B ou nd by D E N V E R B O O K B IN D IN G C O ., 27 15 - 17 fh S t. , D e n v e r, C ol o 80 211
ER 1282
CALCULATION OF VAPOR PRESSURES USING THE FROST-KALKWARF EQUATION
I AND
GAS IMPERFECTIONS DETERMINED FROM THE HEAT OF VAPORIZATION AND VAPOR PRESSURES
by
Jorge Ordonez
ProQuest N um ber: 10781071
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ER 1282
A Thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fu lfillm e n t of the require
ments for the degree of Master of Engineering in Chemical and Petroleum- Refining Engineering.
Signed:
Student f
Golden, Colorado Date: P U . C , 1970
Approved : {
.A rth u Thesis M /is or
James H. Gary id, CPRE Department
Golden, Colorado Date : ^ , 1970
i i
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DEDICATION
To my w ife, Martha, and my children, Jorge Alfredo and Edwin Dereck
ill
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ABSTRACT
A computer program for the Frost-Kalkwarf Vapor pressure equa- t i on,
Log P = A + y + C Log T +
was developed, and the vapor pressures and respective Frost-Kalkwarf constants were calculated for ammonia, argon, carbon monoxide, normal hydrogen, krypton, methane, nitrogen, oxygen, para-hydrogen, propane, and xenon. The temperature range investigated was the trip le point to the c r itic a l point. The average deviation in vapor pressure was mea
sured for every liq u id , and the results were between 0.07% for normal- hydrogen and 0.50% for ammonia and para-hydrogen.
The mid point between the tr ip le and the c r itic a l points was chosen as a reference point in the calculation of the empirical con
stants A, B and C.
The second part of this report was a computer calculation of the second v ir ia l co efficient of ammonia, argon, carbon monoxide, methane, nitrogen, and propane, using the relation of Curtiss and
Hirschfelder,
________ -AH = i + / 3
R (1 - VL/ V ) g ( I n P )/ d(l7rJJ T
The Frost-Kalkwarf equation derivative and the constants B, C, and D, calculated in the f i r s t part of this report, were used in the
i v
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equation.
The results showed good agreement between the calculated values in the region above but near the boiling point for every liq u id , but agreement was not good on points closer to the trip le and c r itic a l point.
v
CONTENTS
General Introduction . . . ... . . . 1
Part I : Calculation of the Vapor Pressure Using the Frost-Kalkwarf Equation . . . 3
Introduction ... 4
Derivation of the Frost-Kalkwarf Equation ... 6
Description of the Problem . . . 10
Computer Program . . . 12
Input Variables . ... 12
Calculation of D and Selection o f the Reference Point to Determine the Empi ric al Constants fo r the Equation. . . . 12
Determination of the Empirical Constants 3 and C . . . 12
Calculation of the Empirical Constant A . . . . 13
Calculation of Vapor Pressures . . . 14
Calculation of Deviations of Vapor Pressure . . . 14
Solution Procedure . . . . 15
Program O u t p u t ... 15
Discussion of R e s u lt s ... 17
M e th a n e ... 20
Propane . . . 20
Argon, Krypton, and Xenon . 24
A r g o n ... 24
K r y p to n ... 26
Xenon . ... 26
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Page
Ammonia . . . 29
Carbon Monoxide . . . 29
Normal Hydrogen . . . . . . 33
Para-Hydrogen . . . 33
Nitrogen . . . 33
Oxy gen . . . 36
Comparison of Results and Conclusions . ... 40
Notation ... . . . 42
Appendix I: Computer Program Notation . ... . . . . 43
Level 1 ... 44
Subroutine SIR ... . . . 45
Appendix I I : Typical Computer Program Listing . . . 46
Appendix I I I : Computer Program Output fo r Every Liquid . . . . 50
Methane ... . . . . . . 51
P ro p a n e ... 55
Argon . . . . . . 58
Krypton . . . . . . 61
Xenon . . . ... . . . 64
Ammonia ... 67
Carbon Monoxide . . . 70
Normal-Hydrogen ... 73
Para-Hydrogen . . . 76
Nitrogen . . . 79
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Part
Page
Oxygen ... 83
Literature C i t e d ... . 86
I I : Gas Inperfections Determined from the Heat of Vaporization and Vapor Pressures ... 88
Introduction . . . . . . 89
Derivation of Equations . . . 91
Description of the Problem . . . 94
Computer Program . . . ... . . . 95
Input Variables . . . 95
Solution Procedure . . . ... . . . 95
Program O u tp u t... . . 95
Discussion of Results . . . ... . . . 96
Ammonia ... 96
A r g o n ... 96
Propane . . ... 100
Methane . . . 105
Nitrogen ... . . . 105
Carbon M o no xide... 109
Conclusions ... 114
Notation ... . . . 115
Appendix I: Computer Program Notation . . . 116
Appendix I I : Typical Computer Program Listing ... . 118
Appendix I I I : Computer Program Output fo r Every Liquid . . . 121
v i i i
Ammonia Argon Propane
Methane . . . Nitrogen
Carbon Monoxide
Literature Cited
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FIGURES
Figure Page
1. Low temperature range of methane.
Observed minus calculated vapor pressure versus temperature. . 21 2. High temperature range of methane.
