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<5 NBS TECHNICAL NOTE 693

NBS

Pubii - cations

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Predicted Values of the Viscosity and

Thermal Conductivity Coefficients

of Nitrous Oxide

QC

100

U5753

no. 693

1977
NATIONAL BUREAU OF STANDARDS

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a Located at Boulder, Colorado 80302.
Predicted Values of the Viscosity and

Thermal Conductivity Coefficients

of Nitrous Oxide

m* t * iwi

1

r

Howard J. M. Hanley

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

* L T J /

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Betsy Ancker-Johnson , Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued March 1977
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 693

Nat. Bur. Stand. (U.S.), Tech Note 693, 64 pages (March 1977)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON"; 1977

For sale by the Superintendent ot Documents, U.S. Government Printing Office , Washington, D.C. 20402

Order by SD Catalog No C 13. 46:693). Price $1.20 (Add 25 percent additional for other than U.S. mailing).
CONTENTS

Page

Abstract 1

1. INTRODUCTION 1

2. DATA SURVEY 1

3. PREDICTION METHOD 2

3 .

1

Summary 8

3.2 Kinetic Theory 8

3.3 The Behavior of the Thermal Conductivity Coefficient in the

Critical Region 9

4. COMPARISONS WITH DATA 10

5. TABLES 12

5.1 Assessment of Accuracy 12

6. MIXTURES 31

7. CONCLUSION 31

8. ACKNOWLEDGMENTS 31

9. REFERENCES 48

APPENDIX. Application of Statistical Mechanics 50

Note on Units

The results in this paper are given in British Engineering units at the

specific request of the sponsor. Selected results, however, are also given

in SI units to conform with the policy of the National Bureau of Standards

to promote the SI system.

in
LIST OF FIGURES

Page

Figure 1. Top curve: 1 atm viscosity percent deviation plot

between values predicted from equation (20) and the

correlation equation (31) . Percent deviation is defined

as n[eq. (20)] -n[eq. (31) ]*100.0/n[eq. (31)]. Bottom

curve: 1 atm thermal conductivity deviations between

equation (21) and equation (31) 14

Figure 2. Dense gas and liquid isotherms for the thermal conductivity coefficient calculated from equation (21) (curves)

compared with the data of reference [19] (open and filled

circles) 16

Figure 3(a). Viscosity coefficient of dilute N„0: comparison between

kinetic theory values and equation (31) 58

Figure 3(b). Plot of the second virial, B, versus temperature. Data,

filled circles, from reference [30]. The curve is from

equation (9A) , while the triangles are results discussed

in the text 58
PREDICTED VALUES OF THE VISCOSITY AND THERMAL CONDUCTIVITY

COEFFICIENTS OF NITROUS OXIDE"

Howard J. M. Hanley

The viscosity and thermal conductivity coefficients of nitrous

oxide are calculated for temperatures between 180 and 900 K (330 to

1600°R) for pressures to 23 MPa (^ 3500 psi) . Tables of values are presented. Two mixtures with carbon dioxide are also discussed. These

transport coefficients (for the pure fluid and for the mixtures) were

predicted from thermodynamic data. Details of the prediction procedure

are presented. Estimates of the accuracy of the tabular values are + 6% for the viscosity and + 8% for the thermal conductivity.

Key words: Carbon dioxide; corresponding states; mixtures; nitrous oxide;

prediction; thermal conductivity; transport property; viscosity.

1. INTRODUCTION

In this report we discuss the transport coefficients — the viscosity

coefficient (f)) and the thermal conductivity coefficient (A) — of nitrous oxide,

N~0. Tables of values are presented from 0.1 to 23 MPa (approximately 15 to

3500 psi) for temperatures between 180 and 900 K (approximately 330 to 1600°R)

.

Two selected' mixtures of nitrous oxide and carbon dioxide are considered briefly,

Nitrous oxide is a common substance yet the transport coefficients have not

been measured over a wide temperature and pressure range. In fact, the data are so scarce we prefer to obtain the coefficients by calculation [1]: the

limited data are used only to check the procedure where possible.

2. DATA SURVEY

A bibliography of data sources for the thermodynamic and transport

properties of nitrous oxide was prepared by the Cryogenic Data Center, National

Bureau of Standards, Boulder, Colorado. The appropriate references for the

Contribution of the National Bureau of Standards, not subject to' copyright

.
viscosity and thermal conductivity are as follows: viscosity coefficient , references [2]-[8]; thermal conductivity coefficient , references [9]-[23].

