ANALYSIS OF A LOW PRESSURE

P-V-T APPARATUS

by .

FERNANDO C. VIDAURRI JR., B.S. in CH.E,

A THESIS

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Technological College in Partial Fulfillment of

the Requirements for the Degree of

MASTER OF SCIENCE IN CHEMICAL ENGINEERING

Approved

/

Accepted

May, 1965 ^

No. 5"'-

ABSTRACT

This investigation was concerned with the develop­

ment of a workable low pressure pressure-volume-temp­

erature (P-V-T) apparatus and possible forms of correla­

tion of data obtained from such an apparatus. An apparatus

of this type would be particularly suitable to use in the

investigation of properties of mixtures. In an effort to

obtain a more precise approximation of zero pressure data

(from which second virlal coefficients may be obtained),

an apparatus was specially constructed to get data from

atmospheric pressure down to 100 mm. Hg.

It was found that the second virial coefficients of

gases could not be determined by expanding the gases from

atmospheric pressure to lower pressure. This occurred

because the relative error associated with the pressure

measurements was too great over this range.

Error analysis revealed that second virial coefficients

having a value greater than 10 cm- /gmol could be determined

with less than 10 per cent error if the expansions were

made from three atmospheres dov.T' to 500 mm. Hg.

An apparatus was proponed to operate in this pressure

range. The apparatus involves the reading of mercury

menisci and is restricted to the measurement of properties

of materials that are in a j aseous state at three atmos­

pheres and 70°C.

il

ACKNOWLEDGEMENT

The author expresses his sincere appreciation to

the members of his Committee for their help in the writing

of this thesis. Especial acknowledgement is given to

Dr. H, R. Heichelheim, Chairman of the Committee, for

his guidance and encouragement, and to Dr. A. J. Gully^

for his assistance with the section on error analysis.

I ".i 1

TABLE OF CONTENTS

Pac e

ABSTRACT li

ACMOWLEDGEI^ENT iii

LIST OF TABLES v

LIST OF ILLUSTRATIONS vi

I. INTRODUCTION 1

Purpose and Scope 1

Review of Previous Research 3

Theory of Burnett Apparatus 3

Development of Compressibility Equations 6

II. APPARATUS AND PROCEDURE 9

Apparatus 1 9

Apparatus 2 13

III. EXPERIMENTAL RESULTS 15

Untreated Data 15

Treatment of Data 18

Discussion of Experimental Results . . . . 20

Error Analysis 25

Proposed Apparatus 31

IV. CONCLUSIONS 3^

LIST OF REFERENCES 35

APPENDIX 36

IV

LIST OF TABLES

Table Page

1A. Experimental Data 16

1B. Experimental Data 17

2. Second Virial Coefficients of Mixtures

of Air and V/ater Vapor at 30^0 21

3. Experimental Second Virial Coefficients . 23

h. Second Virial and Interaction Coefficients of Air and Water Vapor . . 37

5. Relative Percent Error in Pressures and Pressure Ratios 3S

6. Relative Error of Seconci Virial Coefficeints 39

V

LIST OF ILLUSTRATIONS

Figure Page

1. Schematic of Burnett Apparatus 3

2. Apparatus 1 10

3. Apparatus 2 13

h. Pressure Ratio vs_. Pressure, Plot of

Data from Run 1 19

5. Schematic for Average Error Model . . . . 27

6. Proposed Burnett Apparatus 32

VI

CHAPTER I

IKTRODUCTIOT!

Purpose and Scope

A method of obtaining better values of the second

virial coefficients of individual species of molecules

and of the interaction coefficients of mixtures of the

species could improve low pressure thermodynamic data

obtained from statistical mechanics considerations.

Particular groups of well-kno\\m data would probably

not be improved, but better molecular models could be

built by back-calculation, and these better models used

in turn to generate accurate thermodynamic data.

2

Burnett developed a method of measuring the com­

pressibility of gases by successive expansions from

higher to lower pressures. A typical Burnett apparatus

is constructed to obtain data from 120 atmospheres down

to around 10 atmospheres at various isotherms. Modified

Burnett apparatus have been used to get compressibility 7

data down to atmospheric pressure. Zero pressure data

are obtained by passing a curve through isothermal pressure

points and extrapolating from this curve to zero pressure.

Zero pressure data are of interest because the compres­

sibility factor Z is equal to unity. This simplifies the

virial equations that are nsed to describe the system.

1

2

This investigation \ras m.ainly concerned with the

development of a workable low pressure pressure-volume-

temperature (P-V-T) apparatus and possible forms of

correlation of data obtained from such an apparatus.

An apparatus of this type would be particularly suit­

able to use in the investigation of properties of mix­

tures. In an effort to obtain a more precise approx­

imation of zero pressure data, an apparatus was specially

constructed to get data from atmospheric pressure (760

mm. Hg) down to 100 mm. Hg.

Review of Frevions Research

Theory of Burnett Apparatus' o

In the Burnett method of obtaining compressibility

data, a test gas is contained at a measured pressure in

one chamber of a double chambered vessel.

-Hxl vacuum

Figure 1 . Schematic of Burnett Apparatus

The second chamber is evacuated and closed off. The

sample is expanded to fill both chambers and allowed to

come to thermal equilibrium:. The pressure is measured,

then the second chamber is closed off from the first

chamber and evacuated. The repetition of the procedure

results is a series of pressure measurements of decreasing

magnitude that approach zero pressure. The procedure is

repeated until the minimum pressure that can be obtained

and measured in the system by a reasonable number of

expansions is reached.

