ESTIMATION OF SATURATED LIQUID HEAT CAPACITIES ABOVE T H E BOILING POINT R O B E R T C. R E I D A N D J A Y
E. SOBEL
Department of Chemicul Enp.inPPring, .MarJachusetfi Institutr of Technolog), Cambridge. .MasJ.
A technique for estimating the heat capacities of saturated liquids at reduced temperatures from 0.70' to 0.95' is presented. By resolving an equation first suggested by Watson for the heat capacity of a saturated liquid as a function of the two-parameter law of corresponding states, and expanding it with a three-parameter correlation with the critical compressibility as the third parameter, it has been possible to calculate the heat capacities in the critical region with a satisfactory degree of accuracy. Experimental and calculated values of the heat capacities of 15 compounds are compared over the temperature range 0.70' T, 0.95'.
<
maintaining saturated conditions, (2) liquid a t T , is vaporized and expanded isothermally to a zero pressure, ideal gas state, ( 3 ) as a n ideal gas. the material is cooled from T2 to T I ,and (4) the material is compressed isothermally a t T , to a saturated vapor and condensed. Since the cyclic change of enthalpy is zero, then by writing the enthalpy changes for steps 1 to 4 and equating to zero, EVERAL
(HT2 - H T ) )S L = A H ~ T, A HOT*f (fro - Hsa)TI -
+
( H o - H s a ) ~ ~Cpo(T2 - T I ) Letting
7'2
approach T I ,then
( d H ) , L = --AH,
- d(Ho - H s z )
+ CPodT
(1)
Before proceeding further, the definitions of liquid phase heat capacities must be considered. There are three in general use: Cp,, CsL: and ( d Q / d T ) s L . T h e first represents the change in enthalpy with temperature a t constant pressure ; the second shows the change in the enthalpy of the saturated liquid with temperature, ( d H s L l d T ) ;and the third is the heat necessary to effect a temperature change if the liquid is held in a saturated liquid state. I t is the third form which is usually measured in the laboratory. They are related as follows:
Except near the critical point. all the forms are in close agreement numerically. [This statement is not valid for comparable expressions for the saturated vapor, as the volume of the vapor is a large value. I n some cases. in fact, ( d Q / d T ) s c may even b t negative.] Choosing ( d Q / d T ) , L as the desired quantity, then combining Equations 1 and 3,
(dQ dT),L
E CsL'
C P o-
1
dAHL -~ dT
Equation 4 written in reduced form has considered AH" to be only a function of T,, and (H' - H ) to be a function of 1; and P, for any particular substance. \Vatson (26) first suggested such a form. although he solved for (dH,d T ) s Lrather than (dQ 'dT),L. I n the light of recent work expanding the la\\ of corresponding states to include, besides reduced temperatures and pressures, a third correlating parameter such as the acentric factor (76, 77), the critical compressibility factor (9: 74). or Riedel's factor (78, 79) it is inevitable that \Vatson's equation be resolved using a three-parameter law of corresponding states. This present paper accomplishes just such an end. T h e volumetric properties of the gas a n d liquid phases icere assumed to be functions of reduced temperature, reduced pressure. and the critical compressibility factor and the tabulated thermodynamic functions of Lydersen, Greenkorn, and Hougen were used in the calculations ( 9 , 13). T o obtain the tkvo partial derivative terms of ( H o - H s o )' T , in Equation 4, the tabulated values of (H' - H ) given by Lydersen, Greenkorn, and Hougen (9! 73) as functions of T,, P,, and 2, were plotted a t constant T r (or P,) as functions of P,(or T,) for various values of Z , and graphically differentiated at the limit \vhere the constant T , (or P,) curve intersected the generalized saturation curve. T o obtain the temperature derivative of the latent heat of vaporization the Klein correlation for the latent heat of vaporization as a function of reduced boiling temperature and pressure (72) was combined with Watson's temperature function (26) and differentiated to give
T h e vapor pressure derivative was determined from the Clapeyron equation with AHt expressed as discussed above and AVa determined from ( R T P ) ( l Z t ) . T h e reduced form of this derivative is given by the following expression :
T h e saturated liquid molal volume was determined from the reduced saturated liquid density by means of the following relationship :
vc
lid&.=PSL.
