MATEFU Summer School on Superconductors for Fusion
CICC Thermo-Hydraulics
MATEFU Summer School on Superconductors for FusionJune 17th-22nd, 2007, Rigi-Kaltbad, Switzerland
Luca Bottura ‘CICC thermo-hydraulics’ slide no 2
MATEFU Summer School on Superconductors for Fusion
Plan of the lecture
Forced flow equations Scaling and optimal cooling conditions
Pressure drop A porous media analogy (part I)
Heat transfer A porous media analogy (part II)
The effect of a cooling channel Flow instabilities
Luca Bottura ‘CICC thermo-hydraulics’ slide no 3
MATEFU Summer School on Superconductors for Fusion
The cooling circuit
pm.
wpump
qcoil.
qHX
.
€
˙ q HX ≥ ˙ q coil + w pump
Luca Bottura ‘CICC thermo-hydraulics’ slide no 4
MATEFU Summer School on Superconductors for Fusion
V. Arp, Adv. Cryo. Eng., 17, 342-351, 1972.Single-phase, force-flow in pipes cooling rationale
Why forced flow ? Heat transfer rates comparable
to pool boiling can be obtained with reasonable flow rates (Re≈105)
No heat transfer crisis (nucleate to film), but rather a smooth transition
Flow instabilities due to vapor lock or choking flow can be avoided, or at least minimized
The operating temperature can be optimized in a wider range than a pool of boiling helium
Minimum helium inventory (and thus associated cost)
The heat removal rate can be directly controlled, acting on the massflow
Luca Bottura ‘CICC thermo-hydraulics’ slide no 5
MATEFU Summer School on Superconductors for Fusion
Single-phase, force-flow in pipes governing equations
€
∂ρ∂t
+∂ρv
∂x= 0
€
∂ρv
∂t+
∂ρv 2
∂x+
∂p
∂x= −
2 f
Dh
ρv v
€
ρc v
∂T
∂t+ vρc v
∂T
∂x+ φρc vT
∂v
∂x=
2 fρ v v 2
Dh
+wh
ATwall − T( )
mass
momentum
energy
frictional pressure drop
convection heat transfer
inertia
pressure work and Joule-ThomsonHelium is a compressible fluid, with relatively low densityCooling is close to the critical (or pseudo-critical) line, where properties change considerably (up to orders of magnitude)
Luca Bottura ‘CICC thermo-hydraulics’ slide no 6
MATEFU Summer School on Superconductors for Fusion
Helium properties
The variations are large in the range of interest (T ≈ 4… 6K and p ≈ 3… 10 bar )They can lead to non-linear responses and instabilities
Luca Bottura ‘CICC thermo-hydraulics’ slide no 7
MATEFU Summer School on Superconductors for Fusion
Single-phase, force-flow in pipes steady state
mass
momentum
energy
A = 1 cm2
Dh = 1 cmdm/dt =10 g/s
T = 4.5 Kp = 5 bar
ρ = 135 Kg/m3
cv = 2500 J/Kg Kf = 0.03
= 1dρ/dx = 1 Kg/m3 / m
dT/dx = 0.1 K/m
€
˙ m = const
€
dp
dx= −
2 f
Dh
˙ m ˙ m
ρA2+
˙ m 2
ρ 2A2
dρ
dx
€
˙ m c v
dT
dx= ˙ ′ q +
2 f ˙ m 2 v
ρADh
−˙ m
ρφc vT
dρ
dx
≈ 400 ≈ 0.5Pa/m
≈ 0.03≈ 2.5 ≈ 0.5W/m
Luca Bottura ‘CICC thermo-hydraulics’ slide no 8
MATEFU Summer School on Superconductors for Fusion
Single-phase, force-flow in pipes a practical approximation
mass
€
˙ m = const
momentum
€
dp
dx≈ −
2 f
Dh
˙ m 2
ρA2
energy
€
˙ m
d h +v 2
2
⎛
⎝ ⎜
⎞
⎠ ⎟
dx= ˙ ′ q
€
h ≈1
˙ m ˙ ′ q dx
0
L
∫ =˙ q
˙ m
€
p ≈ −2 ˙ m 2
Dh A 2
f
ρdx
o
L
∫ ≈ −2 ˙ m 2
Dh A2
f
ρL
Luca Bottura ‘CICC thermo-hydraulics’ slide no 9
MATEFU Summer School on Superconductors for Fusion
Main scaling rules
€
Dh =4 A
w p
Assume:
€
h ≈ c pΔT
Then:Minimize
mass-flow
Maximize the flow cross
section
Operate close to
the critical
line
Operate at high
pressure and low
temperatureMinimize mass-flow and maximize flow cross-section
€
p∝f
ρ
˙ m 2
A3
Pressure drop
Hydraulic diameter Enthalpy
€
T ∝˙ q
c p ˙ m
Temperature increase
€
w pump ≥ Δp˙ m
ρ∝
f
ρ 2
˙ m 3
A3
Pump work
Luca Bottura ‘CICC thermo-hydraulics’ slide no 10
MATEFU Summer School on Superconductors for Fusion
Scaling of pressure drop
€
p∝f
ρ
˙ m 2
A3
L = 100 mA = 0.5…2 cm2
Dh = 0.5…2 mm pout = 3 barTin = 4.5 K
1/A3
CICC friction factorHelium properties
m2.
