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Ind. Eng. Chem. Res. 1996, 35, 844-850
Possibility of Synergism in the Joule-Thomson Effect Jaime Wisniak* and Hanan Avraham Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel 84105
The possibility of synergistic effects in binary gaseous mixtures was investigated for the differential and integral adiabatic Joule-Thomson effects. The main purpose was to find the conditions for a synergistic effect in the integral Joule-Thomson effect, using the van der Waals, Redlich-Kwong, Soave, and Peng-Robinson equations of state and the variation of the excess molar enthalpy with pressure. Synergism was found to be possible in very few systems. The conditions for the presence of a synergistic effect were found using the plane of the reduced isothermal Joule-Thomson effect over all pressure ranges. The passing of a fluid through a porous plug is always accompanied by a decrease in pressure and a change in temperature. In an adiabatic process the intensity of the phenomena is measured by the derivative of the temperature in relation to the pressure at constant enthalpy, and is called the adiabatic Joule-Thomson effect µ, defined as
µ ) (∂T/∂P)H
(1)
Depending on the operating conditions the value of µ can be positive or negative. The locus of points where µ ) 0 is called the inversion curve. When the process is carried on isothermally, the derivative of the enthalpy in relation to the pressure at constant temperature is called the isothermal Joule-Thomson effect φ and is defined as
φ)
(∂H ∂P )
T
(∂V ∂T)
) -µCP ) V - T
P
(2)
The effects defined by eqs 1 and 2 are called differential Joule-Thomson effects. The integral Joule-Thomson effect is usually defined as the difference between the upstream temperature T0 and the downstream temperature T1:
∆TH ) T0 - T1
(3)
From the industrial viewpoint the integral JouleThomson effect is the most interesting one because it has units of temperature and expresses the overall cooling (or heating) effect that takes place when a fluid passes through a nozzle. The definitions given above are valid for a pure gas and for a mixture of gases. It is of great interest to know if the integral Joule-Thomson effect for a mixture can be larger than the one calculated from the linear combination of the effects of the pure constituents (synergism) and, in particular, if the integral JouleThomson effect has a minimax value where it is smaller than the lowest pure component effect or larger than the largest individual effect. In order to explore the possibility of a synergistic effect, we define a deviation function δ as follows: n
δ)
yi∆THi - ∆THM ∑ i)1
(4)
* To whom correspondence should be addressed. E-mail:
[email protected].
0888-5885/96/2635-0844$12.00/0
where ∆THi represents the integral Joule-Thomson effect for pure component i and ∆THM that of a mixture of composition yi. For most pure gases the integral Joule-Thomson effect ∆THi is negative; hence a positive value of the deviation function δ will indicate the presence of a synergistic effect. As defined, the deviation function δ cannot show the difference between a minimax effect and a simple synergistic effect. Very few authors have considered the possibility of a synergistic or a minimax effect. Gunn et al. (1966) investigated the inversion curve of pure gases and their mixtures and found that the inversion pressure of a mixture can be substantially larger than the inversion pressure of the pure components. They suggested that this phenomenon is typical of mixtures of gases whose reduced temperatures lie on different sides of the maximum of the generalized inversion curve. Gustaffson (1970) investigated theoretically the possibility of a synergistic effect using the van der Waals (VW) equation of state (EOS) to calculate the deviation function δ at low pressures and concluded that this