APPLICATION OF A REDUCED VAPOR PRESSURE EQUATION T O
N0N H Y 13, ROCA RBO N SU BSTA NCES ENRIQUE G .
REYNES AND GEORGE THODOS
The Technological Institute, ,Vororthzuestern Cnnioersity. Evanston, Ill.
The Frost-Kalkwarf vapor pressure equation expressed in reduced form has been applied to substances other than hydrocarbons to include elements, inorganic compounds, and organic derivatives. The vapor pressure data for over 30 substances, ranging in complexity from hydrogen to the aromatic derivatives, were analyzed with the aid of a digital computer to obtain the corresponding vapor pressure coefficients @ and y. This group includes both polar and nonpolar substances. This investigation shows that the same linear correlation between ,6 and y found earlier to exist for hydrocarbons i s also applicable to the substances considered in this study, as long as hydrogen-bonding effects are absent. In view of this generalization, the reduced vapor pressure equation for polar and nonpolar substances exhibits no hydrogen-bonding effects, and only one reduced vapor-pressure point i s necessary to define completely the vapor-pressure function for these substances. This reduced equation was tested with 10 arbitrarily selected substances covering the range of vapor pressures between the triple point and the critical point and was found to be reliable.
HE VAPOR PRESSURI? FCNCTION between the triple point and T t h e critical point has been shown to be nonlinear when the logarithm of pressure is plotted against the reciprocal absolute temperature (20, 30). Exacting vapor pressure measurements have shown that this function in the low pressure region curves downward and reverses in curvature at moderate pressures ( 7 76). Frost and Kalliwarf (2.9: 67) used the Clapeyron equation, in which they assumed that the van der Waals equation applied, and that the latent heat of vaporization varied linearly Xvith temperature, to develop the semitheoretical expression:
lo: P
=
A
+ Br + Clog T + D 7
~
P T?
(1)
Perry and Thodos (87) have established the coefficients A , B, C: and D of this equation from four reliable vapor pressure points for the normal paraffins u p to n-dodecane. H a m r i n and Thodos (42) used Equation 1 in reduced form to represent the vapor pressure behavior for the inert gases. This reduced vapor pressure equation,
has been extended to define the vapor pressures of the hydrocarbons, where cy = A - log P, Clog T,. p = B / T , . y = C, and D = 0.1832 Thus a direct correspondence exists between y and C and the constant p its readily obtainable from the value B and the critical temperature of the substance. At the critical point, Equation 2 reduces to give the folloi\ ing generalized condition,
+
o!
+ p + 0.1832
= 0
(3)
F r o m the detailed 'studies on the vapor pressures of the aliphatic, naphthenic, and aromatic hydrocarbons (72, 84, 707, 7 70), the available values of B and C were used to produce 13
and y (97). These values \\ere found to correlate linearly:
Substituting Equations 3 and 4 into Equation 2 and rearranging yields the follo\l-ing reduced vapor pressure equation. This equation. which requires a single constant for complete definition, is:
Equation 5 defines propell! the vapor-pressure behavior of the hydrocarbons ( 9 7 ) . Constants
and y
T h e present investigation is concerned with the application of Equation 5 to substances other than the inert gases and the hydrocarbons (42. 97) and includes elements. inorganic compounds, and organic derivatives. Vapor pressure data for over 30 polar and nonpolarsubstances were analyzed according to the procedure proposed by Sondak and Thodos (770) in order to establish constants B and C.
P
1
This method required plotting Y = {log - - D [$2 Pb T T /log us. = and obtaining the slope, - &],/log
x
[f
B, and intercept C of the resulting straight line. I n order to calculate the values of Y and X,a program was prepared which utilized the facilities of an IBM-650 digital computer. Typical plots are presented in Figure 1 for carbon tetrachloride, which is nonpolar, and in Figure 2 for sulfur dioxide, which has a polarity of 1.76 debye units. T h e substances included in this VOL. 1
NO.