We have correlated the transport coefficients of several simple fluids

[Ar, 0„, N„ , CH, , C~H,] in a series of papers [24]. Criteria were set up to

evaluate data critically, and an equation to represent the coefficients was

proposed. Unfortunately, this procedure cannot be applied to nitrous oxide at

this time because suitable data are not available for analysis. We have been

able to find only two data sets for the dense gas and liquid, those from

references [19] and [23], for the thermal conductivity. The other references

cover the dilute gas only.

As remarked, therefore, we decided to present tables based on a prediction

procedure, which will be outlined in the next section.

3. PREDICTION METHOD

The method used is an application of a procedure introduced in detail in

reference [1]. An outline is given here.

As is well known, if classical two-parameter corresponding states applies

to two fluids designated a and o, one can write the viscosity of one at a density

(p) and temperature (T) , in terms of the other:

n (p,t) = n (p , t )(-*) -« (-«) (i)

lp o

Also for the thermal conductivity,

1/2/ c\2/3/^cvl/2

»

(pvv| (7) (7) Vp.t) = up„_. t„ j\~\ \-a 1^1 (2)

In the above equations, M is the molecular weight, and p and T are the

density and temperature respectively, for fluid o to correspond to the density

and temperature for fluids a. The variables can be expressed in terms of the

critical parameters, p° and T

C

; thus, p^ = p(p

C

/p*~) and T = T(T /T ). The

CXjO OCX \X y O (JUL
coefficients r| and A also have to be scaled as shown in equations (1) and (2). o o It is common practise to calculate the coefficients for a given those for

o — which is then treated as a reference fluid.

Equations (1) and (2) can be extended for a mixture, x, if it is assumed

that the mixture can be represented as an equivalent pure fluid — the one-fluid

approximation — and that the mixture is "conformal," that is, all components

obey the two-parameter corresponding states law. Hence one can write

1/2/ c\2/3/_c\l/2

V?>T) " Vpa,oTa,o' ,«(?) \tl \f>

;

and similarly for the thermal conductivity, A . Equation (3) can be derived

under well-defined assumptions [25] . The derivation further yields a set of

mixing rules:

Ct p

and

T

C (p

C

) 1 = Z Z x x T

C (p

C

) 1

(5) x

Kx „ a MP ag

where x is the mole fraction of a. The mixing rule used in this work for the mass is M = Z x M (which differs slightly from that proposed in reference [25])

The a$ variables of equations (4) and (5) are given by

T

c = F (t° T

C

) ly/2 (6)

and

/ c \ - l i (paS } = ^a3

1 , c ,-1/3 . 1 . c .-1/3

2

(paa> +

2

(p 33} (7)
where E, a and \\l D are binary interaction parameters, which are best obtained

CXp Ctp from experiment.

The mixing rules, equations (6) and (7), are consistent with the mixing

rules derived for thermodynamic properties using the one-fluid approximation.

If, therefore, the binary interaction parameters are assumed to be those valid

for thermodynamic properties , one has a procedure to predict the viscosity and

thermal conductivity coefficients of mixture x, given the equivalent

coefficients of the reference fluid.

Equations (l)-(3) are not general in that two-parameter corresponding

states is invalid for polyatomic fluids, or for mixtures containing polyatomic

molecules. The situation is, of course, paralleled with respect to the

thermodynamic properties and has been discussed extensively in this latter

context. For example, a recent approach due to Leland [26] and to Rowlinson [27],

is of interest here. According to Leland and Rowlinson a third parameter,

co — which can be taken as the Pitzer acentric factor — is introduced, but the

framework of simple two-parameter corresponding states is preserved, if one considers the scaling functions

f = (T

C /T

C

) ; h (P

C /P° ) <J> (8)

aa,o aa o aa,o aa,o o aa T aa,o

for fluid a with respect r to o. The terms 6 and 6 are called shape factors

aa,o aa,o

and are weakly varying functions of temperature and density:

6 = 1 + (co -co )F(T,p); <J) = 1 + (co -co )G(T,p) (9)

aa,o aa o ' ' Yaa,o aa o

where co and co are acentric factors for fluids a and o, respectively. Using

aa o the shape factors, the compressibility factor (for example) of a would be given

by the relation Z (p,T) = Z (ph , T/f ). With the appropriate expression 3 a o aa,o aa,o

for the Helmholtz free energy, the thermodynamic properties of a are completely

defined in terms of the properties of o.

