Generalized Treatment of Self-Diffusivity for the Gaseous and Liquid

helium-3, permitted the development of a unique relationship between a)6/TR vs. pR for reduced .... To test the validity of eq 3, self-diffusivity mea...
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17

Ind. Eng. Chem. Fundam. 1983, 22, 17-26

Generalized Treatment of Self-Diffusivity for the Gaseous and Liquid States of Fluids Huen Lee and George Thodoso Northwestern University, Evanston, Illinois 6020 1

Selfdiffustvitymeasurements reported in the literature for 27 fluids have been critically reviewed and used to develop a method for predicting this transport property for all fluid conditions includin the liquid state. In this treatment, a dimensional analysis produced the selfdiffusivity parameter, 6 = fU1'2/P,1'2v~'6,which, with the exception of helium-3, permitted the development of a unique relationship between a)6/TR vs. pRfor reduced densities, pR < 1.0. For pR > 1.0, the dimensionless ratio, G = (X' - X)/(X' - 1) has been adopted for a generalized treatment with a)6/TR where X = pR/TR0,',and which becomes constant along the solid melting curve to define X' at the triple-point temperature. For pR < 1.0, the unique relationship a)6/TR = 0.77 X 10-5/pRproduces an average deviation of 0.51 % (519 points) for conditions representing the dilute and dense gaseous states. For pR > 1.0, which includes the liquid state, the involvement of G gives rise to an average deviation of 17.23% (526 points) for the fluids investigated,

Inadequate experimental self-diffusivity measurements for the liquid state of pure fluids have been highly instrumental for the lack of a development of a corresponding-states treatment for this transport property. In this regard, sufficient data appear in the literature for the dilute and dense gaseous states to assist in the formulation of an approach capable of predicting this transport property for fluids existing at these conditions. To obtain such measurements, the radioactive tracer technique employed by Robb and Drickamer (1951) and the diaphragm cell method first introduced by Northrop and Anson (1928) constituted approaches used exclusively before 1960. These methods represented essentially the only means for obtaining self-diffusivities for the dilute, dense gaseous and liquid states. Self-diffusivity measurements obtained by these methods were difficult to establish and therefore the results reported were not found to be of sufficient accuracy to assist in the formulation of a generalized approach. The recent introduction of the NMR method overcomes a number of these limitations and therefore the application of this experimental technique has received extensive utilization in the past 20 years. Consequently, self-diffusivities now become available which are considered not only reliable, but which also properly accommodate the dilute and dense gaseous states,which extend properly into the saturated and compressed liquid regions of these fluids. Of particular note is the work of Jonas et al. (1979), Harris and Trappeniers (1980), and Hartland and Lipsicas (1964), who employ the NMR approach to establish reliable self-diffusivities for a number of substances of varying complexity. The present study was undertaken, after recognition of the availability of this information, to explore the possibility of representing this transport property in a generalized manner over the complete fluid state. In the present study, kinetic theory generalizations have been applied which constitute the basic structure upon which the present analysis has been based. However, these generalizations have been extended through dimensional analysis arguments to develop a comprehensivetreatment which extends well into the liquid region. General Background The coefficient of self-diffusion represents the limiting form for the coefficient of binary diffusion. Using the results of kinetic theory for a hard-sphere model (Hirschfelder et al., 1964), the self-diffusion coefficient at moderate pressures becomes 01 96-4313f 831 1022-0017$0 1.50f 0

where s2(1i1)*( T*)represents the collision integral which depends on the force law of the molecular interaction. Lennert and Thodos (1965) applied the hard-sphere model used in the Enskog theory to calculate self-diffusivities for argon, nitrogen, and carbon dioxide at elevated pressures. Their calculated values predict this transport property with deviations of as much as 16.4% (11points) for carbon dioxide. By assuming that molecular diameters are temperature and pressure dependent, Dymond (1974) introduced corrections to the hard-sphere model of Enskog (1922) to predict self-diffusivities for liquid methane and carbon tetrachloride. However, with this approach, the predicted self-diffusivity values for argon did not provide good agreement with corresponding experimental measurements. These results indicate that such a modification to the Enskog theory inadequately accounts for the liquid-state behavior of structurally simple substances. Jonas et al. (1979), in treating their self-diffusivity measurements for liquid methylcyclohexane in the reduced density range 2.5 < pR < 3.3, pointed out that the hardsphere model must necessarily be restricted to the lower end of the density range which was examined in their experimental work. This brief literature review indicates that, although considerable effort has been successfully directed toward understanding the behavior of the dilute gaseous state for simple fluids, the corresponding dependence associated with dense gases and liquids has not yet been properly resolved. For example, the behavior of the self-diffusivity of liquids near the triple-point region has yet to be properly assessed, especially since the significant degree of coupling between rotational and translational molecular motions in this region significantly influences the magnitude of the self-diffusivity associated with real fluids.

