In the given report, a comprehensive design of the working of a miniature Stirling
Cooler is presented. The motivation of the study is to determine the optimum geometrical
parameters of a cryo-cooler such as compressor length, regenerator diameter, expander
diameter, and expander stroke. In the first part of the study, an ideal analysis is carried out
using the Stirling Cycle and basic thermodynamics equations. Using these equations, rough
geometrical parameters are found out.
In the second part of the study, a more comprehensive Schmidt’s analysis is carried
out. In this analysis, pressure and volume variations are considered sinusoidal and based on
these, various equations regarding efficiency and COP are derived. Various graphs are
generated in CHARTGO (www.chartgo.com) plotting Refrigeration and Work done w.r.t to
various geometrical parameters. With the help of these graphs, the net refrigeration obtained
is calculated for a given geometry of cooler. This model provides a more accurate picture of
the cooler. However in this analysis, regenerator efficiency is considered 100 % which is not
true in practical cases.
In the third and final part of the study, optimization of regenerator is carried out. This
part is based on various looses taking place inside a regenerator are considered and accounted
for. These looses are minimized using an iterative cycle and optimum regenerator dimensions
are obtained. Thus the geometrical results obtained from the third part of the study are
expected to be most accurate as it accounts for most of the losses taking place inside a cryocooler.
(II)
TABLE OF CONTENTS
Acknowledgement I
Abstract II
List of Figures III
List of Table IV
Nomenclature VI
Chapter 1: INTRODUCTION OF IDENTIFIED INDUSTRY 1-5
1.1 About Company 1
1.1.1 Company information 1
1.1.2 Products of Company 2
Chapter 2: IDENTIFICATION OF PROBLEM 6-7
2.1 Problem Definition 6
2.1.1 Problem Statement 6
2.2 Detail Description of Problem 7
Chapter 3: OVERVIEW OF STIRLING ENGINE 8-15
3.1 Introduction 8
3.2 Operating Principal of Stirling Cycle 9
3.2.1 Thermodynamic of Stirling Cycle 11
3.3 Stirling Engine Classification 12
3.4 Basic Components of Stirling Engine 14
Chapter 4: OVERVIEW OF STIRLING COOLER 16-18
4.1 History of Stirling Cooler 16
4.2 The Stirling Cycle as a Refrigerator 16
4.3 Analysis of Stirling Cooling Cycle 17
4.3.1 Work Input to Ideal Stirling Cycle 17
4.3.2 Heat Flow in a Ideal Stirling Cycle Refrigerator 18
4.3.3 Performance of a Ideal Stirling Cycle Refrigerator 18
Chapter 5: ANALYSIS, DESIGN, & ASSEMBLY OF WORK 19-42
5.1 Ideal Analysis of Stirling Cycle 19
5.2 Schmidt’s Cycle and Its Analysis 23
5.3 Simulation With Schmidt Cycle 24
5.4 Design and Assembly Work 29
Chapter 6: SHORTCOMINGS OF SCHMIDT ANALYSIS &
SCOPE FOR FUTURE WORK 43-55
6.1 Shortcomings of Schmidt Analysis 43
6.2 Scope for Future Work 43
6.2.1 Regenerator Optimization 44
6.2.2 Optimization Analysis 45
6.2.3 Optimization Procedure 51
Chapter 7: COSTING AND CONCLUSION 56-58
7.1 Bill of Material and Costing 56
7.2 Conclusion 58
REFRENCES 59
LIST OF FIGURES
Fig No. Fig Name Page No.
1.1 Oil Sealed Rotary High Vacuum Pump 2
1.2 High Vacuum Booster System 2
1.3 Fluid Coupling 3
1.4 Impregnation System 3
1.5 Water ring Vacuum Pump 4
1.6 Diaphragm Pump 4
1.7 Vacuum cum Pressure Pump 5
3.1 Modern Stirling Engine 9
3.2 Stirling Engine And Its Components 10
3.3 P-V Diagram of Stirling Cycle 11
3.4 Alpha Type Stirling Engine 13
3.5 Gamma Type Stirling Engine 14
4.1 Schematic diagram of Stirling Cooler 16
4.2 P-V &T-S Diagram of Stirling Cooling Cycle 17
5.1 Volume Displaced by Piston-1 at Different angles 25
5.2 Volume Displaced by Piston-2 at phase angle 00 25
5.3 Combined effect of Piston-1 & Piston-2 at phase angle 00 26
5.4 Volume Displaced by Piston-2 at phase angle 450 26
5.5 Combined effect of Piston-1 & Piston-2 at phase angle 450 27
5.6 Volume Displaced by Piston-2 at phase angle 900 27
5.7 Combined effect of Piston-1 & Piston-2 at phase angle 900 28
III
Fig No. Fig Name Page No.
5.8 0.25 HP, 1440r.p.m Motor 29
5.9 2-D Drawing of Regenerator 30
5.10 3-D View of Regenerator 30
5.11 Drawings for Body 31
5.12 Drawing for Diaphragm 32
5.13 Drawing of Diaphragm Part 33
5.14 Drawing for Bearing Cover 34
5.15 Drawing for Eccentric Part-1 35
5.16 Drawing for Eccentric Part-2 36
5.17 Drawing of Shaft 37
5.18 Drawing of Side Cover 38
III
LIST OF TABLES
Table No. Table Name Page No.
