matefu summer school on superconductors for fusion cicc thermo-hydraulics [email protected]...

MATEFU Summer School on Superconductors for Fusion

CICC Thermo-Hydraulics

[email protected]

MATEFU Summer School on Superconductors for FusionJune 17th-22nd, 2007, Rigi-Kaltbad, Switzerland

Luca Bottura ‘CICC thermo-hydraulics’ slide no 2


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



The cooling circuit

pm.

wpump

qcoil.

qHX

.

€

˙ q HX ≥ ˙ q coil + w pump



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



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)



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



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



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



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



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.



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


m3.



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


q = 10 W

1/m.

m2.

q = 0 WA = 1 cm2

Dh = 1 mm

q = 0 W

q = 10 W



€

T ∝˙ q

c p ˙ m

L = 100 mA = 0.5 cm2

Dh = 0.5 mm pout = 3 barTin = 4.5…6.5 K


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



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



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



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


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



Residence time

€

tresidence =L

v= L

Aρ

˙ m

L = 100 mA = 1 cm2

Dh = 1 mm pout = 3 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



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



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



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



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 ?!?



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 ?



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



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



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



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



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


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.



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



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



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



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




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




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




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



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



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



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



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)



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



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



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



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



Thermo-syphon experiment




Thermo-syphon low crisis

Flow crisis

Stable flow




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



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 ?



…and where to find out more

S. Van Sciver, Helium Cryogenics, Plenum Press, 1986.

B. Seeber ed., Handbook of Applied Superconductivity, IoP, 1998.

matefu summer school on superconductors for fusion cicc thermo-hydraulics [email protected]...

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