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Theoretical study on a Miniature Joule–Thomson & Bernoulli Cryocooler
Cryogenics 44 (2004) 801–807 www.elsevier.com/locate/cryogenics
Theoretical study on a Miniature Joule–Thomson & Bernoulli Cryocooler L.Y. Xiong 1, G. Kaiser *, A. Binneberg Institut f€ ur Luft- und K€altetechnik, HB K€alte- und Tieftemperaturtechnik Bertolt-Brecht-Allee 20, D-01309 Dresden, Germany Received 15 October 2003; received in revised form 22 October 2003; accepted 10 March 2004
Abstract In this paper, a microchannel-based cryocooler consisting of a compressor, a recuperator and a cold heat exchanger has been developed to study the feasibility of cryogenic cooling by the use of Joule–Thomson effect and Bernoulli effect. A set of governing equations including Bernoulli equations and energy equations are introduced and the performance of the cooler is calculated. The influences of some working conditions and structure parameters on the performance of coolers are discussed in details. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Cryocooler; Joule–Thomson effect; Bernoulli effect; Microchannel
1. Introduction The architecture of the miniature cryocooler discussed here was initially presented by Kaiser et al. [1]. As shown in Fig. 1(a), this cooler consists of three elements, a compressor, a microchannel-based recuperator and a microchannel-based cold heat exchanger. The depths of microchannels for such a cooler have the magnitude of several to tens of micrometres. In their contributions, a thermodynamic analysis based on the ideal gas model has been carried out to study the performance of a possible layout operating with helium and neon. Because the cooling by Joule–Thomson (J–T) effect does not occur for ideal gases, the cooling results only from Bernoulli effect. For this reason, they called the cooler a Bernoulli Effect Cryocooler. The research work presented in this article is a further investigation into the miniature cryocooler mentioned above. A thermodynamic analysis based on the real gas model has been carried out at this time. The J–T effect is evaluated in addition to Bernoulli effect. The results show that J–T effect plays an important role in cooling, *
Corresponding author. Tel.: +49-0351-4081-603; fax: +49-03514081-635. E-mail address: [email protected] (G. Kaiser). 1 Guest researcher from Department of Refrigeration and Cryogenic Engineering, Xi’an Jiaotong University, 710049, PR China. 0011-2275/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2004.03.023
but at the same time, the Bernoulli effect should be considered seriously. As compared with a traditional J– T Cryocooler, an isothermal expansion process (under ideal conditions) included in the thermodynamic cycle of this new cooler takes the place of the isenthalpic expansion process that is included in a traditional J–T Cryocooler cycle. Secondly, the use of microchannels has the interesting characteristics of high surface area to volume ratio, high heat transfer coefficients, high friction factors and high pressure gradients. Since the pressure drops due to the viscous resistance are greatly amplified in microchannel flow, the J–T expansion occurs over the whole flow channels. While for a traditional J–T Cryocooler, the high pressure drop happens mainly in the throttling valve. Finally, the role of the Bernoulli effect corresponding to this new cooler is more important than it is for a traditional J–T Cryocooler. For a traditional J–T Cryocooler, the variations of the flow velocities due to the Bernoulli effect are small and usually can be ignored. But for the cooler discussed here, the variations of the flow velocities become considerable. The Bernoulli effect influences the cooler performance in two ways: direct way and indirect way. In the direct way, the increase of the kinetic energy due to the pressure drop results in a potential cooling; in the indirect way, on account of the amplified viscous effect in microchannel flow, a small velocity increment may result in an amplified pressure drop, hence enhances J–T
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L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
Nomenclature A cross-section area (m2 ) AMTD arithmetic mean temperature difference in recuperator (K) COP coefficient of performance C cross-sectional average velocity (m/s) D hydraulic diameter of the channel (m) f Darcy friction factor h enthalpy of fluid (J/kg) L length (m) m_ flow rate (kg/s) Nu Nusselt number (Nu ¼ aD=k) P pressure (Pa) q0 heat transfer rate per unit length along the channel (W/m) s perimeter of the channel cross-section (m) T temperature (K) t depth of the channel (m)
W z
width of the channel (m) streamwise coordinate (m)
Greek letters a heat transfer coefficient, (W/m2 K) k thermal conductivity (W/m K) q density (kg/m3 ) Subscripts and superscripts s solid microstructure h high pressure r recuperator t throttling channel f fluid l low pressure c cold heat exchanger 0 hot end of the recuperaor
Fig. 1. Illustration of a MJTBC––schematic drawing: (a) initial layout; (b) new layout.
expansion and cooling. Although the main driving mechanism for cold generation is J–T expansion, the role of Bernoulli effect can not be ignored in the study of this microchannel-based cooler. From this point of view, we refer to such a cryocooler as a Miniature Joule– Thomson & Bernoulli Cryocooler (MJTBC) in this article.
2. Construction and operation of MJTBC Fig. 1(b) shows a new layout of the MJTBC. In the new layout, the straight microchannels are used to replace the microchannels with two 90° bends shown in Fig. 1(a), so that the additional pressure drops can be avoided that caused by the flow resistance due to bends. For the purpose of improving the cooling capability of
the new cooler, the narrow throttling microchannels are used to form the flow channels inside the cold heat exchanger and possibly inside a small part of the recuperator near the cold heat exchanger. The working processes of a MJTBC are plotted in. Fig. 2. A general ideal thermodynamic cycle is a–b–c–d. The process a–b is an isothermal compression process occurring in the compressor. The input power requirement in this process can be calculated with the following equation: Ph0 _ h0 ln Win ¼ mRT ð1Þ Pl0 The process b–c and the process d–a represent the heat exchange processes happening in the recuperator. The process c–d is an isothermal J–T expansion process taking place in the cold heat exchanger, where the heat is
2
Nitrogen
Ph0
c c'
1
d' d
b
Pl0 a' a
0 -100
0
100
200
300
h (kJ/kg)
Fig. 2. Schematic view of working processes of a MJTBC.