Observed minus calculated vapor pressure versus temperature. . 22 3. Propane. Observed minus calculated vapor pressure versus
te m p e ra tu re ... ... 23 4. Argon. Observed minus calculated vapor pressure versus
temperature . . . 25
5. Krypton. Observed minus calculated vapor pressure versus
te m p e ra tu re . 27
6. Xenon. Observed minus calculated vapor pressure versus
temperature . . . ... . 28 7. Ammonia low temperature range.
Observed minus calculated vapor pressure versus temperature. . 30 8. Ammonia high temperature range.
Observed minus calculated vapor pressure versus temperature. . 31 9. Carbon monoxide. Observed minus calculated vapor pressure
versus temperature ... . . . 32 10. Hydrogen. Observed minus calculated vapor pressure versus
te m p e ra tu re ... ... 34 11. Para-hydrogen. Observed minus calculated vapor pressure
versus temperature . . . ... . . 35 12. Nitrogen. Observed minus calculated vapor pressure versus
temperature... ... 37 13. Oxygen low temperature range. Observed minus calculated
vapor pressure versus temperature... . 38 14. Oxygen high temperature range. Observed minus calculated
vapor pressure versus temperature . . . ... 39
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15. Ammonia. Comparison of calculated and Davies second v ir ia l coefficient ... . . . . 16. Argon. Comparison of calculated and Gosman, McCarty
and Must second v ir ia l co efficient ... . 17. Propane. Calculated second v ir ia l coefficient versus
temperature ... . . . . . 18. Methane. Comparison of the calculated, the Thomas
and Steenwinkel, and the Byrns, Jones and Stave!ey second v ir ia l coefficient ... « 19. Nitrogen. Comparison of calculated and Strobridge
second v ir ia l coefficient ... . 20. Carbon Monoxide. Calculated second vira l coefficient
versus temperature ... . . . .. ...
Page
c 97
. 101
. 104
. 106
. no
. 113
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TABLES
Table Page
1. C r it ic a l, Triple and Reference Points used in
calculation fo r each l i q u i d ... . . . 18 2. Ammonia. Comparison of calculated and Davies
second v ir ia l co efficient between 273.6 K and 373.16 K . 92 3. Argon. Comparison of calculated and Gosman »
McCarty and Must second v ir ia l coefficient
between 90 K and 150 K . ... ... 102 4. Methane. Comparison of calculated and Thomas
and Steenwinkel, Byrns, Jones and Stavely, and Curtiss and Nirschfelder second v ir ia l c o e ffi
cients at d iffe re n t temperatures ... 107 5. Nitrogen. Comparison of calculated and Stro
bri dge second v ir ia l co efficient between the
boiling point and 120 K . . . I l l
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ACKNOWLEDGMENTS
Appreciation is expressed to Dr. A. J. Kidnay fo r his continual help and guidance throughout this work, and to A. R. Zambrano for his assistance at a ll stages of the investigation. Also, the author wishes to thank Dr. J. H. Gary and Dr. G. B, Lucas for th e ir p a rti
cipation on the thesis committee. Acknowledgment is made to the Ministerio de Minas y Petroleos de Colombia fo r providing financial assistance.
x i i 1
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GENERAL INTRODUCTION
This engineering report is divided into two parts:
1. Calculation of vapor pressures using the Frost-Kalkwarf equation.
2. Gas imperfections determined from the heat of vaporization and vapor pressures.
The aim of the f i r s t part was to test the Frost-Kalkwarf equation on several liquids in the range from the trip le point to the c ritic a l point.
A substantial number of equations have been proposed for the cal
culation of vapor pressures, most of them in the region between the boiling point and the c r itic a l point. For lack of a better name, such equations could be called predictors. According to M ille r's (1) c rite rio n , calculated vapor pressures showed to be within 10% of experimental ones in either or both of the ranges : the boiling point to the c r itic a l point, or 10 - 1500 mm. of Hg. These two ranges are considered separately since some predictors are very good in one range but poor in the other.