effect is possible. He also analyzed the criteria for this effect to happen using the reduced isothermal JouleThomson effect at zero pressure. Sobanski and Kozak (1990) investigated the adiabatic Joule-Thomson effect of mixtures containing N2, NH3, SO2, and the refrigerant gases R114, R170, R12, and R22. They used the Redlich-Kwong (RK) EOS and concluded that a synergistic effect is possible and that it may be present in systems in which the critical temperatures of the components are very different. They also concluded that the specific heat has an essential influence on the possibility of a minimax value. Wisniak (1994) investigated the possibility of synergistic effects by calculating the deviation function δ with the VW, RK, Soave, and Peng-Robinson (PR) equations of state, in the low pressure range. He concluded that a synergistic effect is possible, although it occurred only in one system and was very small. Wisniak (1994) also studied the adiabatic Joule-Thomson effect in the low pressure range, where only the second virial coefficient is considered, and concluded that none of the four equations of state was capable of predicting an extremum. This paper continues the work done by Wisniak and corrects some mistakes that appear in the work of Gustaffson. Synergism in the Joule-Thomson Effect According to Gustaffson (1970) the deviation function δ can be estimated using the second virial coefficient B © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 845
expression for the isothermal Joule-Thomson coefficient:
Table 1. The Deviation Function δ at Low Pressure Expressed by Four Cubic Equations of State: Binary Systems (Wisniak, 1994)a
if δ ) K(P - P0)
KVW ) (Cp0,2 - Cp0,1)
y1y2 1 where y1 ) Cp0,m 1 + (Cp0,2/Cp0,1)1/2
(
) [( ) [(
b22 b11 Cp0,1 Cp0,2
) (
Cp0,2 2 a11 RT Cp0,1
(
1/2
- a22
∂V φ)V-T )V∂T P
( )
)]
Cp0,1 Cp0,2
1/2 2
b22 b11 KRK ) (Cp0,2 - Cp0,1) Cp0,1 Cp0,2 2.5 RT3/2
(
(aR)22
]}
Cp0,1 Cp0,2
1/2
1/2
Cp0,1 - a22 Cp0,2
1/2 2
b22 b11 Cp0,2 1/2 2 (aR)11 Cp0,1 Cp0,2 RT Cp0,1 2 Cp0,2 1/2 Cp0,1 1/2 1 (aR)11 - (aR)22 × RT Cp0,1 Cp0,2
KSoaveb ) (Cp0,2 - Cp0,1)
[
) ( )] ) {[ ] ] [ ]}
Cp0,2 a11 Cp0,1
{[
{β11[(aR)11Tr1]1/2 - β22[(aR)22Tr2]1/2}
a VW ) van der Waals; RK ) Redlich-Kwong; PR ) PengRobinson. b There is no difference between the Soave and PR equations for the deviation function.
to yield
y1y2 2 δ ) -(P - P0) T × Cp0,m
[(
Cp0,2 Cp0,1 d 1 B + B22 - 2B12 dT T 11 Cp0,1 Cp0,2
)]
(5)
The deviation function δ, as given by eq 5, has a sign opposite that reported by Gustaffson (1970) and Wisniak (1994). This difference in sign leads to the presence of a depression effect instead of a synergistic one, as reported in the above two publications. Table 1 gives the equations developed by Wisniak (1994) for the deviation function δ using the four cubic equations of state, with their correct sign. The deviation function δ can be also calculated by using the derivative of the volume: In that case the isothermal Joule-Thomson coefficient is calculated at every temperature or pressure and afterward a numerical integration is done on the isothermal JouleThomson coefficient, as indicated by
∆TH )
1 Cp0
∫PP φ dP 1
0
(6)
After integration the deviation function δ is calculated using eq 4. If we write a generalized cubic EOS as
P)
aR RT V - b V2 + ubV + Wb2
(7)
where b, u, V, W, R, and a are parameters that depend on the EOS selected, we can develop the following
RT a dR - T V - b D dT aR RT - 2 (2V + ub) 2 (V - b) D
(8)
with
D ) V2 + ubV + Wb2
(9)
