^ecUnic&l vZote

PB 151367

9lc. 5

THERMODYNAMIC PROPERTIES OF

HELIUM AT LOW TEMPERATURES

AND HIGH PRESSURES

BY D. B. MANN AND R. B. STEWART

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

technical ^Ttote

May 1959

THERMODYNAMIC PROPERTIES OF HELIUM AT

LOW TEMPERATURES AND HIGH PRESSURES

by

D. B. Mann and R. B. Stewart

The data presented in this report is part of a study

made under contract with Bureau of Aeronautics,

Department of the Navy.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D. C.

Price $1.25

1.

ACKNOWLEDGEMENTS

The data presented in this report is part of a study made under

contract with Bureau of Aeronautics, Department of the Navy. The

substance of this report was presented for the Degree of Master of

Science, Mechanical Engineering at the University of Colorado (1959)

under the title of "Thermodynamic Properties of Helium at Low Temperatures and High Pressures" by Douglas B. Mann.

Sections of this data have been submitted to the American Society

of Mechanical Engineers for presentation at the National Heat Transfer

Conference atStorrs, Connecticut.

2.

TABLE OF CONTENTS

INTRODUCTION

SYMBOLS

PRESENT DATA

MORE RECENT DATA

CORRELATION OF DATA

CALCULATION OF ISOBARS

CALCULATION OF ISENTHALPIC CURVES

LOUNASMAA'S EXPERIMENTAL AND DERIVED DATA

CONCLUSIONS

BIBLIOGRAPHY

APPENDIX

Derived Functions and Diagram Construction

Associated Thermodynamic Property Charts

PAGE

8.

7.

8.

9.

9.

13.

17.

17.

24.

26.

28.

29.

34.

3 .

LIST OF FIGURES

FIGURE PAGE

1 Specific Heat at Saturation of Liquid Helium 12.

Illustrating the Method Used to Determine the

Absolute Entropy of Liquid Helium at the Normal

Boiling Point

2 Specific Heat at Saturation of Liquid Helium 14.

Illustrating the Method Used to Calculate

/C sat. d T Employed to Determine the Absolute

Enthalpy of Liquid Helium at the Normal Boiling

Point

3 Specific Volume of Saturated Liquid Helium 15.

Illustrating the Calculation of /VdP Used in

Determining the Enthalpy at the Normal Boiling

Point

4 Enthalpy-Entropy Diagram of Helium 21.

5 Temperature -Entropy Diagram of Helium 22.

6 Pressure-Temperature Diagram of Helium 23.

7 Isotherms of Helium as Functions of Volume 30.

and Entropy

8 Slopes of Isotherms of Figure 7 Illustrating the 32.

Method Used to Determine Isobars

9 Pressure-Temperature Diagram of Helium 33.

Illustrating the Method Used to Determine

Enthalpy Values

10 Pressure-Specific Volume Diagram for Helium 35 o

11 Pressure-Internal Energy Diagram for Helium 36.

to 1 00 psia.

4.

FIGURE PAGE

12 Pressure -Internal Energy Diagram for Helium. 37.

to 100 Atmospheres.

13 Specific Heat at Constant Pressure of Helium Gas 38.

14 Compressibility Factor Z for Helium Gas. 39.

5.

LIST OF TABLES

TABLE PAGE

I Pressure Reference Points 16.

II Comparison of Values of ( j^j 19.

Ill Entropy of Fluid Helium 31.

6.

THERMODYNAMIC PROPERTIES OF HELIUM AT LOW

TEMPERATURES AND HIGH PRESSURES

by

Douglas B. Mann and Richard B. Stewart

ABSTRACT

The most comprehensive investigations of the

thermodynamic properties of normal helium I and II were published over a period of the last thirty years

by W. H. Keesom. A culmination of most of this

work was published by Keesom in 1942. Since that

time, numerous investigators have found several

discrepancies in this original work in the temperature range below 20 K. Recent interest in the compressed liquid region, an area not covered extensively

by Keesom' s work has resulted in new experimental

data. These investigations and corrections to Keesom's

work have been published, not as additions to the

original Keesom work, but as separate and sometimes

isolated articles of very narrow interest and not in

readily usable form. Also the new data have little

value until their relation to existing data is established.