It is a logical step to introduce shape factors into the viscosity and

thermal conductivity expressions: for fluid a, therefore
n (p,t) = n (p' , t' ) FHn (io) a o a,o a,o a,o

where

p' - Ph ; T' = T/f ; FHn = -£ ) h"2/3

f l/2

( , a,o aa,o a,o aa,o a,o \M / aa,o aa,o fe)'"

Similarly,

A (p,T) = X (p' , T

T ) FH

A (12) a

K ' o a,o a,o a,o

where

a,o \M V/ "aa,o aa,o /

aa,o

FH* - U h"2/ 3 £

l/2_

(13)

And for mixtures,

\( P' T) = % ( Tx,o

} <o (14)

A (p,T) = A (p' , T' ) FH

X (15) x o x,o x,o x,o

For mixing rules for h and f we take

x,o x,o

h =ZExx h D (16)

x,o

a

a 3 a3,o

f h =EZxxD f h D (17)

x,o x,o _ a 3 a3,o a3,o

with
1/2

fa(3,o " ^ag

(f

aa,o

f 3g,o ) (18)

and

a(3,o r ag

3l

;,oJ

1 ,1/3 1 ,1/3 I 3

2 haa,o +

2

h *« -1 (19)

An evaluation of equations (10), (12), (14) and (15) is discussed in

reference [1]. In particular we were interested if these equations could be

used to predict the transport coefficients of a or x, given values for the

reference fluid. Specifically, one might hope that equations could be used

successfully if the shape factors, 9 and (J), and the mixing interaction

parameters, £ R and \\) R

, were taken from a fit of thermodynamic (PVT) data.

Unfortunately, the equations could not represent data satisfactorily if the

above constraint was imposed.

We have, however, been able to propose a possible modification; we suggested

that one should consider expressions of the form

n (p,T) = n (p' , T» ) FHn X

n (p,T) (20) a

K ' o a,o a,o a,o a,o

and

A (p,T) = A (p

1 , T' ) FH

A XX (p,T) (21) a o a,o a,o a,o a,o

with corresponding expression for the mixture. Note that equations (20) and (21)

differ from those introduced earlier byJ the factor X or X„ . Based on how

a,o a,o

equations (20) and (21) represent experiment, it turns out that X should be

unity if fluids a and o follow classical two-parameter corresponding states,

as in the case of the rare gases and their mixtures [25], but should be a function of temperature and density otherwise.

No formal equation is available for X but a possible expression was determined in reference [1] . The expressions are based on a study of the

Modified Enskog Theory. This theory has been discussed in detail in our
previous work [28]. Following the arguments that are given in full in

reference [1], we obtained for the pure fluid (with corresponding expressions

for the mixture)

X^ (p,T) = Q G^ : X* (p,T) = Q G* (22)

ot,o ot,o ot,o a,o 0L,O 0L,0

Q is function of p and T defined as a,o

= [(bp) /(bp)

n

] [1 - 1/exP (pC/p)3]

(23)

where b is a term given by b = B + TdB/dT, with B the equilibrium second

virial coefficient. G and G are ratios given, respectively, by

-,a,o, a,o b

G

n = [] n

/[]

n and []/[]. The bracket expressions are [28]

a,o a o a o

[] n = [1/bpx + 0.8 + 0.761 bpx]

(24)

[] A = [1/bpx + 1.2 + 0.755 bpX ]

with

PR \3TJp !;^ - i < 25 >

for fluids a or o. R is the gas constant.

Notice that the calculation of the correction factors X and X require only

thermodynamic information for fluid a or x with respect to the reference fluid;

for example, b [Equation (23)] can be obtained from the second virial

coefficient of the reference fluid, B , according to the expression

o

(h B ) b (T) = h B + T d aa '° - (26) a aa,o o dT

and similarly for the term bpx [equation (25)].
3 . 1 Summary

To summarize the calculation procedure. To calculate the coefficients of

a fluid, a, or mixture, x, one needs (a) the viscosity and thermal conductivity

coefficients of a reference fluid, (b) the shape factors and, if necessary, the

mixing parameters of the fluid with respect to the reference fluid; these

quantities to be obtained from a fit of thermodynamic data. The viscosity and

thermal conductivity of a or x follow from the equations

n (P,T) = n (p

1 , T' ) FHn x

n

(20) a

VK ' 'o Ka,o a,o a,o a,o v '

\ (p,T) = A (p , T

? ) FH

A X

A (21) a o a,o a,o a,o a,o

n (p,t) = n (p' , t' ) FHn x

n

(27) x o x,o x,o x,o x,o

X (p,T) = X (p' , T' ) FH

A X

A (28) x o x,o x,o x,o x,o

In this work a is, of course, nitrous oxide. Methane was chosen as the

reference fluid and numerical values of the transport coefficients were those

of the correlation of Hanley, Haynes and McCarty [24]. The equation of state

for methane used here is due to Goodwin [29].

