(1967) requires one parameter specific to each binary molecular system. The fitting thus obtained is probably the best of the models so far developed. The reaction model of Kiehe and Bagley (1967) contains two parameters ylecific to each binary molecular system. The fitting of free energies for various systems seems to be excellent, but inferior results are obtained for heats of mixing. Khile the calculations of the present work are all a t 30"C, the usefulness of the energy parameters reported here can be estended b y means of the Helmholtz equation
I n this manner the escess free energy and phase equilibrium properties can be estimated a t other temperatures if not far from the ba5e value. The generalization of the lattice theory to account' properly for change of volume with temperature and with mixing of liquids should prohably best, be accompliahed in the framework of the cell theory. The present, study has shown that practical useful results can be obtained entirely in terms of groui) properties that are physically meaningful. The u~efuhiessof the group concept will be extended when re-espresaed in cell theory. Acknowledgment
The authors thank R. 11. Baer for making available the nonlinear regression computer program and T. IT. Lee for valuable discussions. Nomenclature
E G
fl k
= = = = =
configurational energy, cal Gibbs free energy, cal degeneracy of configuration enthalpy, cal Boltzmann's constant
N A
= number of molecules
A
N,,
number of contacts between segments of types i and j number of i groups in molecule A canonical partition function number of segments or groups on molecule A entropy, cal/"K T = absolute temperature, OK wL, = exchange energy, w,, = e,, - I/t(e,, E,,), cal/gmole x = mole fraction z, = number of contact points of segment i E%, = energy of interaction between an i segment and a j segment, cal/g-mole = naA = Q = ?'A = S =
+
SCPCRSCRIPTS E = a n excess property 0 = pure liquid
*
=
for an athermal solution
literature Cited
Barker. J. A,. J . Chem. Phvs. 20. 1526 11952). Chao, K. C., Robinson, R: L., J r . , Smith, A I . L., Kuo, C. M., Chem. Eng. Progr. S y m p . Sei. 63(81), 121 (1967). Florv P. J., "Piinciple- of Polymer Chemistry," p. 497, Cornel1 U&ersity, Ithaca, N. Y., 1953. Goates, J. R., Snow, R. L., James, 31.R., J . Phys. Chem. 65, 355 (1061 \ \ I Y " I , .
Goates, J. R., Snow, R. L., Ott, J. B., J . Phys. Chem. 66, 1301 ( 1962). Guggenheim, E. A , , Proc. R o ~ JSOC. . A183, 213 (1944). Renon, H., Prausnitz, J. lI.,Chem. Eng. Sei. 22, 299 (1967). Saviiii, C. G., Rinterhalter, D. R., Van Kess, H. C., J . Chem. Eng. Data 10, 168 (1963). Van Ness, H. C.. Sociek, C. A., Kochar, Ir;. K., J . Chem. Ena. Data 12; 346 (i667aj. ' Van Kess, H. C., Socjek, C. A , , Peloquin, G. L., Machado, R. L., J . C h m . Eng. Data 12, 217 (1967b). Wiehe, I. A., Bagley, E. B., IND.EKG.CHEM.FIXDAM. 6 , 209 (1967). RECEIVED for review February 24, 1970 ACCEPTEDAuguat 3, 1970 Division of Industrial and Engineering Chemistry, Symposium 011 Enthalpy of Nixtureh, 159th AIeetiiig ACS, Houston, Tex., February 1970.
Effect of Pressure on Heat Capacity Nitrogen-Trifluoromethane System Stephan M. Balaban' and Leonard A. Wenzel Lelzigh University, Bethlehem, Pa. 18016
A Workman-type heat exchange calorimeter was used to study the effect of pressure on the heat capacity of two mixtures of nitrogen and trifluoromethane. Pressures ranged from 1 to 140 atm and temperatures from -20" to 60°C.
THE
effect of pressures on the constant pressure heat capacity of a gas may be calculated if sufficiently accurate P-I'-T data are available. Since the second derivative of such data is required, the error involved may be considerable. This work Present address, 1Loii.aiito Co., 800 Korth Lindbeig Blvd ,
St. Louis, 310.63166.