The calculation of vapor pressures is rendered d if f ic u lt by the existence of several constants in every equation. The more numerous the constants in^the equation, the more d if f ic u lt i t is to use the equation but the better the results. However, the need is for an
equation that is practical as well as accurate, and as the Frost-Kalkwarf equation has only three empirical constants and covers the range between the
1
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trip le and the c r itic a l points, i t could be a good predictor. In order to test the equation, eleven liquids: ammonia, argon, carbon monoxide, normal-hydrogen, krypton, methane, nitrogen, oxygen, para-hydrogen, pro
pane and xenon were studied, and a computer program was developed to calculate the vapor pressures and the average deviations for every liq u id .
2. The second v ir ia l coefficient of many gasses may be obtained by using the Clausius-Clapeyron equation together with available experimental
heats of vaporization and vapor pressure data. The accuracy of the calculated second v ir ia l co efficient in some cases is comparable to that obtained from good direct P.V.T. measurements. The equation used in the calculation is:
PV = -AH - \ + Æ
RT R (1-VL/V) [d(ln P)/d(l/TJ] ~ 1 V
which is called the relation of Curtis and Hirschfelder. In this case the Frost-Kalkwarf equation is used to evaluate the derivative and a computer program was developed for the calculations. Values of temperature, vapor pressure, liquid volume, vapor volume and heat of vaporization were available for the following six liquids: ammonia, argon, carbon monoxide, methane, nitrogen and propane.
The relation of Curtiss and Hirschfelder is based on the fact
that at s u ffic ie n tly low pressures the contribution of third and
higher v ir ia l coefficients is negligible, and the true second v ir ia l
coefficient can be obtained.
ER 1282
PART I
CALCULATIONS OF VAPOR PRESSURES USING THE
FROST-KALKWARF VAPOR PRESSURE EQUATION
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INTRODUCTION
Numerous equations both empirical and theorical have been pro
posed for relating the vapor pressure to the absolute temperature.
The simplest theoretical relation is the fam iliar Clausius-Clapeyron equation:
Log P = A - (B/T)
where A and B are empirical constants. This equation is derived from the thermodynamically exact Clapeyron equation:
dP/dT = AH/TAV
where AH is the heat of vaporization andAV the corresponding increase in volume on converting liquid to vapor. The derivation of the Clau
si us-Cl apeyron equation assumes: a) that AH is constant and independent of T, b) that the vapor is an ideal gas, and c) that the liquid volume is negligible. This equation predicts lin e a r lin ity in a plot of Log P versus 1/T.
Thodos (2) in a careful survey of empirical vapor pressure equa
tions for saturated hydrocarbons has shown that the plot of Log P versus 1/T is not quite lin ear but is re a lly "S" shaped, with a rever
sal of curvature between the boiling point and the c r itic a l point. The
"S" shape is inherent in the empirical equation of Cox (3) which is:
Log P = An(l - Tb/T)
where P is the vapor pressure in atmospheres, Tb is the normal boiling
4
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point, and
An = Ac 10 Exp(l - T » (F - Tr )
Here Tr is the reduced temperature and Ac , and F are constants.
In order to account for the S shaped form in the plot of Log P versus 1/T, Frost and Kalkwarf (4) have developed the following equa- t i on :
Log P = A + S. + C Log T + 2fL
and the purpose of this report is to test i t on a computer program,
taking the average deviation on vapor pressure as the best measure
or performance.
ER 1282
DERIVATION OF THE FROST-KALKWARF EQUATION
According to Frost and Kalkwarf (4) over the greater part of the temperature range (lower temperatures) the log P versus 1/T plot is convex upwards. This has been explained quite well as due to a de
crease in AH as T increases. In p artic u lar, i f i t is assumed that AH fa lls lin e a rly as T increases but that the other assumptions of the Clausius-Clapeyron equation are sa tis fie d , the Rankine equation results:
log P = A - (B/T) + C log T 1.
Now the reversal in curvature occurs at higher temperatures, between the boiling point and the c r itic a l point. This is a region of high vapor pressure where deviations of the vapor from the ideal gas law would be most noticeable. I t is only lo g ic a l, then, to con
sider the effe c t of such deviations. Being unable to integrate the equation in complete generality, an approximate solution u tiliz in g the van der Waal s' equation w ill be obtained which contains a f i r s t - order correction for the nonideal gas behavior, such that the resu lt
ing equation should be good up to a reduced temperature of perhaps 0.8 or more.
van der Waal s' equation
2 .
may be put in the form:
3.
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I f V in the correction term A/V2 is approximated by RT/P and i f the liq uid volume is to a f i r s t approximation equal to b, then the change in volume of vaporization is approximately
V = RT
aP
r 2 j 2 4.
