where ix end sl ON the liquid levels on the inside and outside respectively, referred to

ION arbitrary point. The difference in the vacuum capacitance, and the (small) vertical

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

While the outside and inside are iv equilibrium.

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

This Wes obviously true for the ins10; capacitor, and wes assured for the outside capacitor

by placing it inside a copper tube whose lover end was immersed in the liquid. Any re-

. sidual radiation leak to the capacitor could be absorbed by Ow high conductivity of the

superf!"41 helium film.

The Capacitance dif4rence was meaCred directly by the use of the system shown in

Fig. 9, which incorporates the use of a emmercial capacitance bridge of the ratio-trans-

former 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 leve'; difference between the inside and the outside, the

desired capacitance difference was set on the bridge, then the corresponding level dif-

ference was approximated by manual adjustment of tit, 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 controlled offset signal to form the velocity command

for the motor-controller system. 7, digital volt-meter at this point served to monitor the

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

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

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 Co 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 equili-

brium, 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 Al at this value, and the bridge controls changed back so that AC(0) could be meas-

ured.

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

and polerizability found in NBS Technical Note 631 (261, the liquid densities taken from

MOTOR

CONTROLLER

----ITACHOMETER

II

VELOCITY COMMAllimlDVM

SIGNAL

CONDITIONING 14---- MANUAL OFFSET

ERROR SIGNAL

PHASE SENSITIVE

DETECTOR

1 DETECTOR

CAPACITANCE BRIDGE

EXT. STD. UNKNOWN

1

VACUUM CAN

MOTOR

I

REDUCTIONGEAR

'""."1"Inirrk"1"1116

TORQUELIMITER

LEADSCREW

Figure 9. A schematic drawing of the system used to maintain a constantdifference between the inside and outside levels.

22n8

the tort 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 Clausius-

Mossotti equation as found in the above reference, which had the effect of making the

polarizability too small by a factor of 4n/3M = 1.047 (M = molecular weight). With this

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

1.05736 and the measured value of 1.05740. It was found that while the values of s and t'

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

thousand over our temperature range. Therefore, the calculated value of t t' = 0.05714,

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

value, was used for all temperatures. Cv/i 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 z2-z1 actually used in equation (21),

cmz2 - zi = -31.27 7 (Ac(22-zi) - AC(0)) (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 f.'sirable to keep equilibration times short, so that equilibrium

points were taken only where the inside level was still in the narrow section, thus limit-

ing the comparison to the lower 2/3 of the inside capacitor and the lower 1/3 of the out-

side 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

coaxiality.

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 z2 was 25 cm below the outside level, z1.

229

3.5 Mass Flew 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

macs 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 Vo, Vi and Vb respectively. The

areas of the inside liquid surface, the outside liquid surface, and the dewar are desig-

nated as Ai'

Ao

and Ad.

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 = 0

During the small time interval dt, the volume added inside is (Vi-Vb)Aidt. If we designate

Ve(< 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 -

VeAddt. (The correct area is Ad and not Ao because the volume displaced by the vacuum can

remains constant during the equilibrium point.) The volume added outside has two contribu-

tions: the added volume on the surface area of VoAodt, and the volume that has been

vacated by the movement of the can = Vb(Ad - Ao)dt. The result is

V0A0 + Vb(Ad - Ao) + (Vi - Vb)Ai - VeAd = 0 (26)

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

outside remains constant, therefore

Vi = Vo (27)

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

V = (Vi - Vb)Ai /Ax (28)

where Ax

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

V =Ai Ad

(Vb-V

e)A- ATA

o

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

24 r)

(29)

r

Figure 10. A schematic diagram of the main variables of the experiment.

31

In order to cover a wider range in flow rates, the inside helium space was construct-

ed with a wide cross-section for the top one-thi.d 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. m 0.160 cm, and an i.d. determined to be 0.1149 ± 0.0006 cm. It had a length

(14 of 60 cm 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 mall curvature of the tube could have large efects 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 we 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 - Ps/Po) to correct for the water

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

pressure of water at the ambient temperature). The idea* 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 (1 of dry gas.

The data were analyzed by using the laminar flow equation for a straight tube applied

to the lowest RI 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

f n2 D5' 32 n2 2Po

4P,,0

Re = -r2-nUn

(30)

where Po

44 the mass density of dry gas at ambient temperature and pressure. The viscos-

ity n 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

WHITE' 3 FORMULA

1

1.6

1.4

1.2

1.0

0.8

o He Gas

O N2 Gas

100

Re11V D

1000

Figure 11. The results of the room temperature gas flow measurementson the flow tube.

33

Table 1.

Node Number

Inside Interface Location

Narrow Wide

Outside

Interface

Location

Vacuum Can 1 3

Support Tube 2 4

Ai (narrow)

Ai

(wide)

Ao (VC)

Ao (ST)

Adewar

4.96 cm2

78.4 cm2

Adewar- 105.2 cm

2

Adewar - 2.8 cm2

219.4 cm2

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

food. The principle uncertainty in its meaurement 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 locetion 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 dower, in an effort

to reduce buildup of frozen air during the long run.

The only problem which could not be definitely eliminated was 4 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 mo way to mmure their concentration, nor are we aware of any data on the sizes

of particles that might be expe_ed 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 (N I%) 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, but 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 mcuum.

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' (not to be confused with the velocities "n'

vs'

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'

d(Vol)IT,p = v dm + v' dm' = 0

dUlT,P,Vol

= u dm + u' dm' (31)

where dm and dm' are the increases in the 1:,quid and vapor masses. Combining these

equations gives

dUl_ v'u - vu'

T,P,Vol VT (32)

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' = 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 2.1:svp

and the definition of dM in terms of the tubecu

cross-section and the average veld t'

dM = pVAxdt (34)

we find

= [pVh - VT (g) eve)Axdt (35)dUIT,P,VolV

In the second part of the energy balance calculation we express dU 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

JnAxdt 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 4, using eqqations (10b) and

(19). Equating this to the previous equation gives

+ (pVh 4)entAx = (pVh2 - VT.UT(")

svp,T211tx(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 q2 by

q2 = - - VT.(0csvp,T2 (37)

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

to be true in most cases, we have hent = h2, and so gent= 42 In any case, by application

of eq. (20), we can find 4(x) at some location x, if we know q2 and the temperature

and pressure at x. For our case the relation is

4(x) = q2 + p42-h(x), (38)

We shall quote all our results for the energy transport as values of As As defined, the

30

36

sign convention of q2 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 vo;ume 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 0/Ax fer the larger

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

accuracy of q2. It is the quantity 42 that is most significant (not 0/Ax) because it is

the quantity to be compared to the 0/Ax of section 2.

This heat of condensation also introduced an extra limitation in the range of veloc-

ities 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 (q2). Therefore, it was not possible for

eq (37) to be satisfied with positive values of 0 if V was greater than some particular

value. When encountered, this limiting value of V is indicated on the graphs of q2 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 fcand 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

correct only if we have accounted

extraneous heat flows between the

plained the precautions taken and

were small.

that have been given for determining the heat flow are

for all the energy exchanges, i.e., only if there are no

inside and the outside helium vessels. Below are ex-

the tests made to ensure that these extraneous heat flows

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 ti 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 sug-

gested 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 37

space were first thermally "tempered" to the base-plate and were of low thermal conduc-

tivity 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 dawn enough to fill with

Therefore the inside space was filled before cool-down with hydrogen gas at a

pressure of a few Torr.* This xted 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 cm3 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 10-5

Torr during data taking, except for jumps of an order of magnitude (which

lasted 1-2 min es) 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 hydro-

gen 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 T2 = 2.31 K, pro-

viding 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 misleaL-

ing 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 en the

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

Assuming that the thermal conductance linking the inside and the outside is tempera-

ture independent, the three data points that were taken yielded a value for the thermal

conductance of 1.5 x 10-3

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

of unchanged indicated pressures (2 - 5 x 10'5 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 W remained at AT = O.

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 q2 be corrected by the addition of -0.019 W/cm2.

Since data could not be taken at low enough AT's to distinguish between the two alter-

natives, we have not made this correction to the data, and instead view this as an upper

limit to the possible systematic error in 42.

*1.0 Torr - 133.3 Pa.32

38

3.9 Error Estimates

The errors of this data are not always as email, 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 desirnble to emphasize a

broad range for the measurements rather than concentrate on 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 P.

A pert 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 respec-

tively. The standard ueviations 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 dif-

ference. 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 devia-

tion) 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, respec-

tively, 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 system-

atic 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

possible systematic error of N, 2.8 x 10-2 x (z2-zI)Pa, where z is expressed in centimeters.

This error becomes significant only for the large AT., small V, data at the highell. tempera-

tures.

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 eir

Table 2. Estimated Errors

Rand mo Error (10)

Systematic Error

MeasuredQuantity(Units)

Temperature

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) 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 ^.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

q2 (W/cm2) 0.01(42+0.047 V) 0.01(42+0.030 V) 0.01(42+0.013 V) 0.01(42+0.004 V)

0.01(42+0.047 V) 0.01(42+0.030 V) 0.01(42+0.013 V) 0.01(42+0.004 V)

4034

the calibrations at 1.92 ,..nd 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 the 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 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 syste-

matic error of about 1 mK in mean temperature rn the T58 scale.

The random error in V is determined from he standard deviation in the voltage-to-

speed conversion (1.5%), the estimated accul y to which the voltage fluctuations could be

averaged by eye, (1%), and the non-uniformity in the geometry (1%). Combined in quad-

rature, 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 q2 (both random and systematic) depended rather strongly on the

conditions of the mRasurement, oecause of the sometime 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 4, is given by 0 (V in cm/s), where u 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 utlue of V (70 a/s), this corresponds to errors of 6.033, 0.020, 0.009 and

0.003 W/cm2. When V is small, and q2 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.

4135

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 42. 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 IAPI vs IVI 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 oily 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

vs is not expected to remain valid for our conditions, it suggests the possibility that vs

might be a function mainly of p; this would make V also a function of Ay, at least if ps/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

phy = 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 (1, 2150 Pa) do we see much effect of the latter on V.

3644

1000

100

10

T2.100 K

is a AT..0190

o AT..0031

10

IV! (cm/s)

100

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

3743

1000

100

a.

a.04

a.

10

1

=PM

Tal . 919 K

0 AT"' .0470

e o C.0193

A La.0031

1

1 1 1 1 tl ill

Iv1(cm/s)

100

Figure 13. The pressure difference vs the net fluid velocity at 1.919 K.

The dashed [solid] line is eq (41) [(42)].

38 44

'1000

100

10

i

T1.650 K

o 0..0190

MED

%me

/I I 11111 I 1 1111 11

rat

=MI

10

lv l(cm /s)

100

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

39 45

1000

Tal.395 K

0 AT=.0548

$ a Als.0197

A a ATs.0013

00 000

A

=NI

10 100

IvI(cm/s)

Figure 15. The pressure difference vs the net Quid velocity at 1.395 K.The dashed [solid] line is eq (41)1(42)3.