568 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
involves the construction of a Workman-type heat exchange calorimeter to find the effect of pressure on the constant pressure heat capacity, compare the pure component data with existing equations of state, and investigate mixing rules between a polar and nonpolar gas mixture. Nitrogen is used to test the accuracy of the method; mixtures of trifluoromethane and iiitrogen Lire used to study mixing rules.
TEMP CONTROL
HEAT C O M P R E S SOR EXCHANGER
I STORAGE
SUPPLY
d
Figure 1, Schematic diagram of heat-exchange calorimeter flow system Apparatus
Constant' pressure heat capacity can be measured by several methods, such as those outlined by Partington and Shilling (1924) and Masi (1954). Two of the more popular methods use a heat input supplied by either an electric heater or latent heat, to raise the gas temperature. T o calculate the heat capacity one must measure the electric power or latent heat', the flow rate of the gas, the inlet and outlet temperatures, and the heat leak to a high degree of accuracy. The ratio of the heat capacity a t high pressure to the heat capacity at 1 a t m can be measured using a method originally devised by K o r k m a n (1930, 1931). A high pressure gas is passed through the tube side of a heat exchanger, the pressure is reduced, and the low pressure gas ii: sent' through the shell side. The flow rates on the shell aiid tube sides are the same. Therefore, a n energy balance around the heat, exchanger (Figure 1) yields
2 125 214
35455-
--
CCIS
This is the basic method used in this work. As applied here, improvemeiits ox'er Workman's apparatus design include a gas recirculation system and a simplified heat exchanger design. The apparatus shown in Figure 1 recirculates gas through the heat exchanger and two coiistaiit temperature baths. A two-stage Corbliii diaphragm compressor is used to provide constant gas f l o ~and prevent oil contamination. The high pressure gas leaving the compre temperature bath. The bath con containing a liquid in which enough high pressure tubing is immersed to alloil- the exit gas to be within 1°C of the bath temperature. The bath liquid, a mixture of ethylene glycol and water, is highly agitated. X Bayley controller maintains the liquid temperature by means of n nickel resistance thermometer and adjusting voltage to an electric heater. To offset the heat inputs to the bath liquid by the circulating gas aiid the electric heater, a steady flov of cooling water a t high temperature< or liquid nitrogen a t low temperatures is used. The temperature-controlled high pressure ga:, stream enters the tube bide of the heat exchanger shown in Figure 2. The heat exchanger assembly is placed inside an evacuated vessel filled with Perlite insulation for thermal isolation. The vessel wall.; are also thermally controlled to reduce the temperature difference from the exchange to ambient. The temperature difference between the inlet and outlet gas stream is measured by a 10-junction copper-constantan
0
c T-3+--
S-L--
~-
0-5
"
5
---
Figure 2. Construction details of heat-exchange calorimeter
thermopile using a Lee& & Sorthrul) Ihell side of the heat exchanger. Here too a thermopile ii used t o measure the tempPratuw difference between inlet and outlet gas -tream, and single thermocouples are used to measure individual qtreani teniperature.. The pressure drop of the low Iwe-ure fluid pn+iiig through the exchanger i? nipasurecl Jrith n maiionieter. -1ftrr the gas leaves the heat eschanger, it p s w s through an H U S iliary exchanger and flowmeter before returning to thc compressor. Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
569
Again, solving for CPHPICPLP
I 16
I14
.
I12 I10
+=
z'
108
+106
loo
__
Figure 3.
600
1000 1200 PRESSURE PSIA
800
1400
1600
1800
Experimental data for nitrogen at
CPHPICPLP=
2000
60.0"C
Data Collection
=
+
+
C P ~ ~ ( A T H PIIHPAPHP)
+
C P ~ ~ ( A T L PIILPAPLP) (2) Solving for CYHp/CpLp
=
-
ATLP
-
+ CPLP, ATHP,ATLP) (4)
On the other hand, when the high pressure gas is heated the energy balance becomes
Q'
=
Cp,, (- ATHPf PHPAPHP) f
"
.I.