The ideal gas lav/ with compressibility factor

correction Z for deviation from ideality is PV = ZnRT,

where

P = absolute pressure of gas

V = volume occupied by the gas

n = number of moles of gas present

R = universal gas constant, vrith units corresponding to units of P, V, and T

T = absolute temperature of gas

Z = f(P,T,composition) = compressibility factor correction for deviation of the volumetric properties of the gas from an ideal gas at the same conditions.

Initially, n moles of test gas occupy volume V.

at pressure P and temperature T. The compressibility

factor has a specific value Z^ at these conditions.

(1) P^V, = Z^n^RT

After expansion, the test gas is at pressure P. and

and occupies volume (V. + Vp).

(2) P (V + V^) = Z-i RT

Dividing equation (2) by equation (1) we obtain

(3) Pi(V^ + Vg) ^ Z n RT

Po^l Z^n^RT

(V^ + Vp) = N = a p p a r a t u s c o n s t a n t Vi

S u b s t i t u t i n r : N in equa t ion (3) and r e v / r i t i n g r e s u l t s in

(k) ?^ = P Q ^ I / N Z Q •

Chamber 1 i s now a t p r e s s u r e V^ and

(5) P^V^ = Z^n^RT .

Expanding again we obtain

(6) P2(V^ + V^) = Z n-iRT .

Dividing equation (6) by equation (5) gives

(7) P2 = ^1^2

NZT" The substitution of equation (4) in equation (7)

results in

P = P0Z2 2 ~"5 •

N ^

Generally, after J expansions

(8) p / = P Zj ^ Zo •

The apparatus constant N is the zero pressure inter-PT 1 cept of the curve obtained by plotting !;"•' against PT. Pj ^

For isothermal operation we observe that (9) Lim Pj.i = ^ ^

Pj-^0 - ^ ,

If there is a slight variation in the temperature of the

apparatus, the apparatus constant is obtained from

Lim (Pj-1/Tj.i) ^ j ^ Pj-^^ (Pj/Tj)

Development of Compressibi]ity Equations

The compressibility factor Z can be expressed as

an analytic function of either the pressure or molar

11

volume at constant temperature. Kamerlingh-Onnes''

proposed that the compressibility factor Z could be

expressed by virial equations of the form

(10) Z = PV/RT = 1 + B'/V + C»/V2 + D'/v3 + .•• 1 0

and Holborn and Otto proposed

(11) Z = PV/RT =1 + BP + Cp2 + DP3 + ... ,

The relationship between the constants of the two

equations are: B = B»/RT5 C = (C»-B'2)/(RT)2; etc.

The adjustable parameters, B', C , D', ••• are

temperature functions usually called virial coefficients.

B' is a measure of deviation from ideality due to binary

molecular interaction; similarly, C of ternary and D'

of quaternary molecular interactions.

At low pressures, the higher order terms become o

neglible and equation (11) reduces to (12) Z = 1 + BP .

Equation (7), written as a general equation with slight

temperature variations is

(13) Pj-l/Tj.iZj.^ = N . Pj/TjZj

Subs t i tu t ing equation (12) in equation (13) we get

(1^) P j , i / T j , i ( 1 + BPj . i ) = N . Pj /Tj(1 + BPj)

Rearranging equation (1^) r e s u l t s in

(15) P j , l Tj = N + B(N - Tj )Pj_^ , PjTj-1 Tj_i

For isothermal operation, equation (15) reduces to

(16) Pj-i/Pj = N + B(N - DPj.-, ,

where B is the second virial coefficient.

The virial coefficients, obtained from P-V-T data

are used in the determination of low pressure thermo­

dynamic data. At lov/ pressures, a workable theory of

thermodyanmic properties of gas mixtures has been devel-

oped by statistical mechanics. The theory is expressed

in the follov/ing form:

Uj_ = g9(T) + RTln(XjLP) + p jZxj_x^Bik(T)

- ?2"xkBii,(T)_] + ... .

gj_(T) = zero pressure reduced free enthalpies, calculated from spectroscopic constants of the atomic or molecular species in question

^i? ^k ~ ^^^ fractions

Bj_| (T) = B; j (T) = second virial coefficient if i=k = interaction coefficient if i/k

(expresses effects of intermol-ecular forces)

Uj_ = chemical potential

All thermodynamic properties of a homogeneous phase

can be determined if the' chf-mical ] otentials are expressed

8

as functions of temperature, pressure, and mol fractions.

The following list illustrates how some of the more com­

mon thermodynamic properties are obtained from chemical

potential data:

g =Zx.u. = specific free enthalpy

V = 9g/t5p = specific volume

s = -<Pg/5T = specific entropy

h = a(g/T)/d(1/T) = specific enthalpy .

CHAPTER II

APPARATUS AKD PROCEDURE

Two low pressure P-V-T glass apparatus were built,

and modifications were tried on each apparatus. Although

data were obtained only from the second apparatus, the

first apparatus will also be presented to show the evolu­

tion of the final design.

Apparatus 1

In this particular variation of the Burnett method,

mercury was used to trap the same fraction of V-, + V2

by raising the mercury level slov/ly above point D. The

mercury level was raised on up past capillary E. The

valve to the vacuum was slowly opened and the test gas

in volume Vp was slowly evacuated. As the evacuation

proceeded, mercury was pulled in from the reservoir so

that the height of the merc iry plus the test gas pressure

above the mercury on one side of the apparatus was equal

to the corresponding pressure on the other side. The

gas pressure above level B was measured until it ap­

proached 0.0001 mm. Hg, l^en this degree of vacuum was

obtained above level B, the mercury levels were adjusted

by the use of the mercury in the reservoir so that the

level A was at a fixed reference level. The pressure

represented by the difference in the heifrhts of mercury

9

Figure 2. Apparatus 1

10

Modification 1

L

V I

lSt3

7~T = r=z= c2..