40

46

A related, but not-equivalent, observation is that neither vs nor vn becomes Solely a

function of AP in this same limit. This comparison is not quite as clear-cut as for V vs

AP, becalme vs and vn (unlike V) can vary significantly along the flow tube, as can be

seen from the expressions obtained from ego (1, 2, and 19).

vn= V + V

s=

psPn

psT (39)

The variation of both 4 and T can cause substantial changes in vn and vs. Nevertheless,

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 vn

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

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

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

where there is a good correlation of vn or vs with AP (e.g., vn at Ti 2.10, vs at

I 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 veloc-

ities. For an ordinary fluid, the flow is known to be turbulent for Reynolds numbers

greater than 2 - 4 x 103. The most reasonable choice (but not the only one!) of a coun-

terpart for He II might be the "total" Reynolds number, defined by

oV0Re = (40)

"n

For all our conditions, this is equal (to within 15%) to 1.15 x 103 V, for V expressed in

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

5 - 85 x 103

, all apparently above the threshold for classical turbulent flow. If the

inside surface of the flow tube were smooth enough, Vitt know that the pressure drop for an

ordinary fluid should be given by the "Blasius Formula"

4LAPB = 0.079 Re-(1/4) ir [(1/2)OV

2] (41)

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

thb main trend of the data is reproduced. In the high velocity region, the formula under-

estimates 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 47

1110

(cm/s)

100

Figure 16. The pressure difference vs the average superfluid velocity at2.100 K. Cf. Fig. 12.

42 48

1000

100

10

1

/NM

/NM

Imm

T1.395

0 AT.0641300

0 AT-.0197 II0

0

00 0

0CO

00

o

00

0

0

1 10

170 (cm/s)

100

Figure 17. The pressure difference vs the average normal fluid velocityat 1.395 K. Cf. Fig. 15.

43 49

This result is very appealing, because it suggests that the two-fluid dynamics re-

dWces to ordinary fluid dynamics in some limiting cases. It suggests, even if only fur-

ther experiment can provis, that the pressure drop should change with L and 0 in the same

we as the ordinary fluid results eq (41 ). It might be suggested that the superfluid

fraction has been reduced to zero by th t. 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 hydro-

dynamics.

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

to a constant friction factor formula, defined by

4LAPc = f [(1/2)pV

2]

D- (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. 16-21.

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 aide to specify what condition it is that determines just where that region starts.

Howaver, in all tne 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 pV2), then it would represent about a 6% correc-

tion 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 Deatty

We have shown in sections 2 and 3 thnt the quantity 42 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 21, assuming the other conditions are matched,

our experimental result for 42(V), V > 0, would represent the heat absorbed tv the in-

coming fluid, and that the result for 42(V), V < 0 would represent an additional heat

conducted away by thc departing fluid.

44 50

-60 -40 -20

V (cm/s)

20

Figure 181 The data of Fig. 12, plotted as deviations from eq (42).

Si'

IanATI.0031

A AT,0193

a ATI,0467

.60 -40 -20

V (cm/s)

0 20 40

Figure 19, The data of Fig. 13, plotted as deviations from '4 (42)1

52

1 1 < < 1

.60 -40 -20 0 20 40

V (cm /`)

Figure 20, The data of Fig, 14, plotted as deviations from eq (42),

53

:Yid(

T1,395

61,0013

A AT2.0197

AT,0548

(0/1)

Figure 21, The data of Fig, 15, plotted le devietIoni frame' (42),

The results for 42 are shown in Figs. 22-25, where they are plotted as a function of

V. The symbols indicate the nominal values for AT, but the explicit dependence on AP is

not indicated; it can be inferred from the value of V.

Constructing a correlation for 4- 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 deneicy 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 leg (20)j gives us an expression that relates 4(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 4 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 Ovs/Ot = 0.

This mounts to setting 4 equal to some simple function of Vp.

We used Vinen's expression Oh

Ns= Apspn vn-vs I - ve) (en vs) (43)

and took the values for A from his graph. The value of vn (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 x2 fixed at T2, and 4 given

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

values of q2 are displayed as the solid lines on the graphs of 4 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 e 0 data, the

poet- spots being T = 1.395 ana T = 2.100, where the calculated values are all about 10%

high. Second, for the higher l.'s they do a reasonable job of predicting the slope of 42,

near V = 0. Finally, one curve (Fig. 25) does 20 through zero (albeit much more steeply)

about where the data reverses sign, at the lowest 1. and AT. This last .eature, and the

other sharp drop at negative V, are found only 1% the conditions in which AP becomes

large enough to cause Ap to reverse sign; we have tht interesting possibility of a heat

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

this current reversal is not quite compelling. The data had a small &latter, 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

beng tat for both V : 0 and V > 0 the observed q2 Is less than expected. As originally

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

should remain valid when V 0 1. These results lead us to conclude otherwise.

r 94

T'2.100 K

0 r' .0190

A es ,0031

0

-80 -60 -40 -20 '0 20 40 60

V (cm/s)

Figure 22, The useful heat current density vs the net fluid velocity at 2.100 K.

The solid lines are the calculations that assume mutual friction is

unaltered by the net fluid flog',

T14919 K

ATE,0470

o A1'12,0192

a ATc,0031

-60 -40 -20

V(cm/s)

Figure 23, Lo useful heat current density vs the net fluid velocity at 1,919 K.

The solid lines are the calculations that assume mutual friction is

unaltered by the net fluid flow,

5

1,0

0,8T21 , 6 60 K

o AT2.0190

0 00 0

-60 -40 -20 0 20 40 60

V(cm/s)

Figure 24. The useful heat current density vs the net fluid velocity at 1,650 K.

The solid line is the calculation that assumes mutual friction is

unaltered by the net fluid flow,

0,5

0,4

0.3

0.1

J

T1,395 K

o ATi,0548

o AT2.0197

a ATE 0013

cP

0

0

00

0

0

0

o0

0

00

a a 4

-60 -40 -20 0 20 40

Y(cm /s)

Figure 25. The usiful heat current density vs the net fluid velocity at 1.395 K,

The solid lines are the calculaWns that assume mutual friction is

unaltered by the net fluid flow, 59

These results seem to be suggesting that we might retain Vinen's identification of

the source ofmutual 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 aeraunts over and above that generated by the mean counterflow of the two components.

5. ENHANCEMENT OF HEAJ 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, such more heat can be transported by forced convection than by "natural" convec-

tion.

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 q2 = pVCDAT; for the departing fluid (V < 0), we get 42 :11 O. The positive velocity

prtion of the enthalpy-rise-heat-current is given on the graphs (Figs. 26-29) of 42 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 an 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 counterfloy, 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 42(T = 1.92, AT = 0.047,

V = 0) = -1.11 W/cm2, the best we can do (ignoring the temperature dependence of the propor-

tionality constant in equation (P)) is

0300)1/3(1.11) x 2 x(-E

7OX7= 4.1 W/cm

2

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

(the lower limit) for the same AT is

2.1

1.8 FCfl dT = V x (0.21 J/cm

3)

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 to

0 0

1.

0.

0.2

0

T-2.100 K

00

0

a

06?

A

oo

43

A

O0

asa

.

o0

oo

0

o

0o e.0190A AT=.0033

0

o

1 1 I 1 1

10 -60 -40 -20 0 20 40 60

V (cm/s)

Figure 26. The data of Fig. 22. The lines are the enthalpy rise values for thedifferent AT's.

61

2.0

0,8

0.4

a

A

T1.919 K

a g',0470

0 AT', 0192

a AT*40031

oo

a

a

0

00

aa

V

a

0

0

00

Li

00 0000

C11°°

-80 40 -40 -20

V(cm/s)

0 20

J40 60

Figure 27. The data of Fig, 23. The lines as the onthalpy rise values for the

different AT's,

1.0

0.8

0.6

N

X

N

I 0.4

0.2

0

I I 1

AO -60 -40 -20 20 40 60

T1.650 K

o ATo.0190

0

0

0

0

0

0

V(cm/s)

Figure 28. The data of Fig. 24. The 1 is are the enthalpy rise values for the

different AT's.

63

0,4

0,3

0.2

aI

0.1

T1.395 K

ATm.0548

o 4411.0197

a AT=.0013

0

00

1

0a

DC

000

oo

00

0

a a

-0.1 _J

.80 -60 .40 .20

=1J20 40 60

V(cm /s)

Figure 29, The ckta of Fig. 25, The lines are the enthalpy rise values for the

diffednt'AT's.

APV 3.6 x103 Pa 3.6 x 103 -1.7%

pVIC dT 151-37&?p

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 q2 show the transition from this purely classical

and predict2ble 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 6 5

. REFERENCES

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

Helium II, I" Physic. 473 (1972).

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

III," Physic. 77, 487 (1974)

00 W. de Haas, A. Hartoog, H.v. Beelen, R. de Bruyn Ouboter and K.W. Taconis, "Dissipa-

tion in the Flow of He II," Physics 75, 311 (1974).

[4] W. de Haas and H. van Beelan, " A Synthesis of Flow Phenomena in He II," Physica 83B,

129-146 (1976).

[5] J. Wilke, The Properties of Liqu ariSolidlieliiiim Oxford University Press (1967).

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

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

of Chicago (1987).

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

[9] W.F. Vinen, "Mutual Friction in a Heat Current in Liquid Helium-I1," I - Proc. Roy.

Soc., mg, 114 (1957); II - Proc. Roy. Soc. A240, 128 (1957); III - Proc. Roy. Soc.

AAR, 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).

[11] D.R. Ladner, R.K. Childers and J.7. Tough, "He II Thermal Counterfiow at Large. Heat

Currents," Phys. Rev. 813, 2919 (1976).

[12] V. Arp, "Heat Transport Through Helium II," Cryogenics 10, 96 (1970).

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

Limit Between 0.01 and 3 Bar," J. Low Temp. Phys. 21, 447, (1975).

[14] G. Krafft, "Kuhlung Langer Rohrsysteme mit Superfluidom Helium," Kern

Forschungszentrum Karlsruhe Report KFK1786, (1973).

(15] R.K. Childers and J.T. Tough, "Superheating in He II," Low Temp. Phys. LT13, 1, 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 1, p.

30 (19E7).

[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

He4 Scale of Temperatures," J. of Res. NCS 64A, 1 (1960).

