LAMS-2673

“(P!5w

~lC-14 REPORT COLLECTION

REPF?CXILJCTION

COPY

LOS ALAMOS SCIENTIFIC

OF THE UNIVERSITY OF CALIFORNIAo LOS

LABORATORY

ALAMOS NEWMEXICO

—

SURFACE FITTINGOF THE

OF LIQUIDAND GASEOUS

. . . . . . -.

. .

EQUATION OF STATE

NORMAL HYDROGEN

. . .

—

.

-.

. .

— -- .- ● ~L..._-_. . . . .*.

. . .
LEGAL NOTICE

This reportwas preparedas an accountof Government sponsoredwork. Neitherthe UnitedStates,nor the

Commission,nor any personactingon behalfof the Commission:

A. Makes anywarrantyor representation, expressed

or implied,withrespecttotheaccuracy,completeness,or

usefulnessof the informationcontainedin thisreport,or

thattheuse of any information,apparatus,method,or process disclosedin thisreportmay not infringeprivately

owned rights;or

B. Assumes any liabilities withrespectto the use

of,or fordamages resultingfrom the use of any information,apparatus,method,or process disclosedin thisreport.

As used in theabove,“personactingon behalfof the

Commission” includesany employeeor contractorof the

Commission,or employeeof suchcontractor,to theextent

thatsuch employee or contractorof the Commission,or

employee of such contractorprepares,disseminates,or

providesaccessto,any informationpursuantto his employmentor contractwiththe Commission,or hisemployment with such contractor.

Printedin USA Price $2.00. Availablefrom the

Officeof TechnicalServices

U. S. Departmentof Commerce

Washington25,D. C.
LAMs-2673

PHYSICS

(TID-4500,17thEd.)

I-DS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITYOF CALIFORNIA LOS ALAMOS NEW MEXICO

REPORT WRITTEN. April1, 1962

REPORT DISTRIBUTED: April4, 1962

SURFACE FITTING OF THE EQUATION OF STATE

OF LIQUID AND GASEOUS NORMAL HYDROGEN

by

R. K. Zeigler

P. C. McWilliams

W. E. Keller

ContractW-7405-ENG. 36 withthe U. S. Atomic Energy Commission

All LAMS reports are informal documents, usually prepared for a speciel purpose ad primarily prepared for uee within the Labemtory rather than for

geneml distribution. This report bse not been edited, reviewed, or verified

for acourscy. All LAMS reports express the views of the authors ae of the

time they were written end do net necessarily refieet the opinions of the Los

Als.mos Scientific LAoratoxy or the find opinion of the suthors on the subjeot.

1

ABSTRACT

This report describes various conrputationsl techniques employed to

describe the equation of state of normal hydrogen data. Using the following functional form, the differences between the observed snd cal.culated values sre in general within 5’j%:

‘2

Pressure = PIV [1- (p3/v)p4] +P5T/(V - P~ ‘t-P7/v)

where P, P,. ..P 12 7

are the resulting parameters, V = volume, T = temperature.

1. Inlxroduction

~ the fsll of 1959, the Statistical Section of the l%eoreticsl

Division assisted in fitting a multiple regression surface to smoothed

hydrogen data. The resul’t of this endeavor appeared in LAMS-2357.*

There were several reasons to contxhu.e the work: 1) foremost, raw data

rather than smoothed data should be used; 2) the surface fit in LAMS2357 was divided into several regions; the ditisions were for the convenience of the equations, rather than natursl data divisions; 3) since

there were divisions, discontinuities existed at the boundaries of the

divisions; and k) the results were not as good as desired.

2* Representation of Data

The raw data represent an Smal.gsnwtion of information from some six

different sources (see Table 1). There we 4 vslues or varisbles involved: temperature (T) in ‘K, pressure (P) in atm., volume (V) in

cc/m, and compressibility (Z). It shouldbe noted that Z = f(T,V). It

should also be noted that at various times different coxibinationsof the

variables were used in attempting to solve the problem.

*

Sherman, Vogel, Moore, Schuch, end Goldman, Ias Alsmos Scientific

Laboratory Report, LAMS-2357, Septefier 25, 1959.

5
3. Two-Dimensional Approach

The approach used primarily was the so-cslled two-dimensional (2-D)

approach, suggested by W. E. Keller of Group CMF-9 of this Laboratory.

The isotemperature xuns illustrate the techniqye. Suppose we hold temperature fixed; then there axe several data points for this temperature.

‘Ihenfit (in two-dimensions) some ftmction, say Z = A P + B P2 + C $

where A, B, end C sre constants determined by a least squares fit. Since

this can be done for several temperatures, we msy write Z = A(T) P +

B(T) P* + C(T) ~. Now if a “nice” functional relation can be found for

A(T), B(T), and C(T) in term of T, the problem is solved, for en ensuing

function Z = f(P,T) follows.

