A Critical Survey of the Major Methods for Measuring and Calculating Dilute Gas Transport Properties

A Critical Survey of the Major Methods for Measuring and Calculating Dilute Gas Transport Properties A. A. WESTENBERG Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland I. Introduction- . . . . . . . . . . . . . . . . . . . . . . II. Viscosity . . . . . . . . . . . . . . . . . . . . . . . A. Experimental Techniques . . . . . . . . . . . . . . . B. Experimeatal Data . . . . . . . . . . . . . . . . . . C.Theory.. . . . . . . . . . . . . . . . . . . . . . 111. Thermal Conductivity . . . . . . . . . . . . . . . . . . A. Experimental Techniques . . . . . . . . . . . . . . . B. Experimental Data . . . . . . . . . . . . . . . . . . C.Theory.. . . . . . . . . . . . . . . . . . . . . . IV. Concentration Diffusivity . . . . . . . . . . . . . . . . A. Experimental Techniques . . . . . . . . . . . . . . . B. Experimental Data . . . . . . . . . . . . . . . . . . C.Theory.. . . . . . . . . . . . . . . . . . . . . . Symbols ....................... References . . . . . . . . . . . . . . . . . . . . . . .

253 255 255 259 261 272 272 276 278 283 283 287 290 299 300

I. Introduction This review is concerned with those dilute gas transport phenomena which are fundamental molecular properties independent of the particular fluid dynamic environment in which they may be operating. Only the three most commonly important will be discussed, namely, shear viscosity, thermal conductivity, and ordinary concentration diffusivity. Since these will be regarded from the molecular viewpoint, concepts such as eddy diffusivity are excluded from consideration. Restriction to dilute gases means that densities are assumed low enough so that only binary collisions are of importance. Thus no high-pressure effects are included, and viscosity and thermal conductivity are assumed in253

254

A. A. WESTENBERG

dependent of pressure. T h e three basic gas transport properties underlie all heat transfer phenomena, either directly or combined with other gas properties in the form of well-known dimensionless groups (Prandtl number, Schmidt number, etc.), or in an unknown way in empirically derived heat transfer coefficients. For each transport property a discussion is given, first, of the major experimental techniques which have been used for its measurement. Where it is appropriate, emphasis is placed on the newer methods or those which show the most promise of adaptability to higher temperature, better accuracy, reactive gases, etc. Second, a summary of experimental data (particularly at high temperatures) is given for the gases which seem most likely to be of importance in some application, or which are of special intrinsic interest for some reason. Where the data seem too extensive for inclusion here, the best sources and compilations are indicated. Third, the theory necessary for the prediction of transport properties where no experimental data are available is summarized briefly, again with emphasis on the newer developments. An effort has been made throughout to present the data and theory in as compact and practical a form as possible for the use of the scientist or engineer interested in getting numerical values and applying them. I t is hoped that this has been done without sacrificing too much in the way of rigor and reliability, and without losing track of the limitations and assumptions which are involved at various points. Transport properties are so widely used, and the literature on them is so voluminous, that no attempt or claim is made for an exhaustive coverage. Such coverage of experimental data is provided by the work and publications of the Thermophysical Properties Research Center (TPRC) at Purdue University (I, 2). T h e basic transport theory is covered in great detail in the well-known treatise of Hirschfelder, Curtiss, and Bird (3) and all workers in this field are indebted to these authors. T h e present writer had occasion to prepare a much more restricted review of transport properties for flame applications ( 4 ) which was in need of updating. Most of the rest of the material, especially that on experimental methods and the newer techniques, has been available only in widely scattered journals. Thus the present review is an attempt to bring a wide variety of information together in one place. While there is unavoidably some duplication with the above sources, it is hoped that this has been minimized. Coverage of the literature is only as complete as the writer (actively engaged in certain phases of transport property research) has found it convenient and necessary to “keep up with” in his own work. No citations later than about September 1964 are included.

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255

11. Viscosity

A. EXPERIMENTAL TECHNIQUES Of the several techniques which have been used for gas viscosity measurement only two are suitable for use over wide ranges of temperature (and pressure) and are accurate enough to discuss here. A third method is the rotating cylinder device whereby the viscous drag force set up by a constantly rotating cylinder causes an angular deflection of a concentric cylinder which is simply related to the gas viscosity. While data on air, which are probably the most accurate (f0.003 yo) available, have been obtained by this method (4, it does not seem feasible to adapt it to high temperatures because of the need for bulky moving parts. T h e two most useful techniques at the present time are the capillary flow viscometer and the oscillating disk. Both are very old (> 100 years) in principle, and their modern usage embodies refinement and careful attention to detail.

1. The Capillary Flow Method T h e basis of this very simple approach is to measure the pressure drop across a length of uniform bore capillary tubing for a fixed Poiseuille flow of the test gas. One convenient form of the governing equation is that the viscosity 9 is given by

where r and L are the capillary radius and length, respectively, p is the gas density evaluated at the capillary temperature and average pressure between inlet and outlet, A P is the pressure drop, and m is the constant 4s/r) is the small slip correction to the mass flow rate. T h e factor (1 simple Poiseuille relation, the quantity s being calculable from the pressure, temperature, and an approximate value of the viscosity. T h e factor c is a small “kinetic energy” correction (an inlet effect) which is best determined empirically. Thus for a capillary of given dimensions one can, in principle, measure A P a t a series of values of m and extrapolate to m = 0 to get the true value of 7. I n practice, measurements of d P at low m are uncertain and the best results (6) make use of an empirically derived value of c. c is found to be a constant for flow rates m low enough so that laminar (Poiseuille) flow is preserved, i.e., for Reynolds numbers

+

256

A. A. WESTENBERG

below about 1600. T h e various corrections noted are small and their importance obviously depends on the accuracy sought by the experimenter. I n measurements where only errors greater than 1-2% are considered important, the above corrections are often ignored. T h e foregoing description implies that the capillary flow technique is an absolute method for viscosity determination, as indeed it is under appropriate conditions. At high temperatures especially there may be corrections necessary for expansion of the capillary tube radius, and these may introduce uncertainties in an absolute determination. Note that the radius enters to the fourth power in Eq. (1). This, and other practical difficulties, means that the capillary method is often utilized only as a relative measurement device with a calibration against some standard gas whose viscosity is assumed known. T h e data of Bearden (5) on air at room temperature have been used for this purpose. For high temperatures the data of Vasilesco (7) on N, are often employed, although it would now seem more appropriate to make use of the recommended values (see Table I) from the TPRC compilation (I) which represent the best fit to all the available data. T h e most refined applications of the capillary flow method which have been (and are being) carried out appear to be those of Ross and his associates at Brown University (6, 8). For gases where the reliability of the required PVT data for computing the density in Eq. (1) is sufficient, the accuracy claimed for these viscosity results is about f0.2%, with a precision of *0.05%. Additional comment on the subject of experimental accuracy is made later in Section I1,B.

2. The Oscillating Disk Method T h e principle of this approach is to observe the viscous damping effect of the gas on a disk suspended in it (usually between two fixed plates) and set into oscillation. While this is, as previously noted, an old idea going back to J. C. Maxwell, its modern development into a precise and accurate technique is largely due to the work of Kestin and his associates at Brown University during the last 10 years. Attainment of its present status as a precision method had to await the development of a suitably detailed theory describing the motion of the disk in the fluid. T h e equations are quite complicated and will not be reproduced here. T h e basic theory is reviewed in a paper by Kestin and Wang (9). Besides the more or less 'straightforward corrections for such things as the damping due to the torsion of the suspension wire, particular attention had to be paid to the edge correction for finite disk size. I t was the solution of the edge correction problem which removed

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

257

the main theoretical obstacle to the successful application of the technique. T h e edge correction theory of Kestin and Wang (9) applies to values of the boundary layer thickness either large or small compared to the separation of the fixed plates. This was later extended (10) to intermediate values of this ratio, but the theory still only permitted the oscillating disk to be used as a relative method requiring calibration. T h e importance of the theory was that it pointed to the logical way in which the edge correction should be incorporated into the calibration. T o use the disk for absolute viscometry the edge correction had to be derived from first principles, and this was accomplished by Newel1 (11) -at least for boundary layer thicknesses large compared to the fixed plate separation. This absolute theory was verified to a high degree of precision in measurements by Kestin and Leidenfrost (12) on air and N,, so that the oscillating disk method may now be used for both relative and absolute viscometry. Most of the precise applications of the method thus far have involved studying the effect'of high pressures (to 70 atm) on the viscosity of common gases (12), which is not of direct interest in a review of dilute gas phenomena. The method is potentially of use at high temperature, however, and results with an all-quartz instrument now being assembled at Brown University will be awaited with interest. T h e oscillating disk method is now claimed (13) to have an accuracy of something like &0.2y0, although earlier estimates (12) were considerably more optimistic. T h e precision of the measurements can be as good as *0.03 yo.

3 . Methods for Dissociated Gases The techniques described above are, of course, intended primarily for use with stable gases and their mixtures. Measurements in reactive gases, particularly simple dissociated systems consisting of atom-diatomic molecule mixtures, have become of increasing interest and technical importance, and the special techniques required in these cases deserve separate mention. Actual examples of this kind are very scarce, however. T h e only reported researches of this type have made use of the capillary flow meter. T h e original work was that of Harteck (14) on the viscosity of H-H, mixtures. This is now mainly of historic interest because the H, contained 2-3% of H,O vapor which complicated the analysis undesirably, and also because certain errors of interpretation apparently were made. Harteck's paper is significant for its pioneering nature,

258

A. A. WESTENBERG

however, and seems to be the first literature report1 of the effusion-type gauge for measuring labile atom concentrations which is now known as the Wrede-Harteck gauge. T h e analysis in Ref. (14) was subsequently repeated by Amdur (15), but this also has been superseded by the very recent work of Browning and Fox (16). These authors have redetermined the viscosity of H-H, mixtures using dry H, (-0.1 yo H,O) in a Pyrex capillary (actually a tube of 7 mm1.D.) flow device. T h e H, was dissociated to the extent of about 60% at total pressures of about 1 mm Hg by means of an R F electrodeless discharge. T h e resulting mixture then flowed through the capillary with the pressure and the atom concentration being measured at both ends. T h e atom concentrations were measured with Wrede-Harteck gauges. Conditions were such that only a small loss (a few per cent) of atoms was suffered by wall recombination during passage through the tube. T h e discharge was then turned off and the pressure drop measured with pure H, at the same total mass flow rate as before. The viscosity of the mixture at the average composition between inlet and outlet can then be shown from Eq. (1) to be related to that of pure H, by

where P, and P , are the pressures at the inlet and outlet of the capillary, and ZH is the average mole fraction of H atoms between inlet and outlet. In deriving Eq. (2) the slip factors are assumed the same for the mixture and pure H, , and the second term of Eq. (1) is neglected. T h e method is seen to be a relative one of attractive simplicity. It is, of course, not so simple in practice because of the necessity of measuring the atom concentration reliably. T h e Wrede-Harteck gauge works best for H atoms at large concentrations where the pressure differentials across the effusive leak are large. For heavier atoms the mean free path is smaller and the required leak size becomes inconveniently small. Other detection techniques, particularly electron spin resonance, may prove very useful in extending the capillary technique to other dissociated gases. Some implications of the Browning-Fox work are examined subsequently.

Harteck (14) includes an interesting (and rather forlorn) footnote to the effect that Wrede had privately informed him of his independent and prior invention of a similar 'device for atom concentration measurement-hence the name Wrede-Harteck gauge.

