The influence of the density on the viscosity coefficient of gases

Coremans,

Physica

J. M. J.

Beenakker,

J. J. M.

26

653-663

1960

THE INFLUENCE

OF THE DENSITY ON THE VISCOSITY COEFFICIENT OF GASES

by J. M. J. COREMANS Suppl.

No. 117r to the Communications

and J. J. M. BEENAKKER from the Kamerlingh Nederland.

Onnes

Laboratorium,

Leidfn,

Synopsis A corresponding dependence showed

a very

An empirical viscosity

states

treatment

of the viscosity simple relation

coefficient

based

coefficient

temperature is given

for simple

on molecular

of several

units is given

gases. The

increase

for the density in the viscosity

dependence. that makes

it possible

to predict

gases over a large range of densities

the increase

of the

and temperatures.

1. Introduction. In the course of our experiments on the density dependence of the viscosity coefficients of helium gas at hydrogen temperatures it was of interest to know the importance of quantum effects. At present, however, no quantum mechanical theory at higher densities is available, while the Enskog theory is, furthermore, only approximate. Hence, it seemed important to compare our data with those of other gases in a corresponding states treatment. In principle one can use two methods: a) The data may be expressed in critical quantities, b) Molecular constants can be used for the reduction factors. The first method has the disadvantage that an undetermined part of the quantum effects is already taken into account, as the critical quantities are strongly influenced by quantum effects. The second approach is more promising. In the course of this work we found that it was possible to derive an empirical formula from the existing viscosity data at higher densities, in which the relative increase of the viscosity coefficient is expressed in terms of a reduced density and the temperature. A further analysis showed that it is possible to extend the Enskog formula in a semi-empirical way. 2. A summary of the experimental different gases, we first of all checked different sources. In general it appears experiments have been performed by -

In treating the results for the the agreement between the results of to be rather good. The most extensive the group of M ic hels i)2)3)4). From

data.

653 -

654

J. M. J. COKEMAXS

our analysis

ANI)

it became clear that these data are very reliable,

their results agreed with the majority Michels c.n. always report their data density is the most appropriate use Michel s’ data throughout perimental data : 1) N2:

2) A: 3) CH4:

4) coz:

5) Hz:

J. J. hf. HEES.1KKEII

of the other as a function

as in all cases

data. Furthcrmorc, of density. As thcl

quantity to use, ne found it convenient to our work. WC analyscd the following VS-

‘Michels e.a.l), Lazarre and Vodars), Koss e.a.6), Kcstin e.a. 7) Sibbitt e.u.s), Iwasakig) and Makitalo). All results agreed well with each other. The older measurcmcnts by 13o y dll) showed too large a spread to be taken into account. We used the isotherms of Michels at 25” and 75°C and pressures up to 1000 atm., and that of Ross at -50°C to 750 atm. Michelse.a.“), Makitalo) and Kcstin e.a.7). Hcrc the agrecmcnt is reasonable. We used the isotherms of Mic he1 s at 0” and 75°C. Kusslz), Ross e.a.6) and Carrls). There is a good agreement. We used isotherms of Kuss at 25” and 100°C. The measurements by Ross at -50°C were not used as not enough data tverc available at lower densities to enable an extrapolation to zero density. Michels c~.a.3), Warburg and \‘on Babola), PhilipsI”), Stakelbeckl6) and Comings and Eglyl7). All the results agree well with each other. S c hr o e r and Beck e r 1s) performed measurements with a rolling sphere: their data differ greatly from that of the others showing a larger dependence on the density. We used the isotherms of Michels at 0” and 75”C, but limited the pressure range to 500 atm. so as to remain in the same region of reduced densities for which there are data for the other gases. Michels e.a.4), Kestin e.a.7) and Gibsonlg). The data agree very well. Again we omitted the results of Boydll). Furthermore, there are data of Kussl”) which differ appreciably from those of the other authors. We used isotherms of M i chel s at 25 and

75°C to pressures up to 1000 atm. Michels e.a.4). Here we only used the isotherm at 25”C, as thcrr 6) D,: appeared to be no appreciable difference with the results on Hz. Ross e.a.6), Kestin e.u.7) and Coremans20). Only the first work 7) He: covers a large density range. We used the isotherm of Ross at -25°C to pressure of 750 atm. For the density we used data as given in the original papers, with the exception of CH4, where no densities were given. Here we used density data of Kvalnes and Gaddy21). In fig. 1 the viscosity isotherms for most of the gases used in our analysis are plotted. 3. Reduction with critical units. For the sake of completeness we will give here the results of the corresponding states treatment using critical

THE

_____ parameters. atures felder,

INFLUENCE

OF THE

DENSITY

ON THE VISCOSITY

Table I gives a survey of the critical

pressures

655

PC,and the temper-

Tc,as used by us. These data were taken from the book by HirschCurtiss

and Bird,

table

4, l-222).

