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Q NBS TECHNICAL NOTE 1002

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U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Measurements of

Combined Axial Mass and

Heat Transport in He I

753
NATIONAL BUREAU OF STANDARDS

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2 Located at Boulder, Colorado 80302.
Measurements of Combined Axial Mass

and Heat Transport in He I

H Bureau if Standard*

3 »78 ~pC£

id, 1 602.

l ^

Warren W. Johnson

Michael C. Jones

-t T^oW^1 e^

i^V , h?. /oo

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80303

**~ *\ ^V J

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

S. NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1978
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1002

Nat. Bur. Stand. (U.S.), Tech. Note 1002, 84 pages (February 1978)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1978

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

Stock No. 003-003-01889-9 Price $2.40 (Add 25 percent additional for other than U.S. mailing)
CONTENTS

Page

1. INTRODUCTION AND BACKGROUND 1 1.1 The Two-Fluid Model 2 1.2 Counterflow ; 3 1.3 Limits to the Heat Current for Practical Applications 5 1.4 Raising Those Limits Through Forced Convection 5

2. THERMODYNAMICS OF HEAT TRANSPORT 6 2.1 Conduction and Convection 6 2.2 The Classical Analog 6 2.3 The He II Energy Equation 10

2.4 A Note on the Experiment 11

EXPERIMENTAL METHODS 12

3.1 Introduction 12 3.2 The Temperature Measurement Systems 15

3.3 Thermometer Calibration and Performance 16 3.4 Pressure Difference Measurements 18 3.5 Mass Flow Rate 24

3.6 The Flow Tube 26

3.7 The Energy Flow Measurement 29

3.8 Extraneous Heat Flows 31 3.9 Error Estimates 33

4. RESULTS AND DISCUSSION 35

4.1 The Net Fluid Velocity 35

4.2 The Heat Current Density 44

5. ENHANCEMENT OF HEAT TRANSPORT BY FORCED CONVECTION 54

6. CONCLUSIONS 59

7. ACKNOWLEDGMENTS 59

8. REFERENCES 60

9. NOMENCLATURE 62

Appendix 1 54

Appendix 2 76

m

MEASUREMENTS OF COMBINED AXIAL MASS AND HEAT TRANSPORT IN He II

Warren W. Johnson and Michael C. Jones

An experiment was performed that allowed measurements of both axial mass and heat transport of He-II in a long tube. The apparatus allowed the pressure

difference and the temperature difference across the flow tube to each be independently adjusted, and the resulting steady-state values of net fluid velocity

and axial heat transport to be measured. For the larger Reynolds numbers, it was found that the relation between pressure difference and net fluid velocity

was nearly indistinguishable from that of an ordinary fluid in turbulent flow.

The axial heat transported was found to be suppressed from the values that were calculated by assuming that "mutual friction" was unchanged by the net fluid flow,

but it was always found to be larger than the "enthalpy rise" value. Taking this

second value as a lower limit, it is shown that a mild extrapolation of these

results suggests that (in appropriate circumstances) forced convection would allow

much greater heat to be transported in long cooling channels than could be transported by "natural" convection alone.

Key words: Axial heat transport; forced convection; helium II; measurements;

1.4 - 2.1 K; pressure drop.

1. INTRODUCTION AND BACKGROUND

This is the first report of the results of an investigation of steady-state combined

heat transport and fluid flow of the superfluid phase of helium 4 (known as He II) through

a long tube.

The first experiments of this type, both at Leiden have been reported only in the last

few years [1-4]. This is the first to be done in this country, and the first to investigate, with an unambiguous method, flow in which both fluid components are turbulent.

There are two motivations for this work, one scientific and one of an applied nature.

A considerable amount is known about the extraordinary hydrodynamics of He II in certain

idealized situations (i.e., where the velocities are small, and laminar or potential flow

is taking place); rather little is known about the hydrodynamics of He II when the flow is turbulent, and so new types of experiments in this area, are of fundamental interest. From

the standpoint of applications, He II is of interest mainly for its excellent heat transfer

properties, which can be utilized in the construction of superconducting devices. This

experiment is the first to attempt to determine the effect of forced convection, i.e., mass

flow, on the heat transported by the fluid. The term heat transport in the context of this

paper is used to mean heat transported in the axial direction by the fluid contained in a long tube which connects a cold fluid reservoir to a hot fluid reservoir. This is not to

be confused with the more familiar problem of heat transfer from a solid wall into the

liquid.

The measurements were made on a tube of length 60 cm and an inside diameter of

10.1149 cm. They covered a range in fluid flow (expressed as the "total" Reynolds number)
The results at small net fluid velocities show that the usable heat transported by

the fluid is predicted by an application of Vinen's expression for mutual friction. The

measured usable heat transport falls considerably below this calculation at the larger

velocities, but is always greater than the calculated enthalpy rise (the result for an ordinary fluid). It is argued that this latter value should be a lower bound for the heat

transport, in which case an extrapolation suggests that forced convection might be expected

to increase greatly the heat that can be removed from a heat source by liquid helium II.

1. 1 The Two-Fluid Model

All of our understanding of the remarkable properties of He II is based on the two- fluid model. It supposes that He II can be treated as a mixture of two interpenetrating

components : the superfluid component and the normal fluid component. (As is common, we

will ofteo use the term "superfluid" to signify both the superfluid component and the

mixture of two components, trusting that the context will indicate which meaning is intended). The relative fractions of the two components is determined almost entirely by the

temperature, the superfluid fraction changing from 0% to 99% in the temperature interval of

2.17 to 1.05 K. The extraordinary hydrodynamics comes from the fact that each component

can have quite different and independent velocities. This is expressed by

P = P

s +

P n 0)

PV = p

s v

s + pn

v

n

, (2)

where p is the total mass density, p , p are the super- and normal-component densities,

v , v are the super- and normal -component velocites and V represents the velocity at which

mass is actually transferred. Mass conservation is expressed as

!f + $-(p$) = . (3)

Other vital assumptions of the model are that: the superfluid component carries no entropy, the normal component has a viscosity n and satisfies the usual boundary conditions of an ordinary fluid, and that the superfluid component, at least under some conditions, can flow like an "ideal fluid", i.e., a fluid with no viscosity and a "slip" boundary condition at the walls. From these and a few other considerations can be derived the

following equations - neglecting dissipation - that are supposed to govern the flow. [5]

L (ps) + $• (psvj = (4)

8v

s 8v

n Ps "8t

+ Pn TRE

+

p

s (V$jvs + Pn(Vy*n

= - ^p + n

n v

2 v

n (5)
9v

"at

+ VVV ~ ^ (6)

% = - s$T + - ^P (7)

The symbols P, T, s, u stand for pressure, temperature, specific entropy, and specific

Gibbs potential or chemical potential. (Throughout this report, specific thermodynamic

quantities will refer to unit mass). Equation (4) represents entropy conservation,

where all the entropy flow is associated with the normal velocity. Equation (5) is the

analogue of the Navier-Stokes equation (momentum conservation). Equation (6) is the

equation of motion of the superfluid component, and eq (7) is a well-known thermodynamic

identity.J The quantities H v ssnn *vv and v *vv are usually neqlected in the so-called linear j d hydrodynamics.

A great variety of experiments done with nonturbulent flows can be quantitatively

analyzed with these equations, thus firmly establishing the two-fluid model. The reader

unfamiliar with He II may consult Wilks [5], or several other texts [6-8], for a thorough

review of the hydrodynamics in non-turbulent flows.

The difficulties arise with the nonlinear types of flow, particularly the turbulent

ones. For a purely classical fluid, i.e., one that obeys the Navier-Stokes equation,

turbulent flow problems still cannot be solved from first principles; most of what we know

is empiricism based on a large body of experimental data. In He II, with extra variables,

and where the possibility exists of still unknown terms in the equations of motion, we must

rely even more on experimental results. As with classical fluids, it appears that the

majority of practical applications of He II will probably involve turbulent flows.

1 . 2 Counterflow

There is one type of turbulent flow of He II that has received a lot of attention; it

is termed "counterflow" or "internal convection". Its experimental realization is exactly

like a thermal conductivity experiment, that is, it is a tube filled with He II that con- nects a heat source to a heat sink. The apparent thermal conductivity is always very high,

usually much higher than any other known substance; it was the discovery of this property

that first signaled the existence of a very unusual type of liquid. For small heat flows,

2 typically much less than 0.1 W/cm , the temperature gradient is extremely small and all

the facts are fully in accord with the equations of motion listed above. The origin of the enormous thermal conductivity is the ability of the two fluid components to flow "through"

each other, essentially on an atomic scale, hence the term "counterflow". Entropy, hence

heat, in the form of the normal component, is able to flow away from the heat source un- impeded except by its viscous resistance with the walls; the requirement that the net amount of fluid flow be zero is met by return flow of the super component. It is vital to note that the flow of heat, i.e., energy in the form of thermal agitation of the helium

atoms, obeys a hydrodynamic equation of flow, rather than the usual diffusive one; this is

in fact the main reason why a two-fluid model works. More detailed discussion of this

point may be found in the texts [5-8].
At some critical heat current (or its equivalent in counterflow velocity) the character of the counterflow changes in that there arises a finite interaction between the

normal and super components, termed "mutual friction", that impedes the counterflow. The

temperature gradient is no longer proportional to the heat current q; instead it rises much

faster, approximately as

1 - <

3

• < 8>

The nature of mutual friction was revealed in a classic series of papers by Vinen [9], He found that it is produced by a unique type of turbulence in the super component only, a type associated with a tangled mass of quantized vortex lines. These lines are the only

locations where the super component has non-zero vorticity w = [ V x v J. They are a struc- ture in the super component; but they also have a drag force exerted on them by the normal

component, thus they provide the means of interaction between the two components. Vinen

argued that the force of mutual friction F ought to be of the form

sn

F = PL (v -v j (9) sn

H o\ n s

J

where p is a temperature-dependent constant, and L is the length of vortex line per unit

volume. This mutual friction force changes the equation of motion for the superfluid (eq.

6) to

It \ = - * + rs hn

• TM

Further, he deduced a dimension argument to conclude that

(\ i V [ lVv

s|

" v oJ

L o

°

' '

' '-i

'' ;:

where v is small. Since v -v « q, combining eqs. (9,10 and 11) gives the experimental

result, eq (8). The constant of proportionality is approximately Ap /s(p sT) 3 where A, the Gorter-Mell ink constant, is a weak function of temperature and is about 100 cm. s/g at

1.9 K.One feature of this result is particularly remarkable, and it is the lack of any

explicit geometrical dependence; the turbulence intensity, L , is only a function of the

heat flux per unit area. No serious challenge has been raised to the main features of Vinen' s analysis. The

most recent work [10,11] has shown that there are some additional features of the flow, at

-2

least for small diameters (< 10 cm), and there are reports of different values for the

constants (see the review of Arp [12]), but by and large the results of Vinen can still be

recommended for large diameters (> 10 cm).
1 .

3

Limits to the Heat Current for Practical Applications

There must be some limit to heat that can be transferred by internal convection, and

the weight of the evidence suggests that this is when the temperature and the pressure near

the heat source assume values such that the vapor is the more stable phase, i.e., when the

liquid is superheated [12-16]. In ordinary liquids, a small superheat of the liquid near a heated surface causes the formation of small vapor bubbles (nucleate boiling), but in He II this does not happen. Instead, at some modest value of superheat, which might depend on apparently extraneous variables, such as the level of mechanical vibration [15], it appears

the He II goes directly into the film boiling stage, where the heated surface is coated by

a layer of the (very low thermal conductivity) vapor.

For all the applications one can envisage, the onset of film boiling would be un- desirable, perhaps disastrous. Therefore, we arrive at the following tentative prescription for the maximum allowed design heat current: from the geometry, the distribution of

heat sources, and from eq (8) calculate the temperature distribution; if the liquid is

superheated at any point (the pressure must be known and the phase diagram consulted) then

the cooling channel must be expanded, or heat intensity reduced. It remains to determine

an appropriate safety factor.

Examining eq (8) we see that for a uniform channel, with a heat source only at the

end,

q

oc ( T /L)

1/3 wax v max

At a typical value of AT =0.05 K and a length of 60 cm, the maximum heat current is about

2 1.0 W/cm . Because of the cube root dependence, and the limitations on AT (T,. < 2.17 K)

it is obvious that q cannot be much bigqer than this, ^max 3M

1 .4 Raising Those Limits Through Forced Convection

It is well known that total heat that can be transported through a long tube by a classical fluid can usually be greatly increased if the convection is forced by a pump. That this ought also to apply to He II was the motivation for this experiment.