Table 2 reports the numerical values of the deviation function δ, calculated using eqs 4 and 6-9, for four different equations of state, using the mixing rules suggested by the authors of the equations (Reid et al., 1977). The calculations were made under the assumption that the specific heat of all the components is ideal, and the mole fraction yi was calculated as indicated in Table 1. Critical constants, acentric factors, and specific heats were taken from Reid et al. (1977). From Table 2 it can be seen that the deviation function δ is negative for all except one of the mixtures, meaning that there is little possibility of synergism, no matter what is the choice of calculation. Similar results were reported by Wisniak (1994). It can also be seen from Table 2 that there is a possibility of synergism in systems that have close critical properties, like N2 + CO. From Table 2 it is also evident that the largest depressive effect occurs in systems containing He or H2, when compared to other systems. For example, the depressive effect is larger in the system Ar + He than in the system Ar + O2. Few experiments have been performed to determine the integral Joule-Thomson effect itself, and the possibility of a synergistic effect has not been mentioned. Koeppe (1959) calculated the integral Joule-Thomson effect of five mixtures: N2 + O2, Ar + O2, Ar + N2, N2 + H2, and N2 + He. His results showed that no synergistic effect was present and that the largest depressive effect occurred in mixtures containing He or H2. Zemlin (1971, 1972) calculated the integral JouleThomson effect of the six mixtures N2 + CO, CO2 + C2H4, CO2 + CH4, CH4 + C2H4, N2 + CO2, and C2H4 + N2 over a wide range of pressures (0.1-12 MPa) and temperatures (273-330 K). From his results it can also be seen that no synergistic effect occurred over the range of the variables studied, and that even a depressive effect was present in the mixture CO2 + C2H4. It can then be seen that theory agrees with experimental results indicating that the possibility of synergism is low. Wisniak (1994) has developed equations to calculate the adiabatic Joule-Thomson coefficient at zero pressure and used them to calculate the mole fractions for which the effect shows an extremum value, under the assumption that Cp01 ) Cp02 ) Cp0m. Table 3 shows the results of using eq 8 with the RK EOS, for calculating the adiabatic Joule-Thomson coefficient in the pressure range (0-20 MPa). It is again evident from this table that the possibility of synergism is low. Some of the mixtures approached a synergistic effect as the pressure increased (N2 + He), and in others the synergistic effect diminished as the pressure was decreased (N2 + O2). It can also be seen that the system Kr + NO shows a maximum (synergistic) effect at a pressure of 20.2 MPa.
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Table 2. The Deviation Function δ (K) for a Pressure Drop of 0.202 MPa Calculated Using the Numerical Derivative of the Volume system
T (K)
VW
RK
Soave
PR
N2 + O2 Ar + O2 N2 + CO CO2 + C2H4 Ar + He N2 + He CH4 + C2H6 Kr + NO N2 + H2
300 300 300 300 300 300 300 300 300
7.9 × 10-4 -0.169 -1.6 × 10-3 -0.106 -1.874 -0.996 -0.273 -0.941 -0.642
-5.1 × 10-3 -0.158 -2.6 × 10-3 -0.092 -1.998 -1.034 -0.550 -1.142 -0.780
-8.6 × 10-3 -0.169 -2.9 × 10-3 -0.042 -0.599 -0.097 -0.609 -1.139 -0.946
-4.6 × 10-3 -0.210 -3.1 × 10-3 -0.148 -1.218 -0.402 -1.416 -1.378 -1.084
Table 3. Effect of Pressure on Synergism in the Adiabatic Joule-Thomson Effect µ (K‚atm-1) Calculated with the RK Equation of State system
P (MPa)
µ1a
µ2a
µma
(µ1 + µ2)/2b
N2 + He
0.1 1.0 2.0 4.0 10.1 20.2 0.1 1.0 2.0 4.0 10.1 20.2 0.1 1.0 2.0 4.0 10.1 20.2
0.318 0.315 0.310 0.296 0.232 0.107 1.134 1.189 1.259 1.433 1.821 0.312 0.318 0.315 0.310 0.296 0.232 0.107
-0.077 -0.077 -0.077 -0.077 -0.078 -0.078 0.445 0.453 0.462 0.481 0.514 0.335 0.380 0.383 0.385 0.386 0.360 0.217
0.050 0.046 0.042 0.034 0.011 -0.017 0.713 0.736 0.763 0.822 0.958 0.368 0.349 0.348 0.346 0.339 0.290 0.155
0.121 0.119 0.117 0.110 0.077 0.015 0.790 0.821 0.861 0.957 1.168 0.323 0.349 0.349 0.347 0.341 0.296 0.162
Kr + NO
N 2 + O2
a Pure components 1, 2: m mixture. b Equimolar average of the pure constituent Joule-Thomson effects.
