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Investigation of superconducting cable cooldown
An analytical expression for a longitudinal wall temperature profile emerging with the cooldown o f a superconducting cable has been obtained. The method used assumes a zone of steady temperature profile travelling along the line at a constant velocity. A comparison with expertment ts made.
Investigation of superconducting cable cooldown N.T. Bendik and N.I. Glukhov
Nomenclature T
temperature, K
o
density, kg m -3
d
~0
ratio of the capacitance of coolant to that of substrate wall
hydraulic diameter of cable, m
r~
dimensionless time
g
acceleration of gravity, m s -2
t
dimensionless longitudinal co-ordinate
w
velocity, m s -1
X
V
velocity of stationary temperature profde, m s -1
dimensionless longitudinal co-ordinate in a moving co-ordinate system
/x~7
temporal width of the heat transfer zone
3'
mass of the one meter, kg m - '
a
heat transfer between the wall and flow, W m -2 K
At
co-ordinate width of the heat transfer zone
0
dimensionless temperature
II
flow-wetted perimeter, m
v
C
capacitance, J kg -1 K -1
dimensionless velocity of stationary temperature profile
F
cross-section area, m 2
a, b
G
coolant flow rate, kg s -1
constant coefficients in approximating expressions, m
r
time, s
Subscripts
x
longitudinal co-ordinate, m
o
initial value
X
coolant heat conductivity coefficient, W m -~ K -~
w
wall
u
dynamic viscosity coefficient, m 2 s-~
g
coolant
Re
Reynolds number
Cu
copper
Pr
Prandtl number
Fe
stainless steel
State-of-the-art and objectives In order to determine the temperature fields arising with the cooldown of the conduit, a one-dimensional description of heat transfer is used: 1'2 aTg
aT~
I(FCp)g a7 + (FC¢op)g aX (FCP)w aTw = 0r
~lI(Tg -
at
= Tw - Tg
(3)
(FCP)w c)Tw _ Tg - Tw
eII(Tw - Tg)
(1) Tw)
with the introduction I of dimensionless variables
n(x, r) =,
On
Ho (FCO)gSt d Ho ; t(x, r) = 4(FCP)w
dx (2)
where St = a/(Cp~o)g is the Stanton number and Ha = f~ cold d r is the integral parameter o f homochronism. The set of equations (1) is transformed into:
The authorsare at The KrzhizhanovskyPower Engineering Institute, Moscow. Paperreceived 12 November 1980.
Equations (3) is solved by a Laplace transform with subsequent numerical integration. In the general case, the reverse transition from dimensionless variables 77 and t to dimensional r and X is also made by numerical methods. If all the values under the integral sign in (2) are assumed constant and correspond to the beginning of the cooldown process, then the dimensionless variables n and t takes the form: ,7 =
~lTg0II
(FCop)w
r; ~ =
elT~o rI
(CCg)o
x
(4)
With the substitution of these variables, ( l ) is transformed into (3):
0011-2275/81/006361-O6 $02.00 © 1981 I PC Business Press Ltd. CRYOGENICS. JUNE 1981
361
~zno + a~- = ~(Ow-Og)
/ a°w-- A(0g
an
whereA - a
0w) Cwo.
a
OtlTgo Cw ;0g w = (FCp)~ (FCoP)w ' a
alTgo
_
(5)
Co Go
alTgo C~ G =
(6)
Tg.w - T~o Two Tgo
(7)
x~
?,go
in laminar conditions of the coolant flow, and
-I go
Xgo
rwj
(8)
width of the heat transfer zone is the distance between the conduit sections with temperatures equal to e and (1-e). The present work updates the results of the earlier report 6 and describes integral relations which permit one to calculate not only the width of a steady heat transfer zone, but also the wall and coolant temperature profiles themselves for any approximation of temperature dependences of wall heat capacity and heat transfer coefficient. In addition, analytical expressions have been obtained to calculate steady profiles of wall and coolant temperature with a quadratic approximation of the temperature dependence of wall heat capacity Cw and a linear variation of heat transfer coefficient a with temperature. The effect of various factors upon the width of a steady heat transfer zone are studied.
in the turbulent mode and G = const.
The results are compared to the literature, 3,6 as well as to the experimental ones obtained on the 1SPK-M installation.
A numerical computer program is used in (3) to solve (5).
T h e basic relations
,
It follows from the analysis of the theoretical and experimental studies, l-s that in a sufficiently long conduit under cooldown one can distinguish completely cooled and completely uncooled sections in between which there is a zone of coolant heat transfer to the conduit wall, with the zone propagating along the conduit at a certain speed, o. The dimensionless wall temperature, Ow, in the heat transfer zone varies from 0 to 1. We shall call Ow(~") temperature profile. According to the literature 6, some time after the beginning of the conduit cooling by a constant flow of refrigerant a steady temperature profile should form which travels at a constant speed v = d~/d~ = const
(9)
Another system of coordinates moving at speed o is introduced in this work. The old and new systems are related as ~ - m7 +K,
X =
(10)
where K is a constant excluded from further calculations. Taking into account (10), the set of equations (5) takes the form:
i 1 - ~pv)dos = B(O =
A(O w -
0g).