The first task in such a program was to reformulate some of the equations of two fluid

hydrodynamics in such a way that the transfer of energy away from a heat source by both

internal convection and "real" or forced convection could be treated in a consistent

manner. This is handled in section 2. The design and performance of an experiment to measure the relevant quantities for a long tube is considered in the sections 3 and 4. Because of the experiment's exploratory nature, and the resulting uncertainties, it

turned out that the apparatus was not quite capable of reaching the region of flow where

the forced convection would increase the total heat transported dramatically. However, the

measured values of the relevant quantities are such that we can, with reasonable certainty

extrapolate their behavior a modest amount, and conclude that, in certain circumstances, the

heat transported per unit cross-sectional area of cooling channel can be increased to at least

ten times the value allowed by counterflow. This conclusion is discussed in section 5.
2. THERMODYNAMICS OF HEAT TRANSPORT

2.

1

Conduction and Convection

The case we wish to treat is when both conduction and convection are important for

the axial heat transport. This is not a common case for any ordinary cryogenic fluid. For ordinary fluids, the thermal conductivity is so small that it is of significance only

for the "heat transfer coefficient," i.e., the transport of heat through the boundary

layer near the wall. Axial energy transport is ordinarily dominated by the physical

movement of hot fluid, whether by natural (gravity-driven) convection or forced convection.

In He II, the counterflow process seems to share some attributes of both conduction

and ordinary convection, and we also wish to consider the effect of forced convection;

thereforesa consistent treatment of the whole process is vital.

2.2 The Classical Analog

It is illuminating to consider a somewhat idealized version of this same problem for

a classical fluid. The situation is shown in Fig. 1. Heat is generated at a point source

in the middle of a channel connecting two heat sinks. We wish to know the relationship

between the heat 0", the impressed average fluid velocity V, and the temperature rise AT.

— T _ o

Q"(AT,V)

Figure 1. A schematic diagram of a long cooling channel with a localized heat source, Q".

We can derive a differential equation that governs the axial energy transport from the

energy conservation equation of the fluid. In the form we will find particularly useful,

it is

M

8 [p(u+V2 /2)j + $ • jp^[h+V2 /2] + q - (12)
Here u is the specific internal energy, h = u + P/p is the specific enthalpy, v is the

velocity vector, and q is the heat current density vector. This equation is derived in

many texts on fluid mechanics, e.g., Ref. [17], p 184. One term, the viscous energy flux,

has been dropped because it can be shown to be negligible for the flows of interest to us. In this equation, the first term in square brackets is the total energy density

(internal plus kinetic); this implies that the term in curly brackets is the total energy

flux vector, j , for a fluid. This form of the energy equation is particularly useful

because it requires no explicit calculations of the rate at which flow work is degraded to thermal energy, nor has any assumption been made about the functional dependence of heat

conduction.

In steady fully-developed turbulent flow through a tube, if we express V, h, and q as sums of steady and fluctuating components, the equation may.be time-averaged, as shown

for example in Ref. [18]. If we now integrate the equations over the cross-sectional area

A of the tube, we get

d P Vh\ + d q

- = 1 dfj(x) (13)

dx \

*

' dx " , * A^ dx

where angular brackets indicate cross-sectional averages, dQ(x)/dx is the local rate of

heat addition per unit length, x is the axial coordinate, and we have taken the cross- sectional area to be a constant. The quantities V h, and q , in eq (13) must now be

interpreted as time-averaged quantities: q is now no longer just the molecular energy

flux, but also contains a turbulent energy flux (the time correlation of the enthalpy

fluctuations and the axial velocity fluctuations).

We can use well-known thermodynamic relations to rewrite h in terms of the specific

heat at constant pressure C , and other quantities

3h = /3h\ 9T /9h\ 3P ~

r 9T 1 3P (14)

3x VaTjp 9x l9Ti T

c)x LP 3x p 3x

where the second equality is an excellent approximation because we can neglect the com- pressibility of the liquid. In a long tube, with no wall heating, the axial temperature

and pressure gradients are not a function of transverse position, so that the first term

of eq (13) becomes

PVC

p R " £•

where

V

-= (y\
Let us assume that the liquid has the usual form for the molecular thermal conductivity k,

and that it is much larger than any turbulent mixing contribution. Then for the configuration of Fig. 1

,

«%%*<*h (-"$*>- x. (15)

where 6 is the Dirac delta function which represents the point heat source located at x . The appropriate boundary conditions are the values of the temperature at the ends of

the tube and at the point heat source. The value of Q"/A , where A is the cross-sectional area of the tube, is found by solving for T(x), and then calculating the discontinuity of the energy flux, i.e., the discontinuity of -k(8T/8x).

When vthe pressure gradient is small enough, we can obtain an explicit solution for T

in terms of an exponential. The character of the solutions is shown in Fig. 2.

TQ+AT

V:=o..-*"

K V>0

J

1 ••..

v»o

1

N

\ \ ••• \ »

Figure 2. The temperature of the fluid in the channel of Figure 1,

for various values of the impressed fluid velocity V.

It is apparent that the heat transported away from the source can be divided into an up- stream part and a downstream part. In the configuration of Fig. 3, which is essentially

the geometry of our experiments, we find we can measure the contribution of each part

separately, thus obtaining a more complete picture of the whole process.

Q'(AT,V)

Q"(AT,V)=Q'(AT,V)+Q'(AT,-V)

Figure 3. Schematic diagram of a configuration that allows us to measure separately the heat removed by each half of the cooling channel shown in

Fig. 1, and the equation that relates the results for the two configurations.
The calculated result for the heat absorbed in this second configuration is shown in

Fig. 4.

Figure 4. The expected results for Fig. 3 for two different temperature

rises at the heat source, if the fluid has a high thermal conductivity.

Asymptotically, the heat removed is given by the enthalpy rise of the fluid.

The solutions display the appropriate values in the two limiting cases. For V = 0, we recover the pure conduction equation, with the result Q" = 2Q' = 2k(AT/x )A . For the

case V >> 0, we find that the upstream part approaches asymptotically the value

pV(Ah)A - pVC (AT)A and the downstream part approaches zero, i.e., we obtain the

A P X familiar result that, if axial conduction is small except near the heat source, then the

heat transported away from the source is given by the enthalpy rise of the fluid.

There is another representation for the energy flux vector which we will find useful

It is found by using the definition of the chemical potential

u = u+lp sT = h - sT (16)

to obtain

J u = pV(u + V

2 /2) + [pstT + q] (17)

the term in square brackets can be identified as T times the total entropy flux. If there

is no mass flow, then the heat and energy fluxes are the same, and the entropy flux differs only by a factor of T, so that the distinctions between the three quantities tend to become lost. With mass flow present, it is obviously important to maintain the distinctions between them [191.
2.3 The He II Energy Equation

As discussed in Section 1, eqs (1-6) are often presented as the basic two-fluid model

equations. Also discussed were the counter-flow experiments that necessitate the replacement of eq (6) by eqs (9 & 10). At least one other modification of the two fluid equations is required.

For the cases we wish to consider it is obvious that entropy is being increased by

the heat transport, so that the equation for entropy conservation, eq (4), is not correct. While explicit modification of this equation has been proposed [20], we have preferred to use the approach of finding the total energy flux, as was done above for the

classical fluid case, and thus we replace equation (4) with an energy equation.

An explicit energy equation has been derived for He II from several different viewpoints (Ret- [8], p 28; Ref. [5], p 168). To a good approximation it is

a [pu] + $• {p% + psTvJ = (18a)

5T n

or

a [pu] + $• {p^h + psT(v -V)} = (18b)

Bl n

All dissipative flux terms have been dropped: by comparison with the ordinary fluid

equation we expect the normal fluid viscous stress part to be small, the irreversible heat

flux term is always 3-5 orders of magnitude smaller than the heat flux defined below, and

numerical estimation of the remaining terms suggests they are even smaller.

By comparison with eqs (12) and (17), we see that the same relations hold, as in the classical case with the new identifications of the total entropy flux as

psTv

n

and the heat flux as

q = P sT(v

n

- H) = p

s sT(v

n

- v

s

) (19)

where the second equality follows from eqs (1) and (2). In many treatments of He II, the heat flux has incorrectly been identified with the

entropy flux, probably because there have been almost no previous experiments on heat flow

where V i 0. The correct identification for the heat flux has been stated recently [21];

evidently the argument just given is the explicit justification for it.

10
We can use the same steps as before to derive a one-dimensional, steady state, energy

conservation equation. It is

§_ [pVh + q] = 1 3p(x) (20)

3x A~^ 3x

where it is understood that V and q are the cross-sectional averages. Again the right

hand side represents the external heat sources.

As in the classical case, this form of the energy equation does not require any detailed knowledge of the hydrodynamic mechanisms that give rise to the pressure and temperature gradients; it only requires their measured values. But q represents the total heat

flux from whatever microscopic mechanism, be it molecular or eddy conduction, or "internal

convection."

The experimental problem then is straightforward: what is the functional dependence

of V and q on the pressure and temperature gradients and on the geometry? Note that be- cause of the hydrodynamic character of q for He II, we must not expect it to be solely a function of the temperature gradient. Ideally we would like to compute these from the two fluid equations, but that is not practical for the reasons discussed in section 1.

2.4 A Note on the Experiment

The experiment was done using essentially the configuration of Fig. 3. Not only does

this allow the separate examination of the upstream and downstream components of the heat

transported, it also allows the net flow velocity V to be measured by a straightforward

volumetric technique. This is important because there is no conventional flowmeter that

we can be confident will work in its usual manner [22]. Vortex-shedding and turbine

meters depend on the shed or bound vorticity having particular values. We know that

vorticity, at least in the superfluid component, is not generated in the conventional

manner, e.g., from the high shear in the boundary layer, but rather its main source is apparently the relative velocity between the normal and super components, and so we have

no good reason to expect the conventional values. Orifice meters, pitot tubes and venturi

meters all depend on a surface of flow separation being in a certain position, but flow

separation is really the same as a bound vortex sheet, and so the same objection applies

to these meters.* The only type of flow meter that we can be reasonably certain would

work in He II is an acoustic one; unfortunately it seems nearly impossible to build one

for small diameter tubes.

Work done in this laboratory [23] has suggested that a venturi meter might still work

in some reasonable manner in He II, but no direct test could be made in that experiment.

11
3. EXPERIMENTAL METHODS

3. 1 Introduction

In the previous section it was shown that the axial transport problem is essentially

the determination of the functional relations between the two fluxes and the two gradients.

Then the energy and mass conservation equations can be used to determine the effect of distributed heat sources, of changes in flow tube cross-section, etc. We found it convenient

to choose the two fluxes to be the net average velocity V and the average heat current

density q, rather than the normal- and super-fluid velocities v and v . The gradients

that drive the fluxes were taken to be the pressure and temperature gradients. In general, any two of the four quantities can be chosen to be the independent variables, whose

values, and the hydrodynamics, then determine the two remaining (dependent) variables.

The general plan for the experiment was to use several feedback systems to maintain a fixed temperature difference and pressure difference between the ends of a tube filled with

liquid helium — the independent variables for this experiment. The tube was surrounded by

an insulating vacuum, so that no heat was transferred through its walls. The resulting

steady state values of overall mass flow and energy flow through the tube -- the dependent

variables — were then measured.

A schematic diagram of the experimental apparatus is shown in Fig. 5. In Fig. 6 we show a photograph of the apparatus with vacuum can removed. The flow tube (FT) was the

only connection between the outside helium bath (0) and the inside helium vessel (I). The

inside vessel (I) was surrounded by a vacuum maintained by the stainless steel vacuum can

(VC). This inside vessel was constructed in two parts, an upper section with a large horizontal cross-sectional area, and a lower section with a small horizontal cross-sectional

area. The temperatures of and I were sensed by identically mounted carbon resistors

(TCO, TCI), which controlled the voltages applied to the two resistance heaters (IH, OH)

so as to maintain a fixed temperature in each space. The actual temperatures in and I were determined from the germanium resistance thermometers Tl , T2 and TB. The entire assembly of vacuum can (VC), inside helium vessel (I), and associated

instruments, was surrounded by a copper heat shield (HS) and was suspended by a single

support tube (ST). This support tube also acted as the pumpout tube for the vacuum space,

and as a conduit for the electrical leads. An exterior motor-driven mechanism and a sliding seal for ST allowed adjustment of the vertical position of VC in the helium bath 0, and

hence adjustment of the hydrostatic pressure difference applied across the flow tube (FT).

