DENSE GAS DIFFUSION COEFFICIENTS FOR THE METHANE-PROPANE S Y S T E M D . E. WOESSNER, B. S. S N O W D E N , J R . , R. A. GEORGE, A N D J . C. M E L R O S E
Field Research Laboratory, Mobal Research and Development Corp., Dallas, Tex. 76221 The NMR spin-echo method of measuring self-diffusion coefficients is applied to the methane-propane system in the dense gas region. Data for the pure components and for mixtures having 0.3, 0.5, and 0.7 mole fraction of methane are reported for pressure and temperature ranges of 1765 to 2 5 1 5 p.s.i. and 77" to 196" F.. Binary diffusion coefficients are derived from these data and compared with values computed from dillute gas molecular theory extended into the dense gas region by using the Enskog correction factor.
NTERDIFFUSIOX in a single-phase binary gas mixture a t
I high pressures is a n important transport process for which
few experimental data are available. Such data have a number of eiigineeriiig applications and are needed in order to test, and develop theoretical approaches to the diffusion process. Contributions to the prohlem have been made by Slattery and Bird (1958), who developed a correlation based on concepts drawn from transport theory, and by Berry and Koeller (1960), who reported a n experimental study of the methaneethane system a t a single composition. I n this paper we present experimental results n-hich describe the interdiffusion process in the methane-propane system at pressures between 1750 and 2500 I1.s.i. and for the temperature range from 77' to 196' F. Experimental nieasurenieiits of the self-diffusion coefficients for both components, D1and D2,were obtained by the nuclear magnetic resonance spin-echo method (Carr and I'urcell, 1954). This technique provides a powerful means for investigating interdiffusion in dense gas systems, as well as in liquids. D 1 and D2 are also referred to in the literature as tracer-diffusion coefficients. I3inary diffusion coefficients, D12, were calculated from the experimental data, using the relationships (Bearman, 1961) between the various diffusion coefficients and the corresponding friction coefficients. This calculation requires a suitable mixiiig rule for the friction coefficients (Bearman, 1960) and a n appropriate thermodj-namic factor to account for the concentration dependence of the activity coefficients. As a n adjunct to the experimental study, the binary diffusion coefficient and the self-diffusion coefficients for the pure component systems were calculated. These calculations are based on dilute gas transport theory (Chapman and Cowling, 1939; Hirschfelder et al., 1954) and the assumption of the Lennard-Jones intermolecular potential. The gas density required in the molecular theory expression for the diffusion coefficients is computed from the two-constant equation of state proposed by Redlich and Kwong (1949). This equation is also used to evaluate the so-called Enskog correction factor (Chap:man and Cowling, 1939; Hirschfelder et al., 1954). This factor provides, in the form of a thermodynamic expression, a n approximate correction for the effect of finite molecular size on the frequency of binary collisions. The resulting expression for the diffusion coefficients may he applied to dense gas systems. For the pure component systems agreement with the experimental results is good. The calculation provideis, therefore, a useful correlating technique for interpreting the composition dependence of the
binary diffusion coefficient for a dense gas system. The diffusion coefficients calculated by this means are referred to by the designation DGMT, indicating dense gas molecular theory. Thus, this designation is restricted, in the present discussion, to the results of calculations in which the Enskog thermodynamic approximation is assumed, and both the Enskog factor and the fluid density are computed from the Redlich-Kwong equation of state. Experimental
Diffusion Cell. A cross-section drawing of the diffusion cell constructed of nonmagnetic materials is given in Figure 1. This cell is designed to fit between the pole caps of the Varian 12-inch electromagnet in which the pole caps are 12 inches apart. Therefore the combined width of the diffusion cell and the gradient coils attached to the cell body is slightly less than this magnet, gap (Figure 2 ) . The cell body dimensions are 23 X 33 X 18 inches. The cell is designed to contain a 1-cc. sample of fluid within the radiofrequencp Figure I ) , to permit pressure studies up to detection coil (D, 2500 p.s.i., and to allow the sample to be maintained up to temperatures of 200' F. The temperature control was provided by circulating white mineral oil through interconnecting channels in the brass cell body. Initially the coil form (C, Figure 1) \vas constructed from a molded Teflon I rod. It was found in our studies of methane that large quantities of methane are absorbed in the Teflon. As expected, the amount of absorption increased with temperature and pressure as well as fluid residence time. This absorbed methane provided 10 to 20'55 of the detectable signal when the sample was left in the cell for 8 hours. After the absorption was observed in two different Teflon coil forms, the problem was remedied by constructing the coil form of glass. Another problem initially encountered in our diffusion cell was convection of the fluid within the coil. This problem is eliminated by placing a glass wool baffle in the sample cell (0, Figure 1). The volume occupied by this glass wool is only 1 or of the total sample volume. 13ecause the average distance of the molecules from the glass wool fibers is much larger than the distance the molecule moves by diffusion during the pertinent time of the X l I R spin-echo measurement ( 2 r ) , the insertion of glass rvool has no effect on diffusion. Experimental evidence that, the glass wool does not affect our NhlR diffusion measurement is provided by comparing the diffusion coefficients at room temperature obtained with and without the glass wool. Within experiVOL.