This thesis is a compilation and correlation of

the present data on the thermodynamic properties of

helium below 20 K. The existing data were compared

to the new data and the errors resolved by selection

based on estimated accuracy, experimental apparatus,

and technique of the investigator.

Methods of property diagram construction are also

explained, and the results are presented in the form of

temperature -entropy and enthalpy-entropy diagrams.

Pressures to 100 atmospheres, temperatures from K

to 20 K and specific volumes from 5 liter s/kgm. to

800 liters/kgm. are presented.

7.

SYMBOLS

The following symbols were used in the thesis:

C . Specific heat of liquid helium at saturation

S c3l t „ . q conditions, joules/gram- K.

h Enthalpy of helium at zero degrees Kelvin,

calories/gram.

h Enthalpy of helium, calories/gram.

P Pressure, atmospheres,,

s Entropy of Helium at zero degrees Kelvin,

calories/gram- K.

s Entropy of helium, calories/gram- K.

T Temperature, degrees Kelvin.

v. Specific volume of helium, liters/kilogram.

p

Density of Helium, grams/cm .

p

Critical Density of Helium, 0. 0693 grams/cm

i P

p-yf Reduced Density of Helium = =-

3

INTRODUCTION

The compilation of the thermodynamic properties of helium

involves the summarization and correlation of large quantities of

sometimes isolated information, both theoretical and experimental,

into a composite form. The properties of vapor, liquid and solid

phases are not usually investigated by one single experimenter or

team; and in general each investigation concerns only an isolated

region dictated by technical demands or the investigator's particular

curiosity.

New experimental information must be correlated carefully with

existing data so that the result reflects an accurate compilation of the

properties under investigation.

The object of this paper is to construct diagrams of properties

for helium which incorporate recent experimental findings in the region

of the compressed liquid state.

The resulting diagrams will include the thermodynamic properties

of pressure, volume, temperature, enthalpy and entropy for pressures

up to 100 atmospheres, specific volumes from 5 to 800 liters per kilogram and temperatures from 3.0 K to Z0 K.

PRESENT DATA

The greatest quantity of technical information prior to 1952 on

the physical properties of helium in the region under investigation

comes from two sources.

1) A compilation by W. H. Keesom of the theoretical and

experimental investigations performed at the University of

Leiden, Holland. This work is very extensive and covers the

temperature range from K to 400 K and pressures from 0. 001

atmospheres up to 200 atmospheres.

2) An investigation of the Joule -Thomson effect in helium by

J.„ Zelmanov' ', This work resulted from discovery of discrepancies found in Keesom's work by Peter Kapitza while he was

constructing a helium liquefier. Kapitza suggested that an error

existed in the Joule -Thomson inversion point values of Keesom's

at temperatures below 20°K. Zelmanov, a colleague of Kapitza,

experimentally investigated this temperature region in respect

9.

to the Joule -Thomson effect and published a revised temperature-entropy diagram of helium below 20 K, incorporating

his findings^" ', The Zelmanov data covered the temperature

range from K to 20 K and pressures from 0. 5 atmospheres

to 60 atmospheres.

MORE RECENT DATA

( 3) (5)

R. W. Hill presented new experimental data in the region

from the critical pressure of 2.2 atmospheres to 100 atmospheres and

at temperatures from 2. 5 K to 20 K. These data from Hill were given

in terms of temperature, specific volume and entropy. To make this

information more useful, it v/as necessary to determine pressure and

enthalpy values from Hill's data and plot these derived results using

the existing Keesom-Zelmanov diagram as a reference.

Some months after these data had been received and the derived

functions calculated and plotted, the author secured a copy of a doctoral thesis by O. V. Lounasmaa''' in which the data of Hill, Lounasmaa's thesis supervisor, were presented in addition to experimental

values of pressure. These data covered the same temperature, specific volume and pressure ranges as Hill's data.