The difference between the inside and the outside levels was sensed continuously by the

coaxial capacitors CI and C2. This signal was used to control that motor so as to maintain

a fixed level difference. Of course, steady state flow could be maintained only for the

length of time it took for the inside vessel (I) to be filled or emptied.

The first dependent variable, the mass flow through the tube, was determined by noting

the average value of the vertical speed of the vacuum can (VC) that the motor-controller

system had set in order to maintain the fixed level difference. The second dependent

12

i
HS\

--/

z&:

ttSs mm

S£So%5'-J

ST

VC

Vacuum

FT. TIB^J!^

^1

Figure 5. A schematic diagram of the apparatus

13
Figure 6. A photograph of the apparatus with the vacuum can removed

14

am
variable, the rate at which energy flowed through the tube, could be determined by noting

the value of the heat being supplied to the inside heater (IH) by the inside temperature

control system.

The inside heater consisted of 60 cm of 0.04 cm diameter manganin wire wound around

the bottom end cap of the inside capacitor, where it would always be immersed in the

liquid. The heater current was brought through the vacuum space by superconducting wires,

which led from a heat sink on the base plate to the inside space. Their low thermal con- ductivity allowed the current to be transmitted without adding a significant thermal link

through the insulating vacuum. The (cold) resistance of the heater was measured to be

664 ohms, of which less than 1 ohm was the room temperature to helium temperature lead

resistance.

Data were taken at four temperatures (1.39, 1.65, 1.92 and 2.10 K) , which span a range

in the normal fluid component fraction from 7 percent to 74 percent. Time permitted data

to be taken for only a single tube (diameter = 0.115 cm, length = 60 cm), and at only a few, but widely-spaced AT's. Fifteen or twenty points were taken at each AT, covering a range of approximately two orders of magnitude in pressure difference.

All of the data presented in this report were taken during a single thirty-five day

period. They are presented in tabular form in Appendix I, and in graphical form in the

results and discussion sections. During this long run, the liquid helium in the outer

helium bath was replenished every two or three days, and the apparatus was never allowed to warm up. Two of the main variables, temperature and temperature difference, were varied

from day to day in an irregular manner, and also there was enough duplication, so that we

can conclude that there was no significant progressive change in the data throughout this

long run. Some data taken during several preliminary runs were also reproduced (to within

about 2% in the dependent variables) during this long run.

3.2 The Temperature Measurement Systems

The two independent temperature control systems had identical carbon resistor temperature sensors. Each resistor mounted in the vacuum space was filed flat on opposite sides

down to a thickness of about 2 mm, wrapped in a layer of cigarette paper saturated in varnish and then wrapped in copper foil, whose loose end was varnished onto a copper piece

in contact with one of the helium spaces. The resistor was connected as one arm in a dc

Wheatstone bridge, whose null detector fed a signal to a temperature controller. Power

dissipation in each resistor was 10 -10 W. The outside controller was of local design.

The inside controller was a commercial temperature controller used in the proportional

mode. The error signal from the inside null detector was attenuated greatly (100:1) so that the control fluctuations were quite weak. This meant that the fixed offset had to be

adjusted manually for each change in conditions, but this was necessary to ensure that the

voltage on the inside heater could be read with sufficient precision.

Judging from the null detectors and the germanium resistor thermometers, short term

fluctuations (minutes to seconds) in the two temperatures were less than 0.0002 K. The

very high effective thermal conductivity of He II makes this possible even in a large

15
system with fluctuating heat loads. However, there were significant long term drifts in

the carbon resistors, occasionally as large as 0.0001 K/hour. These were not seriously

damaging to the measurements, because readings from a set of germanium resistance thermometers were used to determine the actual temperatures for almost every data point.

There were a total of five germanium resistance thermometers mounted on the apparatus.

Two were mounted in the vacuum space near the carbon resistors, and were intended to measure the temperatures of the inside helium vessel (T2) and the outside helium bath (Tl).

Two others were mounted on the flow tube itself in an attempt to measure the fluid temperature at points near the ends; TIB was mounted 1.5 cm from the inside end and TOB 1.5 cm from the outside end. A fifth thermometer (TB) was laid on the bottom of the helium dewar

and also was intended to measure the outside helium temperature. Tl and T2 were inserted

into vacuum grease-coated thermometer wells. Their leads were thermally tempered to copper

surfaces bys

5 cm of 28 gauge copper wire (for thermal contact) and then by 5 cm of 36 gauge

manganin wire (for isolation from any residual heat being conducted from room temperature).

All leads (36 gauge manganin) then ran 10-20 cm to a second tempering station on the base

plate, which accepted the main heat leak, and then up the inside of the support tube to room temperature. TIB and TOB were mounted in similar fashion on a small copper fitting,

which was soldered onto the outside of the stainless steel flow tube. The contact area between the fitting and the tube was approximately 0.15 cm x 0.5 C, where C = circumference = 1.6 cm. A typical installation is shown in Fig. 7. All the (four terminal) germanium resistors were placed in series, with two oil filled

standard resistors and a reversing constant current source. A 5 1/2 digit high impedance

digital voltmeter was used to measure the dc voltage drop for current in each direction and

their average was used to determine the resistance. 1 |jA of current was used throughout

_o the experiment, thus power dissipation in each resistor was always less than 10 W. One possible source of systematic error in the temperature measurements was discovered

to be the placement of the inside heater. It had originally been a resistance wire var- nished onto the outside of the copper walls of the inside helium vessel. Later calculations indicated that, at contemplated heating levels, the thermal boundary resistance

(Kapitza resistance) might cause a significant temperature difference between the walls and

the liquid, thus biasing the inside temperature measurement. This problem was solved by

moving the resistance wire to the inside of the vessel, where it was wrapped loosely around

the bottom of the inside level gauge.

3.3 Thermometer Calibration and Performance

At least once each day, usually more often, a series of "equilibrium" points were

taken. The procedure was as follows: first, the inside heater was shut off entirely, and

the motor-controller system (see pressure control section below) altered to maintain the vacuum can position constant relative to the outside level, and then, when the inside had come to equilibrium with the outside, the offset between the level gauges was measured, the

thermometer resistances recorded and the vapor pressure of the bath recorded. This was usually repeated for several nearby temperatures. Selected equilibrium data for the entire

16
co

+->

n3

+->

S0)

+-> (DEOE

s_Ec

•aE

sa>CD

03O

Q.

s_=3en

17
long run were combined to determine a single resistance vs. He vapor pressure for each

resistor. The Tr

fi temperature scale [24] was then used to convert the vapor pressures to temperatures.

The vapor pressure measurement was made by a commercial capacitance manometer with a 0-1000 mm Hg pressure head. Precautions were taken to avoid several known sources of

systematic error: refluxing vapor, pumping pressure drops, thermo-acoustic oscillations,

and zero shifts in the pressure gauge. The mobile He II film is known to cause refluxing

vapor which biases pressure measurements made with capillary tubes connected to vapor bulbs

[25] therefore the "bath pressure" was measured. The pressure probe was a 3/16 inch

diameter tube (P of Figure 5.) whose open end in the vapor space penetrated 5-10 cm below

the bottom of the dewar radiation shield, (R of Figure 5.), thus the mobile film was probably evaporating from the dewar walls below the shield, and at any rate the estimated vapor

flux would-be too small in this situation to cause a measurable pressure drop. No change

was observed due to vertical motion of the probe, as long as its end was below the bottom

of the radiation shield. This location for the probe also prevented any bias due to the

pressure difference across the radiation shield caused by the steady flow of evaporating

vapor. The probe was stuffed with coarse "brass wool" to damp out any thermo-acoustic

oscillations that might bias the manometer. It was found that the zero of the manometer

drifted 0.2-0.4 mm Hg/day, and was also a function of the absolute pressure (not solely the

pressure differential). Therefore the pressure- to- resistance calibration was based on only those equilibrium points which followed a zero-adjustment of the manometer done with

both ports of the pressure head pumped to a good vacuum. Most equilibrium points were not taken under this condition, so the pressure recorded on the data sheets is usually subject

to some unknown zero-shift error: of course, all these equilibrium points are still useful

for consistency checks between the resistors and for checking at a single point by occasional measurements of local atmospheric pressure, which were compared to a mercury barometer; the two always agreed to within one part in 1000. The measurements were all con- verted to units of mm Hg at 0° C and standard gravity (1 mm Hg = 133.32 Pa). The manufacturer's calibration indicated that the scale remained linear throughout its range to better than 1 part per 1000; it was not attempted to verify this in the pressure range of

the vapor pressure measurements.

3.4 Pressure Difference Measurements

The pressure difference between the two ends of the flow tube was determined from the

following equation,

AP = P SV(T

2

) - P^CT^ + pg(z

2

- Zl ) (21)

where, the subscripts 1 and 2 represent the outside and the inside helium vessels re- spectively P. w is the saturated vapor pressure at the indicated temperature, p is the

S V *\ helium density, g is the local acceleration of gravity (= 979.60 cm/s ), and z is the

18
liquid level height. The presence of the first term is dictated by the fact that the

pressure of the vapor above the liquid is different for the two sides. This effect is quite large at the higher temperatures, where the vapor pressure curve has a large slope,

but could be quite accurately measured because of the way that the temperature calibrations

were taken (see above). Because the resistances of the thermometers were calibrated

against the manometer measuring the vapor pressure, those calibrations give the vapor

pressures directly. The resulting difference in vapor pressure should be far more accurate

th'an absolute pressure.

The second term in the pressure difference equation is due to the hydrostatic pressure

difference. The liquid level difference was determined from the difference in capacitance

of the two matched capacitance level gauges, C-, and C~ of Fig. 5. The capacitors, illustrated in Fig. 8, consisted of two vertical concentric stainless

steel tubes and an end cap assembly. Designed as a three terminal capacitor, the inside

tube (o.d. = d-j = 0.6407 ± 0.0003 cm) acted as the "hi" electrode, the outside tube (i.d. =

d„ = 1.733 ± 0.001 cm) acted as the "lo" electrode, and the grounded end caps acted as the

guard electrodes for the outer tube. With such a configuration, the capacitance in the end

regions is not sensed.

The capacitance per unit length for infinitely long coaxial tubes in a vacuum is given

by

C 2ne 0.5563 — roo , v _ o _ cm (22)

£ ln(d

2 /d

1

) ln(d

2 /d

1

)

The active length £ of the outside tubes was 30.480 and 60.960 cm for the inside and outside capacitors respectively. The dimensions imply a calculated capacitance (at room tem- perature and in a vacuum) of 17.04 and 34.08 pF, which can be compared to the measured

values for these conditions of 17.071 and 34.116 pF, which are higher by 2 and 1 parts per

thousand respectively. This is within the estimated error of the measurement of the dimensions, and demonstrates the good match. The values actually used in the subsequent calculations are values derived from the measured low temperature values of C (measured in low

pressure He vapor, and corrected to vacuum values), and the values of £ given above multiplied by 0.9970 to correct for the thermal contraction.

When a capacitor is partially filled with liquid, its measured capacitance should be

C C(z) =-| [sz + e'(£-z)] (23)

where z is the liquid level height referred to the bottom of the active section, and

& and e' are the dielectric constants of the liquid and saturated vapor respectively. If both capacitors r have the same value of C /£, then their difference will be

v

C AC'

(

z 2- z i*)=<r^'>(z 2- z i) tiC <°> (24)

19
TO CAPACITANCE BRIDGE

SILVER

SOLDER

PHENOLIC

INSULATORS

STAINLESS STEEL

ELECTRODES

GROUNDED

END CAPS

INLET HOLES

FOR HELIUM

EPOXY

CEMENT

PHENOLIC

INSULATORS

SILVER

SOLDER

1 cm FOOT

Figure 8. A drawing of the level sensing capacitor,

20
where z~ and z-, are the liquid levels on the inside and outside respectively, referred to some arbitrary point. The difference in the vacuum capacitance, and the (small) vertical

offset, are contained in the factor AC(0), which is the capacitance difference measured

while the outside and inside are in equilibrium.

The vapor inside the capacitor was assumed to be at the temperature of the liquid.

This was obviously true for the inside capacitor, and was assured for the outside capacitor

by placing it inside a copper tube whose lower end was immersed in the liquid. Any re- sidual radiation leak to the capacitor could be absorbed by the high conductivity of the

superfluid helium film.