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Figure 2.
*‘
Figure 1. A.
B. C.
D. E. F. G.
M
Cross section of diffusion cell
Brass cell body channeled for thermal control fluid Packing nut Glass coil form Copper coil BNC cable connector Electrical connection Teflon packing
H.
Teflon gasket
1. J. K. 1.
Copper packing washer Thermal control inlet
M.
Thermal control exit Sample entrance Sample exit Thermocouple well
N. 0. Sample cell with glass wool baffle
mental error these room temperature measurements are exactly the same. Sample Preparation. FLUID MIXIKG- 4 s AXALYSIS. ~ In this series of experiments, ordinary protonated methane (Matheson Chemical Co. ) and protonated propane (Phillips pure grade) are used with a specially synthesized sample of completely deuterated propane obtained from hIerck Sharp and Dohnie. Gas chromatographic analysis shows both the methane and protonated propane purity to be better than 99%. Reported analysis of the deuterated propane specifies greater than 98% isotopic purity. Fluid samples are handled in the equipment depicted schematically in Figure 2. To handle the fluids and take samples, a number of stainless steel pressure cylinders are required. Autoclave valves and +-inch flexible stainless steel capillary line connect the various components of the flow system. A Ruska piston type volumetric pump ( I , Figure 2 ) is used to make fluid transfers as well as to maintain a desired fluid pressure in the diffusion cell. The pressure is monitored by a Heise Bourdon tube pressure gage which has a precision of = t 5 p s i . A standard circulating system containing white mineral oil permits temperature control (estimated to be within f0.3’F.) of the fluid in the diffusion cell at any desired temperature up to 200°F. The temperature of the fluid within the cell is measured by means of a copper constantan thermocouple and “read-out” potentiometer whose precision is f0.5’ F. A typical fluid mixing and transfer operation to obtain a desired sample in the diffusion cell is as follows: Single-phase 780
I&EC
FUNDAMENTALS
Schematic of apparatus
A.
Diffusion cell
J.
B.
Electromagnet
K.
Cylinder for sample
C.
Magnetic field gradient coils Circulating bath for
1.
collection Cylinder for combined
D.
temperature control
E. F. G. H. 1.
Spin-echo apparatus Oscilloscope Thermocouple potentiometer
Thermccouple leads
sample
M. Cylinder for propane N. Cylinder for methane 0. Vacuum pump
Pressure gage
P.
