Transcript
1
27
2Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Earlier Lecture• A Cryocooler is a mechanical device operating in a
closed cycle, which generates low temperature.
• It eliminates cryogen requirement, offers reliable operation and is also cost effective.
• Heat exchangers can either be regenerative or recuperative type depending upon heat exchange.
• Recuperative Type: J – T, Brayton, Claude.
• Regenerative Type: Stirling, GM, Pulse Tube.
Topic : Cryocoolers
• Ideal Stirling cycle
• Working of Stirling Cryocooler
• Schmidt's Analysis
• Conclusions
3Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Outline of the Lecture
4Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
History• A well developed and a most commonly used
Cryocooler is the Stirling Cycle Cryocooler.
• This cycle was first conceived by Robert Stirling in the year 1815. It was an engine cycle and was aimed to produce work (engine).
• The important events that occurred in the history of cryocoolers are as given in the next slide.
The Chronology
5Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Year Event1815 Robert Stirling – Stirling Engine
1834 John Herschel – concept of using as a cooler
1861 Alexander Kirk – The concept into practice
1873 Davy Postle – Free Piston system
1956 Jan Koehler – First commercial machine for air liquefaction
1965 Jan Koehler – Nitrogen Liquefaction
6Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
An Ideal Stirling CyclePr
essu
re –
p
Volume – V
• Consider a p – V chart as shown in the figure.
• 12: Isothermal compression at TC.
CQ2
11 1 2 2p V p V=
1 2 CT T T= =
2
1
lnCVdQ dW TV
= = −ℜ
7Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
An Ideal Stirling CyclePr
essu
re –
p
Volume – V
• 23: Constant volume heat rejection.
• 34: Isothermal expansion.
CQ
EQ
RTQ
2
3
4
1
2 3V V=
( )V E CdQ C T T= + −
3 3 4 4p V p V=
3 4 ET T T= =
4
3
lnCVdQ dW TV
= = −ℜ
8Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
An Ideal Stirling CyclePr
essu
re –
p
Volume – V
• 41: Constant volume heat absorption.
CQ
EQ
RTQ
2
3
4
1RTQ
4 1V V= ( )V C EdQ C T T= − −
E
C E
QCOPQ Q
=−
4
3
2 4
1 3
ln
ln ln
E
C E
VTV
V VT TV V
+ℜ
=
−ℜ −ℜ
9Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
An Ideal Stirling Cycle
4
3
2 4
1 3
ln
ln ln
E
C E
VTV
V VT TV V
+ℜ
=
−ℜ −ℜ
32
1 4
VVV V
=
Pres
sure
–p
Volume – V
CQ
EQ
RTQ
2
3
4
1RTQ
E
C E
TCOPT T
=−
COP(Stirling) = COP(Carnot)
10Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Stirling & Carnot CyclesPr
essu
re –
p
Volume – V
2
3
4
1
Tem
pera
ture
–T
Entropy – s
2
3 4
1TC
TE
Stirling Cycle
11Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Stirling & Carnot CyclesPr
essu
re –
p
Volume – V
2
3
4
1
Tem
pera
ture
–T
Entropy – s
2
3 4
1TC
TE
5
6
5
6
Carnot Cycle Stirling Cycle
.
12Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Ideal Stirling Cycle
Tc
TE
Tc
Compression
Pres
sure
–p
Volume – V
Regenerative Cooling
Expansion
Regenerative Heating
CQ2
1RTQ
3
EQ 4
RTQ
PistonRegeneratorExpander
1
2
3
4
1
CQ
EQ
13Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Ideal Stirling CycleTi
me
Displacement
Tc
TE
Tc
Compression
Regenerative Cooling
Expansion
Regenerative Heating
PistonRegeneratorExpander
1
2
3
4
1
1
2
3
4
1
1
2
3
4
1
0E CV V= =,maxEV ,maxCV
14Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Ideal Stirling Cycle• As mentioned in the earlier
lecture, the characteristics of a Stirling cycle are• High frequency.
• Regenerative heat exchanger.
• Phase difference between the piston and the displacer motions.
Tim
e
Displacement
1
2
3
4
1
1
2
3
4
1
0E CV V= =,maxEV ,maxCV
15Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Actual Stirling Cycle• In actual Stirling cycle the
discontinuous motion can not be achieved. In view of this sinusoidal motion may be implemented.
• This motion is realistic and can be achieved using a Crank or gas spring mechanism.
Tim
e
Displacement
1
2
3
4
1
1
2
3
4
1
0E CV V= =,maxEV ,maxCV
16Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Actual Stirling Cycle• In reality, the actual working cycle will be different
from Ideal Stirling Cycle in following ways.
• Discontinuous motion, difficult to realize in practice.
• Presence of void volume or dead space (not swept by piston or displacer), pressure drop.
• Ineffectiveness in heat transfer or regeneration.
• Non isothermal compression and expansion.
17Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Stirling Cryocooler – Types• Depending upon the relative
arrangements of piston and displacer/piston, various types of Stirling Cryocoolers are possible, namely• α type Stirling Cryocooler.
• β type Stirling Cryocooler.
• γ type Stirling Cryocooler.
ExpanderTypeα
Compressor
Regenerator
Typeβ Typeγ
18Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Stirling Cryocooler – Types• Two Piston arrangement
(α type)• whose drive mechanisms may
be mounted on same crank shaft.
• Integral Piston & Displacer arrangement (β type)
• The piston and displacer are housed inside same cylinder.
ExpanderTypeα
Compressor
Regenerator
Typeβ Typeγ
19Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Stirling Cryocooler – Types• Split Piston & Displacer
arrangement (γ type)• The compression space is
divided.