B

To vacuum

McLeod 'Pressure Gage

To .*-vacuum

Modification 2

(2®: To atmosphere

11

B - A was then equal to the pressure of the test gas in

V-| . A cathetometer was used to measure both levels to

the nearest 0.01 mm.. After the measurements, the mer­

cury in volume V was quickly lowered. The capillary E

was used to keep a level above E until the level B was

lowered to C. This was to prevent the test gas from

V. from bubbling up through the mercury. The falling:

mercury could carry some small p as bubbles v/ith it and

trap them in the lower part of the apparatus. The gas

bubbles could also carry small mercury droplets upwards

and deposit them on the upper walls of the apparatus.

In either case, the subsequent fraction of test gas

removed would not be the same as before. V and Vp

were allowed to come to the same pressure, then the

procedure of trapping the sam.e fraction of test gas

by raising the mercury level above roint D was repeated.

In modification 1 of apparati:;s 1 , the mercury level

in Vp was to be quickly droiped b" tbe use of a quick-

action large bore piston reservoir arrangement. The

adjustment of level A to thp same level for each expan­

sion was to be achieved by a small piston \^±th fire

thread movement. This modification of apparatus 1 was

inoperable because the piston assemblies cound not hold

the desired degree of ve.cuui-.

In the second modificat io) or apparatus 1 , "I'he pres­

sure above the reservoir li \'ns ijse=l to inject and v/ith-

12

draw mercury from the right side of the apparatus. This

modification featured a lar^e bore stopcock for the rapid

lowering of mercury and a fine ca]dllary bypass for the

fine adjustment of level A. This modification was demol­

ished on the initial trial run. The author has hypothe­

sized that the mercury level on the left side dropped

through the capillary before the mercury in volume Vp

could get down past point D. Subsequent calculations

shov/ed that this was possible if the diameter of the

capillary was greater than 0,^ m.m... Since there was a

vacuum above the mercury in volume V2, the test p:as

passed through the mercury to equalize the pressures in

both sides of the apparatus. The gas carried some of the

heavy mercury with it and blew out the top of the Vp

chamber. The apparatus was also broken in the vicinity

of point C. It is possible that the failure was due

to a structural defect at this point. An initial break

at this point would have the same effect on the apparatus

as the previously described mechanism.

The building of apparatus 1 disclosed that a piston-

type mercury reservoir could not be made to hold a vacuum

unless an elaborate design v;as used. It also revealed

that the capillary design v/as potentially unstable. This

information was considered in the design of apparatus 2.

13

To vacuum ^ — -

To atmosphere

(EdDz:

Apparatus ^

vacuum

Figure 3. Apparatus 2

In apparatus 2, the pressures in volume 1 and vol­

ume 2 were equalized by the use of a three-way stopcock E

before the mercury level was lowered to point C. The

stopcock was then placed in the position shown before

the next trapping of gas sample. The fraction of test

gas retained by the apparatus was originally 0.94, and

It required a series of about -',0 expansions to FO from

700 mm Hg to 100 mm Ug pressure. Since the time involved

in obtaining the readings was about 30 minutes per expan-

1^

sion, it was decided to reduce the fraction retained.

The apparatus was modified so that the fraction retained

was 0.84. This lowered the required number of expansions

required to span the desired range to 12.

The apparatus 2 was also run backwards, that is,

expanding from Vp into evacuated V^. The fraction of

the test gas retained was 0.16. With this method, only

one point could be obtained from an initial pressure.

Different data points were obtained by starting at

different initial pressures. The time required to get

one data point was one hour.

Apparatus 2 worked satisfactorily except for the

stopcocks. They required an excess amount of stopcock

grease to hold a vacuum, and some of this grease was

then picked up by the mercury. Teflon stopcocks are

recommended as replacements.

CHAPTER III

EXPERIMENTAL RESULTS

Untreated Data

The experimental data are presented in Tables 1A

and IB. The test gas for all the runs was air that was

dried by passing it through an eight inch high column of

Drierite. The temperatures were measured by a thermo­

meter taped on the outside of the apparatus. Clear

tape was used to insulate the thermometer and apparatus

from the atmosphere. It was assumed that the thermometer

gave the true reading inside the apparatus and all the

apparatus was at the same temperature. In runs 6 and 7,

the thermometer was calibrated in I C intervals and the

temperature was approximated to the nearest 0.5^* Iri the

other runs, a thermometer calibrated to the nearest 0.1^C

was used and the cathetometer was used to approximate the

temperature to the nearest 0.01^. The visible trend in

the temperature measurements is due to the lowering of

the ambient temperature during the time the experiments

were being run.

5

16

Table 1A. Experimental Data

P = pressure, mm Hg

T = temperature, °C

Run 1 Run 2 Run 3

703.05 587.37 ^90.29 J+09.40 3^1.5« 285.21 238.05 198.57 165.92 138.^^ 115.27 96.51

26.25 26.41 26.32 26.20 26.17 26.09 26.01 25.92 25.90 25.83 25.79 25.70

679.32 567.22 ^73.^0 395.32 329.92 275.03 229.76 191.57 159.89 133.67 111.33 93.155

26.73 26.65 26.61 26.56 26.48 26.31 26.17 26.02 25.81 25.68 25.53 25 .^^

698.72 583.7^ ^87.50 ^07.13 339.67 283.53 236.^6 197.55 16^.78 137.18 11^.63 95.50

25.21 25.33 25.^1 25.35 25.28 25.07 24.92 24.80 2J+.63 24.46 24.28 24.12

Run k Run 5 I I

24.76 > 24.65 2^.51 2^.47 2J+.38 2^.21 24 . l i f 24.09 23.95 23.81 23.57 23.^0

^ 691.16 576.92 kb^ .71 4o i . 96 335.50 281.23 23^.^7 195.81 163.13 136.32 113.^8 9^.935