Luca Bottura ‘CICC thermo-hydraulics’ slide no 11
MATEFU Summer School on Superconductors for Fusion
Scaling of pump workL = 100 m
A = 0.5…2 cm2
Dh = 0.5…2 mm pout = 3 barTin = 4.5 K€
w pump ≥ Δp˙ m
ρ∝
f
ρ 2
˙ m 3
A3
1/A3
CICC friction factorHelium properties
m3.
Luca Bottura ‘CICC thermo-hydraulics’ slide no 12
MATEFU Summer School on Superconductors for Fusion
Temperature increase: strange…
L = 100 mA = 0.5…2 cm2
Dh = 0.5…2 mm pout = 3 barTin = 4.5 K
1/A3
€
T ∝˙ q
c p ˙ m
CICC friction factorHelium properties
q = 10 W
1/m.
m2.
q = 0 WA = 1 cm2
Dh = 1 mm
q = 0 W
q = 10 W
Luca Bottura ‘CICC thermo-hydraulics’ slide no 13
MATEFU Summer School on Superconductors for Fusion
€
T ∝˙ q
c p ˙ m
L = 100 mA = 0.5 cm2
Dh = 0.5 mm pout = 3 barTin = 4.5…6.5 K
CICC friction factorHelium properties
q = 10 W
Temperature increase: surprise !?!
A temperature variation appears in steady state, depending on the helium inlet temperature and pressure (initial state), and on the pressure drop (thermodynamic process) !
This effect did not appear in the scalings
Luca Bottura ‘CICC thermo-hydraulics’ slide no 14
MATEFU Summer School on Superconductors for Fusion
Joule-Thomson expansion effect
S. Van Sciver, Helium Cryogenics, Plenum Press, 1986.
Helium pipeD = 4.8 mmdm/dt = 0.98 g/sq’ = 0.074 W/m
Helium pipeD = 4.8 mmdm/dt = 3.0 g/sq’ = 0.062 W/m
pin = 4.2 barp = 3.2 barTin = 9 K
pin = 10 barp = 9 barTin = 9.5 K
Luca Bottura ‘CICC thermo-hydraulics’ slide no 15
MATEFU Summer School on Superconductors for Fusion
Joule-Thomson effect
D.S. Betts, Cryogenics, 16, 3-16, 1976.S. Van Sciver, Helium Cryogenics, Plenum Press, 1986.
Helium inversion curve
coolingheating
h=const
The J-T effect is significant at large p, and in the proximity of the pseudo-critical line
3 bar
20 bar
Can we exploit it ? WARNING: max T in the coil !
