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Cooling effect at one end of a hollow tube with a pulsating pressure in the inside gas, was first observed by Gifford and Longsworth in early sixties. This marks the inception of one of the most promising cryogenic refrigerators known as 'pulse tube refrigerator' (PTR). Cryogenics is the science of low temperature. Cryogenics refers to the entire phenomenon occurring below -150°C or 123K. Cryogenic engineering involves the design and development of systems and components which produce maintain, or utilize low temperatures. Cryocoolers are devices which produce the required refrigeration power at low temperature. The pulse tube refrigerator has been investigated for cooling various types of sensitive sensors such as infrared detectors for missiles, military aircrafts, tanks, night vision equipment and SQUIDs (super conducting quantum interference devices).

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1.1 Introduction
Cryogenics is the science of low temperature. Cryogenics refers to the entire phenomenon occurring below -150°C or 123K. Cryogenic engineering involves the design and development of systems and components which produce maintain, or utilize low temperatures. Cryocoolers are devices which produce the required refrigeration power at low temperature. There is an increasingly strong need for cryocoolers for various applications. For military purpose, almost all infrared detectors, thermal imagers and night vision devices need some form of cryogenic cooling. The space borne infrared sensors require cryocoolers of particularly high reliability and long life time. In the electronics, cryogenic cooling of the chips of semiconductors brings about significant improvement of performance by drastically reducing the thermal noises. Cryocoolers are main component of the ultra clean high vacuum cryopumps used for high reliability in the manufacture of semiconductors and other thin films. In the fields of medicine and cryobiology, cryocoolers are used for cryogenic surgery and conservation. Cryocoolers are also used as small scale helium liquefiers for the applications such as the cooling of super conducting magnets.
Low cost and high reliability are the crucial factor for the successful applications of cryocoolers in these important domains.
Cooling effect at one end of a hollow tube with a pulsating pressure in the inside gas, was first observed by Gifford and Longsworth in early sixties. This marks the inception of one of the most promising cryogenic refrigerators known as 'pulse tube refrigerator' (PTR). Due to the absence of moving parts in the cold temperature region, and the associated advantages of simplicity and enhanced reliability, the pulse tube system has become one of the most researched topics in the area of cryogenic refrigeration. The pulse tube refrigerator has been investigated for cooling various types of sensitive sensors such as infrared detectors for missiles, military aircrafts, tanks, night
vision equipment and SQUIDs (super conducting quantum interference devices). The main advantage of the new device, as compared to Stirling and Gifford-McMahon systems, is its reliability to reach very low temperature rapidly with less energy consumption, in miniature scale designs and without moving parts in the low temperature zone. Recently their range of use is being extended to various types of super conducting magnet applications.
Cryogenic temperatures are achieved and maintained by one or more refrigerating units known as 'cryocoolers'. Cryocoolers can be classified as either recuperative or regenerative cryocoolers.
1.2.1 Recuperative Cryocoolers.
The recuperative coolers use only recuperative heat exchangers and operate with a steady flow of refrigerant through the system. Figure 1.1 shows the schematic of three common recuperative cryocooler cycles. The Joule-Thompson (JT) cryocooler shown in figure 1.1(a) is very much like the vapour compression refrigerator, except for the addition of the main heat exchanger to cover the long temperature span. In vapour compression refrigerators the compression takes place below the critical temperature of the refrigerant. As a result liquefaction at room temperature occurs in the condenser. Expansion of the liquid in the JT capillary, orifice or valve is relatively efficient and provides enough temperature drops that little or no heat exchange with the returning cold, expanded gas is required. Nevertheless, the irreversible expansion of the fluid in the JT valve is less efficient than a reversible expansion with an expansion engine or turbine. Thus the Brayton cryocooler shown in figure 1.1(b) offers the potential for higher efficiency with sacrifice in simplicity, cost, and possibly reliability. The Claude system combines the JT and the Brayton cycle shown in figure 1.1©. It is used primarily for gas liquefaction where liquid may damage the expansion engines or turbines. The use of valves in compressors or the high pressure-ratios needed for recuperative cryocoolers limits the efficiency of the compression process to about 50% and significantly limits the overall efficiency of recuperative refrigerators.
1.2.2 Regenerative Cryocoolers.
Pulse k,be
or J
(b) Pu\m Tube © Giffofd-ycMahon
Figure 1.2 Regenerative cryocoolers.
A regenerative cryocooler has at least one regenerative heat exchanger or regenerator and operates with oscillating flow and pressure. In a regenerator, the incoming hot gas transfers heat to the matrix of the regenerator where the heat is stored for a half cycle in the mass of the matrix. In the second half of the cycle the returning cold gas, flowing in the opposite direction through the same channel, picks up heat from the matrix and returns the matrix to its original temperature before the cycle is repeated. At equilibrium one end of the regenerator is at room temperature while the other end is at the cold temperature. Very high surface areas for enhanced heat transfer are easily achieved in regenerators through the use of stacked fine-mesh screen or packed spheres. Stirling Refrigerator.
The compressor in the Stirling refrigerator is a valve less type. It creates an oscillating pressure in the system where the amplitude of oscillation is typically about 10 to 30% of the average pressure. In order to provide high power densities and keep the system small, the average pressure is typically in the range of 1 to 30 MPa and frequencies are in the range of 20 to 60 Hz. Helium is almost always used as the working fluid in the regenerative cycles because of its ideal gas properties, its high thermal conductivity, and its high ratio of specific heats.
A pressure oscillation by itself in a system would simply cause temperature to oscillate and produce no refrigeration. The second moving component, the displacer, is required to separate the heating and cooling effects by introducing motion of the gas in the proper phase relationship with the pressure oscillation. When the displacer in figure 1.2(a) is moved downward, the helium gas is displaced to the warm end of the system through the regenerator. The piston in the compressor then compresses the gas, and the heat of compression is removed by heat exchange with the ambient. Next the displacer is moved up to displace the gas through the regenerator to the cold end of the system. The piston then expands the gas, mow located at the cold end, and the cooled gas absorbs heat from the system before the displacer forces the gas back to the warm end through the regenerator. There is little pressure difference across the displacer (only enough to overcome the pressure drop in the regenerator) but there is large temperature difference.
