A model for the parametric analysis and optimization of inertance tube pulse tube refrigerators

1. A MODEL FOR THE PARAMETRIC
ANALYSIS AND OPTIMIZATION OF
INERTANCE TUBE PULSE TUBE
REFRIGERATORS
C. Dodson^'^, A. Lopez^'^, T. Roberts', and A. Razani^'
^Air Force Research Laboratory
Kirtland AFB, NM, 87117-5776
^The University of New Mexico
Albuquerque, NM 87131
^Sandia National Laboratories
Albuquerque, NM 87185-0933
ABSTRACT
A first order model developed for the design analysis and optimization of Inertance Tube
Pulse Tube Refrigerators (ITPTRs) is integrated with the code NIST REGEN 3.2 capable
of modeling the regenerative heat exchangers used in ITPTRs. The model is based on the
solution of simultaneous non-linear differential equations representing the inertance tube,
an irreversibility parameter model for the pulse tube, and REGEN 3.2 to simulate the
regenerator. The integration of REGEN 3.2 is accomplished by assuming a sinusoidal
pressure wave at the cold side of the regenerator. In this manner, the computational power
of REGEN 3.2 is conveniently used to reduce computational time required for parametric
analysis and optimization of ITPTRs. The exergy flow and exergy destruction
(irreversibility) of each component of ITPTRs is calculated and the effect of important
system parameters on the second law efficiency of the refrigerators is presented.
KEYWORDS
Pulse tube refrigerators, regenerators, REGEN 3.2, distributed inertance tube
model, exergy, efficiency, optimization
CP985, Advances in Cryogenic Engineering: Transactions of the
Cryogenic Engineering Conference—CEC, Vol. 53, edited by J. G. Weisend II
2008 American Institute of Physics 978-0-7354-0504-2/08/$23.00
685

2. FIGURE 1. Inertance Tube Pulse Tube Refrigerator (ITPTR)
INTRODUCTION
Pulse Tube Refrigerators (PTRs) play an important role in satisfying the need for
cryogenic cooling of space-based infrared detectors as well as electronics requiring coolers
with high reliability, low vibration, and high efficiency [1]. Usually three types of phase-
shifting processes exist on PTRs that control the phase shift between the mass flow rate and
pressure [1]. The more conventional are Orifice Pulse Tube Refrigerators (OPTRs) where
the mass flow rate and pressure are in phase at the orifice. In Double Inlet Pulse Tube
Refrigerators (DIPTRs) a bypass valve between the warm end of the pulse tube and the
warm end of the regenerator is used to provide a proper phase-shifting mechanism.
In Inertance Tube Pulse Tube Refrigerators (ITPTRs), which are the focus of this
study, the phase shifting is provided by an inertance tube replacing the orifice [2].
FIGURE 1 shows the important components of ITPTRs. A review of these phase-shifting
mechanisms is given by Radebaugh [1]. The thermodynamics of PTRs have been under
study by several investigators to just name a few [3-6].
In first-order models, usually used in design analysis and parametric studies of
ITPTRs, a lumped parameter approximation is used to take into account the inertance,
compliance, and the fluid flow resistance associated with oscillating flow in the inertance
tube. In the present study, a previously developed distributed model of the inertance tube
and a first order model of the pulse tube modified for pulse tube losses are integrated with
the REGEN 3.2 code to model the regenerator. The distributed component method used in
this study divides the inertance tube into several sections and applies the lumped parameter
model to each section. In addition, the pulse tube is modeled using a parameter to quantify
the losses in the pulse tube itself [2]. The emphasis in this study is the integration of the
model for the regenerator using REGEN 3.2 and the first order model for components to
the right of regenerator given in FIGURE 1.
MODELING OF THE ITPTR
The inertance tube is modeled by 2« +1 ordinary differential equations (ODEs) using
the distributed model of the inertance tube and is given as follows:
dP.
drills.
(1)
(2)
m ,=V.
n+l ir
dP„,,
dt
J = ...n
(3)
686