Recent efforts to agree on an international vapor pressure-temperature scale for helium have resulted in several articles on this

temperature scale. Berman and Mate'*' and Van Dijk and Durieux'* ' have presented extensive data on temperature, vapor pressure, saturation densities and heats of vaporization.

Co Domb and J. S. Dugdalev *"' have recently presented a summation

and extension of experimental data dealing with the solid phase for

pressures ranging from 25 atmospheres to 225 atmospheres and temperatures from 0°K to 6°K.

CORRELATION OF DATA

The experimental data of Hill and Lounasmaa and the derived

results of this paper include the region bounded by the KeesomZelmanov properties of saturated and slightly compressed liquid and

the data of Domb and Dugdale concerning the properties of the solid

phase.

10.

The temperature -entropy coordinates of thermodynamic properties

of helium are commonly used by engineers for investigation of liquefaction or refrigeration cycles and the design of heat exchange equipment. The properties of the compressed liquid phase can be plotted

on this coordinate system to show the general behavior trends, but

the plot proves unsatisfactory for calculation purposes because at high

pressures, the slopes of the constant temperature, pressure and

enthalpy lines are approaching a common value, making interpolation

difficult and inaccurate.

The enthalpy-entropy plot is generally used to express the properties of compressed fluids^ 1

' and has the advantage of large differences in the slopes of the isotherms, isochores, and constant enthalpy

lines.

It was therefore decided to express the results of this compilation

on both coordinate systems; thus this new information will be useful

for cycle analysis by plotting the data on temperature -entropy coordinates, while calculations in the compressed liquid state can be made

by referring to the data plotted on enthalpy -entropy coordinates.

When constructing the above plots, it was necessary to establish

the entropy and enthalpy of some fixed point as a reference. The

data of the several investigators could then be referred to this fixed

point for correlation. The normal boiling point of liquid helium was

selected as the fixed reference point even though no such point is

contained in the solid-liquid equilibrium data of Domb and Dugdale.

The entropy difference between the solid-liquid equilibrium at the

normal boiling point temperature and the saturated liquid at the same

temperature was determined in the following manner.

A presently agreed upon interpretation of the Nernst-Simon Statement of the Third Law of Thermodynamics^ '

' ' states that the

absolute entropies of the solid at K and 25 atmospheres pressure

and of the saturated liquid at zero degrees Kelvin and zero pressure

are identically equal to zero. Therefore, we have only to establish

the entropy difference between the saturated liquid at zero temperature

and pressure and the saturated liquid at the normal boiling point of

4. 21 K and one atmosphere pressure in order to determine the

absolute entropy of the liquid at the normal boiling point.

11.

If the specific heat of the saturated liquid is known, the entropy

difference between any two temperatures can be expressed as

S 2 -S, =/

T2C

-ff dT \ T

(1)

Therefore, when T, is equal to absolute zero and T2 is equal to 4. Z1°K

the integral will be equal to the entropy difference between absolute

zero and the normal boiling point.

The values of C from the data of Kramers, Wasscher and

. q . sat.

Gorter were used to calculate entropy differences below 1.9°K,

while the values of C . between 1. 9°K and 5. 05°K are from data of

S cL t 9 Hill and Lounasmaa' '. Figure 1 is the plot of C f /T versus temperature. The phase change at 2. 1'74°k' ' indicated on the curve by a

discontinuity at that temperature is the Lambda point transition. This

is a second order phase change with no discontinuity in entropy or

volume and therefore does not affect the calculation.

The entropy difference between zero degrees Kelvin and the normal

boiling point, S-S , as determined from this plot by graphical integration was 0. 830 calories/gram°K. This value is in agreement with

the value of Lounasmaa'"'.

We may now plot the data of Zelmanov and Domb et. al. on

temperature -entropy or enthalpy -entropy coordinates with a common

entropy scale.

Because the Hill data overlaps the Zelmanov data, it was a simple

matter to establish a common reference point between the two.