absorbed from the load at the load temperature Tc . For the ideal cycle, the cooling power can be evaluated by summing up the enthalpy decrease during the compression at the ambient temperature Th0 and the kinetic energy change between the inlet and the outlet of the gas flow as follows (without considering the heat loss due to axial heat conduction): _ a hb Þ þ Qideal ¼ mðh c
m_ 2 ðC Cb2 Þ 2 a
803
The real cycle of the MJTBC is a0 –b–c0 –d0 instead because of the requirement of the heat transfer temperature difference to drive the heat flow. The compression process a0 –b and the expansion process c0 –d0 are not isothermal processes indeed, which will lead to an additional heat transfer loss in the recuperator as well as to a degradation in the cooler performance. The decreased cooling power due to the heat transfer temperature difference between two streams flowing in the _ a ha0 Þ, which is referred to as an recuperator is mðh energy flow loss in some articles. In order to reduce the energy flow loss, the heat transfer temperature differences should be small. But to transfer the heat at small temperature differences requires a larger surface and a higher pressure drop.
K T h0=300
K
T=120 K
T C=140
P (MPa)
3
T=100 K
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
ð2Þ
where the enthalpy decreased through the isothermal compression is the result of the J–T effect, while the kinetic energy change results from Bernoulli effect. Similarly, we can evaluate the cooling power based on any cross-section of the recuperator by using the corresponding enthalpy and velocity parameters on that cross-section. The influence of the Bernoulli effect can be demonstrated by the following example based on equation (2). Assume the high pressure channel and the low pressure channel have the same geometric dimensions, the nitrogen gas was compressed isothermally at the ambient temperature of 300 K from 0.1 to 1.5 MPa (ha ¼ 311:409 kJ/kg, hb ¼ 308:350 kJ/kg), and then expanded inside the MJTBC. Through the expansion, the velocity increased from 6.65 to 100 m/s. As a result, the contributions of J–T effect and Bernoulli effect to the cooling power can be evaluated by ha hb ¼ 3:059 kJ/kg and 0:5ðCa2 Cb2 Þ ¼ 4:978 kJ/kg respectively. It seems that if the gas experiences a large velocity increase through the expansion inside the MJTBC, the role of the Bernoulli effect may become very important. However, we can see later that at the most places of the MJTBC, the gas velocity increase very slowly during the expansion. The J–T effect is consequently dominant in most regions. Furthermore, to increase the exit velocity Ca requires more mass flow rate, resulting in larger heat transfer temperature difference when considering the real thermodynamic cycle of the MJTBC, which will weaken the cooling ability of the cooler.
3. Theoretical analysis The theoretical analysis is conducted based on onedimensional flow and heat transfer simulation in the streamwise direction z. The coordinate z is counted positive in the direction of flow from the high pressure end to the low pressure end. The pressure distributions as well as the velocity distributions and the temperature distributions are governed by the Bernoulli equations and the energy equations. The Bernoulli equations are: dPh f qCh2 d qCh2 þ ¼ ð3Þ dz dz 2Dh 2
dPl dz
f qCl2 d þ ¼ dz 2Dl
qCl2 2
ð4Þ
The Energy equations are: dCh dhh þ m_ ah sh ðT s Thf Þ dz dz d dT f kfh Afh h ¼ 0 dz dz
_ h mC
_ l mC
ð5Þ
dCl dhl d dT f kfl Afl l ¼ 0 þ m_ al sl ðT s Tlf Þ dz dz dz dz ð6Þ
d dT s ks As ah sh ðT s Thf Þ dz dz al sl ðT s Tlf Þ þ q0 ¼ 0
ð7Þ
Here we assume that there is no heat leak from surroundings, and the solid structure temperature of the cold heat exchanger is a constant and is equal to the load temperature Tc . The flow in microchannels is generally a laminar flow as the mass flow rate employed is generally low due to high pressure drop experienced. The Darcy friction factor f and the Nusselt number Nu (used to determine the heat transfer coefficient a) applied
2.0
AMTD=0.42 K COP = -0.01 % 1.0
Pressure Temperature Velocity 0.0 0
10
120
320
100
280
80
240
60
200
40
160
20
30
40
50
60
120 0 70 80
L (mm)
Tc=160 K Cl0= 100 m/s Lr=50 mm Lc=Lt=10 mm th=tl=20 µ m tt=10 µ m
1.5
AMTD=1.52 K COP = 0.21 % Pressure 1.0
Temperature
360
120
320
100
280
80
240
60
200
40
160
20
Velocity (m/s)
ð9Þ
20
(a)
Pressure (MPa)
Qc Win
360
0.5
2.0
The input power is calculated approximately by Eq. (1). The coefficient of performance (COP) defined by the heat removed at low temperature divided by the work done at high temperature is given by:
Temperature (K)
1.5
Lr
COP ¼
Tc=160 K Cl0= 100 m/s Lr=50 mm Lc=Lt=10 mm th=tl=10 µ m tt=10µ m
Temperature (K)
to the laminar flow are taken from the correlations which can be found in Kakac et al. [2]. For the calculation of the heat conduction in solid structure that may be made from Pyrex glass, a specific thermal conductivity of ks ¼ 1 Wm1 K1 is used. For the prediction of the thermodynamic properties of gases, the Peng–Robinson equation of state is used, while the viscosity and the thermal conductivity of gases are estimated by using the Lucas-corresponding-states method and the Chung-method respectively [3]. Eqs. (3)–(7) are solved by finite difference method. The cooling power is calculated by integrating q0 through the length of the cold heat exchanger: Z Lr þLc Qc ¼ 2 q0 dz ð8Þ
Velocity (m/s)
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
Pressure (MPa)
804
0.5
Velocity
4. Results and discussion
0.0 0
The performance data of a MJTBC operating with nitrogen gas has been calculated. In the calculation, the inlet temperature of the recuperator is Th0 ¼ 300 K; the high and low pressure channels have the common width of W ¼ 50 lm; and the throttling channels for both streams have the same depths of tt ¼ 10 lm. The other thermodynamic parameters and dimensions of structures are studied as variables influencing the performance of the coolers, they are given as follows: Pl0 ¼ 0:1; . . . ; 0:4 MPa;
Cl0 ¼ 25; . . . ; 200 m=s
Tc ¼ 100; . . . ; 220 K; th ; tl ¼ 10; . . . ; 40 lm Lr ¼ 40; . . . ; 140 mm; Lc ; Lt ¼ 5; . . . ; 40 mm The mean heat transfer temperature difference of the recuperator plays an important role in the analysis of the performance of the MJTBC. The arithmetic mean temperature difference (AMTD) defined by AMTD ¼ R DT dz=L is adopted for this purpose. r Lr In order to make a comparison between the initial layout and the new layout, the profiles of the pressure, the velocity and the temperature are plotted for both layouts as shown in Fig. 3(a) and (b), in which the interface between the recuperator and the cold heat exchanger is indicated by a vertical dot line located at the place L ¼ 50 mm. In the new layout, due to the reduction of the channel depth in the region of throttling channels, the velocities jump to higher values as the gas
(b)
10
20
30
40
50
60
120 0 70 80
L (mm)
Fig. 3. Pressure, temperature and velocity profiles: (a) initial layout; (b) new layout.