4.1 History of Stirling Cooler 16
5.1 .P-V-T values for Stirling Cooling Cycle 20
5.2 Capacity & C.O.P of Cooler 22
6.1 Stirling Cycle Refrigerator Design Parameters 53
6.2 First Estimate of losses in Stirling Cooler 54
6.3 Calculated Regenerator Parameters From The Optimization
Procedure
55
7.1 Cost Estimation of Cooler 56
IV
NOMENCLATURE
1) Aff = Fluid axial free flow area
2) Am = Matrix thermal conduction heat transfer area
3) Ar = Total regenerator frontal area
4) As = Matrix total heat transfer area
5) Cc = Cold fluid heat capacity rate
6) Ch = Warm fluid heat capacity rate
7) Cm = Matrix heat capacity rate
8) Cmax = the larger of Cc and Ch
9) Cmin = the smaller of Cc and Ch
10) Cp = Specific heat at constant pressure
11) Cv = Specific heat at constant volume
12) Dd = Displacer diameter
13) Dh = Hydraulic diameter (Dh = 4*rh)
14) Dr = Regenerator diameter
15) d = Screen wire diameter
16) f = Fluid coefficient of friction
17) fr = Frequency
18) G = Mass flow rate per unit free flow area
19) h = Heat transfer coefficient
20) Ie = Regenerator thermal inefficiency
21) Kf = Fluid thermal conductivity
22) Km = Matrix thermal conductivity
23) L = Regenerator length
24) M = Molecular weight
25) Mf = Mass of the fluid
26) Mm = Mass of the matrix material
27) m, = Mass flow rate
28) Pmax = Maximum cycle pressure
29) Pmin = Minimum cycle pressure
30) Ps = System pressure
31) Pa = Pressure ratio
32) pso = System pressure amplitude
33) Qc = Heat conduction
34) Qh = Heat transferred between fluid and matrix material
35) Qnet = Net cryocooler refrigeration
36) Qreg = Regenerator thermal losses
37) Ta = Ambient room temperature
38) Tcold = Cold end temperature
39) Tw = Warm end temperature
40) t = Time
41) Ve = Expansion space volume
42) Vcs = Compression space volume
43) Vm = Volume occupies by the matrix material
44) Vr = Regenerator total volume
45) Vrv = Regenerator void volume
46) veo = Expansion space volume amplitude
47) Wpv = Gross mechanical refrigeration produced
48) W = Mechanical power
49) Xd = Displacer position
50) Xp = Compressor piston position
51) S = Expander stroke
52) Pmean = Mean cycle pressure
53) p = instantaneous cycle pressure
54) P max = maximum cycle-pressure
55) P min = Minimum cycle-pressure
56) Pmean = Mean cycle-pressure
57) W = Work Done by compressor
58) Qe = heat transferred to the working fluid in expansion space
59) Qc = heat transferred in the compression space
60) COP = coefficient of performance of the cryocooler
61) R = characteristic gas constant of the working fluid
62) TC = Temperature of the working fluid in the compression space
63) TD = Temperature of the working fluid in the dead space
64) TE = Temperature of the working fluid in expansion space
65) VC = swept volume of the compression space
66) VE = swept volume of the expansion space
67) VD = swept volume of the dead space
68) x = VC/VE, swept volume ratio
69) y = VD/VE, dead volume ratio
70) t = TC/TE, temperature ratio
71) VT = (VE + VC + VD)
72) ?? = angle by which volume variations in expansion space lead those in the
compression space
73) A = a factor [t2 +2*t*x*cos(??)+x2]0.5
74) S = (VD*TC)/ (VE*TD)
75) B = a factor (t+x+2*S)
76) ?? =A/B = [ t2 + x2 + 2*t*x*cos(??) ]0.5/(t+x+2*S)
77) ?? = atan[{x*sin(??)}]/(t+k+2*S)
V
INTRODUCTION OF IDENTIFIED INDUSTRY CHAPTER 1
CHAPTER 1 Page 1
CHAPTER-1
INTRODUCTION OF IDENTIFIED INDUSTRY
1.1 ABOUT THE COMPANY
1.1.1 COMPANY INFORMATION[1]
INTRODUCTION OF IDENTIFIED INDUSTRY CHAPTER 1
CHAPTER 1 Page 2
Parag Engineering is an ISO 9001-2008 quality certified company. It is a most trusted and
quality manufacturer of Oil Sealed Rotary High Vacuum Pump, Water Ring Vacuum Pump,
Monoblock Vacuum Pump, Fluid Couplings and Impregnation Systems since 1980.
1.1.2 PRODUCTS OF THE COMPANY
A. OIL SEALED ROTARY HIGH VACUUM PUMPS
Fig 1.1: Oil Sealed Rotary High Vacuum Pump
B. MECHANICAL HIGH VACUUM BOOSTER SYSTEM
Fig. 1.2: High Vacuum Booster System
INTRODUCTION OF IDENTIFIED INDUSTRY CHAPTER 1
CHAPTER 1 Page 3
C. FLUID COUPLING
Fig.1.3: Fluid Coupling
D. IMPREGNATION SYSTEM
Fig.1.4: Impregnation System
INTRODUCTION OF IDENTIFIED INDUSTRY CHAPTER 1
CHAPTER 1 Page 4
E. WATER RING VACUUM PUMP
Fig.1.5: Water ring Vacuum Pump
F. DIAPHRAGM PUMP
Fig. 1.6: Diaphragm Pump
INTRODUCTION OF IDENTIFIED INDUSTRY CHAPTER 1
CHAPTER 1 Page 5
G. VACUUM CUM PRESSURE PUMP
Fig.1.7: Vacuum cum Pressure Pump
IDENTIFICATION OF PROBLEM CHAPTER 2
CHAPTER 2 Page 6
CHAPTER 2
IDENTIFICATION OF PROBLEM
2.1 PROBLEM DEFINITION
2.1.1 PROBLEM STATEMENT
‘To design the Stirling Cooler.’
2.1.2 Problem Summary
Parag Engineering Ltd. is a firm which produces the various types of
pumps and fluid coupling. With our discussion with our external guide Mr.
Gopal Patel and our internal guide Prof. Nilesh Pancholi, we have selected the
project to design the Stirling cooler. The Stirling cooler is a device or
refrigerator which can produces below zero temperature (<0 degree).
The basic goal of the development of Stirling Cooler is the reduction of
environmental impact caused by the refrigerators. The various aspects have
been taken into account to further minimize the environmental impact are as
follows:
‘ No CFCs, HCFCs and HFCs.
‘ Use of reusable components and/or recyclable materials.
‘ Designed for long life and high reliability.
‘ Designed for easy dismantling.
‘ Minimum use of harmful and non-renewable materials.
2.2 DETAILED DESCRIPION OF THE PROJECT
A. No CFCs, HCFCs and HFCs.
Until recently CFC’s were used in refrigerators as a refrigerant. As these
substances have very high Ozone depletion potential they will be phased
IDENTIFICATION OF PROBLEM CHAPTER 2
CHAPTER 2 Page 7
out under the Montreal Protocol. Some of the hydrocarbons like propane
or isobutene were tried as a refrigerant. [2]
B. Use of reusable components and/or recyclable materials.
The cooler is designed to work under the normal circumstances for a life
of 10-15years.Although PUR foam insulated cabinets can contribute to
the reduction of the loading capacity of refrigerator on short term, on the
long to medium term,PUR foam insulation which is currently used is
unfavourable for several reasons, an important one being the dismantling
phase of refrigerators.PUR-foam is a thermoset which cannot be
recycled.PUR residues on plastic components will render the recycling of
these components impractical.[2]
C. OTHER APPLICATION OF STIRLING REFRIGEARTION
TECHNOLOGY:
It is a simple domestic refrigerator with single temperature. It can be also
used as domestic refrigerators and freezers. It is also useful in business
market (hotels, retails, etc.), cool boxes and industrial market.[2]
2.3. SPECIFICATIONS OF THE COOLER:
‘ Inlet Temperature = 25??C
‘ Inlet Pressure = 1 atm.
‘ Evaporator Temperature = 5??C
‘ Input Motor Rating = 0.25HP,AT 1440 R.P.M
‘ Capacity of cooler = 0.3 tonnes
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 8
CHAPTER 3
OVERVIEW OF STIRLING ENGINE
3.1 INTRODUCTION
A Stirling engine is a heat engine which was invented by Robert Stirling in 1918. It is
based on gas properties and thermodynamic laws and principles.
The engine uses an external heat source in contrast with combust engines so there is no
explosion inside the cylinder while working. The gas is expanded and compressed
cyclically and continuously to produce motion to transforming energy. Fluid gas remains
inside the system and it is displaced from the hot side to the cool side and vice versa when
the engine is operating. There is no exhaustion like normal petrol engine, the engine
works very quietly.
The compressible gas can be air, hydrogen, helium, nitrogen or even vapour depending on
the design of the engine. Any source of heat can power the engine, from solid coal to oil
and solar energy, only the heat source must be adjusted to the engine. For example in a
solar energy model the solar concentrator and absorber have to be integrated with the
heating part of the cylinder. The Stirling engine was invented as a safer alternative for
steam engines of the time, when steam engines had poor quality and often caused
explosion because of uncontrollable pressure elevation and primitive technology.
This engine offers the possibility for having high efficiency with less exhaust emissions
in comparison with the internal combustion engine.
