Ch. 5 Cryogenic heat transfer
Conduction in solids
The principal new problems associated with conduction heat transfer at cryogenic T
are those of variable thermal properties and low-T insulation. Usual thermal
diffusion eq. can be rewritten to account for property variation. Numerical calculation
is simpler with the transformed diffusion eq.
Low temperature insulation
Superinsulation with highly polished metallic coatings as many as 150 layers.
Interfacial phenomena
Associated with both pressurization and stratification phenomena in cryogenic
vessels containing coexistent liq. and vap. phases.
Heat Conduction in Cryogenics
The strong dependence of thermal conductivity on temperature makes
it necessary to integrate the expression
T2
k (T )dT
T1
There are two integration approaches:
1) Integral approach
2) Polynomial approach
Let’s compare these methods with the assumption of a mean thermal c
onductivity
3) Km dT
Integral Approach
T2 X2
dX
Q= - k (T )dT / A( x)
X1
T1
T2 T1 X2
Q [ k (T )dT k (T )dT ] / dX / A( x)
0 0 X1
Q = -G(2 - 1)
X2
Where G = 1 / dX and 2 , 1 are integrals
A( x)
X1
For uniform cross-sectional area, G = A/L =0.0001m2 / 1 m
Assume T2 = 300 K and T1 = 10 K
Integral Approach (Cont.)
Q = G (2 - 1 )
G = A/L = 0.0001 / 1
For Stainless Steel:
2 = 3000 W/m
1 = 3.6 W/m
Q = (3000-3.6)*0.0001/1 = 0.3 W
Polynomial Approach
The thermal conductivity of stainless steel can be fitted by the following
polynomial equation
k(T) = 2.0E-7 T3 – 3. 0 E-4 T2 + 1.2 E-1 T – 0.388
T2 = 5.00E-8 T4 – 1.00 E-4 T3 + 0.060 T2 - 0.388T + C T2
k (T )dT
T1
]T1
Q = 2989 W/m *0.0001 / 1
Q = 0.299 W
Mean Thermal Conductivity Approach
16
B Conduction from 300K to 10K
14 B in a 1 m long stainless steel
rod with 0.0001 square meter
12
cross section.
10
B
8
Average conductivity of
stainless steel between 300K
6 and 10K is 7.8 W/mk
B
Stainless Steel Q = Km A dT / L
4
Q = 7.8 (0.0001) 290 / 1 W
2
B
= 0.228 W
B
0
0 50 100 150 200 250 300
Temperature (K)
Heat Conduction in Support Members
Conduction in support member
Q = -km As (Th – Tc) / L (1)
Since the support member must support the weight of the cryogenic
system and the imposed acceleration loads, the required cross-sectional
area for a tension member is given by
As = F fs / Sy (2)
where F is the design load on the member, fs is the safety factor desired,
and Sy is the yield strength of the support member. In substituting Eq (2)
into (1),
Q = - F fs (Th - Tc) / L (Sy / km) (3)
Heat Conduction in Support Members
Contact Conductance
Experimental data have shown that the thermal conductance of
metallic pressed contacts increases according to a simple power law
function of temperature and can be described by the relation
k (T) = Tn
Where n typically ranges from 0.75 to 2.5
Thermal conductance also increases asymptotically with increasing
applied force. Other contributing factors include surface finish, the
presence of oxide layer and Kaptiza resistance.
Summary of Th
ermal Contact
Literature
Thermal Contact Conductance as a fun
ction of Temperature & Applied Force
Kapitza Conductance
hk = q / Ts
Phonon Radiation Limit
Treats Phonon radiation as Photon radiation
qphonon=[(T+T)4 – T4]
qphonon=4T3 T [1 + (3/2) T/T + (T/T)2 + (T/T)3/4]
For small T
hPk = 4 T3
where = 4/10 h (kB/D)2 (3N/4 V)2/3
Acoustic Mismatch Theory
Analogous to classical acoustics or bound
ary scattering in optics.
It accounts for the finite reflection at the b
oundary between the two media.
hAk = (16 4 /5) R F L cL T3 / M D3
where F is a constant, R is the gas consta
nt, L and cL are density and sound speed
of the liquid
Comparison Between Theory and Experimental Data
Heat Transfer in Single Phase
Cryogenic Fluids
Heat Transfer in Fluid Flow
• With the exception of heat transfer in He II, conve
ctive heat transfer in cryogenic fluids is not much
different than that for room temperature fluids.