Few experiments have reported the Joule-Thomson effect of mixtures as a function of the concentration, and in some of them a synergistic effect is seen. Roebuck and Osterberg (1938) calculated the adiabatic JouleThomson effect of the mixture He + N2 over a wide range of temperature and pressure with four different mole fractions of He. No synergistic effect was seen in their results. Roebuck and Osterberg (1940) calculated the adiabatic Joule-Thomson effect of the mixture He + Ar over a wide range of temperature and pressure, with four different mole fractions of He. A synergistic effect was detected which increased with pressure until a maximum value was reached. The temperature at which this maximum effect was first seen varied with pressure; at higher pressures this effect appeared only in the low temperature range. Charnley et al. (1955) measured the isothermal Joule-Thomson coefficient of the mixtures CO2 + N2O, CO2 + C2H4, C2H4 + N2O, and N2 + N2O at zero pressure and in the temperature range 273.15-318.15 K. A synergistic effect was evident only in the C2H4 + N2O mixture. At the temperatures of 273.15 and 298.15 K there was even a maximum effect. Vortmeyer (1966) investigated the adiabatic Joule-Thomson coefficient of the mixture CH4 + H2 at zero pressure and in the temperature range 150-350 K. No synergistic effect was evident. Budenholzer et al. (1939) measured the adiabatic Joule-Thomson coefficient of the mixture CH4 + C2H4 at pressures lower than 6.9 MPa and in the temperature range 294-366 K. From their experiments it is evident that a synergistic effect is possible and that it can be maximal. The larger the pressure, the larger the synergistic effect. Strakey et al. (1974) measured the adiabatic Joule-Thomson coefficient of the mix-
Figure 1. Excess molar enthalpy for the system methane + carbon dioxide at 313.15 K.
ture Ar + CO2 in the temperature range 153.15373.15 K at pressures up to 19 MPa, and from their work it is clear that a synergistic effect is possible at zero pressure. The Isothermal Joule-Thomson Effect and the Excess Property As with every other physical property we can define an excess isothermal Joule-Thomson coefficient. We can start with volume, in this way: n
Vmix ) ∆VE +
yi V i ∑ i)1
(10)
In this case the sign of ∆VE will show if there is synergistic effect or not. Replacing the volume in eq 2 by eq 10 gives
( )
φ ) Vmix - T
∂Vmix ∂T
P
n
) ∆VE +
yiVi ∑ i)1
[ ] ∂(∆VE)
T or else
∂T
n
P
[ ]
φ ) ∆VE - T
yi ∑ i)1
-T
∂(∆VE) ∂T
P
( ) ∂Vi ∂T
(11)
P
n
+
yiφi ∑ i)1
(12)
so now we can define the excess isothermal JouleThomson coefficient as follows:
[
∆φE ) ∆VE - T
]
∂(∆VE) ∂T
P
(13)
Since few data are reported for ∆VE, it is more conve-
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 847
Figure 2. Reduced isothermal Joule-Thomson effect for nitrogen in the low reduced pressure range, calculated using the RedlichKwong equation of state.
Figure 3. Reduced isothermal Joule-Thomson effect for nitrogen in the high reduced pressure range, calculated using the RedlichKwong equation of state.
Figure 4. Variation of the reduced isothermal Joule-Thomson effect with temperature, for mixtures containing water, at 101.3 kPa (Pr ≈ 0.008).
nient to use the excess enthalpy instead of excess volume, and use eq 2 to get
∆φE )
[
]
∂(∆HE) ∂P
T
(14)
Many experiments had been done lately to determine excess enthalpies of gas mixtures. Naumowicz and Woycicki (1984, 1986, 1992) measured ∆HE of mixtures containing NH3 and Ar, N2, CH4, or C2H6 at atmospheric pressure. Smith et al. (1983, 1984) measured ∆HE of mixtures containing H2O and H2, CH4, CO, CO2, C5H12, C6H14, C7H16, or C8H18 at atmospheric pressure. Wormald and Colling (1983,1993) measured ∆HE of mixtures containing H2O and N2 or Ar at pressures up to 13.3 MPa and temperatures up to 698.2 K. Richards et al.
