pISSN 1229-3008 eISSN 2287-6251

Progress in Superconductivity and Cryogenics

Vol.19, No.4, (2017), pp.12~17 https://doi.org/10.9714/psac.2017.19.4.012

Schmidt cycle analysis in the quest of designing stirling cryocooler

Debajyoti Roy Chowdhury, Nathuram Chakraborty, and Swapan Chandra Sarkar*

Centre for Rural & Cryogenic Technologies, Jadavpur University, Kolkata 700032, India

(Received 8 August 2017; revised or reviewed 14 November 2017; accepted 15 November 2017)

Abstract

Design of Reverse Stirling Cycle based refrigerator can be predicted by Schmidt theory as a useful tool and by experiment it is
found that for practical purposes the power and efficiency predicted by this analysis are about 35% of the actual values. Therefore,
appropriate provision is to be made for getting the realistic result with the minimum deviation. The present paper first investigates
the suitability of application of Schmidt design analysis for standard ZIF-1002 and PLN-106 Single cylinder Cryogenerator model.
As the result is found to be optimistic, the same design procedure is applied for the design of a separate Cryogenerator for
generating a cooling effect which is sufficient to produce 7 kg per hour liquid nitrogen using an indigenous condenser of 80%
effectiveness. The paper describes all the details of the design methodologies and relevant results are found to be satisfactory.

Keywords: reverse stirling cycle, liquefaction, stirling cryocooler; schmidt analysis; regenerator

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1. INTRODUCTION displacer / expander and condenser.

These components are hermetically sealed as a single
The gas to be liquefied is contacted to the refrigerated unit. Expansion zone is situated above the expander
wall of the condenser. The gas is liquefied by the (displacer) and the compression zone is under the expander
refrigerating effect produced in the inner side of the and above the piston head.
condenser by the repeated compression and expansion of The squirrel cage motor rotates the crank shaft which is
the refrigerant gas such as hydrogen or helium and after converted into the reciprocating motions of the piston and
several cycle of compression and expansion, a low of the expander. Both the piston and expander are coupled
temperature in the range of 50K to 60K is achieved. The in the same shaft. The expander (displacer) unit is in phase
refrigerating effect is imparted to the incoming purified advance of 70-80 degree of the piston.
Nitrogen gas which in turn gets liquefied at the outside wall Several analytical and computerized models for Reverse
of the condenser at 77K and thus further decrease in Stirling Cycle thermodynamic analysis are available in the
temperature leading to solidification of liquid nitrogen is literature [1-6]. Schmidt analysis is adopted as it is widely
not possible. The nitrogen gas and the refrigeration gas used for initial sizing of engine and its produces closed
never come in contact with each other as they flows in two form solution for the engine performance which can be
different circuits. There is a temperature gradient of around easily manipulated by the designer though its efficiency is
17 K between the inner side and outside of the condenser. temperature dependent.
The liquefied gas thus flows down the piping of the The present paper describes the methodology of design
liquefier as shown in Fig. 1. The main units of the
of an indigenous Cryocooler based on Reverse Stirling
Cryorefrigerator are compressor, cooler; regenerator,
Cycle applying Schmidt analysis.

2. CRYOCOOLER DESIGN

The important assumptions on which cryocooler design

is based on Schmidt are as follow:

a) The working fluid behaves as ideal gas.

b) Regenerator effectiveness is 100%.
c) Uniform instantaneous pressure throughout the
system is assumed.
d) Constant mass of working gas is assumed thus
ensuring no leakage of gases.
Fig. 1. Cryogenic system for gas liquefaction process.
e) Sinusoidal volume variation in the working space is
* Corresponding author: scs@cal2.vsnl.net.in to be ensured.
 Debajyoti Roy Chowdhury, Nathuram Chakraborty, and Swapan Chandra Sarkar