The capacitance difference was measured directly by the use of the system shown in Fig. 9, which incorporates the use of a commercial capacitance bridge of the ratio-transformer type. The outside capacitor was connected to the "unknown" ports of the

bridge, and the inside capacitor was connected to the "external standard" ports of the

bridge. With the appropriate settings, the indicated capacitance at balance is equal to the difference of these two capacitors.

In order to maintain a fixed level difference between the inside and the outside, the

desired capacitance difference was set on the bridge, then the corresponding level difference was approximated by manual adjustment of the vertical level of the vacuum can, and

then the level control feedback system indicated in Fig. 9 was activated. The signal

conditioner produced a signal derived from the error signal and its time integral. This was combined with a large manually controllec offset signal to form the velocity command

for the motor-controller system. A digital volt-meter at this point served to monitor the

vertical speed of the vacuum can. A gain-o T"-one amplifier (not shown) was used to float

the velocity command voltage so as to meet the input requirement of the motor-controller.

When running smoothly, the system was capable of maintaining a set capacitance difference

within 30-50 ppm. The main trouble encountered was rough movement in the pulley used to counterbalance the weight of the moving system. It was quite obvious when the system

became stuck, and so the data could be retaken.

The configuration of Fig. 9 was altered slightly for the equilibrium data points.

First, as mentioned earlier, the inside heater was turned off, then the controls on the

bridge changed so that the off-balance signal was determined by the difference of C from

a set value, and then the control system was activated. This had the effect of maintaining

the outside level fixed relative to the vacuum can. When transients had died away, this

allowed measurement of several parameters with the inside and outside at complete equilibrium, i.e., no mass or heat flow, no temperature or pressure difference. As mentioned

earlier, the temperatures were recorded as a check on their stability. Also measured was

the small downward velocity necessary to maintain a fixed outside level, because this

allowed determination of the evaporation rate of the outside He bath. Then the velocity

was fixed at this value, and the bridge controls changed back so that AC(0) could be measured.

The dielectric constants were at first computed from the Clausius-Mossotti equation

and polarizability found in NBS Technical Note 631 [26], the liquid densities taken from

21
f

MOTOR

CONTROLLER

VELOCITY COMMAND

DVM

SIGNAL

CONDITIONING MANUAL OFFSET

ERROR SIGNAL

PHASE SENSITIVE

DETECTOR

* DETECTOR

CAPACITANCE BRIDGE

EXT. STD. UNKNOWN

9 9 9

TACHOMETER

*- MOTOR

REDUCTION

GEAR

TORQUE

LIMITER

LEAD

SCREWd

->

n r

VACUUM CAN,

1 n

±h

Figure 9. A schematic drawing of the system used to maintain a constant

difference between the inside and outside levels.

22
the text of Donnelly [7], and the vapor densities from the ideal gas equation of state. They could also be derived from full vs. empty measurements, which were found not to agree

with the calculations. The reason was found to be inconsistent units in the ClausiusMossotti equation as found in the above reference, which had the effect of making the

polarizability too small by a factor of 47t/3M - 1.047 (M = molecular weight). With this

correction, the agreement was excellent between the calculated value of e at 1.92 K of

1.05736 and the measured value of 1.05740. It was found that while the values of e and e

1

varied significantly with temperature, their difference was constant to within a part per

thousand over our temperature range. Therefore, the calculated value of £ - s

1 = 0.05714,

appropriate for the highest temperature, where other results depend most sensitively on its

value, was used for all temperatures. C 1% was taken to be 0.5599 pF/cm, the average of

the values measured for the two capacitors. Combining all these factors leads to the

expression for ~L<f z \ act ually used in equation (21),

z

2

- h

= -31.27 "pF ^AC(z

2 -z1

) - AC(0)J (25)

where AC(0) was slightly different for the different temperatures.

This single simple calibration served for the entire long run, because the capacitors

proved to have excellent stability and linearity, as determined by the regular measurement

of AC(0). By varying the point at which the outside level was fixed during an equilibrium

point, the two capacitors could be compared with each other over a significant fraction of

their lengths. It was desirable to keep equilibration times short, so that equilibrium

points were taken only where the inside level was still in the narrow section, thus limiting the comparison to the lower 2/3 of the inside capacitor and the lower 1/3 of the outside capacitor. Within this range, AC(0) was found to be essentially independent of

height and time. Expressed as a height difference, via equation (25), the standard deviation

of AC(0) for all the equilibrium points was 0.01 cm. It is thought that the good stability

and linearity of the capacitors is due in part to several design features arrived at by

trial and error. They are: (1) a mounting for the inside tube that allows some small

axial movement, which prevents compression and bowing of this long slender column, and (2)

the rather wide gap, whose proportions appear to be about the optimum compromise between

sensitivity to level change and insensitivity to dimensional errors, particularly lack of

coaxial ity.

A remark should be made about the limitations that the experimental method placed on the pressure difference measurements. They are due to the vapor pressure difference caused

by the temperature difference between the inside helium vessel and outside bath [the first

term in eq (21)]. If this term was large, then the liquid level difference had to be

large and of opposite sign for those flow regimes in which the total pressure difference

was small (essentially flows for which V was small). For an extreme example, at T = 1.92 K with AT = 0.047 K and V = 0, the inside level z~ was 25 cm below the outside level, z-j.

23
3. 5 Mass Flow Rate The volumetric flow rate through the flow tube was determined from the geometry of the

apparatus and from the vertical speed of the vacuum can that was necessary to maintain a fixed level difference between the inside and the outside helium spaces.

The functional relation between these quantities was determined by application of the mass conservation relation to the geometry sketched in Fig. 10. The upward velocities of

the outside liquid surface, the inside liquid surface, and the vacuum can (all with respect

to some fixed point in the laboratory), are designated by V , V. and V. respectively. The

areas of the inside liquid surface, the outside liquid surface, and the dewar are designated as A.no, A and A .. d The small amounts of mass entering or leaving the vapor phase can be ignored, so that

mass conservation can be expressed in terms of the liquid volumes as

volume added outside + volume added inside

+ volume evaporated =

During the small time interval dt, the volume added inside is (V.-V.)A-dt. If we designate

V (< 0) as the velocity of the vacuum can during an equilibrium data point, and assume that

the evaporation rate is not time dependent, then the volume evaporated in time dt is -

V A .dt. (The correct area is A . and not A because the volume displaced by the vacuum can

e d do r J remains constant during the equilibrium point.) The volume added outside has two contributions: the added volume on the surface area of V A dt, and the volume that has been

o o vacated by the movement of the can = V,(A , - A )dt. The result is

V A + V. (A . - A ) + (V. - V. )A. - V A , = (26)

o o b

v d o i b

y

i e d

But under the conditions of the experiment, the level difference between the inside and the

outside remains constant, therefore

V.10= V (27)

The average velocity of net fluid flow through the flow tube (V) is expressed as

V = (V. - V

b )Ai /Ax (28)

where A is the cross-sectional area of the tube. Combining the above equations gives

X 1

As defined, V is positive for flow toward the inside.

24
4JC

cuE

'r—s-cdQ.X

cd

CD -C

4-> 4-O

to

CD

fO

S->E

CDE

s-

+-> fOE

CDo

to

CD

s-3

CT>

25
In order to cover a wider range in flow rates, the inside helium space was constructed with a wide cross-section for the top one-third of the inside capacitor, and a narrow cross-section for the bottom two-thirds. In order to allow large level differences for

inflow it was often necessary to have the outside level well above the top of the wide

section of the inside space, hence well above the top of the vacuum can. In all then,

there were four different combinations of inside and outside surface areas for which data

could be taken; the parameters, and the "mode" identification numbers used on the data

sheets, are listed in Table 1.

3.6 The Flow Tube

The flow tube used in this experiment was a commercial quality stainless steel tube

with an o.d. = 0.160 cm, and an i.d. determined to be 0.1149 ± 0.0006 cm. It had a length

(L) of 60 fcm and was wound in two turns of 10 cm diameter. The inside end was mounted

about 3.5 cm above the outside end. The ends were cut off square.

The inside diameter -was determined by room temperature gas flow measurements performed

after the experiment, but while the tube was undisturbed in its mounting on the apparatus.

The results seemed inconsistent until it was realized that, in laminar flow, even the

rather small curvature of the tube could have large effects on the gas flow. Helium gas

and nitrogen gas were used to cover a range in Reynolds' number (Re) from 55 to 1930. The

volumetric flow rate of gas was measured by a wet test meter whose calibration against a bell prover had a standard deviation of 0.6%. The pressure drop (AP) across the flow tube

was measured by the commercial capacitance manometer used in the experiment. The

volumetric flow rate was multiplied by the factor (1 - P

S /PQ

) to correct for the water

vapor added by the wet test meter (where P is the ambient pressure, and P is the vapor

pressure of water at the ambient temperature). The ideal gas equation of state was used,

the temperature taken to be ambient, and the density of the gas taken to be the mean of the

values calculated for the ends of the tube. The fractional pressure drop never exceeded

0.18. Thus we obtain the equivalent volumetric flow rate Q of dry gas. The data were analyzed by using the laminar flow equation for a straight tube applied

to the lowest Re data to deduce a diameter D. This was then used to calculate the (Fanning)

friction factor f as a function of Re. The equations used were

t

+ So) f = f2 ££[i + 3!L)

4P o

where P is the mass density of dry gas at ambient temperature and pressure. The viscosity q was taken from [26] and [27]. We have plotted in Fig. 11 the experimental friction

factor divided by the expected values for a straight tube in laminar flow; the solid line

26
CO go

ro ro CD CD

QJ CNJ O D oo 8 oo

CO -P

O)

s_

to

ro2o

CD

CD

S-

+-> ro

ScuQ.E

CD

+-> Eoo

cu

4-> CD jQ

4- 3 O -l->

00 S

+->

O

CO

<D CD

S- -£=

+-> CD I— O

CO

S-3

CD

C\J

C\J o

C\J

CO c£) COO

q. Lj

D L6J1S,

27
Table 1.

Mode Number

Inside Interface Location

Narrow Wide

Outside

Interface

Location

Vacuum Can

Support Tube

i (narrow)

A. (wide)

A

Q

(VC)

A

Q

(ST)

dewar

4.96 cm

78.4 cm

A. - 105.2 cm' dewar

A , - 2.8 cm dewar

219.4 cm

2

is the formula of White [28] for curved tubes calculated for our diameter of curvature D . The good agreement makes us confident that the correct value of the diameter has been

found. The principle uncertainty in its measurement is the uncertainty in the volumetric

flow measurement and its corrections. The deviations from the calculated values over the

entire range of the measurement indicate a standard deviation for D of about 0.5%.

Precautions were taken to prevent or reduce any effects on the flow due to the

presence of frozen air. The first was the protected location of the outside end of the

flow tube; it was in a chamber which was recessed about one inch into the bottom of the

base plate (see Fig. 7.). Such a location appears inaccessible to condensing air or falling air particles. Of course, only helium gas was allowed to be present in the dewar

during the cooldown. Another precaution was the addition of a filter (which used ordinary

chemical filter paper) onto the transfer line used to fill the helium dewar, in an effort

to reduce buildup of frozen air during the long run.

28
The only problem which could not be definitely eliminated was a cumulative buildup of

those particles small enough to remain in suspension in the liquid helium; they might be

expected to accumulate from air introduced during the insertion of the transfer tube, etc. There was no way to measure their concentration, nor are we aware of any data on the sizes

of particles that might be expected to remair in suspension. However, the lack of any

significant cumulative shifts in data points that were repeated during the 35-day run, we take to be good evidence that suspended solic particles were not a significant problem.

See, for example, the good reproducibility (^ 1%) of the T = 1.39, AT = 0.019 data taken

at the very beginning and end of the long rur (e.g., data points 9 422 and 10 515).

That evidence is reinforced by a mishap that took place early on October 1. A technician working nearby accidentally knocked off a large rubber vacuum hose used to pump

on the helium dewar. At the time, the liquic helium was probably warm enough to be at a vapor pressure above atmospheric pressure, bit still, in the 30-60 seconds it took him to replace the hose, we would guess that more air should have been introduced than during all the previous activities. Again, no significant changes in the repeated data can be seen following that date.

3.7 The Energy Flow Measurement

The inside helium vessel and the flow tube were surrounded by an insulating vacuum. Thus the energy flow through the tube could be determined from an energy balance

calculation. The essential elements of the situation are shown in Fig. 10. The unusual

feature of this calculation turns out to be the large influence of the vapor that is present.