6 K
4 K
Luca Bottura ‘CICC thermo-hydraulics’ slide no 16
MATEFU Summer School on Superconductors for Fusion
Optimal cooling conditions
Optimal cooling is reached extracting the maximum amount of heat under a given temperature headroom and with the minimum cryogenic load
Best (coil) cooling conditions are obtained at the lowest practical suction pressure (high cp) and at the lowest required dm/dt (low wp)
pout = 3 bar
Tin = 4.5 Kpin = 12 bar
Tin = 4.5 Kpin = 6 bar
h ≈ 20 J/g
Tmax = 6 K
Tmax = 6 K
h ≈ 7 J/g
H. Katheder, Cryogenics, 34, 595-598, 1994.
€
˙ q = Δh 2Δpρ
f
Dh A2
L
Luca Bottura ‘CICC thermo-hydraulics’ slide no 17
MATEFU Summer School on Superconductors for Fusion
Residence time
€
tresidence =L
v= L
Aρ
˙ m
L = 100 mA = 1 cm2
Dh = 1 mm pout = 3 bar
Tin = 4.5 Kpin = 4 bar
Smooth tube friction factorHelium properties
q = 10 W
time
pipe exit m ≈ 10 g/sv ≈ 0.8 m/stresidence ≈ 120 s
.
t=120 s
Luca Bottura ‘CICC thermo-hydraulics’ slide no 18
MATEFU Summer School on Superconductors for Fusion
Pressure drop - early findings
“triplexed”
“fluted”
M. Hoenig, Pressure Drop Characteristics for Cabled Conductors, MIT-PSFC Memorandum, April 28, 1976.
CICC’s have significantly larger pressure drop (3 times) than pipes with the same cross section and hydraulic diameter
€
p = − fUS
2ρv v
Dh
L
Luca Bottura ‘CICC thermo-hydraulics’ slide no 19
MATEFU Summer School on Superconductors for Fusion
Katheder’s correlation
H. Katheder, Cryogenics, 34, 595-598, 1994.
Typical error of fit data in the range of 30 to 70 %
€
p = − fEU
ρv v
2Dh
L
€
fEU =1
v 0.72
19.5
Re 0.88+ 0.051
⎛
⎝ ⎜
⎞
⎠ ⎟
20 < Re < 105
€
fUS =fEU
4
Luca Bottura ‘CICC thermo-hydraulics’ slide no 20
MATEFU Summer School on Superconductors for Fusion
The CICC as a bundle of ducts
M.A. Daugherty, S.W. Van Sciver, Adv. Cryo. Eng.
€
p = − fUS
2ρv v
Dh
L€
f l =6.6
Re D
€
f t =0.079
ReD0.25
Star d
uct
p
The approximation of a CICC as a parallel of independent channels has some success in predicting the overall friction factor, but it is not practical for design purposes
Luca Bottura ‘CICC thermo-hydraulics’ slide no 21
MATEFU Summer School on Superconductors for Fusion
CICC’s friction factors survey
R. Zanino, L. Savoldi Richard, Cryogenics, 46, 541-555, 2006.
€
p = − fEU
ρv v
2Dh
L
Large database of CICC’s with different: Cross section Void fraction Cabling pattern
Shows that the data does not correlate well with a Reynolds number defined on the hydraulic diameter
A parameter is missing
in the analysis ?!?
Luca Bottura ‘CICC thermo-hydraulics’ slide no 22
MATEFU Summer School on Superconductors for Fusion
Are CICC’s porous media ?
NMR image of fluid density in a packed bed of 1 mm spheres in a round and a square pipe (Manz, Phys. Fluids, 1999)
Metal-foam co-sinthered filled tubes for high performance heat exchangers (Lu, IJHMT, 2006)
Computer generated random fiber web (Koponen, Phys. Rev. Lett., 1998)
CICC’s look a lot like porous mediaDo they also behave alike ?