Dimensional Analysis Self-diffusivity is strongly dependent on the state properties of temperature, pressure, and volume associated with a fluid. In addition to temperature and density, which uniquely define the state condition, dimensional analysis includes the influence of the critical properties 2, =

aPvbT~P~v,eMfRg

0 1983 American Chemical Society

(2)

18

Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983

Table I. Sources of Self-Diffusivity Measurements for Carbon Dioxide sources T,K

P ,d c m 3

-

Amdur et al. ( 1 9 5 2 ) Becker et al. ( 1 9 5 3 ) Duffield and Harris ( 1 9 7 6 ) Ember et al. ( 1 9 6 2 ) Kerr ( 1 9 7 0 ) O'Hern and Martin ( 1 9 5 5 ) Pakurar and Ferron ( 1 9 6 5 ) Robb and Drickamer ( 1 9 5 1 ) Robinson and Stewart ( 1 9 6 8 ) Schafer and Reinhard ( 1 9 6 3 ) Takahashi and Iwasaki ( 1 9 6 6 ) Timmerhaus and Drickamer ( 1 9 5 1 ) Timmerhaus and Drickamer ( 1 9 5 2 ) Wendt et al. ( 1 9 6 3 ) Winn ( 1 9 5 0 ) Winter ( 1 9 5 1 )

194.8-3 62.8 293.15 298.15-307.93 296-1680 296-1506 273.15-373.15 1103-1944 273.51-3 17.85 273.77-295.67 233.1 5-513.1 5 298.15-348.1 5 295.6-297.1 272.9-323.2 248-362 194.65-353.15 273-318

no. of points

0.00148-0.00283 0.0298-0.1 57 0.019-0.604 0.000319-0.00182 0.000356-0.001 8 2 0.000468-0.7301 0.000276-0.000486 0.0157-0.998 0.7 86-0.992 0.00105-0.00233 0.0100-0.828 0.00092-0.0618 0.792-1.14 7 0.00149-0.002 18 0.00153-0.00287 0.0017 0-0.0 0 19 8

4

11 20 16 53 42 25 66 7 12 99 16 24 9

4 4

Using mass, length, time, and temperature dimensions, it can be shown that

t

5-

which upon rearrangement of eq 2, results in the expression

4-

(3)

3-

In eq 3, 6 = i W / 2 / P , 1 h ~and / 6 z, = P,u,/RT,. Using eq 1, it can be shown that the product a>p is dependent on temperature only, and therefore, it follows that a plot of (ad6vs. TR should establish the exponent a associated with TR in eq 3 to represent the dilute gaseous state. To test the validity of eq 3, self-diffusivitymeasurements for carbon dioxide were used preliminarily to examine the pattern prevailing for the dilute and dense gaseous states. Table I reports the sources of self-diffusivities for carbon dioxide in the gaseous state. This information has been extended to include measurements available for the liquid state of this fluid. Figure 1 presents the self-diffusivity data of seven literature sources for dilute gaseous carbon dioxide as log (Dp& vs. log TR. Of particular note are the moderate temperature measurements of Amdur et al. (1952) and Winn (1950) and the high-temperature values reported by Ember et al. (1962) and Kerr (1970) which extend up to temperatures of 1680 K. This figure, which includes the other four sources of data, indicates that the resulting relationship mbe expreased linearly with a slope of unity and establishes the exponent of TR as a = 1. This figure also includes the dilute gas theory relationship calculated with the aid of eq 1and utilizes the force constants C / K = 216 K and Q = 3.876 A evaluated by Kennedy and Thodos (1961) from dilute gas viscoSity data for carbon dioxide. The fact that the slope of Figure 1is unity sugof eq 1 is temgests that the collision integral Q(*J)'(P) perature dependent through the relationship, f P ) ' ( T * ) = a / PIz.This temperature dependence for Q(lJ)'(P )is consistent with the linear behavior exhibited in Figure 1 for carbon dioxide. The disparity encountered between the self-diffusivity of carbon dioxide and that predicted by the Enskog theory for dilute gases has also been noted in the experimental work by Pakurar and Ferron (1965). To investigate the influence of density on self-diffusivity for conditions other than the dilute gaseous state, eq 3 must be rearranged and expressed as (4)

A plot of log a)6/TR vs. log pR for carbon dioxide is shown in Figure 2. Fortunately, sufficient information

+ Amur. Irvine. Moron and RDU (19521 +

x Pakuiar and Fenon 11965) 0 Sckfer and Rainhord (19631 0 Wendt. Mundy, Weinman ond Mason D

*z h

Ember, Ferron and Wohl (19621

4 Ken (1970) 11%3)

Wl"" (19501 Winter (1951)

2-

a

v

,' Carbon Dioxide (8=0.01763) T,=304.19 K P,=72.85atm M.44.01 pe=0.468g/cm3

0.6 0.5

0'3'

d4

'