As in the Rankine equation, i t is assumed that AH is linear in T, s p e c ific a lly ,
AH = AH0 - C'T 5.
where AH0 and C are considered empirical constants. Substitution of equations 4. and 5. in the Clapeyron equation
gÇ = AH/TAV 6.
yelds
s . MAH. - C ' T I O * | ^ ) RT2
or
" - S i f w f e i »
or
d(ln P) = / AH0 _ + aPAH j dT
I rt 2 RT R3 T4 / 9.
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8
Equation 9. may be integrated i f P in the small las t term is given as a function of T. To a su ffic ie n t approximation, consistent with our previous assumption, this P may be represented by
which is equivalent to the Clausius-Clapeyron equation. With this substitution, the la s t term in equation 10. integrates as follows:
where i t has been assumed that the AH of equations 11. and 10. and the la s t term o f 10. is constant, this being permissible because this is a firs t-o rd e r correction. In the integrated result in equation 1 1 ., P has been reintroduced using equation 10. Since RT/AH is of the order of magnitude of o . l , the la s t two terms of equation 11. w ill be neglected.
Integration of the equation 11. then results in
p = p0 e “ AH/RT 10
dT
= aP 1 + 2RT + 2R2T2
r
2T2 AH 1 ^ 7 ? + const. 11
In P = A' - aP
R2T2 12
or
log P = A + ^ + C log T + 2P 13
ER 1282
9
where A, B and C are empirical constants, B and C being negative, and
D is related to van der Waal's a by D = a/2.30259R2e
ER 1282
DESCRIPTION OF THE PROBLEM
The development of a computer program fo r the Frost-Kalkwarf vapor pressure equation:
log P = A + Ê.+ C log T + D Ü
T T2
includes two main parts :
a) the calculation of the constants A, B and C
b) the calculation of vapor pressures and the average deviation from experimental values, using the calculated constants.
The constant D is not calculated in the same way as the former constants, because:
0 = a/2.30259 R2
and a is the van der Waal s' constant, a = 27 R2 Tc2/ 64 Pc
Thus, for the general case applicable to the equation:
D = 0.1832 T c 2/P c
The evaluation of the constants A, B, and C is carried out by u tiliz in g a reference point (P ], T-j) included in a set of vapor pres
sure measurements, to obtain the normalized expression:
+ C Log I
Log P] t - D p Pi 1 1
T2 ‘ T p = B
t ' ti
10
E 1282 11
When this equation is rearranged, the following lin e a r relationship results:
log V D = B 1 1
T - T + C
Log log I
1
which can be conveniently expressed in the following form:
Y = BX + C
where the vapor pressure modulus is Y = log £ - d
/ log Ti and the temperature modulus is
X = 1 1 1
Within each set of vapor pressure measurements, a reference point
(T]»?]) is selected and values of Y and X are then calculated for the
other measurements in the set. The resulting values of Y and X are
plotted to produce the lin ea r relationship wanted. The slope of the
straight line is B and its intercept is C. Using these constants and
the reference point (T-j, P-j ) in the Frost-Kalkwart equation, a value
of A can be found. Since the vapor pressure appears on both sides
of the equation, the calculation of vapor pressure involves a t r ia l
and error calculation using the calculated constants and the chosen
data. On the basis of the experimental and calculated pressures an
average deviation can be obtained.
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COMPUTER PROGRAM
The solution of Frost-Kalkwarf vapor pressure equation was pro
grammed for a CDC 809 computer, and a description of the notation, a lis tin g of the program it s e lf , and the runs fo r every liq u id are given in appendices I , I I , and I I I , respectively. A description of the main steps in the program is presented.
Input Variables
Every computer program had the following input: the experimental temperature and vapor pressure data and th e ir dimensions for the liquid under study, the c r itic a l temperature and pressure, the universal con
stant R, and the constant 0.1832 used for evaluating the constant D.
Calculation of D and Selection of the Reference point to Determine the Empirical Constants for the Equation
The aim of the f i r s t set of calculations is to obtain the constant D, which is a function of the c r itic a l temperature and the c r itic a l pressure. Then, the computer program calculates the empirical constants B, and C, using a reference point (P (n), T(n)) to perform the calcula
tions. The reference point was selected by successive tr ia ls u n til the best constants were obtained, as judged by the error. In th is case the middle temperature point between the tr ip le and the c r itic a l point gave the best results.