Luca Bottura ‘CICC thermo-hydraulics’ slide no 23
MATEFU Summer School on Superconductors for Fusion
Basics on porous media - 1/3 Definitions:
Solid phase = strands in the cable Pores = interstices for the helium flow Porosity (relative amount of void) = void fraction
Specific surface S = wetted perimeter per unit strand area
Equivalent particle diameter
Hydraulic diameter
For round strands:
€
Dp =6
S
€
Dp =3
2 f dead
Ds
€
Dh =ϕ
1−ϕ( ) f dead
Ds =2
3
ϕ
1−ϕ( )Dp
Luca Bottura ‘CICC thermo-hydraulics’ slide no 24
MATEFU Summer School on Superconductors for Fusion
Basics on porous media - 2/3 Intrinsic fluid velocity V
Intersticial 3-D flow field Averaged on a length scale
smaller than the pore size, but larger than the molecular length scale
Average fluid velocity v Averaged over a volume Vf
larger than the pore size, including fluid only
1-D flow field if L >> Dh
Seepage velocity v Averaged over a volume V
larger than the pore size, including fluid and solid:
v = v
V
v
Luca Bottura ‘CICC thermo-hydraulics’ slide no 25
MATEFU Summer School on Superconductors for Fusion
Basics on porous media - 3/3
Conservation balances for a CICC
Conservation balances for a porous medium
Darcy Forcheimer drag forceWall friction
Heat conduction Effective conductivity includes composite heat conduction and thermal dispersion
Luca Bottura ‘CICC thermo-hydraulics’ slide no 26
MATEFU Summer School on Superconductors for Fusion
Permeability
Depends on: characteristic dimension
of solid phase Dp
geometry of solid phase (e.g. spheres, fibers, foams, …)
porosity
Typical range of values for a CICC with 1 mm strand and 40 % void:
K ≈ 1 x 10-9 … 4 x 10-9 m2
packed spheres
fiber beds
Luca Bottura ‘CICC thermo-hydraulics’ slide no 27
MATEFU Summer School on Superconductors for Fusion
Pressure drop Momentum balance:
Derive a friction factor
Can be fit to existing data and correlations by appropriate choice of K and cF
packed beds
K = 4 10-9 m2
cF = 0.03
€
∂p
∂x= −
μ
Kvϕ − cF
ρ
K1/ 2vϕ
2 = − f2ρv v
Dh
€
f =a
Re+ b
€
a =ϕDh
2
2K
€
b =ϕ 2DhcF
2K1/ 2
Luca Bottura ‘CICC thermo-hydraulics’ slide no 28
MATEFU Summer School on Superconductors for Fusion€
Nu =hDh
kF
= 0.0259 Re 0.8 Pr 0.4 Tw
Tb
⎛
⎝ ⎜
⎞
⎠ ⎟
−0.716
€
Nu =hDh
kF
= 0.023Re 0.8 Pr 0.4
Heat transfer coefficients in pipes
Estimates initially based on Dittus-Boelter correlation:
Modified by Giarratano, Arp and Smith to adapt the correlation to supercritical helium:
P.J. Giarratano, V. Arp, R.V. Smith, Cryogenics, 11, 385-393, 1971.
Luca Bottura ‘CICC thermo-hydraulics’ slide no 29
MATEFU Summer School on Superconductors for Fusion
Transient heat transfer in pipes
The heat transfer coefficient varies as t-1/2 in the first ms of a heat pulse
The amplitude and duration of the transient depends on the mass-flow
4000
2000
4 8 12 16 20 24
h (W/m2K)
00
t (ms)
h (W
/m2 K
) 200
t (ms)10.1 10 1000.01
1000
1000
010
0010
0
No flow
Re=1.2 105
P.J. Giarratano, Trans. ASME, 105, 350-357, 1983W.B. Bloem, Cryogenics, 26, 300-308, 1986
Luca Bottura ‘CICC thermo-hydraulics’ slide no 30
MATEFU Summer School on Superconductors for Fusion
Transient heat transfer in pipes
H. Kawamura, Heat Mass Transfer, 20, 443-450, 1977
The thermal boundary layer (of thickness BL) needs a time BL
2k/ρcp to fill-up During this time the excess
heat flux into the boundary layer appears as a transient heat transfer coefficient:
The transition from transient to steady-state takes place when the thermal boundary layer is fully developed:
