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Contents

1. Introduction ...................................................................................................................................... 4

1.1 Application Of Cryocooler: ...............................................................................................................5

1.2 Pulse Tube Refrigerators: ..................................................................................................................7

1.3 Components Of Pulse Tube Refrigerator:................................................................................... 12

1.4 Applications And Advantages Of Pulse Tube Cryocooler....................................................... 13

2. Literature Review .......................................................................................................................... 15

3. Aim Of The Present Work ............................................................................................................ 17

4. Thermodynamic Study Of Single Orifice Pulse Tube Cryocooler ............................................ 18

5. Generated Code In SCILAB And Results. .................................................................................. 22

5.1 Generated Code For Thermodynamic Study Of Single Orifice Pulse Tube Cryocooler In

Scilab ............................................................. 22

6. Phasor Analysis .............................................................................................................................. 29

7. Conclusion ....................................................................................................................................... 34

8. References ....................................................................................................................................... 35

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List of figures

Figure 1Classification of cryocoolers ........................................................................................................ 4

Figure 2 Classification of pulse tube refrigerators .................................................................................... 8

Figure 3(a) (b) Stirling type & G-M type refrigerators ............................................................................. 8

Figure 4 Schematic diagram of basic pulse tube refrigerator (BPTR) ...................................................... 9

Figure 5 Schematic diagram of orifice pulse tube refrigerator (OPTR). ................................................. 10

Figure 6 Schematic diagram of Stirling type double inlet pulse tube refrigerator. ................................. 11

Figure 7 Schematic diagram of G-M type double inlet pulse tube refrigerator. ..................................... 11

Figure 8 Schematic diagram of the inertance tube pulse tube refrigerator. ............................................ 12

Figure 9 orifice Pulse Tube Refrigerator (OPTR) with the Working fluid assumed to be Helium gas .. 18

Figure 10 Code generated in SCILAB for Thermodynamic analysis ..................................................... 22

Figure 11 Result obtained from thermodynamics energies ..................................................................... 23

Figure 12 Pressure Variation within The Pulse Tube (Pt), The Compressor (Pc) .................................. 26

Figure 13 Mass Flow Rates Through The Regenerator (Mreg) And Pressure (P).................................... 26

Figure 14 the mass flow rate through hot heat exchanger (mhhx) and pressure (P). ................................ 27

Figure 15 Mass Flow Rate through Pulse Tube (Mpt) And Pressure (P) ................................................. 27

Figure 16 Mass Flow Rate Through Cold Heat Exchanger (Mchx) And Pressure (P) ............................. 28

Figure 17 Mass Flow Rate through Cold Heat Exchanger (Mchx) And Mass Flow Rate through Hot Heat

Exchanger (Mhhx) ..................................................................................................................................... 28

Figure 18 Phasor diagram ........................................................................................................................ 30

Figure 19 Code For phasor diagram in SCILAB .................................................................................... 31

Figure 20 Results obtained through CODE ............................................................................................. 32

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List of tables

Table 1 Design Data for Adiabatic Model .............................................................................................. 24

Table 2 Operating condition for Adiabatic Model .................................................................................. 24

Table 3 Fluid properties for Adiabatic Model ......................................................................................... 25

Table 4 Comparison of results obtained from both the codes ................................................................. 33

Table 5 Phase angle of various parameters with pressure vector ............................................................ 33

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1. Introduction

Cryogenics is the science and technology associated with the phenomena that occur at very low

temperature, close to the lowest theoretically attainable temperature. In engineering, cryogenics can be

best described as an application which operates in the temperature range from absolute zero to about

123K(-150C). In particular, this includes refrigeration, liquefaction, storage and transport of cryogenic

fluids, cryostat design and the study of phenomena that occur at these temperatures.

Device which is used to produce cryogenics low temperature is called as cryocoolers. The term

cryocooler has generally been used for refrigerator of small and intermediate size, which is capable to

obtain and maintain temperature below 123K.Cryocoolers, invented in early 1960, and are mainly

classified in to two groups.