Piston displacement volumetric pump
Q. R.
Flexible high pressure line Vacuum manometer Filter
displacements of methane and propane are made from their respective storage cylinders (AI and iV, Figure 2 ) to a mixing cylinder ( L ) by injecting mercury into the bottom of the storage cylinders with the Ruska volumetric pump. Prior calculations based on the density of the individual fluids permit the approximation of a desired fluid mixture for study. After appropriate volumes of the fluids are transferred to the mixing cylinder ( L , Figure 2 ) , the sample mixture is made single-phase by increasing the pressure within the mixing cylinder (via injection of mercury to reduce the effective hydrocarbon volume). The mixing cylinder is then repeatedly inverted to obtain a high degree of intimate contact of the confined fluids. When the pressure stabilizes after agitation, the fluid sample is assumed to be at equilibrium. The fluid mixture is considered ready for transfer to the diffusion cell when equilibrium is obtained a t a pressure several hundred pounds per square inch above the saturation pressure of the fluid mixture. Before transferring the combined mixture to the diffusion cell, a low pressure sample is displaced from the mixing cylinder, L , into an adjacent sample cylinder ( K , Figure 2 ) for later gas chromatographic analysis. The combined mixture is flashed into the evacuated diffusion cell. Homogeneity of the sample in the diffusion cell is ensured by purging approximately 3 volumes of the single-phase mixture through the cell. This procedure was proved adequate by carrying out preliminary mixing and transfer operations with mixtures of ordinary methane and propane. I n these preliminary tests as well as in all our experiments with fluid mixtures, samples for conipositional analysis were taken from the combination cylinder, L , and also, after the sample transfer had been made, from the diffusion cell, A . The conipositional analyses of the protonated fluids are
pulse
pulse
r= E exp ( -27Y 2 0
1
7=
G2
T3
D)
n u c l e a r magnetogyric ratio
90' and 180°pulses magnetic field gradient applied by gradient coils self - d i f f u s i o n coefficient
T = t i m e interval between the
G D
= =
Et
E (GtO)
I
r 0.5T
1. OT
I
I 1. 5T
2.0T
2.5T
t, i n u n i t s of T
Figure 3.
NMR spin-echo measurement of self-diffusion coefficient D
performed on a Model C-40 gas chromatograph (Instruments, Inc.) Ivith a thermistor detector. Our analyses indicate that there are less than 0.5y0 contaminants in each of the combined mixtures studied. These contaminants, mainly nitrogen and ethane, are normalized out of the compositions reported in this paper. In mixtures of protonated methane with deuterated propane, the amount of methane and contaminants is determined in the standard manner described above and the amount of deuterated propane is determined by difference. Even when composition is determined by this difference method, the accuracy of component composition is estimated to be to within f 0 . 1 mole yo. Spin-Echo N M R Measurement. Spin-echo nuclear magnetic resonance measurements of self-diffusion of the type employed in this work were first described by Carr and Purcell (1954) and later utilized by many workers, including McCall and Douglass (Douglass and McCall, 1958; RIcCall and Douglass, 1967). Thespin-echo measurement of self-diffusion utilizes the amplitude of the spin-echo following the 90' and 180' pulse sequence. The 90' pulse essentially labels the protons (hydrogen nuclei) in the methane or the deuterons of the deuterated propane, allowing the detection of the consequence of their motion on the spin-echo following the 180' pulse. If the two pulses in the pulse sequence are separated by some time r (see Figure 3 ) , a spin-echo will occur a t time 27 following the 90' pulse. The amplitude of this spin-echo is determined by relaxation processes and the combined effects of molecular diffusion and variation in magnetic field strength over the sample. We utilize the dependence of the spin-echo amplitude on the magnetic field gradients to determine the self-diffusion coefficient. Carr and Purcell (1954) derived the following expression, in Tvhich relaxation effects cancel out,
The magnetic field gradient, G, is a function of the current and the geometry of the gradient coils as described by Carr and Purcell (1954). These authors have shown that the shape of the spin-echo has the form
where quantities not previously defined are
A ( t - 27) Ji
= signal amplitude at time t = first-order Bessel function
C
=
d
= diameter of sample
constant
This shape function goes through a series of null points with time. For a given value of d, G can be evaluated by use of the above expression and the time interval between the nulls on either side of the spin-echo peak a t t = 27. I n this manner G values were determined for various currents in the gradient coils. It was found that G is directly proportional to the current in the coil. Determining this proportionality constant allows the direct evaluation of G by monitoring the current in the gradient coils. The value of the pulse interval can be measured directly by use of an electronic counter. From an error analysis on Equation 1 it was found that an E/Eo ratio approxiniately equal to 1/3 gives the least error in the diffusion coefficient resulting from random errors in reading E and EO on the oscilloscope. These random errors arise from the electronic noise in the apparatus. I n this experiment several combinations of 7 and G values were used in the diffusion measurements. The accuracy for the measured self-diffusion coefficient is il.Oyofor the proton measurements and =t5.0yofor the deuteron measurements.