'

1.b

d6 0:8'

I

I

2

3

I

I

l

4 5 6

1

1

1

,

810

TR

Figure 1. Relationship between (SpR)*6 vs. TRfor carbon dioxide developed from experimental measurements and comparison with values calculated from dilute gas theory-eq 1.

is available for this fluid to consider densities from the dilute gaseous region into the dense gaseous and liquid states. This method of correlation brings out the rather interesting fact that 2)6/TR is linearly dependent on pR up to p R = 1.0 which defines the dense gaseous region. For pR > 1.0, this linear dependence no longer holds, but instead a)6/ TR rapidly decreases with increasing values of pR. The high-density region of carbon dioxide presented in Figure 2 includes measurements from the experimental studies of OHern and Martin (1955),Robb and Drickamer (1951), Robinson and Stewart (1968), Takahashi and Iwasaki (1966), and Timmerhaus and Drickamer (1952). With the exception of the data of Robb and Drickamer (1951) and Timmerhaus and Drickamer (19521, the measurements from the remaining sources uniquely extend the dilute gaseous behavior up to pR = 1.0. In their experimental studies with carbon dioxide, Takahashi and Iwasaki (1966) found their measurements to be in disagreement with the results advanced by Robb and Drickamer (1951) and Timmerhaus and Drickamer (1952). These two experimental studies conform to the pattern predicted by the Enskog dense gas theory. The proper extension of the sharply decreasing a>6/TRvs. p R relationship in Figure 2 suggests that a different approach must be utilized for this region in order to accommodate the self-diffusivity be-

Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 19

Carbon Dioxide (S=0.01763) T,=304.19 K pC= 72.85 atm M =44.01

re=0.468 g/cm3

,001

+

Winn

(1950) (1951)

Winter

0.101

--

.. t . $T

0.06--

T

0.04-

I I I l l

I

I

1

I

I I I I I

I

1

I 1 1 1 1 1 1

I

I

I

1 I I l l 1

I

t

i

20

Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983

*Oo0

jld5 -

600

-

400 -

0.77X

200 -

PR

100 1 -60 -

-

40

-

20 -

Ds

IO 1

TR

6:-

-

4-

-

+ Hellurn-3

2-

x

Hydrogen

A

Neon

*

Argon

o Krypton

1.01

-

0.6-0.4-

o o

Xenon Nitrogen Methane

v

Ammonia

0.2 0.10:

--

-

0.06 I I

I

I I I

0.001

I

I

0.003

I

I 1 I I l

0.01

I

I

I

I 1 1 1 1 1

010

0.03

I

I

I 1 1 1 1 1 1

0.3

0.6 1.0

I

2

I

/

4

PR

Figure 3. Dependence of a ) 6 / T R on pR for a number of fluids in their dilute, dense gaseous, and liquid states. pR < 1.0 does not project beyond reduced densities, pR

>

1.0. This departure appears to be unique but specific and shows that for each fluid there i s an asymptotic approach of a)6/TRto zero as pR approaches the limiting value, +*. For carbon dioxide this value is pR* = 3.311. Ordinarily, the range of reduced densities associated within the interval 1.0 < pR < pR* includes the saturated liquid state from the critical point to the triple point, but it can also extend to densities in the dense gaseous and compressed liquid regions. A preliminary assessment of the nature of p ~ shows * this to be a value which is greater than the triple point normalized liquid density. Attempts to relate the nature of this limiting value to the viscosity dependence of dense gases and liquids follows the freevolume concept advanced by Doolittle (1951) given by the expression log I.L = A

+ B-u -u0u g

where uo is the molar volume of the liquid extrapolated to 0 K. The form of eq 6 requires that the viscosity of the liquid state approach infinity as the free volume, u - ug,

tends to zero. Hildebrand (1971) subsequently extended Doolittle’s approach to correlate fluidity and self-diffusivity while Cohen and Turnbull (1959) used theoretical arguments to relate the self-diffusivity of hard spherical molecules with the free-volume and Macedo and Litovitz (1965) extended their approach to include an activation energy needed for molecular flow to occur. Ertl and Dullien (1973b) modified the Hildebrand equation by introducing the exponent m to define self-diffusivity for the liquid state as (7)

where m > 1 and uo is the molar volume at which viscous flow ceases. Since viscosity and self-diffusivity are interrelated through the Stokes-Einstein equation, R T I D p = 6 ~ rfor , the liquid state of hard spheres, studies are reported in the literature wherein the product a)p is only temperature dependent. Attempts to isolate the combined involvement of these transport properties are presented by a number of investigators.

