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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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19 | P a g e
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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29 | P a g e
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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30 | P a g e
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
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31 | P a g e
Figure 19 Code For phasor diagram in SCILAB
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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
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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
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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.
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35 | P a g e
8. References
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as an efficient and reliable Cryocooler,
Proc. Institution of Refrigeration (London) 1999-2000.
[2] Gifford, W.E. and Longsworth, R.C. Pulse tube refrigeration,
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[3] Mikulin, E.I., Tarasow, A.A. and Shkrebyonock, M.P. Low
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[8] Storch, P.J. and Radebaugh, R Development and experimental
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[9] L.Mohanta and M.D. Atrey, Phasor Analysis of Pulse Tube
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