1. Recuperative

2. Regenerative

Figure 1Classification of cryocoolers

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I. Recuperative Cryocooler:

The flow of the working fluid in this type of cryocooler is unique and hence they are analogous

to direct current electrical systems. The compressor and expander have separate inlet and outlet valves

for maintaining the flow direction. In rotary motion of components theres a maximum chance for back

flow because of which valves are necessary when the system has any rotary or turbine component. The

efficiency of the cryocooler depends a lot on the working fluid because it forms an important part of the

cycle. The main advantage of recuperative cryocooler is that, that they can be scaled to any size for

specific output. Joule Thomson cryocooler and Brayton cryocooler are few of the examples of

recuperative type cryocooler.

II. Regenerative Cryocooler:

The flow of working fluid in this type of cryocooler is oscillatory and hence have an analogy to

alternative current electrical system. The working fluid inside this type of cryocooler oscillates in cycles

and while passing through the regenerator exchanges heat with the wire mesh present within the

regenerator. The regenerator takes up the heat from the working fluid in one half on the cycle and returns

the same in the other half. The wire mesh used in regenerator are very efficient because of their very high

heat capacity and low heat transfer losses, but these cryocoolers cannot be scaled up to large sizes. The

phase relation between mass flow and pressure variation is responsible for the cooling effect produced.

The oscillating pressure can be produced with or without the help of valves as in Stirling and Pulse type

cryocooler, and Gifford McMahon type cryocooler respectively.

1.1 Application of Cryocooler:

The major applications of cryocoolers are summarized below [1].

Military

(i) Infrared sensors for night vision & missile guidance

(ii) Infrared sensors for satellite based surveillance

(iii) Gamma-ray sensors for monitoring nuclear activity

(iv)Superconducting magnets used in mine sweeping

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Environmental

(i) Infrared sensors used in satellites for atmospheric studies

(ii) Pollution monitoring infrared sensors

Commercial

(i) Cryopumps for semiconductor fabrication

(ii) Cellular-phone base stations using superconductors

(iii) Superconductors used in voltage standards

(iv) Superconductors used in high-speed communications

Medical

(i) Cooling of superconducting magnets used in MRI

(ii) SQUID magnetometers for heart and brain studies

(iii) Liquefaction of oxygen

(iv)Cryogenic cryosurgery and catheters

Transportation

(i) LNG for fleet vehicles

(ii) Superconducting magnets used in maglev trains

(iii) Infrared sensors used in aircrafts night vision

Energy

(i) LNG for peak shaving

(ii) Superconducting power applications like motors, transformers etc.

(iii) Thermal loss measurements infrared sensors

Police and Security

(i) Infrared sensors used in night-security and rescue

Agriculture

(i) Storage of biological cells and specimens

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Because of the various special application of the cryocooler as mentioned above, the demands for

high performance reliability, low vibration, efficiency, long life time, small size and weight have become

an important aspect for the improvement of the cryocoolers. The regenerative cryocoolers have higher

efficiency than that of recuperative cryocoolers due to smaller heat transfer loss, both Stirling cryocoolers

and Gifford-McMahon (G-M) cryocoolers have an expansion devices (i.e., moving parts) at their cold

ends. The moving parts in the cold end are needed in order to adjust the phase angle and to recover the

energy flow, which result in the decrease in reliability of the system and shorten the life times of the

cryocoolers. The absence of such moving parts in the pulse tube refrigerator/cryocooler at their cold end

and thus have an advantages over other cryocoolers due to its simplicity and is hence more reliable in

operation.

1.2 Pulse Tube Refrigerators:

Working Principle:

The pulse tube refrigerators (PTR) are capable of cooling to temperature below 123K.Unlike the

ordinary refrigeration cycles which utilize the vapor compression cycle as described in classical

thermodynamics, a PTR implements the theory of oscillatory compression and expansion of the gas

within a closed volume to achieve desired refrigeration. Being oscillatory, a PTR is a non-steady system

that requires time dependent solution. However like many other periodic systems, PTRs attain quasi-

steady periodic state (steady-periodic mode). In a periodic steady state system, property of the system at

any point in a cycle will reach the same state in the next cycle and so on. A Pulse tube refrigerator is a

closed system that uses an oscillating pressure (usually produced by an oscillating piston) at one end to

generate an oscillating gas flow in the rest of the system. The gas flow can carry heat away from a low

temperature point (cold heat exchanger) to the hot end heat exchanger if the power factor for the phasor

quantities is favorable. The amount of heat they can remove is limited by their size and power Used to

drive them.