flows into the throttling channels, resulting in a higher pressure drop and expansion ratio through the cold heat exchanger. The gas inside the cold heat exchanger expands from 1.02 to 0.66 MPa in the new lay out, while it expands from 1.27 to 1.14 MPa in the initial layout, which means that with the new layout more heat can be absorbed during the J–T expansion process. Therefore, the result shows that the COP for this layout is larger than that for the initial layout, although the AMTD of the recuperator for the new layout has a larger value. The temperatures of flow inside the recuperator in the initial layout stay in a lower level than that in the new layout, this is because the channels of the recuperator in the initial layout are more narrow than that in the new layout, resulting in higher pressure drops that cause J–T expansion. Fig. 3(a) and (b) also show that in the high pressure channel of the recuperator and in the channels of the cold heat exchanger the variations of velocity are very small, which means that the Bernoulli effect is insignificant in those regions. However, as the gas flows inside the low pressure channel of the recuperator the velocity increases very quickly. Especially inside about 1/3 part
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
of the low pressure channel that is close to the hot end of the recuperator, the velocity increases dramatically as the pressure decreases quickly. The increase of the kinetic energy due to the Bernoulli effect makes the low pressure stream capable of absorbing more heat from the high pressure stream. In other words, the Bernoulli effect functions in this region in a way like pre-cooling for the hot high pressure stream. Another point must be clarified here is about the meaning of the negative COP appeared initially in Fig. 3(a). For the solution of the Bernoulli and energy equations, the solid structure temperature of the cold heat exchanger is assumed to be constant and equal to the load temperature Tc , so a negative COP or a negative cooling power means that external cooling is needed to satisfy the applied boundary conditions. The relation between the exit pressure Pl0 and the cooler performance is plotted in Fig. 4(a) and (b). In the case of constant exit velocity, the mass flow rate and the total heat transfer rate increase with the increase of the exit pressure at a given exit velocity Cl0 . As a result, with the increase of the exit pressure, the AMTD of the recuperator increases directly, while the cooling power and the COP decrease. However, if we keep the mass
Tc=160 K Lr=50 mm th=tl=20 µ m
0
8
Cl0=100 m/s Lc=Lt=10 mm tt=10 µ m
6
Qc COP AMTD Ph0
-1
4
2
-2
AMTD (K) &Ph0 (MPa)
0
-3 0.1
0.2
0.3
0.4
flow rate invariable by decreasing the exit velocity under a higher exit pressure, the results shown in Fig. 4(b) indicate that the variations of the AMTD, the cooling power and the COP are very small. Fig. 5 shows the effect of the exit velocity on the performance of the cooler. The exit velocity Cl0 influences the performance of the cooler in two ways. First, a higher velocity results in a larger pressure drop due to the viscous effect. Second, as the mass flow rate is proportional to the exit velocity, the total heat transfer rate varies directly with the exit velocity. The heat transfer temperature difference in the recuperator consequently increases as the exit velocity increases. The increase of the pressure drop leads to an increase of the expansion ratio as the gas flows through the cold heat exchanger, resulting in more cooling power due to the J–T effect, while the increased AMTD leads to a degradation in the cooler performance. Therefore, an optimal exit velocity for either the COP or the cooling power can be found when the two opposite effects match well. Fig. 6 shows the variations of the performance data with the cold end temperature Tc . The inlet pressure decreases as the cold end temperature decreases because the gas has a lower viscosity at lower temperature, which leads to a reduction of the enthalpy change during the expansion process. At the same time, the AMTD of the recuperator increases with the decrease of Tc . As a result, the cooling power and the COP become smaller when the cold end temperature stays at a lower temperature level. Fig. 7 shows the influence of the recuperator length on the performance of a MJTBC. For a recuperator with a given cross section perimeter, the heat transfer surface area inside the recuperator is dependent on the length of the recuperator Lr , hence a small AMTD is expected if the recuperator is long. If the length of the cold heat exchanger is fixed, the inlet pressure increases as the length of the recuperator increases, elevating the pressure level while degrading the velocity level in the
Pl0 (MPa)
(a) 1
8
6
Qc COP AMTD Ph0
-1
Tc=160 K Lc=Lt=10 mm Lr=50 mm th=tl=20 µ m tt=10 µ m Cl0=100, 50, 33, 25 m/s
-2
2
-3
0.4
Tc=160 K Lr=50 mm th=tl=20 µ m
4
Qc COP AMTD Ph0
Pl0=0.1 MPa Lc=Lt=10 mm tt=10 µ m
3
0.2
2
0.0
1
0 0.1
(b)
4
Qc (mW) & COP (%)
0
AMTD (K) &Ph0 (MPa)
Qc (mW) & COP (%)
0.6
AMTD (K) & Ph0 (MPa)
Qc (mW) & COP (%)
1
805
0.2
0.3
0.4
Pl0 (MPa)
Fig. 4. Effect of the exit pressure on the performance of the cooler: (a) constant exit velocity; (b) constant mass flow rate.