Determination of the Empirical Constants B and C
The subroutine STR(Y, X, N, A, B) f it s the best straight lin e through
12
ER 1282 13
experimental points by using the Least Squares method. In the plot of Y versus X, A is the intercept point that corresponds to the empi
rical constant C, and B is the slope of the straight lin e . Calculations of the Empirical Constant A
The values of B, C, and D obtained in the f i r s t part of the com
puter program together with the experimental values fo r every tempera
ture and vapor pressure are placed in the A equation, which gives us (J) values of A. All of these values must be refined to obtain the best value of A, from which we obtain an average value of A; then, the square of the deviation of average A, and then the standard deviation of A. A normal distribution is assumed fo r the A values so that a
confidence interval can be obtained. In this case, a fte r several t r ia ls , a confidence interval of 0.95 was chosen on the basis of the s ta tis tic a l table of Pearson and Hartley (5 ). The values ± 1.96 were used to calcu
late the lim its of the confidence in te rv al. Thus A] = AA - 1.96 x SS/SQRTF(XN)
and
A2 = AA + 1.96 x SS/SQRTF(XN) are used to give the 95% values.
The next step is to reach the fin a l A value, which is equal to the average A from the 95% confidence inte rv al. I t is calculated from
DA = SUM/M where
M = N - K
ER 1282 14
Calculation of Vapor Pressure
The calculation of vapor pressures is a t r ia l and error process since the vapor pressure appears on both sides of the equation. The f i r s t approximation is to assume that the Frost-Kalkwarf equation is composed by only three terms, so that
Log CCP = A + B + c log T(J)
or
Ln CCP = 2.30259 (A + JL_ ) + C x In T(J) T(J)
or
CCP = EXPF 2.30259 (A + JL_ ) + C x Ln T(J) T(J)
then
CP(J) = CCP Z, = CP(J)
Z 2= EXPF Ln (CCP) + 2.30259 x D x CP(J) T(J)2
Z, is the f i r s t approximation, and Z 2 is a corrected value basis on Z ,th a t must converge to the true value of CP(J).
By means of continuous t r ia ls , incrementing CP(J) b it to b it , a value fo r vapor pressure is obtained.
The ite ra tiv e process in the computer program to calculate the vapor pressure was increased fo r some species to get a better f i t to the experimental data.
Calculation of Deviations of Vapor Pressure
A fter the vapor pressure values have been calculated, the final
ER 1282
15
step is the calculation of the average deviation. For this calcula
tion the equation
DP(J) = ABSF Q p(J) - CP(JT] x 1 ^
is used, where P(J) is the experimental vapor pressure and CP(J) is the calculated vapor pressure.
Solution Procedure
The solution procedure begins as soon as the input variables have been defined and involves several steps : a) Calculation of D, b) Determination of the empirical constants B and C, c) Calculation of the empirical constant A, d) Calculation of vapor pressure, and e) Calculation of deviation of vapor pressure. The calculation of D, PT and RT are made once fo r each computer run. After these in it ia l steps, the program begins the re p e titiv e process using a ll the data.
In order to choose the best reference point several tr ia ls were made with d iffe re n t points such as the boiling, c r it ic a l, and tr ip le points, and fin a lly with the middle temperature point. The la s t one proved to be the most accurate point, perhaps because the independent variable X, or temperature modulus, is only a function of temperature, and the middle point among a group of random points is always the best way to plot a straight lin e .
In the calculation of A, a 95% confidence interval was used be
cause i t proved to be the most convenient fo r the data.
Program Output
The program prints out the number of experimental points within a 95% confidence in te rv a l, the lower and higher A lim its , the con
stants A, B, C, and 0, the calculated vapor pressures and th e ir ab
ER 1282 16
solute differences with respect to the experimental vapor pressures,
and fin a lly , the average deviation in vapor pressures for each liq u id .
The results can be found in appendix I I I .
ER 1282
DISCUSSION OF RESULTS
The vapor pressures were calculated fo r eleven liquids: methane, propane, argon, krypton, xenon, ammonia, carbon monoxide, normal- hydrogen, para-hydrogen, nitrogen and oxygen »
The temperature and the empirical constant B are in kelvin de
grees fo r a ll the liquids; the vapor pressure is in atmospheres and the empirical constant D is in °K^/atm fo r propane, argon, krypton, xenon, ammonia, carbon monoxide, normal-hydrogen, nitrogen, and para- hydrogen, and are in mm of Hg and °K^/mm of Hg respectively fo r methane and oxygen„
A l i s t of the c r it ic a l, the t r ip le , and the reference points used in the calculation fo r each liqu id is given as follows: (Table 1 .)
17
ER 1282 18
Table 1. C r itic a l, Triple and Reference Points used in calculation
for each liq u id .