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Classification:

The following flow chart describes the various types of Pulse Tube Refrigerator.

Figure 2 Classification of pulse tube refrigerators

Types of Pulse Tube Refrigerators:

Pulse tube refrigeration systems can be classified as either a Stirling type or a GM type according

to the method of pressurization and expansion as shown in Fig. 3 (a) and (b). For a Stirling type pulse

tube shown in Fig.3 (a) a piston cylinder apparatus is directly coupled to the hot end of the regenerator

so that the pressure fluctuations are directly generated by the piston movement.

Figure 3(a) (b) Stirling type & G-M type refrigerators

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In Stirling type PTR, the frequency of the compressor is the same as that of the pulse tube. The

heat of compression by the compressor must be removed to the environment by a heat exchanger between

the compressor and the regenerator, commonly known as after cooler or precooler. These aspects are the

same in Stirling type and G-M type refrigerators. These are used for PTRs in the higher temperature

ranges of about 50K.The typical operating frequency of Stirling type PTR is 10-120Hz, which is higher

than that of a GM type pulse tube as shown in Figure 3 (b).

Because of this high operating frequency and the absence of valve losses, Stirling type pulse tube

systems generally produce higher cooling powers than GM type pulse tube. However the rapid flow

oscillation of fluid heat exchange required in Stirling type pulse tube refrigerators limits their

performance at low temperatures, such as at 10K and below. In this range, the longer time allowed for

thermal diffusion by the slower frequency GM type pulse tube refrigerators provides a higher efficiency

option. The G-M type pulse tube refrigerator distributes high/lowpressure gas into the pulse tube and

other components by use of a valve system. Generally a rotary valve or solenoid valve is used in G-M

type cryocooler. 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 hertz (1-5Hz).

The valve system separating the compressor and the pulse tube system provides the possibility of

eliminating vibration problems caused by the compressor and permits remote location of the compressor

from the cold head. Figure 4 shows the main components of Stirling type BPTR. It is composed of six

components: compressor, after cooler, regenerator, cold heat exchanger, pulse tube and warm heat

exchanger. In a BPTR, the oscillatory pressure waves impose a shuttling effect to the working fluid in

the

Figure 4 Schematic diagram of basic pulse tube refrigerator (BPTR)

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Pulse tube. The shuttling effect creates an energy interaction between the pulse tube wall and the working

fluid. This is known as surface heat pumping process [2]. Thus, the BPTR achieves refrigeration through

the surface heat pumping process between the working fluid and the pulse tube walls. BPTRs have

relatively low coefficients of performance and can typically reach a cold end temperature of the order of

124K.

The second type of the PTR is the Orifice Pulse Tube Refrigerator (OPTR), shown in Figure 5.

OPTRs are significantly better than BPTRs, and are among most widely refrigerator until the mid-1990s

in Striling type PTR [3]. The schematic configuration of an OPTR can be viewed as a modification of

the BPTR. This modification is made by including an orifice valve and a surge volume at the warm end

of the BPTR, as depicted in Figure 4. Additional components create an advantage of in-phase relationship

between the mass flow and the pressure within the pulse tube to enhance the heat transport mechanism.

But the mass flow through the regenerator is increases leading to degradation of regenerator performance.

This Drawback is removed by adding a second orifice i.e. double inlet PTR.

Figure 5 Schematic diagram of orifice pulse tube refrigerator (OPTR).

In the double-inlet pulse tube refrigerator (DIPTR) [4] the hot end of the pulse tube is connected

with the entrance (hot end) of the regenerator by an orifice adjusted to an optimal value shown in Figure

6 and 7 for Stirling type and GM type DIPTR respectively. The double inlet is a bypass for the regenerator

and hence reduces the cooling power. In addition, the valve is a dissipative device, which leads to a

deterioration of the performance. However, both these disadvantages are overcome by the fact that the

double inlet reduces the dissipation in the regenerator. As a result, the performance of the overall system

is improved significantly.

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Figure 6 Schematic diagram of Stirling type double inlet pulse tube refrigerator.

Figure 7 Schematic diagram of G-M type double inlet pulse tube refrigerator.

The fourth and the most recently invented PTR is the inertance tube pulse tube refrigerator shown

in Figure 8. In this type of PTR the orifice valve is replaced by a long inertance tube having very small

internal diameter and adds reactive impedance to the system. The implementation of this inductance

generates an advantageous phase shift in pulse tube and produces an improved enthalpy flow. Studies

show that use of the inertance tube is significantly beneficial for large-scale pulse tubes operating at

higher frequencies.