Thus it is necessary to take t'wo lines of data, as illustrat'ed in Figure 3, and take the average between them. One line of data is t,aken while the high pressure gas is heating and the other while it is cooling. The average line should be independent of errors and should represent the heat capacity ratio. If this is t'rue, the average line should extrapolate to intercept the pressure axis a t 1 a t m when the ratio of C p Z p / c p , at.& = 1.0. Dat'a are taken to keep the high pressure AT constant with the same average temperature a t each pressure point. The average temperature of the high pressure stream is used as absolute temperature of that particular heat capacity measurement. This is valid if the heat capacit'y is linear over the temperature range of the ATHP. For this reason, in the region far from the critical, the ATHPis allowed to be as much as 10°C, while close to the critical it is reduced to 2°C.
ATHP
f(Q, PLP, A P L P , PHP,QHP,
~
f'(Q', PLP, APLP,PHP, APHP, CpLp,ATHP, A T L ~ ) ( 7 )
f'(Q',
CPHdCPLP
-
The heat leak terms, Q and Q', depend upon the temperature levels of the exchanger and of the surroundings. Since these may be kept a t constant levels for the two modes of operation, Q and Q' can be held equal. Temperature levels, Joule-Thomson coefficients, and pressure drops are also the same for the two cases. Therefore the correction functions, f (Q, . . . ) and f'(Q', . , . ) in Equatioiis 4 and i are essentially equal. Balaban (1966) has shown that small thermocouple errors can be included in the correction function, f(Q, . . . ) and
Equation 1 is idealized and does not include heat leak or Joule-Thompson effects arising from pressure drops. These effects can he included. \Then the high pressure gas is cooled the energy balance may be written as
Q
ATLP ATHP
_ I ~ . - - - - _ ~ - ~ - - A 400
+
C P ~ ~ ( - A T L P PLPAPLP) ( 5 )
k
30t
200
400
600
800
1000
1200
PRESSURE P S l A
Figure 4.
Experimental data for trifluoromethane
at 57.1 "C
D a t a points corrected for minor temperature variations
570 Ind.
Eng. Chern. Fundam., Vol. 9, No.
4, 1970
I(
IO
Experimental Results
Nitrogen is used to test the apparatus and to compare this type of heat capacity measurement with available data. A t each chosen temperature, two lines of data are taken to find the average line, which should be independent of errors. Measurements at -22.1", l o , 18.9', 39.5", and 60°C and at pressures from 250 to 2000 psia are obtained. The temperature of each point along a given line is not quite identical t o the average temperature, because of the difference in pressure a t the various points, which means a change in heat capacity of the high pressure gas and, therefore, a change in AT. Each point is corrected to the average temperature by employing a n equation of state. The virial equat,ion of state, which later proved to agree with the results to better than 0.3%, is used. The maximum correction employed is 0.5%; thus, the maximum error expected by using this equation is 0.0015%. Another method, which is to calculate the change in heat capacity with temperature using the raw data measurements, can be used to correct the data to ;he average temperature without relying on a n equation of state. An example of the data taken for nitrogen a t 60.0'C is shown in Figure 3. Points of AT1 .,,/ATH~, a t intervals of 25 or 50 psia, are found on the upper and lower data lilies using the Lagrangian interpolation bet'ween the three nearest data points. The average values between the upper and lower lines are then found a t these intervals and represent the quantity C p , , l C p , atm. The results for trifluoromethane are shown in Figure 4. A few points can be compared with the data of Mage et al. (1963) using a flow calorimeter. A heat capacity of 0.2673 Btu/lb "F a t 588 p i a arid 0.4"C is reported. TTsing the zero pressure heat capacities of Goff and Gratch (1950) and the virial equation of state to calculate the small correction to 1 atm, a 1-alue of CpSs8ns,a/Calatm = 1.078 is calculated. Using interpolated data from this heat exchanger calorimeter, a value of 1.076 is found. -4Mage data point a t 1176 psia and 0.5"C gives a ratio of 1.158, compared with a value of 1.156 from this experiment. The data from this paper are compared with the heat
Table 1. Experimental Heat Capacity Ratios for Nitrogen at 1.0"C and Calculated by Michels and Bloomer and Rao Pressure, PSIA
Exptl.
