62
Va dh 0:00075 6 6 0:00129 L
0:15 6 Va 6 0:80;
Experimental data also demonstrate that the calculated pressure drop using the conventional steady flow friction factor is significantly deviated from the experimental data.
[1] Urieli I, Berchowitz DM. Stirling cycle engine analysis. Bristol: Adam Hilger Ltd; 1984. [2] Kays WM, London AL. Compact heat exchanger. 3rd ed. McGraw-Hill; 1984. [3] Miyabe H, Takahashi S, Hamagushi K. An approach to the design of Stirling engine regenerator matrix using packs of wire gauzes. In: 17th Intersociety of Energy Conversion Engineering Conference, 1982. [4] Zhao TS, Cheng P. Oscillatory pressure drops through a wovenscreen packed column subjected to a cyclic flow. Cryogenics 1996;36(5):333–41. [5] Tanaka M, Yamashita I, Chisaka F. Flow and heat transfer characteristics of the Stirling engine regenerator in an oscillating flow. JSME Int J 1990;33(2):283–9. [6] Hsu CT, Fu H, Cheng P. On pressure–velocity correlation of steady and oscillating flows in regenerators made of wire screens. Trans ASME J Fluids Eng 1999;121(1):52–6. [7] Helvensteijn BPM, Kashani A, Spivak AL, Roach PR, Lee JM, Kittel P. Pressure drop over regenerators in oscillating flow. Adv Cryog Eng 1998;43B:1619–26. [8] Ju Y, Jiang Y, Zhou Y. Experimental study of the oscillating flow characteristics for a regenerator in a pulse tube cryocooler. Cryogenics 1998;38(6):649–56. [9] Lee JM, Kittel P, Timmerhaus KD, Radebaugh R. Useful scaling parameters for the pulse tube. Adv Cryog Eng 1996;41:1347–55. [10] Nam K, Jeong S. Measurement of cryogenic regenerator characteristics under oscillating flow and pulsating pressure. Cryogenics 2003;43(10):575–81. [11] Bhagoria JL, Saini JS, Solanki SC. Heat transfer coefficient and friction factor correlations for rectangular solar air heater duct having transverse wedge shaped rib roughness on the absorber plate. Renew Energy 2001;25(3):341–69.
Investigation on the pressure drop characteristics of cryocooler regenerators under oscillating flow and pulsating pressure conditions Sungryel Choi, Kwanwoo Nam *, Sangkwon Jeong
1
Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejon 305-701, Republic of Korea Received 4 June 2003; received in revised form 31 October 2003; accepted 17 November 2003
Abstract This paper proposes a new oscillating flow model of the pressure drop in oscillating flow through regenerator under pulsating pressure. In this oscillating flow model, pressure drop is represented by the amplitude and the phase angle with respect to the inlet mass flow rate. In order to generalize the oscillating flow model, some non-dimensional parameters, which consist of Reynolds number, Valensi number, gas domain length ratio, oscillating flow friction factor and phase angle of pressure drop, are derived from a capillary tube model of the regenerator. Two correlations in the model are obtained from the experiments for the twill square screen regenerators under various operating frequencies and inlet mass flow rates. It is found that the oscillating flow friction factor is a function of Reynolds number while the phase angle of pressure drop is a function of Valensi number and the gas domain length ratio. Experiment also shows the effect of the mesh weave style on the oscillating flow friction factor and the phase angle. Proposed oscillating flow model can accurately describe the amplitude and the phase angle of the pressure drop through the regenerator. 2003 Elsevier Ltd. All rights reserved. Keywords: Friction factor; Oscillating flow; Phase angle; Pressure drop; Regenerator
1. Introduction Recently, cryocooler studies for Stirling or pulse tube refrigerators have been profoundly done due to great interest of environmental issue and demands for efficient future energy system. As a regenerator separates the warm and cold spaces of such a regenerative cryocooler, the accurate prediction of the pressure drop through the regenerator is very important for the optimal design. Most previous researches [1–3] have adopted correlations of the pressure drop based on the unidirectional steady flow experiments. Miyabe et al.Õs studied the steady flow pressure drop over the wire screen regenerator and presented the following friction factor [3]. fM ¼
33:6 þ 0:337; Rel
fM ¼
DP ; 0:5qu2 n
u ¼ u0 =b
ð1Þ
*