Ind. Eng. Chem. Fundam.,

Vol. 22, No. 1, 1983 21

Table 11. Critical Constants and Self-Diffusivity Parameters for Fluids Included in This Investigation

M

ZC

3.016 20.183 39.948 83.800 131.300

0.301 0.307 0.291 0.292 0.287

Monatomic 3.3105 44.45 150.86 209.41 289.75

1.132 26.86 48.34 54.18 57.64

0.04145 0.484 0.536 0.908 1.099

72.38 41.79 74.56 92.29 119.50

0.04604 0.02864 0.02501 0.02865 0.0 280 3

n-hydrogen nitrogen

2.016 28.013

0.305 0.289

Diatomic 33.3 126.15

12.81 33.54

0.0310 0.3141

65.0 89.2

0.01224 0.0 2 16 6

methane ethylene ethane propane n-pentane n-hexane cyclohexane benzene n -heptane n-octane

16.043 28.054 30.070 44.097 72.151 86.178 84.16 78.115 100.20 114.23

0.288 0.276 0.285 0.281 0.262 0.260 0.273 0.271 0.263 0.259

Hydrocarbons 190.555 282.40 305.4 369.8 469.6 507.4 553.2 562.09 540.2 568.83

45.39 49.70 48.2 41.9 33.3 29.3 40.2 48.34 27.0 24.54

0.160 0.218 0.203 0.217 0.237 0.233 0.273 0.302 0.232 0.232

100.0 129.0 148.0 203.0 304.0 370.0 308.05 258.7 432.0 492.0

0.01281 0.01309 0.01227 0.01225 0.01256 0.01242 0.01224 0.01241 0.01 226 0.01232

ammonia carbon dioxide carbon disulfide carbon tetrafluoride chloroform carbon tetrachloride fluorobenzene chlorobenzene bromo benzene iodobenzene

17.031 44.01 76.131 88.005 119.378 153.823 96.10 112.56 157.01 204.01

0.242 0.274 0.293 0.277 0.293 0.272 0.263 0.265 0.263 0.265

Miscellaneous 405.5 111.3 304.19 72.85 78.0 552.0 227.6 36.9 54.0 536.4 45.0 556.4 44.9 560.09 44.6 632.4 670.0 44.6 721.0 44.6

0.235 0.468 0.448 0.629 0.500 0.557 0.357 0.365 0.485 0.581

72.5 94.04 170.0 140.0 239.0 276.0 269.0 308.0 324.0 351.0

0.01102 0.01763 0.01369 0.02514 0.01550 0.01709 0.01 38 2 0.01340 0.01518 0.01618

helium-3 neon argon krypton xenon

Q

TC,K

P c , atm

s

p C , g/cm3 u c , cm3/g-mol

6 = MI/Z/~cI12vcS/6,

Dymond (1974) has adjusted the Enskog dense-gas theory for a hard-sphere model to relate the self-diffwivity with both temperature and molar volume as

O*jld'

ole-[

'%,+,

Cyclohexane (6=001224) p~=02732~/cma

Tc=553.2K

008

006:

TR

004-

313 0566 333 0602 358 0647 383 0692

003 -

Upon making adjustments to the Enskog theory, Dymond and Brawn (1977) subsequently used experimental measurements and utilized a molar volume, uo, for closed packing to define the viscosity as

002

-

P.

PR/T,O

10

Figure 4. Temperature dependent relationships between 3361 T R and p R for liquid cyclohexane and their collapse to a single dependence when related to PR/TR'.~',

In their treatment, Dymond and Brawn insisted that the molar volume for closed packing, uo, is temperature dependent. On the other hand, Chhabra et al. (1980) combined the Hildebrand relationship for viscosity with the Dullien expression for self-diffusivity to obtain the mean momentum transfer distance between molecules as

and express the self-diffusivity as

where x is a weakly temperature dependent parameter for the liquid state. Contrary to the general temperature-independent pattern exhibited by the fluids included in Figure 3 for p R < 1.0, the recent self-diffusivity measurements by Jonas et al. (1979) for liquid cyclohexane at 313,333,358, and 383 K produced isothermal relationships, specific to each in-

dividual temperature, when related as a ) 6 / T R vs. p R as shown in Figure 4. This behavior is consistent with the parameter x which was developed by Chhabra et al. (1980). In order to present a single temperature-dependent relationship for liquid cyclohexane, a ) b / T R was related to the combined variable, pR/ TRo.Ipresented in Figure 4. If the correlational approach of a ) 6 / T R vs. p R / T R o " presented in Figure 4 for cyclohexane is applied to 23 other fluids in their liquid state, relationships specific to each fluid result in the curves as shown in Figure 5. The pattern emerging from these relationships still points to the existence of a common convergence point at p R = 1.0 where a ) 6 / T R = 0.77 X loa for all fluids and a specific, but different, asymptotic approach to (pR/ TR0.')* for each fluid. Nature of the Parameter pR* The exact nature of pR* is still undefined. A preliminary consideration of these asymptotic values might indicate that pR* should be associated with the liquid density of the fluid at its triple point. However, a careful study of such a density value with a ) 6 / T R did not properly match

