the separation factor. For the conditions investigated, the separation factor was a maximum when the volume ratio wak about 3.5. A minimum volume of surfactant solution was necessary to form the required emulsion; however, the separation factor was essentially constant when the ratio of volume of surfactant solution to the feed volume exceeded approximately 0.35. Surfactant concentration also affected the separation of the two components, with the maximum separation factor occurring at a surfactant concentration of 1 wt %.
Literature Cited
Agrawal, J. P., Sourirajan, S., Ind. Eng. Chem., 61, 62 (1969).
ii; G: G:;f ~ ~ & ~ . & ~ ~ $ Develop., & ~ ~ ~lo, ~213; (1971b). 8. Li, N.N., U.S. Patent 3,410,794 (November 12, 1968).
hfcAuliffe,C., J . Phys. Chem., 7% 1267 (1966). Shah, N. D., M S thesis, University of North Dakota, Grand Forks, N.D., 1970. Wallace, R. &I.,Ind. Eng. Chem. Process Des. Develop., 6 , 423 (1967). RECEIVED for review April, 27, 1971 ACCEPTED October 22, 1971
Dichlorodifluoromethane-Specific Critical Region
Heat at
Arvind S. Kenkare' Department of Engineering Science, Oxford Ilniversity, Oxford, England
The specific heat at constant pressure for dichlorodifluoromethane in the thermodynamic critical region i s evaluated from a Martin-Hou type equation of state available for that substance. No experimental data are obtainable for checking these values in the critical region, but peaks in the predicted values of the specific heat, similar to those observed experimehtally for carbon dioxide, are obtained at the critical point and at the transposed critical temperature for pressures away from the critical.
T h e well-known enhancement in heat transfer in natural convection in t h e thermodynamic critical region first observed by Schmidt (1951) has been attributed to the large increases of specific heat and the compressibility in t h a t region. T h e phenomenon has practical applications in engineering, typical instances being the cooling of gas turbine blades, boiling water nuclear reactor heat transfer, and space technology. It is essential, therefore, t h a t the specific heat of the nearcritical fluid be known before any attempt is made to correlate experimental data b y an adequate heat-transfer theory. However, no experimental data on specific heat were available for dichlorodifluoromethane ( D C D F N ) (Freon-12, D u Pont) which was the heat-transfer agent used by the author in a separate research investigation (Kenkare, 1967). Practical problems encountered in accurately carrying out enthalpy measurements in the critical region have inhibited the experimental determination of the specific heats of fluids in that region. The research work described (Kenkare, 1967) derives for D C D F M the specific heat in the critical region by the use of well-known thermodynamic relationships. An equation of state of the Martin-Hou type (Martin and Hou, 1955) has been successfully correlated in the critical region for this substance b y McHarness e t al. (1955) using the P-V-T d a t a available in that region. This paper describes the method b y which the specific heat C, is derived from the P-V-T equation of state with well-known thermodynamic relationships. Although no direct experimental values of the specific heat Present address, The Hatfield Polytechnic, Hatfield, Herts., England.
are available to confirm the validity of these predictions, the derivation gives large peaks in specific heat values in the critical region, which are similar to those experimentally observed for carbon dioxide (Koppel and Smith, 1960). Calculation of Speciflc Heat C,
T o evaluate the specific heat of D C D F M in the thermodynamic critical region on either side of the critical point, the classical thermodynamic relationships between the specific heats are used, in terms of pressure, volume, and temperature (Zemansky, 1957). Thus,
-(g) (%)*
=
The difference in the two specific heats is given by (3) which reduces to R for the ideal gas equation PV = RT. But as the equation of state for D C D F M in the critical region is quite complex, i t is necessary to go back to Equations 1-3 for evaluating C, at any temperature and pressure in this region. It is possible to solve the above equations, as the P-V-T relationship is available in a n equation of state form for D C D F M . Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 1 , 1972
63
Integrating Equation 2 ,
(4) where Cv0 is the constant volume specific heat under ideal gas or zero pressure conditions, and V ois the specific volume under the same conditions. It is obvious that Vo at zero pressure is infinite. Combining Equations 3 and 4, C,
C,
=
Cvo
+
sav
Tr$)v
Crr0- JTk2e-kT
=
[(v -
dV
CB
~
+ T ( g ) v (") dT
P
(5)
measured below the critical density and have a n average deviation of 2.2% from the 50 points (on 7 isometrics) above the critical density (hIcHarness et al., 1955), although better experimental data from Michels (1956), who in a communication to McHarness et al. (1955), gave the maximum deviation of the equation of state from his measurements as 0.6'% and a n average deviation of 0.2y0from 38 points ranging in pressure from 104 to 327 psia and in temperature from 122" to 233°F. This excellent agreement confirms the accuracy of the equation of state. By calculation of the integrals and differentials in Equation 7 , the value of C, is obtained.
+ 2(v ___ - b)2 c 3
b)
' (V
and Equation 5 then reduces to
5(A5
+ B6T +
'
(V
(9)
--J
- b)4
Specific Heat at Zero Pressure CV,for Dichlorodifluoromethane
The Martin-Hou equation of state M a r t i n and Hou, 1955) has been used by McHarness et al. (1955) to represent the P-V-T data for DCDFM on the basis of experimental data for about 114 points in the critical region and is satisfactory to 1.5 times the critical density. The derivation of this equation and the methods used to fit it to experimental data are described elsewhere (Martin and Hou, 1955). This equation of state for DCDFRf fitted by researchers is given by
RT V - b
- b)*
4 Ad ' (u - b)S
The constant J in the above equation is a factor used to convert work units (psia X ft3/lb) to heat units (Btu/lb) and is equal to 0,185053.