The fixed point for enthalpy was also selected at the normal

boiling point of liquid helium. This value of 2. 39 calories per gram

was calculated using the relation

/.T rP

h-h =/ CsatdT+y VdP

T° P°

(2)

Page 12,

II

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-LAME!DA T RANS TION

1

/

L SATURATED LIQUID HELIUM E

-* ^SATURATED LIQUID HELIUM I

\ \ \ ^

^-*WRS AS

^

> *. t I

1 TEMPERATURE, (°K)

Figure 1. Specific Heat at Saturation of Liquid

Helium Illustrating the Method Used

to Determine the Absolute Entropy of

Liquid Helium at the Normal Boiling

Point.

13.

Figures 2 and 3 show the plots of these two additive functions.

The graphical integration results in the required value of enthalpy at

the normal boiling point.

The properties of helium given by Hill' '(Temperature, specific volume, entropy) do not by themselves form a thermodynamically

complete set of data necessary for a temperature -entropy or enthalpyentropy diagram. Thus it was necessary to determine pressure and

enthalpy using suitable the rin o dynamic relations and procedures

described below.

CALCULATION OF ISOBARS

With the properties of temperature, specific volume, and entropy,

it is possible to determine the lines of constant pressure on the two

selected coordinate systems from the Maxwell relation

dS\ ( dP\

which for neighboring states becomes

APV*(M)t AT*

(4)

Only the pressure differences along the isochores of 5, 5.5, 6, 7 and

8 liters per kilogram were used for this purpose, as the Zelmanov data

adequately described the isochores for larger values up to 800 liters

per kilogram. The data for the isochores of 7 liters per kilogram to

70 liters per kilogram of Hill and Zelmanov coincided with those

calculated and therefore established the compatibility of data.

Pressure reference points were required for the isobars as the

integration resulted in pressure differences only. Table I shows these

points.

Sample calculations included in the appendix illustrate in detail

the methods employed in calculating the isobars.

Page _^±

'LAME3DATRANS TION

|

/

/

I

SATURATED LI HELIUM H3UID ^SATURATED LIQUID

\ HELIUM I t?

$

^

t-SC: s at %/c sdT A

TEMPERATURE, (°K)

Figure 2. Specific Heat at Saturation of Liquid

Helium Illustrating the Method Used to

Calculate /csat dT Employed to Determine

the Absolute Enthalpy of Liquid Helium at

The Normal Boiling Point.

?age 15,

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V X V XX\ X

v $$$$

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0.5 I ^ 1.5 PRESSURE .(atmospheres)

2.5

Figure 3. Specific Volume of Saturated Liquid Helium

Illustrating the Calculation ofjv dp Used

in Determining the Enthalpy at the Normal

Boiling Point,

16.

TABLE 1

PRESSURE REFERENCE POINTS

Liquid

Specific volume Satur<ition pressure

( Liter s/Kgm. ) (Atmospheres) Source

5.00 68.8 (Solid)

(2) Domb, Dugdale

5.25 48,0 (Solid) ?i 5.50 33.0 (Solid) ii 6.00 15.0 (Helium II) Keesom(6 ) 6.50 4.7 (Helium II) ii •o (1) Be rman ' . Aelmanov

7,00 0.237 (Helium I)

8.00 1.0 (Helium I)

17.

CALCULATION OF ISENTHALPIC CURVES

As stated above, the normal boiling point of 4. 21 K and one

atmosphere saturated liquid is considered as the fixed reference'

point for enthalpy. The enthalpy change can be described by the

re.] ation

dh =TdS + VdP

(5)

Because enthalpy reference points were not available except at or near the saturated liquid condition, it was necessary to establish

values of enthalpy for several pressures in the compressed liquid

region. This was accomplished by using an isentropic process and

the calculation of enthalpy differences with changes in pressure at

constant specific volume,

dh = VdP (s=c)

(o)

The enthalpy of the normal boiling point serves as the datum, and

enthalpy values were determined at various pressures along an isentropic through this point up to and including 100 atmospheres pressure.