The Stirling engine has high performance in many applications and is suitable where:
‘ Multi-fueled characteristic is required;
‘ A very good cooling source is available;
‘ Quiet operation is required;
‘ Relatively low speed operation is permitted;
‘ Constant power output operation is permitted;
‘ Slow changing of engine power out-put is permitted;
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 9
‘ A long warm-up period is permitted.
Fig.3.1-Modern Stirling Engine
Modern models of Stirling engine have a relatively high efficiency and can be run even at
low temperature, (figure1) shows a new modern engine which can be run by the heat of a
cup of coffee. (Prof .T. Sundararajan, UT of Madras India.)[3]
3.2 OPERATING PRINCIPLES OF STIRLING ENGINE
In its simplest form a Stirling engine consists of a cylinder containing a gas, a piston and
a displacer. The regenerator and a flywheel are other complimentary parts of the engine.
When heat part of cylinder is heated up by an external heat source (figure2), the
temperature rises and gas expands proportional in to the temperature of the heat side.
Total volume is constant and limited by a piston thus expanded gas pushes the piston
down, so the volume of the pressured gas is increased and the gas loses its pressure and
temperature, then the piston backs to the heat side and compresses the gas by momentum
force of the flywheel, when it reaches near its up limit the displacer also pushes the
cooled gas to the heat side of the cylinder so that the gas is compressed and it can be
prepared to do another cycle. The expanding gas pushes the piston down again to produce
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 10
mechanical energy for doing work, this cycling will continue till an external heat source
is available.
The flywheel and the regenerator have great roles in the engine’s performance. The flywheel
converts the linear movement of a working piston to rotary movement, it gives
needed momentum for the cycle procedure. Regenerator takes heat from gas in the
expansion phase and releases heat to the gas in the compression phase, improving the
engine’s efficiency considerably. A Stirling engine and its components are shown in
(figure2) below.
Fig.3.2: Stirling Engine And Its Components
The cycle of a Stirling engine has four phases; heating, expansion, cooling and compression.
Short explanation of each phase is given in the following:
‘ Heating: Heat source provides thermal energy to the engine so that it raises
pressure and temperature of gas.
‘ Expansion: in this phase the volume increases, but the pressure and temperature
decrease, mechanical energy is produced from heat energy during this phase of
cycle only.
Heating
source
Working
piston
Flywheel
Heat
regenerator
Displacer
piston
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 11
‘ Cooling: the gas is cooled and temperature and pressure decrease, so the gas is
prepared to be compressed during this cycle.
‘ Compression: the pressure of gas increases whereas its volume decreases; a part of
produced mechanical energy is used for processing of this phase, because it needs
an amount of work to be done.
Fig. 3.3: P-V Diagram of Stirling Cycle
Looking at the graph (figure 4) of Stirling cycle one can see that, the volume is constant
in heating phase (1-2) and cooling phase (3-4) while during Expansion (2-3) and
Compression (4-1) volume is varying but temperature is constant. (Pierre Gras, January
07, 2009).[3]
3.2 Thermodynamic of Stirling cycle
Heat supplied from the hot source:
Similarly, heat rejected to the cold sink:
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 12
Therefore;
And as the cycle efficiency, therefore
For the constant volume process 1-2,
Therefore;
This shows that the Stirling cycle has the same efficiency as the Carnot cycle.
The work ratio for the Stirling cycle is calculated as:
I.e. since
As formula shows energy of a cycle depends on pressure and volume, so any changes in
these two main parameters changes output power of engine. In simple words it can be
said, temperature of hot side of engine causes pressure to rise and pushes the piston move
down, piston’s moving down changes volume thus it makes ??T and ??P inside cylinder
that forces engine to run.
Early Stirling engines were inefficient compare to other heat engines. But now its
sophisticated models are enough efficient and competitive with internal combustion engines,
the new ones can be run at either high or low temperature heat in almost all
circumstances. [3]
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 13
3.3 Stirling engine classification
Several types of Stirling engines have been introduced for different purposes, the most
known and practical models are Alpha, Beta and Gamma. The working mechanism of all
the three is the same and based on gas expansion at higher temperature and thermodynamic
laws, but each type has individual designation, short explanation for each one
comes in the following.
‘ The Alpha type is the simplest design (figure 5) of Stirling engine, easy to
maintain and repair. It however does use more material to built, and efficiency
may be lower. Hence, it is most useful for stationary or having large engines.
Fig.3.4:Alpha Type Stirling Engine
‘ The Beta type, which this study is based on, has more complicated design and
more difficult to maintain or repair it, however it needs lees component to be built.
Its efficiency is lightly higher than others. Hence, it is most useful for mobile or
small application like laboratory works.
‘ A gamma Stirling is simply a beta Stirling in which the power piston is mounted
in a separate cylinder alongside the displacer piston cylinder, but is still connected
to the same flywheel. The gas in the two cylinders can flow freely between them
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 14
and remains a single body. This configuration produces a lower compression
ratio but is mechanically simpler and often used in multi-cylinder Stirling engines.
Fig.3.5: Gamma Type Stirling Engine
3.4 Basic Components [3]
A Stirling engine consists of a number of basic components, which may vary in design
depending on the type and configuration. The most basic are outlined as follows:
‘ Power Piston and Cylinder
This consists of a piston head and connecting rod that slides in an air tight cylinder. The
power piston is responsible for transmission of power from the working gas to the
flywheel. In addition, the power piston compresses the working fluid on its return stroke,
before the heating cycle. Due to the perfect air tight requirement, it is the most critical
part in design and fabrication.
‘ Displacer Piston and Cylinder
The displacer is a special purpose piston, used to move the working gas back and forth
between the hot and cold heat exchangers. Depending on the type of engine design, the
displacer may or may not be sealed to the cylinder, i.e. it is a loose fit within the cylinder
and allows the working gas to pass around it as it moves to occupy the part of the cylinder
beyond.
‘ Source of Heat
The source of heat may be provided by the combustion of fuel, and since combustion
products do not mix with the working fluid, the Stirling engine can run on an assortment
of fuels. In addition, other sources such as solar dishes, geothermal energy, and waste
OVERVIEW OF STIRLING ENGINE CHAPTER 3
CHAPTER 3 Page 15
heat may be used. Solar powered Stirling engines are becoming increasingly popular as
they are a very environmentally friendly option for power production.
‘ Flywheel
The flywheel is connected to the output power of the power piston, and is used to store
energy, and provide momentum for smooth running of the engine. It is made of heavy
material such as steel, for optimum energy storage.
‘ Regenerator
It is an internal heat exchanger and temporary heat store placed between the hot and cold
spaces such that the working fluid passes through it first in one direction then in the other.
Its function within the system is to retain heat which would otherwise be exchanged with
the environment. It thus enables the thermal efficiency of the cycle to approach the
limiting Carnot efficiency.
On the flip side, the presence of regenerator (usually a matrix of fine steel wool),
increases the ‘dead space’ (upswept volume). This leads to power loss and reduces
efficiency gains from the regeneration.
‘ Heat Sink
The heat sink is typically the environment at ambient temperature. For small heat engines,
finned heat exchangers in the ambient air suffice as a heat sink. In the case of medium to
high power engines, a radiator may be required to transfer heat from the engine.