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259

B. EXPERIMENTAL DATA By far the most comprehensive compilation of gas viscosity data presently available is incorporated in Volume I1 of the Data Book assembled by the Thermophysical Properties Research Center (TPRC) at Purdue University (I). This is a continuing effort which periodically brings the compilation up to date. All of the published experimental data for a given gas are examined, those which seem clearly erroneous are discarded, and the rest are averaged to give a set of recommended values as a function of temperature. These are given both as tables and in terms of an empirically fit power series equation of the form

+ ulT + a,T2 + a3T3+ a4T4+ a5T5

7 ( T ) = a,,

(3)

These would appear to be generally the most reliable values for viscosity within the temperature range of the experimental data, unless some other evidence is available. T h e T P R C tables of recommended values are actually extrapolated somewhat beyond the experimental temperature range using a suitable theoretical model, discussion of which is postponed to Section II,C. There would be no point in repeating the T P R C tables here nor does space permit, but it may be convenient to give the TPRC coefficients used in.Eq. (3) for a number of common gases. This is done in Table I, with the experimental temperature ranges and the limits within which the empirical equation fits all the data. TABLE I RECOMMENDED~ COEFFICIENTS FOR COMPUTING DILUTEGAS VISCOSITIES POWER SERIES EQUATION 7 (ppoise)

+ a,T + a,TZ + u3T3i- a4T4+ aST5 (Tin

= N,,

EMPIRICALLY FiTTED

TO THE

Precision of Exptl. temp. range fit to exptl. data ("I) Gas (OK)

N, 0,

Ar COz CO

H, H,O

so, C1,

a

220-1 850 22C1700 100-1300 220-2070 250-1380 200-1400 100-680 4W1480 200-1100 300-780

+2 f 2 f 2 +2 *I i-1.5 *3 55 f 2 *I

a1

10-4

10-7

7.4582 7.2345 9.4179 5.4646 5.9427 5.8765 4.714 2.9347 0.97746 4.6919

-5.7171 -5.8131 - 9.0267 2.0920 -2.5895 -2.7335 -9.9784 0.92109 10.818 -0.3716

2.9928 3.2167 5.9161 -5.3235 0.6456 0.6558 16.1827 -0.41 666 - 13.426 0.4054

~

~

From Thermophysical Properties Research Center Data Rook ( I ) .

a,

a4

a3

10-1

~~

4.020 I 5.2239 -9.2435 56.092 5.4384 23.64 2.105 9.65 33.469 1,4329

OK)

AVAILABLE EXPERIMENTAL DATA

~~

Air

FROM T H E

10-1' -~

~~

10-14 ~~

-6.2524 0 -7.0913 0 - 14.4572 0 3 1.325 -6.0505 0 0 0 0 128.024 38.164 0 0 51.116 0 0 0

-

260

A. A. WESTENBERG

T h e TPRC recommendations give equal weight to essentially all reported data, except for some which are deemed clearly out of line with the others. Thus it is of interest to compare these averaged values with those obtained directly in the best modern applications of the oscillating disk and capillary techniques. It is also important to compare the latter two techniques with each other. Since, in the opinion of this reviewer, the most careful recent work is being done at Brown University by Kestin and co-workers with the oscillating disk and Ross and co-workers with the capillary method, samples of their data are given in Table I1 together with the TPRC values. (Most of these Brown TABLE I1 ABSOLUTE VISCOSITfES (AT 1 ATM) MEASURED BY MOSTCAREFUL RECENTAPPLICATIONS OF THE OSCILLATING DISK CAPILLARY TECHNIQUES, WITH THE TPRC RECOMMENDED VALUES

SOME COMPARISONS OF THE AND

Gas ~

N* Ar

H 2

From From From From

Temp. ("K)

TPRC value

298 223 373 298 223 373 298 223 423

177.2 141.2 209.6 225.9 177.6 267.1 89.4 71.4 112.2

Best oscillating disk result

Best capillary result

--.. -

. .-

~

~

178.0" 226.4" 89.8", 89.4d

Kestin and Leidenfrost (12). Flynn et al. (6). Barua et al. (8). Kestin and Nagashima (13).

University data are too recent to have been included in the T P R C averages.) Comparing first the two sets of direct experimental data, one notes that the agreement between them is remarkably good, which would seem to establish the viscosities of these gases within very close limits. There is a consistent small discrepancy in that the oscillating disk results are 0.3-0.5yo higher than the capillary results. T h e accuracy of both methods is claimed to be better than this, so there is clearly

SURVEY OF DILUTEGASTRANSPORT PROPERTIES

26 1

still some unexplained difficulty in one or both methods. T h e two oscillating disk values for H, at 298°K were taken with the same instrument (but different suspension systems) in 1959 and 1963. They differ by about 0.5 yo, which indicates some unknown long-term error, Gas impurities probably do not account for it. T h e T P R C values in Table I1 agree very well at room temperature, with somewhat greater discrepancies at the lower and high temperatures. There is little way of judging the reliability of the TPRC values at the highest temperatures since they are based on only one or two investigations. C. THEORY For interpolating or extrapolating beyond the temperature range of the existing experimental data on viscosity a suitable theory is necessary. Various empirical relations have been proposed for this purpose, but the only extrapolation method there is any point in discussing here is that based firmly on the Chapman-Enskog kinetic theory of dilute gases. This has been outlined in so many places that only a very brief and compact description will be given here. T h e theory is given in great detail in the monumental treatise of Hirschfelder, Curtiss, and Bird (3) and practically all other writings in this field rely heavily on this reference. T h e Chapman-Enskog theory as such applies strictly to molecules with spherically symmetric force fields which undergo elastic collisions (no internal degrees of freedom). As a practical matter, these restrictions may be considerably relaxed in many cases, and these are noted in the appropriate places. 1. Pure Nonpolar Gases

In the first approximation (the only one we shall consider) the Chapman-Enskog theory gives for the viscosity of a dilute pure gas 7=

26.69(MT)’I2 ,zgcz,z,*

(4)

where 7 is in p poise, M is molecular weight, T is the absolute temperature in OK, (T is a size parameter (“collision diameter”) of the molecule in angstroms, and 5 2 [ 2 ~ 2 ) * is one of a group of so-called collision integrals. This integral is temperature dependent and involves exlicitly the potential energy of interaction between the molecules. It (or its equivalent) has been calculated and tabulated for several assumed forms of this potential function, T h e * superscript denotes that it is a

262

A. A. WESTENBERG

“reduced” integral, i.e., an integral taken relative to its value for a simple rigid sphere model. The true potential energy of interaction between two molecules cannot be calculated from first principles (i.e., quantum mechanics) for any but a very few simple cases. For any practical use, the necessary procedure is thus to assume a functional form for the potential and fit the adjustable parameters of the function to experimental data. T h e potential will approach reality in varying degree depending on the function chosen, but for practical transport calculations this aspect is less important than that the function possess enough flexibility to allow g od data fitting. This basis for extrapolation and interpolation is a semiempirical one in which theory and experiment are mutually indispensable. There are four classical spherically symmetric potential functions which have’been used most widely in transport property theory. T h e simplest are the purely repulsive forms, the inverse power potential V ( Y ) = d/r6

(5)

and the exponential repulsive potential

v ( r ) = A exP(--rlP)

(6)

where r is the intermolecular distance and d, 6 , A, and p are adjustable parameters. Since no allowance is made in either of these for the attractive forces operating at relatively large distances r , it would be expected that they would be especially applicable at high temperatures where collisions become more penetrating. Expressions for the viscosity in terms of the collision integrals for the inverse power potential are available [see Hirschfelder et al. (3, p. 547)] as are those for the exponential potential (17). Extensive use has not been made of either one in the case of viscosity, however. By far the most widely used potential function is the Lennard-Jones ( 12-6) form y ( r ) = 4€[(O/Y)’2- ( C / Y ) 6 ]

(7)

where the parameter E represents the depth of the potential “well” and is a size parameter which mathematically is the value of r at which rp = 0. T h e Y-’I term is an attractive term while the r-12 represents repulsion, although the latter exponent is usually too strong a dependence on Y. Collision integrals 12‘2*2)*for use in Eq. (4) have been tabulated by Hirschfelder, Curtiss, and Bird (3, pp. 1126-1 127) as a function of the reduced temperature T* = k T / c , k being the Roltzmann 0

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

263

constant. T h e Lennard-Jones form is thus a two-parameter ( u and r / k ) fitting function as were the purely repulsive potentials. T h e final function to be mentioned here is the modified Buckingham (Exp-6) potential

which is a three-parameter function. E is the well depth as before, I, is the value of I at the energy minimum (bottom of the well), and (Y measures the steepness of the repulsive part. Since a is adjustable it can be made to represent the repulsion somewhat more realistically than the Lennard-Jones (12-6) form, although the main virtue of the modified Buckingham potential is that it has three parameters to facilitate data fitting. The data seldom justify the extra elaboration, however. Collision integrals for this potential are tabulated by Mason (18) and are also given in the book by Hirschfelder, Curtiss, and Bird (3, pp. 1164-1 171) as a function of T* for various a. As noted above, the Lennard-Jones (12-6) potential has been used much more extensively than the others for data fitting and is the only one we shall discuss further in connection with viscosity. For extrapolation to temperatures beyond the experimental range using Eq. (4) and the tabulated integrals Q(2,2) *, the most reliable potential parameters (lacking other information) are those derived from the viscosity data themselves. T h e most recent and extensive determination of parameters for the Lennard- Jones ( I 2-6) potential from experimental viscosity data are those compiled by Svehla (19). T h e selections given in Table I11 are taken from this source. Somewhat unfortunately, these parameters were not determined by fitting the T P R C recommended values, but since Svehla used most of the same original data there would probably not be much difference. Note that some of the gases in Table I11 are polar, a fact which may be ignored for the present. There are ways of estimating potential parameters from other gas properties (second virial coefficients, critical constants, etc.) which are summarized in Svehla’s report (19) and used by him to compute viscosities for many (rather exotic) gases for which the necessary direct viscosity data are lacking. This type of estimate is so uncertain that this reviewer prefers not to go into the details here. Until viscosity measurements are made, however, Svehla’s values are probably better than nothing for these more unusual gases. T h e experimental temperature range given in Table I11 over which the fitting procedure was carried out in each case is an important item to note. It is notoriously difficult to fit a pair of parameters u and e/k uniquely to a set of experimental data, and the wider the temperature

A. A. WBTENBERG

264

TABLE 111 LENNARD-

POTENTIAL PARAMETERS DETERMINED FROM EXPERIMENTAL Vlscosrr~DATA'

JONES ( 12-6)

c/k

Gas

.(A)

(OK)

Air Ar

3.71 3.54 4.30 5.95 5.39 4.18 3.63 3.76 3.69 3.94 4.03 5.35 5.95 4.22 3.36 3.34

79 93 508 323 340 350 482 149 92 195 232 412 399 316 113 345

Br8

CCI, CHClt CH,Cl CHaOH CHI

co co*

C,H, CIH' n-CIH14 CI* F P

HCl

Exptl. temp. range (OK)

190-1850 210-1380 280 -870 290-760 270-620 250-570 370-580 190-770 90-550

270-1680 290-520 400-580 390-580 290-770 90-470 270-520

Gas

.(A)

elk (OK)

Exptl. temp. range(OK)

4.21 2.83 2.64 3.62 2.55 5.16 3.66 2.90 3.49 3.80 3.83 2.82 3.47 4.11 4.05

289 60 809 301 10.2b 474 179 558 117 71 232 33 107 335 231

290-520 200-1100 410-1770 270-370 390-790 280-370 200-710 200-1280 130-1700 240-740 60-1100 190-1290 260-1100 280-550

Taken from Svehla (19).