The energy balance is done in two parts; first we calculate the change in total

energy of the helium in the inside vessel (the control volume) from thermodynamics applied

to a small change in its mass. The conditions of the change are that the temperature, and

hence the vapor pressure, remain fixed and that the total volume is fixed, even though the

total mass is changing. Designating the specific volumes of the liquid and vapor as v and

v

1 (not to be confused with the velocities v v , V), and the specific internal energies

as u and u' , we find that the increases of total mass M, of total volume, and of total

energy U are given by

dM = dm + dm 1

d(Vol)|

T p = v dm + v

1 dm 1 =

dU| t p w i = u dm + u

1 dm 1 (31)

where dm and dm 1 are the increases in the Vquid and vapor masses. Combining these

equations gives

dU

V u

• vu T.P.Vol \T - v dM (32)

29
We can use the definition of the chemical potential, eq (16), and the well known fact that

the chemical potentials of the liquid and vapor are equal in equilibrium, to find that

u

1 = u - P(v' - v) + T(s' - s) (33)

Substituting into the previous equation and using the Clausius-Clapeyron equation for the

slope of the vapor pressure curve (gTJcyn anci tne definition of dM in terms of the tube

cross-section and the average velocity

dM = pVA

xdt (34) we find

du| TiP>Vol

= [ P Vh - VT (^) svp

]A

x dt (35)

In the second part of the energy balance calculation we express dll in terms of the external sources of energy, i.e., in terms of the energy that flows through the boundaries of the

control volume. They are the heating rate Q of the internal heater and the energy flux

j A dt that enters through the flow tube. As discussed in section 2, this energy flux can

be expressed in terms of the enthalpy flux and the heat current q, using equations (13b) and

(19). Equating this to the previous equation gives

Q * (PVh q) ent

A

x = [pVh2

- VT

2 (|?) svpJ2]Ax

(36)

The energy flux on the left hand side is to be evaluated at the entrance of the flow tube

to the inside helium vessel. The energy change on the right hand side is to be evaluated

at the temperature and pressure that prevail within the inside vessel, whose values are indicated by the subscript 2. Let us define the heat flux density cu by

<2 = - ]t - ^(ItLpJ, < 37 > 2\5T/svpJ

2

If the temperature and pressure are essentially continuous at the entrance, which we expect D

to be true in most cases, we have h . = hp, and so q . = cu. In any case, by application

of eq. (20), we can find q(x) at some location x, if we know q^ and the temperature

and pressure at x. For our case the relation is

q(x) = q

2 + pvL-h(x)) (38)

We shall quote all our results for the energy transport as values of q 2

. As defined, the

30
sign convention of q„ is consistent with the one for V [eq. (29)], so that heat flow away

from the (higher temperature) inside vessel has a negative sign.

The second term of eq (37) is readily interpreted as the heat of condensation (or

evaporation) that must be absorbed by the liquid in order to change the volume of vapor.

Its presence is due to the particular configuration used to perform the experiment. At the

higher temperatures of this experiment, it becomes nearly as large as Q/A for the larger

values of V, and thus its presence causes a significant deterioration in the measurement

accuracy of q„. It is the quantity q ? that "^s most significant (not Q/A ) because it is the quantity to be compared to the Q/A of section 2. This heat of condensation also introduced an extra limitation in the range of velocities for which data could be taken. For fixed AT, it was often found that for inflow

(V > 0), when this term acts like an extra heat source, that its magnitude increased more rapidly with V than the heat conducted out (q„). Therefore, it was not possible for

eq (37) to be satisfied with positive values of Q if V was greater than some particular

value. When encountered, this limiting value of V is indicated on the graphs of q„ vs. V (figs. 22-25) by a vertical bar.

We have neglected the kinetic energy terms in the total energy balance because they

can be shown to be quite small for all the conditions encountered in this experiment. We

have also left out gravitational potential energy terms, because a careful accounting of

them found that they just cancelled the work done on the fluid by the movement of the

container. In Appendix 2, we rederive eq (36) with a full accounting of potential energy

terms and the work done on the fluid in raising or lowering the vessel.

3.8 Extraneous Heat Flows

The energy balance equations that have been given for determining the heat flow are correct only if we have accounted for all the energy exchanges, i.e., only if there are no extraneous heat flows between the inside and the outside helium vessels. Below are ex- plained the precautions taken and the tests made to ensure that these extraneous heat flows were small.

The poor thermal conductivity of the stainless steel walls of the vacuum can dictated

that it be surrounded by a heat shield, HS of Fig. 5. This took the form of a thin (0.04

cm) copper sheet fitted to surround the sides and top of the vacuum can. Its lower end was

always immersed in the liquid, and its upper end was soldered to the support tube ^ 10 cm above the top of the vacuum can, thus intercepting the heat conducted down the support

tube. On the inside of the support tube, the lower 10 cm was stuffed with coarse brass

wool to act as a radiation shield.

The inside helium space was supported by three long, thin-wall stainless steel legs

that attached to the bottom (or base plate) of the vacuum can. The rather close fit suggested the precaution of mounting three sharpened nylon screws on the legs just below the

wide section of the inside space; they could be extended so as to maintain a fixed spacing

between the inside space and the vacuum can. All electrical leads that went to the inside

31
space were first thermally "tempered" to the base-plate and were of low thermal conductivity wire.

It was not practical to maintain a good vacuum at all times in the vacuum space,

because this prevented the inside helium space from ever cooling down enough to fill with

liquid. Therefore the inside space was filled before cool-down with hydrogen gas at a pressure of a few Torr.* This acted as a thermal exchange gas, as long as the vacuum can remained above * 10 K. Once the vacuum can was cooled below * 3 K, the hydrogen gas was frozen to the walls or adsorbed onto about 3 cm of "molecular sieve" that was present.

After the first transfer was completed and the vacuum gauge outgassed, the seals were checked by monitoring the vacuum space for several hours with a He leak detector; it was checked again several days later. The vacuum was monitored throughout the long run by

a Phillips-type vacuum gauge mounted on the top of the support tube. It registered

2-5 x 10 * Torr during data taking, except for jumps of an order of magnitude (which

lasted 1-2 minutes) that were caused by the sudden withdrawals of a good fraction of the vacuum can from the liquid; this was almost certainly due to slight desorption of hydrogen from the suddenly warmer walls.

The thermal isolation was checked directly at the beginning of the run by measuring

the total thermal conductance between the inside and the outside at temperatures above the

lambda point. This was done by maintaining a fixed outside temperature T

2 = 2.31 K, providing a small heat input with the inside heater, and waiting 1-2 hours for the temperature

to equilibrate at a value 0.5-1.5 K higher than the outside temperature. The vacuum can

was completely immersed for these measurements, but the higher temperature of the inside

prevented liquid from entering. (It was found that this type of test can be very misleading if there exist any pockets on the inside that can trap liquid; the relatively large

latent heat of the liquid that is being evaporated on the inside and condensed on the

outside can cause a large and long-lived false conductivity).

Assuming that the thermal conductance linking the inside and the outside is temperature independent, the three data points that were taken yielded a value for the thermal

conductance of 1.5 x 10 W/K. This value is about 1 1/2 orders of magnitude larger than

was calculated for conduction through the legs, electrical leads, etc. It was found that

conduction through the low pressure hydrogen gas could account for the conductance, if the

indicated pressures at the Phillips gauge were taken at face value. Still, this value for

the conductance was low enough so that we could choose to ignore the unexpected persistence

-5

of unchanged indicated pressures (2 - 5 x 10 Torr) at the lower temperatures, where the

calculated vapor pressure of hydrogen should be considerably smaller.

These same measurements indicated that a heat input of 2 x 10 W remained at AT = 0. This could represent a real heat leak (e.g., incomplete radiation shielding), or it could

reflect a small temperature dependence in the thermal conductivity. This heat leak, if

real, would require that all values of cu be corrected by the addition of -0.019 W/cm . Since data could not be taken at low enough AT's to distinguish between the two alternatives, we have not made this correction to the data, and instead view this as an upper

limit to the possible systematic error in q 2

. *1.0 Torr - 133.3 Pa. 32
3.9 Error Estimates

The errors of this data are not always as small, nor as accurately known, as might be

wished, primarily because of the exploratory nature of the experiment. The method was new, the results could be only crudely anticipated, and it seemed more desirable to emphasize a broad range for the measurements rather than concentrate cm their accuracy. Given these

conditions, they seem satisfactory. The errors for quantities given in Appendix 1 vary,

depending on the conditions, and are summarized in Table 2. A part of the random error in the temperature and pressure difference measurements

could be evaluated quite reliably from the variation of the equilibrium point data. This

variation should include the effect of electronic noise, intrinsic sensor resolution, drift

in the sensor characteristics, and inaccuracies in the data reduction. Including all the

equilibrium point data, we find that the temperature difference had a standard deviation of

(4, 3, 2, and 4) x 10" 5 K at the four temperatures of 2.10, 1.92, 1.65 and 1.39 K respectively. The standard deviations for the height difference were 0.016, 0.012, 0.009 and

0.011 cm respectively. Through eq (21), these figures imply a standard deviation for the

pressure difference of 0.4, 0.2, 0.1 and 0.1 Pa respectively.

For the regular data points there may exist an extra source of random error that is not included in the figures given above; it is the fluctuations that might be introduced by

the control systems that are used to maintain, a "constant" temperature and pressure difference. The short term fluctuations in the altitude and temperature differences (time

constants of a few seconds) were judged by the error signals of the feedback systems to be

rarely more than 0.02 cm and 0.0001 K. However, because it is the time-average values that

count, it is the longer term fluctuations (the drift during the 3-5 minutes it took to record the data) that determine the real error. For the altitude difference, we estimate

that this source .of error was negligible. Unfortunately, the temperature sensors of the

control system (but not the temperature measuring system) were found to drift. On bad

days, these drifts caused progressive shifts in the measured AT of 2-3% over the course of

the day. In the worst cases, the rate of drift suggests that AT should change by an amount

somewhat less than the errors given above. Somewhat arbitrarily then, we increase the error estimates of AT by 50%. Our final estimate of the random error (one standard deviation) in AT is (6, 4, 3 and 6) x 10" 5 K, and in AP is 0.6, 0.3, 0.2 and 0.1 Pa, respectively, for the four mean temperatures.

The systematic error in temperature difference is determined by the systematic error

in the pressure measurement during calibration as discussed below. This leads to a systematic error in temperature difference of between 0.5% and 0.7%.

The pressure difference is subject to systematic error, due to uncertainties in the

capacitance-to-altitude conversion. Estimated at 2 parts per thousand, it implies a -9

possible systematic error of *v 2.8 x 10 x (z

? -z,)Pa, where z is expressed in centimeters.

This error becomes significant only for the large AT, small V, data at the higher temperatures.The errors in the mean temperature are determined by the errors in the vapor pressure

measurements used to calibrate the thermometers. We could evaluate the random errors in

33
Table 2. Estimated Errors

Random Error (la)

Systematic Error

Measured Temperature

Quantity

(Units) 2.100 1.919 1.650 1.395

T (K) 0.2 mK 0.2 mK 0.3 mK 0.4 mK

1.1 mK 1.1 mK 1.1 mK 0.8 mK

AT (K)

s 0.06 mK 0.04 mK 0.03 mK 0.06 mK

0.005 AT 0.006 AT 0.006 AT 0.007 AT

Az (cm) 0.01 cm 0.01 cm 0.01 cm 0.01 cm

0.002 Az (cm) 0.002 Az 0.002 Az 0.002 Az

AP (Pa) 0.6 Pa 0.3 Pa 0.2 Pa 0.1 Pa

0.03 Az (cm) 0.03 Az 0.03 Az 0.03 Az

V (cm/s) 0.02 V 0.02 V 0.02 V 0.02 V

0.02 V 0.02 V 0.02 V 0.02 V

q

2 (W/cm2

) 0.01(q

2 +0.047 V) 0. 01(q2 +0. 030 V) 0. 01(q

2 +0. 013 V) 0.01(q

2+0.004 V)

0.01(q

2 +0.047 V) 0.01(q2 +0.030 V) 0. 01(q2+0. 013 V) 0.01(q

2+0.004 V)

34
the calibrations at 1.92 and 1.39 K, from the standard deviations for data taken on several

different occasions; they were 0.2 and 0.4 mK respectively. A few simple tests indicated

that systematic errors due to placement of i: h e pressure probe could not be too much larger

than this. The absolute calibration of the pressure gauge could not be confirmed in the

region of interest; if we take the manufacturer's estimate and increase it by a factor of

three, we arrive at a systematic error in pressure of about 0.3%. This implies a systematic error of about 1 mK in mean temperature on the Tro scale.