f) Temperature gradient in the heat exchanger is zero. Equation (2) and equation (3) result into formation of
g) Constant temperature of cylinder wall and that of equation (4),
piston are assumed and cylinder content are well
mixed. ∆Q = (1 − E )mg Cv (T2 − T3 )
h) Temperature of the working fluid in the ancillary (4)
space is constant.
Assuming ideal gas behaviour of the refrigerant gas, the
i) Machine speed is constant and steady state is reached.
equation becomes,
Cryocooler performance is quite sensitive to regenerator
effectiveness [7]. If the regenerator does not have an
Qideal = mg T3 ( S 4 − S 3 )
(5)
effectiveness of 100%, the temperature of the gas leaving
the regenerator during cool down will be higher than the S4 and S3 are the entropies at point 4 and 3 respectively
desired refrigeration temperature. The net results of this as shown in Fig. 2.
decrease in effectiveness are that less energy is absorbed in For an isothermal process, the change in entropy can also
the refrigerator because more energy is required to cool the be expressed as the ratio of the specific volume as
refrigerant to the desired refrigerant temperature. This loss V4
in refrigeration can be expressed in terms of actual energy, ∆S = m g RT3 ln( )
V3 (6)
Q actual, absorbed in the regenerator during the constant
volume cool down process of the refrigerant depicted by
step 2-3 of Fig. 2. 3
For helium Cv = ( ) R and for a well-defined machine
2
Qactual = Qideal − ∆Q (1) V4
( ) = 1.24
V3
where Qideal is the heat absorbed in a regenerator of 100% Therefore, the fraction of the cold lost due to inefficiency
effectiveness and ∆Q represents the energy which is not of the regenerator is
absorbed in the regenerator due to deviation of its
effectiveness from 100%. ∆Q (1 − E )Cv (T2 − T3 )
=[ ]
The effectiveness of the regenerator is given by Qideal V
RT3 ln( 4 )
V3 (7)
Qactual (Qideal − ∆Q)
E= =
Qideal Qideal (2) Refrigeration effect loss due to 1% decrease in
effectiveness for a Cryocooler operating between 60K and
Q ideal is given by 300K with helium as the working fluid, can be given with
Qideal = m g C v (T2 − T3 ) the help of equation (7) as follows
(3)
3
In equation (3), mg and CV denote mass of the refrigerant (1 − 0.99) (300 − 60)
∆Q 2
(working gas) flowing through the regenerator and specific =[ ] = 0.279
heat at constant volume respectively. Qideal 60 ln 1.24 (8)

Therefore, it reveals that there will be a loss of 27.9% in

refrigerating effect due to 1% decrease in effectiveness.
Following the same calculation, the effectiveness of the
regenerator at which the refrigerating effect altogether
vanishes can be found out.
Thus
3
(1 − E ) (300 − 60)
∆Q 2
=[ ]
Qideal 60 ln 1.24
3
(1 − E ) (240)
Fig. 2. Stirling Refrigeration Cycle (a) P-V diagram, or, 1 = [ 2 ]
(b) T-S diagram. 60 ln 1.24
 Schmidt cycle analysis in the quest of designing stirling cryocooler

for which E = 0.964.

Therefore, regenerator of effectiveness 96% would
result in zero refrigerating effect which clearly explains the
crucial role played by regenerator in the development of
Reverse Stirling cycle based cryocooler.
For the present analysis, the regenerator having
effectiveness of 100% is used. However, it is established
that the prediction for the overall design parameters from
Schmidt analysis is 35% efficient considering even perfect
regeneration and isothermal compression and expansion
[8].

2.1. Effect of principal design parameters on cold

production
By Schmidt Cycle analysis,

1 A − B 1/ 2 K sin α Fig. 3. Design calculation based on optimum design charts:

QE = PmaxVT ( )( ) (9) (a) Qmax x 10-3 vs. T (b) (Swept ratio) K vs. (Temperature ratio)
k +1 A + B A + ( A 2 − B 2 )1 / 2 T (c) (Phase angle) α vs. (Temperature ratio) T.

A = T + K + 4 XT /(T + 1) (10) TABLE1

COMPARISON OF DIFFERENT PARAMETERS GENERATED FROM SCHMIDT
ANALYSIS FOR ZIF-1002 PLN-106 CRYOGENERATOR STANDARD MODELS.