328
I&EC FUNDAMENTALS
(7)
.. .. . ,, . ,. .~. .. ..-. .. ,_. . , . . .-~. .
40
ii
..
I
20
0.6' 0
0.5175
IO 8
-
m
n -
L
6
If
4 . .. , . . I .. . . . .. .... . _.* ~ _ L .... . . .. . .-
I I1
~
.
~
~_. i
. ~.-~-
[:qFp
3 - 2
__L_
2 -+ 07 08 09 Reduced Temperoture,Tr Figure 2. Variation of estimating function $z with Tbr and T ,
1
0.7
0.8
0.9
Reduced T e m p e r a t u r e , Figure 1 .
Effect of T, on
Tr
$1
T h e final result is given in Equation 8. O
E
20
t;l T h e $ functions may be calculated from the definitions given in the nomenclature; for convenience. all except $ 0 are plotted against the independent variables in Figures 1 to 4. AZ, is the difference between the compressibility factors of the saturated vapor and liquid. I t was found to be nearly independent of Z, and could be expressed explicitly in terms of the reduced vapor pressure as shown in Figure 5. T h e reduced saturated liqukl density has been correlated by Lydersen et ~ l as. a function of the reduced saturation temperature a n d Z,, as shown in Figure 6. 7'0 obtain the heat capacity a t zero pressure, C p o , it is preferable to use experimental values or those calculated from spectroscopic d a t a a n d statistical mechanics. For example. in the latter method, A P I Project 44 ( 7 ) has extensive tabulation of C p o for hydrocarbons Where, however, such values are not available, reasonably good estimation methods for C,' have been drscribed (70. 78).
10 8 6
Reduced Temperature, Tr
II
Figure
3. Effect of T , on $a
Discussion
T h e critical compressibility was chosen as the third correlating parameter in preference to Pitzer's acentric factor or Riedrl's factor, since it \\?as desired to obtain a method valid near the critical point. In this region: Hooper a n d Joffe (8) found Z , correlations preferable. Also it is generally accepted that the critical compressibility method is more applicable to slightly polar substances and many of the reported high pressure-high temperature liquid heat capacity d a t a to be used for testing the estimation techniques were reported for polar materials. I t might have been more expedient to utilize Lydersen, Greenkorn, and Hougeri charts in a somewhat different
Tbr
0.725 0.70
0.675 0.650 0.625 0.60 0.575
I 0
$
0
0.7
0.8
0.9
Reduced Temperature, Tr Figure 4. Variation of estimating function +q with T,,, and T , VOL.
4
NO. 3
AUGUST 1965
329
Table I.
Comparison of Calculated and Experimental Liquid Heat Capacities -vo. of
Compound LVater Ammonia $-Xylene Isopropyl alcohol 1-Butene n-Butane cis-2-Butene n-Propane n-Propene 1-Pentene n-Pentane n-Heptane Methane Ethane Kitrogen
References Sat. liq. Ideal gas (7) (7) (7)
(77)
(25) (3) (5)
(6) (7) (7) (7)
(22) (20) (2.3) (27) (2) (22) (20) (4) (27) (28) (27)
(7) (7) (7) (7) (7) ( 7) (7) (7)
manner. I n these charts (H' - HsL) is tabulated as a function of reduced pressure, reduced temperatlire, and the critical compressibility factor. From such tables, the term dAH, dT in Equation 4 could be eliminated and H,, replaced by HsL This technique was tried but yielded results of much less accuracy. T o obtain (H' - HsL) in the first place. it had to be determined in two parts-i.e , (H" - H s o ) ( H s b- HSL). T h e latter part was apparently obtained from the Clapeyron equation assuming that a single function related saturated values of P, and TTat any given Z,. This leads to some error; the described Klein-LVatson method is more accurate. Xear the critical point the contribution of the preswre correction term in Equation 8 is of the same order of magnitude as the saturated liquid heat capacity. Since this term is proportional to the reduced pressure. any error in estimating the reduced pressure leads to a comparable error in the calculated heat capacity. For this reason it is desirable to employ experimental data for the reduced vapor pressure of the liquid. If such data are not available. it is possible to employ one of the generalized techniques for estimating reduced vapor pressure. I n this case accuracy of the calculated heat capacity may be limited by the accuracy of the pressure-estimating equation.