27.61 27.5^ •27.^9 27.^0 27.28 27.16 27.03 26.89 26.61 26.35 26. U8 25.86

I •

659.02 24.76 - 691.16 27.61 J y 5^9.76 24.6^^ ^76.QP P7-S4 f • ^59.10 383.11 320.03 266.82 222.80 185.755 155.26 129.^0 108.035 90.30

Note: All runs with same apparatus constant. j

Test gas was dried air. j I n

17

Table 1B. Experimental Data

P = p re s su re , mm Hg

T = temperature , °C

Run 6 Run 7 Run 8

70^.22 587.90 ^90.88 ^U9.^6 3^2.29 285.^8 238.11

26.6 26.65 26.55 26 .^ 26.35 26.35 26.35

•

682.28 6^2.58 605.62 570.50 537.26 505.97 ^76.93 ^^9.66 ^23.^5 398.85 375.76 35^.18 333.5^ 31^.37 295.98 278.88 262.75 2^7.67

26.5 26.2 26.0 26.0 26.0 26.0 26.0 26.0 25.8 25.5 25.5 25.5 25.0 25.0 25.0 25.0 25.0 25.0

623.39 102.71 616.63 101.39 ^86.18

80.205 357.59

58.625 3^2.05

56.3^5 ^^9.^-7

7^.135 200.905 33.015

230.^6 38.185

30.01 29.59 27.91 27.79 28.15 28.00 29.30 29.13 28.98 28.91 28.68 28.51 28.^2 28.29 27.58 27.50

Note: All runs with different apparatus constants. Run 8 is a series in which the apparatus was operated backwards.

Test gas was dried air.

y «

18

Treatment of Data

The coefficients of equation (15)

Pj-1Tj = H + BKPj_i - BPj_T Tj/Tj.i ,

of the type y = a + bx.| + 0x2 , v/ere determined

from the data by a curve fit using the criterion of mini-9 1^

mum sum of squares of residuals. ' A point weighting

factor equal to the average deviation of the first three

points from the fitted equation divided by the deviation

of the particular point was used. This was done to give

less credence to points that v/ere far removed from the

general trend of the data. No discernable trend was

found in either the initial coefficients or in the co­

efficients obtained by using the weighted data.

The coefficients of equation (16)

Pj-l/Pj = K + 3(K-1)Pj_^ ,

of the type y = a + bx , were similarly determined

and the same type weighting factors previously described

were then used. Again, no trends v/ere found in either

the initial or the weighted coefficients. The computer

program used in determining the least squares straight

line is presented in the Appendix.

Data obtained from run 1 are used in Figure h to

demonstrate the scatter that war. rmmd in all the runs.

Figure ^ is a plot of Pj-i/lj against T i_i , with rj_>,

19

o

o DC

if)

O L.

Q_

Figure . Pressure Ratio vs. Pressure, Plot

of Data from Run 1.

.2000

.1990 -

.1980

.1970

.I960

.1950

.19^0

.1930

0.0 0.2 0.^ 0.6 0.8 1 .0

Pressure, atmospheres

20

in atmosphere units and corrected for the density of

mercury at 26* 0.

Discussion of Experimental Results

Second virial coefficients for air and water vapor

are reported by the International Joint Committee, on

Psychometric Data° in units of cubic centimeters/gram-

mole. Reciprocal atmospheric units are converted to

these units by multiplying by RT, where R = 82.06

cc-atm(OK)-''(gmole)"'' and T is in ^K. Table h in the

Appendix gives the value of the second virial and

interaction coefficients for air and water vapor at

various temperatures. Table 2 gives the values of •

second virial coefficients that were calculated for j i

different mixtures of air and v/ater vapor at 30°C. ;

The second virial coefficients of air-water vapor mix- |

1 ^ J

tu res -^ were ca lcu la ted from ; (17) B^ixture = 3^, (x^ )2 + 2612X1X3 + BjsCxj)^

where

^mixture - second virial coefficient of the mixture

B-j B22 = second virial coefficients of components ' 1 and 2

B-12 = interaction coefficient between components j 1 and 2 |

x-j = mol fraction of coiaponenr. 1 in the mixture ;

X2 = mol fraction of co iponent

«

• - .

21

Table 2

Second V i r i a l Coeff ic ients of

Mixtures of Air and Water Vapor a t 30°C

Mol f r a c . a i r Wt. f r a c . a i r Second v i r i a l coef.

cc/gmol

0.0 0.0 107^.0

0.10 0.152 876.2

0.20 0.-287 698.7

0.30 0.^08 5^1.^

0.^0 0.518 ^0^.3

0.50 0.617 287.5

0.60 0.707 190.0

0.70 0.790 11^.5

0.80 0.866 58.^

0.90 0.935 22.5

1.00 1.000 6.9

Calculated from equation (17), where

B ^ =6.87 cm^/gmol

B22 = 107^ cm^/gmol 3

3 2 ~ 3^.5 cm /gmol

22

The experimental values of the second virial coeffi­

cients are presented in Table 3- Also presented in this

table are the correlation coefficients for the fit of the

data to straight lines, the slope of the lines, esti­

mated deviation of the slopes, and t_ values obtained

by comparing the calculated slopes to zero slopes.

None of the slopes was statistically significantly

different from zero. The slopes were also found to be

statistically indistinguishable from each other, but

these data are not presented.