In practice, motion of the piston and the displace!" is almost always sinusoidal. The correct phasing occurs when the volume variation in the cold expansion space leads the volume variation in the wann compression space by about 90 degree. With this condition the mass flow or volume flow through the regenerator is approximately in phase with the pressure. In analogy with the AC electrical systems, real power flows only with current and voltage in phase with each other. Without the displacer in the Stirling cycle the mass flow leads the pressure by 90 degree and no refrigeration occurs. Though the moving piston causes both compression and expansion of the gas, net power input is required to drive the system since the pressure is high during compression process. Likewise the moving displacer reversibly extracts net work from the gas at the cold end and transmits it to the warm end where it contributes some to the compression work. Gifford-McMahon Refrigerator.
Because the pressure oscillates everywhere within the Stirling refrigerator, excess void volumes must be minimized to maintain a large pressure amplitude for a given swept volume of the piston. Thus oil removal equipment cannot be tolerated, which means that the moving piston and displacer must be oil free. In the mid 1960s Gifford and McMahon showed that pressure oscillations for cryocoolers could be generated by the use of a rotary valve that switches between high and low pressure sources, the Gifford-McMahon refrigerator, shown in figure 1.2© has the same low temperature parts as the Stirling refrigerator. The irreversible expansion through the valve significantly reduces the efficiency of the process, but the advantage of this approach is that it allows for an oil lubricated compressor with an oil removal equipment on the high side to supply the high and low pressure sources. Pulse Tube Refrigerator.
The pulse tube refrigerator, first conceived in the mid 1960s, was of academic interest until 1984. Since then, improvement in its efficiency has occurred rapidly. Unlike the Stirling or Gifford-Mc-Mahon refrigerators, it has no moving parts at the cold end region. One variation has also been developed with no moving parts in the entire system. The lack of cold moving parts has allowed it to solve some of the problems associated
with cryocoolers in many different applications, such as vibration and reliability. Initially its operating principles were not well understood. The oscillatory flow inside the pulse tube and the associated complex thermodynamic processes are responsible for it. As the level of understanding grew gradually, modifications and improved designs yielded much improved efficiencies. It has become the most efficient cryocooler for a given size. It is also suitable for a wide variety of applications from civilian to government to military and from ground equipment to space systems.
The moving displacer in the Stirling and Gifford-McMahon refrigerators has several disadvantages. It is a source of vibration, has a short lifetime, and contributes to axial heat conduction as well as shuttle heat loss. In pulse tube refrigerator, as shown in figure 1.2(b), the displacer is eliminated.
The proper gas motion in phase with the pressure is achieved by the use of an orifice and a reservoir volume to store the gas during a half cycle. The reservoir volume is large enough that negligible pressure oscillation occurs in if during the oscillating flow. The oscillating flow through the orifice separates the heating and cooling effects just as the displacer does for the Stirling and Gifford-McMahon refrigerators.
The orifice pulse tube refrigerator (OPTR) operates ideally with adiabatic compression and expansion in the pulse tube. The four steps in the cycle are as follows.
(1) The piston moves down to compress the gas in the pulse tube.
(2) Because this heated, compressed gas is at a higher pressure than the average pressure in the reservoir, it flows through the orifice into the reservoir and exchanges heat with the ambient through the heat exchanger at the warm end of the pulse tube. The flow stops when the pressure in the pulse tube is reduced to the average pressure.
(3) The piston moves up and expands the gas adiabatically in the pulse tube.
(4) This cold, low pressure gas in the pulse tube is forced toward the cold end by the
gas flow from the reservoir into the pulse tube through the orifice. As the cold gas
flows through the heat exchanger at the cold end of the pulse tube it picks up heat
from the object being cooled. The flow stops when the pressure in the pulse tube
increases to the average pressure. The cycle then repeats. The function of the
regenerator is the same as that in the Stirling and Gifford-McMahon refrigerators, in that it pre-cools the incoming high pressure gas before it reaches the cold end.
The function of the pulse tube is to insulate the process at its two ends. That is, it must be large enough that gas flowing from the warm end traverses only part way through the pulse tube before flow is reversed. Likewise, flow in from the cold end never reaches the warm end. Gas in the middle portion of the pulse tube never leaves the pulse tube thus fonning a temperature gradient that insulates the two ends from each other. Roughly speaking, the gas in the pulse tube is divided into three segments, with the middle segment acting like a displacer but consisting of gas rather than a solid material. For this gas plug to effectively insulate the two ends of the pulse tube, turbulence in the pulse tube must be minimized. Thus, flow straightening at the two ends is crucial to the successful of the pulse tube refrigerator. The pulse tube is the unique component in this refrigerator that appears not to have been used previously in any other system. The compressor for the pulse tube refrigerator can be a valve less type, sometimes referred to as a Stirling type compressor. The pulse tube refrigerator can be driven with any source of oscillating pressure. It can be, and often is, driven with a valved compressor like that for the Gifford-McMahon refrigerator.
The first report of pulse tube refrigeration by W.E. Gilford and R.C. Longsworth inl963 was enough to excite many researchers due to the potential of high reliability inspite of its simplicity. For several decades then, many researchers have concentrated their efforts on improving the performance of pulse tube refrigerators in various ways. As a result, different configurations of pulse tube refrigerators have been introduced. Several representative configurations are detailed below.
1.3.1 Basic Pulse Tube Refrigerator.
Figure 1.3(a) shows a schematic diagram of a basic pulse tube refrigerator. A basic pulse tube refrigerator consists of a compressor, after cooler, regenerator, cold heat exchanger, hot heat exchanger and pulse tube. The periodic pressurization and expansion produced by the compressor causes the gas to flow back and forth through the regenerator and pulse tube. Figure 1.3(b) depicts the cooling mechanism of a basic pulse tube refrigerator. During the compression process, pressurized gas moves towards the hot heat exchanger located at the closed end of the pulse tube. The gas in the pulse tube experiences near adiabatie compression and associated temperature rise. The gas at the boundary layer exchanges heat with the tube wall. Heat transfer, through the hot end heat exchanger wall, cools the gas in the hot heat exchanger. During the subsequent expansion process, the depressurized gas moves towards the cold heat exchanger. The gas element experiences near adiabatie expansion and an associated temperature drop. The wall releases heat to the gas. The net heat transfer between the gas and the pulse tube wall thus shuttles heat from the cold end to the warm end. However the net, amount of heat transferred is relatively small and disappears when the temperature gradient in the wall becomes sufficiently large to match the temperature excursions developed in the gas during the compression and expansion processes. This is the so-called 'surface heat pumping theory' that explains the cooling mechanism of the basic pulse tube.
