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On the cooling of forced flow cable conductors for superconducting magnets
An exact analytical solution is presented for the cooling of forced-flow, long, cable~conductors for large superconducting magnets. It is shown that, when the temperature profiles for the coolant fluid and the cable material exhibit a wave-front like pattern in their propagation along the conduit, the cooling time for the cable can be determined with a very simple analysis.
On the cooling of forced flow cable conductors for superconducting magnets M.N. E)zi§ik, Ch. Trepp and H.P. Baumann
Nomenclature A
cooling surface area for the material per unit length of the cable-conductor
~Cm
specific heat for the fluid and the material respectively mass flow rate of fluid through the cableconductor, (ie, kg/hr)
Gf h
heat transfer coefficient
L
length of the cable-conductor
m-
t
temperature of the material to be cooled inlet temperature of the coolant fluid initial temperature of the material to be cooled
Of(x, t ) -
Tf(x, t) - Tfo , dimensionless fluid Tmo- Tfo temperature
mass of the material to be cooled per unit length of the cable hA GfCf '
temperature of the coolant fluid
distance along the cable-conductor
hA time 1 MmCrn '
Mm
n
Tf(x, t) Tin(x, t) Tfo Tmo
Om(X, t) =
Tm(x, t) - Teo ,dimensionless material Tmo- Tfo temperature
1
,,+1~
'en~'m'l time
The cable conductors developed for use in large superconducting magnets 1 are cooled by the forced flow of supercritical helium through the narrow, uneven passages inside the cable. Some studies have been reported 2-4 on the determination of the friction factor associated with flow through such interstices, because forced flow through narrow passages hundreds of meters long causes high frictional losses, hence high pumping powers are required. The determination of the time required for the cooling of cable conductors by forced flow, from a given initial temperature to a specifiod temperature, is of interest for operation purposes. This work is concerned with the development of an analytical method of solution for the cooling of long cable conductors by forced flow through the cable. The authors are at the Institut for Verfahrens-und Kaltetechnik, ETH-Zentrum, 8092-Zurich, Switzerland. The Permanent address of MNO is the Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, North Carolina 27650, USA. Paper received 13 March 1980.
~=nx
dimensionless axial distance
rl= mt
dimensionless time
Analysis We consider forced flow through the interstices of a long cable conductor which is initially at a uniform temperature Tmo- The coolant fluid enters the cable at the origin of the axial coordinate x = 0, at time t = 0 with a constant inlet temperature Tfo and a constant mass flow rate of Gf kg h "1. The heat losses from the outer walls of the cable to the surrounding environment is considered negligible in comparison with the heat transferred to the coolant. The duration of cooling is much longer than the transient time for the coolant fluid entering the cable to travel from the inlet to the exit. The Biot number associated with convective heat transfer between the cable wires and the coolant fluid is less than about 0.1, so that the lumped system analysis s can be applied to formulate the transient heat transfer problem.
Let, Tf(x, t) and Tin(x, t) be the temperature distributions in the coolant fluid and the cable material respectively along the cable. The differential equations governing these tempera-
0011-2275/80/090539-03 ~02.00 © 1980 IPC Business Press CRYOGENICS. SEPTEMBER 1980
539
tures are determined by writing energy balance equations over a differential length along the cable for the material inside the cable (ie, wires) and the coolant flow. The resulting equations are summarized below.