27
Experimental Results
where y = nuclear magnetogyric ratio = time interval between the 90' and 180' pulses G = magnetic field gradient applied by gradient coils D = self-diffusion coefficient E = spin-echo amplitude a t time 2r when G # 0 EO= spin-echo amplitude a t t,ime 2r when G = 0
T
The self-diffusion coefficients for pure methane and for pure deuterated propane were measured a t 77', 136", and 196'F. and pressures of 1765, 2015, 2265, and 2515 p.s.i. To facilitate closer comparison of the experimental results with those reported in previous studies (Dawson et al., 1969; Robinson and Stewart, 1968), the self-diffusion coefficients for pure methane and pure protonated propane were also VOL.
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Table 1.
Self-Diffusion Coefficients for Pure Methane
A.
Comparison with
DGMT Results
D H ,Sq. Cm. per Day At 77' F. Pressure, P.S.I.
At 136' F.
At 196' F.
--
1765 2015 2265 2515
Obsd.
DGMT
Obsd.
124.8 107.3 94.2 84.4
125.7 107.1 92.9 82 .O
165.2 142.4 124.4 113.9
DGMT
Obsd.
DGMT
210 .o 180.4 159.8 141.7
164.1 141.6 124.0 110.1
204.7 177.3 156 .O 139.2
B. Comparison with Literature Results D I I ,Sq. Cm. per Day At 77' F. Obsd.
a
Dawson et al. (1969), temp.
~~
=
At 177' F.
___-
DKK a
DGhlT
Pressure, p.s.i.
Obsd.
DKK a
DGMT
74.8O and 177.2O F .
~
Table 111.
Self-Diffusion Coefficients for Pure Propane
Table II.
A. Comparison with DGRfT Results (C3D8) D2z9 S q . Cm. per Day
____ Pressure, P.S.I.
1765 2015 2265 2515
At 77O F.
At 136' F.
-_______ Obsd.
8.9 8.8 8.4 8.5
DG-MT
Obsd.
11.0 10.8 10.7 10.5
12.6 11.9 11.8 11.2
Mole Fraction CHI
__-___At 196" F.
DGMT
Obsd.
13.7 13.4 13.1 12.9
18.1 16.8 16.2 16.0
DGMT
17.2 16.7 16.3 15.9
B. Comparison with Literature Results (C3H8) 0 2 2 ,
S q . Cm. per Day
A t 77' F.
619 1233 2470 a
Obsd.
RS"
...
.. .
8.6
10.4 8.6
...
At 186' F. DGhlT
Obsd.
RSa
DGhlT
12.9 . 12.3 11.5
24.4 20.4 15.7
26.6 20.2 15.7
21.1 19.2 16.8
~~
~~
Robinson and Stewart (1968); temp.
=
~
73.0' and 186.3' F.
measured under conditions comparable to those used by previous investigators. I n Table I the experimental self-diffusion coefficients for pure methane, Dll, are compared with the calculated DGMT values and (in part B ) with the data of Dawson et al. (1969). Table I1 gives a similar comparison of D22, the self-diffusion coefficients for propane. Included in Table I1 (B) are the data of Robinson and Stewart (1968). The results of Dawson et al. (1969) for methane were determined by the NMR spin-echo technique, whereas Robinson and Stewart (1968) used the radioactive tracer, (CH3)Q4H2. The results of the present study are in good agreement with the previous work. The DGMT values also agree well with experiment, except for slight deviations in the case of propane a t the lower temperatures. Self-diffusion coefficients for methane, D1, and deuterated propane, Dz,were also measured in mixtures of these two components a t the temperature and pressure conditions indicated above. The compositions studied were approxiand 0.7 mole fraction of methane (Table 111). mately 0.3,0.5, Since the DGMT calculations lead to values of the binary diffusion coefficient, D12, rather than D1 and Dz, a direct l&EC
FUNDAMENTALS
D 2 , Sq. Cm. per Day
D I , Sq. Cm. per Day Pressure, P.S.I.