^ llllllill


hot heat exchanger
* QH "
pulse tube
cold heat exchanger
(a) basic pulse tube refrigerator
hot heat exchanger
cold heat excrar:;-;-
position (b) surface heat pumping Figure 1.3 Schematic diagram of a basic pulse tube refrigerator and its cooling mechanism.
However, there is a severe limitation in the pulse tube refrigerator that is related to mass flow and pressure wave. The phase difference between the pressure and the mass flow rate in a basic pulse tube refrigerator is 90°. In other words, when the pressure becomes the maximum, the mass flow rate becomes zero at the warm end of the pulse tube. Since the phase difference between the pressure or temperature and mass flow rate is 90 degree, the net enthalpy flow in to the hot heat exchanger must be zero and therefore the cooling power for the basic pulse tube refrigerator is also zero. Thus the cooling mechanism of a basic pulse tube refrigerator is only related to the heat transfer between the gas and the pulse tube wall, which is called the surface heat pumping mechanism. Additional cooling power can be achieved by changing the phase between the pressure and mass flow rates. By placing an orifice valve and a reservoir after the hot heat exchanger, it is possible to reduce the phase difference between the pressure and mass flow rate to a value less than 90 degree. This important advance in pulse tube configuration was achieved by Mikulin in 1984 and is referred to as orifice pulse tube refrigerator.
1.3.2 Orifice Pulse Tube Refrigerator.
In figure 1.4, an orifice valve and reservoir have been added at the end of the hot heat exchanger. The reservoir is large enough to be maintained at nearly constant intermediate pressure during operation. The valve and the reservoir cause the gas to flow through the orifice valve at the points of maximum and minimum pressures. Therefore the reservoir improves the phase relationship between pressure and gas motion. The orifice pulse tube creates refrigeration through PV-work as well as surface heat pumping. The gas column in the pulse tube acts like the displacer in the Stirling cycle refrigerator. This PV-work is transferred from the compressor to the cold heat exchanger through the regenerator. It is continuously delivered from the cold end to the hot end of the pulse tube with associated pressure changes. Then this PV-work is dissipated in the valve and transferred as heat in the hot heat exchanger. As a result, both the PV-work transferred by the gas column and the surface heat pumping near the pulse tube wall affects the cooling performance.
(^) orifice valve
hot heat exchanger QH
pulse tube
cold heat exchanger
Figure 1.4 Schematic diagram of an orifice pulse tube refrigerator.
The disadvantage of the orifice pulse tube is the fact that a large amount of compressed gas that produces no actual refrigeration, must flow through the regenerator. This decreases the refrigeration power per unit of compressed mass and therefore increases the regenerator loss. The larger the mass flow rate in the regenerator is, the smaller the effectiveness of the regenerator will be, and the larger the pressure loss will be. Both of these effects cause a reduced performance.
1.3.3 Double-Inlet Orifice Pulse Tube Refrigerator.
Matsubara and Gao attempted a modification to overcome the disadvantages of orifice pulse tube refrigerator by adding a second orifice valve between the compressor and the hot heat exchanger. The second orifice valve, or bypass valve, helps to pressurize the pulse tube without bringing all the required gas through the regenerator.
w E j

(X) orifice valve

by-pass valve hot heat exchanaer

pulse tube
co!d heat exchanger
Figure 1.5 Schematic diagram of a double-inlet orifice pulse tube refrigerator.
1.3.4 Stirling Type and G-M Type Pulse tube Refrigerator.
Pulse tube systems can be classified as either a Stirling type or a G-M type according to the method of pressurization and expansion. For a Stirling type pulse tube as shown in figure 1.6, a piston cylinder apparatus is connected to the system so thSat the pressure fluctuations are directly generated by the piston movement. The typical operating frequency is 10 to 100 Hz, higher than that of a G-M type pulse tube. Because of this high operating frequency and the absence of valve losses, Stirling type pulse tube systems generally produce higher cooling powers than G-M type pulse tubes. However, the rapid heat exchange required in Stirling type pulse tube refrigerators limits their performance at lower temperatures, such as 10K and below.
Q, *
hot heat exchanger
* Qu *
pulse tube
cold heat exchanger
Figure 1.6 Schematic diagram of a Stirling type basic pulse tube refrigerator.
The G-M type pulse tube refrigerator distributes high/low pressure gas into the pulse tube and other components by use of a valve system. The periodic opening/closing operation of the high/low pressure valves produces a pressure pulsation in the system. Because of the limitations associated with the valve operation, a typical G-M type pulse tube operates at frequencies of a few Hz. The valve system separating the compressor and pulse tube system provides the possibility of eliminating vibration problems caused by the compressor and permits remote location of the compressor from the cold end.
Figure 1.7 Schematic diagram of a G-M type basic pulse tube refrigerator.
There is a wide range of applications such as cooling of infrared sensors for military, space and commercial applications. Other applications include cooling for cryopumps, super-conducting electronics and power systems, semiconductor electronics, gas liquefaction and cryosurgical devices. Some of the major applications of PTR's are listed below.
Military Infrared sensors for missile guidance and tactical applications.
Infrared sensors for surveillance(satellite based)
Police and Infrared sensors for night vision and rescue.
Environmental Infrared sensors for atmospheric studies of ozone hole and green house effects.
Infrared sensors for pollution monitoring.
Commercial Cryopumps for semiconductor fabrication.
High temperature superconductors for cellular-phone base stations. Superconductors for voltage standards. Semiconductors for high speed computers.
Infrared sensors for NDE (non-destructive evaluation) and process monitoring.
Medical Cooling of SC magnets for MRI (magnetic resonance imaging)
SQUID magnetometers for heart and brain studies. Liquefaction of oxygen for storage at hospitals and home use. Cryogenic catheters and cryosurgery.
LNG for fleet vehicles. SC magnets for maglev trains.
Infrared sensors for thennal loss measurements. SC power applications, (motors, transformers, etc.)
Agriculture and Storage of biological cells and specimen. Biology
In the course of the development of the pulse tube refrigerator, continuous efforts have been devoted to the understanding of the refrigeration mechanism. A variety of theories have so far been proposed.