The random error in V is determined from the standard deviation in the voltage-tospeed conversion (1.5%), the estimated accuracy to which the voltage fluctuations could be

averaged by eye, (1%), and the non-uniformity in the geometry (1%). Combined in quadrature, this gives a random error of 2%. An estimate of the error in geometry suggests

that each mode might have a systematic error as large as 2% of V. The relative error in q 2 (both random and systematic) depended rather strongly on the

conditions of the measurement, because of the sometimes large value for the vapor's latent

heat correction (second term of eq (37)). Assuming an error in V of 2%, we find that the error in q 9 is given by aV (V in cm/s), where a has the values of (4.8, 3.0, 1.3 and 0.4)

-4 2 x 10 W/cm for the temperatures of 2.10, 1.92, 1.65 and 1.39 K respectively. At the

most extreme value of V (70 cm/s), this corresponds to errors of 0.033, 0.020, 0.009 and

0.003 W/cm . When V is small, and q ? is large, this error is not too significant, and we must include the estimated random and systematic errors of the first term in eq (37) of

1% and 1% respectively.

35
4. RESULTS AND DISCUSSION

Data were collected at four values of the mean of the outside and the inside tem- perature. Using the formulas given in section 3, each data point was reduced to give

values for the actual pressure and temperature differences between the ends of the flow

tube and the resulting steady state values for the net fluid velocity V and the heat flux

density cu. These results are presented in tabular form in Appendix 1, along with some other data and derived quantities of interest.

4.1 The Net Fluid Velocity

Nearly all the results for the net fluid velocity V (the actual mass flow rate di vided by the total density and the flow tube cross-sectional area) are shown in Figs. 12-

15. The absolute values of V have been graphed there as a function of the absolute value

of the pressure difference, so that data for both directions of flow are superimposed.

The value of the nominal temperature difference is indicated by the symbols; the measured

temperature differences may differ from the nominal by as much as 3%, due to the drifts

discussed earlier.

More than half of the results might be summarized in a very simple statement: they

are largely indistinguishable from those of an ordinary fluid in fully developed turbulent

flow. This statement applies for those flows at the larger velocities or Reynolds num- bers, which are the ones most likely to be of interest for applications.

These graphs of JAP I vs JVJ reveal quite distinctly one of the important results of

this experiment: V depends primarily on AP -- its dependence on AT is significant only at the lower velocities. This result is only to be expected for an ordinary (single-phase)

fluid, because the temperature gradient does not appear in the equation of motion of the

fluid (we are excluding indirect effects due to buoyancy forces). In contrast, the tem- perature gradient appears explicitly in the (simple) superfluid equation of motion^

eq (6), as part of the chemical potential gradient. Even though that simple equation for

v is not expected to remain valid for our conditions, it suggests the possibility that v might be a function mainly of Ap; this would make V also a function of Ap, at least if p /p

is not too small. This possibility is completely excluded (for our conditions) by the

weak dependence of V on AT. To make a numerical comparison, we can use the expression

pAp = AP - psAT

which we have included in the data listing in Appendix 1. We find that under most con- ditions, the temperature difference dominates the chemical potential difference, and that

neither one has much influence on the net mass flow. As a typical example, we can examine

the data for T = 1 .92, AT = 0.019; not until AP is about 50 times smaller than

psAT (^ 2150 Pa) do we see much effect of the latter on V.

35
1000 1 I I I I I I

1

1

I f

T=2.100 K

AT=.0190

• o AT=.0031

100

CM

10

I I I 1 1 1 100

Figure 12. The pressure difference vs the net fluid velocity at 2.100 K, The dashed [solid] line is eq (41) [(42)].

37
1000

I CM

Q.

100

10

T~f 1 I I I I I I

T=1.919 K AT=.0470

• o AT=.0193

a A AT=.0031

I I I I I I I I 100

V| (cm/s)

Figure 13. The pressure difference vs the net fluid velocity at 1.919 K. The dashed [solid] line is eq (41) [(42)].

38
1000 1—I I I I

T=1.650 K

• o AT=.0190

100

CM

Cl10

Figure 14. The pressure difference vs the net fluid velocity at 1.650 K, The dashed [solid] line is eq (41) [(42)].

39
Q.

C\J

uuu 1 1 1 1 1 1 1

1

1 i i i i it/. / -4 T = 1. 395 K / JL

J /-

D AT= .0548 J /

• AT= .0197 J

/

— A AT= .0013

ft

w

/

100 — 7/ _

7 =

— */

—

9

A . /

r

—

n

% P

10

i 1

D

A

A A A

/

i

/im

th

f m

•

i

i l I 1

i

i i i i 1

10

V|(cm/s)

100

Figure 15. The pressure difference vs the net fluid velocity at 1.395 K. The dashed [solid] line is eq (41) [(42)].

40
A related, but not equivalent, observation is that neither v nor v becomes solely a

s n J function of AP in this same limit. This comparison is not quite as clear-cut as for V vs AP, because v and v (unlike V) can vary significantly along the flow tube, as can be seen from the expressions obtained from eqs (1, 2, and 19).

v = V +-9=

n psT

v = v- m-s.

s p

s psT (39)

The variation of both q and T can cause substantial changes 3 in v and v . Nevertheless,

n s

' the average of the velocities at the ends of the flow tube, plotted against the overall

pressure difference, ought to give a good indication of whether or not the local value of

velocity is solely a function of AP. Such plots of the two worst cases of correlation

between v or v and P are shown in Fig. 16 and 17. The data for a particular AT fall on

n s 3 r two lines, whose difference is correlated with the sign of AT (relative to AP). These

worst cases are also the cases where v and v differ the most from V. In those cases

n s where there is a good a correlation of v or v with AP (e.q., v at T = 2.10, v at

n s

v M ' n ' s T = 1.39, or all the small AT data) then it is also true that these velocities do not differ significantly from V. The final major observation to make is that the numerical results for V vs P are largely indistinguishable from what we would expect for an ordinary fluid at these velocities. For an ordinary fluid, the flow is known to be turbulent for Reynolds numbers

3 greater than 2 - 4 x 10 . The most reasonable choice (but not the only one!) of a coun- terpart for He II might be the "total" Reynolds number, defined by

Re=£TM

In

(40)

For all our conditions, this is equal (to within 15%) to 1.15 x 10 V, for V expressed in

cm/s. Then the range in velocities that we could cover corresponded to a range in Re of

3

5 - 85 x 10 , all apparently above the threshold for classical turbulent flow. If the

inside surface of the flow tube were smooth enough, we know that the pressure drop for an ordinary fluid should be given by the "Blasius Formula"

AP

B = 0.079 Re" (L/4) ^ [(l/2)pV2

] (41)

This formula is shown on each of the graphs (Figures 12-15) as a dashed line. We see that

the main trend of the data is reproduced. In the high velocity region, the formula underestimates the pressure drop, but for an ordinary fluid, a rather modest surface roughness

would be capable of making up the difference. We have no data on the surface roughness of our flow tube, so that we are not able to determine if this accounts quantitatively for

the difference.

41
1000

100

I CNJ D10

TT 1 1 I 1 III J—

I

T=2.100

• o AT=.0190

o •

co •

CD

J I

I I I I I I 1 J I I I I I I 10

|v | (cm/s)

100

Figure 16. The pressure difference vs the average superfluid velocity at

2.100 K. Cf. Fig. 12.

42
1000

100

Q10

1 I I I

I I 1

1

1 I I

I I I I 1 1—

I

T=1.395

AT=.0548

• o AT=.0197

D _ ©

• D

• o

O D

•

CD O D

o n

D

D

I I

1 I I I I M I I

I I I I I I 10 100

n' (cm/s)

Figure 17. The pressure difference vs the average normal fluid velocity

at 1 .395 K. Cf. Fig. 15.

43
This result is very appealing, because it suggests that the two-fluid dynamics re- duces to ordinary fluid dynamics in some limiting cases. It suggests, even if only further experiment can prove, that the pressure drop should change with L and D in the same way as the ordinary fluid results eq (41.). It might be suggested that the superfluid

fraction has been reduced to zero by the large flow velocities but that can be ruled out

on a number of grounds, including the results reported in the next section, which would

display no heat transport above the "enthalpy rise" value if the heat current were only

the thermal conduction of He I. Actually, it presents a difficult problem, because there

is no obvious way to derive such results from the current versions of two-fluid hydrodynamics.

For the purpose of a more sensitive comparison of the data, we have made a rough fit

to a constant friction factor formula, defined by

\

AP

C = f Jt [(l/2)pV2

] (42)

where f has the values 0.0062, 0.0062, 0.0067 and 0.0075 for the temperatures 2.10, 1.92,

1.65 and 1.39 K respectively. This formula is plotted as the solid line in figures 12-15.

The fractional deviations of the data from this formula have been plotted in Figs. 18-21.

This plot gives a more exact impression of the data coverage, scatter and departure from a simple behavior.

We have not yet found any simple correlation for the lower velocity data, nor have we

been able to specify what condition it is that determines just where that region starts. However, in all the conditions encountered in this experiment, this ignorance is in re- gions where the pressure drop is small enough so that it may be mainly of academic

interest.

No correction was made for end effects, because we have no experience that indicates

that they should be made. If such an extra pressure drop were present, of about the same

size as for ordinary fluid flow (say 1/2 pV ), then it would represent about a 6% correction to the data.

If the change due to the curvature of the flow tube were about the same for He II as for an ordinary fluid in turbulent flow, then there would be about 1% extra pressure drop

at 10 cm/s, and about 12% extra pressure drop at V = 70 cm/s [29].

4.2 The Heat Current Density

We have shown in sections 2 and 3 that the quantity cu represents the useful heat

(per channel cross-sectional area) that can be removed from a heat source by the fluid in

a cooling channel. We showed that for a heat source which is a small heated section in

the middle of a cooling channel of length 2L, assuming the other conditions are matched,

our experimental result for q 2 (V), V > 0, would represent the heat absorbed by the incoming fluid, and that the result for q2 (V), V < would represent an additional heat

conducted away by the departing fluid.

44
o^Jcvj

o cu

C\J E

o

s- 4-

l/> c

o

•I

+J o >

T3

CO

(/) -a

<u

-t-> E +j

CJ o —

oCNJ n CM

en

4-O

+J-a

ai

CO

s-CD

dxs

dV

D dX8

dV- dV

45
ooCM

CM

"3"

cr

aio

1-0 o

•I

>

-a

in

-a

<u

4->

4-> o oCMOI i—i

ro r^

m OO <T> UD .—i o i—i

<zr oi o o o .—i

II II II

ii 1— I— 1— 1— < <1 <1

<4- O

03

+->

cus-en oI

00 COI dxa

dV 5 dxs

dV- dV

46
o<3-

oCO

E

CM

cr

CDo

s-

<+-

o

+-> >

-o

-a

o

Q.

oCM

O

I

m

-o

cu

oCM

s-en

O

dxa

dV

5 3xi

dV- dV

47
:$

en r^ CO

LO i—

i

CD <3-

ex. o i—

i

un en o o o

i—

i

II II ii

II 1— 1— I— 1— < <] <l

o—

o

oCM

coo o oCMoCMoI oI o

CO o

I

dxa

dV

5 dx9

dV- dV

E

o

CM

cr

o

re>

CD -o

00

-o

CU +J

-t-> O

oCM 4-

1 o

<a +J

<o T3

CU -E

1— o<d- r—

1 CM

CD

S-

=5Ol

48
The results for q

2 are shown in Figs. 22-25, where they are plotted as a function of

V. The symbols indicate the nominal values ^or AT, but the explicit dependence on AP is not indicated; it can be inferred from the value of V. Constructing a correlation for this data is not so straightforward as for the net fluid velocity. In that case, mass conservation required that V be a constant, so that

once it was found that AP is mainly a function of V, we can infer that the local and

average pressure gradients are nearly the same. In this case, the temperature gradient

and the heat current density can turn out to be strong functions of position along the

tube.