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Figure 8 Schematic diagram of the inertance tube pulse tube refrigerator.

1.3 Components of Pulse Tube Refrigerator:

Compressor

The main function of the compressor is to supply gas pressurization and depressurization in the closed

chamber. Electrical power is applied to the compressor where this electrical work is converted into the

mechanical energy associated with sinusoidal pressure waves, resulting in gas motion. In an ideal

compressor, the electrical power provided to the compressor must be equal to f PdV, where the

integration is performed over an entire Cycle, P is the compressor pressure, and f is the compressor

frequency. In an actual system, however, the above-mentioned power (the PdV power) is always less

than the actual measured electrical power due to the associated irreversibilities. Usually reciprocating

nature of compressor is used in case of Stirling model; it may also be a dual opposed piston type.

After cooler

The function of the ideal after cooler is to extract all the heat that is generated in the compressor volume

during the gas compression and dispose to environment. This minimizes the warm end temperature so

that the regenerator can work more efficiently and supply low temperature working fluid to the system.

Typically, these types of heat exchangers are assembled using copper wire mesh screens that are

directly in contact with the housing wall.

Regenerator

The regenerator is the most important component in pulse tube refrigerator. Its function is to absorb the

heat from the incoming gas during the forward stroke, and deliver that heat back to the gas during the

return stroke. Ideally, PTC regenerators with no pressure drop and a heat exchanger effectiveness of

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100% are desired, in order to achieve the maximum enthalpy flow in the pulse tube. The performances

of the real regenerators are of course far from ideal. Stainless steel wire screens are usually selected as

the regenerator packing material, since they offer higher heat transfer areas, low pressure drop, high

heat capacity, and low thermal conductivity.

Cold Heat Exchanger (CHX)

CHX can be best viewed as the equivalent of the evaporator in the vapor compression refrigeration

cycle. This is where the refrigeration load is absorbed by the system. This is the junction of the

regenerator and pulse tube. Copper wire mesh screens are used to exchange heat with the housing wall,

and thereby receive the applied heat load.

Pulse Tube

The pulse tube is the most critical component of the whole refrigerating system. The main objective of

the pulse tube is to carry the heat from the cold end to the warm end by an enthalpy flow. By imposing

a correct phase difference between pressure and mass flow in the pulse tube by phase shifting

mechanisms, heat load is carried away from the CHX to the WHX. Physically, the pulse tube is simply

a hollow cylindrical tube made up of stainless steel with an optimum thickness to enhance the surface

heat pumping.

Hot Heat Exchanger (HHX)

Hot end exchanger is where the gas rejects heat of compression in every periodic cycle of operation.

Upon receiving the enthalpy flow from the pulse tube, the heat load at a higher temperature is rejected

to the environment. Usually, air cooling or water cooling system is used to take away the heat from the

hot end exchanger.

1.4 Applications and advantages of Pulse Tube Cryocooler

Pulse tube cryocoolers are compact and light weight making them a mouthwatering prospect in the

space applications

There are no moving parts on the cold side of the cryocooler which make them vibration less in the

colder end and hence lesser wear and tear which would help in smoother functioning of the

cryocooler.

They are used for the cooling of infrared sensors in the defense sector.

They are of utmost importance in the cooling of superconducting magnets in MRI which is important

in the medical industry

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Design parameters for OPTR

The design data is a very important parameter in studying about the pulse tube cryocooler,

different lengths of pulse tubes, regenerators and materials would result in different working

efficiencies of pulse tube Cryocoolers and hence it needs to be taken care of.

For better performance of an OPTR it has been observed practically that for better performance of an

OPTR:-

Reducing dead volumes of OPTR as much as possible in connecting tubes, cold end and hot end

spaces and compression chamber in order to reduce thermodynamic losses.

Using a high efficiency regenerator to reduce heat loss.

Designing an adequate flow straightening device at both ends of pulse tube to reduce turbulence

loss in pulse tube

Good machining of pulse tube and using thin walled pulse tube to reduce fluid frictional losses.