1.5.1 Surface Heat Pumping Theory.
In their initial work, W.E.Gifford and R.C.Longsworth suggested that only the gas element traveling between the cold and warm exchangers be responsible for the refrigeration effect. This lead to the concept of pressure ratio below which the pulse tube could not work. But the fact that the pulse tube provides refrigeration performances at very low-pressure ratios implies that heat is pumped from the cold end to the warm end of the pulse tube step by step, provided that there exist proper thermal interactions between the gas elements and the tube wall. W.E.Gifford and R.C.Longsworth described this effect as the surface heat pumping effect.
For this effect to take place, there should be (1) Reciprocating relative movement between the fluid and the tube wall. (2) Energy change in the fluid and (3) Moderate thermal contact between the fluid and the tube wall.
Figure 1.8 Schema of the surface heat pumping effect.
pulse tube
Figure 1.8 is a schema for the surface heat pumping effect. The mechanism can be visualized as by examining the physical behavior of one of the gas elements near the tube wall, as indicated in the figure. Suppose that at the beginning of the cycle the temperature of the gas element is in equilibrium with that of the adjacent tube wall. As a result of pressurizing the tube by supplying gas from the left end, the gas element is displaced from position 1 to 2 accompanied with an increase of temperature in it. Because of the moderate thermal contraction between the fluid and the tube wall and the short time of displacement, heat exchange between the gas element and the tube wall in this process remains very poor. After the compression process, there is a quiescent period when the pressure is almost constant in the pulse tube. The gas element with its temperature higher than that of the adjacent tube wall, transfers heat to the tube wall and moves slowly from position 2 to 3 as a result of the slight contraction of the gas being cooled by the tube wall. Its temperature gradually approaches that of the adjacent tube wall. During the expansion process the gas element is displaced from position 3 to 4 with a concurrent temperature drop but without much heat exchange with the tube wall. This process is approximately the reverse of the supplying process except that the starting temperature T3 is lower than T2. Consequently T4 is lower than T[. During the low pressure quiescent period that follows the exhausting process, the gas element absorbs heat from the adjacent tube wall and moves from position 4 to 1 due to the slight expansion of the gas being heated by the tube wall. When it arrives in position 1, its temperature gets once again in equilibrium with the temperature of the tube wall at this position. Thus the cycle is finished. It can be seen that during this cycle the gas element takes heat from the tube wall at positions between 1 and 4 and gives up to the tube wall between positions 2 and 3. In other words it transports a certain amount of heat from one part of the tube wall to another part which is closer to the closed end. There are many such gas elements that work synchronously in nearly the same way at different positions in the pulse tube. So heat is pumped from the cold end to the hot end in relays, which provides a certain capacity of refrigeration at the cold end while the hot end is maintained at room temperature by displacing heat to the environment.
1.5.2 Enthalpy Flow Theory.
While the surface heat pumping theory examines the physical behavior of one gas element, the enthalpy flow theory proposed by R.Radeburg soon after the appearance of the orifice pulse tube refrigerator studies the time-averaged thermodynamic effect of the pulse tube as a control volume.
Consider a cross section of pulse tube, for an ideal gas the enthalpy flow rate through this section is
H = mh = mcpT (1.1)
where m is the mass floe rate, h is the specific enthalpy of gas, cp is the specific heat of gas at constant temperature, T is the gas temperature. The time average of the enthalpy flow rate over one cycle of period r is
l^H^ = {cPLT)\mTdt (1.2)
m = pAptu (1.3)
where p is the density of the gas, u is the local gas velocity, A is the cross sectional area of pulse tube, and for ideal gas
P = jf 0-4)
where P is the pressure, the time average enthalpy flow rate can be rewritten as
lH\ = (cpApt/Rz))uPdt (1.5)
\ ' 0
If the cyclic variations of pressure and velocity can be approximately regarded as sinusoidal, i.e.
P = Pm,+PAsm(m) (1.6)
u = u , sin(&> -<p) (1.7)
where Pav is the average pressure which is not a function of time, PA is the amplitude of pressure variation, 114 is the amplitude of variation of velocity, co = lirf is the angular
H) = (H:
\ t \ ' max
when <j) = 0 when $ = 7i/ 2
Figure 1.9 Energy balance of pulse tube.
So the phase shift angle </> is a crucial factor. When ^ is between 0 and^/2, (H) is
positive, which means that there is an enthalpy flux from one end of the pulse tube to the other, as shown in figure.
Consider the control volume shown in this figure, the first law of thermodynamics says that at steady state IH\ is the same at any cross section of the pulse tube, provided there
is no energy exchange between pulse tube and the surrounding environment. When the temperatures of the cold and hot end heat exchangers, Tc and Th, are all constant, the first law gives
Qf={HJ-Ql (1-9)
QH=(H) (
where Qf is the net refrigeration power, Q{ is the loss of cooling, Qh is the heat rejected at the hot end heat exchanger. From equation (1.9) it can be seen that the refrigeration
capacity at the cold end comes from the enthalpy flux \H) in the pulse tube and is therefore dependent on the phase shift angle (f>.
1.5.3 Thermo acoustic Theory.
Although the enthalpy flow theory seems to be of the nature of classical thermodynamics, it probably originated from the thermo acoustic theory, which had been developing for a long time.
When there is sufficiently large axial temperature gradient along a tube, the fluid in the tube becomes unstable and begins to oscillate spontaneously. The thermally induced spontaneous oscillations, known as Taconic oscillation in cryogenics, have been studied for over two centuries. A theoretical breakthrough was achieved by N.Rott in the 1970's. He discussed the stability of the standing waves using linearized equations of fluid dynamics.
The isothermal model is very useful for analysis of Stirling refrigerators because of its simplicity, but to date it has not been applied to pulse tube refrigerators. In this paper, an isothermal model for pulse tube refrigerators is introduced. This model is based on the premises that a pulse tube refrigerator can be considered to be a type of spit Stirling refrigerator, and the gas in the pulse tube can be divided into three parts. The main assumptions of the isothermal model are that the gas in the middle part of the pulse tube is adiabatie and the gas in the other part of isothermal. Though the present isothermal model is more complex than the isothermal model for Stirling refrigerator, it is much simpler than a nodal analysis. The result from the isothermal model is compared with those of the nodal analysis. The gross refrigeration power and the input power using the former are =20% lower than values obtained using the latter. The pressure ratio and the average mass flow rate are =5%lower using the isothermal model.