The zeroth order solution is introduced into (6) for 0m(~', r/') under the integral signe to obtain the first order solution as
OOm) (~, r/) = e -n [1 + 7/(1 - e'~)]
The material inside the cable:
aOm(x, t) at + mOrn(X, t) = mOf(x, t), for t > 0
(la)
Ore(x, t) = 1
(lb)
for t = 0
(9)
This first order solution is used in (6) to obtain the second order solution as 0~) (~,r/) =e-n I 1 + ~ 1 -e'G) + lr/2 [1 -e-~(1 +~)]1
The coolant fluid: (10)
aOf(x, t)
- at
+ nOf(x, t) = nOm(x, t), for t > 0
Of(X, t) = 0
(2a) (2b)
Proceeding in this manner to obtain higher order solutions, it can be shown that the exact solution for 0m(~, 77) can be expressed as an infinite series in the form
where various quantities are defined as hA hA m - Mm----~m,n = ~
Ore(x, t) =
(3a)
Tin(x, t) - Tfo Tm° _ Tf°
(3b)
Tf(x, t) - Tfo Trno- Tfo
Of(x, t) -
Om(~'n)=l-e-(l~"O ~ r=l
In the following analysis we develop an explicit exact solution to these equations in the form of the infinite series described. Equations (1) and (2) are formally integrated to yield respectively t
/=0
f
Of(x, t ') e rat' dt ,
(4)
1+
= iT
=
T
(12)
T h e t e m p e r a t u r e difference, 0 m (~, 7"/) - 0 f ( t , 7"/), b e t w e e n t h e material and the fluid temperatures is obtained by subtracting (11) and (12).
Om(~, r/) - Of(t, rl) = e-(~' rO ~-~ (~r/)r r=o (r!)2
(13)
For comparison purposes, we also present below the integral form of the solutions for Om(t, 77) and Of(G, r/) based on the analysis by Hausen6.
0
x
f
(11)
(3c)
The above system of equations (1) and (2) is similar to that considered in reference 6, page 332, for the heat transfer problem associated with heat exchangers and their solution has been developed in the form of integrals involving Bessel functions of imaginary argument.
Of(x, t) = ne -nx
~/
~
Similarly, it can be shown that the solution for the fluid temperature 0f(t, r/) is given by
Of(t,r/) = l - e -(~+n)
Orn(X,t) = e-mr+me-mr
r/' 7.
Om(x', t) e nx' dx'
(5)
Om(t,r/)= 1 -
f
e"(~+n) Jo [2i(tg') y'] dr/'
(14)
,i
0
0
Equation (5) is introduced into (4) 0f(t, 77) = Om(~, r/)
=
e-n + e-(n+O n f
f
0
0
Om(t',r/')e(n'+~')d~'dr/'
(6)
= nx,
71 = mt
e-(~'+n) Jo [ 2i(t #')~ ] d~'
where Jo (z) is the Bessel function of order zero and i = x/-1. By noting that the series form of Jo(z) function is given by
Jo(z) = 2_., (-1) r (7)
Equation (6) is now solved by the method of successive approximations. That is, the zeroth order solution is chosen by neglecting the integral term in (6) to yield
,=o
540
(8)
(r!) 2
(16a)
the function Jo [ 2i(t#') ~ ] can be expressed as
(tn)r
Jo [ 2i(t#') ~1 = ~ 0(m °) (~, r/) = e "n
(15)
._~] 2 r
where we deFmed
t
f 0
(16b)
r=O
CRYOGEN ICS. SEPTEMBER 1980
and material, over a distance Ax about the wave front are related approximately by the following relation
Results a n d discussion
To illustrate the implications of the analysis of the cooling of a cable conductor by forced flow through the cable we consider the following example. A 216 m long cable-conductor for a large coil test facility is to be cooled from its initial temperature 300 K by force flow of supercritical helium through the interstices of the cable wires. Helium enters the cable at 200 K at a pressure of 15 atm. The cooling is to be continued until the helium temperature at the cable exit becomes 250 K. In this problem we are concerned with the determination of the cooling time required for the helium to attain the above specified temperature at the cable exit. For this type of configuration, the material