Shape factors for nitrous oxide with respect to methane, equation (8), were obtained by analyzing the PVT data of Couch, Kobe and Hirth [30]. The

procedure is due to McCarty [31] and will not be described here.

3.2 Kinetic Theory

For consistency, the transport coefficients of nitrous oxide for all

pressures and temperatures will be calculated by the method summarized in

section 3.1. However, alternative approaChs are possible for the dilute gas

since some data are available. In previous work on the dilute gas [32-34], we
have used an empirical fit of data and/or calculated values via statistical

mechanics and kinetic theory. This latter approach has several advantages,

one of which is that statistical mechanics and kinetic theory allow an assess- ment to be made of systematic error by comparing independently measured

properties through the pair potential function. For example, in references [32-34], we showed the transport properties of the rare gases, and oxygen and nitrogen

were consistent with their respective second virial coefficients. As a matter

of interest, nitrous oxide is briefly discussed along these lines in the Appendix.

3.3 The Behavior of the Thermal Conductivity Coefficient in the Critical

Region

It is now well known that the thermal conductivity behaves anomalously in

the vicinity of the critical point: in fact the coefficient will approach

infinity as the density and temperature approach their critical values. An

account of this phenomenon is given in reference [35], and it is considered in our previous data correlations of argon, oxygen, nitrogen and methane [24].

Undoubtedly, nitrous oxide will also display the anomaly.

The anomaly can be calculated directly and is discussed in reference [35]

but it is included indirectly in our corresponding states procedure as follows.

The thermal conductivity is given by equation (21)

A (p,T) = A (p , T ) FH

X XX (p,T) (21) a o a,o a,o a,o a,o

where a E NO and o E CH , , but A can be separated into two terms, the

conductivity in the absence of a critical point anomaly, A', and the anomaly

itself, A : o

A = A' + A

c (29)

o o o

Hence

A = A' FH

A X

A + A C FH

A XA (30) a o a,o a,o o a,o a,o
n Since we have incorporated r A in the methane correlation, A will have an

o a equivalent contribution.

4. COMPARISONS WITH DATA

Comparisons between our calculated values of the transport properties of

nitrous oxide and data are restricted to the dilute gas region, and to one data

set for the thermal conductivity [19]. First the dilute gas.

We attempted to assess the accuracy of the dilute gas viscosity data using

the arguments discussed in reference [24]. Data from references [6] and [7] were selected as the most reliable with an estimated accuracy of + 3%.

The dilute gas thermal conductivity data were also evaluated as far as possible. Data from references [10], [14] and [18] were selected as the more reliable with an estimated accuracy of + 6%.

Following the procedure of our previous work [24], the selected viscosity

and thermal conductivity data were fitted to an empirical function for

computation convenience

n° = GV(1)T X + GV(2)T 2/3 + GV(3)T 1/3 + GV(4) + GV(5)T 1/3

+ GV(6)T 2/3 + GV(7)T + GV(8)T4/3 + GV(9)T 5/3

(31)

and similarly for A°, but with coefficients GT(i)(i = 1...9) replacing GV(i)

in equation (31). n and X refer to the dilute gas viscosity and thermal

conductivity coefficients, respectively. Values of GV and GT are listed in

table 1.

10
Table 1. Values of the coefficients of equation (31) for the dilute gas

viscosity, n, , and the dilute gas thermal conductivity, A . Units: temperature

in K, n. in ug/(cm*s) and A in mW/(m*K).