At these derived enthalpy points, changes in enthalpy at constant

pressure were calculated from equation (5) by

dh = TdS (p= c)

(7)

LOUNASMAA'S EXPERIMENTAL AND DERIVED DATA

The doctoral thesis of Lounasmaa, "Specific Heats at Low Temperatures" '' reported in detail the very comprehensive and thorough

experiments conducted by Lounasmaa to determine the specific heats

and equation of state of liquid helium. The experimental area of investigation included temperatures of from 2. 5°K to 20°K, pressures

from 2 atmospheres to 100 atmospheres and specific volumes from

4.81 liters per kilogram to 72. 15 liters per kilogram. This paper

came to the authors 1 attention after the calculation of isobars had been

completed and therefore served as a valuable comparison between

derived and experimental data.

18.

The experimental measurements of Lounasmaa of P, v, T and

specific heat eliminated the differentiation step in deriving the pressure

differences at constant volume as performed by the author. Since this

process of determining the slopes of isotherms on specific volume

entropy coordinates is inherently inaccurate, the derived pressure

functions of the author are considered in error when found in conflict

with the experimental pressure data of Lounasmaa.

Table II shows the comparison of the calculated slopes of V dT'V

as determined by the author and experimentally by Lounasmaa. The

specific volume is 6 liters per kilogram.

The experimental data of Lounasmaa for the functions of pressure

and the derived functions of enthalpy are believed to be of greater accuracy

than the derived functions of the author, and therefore the temperatureentropy and enthalpy -entropy diagrams were plotted using this more

accurate data.

19.

TABLE II

COMPARISON OF VALUES OF VdT/v FOR

Temperature

(°K)

dT/

(a

Mann)

./°K)

3. 56*

4. 36

4. 84

5. 22

5. 52

5. 94

6. 20

6. 38

6. 62

6. 78

d£\

at) y Lounasmaa

(atm./°K)

Error

3.0

3.5

4.0

4.5

5.0

6.0

7.0

8.0

10.0

12.0

3.94

4,50

4.97

5.37

5.70

6. 19

6.51

6.72

6.95

7.05

-9. 6

-3. 2

-2. 7

-2. 8

-3. 2

-4. 1 -4. 8

-5. 1 -4. 8

-3. 8

* extrapolated value

20.

Figure 4 is an enthalpy -entropy diagram for helium. The graph

was not extended to low pressures and high temperatures as the

Zelmanov temperature-entropy diagram is adequate in this region.

Figure 5 is the plot on temperature entropy coordinates of all

data discussed in this paper.

An interesting plot of the Lounasmaa and Domb et. al. data is

shown in Figure 6. The isochores of Lounasmaa terminate with

respect to data very near the saturated liquid for solidification data of

Domb. But to extrapolate the Lounasmaa isochores to an intersection

with the saturation boundary necessitates a rapid change in slope of

the isochores and in the case of the higher pressures the slope of the

isochores must change from positive to negative. This suggests an

error either in Domb's saturation data or in Hill's data near saturation.

The origin of the various data plotted in Figures 4 and 5 is as

follow s

&') Saturated Liquid for Vaporization. Values of

entropy from 0°K to 1.9°K are from the graphical

integration of C versus T of this paper and the

sat.

~~T~~

data of Kramers et. al. From the lambda point to

5. 05°K the entropy values of Hill and Lounasmaa' ' were used. The critical point entropy was found

consistent with the value indicated by Zelmanov

2) Saturated Vapor. The heats of vaporization of

Bermam ' were used to calculate this boundary.

Data of Berman was also used to determine the densities of the liquid and gases at saturation.

3) Saturated Liquid for Solidification. The Domb and

Dugdale data were used for this bo\mdary as well as

the Saturated Solid Boundary.

4) Lambda Transition. From Keesom

5) Isobars. Pressures of 0.5 atmospheres to 3 atmos-

( 1 ?

pheres were from Zelmanov' '. Pressures from 3 atmospheres to 100 atmospheres were from Lounasmaa'^,

page zi.

1.0 1.5 ENTROPY (Cal/gm °K)

2.0 2.5

Figure 4. Enthalpy-Entropy Diagram of Helium

Page 22.