* Quantum mechanical value. range of the data the more reliably can the parameters be established, T h e accuracy of the data is also very important in this connection. It follows, then, that potential parameters determined from accurate data taken over a wide temperature range should be more reliable for extrapolation outside that range than parameters derived from only a small range of data. Svehla's report (29) makes no distinction in this regard and generally tabulates viscosities over the range 100-5000"K, so that his high-temperature entries more than a few hundred degrees above the experimental limit indicated in Table I11 must be regarded as increasingly unreliable. One's skepticism is a direct function of one's need for accuracy, of course. There is generally no way of estimating the reliability of a viscosity determined from theory and parameters used outside their experimental range. Some idea of this is possible in a few cases, however, as pointed out by Amdur and Ross (20) in a paper which contains a useful discussion of the difficulties of such fitting and extrapolation procedures. When reliable intermolecular potentials are available from molecular beam

SURVEY OF DILUTE GAS TRANSPORT PROPERTIES

265

scattering experiments, it is possible to calculate viscosities corresponding to quite high temperatures (several thousand degrees) and yet which overlap direct measurements of viscosity at lower temperaturesgenerally about 1000°K. T h e scattering experiments are usually interpreted in terms of a simple inverse power potential as in Eq. ( 5 ) , since the small angle scattering involved is such that attractive forces are negligible. A comparison is given in Table IV of some viscosities taken TABLE IV OF VISCOSITIES CALCULATED FROM LENNARDJONES( I 2-6) POTENTIAL COMPARISON USINGVISCOSITY-DERIVED PARAMETERS W I T H THOSE CALCULATED FROM INVERSE POWERREPULSIVE POTENTIAL USINGPARAMETERS DERIVED FROM MOLECULAR BEAMSCATTERING"

Viscosity ( p poise) Temp. Gas -

He Ar

N*

a

(OK)

Calc. from Lennard-Jones (12-6) potential and viscosity parametersb

Calc. from inverse power potential and scattering parametersC ~~

~

1000 3000 5000

1000 3000 5000 lo00 3000 5000

443 (900) 1252) 525 1071) 1488) 391 (81 1) 1127)

440 1060 1630 560 1210 I860 380 850 I320

Values in parentheses are extrapolated beyond experimental temperature range. Svehla (19). Amdur and Ross (20).

from Svehla ( I 9) and those derived from the beam scattering potentials (20). T h e excellent agreement at 1000°K in all cases where both the viscosity-derived and the beam values are within their experimental range indicates that the latter are probably quite reliable. T h e agreement at the higher temperatures then becomes considerably poorer, illustrating the dangers of extrapolating the viscosity-derived Lennard- Jones values much beyond the temperature range of the original data.

2. Pure Polar Gases T h e complication introduced by polarity to the theoretical treatment of viscosity has its roots in the fact that the basic requirement of a

266

A. A. WESTENBERG

spherically symmetric potential function is no longer met. Strictly speaking only monatomic gases have truly spherically symmetric potentials, although most of the other nonpolar species dealt with u p to now may be so considered for all practical purposes. Molecules with appreciable dipole moments, however, would clearly interact with potentials which are angular dependent. T h e numerical evaluation of collision integrals is then a much more involved task. A method of bypassing this problem has been outlined by Monchick and Mason (21). By assuming that the relative orientation of two molecules does not change much during the important time of “closest approach” in a collision, it is possible to regard each such collision as being governed by a central force potential and then average over all possible fixed orientations. All are regarded as equally probable. T h e common potential used for polar gases is the Stockrnayer function

where p is the dipole moment and 5 a function of the angular orientation between the colliding molecules. T h e Stockmayer potential is seen to be just a Lennard- Jones (1 2-6) potential with an added angular-dependent term. The Monchick-Mason approach is thus to hold 5 fixed so that Y ( Y ) is a function only of I, which is a tractable collision integral problem, and then average over all angles. T h e resulting averaged integral is a function of T* and the dipole moment and may be used directly in Eq. (4)for Q(2,2)*. With the dipole moment assumed known, the fitting procedure for viscosity data is a two-parameter (cr and elk) problem as in the nonpolar case. Table V shows the potential constants determined in this way for various polar gases and compares them with the values from Table 111 which were obtained from the ordinary Lennard-Jones (12-6) potential ignoring polarity. Most of the same experimental data were used for both sets. T h e interesting fact emerges that in most cases the cr and e / k are not vastly different from the two determinations, which presumably means that the viscosity is not very sensitive to long-range dipole interactions at these ordinary temperatures. This is clear when one notes that the averaged integrals (Q(2v2)*) at T* greater than about one are nearly independent of dipole moment and are about equal to the Lennard-Jones (12-6) integrals Q ( 2 , 2 ) * . Both sets of parameters fit the experimental data to about the same precision over the available temperature range, so there is no advantage of the more complicated potential in this regard. One concludes, therefore, that for most practical purposes the usual fitting procedure with the nonpolar collision integrals

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

267

TABLE V COMPARISON OF VISCOSITY PARAMETERS FOR POLARGASESDETERMINED FOR THE AVERAGED STOCKMAYER (12-6-3) POLARPOTENTIAL AND THE LENNAREJONES (1 2-6) POTENTIAL IGNORING POLARITY

Gas 1.85 1.47 1.08 0.42 1.63 I .01 I .87 1.70

H10 NH, HCI HI

so*

CHCI, CH,CI CH,OH

Lennard- Jones ( I 2-6)“ 0 (-4 4 k (“K)

Averaged Stockmayer (12-6-3)b 0 (4 flk (OK)

2.64 2.90 3.34 4.21 4.11 5.39 4.18 3.63

2.52 3.15 3.36 4.13 4.04 5.31 3.94 3.69

809 558 345 289 335 340 350 482

775 358 328 313 347 355 414 417

Svehla (19).

* Monchick and Mason (21). may be used just about as well for polar gases. Extrapolation to high temperatures would presumably be just as reliable since increasingly penetrating collisions should be affected less by polar interactions. T h e reverse would be true for very low temperatures.

3 . Mixtures T h e kinetic theory relations for computing the viscosity of a binary mixture are given in Hirschfelder et al. (3) in the form 1

+z

7 m i 1 = __-

x + y

,--,

where x1 and x, are mole fractions, M , and M , are molecular weights, 7, and q2 are the viscosities of the pure components, v12 is given by 26.69[2M1M2T / (M, 7 1 2 (CLPOiSe)

=

D2 p2.2,* 12

12

+ M,)]

/2

268

A. A. WESTENBERG

and A:, is a ratio of collision integrals Q:2,.a’*/Q:i*1)*. T h e subscript “12” on the various quantities indicates a quantity characteristic of interaction between the two unlike components of the mixture and introduces the idea of “combining rules” for the first time. These are usually used i n the form 012

= (01

+4/29

(+)I,

=

[(4w‘/k)211’2

(12)

so that they represent mean values of the potential parameters for the means that it is evaluated at mixtures. T h e collision integral Q:i*2)* T* = T / ( E / .~T)h ~e hypothetical ~ mixture viscosity q12 is an important quantity in connection with diffusion as will be discussed later. AT, depends only very weakly-on the potential function employed and the temperature, and rarely varies more than a few per cent from the value 1.1. It is tabulated for the Lennard-Jones (12-6) potential in Hirschfelder et al. (3), for the modified Buckingham (Exp-6) potential in Mason(18), and for the averaged polar Stockmayer potential in Monchick and Mason (21). Extensive comparisons of mixture viscosities computed from Eq. (10) with‘ the experimental values have been published(22) and it is not necessary to repeat them here. It may be of interest, however, to give one such comparison with the most recent, and presumably most accurate, experimental data on the rare gas mixture He-Ar obtained by Iwasaki and Kestin (23) using the oscillating disk method. Table VI gives the results at several mixture compositions. The theoretical calculations were performed with Eq. (10) using the experimental TABLE VI OF THEORETICAL AND EXPERIMENTAL VISCOSITIES OF COMPARISON He-Ar MIXTURES AT 293°K AND 1 ATM

v m l x ( p poise)

Argon mole fraction

Calc. from Eq. (10)

Exptl. value4

1 .oo 0.80 0.63 0.37 0.14 0.00

227.08 230.40 231.84 221.69 -

-

222.75 227.07 230.95 231.61 220.27 196.04

From Iwasaki and Kestin (23).

SURVEY OF DILUTE GASTRANSPORT PROPERTIES

269

viscosities for the pure gases (23). the u12 and (c/k)lz from the pure gas force constants given in Table 111, i.e., Svehla's fitted values (19), and the combining rules in Eq. (12). T h e theoretical and experimental values agree extremely well. It will be noted that this is a case where the mixture viscosity goes through a maximum with composition. Hirschfelder et al. (24) have discussed the conditions for the occurrence of a maximum and calculated a number of specific examples. Mixtures of components having comparable viscosities but widely different molecular weights are most apt to show a maximum. Minima are also theoretically possible but have never been observed. T h e averaged Stockmayer potential treatment for polar molecules has been extended by Mason and Monchick (25) to binary mixtures and extensive comparison made with experiment. T h e results were generally satisfactory, but since the only experimental data available were for polar-nonpolar mixtures the test was not very significant as the important unlike interaction was essentially nonpolar. It is likely that the simpler nonpolar theory would work as well. T h e rigorous binary Eq. (10) is obviously quite a complex relation to apply, and the generalization to more than two components becomes almost prohibitively involved (3). Brokaw (26) has derived approximate forms of the equations which are useful, especially in multicomponent mixtures. T h e first approximation is the only one offering any appreciable simplification and has the form for a v-component mixture

which is closely related to an empirical relation proposed by Buddenberg and Wilke (27). Use of Eq. (13) usually gives ymlxwithin a few per cent of the rigorous Eq. (10). 4. Labile Atoms and Radicals

All of the foregoing treatment has implicity assumed that the two partners in a collision can interact along only one potential curve. I n the case of labile atoms or radicals possessing unpaired electron spins, however, multiple interaction potentials are possible depending on the spin orientation. When two H atoms collide, for example, their spins may be opposed so that the normal ' C attractive (binding) potential is followed, or their spins may be parallel and repel each other in the

270

A. A. WESTENBERG

4

state. T h e two possibilities have statistical weights of and $, respectively. I t has been shown (28,29)that in such a multiple interaction case all of the usual kinetic theory transport formulas are still valid providing the collision integrals used are averaged over the different interaction potentials, each one being weighted by its statistical weight or probability. T h e viscosity of atomic hydrogen has been computed a number of times using various theoretical potentials. Clifton (30) employed a suggestion of Hirschfelder and Eliason (31) to derive an effective rigid sphere collision diameter from an H-H inverse power potential. Weissman and Mason (32) and Vanderslice et al. (33) have computed viscosities more rigorously using quite precisely known potential functions fitted to combined spectroscopic and theoretical data. Both treatments are essentially equivalent and based on the same potential data, although Weissman and Mason (32) cover the low-temperature range (up to 1000°K) and Vanderslice et al. (33) cover 1000-5000°K. Since all ionization and electronic excitation is neglected, however, the viscosities in the latter paper above about 5000°K are unrealistic in the sense that an actual sample of H would have a different viscosity, but the tabulated values should be quite reliable for pure ground state H at very high temperatures (even if only hypothetically obtainable). This is a subtlety which occurs generally in the theory of transport properties in reacting, or excited gases, i.e., any situation where a gas can change its essential composition-usually as a function of temperature. One really deals with a mixture in this case, and one of shifting composition. But calculations for a given chemical constituent in a particular energy state may still be correct even though an actual sample of the gas would contain other constituents and states as well. This is most significant for thermal conductivity and is discussed further in that context. Viscosities of pure ground state H calculated as described above are included in Table VII (mole fraction = l.O), and should be very reliable since they are based on well-established potential functions. All real cases will be mixtures of H-H, , of course, which may or may not (depending on how they are obtained) have H/H, ratios corresponding to chemical equilibrium at a given temperature and pressure. Regardless of equilibrium considerations, it is possible to calculate the viscosity of H-H, mixtures at various compositions and temperatures from Eq. (10) providing information on H-H, interaction is available so that T~~ (and A,*) may be computed from Eq. ( 1 I). T h e viscosity of pure H, is also needed, of course, and is obtained from experiment if available, or else it is calculated. A11 of the authors (30,32,33) rely on Margenau’s calculation (34) of the H-H, interaction, with appropriate weighting of SX

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

27 1

TABLE VII ( p poise) OF GROUND STATEH-H, MIXTURES COMPUTED FROM VISCOSITY THEORETICAL POTENTIAL FUNCTIONS

Mole fraction of H atoms

Temp.