Michels
Bloomer-Rao
292 580 1156 1747
1 034 1.075 1.153 1.222
1.040 1.079 1.159 1 231
1 036 1.075 1,154 1.231
Table II. Errors Associated with Experimental Measurements
Yo
Magnitude of Error
Measurement
Pressure, p i a Absolute temp, "C Temp. diff., "C Effect of flou rate Using eq. of state to temperaturecorrect data Total
Error in C m P I C m a t m Away from Near critical critical
1 0 1 0 004
0 0 0 0
02 01 1 1
0 0015 ___ 0 3
0 0 0 0
1 5 4 1
0__ 3 1 4
capacities of Michels et al. (1951), who calculated the heat capacities by taking the second derivative of their P-1'-T data, in Table I. An accuracy of 170 for llichels' values is quoted. This table also compares the present data with those reported by Bloomer and Rao (1952) from their equation of state based on several sources. .in example of the data obtained for trifluoromethane close to the critical region is shown in Figure 4.Figures j l 6, and 7 give some of the heat capacity ratios calculated from Figure 4 and other data for trifluoromethane and two mixtures of trifluoromethane and nitrogen. Errors
Table I1 gives the maximurn error of the measurements used in this equipment and its effect on the heat capacity ratio, CpHp/ICpLp, h w a y from the critical region, an error
PRESSURE PSIA
Figure 5. Heat capacity ratio vs. pressure for 8.94 mole in nitrogen
yo trifluoromethane
- Experimental
- - - Calculated b y Martin-Hou equation Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
571
,
/
1,- 1 '
f
393'C
)O PRESSURE P S I 1
Figure 6. Heat capacity ratio vs. pressure for trifluoromethane
9.75 mole %
nitrogen in
- Experimental
_ . _Calculated by Martin-Hou equation The CC /, , Btm ratios for nitrogen are calculated using the virial equation of state:
ao!
70
I-
pV- - 1 + B ( T ) + C ( T ) + W T ) + ~
I
iI
I
RT
~
V
(9)
~
112
V3
the hlartin-Hou equation (hIartin et al., 1955, 1959) :
RT
p=V - bo
Fz ( T ) +-+(V -
F3IT) (V - bo)3
+
F4T) (V - bo)4
+-(VF -3 ( T )
(10)
where
F,(T)
=
A,
+ B,T + CZe--JCTITo
(11)
and the Bloomer-Rao equation: PRESSURE PSlb
Figure 7. methane
Heat capacity ratio vs. pressure for trifluoro-
-- Experimental
___
Calculated by Martin-Hou equation
of 0.5% appears to be conservative, while near the critical region, 1 to 2% should be a fair estimate. Equations of State
Comparisons are made between the experimental data, and the values calculated from equations of state using literature values of zero pressure heat capacities. Since most equations of state are in the form of P = j ( V , T ) )heat capacities may be found using the equation:
572 Ind.
Eng. Chem. Fundam., Vo!. 9,
No. 4 , 1970
(bRT V3
U)
+
4: ( - + - 2
~c 3 )(I+V 3 eYYJ/ vVz9
af2 +
(12)
using the zero pressure heat capacities calculated by Goff and Gratch (1950). The virial equation truncated after the third term using the e l k ' and u values of Hall and Ibele (1954) predicts higher heat capacity ratios which, nevertheless, remain within an average deviation of 0.2% of the experimental data. The Bloomer-Rao equation predicts heat capacity ratios slightly higher than those of the virial equation. The Martin-Hou equation using the constants of Smith et uZ. (1962) predicts nitrogen values as much as 1 to 27, lower than the experimental data. These comparisons for nitrogen a t 60.0' and -22.1OC are shown in Table 111. The region of data for trifluoromethane extends into the critical area, making any analysis with the virial equation fruitless.
Table 111. Comparison of Corrected Experimental Heat Capacities and Those Calculated from Equations of State Nitrogen, 6O.O0C
-~
CpHp/CpI Pressure, PSlA
300 400 500 600 7 00 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900
M-H
Exptl.