Corresponding author. Fax: +82-42-869-8207. E-mail addresses: [email protected] (K. Nam), skjeong@ kaist.ac.kr (S. Jeong). 1 Fax: +82-42-869-3210. 0011-2275/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2003.11.006
Here, u0 is frontal flow velocity. Since the regenerator in the cryocooler typically experiences bidirectional oscillating flow and large pulsating pressure conditions, it is clear that there is a fundamental discrepancy between the analysis based on the steady flow correlation and the real phenomenon in the combined oscillating flow and pulsating pressure conditions. Therefore, the recent efforts have been made to predict more accurately the pressure drop characteristic of the regenerator in cryocoolers. Some researchers [4–6] performed the pressure drop experiments subjected to oscillating flow in the regenerator and presented the maximum or the cycle-averaged friction factors for low operating frequencies (under 10 Hz). Validities of their results for cryocooler regenerators are still questionable because the oscillating flow was not combined with pulsating pressure and the frequencies were limited to low range. In case of the oscillating flow with pulsating pressure conditions, Helvensteijn et al. [7] presented the friction factor of the regenerator for 30 through 70 Hz operations, but the mass flow rate in the regenerator was not directly measured but calculated indirectly from the
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S. Choi et al. / Cryogenics 44 (2004) 203–210
Nomenclature Aff ADP D dc dh dw ev fosc fF i L l Ma _ M m_ n P p R Re r T t U
free flow area (¼ ev pD2 =4) amplitude of pressure drop diameter of regenerator diameter of capillary tube hydraulic diameter of screen wire diameter of screen porosity oscillating flow friction factor steady pffiffiffiffiffiffiffi flow friction factor 1 length of regenerator mesh distance Mach number amplitude of mass flow rate mass flow rate number of packed screens pressure pitch gas constant Reynolds number (¼ U1 dh =v) radial direction temperature time amplitude of axial velocity component
u Va v z
Greek letters b opening area ratio c specific heat ratio DP pressure drop e expansion parameter l viscosity m kinematic viscosity (¼ l=qm ) q density /DP phase difference between the pressure drop and the inlet mass flow rate x angular frequency Subscripts 1 warm-end of regenerator 2 cold-end of regenerator avg average value between both sides of the regenerator m mean value in one cycle Superscript dimensionless quantities
pressure data in the reservoir and their computational model. Ju et al. [8] measured the mass flow rate and presented the correlation of the maximum friction factor only at 50 Hz. Relationship of the phase shifts between the velocity and the pressure wave was also presented. Flow patterns in cryocooler regenerators are very complex due to irregular geometry and flow conditions. This complexity makes it impossible to solve full Navier–Stokes equation under exact geometric boundary conditions. Unlike steady and purely oscillating flow, the phase shift and the amplitude attenuation of pressure exist through the regenerator for oscillating flow under pulsating pressure. In this situation, it is hard to define a conventional friction factor because the instantaneous mass flow rate is different along the regenerator. In previous works [7,8], the friction factor was defined as follows. f ¼
ADP dh 2qu2avg L
axial component of velocity Valensi number (¼ dh2 x=v) radial component of velocity axial direction
drop and the mean velocity have often different phase angles. Therefore, a new model is required to describe the pressure drop characteristic more precisely in such a regenerator. In this paper, a new pressure drop model is proposed to represent the pressure drop with two variables, i.e. the amplitude and the phase angle with respect to the oneside mass flow rate. The dimensionless parameters are derived from a capillary tube model of regenerator and experiments are performed to obtain functional forms of the parameters. The experimental data are presented for twill square screen regenerators under various operating frequencies and mass flow rates. Experiments are also carried out to examine the effects of the screen weave style and to ensure the reliability of the experimental measurements. Finally, the pressure drop data from the present oscillating flow model are compared with those from the steady flow friction factor.