Carefully designing dimensions of pulse tube to match with compressor and regenerator for given

pressure and cold end temperature

Parameters dependence on performance of OPTR:

For fixed volume of pulse tube, performance of OPTR increases with increasing diameter of pulse

tube.

Small pulse tube causes flow to easily become turbulent and reduce heat pumping effect due to

gas dispersion and back mixing

Performance of OPTR increases with decreasing length and volume of pulse tube. However

optimum frequency increases with decreasing pulse tube volume

For fixed pulse tube length optimum frequency decreases with increasing diameter and volume

Optimum operating frequency decreases monotonically with increasing pulse tube volume

Pressure flow phase angle is hence a key design parameter for optimizing a pulse tube cryocooler.

High efficiency cryocooler requires an optimal phase angle that minimizes viscous dissipation

losses in the regenerator and maximizes the acoustic power flow in the pulse tube. This will result

in greater cooling capacity for a given amount of acoustic power given to the compressor. The

value of this phase angle is basically driven by valve flow area and reservoir volume for fixed

fluid flow rate.

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2. Literature Review

In 1963 Gifford and Longsworth [5] discovered the Basic Pulse Tube Refrigeration technique

where a very simple effect i.e. oscillation of working gas (pressurization and depressurization) makes it

possible to construct very low temperature refrigerators without the use of low temperature moving parts

or the Joule-Thomson effect. The design was put forward using a hollow tube with one end closed and

the other open with the closed end responsible for heat exchange at ambient temperature and the open

end serving as the cold end. A thermodynamic model of BPTR was put forward by de Boer [6] with

various improvements by taking into account the gas motion during the cooling and heating steps that

result in more accurate temperature profiles.

The first improvisation to the basic pulse tube refrigerator was made in 1984 by Mikulin et al. [7]

where they installed an orifice and reservoir at the top of the pulse tube to allow some gas to pass into

and out of a large reservoir volume. This configuration of the pulse tube refrigerator was given the name

as the Orifice Pulse Tube Refrigerator. An analytical model for OPTR was put forward by Starch and

Radebaugh [8] who made a simple expression for the gross refrigeration power, which agrees with

experiments

In the later works, the mass flow rates were made analogous to AC current flow. This concept led

to phasor representation of the mass flow rates and the pressure wave. L.Mohanta and M.D. Atrey in

2011 [9] came forward with one such phasor diagram representing the various mass flow rates. This

phasor representation of the mass flow rate became a prominent way of analyzing the phase difference

between the mass flow rates and the pressure variation.

Hoffman and Pan [10] studied the phase shifting in Pulse Tube Refrigerator and worked on the

Phasor representation of the mass flow rates and pressure oscillation. They studied the phase Relation for

different configuration of the pulse tube refrigerators and experimentally concluded the optimum phase

relation for the same.

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These phasor represented each mass flow rates as a vector quantity and just plotted the Governing

equations. There were no concrete relation as to how the exact phase difference can be obtained. They

merely served as a method to cross verify the experimental/analytical works.

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3. Aim of the Present work

After going through the various literature, we decided to work on the following topics related to

Orifice Pulse Refrigerator (OPTR):

1. To study the thermodynamic phenomenon occurring within the OPTR and derive the equations

for various mass flow rates and the pressure variation.

2. To develop a SCILAB code from the governing thermodynamic equations, so as to get the exact

Variation of mass flow rates and the pressure oscillation within the OPTR.

3. To plot the mass flow rates and pressure oscillation on a phasor, so as to visualize the dependence

Of one quantity on the other and study the phase relationship.

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4. THERMODYNAMIC STUDY OF SINGLE ORIFICE PULSE TUBE

CRYOCOOLER

The study of the working process of the pulse tube refrigerator becomes very complex due to the

oscillating flow and due to the presence of the regenerator, orifice-reservoir and the double inlet valve.

Compression and expansion of the gas column inside the pulse tube is the reason behind the cooling

effect observed at the cold end of the pulse tube. The compression and expansion process of the working

gas within the pulse tube lies between adiabatic and isothermal processes.

Liang et al [12] was the first to attempt solving the working mechanism of pulse tube refrigerator by

analyzing the thermodynamic behavior of the gas element as adiabatic process. The following

assumptions are made in conjunction with the adiabatic behavior of the working gas:

Hot-end heat exchanger, cold-end heat exchanger and the regenerator have been assumed to be

perfect, which means that there will be a constant temperature gradient between its hot end and its

cold end and the heat exchangers will work at constant temperature at steady state.