9r

7-

5 :

lck

P

250

)

Fig. 1. Viscosity v n

Nz A CH4

A

0

COs

o

Hz

X +

amaqot

isotherms

Michels Kuss

of several

e.a. I), Ross

Michels

500

gases.

e.a. 6)

e.a. 2) 12)

Michels

e.a.

3)

e.a. e.a.

3)

D2

Michels Michels

He

Coremans

4)

20)

We made plots of the relative viscosity qn(= q/qo, in which 71 and ~0 are the coefficients of viscosity in the gas and at zero density resp.), as a function of the reduced pressure PR(= P/PC)from the data as mentioned above. As results are not available over a large range of reduced temperatures TR(==T/T,)for any of the gases, it appeared advisable to derive a plot of qn versus TR at several values of P R from these graphs. This is shown in fig. 2. From this figure it becomes clear that all the data conform rather

656

J.

iU. J. COREMANS ._.____

AIGD J.

J.

hI. BEENAKKEK

well to such a corresponding states treatment. We like to point out here that the results for argon deviate a bit from the drawn curve, a fact that will also be noticed Our low

in later

treatments.

temperature

helium

data

are

in good

agreement

with

this

treatment.

33.5

126. I

48

151

45.8 72.85

304.2

190.7

12.8

33.3 38.4

16.4 2.26

Icig. 2.

r/ILas a function

of T IL at several

values

Kg

Michels

‘,

.\

Michels

i

Cl14

KUSS

;i

5.3

of lulL on a tloublc ~.a. I)

logarithmic

scale.

e.a. ‘i)

e.a. ‘) l”)

0

CO-

Michcls

1)

Hn

Michels

e.a. J,

X

TI? Hc

Michels Corcmans

e.a. 4, 20)

-I-

lioss

e.a. 3)

4. Reduction with molec’ular units. To obtain a corresponding states treatment in molecular units it is convenient to express the relative viscosity, qn, as a function of the reduced temperature T*(== kT/e) and the fraction of the volume occupied by the molecular core: &cl* = ~7c~z(0/2)3.

THE

INFLUENCE

OF

THE

Here E/Kand cr are respectively diameter

of the molecules

VISCOSITY

657

the depth of the potential

well and the

DENSITY

ON

THE

in terms of the Lennard

Jones

(6-12)

potential;

K the Boltzmann constant, and n is the number density. Table II gives a survey of the molecular parameters used. They were taken from Hirschfelder e.a.ss), table I A, with the exception of the data denoted by an asterisk, which were obtained Bureau of Standards 2s). TABLE Molecular

from reports

of the National

II

reduction

factors

Elk “K

Gas

9 1.46*

NZ A

3.681*

119.5*

3.421*

CHI

148.2

3.817

CO2 HZ

200*

3.952*

37.00

2.92%

1%

37.00

2.928

He

10.22

2.556

Again plots were made of ?‘jRas a function of d* for the different isotherms. From these curves smoothed values were derived for ?‘jn as a function of d* at these values of T*. These values are shown in table III and fig. 3, where the values of (?jn - 1) versus T* are given on a double logarithmic scale. TABLE Gas _-

N2

Author -Michels Ross

A CH4

1) s)

Michels Kuss

2) =)

CO2

Michels

3)

HZ

Michels

4)

Dz He

Michels

4)

Ross 6) C 0 r e m a. II s 20)

T*

I

III

d* = 0.04

I

0.2

I

I

1.2

0.4

0.8

1.220

2.59

1.195

1.685 1.62

1.222

1.71

2.71

1.125

1.307

2.00

3.395

1.100

1.275

1.855

2.98

1.013

1.098

1.266

1.813

2.65 1.74

1.013

i ,089

1.241

1.754

2.90 _

1.022

1.110

1.300

1.975

3.19

8.05 9.41

1.006

1.040

1.109

1.388

1.003

1.028

1.089

1.344

1.92 _

8.05

1.008

1.046

1.119

24.29

1.005

1.032 1.122

1.09

1.405 -

1.94 -

3.26

1.013

1.071

3.80

1.014

1.080

2.44

1.014

i ,087

2.28

1.018

2.91 2.12

1.015

2.00

-

1.024

2.43

-

First of all, one can see that the data agree rather well with this type of treatment, again with the exception of the A data. Even our low temperature data lie practically on the curves. From this plot, furthermore, it becomes clear that the main temperature dependence is nearly equal for all values of d*. Hence it is possible to write for the density dependence of the coefficient of viscosity (7~ -

1) = T*“*f(d*)

with cc = -

0.59.