The evaluation of the differential is facilitated by putting
p=-
+ C3eckT)
B2T + C2e-kT + A2 + (V - b)' A3 + B3T + C3ecAT + (VA4- b) (V - b ) a
The values of the heat capacities of gaseous DCDFAI have been calculated by Masi (1952) from spectroscopic data with a n anharmonicity correction based on his experimentally determined heat capacities. The results obtained in this way are reported to be better than 0.15%. AIasi's valuer for C, h a w been converted to corresponding values for Ci , by subtracting the gas constant R, which for D C D F N is
+
A5
+
+ BET+ C6eckT ___-
(V -b)5
(8)
where the constants are:
R
=
0 088734
b
=
0 0065093886
A B = -3 409727134 BB= 1 59434848 X Cz = -56 7627671
At B3
= =
C3 =
A4 A5
=
Bs
=
C5 k
=
=
=
0,06023944654 -1 879618431 X 1 341399084 - 5 48737007 X 0 3 46883400 X loc9 - 2 54390678 X 0 007897014279
Values calculated by this equation show a n average deviation of 0.95% in presiure from all 114 points (on 23 isometrics) 64
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 1 , 1972
Figure 1. Predicted isobaric specific heat for DCDFM in critical region
0.01642 13tu/lb°F. The following empirical equation fits Masi’s converted values from - 150” to 700°F with an average deviation of 0.06%. Thus, according to XlcHarriess e t al. (1955),
Cv,
=
+ 3.32662 X T - 2.413896 X 10-7T2 + 6.72363 X 10-I1Ta Btu/lb°F
0.0080945
(10)
From Equation 10, when the pressure is zero, CV, is purely dependent on T . Curves showing C, against T for different pressures are given in Figure 1 with this method. The specific heat reaches high values in the critical region, the peak value falling off with pre9sure on either side of the critical. The temperature a t which a peak is attained is called the transposed critical temperature which varies with the pressure. I n particular, the specific heat reaches extremely high values a t the thermodynamic critical point-in t h e case of D C D F M , the predicted increase using Equation 9 being several orders of magnitude. As the specific heat has been derived thermodynamically from a n equation of state, the accuracy of the predicted value should depend ou the accuracy of the equation of state. T h e physical explanation for the sudden increase is the infinite gas expansion at coiistaiit pressure at the critical point, and the specific heat which includes the energy necessary for the expansion of the gas when working against external pressure is therefore infinitely large. Discussion
As no numerical procedures involving the computation of first and second derivatives of volume in the critical region a r e used, the computational errors are thereby greatly reduced in the calculation of the specific heat b y the method described in this paper. An experimental check of enthalpy values for D C D F M is, however, necessary near the critical point from which data the specific heat is obtained either b y differentiation of the enthalpy curve or by a finite difference method, though the former is likely to give more accurate values. D a t a obtainable from the D u Pont Co. (1956) for D C D F M illustrate the difficulty of obtaining specific heat values from the temperature-enthalpy curve. According to this data, the curve is not accurate in the critical region. T h e method was also used for predicting the specific heat values of carbon dioxide to compare these with experimental values as given b y Koppel and Smith (1960). A Martin-Hou equation of state is available for carbon dioxide (Martin, 1963) , and the comparison between theoretical and experimental values for this substance is shown in Figure 2. T h e close agreement between theory and experiment for carbon dioxide suggests that values of the specific heat for D C D F h l in the critical region predicted b y the method described in the paper may be adequate, especially in view of the scarcity of experimental data for specific heat values. T h e method could also be extended to substances whose P-V-T data are available in a Martin-Hou type equation of state.
Figure 2. Isobaric specific heat for carbon dioxide in critical region, deduced from experimental values of enthalpy. Values of specific heat predicted from Equation 9 areshown by marked points
Nomenclature
C,
= specific heat a t constant pressure, Btu/lb°F C, = specific heat a t constant volume, Btu/lb”F J = factor to (convert work units to heat units P = pressure, psia T = absolute temperature, OR V = specific volume, ft3/lb
Subscripts
c r 0
=
thermodynamic crjticai point
= reduced = value a t zero pressure or ideal gas conditior
literature Cited
Du Pozt Co., “Thermodynamic Properties of Freon-12 Refrigerant, Wilminghn, Del., 1956. Kenkare, A. S., Heat Transfer in Fluids in the Thermodynamic Critical Region,” D.Phi1 thesis, University of Oxford, oxford, England, 1967. Koppel, L. B., Smith, J. AT., J. Chem. Eng. Data, 5, (4),437-40 (1960). Martin, J. J., J . Chem. Eng. Data, 8 (3), 311-14 (1963). Martin, J. J., Hou, Y. C., AZChE J., 1, 142 (1955). Masi, J. F., J . Amer. Chem. Soc., 74,4738 (1952). McHarness, R. C. Eisemann, B. J., Martin, J. J., Refrig. Eng., 63, 32 (1955).
Michels, A., University of Amsterdam, Amsterdam, Holland, private communication, 1956. Schmidt, E. H. W., “Heat Transmission by Natural Convection a t High Centrifugal Acceleration in Water-cooled Gas Turbine Blades,” Inst. ilfech. Eng. and ASME, Proc., General Discussion on Heat Transfer, 1951, p 361. Zemansky, M. W., “Heat and Thermodynamics,” McGraw-Hill, NewYork, N.Y., 1957, p 251. RECEIVED for review June 1, 1971 ACCEPTEDDecember 8, 1971
Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 1 , 1972
65