ER 1282
O J o O O o U f> O C O cn o o
s- o C O < 3- C O C O o co C O C M 0)0 o + -> 3 o C O O C O o o o C O 3= C O o
c < />
A tm
1.•r— V)
O Qi
C U $- a 0- O < D C $-
£ 5
C h C M o C O C M C M in
■ 8 C M
O' C M C O id*
< u < a o o O O o o o o o o o
4- $- < u aj M 1 , o — C CM O C C M O C L O O o o co « d - o o o o o o sz in o o
cr Q- o o
E C O C O 'd- o C O C O in in o
< u
h- C O
C M C O C M
CM O co o CM C M o o < d *
£ 3
(/) 5 8 co in O) O) C O
o>
m o
V ) C M o in in r>. C O C M E C M o
< u «C '— C O rs. C O o o o E r- in 4 -> S -
C Û- o o o o o o o o o h s.
a_ < o u
5 5 o_ < o
o o o o o o o C O o co
C O
•f— & - , o C O in o C M vl- m o in , C O o CO
$- < u ^ in r-x CD CO < d - O) C O CO CO
1 — CL o o
E O) CO in in C O C O CO CO vt- o
<D
C O C O r— CO O) CO CO in O)
f— r— r—
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3 C O C O C O in in 1^ in =c o o
C Z) -B CM
4-> </) CO vT Cs. Vl- CM CM co E CM 00
5 £ O o_
O-
S i
<C < d - vl- in in C O C O
E£ 1^
CO § s
4->
tOO o o o o O in co o o o
•r- &_ , o in in o CM O) o , C O C O
& - a > ^ o CO C O C O
Ch ChC h CM 5^ CO o
(_) CL o • o
s o o O ) C T ) in CM CM CM co Vj-
< u
1 — C O in o
CM 83 Vf o C O co CO CM in Ch
C D c
*o C D
■a C
X o 0 1
o s- O)
c "O o o >1 $-
z 3= ■a c
0 1 c ra >> 0 1 01
■a c o •r— c C h c c
•r—
(0c -M c c o
toi o C D < d
3 CL o Q. o o JO E
to$- C h
cr o O ) >> c $ L i- t- 4-> >» 4-»
•r— & - s ~ & - C D < a o rd X 0 1
Q. < 5Z X <C o 2T a. z o z
ER 1282 20
Methane
The calculated 61 data points were compared with the experimental data of Armstrong, Brickwedde, and Scott (6) between the tr ip le and the c r itic a l points. Figures 1. and 2. show the two ranges of methane deviations of vapor pressure, fo r low and high temperatures. The low temperature range plot (Figure 1) from 90.0 °K to 165.0 °K shows the observed minus calculated vapor pressures in millimeters of Hg ve r
sus temperature in °K; the deviations are both positive and negative, with negative deviations no greater than 2.569 mm of Hg. Figure 2
is a plot of the observed minus calculated vapor pressures in mm o f Hg versus temperature in °K fo r the region between 165.0 °K and 191.06
°K. From 165.0 °K to 180.0 °K the maximum negative deviation o f 7.217 mm Hg occurs at 173.0 °K, and from 180.0 °K to the c r itic a l point (191.06 °K) there is a constant and very high positive deviation, with a value of 105.584 mm of Hg at the c r itic a l point.
The average deviation in the whole vapor pressure range is only 0.10%, which is an indication of the good behavior of the equation 13.
for methane.
Propane
The experimental 19 data points were taken from that of KuToor, Newitt, and Bateman (7) between the tr ip le and the c r itic a l points.
The reference point for propane was taken as the boiling point, since
this is the middle point of the data. Figure 3 ., which is plot o f
the observed minus the calculated vapor pressure in atm x 10~^ versus
ER 1282 21
CM
I O
CO
SB turn ‘ ("3TB0)a - ( * sqo ) i
90
10 0 U Ô Ï2 Ô 13 0 Ï4 Ô Ï5 Ô 16 0 1 6 5 T e m p e r a tu r e , °K F ig u r e 1 . M e th a n e . Lo w te m p e r a tu r e r a n g e .O b se r v e d v a p o r p r e ss u r e m in u s c a lc u la te d v a p o r p r e ss u r e v e r su s te m p e r a tu r e .
('3 T B 3 )d - ( esqo)d
ER 1282
r» m
o
mes
o
mo o
enoo o
eso
eso
es eso o es
es en
es
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ER 1282 24
temperature in °K, shows f i r s t a positive deviation from the trip le point (189.5 °K) to 281.44 °K followed by a negative deviation from 281.44 °K to 330.70 °K, with a maximum negative deviation at 317.42 °K}
the maximum deviation is 0.52 atm at the c r itic a l point.
The behavior of equation 13. is considered reasonable since the average deviation for a ll the data is 0.43% I t is possible to define three d efin ite regions for the methane and the propane. The f i r s t one covers approximately 60% of the whole range of temperature and is a region of very low deviations, the boiling and the reference points are included in the f i r s t part.
The second region covers approximately 25% of the whole range of temperature and is characterized by a s lig h tly higher negative deviation.
The third and la s t one is characterized by a sudden and very marked positive deviation reaching a maximum at the c r itic a l point.