0
-0.2 50
100
150
200
Cl0 (m/s)
Fig. 5. Effect of the exit velocity on the performance of the cooler.
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
1
Qc COP AMTD Ph0
Pl0=0.1 MPa Lc=Lt=10 mm tt=10 µ m
3
4
3
0
2
-1
1
90
120
150
180
2
Cl0=100 m/s Tc=160 K th=tl=20 µ m
Pl0=0.1 MPa Lr=50 mm tt=10 µ m
2
0
1
0 0
210
3
1
-1
0
-2
4
Qc COP AMTD Ph0
AMTD (K) & Ph0 (MPa)
Cl0=100 m/s Lr=50 mm th=tl=20 µ m
Qc (mW) & COP (%)
Qc (mW) & COP (%)
2
AMTD (K) & Ph0 (MPa)
806
10
20
30
40
Tc (K)
Lc = Lt (mm)
Fig. 6. Effect of the cold end temperature on the performance of the cooler.
Fig. 8. Effect of the cold heat exchanger length on the performance of the cooler.
Qc COP 3
AMTD Ph0
0.3
2 0.2 1
0.1
0.0 40
60
80
100
120
140
0 160
Lr (mm)
Fig. 7. Effect of the recuperator length on the performance of the cooler.
cold heat exchanger. As a result, the pressure drop and the expansion ratio of the flow inside the cold heat exchanger decrease. As displayed in Fig. 7, when the length of the recuperator is less than a certain value, the cooling power and the COP increase with the increase of Lr , but they decrease slowly if Lr exceeds that value. Fig. 8 shows the effect of the cold heat exchanger length Lc on the performance of the cooler. The cold heat exchanger length influences the pressure drop of the flow inside the cold heat exchanger and hence affects the cooling performance due to the J–T effect. The longer the length of the cold heat exchanger is, the larger the cooling power or the COP is. However, it is not rational to design a MJTBC with a very long cold heat exchanger because more input power is needed to satisfy the requirement of high pressure ratio in such case, and it is not an economical way for the COP increases more and more slowly as the Lc increases. Moreover, for a low power cryocooler in most cases the length of the cold
0.4
5
Qc COP AMTD Ph0 εc
0.3
0.2
Cl0=100 m/s Pl0=0.1 MPa Tc=160 K Lr=50 mm Lc=Lt=10 mm tt=10 µ m
4
3
0.1
2
0.0
1
-0.1
AMTD (K) & Ph0 (MPa)
0.4
Pl0=0.1 MPa Lc=Lt=10 mm tt=10 µ m
heat exchanger should be limited within a certain range to match the cold load interface. Fig. 9 shows the influence of the depths of flow channels of the recuperator on the performance of a MJTBC. Here the high pressure channel has the same depth as the low pressure channel. If the exit pressure Pl0 , the exit velocity Cl0 , the hot end temperature Th0 and the channel width are constant, the mass flow rate and the total heat transfer rate of the recuperator are approximately proportional to the low pressure channel depth tl . As tl increases the AMTD of the recuperator increases because more heat flow need to be transferred. The pressure drops of the flow inside the channels of the recuperator decrease with the increase of tl mainly due to the viscous effect, but the pressure drop of the flow inside the cold heat exchanger (where the channel depths do not change) increases with tl because of the increases of the mass flow rate, and simultaneously the expansion ratio in the cold heat exchanger ec increases. As a result, the cooler has a better performance when the channel depth is around a certain value.
Qc (mW) & COP (%)
Cl0=100 m/s Tc=160 K th=tl=20 µ m
4
AMTD (K) & Ph0 (MPa)
Qc (mW) & COP (%)
0.5
0 10
20
30
40
tl = th ( m)
Fig. 9. Effect of the recuperator channel depths on the performance of the cooler.
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
5. Conclusion In this article we have developed a theoretical model to predict the spatial distributions of the pressure, the temperature and the velocity along microchannels in MJTBCs. Based on this model the performance of the cooler operating with nitrogen gas has been calculated. The influences of some working conditions and some structure parameters have been studied. The working conditions studied include the exit pressure, the exit velocity and the cold end temperature, while the structure parameters studied include the length of the recuperator, the length of the cold heat exchanger and the depths of channels inside the recuperator. More details about the influences of the structure parameters should be studied in order to find a way to improve the performance of coolers. And further investigations concerning the use of other gases or gas mixtures are suggested. Although the initial work was first
807
conducted 5 years ago, the work to be carried on is far from enough until recently. And there is no experimental work conducted up to now, it will be our next step.
Acknowledgement This research is sponsored by the BMWA Program through contract P 0006903PBN2B.
References [1] Kaiser G, Reißig L, Th€ urk M, Seidel P. About a new type of closedcycle cryocooler operating by use of the Bernoulli effect. Cryogenics 1998;38(9):937–42. [2] Kakac S, Shah RK, Aung W. Handbook of single-phase convective heat transfer. New York: John Wiley & Sons; 1987. [3] Poling BE, Prausnitz JM, O’Connell JP. Properties of gases and liquids. 5th ed. New York: McGraw-Hill; 2000.