Figure 5. Variation of the reduced isothermal Joule-Thomson effect with temperature, at two levels of the reduced pressure.
(1981) measured ∆HE of mixtures containing H2O and N2 or C7H16 at atmospheric pressure. Wormald (1977) measured ∆HE of the mixtures of CH4 + C6H6, CH4 + C6H12, and C6H6 + C6H12 at atmospheric pressure. Barry et al. (1982) measured the property ∆HE of the mixture CH4 + CO2 at pressures up to 4.6 MPa while Zemlin (1972) measured the property ∆HE of the mixtures CO2 + C2H4, CO2 + CH4, CH4 + C2H4, N2 + CO2, and C2H4 + N2 at pressures up to 8 MPa. In the experimental range of temperatures and pressures studied, the value of ∆HE was always positive and increased with pressure. In other words, the derivative of ∆HE with pressure was positive. In Figure 1 experimental values of ∆HE for the mixture CH4 + CO2 (Barry et al., 1982) are plotted as a function of pressure for an equimolar mixture. One can see from this figure that the derivative of the excess molar enthalpy with pressure is positive, and also that this derivative is approaching zero as the pressure decreases. As can be seen from eq 2, the relation between the two differential Joule-Thomson coefficients is (-Cp). If Cp1 ) Cp2 ) Cpm, there will be a depressive effect in the adiabatic Joule-Thomson coefficient. This also means that the specific heat has an essential influence on the possibility of synergism. It is evident that the value of ∆φE (eq 14) decreases as the pressure decreases. This result should be expected because the value of an excess property approaches zero as the mixture approaches ideal gas behavior at low pressures. The Reduced Isothermal Joule-Thomson Effect Francis and Luckhurst (1962) claimed that the corresponding states theorem is capable of describing the isothermal Joule-Thomson coefficient but not the adiabatic Joule-Thomson coefficient because of the specific heat. They calculated the relation φ0/Vc from experimental data on the second virial coefficient. Gustaffson (1970) continued their work and found a generalized curve which describes much of the available experimental data on the isothermal Joule-Thomson coefficient at zero pressure. Figure 2 shows the variation of the reduced isothermal Joule-Thomson coefficient at low pressures and in the reduced temperature range 0.5-5.5, calculated using the RK eq of state for N2 gas. In Figure 3 one can see the curves of the reduced isothermal JouleThomson coefficient at higher pressures and in the same reduced temperature range. A minimum in these
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
curves at every pressure is necessary in order that for every pressure there be two points for which φ ) 0 (or µ ) 0) to describe the inversion curve. For reduced pressures lower than 1 no minimum is evident (no phase change is considered). It is also seen that the value of φ/Vc is approaching infinity as the temperature is getting lower. No difference can be seen between the two curves for reduced pressures between 0.001 and 0.01. As the pressure increases, the minimum is moving slightly right and going up. Analysis of the inversion curve shows the same result. For increasing pressures, the two reduced temperatures are getting closer until they coincide at the maximum inversion pressure. Beyond this pressure the value of φ is always positive. The values of φ/Vc increase with pressure until the curve crosses the zero axis for