€
hBLΔT =kρc p
πt
€
hBLΔq =πkρc p
4 t
€
h = max hST ,hBL{ }
€
Z =hST
2 t
kρc p
∝ δBL2 t
qw=const Tw=const
Luca Bottura ‘CICC thermo-hydraulics’ slide no 31
MATEFU Summer School on Superconductors for Fusion
Steady-state heat transfer in CICC’s
Y. Wachi, et al., IEEE Trans. Appl. Supercon., 5(2), 568-571, 1995.
Dittus-Boelter-Giarratano
Lower laminar limit (Nu=4.36)
Direct measurements are very sparse
The scattering of data is large
Nonetheless, it seems that direct measurements as well as indirect determinations (e.g. based on stability) point to the fact that heat transfer in CICC’s is much enhanced with respect to the value expected from pipe correlations
Luca Bottura ‘CICC thermo-hydraulics’ slide no 32
MATEFU Summer School on Superconductors for Fusion
Thermal dispersion - 1/4
The meandrous flow induces mixing of fluid at the length scale of the pores, much larger than the molecular one
Exchange of mass m under a temperature difference T between two fluid elements originally at a distance results in a heat exchange
q = m cp T
and equivalent conductivity
k ≈ q V / T 2 = m cp V / 2
mixingm
T1
T2
T
Luca Bottura ‘CICC thermo-hydraulics’ slide no 33
MATEFU Summer School on Superconductors for Fusion
Thermal dispersion - 2/4
The mechanism of heat transfer caused by mixing at the pore scale is called thermal dispersion
The resulting effective conductivity is anisotropic (different transport in longitudinal and transverse direction) and can be much larger than the fluid (molecular) conductivity
The enhancement is proportional to the Peclet number of the flow
transversek e
ff /
k flu
id
Pe
longitudinal
k eff /
k flu
id
Pe
Metzger, et al.IJHMT, 2004
Metzger, et al.IJHMT, 2004
Luca Bottura ‘CICC thermo-hydraulics’ slide no 34
MATEFU Summer School on Superconductors for Fusion
Thermal dispersion - 3/4
Nield, Bejan (1992) Empirical fit to
measurements
Hsu, Cheng (1990) Averaging of mass and heat
exchange along the flow streamlines
Bo-Ming (2004) Fractal model of the
tortuous flow
… and others
Luca Bottura ‘CICC thermo-hydraulics’ slide no 35
MATEFU Summer School on Superconductors for Fusion
Thermal dispersion - 4/4
Ahe = 3.5 (cm2)
dm/dt = 2 … 8 (g/s) v = 1 … 10 (cm/s)Pe = 50 … 500 (-)
Thermal conductivity is enhanced by a factor 5 to 50
LCJ conductor
relevant
range
Luca Bottura ‘CICC thermo-hydraulics’ slide no 36
MATEFU Summer School on Superconductors for Fusion
Heat transfer coefficients Internal heat transfer hint, between the solid and fluid phases at the level of the pore
Wall heat transfer hwall, between the porous medium and the pipe wall/channel boundary
hint
hwall
hwall
Luca Bottura ‘CICC thermo-hydraulics’ slide no 37
MATEFU Summer School on Superconductors for Fusion
Dittus-Boelter
Porous media correlations
factor ≈
3…5
Internal heat transfer
Large uncertainty on relevant geometric parameters (geometry, porosity, form drag factor)
Always well aboveDittus-Boeltercorrelation !
relevant range
Luca Bottura ‘CICC thermo-hydraulics’ slide no 38
MATEFU Summer School on Superconductors for Fusion
Dittus-Boelter
relevant range
thermal dispersion
Wall heat transfer
Thermal dispersion effect at low Re:
Values consistently in excess ofDittus-Boelter
correlation
€
h porous
h free≈
kT
kF
Porous media correlations
Luca Bottura ‘CICC thermo-hydraulics’ slide no 39
MATEFU Summer School on Superconductors for Fusion
A parallel cooling channel
A cable with a central cooling channel has a significantly lower pressure drop per unit length Potential for larger mass-flow, lower temperature increase, lower residence time and lower pumping power
NET cable
CEA cable
M. Morpurgo, Particle Accelerators, Gordon and Breach, 1970.
R. Makeawa, et al., IEEE Trans. Appl. Supercon., 5(2), 741-744, 1995.
What has been will be again, what has been done will be done again; there is nothing new under the sun.