• These systems have variable dead volume in compression space due to the movement of displacer.
ExpanderTypeα
Compressor
Regenerator
Typeβ Typeγ
20Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Design Parameters• The various design parameters of
a Stirling Cryocooler are as follows.
• Evaporator temperature (TE)• Condenser temperature (TC)• Compression Volume (VC)• Expansion Volume (VE)• Regenerator Volume (VR)• Pmax, Pmin, Pavg.• Phase angle (α)• Crank angle (ø)
ExpanderTypeα
Compressor
Regenerator
TC TEVE
VC
VR
21Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Schmidt’s Analysis• In the year 1861, Gustav Schmidt, a German
scientist, presented a Stirling Cryocooler analysis.
• This analysis is based on a realistic cycle and is assumed to provide a first guess of dimensions. The following are the assumptions.
• Perfect isothermal compression, expansion.
• Harmonic motion of piston and displacer.
• Perfect regeneration.
22Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
• The non – dimensional parameters in the Schmidt’s analysis are
• Swept volume ratio :
• Temperature ratio :
• Dead volume ratio :
Schmidt’s Analysis
C
E
VkV
=
C
E
TT
τ =
D
E
VXV
=
23Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
• Expansion volume variation :
• Compression volume variation :
Schmidt’s Analysis
( )1 1 cos2e EV V φ= +
( )1 1 cos( )2c CV V φ α= + − C
E
VkV
=
( )1 1 cos( )2c EV kV φ α= + −
Volu
me
–V
Angle – ø
VC
VE
α
24Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
• Let the instantaneous pressure in the system be same throughout the system, p.
• Also, Te and Tc are assumed to be constants as TEand TC respectively.
• Let MT be given as shown.
Schmidt’s Analysise e c c d d
Te c d
p V p V p VMRT RT RT
= + +
2E
TC
KVMRT
=
2E
C
KVRT
=
25Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Schmidt’s Analysis( ) ( )1 cos 1 cos( )
2 2 2CE D E E
C E E D C
T kpV V T KVRT T V T RT
φ φ α+ + − + + =
C
E
TT
τ = D
E
VXV
=2
E Cd
T TT +=
21
XS ττ
=+
( ) ( )1 cos 1 cos( ) 2K k Sp
τ φ φ α= + + + − +
2 2( cos ) ( sin )A k kτ α α= + + 2B k Sτ= + +AB
δ =
sintancos
kk
αθτ α
=+
26Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Schmidt’s Analysis
[ ]cos( ) 1Kp
B δ θ φ=
− +
[ ]min 1Kp
B δ=
+
[ ][ ]11ratiop
δδ
+=
−
@ φ θ=
[ ]max 1Kp
B δ=
−@ φ θ π= −
• Substituting, A, B, θ and δ in the mass equation and rearranging, we get
27Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Schmidt’s Analysis• Mean pressure
2
0
1 ( )2mp pd
π
θ φπ
= −∫
max11mp p δ
δ−
=+
0.52
sin
1 1m E
E ep VQ pdV π δ θ
δ= =
+ − ∫ 0.52
sin( )
1 1m E
C cp V kQ pdV π δ θ α
δ
−= =
+ − ∫
E
T
QCOPW
= E
C E
TT T
=−
E
C E
QCOPQ Q
=−
28Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Losses• In the earlier slide, we saw the cooling effect
based on Schmidt's analysis.
• But, in an actual system, there are many losses. Few of them are as listed below.
• Ineffectiveness of regenerator.• Pressure drop in system.• Solid conduction losses.• Shuttle conduction losses.• Losses in power input.
29Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Losses• Considering the above mentioned losses, the net
cooling effect and gross power required is given by the following correlations.
• Qnet = QE – Σ(losses).
• Wtotal = WT + Σ(losses).
• In general, QE calculated from Schmidt's analysis, in which 60 – 70% are considered as losses, while losses in power input is due to mechanical efficiency.
30Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Summary• A Stirling Cycle was first conceived by Robert
Stirling in the year 1815.
• COP(Stirling) = COP(Carnot).
• In reality, the actual working cycle has discontinuous motion, pressure drop, ineffectiveness and non isothermal processes.
• Depending upon the relative arrangements of piston and displacer/piston, α, β, γ are the different types of Stirling cryocooler.
31Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Summary• Gustav Schmidt presented a Stirling Cryocooler
analysis in the year 1861, it is assumed to provide a first guess of dimensions.
• The net cooling effect and gross power required is given by the following correlations.
• Qnet = QE – Σ(losses).
• Wtotal = WT + Σ(losses).
32Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
• A self assessment exercise is given after this slide.
• Kindly asses yourself for this lecture.
Self Assessment1. A Stirling cycle consist of two _______ processes.2. In an isothermal process, dQ is given by _______.3. In a constant volume process, dU is given by ____.4. COPCarnot and COPStirling are _____.5. COP of Stirling cycle is _______.6. In an actual Stirling cycle, the discontinuous
motion is approximated to _______ motion.7. The volume not swept by piston/displacer is ____.8. In a _____ type unit, the piston and displacer are
housed inside same cylinder.9. In Schmidt's analysis, instantaneous pressure is
assumed to be _____.33Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Answers
34Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
1. Isothermal and Constant volume
2. m
3. R
4. Equal.
5. M
6. Sinusoidal
7. Void volume
8. Beta
9. Constant
[ ]2 1ln /CdQ dW T V V= = −ℜ
( )V E CdU C T T= + −
( )/E C ET T T−
35Prof. M D Atrey, Department of Mechanical Engineering, IIT Bombay
Thank You!
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