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a NBS TECHNICAL NOTE 675

U. S. DEPARTMENT OF COMMERCE/National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

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THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

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- Located at Boulder, Colorado 80302.
Heat Transfer and Flow of Helium

in Channels-Practical Limits for

Applications in Superconductivity

M. C. Jones

W. W. Johnson

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

a

OV BTANTDAPPH

APR 2 1976 na i ace -#ef

,j c /oo

753^

/?76>

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

US NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued January 1976
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 675

Nat. Bur. Stand. (U.S.), Tech Note 675, 45 pages (Jan. 1976)

CODEN: NBTNAE

For saie by the Superintendent of Documents, U.S. Government Printing Office , Washington, D.C. 20402

(Order by SD Catalog No. C13.46:675) $1.30
CONTENTS

Page

Foreword v Introduction 2

Thermodynamic and Transport Properties of Helium 3

Heat Transport to Helium I Above the Critical Pressure 5

a. Pure forced convection 6

b. Mixed convection 16

i. Vertical channels 16

ii. Horizontal channels 19

Heat Transfer to Helium I Below the Critical Pressure 21

Heat Transfer to Helium II 28

Oscillations in Forced Flow Helium Cooling Systems 32

Acknowledgments 36

Nomenclature 36

References 38

LIST OF FIGURES

Fig. 1. Phase diagram of Helium-4 4

Fig. 2. Heat transfer coefficients for supercritical helium [16] in

a vertical tube 2.13 mm diameter, downward flow 9

Fig. 3. Heat transfer coefficient vs. heat flux at z/d = 45 for

fixed inlet conditions 10

Fig. 4. Ratio of experimental heat transfer coefficient to

theoretical by equation (1) 11

Fig. 5. Prandtl number, viscosity and specific volume vs. enthalpy

at 2.5 atm pressure 14

Fig. 6. Convective heat transfer regimes: 1, data of ref. [14];

2, data of ref. [15]; 3, data of ref. [16J;

4, data of ref. [20] 20

Fig. 7. Wall temperature profiles for sub-critical helium showing

transitions to film boiling; 1.1 atm pressure 23

Fig. 8. Wall temperature profiles for subcritical helium showing

transitions to film boiling; 2 atm pressure 24

Fig. 9. Correlation for transition to film boiling in forced

convection flow in channels 25

i i i

Heat Transfer and Flow of Helium in Channels - Practical Limits for Applications in Superconductivity*

M. C. Jones and W. W. Johnson

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302 U.S.A.

ABSTRACT

Heat transfer and fluid mechanics of helium flowing in channels are reviewed,

Emphasis is placed on observed or anticipated limits of operation which might be

expected to apply in applications of superconductivity. Topics included are:

the high-heat-flux degradation of heat transfer and possible effects of buoyancy

forces in supercritical helium; transition to film boiling in subcritical

helium; limiting heat currents in helium II; and the possibility of oscillations

in forced flow helium systems.

Key Words: Buoyancy; channels; film boiling; forced convection; helium I; helium II; mixed convection; nucleate boiling; subcritical; supercritical;

transition; turbulent flow.
Introduction

The development of large superconducting devices is intimately related

to the fluid mechanics and heat transfer characteristics of cryogenic helium.

In the earliest successfully developed magnets for bubble chambers and

accelerator beam transport and focussing, the main function of the helium

was to cool the conductor matrix down, to stabilize it against flux jumps,

and to provide a heat sink for the relatively low losses which occur in

charging. The success of this phase of development of superconducting technology is attested to by the existence of several such devices with 1000 hours or more of routine operation behind them [1,2]. The wide range of applications

under consideration for the future, however, demands much more of the helium

as a heat transfer medium, and will exercise the ingenuity of designers to

the full. The simple expedient of immersing a device in a bath of liquid

helium at a temperature close to 4 K will not suffice or may simply be

impractical

.

Our research philosophy at the National Bureau of Standards has been to

explore as far as possible all modes of application of helium as a heat

transfer medium in order to preserve as many options as possible for the

designer. As a consequence we are interested in all phases of helium

and we are particularly interested in exploring the possible boundaries

of operation imposed by the thermodynamic and transport properties of helium.

In this paper, after a brief discussion of the relevant properties

of helium, we consider some important characteristics of helium flowing in

channels, since we anticipate that this mode of cooling in some form or other will be preferred over natural convection in future large devices.

First we discuss heat transport to helium I above the critical pressure,

then heat transport to helium I below the critical pressure. We then

discuss some possibilities for cooling to lower temperatures by means of

helium II. In the final section we consider the problem of flow stability

and oscillations in channels cooled by forced flow of helium.
Heat transfer under steady state conditions is the traditional approach

to technology of cooling superconducting devices and is the subject of

most of the following material. But we should like to take this opportunity

to urge that future research concentrate more on transient conditions; it

appears that performance limits may often only occur during such unsteady

circumstances, as for example in electrical power network faults or in

necessary pulsed operation. No work to date has been done under flow conditions,

but measurements in static liquid helium have already shown that under pulsed

heating orders of magnitude increase in heat flux may be sustained for a given

wall temperature rise [3,4,5].

Thermodynamic and Transport Properties of Helium

With reference to the phase diagram, figure 1, the first point to make

is that, with critical temperature and pressure 5.20 K and 0.2275 MPa

(2.245 atm) respectively, we can hardly avoid the near-critical region in

cooling superconductors. At pressures greater than critical, the fluid

undergoes a smooth, albeit steep, transition from liquid-like density to

gas-like density as the temperature increases and crosses the so-called

transposed critical, or pseudocritical , line. This is the dashed line

in the figure and is the locus of maxima in the specific heat at constant

pressure. At sub-critical pressures, we have to contend with the vaporliquid phase boundary and the associated boiling and two-phase phenomena.

This proximity to the critical point, either at super-or sub-critical

pressures is no disadvantage from a heat transfer point of view, but the

processes involved in the transport of heat are complex and are perhaps

best described by reference to experimental results. As we shall see,

there are boundaries to the regimes of good heat transfer and these must

be established experimentally.

In the supercritical pressure region we have to admit to some un- certainties in transport properties which cloud the interpretation of

experimental results somewhat. First of all, for the viscosity [6] of

dense helium between 4 and 20 K the experimental uncertainty is rather

larger than for most fluids (possibly as much as 8%). Secondly, until

quite recently, no experimental results were known to us on the thermal

conductivity in this range. McCarty [6] has published interim estimates

made by Roder based on published data above 20 K and below 4 K [7,8],
GO

3=

Q-GOo

SS

55

8

TO

E

4-O

E

fO

sCDto

03

0>

BdlAI '3dnSS3Ud
and on the thermal conductivity of hydrogen. Recently, we have become aware of new measurements, yet to be published, by Kellner [9]. We have

the provisional reassuring news that the new measurements are reasonably

close to Roder's estimates. Nevertheless, when interpreting heat transfer

results it is important to quote the source of data used for properties

and to bear in mind these uncertainties. In all of our work on helium I discussed below, we have used the data on properties published by McCarty.