Theoretical study on a Miniature Joule–Thomson & Bernoulli Cryocooler L.Y. Xiong 1, G. Kaiser *, A. Binneberg Institut f€ ur Luft- und K€altetechnik, HB K€alte- und Tieftemperaturtechnik Bertolt-Brecht-Allee 20, D-01309 Dresden, Germany Received 15 October 2003; received in revised form 22 October 2003; accepted 10 March 2004
Abstract In this paper, a microchannel-based cryocooler consisting of a compressor, a recuperator and a cold heat exchanger has been developed to study the feasibility of cryogenic cooling by the use of Joule–Thomson effect and Bernoulli effect. A set of governing equations including Bernoulli equations and energy equations are introduced and the performance of the cooler is calculated. The influences of some working conditions and structure parameters on the performance of coolers are discussed in details. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Cryocooler; Joule–Thomson effect; Bernoulli effect; Microchannel
1. Introduction The architecture of the miniature cryocooler discussed here was initially presented by Kaiser et al. [1]. As shown in Fig. 1(a), this cooler consists of three elements, a compressor, a microchannel-based recuperator and a microchannel-based cold heat exchanger. The depths of microchannels for such a cooler have the magnitude of several to tens of micrometres. In their contributions, a thermodynamic analysis based on the ideal gas model has been carried out to study the performance of a possible layout operating with helium and neon. Because the cooling by Joule–Thomson (J–T) effect does not occur for ideal gases, the cooling results only from Bernoulli effect. For this reason, they called the cooler a Bernoulli Effect Cryocooler. The research work presented in this article is a further investigation into the miniature cryocooler mentioned above. A thermodynamic analysis based on the real gas model has been carried out at this time. The J–T effect is evaluated in addition to Bernoulli effect. The results show that J–T effect plays an important role in cooling, *
Corresponding author. Tel.: +49-0351-4081-603; fax: +49-03514081-635. E-mail address: [email protected] (G. Kaiser). 1 Guest researcher from Department of Refrigeration and Cryogenic Engineering, Xi’an Jiaotong University, 710049, PR China. 0011-2275/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2004.03.023
but at the same time, the Bernoulli effect should be considered seriously. As compared with a traditional J– T Cryocooler, an isothermal expansion process (under ideal conditions) included in the thermodynamic cycle of this new cooler takes the place of the isenthalpic expansion process that is included in a traditional J–T Cryocooler cycle. Secondly, the use of microchannels has the interesting characteristics of high surface area to volume ratio, high heat transfer coefficients, high friction factors and high pressure gradients. Since the pressure drops due to the viscous resistance are greatly amplified in microchannel flow, the J–T expansion occurs over the whole flow channels. While for a traditional J–T Cryocooler, the high pressure drop happens mainly in the throttling valve. Finally, the role of the Bernoulli effect corresponding to this new cooler is more important than it is for a traditional J–T Cryocooler. For a traditional J–T Cryocooler, the variations of the flow velocities due to the Bernoulli effect are small and usually can be ignored. But for the cooler discussed here, the variations of the flow velocities become considerable. The Bernoulli effect influences the cooler performance in two ways: direct way and indirect way. In the direct way, the increase of the kinetic energy due to the pressure drop results in a potential cooling; in the indirect way, on account of the amplified viscous effect in microchannel flow, a small velocity increment may result in an amplified pressure drop, hence enhances J–T
802
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
Nomenclature A cross-section area (m2 ) AMTD arithmetic mean temperature difference in recuperator (K) COP coefficient of performance C cross-sectional average velocity (m/s) D hydraulic diameter of the channel (m) f Darcy friction factor h enthalpy of fluid (J/kg) L length (m) m_ flow rate (kg/s) Nu Nusselt number (Nu ¼ aD=k) P pressure (Pa) q0 heat transfer rate per unit length along the channel (W/m) s perimeter of the channel cross-section (m) T temperature (K) t depth of the channel (m)
W z
width of the channel (m) streamwise coordinate (m)
Greek letters a heat transfer coefficient, (W/m2 K) k thermal conductivity (W/m K) q density (kg/m3 ) Subscripts and superscripts s solid microstructure h high pressure r recuperator t throttling channel f fluid l low pressure c cold heat exchanger 0 hot end of the recuperaor
Fig. 1. Illustration of a MJTBC––schematic drawing: (a) initial layout; (b) new layout.
expansion and cooling. Although the main driving mechanism for cold generation is J–T expansion, the role of Bernoulli effect can not be ignored in the study of this microchannel-based cooler. From this point of view, we refer to such a cryocooler as a Miniature Joule– Thomson & Bernoulli Cryocooler (MJTBC) in this article.
2. Construction and operation of MJTBC Fig. 1(b) shows a new layout of the MJTBC. In the new layout, the straight microchannels are used to replace the microchannels with two 90° bends shown in Fig. 1(a), so that the additional pressure drops can be avoided that caused by the flow resistance due to bends. For the purpose of improving the cooling capability of
the new cooler, the narrow throttling microchannels are used to form the flow channels inside the cold heat exchanger and possibly inside a small part of the recuperator near the cold heat exchanger. The working processes of a MJTBC are plotted in. Fig. 2. A general ideal thermodynamic cycle is a–b–c–d. The process a–b is an isothermal compression process occurring in the compressor. The input power requirement in this process can be calculated with the following equation: Ph0 _ h0 ln Win ¼ mRT ð1Þ Pl0 The process b–c and the process d–a represent the heat exchange processes happening in the recuperator. The process c–d is an isothermal J–T expansion process taking place in the cold heat exchanger, where the heat is
2
Nitrogen
Ph0
c c'
1
d' d
b
Pl0 a' a
0 -100
0
100
200
300
h (kJ/kg)
Fig. 2. Schematic view of working processes of a MJTBC.