4. Results of the Two-Dimensional.Approach

A. Isotemperature Runs. In 2-D, the results were qyite good for

a(T) p2 ~ b(T) P +

the various temperatures. The expression Z = e

2 4

c(T) P + d(T) # -t-f(T) P fits the data extremely well. unfortuns$ely, the coefficients a(T), b(T), c(T), d(T), and f(T) were not

well behaved. Thus, an attempt to extend this into 3-D was rather

unsuccessful, resulting in some errors of magnitude of 25%.

B. Isochore RU.tlSe Again, the 2-D fits were rather good. In fact,

they were better thsn the isotemperature runs. (It should be noted

that the T,P surface is much better to work with than the Z,P surface). The equation used here for the 2-D mns was P = A(V) +

B(V) T + C(V) T2. Here the coefficients A(V), B(V), and C(V)

6
were qpite well.behaved,

The evolving e~ation is

&lLowing a smoother transition into 3-D.

as follows:

Pressure =

-+

+

pl + p# + p3/@ + P4/V3 -I-P5/V4 + P6T + P7T/V

P8T/V2 + P9T/@ + PIOT/V4 + P~T2 + PUT2/V

p#/$ + p14@/V3 + P15T2/V4,

(1)

where Pl, P2,.0.P15 we the resulting psmmeterse

It shouldbe noted

in LAMS-2357. The

that this equation is very similar to that used

fundamental difference in approach is the division of data. Here the data have been divided into 2 classes:

class 1 = GAS plus FLUID, class 2 = LIQUID. Obviously, this represents a “logical” data division. IUrther, the progrwn was rewritten to minimize the

percentage deviation of

the fitted values.

percentage of error. Table 2 shows the

egyation (l). Also IBM Listing 1 enumerates

5* Boulder Egpation

From a writeup by R. D. Goodwin, C!EL,National Bureau of Standards,

Boulder, Colorado, entitled Approximate Wide-Range E~ation of State for

~nY Me fo~owi~ P~~eters we~ detemed by the method of least

Sqylsxes:

7
‘2

Pressure = PIV [1 - (P3/V)pk] +P5T/(V-p~+p#J),

where P,, PO,...P7 sxe the retit~ par~~rs”

.& G I

Again, the data were divided into

suits for the GAS-FLUID data were

less than 5$. See Table 2 for an

data. Also see IBM Listing 2 for

tie forementioned

very gratifying;

exact percentage

a point-by-point

two groups. The remaximumerror was

deviation for these

enumeration of the

results. Unfortunately, this tiction does not seem to fit LI@ID data

alone. A fizrtherrun was made fitting ALL data (see Listing 3 and Table

2). It should be noted that the e~ression does fit all data z%?latively

well.;the large percentage errors here are attributed

I@ the present, this represents the best U.

6. Jl@anation of Table 3

Table 3 enumerates the

eqpations converged for the

and the IBM Listings

parameter vslues to which

three runs made. A brief

IBM listings follows: The final values

parameters are given on the 5th ~ge of

Listings 2 and 3.

fitted curve, the

etc. are listed.

the temperature.

tothe liqpid area.

the least sqyares

qplanation of the

and standard deviations of the

Listing 1 and the 3rd pages of

Then the data points, the corres~nding points on the

deviations between fitted and actual dependent variables,

The first independent variable is the volume, the second,

For a more detailed

computational least squares iterative

8

explanation of the listings and the

procedure, refer to IA-2367,The
Solution of the General Ieast Squares Problem with Special Reference to

High-Speed Computers, October 15, 1959, by R. H. Moore and R. K. Zeigler

of this Laboratory.

7* ~

The Boulder equation ties an adequate over-dll.job, and does particularly weld.for the bulk of the data, namely the GAS-FLUID data. It

should be noted that most of the points with maximum deviation were near

the “criticsl point”.

9
TABLE 1

DATA SOURCE LIST

The following were the data sources used for this work. References are

for Normal Hydrogen.

1.

2.

3*

4.

5*

D. White, A. S. lZriedman,and H. L. Johnston, J. Am. Chem. Sot. —72,

3927 (1950). [P ~ T along saturated vapor curve from 15°K to

critic~ mint (33.2°K)1

E. Mathias, C. A. Crommelin and H. Kamerlingh Onnes, Comm. Phys.

Lsb. Univ. Leiden 154b (1921.)as corrected by H. W. Woodley, R. B.

Scott, and F. G. Brickwedde, NBS Research Paper 1932 (1948).

(Density of saturated vapor from 14°K to critical pint).

H. L. Johnston, W. E. Keller, snd A. S. Friedman, J. Am. Chem. Sot.

~, 1482 (1954). (PVT of liq@d from 20°K to critical point, up to

100 atm.).

H. L. Johnston, D. White, H. Wirth, J. Swanson, L. Jensen, and

A. S. Friedman, Wright Field, ADC TR-26L25 (1953). (PVT of gas

from critical point to 300°K from 20 atm to 100 atm.).

D. White, A. S. Friedman, and H. L. Johnston, TR-26L12(1953).

(PVTof gas from 20°K to 300°K, P = O to 20 atm.).

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