(W 200” 400” 600” 800” 1000b 2000b 30006 4000b 5000b 10,000b 15,000*

.-

0

200 318 439 558 676 1250 1830

0.2

-

198 325 45 1 516 699 1310 1920

0.4

192 324 453 5 79 703 1330 1960

0.6

-

181 313 440 563 682 1300 1930

0.8

165 290 409 523 63 I 1200 1800

1 .o 46.2 15.3 100 122 142 253 356 453 542 1030 I540

Weissman and Mason (32). Vanderslice et al. (33).

the “perpendicular” and “triangular” configurations, and fitting to either the Lennard-Jones (12-6) or the modified Buckingham (Exp-6) potentials. T h e tabulation of Vanderslice et al. (33) for H-H, viscosities at various compositions from 1000-5000°K is convenient, and selections from this work are given in Table VII. As noted, these are forgroundstate mixtures only. Recalling the measurements of the viscosity of H-H, mixtures recently reported by Browning and Fox (16) and discussed in Section II,A,3, it is apparent that a significant comparison of theory and experiment is possible. Indeed, Browning and Fox do just that in several different ways with good results. One could either compare the H-H, mixture viscosities as computed above directly with experiment, or use the experimental mixture viscosities to compute back to a viscosity of pure H. T h e latter comparison is made in Table VIII. T h e H viscosities listed as “Exptl.” were obtained by Browning and Fox from their mixture measurements and H-H, interaction parameters from theory using the Lennard- Jones ( 1 2-6) potential. They actually used Clifton’s values (30) of ml, = 2.75 A and ( e / k ) l , = 32.3”K. T h e H viscosities labeled “Theor.” were taken from Weissman and Mason (32) who had originally computed them as described above at the appropriate temperatures for comparison with Harteck’s early data (14). (Browning and

A. A. WESTENBERG

272

TABLE VIII COMPARISON OF VISCOSITIES OF PUREGROUND STATEHYDROGEN ATOMSOBTAINED FROM MEASUREMENTS ON H-H, MIXTURES AND WHOLLY FROM A THEORETICAL H-H POTENTIAL V H ( P poise)

4

Temp. (OK)

(from H-H, mixture data")

Exptl.

Theor.b

190-195 273 313

44.3 (190°K) 57.0 70.9

45.1 (195°K) 57.6 71.7

Browning and Fox (16).

* Weissman and Mason (32).

Fox give essentially the same values for H viscosity computed wholly from theory.) T h e agreement in Table VIII is indeed excellent and is a remarkable demonstration of the successful meshing together of a great deal of theory and experiment from widely scattered sources. Appropriately averaged collision integrals for calculating transport properties in other systems involving atoms have been published by Yun and Mason (35).Integrals for ground state interactions of N- N, 0-0, N - 0 , N-N,, 0-0,, 0 - N , , N,-N,, 0,-0,, and N,-0, are given over the temperature range 1000-5000"K. Some of these get very complicated, the 0-0 case involving evaluation of eighteen different potential curves, for example. III. Thermal Conductivity

A. EXPERIMENTAL TECHNIQUES Most of the published data on gas thermal conductivity have been obtained in steady state systems subject to simple solutions of the appropriate equation of heat conduction. These solutions are discussed i n detail by Jakob (36). Three such techniques which are best adapted to high temperatures are described in this section. I n addition, brief mention is made of a nonsteady state technique using a shock tube which, at lease in principle, may prove valuable for measurements at very high temperatures, and also a recent method for obtaining data on Prandtl numbers directly without separate determination of both

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

273

thermal conductivity and viscosity. Finally, a few remarks on dissociated gas systems are made. 1. The Concentric Cylinder Method

This is the oldest and most common technique first used by Schleiermacher about 80 years ago. T h e apparatus in its simplest form is a cylindrical tube containing a coaxial, electrically heated wire (generally platinum). T h e gas to be measured is allowed to fill the annulus between wire and tube, and the whole system is immersed in a thermostat. T h e basic governing equation is

where Q is the steady state heat output of the wire of length L , T, and T, are the wire and cell wall temperatures, and r , and re are the wire and cell radii, respectively. I n practice the difference between T , and T, is kept very small so that the thermal conductivity h may be considered to be obtained at the mean temperature. For precise work a number of corrections (37) must be made, e.g., for end effects and radiation heat transfer (Q must be the heat output transferred to the gas only), so that the method can become complicated. T h e difficulties all become greater at high temperature as usual, especially those of providing uniform temperature around the cylinder and avoiding free convection effects. I n view of the difficult corrections, the concentric cylinder method is often used as a relative measurement with some suitable calibrating gas, usually dry air. For this purpose the data of Kannuluik and Carman (38) are often used. T h e precision attainable in the best applications of the concentric cylinder method is about f0.I yo. Accuracy is more difficult to assess. Johnston and Grilly (39)estimate an over-all accuracy for their measurements of +0.5y0 in the temperature range 80-380°K, but since their data on helium, for example, disagree by more than 0.5% with the later results of Kannuluik and Carman (40)this may be too optimistic an estimate. Perhaps f1 yo at moderate temperatures would be a more realistic guess as to the accuracy of the best absolute measurements of thermal conductivity by this technique. At high temperatures, the accuracy rapidly becomes considerably more questionable. Comparisons with other techniques to be given later will illustrate this point.

274

A. A. WESTENBERG

2. The Line Source Flow Method This is a technique of much more recent origin developed by Westenberg and de Haas (41), although it is based o n an old idea (42). Instead of a static system, it makes use of a fine electrically heated wire (the line source) immersed in a jet of the test gas under laminar flow conditions. Although other ways of utilizing this basic arrangement are possible, the most advantageous involves a measurement of the half width at half maximum y I , of the thermal wake a distance z downstream of the source. Then the equation for the gas thermal conductivity is

where U is the gas velocity, and cD and p are the gas specific heat (at constant pressure) and density, respectively. T h e method thus requires only measurements of distances, the velocity U , and relative temperature differences (with a fine thermocouple) to get the wake width. Elimination of the requirement for absolute temperature differences and heat input rates, as is necessary in Eq. (14), is the great advantage of this technique over the concentric cylinder method-particularly for high temperatures. T h e need for accurate knowledge of U is a disadvantage, since this can be difficult to measure at the low velocities of 50-500 cm sec-1 which are used. U cannot be calculated well enough from the jet geometry and total flow rate, but fortunately a low-speed anemometer developed by Walker and Westenberg (43) is well suited for this purpose. T h e line source method is an absolute technique that is especially useful at moderately high temperature (300-1200°K). It is not capable of the precision attainable with the static method. Its accuracy, however, may be comparable, and is limited mainly by the accuracy of the velocity measurement, i.e., 2-3 yo. 3. The Long Hot Wire Method An interesting variant of the conventional concentric cylinder method has been described by Blais and Mann (44). This is a steady state device in which free convection and large temperature gradients are deliberately allowed to occur, in contrast to the usual condition. If the cell is made very long, however, in the central region of the heated wire the axial gradient can be made much smaller than the radial gradient. T h e convection effect is then negligible and Eq. (14) can be used essentially in its simple form except that the derivative at the wire d(Q/L)/dT, is substituted for Q/L(T,, - T,.).Q/L is plotted against T , and the derivative

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

275

determined numerically at various temperatures. Radiation corrections to the total electrical heat input can be determined by measurements in vacuum. Since the central wire can be made very hot, data to quite high temperatures ( M 2000°K) have been obtained by this method. With some gases, contact with the hot wire can lead to dissociation and falsification of the results. This is a difficulty which would occur with most methods using hot metal surfaces, however, and in favorable cases it would appear that the long hot wire device should be exploited much more extensively than it has. 4. The Shock Tube Method

A nonsteady state approach to the problem of measuring thermal conductivities at very high temperatures was first described by Smiley (45) and later used by others(46,47). T h e idea is to measure the temperature rise as a function of time of a plane end plate suddenly exposed to the hot gas behind a reflected shock wave. Relatively simple heat conduction considerations then allow the gas thermal conductivity to be determined. A thin film resistance thermometer gauge is used for the temperature rise measurement. At high temperatures dissociation may complicate the interpretation, but this technique has been used for measurements in argon to the highest temperature yet attained in thermal conductivity research (8600°K).

5 . The Prandtl Number Method This special technique devised by Eckert and Irvine (48) measures the Prandtl number c , , ~ / hdirectly rather than either 77 or h separately, which has definite advantages in certain connections. T h e method relies on a measurement of the adiabatic “recovery” temperature attained by a flat plate in a high (subsonic) velocity stream of the test gas. T h e recovery temperature, in turn, may be shown to be a simple function of the Prandtl number only. In practice the flat plate is adequately approximated by one junction of a differential thermocouple, the other junction being located upstream of the nozzle to measure the stagnation temperature of the gas. T h e technique yields the Prandtl number at some average temperature between the static and recovery temperatures, but this is not a great disadvantage because the Prandtl number is very insensitive to temperature. Clearly, separate thermal conductivities determined in this way can be no more accurate than the viscosities

276

A. A. WESTENBERG

themselves are known. T h e techcique is fairly precise and is capable of use over a considerable temperature range.

6 . Methods f o r Dissociated Gases T h e effects of gas dissociation on thermal conductivity are similar but somewhat more complex than on viscosity because the dissociation process involves energy. This fact contributes to the effective thermal conductivity in addition to the mixture complication noted for viscosity in Section II,C,4. This is considered somewhat more fully later in Section III,C,5. But as far as experimental measurement is concerned, the same techniques described above for undissociated gases should be applicable provided the dissociated gas is in chemical equilibrium at the temperature and pressure of measurement. Otherwise the effects of the necessary surfaces and instruments on the reacting mixture may be exceedingly difficult to assess. This is the main problem with experiments in discharge-generated atom-molecule systems which are far from equilibrated at the usual low temperatures employed. But in special systems such as the N,O, z? 2 N 0 , case studied by Coffin and O’Neal (49), or high-temperature shock tube studies on diatomic gases (46), chemical equilibrium may be attained and the usual measurements undertaken. T h e composition is then calculable from thermodynamics and no special techniques for measuring labile species are necessary.