___
0.306
0.704
0.500
77" F. 56.6 49.2 43.4 40.5 At 136" F. 93.8 79.4 67.8 60.7 At 196" F. 130.2 113.6 101.9 88.9
0.306
0.514
0.704
14.1 13.4 13.6 12.9
21.3 19.4 18.9 17.6
37.8 31.2 33.0 30.5
21.8 20.8 19.8 18.4
33.5 33.0 34.0 30.3
55.8 48.4 43.8 37.5
36.0 30.3 28.2 27.2
52.4 43.6 39.2 35.7
76.0 71.5 65.1 57.1
At
1765 2015 2265 2515
20.0 19.8 19.0 18.5
30.1 28.3 26.5 25.7
1765 2015 2265 2515
31.1 29.2 28.1 26.2
53.3 45.8 42.0 38.9
1765 2015 2265 2515
50.8 45.1 40.9 38.6
87.8 74.1 64.9 58.7
a
782
Self-Diffusion Coefficients for Mixtures of Methane and Deuterated Propane
Interpolated.
comparison of the experimental data with such values is precluded. However, the D1 and D2 data can be used to compute values of DlZwhich are plotted as points in Figures 4 to 6. The corresponding values of D12 provided by the DGMT calculations are shown as curves in these figures. Calculation of Dlz
Mixing R u l e for Friction Coefficients. The diffusion coefficients, D1, Dz,and D12, are interrelated (Bearman, 1961) by expressions which involve the corresponding friction coefficients, (11, (22, and ~ I Z , Zl.rll+
~ d - 1 2=
Pk T
Di
(3%)
Figure 4.
Binary diffusion coefficients at
77" F. information from the volumetric data for methane-propane mixtures reported by Reamer et al. (1950), a different procedure is adopted. Since it may be anticipated that a12 will differ only slightly from unity, values of this parameter are derived from solution theory. Everett and Munn (1964) and Kohn (1967) have shown that the Flory-Huggins equation (Flory, 1942; Guggenheim, 1952; Huggins, 1942) gives a good representation of the dependence of the activity coefficients on composition for mixtures of normal paraffin hydrocarbons in the liquid state. The relevant parameters of the theory are the effective ratio of molecular sizes, r , the coordination number, z, and the interchange energy, w12,defined as
2515
X
w12
I 00
I
I
I 02
06
04
08
10
MOL FRACTION, M E T H A N E ---f
Figure 5.
Binary diffusion coefficients at
136" F.
Here, methane and propane are denoted by subscripts 1 and 2, respectively, x is the mole fraction, P is the molecular volume, k is Boltzmann's constant, and T is the absolute temperature. The factor a 1 2 is the required correction for the nonideality of the gas mixture. If the activity of a component of the mixture is denoted by f i (i = 1, 2 ) , cy12 is given by
For diffusion processes in the liquid phase, Bearman (1960, 1961) has suggested the mixing rule, 612
= (611622)1'2
(5 )
This provides a suitable approximation, particularly if the solution can be described as regular in the usual thermodynamic sense. Use of Equation 5 with Equations 3 yields the relationship, D12 = a 1 2 (XiDz x2D1) (6 1
+
McCall and Douglass (1967) have discussed the application of this expression to liquid-phase self-diffusion measurements. Nonidealily Correction Factor. I n any exact evaluation of the factor ~ 1 2 ,defined by Equation 4, activity data are required. Although it would be possible to obtain this
=
h (E1
+
€2
- 2E12)
(7)
Here, the energy parameters, el, c2, and t12, can be identified with the well depths of the Lennard-Jones intermolecular potential functions characterizing the three types of molecular interaction. The Flory-Huggins expression has in fact been applied to activity data for methane-propane mixtures a t low temperatures by Cutler and Morrison (1965). The best fit of the theory to these data was obtained with values of r = 1.79 and zw12/k = 52' K. If the Flory-Huggins expression is differentiated with respect to the mole fraction of the smaller component (methane), the following is obtained,
Figure 7 shows plots of this expression using the values of r and zu12 found by Cutler and Morrison. This value of zw12 is consistent with the values of el, €2, and €12 which were used in obtaining the DGRIT results given in Tables I and 11, if the coordination number is assumed t o have a value of z = 4. The results shown in Figure 7, therefore, confirm our expectation that parameter a12 will differ only slightly from unity for the thermodynamic states of interest. Application of Mixing Rule. Assuming the values of a12 given in Figure 7 , Equation 6 is used to calculate values of the binary diffusion coefficients from the data in Table 111. For each of the three temperatures studied, the resulting D12 values are plotted as points in Figures 4 to 6. The agreement with the DGblT calculations, represented by the curves VOL.