absorbed from the load at the load temperature Tc . For the ideal cycle, the cooling power can be evaluated by summing up the enthalpy decrease during the compression at the ambient temperature Th0 and the kinetic energy change between the inlet and the outlet of the gas flow as follows (without considering the heat loss due to axial heat conduction): _ a hb Þ þ Qideal ¼ mðh c
m_ 2 ðC Cb2 Þ 2 a
803
The real cycle of the MJTBC is a0 –b–c0 –d0 instead because of the requirement of the heat transfer temperature difference to drive the heat flow. The compression process a0 –b and the expansion process c0 –d0 are not isothermal processes indeed, which will lead to an additional heat transfer loss in the recuperator as well as to a degradation in the cooler performance. The decreased cooling power due to the heat transfer temperature difference between two streams flowing in the _ a ha0 Þ, which is referred to as an recuperator is mðh energy flow loss in some articles. In order to reduce the energy flow loss, the heat transfer temperature differences should be small. But to transfer the heat at small temperature differences requires a larger surface and a higher pressure drop.
K T h0=300
K
T=120 K
T C=140
P (MPa)
3
T=100 K
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
ð2Þ
where the enthalpy decreased through the isothermal compression is the result of the J–T effect, while the kinetic energy change results from Bernoulli effect. Similarly, we can evaluate the cooling power based on any cross-section of the recuperator by using the corresponding enthalpy and velocity parameters on that cross-section. The influence of the Bernoulli effect can be demonstrated by the following example based on equation (2). Assume the high pressure channel and the low pressure channel have the same geometric dimensions, the nitrogen gas was compressed isothermally at the ambient temperature of 300 K from 0.1 to 1.5 MPa (ha ¼ 311:409 kJ/kg, hb ¼ 308:350 kJ/kg), and then expanded inside the MJTBC. Through the expansion, the velocity increased from 6.65 to 100 m/s. As a result, the contributions of J–T effect and Bernoulli effect to the cooling power can be evaluated by ha hb ¼ 3:059 kJ/kg and 0:5ðCa2 Cb2 Þ ¼ 4:978 kJ/kg respectively. It seems that if the gas experiences a large velocity increase through the expansion inside the MJTBC, the role of the Bernoulli effect may become very important. However, we can see later that at the most places of the MJTBC, the gas velocity increase very slowly during the expansion. The J–T effect is consequently dominant in most regions. Furthermore, to increase the exit velocity Ca requires more mass flow rate, resulting in larger heat transfer temperature difference when considering the real thermodynamic cycle of the MJTBC, which will weaken the cooling ability of the cooler.
3. Theoretical analysis The theoretical analysis is conducted based on onedimensional flow and heat transfer simulation in the streamwise direction z. The coordinate z is counted positive in the direction of flow from the high pressure end to the low pressure end. The pressure distributions as well as the velocity distributions and the temperature distributions are governed by the Bernoulli equations and the energy equations. The Bernoulli equations are: dPh f qCh2 d qCh2 þ ¼ ð3Þ dz dz 2Dh 2
dPl dz
f qCl2 d þ ¼ dz 2Dl
qCl2 2
ð4Þ
The Energy equations are: dCh dhh þ m_ ah sh ðT s Thf Þ dz dz d dT f kfh Afh h ¼ 0 dz dz
_ h mC
_ l mC
ð5Þ
dCl dhl d dT f kfl Afl l ¼ 0 þ m_ al sl ðT s Tlf Þ dz dz dz dz ð6Þ
d dT s ks As ah sh ðT s Thf Þ dz dz al sl ðT s Tlf Þ þ q0 ¼ 0
ð7Þ
Here we assume that there is no heat leak from surroundings, and the solid structure temperature of the cold heat exchanger is a constant and is equal to the load temperature Tc . The flow in microchannels is generally a laminar flow as the mass flow rate employed is generally low due to high pressure drop experienced. The Darcy friction factor f and the Nusselt number Nu (used to determine the heat transfer coefficient a) applied
2.0
AMTD=0.42 K COP = -0.01 % 1.0
Pressure Temperature Velocity 0.0 0
10
120
320
100
280
80
240
60
200
40
160
20
30
40
50
60
120 0 70 80
L (mm)
Tc=160 K Cl0= 100 m/s Lr=50 mm Lc=Lt=10 mm th=tl=20 µ m tt=10 µ m
1.5
AMTD=1.52 K COP = 0.21 % Pressure 1.0
Temperature
360
120
320
100
280
80
240
60
200
40
160
20
Velocity (m/s)
ð9Þ
20
(a)
Pressure (MPa)
Qc Win
360
0.5
2.0
The input power is calculated approximately by Eq. (1). The coefficient of performance (COP) defined by the heat removed at low temperature divided by the work done at high temperature is given by:
Temperature (K)
1.5
Lr
COP ¼
Tc=160 K Cl0= 100 m/s Lr=50 mm Lc=Lt=10 mm th=tl=10 µ m tt=10µ m
Temperature (K)
to the laminar flow are taken from the correlations which can be found in Kakac et al. [2]. For the calculation of the heat conduction in solid structure that may be made from Pyrex glass, a specific thermal conductivity of ks ¼ 1 Wm1 K1 is used. For the prediction of the thermodynamic properties of gases, the Peng–Robinson equation of state is used, while the viscosity and the thermal conductivity of gases are estimated by using the Lucas-corresponding-states method and the Chung-method respectively [3]. Eqs. (3)–(7) are solved by finite difference method. The cooling power is calculated by integrating q0 through the length of the cold heat exchanger: Z Lr þLc Qc ¼ 2 q0 dz ð8Þ
Velocity (m/s)
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
Pressure (MPa)
804
0.5
Velocity
4. Results and discussion
0.0 0
The performance data of a MJTBC operating with nitrogen gas has been calculated. In the calculation, the inlet temperature of the recuperator is Th0 ¼ 300 K; the high and low pressure channels have the common width of W ¼ 50 lm; and the throttling channels for both streams have the same depths of tt ¼ 10 lm. The other thermodynamic parameters and dimensions of structures are studied as variables influencing the performance of the coolers, they are given as follows: Pl0 ¼ 0:1; . . . ; 0:4 MPa;
Cl0 ¼ 25; . . . ; 200 m=s
Tc ¼ 100; . . . ; 220 K; th ; tl ¼ 10; . . . ; 40 lm Lr ¼ 40; . . . ; 140 mm; Lc ; Lt ¼ 5; . . . ; 40 mm The mean heat transfer temperature difference of the recuperator plays an important role in the analysis of the performance of the MJTBC. The arithmetic mean temperature difference (AMTD) defined by AMTD ¼ R DT dz=L is adopted for this purpose. r Lr In order to make a comparison between the initial layout and the new layout, the profiles of the pressure, the velocity and the temperature are plotted for both layouts as shown in Fig. 3(a) and (b), in which the interface between the recuperator and the cold heat exchanger is indicated by a vertical dot line located at the place L ¼ 50 mm. In the new layout, due to the reduction of the channel depth in the region of throttling channels, the velocities jump to higher values as the gas
(b)
10
20
30
40
50
60
120 0 70 80
L (mm)
Fig. 3. Pressure, temperature and velocity profiles: (a) initial layout; (b) new layout.