B. EXPERIMENTAL DATA T h e TPRC compilation ( I ) is not nearly as extensive for thermal conductivity as it is for viscosity, although presumably there will be additions to it in due time. Only a few common species in the gas phase are presently available. It is of interest to compare some results obtained by different methods where possible. Table I X (50-52) shows several sets of data obtained by various workers using the same basic concentric cylinder method, as well as data from the line source flow technique (41). T h e agreement over a fairly wide temperature range is generally within 5 % , which is an indication of the best that can be expected of the available techniques at the higher temperatures. Table X (53-57) gives a summary of the thermal conductivities for those pure gases which have been measured over at least a few hundred degree temperature range. T h e values given are, in some cases, taken from smoothed plots of several different investigations, and represent the best available data in the opinion of this reviewer. The extremely

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

277

TABLE IX SOMEDATAON THERMAL CONDUCTIVITY (cal cm-' sec-I OK-' x lo6) OBTAINED BY VARIOUS WORKERS B Y T w o DIFFERENT TECHNIQUES

NZ Reference

COZ

~-

~~

300°K

500°K

1000°K

300°K

500°K

1000°K

6.2 6.1 6.1

9.5 9.5 9.1

15.4 15.7

15.1

4.2 4.0 4.0

8.2 7.9 7.7

16.5 16.9 16.1

6.1

9.2

15.7

3.9

7.8

16.7

Conc. cyl. 50 51

52 Line source 41

TABLE X SELECTED EXPERIMENTAL DATAON PUREGAS THERMAL CONDUCTIVITIES'

Temp. (OK)

300 400 500 600 700 800 900

1000 1100

1200 1300 1400 1500 2000

N* (Ref. 41)

(Ref.

6.1 7.6 9.7 10.6 12.1 13.5 14.7 15.7 16.5 17.1

53)

CO, (Ref. 41)

CO (Ref. 54)

3.9 5.8 7.8 9.7 11.7 13.5 15.2 16.8 17.9 18.6 -

6.0 7.7 9.2 10.6 -

-

6.3 8.2 9.8 11.3 12.7 14.1 15.5 16.9 18.3 19.6 -

-

0,

-

-

-

-

-

-

-

-

-

-

-

-

(Ref.

H,O (Ref.

Ar (Ref.

54)

55)

56)

43.4 54.5 65.0 75.2 85.5 -

6.2

4.2 5.3 6.3 7.2 8.1 8.8 9.5 10.3 10.8 11.3 11.9 -

Ha

8.5 11.1

13.9 16.8 20.0 23.4 -

-

-

-

-

Values are in units of cal cm-' sec-I OK-' x lo5.

-

-

He (Refs. 40.44) 36.1 43.5 50.4 56.2

SO, (Ref. 57)

2.3 3.4 4.7 6.0 7.3

-

99.1 106 113 119 153

278

A. A. WESTENBERG

limited extent of Table X indicates clearly the great lack of data over a wide temperature range in this field.

C. THEORY Even more than was the case with viscosity, one must rely heavily on theoretical computation of thermal conductivities for practical applications. T h e situation is unpleasantly complicated by the fact that energy can be carried in the internal degrees of freedom of polyatomic molecules, as will be evident in the following discussion.

1. Pure Monatomic Gases For a pure monatomic gas (denoted by superscript zero), the rigorous kinetic theory gives for the thermal conductivity in first approximation

where Xo is in units of cal cm-' sec-' OK-l and the other symbols are the same as defined for Eq. (4). T h e intermolecular potential parameters for use in predicting Xo are not usually obtained by fitting the experimental data themselves. This can be done in principle, of course (for monatomic gases), but by far the more common procedure is to make use of viscosity measurements as a function of temperature, which are easier and usually more reliable. One of the imporfant results of the rigorous kinetic theory is that viscosity and monatomic thermal conductivity are related by the expression

where R is the ideal gas constant (1.987 cal mole-' OK-'). Thus the same collision integrals apply to both transport properties, and potential parameters determined from viscosity data (see Table 111) may be used to predict monatomic thermal conductivity. In the temperature range where the viscosity was measured, of co-urse, Eq. (17) may be used to obtain Xo directly with no need for potential parameters. This relation between ho and 71 is well established.

2 . Pure Polyatomic Nonpolar Gases So far the discussion has been confined to pure monatomic gases for which the theory is relatively simple and clean cut. Most gases, however,

SURVEY OF DILUTEGASTRANSPORT PROPERTIES

279

are polyatomic (taken to include diatomic species), and the effects of internal degrees of freedom on heat transport must be considered. This involves the so-called Eucken correction to the monatomic thermal conductivity. A particularly illuminating discussion of this problem for nonpolar gases has been given by Hirschfelder (58) using relatively simple arguments. The whole subject has since been explored with considerably more rigor and elegance by Mason and Monchick (59). The important features of their work will be referred to at appropriate points later, but first we shall make use of Hirschfelder’s simpler approach. A pure polyatomic gas is considered to be a “reacting” mixture of species representing all of the possible internal energy states (i.e., vibrational and rotational, but only the ground electronic state) of the molecule. In order to define a thermal conductivity at all for such a “mixture,” i.e., to equate the heat flux to the product of h and the temperature gradient, two important assumptions are necessary. First, it is assumed that the distribution of “species” among the internal energy states is that characteristic of local thermodynamic (Maxwellian) equilibrium, so that it is a function only of the local temperature. Any situations where vibrational or rotational temperatures are different from the ordinary translational temperature would thus violate this condition. T h e effect of a temperature gradient may be viewed as setting up concentration gradients in the species in various internal energy states. Therefore, the heat flux carries terms due to the diffusion of these different species in addition to that due to the molecules with different translational energy which is accounted for by ho. The second assumption necessary is that all the different internal energy state species have the same self-diffusion2 coefficient D. This is a good assumption except where different electronic states are involved, which is the reason for excluding electronic excitation from this treatment. If h is the thermal conductivity of a polyatomic gas and ho the hypothetical value the thermal conductivity would have if the molecules had no internal degrees of freedom (i.e., ho is the translational thermal conductivity), then under the foregoing assumptions it can be shown that A/ho = 1

-

6,

+ (2/5)6,C,M/R

(18)

where 6, is a dimensionless parameter defined by

* As Mason and Monchick emphasize (59),this is not strictly the self-diffusion coefficient which would be measured in an isotopic tracer experiment, but for practical calculations little error would be incurred by using this value.

280

A: A. WESTENBERG

and N is the number of moles per unit volume. T h e factor 5R/2is the value of cpM for a monatomic gas, so that 6, may be thought of as a Lewis number for the fictitious gas regarded as monatomic. 6, depends on the potential function used to approximate the gas behavior, but for any model it has nearly the same value. Thus for rigid spheres 6, = 4/5, while for both the Lennard- Jones (12-6) and modified Buckingham (Exp-6) functions 6, is weakly temperature dependent and varies from 0.88 to 0.90 over a wide temperature range. Thus Hirschfelder has assigned an average value of 6, = 0.885 as closely approximating all realistic potentials. Using this value in Eq. (1 8) gives h/ho = 0.115

+ 0.178~,M

(20)

(c, in cal gm-' O K - ' ) as the final expression for the modified Eucken correction. (The original Eucken correction corresponds to 6, = 8.) This relation is extremely useful in computations on polyatomic gases. T h e essence of the Mason-Monchick contribution (59) to the polyatomic gas problem was to remove the necessity for assuming equilibrium between the internal and translational degrees of freedom. By use of the formal kinetic theory taking account of inelastic collisions, they were able to incorporate the rotational and vibrational (only the former is usually important) relaxation times into an Eucken-type correction to the monatomic thermal conductivity. T h e modified Eucken correction of Hirschfelder described above was then obtained as a first approximation to the more rigorous theory. The nonequilibrium internal energy expressions lead to better agreement with experiment at low temperatures and are most important in this region. At high temperatures the more elaborate theory differs very little from the simple modified Eucken correction for practical calculations and the latter is usually adequate. But the Mason-Monchick treatment represents an important advance because it puts the simpler theory on a much firmer basis. T o summarize: the most satisfactory method of predicting hightemperature thermal conductivities of pure nonpolar polyatomic gases is to use viscosity data to compute the quantity A* by means of Eq. (17). If no viscosity data are available at the temperature desired, then suitable potential parameters derived from experimental viscosities at lower temperatures-such as the Lennard- Jones parameters given in Table III-should be used in Eq. (16) to compute ho. Then Eq. (20), with the appropriate cp data at the desired temperature, will yield the value of A. Table XI gives some examples of common gases treated in this way. Experimental viscosities and specific heats were taken from Svehla's compilation (I 9). The general agreement with measured h values is very good.

SURVEY OF DILUTE GAS TRANSPORT PROPERTIES

28 1

TABLE XI COMPARISON WITH EXPERIMENT OF THERMAL CONDUCTIVITIES CALCULATED FROM VISCOSITIES USINGTHE MODLFIED EUCKENCORRECTION^ ~~

300°K

Ar N* 0,

H*

co

CO, Air

4.3 6.4 6.5 45 6.4 4.0 6.5

500°K

4.2 6. I 6.3 43 6.0 3.9 6.3

6.3 9.5 10.0 64 9.6 8.0 9.5

1000°K

6.3 9.1 9.8 65 9.2 7.8 9.1

10.1 16.0 17.9

10.3 15.7 16.9

-

-

-

-

16.1 16.3

16.8 16.2

A in units of cal cm-I sec-l OK-' x los.

3 . Pure Polar Gases T h e complications due to polarity are worse for thermal conductivity than for other transport properties because of the possibility that internal (i.e., rotational) energy may be transferred in resonant collisions without affecting translational energy. Physically, this may be pictured as a grazing collision being converted to an apparent near head-on collision. This is because one molecule transfers a rotational quantum of energy to the other, so that it appears as though the first molecule undergoes a more pronounced distortion in its trajectory than it really does. T h e net result is to lower the effective diffusion coefficient for internal transfer and thus the thermal conductivity. This phenomenon has been treated by Mason and Monchick (59) for various types of polar molecules. Actually the resonance correction is rather small except for light linear molecules like HCI, or the nonlinear H 2 0 , where a rotational quantum represents an appreciable amount of energy. Baker and Brokaw have recently reported (60) an interesting test of the importance of the resonant exchange process for H,O, D2 0 , and mixtures of these two. T h e results were somewhat conflicting, but seemed to indicate that exact resonance in the rotational exchange may not be necessary. This would imply an essentially classical, rather than quantum mechanical, exchange in grazing collisions. In any case, the necessary data on rotational relaxation times are usually not available for application of the theory, and it must suffice to treat a polyatomic polar gas as described in the previous discussion where the polarity is ignored.

282

A. A. WESTENBERG

4. Inert Mixtures T h e theoretical treatment of even monatomic gas mixtures is a formidable and cumbersome exercise in its full rigor [Hirschfelder et al. (3, Chap. S)], although fully developed. For polyatomic gas mixtures, the Mason-Monchick treatment has been formally developed ( 6 4 , but is hardly suitable for practical calculation. T h e most nearly rigorous approximate, but useful, theory for polyatomic gas mixtures is that of Hirschfelder (62). Assuming full equilibrium of internal and translational degrees of freedom, the mixture relation is

where h,O is the translational thermal conductivity of the mixture-a complex function of composition, molecular weights, diffusion coD,, and D,,are selfefficients A$’, and collision integrals (63)-and diffusion and binary diffusion coefficients, respectively. T h e pure gas conductivities A, would presumably be available from experiment or the computation methods discussed previously. It is also necessary to have all the binary diffusion coefficients D, for the various pairs in the mixture. T h e hypothetical monatomic conductivities h+ocan be obtained from viscosities by way of Eq. (17) and the D,,by means of Eq. (19). Thus it is possible to calculate A,, with all experimental input dataeither direct or by means of well-established kinetic theory relations. This type of calculation has been compared with experimental mixture (41) and H,O-0, (53) (ignoring the polar data for both N,--0, nature of H,O) over a range of temperature with fair success. While the resonance correction previously noted for H,O should affect the H,O -0, mixture calculations, the agreement with experiment is good without considering it.

