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 .