B = (T 2 + 2TKCosα + K 2 )1/ 2 (11)

TE
T
45K
6.7
50K
6
55K
5.4
60K
5
65K
4.6
Qma
72.5*10-3 72.8*10-3 75*10-3 74.8*10-3 75.4*10-3
Pmax = Pmin ( A + B) /( A − B) (12)
K
x
3.6 3.5 3.2 3 2.85
Z α 108° 107.2° 106.2° 105.8° 104.4°
VT = VC + VE = (1 + K )VE (13) I
F
A 12 11.21 10.3 9 9.09
B 6.55 5.98 5.45 5.77 4.77
1 Pmax 3.45*105 3.33*105 3.29*105 4.63*105 3.25*105
W = (T − 1)QE (14) 0
VE
134.22 136.36 142.099 149.6 155.32
0 *10-6 *10-6 *10-6 *10-6 *10-6
From the above equations, it is noted that QE (cold 2 617.41 614.83 596.81 598.4 598.41
VT
produced in the expansion space) depends on the following *10-6 *10-6 *10-6 *10-6 *10-6
483.19 478.20 454.71 448.8 442.97
independent design parameters i.e. Temperature ratio of the VC
*10-6 *10-6 *10-6 *10-6 *10-6
compression space to that of the expansion space (T), W 18.99kW 16.66kW 14.66kW 13.33kW 11.99kW
swept volume ratio (K), Dead volume ratio of the total T 6.7 6 5.4 5 4.6
internal volume of heat exchanger, ducts, ports to the Qma
72.5*10 -3
72.8*10 -3
75*10 -3
74.8*10 -3
75.4*10-3
expansion volume (X), phase angle (α), maximum and x
K 3.6 3.5 3.2 3 2.85
minimum pressure of the refrigerant ( Pmax and Pmin ), α 108° 107.2° 106.2° 105.8° 104.4°
speed of the engine (N) and expansion space volume A 12 11.21 10.3 9 9.09
P
( VE ). L
B 6.55 5.98 5.45 5.77 4.77
N Pmax 3.45*105 3.33*105 3.29*105 4.63*105 3.25*105
The effect of the four principal design parameters T, K, 121.49 91.186 94.44* 397.72* 394.55*1
VE
Q
X and α to produce E is not known initially but these
1
0
*10-6 *10-6 10-6 10-6 0-6
558.84 410.337 396.66 397.72 394.55
6 VT
important design parameters must be determined optimally *10-6 *10-6 *10-6 *10-6 *10-6
437.35 319.151 302.22 298.29 292.07
in advance with optimum charts [9, 10]. Chart as in Fig. VC
*10-6 *10-6 *10-6 *10-6 *10-6
3(a-c) along with other design guidelines [11, 12] are used W 12.54kW 10.99kW 9.68kW 8.8kW 7.91kW
in the subsequent design calculation for Stirling
refrigerator. The results are then verified with existing dimensions
available in the manual of those machines [13-15].

3. SCHMIDT THEORY IN THE DESIGN OF 3.1. Refrigerating effect evaluation of Cryogenerator units
CRYOGENERATOR UNITS
3.1.1. Cooling capacity of Model ZIF-1002:
Based on the Schmidt analysis, computations are done on Refrigerating capacity required per cycle at 77K for
different standard units such as ZIF-1002 and PLN-106. ZIF-1002 Cryogenerator model is 47.94 J, for operating
 Debajyoti Roy Chowdhury, Nathuram Chakraborty, and Swapan Chandra Sarkar

speed of 1460 rpm and refrigerating ability of 4200 kJ/hr TABLE 3

COMPARATIVE STUDY OF THEORETICAL AND ACTUAL VALUES OF CRANK
[14]. Now, QE = heat lifted in the expansion space
ANGLES AND POWER.
(Refrigeration produced) = 47.94/0.35= 136.97 J/cycle.
Cryogenerator
ZIF-1002 PLN-106
model
3.1.2. Cooling capacity of PLN-106:
Actual crank displacement
Cooling load required per cycle at 77K for PLN-106 70 70
angle (degree)
Cryogenerator model is 31.86 J, for operating speed of
Theoretical crank
1450 rpm and refrigerating ability of 2772 kJ/hr [15]. QE = displacement angle (degree)
104.4 104.4
heat lifted in the expansion space (Refrigeration produced)
Actual power of the motor in the
= 31.86/0.35= 91.034 J/cycle. installed machines (kW)
17 11