Working fluid has been assumed to be an ideal gas.

Viscous effect of the gas has been neglected.

The variation of mass flow rates, pressure and temperature has been assumed to be sinusoidal.

There is no phase difference between the pressure and the temperature throughout the working space

of the pulse tube refrigerator.

There is no length wise mixing or heat conduction.

Figure 9 orifice Pulse Tube Refrigerator (OPTR) with the Working fluid assumed to be Helium gas

The pressure variation within the pulse tube has been assumed to be sinusoidal, so the pressure variation

at any instant within the pulse tube is computed with the help of the following equation

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i.e.

= + 1 sin()

Where, = 2

Now in order to calculate mass flow rate, pressure and temperature as a function of time and

position in the system, the governing equations are applied to all of the discrete volumes. These equations

include the ideal gas law, the mass conservation equation and the energy conservation equations.

Substituting the ideal gas law into the mass conservation equations for the regenerator gives

(

)

=

(

) =

=

Where; =

ln

As the temperature profile within the regenerator has been assumed to vary linearly along its

length, so instead of average temperature we have to take the logarithmic mean temperature of the same.

Since the temperature at the hot-end heat exchanger and the cold-end heat exchanger has been assumed

to be constant so similarly proceeding we can get the mass flow rates at the hot-end heat exchanger and

the cold-end heat exchanger as:

(

)

=

(

) =

=

(

)

=

(

) =

=

For determining the mass flow rate within the pulse tube we assume energy conservation equation

instead of mass conservation as the temperature is known to vary along with the pressure which is

sinusoidal in nature. Hence applying the energy conservation equation in the pulse tube we get,

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(

)

=

Using the above mentioned equation and the ideal gas law, we get,

= ( )

Or,

=

+

Combining these equations we get mass flow rate at the cold end heat exchanger as,

=

+

Where;

= +

+

As we can see from the diagram shown above,

o hm m

Also we can express the mass flow rate at through the orifice as,

= ( )

Where, Z is the impedence and is given by the formula;

=

The pressure variation within the pulse tube is already know but the pressure variation of the

compressor is still not known, which can be found by assuming that the mass flow rate within the

regenerator is directly proportional to the pressure difference between the compressor and the pulse tube

i.e.

( )reg reg cp tm C P P

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Where, Creg is calculated using Erguns law for laminar flow and is mathematically given by the

Following formula:

=

22

4 150

3

(1 )2

The results obtained on doing the analysis were plotted in GNUPLOT and the code has been generated

in the SciLab software .

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5. Generated code in SCILAB and results. 5.1 Generated code for Thermodynamic study of single orifice pulse tube cryocooler in

Scilab

Figure 10 Code generated in SCILAB for Thermodynamic analysis

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The results obtained on executing the code shown above were as follows:-

These results are obtained on running the code for t=0 to 1 second at an interval of 0.1 second

Figure 11 Result obtained from thermodynamics energies

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a) Design data:

Table 1 Design Data for Adiabatic Model

b) Operating condition:

Table 2 Operating condition for Adiabatic Model

Components Parameters

Regenerator Length (Lreg) = 0.3 m

Diameter (Dreg) = 0.032 m

Porosity (ev) = 0.7

Hydraulic Diameter (Dh) = 0.04 mm

Pulse Tube Length (Lpt) = 0.8 m

Diameter (Dpt) = 0.02 m

Volume (Vpt) = 0.00025 m3

Cold-end Heat Exchanger Volume (Vchx) = 0.00002 m3

Hot-end Heat Exchanger Volume (Vhhx) = 0.00002 m3

Orifice Diameter (D0) = 1 mm

Reservoir Volume(Vr)=0.007 m3

Operating Parameters Values

Average Pressure 10.5bar

Oscillating Pressure 2bar

Frequency 2 Hz

Cold- End Heat Exchanger 100K

Hot-End Heat Exchanger 300K

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c) Fluid properties:

Physical Condition Physical Properties

Temperature(200K)

Pressure(10 bar)

Dynamic Viscosity()=15.21x10-6 Ns/m2

Density ( ) =2.389Kg/m3

Specific Heat Capacity at Constant

pressure(Cp)=5193.0J/KgK

Gas Constant(R)= 2074.6J/KgK

Adiabatic Constant()=1.67

Table 3 Fluid properties for Adiabatic Model

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A. Graph representing the pressure variation within the pulse tube (Pt), the compressor (Pc)

Figure 12 Pressure Variation within The Pulse Tube (Pt), The Compressor (Pc)

B. Graph representing the mass flow rates through the regenerator (mreg) and pressure (P).

Figure 13 Mass Flow Rates Through The Regenerator (Mreg) And Pressure (P).