[25] S.G. Sydoriak and R.H. Sherman, "The 1962 HPScale 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 63i (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] C.M. White, "Streamline Flow Through Curved Pipes," Proc. Roy. Soc. A123, 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

fnolish

A Gorter-Millink constant

Ad, A Ao horizontal cross-sectional area

Ax cross-soctional Area of the flow tube

Cp isobaric heat capacity per unit mass0u,

0v , C

1,C2

capacitance

O diameter of the flow tube

E total energy of the helium

g gravitational constant

h' hl, h2, hent enthalpy per unit mass

Ju (4) the energy (total energy) current density

active capacitor length

flow tube length

Lo length of quantized vortex line per unit volume

M (m, m') the total (liquid, vapor) mass of helium within the inner vessel

P' P11 P2' Psv pressure

Po'

Ps (for gas flow measurements) the ambient pressure, the partial

pressure of water vapor

0 heat current or rate of heat addition

q, q1, 42' gent heat current density

Re Reynolds' nomlsor

s, s', si, s2 entr "oy per unit mass

T, Ti, 12, Tent temperature

U total internal energy of the helium within the inner vessel

u, u', ul, u2, uent internal energy density per unit mass

v, volume per unit mass

vn, vs normal and super component velocities (cross-sectional average)

V net fluid velocity (cross-sectional average)

Vb, Ve' Vi, Vo vertical velocities

x axial coordinate

zl, z2, zb, zent altitudes

Greek

A (indicates the difference of two values)

c, dielectric constant

q, qn coefficient of viscosity

p, pn, ps Miss per unit volume

p Gibbs potential per unit mass (chemical potential)

w vorticity

0 volumetric flow rate

Subscripts

1 (2)

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

APPENDIX 1

The following tables give the important results for almost every data point taken

during the long rur. Listed are the average and the difference of the temperatures of tte

inside and the outside helium vessels (TMEAN,DT), the altitude difference calculated from

the preceed4 two quantities (DP), the velocity of net fluid flow through the flow tube

(V), and the nominal heat current density at the insi& 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 out de; a positive pressure difference, that the inside end of

the flow tube is at a higher p.,!ssure 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 T58 Ne4 vapor pressure

scale), the pressure difference is given in pascals (= 1 newton /meter2 = 10 dyne/centi-

meter2), the altitude difference in centimeterJ, and the velocities are given in centi-

meters/second. The reader should be alert to a practice that is cos-manly found in the

He II literature and this paper, and that is the mixing of MKS and cgs ,nits. Therefore,

when various quantities are multiplied together, further multiplicatioh by powers of ten

may be necessary to maii.tain consistent units.

Each data point is identified tqf numbers found in the ninth column of the table; the

month and day on which a data-taking s.-ision was star:=44 4' iocilated by the numborr under

DATE , the sequence number for that data point is irtc;; n- .016er XN, and the mode number

that indicates the location of the liquid interfaces given under MO (see Table 1). The

data points have been sorted: first by TMEAN, tnen by whether they are regular data points

or equilibrium points (Q2 = 0) and finally by chronolojical 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, v.:lich is given by AP - p;AT. Also tabulated is Ql, (Ql E 41), the nominal heat

flux density at the outside end of the flow tube, defined by Ql = Q2 + (z2-zi)pV. Finally,

we have tabulated the nominal values for the normal- and super-fluid velocities at the two

ends from the formulas

64

I 0

VN1 = V +

VS1 = - PSsl

, etc.

Ps 1

The thermodynamic properties were taken from a power-law interpolation of tables A and E of

Putterman [8].

4611 AN

K)

10035

41634j$.1 032

010029

0,0034

i1C034

t30036010021

1014051

01

IKI 1CM)

.31913.14.46

53011 14.34.31497 46511

601597 14685

64150 0.610

4411174

031814 .67

631493 1466

11412 1401 4

JP

IPA)

61-11413A261

-.3

44115

101,4

"1609.4

it1111C454 .31145 11sel 6P:so

4616433 601441

140454 011113 list1 .0802,1101 01463 10.Z3 :stel

4014454 01907 .1e I 46 .45811014011 641933 146.!

414481 01943 1360 03.1:101000 01841 11041 41.5

4510064 01b99 11.:: 4185

,bliob, 641899 11601 4386

2010471 01336 146)4 "01

201;1036 01ill 140803 .7

2614:44 010197 1,1.84:o '4363

2614444 01497 61143 Z164

2610046 Otil7 1781,A 32622610046 801191 12,4 hitt

251004* .31495 61484: . r2614444 0150 1S*01 14,12614043 010b 13893 Le?MOM 621515 6B855 1085240047 saA90 14.1t do('

261045 1:1690 1486 6181

2610037 60013 1S800 .16

2.100,1) 631946 24131 -Web2.10036 0/893 65.0 111/1

2610073 630332 2E1 .0

2610069 .J0336 6160 116s

261404 60C329 63.29 -1;9/

26/00bi 600334 618.11 11:61

1610071 .30326 2.56 .1261005 0032? 4.5MOO?. 630324 411

2.171 $3041 30:3 0782

2610007 J0307 19.32 31084

2610606 $441;2 1180 2240161600 830314 2.1.1 ;I5261g72261007/

614344 6Z843

.20307 -.2.

'44

.11;81

Appendix 1. Experimental Data.

V 02 3108 0110 48111NNM0

1CM15) (W/CM21 100 IKI

4.43

31655

31681D630

46725864?.48654

0.00

0.,0

31.ar

-sz.4

6497.6533

1114.6448

.001

.1154

1499

604468050

'6154

.30095

*00334

04021:3030

.01053

.0. .4

.03623

03316

633151

.603043

.4343

.018 ?'

.01212

6017:1

6%104

.01952

.31611

401832

641795

902021

.31262

.01762

317 7 1

90 4 4911 9 3

91710 1

91713 3

1014 391715 3

91716 1

920 3 1

920 4 4

920 5 32t$15

-0.2621856

97d9.4199

036

6323380,3106

00013

641346

alass.3147?

920 6 4

920 1 3

920 5 462584,

.A.93e1.24

.6230

6208-.272

620306

.43306

.00221

41085

.01865

011825

920 9 4

32310 3

92011 4.8248 00221 631825 52012 3

21,36 68259 30221 .01825 92013 10103 .1497 03106 .31908 92014 1

4.04 8496 .30015 601644 92110 110873 -.b67 603028 641470 92111 1

615.5* 8319 03084 .01686 92112 119427' .717 .11 01438 92113 1

-.297 .03136 .01696 92114 10.00 .8496 802004 .01629 92115 1

11.93 8649 .40)021 01510 92116 11100 -.360 .00071 601682 9211? 114.0 .627 .830023 .01527 92118 1

69846 .6388 00052 611674 92119 1;944 8497 *03061 601624 92120 1

0.03 68497 9.40065 01648 10 2 3 137.32 '926 .00052 .01170 10 2 4 4

-36,43 8140 800394 401714 10 2 5 3

0.04 820 800030 *00224 IC 210 1

.11.64 8222 100045 *00143 10 211 11(8 8202 800047 800102 1P '12 1

"4145 8212 83004? 801173 10 [44 1

0J0 -.289 .00205 10 214 15471 ,040 601354 e;..0172 10 215 1

9(!bS .03057 804164 10 216 130.61 .1;94 80005i 800143 10 217 3

'iloMV

49.12

qtsid

8021

66056.121

811022

64434003044

.00285

.4:1ro

.00156

10 210 3

14 219 3

10 220 3Gw 6820 101056 *0449 IC 221 1

'1,o1; 810 830002 040264 10 222 1

RONU

1PAI

01 VN2

ICN/S1

VN1

ICN/SI

VS2

ICN/S/

VS1

ICN/SI

354264 .6497 1.24 1632 4607 3.66

4673.7 '0164 35622 37.11 46.69 38076.338063

3513.36096.411

38.10'1424

40610

.1.32

3666S447

.31.16

3.69

30134 1,417 0462 .74.47 71609 10119.3164A 16201 51640 61666 50647 4915632500 16045 4000 91634 4803 .40680

330365 6499 1624 .1632 4.67 3670

44265 6494 '1623 1631 4.66 36;6

362962 6202 29.55 31.13 40.03 33617342462 6330 33621 39602 1.31 26,66.339465 6253 24610 23.1.8 33.91 26,03430,0 *,762 27,76 41630 29632 216633529.9 6293 19675 20.81 28.61. 2305.M0.0 6766 26637 20602 2303 .20,3/46050 063 27645 0695 '2449 1627.3474 0011 21691 23611 14636 1569110652 005 22683 4600 19677 166914476 a 6690 22601 -23.21 /86111 16611153461 649? .1.24 1631 4691 3.69

35311.4 6496 1623 1631 4606 LAT53644 6345 10.02 15182 23.1.8 19.29349143 663$ 16635 .1762? /2645 104554566 6323 17645 13.1.1 26.30 21.66

3164360 6660 18.05 1949 .15620 13616391164 6496 1624 1631 4686 3660

.3924.8 6364 11.32 11591 19.29 1500349766 6601 12660 13,37 8625 .7,32

352160 6409 9.10 9650 18482 13670

349289 .501 10642 114:0 4665 9,14330266 .1497 .1.24 1612 4687 3.69

9424 ; )7 1624 1632 4667 3.60

3665,2 1163 35601 3640 46.39 36533744 901 .37634 49630 '33662 3063161661 .8205 .04 04 2.41

0960 '6265 1262S 12637 ..9683 69659204 624S 9.66 904 12.73 12.31

60969 1268 106SS 10665 8102 101360465 6209 .674 67S 2.1.1 2.29611,1 6270 4695 4.99 86/9 7691

546,0 '8254 662? 6632 3638 .3.34464,0 8119 .30693 '31621 29691 .2961225085 *6191 90639 5694 8.62 5669333584 6176 49613 069657 49607 4702-515.6 .6215 26649 .2603 25614 2464056014 .6286 03 04 248 2.27

37,4 242 17,62 1/677 15661 0,24

HEIN 07

(111 10

4016907 600005

.2e1t494 .00004

4011039 .3J002

.2801973 8100140

2,11074 ,Jo w.

2.01010 .0044

609062 003063

609063 .00003

601017 .444061

Appendix 1 (continued). Experimental Data.

31 UP i Gi

I641 RAI 101/S1 111/6421

81 0 0810 0.000

.i1 .2 0.01 0.000

880 $1 0800 0.040

03 00 0000 6.310

.00. ,9 0,01 443,10

O3 0 0404 6,000

5,14.813 it0b2 000 MOO.112 ''..! ma 0.460.:1 .,0 0.04 0.016

102384 .12883 0.51 to2 we1; ...Al

1.92413 802018 -1507 bl 001 .9611.92411 e32941 1804 -31.e :404 1.1611.92467 .02867 1265 41,[ 24814 0004

1091935 .01.728 2503 of/ 6.03 -1.108

1091942 .01.721 tai 11.0 1504 14611091941 .04714 27.72 25.1 21.94 1.448

1o91141 .1.011 1201 14111 '45823 8545

1.91948 .04716 '310/3 63.7 33,87 1.635

1.91944 114701 .3c.et 61.5 :w4 1.5691891957 214706 28166 35s4 0.0 10081.91952 844702 -1786 1361 5.21 84601.91969 04464 .26,22 .162 11191 10191891954 .31.094 .2507 t.4 J. J0 -1.109

1.91906 234694 640 10081.91943 .146t7 24801 150 7,13 164141691943 .31.687 2420 1526 220 1.0131091943 134688 26.6 411 11.07 10861.91943 804666 2609 411 10.99 1062

POO 011B OATENNMO RONU

10 10 IPA)

.J1171 .44051 917 1 1 5.6

.030b7 .0020! 917 2 1 428

86013 800294 117 3 1 3 :..10049 ei16251 t.i17 6 1 .'s

03069 .00233 917 0 1 913.0J014 .03123 917 6 1 916

01011 e44666 91711 1 769761

.4006 106054 41712 1 561B31024 I40129 9111? 1 1,6

.15139 .02685 9 2 2 1 32764

.03137 .02702 9 2 3 1 3296.5

$31117 .02572 9 2 4 1 33440131257 .827211 9 2 5 1 21749

.00231 204438 910 1 1 530901101136 104242 910 2 4 32168$00120 604142 910 3 4 53324.01011 404569 110 4 4 5110.4.4019 .03940 910 5 4 31240016e .041412 910 6 4 917189

03109 604624 910 7 4 5334.1

30767 .31.350 910 1 4 9117.9

40136 604264 910 9 2 6527146

.33226 .64412 91010 1 27960

.00226 44410 911 1 2 52116444455 911 2 2 525663

100298 .04455 911 3 1 25663600161 .04276 911 4 1 527448

100161 .04276 911 5 2 527468

60 0.000 0600 0.00 600 611

31 UP i Gi

I641 RAI 101/S1 111/6421

81 0 0810 0.000

.i1 .2 0.01 0.000

880 $1 0800 0.040

03 00 0000 6.310

.00. ,9 0,01 443,10

O3 0 0404 6,000

5,14.813 it0b2 000 MOO.112 ''..! ma 0.460.:1 .,0 0.04 0.016

01 VNI VNI VSI Vit

14/0421 (ous) icmis) MOD IC9091

0.000 0.90 0.00 0001 1111 .