+
Comparison of Calculated with Experimental Results
Reduced T e m p , , Range, a C. 0.70to 0 . 9 4 0 . 7 0 to 0 94 0.70 to 0.927 0 . 7 0 to 0.931 0 . 7 0 to 0.85 0 . 7 0 to 0.92' 0 . 7 0 to 0.856 0 . 8 0 to 0.915 0 . 8 0 to 0.943 0 . 7 0 to 0 . 8 2 0 . 7 0 to 0.81 0 . 7 0 to 0 . 9 5 0 . 7 0 to 0 . 9 4 0 . 7 0 to 0 . 9 5 0 . 7 0 to 0.927
% Deviation
Data Points
Au.
1.04 2.35 1.75 2.76 2.31 1.02 2.00 2.58 3.10 2.36 1.58 1.20 1 .os
7
7 6 6 4
6 5
1.19
2.21
the calculated saturated heat capacity of methane, C,,', is plotted as a function of temperature in Figure 7 and compared with experimental values. Of the 83 points calculated, only two showed deviations greater than 5y0from experimental data. For ammonia, Z , = 0.243, and nitrogen, Z, = 0.29, the errors were 5.15 a n d 6.7?&, respectively. These errors can be attributed to the use of a single curve to estimate the term
3
aTC
[-TIp, (Ho
-
Hsl)
for both values of Z,. T h e calculated data for this term were scattered about this line and if two overlapping lines had been drawn to represent it, this would have reduced the maximum error for each compound below 5%. T h e increased difficulty of reading the two curves did not justify this action. Example
T o illustrate the use of Equation 8, the saturated liquid heat capacity of isopropyl alcohol a t 204' C. is estimated. T h e critical properties are T , = 508.8' K. (73),P, = 53.0 atm. (73). and V , = 219 cc.,'gram mole (75). T h e normal
T h e heat capacities for 15 saturated liquids were calculated over a temperature range from the normal boiling point to as close to the critical point as experimental data existed. T h e critical compressibilities of the liquids ranged from 0.23 (water) to 0.29 (nitrogen and methane). A total of 83 separate points were calculated a n d compared directly to experimental data (Table I). To illustrate the use of Equation 8,
3 h
-I
N
'>
N
v
\
2
-
'
01 0 2 0 3 0.4 0 5 0.6 07 0.8 0.9 Reduced Vapor Pressure, Pvpr Figure
330
I&EC
5.
1 /(Z, - ZL) as function of P,p,
FUNDAMENTALS
Max. 2.86 5.15 4.40 4.96 4.93 2.17 4.13 3.56 4.56 3.91 2.59 2.24 1.65 2.71 6.70
Reduced Temperature, Tr Variation of p s L , with T,
Figure 6.
g3], 1 2
$0
= In
$2
=
A
Pb.[ 1 -
dimensionless
(0.38)(1.04) RTb,(l (1 - T b , ) ’ . 3 8
T,)-0.62
-
, cal.,’gram
mole
”K
EXPERIMENTAL DATA A
CALCULATED FROM
SUPERSCRIPTS = ideal gas state
SUBSCRIPTS b
130 Figure 7. methane
140
150
160 TEMPERATURE
180
170
190
OK.
Calculated and experimental values of
ClsL
for
boiling point is 355.5’ K. ( 2 4 ) . T h e critical compressibility factor is thus (53)(219) ’(508.8)(82.07) = 2, = 0.278. T h e only additional d a t a needed are the reduced vapor pressure 273);508.8 = 0.931. For isopropyl alcohol a t T , = (204 a t T , = 0.931 the reduced vapor pressure is P, = 0.519 ( 6 ) . From API Project 44 ( 7 ) : C,” = 31.1 cal./gram mole K. a t this temperature. To use Equation 8>
+
1
/ 1 \ r
11’2
(from Figure 1) = 11.3 a t T , = = 15.0 a t T , = (from Figure 2) $ 3 (from Figure 3) = 6.4 a t T , = +b4 (from Figure 4) = 1.63 a t T,= 1, AZ (from Figure 5) = 1.68 a t P, = psi, (from Figure 6 ) = 1.85 a t T , = $?