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C. Graph representing the mass flow rate through hot heat exchanger (mhhx) and pressure (P).

Figure 14 the mass flow rate through hot heat exchanger (mhhx) and pressure (P).

D. Graph representing mass flow rate through pulse tube (mpt) and pressure (P):

Figure 15 Mass Flow Rate through Pulse Tube (Mpt) And Pressure (P)

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E. Graph representing mass flow rate through cold heat exchanger (mchx) and pressure (P):

Figure 16 Mass Flow Rate Through Cold Heat Exchanger (Mchx) And Pressure (P)

F. Graph representing mass flow rate through cold heat exchanger (mchx) and mass flow rate

through hot heat exchanger (mhhx) and showing the phase difference between them in the zoomed

view on right top corner:

Figure 17 Mass Flow Rate through Cold Heat Exchanger (Mchx) And Mass Flow Rate through Hot Heat Exchanger

(Mhhx)

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6. PHASOR ANALYSIS

There are several methods adopted to analyze PT cryocooler amongst which phasor analysis is

one of the easiest method to verify the consistency and check the obtained result. A phasor diagram for

a pulse tube refrigerator (PTR) is a vectorial representation of mass flow rate, pressure, and temperature

at different locations as a function of time. Phasor or phase vector is a way of representation of a sinusoidal

function whose amplitude (A), phase () and frequency () are time-invariant. It can be called as a sub

set of a more general concept called analytic representation.

In the present work, phasor analysis of pulse tube cryocooler is presented based on the law of

conservation of mass. The phasor analysis helps to understand the importance of phase difference

between mass flow rate at the cold end and the pressure pulse in the pulse tube. The refrigerating effect

for different types of PTR strongly depends on the phase shift arrangement and also on the phase

difference. Refrigeration effect can be obtained as follows.

=< >=1

2

| | cos

Where, is phasor angle.

Q = refrigeration effect

Thus we can see that for given design and operating parameters more refrigeration effect can be

obtained by lowering the phasor angle .Hence theoretically should be zero to get maximum refrigeration

effect. Thus numerical value of would be indicative of the losses in system .Now basically phasor angle

is the difference of phase of mass flow rate in each component with pressure .In phasor analysis adiabatic

process is assumed in PT and isothermal process in all other parts. We have previously assumed mass

flow rate through orifice is proportional to pressure difference and thus it would be in phase with pressure

lying on horizontal axis. Now all the vertical components for each different component is dependent on

its respective operating parameters and thus makes phasor angle useful for determining various operating

parameters while designing cryocooler for required refrigeration effect. Here the phasor analysis is done

at steady state to obtain the final refrigeration effect.

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Figure 18 Phasor diagram

Notation:

hh

c

Tm A

T 1 hhx

h

PVB

RT

1 pt

c

PVC

RT

1 chx

c

PVD

RT

1 reg

mean

PVE

RT

1 eq

mean

PVF

RT

From the phasor diagram obtained earlier, a code was generated to calculate the parameters using the

lengths in the phasor diagram and the results obtained were very close to that obtained using the code

written earlier using the thermodynamic analysis. The code for the same is shown below

31 | P a g e

Figure 19 Code For phasor diagram in SCILAB

32 | P a g e

On executing the above given code we obtain the phase angles between the various parameters and the

pressure vector. We also obtain the values of different parameters from analyzing the phasor diagram.