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90

80

70

60

50

40

30

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10

/

v/ / 1 V /f

/

/ /

to / 7 / / /

/ / /

to/ f 7 //

SOLID y w

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HELIUM n , ,' 3—

' i—''

, _4—

r^tz ~~ .2 r

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(0

0>

TEMPERATURE,(°K)

15 20

o MINIMUM PRESSURE VALUES OF L0UNASMAA.

a —SOLID DENSITIES CORRESPONDING TO

DENSITIES OF LOUNASMAA.

« —LAMBOA TRANSITION DENSITIES CORRESPONDING TO LOUNASMAA. Figure 6. Pressure-Temperature Diagram of Helium

24

6) Isochores. Specific volumes of 5 liters per kilogram to 70 liters per kilogram were from Lounasmaa

and from 70 liters per kilogram to 800 liters per kilofram were from Zelmanovv '.

7) Enthalpy. Below 2 atinospheres the temperature,

pressure, entropy, enthalpy, and density data of

Zelmanov were used and from 2 atmospheres to

100 atmospheres the data of Lounasmaa* ' were used.

CONCLUSIONS

The processes of plotting new experimental data and calculated

values derived from this new data using existing reference material

subjects the final result to some inherent form of error. The early

work of Keesom below 3. 5°K is subject to error because of changes

in the accepted values of the vapor pressure -teinperature scale for

helium and the less accurate methods of measurement that existed

(7) when his data was published, ' The Keesom data used in this paper

was in general taken from his data of physical properties near the

normal boiling point and is considered to be within his estimate of 3<f *

The Zelmanov study of the Joule-Thomson effect and his resulting

temperature entropy diagram are estimated by Zelmanov to have an

accuracy of within 3 per cent. Although in plotting one point it was

found that a discrepancy of 25 per cent with Hill's data occurred at

a pressure of 60 atmospheres and a specific volume of 6 liters per

kilogram. Because this point was at the fringe of the Zelmanov area

of investigation and becatise all other data of Zelmanov agreed to within 3 percent of the Hill data, the point was disregarded.

The work of Domb and Dugdale was used to establish the reference

pressures when calculating isobars from Hill's data. The reference

points were no longer needed after the data of Lounasmaa was received,

The data of Lounasmaa is considered within his estimate of 1 per

cent because the experimental PVT data was of such scope as to

eliminate inaccurate differentiations in finding derived PVT functions.

All other data cited are in agreement within the error mentioned

above and therefore the overall accuracy of the plotted data of this

paper is estimated to be within 5 per cent and possibly much less.

25,

Future investigations made possible by this paper include the

extension of the temperature and pressure data to values greater than

20 degrees and 100 atmospheres. This could be done by correlation

of the high pressure, high temperature data of Keesom with the data

of this paper s using some type of equation of state. The result would I a very complete extensive composite diagram of state for helium.

26.

BIBLIOGRAPHY

(1) Berman, R. and C. F. Mate, "Some Thermal Properties of

Helium and Their Relation to the Temperature Scale",

Philosophical Magazine , (May, 1958), pp. 461-469.

(2) Domb, C, and J. S. Dugdale, "Solid Helium" , Chapter XI,

Progress in Low Temperature Physics, Volume II, Amsterdam, The Netherlands, North Holland Publishing Company, (1957), 480 pp.

(3) Hill, R. W. , Personal Letter to R. B. Scott, Chief Cryogenic

Engineering Laboratory, Boulder, Colorado, (7 October,

1957).

(4) Hill, R. W. , andO. V. Lounasmaa, "The Specific Heat of

Liquid Helium" , Philosophical Magazine, (Feb., 1957),

pp. 143-148.

(5) "The Entropy Diagram of Helium",

Fifth International Conference on Low Temperature Physics

and Chemistry, Madison, Wisconsin, (26 August, 1957).

(6) Keesom, W. H. , Helium , Ilsevier, (1942), 494 pp.

(7) Keesom, W. H. , and K. Clusius, "Uber die Spezifische

Warme des Flussigen", Leiden Communication, Number

2l9e, Volume 35, (1932), pp. 42-58^

"

(8) Kramers, H. , J. Wasscher, and C. Gorter, "The Specific

Heat of Liquid Helium Between 0. 25 and 1.9 K" , Physic a , Volume 18, Number 5, (1952), pp. 329=338*

(9) Lounasmaa, O. V., "Specific Heats at Low Temperatures",

Thesis submitted for the Degree of Doctor of Philosophy,

University of Oxford, (1958).