3. where the following variables are given as:
n
P, =P,A-
71
X •- (64°
dC,
-a°T.
2rRT, c,
AI
n
rRT,
I
.R,
• , ( 1 - D2)a
,L.
4
n_
Tud^
IXju-
n_
71d'^ p
I
V+A
,V- In
rRT,
4|OT^(/)|
TldjU
(4)
where the variables P^ corresponds to the pressure that comes in from the hot heat
exchanger, A is the area of the cross section of the inertance tube, d is the diameter of the
inertance tube, C, are the capacitance coefficients, / is the length of the inertance tube, «is
the number of times the inertance tube is split, T^ is the hot temperature, R, are the
resistive coefficients, jU is the gas viscosity, p is the gas density, L, are the inductance
coefficients, FJ^is the volume of the reservoir plus the last piece of the unaccounted
inertance tube volume, Oj, 7,, X, Z)j and D2 are the laminar/turbulent coefficients
described in [2]. The mass flow rate from the pulse tube is linked to the inertance tube by
the conservation of mass and energy for the hot heat exchanger.
The NIST Computational Finite Difference (CFD) code known as REGEN3.2 was
used to model flow through the regenerator [7]. The input option of REGEN 3.2 allows
one to specify the temperature inlet to the regenerator, mass flow rate, pressure ratio, and
the phase between the mass flow and pressure at the cold side of the regenerator. This is
convenient for integration of the first order model developed for the inertance tube, pulse
tube, CEIX (cold heat exchanger), and HHX (hot heat exchanger), using an ODE solver
with the REGEN 3.2 to model the regenerator. The procedure for the integrated model for
ITPTRs starts with a sinusoidal pressure input at the cold side of the regenerator;
subsequently, a system of ODEs is solved for the components to the right side of the
regenerator. The output solution to the system of ODEs is the transient mass flow rate at
the cold side of the regenerator. Therefore, the amplitude
Input design parameters of pulse tube, CHX, HHX,
and inertance tube for the ODE system.
Pressure Input to ODE solver to calculate the mass
flow rate and the phase shift between them at the
cold side of the regenerator
Input design parameters for REGEN3.2 to calculate
input power required for the combined system.
i
Changing input design parameters of the regenerator —
I
Changing input design parameters of ODE system
FIGURE 2. Procedure for parametric analysis and optimization of ITPTRs
687