Below the lambda temperature, which decreases smoothly from 2.177 K at 0.0053 MPa (0.0497 atm) to 1.763 K at 3.013 MPa (29.74 atm), as

is well known, we encounter the helium II liquid phase (superfluid). The

implications of the unique superfluid properties are discussed further

below; but briefly, it is no longer possible to assign a unique value of

the transport properties to any particular thermodynamic state [10J. Here

again, these properties are at best \/ery favorable from a heat transfer

viewpoint, but there are boundaries to the existence of such favorable

properties and these must be established.

Heat Transport to Helium I Above the Critical Pressure

Heat transport to fluids near the critical point, and in particular

at pressures above critical, has been a subject of investigation for

many years. Indeed, a review paper was published in 1971 by Hall [11]

containing 57 references to previous work. However, the majority of

experimental papers referred to studies with water and carbon dioxide.

None of the papers quoted dealt with helium although a few dealt with

hydrogen and oxygen. In fact the first useful data on helium were not

available in the open literature until 1971.

The advantages of supercritical helium as a heat transfer medium are

that, with high enough flow rates, heat transfer coefficients comparable

to subcritical nucleate boiling can be realized (^1.0 W/cm *K) without

the attendant possibility of "burn out" which, as will be seen below, can lead to a large temperature rise sufficient to exceed the critical temperatures of all known superconductors. Furthermore, the fluid has almost
as high a heat capacity as subcritical, i.e., liquid helium, to cope

with transient heat release in the superconductor. However, one thing

that is clear from studies with other fluids is that a considerable range

of wall temperatures can sometimes be obtained under seemingly identical

conditions. Most disturbing of all, the phenomenon of temperature "spikes"

along a vertical heated channel is well known. In carbon dioxide and

water these spikes may be of the order of 100 K. This somewhat unpredictable

nature of heat transfer at supercritical pressures should make us cautious

and, one would hope, would lead to some yery careful experimental work

with helium. It is probably fair to say, however, that great strides have

been made towards understanding temperature spikes and so-called deterioration

in heat transfer during the last few years, but unanimity has not yet been

reached and some unexplained phenomena remain [12].

It is convenient to divide the rest of this discussion into two parts.

In part a. we discuss heat transfer which is dominated by pressure drop

forces, i.e., pure forced convection; this appears to include all of

the available experimental data. In part b., we discuss the possible

effects of buoyancy and acceleration, i.e., mixed convection.

a. Pure forced convection.

If the Reynolds number of the flow is high enough, flow should be

dominated by pressure drop forces and results should be independent of

channel orientation. Under these conditions a variety of behavior can be

expected. First, with relatively low heat flux, the wall and bulk fluid

densities are almost equal and the heat transfer coefficient is predictable

by the usual empirical relationships among Nusselt, Reynolds and Prandtl

numbers for turbulent flow, e.g.,

C C

Nu = C

] Re

2

Pr

3

(1)

where C, , C- and C

3 are constants. The heat transfer coefficient in this case is a constant independent of the heat flux for a given thermodynamic

state and we have a linear heat transfer process. The experimental results

of Johannes [13] appear to be of this kind.
In order for this to apply, the thermodynamic state of the helium

must be away from the region of sharp property variation, i.e., temperature

well above or below the transposed critical, or the heat flux must be small.

Whereas the results of Johannes all pertain to the situation where

both bulk fluid and wall temperatures remained below the transposed

critical (i.e., T

R

< T

w

< TTp) those of Giarratano et al . [14], covered

a wide range of thermodynamic states and all situations were encountered,

i.e., Tg < Tw

< TTC , T

B

< T

TC

< T,,, and T

TC

< Tp < T... In this case the

equation

Nu = 0.0259 Re°-

8

Prg4 (^)

' (2)

8 '* (i)

fitted the data with a standard deviation of 8.5%. For a given thermodynamic

state we now have a slightly non-linear heat transfer process, but we note

that for low heat flux and temperature rise equation (2) reduces to

equation (1). We note also that in the experiments of Giarratano et al. the

density ratio P

b /pw did not exceed about 2.5. Equation (1) might be

considered the limiting form of the heat transfer coefficient for near homogeneous conditions.

Equation (2), in this limit, predicts a heat transfer coefficient proportional

to the quantity k

* (C /y) ' for given channel diameter and flowrate.

This quantity is dominated by the behavior of C , and reaches a maximum at

the transposed critical temperature when plotted against temperature at

constant pressure. Thus, if we take a channel with constant flow rate and

inlet conditions below T

TC , the heat transfer coefficient at any point along

the channel should rise with heat input as the fluid temperature approaches

TTC . This can be clearly seen in the data of Johannes. For high heat flux,

however, the wall temperature may be expected at some point to exceed the

transposed critical and we could have a wide variation of properties across

the channel. In the experiments reported by Ogata and Sato [15] density

ratios of about 5 were achieved at 3 atm pressure and in this case heat

transfer coefficients a good deal less than those given by equation (2) were obtained at the highest heat fluxes.
In order to investigate this situation in more detail we performed

experiments on an improved apparatus at 2.5 atm [16]. This is only 10%

above the critical pressure and we were able to obtain density ratios

Pr/Pu u P to 10, when in all cases we had T„ < TTC

and both T

w

< TTf

and Tw > T

TC

. In figure 2 we show a typical example of heat transfer

coefficient profiles for a given flow rate and inlet temperature. The

data are for a 2.13 mm inside diameter tube.

The most significant feature is the variation in heat transfer

coefficient with change in heat flux. In all cases studied the pattern

was the same: the whole profile rises at first with increase in heat flux

and then falls below its original low heat flux level, dropping proportionately

more toward the outlet. If we take a fixed position in the tube and for a given flow rate and inlet temperature we plot the local heat transfer

coefficient against heat flux the result is typified by the two examples

given in figure 3. We have also plotted as dashed lines the limiting

values given by equation (1) using the constants of equation (2) for

comparison. Thus, a maximum in heat transfer coefficient is reached long

before any maximum that would be predicted by equation (1). Since T

R was always less than T

TC

of course the result calculated from equation (1) never in fact reaches a maximum for these data.

In figure 4 all of the results for the thermometers at respectively

5, 7 and 10 cm from inlet are plotted as the ratio of experimental heat

transfer coefficient to the limiting value by equation (1) versus the

ratio P B /pu. This plot clearly reveals the range of behavior we have been

discussing, from the homogeneous or limiting low heat flux behavior of

equation (1) to departures as much as 90% as the stream becomes less and less

homogeneous.