zx~7 =
£n 1 Cwo(Cwo - Cw[~go) e
/q"
v~o
o 1-~o
B A
d0w[
(13)
0
as well as an expression for the temperature profile velocity: 1
f
o B 1-~ov A
d0w = 1
(14)
0
The reference of the new coordinate axix X = 0 was taken at a point on the wall temperature profile corresponding to temperature 0 w = 0.5. After substitution of (13) in (11), we fmd an integral expression relating the dimensionless coordinate X to dimensionless wall temperature 0w: w d0w
i
o., A
- o
A(i --~,o)l
At ~oo~ 1,
(Cwo + CwITgo).2
(12)
(16)
Equations (13) - (15) take the form
0g
=
i
w
e = 1
B ~-
d0w
B d0w ~-
(17)
(18)
--B d0 w A 0
=
The temporal width of heat transfer zone z ~ is the time during which the wall temperature in a certain section of the conduit chantes from (1 - e) to e. The co-ordinate
362
0g = ~ w
(11)
Equation (11) is solved at ~0 < 1 and at constant values of heat transfer coefficient or, heat capacity Cg, helium flow rate G, as well as with linear approximation of the temperature dependence of wall heat capacity Cw(0w). As a result, formulas were obtained defining the temporal width ~ and the co-ordinate width A~ of the steady heat transfer zone:
=
Assuming the flow and wall temperatures are equal on the boundary of the steady heat transfer zone (0w = 0g = 0 at the beginning of the zone and 0w = 0g = 1 at the end), a relation between the dimensionless flow and wall temperatures follows from (11):
0g);
dx d0w dx
First we obtained relations for the calculation of steady profiles of wall and coolant temperature.
w[t -aOw-jo o. Ow
0 w
wl
(19)
"0
Substituting (6) - (8) relating coefficients A and B to wail temperature in (17) - (19), the wall and coolant temperature profiles can be found. This method is much simpler than trying to solve (5).
CRYOGENICS.
J U N E 1981
We can now obtain expressions to calculate the wall and coolant temperature profiles for the following temperature dependences of the heat transfer coefficient and wall heat capacity:
ot/Odrgo = 1 +mOw, Cw/fwImgo = 1 +aOw + BO2w
Ar/-
(20)
l+a+b
3K +
0
+~-0w+~
v = (l+a+b)/O
a
- m(2K + 1)D
c'm+'l}
em+l
From (27) we can f'md the width of the steady heat transfer zone for the cooling of a cryogenic conduit, the wall heat capacity of which corresponds to curve 1 in Fig. 1, in different temperature ranges at a constant heat transfer coefficient (m = 0). We limited the heat transfer zone by e = 0.1. The results are given in Table 1, as well as parameters a and b of parabolas (20) approximating the temperature dependence of the wall heat capacity for each temperature range.
1
2 +
+a-+b)2
(22) (23)
(1 +aOw + bO2w)
A comparison of the results in Table 1 suggests that the smaller the relation (Cw0 - CwlTgo)/(Two -- Tg0) (ie the
(24) 0.5 1-0w
([3
2m
(27)
600[
After the integration of (24) we obtain X = D~n
1-e e
where D and k are found from (26).
Hence relations (17) - (19) are transformed to:
°w
~n
m(K + 1 ) - K
B = 1 +mOw
0g = 0w
]
3K + - DK(m + I) I a(K + 1) x ~n[ (2 - e)K + 1 m(K + 1) - K I (e + 1)K + 1
(21)
l + aOw + bO2w
2
+a(K+l)
2K
+
By incorporating G = const and Cg = const, and substituting (20) in (6)7 we obtain: A = (1 + mow)
1 +a+b 23{[/) l+a+b
2K
K+a(K+ 1)
2 Ln0.5+ 1 a(K + 1) 0w m ( K + I ) - K
0
2
400
1.5K+ 1
Owm + 1
6
500 1
DK(m + I )] £n I OwK + K + I I
a
~
./.~-~
6
~3oo
(25)
50
//~
~"
- 40
3o~
where D = (2 + 3K + 2/a) [re(K+ 1) - KI ;K = 2b/3a ( 2 K + 1)[m + m 2 ( K + 1 ) - K ]
200
(26)
I00
Equations (22) and (25) determine the wall and coolant temperature profiles in a steady heat transfer zone. By using (10) and (14), one can fmd the wall and coolant temperature variation with time, and by setting the value of e, determine the width of the steady heat transfer zone zx~.
"4
I
I0
iO 300
I
I00
200
T, K Fig. 1 Temperature dependences of wall heat capacity and heat transfer coefficient (solid lines) and their approximations (dotted lines): 1 - wall heat capacity, 2 - heat transfer coefficient for turbulent conditions (G = 3.1 7 10--3 kg s-1 ), 3, 4 - heat transfer coefficient for laminar conditions, 5 -approximation of wall heat capacity in the 20-290 K range, 6 - approximation of wall heat capacity in the 90-290 K range
From (10) and (12) we f'md a formula from (25) for the width of a steady heat transfer zone:
The wall heat capacity in different temperature ranges
Temperature range, K (Cwo-Cw[
-
Oo
Calculation of the width of the heat transfer zone
Table 1.
20
Tgo)l(Two-Tgo), J kg -1
40-100 K -1
3.3
a
4.48
b
4) .48
A17
C R Y O G E N I C S . JUNE 1981
4.2
20-200
40-300
100-300
200-300
2.1
1.5
0.9
0.4
12.68
17.16
1.41
0.2
-6.0
-9.6
-0.7
-0.08
5.3
6.4
16.6
76.8
363
less the wall heat capacity varies with temperature), the wider the width of heat transfer zone A~7. At a constant value of the wall heat capacity, a steady heat transfer zone does not appear at all. 6 The wall heat capacity will actually be constant when a = 0 and b = 0 and X = _+oo for any value of 0 w and m: this follows from (25). For the heat transfer coefficient which increases with temperature (m > 0), the width of the heat transfer zone will be smaller than for a constant t~. Table 2 gives the values of Z~rlduring conduit cooldown from 100 to 300 K for different values of m.
(Cw°
Table 3 shows that it is hardly possible to recommend the application of a particular linear approximation as they provide a heat transfer width which is different from the quadratic approximation of the wall heat capacity.