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where
t-
-
-------
--
T77. m
Here the friction coefficient, {23m, is that given by dilute gas theory, but 8 is the molecular volume of the system under the T and P conditions of interest-Le., the dense gas molecular volume. Introducing the compressibility factor, Equations 10 and 11 yield
~
2 (kT ) 2 D..23 {ij"P Y
I
I
96
0 25
Figure 7.
0 50 MOL FRACTION METHANE
I
---+
0 75
I
1 IO0
Nonideality correction factor
where P is the system pressure. Comparing Equation 12 with Equation 3c, it is evident that the Enskog factor provides a second correction factor to the dilute gss friction coefficient, the first being that given by a12. The friction coefficient according to dilute gas theory (Chapman and Cowling, 1939; Hirschfelder et al., 1954) is = ~(nJf,3kT)'12u,,2R2,3 ( i ,= ~ 1, 2 )
in Figures 4 to 6, is not as good as for the pure component systems. Although the calculated curves must be regarded as providing only an approximate description of the effect of composition, the mixing rule given by Equation 5 also represents an approximation. Further studies, both experimental and theoretical, will be required in order to improve the agreement between calculations based on molecular theory and DI2values derived by means of a mixing rule from measurements of D1 and D2. Nevertheless, the present results are felt to be in rather good qualitative agreement with respect to the temperature, pressure, and composition dependences of D12.
(13)
where M is the molecular weight and u is the molecular collision diameter. Q is a dimensionless collision integral dependent on E, the well depth of the intermolecular potential. If the potential is of the Lennard-Jones form, u is the distance a t which the potential vanishes, and R is a tabulated function (Hirschfelder el al., 1954). A convenient approximation for Q is given by (Kim and Ross, 1967)
i + (3)
Rij = 0.703 1
(1.05)
-
The combining rules applicable to the molecular parameters in Equations 13 and 14 are
Dense Gas Molecular Theory
The DGMT model of the interdiffusion process does not constitute a rigorous theory. The discussion given by Chapman and Cowling (1939) and by Hirschfelder, Curtiss, and Bird (1954) shows that the Enskog factor, Y , utilizes a thermodynamic expression to account for the effect of molecular size on collision frequency:
Y =
-
2
+ T ( d Z / a T ) , -- 1 B + T (dB/dT)
(91
Here, 2 is the compressibility factor and B is the second virial coefficient. For a real gas both of these parameters and their dependence on temperature are markedly affected not only by the repulsive force, which determines the molecular size, but also by the attractive force between molecules. Consequently, the Enskog factor must also be influenced by molecular properties other than molecular size. This criticism of the basis of the Enskog factor does not negate its utility as an approximate measure of the departures from the dilute gas theory. For the latter a vast amount of experimental confirmation exists. Hence, if the Enskog factor does not deviate from unity by more than about 20%, it may be used with the dilute gas theory to provide a heuristic calculation with which experimental data can be compared. Indicating a binary diffusion coefficient applicable to the dilute gas region by the superscript a ,we may therefore write the expression
784
I L E C
FUNDAMENTALS
