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51 8 2016 385
*1
Regenerative Heat Exchanger and Its Design Approach
Kazuhiro HAMAGUCHI*1
Synopsis: The regenerator is an indispensable regenerative heat exchanger in Stirling cycle machines, G-M cryocoolers and pulse tube cryocoolers. Its existence is a key factor in enabling these machines to have the highest thermal efficiency and coefficient of performance. A regenerator is composed of a matrix and housing. The matrices are often made of stacks of fine wire mesh screens. However, an optimum balance between flow loss and heat transfer characteristics is required. In this report, based on the theoretical studies of thermal behavior in wire elements and matrices, relationships between design factors are obtained. An approach for designing regenerator used in Stirling cryocoolers is presented.
Keywords: temperature ratio, temperature efficiency, flow loss, reheat loss, indicated COP (Some figures in this article may appear in colour only in the electronic version)
GM
Fig. 1
Fig. 1 Fig. 2
Fig. 2
Fig. 1 Stirling cycle machine.
Fig. 2 Temperature distribution in regenerator.
*1
Received August 10, 2016
191-8506 2-1-1 Meisei University, 2-1-1 Hodokubo, Hino, Tokyo 191-8506, Japan
E-mail: [email protected] DOI : 10.2221/jcsj.51.385
386 TEION KOGAKU J. Cryo. Super. Soc. Jpn. Vol. 51 No. 8 2016
Fig. 1
Fig. 3
2)
Fig. 4
3)
a)
Fig. 3 Effects of normal flow passage at regenerator.
Fig. 4 Effects of tapered flow passage at regenerator.
b) c) d)
200 4000.05 0.03 mm
a) b)
c)
d) e)
b) c)a) d) e)
Cp
1 MPa 2 MPa10 20 K
4.2 K0 70 K
10 K10 K
Gd Dy Ho Er
70 K
Fig. 5(a)
Fig. 5(b)Fig. 5(a)
51 8 2016 387
(a) Vertical stack (b) Parallel stack Fig. 5 Stacking method of wire gauzes.
(1)
(mm2/mm3) dh(mm)
a)
( )
m
14d
φσ −= (1)
dm (mm) b)
dh dh 4
4σφ4 (2)
c)
(2)
P(Pa) fRe f
Re u
2
h
21 u
lrPd
fρ
= (3)
νudRe h= (4)
φ0u
u = (5)
lr (m)(kg/m3) (m2/s) u
u0
(m/s)(3)
m(W/(m2K)) Nu
Fig. 6 Effects of stacking method and porosity on balance values of two characteristics.
ReNu
g
mh
λαdNu = (6)
g (W/(mK)) (4) 4)
Fig. 6Re
30 50 100 VP f/Nu
f/Nu f/Nu
f/Nu
f/Nuf/Nu
f/Nuf/Nu
f/Nu
(5)
5)
(7) f (8)Re
388 TEION KOGAKU J. Cryo. Super. Soc. Jpn. Vol. 51 No. 8 2016
60.1175 +=Re
f (7)
67.033.0 ReNu = (8) (7) (8) (3) (4) (6)
6)
(6) 7)
e (W/ (mK))
( )
−
−+−
+=
sg
g
s
ge112
121ln1
λλ
λλ
λλ
A
A (9)
s g (W/(mK))A
( ) 2141 −−=π
φAAA
−−=
1φ
Fig. 7
dm
Tf*
in
Tf*
out Fig. 7(b) Fig. 7(a)
Fig. 7 Heat storage in a wire element.
Fig. 7(c)
Fig. 7(b)
Tr Bi FoFig. 8
8)
Tr Bi Fo
0fm
d
==
RTTTr (10)
Td /Tfm
R=0
d
mm
2λα dBi = (11)
m (W/(m2K)) dm
d (W/(mK))
2m
d4d
tFo κ= (12)
d (m2/s) t(s)
Fig. 8
dm
Bi Fo TrTr =1
Tr =1
Fig. 9
Fig. 8 Relation between Bi and Fo with temperature ratio Tr.
51 8 2016 389
Ntu CrFig. 10 Ntu
Cr Crs
p
m
CmSNtu α= (13)
m (8)S (m2) m
(kg/s) Cp (kJ/(kgK))
tCmMCCr
p
m= (14)
M (kg) Cm
(kJ/(kgK)) (s)
vr
m
CMMCCrs = (15)
Mr (kg)Cv (kJ/(kgK))
0xfmh
0ff
=
==
−−
=TTTT xlrxε (16)
Tf x=lr
Tf x=0
Tmh
Fig. 9 Control volume of matrix.
Fig. 10 Temperature efficiency versus Ntu with Cr (Crs>100).
9) Crs 100 CrsFig. 10
=0.99 Cr Ntu
2)
Wp( )
PmW Δ=ρp (17)
m P
P f(7)
Qr(W)
Fig. 2Ta Tr Tr
Tr
Ta Ta
Tr
Qr
rpr TCmQ = (18)
m Cp
(18)
Tr
(1- )
Qmax
max
r1QQ=− ε (19)
Qmax ( )rapmax TTmCQ −= (20)
390 TEION KOGAKU J. Cryo. Super. Soc. Jpn. Vol. 51 No. 8 2016
( )( )ε−−= 1rapr TTmCQ (21)
(22)(23)
Cr 0 (23) (22)
221+
=−Ntu
ε (22)
( )+⋅−=−
2tanh11
NtuCrNtuCrε (23)
Qc(W)
LrTTAQ ra
emc−= λ (24)
Am Lr (m2)(m) Ta Tr ( )
e (9)
77 K, 1 W10) Fig. 11
COPi
opi
ocrei WWW
QQQWCOP
Σ−−Σ−−−
= (25)
We Wi (W)Qo Wo Qr
Qc Wp
Qo
Fig. 11 Heat balance in a Stirling cryocooler.
Wo
(25) COPi
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2) (B) 64-624
(1998) 2702-2709 3)
(B) 78-794 (2012) 128-139
4) (B) 70-697 (2004)
2425- 2432 5) M. Tanaka, et al.: “Flow and heat transfer characteristics of Stirling
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6) (2009) 124-129
7) No.145 (1988) 76
8) (B) 49-445 (1983) 2001-
2010 9) H. Miyabe, S. Takahashi and K. Hamaguchi: “An approach to the
design of Stirling engine regenerator matrix using packs of wire gauzes”, Proc. 17th IECEC (1982) 1839-1844
10) 62-719 (1987) 1-7
1950 2 28 1981