Comparison of the results
Experimental installation
To compare the results of the present work with the published data. 3 The results obtained experimentally would include an estimate of the conduit length upon which a steady profile is established. The lower estimate 6 equals the co-ordinate width of the heat transfer zone ~ ' . The upper estimate can be assumed similar to the lower one: this assumption does not contradict the literature.a, s The numerical growth of temperature profiles for cooling a cryogenic copper conduit in the temperature range of 50-300 K 3 showed that the heat transfer width is/x~ = 7.6 for ~ = 100. The copper heat capacity was approximated by Cw = 0.412 x 10 -4 Taw - 0.281 x 10 -1 Taw + 6.51 Tw
- 1 6 2 J k g -1 K -1 Unfortunately, the work omits the value of e limiting the heat transfer zone. The width of a steady zone calculated for this mode of cooling from (27) is:/xr/= 6.4 at e = 0.1 and/~rl = 8.6 at e = 0.05. In the calculation, the wall heat capacity approximation, (20), at a = 6.94 and b = - 0 4 . 0 4 was used. This approximation differs from the true value of the wall .heat capacity by 9% at a maximum and from that used in reference 3 by 2%. The heat transfer coefficient was found from (8). If the heat transfer zone of a steady nature is assumed to have been established across the length ~"= 100, then the results of the numerical calculation and those obtained from (27) agree well.
364
The experimental verification of (25) was made on the 1SPK-M installation ie, a coaxial superconducting cable section where the coolant, helium, flows in the clearance between two cores of superconductor-plated copper pipes. The linear pipe is evacuated. The system is accommocated in a sealed stainless steel pipe, the space between the outer copper pipe and the stainless steel pipe is fflled with helium which penetrates from the annular gap through the apertures along the outer pipe coaxial. The wall temperature was determined from the electrical resistance of the individual cable sections which were measured by means of potential probes mounted on the outer surface of the inner core. The section length was approximately 1 m, the total length of the conduit being 5.6 ram, that of the outer copper pipe 80 × 3.0 mm and the stainless steel pipe 89 x 3.5 ram. The copper and stainless steel heat capacity is given by Skoot. 7 The total wall heat capacity for 1SPK-M is calculated from: Cw = Ccu 7Cu + CFe 7Fe ~'Cu + ')'Fe Table 3. conduit
Art
Linear approximations
A~" during conduit cooldown 0.5
1.0
1.5
2.0
3.0
4.0
10.0
oo
16.6 13.1
10.9
9.6
8A
6.7
5.0
3.0
0
(29)
Steady state heat transfer width of a copper
Temperature interval, K
In reference 6, the width of the heat transfer zone was determined for different temperature intervals with three kinds of linear approximation of the temperature depend-
z~
(28)
The results of the calculations of a steady heat transfer width of a copper coduit derived from (12) and (27) are given in Table 3. The heat transfer coefficient used to evaluate (27) was assumed constant (m = 0).
It should be emphasized that the variability of the heat transfer coefficient does not cause the steady temperature profile to appear. This profile sets in at a variable wall heat capacity which is independent of t~.
0
CwlTg )0~2 ~n 1 -- e e
and differs from (12) only in the numerator under the logarithm sign. This difference is due to the assumption that e is small.
As illustrated in Table 2, the greater the growth of the heat transfer coefficient with temperature, the narrower the heat transfer zone. With an infinite coefficient, there will be a stepped temperature profile (/x~ = 0).
m
+
Cwo (Cw0 - Cw ITgo)
The wall heat capacity of this conduit is given by a parabola with parameters a = 1.41 and b = - 0.7.
Table 2.
ence of the wall heat capacity: one, the approximation line passes through points on the heat capacity curve which outline the temperature interval boundaries, two and three, the approximating lines cross the mean temperature point in the interval considered and points of minimum and maximum heat capacity, respectively. Equation (12) was used for the calculation. At m = b = 0 (27) takes the form
Quadratic approximations
40-200
100-300 40-300
1 4.2
20.2
3.9
2 3.6
15.4
3.3
3 9.2
33.1
11.6
6.5
21.1
7.4
CRYOGENICS. JUNE 1981
The temperature dependence of the wall heat capacity obtained from (29) is shown in Fig. 1. The approximating curves determined from (20), were plotted through the two extreme points of the cooling temperature intervals and through a mean temperature point. The error incurred by replacing the true temperature dependence of the wall heat capacity and by the appoximating parabolas does not exceed 3% in the 100-300 K range and 10% in the 20-300 K range.
4.36 ~kgo/d
Helium heat capacity. Cg in the above temperature inter-
Heat transfer coefficient, a depends on the helium thermo-
The experiments run on cooldown with the 1SPK-M installation were conducted in the temperature ranges from room temperature (290 K) to 90 K-with coolant flow rate of 0.58 x 10 -3 kg s -1 and to 20 K with the coolant flow rate of 1.75 x 10 -3 kg s -1 . The helium flow rate at the cable inlet was maintained constant during the experiment. The gas pressure was 4 atm, and because of low hydraulic resistance it varied little along the conduit, aiTo corresponding to the given cooling conditions is 15.~ W m-2 K-1 for up to 90 K and 8 W m -2 K -1 for cooling up to 20 K. The calculation of c~trgo was made by a formula valid for a laminar flow: =
Figs 3 and 4 show the dependences Of 0w0/) for the final cable sections. For the purpose of comparing calculations, we shall examine the variation of parameters included in (4) - (11), with temperature. vals hardly varies, within -+ 2.5%, and it can be considered constant.