'2;.:141 0444 0800 600 011t:

MOO 0.01 600 OM 611r0.000 61C Q,00 601 Poll,

0.000 0110 00 0.00 01115)

0.000 600 611 0.00 61060.000 MI 0800 600 60140.000 1001 660 0.00 IAN0.500 610 0000 600 MO

4961 4.10 660 3.78 3111

6961 4.18 660 3870 630020 19.65 21.25 29.16 27.11'

1.138 27020 .29,63 21841 41.91

'16108 405 616 11,1.0 4,11

022 9.1.5 11166 11069 16618111 1903 17,61. 17861 14.11

11850 0,97 01 43.07 611.*0667 2616 3693 4637 hal010 00802 0619 31,11 31614

024 19019 2107 31,61. 16416671 .3603 43051 3619 190909 6.36 7,13 17809 1641

.14109 409 611 11.1.0 Mai

11110 .401 611 4,41 4,10'.

1017 11.47 13.22 3610 2031.211 11054 13.30 3811 1414970 5.96 CU 1611 16916968 5149 6.11. 1601 1649

1.161 $31117 .02572 9 2 4 1 33440 020 19.65 21.25 29.16 27.11'

1.92467 .02867 1265 41,[ 24814 0004 131257 .827211 9 2 5 1 21749 1.138 27020 .29,63 21841 41.91

1091935 .01.728 2503 of/ 6.03 -1.108 .00231 204438 910 1 1 53090 '16108 405 616 11,1.0 4,11

1091942 .01.721 tai 11.0 1504 1461 1101136 104242 910 2 4 32168 022 9.1.5 11166 11069 1661

1091941 .04714 27.72 25.1 21.94 1.448 $00120 604142 910 3 4 53324 8111 1903 17,61. 17861 14.11

1o91141 .1.011 1201 14111 '45823 8545 .01011 404569 110 4 4 5110.4 11850 0,97 01 43.07 611.1.91948 .04716 '310/3 63.7 33,87 1.635 .4019 .03940 910 5 4 3124 *0667 2616 3693 4637 hal1.91944 114701 .3c.et 61.5 :w4 1.569 0016e .041412 910 6 4 917189 010 00802 0619 31,11 31614

1891957 214706 28166 35s4 0.0 1008 03109 604624 910 7 4 5334.1 024 19019 2107 31,61. 1641

1.91952 844702 -1786 1361 5.21 8460 30767 .31.350 910 1 4 9117.9 6671 .3603 43051 3619 1901.91969 04464 .26,22 .162 11191 1019 40136 604264 910 9 2 6527146 909 6.36 7,13 17809 1641

1891954 .31.094 .2507 t.4 J. J0 -1.109 .33226 .64412 91010 1 27960 .14109 409 611 11.1.0 Mai

1.91906 234694 640 1008 .00226 44410 911 1 2 521164 11110 .401 611 4,41 4,10'.

1.91943 .146t7 24801 150 7,13 16414 44455 911 2 2 525663 1017 11.47 13.22 3610 2031691943 .31.687 2420 1526 220 1.013 100298 .04455 911 3 1 25663 1.211 11054 13.30 3811 1411091943 134688 26.6 411 11.07 1086 600161 .04276 911 4 1 527448 4970 5.96 CU 1611 1691

1.91943 804666 2609 411 10.99 1062 100161 .04276 911 5 2 527468 6968 5149 6.11. 1601 1649

Appendix 1 (continued) . Experimental Data.

IRAN 01

41 MI

.69i941 04964441939 6046824411411 0467'411931 144614

14911934 0466441933 46511690934 $44664

591125 4463111922 4446229591924 44623is11913 .,.27MM. 446211491924 644628101920 .3024

1.91923

101923 64462(1

1.91922 624414

141314

.30315

04619

111131, 0245?

1.91962 .30315

.00315

04441

1511942

1511942 040315

1591941

1591942 .30315

1,9193b 034301191934 .36313

1691934 .30313

16019* 6503131.91962 .00339

1691437 .00314

1091934 633312

1691939 00301

101939 /40113

1091965 600343

141937 etit1311

101943 634361

1191942 02413691936 .40312

1.91936 76313

1091946 .30322

141953 ....0316

1.91950 .00316

1691947 6611428

1691447. .31931

1.91953 .01926

1.91953 .01923

1.91951 631325

1691955 014261.91955 601322

1.91955 .01931

101966 .31929

LI

ICN1

625621

24,9825412544

bei.

.1638

"45.14

21,98'240122611.25641

25,51-23.40

13.0.24425,412400

'1500612.35

4 le..4

s6.50

1015

.4.67.

1645

604.51

-3.35

61136Z41.t..74'208'056:03-1.17

2611-1.33

-1.,

.160

1.045.153.0

61004'25/0

5.2J

296i96694

19.1211654

1065

45.18

3P

IPA;

6011.0

16J

bell

45141

3174o.2

4906.3

IND' 66.6

4,029e41),d

6.5

..,i

1161

0.5

5062

6,

.1001

*5.6

4.3.8

4;04

.3o031.0

644.6

22.9

-21.9

leJ

14.5

16711.0

1e7,7

.46 J

5.t

30.loS

1.2

-57.3

OM3.4

'216022663

24.6013763

15064315.3

Z64

3.6

V 02

ICNOS1 IN/CN21

644 6148300 64636.5J 162630.00 '10108

1/50 4144900 61368

6144 '1005

25609 630.00 6184

.26/1 42413.06 0491344 100845,94 166616.21 -.ill

4,04 '18106

619 1.2214,24 16436

-058.003

2694

21.23

440

403i .6496

'2142

.25605

'095

::;6946

.21628 63592145 l395

16.17; 636517644 '66416

15$73 .40515,21 6.437

Goia .49611000 .6436

11.05 0153eli .645746E6t 0464

-t. '6 "6406

7647 6473..5o09 '6414

500 .0460.00 "6495

0.34 006(4o17 '07026.82 6301

4644 66450

.i.2. '6956

49634 642C',1oi3 494434649 .52736.65 692950.13 .333

0.00 6552

0.40 .851

otoa

110

4224442594417544221601994

.01200604222

4054300220604463

4414944149104465.4060

.44224

.44164

110274

00144

.03023

44022

.30263

e43414

.00331

.30024

444360043403061061434

03335

64J426

641049

610030

.34034

oG004S

42434600443

60434;

604040

133033

600319

.28016

43419.33494

.04059

.30219

00455.40190

04062603247

.00096

.33092

OTle

IKI

43954414432444365604536

d0.522104393

44463644336

604447

44199(04194

604370407604335

644293

443/3

60261C

602716

e021364024/

40246604276

40243644233

.00265

.40310

4427600346.00260

604307

.00310

.03267

.00325

014282

.00333

604343

.00302

438264

40250400243

.00242

4178001664601040

41641.01007

.01o93

641007

.01712

101778

PATENNNO

911 6 P

911 1 i

911 8 2

911 9 2

91110 3

91111 3

91112 2

91113 1

91114 2

41119 1

91116 1

91111 2

91118 2

91119 1

91120 1

31121 1

91122 1

914 4 1

914 9 1

926 9 1

12811 1

92611 1

92812 1

92813 1

92814 1

92815 1

32116 1

92817 1

92818 1

9E819 1

92820 1

92821 1

12822 1

92823 1

92824 1

92825 1

92026 1

92821 1

92626 1

929 4 1

929 5 3

929 6 3

929 9 1

92910 4

92911 3

92912 4

92913 3

42914 4

92915 3

92916 1

930 3 1

RCMU

IPA)

426142556042666923690936364926721621964169441996442461652461911241724

41934411/4

-.3322:::;

41694

:II:::

-353.3

391.3

34693194260'3820'32960

3690635210

633614

'357.9

'340039062'3410"3500.3464'354/1

'35007

361,6412.9.24663

416466- 2391.0

6193642

2427.22024664297621066,0621644

2167.9

01 VN2

111/CN21 ICN/S1

VNI

ICN/S1

VS2

ICN/S1

V31

ICl/11

10,41 4679 949 4644 4,40

6t,191 .746 49649 .93 .441e41.9 1.36 643 11.31 10.21.1006 405 9659 4661 4611

2,297 42696 34? 70664 66212461 "6104 .10018 "56.30 031.14

1045 474 "5053 4.314 35111

412 62663/ 3246 .22.17 '195116104 04 6693 6530 35991469 24631 2743 1741 m11144442 1644 6.35 16626 16616942 7.36 0.29 16.19 16.6146334 4906 22698 .12639 115191036 1644 2241 61611 114446144 6404 443 406 3511.49/ 2692 3.20 13.06 11.194414 903 1141 1614 6144

.496 4611 446 306 MI66146 621612 .3694 622629 -20511

. -1696 62629 62631 15111

.6356

.4461961/

622691 61113:311

21:361944

12533

19.1196.351 22036401 -22.93 gel: 4942 02129§

6.394 19.53 1904 2244 22.606421 1942 .2614 16669 61641".383 13.96 1641 19644 19.916439 744 4106 614624 166116448 13.20 13.31 16 06 1609.496 62429 2631 1.08 1G6?460 13.42 13.54 69.76 9.61'043 8096 9.03. 1200 12468

'004 '11.0 '11010 7.17 -7.11

.6446 642 646 11631 1144'0079 '803 '9101 '5001 64,91

459 5.29 543 9.26 9.9.6464 46 67634 3630 341'007 3/72 3.75 7.62 7.16

'0195 '2.28 6200 107 1.86

.0096 2634 202 1.89 146se325 22.46 22.65 25.57 2639,362 .2144 28694 665 2964694 662 447 3626 3.15