all temperatures and the value of φ/Vc becomes positive for all temperatures. At higher temperatures all curves look similar because the volume increases until the value of φ (eq 8) approaches the value of b. Parameter b in the general EOS depends only on Vc and Zc so the value φ/Vc approaches a constant value. The same result is true for every pressure, causing all the curves to look similar in the higher temperature range. The minimum of the curves φ/Vc is moving to the left because of the unsymmetrical shape of the inversion curve. The upper branch of the inversion curve is somewhat larger than the lower branch of the same. The value of the higher reduced temperature on the inversion curve is farther from the reduced temperature at the maximum of the inversion curve compared to the lower reduced temperature, and so the minimum is moving right at higher reduced temperatures. In order to find the minimum of the curves φ/Vc, we must first calculate the following derivative:
() ∂φ ∂T
( )
∂2V ) -T P ∂T2
(15)
From eq 20 we find that the minimum occurs when
V ) Vc ) 3b
From the VW EOS we can derive the following relation between the pressure and the temperature at the minimum value of the reduced isothermal JouleThomson coefficient:
Pr ) 4Tr - 1
φ)
( )
and using eq 23 yields the following simple expression for the isothermal Joule-Thomson coefficient for a van der Waals gas at zero pressure:
φ ) Vc
( )
R ∂V 2 ∂T (V - b) ζ
RT aR aR ≈ ) V - b V2 + ubV + Wb2 D
(
(17)
)
R R 2 (V - b)ζ (V - b)2 P R 2RT 6a ∂V + 4 ) 0 (18) 2 3 (V - b)ζ (V - b) V ∂T P
[
]( )
Using eq 16 and simplifying yields
( ){ ∂V ∂T
-
P
[
]}
2 2 3a RT + V - b ζ (V - b)3 V4
(24)
(25)
Calculating the isothermal Joule-Thomson effect from eq 8 and simplifying yields
φ)V-
(16)
2a RT - 2 2 (V - b) V
+
3b - 6b ) 3Vc 2b - 3b
From eq 23 it can be seen that there is a discontinuity in the isothermal Joule-Thomson coefficient when V ) 2b. When V goes from a value larger than 2b to a value lower than 2b, φ varies from -∞ to +∞. In order to find the conditions that will make φ/Vc continuous at zero pressure, we begin with eq 7. When Pr ≈ 0, eq 7 reduces to
dR (1 - TR dT )(V - b)D -V2 + 2bV + b2(W + u)
V ) b ( b(1 + u + W)1/2
(26)
)0
3a RT 2a RT - 4) - 3 3 3 (V - b) V (V - b) V (V - b)
E)
(20)
3V2 + 2bV(u - 1) + b2(W - u) -2(V - b)
(28)
In order to fulfill this condition (sufficient, but necessary), one of the two roots of the numerator of E should equal b so that E will have a finite value for every value of V. Using the positive root of the denominator so that φ is continuous at Pr ) 0, we get
(19)
Finally, substituting ζ for its equivalent we obtain
(27)
When (1 + u + W) g 1, there will be a discontinuity only at the positive root; when (1 + u + W) is less than 1 or equal to 0, then two discontinuity points will be present. In order that no discontinuity be present, l’Hoˆpital’s rule requires that in eq 26 the ratio E of the derivatives of the numerator and denominator be finite:
Hence eq 15 becomes
-
(23)
The roots of the denominator are
where
ζ)
3b - 2V V2 - 3V2 + 3bV )V V - 2V + 2b 2b - V
P
R R 1 V-b ) 2a V - b ζ RT (V - b)2 V2
(22)
Using the VW EOS at zero pressure to calculate the reduced isothermal Joule-Thomson coefficient gives
Using the VW EOS to calculate the minimum together with eq 8 yields
∂V ) ∂T P
(21)
(u - 1)2 - 3W + 3u ) (u + 2)2
(29)
u ) -(W + 1)
(30)
or
which satisfies the condition (1 + u + W) g 0.