22

Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983

Table III. Sources of Self-Diffusivity Measurements for Fluids Included in This Investigation -~ Monatomic helium-3 Flowers and Hunt (1973); Luszczynski et al. (1962) neon Bewilogua et al. (1971); Groth and Sussner (1944); Srivastava and Srivastava (1959);Winn (1950) Amdur and Schatzki (1957); Cini-Castagnoli and Ricci (1960ii,b); Corbett and Wang (1956); argon De Paz et al. (1967); Hutchinson (1949);Mifflin and Bennett (1958); Naghizadeh and Rice (1962); Srivastava and Srivastava (1959); Winn (1950) krypton Benenson et al. (1976); Carelli et al. (1973); Carelli et al. (1976); Codastefano et al. (1978); Durbin and Kobayashi (1962); Naghizadeh and Rice (1962); Schafer and Schumann (1957); Srivastava and Srivastava (1959); Weisman and DuBro (1970); Wendt et al. (1963) xenon Amdur and Schatzki (1957); Benenson et al. (1976); Ehrlich and Carr (1970); Naghizadeh and Rice (1962) Diatomic Amdur and Beatty (1965); Harteck and Schmidt (1933); Hartland and Lipsicas (1964); Lipsicas ( 1962) Krynicki et al. (1974); Winn (1948);Winn (1950); Winter (1950)

n-hydrogen nitrogen

n-heptane n-octane

Hydrocarbons Dawson et ai. (1970); Harris and Trappeniers (1980); Mueller and Cahill(l964); Naghizadeh and Rice (1962); Oosting and Trappeniers (1971); Rugheimer and Hubbard (1963); Winn (1950) Hamann and Richtering (1970) Gaven et al. (1962); Noble and Bloom (1965) Robinson and Stewart (1968) Douglass and McCall(1958) Douglass and McCall(l958) Jonas et al. (1980) Collings and Mills (1970);Mills (1963); Hausser et al. (1966);Parkhurst and Jonas (1975); Rathbun and Babb (1961) Douglas and McCall ( 1958) Douglass and McCall (1958)

ammonia carbon dioxide carbon disulfide carbon tetrafluoride chloroform carbon tetrachloride fluorobenzene chlorobenzene bromobenzene iodobenzene

Miscellaneous McCall and Douglas (1961); O'Reilly et al. (1973); Paul and Watson (1966) see Table I Woolf (1981) Rugheimer and Hubbard (1963) Sandhu (1975) Collings and Mills (1970); McCool and Woolf (1972); Rathbun and Babb (1961) Ertl and Dullien (1973b) Ertl and Dullien (1973b) Ertl and Dullien (1973b) Ertl and Dullien (1973b)

methane ethylene ethane propane n-pentane n-hexane cyclohexane benzene

0

o

Neon

o Argon

o Krypton Xenon o Nitrogen Methane Ethylene Ethane * Propane A n-Pentane v n-Hexane t Cyclohexane Benzene 4 n-Heptane A n-Octane ir

0.002-

0.004

0.003

000101I

1.0

I

I

I

1

1

v Chlorobenzene D Q

1

I

I

1.5

Bramobenzene lodobenzene 1

1

I

20

I

1

1

1

2.5

I

I

I

I

1

4

1

30

I

I

3.5

I

I

I

1

I

4.0

PR

TY Figure 5 . Relationships between DDB/TRand

pR/TRD.lo

for a number of fluids in the liquid state.

the behavior of the experimental results. Instead, there appears to be a strong inference that this asymptotic value represents a reduced density that is greater than the corresponding triple point liquid density. In an attempt

to better understand this behavior, we directed our attention to the postulation proposed by Crawford et al. (1975), who noted that the melting transition can provide considerable information regarding the molecular orien-

Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 23

Table IV. Values of X* and Deviations Obtained with Eq 1 5 u,, limiting molar volume, cm3/gmol

neon

argon krypton xenon

no. of data points

solid (tp)