Argon, Krypton, and Xenon
Argon, Krypton and Xenon were tested between the tr ip le and the c r itic a l points and the data were taken from Bowman, Aziz, and Lim (8 ).
The Frost-Kalkwarf equation showed sim ilar behavior for the three l i - quids.
Argon: 38 data points were tested for argon. A plot of observed
minus calculated vapor pressure of argon, measured on atm x 1(T^ versus
temperature in °K (Figure 4 ), shows two well defined parts. The f i r s t
one covers approximately 85% of the range from the trip le point (83.78 °K)
ER 1282
25
m m
i— !
H
P . P .
< 8
O O
O
00O
r 4 HOT x m v ' ( eo m )d - (• sqo)d
ER 1282 26
to 149.02 °K and Is a negative-positive deviation with very smooth peaks. The second region goes from 142.09 °K to the c r itic a l point
(150.85 °K ), and shows a very sharp downward trend at the c r itic a l point, with a maximum deviation of 0.1729 atm x 1 0 ~ \
The average deviation fo r a ll the data is 0.12% which is good performance fo r any vapor pressure equation.
Krypton: 42 data points were tested for krypton. Figure 5 which is a plot of observed minus calculated vapor pressure in atm x 10~t versus temperature in °K, shows two d is tin c t parts. The
f i r s t one covering approximately 90% of the data, shows both negative and positive deviations from the tr ip le point (115.95 °K) to 202.47 °K with a maximum deviation of 0.396 atm x 10"^ at 129.88 °K. The second part, covering the remaining 10% of the data shows a sharp drop from the 202.47 °K point to 209.05 where the maximum deviation is -1.540 atm x 10-1 and then a rise to the c r itic a l point where the error is -0.9310 atm x 10-1.
The behavior of the Frost-Kalkwarf equation fo r krypton is quite sim ilar to that fo r Argon. The average deviation in vapor pressures is 0.23%, which is good performance fo r the equation.
Xenon: The experimental data is composed of 37 points. Two main regions can also be seen in figure 6. which is a plot of observed minus calculated vapor pressure of xenon in atm x lO"! versus tempera
ture in °K. The f i r s t region covering 95% of the data, consists of
negative and positive deviations with a maximum of 0.589 x 10"! atm
at 280.80 0K. In the second region, the deviations fe ll sharply to
the maximum of -1.647 atm x 10”1 at the c r itic a l point.
ER 1282
rH
O
01 X unv 4 ('3%e3)d - ('s q o )a
rH
I <N
S
12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 2 0 0 T e m p e r a tu r e , °K F ig u r e 5» K r y p to n . O b se r v ed m in u s c a lc u la te d v a p o r p r e ss u r e v e r su s te m p e r a tu r e »
P (O b s. ) - P (C a lC o ) , A tm x 1 0 ”
ER 1282
r-d
-1
- 2 170 190 210
T em perature, °K
230 o 250 270 290
Figure 6. X en on . O b served m in u s calculated vapor pressure versus tem perature.
ER 1282 29
The average deviation fo r the entire range of vapor pressures is 0.15%.
Ammonia
The experimental data consisted of 23 data points taken from Davies (9 ).
Figures 7. and 8. show both the low and high temperature deviation plots for ammonia. Figure 7 is a plot of observed minus calculated vapor pressures in atm x 10"% versus temperature in °K . From the trip le point (195.42 °K)to the reference point (300.00 °K) there is a region of smooth deviations', at 300.00 °K the line makes a sudden change downward with a maximum deviation of -7.293 atm x lO~^ at 340.00 °Ke
The high temperature p lo t. Figure 8, is a plot of vapor pressure difference in atm, versus temperature in °K, and is also s p lit into two regions. The f i r s t region, from 360.0 °K on to
3 8 0 . 0OK,consists of a constant negative deviation with a maximum of -0.14942 atm. The se
cond region shows only positive deviations with a maximum of 3.00518 atm at the c r itic a l point.
The average deviation for ammonia is 0.50% over the whole vapor pressures range.
Carbon Monoxide
The experimental data consisted of 26 points taken from Smeeton Leah(10).
Figure 9. is a plot of observed minus calculated vapor pressure
in atm x 10"*^ versus temperature in °K for carbon monoxide. As was
the case for argon, krypton, and xenon, two typical regions are defined.
P (O b s .) - P (C a lc .) , A tm x 1 0 "
ER 1282 30
8
6
4
2
0
- 2
4
—6
- 8
200 240 280 320 360
T em perature, °K
Figure 7. A m m o n ia low tem perature range. O b served
m in u s calculated vap or pressure versus
tem perature.
P (O b S e ) - P (C a lc .) , A tm .