and the flow conditions considered in this example we have taken A = 0.28 m 2 m "~, h = 221 w m "2K "~, Cm = 401 J kg "1, Cf = 5194 J kg"1, Gf = 0.75 x 10 -3 kg s"1, and M m = 3.6 kg m "1. Then the numerical values of the parameters m and n become m = 0.043 s"~ and n = 15.9 m "~. We solved this problem for the above values o f m and n, by calculating the fluid temperature as a function of time and position along the cable-conductor by using both the results from the present analysis and Hausen's equations. It is found that it would take about 22.3 h for the helium to attain a temperature of 250 K at the cable outlet. The top curves in Fig. 1 shows the results of these calculations for the fluid temperature Tr(x, t), plotted as a function of the distance along the cable-conductor with time intervals of 2.3 h. The lower curves in this figure shows the temperature difference, Tin(x, t) - Tf(x, t), between the material and fluid temperatures as a function of the axial position, with time intervals of 2.3 h. For the specific cooling problem considered here, the temperature profiles shown in this figure, for both the fluid and the material temperatures, can be considered to exhibit a wave-front type pattern in their propagation along, the cableconductor. We now utilize the behaviour of these temperature profiles to develop a very simple analysis for the determination of cooling time as described below. A scrutiny of the temperature profiles shown in Fig. 1 implies that the mean temperature gradients for the fluid
/ // / ////l
ATr(x, t)
~Tm(x, t) -
Ax
-
Ax
(18)
In view of this relation, (la) and (2a) can be combined to obtain the following expression for the velocity of propagation, v, of the wave front dx
v= dt
-
m
n
(18)
Then, for a cable-conductor of length L, the time required for the temperature wave front to reach the cable exit is given by L t = -- =L v
n -m
(19)
We can now apply this relation to determine the time required for the wave front to travel along the entire length of the cable for the example considered above. By setting L = 216 m, m = 0.043 s"~ and n = 15.9 m "1, we obtain t = 79870 s = 22.2 h. which is almost the same as that obtained from the solution of the differential equations. It should be recognized that, this simple analysis based on the concept of propagation of the temperature profile as a wave front is valid when the distance Ax over which the wave front is located is much smaller than the total length L of the cableconductor. This is certainly valid for the case in Fig. 1. We have also examined the temperature profiles as a function of position and time for the same 216 m long cable-conductor for various other combinations of m and n values, such that the ratio m/n varied from 10-3 ms "1 to 10 "~ ms "1. The resulting temperature distributions exhibited profiles similar to that shown in Fig. 1 and the ratio mn "~ appeared to be controlling the slope of the wave front. One of the authors, MNt3, gratefully acknowledges the hospitality extended to him by the Institut f~ir Verfahrens-und Kaltetechnik, ETH, Z~irich, during his stay with a sabbatical leave. This work was supported in part through a grant by the Swiss Institut for Nuclear Research (SIN).
References ..
1
rf (~,1)~
o" 150-
-
2
-
j IO0
ETm ( x , f ) - T
(gt)3
2
-I m =0.043 s n=15.9 rn "l
3
4 50 Fig. 1
IO0 150 Distonce from inlet, rn
2OO
The fluid temperature T f [ x , t) end the temperature difference
Tin(x, t) - Tf(x, t), w i t h t i m e intervals o f 2,23 h
CRYOGENICS.