None of the correlation coefficients is large enough

to assert even with only 10 per cent certainty that a j

correlation exists between a plot of Pj_-|/Pj against j

Pj__. . This can be attributed to the fact that the ,

value of the second virial coefficient that v;as being J

determined was below the sensitivity limit of the j

apparatus. |

The initial plotting of pressure ratios Pj -i/Pj

against pressures Pj_i obtained by the low pressure

Burnett apparatus technique seemed to indicate that

there was too much scatter in the data for them to be

useful in predicting second virial coefficients. This

was verified by the low correlation coefficients obtained 1

in trying to pass various types of curves through the J

data. The investigation vmc there Tore transformed to *

one of obtaining the optimum pressure ranr;e to be used ;

Run

1

2

3

k

5

6

7

8

23

Table 3.

Experimental Second Virial Coefficients

Second v i r i a l coef .

15.83

117.79

.170.80

135.72

9 0 . 0 ^

•193.15

•106.87

281.20

C o r l . coef .

0.02

0 .15

0.27

0.20

0.08

0 .28

0.07

0.06

Slope x10-^

0.127

0.9^6

-1.3 '°8

1 .097

0.720

-1 .567

-0 .266

5.7^8

E s t . dev . of s lope x10-=

2.16

2.08

1 .61

1.79

2.92

2.66

1 .06

37.00

t.

0 .06

0.U5

0.86

0.61

0 .25

0.59

0 .25

0.16

t,-tests were run to see if any of the slopes were significantly different from zero. I

2h

in determining second virial coefficients by this tech­

nique. An estimation of the lim.its imposed on the system

by the precision limits of the pressure measuring instru­

ment was also undertaken. The development of the theore­

tical equations for these studies is presented in the

Error Analysis section.

The optimum pressure range has an upper bound in

the region where the third virial coefficients become

large enough to introduce nonlinearity. The upper

limit depends upon the gas that is being tested, but

there is an optimum upper limit for an apparatus that

is to be used to test various gases. The lower bound j

is fixed by the scatter of data that is introduced by :

errors in the pressure measurements. The precision ;

limits also impose restrictions on the size of the \

virial coefficients that can be investigated by this I t

method.

25

Error ^^alysis

In calculating the net pressu. e in the system, the

difference in the readings of the height of mercury in

each leg must be figured. If we let y be the greater

height reading

(^s) p^et = y - ^ • The variance associated with the pressure is given by

(19) (T^ = OP/ax)2(j2 + i3?/ayf-(ll . p - J

Differentiation of equation (18) to get terms required

in equation (19) yields

(20) ap/ay = i, and ar/ax = -1 . Substitution of (20) into (19) results in

(21) (T^ = (j2 -H 0-2 .

Relative error is defined as

(22) Rel. err. P ^ G"p/P .

Equation (22) gives the relative error associated with

each pressure measurement. Although (5 remains constant

for the apparatus, the relative error changes with the

pressure level.

The pressure ratio is given by

(23) R = P-,/P2 5

and the variance associated with the ratio is given by

Differentiation of equation (23) results in

(25) dn/S?^ = I/P2, and dW^^p " "^1/^2 '

26

Subst i tut ing (25) in (2 +) gives

(26) cTp = CT^^/p^ + i?cr'p2/P2 .

The relative error in the pressure ratio is given by

(27) ^^/n = (1/p2 + 1/p|)^^Vp .

Substituting equation (21) in equation (27)

(28) CTp/R = (1/P2 + 1 A | ) ^ / 2 ( ^ 2 ^ ^2)1/2 ^

^f (Sx ~ (JV' "then (28) reduces to

(29) ( p/R = Cn;/2"(1/Pf + 1/P2)^''^ .

Equation (29) gives the relative error associated with

each pressure ratio.

Table 5 in the Appendix crives the relative errors

of pressures and pressure ratios at different pressure »

levels for two different apparatus constants (fraction l 1

retained levels). Table 5 also contains relative error . J

data for a three manometer system. For these calculations, ' [

the precision limit (0.01 ram.) of the cathetometer was '

used as (J^,

The value of a pressure ratio is dependent upon the

previous value of the ratio and the error associated with

the previous value. This interdependency makes an a

priori analysis of error involved in a least squares

treatment of the data and subseouent error analysis of

the fitted coefficients very difficult, if not impossible.

A model of the system was therefore constructed to obtain

an approximation of the error.

Pressure Ratio

Ys- o '4 L

X X y-

L

:

>h<

D

n Pressure

27

1:, Ip ^ ll

Figure 5. Schematic for Average Error Model

The relative error in a pressure ratio can be deter­

mined from equation (27). The model consists of taking

a weighted average of the relative error in the pressure

ratio over a desired pressure range. The weighting

factor of a point is determined by dividing the sum of

one-half the distance to the previous point plus one-

half the distance to the next point by the total distance

to be used in the weighting procedure.

If k is the fraction of the total volume retained

after each expansion, the value of the pressure after

n - 1 expansions is given by

(30) PJ, = k""" P .

The number of expansions required to span a given pressure

range can be found from

(31) IK = ln{P^/V^)

In k

28

The value of the point P is determined from o

(32) P = P /k . The distance between consecutive points is obtained from

(33) 1„ = P„ - k"P, - l„_i .

The total distance D is

(3^) D = 1/2(1 + l„+i) + t \ .

Distances to be used in the wei hting factors are deter­

mined from

(35) d = 1/2(lj + 1^+^) .

The point weighting factor is equal to d^/D.

Relative error of the pressure r a t io at point P

is determined from

(36) r e l . e r r . P = (1/p2 + W^^^P^)^^^

Equation (36) can be rewritten as

(37) r e l . e r r . P, = O i i l L i L ^ V n . ^ kP^

The general expression for equation (37) is

(38) rel. err. P = (].2n + )1/2^^ n r — — — — — ^ .

The v/eighted average relative error can be found from

(39) d^(k^"j- 1)^^^^

Dk"P ' o

where n is the nearest integer to the value found by

equation (31).