66399 43.97 4635 9694 49046414 4146 4621 41616 4664'051 47,64 50.20 55051 53e17

.466 4145 4341 .36686 .3622

...50. 32.49 34124 40.23 36421.031 659.62 63.05 566604 654,30

666652 -3.62 '4001 3428 3.16

'64351 3.42 4407 3.28 3e15

Appendix I (continued), Experimental Data,

1141111 01 UZ OP V 61 0108 OTI8 OAT ENNNO ROMU (11 VN1 YR VS! 1111

141 (41 (CM) IPA) (CM /S1 (11/CM1) (K) (K) (PAI WW1 101/11 (C11/31 1C11/11 (C11/0

1.11163 401129 7416 "11,d 32442 "4935 431064 41707 930 4 4 2263.9 496

1.91162 .01925 40 14141 '34151 '.364 .00176 401113 930 5 3 063I0 407/

1.91951 .01921 13,07 44.1 37,11 021 .40066 40110 930 6 4 22612 4695

1891961 011919 160 10,4 .it.e/ '.612 440159 401794 931 1 3 '1191141 '4916

1091961 .01914 1304 2.1 1.11 .932 .00067 .01715 931 1 4 '111143 "4651

1691910 41911 746 4711 '2449 '4641 400131 .01794 930 9 3 '111141 "4940

1491161 401911 '14411 145 0.00 's051 44012 401761 93119 1 '1156.0 '4151

1491140 111111 locl 140 0400 44911 .00015 .00246 10 1 4 1 33047 "4451

141944 441116 03426 '114$ 14.94 "4439 404011 440241 10 1 5 1 '376.7 "4410

109114d 041111 "412 2142 '16.24 '1405 .00019 .00243 10 1 6 1 '333.5 "4431

Ist.e61 111917 11s/5 300 6403 '4451 .J0093 .01775 10 34 1 '116540 '4051

611:44 411924 "12454 0249 11114 '.933 .J3067 .01733 10 3 5 1 193411 '4686

1.91911 441911 '446 10449 20467 "4604 4441142 401003 10 36 1 '113510 '.930

1.91963 41914 "11401 110 19417 "1930 410070 401745 10 3 7 1 "111947 .715

1.11963 41926 1.79 lb.'. 1740 .719 .00134 .01101 10 3 1 1 "214349 "4916

1411165 .31921 1003 111. 440 '.053 403094 1011111 10 3 9 1 '11654/ '4153

1.91971 141134 '11421 '940 11t12 '4916 404076 401757 10 310 1 '1111141 '066

1491970 .01129 '402 11.1 '11454 4747 o01115 .01795 10 311 1 '115443 402

1.91974 41913 '11434 "tots 14..1 '.912 .00073 401756 10 312 1 '117145 '.790

1.91970 141424 '94b2 13.4 6,46 094 '40072 4017921 10 313 1 '115646 "4194

1,91974 441923 '10.63 '3., 613 000 4011070 .01761 it 314 1 '216143 '4116

1491913 431920 '94b1 10.3 444 "4416 401105 .017119 10 315 1 '115941 417

1.91973 131931 '10.41 3,7 0.00 '053 44091 401711 10 316 1 '217143 1653

1491971 431325 'Usti '1.2 b109 ',o006 444061 401756 10 317 1 '116147 4114

1.91976 441931 '10435 6.2 "bsli '4021 401104 4317011 10 3111 1 101111 0791.91931 .30317 1.61 .8 PM .498 oij017 .01244 10 321 1 '31643 "4496

01954 a0510 29.0 4150 09614 .031 .00026 460243 10 312 3 96.9 4194

1.91953 00310 11o13 31246 '56.56 4059 .00026 .00244 10 123 3 '4341 '4191

1.91955 .3031b 13494 21345 '49423 '4114 400014 400141 10 314 3 '139 "4119

1.91950 410311 1469 134,3 and! 4201 .00016 .00231 10 315 3 '11949 4.4171

1.91959 .00314 5.19 Mb '32.33 '4154 440011 402311 10 Z26 3 '15541 4319

1.91960 .30317 161 0.0u '.499 403116 .00241 ill 317 1 35544 '4499

690935.40033 .03 6400 OICIO M004.600052 921 1 341 1.110

1.9096i40001 o11 .1 Odi 0.000 40009.40M 9 3 1 1 141 LOH

140614 400003 .01 .3 0.00 0400 401011 '400044 91123 1 '145 0.000

/490500 .00003 -41 0.00 0,030 440010 '4000,50 91124 1 '246 0.100

14913t1 .00001 .80, 10 0.00 0.000 400004'MM 9/125 1 '1.5 0.000

1491221 413062 '.31 40 0.00 0.000 440004 '100051 91126 1 "240 0.000

149451 430006 401 .2 0.00 0,000 400002'400060 91117 1 '541 0.000

1,93911 440411 .01 6404 C.040 '.30001 '400066 91111 1 '149 follf

14437 .33033 .01 .1 0.00 0.030 "401003 400061 91129 1 '345 0.000

21.27 19,11 16,17 14007,06 0,16 3f,06 011$1,9V

23.46 24.72 11411 IfeW1401 3344 41.91 atioW11011 11411 17411 61 0

'21.10 '19439 11,41 '114141,:.

"341 '4411 3411 34

'2021 '2.32 1.19

12.92 13.03 16.11 11614:.

'11410 '11417 14471 '14430.:

'3411 '4407 3411 416.95 OM 24,73 23A1;7

'13494 "15131 '11413 *PAU14.11 14416 21.01 ZIlie

'10419 '21.46 '14433 1311t,"".

"3411 '4407 341$

1466 9.11 16430 1661:

14,93 '11.11 4654 61C6.32 6.64 13i93 11434

'11401 "11473 '5140 1.14

3.99 4411 11449 11411'

'9470 '11411 "2491 "2.16`;

3,52 0107 3.29 3.16.1

1.11 2119 9449 MP0,01 MI '1.11 '1441

'1,19 '1431 °411 .64Y:

69471 1004 19,32 .6(44k.

'51413 '19445 "51434 4.1174

'49476 '50416 41411 144;'31494 49411 '1741ll '3611!:

'33450 '33411 '31436 '31,11'

"2430 '1431 1419 1.11

0400 0.00 0.00 0411?,

0401 1400 0.00 MI

1411

0.00 0400

0.00

04111,,,!

1101 0401

0.00 0.00 Q4011 1411

0.00 0.00 04110 1o01,

hoo hoe Los lisC

1411 0.00 0.00 0411',:

0.00 IMO 601

4111iN Of

4KI IKI

.011271 600002

'.111111 604001

01;141 .30003ii:.:: ':

Iiiiilliol .00003111106. .630041

1640 640001

halm 00001P691910 604001402114 t00342a,...69,231 640001

Z1+91711 .00400401712 .600003p., .

3691771 630000

14691913 -630001

40$1119 '400641691901 '000446

A190981 '0000?

102914 '030044146191 40004

1,91712 100141

161,101 .00001

1692910 .13000o

1694911 6.44J41

1691701 .404002

1.9110, 40002

1.66716 .42941

1.66769 1293 o

1.66792 632940

1449771164976

1.64976

1.64971

1.64977

1464977

1.649N1464976

1444751464911

6644731.64973

1441041049731.64973

14497?1164977

1.64979

66497?6649711.64977

.11i02

.:11197

.2101

.31691

.41867

.41685

Ii1195

.21$93

.31194

0189*.41566

.418f:6

.111:7

10162113

141666

.01881

441161

4167431176

.01874

.01174

Appendix I (continued). Experimental Data.

Li

1CNI

OP

(PA)

V 42

(CN/SI III/C1121

0,00

1K/

0710

IKI

DAMON° RONU

(PA)

41 412

Mae; 1CN/$1VN1

10/11

VUICN/3I

4641 61 600 0.000 .04609 .43046 914 II 267 0.000 600 loll.41.oi,1

60

41

0400100

0.000

4.040

4001043009

.800044

640046914 21914 31

0165

.365

0.000

0.000

0.01 I.01 IIIloll

642 60 0100 0.000 600076 600151 928 11 364 0.000 loll Sellea a Oen 0.000 600056 600089 928 21 161 0.000 040 loll loll6i1 62 0600 0.000 600041 601040 928 31 .67 0.000 loll 041..1 61 1600 0,000 600014 .600002 928 4 1 66 0.000 600 loll 1110$01 I0 040 0.000 603009 s00041 928 51 167 0.000 0401 141600 ell 0.40 0,000 600008 6.00042 926 6 1 265 0.000 0101 1111eCO 61 OM 0.300 .00009 el00041 928 71 0.300 0100601 14 IWO 'MOO .J4011 6.00038 92$ 81 63 0.000 600 Os"4t .1 0.00 0.640 .40016 6.00014 62629 1 363 0,000 hIS loll Ii604 61 0.04 0.000 42003 6.00056 929 11 .60 MOO 0400 loll I,"lei 6.11 0.40 4.000 00002 6.60057 929 21 67 0.000 loll.64 .3 0.00 0.000 40001 6.40058 929 31 .5 0.000 I'll004 a 01G4 0.000 .30004 6.00062 929 71 .2 0,000 Sell loll.01 6.3 0.011 0.300 6.00000 00055 529 6 762 LOU 040 0.00 0611

60? 60 0601 0.300 6.03001 6.00064 930 II 3.8 0.000 0610 1600.1 11 640 0.000 .20005 .60033 930 21 .2 0.000 MI hIS

4.40 .1 0.00 0.000 '00069 'm05? 10 1 3 1 67 0.000 0640 040

5311612 7642,1 6.06 0.030 6.44010 6.00187 10 3 1 4 764361 0.000 0200 0100 hOC6602 645 0.000 6.40004 6.00070 10 32 1 4.1 0.000 0.00 0.00 06106.01 6.1 0.04 6.000 100005 6.04053 10 3 3 1 261 0.000 0400 0100 0406.02 6.4 0.40 0.000 .43004 6640060 10 319 1 262 0.000 0.00 0100 0.006.4! 60. 0.000 6.04041 6.00062 10 320 1 116 0.300 040 040 600

dia u.03 .,660 .20161 42840 1 1 1 .132615 ee660 4619 .1607 1691.5411 4C.5 622.62 6.511 40231 .02865 9 2 8 1 .147963 .e710 .26625 .31630 .2160111.b '461 11.65 .008 .30121 602782 5 2 9 1 .1370.4 ..525 13693 15623 2301

"foli 312 0.00 .4544 .30016 .01837 925 6 1 .92761 6544 649 4600 142lhd; 6133,4 35.76 . 501 .00076 J01785 925 4 105661 .420 29680 31666 3762?

4,4i 1360 36176 .319 .04151 41638 925 33 48961 .5),5 .41.57 .43638 456813G.64 6.514 .30071) .41746 925 9 4 .101968 .457 24.51 26,01. 32616

106 133li 642.12 6.354 .00140 601633 92510 3 .82265 ..524 46.35 48686 4160740 314 602 ..546 .00094 41819 9251/ 1 91669 .1546 4.52 .1602 1662

.4,41 111 0104 6.545 .00094 .01630 92516 1 123.7 6.545 .641 .7602 1662

.7.10 626.1 16.2? 6,546 .63075 601805 92517 1 15461 6.454 11673 12.1.3 19690

62.73 34.2 618.70 .$44$ 123122 41839 62518 1 892.0 6.546 .24604 .25612 .17637.7.10 626.1 16.4/ .0d49 .20076 .01804 92519 1 .95463 6.453 11617 1248 20605

62.71 616156 .0.49 .00123 e0/635 92520 1 .88963 6.546 .23694 .25660 .17624

4.14 626.1 18.57 ..:51 .00078 601601 92521 1 .95164 6.455 11699 1217A 20620

.4.44 3{1 6.04 .1545 .20093 .31322 92522 1 11969 .4145 4651 .7.0Z lotl

66145 15.82 6.554 .20086 401805 92524 1 141.9 .64,3 9621 9674 17647

64134 25.2 61546 ,467 103119 41831 92525 1 97.4 6.547 42640 13697.0.64 610.6 12651 .40085 .01805 9252/ 1 13066 6.494 5685 6616 1461?