b, cm3/ gmol

pR*

X*

13.98 24.98 29.66 37.10

11.92 20.51 24.40 31.52

2.983 2.986 3.112 3.220

3.165 3.205 3.301 3.414

30 14 19 17

60.31 12.57 15.77 13.52

29.31 23.77 29.05

19.33 24.04 19.62 25.38

2.713 3.043 3.058 3.096

2.959 3.261 3.395 3.310

25 9

34.33 4.60

26.45 34.05 39.21 53.17 75.09 91.67 78.64 65.59 106.69 119.90

3.241 3.295 3.321 3.331 3.289 3.332 2.985 3.154 3.345 3.363

3.503 3.641 3.664 3.851 3.704 3.701 3.196 3.383 3.728 3.705

88 60 13 4 7 10 39 35 9 8

11.43 23.82 12.74 36.79 15.28 26.05 1.99 8.36 37.47 34.36

24.10 47.64

3.311 3.470

3.344 3.924

7 29

19.62 24.99

45.79

35.99

3.056

3.368

4

5.02

69.9 88.3

64.79 70.06

3.414 3.07

3.750 3.325

8 27

4.78 9.42

66.49 76.68 79.85 87.21

3.18 3.28 3.30 3.32

3.473 3.633 3.653 3.704

17 15 15 17 526

3.13 5.25 5.64 9.40 17.23

M

TtP, K

Ha

20.183 39.948 83.800 131.300

24.54 83.79 115.94 161.36

14.3 24.5 29.0

Pb EDC Monatomic 14.2 24.3

av% error

Diatomic n-hydrogen nitrogen oxygen carbon monoxide

2.016 28.013 32.00 28.01

13.947 63.14 54.363 68.14

methane ethylene ethane propane n-pentane n-hexane cyclohexane benzene n-heptane n -octane

16.043 28.054 30.070 44.097 72.151 86.178 84.16 78.115 100.20 114.23

87.6 104.0 90.34 86.05 143.43 177.83 279.704 278.683 182.54 216.342

44.01 76.131

216.55 161.3

88.005

86.35

119.378 153.823

209.65 250.55

96.10 112.56 157.01 204.01

231.95 227.85 242.45 241.88

23.97 30.3 30.3

Hydrocarbons 32.0

31.8

44.6 61.0 94.0 111.0 103.2 82.0 129.1 146.4

45.7 61.0 91.1 110.7 99.5 82.5 129.1 146.4

30.94 39.06

77.0

Miscellaneous carbon dioxide carbon disulfide carbon tetrafluoride chloroform carbon tetrachloride fluorobenzene

chlorobenzene bromobenzene iodobenzene

Hildebrand.

Papadopoulos.

28.4 49.0

84.65 94.02 98.10 105.76

Ertl and Dullien.

tation of the solid and fluid states in equilibrium with each other. For the viscosity behavior of the liquid state, Hildebrand (1971), Ertl and Dullien (1973a), and Papadopoulos (1977) utilized the free-volume concept to establish limiting molar volumes, uo, at the onset of infinite viscosity. Values of uo reported by them are presented in Table IV along with solid densities associated with the triple point state. Since the viscosities used by Hildebrand, Ertl, and Dullien and Papadopoulos were almost exclusively confined to those in the saturated liquid state, the limiting values of uo reported represent essentially the terminal state of these saturated liquid curves. A review of their uo values presented in Table IV, and more specifically when compared with the corresponding solid molar volumes, strongly infers that these solid molar volumes at the triple point constitute the basis upon which the free-volume concept should be based. The viscosity measurements of Jonas et al. (1980) for cyclohexane in the compressed liquid state provide additional information relative to the involvement of densities corresponding to the solid melting curve. Values of uo obtained from their viscosity measurements show that these uo values are not constant but vary and depend on temperature. A comprehensive treatment of the solid melting curve from a generalized point of view is beyond the scope of this investigation; however, from a cursory perspective some rather interesting generalizations can be

ti

o Argon

Triple Points

E

Nitrogen

+ Methane

2

1

1

0.4

1

1

1

1 1 '

0.6 OB 1.0

1

2

I

1

3

4

TR Figure 6. Dependence of reduced density on reduced temperature along the solid melting curve for argon, nitrogen, and methane.

formulated regarding the temperature dependence of the density for the solid state. Figure 6 presents the reduced density dependence of the solid state with reduced temperature for argon (Cheng et al., 1973; Stishov and Fedesimov, 1971; van Witzenburg and Stryland, 1968) and for nitrogen and methane (Cheng et al., 1975). With this correlation method, each of these fluids exhibits a relationship that is unique to itself and which increases nonlinearly with temperature. The functional dependence of these relationships can be expressed as where a is a constant unique to each substance. The temperature dependence of density for each of the substances in Figure 6, in the region between the triple point

Ind, Eng. Chem. Fundam., Vol. 22, No. 1, 1983

24

,

,

I

,

I

I

= 1.0, can be expressed in equation form as

,

596

- X lo5 = [0.7094G + 0.1916]2,5

06~10~ = [07094G t01916] T"

04

bs

03

TR 0,2 o Neon o Arpm

010 0.08

0 Krypton

Xenon

4 Nitrogen

0.06 0.05 004

t

x Carbon Tetrafluoride o Chloroform 9 Carbon Tetrachloride A Fiuorotanzena v Chlorobenzene D Bronwbenmne 4 lodobenzene

0.02

'

01

'

02

'

03

'

04

'

'

05

mhans

m Ethylene Efbru) Ropane A n-Psntone v n-Huane t Cyclohexane 8.nzem 4 n-Hep(ane A n-Octone

Carbon Daride

0 Carbon Dhulfide

0.03

O.OlOl* 0

.