13.3 (For TE= 8.8 (For

For both the units, Tc = temperature of the working fluid Theoretical power of the motor 60K) TE= 60K)
in the compression space = 300K, X= Dead volume ratio= (kW) 16.67 (For TE= 10.5 (For TE=
0.5 are assumed. By varying TE= temperature of the 50K) 50K)
working fluid in the expansion space, the different
parameters generated for ZIF-1002 and PLN-106 are Comparison of theoretical with calculated value of swept
presented in Table 1. volume of expansion space, compression space, crank
angle, power requirement with actual value for ZIF-1002
3.2. Design of the expansion zone and compression zone: and PLN-106 are reported in Table 2 and Table 3. As the
Guidelines for the design calculation are available in the results are very optimistic, the same procedure can be
literature [16]. adopted for the design of any new Cryogenerator.

3.2.1. Cryogenerator model ZIF-1002:

4. INDIGENOUS DEVELOPMENT OF THE
∏ ∏ CRYOGENERATOR
VE = DE2 LE = (0.07)2 * 0.03 = 115.45*10-6 m3
4 4 It is intended to develop a Cryogenerator having a
sufficient cooling capacity to liquefy pure nitrogen gas at
∏ 2 ∏ the rate of 7kg/hr. The nitrogen is to be fed to the
VC = DC LC = (0.1016)2*0.052 = 421.58*10-6 m3
4 4 cryocooler at 300K. The actual refrigerating effect required
is calculated by energy balance. Schmidt theory is then
3.2.2. Cryogenerator model PLN-106: applied to evaluate the design load and for fixing different
dimension of the machine.
∏ 2 ∏
VE = D E LE = (0.07)2 * 0.03 = 115.45*10-6 m3
4 4 5. DESIGN OF THE CRYOGENERATOR

∏ 2 ∏ Refrigeration load of the indigenous Cryogenerator

VC = DC LC = (0.08)2*0.052=261.38*10-6 m3 model for liquefaction of 7 kg/hr. liquid nitrogen is
4 4 computed to be 3801.43 kJ/hr. (1055.67 watt) using a
condenser having effectiveness of 80%. Thus, refrigerating
TABLE 2
COMPARATIVE STUDY OF THEORETICAL AND ACTUAL VALUES OF
capacity required at 77K is 43.39 J per cycle. The speed of
COMPRESSION AND EXPANSION SPACE. motor is noted to be 1460 rpm.
Cryogenerator Refrigeration to be produced in the expansion space (QE) =
ZIF-1002 PLN-106
model 43.39/0.35 = 123.98 J/cycle. The temperature of the
Actual swept volume of working fluid in the compression space (Tc) and dead
expansion space calculated volume ratio (X) are as 300 K, 0.5 respectively. Different
115.45*10-6 115.45*10-6
from actual displacing unit
stroke length (m3)
parameters generated for varying temperature of the
working fluid in the expansion space (TE) are reported in
Swept volume of expansion
space predicted by Schmidt 149.6*10-6 99.43*10-6 Table 4.
analysis (m3) The result shows more or less similar trends as predicted
Actual swept volume of by ZIF-1002 and PLN-106. Compressor cylinder diameter,
compression space calculated piston stroke, displacing unit diameter, displacing unit
from actual compressor cylinder
421.58*10-6 261.38*10-6
stroke, crank displacement angle parameters are fixed as
diameter and piston stroke (m3) 102 mm, 52 mm, 70 mm, 30 mm respectively based on the
Swept volume of compression prediction from Schmidt analysis and considering the
space predicted by Schmidt 448.8*10-6 298.29*10-6 required factor of safety for the indigenous built
analysis (m3)
Cryogenerator for producing 7 kg/hr. liquid nitrogen.
 Schmidt cycle analysis in the quest of designing stirling cryocooler