5 . Reacting Mixtures Mention should be made of the so-called “effective” thermal conductivity in a mixture of chemically reacting gases which has come into considerable vogue. Wherever there are chemical reactions in a flow system there are gradients in concentration of the various chemical species, so that energy is transported by diffusion as well as conduction. At any point in such a chemically reacting system the mixture has a certain thermal conductivity which is a true molecular property. I n this sense a chemically reacting system is just another mixture. In certain applications it has been considered useful to define an “effective”

SURVEY OF DILUTE GAS TRANSPORT PROPERTIES

283

thermal conductivity for a reacting mixture in the sense of a quantity which, when multiplied by the temperature gradient, gives the total heat flux. It should be clearly understood that such an “effective” thermal conductivity can only be defined when the reactions are so fast compared to the diffusion processes that the mixture may be regarded as being in chemical equilibrium at the local temperature. This is the shifting equilibrium idea and is quite analogous to that discussed earlier in connection with the Eucken correction for polyatomic gases. T h e diffusive contributions to the total heat flux are thus contained in the “effective” thermal conductivity. It should not be construed that a reacting mixture is somehow fundamentally different from a nonreacting system insofar as the basic concept of thermal conductivity is concerned. Since this review is concerned only with the fundamental transport properties themselves, the various ramifications of the reacting mixture concepts will not be dealt with further. IV. Concentration Diffusivity

A. EXPERIMENTAL TECHNIQUES T h e classic techniques for concentration diffusion measurements in the gas phase are those associated with the names of Loschmidt or Stefan. These are described in some detail by Jost (64),and are discussed briefly below. T h e newer point source flow technique of Walker and Westenberg is then given more extensive treatment, since it is perhaps the best method presently available for use over a wide temperature range. Finally, some special methods applicable to labile atoms are given. 1. The Loschmidt Method

Various refinements of this technique have been used, but basically the apparatus consists simply of a long tube divided into two chambers by a stopcock or diaphragm of some kind. Initially each chamber contains one of the pure gases of the pair being measured. At time zero, the stopcock is opened and the two gases mix by diffusion. For constant temperature and binary diffusion coefficient, the time-dependent one-dimensional diffusion equation is simply ax,jat =

D,,a2xx,lazz

(22)

where x1 is the mole fraction of component 1, and t and z are time and distance along the diffusion cell, respectively.

A. A. WESTENBERG

284

With a cell of total length L, the boundary conditions are at t = 0: x1 = 1

for 0 < z < L/2

x1 = 0

for L/2 < z


and at any t at z = 0 and L:

axl/az = o For large enough time t , Eq. (22) may then be simply solved by separation of variables to give XI

=

# + (2/7r)cos (7ra/L)(- DnZt/L2)

(23)

Analysis of the mixture at some point in the tube at known times then enables the binary diffusion coefficient to be determined. I t is, of course, extremely important that opening the stopcock between the two chambers does not make any disturbance in the gas which would cause mixing to occur by other than molecular processes. Most of the refinements introduced since Loschmidt’s original work have been concerned with this point. T h e technique in various forms has been used in recent years by Boardman and Wild (65), Klibanov et al. (66), Boyd et al. (67), Strehlow (68), Bunde (69), and others. It is capable of precision of about f0.1 yo when used with the greatest care (69). The main disadvantage of the Loschmidt method is that it requires very careful thermostating of a fairly long column which also contains moving parts, and so is inherently unsuitable for high-temperature measurements. There is also some uncertainty about the concentration dependence of the diffusion coefficient measured in this way which is important in certain cases and limits the accuracy of the technique to perhaps f0.5 yo.

2. The Stefan Method T h e method of Stefan has been used for measuring the diffusion coefficient of a vapor in a gas. It is a steady state technique in contrast to the Loschmidt method. One component is liquid and is put in the bottom of a vertical tube. T h e other (gas) component is passed slowly over the open mouth of the tube. T h e liquid evaporates, the vapor diffusing up the tube and out the end where its concentration is kept at essentially zero by the gas stream. T h e vapor concentration ( N & (moles ~ m - ~at) the liquid boundary is determined by the vapor

SURVEY OF DILUTE GAS TRANSPORT PROPERTIES

285

pressure. With these two boundary conditions, the vapor-gas diffusion coefficient is given essentially by 4

2 =

Nuwl)"

(24)

where A and L are the cross sectional area and length of the tube, and as gauged by the rate of fall of the liquid level. This is probably the simplest of all the techniques but it is not well suited for wide temperature ranges. I t is also, of course, restricted to use with volatile liquids. For these reasons it has not been widely used of late, the best recent work being that of Schwertz and Brow (70).

& is the number of moles of liquid evaporating per unit time

3. The Point Source Flow Method This is a steady state flow technique developed by Walker and Westenberg (71), and is analogous to the line source technique used for thermal conductivity measurements. One component (the carrier gas) is passed through a tube ending in a series of precision screens, so that the flow at the exit is slow, uniform, and laminar. T h e second component is introduced through a fine hypodermic tube whose exit protrudes through the final screen, thus furnishing a point source of trace gas. T h e diffusion equation for constant temperature and diffusion coefficient with species 1 present only in trace concentrations is D,,V2x,

-

v Qx,

=0

(25 )

With a flow rate from the point source of Q cm3 sec-l and the boundary condition x1 = 0 at z = 03, the solution to Eq. (25) at an axial distance z downstream from the source may be shown to be D,,

= !2/44Yl)tnz

(26)

where (yl), denotes the maximum concentration existing at the axial position z. T h u s by sampling the gas with a fine probe and measuring its composition (a thermal conductivity analyzer is very handy) at the distance z, the D,, may be determined. This technique is relatively simple, of fair precision (+1-2 yo),and, most important, adaptable to quite high temperature by heating the carrier gas. Since it is a flow method, thermostating difficulties are largely avoided, as are stray convection currents. In its original form the carrier gas was simply passed through a furnace section prior to reaching the screened exit. Temperatures up to 1200°K could be obtained in this way without serious structural problems. Higher temperatures were

286

A. A. WESTENBERG

prevented mainly by loss of integrity of the screens. I n an .adaptation of this technique Ember, Ferron, and Wohl(72) used the burned gas from a flat flame stabilized above a porous plate to attain temperatures up to 1700°K. Another advantage of the point source technique is that it is a tracer method, so that the measured D,, is always representative of essentially zero concentration of the point source gas. This clarifies the concentration dependence problem found in the Loschmidt method.

4. Methods for Dissociated Gases Direct measurements of the diffusion coefficients of unstable atoms and radicals are almost nonexistent, the obvious reason being that it is a very difficult problem to set up experiments of this kind which are not badly obscured by other phenomena-usually chemical reactions. Of the three direct, absolute techniques which have appeared in the literature, the steady state flow method of Walker (73) is perhaps the most promising. A diatomic gas at about 1 mm pressure is passed steadily through a quartz tube and dissociated to a small extent by some suitable means, usually a microwave or radio-frequency electrodeless discharge. As the mixture of atoms and molecules flows down the tube, the atoms recombine on the walls and in the gas phase so that a gradient in their concentration is set up. T h e magnitude of the gradient is a function of the flow velocity, the over-all reaction rate constant, and the atommolecule diffusion coefficient. Assuming one-dimensional flow and an atom loss rate controlled by kinetics which are first order in the atom concentration, the diffusion equation is D12 d2xl/dz2- U dxlldz

-

kxl

=0

(27)

where U is the constant and uniform linear flow velocity and R is the over-all first-order rate constant. The restriction to first-order kinetics is important since otherwise the equation cannot be solved analytically. With boundary conditions x, = ( S C ~at) ~z = 0 and x1 = 0 at z = co, the solution is

Then it is easily shown that a measurement of the slope (S)of a plot of 1 n[x,/(~,)~] versus z at two different velocities permits the chemistry and the diffusion to be separately determined. For example, using U = 0 and U = U it follows that

SURVEY OF DILUTEGASTRANSPORT PROPERTIES

287

T h e main problem is the determination of the (relative) atom gradient. Walker made use of the chemiluminescent reaction of 0 with NO to monitor the 0 photometrically, but other techniques may be used. Electron spin resonance spectrometry seems especially promising for this purpose, as it is quite general for many atoms. Two transient techniques for measuring absolute atom-molecule diffusion coefficients directly have been used. Krongelb and Strandberg (74) observed the decay of 0 atoms with time by means of electron spin resonance in a tube filled with partially dissociated oxygen (no flow) and were able to infer the 0-0, diffusion coefficient. Young(7.5, 76) has used a somewhat similar technique (with spectroscopic detection of the first positive N, bands to measure N atom concentration) to get an approximate value for the N-N, diffusion coefficient at room temperature. T h e experimental data were quite scattered, however. A means of measuring the relative diffusion coefficient of H-H, as a function of temperature has been reported by Wise (77). It is a steady state technique utilizing a (no flow) heated tube with a presumably constant source of atoms at one end and a movable tungsten filament catalytic recombination probe at the other for measuring relative atom concenpations. By making certain assumptions-mainly that the filament catalytic activity for H atom recombination is independent of temperature-the relative diffusion coefficient for H-H, as a function of temperature was determined. I n the absence of any quantitative knowledge of the catalytic activity of the probe, however, it is not an absolute technique and must be based on some other reference value. For dissociated gases especially, it may turn out that the best method for determining diffusion coefficients will be an indirect approach by way of the measured viscosity of atom-molecule mixtures. Since the relation between these two properties involves the kinetic theory, discussion of this idea is postponed until Section IV,C,4. B. EXPERIMENTAL DATA At the time of writing this review the T P R C compilation of diffusion coefficients had not been issued. Until this is available readers may find Table XI1 (78-90) useful, which contains the best values (at 298°K and 1 atm) of binary diffusion coefficients for many common gas pairs. Two significant figures are given in all cases which probably reflects the accuracy of most of them. With a few exceptions the data were obtained by one of the three major techniques described in the previous section. Unless otherwise stated the concentration dependence of the diffusion coefficient has been ignored. T h e room temperature data of Table XI1

A. A. WESTENBERG

288

TABLE XI1 EXPERIMENTAL BINARY DIFFUSION COEFFICIENTS AT 298°K

Gas pair ~~

N,-CO,(trace) NZ-CO Nz-H, Nz-0, N,-HzO N2-C2H4 Nz-CzHs N,-n-C4H,, N,-Ar N,(trace)-He N,-He(trare) 0,-CH,(trace) O,(trace)-CO, 0,-CO(trace) 0,-H,O(trace) 0,-H,(trace) Oz-CJ& O,-Ar Ha-CO HZ-CO, HZ-CHI H2-C2H4 H2-Cd-h H,(trace)-Ar HZ-HZO

0.17 0.22 0.78 0.22 0.24 0.16 0.15 0.096 0.20 0.69 0.73 0.23 0.16 0.22 0.29 0.82 0.080 0.14 0.75 0.65 0.73 0.60 0.54 0.85 0.99

78 65 69 65

70 67 67 67 79 78 78 80 80 80 80

80 81 65

82 67 67 67 67 83 70

AND

Dti

1

ATM

(cmz sec-I)

Ref.