1.030 1.039 1.046 1.053 1.062 1.069 1.077 1.086 1,094 1.100 1.107 1.115 1.120 1.128 1.134 1.140
B-R
1.030 1.038 1.046 1.053 1 ,061 1.068 1.076 1.083 1.090 1.097 1.103 1.110 1.116 1.122 1.128 1.134
__-___
atm Calculated b y
1.030 1.038 1.046 1.054 1.062 1.069 1.077 1,085 1.092 1.100 1.107 1.114 1,121 1,128 1.135 1,142
Virial
Exptl.
M-H
8-R
Virial
1.031 1.039 1.047 1.055 1.063 1.071 1.079 1.086 1,094 1,101 1.109 1.116 1.123 1.130 1.136 1,143
1,045 1.067 1.092 1.105 1,118 1,138 1,155 1,173 1,191 1.209 1,223 1.238 1.255 1.271 1.283 1,296 1.315
1.042 1.058 1.074 1.091 1.107 1.123 1,140 1,156 1,172 1.187 1.202 1.217 1.231 1.245 1.258 1.270 1.282
1.047 1.064 1.082 1.100 1.119 1.137 1.156 1,175 1.193 1.212 1.230 1.248 1.266 1,283 1,299 1.315 1.330
1.046 1,063 1.081 1.098 1.116 1.134 1,152 1,170 1.188 1.205 1.223 1.240 1,257 1,273 1,289 1,304 1.319
The trifluoroniethane constants for the Martin-Hou equation have been evaluated by Hou, using his own P-TI'-T data (Hou arid JIartiii, 1959). The com1)arison between the heat capatit)- ratios calculated from this equation using the zero pressure heat capacities of Hou and Martin aiid these esperimental data is show1 i i i Figure 8. The experimental peak heights are lovier aiid left of the calculated values. The top of the peak occurs a t the critical volume according to the van der K a a l s equation. The 131oomer-Rno equation i. fitted to the data of Hou aiid Martin (1959) using the technique described by Bloomer and Rao (1952) with one esception. The term representing the second virial coefficient B,RT - -4, - CJT2 is fitted to the term representiiig the second virial coefficient in the Martin-Hou equation, .I2 BJ' C2e-"T!To, with a masimum error of 0.1%. The best fit of this equation with the data of Hou and J I a r t i n (1959) gives a standard deviation in pressure of 0.19 compared with 0.04 for the Martin-Hou equation. The heat capacit,ies calculated by the BloomerRao equation are sometimes 10% higher than those calculated by the lIartin-Hou equation or those obtained from the esperimental data.
+
+
Mixing Rules
The ability to predict misture behavior with the MartinHou equation is limited because of the empirical nature of the equation. Smith et al. (1962) used the rule of Lorentz (1881) for the interaction constants -1 and C, the geometric rule for B and T,,and the arithmetic rule for b, to estimate K l uin the second-order mising rule equation
I n general, mising rules for K 1 2may be put in the form
K 1 2=
(K113'
+
rules using Equation 13 and 14 where N = - 1, 0, 1/3, and 1 for A , B , and C are tried. The data taken in this paper are above the critical region, so the C t'erni has little effect oil the calculations. Smith's suggest,ion of the Loreiitz conihination is, therefore, chosen. i2' = 1 appears to offer the best combinatioii for -1 aiid 3. This indicates a strong preference for the trifluoromethane constants that are, in general, an order of magnitude greater t'han those for nitrogen. Mising rules with progren-kely greater S ' s favor the higher numbers.. Nixing rules u5iiig S = 2 are then tried in combination with the rules mentioned previously. Using = 1 for --I, -2' = 2 for B, and S = 113 for C yields a set of mising rules that are approsimately the desired d u e s to the left of the heat capacity peak near the critical region. They predict the peak at the correct pressure, but give coildensation a t the peak rather than a simple maximum as shown in Figure 7 . Conclusions
The heat eschaiige method provides a relatively quick aiid easy m y of determining the effect of pressure on gas and gas mixture heat capacities. Precision is estimated a t 0.5% for nitrogen away from the critical region aiid 1.5y0 for trifluoromethane close to t'he critical region. The iiitrogen heat exchange data agree to an over-all accuracy of 0.2% with the heat capacit'y ratios calculated by bot'h the virial equatioii of state and the Bloomer-Rao equations of state using the zero pressure heat capacities of Goff and Gratch. The MartinHou equation gives nitrogen heat capacity ratios 0.67, lower than the experimental values and trifluoromethaiie heat capacity ratios to within 2% iii the region just below the critical density and within 10% in the vicinity of the critical de 11sit y. Binary mising rules of the type KZf = 1712K11 (KI1.V+ K 2 ? . V ) I/.\2Y11T2K12 YzZK2,where Klz = evaluated 21iN for the Martin-Hou equation and compared with iiiisture data result in having S = 1 for - l i , = 2 for B,, aiid .Y = 1/3 for Ci.Although these rules give a good representation below the critical density, they do not seem adequate for densities near or greater than the critical density.