0.931 0.931. T,,
=
0.700
0.931 0.931, TO, = 0.700 0.519 0.931
Substituting in Equation 8, C , T ~=’ 31.1 - 3.86
15
+ (1.68)(1.63)(0.519)
= 63.5 cal./gram mole
(1.987)(0.278)
” K.
T h e experimental value is 63.72 cal./gram mole and the per cent error is 0.35. Nomenclature
C
=
H
= = = = = = =
AH?
P Q R
T V Z p
= =
>]
1.85
heat capacity, cal./gram mole ” K. enthalpy. cal./gram mole ” K . enthalpy of vaporization, cal./gram mole pressure: atm. heat, cal.,’gram mole gas constant? 1.987 cal./gram mole K. temperature, O K . specific volume, cc.,’gram mole compressibility factor, dimensionless density, g r a m mole, cc.
K. (5)
G
= quantity measured a t normal boiling point = quantity measured a t critical point
L
=
p
= quantity measured a t constant pressure
liquid state
r
=
sL
= saturated liquid = saturated vapor = quantity measured a t constant temperature
su
T
reduced parameter
vapor state
L’
=
up
= vapor pressure
literature Cited
(1) American Petroleum Institute Project 44, “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons,” Carnegie Press, Pittsburgh, Pa., 1953. (2) Averbach: \V. G., Sage, B. H., Ind. Eng. Chem. 42, 110 (1950). (3) Corruccini, R. J., Ginnings, D. C., J . Am. Chem. SOC.69, 2293 (1947). (4) Douglas, T . B., Furukawa, G. T., J . Res. S u t l . Bur. Std. 53, 139 (1954). (5) Ginnings, D. C., Corruccini, R . J., Ind. Eng. Chem. 40, 1990 (1948). (6) Hatch, L. F., “Isopropyl Alcohol,” McGraw-Hill, New York, 1961. (7) Haupt, R. F., Teller, E.; J . Chem. Phys. 7, 925 (1939). (8) Hooper, E. D., Joffe, J., J . Chem. Enp. Datu 5, 155 (1960). (9) Hougen, 0. A , \Vatson, K. M.: Ragatz, R. A . , “Chemical Process Principles,” Part 11, “Thermodynamics,” Wiley, New York, 1959. (10) Janz, G. J., “Estimation of Thermodynamic Propertie. of Organic Compounds,” Academic Press, New York, 1958. (11) Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” LViley, New York, 1936. (12) Klein, V. A., Chem. Eng. Progr. 45, 675 (1949). (13) Kobe, K. A, Lynn, R. E., Chem. Reus. 52, 117 (1953). (14) Lydersen. A . L., Greenkorn, R. A., Hougen, 0. A , , Eng. Expl. Sta., University of \Visconsin, Rept. 4 (October 1955). (15) Mathias, E., “Le Point Critique des Corps Purs,” Paris, 1904. 79, 2369 (1957). (16) Pitzer,K. S.:Curl, R . F., J r . , J . Am. Chem. SOC. (17) Pitzer, K. S., Lippmann, D. Z., Curl, R . F., Jr., Huggins, C. M . , Petersen, D. E., Ibid., 77, 343 (1955). (18) Reid: R . C., Sherwood, T. K.. ”Properties of Gases and Liquids,” McGraw-Hill, New York, 1958. (19) Riedel, L., Chem.-Inp.-Tech. 28, 557 (1956). (20) S a w . B. H.. Ind. E m . Chem. 28. 489 (1936) (21j Sage, B. H., Lacey,-\V. N., Ibth.,-27: 1484 (1935). (22) Schlinger, \V. G.. Sage, B. H., Ibtd., 41, 1779 11949) (23) Ibzd., 44, 2454 (19521. (24) Timmermans, J., “Phvsico-Chemical Constants of Pure Organic Compounds,“ Elsevier, New York, 1950. (25) U. S. Bur. Standards, Czrc. 142 (1923). (26) LVatson. K . M., Ind. Eng. Chem. 35, 398 (1943). (27) \Viebe, R., Brevoot, M., J . Am. Chem. Soc. 52, 611 (1930). (28) Ibid., p. 622. RECEIVED for review April 16, 1964 ACCEPTED November 13, 1964
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