OVERVIEW OF STIRLING COOLER CHAPTER 4
CHAPTER 4 Page 16
CHAPTER 4
OVERVIEW OF STIRLING COOLER
4.1. History of Stirling Cooler [4]
The main events that occurred in the history of Stirling coolers are given in following table:
YEAR EVENT
1815 Robert Stirling-Stirling Engine
1834 John Herschel-concept of using as a cooler
1861 Alexander Kirk-The concept into practise
1873 Davy Postle-Free piston system
1956 Jan Koehler-First commercial machine for air Liquidification
1965 Jan Koehler-Nitrogen Liquidification
Fig.4.1: Schematic diagram of Stirling Cooler
4.2. The Stirling Cycle as a Refrigerator
The ideal Stirling-cycle refrigerator or heat-pump is, in effect, identical to a Stirling-cycle
engine except that the heat absorbing end of the machine now becomes the cold region, and
the heat rejecting end of the machine becomes the hot region. The thermodynamic processes
for a refrigerator are illustrated using a pressure-volume and temperature-entropy diagrams in
Figure 4.2. Because refrigerator/heat-pumps tend to have a smaller temperature difference
between hot and cold regimes than an engine, the pressure-volume and temperature-entropy
OVERVIEW OF STIRLING COOLER CHAPTER 4
CHAPTER 4 Page 17
diagrams appear somewhat squatter in comparison. It should be noted that for the ideal
Stirling Cycle the heat-exchangers, regenerator, and transfer passages are assumed to have
zero volume. The processes are as follows:
Fig. 4.2: P-V & T-S Diagram of Stirling Cooling Cycle
‘ 1-2: Isothermal expansion ‘ the low-pressure working gas expands isothermally at
cold end temperature, hence absorbing heat from the cold space (via the heat
absorbing heat-exchanger) and doing work to the power-piston.
‘ 2-3: Isochoric displacement ‘ the displacer-piston transfers all the working gas
isochorically through the regenerator to the hot end of the machine. Heat is delivered
to the gas as it passes through the regenerator, thus raising the temperature of the gas
to that of the hot space. As the temperature rises, the gas pressure increases
significantly.
‘ 3-4: Isothermal compression ‘ the power-piston does work to the gas and compresses
it isothermally at hot end temperature, hence rejecting heat to the hot space (via the
heat rejecting heat-exchanger). Because the gas is at high pressure, more work is
required for compression than was obtained from the gas during expansion (in 1??2).
The cycle therefore has a net work input.
‘ 4-1: Isochoric displacement ‘ the displacer piston transfers all the working gas
isochorically through the regenerator to the cold end of the machine. Heat is absorbed
from the gas as it passes through the regenerator, thus lowering the temperature of
Note that, unlike the work output from an engine the refrigerator work has a positive
OVERVIEW OF STIRLING COOLER CHAPTER 4
CHAPTER 4 Page 18
value under the sigh conversion used here since a net energy input is the gas to that of
the cold space. As the temperature reduces, the gas pressure drops significantly, and
the system returns to its initial conditions.
4.3. Analysis of Stirling-cycle Refrigerator
4.3.1. Work input to an ideal Stirling-cycle refrigerator[5]
An equation for net work input to an ideal Stirling-cycle refrigerator at-pump can be derived
in exactly the same way as work output for a Stirling-cycle engine.
Note that, unlike the work output from an engine, the refrigerator/heat-pump work has a
positive value under the energy sign convention used here, since a net energy input is
required to move heat from a low to high temperature regime.
4.3.2. Heat flow in an ideal Stirling-cycle refrigerator [5]
Equations for the heat flows into and out of an ideal Stirling-cycle refrigerator/heat-pump
can be derived in a similar way as heat flows in a Stirling-cycle engine .
The main difference is that in a refrigerator/heat-pump the heat flows into the system at a
low temperature (TL) and out of the system at a high temperature (TH).
For a refrigerator refrigerating effect produced is given by
ln (V2/V1)
4.3.3. Performance of an ideal Stirling-cycle refrigerator [5]
The coefficient of performance for any refrigerator is defined as the ratio of heating/cooling
effect to work input, i.e.
for a refrigerator, the refrigeration coefficient of performance is:
Which simplifies to:
=
ANALYSIS, DESIGN, & ASSEMBLY OF WORK CHAPTER 5
CHAPTER 5 Page 19
CHAPTER 5
ANALYSIS, DESIGN, & ASSEMBLY OF WORK
5.1 Ideal Analysis of Stirling Cycle
In the following section we present an ideal analysis of the Stirling cryocooler. We assume
the maximum and minimum pressure in the cryocooler to be 1 and 1.13 bar respectively.
Further assuming ideal gas behaviour and applying ideal gas equations we find out the
approximate dimensions of the compression chamber.
Nomenclature:
a) TC = Temperature in the compression chamber
b) TE = Temperature in the expansion chamber
c) m = Mass flow rate of the helium in Kg/s
d) R = Characteristic gas constant
e) r = Ratio of maximum volume V1 to minimum volume V2
Initial Values:
a) Pressure before compression P2 =1 bar
b) Temperature before compression T2 = 298 K
1) Isothermal expansion process (1-2)
PV = constant
P1V1=P2V2
T1=T2=Th
Heat transfer,
dQ = dW = -RThln(V2/V1)
2) Constant volume regenerative heat transfer (2-3)
PT=constant
ANALYSIS, DESIGN, & ASSEMBLY OF WORK CHAPTER 5
CHAPTER 5 Page 20
P2T2=P3T3
V2=V3=Vmax
Heat transfer,
dQ = = -Cv(Th-Tc)
3) Isothermal compression process (3-4)
PV = constant
P3V3=P4V4
T3=T4=Tc
Heat transfer,
dQ = dW = -RTcln(V2/V1)
4) Constant volume regenerative heat transfer (4-1)
PT=constant
P4T4=P1T1
V4=V1=Vmin
Heat transfer,
dQ = =+Cv(Tc-Th)
After solving all the process equations we get,
Point Pressure Volume Temperature
1 1.050 bar 1665.57 cm3 298 K
2 1.0 bar 1889.4 cm3 298 K
3 0.92 bar 1889.4 cm3 278 K
4 1.13 bar 1665.57 cm3 278 K
Table 5.1.P-V-T values for Stirling Cooling Cycle
ANALYSIS, DESIGN, & ASSEMBLY OF WORK CHAPTER 5
CHAPTER 5 Page 21
Applying equation PV=mRT at any point we get,
M=0.1123 kg
Dimensions of compressor cylinder
In compressor cylinder,
V = ?? *( D2/4) * (L+c)
Where,
D = Cylinder diameter
L = Stroke length
c = Clearance length
r = V2/V1 = 1.13
(L+c)/c =1.13
L/c = .13
c = L/0.13
Now assuming, D/L =1.1
V = [(pi * L3)]*(1.21/4*8.69) = 1.665E-3 m3
L = 147 mm ~ 150mm
D = 160 mm ~ 162 mm
COP = Tc/Th-Tc
= 278 / (298-278)
COP = 13.9
ANALYSIS, DESIGN, & ASSEMBLY OF WORK CHAPTER 5
CHAPTER 5 Page 22
Now,
Net work input for Stirling Cooler,
W= mRln(V2/V1)(Th-Tc)
Here 0.125 HP electric motor is used for work input,
So, W= mRln(V2/V1)(Th-Tc)
=0.125 HP=335.7 kJ/hr
For Stirling Cooler, the refrigeration effect is:
Qr=mRTL ln(V2/V1)
= 1.2557 kJ/sec
But 1TR=3.517 kJ/s
So, Qr = 0.3 ton
Therefor,
Parameter Measure
Refrigeration Capacity 0.3 ton
C.O.P 13.9
Table 5.2 Capacity &C.O.P of Cooler
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5.2 Schmidt’s Cycle and Its Analysis
The classical analysis of the operation of Stirling engines is due to Schmidt (1861). The
theory provides for the harmonic motion of the reciprocating elements, but retains the major
assumption of isothermal compression and expansion and of perfect regeneration. It, thus
remains highly idealized, but is certainly more realistic than the ideal Stirling Cycle. [3]
Assumptions of the Schmidt cycle:
1) The regenerative process is perfect.