4. TABLE 1. Important geometric and flow parameters for the ITPTR system
Component
Input
Regenerator
CHX
Pulse Tube
HHX
Inertance
Tube
Reservoir
Important data
Sinusoidal input pressure, Pr=1.3,1.5,/ = 5QHzJ^ = 300KJ^ = SOK
400 mesh stainless steel, L = .05 ra, D = .02 m
L = .005m,D = .01m
L = .08m,D = .01m, fj^j=l
L = .005m,D = .01m
D = 1,1.5,2 mm, L=l,1.5,2 m, distributed model parameters are a^ =12,D^ = .184 andD^ = .2
50,200cc
of the mass flow rate and its phase with respect to pressure at the cold side of the
regenerator can be calculated. Next REGEN 3.2 is used to calculate the power input to the
regenerator required to drive the system. Parametric analysis and optimization can be
performed using the input pressure at the cold side of the regenerator. FIGURE 2 shows
the flow diagram for integration of the ODE solver and REGEN 3.2 and procedure for
parametric optimization. The parametric iteration presented in FIGURE 2 is continued
until a specific criteria or objective function is met.
RESULT AND DISCUSSION
The procedure described in FIGURE 2 is used for different values of pressure ratios at the
cold side of the regenerator and selected geometries of the inertance tube and fixed
parameters for the pulse tube and CHX and HHX. The amplitude of the mass flow and the
phase shift between the mass flow and pressure at the cold side of the regenerator is then
calculated for input to REGEN 3.2. TABLE 1 shows the parameters used in this study.
Enthalpy flow and exergy flow at the cold side of the pulse tube is determined. The output
of REGEN can be used to calculate the energy, acoustic power, and exergy flow in and out
of the regenerator. Conservation of energy for the CHX can be used to calculate the
cooling capacity and efficiency of the ITPTR. FIGURE 3 shows mass flow rate at the inlet
of the inertance tube and pulse tube as a function of time for two values of Pressure Ratios,
PR=1.3 and PR=1.5, for the inertance tube with a length of one meter and diameter of two
millimeters, the input pressure at the cold side of the regenerator is also shown (right axis)
on the same FIGURE.
L
[m]
1
1.5
2
D
[mm]
1
1.5
2
1
1.5
2
1
1.5
2
n pj
[W]
10.0
28.2
59.0
7.72
21.2
43.2
6.44
17.2
34.2
p
10.0
27.9
58.3
7.75
21.1
42.7
6.50
17.2
33.8
TABLE 2.
Pr=1.3
/3
Upj
66.8
31.0
2.0
71.8
42.0
8.6
74.8
50.6
17.7
n
-15.4
-20.8
-23.9
-13.4
-21.4
-26.6
-8.6
-19.4
-26.6
Results for the ITPTR
m„
Ig/^l
2.46
3.37
5.83
2.46
2.75
AA
2.53
2.53
3.48
" I T
Ig/^l
.25
.71
1.53
.19
.55
1.17
.17
.46
.96
[W]
20.1
57.2
120.8
15.7
43.9
90.9
13.3
36.4
73.9
system
p
20.1
56.9
119.7
15.8
43.8
90.1
13.5
36.4
73.3
Pr =
Upj,
n
70.4
39.9
11.2
74.5
49.8
19.6
76.8
56.8
28.8
1.5
n
-11.1
-16.3
-19.2
-7.4
-15.2
-20.2
-0.9
-11.4
-18.5
nipj
Ig/^l
3.85
4.99
8.07
3.89
4.27
6.37
4.00
4.09
5.27
m,p
Ig/^l
.31
.91
1.96
.25
.72
1.53
.22
.62
1.28
688

5. Pr=1.3, IT-mass
Pr=1.3, PT-mass
Pr=1.5, IT-mass
Pr=1.5, PT-mass
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Time - [sec]
FIGURE 3. Mass flow and pressure for an ITPTR for an mertance tube with length L=l m and
diameter D=2 mm
For the PR=1.3, the phase shift is -23.9 degrees for mass flow lagging the
pressure at the inlet of the inertance tube and its value is 2.0 degrees for the mass flow
leading the pressure at the cold side of the pulse tube. The corresponding values for
PR=1.5 are -19.2 degrees and 11.2 degrees, respectively. FIGURE 4 gives the same results
for a different geometry of the inertance tube with a length of one meter and diameter of
one millimeter.
The effect of the phase shift at the cold side of the regenerator provided by the
inertance tube obtained from the solution of the ODE system on the regenerator
performance is evaluated using REGEN 3.2, as described previously. The performance of
the regenerator is evaluated using the exergetic efficiency of the regenerator defined by [8]
Vre
E EI + ED
(5)
a -0.8
.^
'"^'/'X
/ /
- /
j f c ^ s ^ s ^
/^ "
•
• • > ; .
> Pr=1.5
r^^*-
V
—
/ ' ••
- Pr=1.3, IT-
- Pr=1.3, PT
— Pr=1.5, IT-
Pr=1.5, PT
/ /
. /
: /
mass
-mass _
mass
-mass
..-'
V°^
2.9
2.7
2.6 »
2.5 1
2.4 1
2.3
2.2
2.1
0 0.002 0004 0.006 0.008 001 0012 0014 0.016 0.018 002
Time - [sec]
FIGURE 4 Mass flow and pressure for an ITPTR for an inertance tube with length L= 1 m and
diameter D=l mm
689