(r

'

06

' OS

07

'

09

I

IO

G=-x*-x

x'

I Figure 7. A generalized relationship between I)b/TRand G for the dense fluid state resulting from the correlation of a number of fluids of different complexities.

and T R = 1.0, can be expressed by the approximate relationship (13) which defines the constant a = pR/TR0.lalong the solidmelting curve up to TR = 1.0. For a comprehensive treatment of the dependence of the solid-melting curve for temperatures T R > 1.0, the exponent of eq 13 increases from 0.1 and depends on the temperature. To accommodate the behavior of self-diffusivity in the region between the solid state at the triple point and the critical point, the normalized density-temperature ratio pR

= LYTRO.~

\G)\ f

-

G =

PR

.

x*- x x*-1

=-

\ (A) ~

'R

TRo'l

11Al 'L=l

cp

has been adopted, where the asterisk refers to the solidmelting curve. The ratio ( p R / TRo.l)cp is equal to unity at the critical point ( p R = 1.00 and T R = 1.00) and becomes Xcp= 1.00. The normalized density-temperature ratio expressed by eq 14 presents a new parameter that includes the basic variables needed to accommodate the dense gas, saturated liquid, and compressed liquid states existing within the region between the solid-melting curve and the critical point. Generalized Treatment of Self-Diffusivity for the Dense Fluid State The self-diffusivity behavior of the liquid state for 24 fluids has been considered by relating the dimensionless parameters D6/TR with G. The nature of these fluids, tabulated in Table IV, ranges from the simple monatomics such as neon, argon, krypton, and xenon, to the more complex hydrocarbons including the normal alkanes, cyclohexane, and benzene and a variety of chlorinated and fluorinated compounds. The combined effect of the dependence of a ) 6 / T R upon G is presented in Figure 7 for these 24 fluids and can be conveniently expressed by the single relationship presented in this figure. This relationship, which is bounded between the triple point density of the solid state at G = 0 and the critical point where G

(0 < G < 1.0) (15) TR and assumes the vilues a ) 6 / T R = 0.0161 X at G = 0 and 5 9 6 / T R = 0.77 X at G = 1.0. The application of eq 15, representing the dense fluid state, is limited to G 5 1.0. For values G I 1.0, eq 5 becomes applicable. The intercept a ) 6 / T R = 0.0161 X at G = 0 represents a value that is unique for all fluids along their solid-melting curves in the vicinity of their triple point temperature. Because of its uniqueness, this value can be visualized as representing the self-diffusion parameter for all fluids at the solid-melting state. Self-diffusivities calculated with eq 15 have been compared with corresponding experimental measurements to produce the average deviations presented in Table IV. These deviations range from 1.99% (39 points) for cyclohexane to 60.31% (30 points) for neon. The overall average deviation for all substances is 17.23% (526 points). The excessive deviation encountered with neon could be the result of quantum effects associated with this fluid because the next highest value is found to be with propane with a deviation of 36.79% (4points). The utilization of the dimensionless density-temperature ratio, G, requires the availability of the triple-point solid reduced density, pR*, More often than not, such measurements are not readily available; however, based on limited experimental measurements along the solidmelting curve, some generalizations can be drawn that permit the prediction of the solid density at the triple point. In this regard, the application of the thermodynamically rigorous Clausius-Clapeyron expression

provides a means for calculating solid densities, if the pressuretemperature behavior of the solid melting curve, the heat of fusion, and the corresponding liquid density are all known over the region of interest. Sharma (1980) presents the expression U* 3.888~~~0 (17) for the calculation of u*, the limiting molar volume of solids at their triple point by utilizing zc, the critical compressibility factor and uo, the hard core volume or the characteristic volume at absolute zero temperature. The molar volume of solids at their triple point has been found in this study to be closely related to b, the excluded molecular volume, resulting from the modified van der Waals equation, examined by Pons (1981)

+ % ) ( u - b) = RT (18) where u = ( n- l)uc/(n+ 1). Pons presents values of b for (P

a number of substances. These values have been used in this study to establish a link between solid molar volumes at the triple point, using direct experimental volumes, whenever possible, or by associating them with the molar volumes reported by Hildebrand (1971),Ertl and Dullien (1973),and Papadopoulos (1977). All these molar volumes are presented in Table IV. This table also includes values of b obtained from eq 18 that have been used to develop a relationship with molar volumes u* as shown in Figure 8. Except for chloroform and cyclohexane, the fluids in Table IV exhibit a linear relation with u*, namely u* = 1.21b (19) In the absence of measurements for the solid density at

Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 25

Table V. Survey of Self-DiffusivityMeasurements in the Critical Point Region substance year method reference C.H. 1965 NMR sDin-echo Noble and Bloom (1965) ~.. NMR spin-echo Trappeniers and Oosting( 1966) 1966 NMR spin-echo Hausser e t al. (1966) 1966 NMR spin-echo Hamann et al. (1966) 1966 Bloom (1966) NMR spin-echo 1966 De Paz (1968) tracer 1968 Hamann and Richitering (1970) NMR spin-echo 1970 Cini-Castagnoli et al. (1970) tracer 1970 NMR spin-echo Oosting and Trappeniers (1971) 1971 NMR spin-echo Tison and Hunt (1971) 1971 tracer Carelli et al. (1973) 1973 NMR spin-echo Flowers and Hunt (1973) 1973 NMR spin-echo Duffield and Harris (1976) 1976

+ Ertl and Dullien (1973) o

I

o

f

4

Hildebrand (1971) Papadopoulos (1977) Experimental

" " 50 " " 10 " 0 " " 150

b, cmyg-mole Figure 8. Linear relationship between b and u* resulting from the limiting molar volumes of the free-volume theory and experimental solid-molar volumes at the triple point.

the triple point, eq 19 can be used for a first-order approximation. It should be noted that the value of pR* for establishing the self-diffusivity in the liquid state becomes very sensitive and should be known very accurately in order to predict properly this transport property. The existence of an anomaly for thermal conductivity and viscosity in the vicinity of the critical point has been demonstrated in the past and is reasonably well documented from very careful measurements conducted in this region. However, a recent literature review for self-diffusivity measurements fails to confirm the presence of such an anomaly in this region. Table V summarizes the results of this review and includes work reported since 1965 for a variety of fluids. These measurements were almost all exclusively conducted using the NMR spin-echo method. With the exception of the work of Cini-Castagnoli et al. (1970) and Duffield and Harris (1976), the remaining 12 sources of information point out that no anomaly was noted in their measurements. Since this condition prevails, it can be concluded that the normal behavior of self-diffusivity can be established using eq 5 for p R < 1.0 and eq 15 for p R > 1.0. Both relationships at the critical point produce the self-diffusivity value 0.77 x

a, =

6

Equations 5 and 15 cover the fluid conditions which range from the dilute and dense gas into the saturated and compressed liquid states and approach as limiting values the ideal gas behavior on the one hand and the solid state associated with the melting curve on the other to span the

anomaly reported no no no no

no no

no Yes

no no no no yes

complete fluid state behavior of fluids. These two relationships only require knowledge of physical properties such as pR* and the critical constants for the prediction of self-diffusivity over the complete range of conditions from the dilute and dense gaseous state into the saturated and compressed liquid regions of fluids. No adjustable parameters are needed. Nomenclature a, b = constants for modified van der Waals equation, eq 18 a, b, c , d , e, f, g , = exponents, eq 2 A = constant, eq 8 A , B = constants, eq 6 and 9 C = constant, eq 7 G = normalized density-temperature modulus, eq 14 m = exponent, eq 3 and 7 M = molecular weight n = exponent, eq 18 P = pressure P, = critical pressure, atm r = radius of spherical molecule, Stokes-Einstein equation R = gas constant T = absolute temperature, K Ttp= triple-point temperature, K T* = normalized temperature, T / ( c / K eq ), 1 T , = critical temperature, K TR = reduced temperature, T / T , X = normalized density-temperature ratio, p R / TRo.l X* = normalized density-temperature ratio, pR/TRo.lat the solid-melting curve u = molar volume, cm3/g-mol u* = molar volume of solid state at triple point, eq 17 and 19 u, = critical volume, cm3/g-mol u1= molar volume of saturated liquid state, cm3/g-mol,eq 16 uo = molar volume when viscosity becomes infinite, cm3/g-mol uo = hard core volume at 0 K, eq 17 u, = molar volume of saturated solid state, cm3/g-mol, eq 16 z, = critical compressibility factor, P,u,/RT, Greek Letters CY = constant, eq 2 and 13 6 = self-diffusivity modulus, i W J 2 / P c 1 / 2 us/cm2 ~/6, 2J = self-diffusivity, cm2/s a)* = self-diffusivity for dilute gas theory, e 1 2Jc = self-diffusivity at the critical point, cm1/ s = maximum energy of attraction, Lennard-Jones potential E = mean momentum transfer distance between molecules, eq 10 K = Boltzmann constant Xf = latent heat of fusion, eq 16 p = viscosity ir = pressure, atm ir = constant, 3.14159; Stokes-Einstein equation p = density, g/cm3 p c = critical density, g/cm3 p R = reduced density, p / p c pR* = reduced density for solid state at triple point = collision diameter for Lennard-Jones potential, eq 1 x = characteristic parameter for the liquid state, eq 11

28

Ind. Eng. Chem.

Fundam., Vol. 22,

No. 1, 1983

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Received for review September 11, 1981 Accepted September 8, 1982