2) The instantaneous pressure is same throughout the system.
3) The working fluid obeys the characteristic gas equation, PV=RT.
4) There is no leakage, and the mass of the working fluid remains constant.
5) The volume variations in the working space occur simultaneously.
6) There are no temperature gradients in the heat-exchangers.
7) The cylinder wall, and piston, temperatures are constant.
8) There is perfect mixing of cylinder contents.
9) The temperature of the working fluid in ancillary spaces is constant.
10) The speed of the machine is constant.
11) Steady state conditions are established.
Basic Equations:
Volume of expansion space:
Ve = ?? *VE* [1+ cos (??)]
Volume of compression space:
Vc = ?? *VC *[1 + cos (?? ‘ ??)]
= ?? *x* VE *[1 + cos (?? ‘ ??)]
Using the above equations and performing Schmidt’s calculations we get the following
equations:
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Instantaneous Pressure,
p = P max *(1-??)/ [1 +??*cos (‘?? ??)]
Or,
p = P min *(1+??)/ [1 + ??*cos (?? ‘ ??)]
Therefore,
P max / P min = (1+??)/ (1 – ??)
Mean Cycle-Pressure,
Pmean = Pmax *[(1-??)/ (1+??)] 0.5
So,
Pmax = Pmean *[(1+??)/(1-??)]0.5
And,
Pmin = Pmean *[(1-??)/(1+??)]0.5
Heat Transferred in expansion and compression space,
Qe = [??* P mean * VE * (??) * sin(??) ]/ [ 1+ (1 ‘ ??2)1/2 ]
Qc = [ ?? * Pmean * VE * x * (??) * sin( ??-?? ) ] / [ 1 + (1 ‘ ??2 )1/2 ]
Work done in compressor,
W = Qe ‘ Qc
Coefficient of performance,
COP = Qe/(Qe-Qc)
5.3 Simulation With Schmidt Cycle
A number of graphs made in CHARTGO are presented in this section. The plots which we
obtain enable us to find the optimum parameters which are necessary to obtain the required
refrigeration capacity of miniature Stirling Cooler.
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Figure 5.1: Volume displaced by Piston-1 at different crank angles
Figure 5.2: Volume displaced by Piston-2 at phase angle 0 degree
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Figure 5.3: Combine effect of Piston-1 & Piston-2 at phase angle 0 degree
Figure 5.4: Volume displaced by Piston-2 at phase angle 45 degree
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Figure 5.5: Combine effect of Piston-1 & Piston-2 at phase angle 45 degree
Figure 5.6: Volume displaced by Piston-2 at phase angle 90 degree
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Figure 5.7: Combine effect of Piston-1 & Piston-2 at phase angle 90 degree
To achieve maximum refrigeration capacity it is necessary that compression pocket and
expansion pocket must have more area as much as possible.
From Figure 5.3, Figure 5.5 & Figure 5.7,
It is clear that we can achieve maximum area of compression pocket and expansion
pocket phase angle must be 90o .
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5.4 Design and Assembly Work
In this portion of the project, we have produced 2-D & 3-D view of the following parts to
complete the design procedure.
1] Motor Used for Input
2] Regenerator
3] Body:
4] Diaphragm
5] Diaphragm Part
6] Bearing Cover
7] Eccentric Part-1
8] Eccentric Part-2
9] Shaft
10] Side Cover
1] Motor Used for Input:
Fig.5.8:0.25 HP, 1440r.p.m Motor
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2] Regenerator:
Fig.5.9: 2-D Drawing of Regenerator
(400*400mm steel mesh inside)
Fig.5.10: 3-D View of Regenerator
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3] Body:
Fig.5.11: Drawings for Body
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4] Diaphragm
Fig.5.12: Drawing for Diaphragm
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5] Diaphragm Part:
Fig.5.13: Drawing of Diaphragm Part
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6] Bearing Cover:
Fig.5.14 Drawing for Bearing Cover
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7] Eccentric Part-1:
Fig.5.15: Drawing for Eccentric Part-1
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8] Eccentric Part-2
Fig.5.16.Eccentric Part-2
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9] Shaft:
Fig.5.17 Drawing of Shaft
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10] Side Cover
Fig.5.18 Drawing of Side Cover
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ASSEMBLY PROCEDURE
Following procedure is carried out for assembly work by using CAD Software
PTC CREO 2.0.
STEP-1
STEP-2
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STEP-3
STEP-4
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STEP-5
STEP-6
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STEP-7
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CHAPTER 6
6.1 Shortcomings of Schmidt Analysis
One of the biggest shortcomings of Schmidt cycle is the assumption that the efficiency of the
Regenerator is 100%. In reality the efficiency of the regenerator is expected to be around
98%. This drop in regenerator efficiency will reduce the value of Pmax and increase the value
of Pmin thereby reducing pressure ratio and the refrigeration capacity will be reduced
substantially (from 495 to around 250).
Also the pressure losses in the regenerator due to large flow resistance of meshes is neglected
which in reality can have a substantial impact on the performance of a Stirling cryocooler.
6.2 Scope for Future Work
The future work will consist of incorporating the regenerator efficiency into the given
Schmidt model; by doing so we will be able to obtain more accurate parameters to get the
required refrigeration capacity. Taking regenerator effectiveness into account is expected to
bring the changes in following quantities:
1) Refrigeration capacity ‘ The refrigeration capacity is expected to decrease. This is due to
the fact that the heat transfer between the fluid and the regenerator is not 100% and hence
the temperature of fluid reaching the expansion or the compression chamber is slightly
more or less than the ideal values we assumed in the previous sections. Hence the actual
refrigeration capacity of this cryocooler is expected to be around 250 mW instead of 495
mW.
2) Maximum pressure (P max) -The value of maximum pressure is expected to decrease.
3) Minimum pressure (P min) ‘ The value of minimum pressure is expected to rise.
4) COP ‘ Since the refrigeration capacity is reduced hence, the COP is reduced accordingly.
5) Power input ‘ In order to increase the reduced refrigeration capacity, the power input
has to be increased.
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6) Value of x ‘ From the graph obtained between P max and x in the Schmidt analysis above,
we observe that as the value of P max decreases, the value of x increases correspondingly.
Hence by incorporating the regenerator efficiency we can expect the value of x (i.e. VC) to
increase with corresponding decrease of P max.