flows into the throttling channels, resulting in a higher pressure drop and expansion ratio through the cold heat exchanger. The gas inside the cold heat exchanger expands from 1.02 to 0.66 MPa in the new lay out, while it expands from 1.27 to 1.14 MPa in the initial layout, which means that with the new layout more heat can be absorbed during the J–T expansion process. Therefore, the result shows that the COP for this layout is larger than that for the initial layout, although the AMTD of the recuperator for the new layout has a larger value. The temperatures of flow inside the recuperator in the initial layout stay in a lower level than that in the new layout, this is because the channels of the recuperator in the initial layout are more narrow than that in the new layout, resulting in higher pressure drops that cause J–T expansion. Fig. 3(a) and (b) also show that in the high pressure channel of the recuperator and in the channels of the cold heat exchanger the variations of velocity are very small, which means that the Bernoulli effect is insignificant in those regions. However, as the gas flows inside the low pressure channel of the recuperator the velocity increases very quickly. Especially inside about 1/3 part
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
of the low pressure channel that is close to the hot end of the recuperator, the velocity increases dramatically as the pressure decreases quickly. The increase of the kinetic energy due to the Bernoulli effect makes the low pressure stream capable of absorbing more heat from the high pressure stream. In other words, the Bernoulli effect functions in this region in a way like pre-cooling for the hot high pressure stream. Another point must be clarified here is about the meaning of the negative COP appeared initially in Fig. 3(a). For the solution of the Bernoulli and energy equations, the solid structure temperature of the cold heat exchanger is assumed to be constant and equal to the load temperature Tc , so a negative COP or a negative cooling power means that external cooling is needed to satisfy the applied boundary conditions. The relation between the exit pressure Pl0 and the cooler performance is plotted in Fig. 4(a) and (b). In the case of constant exit velocity, the mass flow rate and the total heat transfer rate increase with the increase of the exit pressure at a given exit velocity Cl0 . As a result, with the increase of the exit pressure, the AMTD of the recuperator increases directly, while the cooling power and the COP decrease. However, if we keep the mass
Tc=160 K Lr=50 mm th=tl=20 µ m
0
8
Cl0=100 m/s Lc=Lt=10 mm tt=10 µ m
6
Qc COP AMTD Ph0
-1
4
2
-2
AMTD (K) &Ph0 (MPa)
0
-3 0.1
0.2
0.3
0.4
flow rate invariable by decreasing the exit velocity under a higher exit pressure, the results shown in Fig. 4(b) indicate that the variations of the AMTD, the cooling power and the COP are very small. Fig. 5 shows the effect of the exit velocity on the performance of the cooler. The exit velocity Cl0 influences the performance of the cooler in two ways. First, a higher velocity results in a larger pressure drop due to the viscous effect. Second, as the mass flow rate is proportional to the exit velocity, the total heat transfer rate varies directly with the exit velocity. The heat transfer temperature difference in the recuperator consequently increases as the exit velocity increases. The increase of the pressure drop leads to an increase of the expansion ratio as the gas flows through the cold heat exchanger, resulting in more cooling power due to the J–T effect, while the increased AMTD leads to a degradation in the cooler performance. Therefore, an optimal exit velocity for either the COP or the cooling power can be found when the two opposite effects match well. Fig. 6 shows the variations of the performance data with the cold end temperature Tc . The inlet pressure decreases as the cold end temperature decreases because the gas has a lower viscosity at lower temperature, which leads to a reduction of the enthalpy change during the expansion process. At the same time, the AMTD of the recuperator increases with the decrease of Tc . As a result, the cooling power and the COP become smaller when the cold end temperature stays at a lower temperature level. Fig. 7 shows the influence of the recuperator length on the performance of a MJTBC. For a recuperator with a given cross section perimeter, the heat transfer surface area inside the recuperator is dependent on the length of the recuperator Lr , hence a small AMTD is expected if the recuperator is long. If the length of the cold heat exchanger is fixed, the inlet pressure increases as the length of the recuperator increases, elevating the pressure level while degrading the velocity level in the
Pl0 (MPa)
(a) 1
8
6
Qc COP AMTD Ph0
-1
Tc=160 K Lc=Lt=10 mm Lr=50 mm th=tl=20 µ m tt=10 µ m Cl0=100, 50, 33, 25 m/s
-2
2
-3
0.4
Tc=160 K Lr=50 mm th=tl=20 µ m
4
Qc COP AMTD Ph0
Pl0=0.1 MPa Lc=Lt=10 mm tt=10 µ m
3
0.2
2
0.0
1
0 0.1
(b)
4
Qc (mW) & COP (%)
0
AMTD (K) &Ph0 (MPa)
Qc (mW) & COP (%)
0.6
AMTD (K) & Ph0 (MPa)
Qc (mW) & COP (%)
1
805
0.2
0.3
0.4
Pl0 (MPa)
Fig. 4. Effect of the exit pressure on the performance of the cooler: (a) constant exit velocity; (b) constant mass flow rate.