2 MAY 1962
127
14
13
12 0
Cardoso ond Ugo
I/ Toriumi ond Hora
10
Y 9
8
I
1
i
I
X
Figure 1. Relationship of Y vs. X for carbon tetrachloride
T g , O K. Tc, K. Acetic acid Acetone Ammonia
391.5 329.7 239.8
594.8 508.7 405,5
Ani 1ine Bromine n-Butyl alcohol Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorine Chlorobenzene Chloroform Ethyl alcohol Ethyl chloride Fluorobenzene Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen sulfide Methyl alcohol Methyl chloride Nitric oxide Nitrogen Nitrous oxide Oxygen Phenol Phosgene n-Propyl alcohol Sulfur dioxide Sulfur trioxide Water
457,6 331.9 390.2
698.8 584.2 561.2 304.2 552.2 133.2 556.4 417.2 632.4 536.6 516.2 460.4 559.8 33.24 363.2 324.6 373.6 513.2 416.3 180.2 126.2 309.7 154.8 692.4 455.2 537.2 430.7 491.4 647.4
52.3 102 49.0 72.85 78 34.5 45.0 76.1 44.6 54.0 63.0 52.0 44.6 12.79 84.0 81.5 88.9 78.5 65.9 64.0 33.5 71.7 50.1 60.5 56.0 50.2 77.8 83.8 218.3
128
I&EC FUNDAMENTALS
I
-8
-7
1
1
-6
-5
-A
Figure 2. Relationship of Y vs. X for sulfur dioxide
A review of Figure 3 reveals that /3 and y are independent of polarity effects. This conclusion can be drawn from the fact that highly polar substances such as sulfur dioxide and methyl chloride follow the pattern exhibited by nonpolar substances such as nitrogen and carbon tetrachloride. I n view of this observation, it is reasonable to assume that polarity does not influence the establishment of constants p and y. I n Figure 3, hydrogen and neon can be seen to deviate slightly from the straight line represented by Equation 4. This behavior can be explained from the fact that these two substances possess excessive quantum deviations. Helium
Vapor Pressure Constants of Substances investigated
Pc, atm. 57.1 46.6 111.3
319.5 81.2 349.9 238.6 405.3 334.4 351,6 285.4 358.2 20.39 206.2 188.2 213.6 337.9 249.2 121.4 77.4 184.7 90.0 315.7 281.4 371 .O 263.2 317.8 373.2
-9
X
study are presented in Table I, along with the references used to enable the calculation of constants B and C. I n this table, the critical temperature and pressure (68), the normal boiling point, and the polarity (724) of each substance are also presented. T h e variation in polarity for these substances ranges from 0 to 2.70 debye units. A plot of values of p us. y on rectilinear coordinates is presented in Figure 3 for all the substances investigated in this study. The available vapor pressure data for bromine, hydrogen bromide, and sulfur trioxide lack internal consistency, and therefore the calculated p and y values cannot be considered reliable. With the exception of a limited number of substances, the majority produced a correlation that is identical to that exhibited by the hydrocarbons (97).
Table 1.
I
I
-10
-11
B
B
=
B/Tc
-2784.0 -2358.0 -1723.2
-4.6809 -4,6353 -4.2495
-3348.6 -2498.8 -4199.5 -1345.4 -1868.0 - 442.0 -2366.9 -1484.6 -2860.3 -2529.4 -2895.3 -1882.3 -2601.9 - 46.619 -1495.4 - 1142.9 -1352.6 -2342.3 -1613.9 - 1252.7 - 399.26 -1249.0 - 457.41 -4566.5 -1909.5 -3550.6 -1962.7 -3870.2 -3021.6
-4.7922 -4.2775 - 7.4763 - 4.4229 -3.3830 -3.3191 - 4,2540 - 3.5588 -4.5229 -4.7141 - 5,6093 -4.0888 - 4.6482 - 1 ,4025 -4.1177 -3.5213 -3.6204 - 4,5644 -3,8771 -6,9531 -3.1637 -4.0335 -2.9556 -6.5956 -4.1952 -6.6099 -4,5570 - 7,8764 -4,6673
r = C
-
4.5543 - 4.4225 4.7059
-
- 4.9405
- 6.7585 -11,0533 - 5.3795 - 3.2460 - 3.4051 - 5.1023 - 3.9254 - 5.5013 - 6.5060 - 5.1449
- 4.6269 - 5.8797 - 0.3667 - 6.2161 - 3.5088 - 3.9571 - 2.7620 - 4.3518 - 10,1405 - 2.9818 - 4.6268 - 2.4673 -10,8061 - 5.0432 8.3252 - 5.5629 -13.6430 - 5.3650
-
Debye Units 1.04 2.70 1.53
1.60 0.40 1.65 0.13 0 0.13 0 0.23 1.58 1.26 1.63 1.76 1.39 0 0.79 2.15 1.10 1.66 1.97 0 0 0.25 0 1.70 1.18 1.53 1.76 0 1.85
References Used 58, 93, 99, 103, 130 25, 28, 96, 101, 114 6, 21, 22, 49, 54, 66, 67, 106, 775 5, 60, 91 92, 120, 127 15, 59 7 , 65, 76, 77, 88, 732 49, 70, 83, 94, 96, 123 3, 18, 79, 72, 79 96, 98, 109, 130, 131 35, 41, 43, 56, 85, 118 78, 94, 129 4, 25, 50, 96, 102 69, 74, 86, 91, 99, 102, 109 7, 38 95, 129 40, 64, 77, 80, 705, 725, 126 44, 111, 112 13, 16, 36, 49, 7 12 9, 32, 62, 83, 112 78, 90, 96, 99, 104 75, 113 37, 45, 57 2, 24, 33, 47, 48, 55, 89 11. 14 2, 24, 27, 46, 119, 121, 128 5, 59 31, 34 78, 90, 99, 104 6, 17, 73, 100, I17 8, 39, 708 26, 52, 53, 63, 81, 122
1
Figure 3.