Experimental results and comparison with calculations
OdTg 0
respectively. The values of the dimensionless length and width of the heat transfer zone are thus quite close to the above estimates of the conduit length across which a steady temperature profile is established.
physical properties which vary with temperature, a(Tw) can be found from (7) and (8). Since the cooldown of a cryogenic conduit is a nonstationary process, one should
~
0.8
2/
k 0.4
(30)
The dependence of Tw(r) obtained in the experiments with the use of al r g o found from (30) and incorporated in . . (4) are reduced to a dlmensxonless form 0w(r/). The thermophysical properties of helium were taken from reference 8. The temporal heat transfer width/xr/versus x co-ordinate at e = 0.1 is shown in Fig. 2. It is evident from the figure that a steady width of the heat transfer zone is virtually established for the final cable sections corresponding to ~"= 1.71 in the 90-290 K range and ~"= 10.28 in the 90290 K range.
0.2
I
I
I
I
-8
-6
-4
-2
0
2
6
4
Fig. 3 E x p e r i m e n t a l (solid line) and calculated (dotted lines) temperatures in the steady heat transfer zone under 1 S P K - M cooldown in the 9 0 - 2 9 0 K range: 1 - wall t e m p e r a t u r e , 2 - c o o l a n t temperature
T
The coordinate width of the steady heat transfer zone for these temperature intervals is z~- = 1.58 and z~- = 9.0,
@
I0
-0.8
f -0.6 6 P
\ <3
\
- 0.4
4
0.2 2
/ 0 0
f
•
I
i
i
]
I
I
2
3
4
xlm
Fig. 2
Variation of temporary width of heat t r a n s f e r zone across
1 S P K - M under c o o l d o w n : 1 - in the 9 0 - 2 9 0 K range, 2 - in the 2 0 - 2 9 0 K range
CRYOGENICS. JUNE 1981
-2
I
-I
0
I
I
I
2
"q Fig. 4 Experimental (solid line) and calculated (dotted lines) temperatures in t h e steady heat transfer zone under 1 S P K - M cooldown in the 2 0 - 2 9 0 K range: 1 - w a l l t e m p e r a t u r e , 2 - coolant temperature
365
alSO take this into account when evaluating the heat transfer coefficient. 1 The estimates made according to the method described by Dreitser et al 2 for turbulent flow show that at the temperature variation rates occurring during cooldown on the 1SPK-M, exerts no essential effect on heat transfer: the parameter of thermal instability, K~g, found from
aTw
d
K g- aT rw
(31)
does not exceed 10 -6 . The associated heat transfer coefficient differs from the steady one by 1% maximum. In the case of laminar flow conditions, there are no recommendations valid for the calculation of conduit cooldown processes which account for the variability of heat transfer in literature known to us, but one can assume that at the same temperature variation rates there will be no essential influence of the non-uniformity on the value of a. The heat transfer coefficient for both laminar and turbulent flow conditions can therefore be calculated from the steady heat transfer formula (7) and (8). In the cooling conditions considered, the gas flow is laminar. The temperature dependence of the heat transfer coefficient is given in Fig. 1 (curve 3). It is well approximated by straight line 4 given by (20) with m = 1.35 in the 90-290 K interval and m = 3.5 in the 20-290 K interval. The error of this approximation is less than 6% in the 100-300 K range and 12% in the 20-300 K range. Generally speaking, the helium flow rate, G, varies along the conduit because of its increased density with decreasing temperature. The helium flow rate in the heat transfer zone is found from: O = G0 - VFg(Pg0 - pg)
(32)
Calculations made using this formula on the results of the results of the experiments show, however, that the flow rate varies by no more than 1.5% as compared to the input which is due to a high linear heat capacity of the wall, and hence low velocity of the heat transfer sone, V. The helium flow rate can therefore be assumed constant.
366
The wall heat capacity. Cw can be found from (29). As mentioned above, the temperature dependence of this heat capacity is approximated by parabolas: Cw/CwlTgo = 1 + 1.8 0w -- 0.88 02w for 90-290 K, and
Cw/Cwlrgo = 1 + 173.1 0w - 89.1 02w Value ~v does not exceed 0.02 at T 2 20 K, and condition (16) is observed. Thus, relations (22) - (26) obtained in the present work fully suit the calculation of the cooldown process on 1 SPK-M. Figs 2 and 3 show temporary wall and coolant temperature dependences 0w(r/) and 0g 07) calculated from (10) and (22) - (26) for the above cases of cooling 1SPK-M. A comparison of the calculated and experimental results show that there is a good agreement between them, and it may be concluded that a steady heat transfer zone does set in on 1SPK-M and that the zone is well described by (22) - (26). The authors are deeply indebted to S.K. Smirnov for prompting discussions and to N.Ye. Komissarzhevsky for assistance in conducting the experiment and processing the results.