29

The computer program that war used to calculate the

weighted relative error is rrecentod in the Appendix.

Allowances were made for i'!ultl])lc: manomnters, and the

following pressure precision limits were used:

(Tp = 0.02if5 nmi. Hg for 2 atm.<P 3 atm.

(Tp = 0.020 mm. Hg for 1 atm.<P = 2 atm.

Cp = 0.01^1 mm. Hg for P = 1 atm. .

The weighted relative error for a pressure range of

1 atmosphere to 100 mm. Hg was ro iid to be 7.5x10"?, and

2.2x10-P for a pressure ranj e of . atmospheres to 500 mm.

Hg. Since the variance of the estimate, (i' ( ), is a

measure of the average deviation of the data points from

their estimated values, it was assu.med that the square of

the weighted relative error \ms equal to the variance of

the estimate.

The variance of the slope is given by

iko) G' (b) = e^(y)/ 'y? ,

where Z'x = ^x - 3cl'x.

The variance of the intercept can be determined from

( 1) (r^(N) = Q2(^)JJ 4- 1/n + x^/Z'x^J ,

where y is the pressure ratio, x the pressure, and n the

number of expansions. The value of' the second virial

coefficient is ,p:iven by

( 2) /S = b/(N-1) ,

and its variance by

(^3) cr| = (<5/9/c?K)%-[; + O / 3 / a b ) ^ ^ ^

30 •

The relative error in the second virial coefficient is

expressed by

( 0 (rel. error yd) = cs'^j + c5§

For a fraction of total volume retained equal to 0.90

and a pressure range of three atraoshperes to 500 mm. Hg,

the expression for rel. error^ is

( 5) (119.5 + 0.m-88/b2)^/2^ 2.2x10-5 .

A pressure range of one atmoshpere to 100 mm. Hg results

in the expression for rel. error^ as

( 6) (98.^ + 0.785/b2)V2x 7.5x10-5 .

Table 6 in the Appendix presents values of relative

error of second virial coefficients that were determined

from the expressions ( 5) and (^6). The data indicate that

the minimum value of the second virial coefficients that

can be determined with a ten per cent tolerance is about

10 cm- /gmol for a three manometer system and 80 cm^/gmol

for a one manometer system. The value for the second

virial coefficient of air at 2^^C is 7 cm3/gmol. Air

was the gas that was studied with a one manometer system.

31

Proposed Apparatus

The distance of extrapolation of the data by the

method that was investigated should be limited to a

maxim.um of one-fifth to one-fourth of the total distance.

The optimum pressure range for obtaining second virial

coefficients was found to be from three atmospheres to

500 mm . Hg. The method can give values of second virial

coefficients with a 10 per cent tolerance if the value

of the second virial coefficient is greater than 10

cm^/gmol.

The results of the error analysis suggested that a

three-manometer system could be used in obtaining reliable

low pressure P-V-T data. A three-manometer system would

allow the upper pressure to be about three atmospheres.

Figure 6 represents one possible design. The percentage

of test gas retained after each expansion should be about

90 per cent. This will require about 15 expansions to

drop the pressure from thrre atmospheres to 500 nmi. Hg.

Some sort of temperature controller is desirable

for the apparatus. The effect of temperature on pres­

sure ratios can be eliminated by correcting for temp­

erature change, but the resulting: second virial coeffi­

cients will be some kixid of mean value over the temp­

erature change. This type arparatu.s v/ill also be limit.ed

to operation below 70* 0, v/h.are the vapor pressure of

32

u (d P«

<

+> -P

a;

m 0) (0

o O

VO

•H

33

mercury starts becoming large enough to affect the measure­

ments.

Teflon stopcocks should be used. They should slow

down the contamination of mercury by the excess stop­

cock grease that is required by glass stopcocks. It is

imiperative that the stopcoclrs hold a vacuum during the

course of the runs.

A lov/ pressure P-V-T apparatus of this type should

prove to be particularly suitable for use in the investi­

gation of properties of mixtures. The apparatus is res­

tricted to the measurement of materials that are in a

gaseous state at three atmospheres and 70^0, and that

have a second virial coefficient greater than 10 cm3/gmol.

CHAPTER IV

CONCLUSIONS

1. It was found that the second virial coefficients of

gases could not be determined by expanding the gases

from atmospheric pressure to lower pressures. This

occurred because the relative error associated with the

pressure measurements was too great. The pressure

measurements were obtained by reading mercury menisci.

2. Error analysis revealed that second virial coefficients

having a value greater than 10 cm^/gmol could be deter­

mined with less than 10 per cent error if the expan­

sions were made from three atmospheres down to 500 mm.

Hg.

3. An apparatus was proposed to operate in this pressure

range. The apparatus is restricted to the measurement

of properties of materials that are in a gaseous state

at three atmospheres and 70°C, and that have a second

virial coefficient greater than 10 cm3/gmol.

3^

LIST OF REFEREKCES

1. Bridgeman, O.C, Physical Reviews. 3^, 527-33 (1929).

2. Burnett, E.S., J. Apjo. Mech., 58, A136-^0 (1936).

3. Epstein, L.F., J. Chem. Physics, 20, I98I-2 (1952) and 21 , 762 (L) TW^B).

k. Goff, J . A . , and S. Gra tch , T r a n s a c t i o n s , ASHVE, 51 , 125-58 (19^f5).

5. Goff, J . A . , and S. Gra tch , T r a n s a c t i o n s , ASHVE. 52, 95-121 ( 1 9 ^ 6 ) .

6 . Goff, J . A . , Hea t ing , P ip ing and Air Cond i t ion ing , 21 , No. 11, 118-28 (10^97.