S E P T E M B E R 1980
5 6
Hoenig, M.O., Montgomery, D.B. Dense supercritical-helium cooled superconductors for large high field stabilized magnets, IEEE Transactions on Magnetics MAG-II 2 (1975) Dresner, L., Lue, J.W. Design of force-cooled conductors for large fusion magnets, Proc Sixth International Conference on Magnet Technology, Bratislava, Czechoslovakia (1977) Hoenig, M.O., lwasa, Y., Montgomery, D.B., Bejan, A. Supercritical helium cooled, cabledsuperconducting hollow conductors for large high field magnets, Prec. Sixth International Cryogenic Engineering Conference, Grenoble, France (1976) Lue, J.W., Miller, J.R., Lottin, J.C. Pressure drop measurements on forced flow cable conductors, IEEE Transactions on Magnetics, MAG-15 1 (1979) 53-55 Ozisik, M.N., Basic Heat Transfer, McGraw-HillBook Company, New York, (1977) Hausen, H., W~rme~bertragungin Gegenstrom, Gleichstrom und Kreuzstrom, Springer-Verlag, Berlin (1950)
541
On the cooling of forced flow cable conductors for superconducting magnets M.N. E)zi§ik, Ch. Trepp and H.P. Baumann
Nomenclature A
cooling surface area for the material per unit length of the cable-conductor
~Cm
specific heat for the fluid and the material respectively mass flow rate of fluid through the cableconductor, (ie, kg/hr)
Gf h
heat transfer coefficient
L
length of the cable-conductor
m-
t
temperature of the material to be cooled inlet temperature of the coolant fluid initial temperature of the material to be cooled
Of(x, t ) -
Tf(x, t) - Tfo , dimensionless fluid Tmo- Tfo temperature
mass of the material to be cooled per unit length of the cable hA GfCf '
temperature of the coolant fluid
distance along the cable-conductor
hA time 1 MmCrn '
Mm
n
Tf(x, t) Tin(x, t) Tfo Tmo
Om(X, t) =
Tm(x, t) - Teo ,dimensionless material Tmo- Tfo temperature
1
,,+1~
'en~'m'l time
The cable conductors developed for use in large superconducting magnets 1 are cooled by the forced flow of supercritical helium through the narrow, uneven passages inside the cable. Some studies have been reported 2-4 on the determination of the friction factor associated with flow through such interstices, because forced flow through narrow passages hundreds of meters long causes high frictional losses, hence high pumping powers are required. The determination of the time required for the cooling of cable conductors by forced flow, from a given initial temperature to a specifiod temperature, is of interest for operation purposes. This work is concerned with the development of an analytical method of solution for the cooling of long cable conductors by forced flow through the cable. The authors are at the Institut for Verfahrens-und Kaltetechnik, ETH-Zentrum, 8092-Zurich, Switzerland. The Permanent address of MNO is the Mechanical and Aerospace Engineering Department, North Carolina State University, Raleigh, North Carolina 27650, USA. Paper received 13 March 1980.
~=nx
dimensionless axial distance
rl= mt
dimensionless time
Analysis We consider forced flow through the interstices of a long cable conductor which is initially at a uniform temperature Tmo- The coolant fluid enters the cable at the origin of the axial coordinate x = 0, at time t = 0 with a constant inlet temperature Tfo and a constant mass flow rate of Gf kg h "1. The heat losses from the outer walls of the cable to the surrounding environment is considered negligible in comparison with the heat transferred to the coolant. The duration of cooling is much longer than the transient time for the coolant fluid entering the cable to travel from the inlet to the exit. The Biot number associated with convective heat transfer between the cable wires and the coolant fluid is less than about 0.1, so that the lumped system analysis s can be applied to formulate the transient heat transfer problem.
Let, Tf(x, t) and Tin(x, t) be the temperature distributions in the coolant fluid and the cable material respectively along the cable. The differential equations governing these tempera-
0011-2275/80/090539-03 ~02.00 © 1980 IPC Business Press CRYOGENICS. SEPTEMBER 1980
539
tures are determined by writing energy balance equations over a differential length along the cable for the material inside the cable (ie, wires) and the coolant flow. The resulting equations are summarized below.