ð2Þ
Here, uavg represents the mean amplitude of Darcian velocity between the warm and the cold-ends of the regenerator. This definition is similar to the steady flow friction factor and presumes that the phase angle of the pressure drop is equal to that of the mean Darcian velocity. However, in cryocooler operation, the pressure
2. Theory 2.1. Steady flow model vs. oscillating flow model Although regenerators used in cryocoolers are subjected to oscillating flow and pulsating pressure condi-
S. Choi et al. / Cryogenics 44 (2004) 203–210
tions, most previous researches have used a pseudosteady flow model involving the pressure drop such as a form of Eq. (3). In this paper, the following expression is designated as the steady flow model. m_ avg DP ¼ ADP ð3Þ _ avg M _ avg expðixtÞ. where m_ avg ¼ M As an alternative method to describe the pressure drop, a new model is proposed as the form of Eq. (4). This model, which is called an oscillating flow model in contrast to the steady flow model, represents the pressure drop in two variables; the amplitude and the phase angle with respect to the mass flow rate at the warm-end of the regenerator. DP ¼ ADP expði/DP Þ
m_ 1 _1 M
ð4Þ
_ 1 expðixtÞ. where m_ 1 ¼ M 2.2. Non-dimensionalization of oscillating flow model In order to generalize the proposed oscillating flow model, the process to find out the relevant dimensionless parameters is required prior to the experiments. A useful technique to find the dimensionless parameters is to consider the governing equations of the flow in small capillary tube. The regenerator in this analysis is assumed to be a bundle of numerous small capillary tubes. We used the following equations as the governing equations (continuity and momentum equations) of the flow in each capillary tube [9]. oq 1 oðqvrÞ oðquÞ þ þ ¼0 ot r or oz ou ou ou oP l o ou q þ r þv þu ¼ ot or oz oz r or or
ð5Þ ð6Þ
The variables are normalized by appropriate reference values as follows. q P u v ; ; P ¼ ; u ¼ ; v ¼ qm Pm U1 U1 dc =L r z r ¼ ; z ¼ and t ¼ t x dc L q ¼
These non-dimensionalized variables are substituted into the Eqs. (5) and (6) and rearranged to give the dimensionless form of the governing equations with the significant dimensionless parameters. oq 1 oðq v r Þ oðq u Þ þe þ ¼0 ð7Þ r or oz ot ou ou ou q þe v þu ot or oz e oP 1 1 o ou þ r ¼ ð8Þ c Ma2 oz Va r or or
205
U1 : ratio of gas diswhere, e (expansion parameter) ” xL placement length to tube length, Va (Valensi number) ” dc2 x=v: ratio of tube inner diameter to viscous ffi pffiffiffiffiffiffiffiffiffiffi diffusion length, Ma (Mach number) ” U1 = cRTm . The expansion parameter can be represented in terms of Reynolds number and the gas domain length ratio defined as dc =L.
dc 1 ð9Þ L Va Consequently, the governing non-dimensional parameters for oscillating flow and pulsating pressure in capillary tube are Reynolds number, Valensi number, Mach number and the gas domain length ratio. To adopt the above non-dimensionalized parameters to wire screen regenerators, the diameter of the capillary tube is set to be the hydraulic diameter of the wire screen defined as follows. ev dc ¼ dh ¼ dw ð10Þ 1 ev
e ¼ Re
Velocity amplitude at the warm-end of the regenerator (U1 ) is calculated as the following formula. _ 1 =ðqAff Þ U1 ¼ M
ð11Þ
Amplitude of the pressure drop (ADP ) can be obtained from the oscillating flow friction factor defined in a similar way to Fanning steady flow friction factor [2]. fosc
ADP dh 2qm U12 L
ð12Þ
In general, the oscillating flow velocity through the cryocooler regenerator is so small that the Mach number is usually very low [9]. Therefore, the oscillating flow friction factor and the phase angle is to have the functional relationships as follows. dh fosc ¼ function Re; Va and ð13Þ L dh /DP ¼ function Re; Va and ð14Þ L From the above functional relationships, the oscillating flow friction factor and the phase angle can be determined by measuring the mass flow rate and the pressure at the ends of the regenerator.