ER 1282
—1
- 2
-3
400
3 90 405.6
380
T em peratu re, °K
Figure 8. A m m o n ia high tem perature range. O b served m in u s calculated
vapor pressure versus tem perature.
P (O b s. ) - P (C a le .) , Ac m x 1 0 “
ER 1282 32
-1
- 2
-3
-4
120 130
68 70 80 90 100 110
T em perature, °K
Figure 9. C a rb o n M on oxid e. O b served m in u s calculated vapor pressure versus
tem perature.
ER 1282
33
The f i r s t one covering about 80% of the whole temperature range is a succession of small negative-positive deviations, with maximum positive deviation of 0.256 atm x 10*^ at 105.69 °K. The second region is a large negative deviation with a maximum of - 4.345 x 10"**
atm a t the c r itic a l point (132.92 °K) The average deviation in vapor pressure fo r carbon monoxide is 0.37%.
Normal-Hydrogen
The experimental data of Weber, O ilie r , Roder and Goodwin (11) was used fo r the 40 points selected. For normal-hydrogen the vapor pres
sures were calculated only from the boiling point to the c r itic a l point. Figure 10. is a plot of observed minus calculated vapor pres
sure in atm x 10"^ versus temperature in °K fo r normal-hydrogen; the maximum deviation, -0.700 atm x 10"^ occurs at 32.500 °K, and the erro r a t the c r itic a l point is 6.200 atm x 10'^.
The average deviation in vapor pressures of 0.07% fo r normal- hydrogen is very good performance.
Para-Hydrogen
Experimental data consisting of 22 points was taken from Roder, Weber, and Goodwin (12). Figure 11. is a plot of the observed minus calculated vapor pressure of para-hydrogen measured in atm x lO"^ versus temperature in °K. For approximately 90% of the whole temperature range, the deviations is small and negative; i t then rises to a maximum posi
tiv e deviation of 11.8 atm x lO~^ at the c r itic a l point (32.976 °K).
The average deviation in vapor pressures is 0.50% which is satisfactory.
Nitrogen
The 66 data points were taken from Strobridge (13).
P (O b s .) - P (C a lc .) » A tm x 1 0 "
ER 1282 34
12
10 8
6
4
2
0
"2
~4
“ 6
—8
-10
“1220 21 22 23 24 25 26 2 7 2 8 29 30 31 32 33
T em perature, °K
Figure 10. N orm al-H yd rogen . O b served m in u s calculated vapor pressure
versus tem perature.
COco
CMCO
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COo
CO
00 CM
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s£>
CM
4J
OS
00
«— I
sr
CM CO 00 CM
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01 X raav ‘ c 31*0)5 - (* sqo)5
ER 1282 36
Figure 12. is a plot of the observed minus calculated vapor pressure of nitrogen in atm x 10"^ versus temperature in 0K. This figure shows f i r s t , a region of very low negative deviations includ
ing about 90% of the data, and then a region of increasing deviations with a maximum deviation of 11.8 atm x 10"^ at 126.00 °K, fa llin g to
a deviation of 8.7 atm x 10"^ at the c r itic a l point (126.20 °K).
The average deviation in vapor pressures for nitrogen is 0.26%, which indicates good behavior of the Frost-Kalkwarf equation.
Oxygen
The 30 data points were taken from Brower and Thodos (14).
Figures 13. and 14. are plots of the observed minus calculated vapor pressures of oxygen in mm. of Hg and mm of Hg x 10% respectively, versus temperature in °K.
Figure 13. shows the low temperature range which is between the tr ip le point and 120.996 °K; i t is a region of negative-positive de
viations, with -3.097 mm of Hg the maximum deviation. This low range consists of about 65% of the whole data.
Figure 14. , the remaining 35% of the data, is the high tempera
ture range. The f i r s t 60% of this range shows a slow positive ris e , followed by a series of very sharp negative-positive deviations, with maximum deviation of 215.08 mm. of Hg at 150.163 °K.
The average deviation in vapor pressure is 0.27% fo r oxygen.
1 0 .0
ER 1282 37
CN
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o
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00
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CN
m
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x m 3 V *(" 3T B 0)d - ('sqo)d
F ig u r e 1 2 . N it r o g e n . O b se r v e d m in u s c a lc u la te d v a p o r p r e ss u r e v e r su s te m p e r a tu r e .
ER 1282 38
o o
0 0 -M
O 3
m
8
hr a a 4 (" 3 % e3 )d - ('sqo)d
F ig u r e 1 3 . O x y g e n . Lo w te m p e r a tu r e r a n g e . O b se rv ed m in u s c a lc u la te d v a p o r p r e ss u r e v e r su s te m p e r a tu r e .
ER 1282 39
m m
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