The results obtained were as follows:-

Figure 20 Results obtained through CODE

33 | P a g e

Table 4 Comparison of results obtained from both the codes

Table 5 Phase angle of various parameters with pressure vector

Mass flow rates in different sections of

the systems

Results obtained by

Phasor analysis

( )

Results obtained by

Thermodynamics analysis

of systems

( )

Mcp (mass flow rate in compressor) 0.0056625 0.004781

Mreg (mass flow rate in regenerator) 0.0044512 0.0038991

Mchx (mass flow rate in cold heat

exchanger)

0.0032161 0.0023905

Mc (mass flow rate through cold end of

pulse tube)

0.0030597 0.0023905

Mhhx (mass flow rate in hot

heat exchanger)

0.0008025 0.0007968

Mo (mass flow rate through orifice) 0.0007984 0.0007968

Mh(mass flow rate through hot end of

pulse tube)

0.0007984 0.0007968

Mpt (mass flow rate through pulse tube) 0.0024136 0.0023905

Angle between various mass flow rates

and Pressure

Phase angle obtained by phasor analysis

( Mcp and pressure) 64.976497

( Mreg and pressure) 57.445336

( Mchx and pressure) 41.862082

( Mc and pressure) 38.481130

( Mhhx and pressure) 1.9312147

34 | P a g e

7. Conclusion

A thorough study of the mass flow rate through the various parts of a GM type single orifice Pulse

Tube Cryocooler (OPTC) was done. This study helped in developing a SciLab code which helped us

generate the time varying graphs for the mass flow rate and pressure oscillation at the various parts of a

GM type single orifice Pulse Tube Cryocooler when a initial working condition and dimensions for the

various components of the OPTC are provided.

The output of the SciLab program was further used to construct a phasor diagram for the mass

flow rates at various sections of the GM type OPTC, which is a convenient way to observe the phase

relationship and hence make necessary adjustments to optimize the output.

The output shown by the SciLab code was within 10% - 15% range of the predicted results of the

phase diagram. It was also observed that the mass flow rate at various sections in a typical GM type

single orifice pulse tube cryocooler varies sinusoidally with time due to sinusoidal variation in pressure.

It was also observed that the Refrigeration effect is a function of cosine of the phase angle and hence

an increase in phase angle tends to reduce the refrigeration effect.

Hence we could conclude that the study of phase angles is very important for achieving the

optimum conditions of operation for the pulse tube to reduce the viscous dissipation effects and smoother

functions of pulse tube cryocooler.

35 | P a g e

8. References

[1] Radebaugh, Ray. Development of the pulse tube refrigerator as an efficient and reliable Cryocooler,

Proc. Institution of Refrigeration (London) 1999-2000.

[2] Gifford, W.E. and Longsworth, R.C. Pulse tube refrigeration, Trans ASME B J Eng. Industry

86(1964), pp.264-267.

[3] Mikulin, E.I., Tarasow, A.A. and Shkrebyonock, M.P. Low temperature expansion Pulse tube,

Advances in cryogenic engineering 29(1984), pp.629-637.

[4] Zhu Shaowei, Wu Peiyi and Chen Zhongqi, Double inlet pulse tube refrigerators: an Important

improvement, Cryogenics30 (1990), pp. 514-520.

[5] Gifford, W.E. and Kyanka, G.H. Reversible pulse tube refrigerator, Advances in Cryogenic

engineering 12(1967), pp.619-630.

[6] De Boer, P. C. T., Thermodynamic analysis of the basic pulse-tube refrigerator, Cryogenics34 (1994),

pp. 699-711.

[7] Mikulin, E.I., Tarasow, A.A. and Shkrebyonock, M.P. Low temperature expansion pulse Tube,

Advances in cryogenic engineering 29(1984), pp.629-637.

[8] Storch, P.J. and Radebaugh, R Development and experimental test of an analytical model Of the orifice

pulse tube refrigerator, Advances in cryogenic engineering 33(1988), Pp.851-859.

[9] L.Mohanta and M.D. Atrey, Phasor Analysis of Pulse Tube Refrigerator, Cryocoolers 16 (2011), 299-

308

[10] Hofmann and H. Pan, Phase Shifting in Pulse Tube Refrigerator, Cryogenics (1999)

[11] Shashank Kumar Kushwaha , Amit medhavi & Ravi Prakash Vishvakarma, Mathematical Analysis of

Pulse Tube Cryocoolers Technology, Global Journal of researches in engineering Electrical and

electronics engineering ,Volume 12, Issue 6, Version 1.0( 2012),pp.21-31.

[12] Study on pulse tube refrigeration, J.Liang, A.Ravex, P.Rolland. Cryogenics Volume 36, issue 2

Pp.101-106.