7. Heichelheim, H.R,, K.A. Kobe, I.H. Silberberg, and J.J. McKetta, J. Chem. and Engr. Data, 7, No. h, 507-9 (1962).

8. Heichelheim, H.R., personal communication.

9. Hoel, P.G., Introduction to Mathematical Statistics, John Wiley and Sons, Inc., Kew York, 19 2" ^

10. Holborn, L., and J. Otto, Z. Phys., 23, 77 (192^).

11. Kamerlingh-Onnes, H.K., Commun. Phys. Lab. Univ. Leiden, 71, (1901), cited by ref. 15-

12. Lennard-Jones, J.E., and W.R. Cook, Proc. Roy. Soc. (London) 115A, 33^ (1927).

13. Pfefferle, W.C., Jr., J.A. Goff, and J.G. Miller, J. Chem. Physics, 23, 509-13 (1955).

1^. Volk, V/., Applied Statistics for Engineers, McGraw-Hill Book Co., Inc., New York, 1958.

15. Zaki, W.N., H.R. Heichelheim, K.A. Kobe, and J.J. McKetta, J. Chem. and Engr. Data, 5? 'o. 3, 3^3-9 (19^0).

35

APPENDIX

36

37

APPENDIX A

Table h

Second Virial and Interaction

Coefficients of Air and Water Vapor

Units, cm^/gmol

A = dry air, B = water vapor, C = interaction

°c 0

10

20

30

^0

50

A

13.2

10 .9

8.81

6.87

5.09

3 . ^ ^

To l .

1 .2

1 .1

0 .95

0.86

0 .78

0.71

B

1830

1510

1260

107^

92^

803

To l .

800

^00

210

116

66

-0

C

^2 .0

39.3

36.8

3^.5

32.3

3 0 . ^

To l .

6.1

6.0

5.8

5.7

5.5

5.^

Note: Affix negative sign on all values of A, B, and C. Tolerances are plus and minus the tabular values.

Tabular values from pp. 121-22 of reference 5.

38

TABLE 5

Relative Percent Error in

Pressures and Pressure Ratios

p

100

500

1000

5000

Three man

100

500

1000

5000

mm.

lomet'

mm.

R e l . % e r r

0 . 0 1 ^

0 .0028

0 . 0 0 1 ^

0 .00028

e r s , CTp = 0 .

0 . 0 2 ^ 5

0 .00^9

0 . 0 0 2 ^ 5

0 .000^9

. p

02'+5

R e l .

R = 0 . 9 0

0 .0209

0 . OO'f 1 8

0 .00209

0 .000^18

0 .0366

0 .00732

0 .00366

0 .000732

% e r r . R

R = 0 . 8 0

0.0221

0 .00^88

0.00221

0 .000^^8

0 .0387

0 .00783

0 .00387

0 .000783

39

Table 6

Relative Error of

Second Virial Coefficients

Slope Second Virial 3 Manometers 1 Manometer Coefficient Rel. Error of Rel. Error of

Second Virial Second Virial Coefficient Coefficient

10"^

10-^

10"^

10"^

1 .17

11.65

116.5

1165.

0.85

0.085

0.0085

0.000882

6.61

0.661

0.0661

0.0067

^0% tolerance 10^ tolerance (§10 cm3/gmol @80 cm- /gmol

^0

APPENDIX B

Computer Program

Least Squares Straight Line for Low

Pressure Burnett Apparatus Data

FORTRAN

DIMENSION P ( 2 0 ) , T(20) , Y(20), YDEV(20), Y0(20), P0(20)

K =

AA =

T05 =

T01 =

TAVG =

READ, ( P ( I ) , T ( I ) , I - 1,N)

P(1) = (P (1 )*13 .5315) / (13 .5955*760 . )

T(1) = T(1) + 273.15

DO 1 J=2,N

T(J) = T(J ) + 273.15

P ( J ) = (P(J )*13 .5315)7(13 .5955*760 . )

1 Y(J) = ( P ( J - 1 ) * T ( J ) ) / ( P ( J ) * T ( J - 1 ) )

RCON = 82 .06

TAVG = TAVG + 273.15

ST = 0 .0

TOL = 0 .0

2 A = 0 .0

B = 0 .0

C = 0 .0

•1

D = 0 . 0

E = 0 . 0

DO 3 J = 2,N

A = A + Y(J )

B = B + Y ( J ) * Y ( J )

0 = 0 + P ( J - 1 )

D = D + P ( J - 1 ) * P ( J - 1 )

3 E = E + Y ( J ) * P ( J - 1 )

F = D - (C*C/AA)

G = B - (A*A/AA)

CA = E - (C*A/AA)

BN = CA/F

R2 = BN*CA/G

R = (ABSF(R2))**0.5

S2Y = ( 1 . - R2)*G/(AA - 2 . )

S2B = S2Y/F

SB = (ABSF(S2B))**0.5

RBP05 = BN + SB*TO5

RBN05 = BN - SB*T05

RBP01 = BN + SB*T01

RBN01 = BN - SB*T01

RINCP = (A - BN*C)/AA

BETA = 3N/(RINCP - 1 . )

SYO = (ABSF(S2Y*(1./AA + (C''^C)/( AA-AA*F))) ) * * 0 . 5

SYP05 = RINCP + T05*SY0

SY]']05 = RINCP - T0^*SY0

h2

BBP = RBPO5/SYNO5

BBN = RBNO5/SYPO5

BETCO = BETA*RCON*TAVG

BBPCO = BBP*RGON*TAVG

BBNCO = BBN*RCON*TAVG

T0L1 = BETCO + BBPCO

T0L2 = BETCO - BBNCO

h FORMAT ( HY, X)