.3sib 16.1 69.63 6.503 40107 .01822 92527 1 .90369 6.553 .15661 .16672 .8611

65464 60.7 12.54 ..559 600079 101799 12928 1 12864 6.494 5688 6619 14620

**oil 12,4 740 6.514 .00099 41818 92529 1 10563 6552 .13.34 .14650 460765.39 67.3 10.64 .6562 640060 601801 92530 1 .92367 6507. 3693 4.11 12631

64.8? 3,4 4.00 .145 644091 .01810 MU 1 .914.1 .6545 4650 .7601 1662

1:

1:11

lotl

01000410Doll0.60100

1696

.20612

23620

1661

3667t

456243149

.30658

1661

1611

19661

.17608

1906. 16696

19691

1611

1%22

. 13.74

13697

469714600

.50712614

1661

O-tt!

Appendix 1 (continued). Experimental Data.

Yp

t$CAN OT 01 OP V 02

IKI (CM) 1P11 1CN/51 IM /M/

-.524

.5606532$556.544

6546.$471

9116444.$479

63944531o529-.095

.547,547.430

'614076 $01174 4141 9ob .5.53

:61417i $11674 4534 .367 1424

1,14907 $01072 4555 7,4 3018'4'1.14 977 $011711 1516 also 6.48

...6141171 501189 401 3.0 060J

.614160 41904 OM 3.1 0600

1614110 501911 3.71 2830 91o5')

1614910 .01911 16,43 314.b 0.61614170 .01911 1010, MO 48,55

664911 641901 1401 219.3 46.44

614991 101901 .27 69.4 u28.9,

1614190 501413 968, 43.0 15.74

614993 $31910 9144 0!.b 6118664917 .J1908 24141 44707 7.54611001 6.61413 46.4 11.1 21016

1063001 601413 66/4 261 01E1

1$19J07 0111! Oa i10)J

1.63216 sJ0063 .42 .1 4$J0

1.61352 610001 id 0.0.

148921 $00000 ta 0.0J

145461 600314 ..1J 4.64

1.64993 .16001 161 .1 4.44

1,64524 610061 $)1 .1 4.10

1664026 $111041 .01 .1 0.00

154024 630001 3,40 04 0.00'

1.64990 610001 ..40 .1 0.30

1.65941 40104 $1,1 .1 0,0J

1.61410 .30002 600 .0 0.00

1665921 400041 .10 60 0.00

664971 $00002 .116 0.00

10,4024 $00103 .01 .4 0.03

1663955 .1,40002 601 -.2 0600

663997 610041 401 -.2 0;00

1639517 .31926 1.40 349 0600

1.39520 ,01926 4,13 00,5 18.71

1.3954 .31925 1.33 45.3 '111625

1.39534 .31940 1,42 364 0.00

1.39539 .01944 8607 14o9 26.24

1639533 41949 0,0 .34.4 26.73

1.39539 .01949 st1.07 .44.4 25.97

1.39536 .31354 4,,3 41,4 30,871.39538 .01954 4.43 J201 27.201639538 .31354 4943 12.9 25.571.39561 .01983 14935 1730 37.15

1639597 6i1)83 10.65 18199 38.861639567 02003 23.16 WO .55e62

0.000

4.000

0.000

0.100

0.3.10

0.010

0.000

3$000

mao0.030

,0.030

0.000

0.000

0.000

0.100

0.000

6231.208-.158

-.231

4182$1866181,1089118-.122

6154e664..045

UM(K)

OTI8

1K1

OATENNMO 1213110 1,11 VN2

111/0421

VNI

101/S1

VS2

ICN/S1

VS1

(CM /S)

.00100 .01818 92532 1 90669 -.552 11$76 -12.63 *3$91 .691

.03086 .01802 92533 1 -920.2 .518 1.55 1.58 9.90 8577

.00095 601814 92534 1 90865 6552 -10.23 1041 240 615

.J0083 .01790 92535 1 .9160 -.523 1116 .25 613 6.03

.00092 .01805 92536 1 -911.4 $544 .50 -7.00 1,62 1.61

.0111, 601892 927 4 1 93161 4546 .51 7.02 1$62 1.61.

.00075 .01844 927 5 4 1198.4 .6214 4 593 4660 52.99 52.19

.00219 .01920 927 6 3 20.3 -.481 -5 .59 62.80 612 619,3006 .41903 327 7 3 109.2 .$499 4 .46 5247 45683 4600$00379 4:60! 927 8 4 .11520 -.246 4 06 43.33 47.91 47.11

630139 $010.7 927 3 6441 -.537 .3 .69 33091 612 2640600081 .01820 92710 3 .99863 +6398 1 .40 20.71 27.32 26.92

31080 .01820 92711 4 99863 .398 1 .87 20605 2605 2631.30272 .01493 92712 3 41562 6419 6 A7 7300 "67.26 '6613$13110 .01894 92713 1 977.7 .$433 1 .34 16.28 23.49 23.14

$0010.. .01870 92714 1 -933.5 -.547 .51 .04 1$13 1$62

$37111 437111 92719 1 8880 546 2 .03 28.92 20462 20629

.03001 .00014 9 2 6 1 .1 0.000 .00 0.00 0.00 601$11047 141417 9 210 1 1.3 0.300 .00 0500 0500 0.00

$03001 0017 925 1 1 .2 0.300 .00 0.00 0.00 0.00

.13000 609316 925 2 1 .1.6 0.000 .00 0.00 0600 0.00

$03004 .00015 925 3 1 43 0.000 .00 0.00 0000 0.00

o0J003 400016 925 4 1 .64 0.100 .00 0.00 0.00 0.00

.40005 .00016 925 5 1 .62 0.000 .30 0.00 0,00 600

.31302 03023 92512 1 .64 0.000 .00 0500 0.00 0.00

.31002 .00020 92513 1 .2 0.100 .00 0.00 0600 600.30004 .40021 92514 1 .166 0,000 IOU 0600 0600 0510

40004 .00018 92515 1 .69 0.000 .00 0,00 COO OM

.J1061 .00170 927 1 1 .7 0.000 .00 0.00 0.00 601

.00020 .00069 921 2 1 1.0 0.300 .00 0.00 0.00 0601

.03009 .00050 927 3 1 1.2 0.000 .00 0.00 0,00 0.00

603009 .00012 :0 1 1 1 .7 0.300 .00 0.00 0.00 0.00

.00011 .00009 10 1 2 1 64 0.000 .00 0.00 MO 601

.30102 601785 9 3 8 1 35746 $231 642 .9625 .69 .69

600101 .01768 9 3 9 1 402.0 .6169 1 $12 11.93 19.33 19.22

.43130 .01786 9 310 1 312.9 6198, .24602 26619 708 17.66

.33102 .01804 9 4 2 1 3604 6231 .8.41 945 .69 .69

.J0102 .01776 9 4 3 1 451.1 6127 19.62 21.14 2608 26.62

.00102 .01776 9 4 4 3 45141 6130 19195 21.49 27.25 27;09

00102 $01716 9 4 5 3 45161 .$127 19637 20687 26.51 26.00

130144 .01818 9 4 6 3 2744 6178 .34.81 .38601 4445 3634610144 63/818 9 4 7 3 274.0 $/30 .31.50 -34.40 2615 6,66600144 .01818 9 4 8 1 214.0 $180 .30.01 .3208 .2621 603.03091 .01794 9 4 9 4 546.4 -.078 31.56 34.01 37.60 37.31

.00167 ,01858 9 410 3 -190.9 -.157 41.18 45.14 38661 38.39,

.00242 .01936 9 411 3 16.7 6152 .55.89 '61.69 55660 55616 .

77,

Appendix 1 (continued). Experimental Data.

:'1141AN

?itIKI

01

IKI

82

(COI

JP

IPAI

V 02

101/51 III/CM2I

0108

I()

0118

IKI

DATENNM6 ROMU

(141

GI VNZ

11/1CM21 ICN/SI

VN1

ICN/S)

VS2

1011$1

VS1

ICN/$1

1:39172 802007 12,73 235,5 43.71 ,44 43016 101806 9 412 4 61310 405/, 38141 41.43 44.14 438683,31969 .02001 16.91 271.0 °48814 8029 .03199 $01886 9 413 3 10584 .110 64941 54811 48810 41814149111 001951 11814 12184 ilstb 8164 .30096 41/12 9 411 4 4960 1199 2509 2101 32421 32.06

4039113 .01964 16.95 271.1 46667 ..,4031 803191 801051 9 416 3 91.0 .141 47111 611.41 46611 46414111115 01961 1501 19584 39856 8152 803016 001782 9 417 4 16116 W3 34.03 31.63 40.02 39014,19141 08/161 7851 13787 42.91 8091 803156 801018 i 418 3 131.1 ,111 3641 39e11 3100 31.411.19131 $31945 1101 4811 4800 1233 83001 801114 9 419 1 36184 0133 1841 9012 .69 110

'611111 .1191V 6,50 62,4 2201 4194 843112 801191 1 410 1 43111 eau 11.69 16.91 23.30 13.161011113 036912 825 260 12814 8188 .00122 .01531 1 411 1 344.0 411 19161 61651 61146 12.19631111 .01911 341 17.1 1341 .226 813104 .01824 9 422 1 319.4 .191 1.01 MO 1341 13417

1009114 111900 148 1181 22.41 1140 100141 801842 9 423 I 30181 .191 11.10 3611 011.19 411111119163 .11914 2844 482 8863 e23() .03115 801835 9 414 I 3114 411 .16 6.06 9.33 9.11

1,11101 41992 1819 130 06655 .216 $00115 801856 9 421 1 3169 .131 14.39 11.19 1.90 1411611164 .02001 1.11 r

.., 4.57 8236 833108 801156 9 416 1 31641 '416 3.11 4.41 1.21 14114019915 .01004 *100 464 4,13 8223 800111 801868 9 421 1 161.6 .111 611.14 13.41 64011 3.43141113 Iii001 100; 3,6 4,jd ,215 .33106 .01064 9 428 I 311.1 431 1011 69.39 .10 .10

1.39114 .11111 2.04 3.6 6800 1238 800115 801911 5 8 5 I 3914 .131 6141 1111 on .11

149111 811111 610J Job 4$03 8229 800098 .01764 9 8 6 1 313.4 6419 6611 10 al Al1019131 .01911 68106 8582 27.08 8181 .00090 .01742 9 8 7 3 444.1 .116 10,41 11.14 17.11 MO1,19131 031910 4045 92s4 21012 6011 .03134 861791 9 8 8 3 2690 0119 631191 63441 11,11 21111619114 835111 5861 5,1 6800 '8326 atmes .05295 9 8 9 I 1032.9 .316 11.16 14.14 di .99