We proceed in a manner similar to that found in section 2.2 The one-dimensional

energy equation [eq (20)] gives us an expression that relates q(x) to T(x), and to the

velocity and the local pressure; the latter two quantities can be taken directly from

experiment. The model building comes in trying to devise a successful second equation

that will allow solution for the unknowns q and T. The most natural choice we have come up with so far is to set the chemical potential

gradient equal to one of the mutual friction expressions using eq (10) with 3v /8t = 0. This amounts to setting q equal to some simple function of Vu. We used Vinen's expression [9],

Ps^ = *PsPn(iVv

S |-»o)VvS

' <«>

and took the values for A from his graph. The value of v (not very influential in the

calculation) was arbitrarily fixed at 0.5 cm/s, a value near his.

The equations were integrated numerically, the T at x~ fixed at T

2

, and q given

various starting values, until the temperature at the other end matched. The resulting

values of cu are displayed as the solid lines on the graphs of q vs V, Fig. 22-25.

The calculations do reproduce some features of the data. First, they do a reasonable

job of doing what they were originally designed to do, i.e., predict the V = data, the

poor spots being T = 1.395 and T = 2.100, where the calculated values are all about 10%

high. Second, for the higher T's they do a reasonable job of predicting the slope of cu, near V = 0. Finally, one curve (Fig. 25) does go through zero (albeit much more steeply)

about where the data reverses sign, at the lowest T and AT. This last feature, and the

other sharp drop at negative V, are found only for the conditions in which AP becomes

large enough to cause Au to reverse sign; we have the interesting possibility of a heat

current flowing toward higher temperatures. Unfortunately, the experimental evidence for

this current reversal is not quite compelling. The data had a small scatter, and were

reproducible, but the possibility of a systematic error (an extraneous heat input) of

about the size needed to cancel the effect, cannot be excluded.

Overall, though, this simple theory does not work very well, the most striking result

being that for both V < and V > the observed cu is less than expected. As originally

formulated, Vinen's arguments that lead to eq (43) clearly suggest that the equation

should remain valid when V ^ 0. These results lead us to conclude otherwise.

49
oID ^

oo o •l~

"~; c:o

CM

+-> o o rtS •r~ «d- S- >>t4-> •^ ^~ o ra o 3 1

+J

CD 3 > E

-o a>

•1

o E 3 3 CM CO ip 00

+->O) 4-> £3 n3

Ol +-> JZ

4-> 00 o 00 o > »P"

+-> •

i/> >> <a 3 *oo ^j i— o N^« !— 3 i— E oo O M- u c r—

cu to -a -o a •!-3 > 4-> <D r— o E -E <4- CM CU +J 1 S- -p

S- aj a; 3 s- c o

cu -M oo .c

f0 ai +->

<L> c

-C •r- >, r- -Q

O ,

3" 3 -a -a 1 4- •r- Q) QJ i— S- to o a; 3 oo -P

O) CD (0 JZ .c c

y- 1— 3 o . 1 CM

CM

CU

S- oCOI ( 2 "i3/M) 6 D

50
oCO ^

co en •rTM ^~

en c • o 1 r4-> U

fO •r- o S_

«d- >><<- +J

•r- i

O fO o 3 f +-> Ol 3 > E

"O CD

•1—• o £ 3 3 CM CO H^ COre

+-> CU -l-> c 03

CU -M -C

+-) CO o to O > •r—

-l-> • >> ro 3 CO +-> t— O ^^ •r" 3 i

e (/) O M-

(J

C i

(1) ro "O > •a a •!-3

+j CO i— o c -C C|_ CM ai +-> I s- +j

s- CU CU 3 s- c o

CD +J CO J=

<o a; +-> CD s=

.c: r- >, i— -O o 3 -o -o 1 ll- •r- CU

CU i— S_

CO

O CU 3 CO +->

a> CU ro -c -C C

h- 1— 3 oCO1 mCSi

cu

s-CD

Ooo o l

(^/Mrt)-

51
oLT> CO kO •!- r— CO +J -r-

(TJ -(-> O

>»-

+-> i_

•i- 4- OO i—

i— <o <D 3 > +->3 O E

•rTM3 CO i— CD M- E

+-> CO

<d to c: oj

CD +J

-£= fO

CO> c:o • >>•>- s

4-> +J

O

•i- (O i—

CO i— 4-

£= 3

(!) UTJ

-O r- -r- fO 3

4-> CJ i— C 4-

CD CD

S_ -C +J

S_ 4-> CD 3 C U CO

•I- CD

ITS (D-P

CD c

-c •<- >, r— -Q

3TIT3

M_ .,_ QJ CD r— Sc/1 O CD 3 CO +->

CD CU (O I— H- 3

CNJ

CD

S_3

(^3/M) 6 D

52
^c

Lf) •i

<Ti CO c - o

•r—

+-> 4-> o

03

5- >><<-

+-> •r- ; o <T3 o 3 i

+-> QJ 13 > E

-o <U

•1

e 3 3 i

to M03

+-> OJ +J c 03

<u +J

-C+J 00

i/>

O > # >i 03 3

+J 1

o

•i

=3 1

W o 4- c 1

OJ ra -o

"O o 3

4-> qj i

c -C UQJ +J

s_ +J

t- OJ QJ 3 S- C u 03 cu +J uo -SZ (O ai +-> O) c

_c •i> i

_Q

r—

Z3 -a "O

M- •r— QJ O) I SCO o QJ

13 to +J

QJ OJ "3 -C -C ST H- 1— rs # UD

CM

CD

5-

Z3en

tn *l- oo CNJ O o o o ( 2 UJ3/M) 2 D53
These results seem to be suggesting that we might retain Vinen's identification of

the source of mutual friction with the presence of vorticity in the superfluid fraction,

if we now suppose that the net fluid flow is capable of generating superfluid vorticity,

in amounts over and above that generated by the mean counterflow of the two components.

5. ENHANCEMENT OF HEAT TRANSPORT BY FORCED CONVECTION

Without understanding quantitatively how mutual friction is increased by mass flow we

can still draw some qualitative conclusions which indicate that in some circumstances at least, much more heat can be transported by forced convection than by "natural" convection.

Our reasoning is based again on investigating the limit as the heat current goes to zero everywhere except near the heater. The energy equation for He II also integrates

immediately to give the now familiar result that the total heat absorbed by the fluid is

equal to the enthalpy difference times the flow rate. For the incoming fluid (V > 0), we get q 2 = pVC AT; for the departing fluid (V < 0), we get cu = 0. The positive velocity

portion of the enthalpy-rise-heat-current is given on the graphs (Figs. 26-29) of cu vs

V. This value should, and does, act as a lower bound for the useful heat that can be

rejected by the heat source. If the extra mutual friction caused by the mass flow has not "killed off" the heat current, then we can expect an extra contribution.

Now let us consider a comparison of "natural" to forced convection. Suppose we want

to use the flow tube from these experiments to cool some localized heat source. First,

let us estimate the best that we can do at 1.8 K with pure counterflow, with two tubes

connected to the source, but no forcing of circulation around the loop. We suppose that

we have a pressurized system so that we can increase the AT to 0.300 K. At best, the heat

current per tube will increase as the cube root of T, and since q 9 (T = 1.92, AT = 0.047,

2 V = 0) = -1.11 W/cm , the best we can do (ignoring the temperature dependence of the proportionality constant in equation (8)) is

i 1/3

(1.11) x 2 x(£|0

Second, how well can we do with forced convection? The enthalpy-difference-heat-current

(the lower limit) for the same AT is

f,

•2.1

^ C dT = V x (0.21 J/cnT)

If we have enough head (2.5 m) to force the flow at 200 cm/s, we get

42 W/cm2

which is 10 times larger. The pump work divided by the heat transported is
o

\o CD .c

4->

s_o4-

10

cd=3 o<d- <o>

CD

i/i •1—s->»

Q. i

oCM

4-> C

OJ

CD .c

+>

cu

SO fO

to

*. cu oCMoI o

OJ

CM

CM

to

CM

s- oCOI ( 2 ^3/M) 6

55
D

D

O

O

O

° o

o

D O

O

<

NZ O CM I—

1

CTi r-~ en oo

i—

1

^3- r-l o

CT> o o o

•

II II II r—1 h- 1— 1—

II < < < 1— D o <

oVO

CU -G

+->

s-o

4-

C/l CD3 o 1

«3- re>

CD

to

s->>Q.

o (O

C\J

+->c

CD

O)

4->

CD

s- o

to ^TMs

CD

I/) C

oCNJ

o^3-

o

I

O

CNJ

KO CNJ COo

«3- o

o00 O I

CD

ro

CNJ

. CTI • •r- l/l u_ —

1—

4- <l O

+-> ro c

4-> CD

ro i_ Tl CD 4-

CD 4-

\— -a

r\J

CD

CD

(^o/Mrt)-

56
oVO CO

4->

S-o<4-

CO

CO3 o i

** (0>

COCO

s->>Q.

03 o J= CM -M

cu

4->

CO

s-

<T5 o COCO '—»

•1— to "— E CO

.CT o 1— ^^> o • COI <*

CM

CD • •p- to u. -1— 4- < o o ro c:

«d- -M CO T3 CO 4-

C0 <_ oI

i— -a

COCM

CO

s-CD

<=?? ( 2 "i3/M) 6 b57
o

O

O

^ B

D

D

o oCO

i %

D O

D

±i- CO r-^ ro

<3- CTi i—

i

LD LO i—

i

o

Ol o o

ro •

II II II 1— h- hi—

i

< < <

°o

<3

<

<

o

O < o

I

OCO

I LO ^- ro C\J o o o o

( z wo/M) 2

t)-

oI

58
AP-V ~ 3.6 xlO

3 Pa ~ 3.6 x 10

3

pVJC dT 0.21 J/cm3 2.1 x 10

b

P

1.7%

If 17% pumpwork were acceptable, then another factor of 3 increase in heat absorbed is

possible.

While this example may not be typical of any prospective application, it does

illustrate how in some situations forced cooling might be of interest.

These experimental results for q« show the transition from this purely classical

and predictable heat transport to the more or less predictable values for zero mass flow -- the pure counterflow regime.

6. CONCLUSIONS

This experimental study enables the following generalizations to be made regarding

combined heat and mass flow in Helium II.

1. Velocity of flow is primarily a function of pressure gradient. Towards the higher

velocities encountered in these experiments, the relationship between velocity and

pressure gradient is independent of the temperature gradient and strongly resembles a classical fluid. At lower velocities a complex relationship exists between V, AT and

AP, which needs further clarification.

2. At zero velocity, axial heat transport typical of the "mutual friction" regime is

again confirmed. Either positive or negative velocities reduce this heat transport

below what would be calculated from the Vinen theory, and, as the magnitude of the

velocity increases, classical forced convection heat transport is approached. For strongly negative velocities there is some evidence that a reversal in the direction

of the heat current takes place as predicted from the Vinen theory. More definitive

experiments would be required to confirm this.

3. In practical situations, axial heat transport far in excess of the usual pure

counterflow values may be achieved by imposing mass flow or forced convection with

the usual classical pressure drop.

7. ACKNOWLEDGEMENTS

Vincent Arp played a vital role in initiating and encouraging this project, and in

defining the problems which it addresses. Jim Siegwarth generously loaned experimental

equipment and expertise. One of us (W.J.) would like to acknowledge support provided

through the National Bureau of Standards Postdoctoral Research Associateship Program in

association with the National Research Council.

59

J—
8. REFERENCES

[I] G.v.d. Heijden, W.J. P. de Voogt, and H.C. Kramers, "Forces in the Flow of Liquid

Helium II, I" Physica 59, 473 (1972).

[2] G.v.d. Heijden, A.G.M.v.d. Boog and H.C. Kramers, "Forces in the Flow of Liquid He II,

III," Physica 77, 487 (1974)

[3] W. de Haas, A. Hartoog, H.v. Beelen, R. de Bruyn Ouboter and K.W. Taconis, "Dissipation in the Flow of He II," Physica 75, 311 (1974).

[4] W. de Haas and H. van Beelen, " A Synthesis of Flow Phenomena in He II," Physica 83B , 129-146 (1976).

[5] J. Wilks, The Properties of Liquid and Solid Helium, Oxford University Press (1967).

[6] W.E. Keller, Helium- 3 and Helium-4, Plenum Press (1969).

[7] R.J. Donnelly, W.I. Glaberson and P.E. Parks, Experimental Superfluidity, University

of Chicago (1967).

[8] S.J. Putterman, Superfluid Hydrodynamics, North-Holland (1974).

[9] W.F. Vinen, "Mutual Friction in a Heat Current in Liquid Helium II," I - Proc. Roy. Soc. A240, 114 (1957); II - Proc. Roy. Soc. A240, 128 (1957); III - Proc. Roy. Soc. A242, 493 (1957); IV - Proc. Roy. Soc. A243, 400 (1958).