0
-0.2 50
100
150
200
Cl0 (m/s)
Fig. 5. Effect of the exit velocity on the performance of the cooler.
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
1
Qc COP AMTD Ph0
Pl0=0.1 MPa Lc=Lt=10 mm tt=10 µ m
3
4
3
0
2
-1
1
90
120
150
180
2
Cl0=100 m/s Tc=160 K th=tl=20 µ m
Pl0=0.1 MPa Lr=50 mm tt=10 µ m
2
0
1
0 0
210
3
1
-1
0
-2
4
Qc COP AMTD Ph0
AMTD (K) & Ph0 (MPa)
Cl0=100 m/s Lr=50 mm th=tl=20 µ m
Qc (mW) & COP (%)
Qc (mW) & COP (%)
2
AMTD (K) & Ph0 (MPa)
806
10
20
30
40
Tc (K)
Lc = Lt (mm)
Fig. 6. Effect of the cold end temperature on the performance of the cooler.
Fig. 8. Effect of the cold heat exchanger length on the performance of the cooler.
Qc COP 3
AMTD Ph0
0.3
2 0.2 1
0.1
0.0 40
60
80
100
120
140
0 160
Lr (mm)
Fig. 7. Effect of the recuperator length on the performance of the cooler.
cold heat exchanger. As a result, the pressure drop and the expansion ratio of the flow inside the cold heat exchanger decrease. As displayed in Fig. 7, when the length of the recuperator is less than a certain value, the cooling power and the COP increase with the increase of Lr , but they decrease slowly if Lr exceeds that value. Fig. 8 shows the effect of the cold heat exchanger length Lc on the performance of the cooler. The cold heat exchanger length influences the pressure drop of the flow inside the cold heat exchanger and hence affects the cooling performance due to the J–T effect. The longer the length of the cold heat exchanger is, the larger the cooling power or the COP is. However, it is not rational to design a MJTBC with a very long cold heat exchanger because more input power is needed to satisfy the requirement of high pressure ratio in such case, and it is not an economical way for the COP increases more and more slowly as the Lc increases. Moreover, for a low power cryocooler in most cases the length of the cold
0.4
5
Qc COP AMTD Ph0 εc
0.3
0.2
Cl0=100 m/s Pl0=0.1 MPa Tc=160 K Lr=50 mm Lc=Lt=10 mm tt=10 µ m
4
3
0.1
2
0.0
1
-0.1
AMTD (K) & Ph0 (MPa)
0.4
Pl0=0.1 MPa Lc=Lt=10 mm tt=10 µ m
heat exchanger should be limited within a certain range to match the cold load interface. Fig. 9 shows the influence of the depths of flow channels of the recuperator on the performance of a MJTBC. Here the high pressure channel has the same depth as the low pressure channel. If the exit pressure Pl0 , the exit velocity Cl0 , the hot end temperature Th0 and the channel width are constant, the mass flow rate and the total heat transfer rate of the recuperator are approximately proportional to the low pressure channel depth tl . As tl increases the AMTD of the recuperator increases because more heat flow need to be transferred. The pressure drops of the flow inside the channels of the recuperator decrease with the increase of tl mainly due to the viscous effect, but the pressure drop of the flow inside the cold heat exchanger (where the channel depths do not change) increases with tl because of the increases of the mass flow rate, and simultaneously the expansion ratio in the cold heat exchanger ec increases. As a result, the cooler has a better performance when the channel depth is around a certain value.
Qc (mW) & COP (%)
Cl0=100 m/s Tc=160 K th=tl=20 µ m
4
AMTD (K) & Ph0 (MPa)
Qc (mW) & COP (%)
0.5
0 10
20
30
40
tl = th ( m)
Fig. 9. Effect of the recuperator channel depths on the performance of the cooler.
L.Y. Xiong et al. / Cryogenics 44 (2004) 801–807
5. Conclusion In this article we have developed a theoretical model to predict the spatial distributions of the pressure, the temperature and the velocity along microchannels in MJTBCs. Based on this model the performance of the cooler operating with nitrogen gas has been calculated. The influences of some working conditions and some structure parameters have been studied. The working conditions studied include the exit pressure, the exit velocity and the cold end temperature, while the structure parameters studied include the length of the recuperator, the length of the cold heat exchanger and the depths of channels inside the recuperator. More details about the influences of the structure parameters should be studied in order to find a way to improve the performance of coolers. And further investigations concerning the use of other gases or gas mixtures are suggested. Although the initial work was first
807
conducted 5 years ago, the work to be carried on is far from enough until recently. And there is no experimental work conducted up to now, it will be our next step.
Acknowledgement This research is sponsored by the BMWA Program through contract P 0006903PBN2B.
References [1] Kaiser G, Reißig L, Th€ urk M, Seidel P. About a new type of closedcycle cryocooler operating by use of the Bernoulli effect. Cryogenics 1998;38(9):937–42. [2] Kakac S, Shah RK, Aung W. Handbook of single-phase convective heat transfer. New York: John Wiley & Sons; 1987. [3] Poling BE, Prausnitz JM, O’Connell JP. Properties of gases and liquids. 5th ed. New York: McGraw-Hill; 2000.