101911/ .31423 2111.45 2;11,5 47,46 8381 603158 604426 9 810 4 1300.6 .101 3404 43.00 41.19 41011.19110 801523 16802 22114 44$54 $104 .30963 $05371 9 611 3 111.1 .311 641000 061110 044.11 43411.11141 41113 2701 34589 49892 8381 803154 804890 9 812 4 134161 .196 36.99 4101 11.11 10.11

1119111 .11111 16.21 3160 53,5i 66059 801128 805391 9 313 3 2480 8406 691113 611431 611.31 0114311.19191 111533 3.i? 1340 3400 $159 03804 805368 9 614 3 90350 8371 639896 11014 034.19 613111119111 101516 .61 580 6800 026 633386 $05268 9 816 1 61032$1 316 01080 614011 $11 Olt

149111 .11115 1469i 61240 43.45 071 .03212 .05063 9 811 4 6116681 .110 11001 1141 34.11 33.11019114 .05521 1827 132.9 248110 8166 80072e .05349 9 118 3 3681 68371 36801 64617 19833 02104

1119111 .0515 61:443 *9285 26830 *8368 803224 45109 9 619 4 11314 8196 1604 1902 19.31 11.19

109111 80025 17.11 Rill lo.14 '027 .30385 .05298 9 9 I I 11186 8311 10891 14831 .91 .49

1.11110 $0913 6:8:9 582 0.00 8327 .00383 .05286 9 9 2 I 103180 8311 10491 14131 691 .99

1,31111 05511 .153 5.3 3.38 *8327 .03377 .05264 9 9 3 I 10310 8321 10,92 14.31 $91 .99

1,19117* $35.97 10.24 661,6 24.32 1067 .00253 .05122 9 9 4 1 1091.1 410 12.01 14.69 15.40 24401139113 $115493 893 71.1 23.44 .8216 1;0654 605318 9 9 5 1 9610 8363 03001 39.29 22880 22134

1,19496 801422 5,45 5,4 0,13 *8323 .30314 805198 9 '11 1 101183 I323 '10884 14.15 895 111

144494 835424 3854 *3983 2;.42 1362 .00253 $05080 .42 1 105589 .240 1819 9.80 21.39 21.01

1139451 .05431 2814 56.0 15,08 8239 .30593 .05253 9 913 1 96883 08355 21.09 34862 0830 188011111511 805431 5.46 5.6 G.00 8324 .03379 805201 9 914 1 101288 .324 10.88 .44821 .96 190

1839443 805432 7805 25,5 17.64 8363 .00250 )05102 9 915 1 114482 8251 5.47 6838 1802 18.42

1.11499 105434 -3.27 56.7 15898 -.255 803557 605251 9 916 1 981e2 8352 2404 31.41 15123 14892

3039541 801435 802 16.6 15.59 *8362 .00260 805118 9 917 I 03519 8208 3.45 3184 16.66 16.40

14391:5 035435 3693 27.6 1201 6210 40517 805251 9 918 1 9185 0.344 21.96 28816 12811 11461.39101 $05437 '5045 5.6 0.00 '0325 800319 805211 i 919 1 101483 8325 10890 14824 .96 .98

1819504 .05433 .684u 781 12807 8351 800271 .05141 9 920 1 102689 8284 .10 837 13.12 12.93

1,39501 805437 4.53 19.0 0,56 ,290 .00471 .05245 9 921 1 100089 *8342 18830 .23856 47.70 7'853

1439505 .05437 6815 *482 9.92 *8351 803296 .05159 9 922 1 102480 8292 1887 2.86 10896 10180

10139101 805439 401 15.4 6865 8390 800445 805241 9 923 1 100488 8340 16811 21855 5117 *5862

1.39156 .05439 6 5148 5.5 3.00 425 40378 .05213 9 924 1 10140 8325 10189 14122 196 .98

1839509 a45440 65.95 162 7.21 043 .00309 805116 9 925 1 102187 .301 4.29 MI 6.23 8112

1839513 605439 5401 12.2 64.66 005 .00426 805231 9 926 1 1008.4 6031 15817 19862 *3895 38061839511 805441 -5.73 1.1 403 8337 .30323 45118 9 921 1 .101988 *8307 6832 8146 5191 5890

1839513 .05441 *5816 1064 014 .00415 .05232 9 928 1 610169 036 14.13 61600 6248 1.19

t",1r)

Appendix 1 (continued). Experimental Data.

MANIIII

01

IKI

01

(CM)

OP

IPAI

0 02

(CM/S) IN/CM2)

0106

IK)

0110

(K)

OATENNMO ROMU

(PA)

01 VN2

IM/C411 (CM/S)

VNI

ICM/S)

VS1

ICM/S)

VII

ICUS)

101512 05443 5841 50 MG 8325 .00375 .05217 9 929 1 1015.8 025 1000 1402 .96 01

109531 801924 1871 480 0800 -.231 .00154 .01903 915 6 1 -357.2 031 841 9.25 .69 869

141534 01917 401 .1C.5 11895 8206 .00121 01141 90 9 1 .40084 -.167 11.43 12326 19851 19.45

1.31529 01415 1.34 4084 .10846 8150 .00199 .01954 91510 1 3114 8190 2402 26.41 1701 17117109344 .00101 s14 1'3 0606 6093 .0104d .00214 91512 1 2285 8093 3451 3.61 01109345 .00125 305 450 1504 003 01150 .00215 91513 1 22.4 8006 .10821 1104 11.11 1107101346 GAM .05 1.1 0.00 093 00005 00105 91516 1 .220 093 3.57 1.61 1111

1.39347 800121 1100 26/0 000 8019 .00014 .00102 91517 3 244.0 001 48817 4803 1416 601109341 00130 12844 178.9 3905 .014 800003 .00091 91516 3 154.1 .001 .3003 39.31 3101 1106,109344 .00127 05 1o2 0.00 095 00003 00011 91519 1 2284 095 '3865 01

101349 06123 .05 1.1 0800 8093 800034 00111 916 5 1 .210 .093 .3150 301 ell 811

101350 00124 1862 2181 12.10 8333 803031 800164 916 6 I 4489 1031 1103 1101 1100 11811

1139352 100124 1.51 230 12.51 017 00045 .00203 916 7 1 -.019 13119 13.27 11847 .11167

.101311 .00123 865 1.1 0.4J 8013 800034 .00110 916 1 I 210 *013 3.50 3461 01 811

101347 0010 891 1282 109 .011 01131 800161 916 1 I 310 e116 6.59 6.62 1.16 106101311 .00113 845 14.4 .9801 8047 .40037 .00115 91610 1 .164 041 1101 10117 1.17 116109341 01116 874 40 104 015 801036 00110 91611 1 .000 064 406 4.17 7.43 7.43

109341 00115 .61 1080 1834 8054 603041 ,0117 91612 1 .100 8055 9,43 .9444 1.111 7.173031341 .00109 5114 114 0030 '5090 8110036 .00163 91113 1 .194 .4190 1144 3.46 07 117

101341 890115 805 1.1 1604 8093 01034 .04111 916/6 1 .2111 093 3451 859 01 011131341 00121 04 4.2 1846 065 .00036 80011t 91617 1 .310 1064 406 101 705 7665

1839141 00124 80 10 7.26 051 03135 .00111 91611 1 011.5 1191 9.44 09.49 .700 7111109353 .00114 07 .314 4.84 .or7 800039 .00171 91611 1 .2584 o046 1.90 /891 5.01 5,01

1819341 .30116 at S.b 3.90 8076 00434 800171 91620 1 1686 1011 .69 .92 .303 3031039351 800114 04 1.1 0.00 8091 .00032 00101 91621 1 200 1091 3841 3.51 .21 .27

109351 .04119 836 30 404 0176 011036 00161 91622 1 25o5 016 1.91 1.92 506 5.06

1.39345'.00114 .21 bib 4.05 075 800033 800165 91623 1 150 o015 4892 6.96 302 .301109353 .00115 651 ish 4.84 8071 1;0036 00166 91624 1 .270 s011 3.14 3.15 6.09 6109

1.39356 .00'19 643 '81 504 067 840037 .00111 11625 1 1484 8166 .1813 80 504 5031.39354 .J0116 ..I 0.00 092 .03030 .00166 91626 1 2085 .092 -3.54 .56 01 di

109347 .00117 606 1s0 0.00 .092 600010 .00022 10 330 I 2019 8092 .303 .55 e27 .17

109315 100141 24.95 3hbs4 56.61 100 600014 .00101 IC 331 3 330.6 010 509 4700 56867 6081.39352 800141 0.42 14.1 28.92 1106 401001 $00094 10 332 3 120 001 .2609 2691 .2693 40.91109367 .00144 9.32 114.8 34.16 ..010 400001) .00103 10 333 3 107.0 .000 3309 34,10 '3409 34.16

1.39354 .0014e -A,Cb 1.4 0.04 099 804010 080090 IL 334 3 2519 1191 301 '304 04 .30

1.39341 .0b134 1,5 thyJ ..096 400009 sU3007 10 5 3 I 23.5 .096 .3867 300 o39 011,39357 .30127 'o4 1.4 ilati ...C94 $10010 .06019 10 5 4 1 220 .8094 3,61 .343 .20 al1.393b0 .J0125 24.97 3)6.4 0,67 .019 .00013 104469 10 5 5 3 33362 .8009 55.92 57100 4601 6041.39357 .00121 15.50 2!3,5 44613 6015 600003 600079 10 5 6 3 200.9 -.001 43652 460 44$11 44031.39357 .10124 9,44 1360 33606 .009 603006 .00079 10 5 7 3 111.5 000 32671 3606 .3301 13.161.39362' 30126 4.54 1340 33693 8014 00000 .00077 10 5 0 3 110.9 .001 3304 3890 3306 33.13

1.39360 .00128 co8, 99l3 .26/83 .007 .00010 .00019 10 5 9 3 75.3 000 1615 1861 21065 8431.39361 .00125 2.155 326: ..15.04 6011 40009 .00077 ID 510 I 9.1 013 11.41 11.55 .15,00 15,40

109364 .4016 604 1.3 0.00 6094 .00009 .00077 10 511 1 .2282 8094 .301 303 Al .10

1.39515 .3141; -1.71 M 0.00 .231 .03095 .01771 10 514 1 35445 8231 '1442 905 .69 461

1,39519 .01924 31,35 -18.1 1 00 6222 MOH 01165 10 515 I 79.3 093 500 6.06 14.46 14011.39515 .01925 ..422 2E63 01 ..118 00111 .01786 ID 516 1 434.9 8216 1'8,75 11855 1201 12061.39522 101920 .9.3 10 9 .6229 600040 101765 10 517 1 6987 .8206 ells 2.16 1107 11.61

1.39511 .01923 -.85 12,4 .6160 -.207 o00112 01703 ID 511 I 41.6 8226 1605 .1706 01.11 01.11

109510 .01917 .1.80 3.6 0.00 -.232 00091 .01776 10 519 I 56.0 031 .44 .101 .69 .69

109518 .31920 2.22 .2.6 6.12 ..234 .30192 p01110 10 520 1 453.1 '017 840 o56 101 107

Appendix 1 (continued). Experimental Data.