6. Regenerator Length - [m] Regenerator Length - [m]
FIGURE 5. Exergy destruction (left) and exergy efficiency of regenerator (right) as functions of
regenerator length
where Ei, E2, are the rate of exergy entering, leaving the regenerator and ED is the rate of
exergy destruction (irreversibility) in the regenerator due to heat transfer and fluid friction.
One goal of this study is to calculate the effect of the geometry of the inertance tube on the
exergetic efficiency of the regenerator. TABLE 2 gives the results of calculations for
different geometries of the inertance tube and a pulse tube length of 8 centimeters. Other
parameters are defined in TABLE 1.
The numerical solution of the ODE system simulating the components to the right
of regenerator produces a mass flow rate that is not necessarily sinusoidal. The phase shift
between the mass flow rate and pressure at any location to the right of regenerator can be
calculated numerically using the following equation.
tan 6
* =
m(^t^ sin cotdt
. J
T
m[t)coscotdt
(6)
where 0 is the phase shift, m(t) is transient mass flow rate and T is the period. The phase
shift and amplitude of mass flow at the inlet to inertance tube and the cold side of the pulse
tube are given in TABLE 2 for the cases studied. In particular, the ODE system provides
the phase shift and the amplitude of mass flow at the cold side of regenerator for the input
to REGEN 3.2 to calculate heat and mass transfer in regenerator. Another method to relate
the phase shift and mass flow at the cold and hot sides of pulse tube is to use conventional
harmonic analysis of pulse tube. The following equations relate the phase shift and
amplitude of mass flow at the cold and hot sides of pulse tube. The solution of these
equations is used to check the numerical solutions of the ODE systems.
- sin Op.
rRT^
Vpa>y
(7)
Vpa>
tan 9„_^ = tan 9„_, + — (8)
690

7. Regenerator Length
FIGURE 6. Second law efficiency and cooling capacity (left) and exergy destruction (right) of the
ITPTR systems as a function of regenerator length
where V is the volume of the pulse tube, p is the amplitude of pressure in the pulse tube,
CO is the angular velocity, y is the specific heat ratio, R is the gas constant and the subscript
PT,c and PT,h denote the cold and hot side of the pulse tube, respectively.
FIGURE 5 shows exergetic efficiency and exergy destruction in the regenerator for
two selected ITPTRs as a function of the length of the regenerator. The mass flow rates
and phase shifts used for the regenerator study are typical of the data generated for
inertance tube of length one meter and diameter of 2 millimeters for the values of Pr=1.3
and Pr = 1.5. FIGURE 6 shows the second law efficiency and cooling capacity of the two
ITPTR systems using an exergy balance on the CHX. The exergy carried into the pulse
tube is calculated from the ODE system and the exergy into the CHX from regenerator is
calculated using REGEN3.2. Comparison of FIGURES 5 and 6 shows the effect of the
other irreversibility in the ITPTR system causing the reduction in efficiency.
CONCLUSIONS
The previously developed first order model of a multisection inertance tube, pulse
tube, CHX, and HHX using an ODE system is combined with the NITS CFD code REGEN
3.2 to model the regenerator. The computational power of REGEN 3.2 is integrated with
the ODE system using a sinusoidal pressure wave at the cold side of the regenerator. The
combined system can be used for parametric studies and optimization of ITPTRs. The
effect of the geometry of the inertance tube as method of controlling the phase shift
between mass flow rate and pressure to design proper regenerator is discussed. A second
law-based criterion for the regenerator is used to evaluate its performance and results for
selected cases are presented.
ACKNOWLEDGEMENTS
The authors would like to express their appreciation to Abbie O'Gallagher and John
Gary of NIST for their assistance with REGEN 3.2.
691

8. REFERENCES
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(2000), pp. 363-379.
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and exergy flow in pulse tube refrigerators. Advances in Cryogenic Engineering, Vol. 51-B, pp. 1572-
1597, 2005.
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10 (1996), pp. 849-857.
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refrigerator. Cryogenics, Vol. 39, no. 1 (1999), pp. 85-92.
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optimization of pulse tube refrigerators," Cryogenics, vol. 47, pp. 166-173, 2007.
7. J. Gary, A. O'Gallagher, R. Radebaugh, and E. Marquardt. REGEN 3.2: User Manual, National Institute
of Standards and Technology, 2001.
8. A.J. Lopez, "Second-law analysis and optimization of cryogenic regenerators using REGEN 3.2", MS
thesis. The University of New Mexico, 2007.
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