Though there are many models which simulate the Stirling refrigerator. Such as nodal analysis and the adiabatie model, which are more accurate than the Isothermal model, is still widely used because it is simple and can represent the real process as occurring to a certain extent. In the pulse tube refrigerator, nodal analysis has been employed, but the isothermal model has not been introduce to date. Because nodal analysis is very complex, requiring a large computer and more CPU time, the present authors have attempted to develop the simpler model.

li I ”( J” -

3 2 :: :
Figure 2.1 schematic diagram of orifice pulse tube refrigerator
Orifice : 0 Water cooler : 1 Pulse tube : 1 ;2
Refrigeration power heat exchanger : 3 Regenerator : 4 Water cooler: 5 Compressor : 5;6 Reservoir: 7
Figure 2.1 is a schematic diagram of an orifice pulse tube refrigerator. The gas in the pulse tube can be divided into three parts, the cold part which flows from the regenerator and expands to give out work, which we define as gas III; the hot part which flows from the orifice and absorbs work which we define as gas I and the middle part which never flows out of the pulse tube and is similar to a displacer into split Stirling refrigerator, which we define as gas II. So we may consider the pulse tube refrigerator to be a type of split Stirling
Refrigerator, the difference being that the solid displacer in a split Stirling Refrigerator is replaced by gas flow through orifice. So the idea of the isothermal Model from Stirling refrigerator can be easily transferred to a pulse tube Refrigerator.
In the isothermal model, it is assumed that:
1 gas II is adiabatic and the other parts of gas are isothermal.
2 There is no gas leakage through the piston seal.
3 The orifice is no ideal jet.
4 There is no field resistance and
5 The gas is an ideal gas.
Mass flow rate at the orifice
K P Pv 2/k Pi
K-l Vi P P
Ao\2 K P_7 K-l Vt
El P
Pressure of the reservoir
dp = RTmo dt Vo
Mass flow rate at section I
mi = mo+ Vi_ dP RTo dT
Mass of gas I
mi =mio + I (-mi) dt
mlO can be obtained by
(mi)min = 0
Volume of gas I
V ii = mi RTo P
Volume of gas IT
V II = Clio P Volume of gas III
Cuo can be obtained from
(Vlll) nun = 0
Average temperature of the regenerator
Which is similar to that for a Stirling refrigerator from volume 6 to gas III, So that pressure in the pulse tube.
Vjl + Y_s + Yj + y3 +Vih
and M can be obtained
Volume change of compressor 6 V&= Ve+0.5 Input power
Gross refrigeration power
Q = l
Pressure ratio
P ” Pmax
Equation (l)-(7) are governing equations of the isothermal model for pulse tube refrigerators. Though it is more complex than the isothermal model for a Stirling Refrigerator. It is still much simpler than the model analysis used in the design of pulse tube refrigerators.
Calculation method
Using the isothermal model, we wrote a computer program which can be run with a personnel computer, and whose running time is only a few minutes. An iterative method is used to sole these equations. First, we assume the volume of gas m, so that pressure P and M can be obtained. Second, M can be obtained. Second, mio, Cuo and Volume of the gas m can be obtained with the calculated P. Then another P and M can be obtained with the calculated volume of gas in. This process is repeated until the convergence solution is arrived at.
Comparison with nodal analysis
A comparison of the results calculated from the isothermal model with those from nodal analysis will be given shortly; detailed information about the nodal analysis is given in reference 5.The present data are as follows
The swept volume of cylinder 6 = 80cm
The dead volume of cylinder 6 = 1.44cmD
The volume of water cooler 5 = 1.2cmD
The volume of refrigeration
power exchanger 0 = 0.2544cmD
The volume of the pulse tube = 27.143cm
The volume of water cooler 1 = 1.866cmD
The volume of reservoir = ldmD
The ambient room temperature = 300k
The refrigeration temperature = 60k
The frequency = 10Hz and
The working medium is Helium.
Implicit real*4(a-h,m,o-z) real*4 kappa
common/ A/tim(361 ),Angle(361 ),m l dot(361 ),dp7bydt(361) common/B/n,nplusl ,nby2
dimension V6(361),ml int(361),ml(361),VI(361),VII(361) dimension S(361 ),C(361 ),P(361 ),p7(361 ),VIII(361) dimension dpbydt(361 ),m2dot(361) dimension m4dot(361 ),m2(361 ),m4(361) dimension viiiold(361),viiinew(361)
dimension PP1 (21 ),PP2(21 ),PP3(21),PP4(21 ),PP5(21 ),PP6(21) dimension PP7(21)
dimension Ori(l l),Pr(l l),Mtot(l l),M0dotout(l l),M2dotout(l 1)
dimension M4dotout(l l),Ref(l l),Pinput(l l),RefpM(l 1)
For every degree, the data is calculated.