3. Experiments 3.1. Regenerator specimens Stainless steel wire screen is widely used as popular regenerator materials for cryocoolers. It can be classified according to the wire diameter and the weave style. The wire diameter and the porosity determine the hydraulic diameter of the wire screen by Eq. (10). Three types of
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S. Choi et al. / Cryogenics 44 (2004) 203–210
Fig. 1. Microscopic photos of #200 plain and twill screen (left: plain, right: twill).
regenerator (#400 twill, #250 twill and #200 twill screen regenerators) are fabricated for the experiment. Also, they are divided into plain square and twill square screens according to the weave style. The magnified pictures of the two kinds of screens are shown in Fig. 1. Plain square is weaved such that the wires are woven in a simple over and below pattern. On the other hand, twill square is weaved such that each wire passes over two wires and then below two. In order to investigate the effects of the weave style, #200 plain screen regenerator is fabricated and compared with the #200 twill. One more #400 twill screen regenerator is made to confirm the reliability and the repeatability of the experimental results in this study. Table 1 shows the specification of the five regenerators as mentioned above. All of the specimens are designed to have the same porosities within 3% error in order to eliminate the uncertainty caused by different porosity. This porosity value is also reasonable in most regenerative cryocoolers. 3.2. Experimental apparatus and measurements Fig. 2 shows the schematic diagram of the experimental apparatus in which the mass flow rates, the pres-
sures, and the temperatures at both sides of the regenerator are measured. A linear compressor generates oscillating flow and pulsating pressure inside the regenerator specimen. The inner diameter and the length of the regenerator housing are 12.25, 85 mm, respectively. The compressor is controlled by an inverter and a transformer, so that the experimental data are obtained with various inlet mass flow rates and frequencies (45–80 Hz). Two fine hot wires (TSI model: 1260A-T1.5) and two piezoelectric pressure transducers (Kistler model 601A) are used to measure the mass flow rate and the dynamic pressure, respectively. Since the diameter of the hot wire is 3.8 lm and the natural frequency of the piezoelectric pressure sensor is 150 kHz, there is no need to compensate for the time lag of the acquired signal. The detailed information about the experimental apparatus can be found in the authorsÕ other research paper [10]. Although the oscillating flow model assumes that the pressure wave and mass flow rate are first order harmonic functions, the actual waves include higher order components. Therefore, only the first order components of the actual waves are taken by means of curve fitting to get the amplitude and the phase angle. Fig. 3 shows the example that compares the measured data with the first order components.
4. Results and discussions 4.1. Results and correlations for twill screen regenerator Fig. 4 shows the results for the #400 twill screen regenerator. The oscillating flow friction factor is a function of Reynolds number, but independent of Valensi number. On the other hand, the phase angle is proportional to Valensi number, but independent of Reynolds number. The vertical scattering of the data is
Table 1 Geometric properties of regenerator specimens (regenerator length ¼ 85 mm, inner diameter ¼ 12.25 mm) Type of wire screen
Number of screens
Wire diameter dw [lm]
Porosity ev
Hydraulic diameter dh [lm]
Mesh distance l [lm]
Pitch p [lm]
Opening area ratio b
#400 #250 #200 #200 #400
1261 1039 691 827 1261
30 38 52 52 30
0.679 0.697 0.679 0.680 0.679
63.5 87.4 110.0 110.5 63.5
33.5 63.6 75 75 33.5
63.5 101.6 127.0 127.0 63.5
0.278 0.392 0.349 0.349 0.278
*
twill twill twill plain twill
One more #400 twill mesh was tested to verify the repeatability of the experiment.