Application of Redlich-Kwong Equation. Equation 9 represents an approximation, expressed in thermodynamic terms, which is intended to account for deviations from dilute gas behavior which have their origin in molecular transport effects. Consequently, the empirical character of Y is clear. Furthermore, its numerical evaluation, by means of a simplified analytical equation of state, requires no more justification than that of agreement with experimental data. The extent to which this agreement is achieved establishes the usefulness of this correlating technique. An example of such an equation of state is that proposed by Redlich and Kwong (1949). Their equation has enjoyed considerable success in a variety of thermodynamic applications (Chueh and Prausnitz, 1967; Joffe and Zudkevich, 1966). According to this equation, parameters 2 and Y are given by
Z=--
P
V-b
a
(B + b)kT3"
Here, a and b are constants which can be related to the critical volume, P,, and temperature, T,, by
b = 0.25997,
(l'ib)
For a single-component system the constant b* is identical with 6. In the case of a mixture, however, the dependence of b* on composition differs from that of b (Thorne, 1939), and hence the two parameters must be distinguished. The mixing rules for a and b which are consistent with the second virial coefficient for a gas mixture (Beatt'ie and St'ockmayer, 1942) are given by the following expressions,
a
=
allz?
b = biiz?
+
2a122122
+
azzzz2
+ 2312~1~2+ bzzxz2
(18b)
These rules have been previously used in connection with analytical equations of state (Boyd, 1950; Joffe and Zudkevich, 1966) and are used in the present work. The Lennard-Jones intermolecular potential parameters uij and eij, appearing in Equations 13 and 14, can be correlated, for single component systems conforming to the principle of corresponding states, by the empirical relations (Hirschfelder et al., 1954), E = O.77kTC (19a) $7ru3 = 0.75rc
(19b)
Using Equations 19 in conjunction with Equations 17 yields the following expressions for ai3 and bij in terms of the molecular parameters,
b,, = 0 . 7 2 6 ~ ~ ~ ~
(20b 1
The generalization of the Enskog Y factor to a mixture of hard sphere molecules has been developed by Thorne (1939). Consistency with this formulation requires that b* = Xlbl"
+ Xzbz*
(21)
where
u12
u12
Selection of Molecular Constants. I n addition to the molecular weight, the molecular constants required in the DGhlT calculations are the parameters of the Lennard-Jones intermolecular potential function, E and u. For a specified set of T , P , and z conditions, the quantities 17,03,Y, and 2 can be calculated using Equations 13 to 16, 18, and 20 to 22, if values of the molecular constants for both components of the binary mixture are assumed. Suitable Lennard-Jones parameters can be derived from the analysis of experimental measurements of dilute gas properties for pure component systems. These properties may be either thermodynamic or transport in character, and results for methane and propane are included in the studies reported by Flynn and Thodos (1962), Tee, Gotoh, and Stewart (1966), and Galloway and Sage (1967). The parameters given by the latter authors for the appropriate temperature range were adopted as initial estimates and used in the equations noted. Equation 12 was then used to obtain DGMT values of D73. I n the case of methane, the Dll values calculated on the basis of the initial estimates of e and u agree well with the experimental data, as indicated in Table I. Therefore, no further adjustments in the Lennard-Jones parameters for
Table IV.
Molecular Parameters for Interdiffusing Species
Molecule
M
u?A.
CHI CaHs
16.042 44.094 52.146
3.75
163
5.00 5 .OO
316
CDs
e/k,
K.