PUNCH h

PUNCH, (Y(J), P(J-1), J = 2,N) •

5 FORMAT (8HB, R2, R)

PUNCH 5

PUNCH, BN, R2, R

6 FORMAT (12HS2Y, S2B, SB)

PUNCH 6

PUNCH, S2Y, S2B, SB

7 FORI-IAT (26HRBP05, RBN05, RBP01 , RBN01 )

PUNCH 7

PUNCH, R3P05, RBN05, RBP01, RBN01

8 FORMAT (29HINCP, BETA, BETCO, T0L1, T0L2)

PUNCH 8

PUNCH, RIP'CP, BETA, BETCO, T0L1 , T0L2

9 ST = ST + 1 .

IF(ST - 1 . ) 3 0 , 3 0 , 3 2

30 DO 31 J = 2,N

YO(J) = Y(J )

^3

31 po(J-i) = P(J-i)

GO TO 19

32 DO 33 J = 2,N

Y(J) = YO(J)

33 P(J-1) = P0(J-1)

GO TO 17

19 DO 12 J = 2,^

YCAL = RINCP + BN*P(J-1)

YDEV(J) = ABSF(YCAL - Y(J))

12 TOL = TOL + YDEV(J)

DEVAV = TOL/3.0

DO 13 J = 5,N

YCAL = RINCP + BN*P(J-1)

13 YDEV(J) = ABSF(YCAL - Y(J))

17 AAA = 0.0

DO 10 J = 2,N

IF(YDEV(J) - 0.000001)15,1^^1^

1^ WT = DEVAV/YDEV(J)

IF(ST - 1 . ) 1 8 , 1 8 , ^ 0

ho CONTIMJE

IF(ST - 2 . ) ^ 1 , ^ 1 , ^ 2

k^ m = WT*WT

GO TO 18

h2 CONTINUE

IF(ST - 3 . ) ^ 3 , ^ 3 , ^ ^

J+3 V/T = WT*WT*WT

kh

GO TO 18

hh WT = WT**6.0

18 CONTIl UE

IF(WT - 3 . ) 1 6 , 1 5 , 1 5

15 V7T = 3 .0

16 AAA = AAA + WT

10

11

Y(J ) =

P ( J - 1 )

GO TO

STOP

END

WT*Y(J)

= WT*P(J-

2

-1)

"a

N = 12

AA = 11

T05 = 2

T01 = 3

TAVG = ;

703.05

587.37

^90.29

^09 .^0

3^1.58

285.21

238.05

198.57

165.92

138.^^

115.27

96.51

.306

.355

26.0

26.25

26.^1

26.32

26.20

26.17

26.09

26.01

25.92

25.90

25.83

25.79

25.70

1+5

Sample Input Data for

Computer Program

Note: AA = N - 1

T05 = tr, Q^ at AA - 2 degrees of freedom

T01 = tQ^oi at AA - 2 degrees of freedom

he

Definition of Computer Output Terms

Y = pressure ratio = P(J-1)/P(J)

X = pressure, in atmospheres = P(J)

B = slope of straight line through data points

R = correlation coefficient

S2Y = average varience of estimate

SB = average deviation of slope

RBP05, RBN05 = 95% confidence range of slope

RBP01, RBN01 = 99^ confidence range of slope

INCP = intercept of straight line through data points

BETA = second virial coefficient, in reciprocal atm. units

BETCO = second virial coefficient, cm- /gmol

T0L1 , T0L2 = 95! confidence range of second virial coefficient

h7

Computer Program

Weighted Average Error Program

FORTRAN

*FA] IDK1 00 -

DIMENSION PL(IOO), D(100)

P1 =

F'K =

13 c = 1.

P = PN/P1

1 c = c - 0.05

A = (L0GF(P))/(L0GF(C))

K = A

PO = P1/C

M = N + 1

PL(1) = PO - C*PO

BA = 1 .

DO 2 1=2,M

BA = BA + 1 .

2 PL(I) = PO - (C**BA)*PO - PL(I-1)

ST = 0.

DO 3 1=2,N

3 ST = ST + PL(I)

DD = 0.5*(PL(1) + PL(M)) + ST

DO h 1=1 ,N

-8

h D ( I ) = 0 . 5 * ( P L ( I ) + PL( I+1) )

smi = 0 . 0

PT = 0 . 0

DO 10 1=1 ,N

PT = PT + 1 .

PON = P1*(C**PT)

I F ( P 0 N - 1 5 2 0 . ) 6 , 6 , 5

5 SIGP = 0 . 0 2 ^ 5

GO TO 9

6 CONTINUE

I F ( P 0 N - 7 6 0 ) 8 , 8 , 7

7 SIGP = 0 . 0 2 0

GO TO 9

8 SIGP = 0 . 0 1 ^ 1 ^

9 R = SIGP/P1

Q = 2.*PT

10 SUM = SUM + (R*D(I)((C**Q + 1 . )** .5 ) ) / ( (C**PT)*DD)

11 F0RMAT(2( 10X15) , 10X1^-^5.3, 10X12, 9XEII+.8)

PTOICH 1 1 , P I , PN, C, N, SUM

I F ( . 6 0 - 0 ) 1 , 1 ,12

12 CONTINUE

PM = p]\j . 1 0 0 .

I F ( P N - 3 0 0 . ) 1 ^ , 1 3 , 1 3

1^ COl JTINUE

STOP

El JD

9

Definition of Computer Output Data

Computer output data represents, in order across

the field, upper limit of pressure range, lower limit

of pressure range, apparatus constant (fraction of total

volume retained), number of e::pansions required to span

pressure range, and the weighted average error associated

v;ith the pressure ratios.

TEXAS TECHNOLOGICAL COLLEGE LUrH' CK. TEXAS

~n

'J .