References 1 2 3
4
5 6 7 8
Koshkin,V.K., Kalinin, E.K., Dreitser, G.A., Yatkho, S.A. Nestatsionarni teploobmon. Mashinostroyenie Publishing House, Moscow(1975) Dreitser,G.A., Kuzminov,V.A. Raschet, razogreva i ohlazhdeniya truboprovodov, Mashinostroyenie Publishing House, Moscow(1977) Baron, A.M., Yeroshenko, V.M., Yaskin, L.A. Studies on cooldown of cryogeniccables Cryogenics 17 (1977) 161-166 Keilin,V.E., Kovalev,I.A., Lelekhov, S.L. Inzhenernofizieheski zhurna127 (1974) 6 Larsen,F.M.lntJHeatMass Transfer 10 (1967) Benkik,N.T., Smlrnov, S.K., Blinkov, Ye.L. Calculation of stationary heat transfer zone in conduit under cooldown Cryogenics 19 (1979) 477--482 Skott, R.B. Technica nizkikh temperatur. Inostrarmaya literatura Publishing House, Moscow(1962) MeCarty,R.D. Thermophysicalproperties of helium-4 from 2 to 1500 K with pressures to 1000 atmospheres. NBS Technical Note 631, (1972)
CRYOGENICS. JUNE 1981
Investigation of superconducting cable cooldown N.T. Bendik and N.I. Glukhov
Nomenclature T
temperature, K
o
density, kg m -3
d
~0
ratio of the capacitance of coolant to that of substrate wall
hydraulic diameter of cable, m
r~
dimensionless time
g
acceleration of gravity, m s -2
t
dimensionless longitudinal co-ordinate
w
velocity, m s -1
X
V
velocity of stationary temperature profde, m s -1
dimensionless longitudinal co-ordinate in a moving co-ordinate system
/x~7
temporal width of the heat transfer zone
3'
mass of the one meter, kg m - '
a
heat transfer between the wall and flow, W m -2 K
At
co-ordinate width of the heat transfer zone
0
dimensionless temperature
II
flow-wetted perimeter, m
v
C
capacitance, J kg -1 K -1
dimensionless velocity of stationary temperature profile
F
cross-section area, m 2
a, b
G
coolant flow rate, kg s -1
constant coefficients in approximating expressions, m
r
time, s
Subscripts
x
longitudinal co-ordinate, m
o
initial value
X
coolant heat conductivity coefficient, W m -~ K -~
w
wall
u
dynamic viscosity coefficient, m 2 s-~
g
coolant
Re
Reynolds number
Cu
copper
Pr
Prandtl number
Fe
stainless steel
State-of-the-art and objectives In order to determine the temperature fields arising with the cooldown of the conduit, a one-dimensional description of heat transfer is used: 1'2 aTg
aT~
I(FCp)g a7 + (FC¢op)g aX (FCP)w aTw = 0r
~lI(Tg -
at
= Tw - Tg
(3)
(FCP)w c)Tw _ Tg - Tw
eII(Tw - Tg)
(1) Tw)
with the introduction I of dimensionless variables
n(x, r) =,
On
Ho (FCO)gSt d Ho ; t(x, r) = 4(FCP)w
dx (2)
where St = a/(Cp~o)g is the Stanton number and Ha = f~ cold d r is the integral parameter o f homochronism. The set of equations (1) is transformed into:
The authorsare at The KrzhizhanovskyPower Engineering Institute, Moscow. Paperreceived 12 November 1980.
Equations (3) is solved by a Laplace transform with subsequent numerical integration. In the general case, the reverse transition from dimensionless variables 77 and t to dimensional r and X is also made by numerical methods. If all the values under the integral sign in (2) are assumed constant and correspond to the beginning of the cooldown process, then the dimensionless variables n and t takes the form: ,7 =
~lTg0II
(FCop)w
r; ~ =
elT~o rI
(CCg)o
x
(4)
With the substitution of these variables, ( l ) is transformed into (3):
0011-2275/81/006361-O6 $02.00 © 1981 I PC Business Press Ltd. CRYOGENICS. JUNE 1981
361
~zno + a~- = ~(Ow-Og)
/ a°w-- A(0g
an
whereA - a
0w) Cwo.
a
OtlTgo Cw ;0g w = (FCp)~ (FCoP)w ' a
alTgo
_
(5)
Co Go
alTgo C~ G =
(6)
Tg.w - T~o Two Tgo
(7)
x~
?,go
in laminar conditions of the coolant flow, and
-I go
Xgo
rwj
(8)
width of the heat transfer zone is the distance between the conduit sections with temperatures equal to e and (1-e). The present work updates the results of the earlier report 6 and describes integral relations which permit one to calculate not only the width of a steady heat transfer zone, but also the wall and coolant temperature profiles themselves for any approximation of temperature dependences of wall heat capacity and heat transfer coefficient. In addition, analytical expressions have been obtained to calculate steady profiles of wall and coolant temperature with a quadratic approximation of the temperature dependence of wall heat capacity Cw and a linear variation of heat transfer coefficient a with temperature. The effect of various factors upon the width of a steady heat transfer zone are studied.
in the turbulent mode and G = const.
The results are compared to the literature, 3,6 as well as to the experimental ones obtained on the 1SPK-M installation.
A numerical computer program is used in (3) to solve (5).
T h e basic relations
,
It follows from the analysis of the theoretical and experimental studies, l-s that in a sufficiently long conduit under cooldown one can distinguish completely cooled and completely uncooled sections in between which there is a zone of coolant heat transfer to the conduit wall, with the zone propagating along the conduit at a certain speed, o. The dimensionless wall temperature, Ow, in the heat transfer zone varies from 0 to 1. We shall call Ow(~") temperature profile. According to the literature 6, some time after the beginning of the conduit cooling by a constant flow of refrigerant a steady temperature profile should form which travels at a constant speed v = d~/d~ = const
(9)
Another system of coordinates moving at speed o is introduced in this work. The old and new systems are related as ~ - m7 +K,
X =
(10)
where K is a constant excluded from further calculations. Taking into account (10), the set of equations (5) takes the form:
i 1 - ~pv)dos = B(O =
A(O w -
0g).