6.2.1 Regenerator Optimization
Regenerator optimization is the process of choosing regenerator design parameters that
maximize system performance. For cryogenics refrigerators, optimization generally refers to
maximizing the available refrigeration by systematically selecting regenerator parameters
such as geometry, type of matrix design, and matrix material that achieve this goal. The
critical parameters affecting the thermal performance of regenerator are the number of heat
transfer units, the fluid heat capacity ratio, the matrix heat capacity ratio, the thermal losses
such as the longitudinal conduction. [7]
These parameters establish the thermal performance of a regenerator because they determine
the temperature difference between the fluid and the matrix, the temperature swing of the
matrix material, and any other irreversible heat transfer processes that contribute to
degradation in regenerator performance. To maximize thermal performance both the NTU
and matrix capacity ratio must be made as large as possible. However, in designing a
regenerator for an actual cryogenic refrigerator the major obstacle limiting the magnitude of
these parameters is the additional requirement to keep the pressure drop and regenerator void
volume small. It is these conflicting requirements that lead to the need for regenerator
optimization in a cryogenic refrigerator and a thorough understanding of the interaction of all
key parameters. Walker (1973) describes the optimization problem for the designer as the
task of satisfying the following conflicting requirements:
1) The temperature swing of the matrix must be minimized. Thus, the matrix heat capacity
ratio must be a maximum. This can be achieved by a large, solid matrix.
2) The pressure drop across the regenerator must be small. The effect of the pressure drop
across the across the matrix is to reduce the magnitude of the pressure excursion in the
expansion space, thereby reducing the area of the expansion-space PV diagram and the gross
refrigeration produced by the refrigerator. The pressure drop is minimized by a small (short),
highly porous matrix.
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3) A third consideration is the void volume. For a fixed-volume refrigerator such as the
Stirling cycle refrigerator, the void volume influences the ratio of the maximum-to-minimum
volume of the working space, which directly affects the pressure excursion in the expansion
space. For maximum refrigeration, the pressure ratio must be large, or the void volume small.
This can be achieved by a small, dense matrix. [7]
6.2.2 Optimization Analysis
To illustrate the conflicting requirements that occur in the optimization of a regenerator for a
cryogenic refrigerator, we shall consider a procedure for maximizing the available
refrigeration in Stirling cycle refrigerator by optimizing typical regenerator design. The
Stirling-Cycle refrigerator operates between an ambient temperature of 300 K and a
refrigeration temperature of 80 K. The objective of the optimization is to maximize the
available refrigeration by determining the values of the key regenerator parameters ‘ such as
the length, matrix material, and porosity required to maximize the performance of the
regenerator. The optimization is performed given the following operating condition:
1) The temperature difference across the regenerator is 300 K to 80 K.
2) The frequency of operation of the cryocooler is fixed.
3) The mean operating pressure of the refrigerator is fixed.
4) The piston and displacer motions are sinusoidal. [7]
The equation expressing the maximum available refrigeration, Wnet is obtained through a
decoupled energy summation of the gross refrigeration produced by the cryocooler, less the
individual losses limiting the net available refrigeration. The decoupled approach has been
shown to provide accurate results by Harris, Rios, and Smith (1971) in their paper on
regenerators for Stirling-type refrigerators. The energy summation is:
Wnet = Wpv ‘ (W??p + Qreg + Qc + ??Q1)
Where the individual losses described above consist of those associated with the regenerator
‘ such as the pressure drop loss W??p, the thermal loss of the regenerator Qreg and the solid
conduction loss through the matrix material Qc ‘ and those associated with other cryocooler
components, ??Q1. The regenerator losses expressed above are defined by the
thermodynamics and dynamic equations. [7]
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GROSS REFRIGEARTION [7]
The gross refrigeration is defined as the cyclic integral of the maximum cycle pressure and
expansion volume variations:
Wpv = ‘ P*dVe
The equations for the maximum pressure and volume variations for Stirling cycle
refrigeration will be presented latter in the report. [7]
PRESSURE DROP LOSS [7]
The pressure drop loss is the loss in refrigeration resulting from the pressure difference
between the compression and expansion spaces. The loss in pressure in the expansion space
is produced by both the frictional pressure drop through the regenerator and the pressure drop
caused by the filling of the regenerator void volume during the pressurization and depressurization
of the regenerator,
??p = ??pf + ??ppv
And the loss in refrigeration is:
W??p = ‘pc*dVe – ‘(pc – ??p)*dVe
Or,
W??p = +’??p*dVe
REGENERATOR THERMAL LOSS
The regenerator thermal loss is expressed in terms of the regenerator effectiveness, fluid
thermal capacity, and the temperature difference across the regenerator.
Qreg = + (1-Er)*m*cp*(Tw ‘ Tc) *??h
LONGITUDNAL CONDUCTION LOSS
The longitudinal conduction loss is defined in terms of the matrix cross-section area-to-length
ratio, thermal conductivity, and the temperature difference across the regenerator. In addition,
the conduction loss occurs over the total period of operation and therefore is expresses as:
Qc = + [Am/L *’ (k*dT) m]*(??h + ??c) (For a balanced heat flow, ??h = ??c)
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Or, Qc = +2*[Am/L*’ (k*dT) m]*??h
Experimental results for wire screen matrices have shown that longitudinal conduction for
many commonly used matrix materials is controlled primarily by the interfacial resistance
between screens. It can be represented by the following equation:
Km = 0.7*(Tm/300 K)
Thus thermal conduction can be expressed in terms of average thermal conductance over the
temperature range from 80 to 300 K as:
Qc = +2*??h*Km*(Am/L)*(Tw-Tc)
OPTIMIZATION EQAUTIONS:
Expressing the optimization equation as the sum of the ratio of the individual energy loss
terms to the gross refrigeration, we obtain the dimensionless equation that defines the terms
to be minimized in order maximize the available refrigeration:
(Wnet + Qreg + W??p + Qc + ??Q1)/Wpv = 1
The relations between the key parameters in the above equation are defined below using the
classical Schmidt analysis for a Stirling Cycle refrigerator. The approach of the Schmidt
analysis is to specify simple sinusoidal motion for the compressor piston and the displacer
and to assume isothermal compression and expansion processes. As the expansion process is
assumed to be isothermal, the maximum refrigeration produced is equal to the gross
expansion work performed:
Qr = m*qr*??c = Wpv
Where qr is the heat transferred to the working fluid per cycle in the expansion space and Ps
is the system pressure, considered here as the maximum working pressure in the refrigerator.
For an ideal Stirling cycle refrigerator with isothermal compression and expansion processes,
the heat transferred to the working fluid is given by:
qr = R*Tc*ln(Pa)
Also, as the motion is considered sinusoidal, a phasor diagram can be used to describe the
variations of the compression and expansion volumes and to assist in the development of the
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equations describing these variations. The advantage of using the phasor notation is that the
amplitude and phase relationship of each of the system operating parameters can be presented
on a phasor diagram that provides physical insight into the mathematics of the Schmidt
analysis (Ackermann, 1981).
GROSS REFRIGERATION EQUATION [7]
The system pressure Ps and the expansion volume derived from Schmidt analysis of phasor
equations are as follows:
Ps = Pmean + pso*sin (??t + ??)
Ve = ??(Ve) max + veo*sin (??t)
From the above pressure equations, gross refrigeration can be determined by substituting the
differentiated expansion volume and the maximum system pressure equation in the gross
refrigeration equation.
Wpv = ‘ [Pmean + pso*sin (??t + ??)]*(??*veo*cos (wt)) dt
And integrating over the complete cycle gives:
Wpv = ??*pso*veo*sin (??)