(Ecclesiastes 1:9)
Luca Bottura ‘CICC thermo-hydraulics’ slide no 40
MATEFU Summer School on Superconductors for Fusion
The drawbacks of a cooling channel
In the presence of a cooling channel the conductor can develop: Pressure gradients (usually negligible)
Flow gradients (relatively large) Temperature gradients (relatively small)pB,
TBpH, THvH
vB
Exchange of energy over a perimeter w
Luca Bottura ‘CICC thermo-hydraulics’ slide no 41
MATEFU Summer School on Superconductors for Fusion
Central cooling hole:temperature gradients
Transverse temperature gradients appear between the helium in the cable bundle and the helium flowing in the cooling hole (expected)
Transverse temperature gradients also appear across the cable bundle (not necessarily expected !)
x=70 mm
x=570 mm x=1050 mm
Courtesy of P. Bruzzone and C. Marinucci, EPFL/CRPP
heater
T-sensors
Luca Bottura ‘CICC thermo-hydraulics’ slide no 42
MATEFU Summer School on Superconductors for Fusion
AH = 20 mm2
AB = 80 mm2
v ≈ 0.6 m/s11.7 5.8
Central cooling hole:diffusion and residence time
A heated helium slug is smeared when travelling downstream (Airy diffusion), and it propagates at a speed given by the area-weighted average of the helium in the hole and in the cable bundle
€
v =v H AH + vB AB
AH + AB
vH = 1.9 m/s4.7 2.3
18.7 9.3vB = 0.5 m/s
Luca Bottura ‘CICC thermo-hydraulics’ slide no 43
MATEFU Summer School on Superconductors for Fusion
Central cooling hole:thermo-syphon
Gravity pressure drop ρ g ≈ 1500 Pa/m Frictional pressure drop 2 ρ f/D v2 ≈ 100
Pa/m A heat load in the cable bundle leads to
an increase of helium temperature w/r to the cooling hole and thus to a significant reduction of the helium density
e.g. T : 4.5 → 6.5 K ρ : 148 → 120 g/l
Tbundle > Thole ρbundle < ρhole The buoyancy lift on the helium in the
bundle increases. If ρbundle is sufficiently small the hydrostatic pressure in the bundle may become smaller than the pressure from the hole
He flows upward in the bundle.
Thermosyphon effect
Courtesy of R. Herzog, EPFL/CRPP
Luca Bottura ‘CICC thermo-hydraulics’ slide no 44
MATEFU Summer School on Superconductors for Fusion
Thermo-syphon experiment
Courtesy of R. Herzog, EPFL/CRPP
Luca Bottura ‘CICC thermo-hydraulics’ slide no 45
MATEFU Summer School on Superconductors for Fusion
Thermo-syphon low crisis
Flow crisis
Stable flow
Courtesy of R. Herzog, EPFL/CRPP
Luca Bottura ‘CICC thermo-hydraulics’ slide no 46
MATEFU Summer School on Superconductors for Fusion
Thermo-syphon analytical estimate
pH
TH
pB
TB
The heating of the cable bundle causes a temperature and density difference:
The flow balances locally under a small pressure difference:
The flow crisis is reached when the flow difference is comparable to the steady state flow:
€
T =˙ ′ q
wh∝
˙ ′ q
˙ m n
€
ρ =βT
€
p << pH ≈ pB
€
˙ ′ q critical ∝f
Dh
1
A2
1
βρ˙ m 2+n
n=0.8
Luca Bottura ‘CICC thermo-hydraulics’ slide no 47
MATEFU Summer School on Superconductors for Fusion
Summary… Cooling conditions can and must be optimized with
care (each W counts !). In general, it is best to use the lowest practical pressure lowest necessary pressure drop and massflow
Correlations in CICC’s (for f and h) are affected by large uncertainties design parametrically In any case, measure !
There’s more… cooling, pressure drop and heat transfer in two-phase, heat transfer in Helium II, the associated cryogenics, and more…
… and there’s work to do for you ! The porous media analogy is exciting: how far can one
get in predicting performance ?
Luca Bottura ‘CICC thermo-hydraulics’ slide no 48
MATEFU Summer School on Superconductors for Fusion
…and where to find out more
S. Van Sciver, Helium Cryogenics, Plenum Press, 1986.
B. Seeber ed., Handbook of Applied Superconductivity, IoP, 1998.