The zeroth order solution is introduced into (6) for 0m(~', r/') under the integral signe to obtain the first order solution as
OOm) (~, r/) = e -n [1 + 7/(1 - e'~)]
The material inside the cable:
aOm(x, t) at + mOrn(X, t) = mOf(x, t), for t > 0
(la)
Ore(x, t) = 1
(lb)
for t = 0
(9)
This first order solution is used in (6) to obtain the second order solution as 0~) (~,r/) =e-n I 1 + ~ 1 -e'G) + lr/2 [1 -e-~(1 +~)]1
The coolant fluid: (10)
aOf(x, t)
- at
+ nOf(x, t) = nOm(x, t), for t > 0
Of(X, t) = 0
(2a) (2b)
Proceeding in this manner to obtain higher order solutions, it can be shown that the exact solution for 0m(~, 77) can be expressed as an infinite series in the form
where various quantities are defined as hA hA m - Mm----~m,n = ~
Ore(x, t) =
(3a)
Tin(x, t) - Tfo Tm° _ Tf°
(3b)
Tf(x, t) - Tfo Trno- Tfo
Of(x, t) -
Om(~'n)=l-e-(l~"O ~ r=l
In the following analysis we develop an explicit exact solution to these equations in the form of the infinite series described. Equations (1) and (2) are formally integrated to yield respectively t
/=0
f
Of(x, t ') e rat' dt ,
(4)
1+
= iT
=
T
(12)
T h e t e m p e r a t u r e difference, 0 m (~, 7"/) - 0 f ( t , 7"/), b e t w e e n t h e material and the fluid temperatures is obtained by subtracting (11) and (12).
Om(~, r/) - Of(t, rl) = e-(~' rO ~-~ (~r/)r r=o (r!)2
(13)
For comparison purposes, we also present below the integral form of the solutions for Om(t, 77) and Of(G, r/) based on the analysis by Hausen6.
0
x
f
(11)
(3c)
The above system of equations (1) and (2) is similar to that considered in reference 6, page 332, for the heat transfer problem associated with heat exchangers and their solution has been developed in the form of integrals involving Bessel functions of imaginary argument.
Of(x, t) = ne -nx
~/
~
Similarly, it can be shown that the solution for the fluid temperature 0f(t, r/) is given by
Of(t,r/) = l - e -(~+n)
Orn(X,t) = e-mr+me-mr
r/' 7.
Om(x', t) e nx' dx'
(5)
Om(t,r/)= 1 -
f
e"(~+n) Jo [2i(tg') y'] dr/'
(14)
,i
0
0
Equation (5) is introduced into (4) 0f(t, 77) = Om(~, r/)
=
e-n + e-(n+O n f
f
0
0
Om(t',r/')e(n'+~')d~'dr/'
(6)
= nx,
71 = mt
e-(~'+n) Jo [ 2i(t #')~ ] d~'
where Jo (z) is the Bessel function of order zero and i = x/-1. By noting that the series form of Jo(z) function is given by
Jo(z) = 2_., (-1) r (7)
Equation (6) is now solved by the method of successive approximations. That is, the zeroth order solution is chosen by neglecting the integral term in (6) to yield
,=o
540
(8)
(r!) 2
(16a)
the function Jo [ 2i(t#') ~ ] can be expressed as
(tn)r
Jo [ 2i(t#') ~1 = ~ 0(m °) (~, r/) = e "n
(15)
._~] 2 r
where we deFmed
t
f 0
(16b)
r=O
CRYOGEN ICS. SEPTEMBER 1980
and material, over a distance Ax about the wave front are related approximately by the following relation
Results a n d discussion
To illustrate the implications of the analysis of the cooling of a cable conductor by forced flow through the cable we consider the following example. A 216 m long cable-conductor for a large coil test facility is to be cooled from its initial temperature 300 K by force flow of supercritical helium through the interstices of the cable wires. Helium enters the cable at 200 K at a pressure of 15 atm. The cooling is to be continued until the helium temperature at the cable exit becomes 250 K. In this problem we are concerned with the determination of the cooling time required for the helium to attain the above specified temperature at the cable exit. For this type of configuration, the material and the flow conditions considered in this example we have taken A = 0.28 m 2 m "~, h = 221 w m "2K "~, Cm = 401 J kg "1, Cf = 5194 J kg"1, Gf = 0.75 x 10 -3 kg s"1, and M m = 3.6 kg m "1. Then the numerical values of the parameters m and n become m = 0.043 s"~ and n = 15.9 m "~. We solved this problem for the above values o f m and n, by calculating the fluid temperature as a function of time and position along the cable-conductor by using both the results from the