N=360 "
Units should be as follows
Pressure should be in Pascal (1 pascal=TN/m2)
Volume should be in m3
Temp should be in Kelvin
If we choose p=1.013e5(l atm),V=22.42e-3m3 and t=273K
R is the Universal Gas Constant
Gas Constant=R/M=8317.0/4.0026 for helium
Input DATA
Freq PP1 Pavg PP2 V60 PP3 VP PP4 V4 PP5 V7 PP6 TE PP7 AO PP8 num= 11
do 410 ip=l,num
pp 1 (ip)=0.2e-6+(ip-1 )*0.1 e-6
Vl = 1.8866e-6
T4=(T0-TE)/log(T0/TE) Gasconst=8317.0/4.0026
A0=0.6e-6 ivar=l
332 weight0=0.75 weight=weight0 error=0.02
333 continue
Time & angle variation
Do 10i=l,nplusl angle(i)=(i-l)
tim(i)=angle(i)/(360.0*freq) s(i)=sin((pi/180.0)*angle(i)) c(i)=cos((pi/l 80.0)*angle(i)) pr(i)=pav+pamp*sin(2.0*pi*freq*tim(l)) prl (i)=pav+pamp*sin(2.0*pi-freq*tim(i)+(pi/l 80)*ph; write(2,*)t-im(i),c(i) 10 continue
STEP 1: Volume variation of the compressor
do 20 i=T,nplusl angle(i)=(i-l)
v6(i)=v65+0.5*v60*(1.0-sin((pi/180.0)*angle(i))) write(l,*)tim(i),angle(i),v6(j) 20 continue sV6=0.0 do 30 i=l,n sV6=sV6+v6(i)
30 continue
STEP2: Calculation of the Masstot of gas (M)using Equn(l 1) To start with assume that p7avg & P=Pavg & VIII initialize do 32 i=l,nplusl VIII(i)=1.0e-6 p7(i)=pavg p(i)=Pavg 32 continue
Calculation of VHIavg
do 34 i=l,n
34 continue
Calculate Initial Total Mass
98 continue
99 continue
do 36 i=l,n
Factor=(V6(i)+V5),T0+(V4/T4)+(V3+Viii(i))/TF Mass=(P(i)/Gasconst)*Factor Smass=smass+mass 36 continue
write(*, *)' mass=', masstot
STEP3:Obtain P(i) from V6(i)
35 do40i=l,nplusl
40 continue SP=0.0 do 50 i=l,n
50 sp=sp+p(i) sp=sp/n ratio=pavg/sp do 51i=l,nplusl p(i)=p(i)*ratio
51 continue
if(jdt.l0)goto 99 massold=masstot do 60 i=l,nplusl write( 1 ,*)tim(i),p(i)
60 continue
Calculation of dp/dt
do 65 j=l,n i=j+l
65 dpbydt(j)=(p(i)-pG))/(tim(i)-timG)) dpbydt(nplus 1 )=dpbydt( 1) do 67 i=l, nplusl write(l,*)tim(i),p(i),dpbydt(i)
67 continue
Calculation of the mass flow thro Orifice AO is the orifice in mm2
72 do 75 i=l,nplus If (p(i).gt.p7(i) then Term l=(P7(i)/p(i))**(2.0/kappa) Term2=(P7(i)/p(i))**((kappa+1.0)/kappa) Term3=2.0*kappa*p(i)*p(i)/(kappa-l .0*gas const M0 dot (i)=-a0*(term3*(terml-term2))**0.5 Else if (p(i).it .p7(i)) then Term l=(P(i)/p7(i))**(2.0/kappa) Term2=(P(i)/p7(i))**((kappa+1.0)/kappa)
Term3=2.0*kappa*p7(i)*p7(i)/(kappa-1.0*gas const *T0) MO dot (i)=-a0*(tenn3*(terml-tenn2))**0.5 Else End if
Write (2,*) tim (i),p(i),mO dot (i) 75 continue
Calculation of dp7/dt Do 80 i= l,nplus 1
Dp7 by dt (i) = mO dot (i)*gas const t0/v7 80 continue
Calculation of p7
Do 82 i=l,nplus 1 Dtx =tim (2)-tim(l) P7(i)=p7(i) + dp7 by dt (i)*dtx
82 continue
If ( go to 72' Do 83 i=l,n plus 1
Write (1,*) tim (i),p(i),p7(i),m0 dot(i)
83 continue
72 calculation of p7avg Sp7=0.0 Do 84i=1.0 Sp7=sp7+p7(i)
84 continue
Write(3,*)'p7avg=',p7avg Mass flow rate at section 1 Do 85i=l,nplusl
Xx=(vl *dpbydt(i)/(gas const*T0) M1 dot(i)=mOdot(i).xx,ml dot(i)
85 continue
Mass flow rate integration
Tstart=tim(i) Tend=tim(J) Dtx=tim(2)-tim(l)
M1 dottot=(-m 1 dot(i)-m 1 dot(j))*dtx/2.0
130 continue
do 140 i=2,nplusl
VIIImin=VIII(i) else end if 140 continue
do 150 i=l,nplusl VIII(i)=VIII(i)-VIIImin VII(i)=VP-VIII(i)-VI(i) write(2,*)tim(i),VI(i),VII(i),VIII(i) 150 continue sVII=0,0
do 170 i=l,nplusl
170 sVII-SVII+VII(i)
CII0-VIIavg/(Pavg**(-1.0/1.66)) Write (*,*)ix,CII0 ,index=',ix,eII0,index if (ix.le.lO)goto 160 155 sviii=0.0 do 171 i=l sviii=sviii+viii(i)
171 continue
VIIIavg=SVIII/nplusl do 172i=l,nplusl
172 viiinew(i)=viii(i)
write(*,*)"VI,VII,VII Averages' write(*,*)vIavg,viiavg,Aviiinew index=index+1 if (
write (*,*)TNDEX GREATER THAN 75,HENCE QUITTING' goto 181 else end if
do 174 i=l,nplus 1
viii(i)=weight*viiiold(i)+(1.0-weight)*viiinew(i) 174 continue
Smass=0.0 do 176 i=l,n
Factor =(V6(i)+V5)/T0+(V4/T4)+(V3+VIII(i))/TE Mass=(P(i)/Gasconst)*Factor smass=smass+mass 176 continue
Write (*,*)'mass=',masstot Massnew=masstot
Percent=(massold-massnew)* 1 OO/massold If(percent.It.O.O)percent=-percent
if (percent .gt.error )go to 98
181 pmin=p(l) Pmax=p( 1) Do 175 i=2 ,n plus 1 If (p(i).it .pmin )then Pmin =p(i)
Else if (p(i).gt.pmax)then Pmax =p(i) Else End if 175 continue
Pratio=pmax /pmin
Write (*,*)pmax ,pmin ,pratio
Calculation of input power =intergral of p(i) dv 6(i)
Sum = 0.0 Do 200 i=l,n J=i+1
Vstart=v6(i) vend=v6(j)
sum =sum+(p(i)+p(j))*(vend -vstart)/2.0 200 continue
Powerin =sum*freq
Write (*,*)'INPUT POWER powerin
Calculation of input power =intergral of p(i) dv 6(i)
Sum = 0.0 Do 200 i=l,n J=i+1
Vstart=viii(i) vend=viii(j)
sum =sum+(p(i)+p(j))*(vend -vstarf)/2.0 210 continue
Ref power =sum *freq
Write (*,*),REFGEN OUTPUT=',ref power
Calculation of avg mass flow rate mO dot =intergral mO dot *dt/tau
Do 205i=l,n J=i+1