~~

H,-Br, HO-CBHB H,-n-C4H,, H,-N,O H&H,OH Hz-(CzHJzo Hz-(CH&CO

0.58 0.34 0.38 0.62

COZ-CZH, C02 C H 4 C02 -H 2 0 COZ-CSH, Cot-CHSCHO

0.15

c0,-co

-

c0,-cs,

COZ-CeHe C0,-Ar COZ-NZO COZ-CZHbOH COz-(CzHs)zO CO,(trace)-D, HZO-CHb HXO-CZH, CO-CZH, He(trace)-Ar He-Ar(trace)

0.44 0.35 0.43 0.16

0.18 0.19 0.086 0.092 0.074 0.061 0. I4 0.1 1 0.79 0.062 0.49 0.28 0.20 0.13 0.75 0.72

84 85 67 86 87 64 64 82 82 82 70 88 88 89 85 65 65 87 64 90 70 70 82 91

91

are very useful as reference values for predicting values at other temperatures by means of the theory to be outlined in the next section. Experimental data at high temperatures are, of course, far more scarce. T h e highest temperature ( m 1700°K) at which any diffusion measurements have been made are for the point source self-diffusion data reported by Ember et al. (72). T h e data of Klibanov et al. (66) for H,O-air and C0,-air to 1500°K were made by a capillary-leak method closely related to the classical Loschmidt technique. T h e data were so imprecise, however, as to be of interest mainly from an historical point of view. T h e moderately high-temperature data of Walker and Westenberg (71, 78, 80, 91) and Westenberg and Frazier (83) were obtained by the point source method. This work covered the nominal range 300-1 150"K, except for certain gas pairs which underwent

TABLE XI11 EXPERIMENTAL DIFFUSION COEFFICIENTS (AT 1 ATM)AT HIGHTEMPERATURE OBTAINED BY

THE

POINTSOURCE~ I E T H O D ~

D , j (cm' sec-I) Temp. ( O K )

-

CH,-0, (Ref. 80)

0,-CO, (Ref. 80)

N,-CO, (Ref. 78)

H,-0, (Ref. 80)

CO-0, (Ref. 80)

H,O-0, (Ref. 80)

0.22 0.37 0.54 0.74 0.96 1.19

0.47 0.69 0.94 1.22 1.52 1.85 2.20 2.58 -

He-Ar (Ref. 91)

He-N, (Ref. 78)

H,-Ar (Ref. 83)

C0,-CO, (Ref. 72)

0.14 1.22 1.77 2.41 3.12 3.89 4.74 5.65 6.62 -

0.86

I .76 2.16

~-

300

400

500 600 700 800 900 lo00 1100 1200 1300 1400 I500 1600 I700 O

$ U r 3

0.23 0.39 0.58 0.80

1.05

1.33 1.62 I .95 -

0.16 0.28 0.42 0.58 0.77 0.97 1.19 1.43 1.68

0.17 0.30 0.44 0.61 0.79 0.99 1.21 1.45 1.70

T h e first gas of each pair is the trace component.

0.82 1.40 2.09 2.88 3.16 4.74 5.79 -

-

0.76 1.27 1.89 2.59 3.38 4.24 5.17 6.17 1.24

1.44

2.14 2.95 3.84 4.80 5.94 7.22 8.65 -

C

2 0

2.50 2.86

3.21 3.57

m M

A. A. WESTENBERG

290

spontaneous ignition at somewhat lower temperatures. The available high-temperature data are given in tabular form in Table XIII. No other published data have extended to temperatures much higher t h i n about 400°K. T h e opportunity for more research in this area is clear.

C. THEORY Of all the transport properties, the determination of diffusion coefficients from theory is probably carried out most often-especially at high temperatures-because of the lack of direct experimental measurements. T h e various approaches to this problem which are based on the rigorous kinetic theory are summarized below. 1. The Basic Binary Lhfision Relations Since considerable use has been made of all four of the common potential energy functions defined by Eqs. (5-8) for diffusion calculations, we shall include all of them in the discussion. The reader is reminded once more that these are meant to apply strictly only to spherically symmetric molecules. Fortunately, the effects of polarity in a polarnonpolar collision are negligible, and the polar-polar case is mentioned briefly later. Internal degrees of freedom are not important in ordinary concentration diffusion since the energy carried in this way is not a factor. Thus no separate discussion of polyatomic gases is necessary. I n the form for practical use, the binary diffusion coefficient (cma sec-l) in terms of the inverse power potential is given by D.

ti

=

I 3 6 x 1 0 - 3 [ ( ~+ ~ ~,)/~~~,l1'2(k/d~~s~~)2/a~312+2/*,, (30) P"[3 - (2/6,)]

where k is the Boltzmann constant, P is the pressure in atmospheres, A(')is an integral (a function of aij) which is available [see Hirschfelder et al. (3, p. 548)], and I'is the gamma function of the indicated argument. di, and 6 , are, of course, the parameters of the potential function in angstroms. This is the first approximation to Ddj, the concentration dependence entering only in the second approximation which we shall not consider. A convenient feature of this potential is that it gives a simple power dependence of D , on the temperature, so that for a given set of data a plot of log D, versus log T should be linear with a slope of (3/2 2/6). For the exponential repulsive potential,

+

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

29 1

where aij = ln(Aij/kT) and pij bring in the potential parameters, and 1(1,1) is a form of the collision integral tabulated ( 1 7) as a function of a. For the Lennard- Jones ( 12-6) potential,

D..= 1.86 x 13

+

1 0 - 3 ~ ~M ~~ )~/ M , M ~ I ~ / T Z

Pop;;.”*

(32)

and for the modified Buckingham (Exp-6) potential

D..= 1.86 x 10-3[(Mi+ Mj)/MiMi]1/2T3/2(33) 1)

P(Ym);p;;

11’

*

T h e collision integrals 52(’”)* for the latter two potentials are available [see Hirschfelder et al. (3, pp. 1 126 and 1164, respectively)]. I t should be noted that any of the equations (30)-(33) may be converted to the form for self-diffusion Dii simply by letting Mi = M i . 2. Potential Parameters from Digusion In order to make use of any of the potential functions for computing binary diffusion coefficients and (especially) their dependence on temperature, it is necessary to have available values of the parameters for the potential characteristic of the particular pair of gases being considered. T h e most reliable way of obtaining these parameters is by fitting the measured diffusion coefficients themselves as a function of temperature. I n this sense the whole procedure is somewhat academic, since in order to make theoretical predictions of a quantity, the quantity itself has to be measured. However, for extrapolation of data beyond the range of temperature over which measurements have been made, the interplay of theory and fitted experimental parameters is most useful. T h e only diffusion measurements which have been made with sufficient precision and over a wide enough temperature range for meaningful potential fitting are those given in Table XIII. Using a least-squares procedure, the data on the nine unlike gas pairs may be fitted about equally well (i.e., with a standard deviation of less than 3 yo) over their measured temperature ranges by any of the four potential functions. T h e potential parameters determined in this way are given in Table XJV. A word of caution is appropriate at this point. I t should be emphasized that, even over a range of measurement of several hundred degrees as in these examples, the individual parameters of a particular potential cannot be very precisely determined in this way. A particular pair of parameters-the Elk and u for the Lennard- Jones (12-6) potential given for 0,--0, in Table XIV, for example-are

TABLE XIV

POTENTIAL PARAMETERS OBTAINED BY FITTING EXPERIMWTAL DIFFUSION DATAOF TABLE XI11 Inverse power repulsion Gas pair

CH4-Ox

0,-co,

NS-CO, Hx-0,

co-0,

HXO-0, He-Ar He-N, He-Ar

d (erg A6 x lolo)

4.56 3.75 20.6 1.62 75.6 358 1.81 22.6 1.68

S 6.83 6.54 7.91 7.13 9.52 11.41 8.12 10.45 1.34

Lennard- Jones (12-6)

Exponential repulsion

A /k (OK x 8.0 6.5 24.5 9.0 105

900

26 350 12.5

p

(A)

0.49 1 0.520 0.432 0.395 0.339 0.261 0.307 0.241 0.371

c/k

'

(OK)

182 213 154 152 91 80 125 69 155

Modified Buckingham (Exp-6)

(A)

s/k (OK)

r, (A)

3.37 3.36 3.52 2.82 3.48 3.33 2.59 2.87 2.76

220 255 184 117 110 100 150 85 112

3.61 3.62 3.78 3.42 3.73 3.57 2.78 3.07 3.37

u

U

17 17 17 12 17 17

17 17 12

?

?

s! i 2

4 E 0

SURVEY OF DILUTEGASTRANSPORT PROPERTIES

293

uniquely determined in that Elk = 213 must be used with u = 3.36. But another pair of values may also fit the data within the experimental error nearly as well. For the 0,-CO, case, either E / K = 200, u = 3.40, or E / K = 240, u = 3.31 will do nearly as well as the values given. Thus one should be wary of inferring that the potential well depth for 0,-CO, interaction has a value of E / K = 213°K. This is only a fitting parameter by the Lennard-Jones for use with u = 3.36 in calculating (12-6) model, and says very little about the "real" interaction potential. A comparison of the diffusion coefficients predicted from the potential parameters of Table XIV using Eqs. (30-33) and the appropriate collision integrals is shown in Table XV as the first four entries under each gas pair. I n the range of experimental measurements, i.e., up to about 1000"K, the agreement between the different models is excellent as it should be since they were all fit to the same data. I n the range 1500-3000°K, which represents extrapolation, the predictions begin to deviate as much as 20% in some cases. Without additional evidence, it is impossible to state which of the predictions is to be preferred. I n the cases of He- N, (78) and He- Ar (91), the necessary additional evidence is available in the form of intermolecular potential energy functions derived independently from molecular scattering experiments (92, 93). These potentials, generally of the inverse power repulsion type, are valid at closer distances of approach (corresponding to higher temperatures), and may be used directly to compute diffusion coefficients at higher temperatures than are easily accessible to measurement. Table XV contains entries for He-N, and He-Ar obtained this way. I n the two cases cited, both the pure repulsion potential predictions (inverse power and exponential) derived from diffusion measurements agree better with the scattering predictions than do the other two potentials. I n view of the extreme scarcity of such scattering data, however, one should not take this as a general rule. At the present time, for lack of better evidence, it is probably best to choose a model on the basis of simplicity for practical computation. T h e inverse power potential is preferable from this point of view, with the exponential and Lennard-Jones (12-6) models nearly as simple. T h e modified Buckingham (Exp-6) has little advantage to offer and is more cumbersome. For high-temperature work especially, the purely repulsive potentials should be adequate for most purposes, since the existence of an attractive term is then less significant.

a

3. Potential Parameters from Viscosity Except for the gas pairs discussed in the preceding section, diffusion data have not been obtained over a wide enough temperature range to

A. A. WESTENBERG

294

TABLE XV PREDICTED DIFFUSION COEFFICIENTS (Cm, SeC-’) USING VARIOUS Temp. Gas pair Potential function He-N,

CO,-N,

He-Ar

CO,-O,

CH4-0,

Hz-0,

CO-0,

L-J M.B. (ExP-6) I.P.R. Exp. R. L-J (viscosity) I.P.R. (scattering) L-J M.B. (ExP-6) I.P.R. Exp. R. L-J (viscosity) L-J M.B. (ExP-6) I.P.R. Exp. R. L- J (viscosity) I.P.R. (scattering) L-J M.B. (ExP-6) I.P.R. Exp. R. L- J (viscosity) L-J M.B. (ExP-6) I.P.R. Exp. R. L-J (viscosity) L-J M.B. (ExP-6) I.P.R. Exp. R. L-J (viscosity) L-J M.B. (ExP-6) I.P.R. Exp. R. L- J (viscosity)

(OK)

300

500

lo00 1500

2000

2500

3000

0.741 0.741 0.745 0.746 0.723 0.174 0.174 0.177 0.178 0.156 0.757 0.755 0.763 0.765 0.756 0.161 0.162

1.77 1.78 1.76 1.76 1.69

5.65 5.66 5.68 5.69 5.34 5.54 1.45 1.46 1.46 1.46 1.27 6.17 6.23 6.25 6.26 5.60 5.98 1.43 1.42 1.44 1.44 1.27 1.95 2.00 1.99 1.99 1.79 6.93 6.99 7.06 7.10 6.27 1.73 1.74 1.76 1.77 1.61