ω Linear compressor
T1 m& 1 P1
T2 Regenerator
Orifice
P2 m& 2
Fig. 2. Schematic diagram of experimental apparatus.
Reservoir
S. Choi et al. / Cryogenics 44 (2004) 203–210
207
7
Pressure drop [bar]
1.0
Real data Sine fit
Va = 0.157 Va = 0.175 Va = 0.192 Va = 0.210 Va = 0.227 Va = 0.245 Va = 0.262 Va = 0.280
6
0.5
5 4
0.0
fosc
3
-0.5 2 -1.0 0
1 90
180
270
360 0
Angle [deg]
(a)
0
5
10
15
20
(a)
Inlet mass flow rate [kg/sec]
0.0008
30
35
40
45
50
0.7 0.6
Real data Sine fit
0.0004
0.5
0.0000
φ∆ P
-0.0004
0.4 0.3 0.2 0.1
-0.0008 0 (b)
90
180
270
360
Angle [deg]
not significant considering the error due to the first order curve fitting used in the data processing. Both the #250 twill and #200 twill screen regenerators have the same tendency as the #400 twill screen. The effects of the gas domain length ratio can be recognized from the experimental data comparisons among #400 twill, #250 twill and #200 twill screen regenerators as shown in Fig. 5. It is clear that the gas domain length ratio has a considerable effect on the phase angle, but little effect on the oscillating flow friction factor. Hence, Eq. (13) and (14) are simplified as the following equations. fosc ¼ functionðReÞ dh /DP ¼ function Va and L
ð15Þ ð16Þ
The oscillating flow friction factor of the tested twill screen regenerators is well fitted by the following correlated equation as shown in Fig. 6(a). 39:52 þ 0:01 ¼ Re
0.0 0.00
Re = 7.5 ~ 15 15 ~ 22.5 22.5 ~ 30 30 ~ 37.5 37.5 ~ 45 0.05
0.10
(b)
Fig. 3. First order harmonic component for #400 twill screen at 70 Hz (a) pressure drop, (b) inlet mass flow rate.
fosc
25
Re
ð17Þ
0.15
0.20
0.25
0.30
Va
Fig. 4. Oscillating flow friction factor and phase angle for #400 twill screen (a) friction factor, (b) phase angle.
As shown in Fig. 5(b), the phase angle (/DP ) cannot be correlated by Valensi number only. Modified phase angle as a new dimensionless parameter with the following definition [11] is introduced to get the proper correlation. /DP /DP ðdh =LÞ
2:62
ð18Þ
Fig. 6(b) shows that the modified phase angle is linearly varied with the Valensi number. Therefore, the phase angle of the twill screen regenerator is well fitted by the following correlated equation as shown in Fig. 6(b). 2:62 dh /DP ¼ 1:32 108 Va ð19Þ L 4.2. Comparison between twill screen and plain screen regenerators As all properties of both #200 twill and #200 plain screen regenerators are identical except for the weave style as shown in Table 1, the difference of their results
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S. Choi et al. / Cryogenics 44 (2004) 203–210 7
7
d h /L=0.00075 (#400 twill mesh) d h /L=0.00103 (#250 twill mesh) d h /L=0.00129 (#200 twill mesh)
6 5
5 4
4
f osc
d h/L=0.00075 (#400 twill mesh) d h/L=0.00103 (#250 twill mesh) d h/L=0.00129 (#200 twill mesh) Correlation curve
6
fosc
3
3 2 2 1 1 0
0 0
20
40
(a)
60
80
0
100
Re 1.4x10 1.0
0.8 0.7
1.0x10 -8
φ
*
∆P
0.6
φ ∆P
Re -8
1.2x10 -8
d h/L=0.00075 (#400 twill mesh) d h/L=0.00103 (#250 twill mesh) d h/L=0.00129 (#200 twill mesh)
0.9
10 20 30 40 50 60 70 80 90 100 110 120
(a)
d h/L=0.00075 (#400 twill mesh) d h/L=0.00103 (#250 twill mesh) d h/L=0.00129 (#200 twill mesh)
8.0x10 -9 6.0x10 -9
0.5 0.4
4.0x10 -9
0.3 2.0x10 -9
0.2 0.1 0.0 0.0
(b)
0.2 0.2
0.4
0.6
0.8
1.0
Va
(b)
0.4
0.6
0.8
Va
Fig. 6. Correlation curve of oscillating flow friction factor and phase angle (a) friction factor, (b) phase angle.