[10] R.K. Childers and J.T. Tough, "Helium II Thermal Counterflow: Temperature -and

Pressure - Difference Data and Analysis in Terms of the Vinen Theory," Physical Rev. B13 , 1040 (1976).

[II] D.R. Ladner, R.K. Childers and J.T. Tough, "He II Thermal Counterflow at Large Heat

Currents," Phys. Rev. BJ3, 2919 (1976).

[12] V. Arp, "Heat Transport Through Helium II," Cryogenics ]0, 96 (1970).

[13] C. Linnet and T.H.K. Frederking, "Thermal Conditions at the Gorter-Mel 1 ink Counterflow

Limit Between 0.01 and 3 Bar," J. Low Temp. Phys. 2J, 447, (1975).

[14] G. Krafft, "Kuhlung Langer Rohrsysteme mit Superfluidem Helium," Kern Forschungszentrum Karlsruhe Report KFK1786, (1973).

[15] R.K. Childers and J.T. Tough, "Superheating in He II," Low Temp. Phys. LT13, 2> 359

(1972).

[16] R. Eaton III, W.D. Lee and F.J. Agee, "High-Heat Flux Regime of Superfluid Cooling in

Vertical Channels," Phys. Rev. A5, 1342 (1972).

[17] L.D. Landau and E.M. Lifshitz, Fluid Mechanics , Pergamon Press (1959).

[18] R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena , John Wiley (1960).

[19] S.R. de Groot, Thermodynamics of Irreversible Processes , North-Holland (1951).

[20] E.F. Hammel and W.E. Keller, "Dissipation and Critical Velocities in Superfluid

Helium," International Conference of Low Temperature Physics - LT10, Moscow, Vol I, p. 30 (1967).

[21] P.H. Roberts and R.J. Donnelly, "Superfluid Mechanics," Ann. Rev. of Fluid Mech. 6,

179 (1974).

[22] R. Meservey, "Superfluid Flow in a Venturi Tube," Phys. of Fluids 8, 1209 (1965).

60
[23] P.M. McConnell, "Liquid Helium Pumps," (U.S.) National Bureau of Standards Internal

Report NBSIR-73-316 (1973).

[24] F.G. Brickwedde, H.v. Dijk, M. Durieux, J.R. Clement and J.K. Logan, "The 1958

He

4 Scale of Temperatures," J. of Res. NBS 64A, 1 (1960).

[25] S.G. Sydoriak and R.H. Sherman, "The 1962 He

J Scale of Temperatures I," J. of Res. 68A , 547 (1974).

[26] R.D. McCarty, "Thermophysical Properties of Helium-4 from 2 to 1500 K with Pressures

to 1000 Atmospheres," (U.S.) National Bureau of Standards Tech. Note 631 (1972).

[27] H.J.M. Hanley and J. Ely, "The Viscosity and Thermal Conductivity Coefficients of

Dilute Nitrogen and Oxygen," J. Phys. Chem. Ref. Data 2, 735 (1973).

[28] CM. White, "Streamline Flow Through Curved Pipes," Proc. Roy. Soc. AJ23, 645 (1929).

[29] H. Ito, "Friction Factors for Turbulent Flow in Curved Pipes," Trans. ASME, J. of

Basic. Eng. , June 1959, p 123.

61
9. NOMENCLATURE

English

A Gorter-Mellink constant

^d' ^i ' ^o horizontal cross-sectional area

A cross-sectional area of the flow tube

C isobaric heat capacity per unit mass

C, C , C-, , C2 capacitance

D diameter of the flow tube

E total energy of the helium

g gravitational constant

h, h-, , hp, h . enthalpy per unit mass

j

u (je

) the energy (total energy) current density

£ active capacitor length

L flow tube length

L length of quantized vortex line per unit volume

M (m, m') the total (liquid, vapor) mass of helium within the inner vessel

P, P], P 2

. p sv pressure

P , P (for gas flow measurements) the ambient pressure, the partial

pressure of water vapor

Q heat current or rate of heat addition

q» Q-|» q

2

» %ut heat current density

Re Reynolds' number

s, s

1 , S-,, S2 entropy per unit mass

T, T-|, T

2

, Tent temperature

U total internal energy of the helium within the inner vessel

u, u

1 , u-j , Up, u . internal energy density per unit mass

v, v' volume per unit mass

v , v normal and super component velocities (cross-sectional average)

V net fluid velocity (cross-sectional average)

V

b

, V , V., V vertical velocities

x axial coordinate

Z T

z 2' z b'

z ent

altitudes

62
Greek

A

£, e

1

n. n

p. p r M

U)Q

n

(indicates the difference of two values)

dielectric constant

coefficient of viscosity

p mass per unit volume

Gibbs potential per unit mass (chemical potential)

vorticity

volumetric flow rate

Subscripts

1 (2) I or > for the outside helium bath (inner vessel)

o (i)

) e, (b, d) for evaporation measurements (the vacuum can, the dewar)

ent at the entrance of the flow tube to the inner vessel

SV, svp along the liquid-vapor coexistence line

u (e) associated with the internal (total) energy density

Superscripts

primed and unprimed quantities distinguish between the vapor and the liquid,

respectively

63
APPENDIX 1

The following tables give the important results for almost every data point taken

during the long run. Listed are the average and the difference of the temperatures of the

inside and the outside helium vessels (TMEAN,DT), the altitude difference calculated from

the preceeding two quantities (DP), the velocity of net fluid flow through the flow tube

(V), and the nominal heat current density at the inside end of the flow tube (Q2).

The sign convention is determined by taking the positive direction to run from the

outside toward the inside. Therefore positive temperature differences indicate that the

inside was hotter than the outside; a positive pressure difference, that the inside end of

the flow tube is at a higher pressure than the outside end. The negative values of Q2

indicate that heat is flowing towards the (cooler) outside, etc. All temperatures are given in kelvins (measured on the Trg He vapor pressure

scale), thev pressure difference is given in pascals (- 1 newton/meter = 10 dyne/centimeter ), the altitude difference in centimeters, and the velocities are given in centimeters/second. The reader should be alert to a practice that is commonly found in the

He II literature and this paper, and that is the mixing of MKS and cgs units. Therefore,

when various quantities are multiplied together, further multiplication by powers of ten may be necessary to maintain consistent units.

Each data point is identified by numbers found in the ninth column of the table; the

month and day on which a data-taking session was started is indicated by the numbers under

DATE , the sequence number for that data point is indicated under NN, and the mode number

that indicates the location of the liquid interfaces is given under MO (see Table 1). The

data points have been sorted: first by TMEAN, then by whether they are regular data points

or equilibrium points (Q2 = 0) and finally by chronological order.

In columns 7 and 8 are given the results for the two auxiliary thermometers TIB and

TOB. They were intended to measure the temperatures near, but not at, the inside and

outside ends respectively. (See the text for an account of the relatively large uncer- tainties in this particular measurement). The temperature of the outside helium bath was subtracted from the indicated temperature at each location to give the listed values of DTIB and DTOB. Several other quantities of possible interest can be calculated from the experimental

results by using known values for some thermodynamic properties and some formulas from

section 2 and 3. We have listed RDMU, the density times the difference in the chemical

potential, which is given by AP - psAT. Also tabulated is Ql , (Ql = q-,), the nominal heat

flux density at the outside end of the flow tube, defined by Ql = Q2 + (z^-z-JpV. Finally, we have tabulated the nominal values for the normal- and super-fluid velocities at the two ends from the formulas

64
VN1 = v + #r > ps i'i

vsi =

p

n 01 & *«

Ps s l'l

The thermodynamic properties were taken from a power- law interpolation of tables A and E of

Putterman [8].

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75
APPENDIX 2

The purpose of this appendix is to prove the assertion that the formulas already given

for the overall energy balance and energy flux -- in particular eq (36) -- are still

correct when the gravitational and work terms are taken into account. Our procedure is as before, i.e., equating the change in energy calculated by two different methods.

In the first method, we calculate the change in the total energy E of the helium

contained within the control volume (the inside vessel), due to the increment in its mass, when the temperature is held constant. It is

dE = dU + dE

grav

(A2J)

where dE is the change in the gravitational potential energy, and dU is again the

change in interhal energy of the helium.

We can prove that the variation of u within the liquid is negligible, by considering

its variation with depth,

du _ (du\ 61 (du\ dP fA2 2

,

For equilibrium within the control volume, with a gravitational field present, we still

have dT = 0, but now dP/dz = -pg, i.e., the pressure varies with depth within the liquid.

However, using the measured values of expansivity and compressibility for the liquid to evaluate the size of the pressure derivative, we find that

du _ 3z " P\

is much smaller (< 1%) than the corresponding variation of the gravitational potential

energy density with depth (= d(gz)/dz = g), so that it can be safely neglected.

Since u everywhere within the liquid has nearly the same value that it has at the

surface, we can repeat the entire argument used to eliminate from dU any explicit dependence on the vapor; we obtain the same equations [eqs (31) - (35)], as long as we realize

that the pressure in eq (33) is the pressure of the vapor. Therefore, eq (35) is still

valid if we evaluate the enthalpy of the liquid at the pressure of the surface, which we designate as P'p.

We may evaluate the change in the gravitational potential, without loss of generality,

by assuming the inside space to have a constant horizontal cross-section A-, and by evaluating the gain or loss of liquid at the surface (altitude z~) and at the bottom of the

inside space (altitude z b

). It is

dE

grav =

9 (z2PA

i dz

2

" VW (A2 - 3)

76
We can express this differently by using eq (28) and the identities dz

2/dt = V- and

dz. /dt = V. , and then combine it with the result for dU, to obtain

dE=(pu2-P'

2

-T

2(!V Vdt r svp,T,

+ pg(z

2

- z

b

) VjjA^t + pgz

2VAxdt (A2.4)

where, as before, T

2

is the inside temperature, and u

2 is the liquid energy density at this

temperature.

The second method is to calculate the energy that passes through the boundary of the

control volume. It is the sum of the electrical heat, the work done on the fluid by the

walls of its container, and the energy that enters through the flow tube.

dE = dQ - dW + j e Axdt (A2.5)

The work term is evaluated simply by the pressure-volume work done on the fluid by the

top and bottom of the inside space, taking into account the hydrostatic pressure difference.

-dW = -P' 2

V

b

A.dt + (p' 2 + g(z

2

- zj) V^-dt

= g(z

2

- z

b

) V

b A.dt (A2.6)

The total energy current j contains, as before, the enthalpy current and the heat

current, but we must also add a gravitational potential energy current, gzent

pV, where z gnt

is the altitude at which the fluid enters the inside space. The enthalpy is to be evaluated at the pressure at which the liquid enters, which 'is, by continuity in the pressure,

the local fluid pressure at the altitude of entry, P' 2 + pg(z

?

- z t

). Collecting terms,

we find that the z„. terms cancel to give

ent a

J'e = (p%t + p, 2 + W + P9z

z) V • (A2 ' 7)

Then substituting into eq (A2.5), and equating it to eq (A2.4), we find that all the

remaining terms containing g cancel, and we recover eq (36).

77
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-1002

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

MEASUREMENTS OF COMBINED AXIAL MASS AND HEAT TRANSPORT IN He II

5. Publication Date February 1978

6. Performing Organization Code

275.05

7. AUTHOR(S)

WARREN W. JOHNSON AND MICHAEL C. JONES

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2750551

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY N&TES

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

bibliography or literature survey, mention it here.) An experiment was performed that allowed measurements of both axial mass and

heat transport of He-II in a long tube. The apparatus allowed the pressure difference

and the temperature difference across the flow tube to each be independently adjusted

and the resulting steady-state values of net fluid velocity and axial heat transport

to be measured. For the larger Reynolds numbers, it was found that the relation between pressure difference and net fluid velocity was nearly indistinguishable from

that of an ordinary fluid in turbulent flow. The axial heat transported was found to

be suppressed from the values that were calculated by assuming that "mutual friction"

was unchanged by the net fluid flow, but it was always found to be larger than the

"enthalpy rise" value. Taking this second value as a lower limit, it is shown that

a mild extrapolation of these results suggests that (in appropriate circumstances)

forced convection would allow much greater heat to be transported in long cooling

channels than could be transported by "natural" convection alone.

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)

Axial heat transport; forced convection; helium II; measurements; 1.4 - 2.1 K; pressure drop.

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