PRESSURE DROP EQUATIONS [7]
The final compression and expansion space pressures are found by including the pressure
drop. The pressure drop has the effect of reducing both the magnitude of the pressure in the
expansion space and the phase angle by which the expansion space pressure leads the
expansion space volume variations. Both effect result in a reduction in the refrigeration
produced by the cryocooler. The total pressure drop is the sum floe frictional pressure drop
across the regenerator, ??pf, and the reduction in the system pressure phasor caused by the
pressurization and depressurization of the regenerator void volume, ??prv. From the analysis
of the phasor diagram we obtain:
??pf = – (??pf) 0*sin [??t + (??+??)]
Where ?? is the angle between the pressure drop phasor and the system pressure phasor.
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And,
??prv = – (??prv) 0*sin (??+??)
Substituting these two pressure drop components, and the differentiated expansion volume
into the pressure drop loss equation and integrating it over a cycle, we obtain:
W??p = + ??*veo [(??pf) 0*sin (??+??) + (??prv) 0 *sin (??)]
And the ratio of pressure drop loss to the gross refrigeration is defined in terms of the
pressure drop amplitude (??pf) o and (??prv) o as:
W??p/Wpv = + [(??pf) o*sin (??+??)]/pso*sin (??) + (??pso) o / pso
REGENERAOR FRICTION PRESSURE DROP [7]
The amplitude of the frictional pressure drop is determined from the Fanning pressure drop
equation:
(??pf) o = f*(G2/2??)*(L/rh)
Where G is the average mass flow rate per unit of the free flow area of the fluid, out of the
regenerator and into the expansion space during the heating flow period and out of the
expansion space and in to the regenerator during the cooling flow period:
G = (me/Aff) = 1/??c*’ (??f) e*w*dt
This equation provides a means for evaluating pressure drop given the flow velocity and
matrix geometry. However, it does not relate the pressure drop to the heat transfer
characteristics of the regenerator as required to facilitate the optimization procedures. To
achieve this, Kays and London (1964) present a useful correlation between the dimensionless
Stanton number and the friction factor. A geometric factor can be defined in terms of Stanton
number and friction factor, or equivalently in terms of the total regenerator NTU:
?? = St*Pr/f = 2*(NTU)*Pr2/3/f*(rh/L)
Where the relationship between the NTU and St is :
NTU = St/2*(L/rh)
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Substitution of this expression into the pressure drop equation produces an optimization
equation for the frictional pressure drop in terms of NTU parameter and the geometric factor:
W??pf/Wpv = [(NTU*pr2/3/??)*(G2/??)*sin (??+??)]/pso*sin (??)
From the above equation we can see that to minimize the pressure drop loss, we must
minimize the NTU/?? ratio or equivalently, for a given geometry, minimize the NTU, which is
opposite to what must be done to reduce the regenerator thermal loss.
REGENERATOR VOID VOLUME PRESSURE DROP [7]
The pressure drop related to the regenerator void volume is the difference between the
amplitude of the system pressure phasor with no void volume and the amplitude of the
pressure with regenerator void volume:
(??prv) = pso ‘ p*so
where the star denotes the pressure amplitude with regenerator void volume. To determine
the void volume pressure ratio in the work loss due to pressure drop equation, the system
pressure amplitude is derived in terms of the mean cycle pressure and the pressure ratio
between the maximum and minimum cycle pressures,
pso = Pmean ‘ Pmin = Pmax ‘ Pmean
which can also be expressed in terms of the pressure ratio:
pso = Pmean*(Pa-1)/Pa+1)
where Pa is the pressure ratio defined as Pmax/Pmin. Substitution of this expression or the
system pressure amplitude in the first equation produces the following expression for the
pressure ratio term in the optimization equation:
(??prv) o/pso = [(Pa-1)/Pa+1) ‘ (Pa*-1)/ (Pa*+1)]/ [(Pa-1)/Pa+1)]
where Pa* denotes the pressure ratio with regenerator void volume.
If we now use Schmidt analysis to define the pressure ratio in terms of the volume and
temperature system parameters, we see that the void volume pressure ratio is a function of the
void volume of the regenerator, and is expressed in terms of void volume to expansion
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volume space. Thus, for a given temperature and compression-to-expansion-volume ratio, the
pressure loss from the regenerator void volume is a function of only void volume ratio,
Vrv/Ve, and increase asymptotically to 1 as the regenerator void volume ratio increases:
W??prv/Wpv = (??prv)o/pso = (Vrv/Ve)/[1/2*(1+Tc/Tw)*(Tw/Tc+Vcs/Ve)+(Vrv/Ve)]
where the subscript are as follows: w is the warm temperature, c is the cold temperature, cs is
the compression space, e is the expansion space, and rv is the regenerator void volume.
6.2.3 OPTIMIZATION PROCEDURE
From the above equations we see that to optimize the regenerator in the Stirling cycle
cryocooler we must determine the regenerator parameters ‘ length, frontal area, matrix
geometry, matrix material, and porosity ‘ that produce the optimum NTU and matrix
capacity ratio, and minimize the longitudinal thermal conduction and pressure drop. Also, the
optimization of these parameters must be consistent with constraints imposed on the
cryocooler design, which for our example are:
1) The temperature difference across the regenerator is 298 K to 277 K.
2) The mean operating pressure of the refrigerator is 1.09 bar.
3) The piston and displacer motions are sinusoidal.
4) The nominal displacer stroke is
With these constraints, and the idealization of the Stirling cycle cryocooler, the optimization
equations are:
1) THE REGENERATOR THERMAL LOSS
Qreg/Wpv = (1-Er)*(Tw-Tc)*(cp) f/qr = Ie*(Tw-Tc) (cp) f/qr
Where, considering an ideal Stirling cryocooler, the heat transferred to the working fluid is
given by:
qr = R*Tc*ln(Pa)
and the regenerator loss ratio is:
Qreg/Wpv = Ie*[(Tw-Tc)/Tc]*[(cp) f/R (ln (Pa))]
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Also, as the refrigeration in a regenerative cycle cryocooler only occurs during one half of the
cycle, the average mass flow rate through the regenerator during each period is:
m = Wpv/qr*??c = 2*Wpv/qr = [2*??*fr*pso*veo*sin (??)]/qr
= [??*fr*pso*Ve*sin (??)]/R*Tc*ln (Pa)
where fr is the frequency of operation of the cryocooler.
2) THE LONGITUDINAL CONDUCTION LOSS
Qc/Wpv = [??*Km*Am*(Tw-Tc)/Lf]/m*R*Te*ln(Pa)
3) THE FRICTIONAL PRESSURE DROP LOSS
W??pf/Wpv = [(NTU*Pr2/3/??)*(G2/??)*sin (??+??)]/pso*sin (??)
4) THE VOID VOLUME PRESSURE DROP LOSS
W??prv/Wpv = (Vrv/Ve)/ [1/2*(1+Tc/Tw)*(Tw/Tc+Vcs/Ve) + (Vrv/Ve)]
The two additional equations required to obtain a solution are:
5) THE NUMBER OF HEAT TRANSFER UNITS
NTU = St*(L/rh)*1/2
Where the Stanton number is derived from Kays and London’s experimental data (1964) as
St*Pr2/3 = 0.68*Re-0.4
6) THE MATRIX HEAT CAPACITY RATIO
Cr/Cmin = (M*cp) m/ (m*cp) f*??h
To illustrate the computational processes involved in optimizing a regenerator we consider
the optimization of a regenerator for a Stirling cryocooler. The cryocooler is required to
produce 0.348 Watts of cooling at 277 K and will use a wire screen regenerator matrix. The
design conditions and thermal properties for this cryocooler are summarized in the 1st table
on the next page.