658

J. AI. J. COKEbIANS ~_____

AND

J. J. M. BEENAKKEK

To analyze the behaviour of f(d*) ,\Vi’111adc a plot of (,/,t ~~- 1) T”O.5” vcrs11s d* (cf. fig. 4). As could be expected, all the experimental data fall practically on the same curve. The agreement is rather good, especially if one takes into account the uncertainty in the molecular constants. It sppcared possible to describe the behaviour of j(d*) by a series expansion in d*:j(d*) =I- Citi” -I- CZECH_I + C3d*3 in which Ci = 0.55, C2 = 0.96 and Cs = 0.61. In fig. 4 this formula for /(d*) is represented by the drawn cur\.e 1. Summarising the relative viscosity of a gas can be rcl~rcscnted by the expression 71~ = 1 + (0.55 d* + 0.96 d*z + 0.61 d*3) F-O.59 (1). To test this empirical relation we made a deviation plot for the isotherms of He, Hz, NB and CH4 at reduced temperatures ranging from 25 to 1.7, which is shown in fig. 5. In this plot we used isotherms which were not taken into account in the derivation of our formula. The deviations appear to be, in general, smaller then 3 $4. Thus expression (1) may be useful in calculating the increase of the viscosity for not too complicated molecules in the moderate density range, for not too low reduced tempcraturcs. As for theoretical work the low density limit of ?yl/ird* is of iml)ortancr

THE

INFLUENCE

OF THE

4-

DENSITY

ON THE

VISCOSITY

659

T

Fig. 4. (q~ -

1) T*o.ss as a function

Michels

q

Nz A CH4

0

CO2

Michels

e.a. 3,

o

Hz’

Michels

e.a. 4,

X

D2

Michels

+

He

Ross

v n

e.a. 1)

Michels Kuss

Ross

of d*. (1)

e.a. 6)

ea. 2, 12)

e.a. 4, e.a. 6)

Coremans

20)

for theoretical work, we tried to estimate the accuracy with which the value of Cr can be determined from the existing experiments. It appeared to be of the order of 10%. For the value of Cs the accuracy is far less, as it depends completely on whether or not one takes a third term into account. This third term was introduced for practical purposes, namely to obtain also agreement at higher densities. Hence not too much physical significance should be attached to it. To obtain an idea of the contribution of the different terms as follows from formula (l), we have drawn the influence of the linear, square and cubic term in fig. 4 (resp. curve 2, 3 and 4). Finally it is interesting to compare this value with the determinations by K es tin e.a. 24). They performed measurements in the low density range only. Of course, a rather high sensitivity is needed, as in this region the increase in the viscosity is of the order of a few percent at the most. From their results they calculated the linear term in the increase of the viscosity with density. We will limit ourselves to those cases where the increase of the viscosity was of the order of 2% and larger, for in the case of

660 smaller

J. M. J. COKEMANS

AND

changes there is a greater

2. 3. 4.

(‘1 vnlws,

calculntctl

(;:I5 h COr

,1* I

isotherms Irom Iormul;~ (1) IZOSS r.a. ‘i) ICOSS C.Q. “) Michcls (‘.u. ‘I) ICOSS e.n. “)

(~xp(~rilll(‘lltdtht;: of I
I (‘1

’ p ill :,,I,. ‘/IL (III:Ls.\~.llW) irll:lr. \..Llil,.)

2.45

0.65

1.031

2.49

0.57

1.026

26.0

~

1.47

0.047

1.020

22.4

~

3.20

Iir x1

frml

600

amagat

300

of some viscosity _. I* 24.29 T* 1.67 7‘* z a.73 7‘* _ 2.71

CH4 Hz 92

errors. In table IV

from these data.

)

P

Fig. 5. Ikviation 1. ifc

HEENAKKEK

chance for systematic

we give the value of Cl determined

-56

J. J. X.

1.71

29.3

0.63

1.031

21.1

/

0.52

1.071

57.7 63.1

0.50

1.077

o:!

3.26 2.54

~

0.465

1.050

Se

1.35

I

0.48

1.070

49.0 ~

31.0

In general there is a good agreement. This is of special importance values were derived mainly from data at higher densities. 5. Some remarks

on the Ens kog /ormula.

As is well known,

as our

Enskog

derived a treatment of the influence on the density of the viscosity coefficient for a hard sphere molecular model. For the increase of the viscosity he obtains an expression of the form: ,‘/R=

l/x + 0.82

+ 0.761422

where x = 1 + 0.625 d* + 0.2869 d*” + . . . and 2 the diameter of the hard spheres. It is clear temperature independent increase of the relative density. It seemed interesting to look wcther the simple

+ ......