• However, care must be taken to ensure the correla
tion is valid for the flow conditions being examine
d and that the fluid properties are evaluated at the
pressure and temperature of the cryogenic fluid
Near Critical Fluid Properties
% Near critical region
Flows with large property variation.
For H2, as the fluid state approaches the critical state, the ratio of the heat transfer
coefficients increases sharply.
Forced-convection processes
In general, usual other correlations
are o.k., but near the critical point, it is
difficult to use those correlations
because of great variations in cp, mu,
k, rho, beta.
% Transition flow
A flow region in which the characteristics of both
laminar and turbulent flow coexist.
There is also a tendency for instability in the flow
pattern.
Very little is known about this flow regime and
no really satisfactory method or correlation
exists for computing its heat transfer
coefficients.
A residual L/D_e influence is observed that is
greatest at the lower range of Re # and
gradually diminishes at higher Re #. ---> Use
McAdams empirical correlation data figure.
Reduced Re # case of heat transfer case.
Heat Transfer in Forced Convection
• Liquid Flow: heat transfer to a turbulent, fully deve
loped cryogenic liquid follows the Dittus-Boelter c
orrelation
Nu = 0.023 Ref4/5Prf2/5
• Gas Flow: heat transfer to a turbulent, fully develo
ped gas
Nu = 0.023 Reb4/5Prb2/5 (Tw/Tb)-0.57-(1.59/x/D)
Heat Transfer in Fluid Flow (Cont.)
Near Critical Point: Fluid properties can vary signifi
cantly near the critical point
For oxygen and carbon dioxide-
Nu = 0.023 Ref4/5Pr(min)f,b2/5
For hydrogen
Nu = 0.0208 Ref4/5Prf2/5[1+0.0146(w/ b)]
Heat Transfer in Natural Convection
Laminar Convection
Nu = Cl Ra1/4
Turbulent Convection
Nu = Ct Ra1/3
Natural convection processes
Nu = f (Ra_n)
Usual non-cryogenic heat transfer relations are satisfactory, even during the orbital flight
test, for LOX and LH2, a low-gravity (a/g as low as 8 x 10e-4) environment.
Pressurized-discharge processes for cryogens
(a) pressurization, including the calculation of the transient T, velocity, and concentration
profiles in the gas space and the flow rate and quantity of pressurant
(b) liquid stratification, including the calculation of transient T, velocity, and concentration
distribution in the liquid
(c) interfacial phenomena, including the study and prediction of mass and heat transfer
rates across gas-liquid and gas-solid interfaces
Should the vehicle design be altered by reversing the relative positions of the LOX and
fuel tanks, a higher pressure would be required in the LOX to supply sufficient suction
head of the turbopumps.
The importance of the
pressurizing system to
flight vehicle weight ; For a
Saturn V S1C stage LOX
tank pressure of about 22
psia at engine cutoff, the
mass of pressurant
remaining in the LOX tank
would be approximately
4500 lb.
Heat Transfer in Vent and Fill Lines
Acoustic Refrigerator
Tc Pump Heat Th
Thermoacoustic Oscillation
Tc Pump Heat
Th
Thermoacoustic Oscillation
Diaphragm
Small
Chamber
Small
Diameter
Tube
Liquid
Helium
Container
Liquid Helium
Thermoacoustic Oscillation (Cont.)
• Lord Rayleigh provided a qualitative explanation fo
r the heat-driven oscillation based on a critical val
ue known as the Rayleigh’s number a few 100 yea
rs ago.
• Nikolaus Rott derived the wave equation and ener
gy equation (stability criteria for helium gas) along
a temperature gradient in a channel (1960’s).
• Gu and Timmerhaus derived the stability criteria fo
r triple point hydrogen (1991).
• Yuan and Spradley determined the stability criteria
for neon gas (1992).
Thermoacoustic Oscillation (Cont.)