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The three assumptions in the 1st table regarding the regenerator geometry draw on previous
experience with Stirling cryocooler and wire screen regenerators. As we will see as we
proceed with the optimization, the requirement for good, first-order assumptions is critical to
the process in order to minimize the number of iterations required to arrive at a solution. The
optimization process is not a numerically automated process that will converge to the solution
based on the constraints and input parameters; instead it will be an interactive process
between the designer and the available regenerator performance data.
To perform the optimization, we begin by assuming some reasonable loss values for the
individual loss ratios, and from these values we determine the NTU, matrix capacity ratio,
and from these values we determine the NTU, matrix capacity ratio, and void volume ratio. If
the values obtained are unrealistic, or if the design can be further optimized to reduce the loss
terms, then additional iterations are required to further optimize the regenerator. If the values
are reasonable, the designer can proceed with the cryocooler overall design to determine
whether the regenerator design meets the other cryocooler requirements. A first estimation of
losses is presented in the 2nd table.
From the values of the 1st and the 2nd table we can proceed to calculate the performance
parameters and losses by first calculating the void volume ratio, and the expansion swept
volume. With these valued in hand, all of the other regenerator dimensions can be determined
and, in turn, the performance parameters calculated.
PARAMETER VALUE
Input parameters
Cycle Stirling
Working Fluid Air
Warm Temperature 298 K
Cold Temperature 277 K
Net Refrigeration(Wnet) 348 mW
Regenerator Material Steel Screen
Screen Mesh 400*400
Wire Diameter (dw) 0.2 mm
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Mean Pressure (Pmean) 1.09 bar
Properties
Prandtl Number 0.75
Regenerator Porosity 0.65-0.72
Matrix Density (??m) 8.7 gm/cm3
Mean matrix thermal conductance (Km) 7.62 MW/cm-K
Mean regenerator air density (??f) 0.005 g/cm3
Mean expansion space air density (??ef) 0.009 g/cm3
Air specific heat (cpf) 4.2 J/g-K
Mean air thermal conductivity (Kf) 4.358E-3 W/m.K
Gas constant (R) 0.319KJ/Kg-K
Assumptions
Swept volume ratio (Vcs/Ve) 0.88
Compression-to-expansion-volume phase angle (??) 900
Regenerator matrix porosity (??) 0.67
TABLE 6.1: STIRLING CYCLE REFRIGERATOR DESIGN PARAMETERS [7]
Loss parameter Value
Regenerator thermal loss (Qreg/Wpv) 0.20
Frictional pressure drop loss (W??pf/Wpv) 0.10
Void volume pressure loss (W??prv/Wpv) 0.51
Thermal conduction loss (Qc/Wpv) 0.05
Additional thermal and mechanical losses (??Q/Wpv) 0.12
Total losses 0.98
Net refrigeration (Wnet/Wpv) 0.02
TABLE 6.2: FIRST ESTIMATE OF LOSSES IN A STIRLING CRYCOOLER [7]
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Parameter Value Formula
Pressure ratio 0.81
Average mass flow rate (m) 0.1123 kg/s m = 2*Wpv/R*Tc*ln(Pa)
Expansion swept volume (Ve) 1.89E-03 m3 Wpv =??/2*fr*pso*Ve*sin(??)
Void Volume (Vrv) 3.49E-03 m3 Vrv = 1.85*Ve
Regenerator volume (Vr) 5.21E-03 m3 Vr = Vrv/??
Matrix volume (Vm) 1.72E-03 m3 Vm = (1-??)*Vr
Displacer diameter (Dd) 0.070 m
Regenerator diameter (Dr) 0.162 m Dr = Dd ‘ 2*t
Regenerator length (L) 0.150 m
Regenerator area (Ar) 3.84E-03 m2
Regenerator free flow area
(Aff)
2.758E-03 m2 Aff = ??*Ar
Mass flow/area (G) 40.7179 kg/m2 G = m/Aff
TABLE 6.3: CALCULATED REGENERATOR PARAMETERS FROM THE
OPTIMIZATION EQUATIONS [4]
COSTING AND CONCLUSION CHAPTER 7
CHAPTER 7 Page 56
CHAPTER 7
COSTING AND CONCLUSION
7.1 Bill of Material and Costing
TABLE 7.1 Cost Estimation of Cooler
Sr No. Part Name Material Qty. Rate (Rs.) Cost(Rs.)
1 Electrical Motor 1440 rpm
025 HP
Crompton
Greaves
1 2500 2500
2 Regenerator Mild Steel +
Steel Mesh
1 720 720
3 Bearing Cover Mild Steel 1 180 180
4 Diaphragm Polypropylene 2 130 260
5 Diaphragm Part Mild Steel 2 70 140
6 Body Mild Steel 1 850 850
7 Eccentric Part Mild Steel 2 90 180
8 Side Cover Mild Steel 2 250 500
9 Shaft Mild Steel 1 260 260
TOTAL
13 Rs. 5590
COSTING AND CONCLUSION CHAPTER 7
CHAPTER 7 Page 57
A. Total Cost = 5590 Rs.
B. Assembly Cost (+) = 560 Rs.
C. Overhead Charges (+) = 610 Rs.
D. Profit (+) = 1000 Rs.
‘ Total = 7750 Rs.
‘ Taxes (+) (4%+2%) = 471 Rs.
‘ Selling Price = 8221 Rs.
COSTING AND CONCLUSION CHAPTER 7
CHAPTER 7 Page 58
7.2 CONCLUSION
In this report we have reproduced the analysis given by Schmidt’s for a Stirling cryocooler.
We changed a few input parameters according to our needs. By assuming the relative ratio of
different losses, we obtained regenerator parameters from optimization equations. With the
help of these parameters we calculated the actual loss’s taking place inside the cryocooler.
The theoretically refrigeration assumed value of 0.3 ton.
At the starting of our project we have assumed the surrounding air at STP .So, the final
calculations are done on that basis throughout. We have assumed the D/L ratio around 1.1
and found out the length and diameter of the cylindrical shell. We have plotted different
graphs for volume vs. crank angles online ON CHARTGO .We made our final assembly and
drawings by using software CREO 2.0 and Autocad-2013.The main problem aroused in the
project is about Regenerator efficiency, which was around 0.02% .So we have done some
.
CHAPTER- Page 59
REFERENCES
[1] www.paragengineering.com
[2] The free piston Stirling Cooling System, by Global Cooling BV, Sunpower Inc.,
Greenpeace, Presented in 19th International congress on Refrigeration Exhibition.
(Page- 13 & 15)
[3] http://www.occc.edu/gholland/Thermo/Stirling_Intro.pdf
[4] www.nptel.iitm.ac.in ‘ Lecture No.27 on Cryocooler by Prof.Adtrey
[5] http://www.nmri.go.jp/env/khirata/ Schmidt Theory for Stirling Cooler-National
Maritime Research Institute
[6] [Graham Walker, Stirling Cryocooler, Cryocooler Part-1, page 95-142
[7] Robert Ackermann, Cryogenic Regenerative Heat Exchangers, Page 147-191