TABLE 4 reported in Table 5 and Table 6 for the indigenous

SCHMIDT DESIGN CALCULATION OF INDIGENOUSLY BUILT cryocooler.
CRYOGENERATOR.
TE 45K 50K 55K 60K 65K

T 6.7 6 5.4 5 4.6 6. RESULT AND DISCUSSION

72.5 72.8 75 74.8 75.4
Qmax
*10-3 *10-3 *10-3 *10-3 *10-3
From the Table 5 and Table 6, it can be inferred that
C K 3.6 3.5 3.2 3 2.85
R design methodology adopted is suitable for Cryogenerator
α 108° 107.2° 106.2° 105.8° 104.4°
Y development based on Reverse Stirling Cycle. There are
O A 12 11.21 10.3 9 9.09
G
minimum deviations of the actual dimension of the machine
B 6.55 5.98 5.45 5.77 4.77
E 3.45 3.33 3.29 4.63 3.25 from the theoretically predicted dimension by Schmidt
Pmax
N * 105 *105 *105 *105 *105 analysis. The power requirement depends on the lowest
E 121.49 123.68 128.62 135.41 139.57
R VE cryogenic temperature to be attained in the expansion space
*10-6 *10-6 *10-6 *10-6 *10-6
A 558.84 556.54 540.21 541.66 537.35 of the refrigerator and it decreases with increase in
T VT
*10-6 *10-6 *10-6 *10-6 *10-6 temperature of the expansion space. The power is also
O 437.35 432.86 411.59 406.25 397.78
R VC accurately calculated by the analysis. Refrigerating
*10-6 *10-6 *10-6 *10-6 *10-6
17.20 15.08 13.27 12.06 10.86 capacity at 77K for liquefaction of nitrogen gas is
W
kW kW kW kW kW computed as 43.39J for the indigenous Cryogenerator.

TABLE 5
COMPARATIVE STUDY OF THEORETICAL AND ACTUAL VALUES OF
COMPRESSION AND EXPANSION SPACE.
7. CONCLUSION
Indigenously built Cryogenerator
The result found is very encouraging for application of
Actual swept volume of expansion
space calculated from actual 115.45*10 -6 Schmidt analysis in the design of Cryocooler. The
displacing unit stroke length (m3) investigating team has successfully attempted an
indigenous development of a Reverse Stirling cycle based
Swept volume of expansion space Cryogenerator which is the main component of the Stirling
133.78*10-6
predicted by Schmidt analysis (m3)
Cycle based liquefaction system for cryogenic gases.
Actual swept volume of Though the liquefaction of nitrogen gas is presently studied,
compression space calculated from the same may be extended for liquefaction of methane or
424.90*10-6
actual compressor cylinder diameter other cryogenic gases excepting hydrogen and helium. The
and piston stroke (m3)
same cryocooler may also be used for studying
Swept volume of compression space superconductivity of materials and cooling
401.32*10-6
predicted by Schmidt analysis (m3) superconducting cables.

TABLE 6
COMPARATIVE STUDY OF THEORETICAL AND ACTUAL VALUES OF CRANK
NOMENCLATURE
ANGLES AND POWER.

Indigenously built Cryogenerator QC = QE = Heat lifted in the expansion space or

Actual crank displacement angle
70 refrigeration produced per cycle.
(degree)
Theoretical crank
A and B are two factors.
104.4
displacement angle (degree) α = Angle by which volume variation in the expansion
Actual power of the motor in the
17 space lead to those in the compression space.
installed machines (kW)
12.06 (For TE= 60K)
Pmax = Maximum cycle pressure
Theoretical power of the motor (kW)
15.08 (For TE= 50K)
Pmin = Minimum cycle pressure
∏ 2 ∏ VT = Combined swept volume
VE = D E LE = (0.07)2 * 0.03 = 115.45*10-6 m3
4 4
VC = Swept volume of the compression space
∏ 2 ∏
VC = DC LC = (0.102)2*0.052 = 424.90*10-6 m3 VE = Swept volume of the expansion space
4 4
VD = Total internal Dead volume
Comparison of actual swept volume of expansion space, V
compression space, crank angle, power used with the X = D = Dead volume ratio
theoretical result obtained from Schmidt analysis are
VE
 Debajyoti Roy Chowdhury, Nathuram Chakraborty, and Swapan Chandra Sarkar

VC [3] W. R. Martini, “Stirling Engine Design Manual,” DOE / NASA /

K= = Swept volume ratio 3194-1, NASA, CR- 168088, 2nd ed., Jan. 01, 1983.
VE [4] A. Schock, “Nodal analysis of Stirling cycle devices,” Proceedings
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T engineering, San Diego, CA, paper 789191, pp. 1771–1779,
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