Three gas properties are important in determining th
e stability criterion of TAO
1) Heat capacity ratio = cp / cv
2) Prandtl Number Pr = cp / k = / (1.77 - 0.4
5)
3) Exponent of temperature power law for viscosity,
where = a T1 -
Thermoacoustic Oscillation Experiment
Thermoacoustic Oscillation Stability
Remedies for Thermoacoustic Oscillatio
n
1) TAO
Damper
3) Porous
Damper 2) Perforated
tube
Thermoacoustic Driven Orifice Pulse Tu
be Refrigerator
Effectiveness of Heat Exchangers
Recuperative HX:
= actual heat transfer / max. heat transfer
= Ch(Thi-Tho) / Cmin(Thi-Tci) = Cc(Tco-Tci) / Cmin(Thi-Tci)
or
= f (Ntu, Cmin, Cmax)
Regenerative HX:
= f (Ntu, Cf, Cr)
Ineffectiveness = 1 -
Effectiveness of Regenerators
Heat and Mass Transport in Regenerators
Heat and Mass Transport in Regenerators
Multi-Phase Heat Transfer
in Cryogenics
Multiphase heat transfer applications in
Cryogenics
• Commercial applications- vapor-compression cyc
les, heat pipes, etc.
• Industrial- LNG plants, separation processes
• Aerospace – thermal storage units, capillary pump
ed loops, liquid propellant systems, etc.
• Cooling of superconductors
Regimes of Boiling Heat Transfer
Film boiling can exist practically with
cryogenic fluids because of their low
saturation T. In fact, it is possible to have a
heat flux in film boiling greater than that of
(q/A)_max and still maintain sufficiently
low surface T to prevent melting.
In practical systems the total rate of heat
xfer from a surface in film boiling will
include an important component of
radiation because of the high surface T.
Multiphase processes : Boiling heat transfer
Pool boiling figures
Watch the critical
heat flux and
superheated T.
Cryogenic Thermal Storage Unit
(CRYOTSU)
Cryogenic Thermal Storage Unit (Cont.)
Capillary Pumped Loops
Three important generalizations from experience
(1) The interfacial T is essentially that of equil. (saturation ) conditions corresponding to
system pressure.
(2) During pressurized discharge of a liquid from a vessel both condensation and
evaporation of the cryogenic propellants at the interface are possible, but usually are not
significant factors.
(3) During self-pressurization of liq. containers, interfacial evaporation occurs and the
system pressure is governed by the vapor-pressure characteristics of the phases at the
interfacial T.
Mass transfer by condensation or
evaporation at a vapor-liquid
interface depends on the relative
rates of heat xfer from each
phase at the interface.
Interfacial evaporation may reasonably be expected in the pressurization of subcooled
LH2, whereas much larger T differences (T_vap - T_liq) are required to cause
evaporation at LN2, LOX, or water interfaces. In the latter systems condensation may
more often prevail. One of the problems associated with pressurization of cryogenic
vessels is the high rate of initial condensation of the pressurant on the internal surfaces,
including the liq-vap interface and the consequent loss in P.
The presence of an insulating material on these surfaces allows for a surface T
response time delay and a fairly rapid reevaporation of a condensed liquid layer.
(k_rho_Cp) is used to discriminate as to the suitability of insulants on whose
surface a condensed layer of liquid will have minimum residence time. The low
value of this property for styrofoam indicates the rapid surface T response that can
be expected for this material.
Stratification in cryogenic vessels
Thermal stratification of a
cryogenic liquid in a vessel
results from external heat
exchange and consequent
nonequilibrium phenomena
within the liquid. The
phenomenon of thermal
stratification is important to
propellant tank design and
operation, as it influences the
selection of venting devices,
insulation, pumps, and tank
structure, among other things.
Thermal stratification layer.
Schlieren photographs indicate
that side-wall heating produces
the greatest amount of thermal
stratification.
Radiation
The principal problem of radiation heat transfer at cryogenic T is the determination of
the radiation properties of surface. Gaseous radiation is less of a problem since most of
the substances that remain as gases at low T are not significant radiators nor absorbers.
New consideration into the treatment of radiation heat xfer in cryogenics; Effect of
condensed gases on the radiation properties of cold surfaces. Such condensate layers
build up complex systems known as cryodeposits on the cold substrate. Much of the
work to date has been for systems of H2O and CO2.
Temperature effect on emissivity ? Homework
Helium II
Two fluid model
Kapitza conductance.
Maximum heat flux --> figures
At heat flux rates in excess of (q/A)_max
a helium II system goes over into film
boiling controlled by ordinary fluid and
thermal phenomena. Under these
conditions the superfluidity effects are
destroyed at the heated surface and are
replaced with those circumstances that
govern the natural convection of a vapor
layer in film boiling. The liquid helium II is
lifted off the surface but becomes an
effective heat sink for the vapor. Since
the greater thermal resistance resides in
the vapor film, the unusually great heat
transportability of the superfluid is no
longer effective in promoting heat xfer.