zx~7 =
£n 1 Cwo(Cwo - Cw[~go) e
/q"
v~o
o 1-~o
B A
d0w[
(13)
0
as well as an expression for the temperature profile velocity: 1
f
o B 1-~ov A
d0w = 1
(14)
0
The reference of the new coordinate axix X = 0 was taken at a point on the wall temperature profile corresponding to temperature 0 w = 0.5. After substitution of (13) in (11), we fmd an integral expression relating the dimensionless coordinate X to dimensionless wall temperature 0w: w d0w
i
o., A
- o
A(i --~,o)l
At ~oo~ 1,
(Cwo + CwITgo).2
(12)
(16)
Equations (13) - (15) take the form
0g
=
i
w
e = 1
B ~-
d0w
B d0w ~-
(17)
(18)
--B d0 w A 0
=
The temporal width of heat transfer zone z ~ is the time during which the wall temperature in a certain section of the conduit chantes from (1 - e) to e. The co-ordinate
362
0g = ~ w
(11)
Equation (11) is solved at ~0 < 1 and at constant values of heat transfer coefficient or, heat capacity Cg, helium flow rate G, as well as with linear approximation of the temperature dependence of wall heat capacity Cw(0w). As a result, formulas were obtained defining the temporal width ~ and the co-ordinate width A~ of the steady heat transfer zone:
=
Assuming the flow and wall temperatures are equal on the boundary of the steady heat transfer zone (0w = 0g = 0 at the beginning of the zone and 0w = 0g = 1 at the end), a relation between the dimensionless flow and wall temperatures follows from (11):
0g);
dx d0w dx
First we obtained relations for the calculation of steady profiles of wall and coolant temperature.
w[t -aOw-jo o. Ow
0 w
wl
(19)
"0
Substituting (6) - (8) relating coefficients A and B to wail temperature in (17) - (19), the wall and coolant temperature profiles can be found. This method is much simpler than trying to solve (5).
CRYOGENICS.
J U N E 1981
We can now obtain expressions to calculate the wall and coolant temperature profiles for the following temperature dependences of the heat transfer coefficient and wall heat capacity:
ot/Odrgo = 1 +mOw, Cw/fwImgo = 1 +aOw + BO2w
Ar/-
(20)
l+a+b
3K +
0
+~-0w+~
v = (l+a+b)/O
a
- m(2K + 1)D
c'm+'l}
em+l
From (27) we can f'md the width of the steady heat transfer zone for the cooling of a cryogenic conduit, the wall heat capacity of which corresponds to curve 1 in Fig. 1, in different temperature ranges at a constant heat transfer coefficient (m = 0). We limited the heat transfer zone by e = 0.1. The results are given in Table 1, as well as parameters a and b of parabolas (20) approximating the temperature dependence of the wall heat capacity for each temperature range.
1
2 +
+a-+b)2
(22) (23)
(1 +aOw + bO2w)
A comparison of the results in Table 1 suggests that the smaller the relation (Cw0 - CwlTgo)/(Two -- Tg0) (ie the
(24) 0.5 1-0w
([3
2m
(27)
600[
After the integration of (24) we obtain X = D~n
1-e e
where D and k are found from (26).
Hence relations (17) - (19) are transformed to:
°w
~n
m(K + 1 ) - K
B = 1 +mOw
0g = 0w
]
3K + - DK(m + I) I a(K + 1) x ~n[ (2 - e)K + 1 m(K + 1) - K I (e + 1)K + 1
(21)
l + aOw + bO2w
2
+a(K+l)
2K
+
By incorporating G = const and Cg = const, and substituting (20) in (6)7 we obtain: A = (1 + mow)
1 +a+b 23{[/) l+a+b
2K
K+a(K+ 1)
2 Ln0.5+ 1 a(K + 1) 0w m ( K + I ) - K
0
2
400
1.5K+ 1
Owm + 1
6
500 1
DK(m + I )] £n I OwK + K + I I
a
~
./.~-~
6
~3oo
(25)
50
//~
~"
- 40
3o~
where D = (2 + 3K + 2/a) [re(K+ 1) - KI ;K = 2b/3a ( 2 K + 1)[m + m 2 ( K + 1 ) - K ]
200
(26)
I00
Equations (22) and (25) determine the wall and coolant temperature profiles in a steady heat transfer zone. By using (10) and (14), one can fmd the wall and coolant temperature variation with time, and by setting the value of e, determine the width of the steady heat transfer zone zx~.
"4
I
I0
iO 300
I
I00
200
T, K Fig. 1 Temperature dependences of wall heat capacity and heat transfer coefficient (solid lines) and their approximations (dotted lines): 1 - wall heat capacity, 2 - heat transfer coefficient for turbulent conditions (G = 3.1 7 10--3 kg s-1 ), 3, 4 - heat transfer coefficient for laminar conditions, 5 -approximation of wall heat capacity in the 20-290 K range, 6 - approximation of wall heat capacity in the 90-290 K range
From (10) and (12) we f'md a formula from (25) for the width of a steady heat transfer zone:
The wall heat capacity in different temperature ranges
Temperature range, K (Cwo-Cw[
-
Oo
Calculation of the width of the heat transfer zone
Table 1.
20
Tgo)l(Two-Tgo), J kg -1
40-100 K -1
3.3
a
4.48
b
4) .48
A17
C R Y O G E N I C S . JUNE 1981
4.2
20-200
40-300
100-300
200-300
2.1
1.5
0.9
0.4
12.68
17.16
1.41
0.2
-6.0
-9.6
-0.7
-0.08
5.3
6.4
16.6
76.8
363
less the wall heat capacity varies with temperature), the wider the width of heat transfer zone A~7. At a constant value of the wall heat capacity, a steady heat transfer zone does not appear at all. 6 The wall heat capacity will actually be constant when a = 0 and b = 0 and X = _+oo for any value of 0 w and m: this follows from (25). For the heat transfer coefficient which increases with temperature (m > 0), the width of the heat transfer zone will be smaller than for a constant t~. Table 2 gives the values of Z~rlduring conduit cooldown from 100 to 300 K for different values of m.
(Cw°
Table 3 shows that it is hardly possible to recommend the application of a particular linear approximation as they provide a heat transfer width which is different from the quadratic approximation of the wall heat capacity.