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 849
From the four equations of state studied here, only the Peng-Robinson EOS satisfies eq 30. Inspection of Figure 2 shows that a discontinuity occurs not only with the VW EOS but also with the RK eq of state, and not only at zero pressure but also at pressures higher than zero. No minimum is seen at reduced pressures lower than 1, for in this pressure range the reduced isothermal Joule-Thomson coefficient approaches -∞. As mentioned before, a minimum value of φ/Vc is necessary in order for the inversion curve to have a second root in the lower temperature and pressure. This minimal point cannot be found because the inversion curve is approaching the saturated vapor state. The VW EOS predicts that a linear relation exists between the reduced pressure and the reduced temperature at the minimum point. Equation 22 shows the same results, as can be seen in Figure 3; that is, the minimum value is moving to the right. In Figure 4 we have plotted the values of φ/Vc at different values of Tr, for mixtures containing water which are very well correlated by the expression
2.88 0.56 φ ) 0.766 + Vc Tr T2
(31)
r
with a coefficient of determination R2 ) 0.963. This curve, like the one for Pr ) 0, found by Gustaffson, has no minimum but it does not go to -∞ as fast as the curve at Pr ) 0. This is also evident from an inspection of Figures 2 and 3, although the RK equation of state shows no difference between the curves for Pr ) 0.01 and 0.001. In Figure 5 the two curves are shown together. Gustaffson (1970) found that if
Tc1 > Tcm > Tc2
(32)
then it must be that
Vc1 < Vcm < Vc2
(33)
for a synergistic effect to be present at Pr ) 0. In addition, it can be seen from Figure 4 that all the curves of φ/Vc at a reduced temperature larger than the one at the minimum value have the same shape. Hence, if the mixture and pure gases have reduced temperatures that are larger than the one at the minimum, the criteria for a synergistic effect will be the same as for Pr ) 0. At reduced temperatures that are lower than the one at the minimum of the curves, the criteria will be the opposite, in order for a synergistic effect to exist. The criteria given by eqs 32 and 33 will be true for every mixture for which the reduced critical temperature of its constituents is on the right branch of the reduced isothermal Joule-Thomson curves at any given pressure. On the left branch of the curves, at any given pressure, the criteria given by eq 33 will change to
Vc1 > Vcm > Vc2
(34)
Conclusions From the above relations we can reach the following conclusions: 1. Synergistic effects in the Joule-Thomson effect are possible in mixtures that satisfy the criteria given by eqs 32 and 33. Most of the mixtures investigated did not fulfill the condition given by eq 33 and hence did not present either a synergistic or a maximal effect. These mixtures, on the other hand, fulfill the condition
given by eq 34, but this condition will lead to synergistic effects only in the high pressure range (Pr > 4) and lower temperatures. Even at these conditions, if the difference between the two reduced temperatures is too high (∆Tr > 2), then the reduced temperatures lie on different sides of the minimum. In this case a synergistic effect will be possible but not a maximal effect. Because the two conditions given by eqs 33 and 34 oppose each other, no gas mixture can fulfill them simultaneously at every pressure, and a gas will have a synergistic effect only in the low pressure range or in the higher pressure range. 2. Mixtures of gases which have substantial differences between their reduced temperatures will not have a maximal effect at any pressure. 3. Because synergistic effects decrease as the temperature decreases and change into a depressive effect, it will be difficult to find a synergistic effect in the integral Joule-Thomson effect. 4. For synergism to be present in the integral JouleThomson effect, one should look for a mixture that fulfills the conditions given by eqs 32 and 33, and that its reduced temperatures be located on the right branch of the reduced isothermal Joule-Thomson effect (Figures 2 and 3) at any given pressure. Nomenclature a ) parameter in EOS b ) parameter in EOS Bij ) second virial coefficient Cp ) specific heat at constant pressure D ) variable defined by eq 7 E ) variable defined by eq 28 K ) variable defined in Table 1 H ) enthalpy ∆HE ) excess molar enthalpy of mixing (J‚mol-1) n ) number of components P ) pressure Pr ) reduced pressure R ) universal gas constant T ) absolute temperature Tc ) critical temperature Tr ) reduced temperature ∆TH ) integrated adiabatic Joule-Thomson coefficient V ) volume Vc ) critical volume ∆VE ) excess molar volume of mixing y ) molar composition Greek Letters R ) function in the Soave and PR equations β ) function containing the acentric factor at the Soave and PR EOS δ ) deviation function defined by eq 4 µ ) adiabatic Joule-Thomson coefficient ζ ) variable defined by eq 17 φ ) isothermal Joule-Thomson coefficient ∆φE ) excess molar isothermal Joule-Thomson coefficient of mixing ω ) acentric factor Subscripts m ) mixture 0 ) zero pressure 1 ) component 1 2 ) component 2
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Received for review May 25, 1995 Revised manuscript received October 25, 1995 Accepted November 17, 1995X IE9503149
X Abstract published in Advance ACS Abstracts, January 15, 1996.