'MEAN

. ,Alil

01

(K)

01

ICM/

JP

IPAI

V 02

(CM /SI IM/CM21

OTOB

IKI

0118

IKI

08TENNMC ROMU

IPRI

41 VN1

114/021 ICN/S1

VN1

(CM/S1

V51

ICN/SI

VS1

101/$1

'.'.:4839511

'14139511

801920 -1.33 10.5 .4687 .219 800104 .01783 10 521 1 34989 8230 11817 14801 4411 81101916 .J835 .1614 13,66 82e2 811624 .01755 10 522 1 371,3 8192 5.09 6816 14114 14145

'.:4039516r .

.01909 1179 3.8 IMO 8231 800095 .01766 10 523 1 354,6 431 1841 9835 Al 1'.,

1840459 800002 801 sill 0.04 0.000 40051 1100067 9 3 2 1 .3 0,000 0800 1.10 0.00 18111831914 .00110 841 .1 0.00 0.300 00031 800049 9 3 3 1 189 0.000 0.00 0.00 0801 0810J839411 00040 861 81 0.40 0.000 '410012 800041 9 3 4 1 'il1 0.000 3,00 0.00 0.00 00001439015 $J3;d1 8.4 e 0.00 0.000 '80)001 400033 9 3 5 1 -.2 0.300 0800 1800 0801 0.001835551 .830333 6;3 81 4843 0,:00 01007 803044 9 3 6 1 A 0.000 0400 0.00 0801 0.00

,1844461,800403 .43 81 u840 0.000 400041 800064 9 3 7 1 .5 0.000 MO 0.00 600 0.11

11136541 836101 oUl .1 0800 08.1JC $13304 04640 9 4 1 1 .1 0.000 0.00 0800 0800 0100

1,46491 .36044 .4 80 0840 0.030 .03053 .00070 9 414 1 1.1 0.000 4.00 0000 0801 0811

1816561 510334 .,.1 81 680.i G.36 ..1.1403 .000411 9 429 1 81 0.000 0.00 0,00 0800 0,00

1846464 804064 0? I3 0.G0 0.000 -.03054 "o06e70 9 5 1 1 14 0.600 0.00 0810 0,00 0,00

1.36556 so4'dd4 .4e 82 .803 0o000 .03003 .00045 9 5 2 1 89 0.000 0.00 0.00 0,00 08111831776 8.10006 .3 e 4 4800 omo ,43003 800065 9 e 3 i -1.1 0.000 0.0 0.00 600 1.04

1842184 801014 ;,2 81 0800 0.030 .03,34 ..06045 9 0 4 1 83 0.000 0.00 0800 0.00 0110

1.36831 81i0;v2 584 .0 0804 4.600 41001 813062 9 9 6 1 .3 0.000 0.00 0.110 0800 0811

1842141 804041 1u1 84 6114 0.000 .00053 800050 9 9 1 1 81 0.000 0.00 0800 0.00 0.00

104465 04.01 6.8 II 0.46 0.039 03051 .60045 9 9 8 1 82 0.300 0.00 0400 0,00 MI'1838561 .361145 .80 .1 0.00 0.000 103003 - .00040 9 9 9 1 -.8 0.300 0.00 0.00 0.01 1411.31765 $84042 541 81 4.33 0.030 .43004 810061 9 910 . 82 0.000 0800 0000 0,00 000

1,40481 .830041 .33 84 080u 0.000 00032 .00021 915 1 1 .3 0.000 0.00, 0.00 0800 600

10400 .3.013 $.; .3 ..J0 0.000 05302 .00018 915 2 1 .20 0.300 0.00 0)00 0.00 800

1.39519 .33440 ..1 .1 0800 MOO .00027 .00062 915 3 1 .0 0.000 0.00 0400 MO off

1839505 -.00".4e .11 .1 0003 1.600 .u0047 .00120 915 4 1 84 0.000 0.00 0400 0,00 110

1.39051 8311001 out .1 6.40 3.010 .33062 .00130 915 5 1 -.0 0.100 08r 0100 0800 800

WOW I00465 011 a MO 0.000 103058 800112 915 6 1 .7 0$000 0., 0.00 IMO III

1.355b, 804000 oul .1 0,43 34400 .00349 .33093 915 7 1 .2 0.000 0.00 0.00 0.00 801

1.39216 .30003 .01 .2 0.04 0m0 .30047 .00093 91511 1 84 0.000 0.00 IMO 0.00 811

1.39242 830000 536.u2 7620.2 .,00 4.010 04006 .06316 91514 1 761980 0.400 0.00 0.00 0.00 810

1.39202 .800004 '684 80 0.00 0.000 .30003 00009 91515 1 .0 0.000 0.00 600 0.00 .00

1.39206 .00042 844 .1 4800 0.000 .03034 .10071 916 1 1 .3 0.000 0$00 0.00 0800 .00

1.39374 $00001 0.1 41 4.30 0.000 .00031 .00072 916 2 1 0.030 0.00 0800 0811 .00

14940 ,34.1 .01 81 4800 0.000 .04030 .03071 916 3 1 -.2 0.000 0.00 0.00 600 .00

1.39293 .30000 .0 0800 0.000 .00028 .00065 916 4 1 .41 0.300 0$10 0.00 0.00 801

1.39293 830341 .443 -.3 0.34 0.700 .43027 00057 91614 1 83 0.000 0,00 0800 0811 .00

109253 .$00341 84 81 u.00 0.000 o40029 .00059 91615 1 .1 0.000 0.00 0.00 0401 011

1.39291 .06441 .6 801 0.010 .03329 00057 91627 1 .1 0.000 0.00 0.00 0800 .00

1839412 00400 Jsd 'e0 6a6 0.000 00025 .00057 9163 1 .0 0.300 0.00 0.00 0.00 .00

1.39274 ".00662 4 sigh 0.006 -.03011 00042 14 328 1 0.000 0.00 0.00 0800 .00

6u

ix l (continued). ioerimental Data,

OF

I

/CO/S) IvICN21

'000

'6046'

142 4410 6:00WID

can OTIS MEM1K)

01614 $10043 16 029 1

003014 !Mita!40019.00004C0361157'.400673'40I12 40049

IC 511It 5.2 1

10 512 1

IG 513 1

RCHU 01 VIII VIII VS2 VII

(PA' 111/01i) ICH/SI ICK/S1 ICH/S1 104/11

16 0.010 0$00 1101. 0101 0$11

4 4000 600i 6460 ..010 601C.

101 0006 0.00 IWO' 'GOO 0411

.7 0.060 0001 1$00 0,00 0,00

14 i0010 0006 140 :0000

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 + dEgrav (A2.1)

where dEgrav is the change in the gravitational potential energy, and dU is again the

change in internal energy of the helium.

We can prove that the variation of u within the liquid is negligible, by considering

its variation with depth,

du _ auN dT (0u) dP27 5Tip 27 3V T 27

(A2.2)

For equilibrium within the control volume, with a gravitational field present, we still

have dT = 0, but now dP/di = -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 _ /007 P319

Tg

is much smaller (< 1%) than he 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 argumek used to eliminate from dU any explicit depend-

ence on the vapor; we obtain the same equations [eqs (31) - (35)1, as long as wc. 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 uwface, ch we

designate as P'2.

We may evaluate the change in the gravitational potential, without loss of gent. ility,

by assuming the inside space to have a constant horizontal cross-section Ai, and by evalu-

ating the gain or loss of liquid at the surface (altitude z2) and at the bottom of the

inside space (altitude zb). It is

dEgrav = 9(z2pAidz2 - zbpAidzb) (A2.3) 4,1)

We can express this differently by using eq (28) and the identities dz2/dt = Vi and

dzb/dt = Vb, and then combine it with the result for dU, to obtain

dE = (put + p'2 - T2() AvdtVsvp,T;)

+ pg(z2 zb) VbAidt + pgz2VAxdt (A2.4)

where, as before, 12 is the inside temperature, and u2 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 enterc through the flow tube.

dE = dQ - ced jeAxdt (A2.5)

The work term is evaluated simply by the p,41$ure-volume Ark done on the fluid by the

top and bottom of the inside space, taking into account the hydrostatic pressure dif-

ference.

-dW ,12VbAidt + (1)12 + g(z2 zb)) VbAidt

zb) VbAidt (A2.b)

The toti) energy current je contains, as before, the enthalpy current and the heat

current, but we must also add a gravitational pottgltial energy current, gzentpV, where zent

is the etitude at which the fluid enters the inside space. The enthalpy is to be evalu-

ated at the p-,ssure at which the liquid enters- which -is, by continuity in the pressure,

the local f1u'c .pressure at the altitude of entry, P'2 + pg(z. - )ent-'

Collecting terms,

we find that the tent terms cancel to give

je (Puent P'2 gent Pgz2)V(A2.1)

Then substituting into eq (A2.5), and equating it to eq (A2.4), we find that all the

retaining terms containing g cancel, and we recover eq (36).

H1111.114A 7-70

.

BIBLIOGRAPHIC DATASHEET

1. PUBLICATION OR REPORT NO.

NBS TN-1002

2. Gov't AccessionNo.

3. Recipient's Accession No.

4. TITLE AND SJBTITLE

MEASUREMENTS OF COMBINED AXIAL MASS AND HEAT TRANSPORT INHe II

5. Publication Date

Februar' 19786. Performing Organization Code

275.05

7.WHIalt(S)_

WRREN . JOHNSON AND MICHAEL C. JONESIL Performing Organ. Report No.

17PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDSDEPARTMENT OF COMMERCEWASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

275055111. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & PeriodCovered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significantbibliography or literature survey, mention it here.)

An experiment was performed that allowed measurements of both axial mass andheat transport of He-II in a long tube. The apparatus allowed the pressure different=and the temperature difference across the flow tube to each be independently adjust-d,and the resulting steady-state values of net fluid velocity and axial heat transportto be measured. For the larger Reynolds numbers, it was found that the relation be-tween pressure difference and net fluid velocity was nearly indistinguishable fromthat of an ordinary fluid in turbulent flow. The axial heat transported was found tobe suppressed from the values that were calculated by assuming that "mutual friction"was unchanged by the net fluid flow, but it was always foun.; to be larger than the" enthalpy rise" value. Taking this second value as a lower limit, it is shown thata mild extrapolation of these results suggests that (in appropriate circumstances)forced convection would allow much greater heat to be transported in long .00lingchannels than could be transported by "natural" convection alone.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the fitat letter of the bray key word unless a propername; soparated by semicolons)

Axial heat transport; forced convection; helium II; measurements; 1.4 - 4.1 K;pressure drop.

It AVAILABILITY I XI Unlarnitecr

El For Official Distribution. Do Not Release to NTIS

kij Order From Sup. of Doe., U.S. Government ecintjulticeWashington, D.C. 20402, SD Caj. No. C13.14D: I t./U

19. SECURITY CLASS(THIS REPORT)

UNCLASSIFIED

21. pr 1. . PAGES

84

20. SECURITY CLASS(THIS PAGE)

UNCLASSIFIED

22. Price

$2.40C1 Order From National Technical Information Service (NTIS)Springfield, Virgini., 2151

b' 4*U.S. Government Printing Office: 1973.777-067/1230 Region 8

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