316
methane (e1 and u l ) were made. For propane, however, the calculated 0 2 2 values were in considerable disagreement with the observed values. Consequently, the value of €2 was adjusted to make the rat'io, e1/cZ, equal t o the ratio of the observed critical temperatures for methane and propane. This adjusted value gave (retaining the initial value of uz) fairly satisfactory agreement with experiment, as s h o m in Table 11. The values of the Lennard-Jones parameters selected in the manner described are recorded in Table IV. Although the well-depth parameters, e, are some 10 t'o 15% larger than those derived from dilute gas properties, it is believed that their use compensates for other inexact features of the DGAIT calculations leading to values of Dij. Even in the case of methane a liquid-phase transport property (the shear viscosity) displays an appreciable departure from the corresponding-states relationship determined from experimental data for the rare-gas substances (Boon e2 al., 1967). Consequently, it is not unexpected that the molecular parameters characterizing the behavior of methane and propane in the dilute gas region require adjustment when used to correlate diffusion coefficients measured under dense gas condit'ions. The molecular theory on which the DGMT calculations are based also provides a means of assessing, a t least approximately, the isotope effect which is involved in using deuterated propane rather than ordinary propane in the tno-component diffusion studies. -4s indicated in Table IV, the values of the Lennard-Jones parameters are assumed to be the same for both species of propane, and only the molecular weight is altered. The binary diffusion coefficients for the methane-protonated propane and the methane-deuterated propane systems are then compared simply by comparing the values of Mij computed from Equation 15a. This shows that the substit,ution of deuterons for protons in the propane molecule has only a slight effect on the binary diffusion coefficient. This is confirmed by the experimental results for the self-diffusion of the two species, as observed in pure component systems (Table 11). Conclusions
Nuclear magnetic resonance spin-echo experiment3 provide a method of measuring the self-diffusion coefficients of both components in a binary dense gas mixture. hleasurements are reported for approximately 0.3, 0.5, and 0.7 mole fraction of methane, as well as the pure components. Temperature and pressure conditions for these data are 77', 136', and 196'F. and 1765, 2015, 2265, and 2516 p.s.i. Assuming a geometric mean mixing rule for the friction coefficients, the measured self-diffusion coefficients for the mixtures, D1 and Dz, are used to calculate binary diffusion coefficients, D12. An approximate molecular theory calculation of D,, (i, J = 1 , 2 ) is developed. This theory uses the Enskog thermodynamic factor, Y , to correct the well-known expression applicable to the dilute gas region. Both the Y factor and the density of the gas mixture are calculated from the Redlich-Kwong equation of state. VOL.
8
NO.
4
NOVEMBER
1969
785
In the case of the pure component systems, agreement between the diffusion coefficients calculated from the approximate theory and those measured by the NMR spin-echo technique is good. For mixtures, it is less satisfactory. The theory does predict, however, a dependence of the binary diffusion coefficient on temperature, pressure, and composition which is similar to that derived from the experimental observations. Acknowledgment
The authors thank J. S. Grisham, Jr., L. Loop, and M. S u n n for their assistance. Appreciation is also expressed to the Mobil Research and Development Corp. for permission to publish this work. Nomenclature
a = attractive force constant in Redlich-Kwong equation aa3 = component of a for ij interaction A = signal amplitude a t time t b = repulsive force constant in Redlich-Kwong equation b,, = component of b for il interaction B = second virial coefficient C = constant d = sample diameter D , = self-diffusion coefficient in a mixture D,, = self-diffusion coefficient in a pure system D12 = binary diffusion coefficient E = spin-echo amplitude at time 27 f z = activity of component i G = magnetic field gradient J1 = first-order Bessel function IC = Boltzniann’s constant AIt = molecular weight of component i P = fluid pressure r = ratio of molecular sizes t = time = temperature I.’ = molecular volume 5, = mole fraction of component i Y = Enskog factor z = coordination number 2 = compressibility factor
r
GREEKLETTERS a12
y
786
= nonideality correction factor = nuclear magnetogyric ratio = intermolecular potential well depth = friction coefficient
l&EC
FUNDAMENTALS
a,j =
molecular collision diameter
= time interval between 90’ and 180’ pulses w12 = interchange energy fi = dimensionless collision integral 7
SUPERSCRIPTS AND SUBSCRIPTS 1 = methane 2 = propane 0 = zero value of magnetic field gradient value of variable applicable to dilute gas region * = = value of b applicable in Enskog Y factor c = value of property a t critical point literature Cited
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