Fig. 5. Comparison of oscillating flow friction factor and phase angle for different mesh number (a) friction factor, (b) phase angle.
4.4. Superiority of the oscillating flow model over the steady flow model implies the effects of weave style. Fig. 7 shows the comparison between their experimental results. The weave style seems to have little effect on the phase angle, but a relatively larger effect appears on the oscillating flow friction factor. The oscillating flow friction factor of the plain screen regenerator is in general larger than that of the twill screen. Since the oscillating flow friction factor has no information about the weave style, an additional non-dimensional parameter may be required to account for the effect of the weave style. 4.3. Reliability and repeatability test In order to ensure the reliability of the experimental results, repeatability test should be performed for two specimens whose all properties are identical. Fig. 8 shows that two identical #400 twill screen regenerators have consistent results. Accordingly, uncertainty due to fabrication procedure is negligible and the experimental results in this study are consistent.
Fig. 9 shows the typical pressure drop and the mean mass flow rate data under oscillating flow and pulsating pressure. The mean mass flow rate in this plot is the averaged value between the mass flow rates at two ends of the regenerator as indicated in Eq. (3). It is clearly observed from Fig. 9 that the phase of the pressure drop is different from that of the mean mass flow rate. The steady flow model expressed by Eq. (3) cannot describe this fundamental phase shifting phenomenon because it is assumed that the pressure drop phase is always equal to the phase of the mean mass flow rate. In the oscillating flow model, the phase of the pressure drop is correlated with respect to the phase of the mass flow rate at the warm-end. Hence, it is not necessary to confine the pressure drop phase equal to the phase of the mean mass flow rate. Fig. 9 clearly shows that the oscillating flow model shall predict the pressure drop better than the steady flow model. We now compare the amplitude of the pressure drop data with the pressure drop which is predicted by the
S. Choi et al. / Cryogenics 44 (2004) 203–210
209
7
5
#200 twill mesh #200 plain mesh
4
6
#400 twill * #400 twill
5 3
f osc
4
f osc
3
2 2 1
0
1 0 0
10
20
30
40
(a)
50
60
70
80
90
100 110
Re
10
15
20
25
30
35
40
45
Re 0.7 0.6
#200 twill mesh #200 plain mesh
0.5
0.4
φ
5
(a)
0.6 0.5
0
φ∆ P
0.3
∆P
0.4 0.3 0.2
0.2
#400 twill * #400 twill
0.1 0.1
0.2
0.4
0.6
0.8
1.0
Va
Fig. 7. Effect of weave style on oscillating flow friction factor and phase angle (a) friction factor, (b) phase angle.
steady flow friction factor. Pressure drop amplitude is readily calculated by the MiyabeÕs steady flow friction factor as indicated in Eq. (1). Here, the flow rate is set to be the mean peak value between the warm and the coldends as mentioned in Eq. (2). Fig. 10 shows the pressure drop calculation from the steady flow friction factor. It is evident that when the Reynolds number increases, difference between the calculated pressure drop and the experimental data becomes large. From these experimental results, we can say that the conventional steady flow friction factor cannot accurately simulate both the amplitude and the phase angle of the pressure drop. Accordingly, the oscillating flow model proposed in this paper is undoubtedly very useful to simulate the pressure drop through the regenerator under actual operating conditions of cryocoolers. 5. Conclusions A new model is presented for accurate description of oscillating flow characteristic through regenerator under
0.1
(b)
0.2
0.3
Va
Fig. 8. Reliability and repeatability test result (a) friction factor, (b) phase angle.