if(mOdot(i). 11.0.0)m0dot(i)=-m0dot(i) if(m0dot(i).lt.0.0)m0dot0)=-m0dot(j) sum=sum+(m0dot(i)+m0dot(j))*dtx/2.0 205 continue
Write(*.*)'MOdot Average=' ,m0dotavg
Calculation of avg mass flow rate m2dot=integral m2dot*dt/tau Mass flow rate at cold end of pulse Tube
Do 215 i=l,nplusl M2(i)=viii(i)*p(i)/(gasconst*TE) 215 continue
Dtx=tim(2)-tim(l) Do 217 i=l,n J=HT
217 continue
M2dot(nplus 1 )=m2dot( 1)
218 continue
Sum=0.0 Do 220 i=l,n J=i+1
If(m2dot(i). 1 t.0.0)m2dot(i)=-m2dot(i) If(m2dot(j).lt.0.0)m2dot(j)=m2dot(j) Sum=sum+(m2dot(i)+m2dot(j))*dtx/2.0 220 continue
M2dotavg=sum*freq Write(*,*)'m2dot Average=',m2dotavg
Calculation of Avg Mass flow rate m4dot=integral m4dot*dt/tau Mass flow rate at the Hot end of the Regenerator
do 225 i=l,nplusl
m4(i)=v6(i)*p(i)/(gas const*T0) 225 continue
dtx=tim(2)-tim(l) do 227 i=l,n
m4 dot (i)=(m4(j)-m4(i))/dtx 227 continue
m4 dot (m plus l)=m4 dot (1) do 228 i=l n plus 1 write (8,*)tim (i),m4 dot (i) 228 continue Sum=0.0 Do 230 i=l,n J=i+1
If(m4 dot (i).it 0.0)m4 dot (i)=-m4 dot(i) If(m4 dot (j).it 0.0)m4 dot (j)=-m4 dot(j) Sum=sum+(m4 dot (i)+m4 dot 0'))* dtx 12.0 230 continue
M4 dot avg =sum*freq
Write (*,*)'M4 dot Avg =',m4 dot avg
Refrigeration power per unit mass flow rate Q/m2 dot KJ/Kg
Ref per mass =Ref power /(m2 dot avg * 1000.0)
Write (*,*)'ref gen per unit mass flow rate =',ref per mass ,'KJ/Kg
Total mass flow print out
Write (*,*)'massold massnew percent index' Write (*,*) massold massnew percent index If (weight .ne.weightO) go to 207 Index 1 = index Error l=error Weight l=weight Pratiol=pratio Powerin 1 =powerin Ref power l=refpower Modotavg 1 =m0dotavg M2dotavg 1 =m2dotavg M4dotavg 1 =m4dotavg refpennass 1 -_refermass wei ght=wei ght-0.05 goto 333 207 continue
Write(*,*)refpowerl ,refpower Write( *, * )powerin 1,powering Write(*,*)m0dotavgl ,m0dotavg Write(*,*)m2dotavg.l ,m2dotavg Write(*,*)m4dotavgl,m4dotavg
Write(*,*)refpermass 1 ,refpermass Write(*,*)weightl,weight Wri te(*,*) error 1 ,error Write(*, *)index 1,index
Do 180 i=l,nplusl
Write(l,*) tim(i),m0dot(i),mldot(i),m2dot(i),m4dot(i) Write(2,*)tim(i),p(i),v(i),vI(i),vII(i),Viii(i),v6(i) Write(3,*)tim(i),p(i),p7(i) 180 continue
Write(4 *^ ***************************************** Write(4,*)'PARAMETERS OF PULSE TUBE'
^ ^ ^| ^ ^£ SjC ^Jc sjc ijc JJC ^ jjc *fc *|c sjc iji ^ sjc ^ 2jC ^ 5J£ ^jC
Write(4^)'V60 V65 ,VP,V1,V3,V4,V5,V7 in cm3' Write(4,300)V60* 1 e6,V65* 1 e6,VP* 1 e6,V7* 1 e6 300 fonnat(2x,4(F12.5,2x)) Write(4,*)'to,te,t4 in K' Write(4,300)t0,te,t4 Write(4,*)'freq=',freq Write(4,*)'period in s=',tau Write(4,*)'p7 avg in pascal=',p7avg Write(4,*)'pavg in pascal=',pavg
Write(4,*)'orifice opening in mm2=',A0*le
write(4,*)'massold(Kg) massnew(Kg) percent Index' write(4,310)massold, massnew, percent, index Mtot(ip)=Massnew 1 310 format(2x,3(E12.5,2x),110) write(4,*)
write(4,*)'VIAverage VIIAverage VIII Average in m3
write(4,310) viavg,viiavg,viiiavg
write(4,*)'Pressure Ratio=',pratio Pr(ip)=Pratio
write(4,*)TNPUT POWER in W=\powerin Pinput(ip)=Pratio
write(4,*)'REFGN OUTOUT in W=',refpower Ref(ip)=Refpower
write(4,*)'M0dot Average in Kg/s=',m0dotavg M0dotout(ip)=m0dotavg write(4,*)'M2dot Average in Kg/s=',m2dotavg M2dotout(ip)=m2dotavg write(4,*)'M4dot Average in Kg/s=',m4dotavg M4dotout(ip)=m4dotavg write(4,*)'Refgn/M2dot in KJ/Kg=',refpermass Refpm(ip)=Refpermass write(4,*)TNDEX values=',index write(4,*)'WEIGHT factor=',weight write(4,*)'ERROR percent=', error 410 continue
do 415 i=l,num
write( 1 ,*)pp 1 (i),Pr(i),-Pinput(i),ref(i),refpm(i) write(2,*)ppl(i),Ori(i),m0dotout(i),m2dotout(i),m4dotout(i) 415 continue end
As seen from the program in the previous chapter, the input parameters are frequency, pressure, swept volume of cylinder, volume of the pulse tube, volume of refrigerator, volume of reservoir, cold end temperature and cross sectional area of orifice.
The following data have been used in our analysis:
Swept volume of cylinder V60=80cm3. Dead volume of cylinder V65=1.414cm3. Volume of water cooler V5=l .2cm3. Volume of regenerator V4=25.158cm3.
Volume of refrigeration power heat exchanger V3=0.2554cm3.
Volume of pulse tube Vp=27.143cm3.
Volume of water cooler Vl=l .886cm3.
Volume of reservoir V7=lcm3.
Ambient room temperature To=300K.
Refrigeration temperature TE=60K.
Working medium is helium.
In order to have a better understanding of the working of the pulse tube refrigerator, we observe the effect of variation of the input parameters on output parameters like pressure ratio, refrigeration power, refrigeration power per mass flow rate and power input.
The cross sectional area of the orifice was varied from 0.2 mm2 to 1.1 mm2. The values obtained are given below.
Table 4.1: Variation of orifice area.