The central question, of course, is whether the departure from

homogeneous condition is predictable. The problem is that in the presence

of such inhomogeneities any application of the well-known semiempirical

theories of turbulent transport involve hypotheses; the prescriptions for
.26

.24

.22

.20

P = 2.5 atm

G = 6.8-7.6g/s-cm2 TBinlet

= 5038 - 5043 D

q= 0.067 W/cmz TB utlet = 5-197 K

egE

a

18

.16

.14

.10

.08

.06

q = 0.029 W/cm*

= 5.I29K

0.260 W/cmz

outlet =5329K

363 W/cm'

outlet = 5.360 K

2 4 6 8 10

LENGTH ALONG TEST SECTION, cm

12

Fig. 2. Heat transfer coefficients for supercritical helium [16] in a vertical tube 2.13 mm diameter, downward flow.
\ 1 1 1 1 \\\\ — \\\\ — \\\\ —

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LU \ Z " \ m __

<3- en /

V„^z/^,\ 1/ i \wV.

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LT> Q£ 1 1 "- i \q4 i

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<d"

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C\J 4- • o

Heat

trans

conditions

ro

en

LOo o mo

CVJ o

». UID/M '1N3I3IJJ303 H3JSNVyi 1V3H

10
2.0

1.0

0.8

0.6

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—

0.2

0.1

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A&

i r

AA ^

AAA

•yw

A A

^ A

-<£-

A

A&

A

A ^ ^

A

A *

A A

*

A A

2.5 atm.

8 10

P

B /P W

Fig. 4. Ratio of experimental heat transfer coefficient to theoretical

by equation (1)

11
calculating turbulent diffusivities all are taken from isothermal

turbulence studies. However, as we shall show below some understanding of

the gross effects may be gained by a proper appreciation of the role of

fluid properties.

To achieve this understanding, let us go back to the simplest of

theories: namely, the old mixing length theory of Prandtl applied to

tube flow. Neglecting the convective terms, the shear stress and heat flux

may be related to local velocity and temperature gradients in fully

developed steady turbulent flow by

T

= p(e

M

+ v) ly (3)

and

1 pC

p

(E

H +

») | * ^H + a » | " < 4 >

Equation (3) defines an eddy diffusivity for momentum transport which,

by the mixing length theory, is given by

e = I

2 I— I (5)

Under classical isothermal conditions, £, in the near proximity of

the wall, is a universal function of the distance y from the wall. Various

semiempirical expressions have been proposed for £ [17], but we shall simply

write I = i(y). Also, for fully developed turbulence, when only pressure

drop and friction forces are present, x and q are linear functions of radius

decreasing to zero at the channel center. In the region close to the wall, we may approximate t and q by their values t w and qw at the wall. In

any case, the variation of t and q across the channel will not be dependent

on fluid properties. Finally, we shall assume, as is customary and has

been born out by experiment, that e H is approximately equal to e».

12
Under the above assumptions and approximations equations (3) and (4)

may be integrated using (5) to give

u +

- h - f . 2 *?(6)

u o / 2 +2

2

(M_ + / ( 1L_) + 4£ (£_) ) \ ^ PW

(HW-H,

+

- ^- f L^==^ , (7)

w qW o / T +2 2

(Z

. i )E- + / (M_) +4£ (£_)

Pr \i S pw

1 12

where we have used the friction velocity u* = (tw/pw

) "to define the

dimensionless velocity u and the dimensionless distance from the wall,

y = yu*/v

w

. Note that for Pr z 1 we have similarity of enthalpy and velocity

profiles rather than temperature and velocity.

In equations (6) and (7) the standard dimensionless profiles are obtained under homogeneous conditions i.e., p = pw

, v = v

w

etc. In general,

however, these profiles could not be evaluated except by numerical means making use of the equation of state at each step and using an empirical

expression for £ . We would also need to determine the friction velocity

1 12

u* = (tw

/pm) by integral mass conservation:

R m = / 2-rrrpudr . (8) o

On the other hand, we can readily find an approximate solution enabling us to pick out the major property effect.

This can be done by noting first of all with reference to figure 5 that by a suitable choice of reference enthalpy the specific volume 1/p

is proportional to the enthalpy when T

w

> T

T

p, at least in some region close

to the wall. Then p/p

w

= vw

/v = Hw/H = 1 + (H

W-H)/(H-H ). We notice also

that, in this region, the viscosity is only a weak function of enthalpy. Thus,

since the integrands appearing in (6) and (7) are strongly weighted towards

small y , we may set v/v^ = 1 and p/pw

= 1. The integrals in (6) and (7) are

now the same as the constant property integrals to first order.

13
10 20 30 40

H (J/g)

Fig. 5. Prandtl number, viscosity and specific volume vs. enthalpy at

2.5 atm pressure.

14
The constant property equivalents of (6) and (7) for T„ « T

w

< TTC are

+

1 ° i + /m?7 (9)

(T,,-T)

w

i ° (X -d+ZvmT7 (10)

Kr

B

Using (8) it is readily shown that u£ a u*, again to first order, and hence

it follows that

«i 2 dy

+

h « W£_ _^ B

(11)

h

l pB CpB R+

/ 2dy+

where <Cp> = (HW-H)/(T

W

-T). Finally, it is well known that the ratio of

the integrals appearing in (11) is (P^d/P^u) ' for 1 < Pr < 20. Then the

ratio of experimental heat transfer coefficient to that calculated by (1)

should be

.0.5

h _ P

h, - P

W

B

<Cp>

Cp

B m">- ' (12)

Summarizing equation (12), there is first a correction Pu/pB due to a thickening of the viscous sublayer resulting from the decreased wall density.

Secondly, there is a correction <Cp>/CpB because of the similarity of enthalpy

(rather than temperature) profiles, and finally there is the well-known Prandtl

number effect.

15
Equation (12) gives an explanation for the departure of the experimental

heat transfer coefficient from that given by equation (1) for the situation

Tg < Tjp < T,|. In this case pw

is always less than pB and at high enough

wall temperatures < Cp > is less than Cp

B

- The Prandtl number at the wall

is again usually less than Pr

g

but the dependence in (12) is weak. On

the other hand, when T

w

is in the vicinity of TTC it is possible for

<Cp> to be greater than Cp

B and the ratio given by (12) to be greater

than 1.0 as is observed. But in fact (12) predicts too low a heat transfer

coefficient by as much as 50%. Shiralkar and Griffith [18] found similar

discrepancies even with accurate numerical evaluation of the integrals in

(6) (7) and (8) in their comparisons with data for water and carbon dioxide.