0.0010
1.0
0.0008
0.8
0.0006
0.6
0.0004
0.4
0.0002
0.2
0.0000
0.0
-0.0002
-0.2
-0.0004
-0.4
-0.0006
-0.6
-0.0008
-0.8
-0.0010
0
50
100
150
200
250
300
Pressure drop (bar)
(b)
0.0 0.0
Mean mass flow rate (kg/s)
0.0 0.0
-1.0 350
Phase angle (degree) Fig. 9. Experimental data of the mean mass flow rate and the pressure drop for #200 plain screen regenerator at 75 Hz.
pulsating pressure conditions. The non-dimensionalized correlations of the model are derived from the experimental results of twill screen regenerators (#400, #250, #200).
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S. Choi et al. / Cryogenics 44 (2004) 203–210
50 Hz 60 Hz
1.50
A∆P (bar)
Acknowledgements
#200 twill mesh
1.75
The authors are grateful to Ministry of Science and Technology for the support of this work through the Dual Use Technology Program (grant no. 99-DU-04-A02). This work is also supported by LG Electronics Inc. and the Combustion Engineering Research Center of KAIST.
70 Hz Miyabe's correlation Oscillating flow model
1.25 1.00 0.75 0.50
References
0.25 0.00 10
20
30
40
50
60
70
80
90
Reavg Fig. 10. Comparison between the amplitude of the experimental pressure drop and the calculated value from the steady flow friction factor.
Oscillating flow friction factor (fosc ) and phase difference between pressure drop and mass flow rate at the warm-end (/DP ) can be expressed as follows by Reynolds number (Re), Valensi number (Va) and the gas domain length ratio (dh =L). fosc ¼
39:52 þ 0:01 Re
/DP ¼1:32 10
8
ð5 6 Re 6 100Þ
dh L
2:62
Va dh 0:00075 6 6 0:00129 L
0:15 6 Va 6 0:80;
Experimental data also demonstrate that the calculated pressure drop using the conventional steady flow friction factor is significantly deviated from the experimental data.
[1] Urieli I, Berchowitz DM. Stirling cycle engine analysis. Bristol: Adam Hilger Ltd; 1984. [2] Kays WM, London AL. Compact heat exchanger. 3rd ed. McGraw-Hill; 1984. [3] Miyabe H, Takahashi S, Hamagushi K. An approach to the design of Stirling engine regenerator matrix using packs of wire gauzes. In: 17th Intersociety of Energy Conversion Engineering Conference, 1982. [4] Zhao TS, Cheng P. Oscillatory pressure drops through a wovenscreen packed column subjected to a cyclic flow. Cryogenics 1996;36(5):333–41. [5] Tanaka M, Yamashita I, Chisaka F. Flow and heat transfer characteristics of the Stirling engine regenerator in an oscillating flow. JSME Int J 1990;33(2):283–9. [6] Hsu CT, Fu H, Cheng P. On pressure–velocity correlation of steady and oscillating flows in regenerators made of wire screens. Trans ASME J Fluids Eng 1999;121(1):52–6. [7] Helvensteijn BPM, Kashani A, Spivak AL, Roach PR, Lee JM, Kittel P. Pressure drop over regenerators in oscillating flow. Adv Cryog Eng 1998;43B:1619–26. [8] Ju Y, Jiang Y, Zhou Y. Experimental study of the oscillating flow characteristics for a regenerator in a pulse tube cryocooler. Cryogenics 1998;38(6):649–56. [9] Lee JM, Kittel P, Timmerhaus KD, Radebaugh R. Useful scaling parameters for the pulse tube. Adv Cryog Eng 1996;41:1347–55. [10] Nam K, Jeong S. Measurement of cryogenic regenerator characteristics under oscillating flow and pulsating pressure. Cryogenics 2003;43(10):575–81. [11] Bhagoria JL, Saini JS, Solanki SC. Heat transfer coefficient and friction factor correlations for rectangular solar air heater duct having transverse wedge shaped rib roughness on the absorber plate. Renew Energy 2001;25(3):341–69.