SI No Orifice area (mm2) Pressure ratio Input Power (W) Refrigeration Power (W) Refrigeration Power/ Mass flow (kJ/kg)
1 0.2 1.49799 55.33803 11.07981 7.75113
2 0.3 1.4659 75.79030 15.12563 9.91248
3 0.4 1.43755 93.03354 18.53474 11.34576
4 0.5 1.42867 115.11232 23.14549 13.18287
5 0.6 1.37415 113.26436 22.40400 12.01510
6 0.7 1.34152 116.82361 22.95918 11.61919
7 0.8 1.30980 116.72742 22.80734 10.97142
8 0.9 1.28059 114.09727 22.16148 10.21472
9 1.0 1.25284 108.18174 20.82829 9.30185
10 1.1 1.23104 106.48635 20.65185 8.85962
Effect of variation of orifice area on pressure ratio:
As seen from fig. 4.1, the pressure ratio decreases with increase in orifice area for most of the values. The stray points are due to approximations in the program. Effect of variation of orifice area on input power:
From fig. 4.2, we can see that input power increases gradually, remains almost constant and decreases slightly. The stray points are due to approximations made in the program.
Effect of variation of orifice area on refrigeration power:
Refrigeration power is affected in the same manner as that of the input power as in fig.4.3. It increases gradually with cross sectional area and remains constant and decreases slightly. The stray points are due to approximations made in the program. Effect of variation of orifice area on refrigeration power/mass flow:
Refrigeration power/mass flow rate increases with cross sectional area of orifice and reaches a maximum point and decreases as shown in fig. 4.4
A comparison of the published results with the results obtained from the program for the variation of the cross sectional area of the orifice is shown in Table 4.2.
Table 4.2: Comparison of published results and program results.
Orifice Area ( mm )
0.2 0.4 0.6 0.8 1
Pressure Ratio
Published 1.49 1.43 1.36 1.30 1.24
Program 1.49 1.43 1.37 1.30 1.25
Deviation (%) 0 0 0.72 0 0.80
Input Power
Published 53.5 91.8 108 111 101
Program 55.3 93.0 113 116 108
Deviation (%) 3.36 1.30 4.62 4.50 6.93
Gross Refrigeration Power
Published 10.6 18.3 22.0 22.2 20.3
Program 11.07 18.5 22.4 22.8 20.8
Deviation (%) 4.43 1.09 1.81 2.70 2.46
Refrigeration Power/mass flow rate(kJ7kg)
Published 7.42 11.1 11.7 10.6 8.89
Program 7.75 11.3 12.0 10.97 9.30
Deviation (%) 4.44 1.80 2.56 3.49 4.61
The program results match with the results published in the paper with a deviation not more than 7%.
Pressure Ratio,
0.4 0.6
Orifice Opening, A (mm2)
Fig 4.1: Pressure Ratio Vs Orifice area
Fig 4.2: Power Input Vs Orifice area
The apparatus for orifice pulse tube refrigerator was developed and all the components were assembled. Due to some technical difficulties, encountered in the testing apparatus, it was not possible to conduct the experiment.
All the technical difficulties encountered can be rectified and experiment can be conducted and the system results can be compared with the predicted values.
The performance and characteristics of an orifice pulse tube refrigerator have been simulated successfully by a numerical method by a personal computer. From the knowledge of the results obtained the input parameters can be varied so as to get an optimum refrigeration power. This method avoids necessary laborious manual calculations.
1. W.E. Gifford andR.C. Longsworth, Pulse-tube refrigeration, Trans. ASME, 1964, p. 264-268.
2. G. Walker, Cryocoolers, Plenum Press, New York and London, 1983.
3. W.E. Gifford and R.C. Longsworth, Surface heat pumping, Adv. in Cryogenic Eng. 1L 1966, p. 171-179.
4. R. C. Longsworth, An experimental investigation of pulse tube refrigeration heat pumping rates, Adv. in Cryogenic Eng. 12, 1967, p. 608-618.
5. E.I. Mikidin, A.A. Tarasov, and M.P. Shkrebyonock, Low-temperature expansion pulse tubes, Adv. in Cryogenic Eng. 29, 1984, p. 629-637.
6. R. Radebaugh, J. Zimmerman, D.R. Smith, and B. Louie, Comparison of three types of pulse tube refrigerators: New methods for reaching 60 K, Adv. in Cryogenic Eng. 31, 1986, p. 779-789.
7. Sh. Zhu, P. Wu, and Zh. Chen, Double inlet pulse tube refrigerators: an important improvement, Cryogenics 30, 1990, p. 514-520.
8. J. Good, S. Hodgson, R. Mitchell, and R. Hall, Helium free magnets and research systems, Cryocoolers 12, 2003, p. 813-816.
9. G. TJmmmes, R. Landgraf, M. Muck, K. Klnndt, and C. Heiden, Operation of a high-Tc SQUID gradiometer by use of a pulse tube refrigerator, Proceedings ICEC 16, 1996, p. 283-286.
Post: #2
Submitted by : -
Rajiv K. Singh
Suryakant N. Wahane
charging and testing of refrigeration system

What do you mean by charging of refrigeration system?

The amount of refrigerant to be charge depends on size and installation and length of connecting pipe.
Direct expansion and condensation need that the outlet from the receiver be covered with liquid making a seal to prevent the gas into the liquid line.
It is generally advised not to have the receiver one third full at desired temp.
A sight glass in liquid line is used in a small size installation for finding the correct charge.

HOW DO WE KNOW TIME OF charging of refrigerator?
During the operation , if any bubble or foam is seen, it indicate insufficient refrigerant in the system ,hence more charge is added until no bubble are observed .
If refrigerator is under charged, the compressor suction starts and delivery become superheated, delivery pressure may be low, there will be a large bubble in the liquid line sight glass.
If refrigerator is over charged , delivery pressure may be high, delivery temp. will be low.
Hence refrigerant quantity to be charged is decided by taking the weight of charging cylinder before and after charging by spring balance.

What are the necessary factors for charging of refrigeration system?
Use of clean oil and moisture free drums for transfer and charging purpose.
The system should be evacuated properly .
Charging line must be purged and it must be made sure that all connections are air tight.
The suction pressure may not be allowed to rise above during charging .
The refrigerant in the liquid form should under no circumstance be introduce to the compressor .
Charging line should be flexible.

What are the generally used methods of charging?
Following are the two methods of charging



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