We should recall that we have assumed the mixing length to be of the same form as for the isothermal case in the absence of any real knowledge of

its behavior in the presence of strong property gradients. Considering all

the approximations involved it might stio ll be useful to determine empirically

the exponents a, b and c in a modified form of equation (12), i.e.

r$e«j&r

As a final point, Shiralkar and Griffith have observed that the

deteriorated heat transfer is strongly dependent on upstream conditions.

They feel that the thickened viscous sublayer is easily perturbed. Perhaps

quantitative prediction is too much to expect without much greater control of

experiment.

Based on our experimental results, a fairly safe conclusion for supercritical

helium seems to be that equation (1) should not be used if P b /p

w

is greater than 2.5,

b. Mixed convection

i. Vertical channels

In steady flow of a fluid in a vertical channel there is a balance

between the forces acting on the fluid. These are inertial, body forces,

pressure gradient forces and shear stresses. In a. above, only the latter

16
two were considered and it is under these conditions that a linear shear

stress distribution exists, falling from a wall value to zero at channel

center. When a third force is present the shear stress distribution can

be entirely different. Thus, when the gravitational force or buoyancy

force is considered, it is possible for this force acting on a thin warmer

layer adjacent to the inside wall to be balanced entirely by the shear

stress at the wall. This can only occur in upward flow with heating or downward flow with cooling. Then, there can be no frictional pressure

drop force acting. As a consequence, the core of the flow -- practically

at constant temperature -- cannot experience any shear stress, which is

then zero all the way to the channel center. In downward flow with heating

it is in principle possible for the buoyancy force on the warm layer to

be entirely balanced by the pressure drop force. This would result in zero shear stress at the wall, but not in the core.

The importance of shear stress in turbulent flow is that the working

of the shear stresses against the mean flow in the proximity of the wall

is what generates the turbulence. The turbulent intensity is a result of

a balance between generation, convection, diffusion and dissipation. Then

the reduction of the shear stress can cause the intensity to decay, with

a deterioration in heat transfer coefficient as a consequence.

It is now generally thought [11,19] that the wall temperature spikes

referred to above, when observed only in upward flow, are due to this

anomalous shear stress distribution caused by the buoyancy forces. A

similar phenomenon is observed in accelerated flows where relaminarization

can take place. Temperature spikes are not generally observed in downward

flow although a broader wall temperature maximum may occur in any orientation

due to the transverse property variation discussed in section a. above.

On the basis of these considerations criteria have been proposed to

signify conditions under which drastically reduced shear stress can be

expected. Thus, Hall [11] gives the criterion

17
Reg'

7 « 8.3 x 10

3

Gr (13)

where the Grashof number Gr is defined as Gr = (p B -pw )g(4b) /pBv B

, b being

the channel half width. Tanaka et al . [19] gave a similar criterion based on

a slightly different model

Re

f

1/8 « 1.55 x 10

3

Gr (14)

where Gr can be Gr = (p B -p

f )gd /p fvf

for the effect of buoyancy or

3 2

Gr = u B (du

B /dz)(pB /p^)d /vf

for the effect of acceleration. In Tanaka 's expressions the subscript f indicates that a property is to be evaluated

at Tx = (Tw

+ T

B )/2. These criteria effect a division of all heat transfer

experiments into two classes: on the one hand pure forced convection,

and on the other pure free convection with mixed convection in some ill

defined region of the line. In figure 6 we have drawn rough regions

encompassing different experimenters results for heat transfer to supercritical helium together with a line representing equation (13). We see that all belong to the forced convection class. The nearest to being mixed

convection are the experiments of Giarratano and Jones at 2.5 atm and it

is noteworthy that the shaded region closest to the line is where all

data from those experiments lie in which experimental heat transfer

coefficients are less than 70% of the values given by equation (1). But we have seen above in this case that another explanation exists for such

departures; therefore we hesitate to ascribe these departures to buoyancy

effects. The two effects will be difficult to separate without simultaneous

radial velocity profiles or variation of the channel orientation or flow

direction.

It is also interesting to note that, for a given channel at a fixed

operating pressure and gravitational acceleration, Grashof numbers obtainable

are practically limited by the density differences achievable in the

region of the transposed critical temperature. So we may ask the question:

under what conditions could we get into a mixed convection region? Assuming

a fixed upper limit to density differences, we can either reduce the Reynolds

18
number by reducing the flow rate, or we can increase the Grashof number

by increasing the acceleration (e.g., by rotation) or the channel dimension

at constant Reynolds number (corresponding reduction in flow rate). In view of practical applications in rotating machinery, or large magnets

with large diameter channels, we strongly urge that future experiments

cover these situations and that both upward and downward flow be investigated,

Ii. all cases, in order to observe temperature spikes a number of thermometers

shou^ be used to measure a complete axial temperature profile since a single isolated thermometer could miss such a phenomenon. Pressure drop

measurements should also accompany these heat transfer measurements in

order to correlate possible observations of temperature anomalies with

shear stress in the fluid. The usefulness of radial velocity profiles

has already been mentioned.

ii. Horizontal channels

Study of horizontal channels under mixed convection conditions does

not appear to be as well advanced as for vertical channels and no studies

have been reported with helium. While one expects no such dramatic effect

as temperature spikes, caution should nevertheless be exercised because a circumferential variation in wall temperature can result from a buoyancyinduced secondary flow.

A criterion similar to those presented for vertical channels has

been developed by Petukhov et al . [21] for a 1% effect of buoyancy on the heat transfer coefficient. With the Grashof number defined as

4 2

Gr = 3nqu9d / |< rv

r tne criterion is given as

Gr = 3.0 x 10" 5

Reg'

75

Pr0,5 [1+2.4 Re~ 1/8 (Pr2/3

-l )] (15)

and is plotted in figure 6 with Pr as a parameter. Interestingly, the

criterion is yery similar to those given for vertical tubes for strong

buoyancy effects, but note that, as defined, the Grashof number here may be

two orders of magnitude larger.

For typical proposed operating conditions of superconducting power

transmission lines operating points fall below the dashed lines representing

equation (15). Such conditions may be difficult to simulate in the

19
O:\J L_l

4-

<D . * S-

<3" 4-o

03 • 4-> 4- <o

a; T3

Sr> 4- <d- o • *» (0 1—1

+j \o

fO 1—

-ai

—i i . r— 4-

d)

S-

</l 4-

O) o E•^ fO

cr>+-> a «3

S- T3

$- #* a> co

4-

</) • •* Cl—1

ro Lf> S- r—

+Ji

—l

+-> • fO 4-

<D a> JC sqj M- > o

•r-

+-> (TJ (J 4-> ai (T3 > -o co r- O <NJ • <£>• C7

y39WHN S(nONA3«

20
laboratory, mainly on account of the large disparity between laboratory

scale measurements, where diameters are measured in millimeters, and full

scale equipment where the diameters may be a factor of 10 larger and

4

Grashof numbers 10 larger. Tests should perhaps be performed on large

scale prototypes.