+
K22fi)l'dy
2li.V
Nitrogen, -22.1 'C CpHp/CpI atm Calculated b y
(14)
The Halsey-Fender rule (Fender and Halsey, 1962), S = -1, the geometric rule, S = 0, Lorentz combination, A- = 1/3, and the arithmetic rule, S = 1, are all ea-er of thir Mixture heat capacity ratios using Smith'. rules predict value$ that are too lon. &\1164combination\ of the four mixing
+
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
573
Nomenclature
M
= mixture
A = Martin-Hou constant A,, = Bloomer-Rao constant B = Nartin-Hou constant Bo = Bloomer-Rao constant B ( T ) = second virial coefficient
o
= zero pressure, reference temperature
1 11 2 22
= component component = component = component
Martin-Hou constant third virial coefficient C, heat capacity, Btu/lb mole O F D ( T ) fourth virial coefficient F ( T ) = function of the Martin-Hou equation K = generalized equation of state constant AI = mass flow rate, lb moles/min N = unspecified exponent P = pressure, psia Q = heat leak, Btu/lb mole R = gas constant T = temperature, O F , “C V = volume, ft3/lb mole Y = mole fraction a = Bloomer-Rao constant b = Bloomer-Rao constant = molecular covolume, cc/g mole bo c = Bloomer-Rao constant d = Bloomer-Rao constant f = a function of C
C(T)
= = = =
GREEKLETTERS = = = = = = =
LY
y
6 A E
p
u
Bloomer-Rao constant Bloomer-Rao constant Bloomer-Rao constant finite difference potential energy of interaction Joule-Thomson coefficient intermolecular separation a t zero potential energy
SUBSCRIPTS = high pressure L P = low pressure
HP
=
1 1 interaction 2 2 interaction
literature Cited
Balaban, S. N., Ph.D. thesis, Lehigh University, Bethlehem, Pa., 1966. Bloomer, 0. T., Rao, K . N., “Thermodynamic Properties of Sitrogen,” Institute of Gas Technology, Res. Bull. 18 (1952). Fender, B. E. F., Halsey, G. D.j Chem. Phys. 36, 1881 (1962). Goff, J. A., Gratch, S., Trans. A.S.JI.E. 72, 741 (1950). Hall, N . A,, Ibele, W. E., Trans. A.S.M.E. 76, 1039-55 (19.54). Hou, 1‘. C., JIartin, J. J., A.I.Ch.E.J. 5 ( l ) , 125-9 (1959). Lorentz, H. S.,Ann. Phys. Chem. 12, 127 (1881). Alage, D. T., Jones, 11.L., Kat.z, D. L., Roebuck, J. R., Chem. Eng. Progr. Synip. Ser. 59 (44), 61-5 (1963). Martin, J. J., HOU,Y. C., A.I.Ch.E.J. 1, 142-31 (1955). Martin, J. J., Kapoor, 11. l l . j De Xevers, N., A.I.Ch.E.J. 5 , 159-60 (1959). hlasi, J. F., Trans. A.S..lI.E. 76, 1067-74 (1954). hlichels, iz., Lunbeck, 11. J., Wolkers, G. J., Physica 17, 801-16 (19.51 ).
Pariingfon, J. R., Shilling, N . G., “Specific Heat of Ga