Comparison of the results
Experimental installation
To compare the results of the present work with the published data. 3 The results obtained experimentally would include an estimate of the conduit length upon which a steady profile is established. The lower estimate 6 equals the co-ordinate width of the heat transfer zone ~ ' . The upper estimate can be assumed similar to the lower one: this assumption does not contradict the literature.a, s The numerical growth of temperature profiles for cooling a cryogenic copper conduit in the temperature range of 50-300 K 3 showed that the heat transfer width is/x~ = 7.6 for ~ = 100. The copper heat capacity was approximated by Cw = 0.412 x 10 -4 Taw - 0.281 x 10 -1 Taw + 6.51 Tw
- 1 6 2 J k g -1 K -1 Unfortunately, the work omits the value of e limiting the heat transfer zone. The width of a steady zone calculated for this mode of cooling from (27) is:/xr/= 6.4 at e = 0.1 and/~rl = 8.6 at e = 0.05. In the calculation, the wall heat capacity approximation, (20), at a = 6.94 and b = - 0 4 . 0 4 was used. This approximation differs from the true value of the wall .heat capacity by 9% at a maximum and from that used in reference 3 by 2%. The heat transfer coefficient was found from (8). If the heat transfer zone of a steady nature is assumed to have been established across the length ~"= 100, then the results of the numerical calculation and those obtained from (27) agree well.
364
The experimental verification of (25) was made on the 1SPK-M installation ie, a coaxial superconducting cable section where the coolant, helium, flows in the clearance between two cores of superconductor-plated copper pipes. The linear pipe is evacuated. The system is accommocated in a sealed stainless steel pipe, the space between the outer copper pipe and the stainless steel pipe is fflled with helium which penetrates from the annular gap through the apertures along the outer pipe coaxial. The wall temperature was determined from the electrical resistance of the individual cable sections which were measured by means of potential probes mounted on the outer surface of the inner core. The section length was approximately 1 m, the total length of the conduit being 5.6 ram, that of the outer copper pipe 80 × 3.0 mm and the stainless steel pipe 89 x 3.5 ram. The copper and stainless steel heat capacity is given by Skoot. 7 The total wall heat capacity for 1SPK-M is calculated from: Cw = Ccu 7Cu + CFe 7Fe ~'Cu + ')'Fe Table 3. conduit
Art
Linear approximations
A~" during conduit cooldown 0.5
1.0
1.5
2.0
3.0
4.0
10.0
oo
16.6 13.1
10.9
9.6
8A
6.7
5.0
3.0
0
(29)
Steady state heat transfer width of a copper
Temperature interval, K
In reference 6, the width of the heat transfer zone was determined for different temperature intervals with three kinds of linear approximation of the temperature depend-
z~
(28)
The results of the calculations of a steady heat transfer width of a copper coduit derived from (12) and (27) are given in Table 3. The heat transfer coefficient used to evaluate (27) was assumed constant (m = 0).
It should be emphasized that the variability of the heat transfer coefficient does not cause the steady temperature profile to appear. This profile sets in at a variable wall heat capacity which is independent of t~.
0
CwlTg )0~2 ~n 1 -- e e
and differs from (12) only in the numerator under the logarithm sign. This difference is due to the assumption that e is small.
As illustrated in Table 2, the greater the growth of the heat transfer coefficient with temperature, the narrower the heat transfer zone. With an infinite coefficient, there will be a stepped temperature profile (/x~ = 0).
m
+
Cwo (Cw0 - Cw ITgo)
The wall heat capacity of this conduit is given by a parabola with parameters a = 1.41 and b = - 0.7.
Table 2.
ence of the wall heat capacity: one, the approximation line passes through points on the heat capacity curve which outline the temperature interval boundaries, two and three, the approximating lines cross the mean temperature point in the interval considered and points of minimum and maximum heat capacity, respectively. Equation (12) was used for the calculation. At m = b = 0 (27) takes the form
Quadratic approximations
40-200
100-300 40-300
1 4.2
20.2
3.9
2 3.6
15.4
3.3
3 9.2
33.1
11.6
6.5
21.1
7.4
CRYOGENICS. JUNE 1981
The temperature dependence of the wall heat capacity obtained from (29) is shown in Fig. 1. The approximating curves determined from (20), were plotted through the two extreme points of the cooling temperature intervals and through a mean temperature point. The error incurred by replacing the true temperature dependence of the wall heat capacity and by the appoximating parabolas does not exceed 3% in the 100-300 K range and 10% in the 20-300 K range.
4.36 ~kgo/d
Helium heat capacity. Cg in the above temperature inter-
Heat transfer coefficient, a depends on the helium thermo-
The experiments run on cooldown with the 1SPK-M installation were conducted in the temperature ranges from room temperature (290 K) to 90 K-with coolant flow rate of 0.58 x 10 -3 kg s -1 and to 20 K with the coolant flow rate of 1.75 x 10 -3 kg s -1 . The helium flow rate at the cable inlet was maintained constant during the experiment. The gas pressure was 4 atm, and because of low hydraulic resistance it varied little along the conduit, aiTo corresponding to the given cooling conditions is 15.~ W m-2 K-1 for up to 90 K and 8 W m -2 K -1 for cooling up to 20 K. The calculation of c~trgo was made by a formula valid for a laminar flow: =
Figs 3 and 4 show the dependences Of 0w0/) for the final cable sections. For the purpose of comparing calculations, we shall examine the variation of parameters included in (4) - (11), with temperature. vals hardly varies, within -+ 2.5%, and it can be considered constant.