In any case, experiments should be performed stressing the measurement

of circumferential temperature distributions in the free convection dominated

region.

Heat Transfer to Helium I Below the Critical Pressure

The highest heat transfer coefficients for helium flowing in channels

are obtained below the critical pressure [22]. This occurs when nucleate

boiling takes place at the wall, or in other words, we are in a wetted-wall

regime. Indeed, at 2 atm we have observed heat transfer coefficients as high as ^ 10 W/cm • K under forced flow conditions, although it must be

admitted that this is a very approximate figure since the temperature

rise was often of the same magnitude as the total estimated systematic

error (0.02 K) . One very significant fact which emerges from our observations and others is that, in the nucleate boiling regime, the heat

transfer coefficient, or the relation between the heat flux q and the

temperature rise, TW-T

B

, is almost independent of the mass flow rate and

vapor mass fraction of helium. It is essentially the same as for unconfined

surfaces, or "pool boiling". The relationship is well represented by the

well-known semiempirical equation of Kutateladze [23]

a V2 4

q Cp.p,

1/2

( (-2- ) = 3.25 x 10

4 [t—^ (t£- ) 3

h_

ki

v gp.

' L ^pw ki

v gp.

0.6

'V *L SKL

[g(f ) 2

(^-) 3/2 ]°' 125

[ p 1/9

]°- 7 (16) \ gpL (ag p

L

) 1/2

2 5 which gives q a AT ' for a given pressure, all properties being evaluated

on the saturation line. This relationship was derived by a dimensional

analysis of the governing equations and a fairly detailed empirical knowledge

of boiling phenomena.

21
Far more important from a practical standpoint, however, is the

transition from this very attractive heat transfer regime to the film

boiling or dry-wall regime; for here large temperature excursions take

place. In figure 7 we show temperature profiles along our 2 mm dia. vertical

test section with downward flow at 1.1 atm pressure. The temperature

excursions are obviously quite unacceptable for any superconductivity

application. In figure 8 the situation for 2 atm pressure, but with all

other conditions essentially unchanged, is seen to be quite different; for

now the excursion may be acceptable for some high field superconductors. It

is clear that a thorough knowledge of this transition and the conditions under

which it takes place are very important, and this was the thrust of our study reported earlier. A similar study was carried out by Ogata and

Sato [24] and we find good agreement with their results.

An interesting and useful point of agreement between these two studies

is that the transition at any given pressure is predictable within the

scatter of the data by a unique relationship between the critical heat

flux q and the local critical quality x . This seems to account implicitly

for the separate effects of mass flow rate and position along the channel

at which transition occurs. We found that our data on transition

supplemented by the low mass flux (natural convection) data of Johannes

and Mollard [25] could be represented by the simple relationship

Ku = 0.031 + 0.078 (l-x

c

) 3 ' 92

(17)

where Ku is the dimension! ess critical heat flux parameter

q /(Ap./ [ag(p.-p

v

)] ' ) of Kutateladze which has been so successful

in correlating transition data for pool boiling. This correlation is shown

in figure 9.

We were particularly pleased that this relationship predicted some

data of Keilin et al . [26] to within 5%, because this data was for a transition under somewhat different conditions than our own. These authors

observed the transition at 278 diameters downstream and local quality of

about 0.4 compared to our conditions of less than 48 diameters and quality

less than 0.2, for similar pressure and mass flux.

22
32

28

24

20

j

8

PRESSURE = I.I atm

MASS VELOCITY = 13 g/s-cnV

QUALITY (a) INLET =-0.07

' TBULK

8 16 24

L/D

32 40 48

Fig. 7. Wall temperature profiles for sub-critical helium showing

transitions to film boiling; 1.1 atm pressure.

23
7.5 PRESSURE = 2atm

MASS VELOCITY = 13 g/s-cm2

QUALITY (§) INLET = -0.05 to -0.03

Fig. 8. Wall temperature profiles for subcritical helium showing

transitions to film boiling; 2 atm pressure.

24
1 1 1 \ ° 1 1 1 1 1 1

^5

CU

CO

cu

as

CO

CVI

— CD Ck8 CO

"73 II

—

ca

cu O0 cu

OS EE ^^^ i—.

oo

SZ OS cu

caA • HBB

cci K^S ^TM COO <

cu>

cu o — oV

—

— o \ (D

—

— o o\QccDoi|

O

— > >>

\p° 1 1 1 1 1 1 1 < $

\< f3

C>l

oi

CNI

Ocoo>coo

-o

0)a

s-o

CDco

-Qoco

toc

fO

s4->

s-o

4-


 • E to O r—

•i- Q)

+->

C

fC c

i— »a a) .c

i- o

S-o c

cr>

CD

CO CNI

CM

25
In a more recent study, Keilin et al . [27] found that their data on transition to film boiling at pressures between 1.1 and 1.5 atm correlated

well on a plot of Ku vs. Fr where Fr is a Froude number, u.//^l. The character1/2 istic length Z was taken as the characteristic bubble diameter (a/ (g(p,-pv

)) ) • (Note that this is the square root of the usual definition for Froude number).

All data on transition were within ±10% of a line given by

Ku = 0.031 Fr0,53

(18)

While these authors do not feel that quality has any explicit influence on the

value of the critical heat flux in their experiments, it is not clear that

their results are inconsistent with our results or with other authors' results. The

difficulty in making precise statements of comparison is that conditions tend

to be concentrated in different areas in different experimental arrangements.

For instance, while Keilin et al . observed the transition to film boiling at

qualities from 0.33 to 0.6, many of our transitions occurred under subcooled

boiling conditions (negative thermodynamic quality) and our range was from

-0.24 to +0.38. We should like to recommend a review of critical heat flux

correlations for helium under flow conditions in channels with all the data in

hand.

In the final figure in this section, figure 10, we have summarized

our subcritical heat transfer results on a q vs. AT plot with shaded areas representing the entire data as labeled. On the same plot for comparison

we have shown supercritical heat transfer at 3 atm calculated from equation

(2) for a bulk fluid temperature of 4.2 K. This plot shows clearly the

unquestioned superiority of the nucleate boiling regime at 2 atm, although

extrapolation to even higher subcritical pressures is dangerous since the

transition heat flux q should go to zero at the critical pressure. This

is inherent in the parameter Ku. If a few degrees AT may be permitted

there is little to choose between film boiling at a subcritical 2 atm

pressure and the supercritical 3 atm pressure, but it is clear that film

boiling at 1 atm should be avoided. Again, referring to the previous

section, AT greater than ^ 2 K is not properly represented by equation

(2) for here the density ratio p,/p„ is beyond the limit for which the

equation was developed.