(2)

= $zw~,~,~,. Here 0 ,,,S, is that this formula gives a viscosity as a function of temperature

dependcncc,

THE

INFLUENCE

OF THE

DENSITY

ON THE

as found by us, could be easily introduced simplest

way is to make the diameter

661

VISCOSITY

in the E nskog

formula.

The

of the hard sphere molecule temper-

ature dependent 8). For this purpose one can compare the viscosity formulae for a real and a hard sphere gas. The low density viscosity coefficient is given classically by the expression 52/nmkT iv11 =

16n&W~s’*(T*)

where Q@YQ*(T*) is the collision integral that takes into account the influence of the intermolecular potential with respect to a hard sphere i.e. for a hard sphere Q(2~2)*(T*) = 1

Fig. 6. q~ -

1 versus

1. Enskog I7 Nz _4 a

theory

d*[Q(2~z)*(T*)]3/2 (form. 2)

Michels

e.a. 1)

Michels

e.a. 2)

0

CH4 COs

Kuss 12) Michels e.a. 3,

o

CO2

o

Hs

Michels Michels

X

nz

Michels

+

He

Ross

q

Ross

e.a. ‘3)

e.a. 4, e.a. 4, e.a. 4, e.a. 6)

Coremans

20)

662

J.

M.

J. COREMANS

AND

J. J. hf. BEEiYii\KKER

At temperatures between T* = 2 and 20, _0(2,2)* is roughly proportional to T4’-O*2. Hence a hard sphere gas should have a diameter proportional to T*-0.1. As the Enskog formula is an expansion in terms of 03, the quadratic term will in this way become proportional to T*-0.6. Thus the general behaviour with respect to temperature will be given correctly at intermediate densities. To make a more quantitative comparison it seems useful to introduce a temperature dependent diameter in the hard sphere

treatment

in the following =

0h.s.

way:

(3)

gL.J. [WJ)*(T*)]t.

Here 0~. J. is the Lennard- Jones diameter. The values of sI)@.2)* (7’“) are tabulated in Hirschfelder, Curtiss and Bird 22), table ILM. Tf one introduces relation (3) into equation (2) one obtains an expression that is now dependent on d*[Q(2,2) *( I’*)] I. To check whether the gases conform to this expression we plotted VR - 1 versus d*Gnr (cf. fig. 6). From our foregoing work it is clear that in this treatment a larger q-cad of the data may be expected, as the temperature dependence of the> linear part of the curve is no longer correct. At intermediate densities however, results are reasonable, as should be expected. The absolute values derived from the Enskog formula are, however, in strong disagrccmcnt with the data. Better results are attained with the Enskog relation by using (l/ 1.2) d*LY (curve 2) or (l/ 1.3) d*LY (curve 3) instead of d*G (curve 1). This means that the diameter that enters in the Enskog formula has not only to be made temperature dependent, but must also be changed slightly. It is clear that such a treatment is an empirical one, which will only give reasonable results at those temperatures where the repulsive part of the potential is predominant, as is the case in the tempcraturc range under consideration. Acknowledgements. It is a pleasurc to express our sincere thanks to Prof. Dr. J. Ross, Dr. E. G. I). Cohen and Dr. P. Mazur for the discussion on different aspects of this paper. This work is part of the research program of the group for molecular physics of the “Stichting voor Fundamentecl Onderzoek der Materie (F.O.M.)” and has been made possible by financial support from the “Ncderlandse Organisatie VOOI Zuiver Wetcnschappelijk Onderzoek (Z.W.O.)“. Keccivcd

3-5-60

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hlichrls,

A. and

2)

Rfichels,

A.,

I(otzcn,

Gibson, A.

and

I<. O.,

Schnurman,

Prw.

my.

SW-. \I’.,

A IW Phys~ca

3)

Michcls,

A.,

Botzen,

A.

and

Schuurman,

W.,

Physica

(1931) ‘0 %I

288. (1954) (1957)

1141. 95.

THE

INFLUENCE

4)

Michels,

A., Schipper,

5)

Lazarre,

F. and Vodar,

6)

Ross,

7)

Kestin,

8) Sibbitt, 9) Iwasaki,

DENSITY

A. C. J. and Rintoul, B., Comptes

J. F. and Brown, J. and Wang,

OF THE

Rendus

ON THE

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VISCOSITY

19 (1953)

663 1011.

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W. L., Hawkins,

ASME

80 (1958)

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16.

‘1)

13) Carr, N. L., Inst. Gas Technol., Research Bull. 23 (1953). E. and Von Babo, L., Ann. Phys. (Leipzig) 17 (1882) 390. 14) Warburg, 15)

Philips,

P., Proc. ray. Sot. A87

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(John

(1912) 48.

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Wiley

V. L., J. Amer. Chem. Sot. 53 (1931) 394.

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