Correlation of constants
p
unexpectedly falls in line despite the fact that this substance also possesses considerable quantum deviations (23, 57). The slight anomaly encountered with oxygen may be oiving to the paramagnetic nature of liquid oxygen (70). Hydrogen-Bonding EflFects
The results presented in Figure 3 point to the existence of compounds that d o not conform to the general pattern exhibited by the majority of substances. For example, the values ol j3 and Y for methyl alcohol and ethyl alcohol are found to deviate considerably from Equation 4. These deviations
7
and y for reduced vapor pressure equation
for the alcohols become less pronounced with increasing molecular weight and become insignificant for n-butyl alcohol. I n addition to the alcohols, substances such as acetic acid, aniline, ammonia, acetone, and water do not follow Equation 4. This anomalous behavior can be attributed to hydrogen-bonding effects present in these substances. I n view of these observations, it can be assumed that polarity does not influence the relationship between p and 7,while hydrogen-bonding contributes significantly in this direction. A continuation of vapor pressure studies, specifically directed to substances possessing hydrogen-bonding effects, deserves further consideration. VOL. 1 NO. 2 M A Y 1 9 6 2
129
Comparison of Results
Equation 5 has been applied to 10 substances included in this study. To evaluate the constant y, the normal boiling point of each substance was used. These calculated y values a r e compared below with the values obtained directly from the interpretation of vapor pressure data when Y is plotted us.
x.
Values Vapor pressure data
Carbon monoxide Carbon tetrachloride Chlorobenzene Ethyl chloride Fluorobenzene Hydrogen sulfide Methyl chloride Nitrogen Nitrous oxide Sulfur dioxide
-3 4051 -5.1021 -5.5013 -4.6269 - 5.8797 -3.9571 -4.3518 -2.9818 - 4,6268 -5.5629
Equation 5 - 3.6850 -4.7980 -5.4609 -4.7621 - 5.3905 -3 5468 -4.3153 -3.0472 -4.4655 -5.5175
T o check the validity of Equation 5 using the ?-values resulting from the normal boiling points, vapor pressures were calculated and compared with corresponding experimental values using a n IBM-650 digital computer. I n this comparison over 100 experimental points ranging from low pressures u p to the critical point were considered. T h e resulting average deviation for these 10 substances \cas 1.12%. Compared with this value: an average deviation of 3.94YGresulted for ille same experimental points when the normal boiling point and the critical point were used to establish the straight line relationship represented by the expression log P = A
+ 5.
Nomenclature
A , B , C, D = constants for Equation 1 P = vapor pressure, m m . of mercury Po = vapor pressure of reference point, m m . of mercury P, = critical pressure, mm. of mercury P R = reduced vapor pressure. P P, T = absolute temperature, K . Tb = temperature of reference point, O K. T , = critical temperature, K. TK = reduced temperature, T T , T
[;
-
X
= temperature modulus,
Y
= vapor pressure-temperature
p
[T: -
P I +Jpog
$]iiiog modulus,
1 log Pb - D
T
a, p, y = constants for Equation 2
Literature Cited
(1) (2) (3) (4) (5) (6) (7) (8)
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130
-
I & E C FUNDAMENTALS
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DYNAMICS OF A FLOW-FORCED HEAT EXCHANGER LOWELL B. KOPPEL School of Chemical Engineering, Purdue University, West Lafayetle, Ind. The partial differential equation and boundary conditions describing the temperature response of a steamwater heat exchanger to an arbitrary variation in water velocity are solved. The solution is compared with the approximate linearized solution for a step change in velocity. It is found that significant error can result in the approximate solution for large magnitudes of step change or for high exchange to heat capacity ratios.
HE EQUATIOXS DESCRIBING the dynamics of flow-forced heat Texchangers have been solved only in their linearized form ( 7 ) . The solutions obtained in this manner m a y be expected to give good accuracy only if the magnitude of the flow disturbance is small compared with the magnitude of the steady-state flow rate. I t is pertinent to seek quantitative measures of the accuracy of these linearized solutions. To this end, a simple case of flow-forced heat exchange, for which a n analytic solution may be found, is considered.
Description of System
Consider a heat exchanger in which one of the fluids remains a t a constant, uniform temperature, independent of the other fluid. Such conditions are approximated in an exchanger with a condensing vapor as one of the fluids or in an exchanger where the flow rate or specific heat of one of the fluids is extremely high compared with that of the other. For convenience,
designate the constant temperature fluid as steam and the other as water. Assume that the water is in plug flow, has constant physical properties, and exhibits perfect radial mixing and negligible back-mixing. I n addition, it may be assumed that the energies of compression and viscous dissipation are small compared with the other energies carried by the water stream. Derivation of Equations
With these restrictions, the energy balance describing the temperature behavior of the water may be written:
Define the dimensionless variables: X I
x = T -
L
V O L 1 NO. 2 M A Y 1 9 6 2
131