(lO)Tsu, To C. , and D. T, Beecher, Thermodynamic Properties

of Compressed Water , ASME, (1957).

(11) Van Dijk, H. , and M. Durieux, "The Temperature Scale in

the Liquid Helium Region", Progre ss in Low Temperature

Physics, Volume II, Amsterdam, The Netherlands, North

Holland Publishing Company, (1957), 480 pp.

27.

(1Z) Zelmanov, J. , "The Entropy Diagram for Helium at Low

Temperatures", Journal of Physics (USSR), Volume

8, Number 3, (1944), pp. 135-141.

(13) "Joule -Thorn son Effect in Helium

at Low Temperatures", Journal of Physics (USSR),

Volume 8, Number 1, (1940), pp. 43-52.

(14) Zemansky, M. W. , Heat and Thermodynamics, Fourth Edition, New York, N. Y. , McGraw-Hill Book Company,

Inc. , (1957), 484 pp.

(15) Clement, J. R. , J. K. Logan, and J. Baffney, "Liquid Helium

Vapor Pressure -Temperature Scale", Physical Review,

Volume 100, (1955), p. 743.

28

APPENDIX

29.

Derived Functions and Diagram Construction

Figure 7 is the plot of isotherms on specific volume -entropy

coordinates used in determining isobars. The data are from Hill,

extended slightly by extrapolation at the lowest temperatures. Table

III shows the data for this plot.

Slopes of the isotherms in Figure 7 were taken using a straightedge. These slopes { *^=

J were then plotted versus temperature,

as example of which is shown in Figure 8. The integration was performed graphically using a temperature increment of 0. 2°K, and the

resulting pressure differences were then based on fixed pressures from

Table I.

Figure 9 is a plot of pressure and temperature showing isochores as determined from Figures 7 and 8. The isentropic used in

determining the enthalpy values is also shown. It was plotted using

values determined from Figure 7.

Page 30.

25

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2.8 3.2 3.6 4.0

Figure 7. Isotherms of Helium as Functions of Volume and

Entropy.

31.

TABLE III* 3 )

ENTROPY OF FLUID HELIUM

Specifi c Volume

liter s/kgm.

Temp.

°K 72. 15 24.05 14.43 10.31 8.02 6.56 5.55 4.81

2.50 0.441 0.370

3.00 0.520 0.426

3.50 0.590 0.481 0.388

4.00 0.656 0.534 0.432

4.50 2. 152 0.874 0.717 0.586 0.475

5.00 2.232 0.937 0.774 0.636 0.516

6.00 2.367 1.769 1.462 1.240 1.051 0.879 0.729 0.597

7.00 2.482 1.883 1.573 1.345 1. 151 0.873 0.815 0.673

8.00 2.580 1.980 1.668 1.438 1.239 1.058 0.895 0.746

10.00 2.746 2. 143 1.828 1.594 1.392 1.205 1.037 0.880*

12.00 2.880 2. 276 1.960 1.724 1.520 1.331 1. 160 1.002*

15.00 3.046 2.440 2. 123 1.866 1.681 1.491 1.319 1. 159*

20.00 3.260 2.654 2.337 2. 100 1.894 1.705 1.533* 1.373*

Extrapolated Values

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Figure 8, Slopes

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Used to Determine Isobars.

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Illustrating the Method Used to Determine

Enthalpy Values

34.

Associated Thermodynamic Properties Charts

The charts presented on the following pages were compiled from

the data of Lounasmaa except in the x temperature and pressure

region of the vapor dome. Data for this region was taken from

Bermanand Mate ' ' and Hill and Lounasmaa' '.

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He lium ( Lounasmaa

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Figure 11. Pressure-Interna.1 Energy Diagram for Helium

- 100 psia

page 37

100

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INTERNAL ENERGY, Joules /gm

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Figure 12. Pressure- Internal Energy Diagram for Helium

- 100 Atmospheres

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