Experimental results and comparison with calculations
OdTg 0
respectively. The values of the dimensionless length and width of the heat transfer zone are thus quite close to the above estimates of the conduit length across which a steady temperature profile is established.
physical properties which vary with temperature, a(Tw) can be found from (7) and (8). Since the cooldown of a cryogenic conduit is a nonstationary process, one should
~
0.8
2/
k 0.4
(30)
The dependence of Tw(r) obtained in the experiments with the use of al r g o found from (30) and incorporated in . . (4) are reduced to a dlmensxonless form 0w(r/). The thermophysical properties of helium were taken from reference 8. The temporal heat transfer width/xr/versus x co-ordinate at e = 0.1 is shown in Fig. 2. It is evident from the figure that a steady width of the heat transfer zone is virtually established for the final cable sections corresponding to ~"= 1.71 in the 90-290 K range and ~"= 10.28 in the 90290 K range.
0.2
I
I
I
I
-8
-6
-4
-2
0
2
6
4
Fig. 3 E x p e r i m e n t a l (solid line) and calculated (dotted lines) temperatures in the steady heat transfer zone under 1 S P K - M cooldown in the 9 0 - 2 9 0 K range: 1 - wall t e m p e r a t u r e , 2 - c o o l a n t temperature
T
The coordinate width of the steady heat transfer zone for these temperature intervals is z~- = 1.58 and z~- = 9.0,
@
I0
-0.8
f -0.6 6 P
\ <3
\
- 0.4
4
0.2 2
/ 0 0
f
•
I
i
i
]
I
I
2
3
4
xlm
Fig. 2
Variation of temporary width of heat t r a n s f e r zone across
1 S P K - M under c o o l d o w n : 1 - in the 9 0 - 2 9 0 K range, 2 - in the 2 0 - 2 9 0 K range
CRYOGENICS. JUNE 1981
-2
I
-I
0
I
I
I
2
"q Fig. 4 Experimental (solid line) and calculated (dotted lines) temperatures in t h e steady heat transfer zone under 1 S P K - M cooldown in the 2 0 - 2 9 0 K range: 1 - w a l l t e m p e r a t u r e , 2 - coolant temperature
365
alSO take this into account when evaluating the heat transfer coefficient. 1 The estimates made according to the method described by Dreitser et al 2 for turbulent flow show that at the temperature variation rates occurring during cooldown on the 1SPK-M, exerts no essential effect on heat transfer: the parameter of thermal instability, K~g, found from
aTw
d
K g- aT rw
(31)
does not exceed 10 -6 . The associated heat transfer coefficient differs from the steady one by 1% maximum. In the case of laminar flow conditions, there are no recommendations valid for the calculation of conduit cooldown processes which account for the variability of heat transfer in literature known to us, but one can assume that at the same temperature variation rates there will be no essential influence of the non-uniformity on the value of a. The heat transfer coefficient for both laminar and turbulent flow conditions can therefore be calculated from the steady heat transfer formula (7) and (8). In the cooling conditions considered, the gas flow is laminar. The temperature dependence of the heat transfer coefficient is given in Fig. 1 (curve 3). It is well approximated by straight line 4 given by (20) with m = 1.35 in the 90-290 K interval and m = 3.5 in the 20-290 K interval. The error of this approximation is less than 6% in the 100-300 K range and 12% in the 20-300 K range. Generally speaking, the helium flow rate, G, varies along the conduit because of its increased density with decreasing temperature. The helium flow rate in the heat transfer zone is found from: O = G0 - VFg(Pg0 - pg)
(32)
Calculations made using this formula on the results of the results of the experiments show, however, that the flow rate varies by no more than 1.5% as compared to the input which is due to a high linear heat capacity of the wall, and hence low velocity of the heat transfer sone, V. The helium flow rate can therefore be assumed constant.
366
The wall heat capacity. Cw can be found from (29). As mentioned above, the temperature dependence of this heat capacity is approximated by parabolas: Cw/CwlTgo = 1 + 1.8 0w -- 0.88 02w for 90-290 K, and
Cw/Cwlrgo = 1 + 173.1 0w - 89.1 02w Value ~v does not exceed 0.02 at T 2 20 K, and condition (16) is observed. Thus, relations (22) - (26) obtained in the present work fully suit the calculation of the cooldown process on 1 SPK-M. Figs 2 and 3 show temporary wall and coolant temperature dependences 0w(r/) and 0g 07) calculated from (10) and (22) - (26) for the above cases of cooling 1SPK-M. A comparison of the calculated and experimental results show that there is a good agreement between them, and it may be concluded that a steady heat transfer zone does set in on 1SPK-M and that the zone is well described by (22) - (26). The authors are deeply indebted to S.K. Smirnov for prompting discussions and to N.Ye. Komissarzhevsky for assistance in conducting the experiment and processing the results.
References 1 2 3
4
5 6 7 8
Koshkin,V.K., Kalinin, E.K., Dreitser, G.A., Yatkho, S.A. Nestatsionarni teploobmon. Mashinostroyenie Publishing House, Moscow(1975) Dreitser,G.A., Kuzminov,V.A. Raschet, razogreva i ohlazhdeniya truboprovodov, Mashinostroyenie Publishing House, Moscow(1977) Baron, A.M., Yeroshenko, V.M., Yaskin, L.A. Studies on cooldown of cryogeniccables Cryogenics 17 (1977) 161-166 Keilin,V.E., Kovalev,I.A., Lelekhov, S.L. Inzhenernofizieheski zhurna127 (1974) 6 Larsen,F.M.lntJHeatMass Transfer 10 (1967) Benkik,N.T., Smlrnov, S.K., Blinkov, Ye.L. Calculation of stationary heat transfer zone in conduit under cooldown Cryogenics 19 (1979) 477--482 Skott, R.B. Technica nizkikh temperatur. Inostrarmaya literatura Publishing House, Moscow(1962) MeCarty,R.D. Thermophysicalproperties of helium-4 from 2 to 1500 K with pressures to 1000 atmospheres. NBS Technical Note 631, (1972)
CRYOGENICS. JUNE 1981