present analysis and Hausen's equations. It is found that it would take about 22.3 h for the helium to attain a temperature of 250 K at the cable outlet. The top curves in Fig. 1 shows the results of these calculations for the fluid temperature Tr(x, t), plotted as a function of the distance along the cable-conductor with time intervals of 2.3 h. The lower curves in this figure shows the temperature difference, Tin(x, t) - Tf(x, t), between the material and fluid temperatures as a function of the axial position, with time intervals of 2.3 h. For the specific cooling problem considered here, the temperature profiles shown in this figure, for both the fluid and the material temperatures, can be considered to exhibit a wave-front type pattern in their propagation along, the cableconductor. We now utilize the behaviour of these temperature profiles to develop a very simple analysis for the determination of cooling time as described below. A scrutiny of the temperature profiles shown in Fig. 1 implies that the mean temperature gradients for the fluid
/ // / ////l
ATr(x, t)
~Tm(x, t) -
Ax
-
Ax
(18)
In view of this relation, (la) and (2a) can be combined to obtain the following expression for the velocity of propagation, v, of the wave front dx
v= dt
-
m
n
(18)
Then, for a cable-conductor of length L, the time required for the temperature wave front to reach the cable exit is given by L t = -- =L v
n -m
(19)
We can now apply this relation to determine the time required for the wave front to travel along the entire length of the cable for the example considered above. By setting L = 216 m, m = 0.043 s"~ and n = 15.9 m "1, we obtain t = 79870 s = 22.2 h. which is almost the same as that obtained from the solution of the differential equations. It should be recognized that, this simple analysis based on the concept of propagation of the temperature profile as a wave front is valid when the distance Ax over which the wave front is located is much smaller than the total length L of the cableconductor. This is certainly valid for the case in Fig. 1. We have also examined the temperature profiles as a function of position and time for the same 216 m long cable-conductor for various other combinations of m and n values, such that the ratio m/n varied from 10-3 ms "1 to 10 "~ ms "1. The resulting temperature distributions exhibited profiles similar to that shown in Fig. 1 and the ratio mn "~ appeared to be controlling the slope of the wave front. One of the authors, MNt3, gratefully acknowledges the hospitality extended to him by the Institut f~ir Verfahrens-und Kaltetechnik, ETH, Z~irich, during his stay with a sabbatical leave. This work was supported in part through a grant by the Swiss Institut for Nuclear Research (SIN).
References ..
1
rf (~,1)~
o" 150-
-
2
-
j IO0
ETm ( x , f ) - T
(gt)3
2
-I m =0.043 s n=15.9 rn "l
3
4 50 Fig. 1
IO0 150 Distonce from inlet, rn
2OO
The fluid temperature T f [ x , t) end the temperature difference
Tin(x, t) - Tf(x, t), w i t h t i m e intervals o f 2,23 h
CRYOGENICS.
S E P T E M B E R 1980
5 6
Hoenig, M.O., Montgomery, D.B. Dense supercritical-helium cooled superconductors for large high field stabilized magnets, IEEE Transactions on Magnetics MAG-II 2 (1975) Dresner, L., Lue, J.W. Design of force-cooled conductors for large fusion magnets, Proc Sixth International Conference on Magnet Technology, Bratislava, Czechoslovakia (1977) Hoenig, M.O., lwasa, Y., Montgomery, D.B., Bejan, A. Supercritical helium cooled, cabledsuperconducting hollow conductors for large high field magnets, Prec. Sixth International Cryogenic Engineering Conference, Grenoble, France (1976) Lue, J.W., Miller, J.R., Lottin, J.C. Pressure drop measurements on forced flow cable conductors, IEEE Transactions on Magnetics, MAG-15 1 (1979) 53-55 Ozisik, M.N., Basic Heat Transfer, McGraw-HillBook Company, New York, (1977) Hausen, H., W~rme~bertragungin Gegenstrom, Gleichstrom und Kreuzstrom, Springer-Verlag, Berlin (1950)
541