26
CMZo

CO

CM

CSJ c 1 1 o

•r— CU +-> o fO c 3

CU CT

s. CU

CU 4- >>

QJ -O

Ss- E cu O «-

s- 00 4-

«3 i- +J -»-> fO -o 4-> to

&- CU

cu j=

<4-00 r— C fO

«o o

S- •r—

4-> •<-> *: •p~

+-> s- <

ro

CU o

s-Q.

r—• 3

ro to U

•r- JC

4-> 4-> •r- •r- S- 3 u

JD c 3 o

CO 00 4- s_ • O to E Q.+J

>> E- •«

SO

to

U CO EE X3' ' 3 e o

c/) ro 4-

en

27
Heat Transfer to Helium II

It appears likely at this time that some applications of superconductivity

will require operation at temperatures below the lambda point, 2.17 K,

of helium. The study of heat transfer in the He II phase (the superfluid

phase of liquid helium) remains largely separate from studies of heat

transfer in all other liquids. The main phenomena of superfluidity are

so different, and the vocabulary that describes them so specialized, that

communication to persons outside the field can be difficult. Before

discussing superfluidity itself, however, we should first mention briefly

one special problem in heat transfer that probably has nothing to do with

superfludity but is ordinarily only detectable and important at He II temperatures, and that is the "thermal boundary resistance" ("Kapitza

resistance") between dissimilar materials. It is characterized by a temperature jump at the interface between two materials which is proportional

to the heat flux passing through that interface.

Most studies have been done on He II - metal interfaces, where the

effect is largest, and where low bulk thermal conductivities do not obscure

it. Its strong temperature dependence (« T ), and the low thermal conductivity

of He I, ordinarily make it unimportant for T > 2.2 K. In He II its magnitude

can become a limiting factor in heat transfer. Perhaps the bigger problem

is that its value is difficult to predict and control. For "clean" Cu-He

interfaces, reported values at T = 1.9 K range from 1.3 to 8.3 cm . K/W [28].

Even surfaces prepared by the same procedure may differ by 50%. This variation

is known to be caused by the great sensitivity of the effect to the exact

condition of the surface, but the detailed effects of, e.g., dislocations,

is still controversial.

One of the most characteristic features of He II work concerns conduction

of heat through the liquid itself; the unique behavior can be grasped once

the main features of the "two-fluid" model are understood. It supposes we

can view He II as consisting of a homogeneous mixture of two fluids, called

the "superfluid" and the "normal fluid". The relative concentrations are determined only by the temperature (and weakly, the pressure), but the

velocities of the two fluids may differ, thus nearly doubling the number of

variables as compared to ordinary hydrodynamics.

28
In most respects, the normal fluid fraction behaves like an ordinary

fluid. On the other hand, the superfluid fraction has the unique properties

of carrying no entropy, and, in certain circumstances, of obeying an Euler

type equation of motion (perfect fluid equation). Both fluids have the

unique property of being accelerated (strongly) by a temperature difference

as well as a pressure difference, the superfluid fraction being driven

toward the higher temperature and the normal fluid fraction being driven

away.

These properties allow heat to be transfered within the stagnant liquid

by a unique type of "internal convection" or "counterflow". The superfluid

fraction flows toward the heat source, absorbs heat by its conversion to

normal fluid, which then flows back toward the heat sink. At low heat

currents (low relative velocities) the two fluids do not interact, so that the two fluids can flow through each other unhindered, and effective

thermal conductivities as measured along the axis of a tube can easily

be many orders of magnitude larger than in good metals. Unfortunately

this excellent state of affairs has a limit, because the two fluids

p begin to interact, at some "critical axial heat flux" (typically .01 - .1 W/cm ) which is dependent on the geometry, by a process which is called "mutual

friction". The temperature gradient then starts to increase strongly,

approximately as the third power of the heat flux. In most practical

situations, i.e., diameters £ .05 cm, T £ 1.5 K, q > .01 W/cm , mutual friction

will be dominant; even so, the effective thermal conductivity is still orders

of magnitude superior to any metal at these temperatures, as can be seen in

figure 11. For quantitative discussions of these topics, the reader can consult

existing reviews [10] and monographs [29,30]. We note that natural convection,

and nucleate boiling are never observed.

For technological purposes, e.g., cryogenic stabilization of superconducting magnets, it is the peak, or limiting, heat flux (also called the

"critical heat flux") that can be accommodated without a large temperature

increase, that is of most interest. Most of the available information

suggests that in He II this limit corresponds to the transition to film

boiling: the wall is no longer wetted and its temperature jumps dramatically.

The simplest assumption is that this occurs when the local temperature (determined by the heat current distribution and mutual friction) becomes somewhat

29
1.0 —

q, W/cm'

0.1 —

8T/8X, K/cm

Fig. 11. The axial heat flow (solid line) along a tube filled with He II at

T = 1.9 K, as a function of the temperature gradient, where "mutual

friction" is dominant. The dashed portion of the curve indicates

the approximate range of the limiting heat flux. The dotted curves are lines of equal effective thermal conductivity.

30
greater than the saturation temperature appropriate to the local pressure,

just as in helium I. In long heated tubes (perhaps length/diameter > 10)

without net axial mass flow, this limiting heat flux turns out to be roughly

independent (± 30%) of length, width, and heat flux distribution at the

2 liquid boundary, and to have a value of about 1 watt per cm of tube cross-section . It should be pointed out that substantial pressurization (subcooling) increases

the limiting heat flux only slightly [31].

One outstanding problem is that of limiting heat fluxes in more open

geometries, such as flat plates or cylinders far from a wall, where the

heat flux need be large only quite near the heated surface. Experiments

have indicated considerably larger peak fluxes, 3-6 W/cm for centimetersized surfaces, and a marked depth dependence (sub-cooling dependence),

in qualitative agreement with the model outlined above [32,33]. We need more experiments where wall temperatures and liquid temperatures near the

wall are measured, and where careful attention is paid to nearby surfaces

that can channel or obstruct the heat currents. The influence of the

surface condition, and the degree of super-heating are also of interest.

Whether the limiting heat flux in long channels can be increased is an important and open question. The most straight-forward, and possibly

only way, i.e., by use of forced convection of the He II, raises a number

of questions.

We can formulate the model situation as follows: net mass flow of

He II through a heated tube is produced by a pressure drop along it, and

sensible heat is absorbed by the temperature rise of the fluid. If such

a situation can be produced, it will undoubtedly increase the limiting

flux: a temperature rise from 1.8 to 2.0 K and a velocity of 100 cm/s

2

corresponds to a heat flux of about 10 watts per cm of tube cross-section.