17.9 17.8 18.3 18.5 16.9 19.1 4.62 4.60 4.93 5.04 4.02 19.6 19.7 21.0 21.5 17.7 20.5 4.59 4.55 5.04 5.19 4.04 6.23 6.38 6.88 7.11 5.67 22.2 23.1 24.3 25.2 19.8 5.48 5.54 5.76 5.89 5.07

25.9 25.7 26.7 27.2 24.5 28.4 6.69 6.49 7.29 7.54 5.82 28.4 28.4 31.0 32.1 25.7 30.6 6.66 6.58 7.54 7.90 5.85 9.03 9.23 10.26 10.77 8.21 32. I 34.0 36. I 38.1 28.7 7.93 7.88 8.44 8.70 7.34

35.0 34.6 36.3 37.2 33.2 39.3 9.05 8.99 10.0

0.164

0.164 0.156 0.226 0.230 0.229 0.230 0.225 0.821 0.818 0.828 0.830 0.830 0.224 0.224 0.224 0.225 0.208

~

POTENTIALS’

~

-

0.440 0.439 0.434 0.433 0.389 1.89 1.90 1.86 1.86 1.77 0.419 0.418 0.412 0.410 0.389 0.581 0.593 0.574 0.570 0.554 2.09 2.07 2.06 2.05 1.97 0.542 0.542 0.538 0.537 0.503

11.1 11.1 11.3 11.3 10.5 11.4 2.86 2.86 2.98 3.01 2.49 12.2 12.1 12.7 12.8 11.0 12.3 2.84 2.81 3.00 3.04 2.51 3.86 3.95 4.11 4.18 3.52 13.7 14.1 14.5 14.9 12.3 3.40 3.39 3.52 3.57 3.15

10.5

7.87 38.5 38.2 42.6 44.6 34.8 32.1 9.01 8.89 10.48 11.15 7.92 12.22 12.45 14.23 14.23 11.1 43.4 46.6 50.0 53.6 38.8 10.72 10.62 11.53 11.99 9.94

~.

“Pressure = I atm:. L-ILennard- Jones ( 1 2-6) ; M.B. (Exp-6) = modified Buckinghan (Exp-6); I.P.R. - inverse power repulsion; E X ~R.. -~ Exponential repulsion. 1

SURVEY OF DILUTEGASTRANSPORT PROPERTIES

295

TABLE XV(continued) Temp. Gas pair Potential function

HZO-0,

H,-Ar

L-J M.B. (ExP-6) I.P.R. Exp. R. L-J (viscosity) L-J M.B. (ExP-6) I.P.R. Exp. R. L-J (viscosity)

(OK)

~

300

500

lo00

1500

2000

2500

3000

0.288 0.286 0.294 0.294 0.214 0.847 0.851 0.861 0.861 0.821

0.692 0.691 0.691 0.690 0.620 2.16 2.15 2.13 2.11 1.96

2.21 4.32 2.20 4.31 2.21 4.35 2.21 4.37 2.08 4.07 7.18 14.2 7.20 14.5 7.27 14.9 7.26 15.1 6.25 12.3

6.96 6.93 7.05 7.11 6.61 23.0 23.9 24.9 25.4 19.8

10.09 10.00 10.24 10.38 9.6 33.3 35.2 36.9 38.3 28.6

13.65 13.48 13.90 14.16 13.3 45.0 48.3 51.0 53.6 38.7

enable reliable determination of potential parameters directly as described. As a consequence, the usual situation is that one is forced to use parameters obtained indirectly from other gas properties. T h e best method is to use potential parameters for the pure gases which have been determined from viscosity data, plus combining rules by which the parameters for the interaction of unlike molecules may be estimated from those for the respective species making up the diffusion pair. Using the general assumption that an unlike potential q i j is related to the potentials for the pure species by

v . = ( v .Ifv3..)I12 3 it may be shown that the following combining rules are obtained: inverse power repulsion:

exponential repulsion:

A. A. WESTENBERG

296

T h e Lennard-Jones (12-6) model is the only one for which extensive viscosity fitting has been done, and pure gas parameters for this potential have been given in Table 111. By using these values for the pure gas parameters in the combining rules given by Eq. (36), the values for aii and eij for an unlike pair may be estimated. These may then be used in Eq. (32) with the tabulated collision integrals to compute D i j . T h e entries in Table XV labeled L-J (viscosity) were obtained in this way. It will be noted that these theoretical values are generally low compared to the other Lennard- Jones entries which were fitted from experimental diffusion data and extrapolated. This seems to be generally true of the viscosity-predicted diffusion coefficients, the discrepancy being as much as 20-30y0 in some cases at high temperature. T h e room temperature viscosity predictions are very good-generally within 5 %-which is the case also for the other gas pairs given in Table XII. It is often very convenient to make use of an approximate analytical expression for the diffusion coefficient as a function of temperature. Thus for values of T* > 3, the Lennard-Jones (12-6) collision integral may be empirically fitted to within 2 yo by the expression Q".l'*

1.12 ___

(p90.17

When this relation is inserted into Eq. (32) one obtains the relation D,,

1.66 x 1 0 - 3 [ ( ~ , R3

+ M,)/M,M,Iw~.~~

Po;,(E1,/k)0.'7

(for

T* > 3)

(38)

This may be used with the potential parameters of Table I11 and the combining rules to estimate diffusion coefficients in good approximation to the exact use of the collision integral tables. It should be emphasized, however, that the approximation breaks down rapidly for T* < 3 and it should not be used in such cases. 4. Digusion Coeficients Determined from Experimental Data on Binary Mixture Viscosities

Recently there has been a revival of interest in the calculation of binary diffusion coefficients indirectly from measurements of the viscosity of mixtures of the corresponding gas pair. This is a rather more direct procedure than the calculation from viscosity parameters and combining rules just discussed. Examination of the relation given as Eq. (10) for the viscosity of a binary mixture reveals that it depends on the pure

SURVEY OF DILUTEGAS TRANSPORT PROPERTIES

297

component molecular weights and viscosities, the composition, the ratio of collision integrals A:, = Q~~2)*/52(11;1)*,and the quantity q12 defined by Eq. (1 I). As previously noted, the A:, is practically independent of temperature and may be taken equal to 1.10 for any realistic potential. Comparison of Eq. (1 1) with Eq. (32) shows that q12 is related to the binary diffusion coefficient by

T h u s a knowledge of the viscosities of the pure components and that of a mixture should allow the binary diffusion coefficient to be determined with good reliability. This sort of calculation has been carried out by Weissman and Mason (94) for a number of simple nonpolar gas pairs and by Weissman (95) for polar (one member only) and polyatomic pairs. T h e agreement of the diffusion coefficients determined in this way with the directly measured values was generally excellent. I t would seem, therefore, that this' approach provides a good method of arriving at binary diffusion coefficients for cases where mixture viscosity measurements may be simpler and more reliable than direct diffusion experiments. A case in point is the determination of the binary diffusion coefficient of H-H, from the measured mixture viscosity. This has been done by Weissman and Mason (32) using Harteck's (14) old data for the viscosity of H-H, mixtures, and by Browning and Fox (16) using their own modern viscosity measurements. Both used essentially the same theoretical values for the viscosity of pure H given in Table VIII, and the DH-H, determined were in close agreement. A comparison of Weissman and Mason's values of D H P H (extrapolated , at the higher temperatures beyond the range of temperature of the original mixture viscosity determinations) with the directly measured values of Wise (77) is given in Table XVI. Since Wise's data were only relative, they have been converted to absolute values using the reference value derived from viscosity at 293°K. T h e agreement is only fair and apparently gets worse at the higher temperatures. T h e need for more direct and indirect measurements in this area is apparent.

5 . Dissociated Gases When the procedure involving mixture viscosity is not feasible, the binary diffusion coefficients of such pairs as 0-0, , N-N, , etc. must be calculated wholly from theory. As noted in Section II,C,4, this type

A. A. WESTENBERG

298

TABLE XVI COMPARISON OF DIRECTLY MEASURED DIFFUSION COEFFICIENTS (AT 1 ATM) FOR H-H, W I TH THOSE OBTAINED FROM MIXTURE VISCOSITIES~

T C'K)

Direct data*

From mixture viscosityC

293 400 500 600 700

2.01 3.42 4.62 5.93 7.24

2.01 3.42

5.02

6.88

8.98

Direct relative data converted to absolute values using viscosity-derived values at 293°K. Underlined values extrapolated beyond range of original mixture viscosity data. Wise (77). Weissman and Mason (32).

of situation may be complicated by the possibility of multiple interaction potentials. T h e averaged collision integrals of Yun and Mason (35) have been used to calculate diffusion coefficients for N-N, , N-N, 0-0, , 0-0, and 0,-0, (96) and for H-HI H,-H, , and H-H, (33)up to very high temperatures. These are the best sources of information of this type at the present time. Apparently no calculations for other labile atoms or radicals have been made. When the species are not in their ground state but are electronically excited there is reason to believe (97) that diffusion coefficients would be abnormally small because the effective interaction cross sections would be large. Only crude estimates for these can be made at present, however.

6 . Polar Gases For binary diffusion in polar-polar pairs, the nonspherical nature of the potential interaction presumably should make the ordinary kinetic theory formulas invalid. T h e Mason-Monchick polar gas theory for mixtures (25) should apply to this case, but no experimental data on a polar-polar diffusion coefficient have been reported. T h e lack of data may be partly caused by the fact that such pairs usually undergo chemical reaction when mixed. Mason and Monchick do give polar gas potential parameters and combining rules applicable to such systems. T h e same paper (25) also presents extensive comparison with experiment for polar-nonpolar cases with quite good results.

SURVEY OF DILUTE GASTRANSPORT PROPERTIES

299

7. Multicomponent Diffusion T h e rigorous expressions for diffusion coefficients in multicomponent mixtures [see Hirschfelder et a(. (3, p. 541)] are extremely complex and unwieldy and have rarely been used. I n the special case where one component is present as a trace in a mixture, its effective diffusion coefficient is related to the various pair diffusion coefficients and the composition by

This expression has been well verified experimentally in two independent studies (98, 99).

SYMBOLS A A A C CV

d

D

I

k

k

L m M N P

Q Q r r

rm

R S

S t

T U v

Parameter in exponential repulsive potential Collision integral ratio Q[*"'/Q"~l' Area Inlet correction Specific heat at constant pressure Parameter in inverse power potential Concentration diffusion coefficient Collision integral tabulated for exponential repulsive potential Boltzmann constant Reaction rate constant Length Mass flow rate Molecular weight Molar concentration Pressure Heat flow rate Volumetric flow rate Radius Intermolecular distance Parameter in modified Buckingham potential Ideal gas constant Slip correction Slope Time Temperature Velocity Velocity

X

X

Y z

z LY

a a

6

4 E

4 T

h

P

P P L7

Q

Q

Mole fraction Quantities in theoretical relations for mixture viscosity Quantities in theoretical relations far mixture viscosity Distance Quantities in theoretical relations for mixture viscosity Power series coefficient Parameter in modified Buckingharn potential Parameter In(A/kT) in exponential repulsive potential Parameter in inverse power potential Dimensionless parameter (4/5)A* Parameter in Lennard- Jones and modified Buckingham potentials Angular orientation function in Stockmayer potential Shear viscosity Thermal conductivity Dipole moment Density Parameter in exponential repulsive potential Parameter in Lennard- Jones potential Intermolecular potential energy function Collision integral

300

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