The implicit assumption, which we take for granted with an ordinary

fluid, is that mass flow is driven by the pressure gradient only . In He II we know that in at least certain special situations, e.g., the "fountain

effect", a large mass flow is driven by a temperature gradient only. We can not predict with any certainty how the temperature gradient will change

the pressure gradients needed to produce a given net mass flow. The

difficulty is that there are large gaps in our knowledge of two-fluid

31
hydrodynamics. For example, the conventional formulations of the theory of

"mutual friction", and almost all of the experiments, consider only the case of zero net mass flow. It would not be surprising if, at the least, some of the empirically determined quantities of that theory are a function

of the net mass flow. What we need to know is how both the heat flow rate

and the mass flow rate along the tube depend on both the temperature gradient

and the pressure gradient, and also how they depend on tube diameter and

temperature. Despite these uncertainties, our best guess is that in

millimeter-and larger-sized geometries and at large mass velocities the

usual benefits of forced convection will be obtained.

A final question concerns pumping the superfluid fraction. A truly

ideal fluid, e.g., could not be set into rotation, hence could not be

centrifugally pumped. In fact, the ideality of the superfluid fraction

usually breaks down for large velocities, so we expect conventional pumps

to work in some fashion [34]. However,' there have been no in depth investigations of pump characteristics in helium II. One speculative possibility

that has occurred to us is to close off one end of a tube with a "super-leak",

the other end being open, net mass flow then being driven through the tube

by the "fountain effect". Such a system was demonstrated long ago with

a point heat source near the superleak [35]. Our guess is that such a "fountain pump" will also work well with a distributed heat source, and

prove attractive from an engineering standpoint, because of the absence of

moving parts.

For a long time it has been known that He II is a superior heat transfer

medium for small heat fluxes. We expect that further research, if care is

taken in understanding the unique properties of He II, will show it to be

a superior medium at high heat fluxes and to have a place in the technology

of superconductivity.

Oscillations in Forced Flow Helium Cooling Systems

To date no systematic experimental study of oscillations in forced

flow helium systems has been reported. Rather, this has been a phenomenon

frequently referred to, but more as a nuisance in the pursuit of other goals,

notably the heat transfer studies reported above, than as a subject for study.

32
We can identify three distinct modes of oscillation in forced flow

systems from a large body of literature on other cryogenic and higher

temperature fluids. These are

a) Density wave oscillations [36,37]

b) Acoustic oscillations [38,39]

c) Pressure drop or Helmholtz oscillations [40].

In a) the mechanism is that of an enthalpy and density disturbance travelling

at the stream velocity. It can be shown that such a disturbance becomes

amplified due to thermal expansion as it travels down the channel. The

resulting pressure drop disturbance is propagated back at sonic velocity,

but arrives at the inlet delayed in time by the residence time of the

fluid in the heated channel. Depending on the nature of the source flow

impedance this pressure drop disturbance can again induce a flow and hence

density and enthalpy disturbance at inlet. If it is larger than the

initiating disturbance a self-sustained oscillation is set up with period

approximately equal to the residence time of fluid in the channel.

The oscillations of type b) are of the "organ pipe" kind with a standing wave pattern set up between flow discontinuities (e.g., a valve).

It appears necessary for such oscillations to be self-sustained that heat

addition be concentrated within a part of the standing wave where velocity

fluctuations are all in the same sense and such as to cause a fluctuating

heat addition in phase with the pressure.

Type c) appears to occur when a reservoir of compressible fluid

feeds a heated channel which exhibits negative differential flow resistance,

a phenomenon characteristic of two phase and supercritical fluids. In this

case, the channel pressure drop is in phase with the reservoir pressure

when the reservoir-channel system oscillates in the Helmholtz (mass/spring)

mode, which is thereby sustained.

Provided the time constant for radial thermal diffusion is small compared

to the period of oscillation, these can all be studied by the application of

classical linear stability theory to the one dimensional conservation equations

of mass, energy and momentum describing flow in the heated channel. But of

equal importance to the dynamics of the channel itself are the external flow

impedances or acoustic impedances at inlet and outlet of the channel.

33
We have made calculations for type a) for supercritical helium systems.

We find that under certain circumstances (e.g., low pressures and proximity

to the transposed critical temperature) these oscillations are a real

possibility in supercritical helium. An example of computed frequency

response for a heated channel at 2.5 atm taken from [37] is given in

figure 12. The "open loop gain" and "phase angle" refer to a velocity perturbation at the entrance of the heated channel. This case is that of a simple uniformly heated circular channel fed from a constant pressure

reservoir via an inlet flow restriction and discharging via a second

flow restriction to a second constant pressure reservoir. The flow

restrictions define the flow impedances external to the channel. With

the open loop gain greater than unity at a frequency where the phase

angle is -it, this system must be unstable. Two-phase helium systems

should be even more susceptible to this type of instability. However, this

system can be stabilized by increasing the upstream flow restriction at the

cost of increased system pressure drop.

Stability computations for acoustic oscillations, type b), are con- siderably more difficult and those that have been reported [39] have not

permitted generalization of the results. We therefore still lack criteria

that can be applied to helium systems.

Arp [41] has studied the possibility of negative differential flow

resistance in helium cooled channels and finds that this is only a possibility

at inlet conditions of below about 3.5 K and below about 3 atm for short

channels with high heat flux. Thus, oscillations of type c) may be of yery

limited possibility in applications of superconductivity, but we are investigating other possibilities for sustaining this mode of oscillation.

In all cases it is necessary to understand that, for a given mode, stability is not determined by helium properties and the parameters of the heated

channel alone. Of equal importance are the flow impedances at either end of

the channel and the external coupling, if any, between hydraulic and thermal

disturbances. These determine the manner in which propagated disturbances

are reflected or transformed.

A thorough understanding of the mechanisms of oscillations and their

stability criteria will enable successful design ensuring stable performance. This should not be left to chance. Experimental verification

of the theoretical methods at our disposal is urgently needed.

34
310NV 3SVHd

-a

•» c c

-C fO o

+-> •- ai^ 4-> c o

cu ir> •!- r- • s_

<d- 4->

C (/>

cn

O c

•3- TCM S-ai i— »-> CD C c a> c

fO E O T- r- Mr— C -i- (O CU •—i s- -C o

S-o

4-

X

cu -ac •

ai

CU TCD ^s.

10 CD Co o

Q. • <A r—

CO

s- .c

4-> >> •r- o 3 c

cu i- 3 cu

cr-i-> cu cu

S- E *+- (0 -a -a

cu

-t-> E 3 O

QELf) O • o o

CM • s- E cu 3 CUE

r-^ to ID -i- i— -a o Eo am ltj o CMo 4-> (0 o

cu 4-> C. a> Er-

+J fO

<0 >

ID 3• cr CM CU

CD

NIV9 dOOl N3dO

35
Acknowledgments

The many contributions of our colleagues Patricia J. Giarratano and

Vincent D. Arp are gratefully acknowledged.

Nomenclature

b channel half width

C constant pressure heat capacity

d tube diameter

g gravitational acceleration

Gr Grashof number, variously defined in text

h heat transfer coefficient

H enthalpy

k thermal conductivity

Ku Kutateladze parameter, defined in text

i mixing length

m mass rate of flow

Nu Nusselt number, hd/k

Pr Prandtl number, uC /k

q heat flux

Re Reynolds number, p<u>d/u

T temperature

u axial velocity component

u* friction velocity

v specific volume

x thermodynamic quality

y coordinate perpendicular to channel wall

Greek

a thermal diffusivity

3 thermal expansivity

e eddy diffusivity

v kinematic viscosity

X latent heat of vaporization

p density

a surface tension

x shear stress

y viscosity

36
Subscripts

B bulk fluid

C critical, referring to boiling transition

f film

H heat

L liquid

M momentum

TC transposed critical

V vapor

W wall

reference value

1 constant property case

Superscripts

+ dimensionless parameters defined in text

37
References

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,

IEEE Transactions on Magnetics, vol. Mag-11 no. 2, p. 467, March 1975.

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New York and London, 1971).

12. See discussion by W. B. Hall, Heat Transfer under Supercritical Pressures,

Proceedings of the 5th International Heat Transfer Conference, Tokyo 1974.

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(Plenum Press, New York and London 1972).

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Guildford, U.K., 1972).

16. P. J. Giarratano and M. C. Jones, Int. J. Heat Mass Transfer 18^, 649 (1975).

17. For example, E. R. Van Driest, Heat Transfer and Fluid Mechanics

Institute, paper XII, (1955) and R. G. Deissler, NACA report no. 1210

(Lewis Flight Propulsion Laboratory, Cleveland Ohio, 1955).

18. B. S. Shiralkar and P. Griffith, ASME J. Heat Transfer 91_, 27 (1969)

19. H. Tanaka, A. Tsuge, M. Hirata and N. Nishiwaki, Int. J. Heat Mass

Transfer 16, 1267 (1973).

20. G. P. Malyshev and V. G. Pron'ko, Inzhenerno-Fizicheskii Zhurnal 23,

814 (1972). English translation; Journal of Engineering Physics 23,

1382 (1972).

38
21. B. S. Petukhov, A. F. Polyakov, V. A. Kuleshov and Yu. L. Sheckter,

Paper NC4.8, Proceedings of the 5th International Heat Transfer

Conference, Tokyo 1974.

22. P. J. Giarratano, R. C. Hess and M. C. Jones, Advances in Cryogenic

Engineering vol. 19 p. 404 (Plenum Press, New York and London 1974).

23. S. S. Kutateladze, Heat Transfer in Condensation and Boiling (State

Scientific and Technical Publishers of Literature on Machinery,

Moscow and Leningrad 1952).

24. H. Ogata and S. Sato, Cryogenics T_3, 610 (1973).

25. C. Johannes and J. Mollard, Advances in Cryogenic Engineering vol. 17

p. 332 (Plenum Press, New York and London 1972).

26. V. E. Keilin, Yu. V. Klimenko and N. A. Kovalev, Cryogenics 9, 36 (1969).

27. V. E. Keilin, I. A. Kovalev, V. V. Likov and M. M. Pozvonkov, Cryogenics

15, 141 (1975).

28. N. S. Snyder, Cryogenics 10, 89 (1970).

29. J. Wilks, The Properties of Liquid and Solid Helium (Clarendon Press,

Oxford, 196TT

30. W. E. Keller, Helium-3 and Helium-4 (Plenum Press, New York 1969).

31. G. Kraft, Karlsruhe report KFK 1786 (1973).

32. D. N. Lyon, Advances in Cryogenic Engineering vol. 10 p. 371 (Plenum Press,

New York and London 1965).

33. D. W. B. Mathews and A. C. Leonard, Advances in Cryogenic Engineering

vol. 19 p. 417 (Plenum Press, New York and London 1974).

34. For an informal experiment which demonstrates this see P. M. McConnell,

NBSIR 73-316 (NBS, U.S. Department of .Commerce, Washington D.C., 1973).

35. F. A. Staas, K. W. Taconis and W. M. van Alphen, Physica 27, 893 (1961).

36. J. C. Friedly, J. L. Manganaro and P. G. Kroeger, NAS8-21014,

NASA Marshall Space Flight Center, Huntsville, Alabama, Oct. 1967.

37. M. C. Jones and R. G. Peterson, J. Heat Transfer Trans. ASME Series C,

97, 521 (1975).

38. R. S. Thurston, J. D. Rogers and V. J. Skoglund, Advances in Cryogenic

Engineering vol. 12 p. 438 (Plenum Press, New York and London 1967).

39. V. S. Krishnan and J. C. Friedly, Proceedings of the 5th

International Heat Transfer Conference, Tokyo, 1974.

40. J. S. Maulbetsch and P. Griffith, Proceedings of the Symposium on Two-Phase Flow Dynamics, vol I, p. 799 EURAT0M Report 4288e,

Brussels, 1969.

41. V. D. Arp, Cryogenics Division, Institute for Basic Standards, National

Bureau of Standards, Boulder, CO 80302, to be published.

39
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1. PUBLICATION OR REPORT NO. NBS TN 675

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4. TITLE AND SUBTITLE

Heat Transfer and Flow of Helium in Channels - Practical Limits for Applications in

Superconductivity

3. Recipient's Accession Nc

5. Publication Date January 1976

6. Performing Organization Code 275.05

7. AUTHOR(S)

Jt/L.—CL.Jone s and W .—SJ,—Johnson

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NATIONAL BUREAU OF STANDARDS

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16. ABSTRACT (A 200-word or less (actual summary ot most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Heat transfer and fluid mechanics of helium flowing in channels are reviewed.

Emphasis is placed on observed or anticipated limits of operation which might be

expected to apply in applications of superconductivity. Topics included are: the high-heat-flux degradation of heat transfer and possible effects of buoyancy

forces in supercritical helium; transition to film boiling in subcritical

helium; limiting heat currents in helium II; and the possibility of oscillations

in forced flow helium systems.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Buoyancy ; channels; film boiling; forced convection;

helium I; helium II; mixed convection; nucleate boiling; subcritical;

supercritical; transition; turbulent flow.

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