GV(1

GV(2

GV(3

GV(4

GV(5

GV(6

GV(7

GV(8

GV(9

GT(1

GT(2

GT(3

GT(4

GT(5

GT(6

GT(7

GT(8

GT(9

.2546211337E+07

-.1670382040E+07

.2898185505E+06

.3474760268E+05

-.1674096936E+05

.1445438546E+04

.8767956930E+02

-.1932828150E+02

.7959594290E+00

-.1550251226E+07

•9682908726E+06

-.2120883669E+06

.1779303821E+05

-.9793550977E+03

•3480491811E+03

-.6424000898E+02

.4582750314E+01

-.1120206739E+00

11
Figure 1 displays a deviation plot between the experimental data fitted to

equation (31), and our estimates from equations (20) and (21), with the pressure

set at one atmosphere. Our representation of the dilute gas viscosity coefficient

can be considered satisfactory, the thermal conductivity less so. for both

coefficients systematic differences are observed but it is not clear whether

they are caused by errors in the procedure, or are due to errors in the data. A

possible reason for the deviations in the thermal conductivity is that our method does not take into account correctly the contribution of the internal

degrees of freedom of nitrous oxide. This problem is under further investigation.

We next compared our predicted conductivities for nitrous oxide with the

dense gas and liquid data of reference [19]. The result is shown in figure 2.

It is seen that the calculated values are within about 5%, or less, of the

measured values.

5 . TABLES

Tables of values for the viscosity and thermal conductivity coefficients

of nitrous oxide were generated from equations (20) and (21) . Tables 2 and 3 give the results in the SI system of units, tables 4 and 5 in engineering units.

For convenience, saturated liquid values are listed separately as table 6 .

5.1 Assessment of Accuracy

An estimation of the accuracy of the tabular values is clearly difficult

and has to be based mainly on how well the prediction method can represent

other fluids for which reliable data are available. For nitrogen, ethane,

propane, butane and carbon dioxide, and their mixtures, we were generally

able to predict the coefficients to within about 6% using methane as the

reference fluid. Here we have represented the dilute gas viscosity data to 3%

and the dilute gas thermal conductivity data to 10%. The dense gas thermal

* The number of significant figures for an entry are more than the accuracy of

the calculations warrant (see Section 5.1). Extra figures are included to

facilitate interpolation if required.

12
conductivity measurements of reference [19] have been represented to within 6%.

Based on the above two factors, and our experience in evaluating data for fluids

similar to nitrous oxide, we estimate the viscosity to be accurate to + 6% and

the thermal conductivity to be accurate to + 8%. In the critical region this

latter figure should be increased to + 20%. As in our previous work, these

assessments refer to an accuracy estimation on a 2a basis.

13
Figure 1. Top curve: 1 atm viscosity percent deviation plot between values

predicted from equation (20) and the correlation equation (31) . Percent

deviation is defined as n[eq. (20)] - n[eq. (31) ]*100. 0/n [eq. (31)].

Bottom curve: 1 atm thermal conductivity deviations between equation (21)

and equation (31)

.

14
1 •

•

•

•

1 1

•

- •

•

—

—

>-

CO

o - o

CO

>

•

•

•

—

—

1

•

m—

—

<

N0I1VIA3Q lN33U3d

15
Figure 2. Dense gas and liquid isotherms for the thermal conductivity

coefficient calculated from equation (21) (curves) compared with the data of

reference [19] (open and filled circles).

16
300

PRESSURE, atm

17
00

p.

CO

4-1 HC

•HXo

en3O

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>>u

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cnoo

CO H>

CD H

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Table 4. The viscosity coefficient of nitrous oxide. Pressure in 10 psi

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Table 6. Saturated Liquid Transport Coefficients for NO

T

°F

127.03

• 99.99

• 50.0

40.0

22.0

• 4.0

5.0

14.0

23.0

32.0

41.0

50.0

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332.64

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409.66

419.67

437.67

455.67

464.67

473.67

482.67

491.67

500.67

509.67

n lb-mass/f f s

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.13464E-03

.12516E-03

.10998E-03

.96711E-04

•90656E-04

.84921E-04

.79464E-04

.74250E-04

.68980E-04

.63901E-04

BTU/hr«ft«°F

.11312E+00

.10210E+00

.84157E-01

.80840E-01

.75034E-01

•69392E-01

.66617E-01

.63863E-01

.61126E-01

.58405E-01

.55536E-01

.52725E-01

59.0

68.0

77.0

86.0

95.9

518.67

527.67

536.67

545.67

555.57

58956E-04

,54078E-04

.49158E-04

.44106E-04

, 38150E-04

.50018E-01

.47495E-01

.45266E-01

.43498E-01

.42357E-01

30
6 . MIXTURES

Equations (27) and (28) were used to generate tables for two mixtures of

nitrous oxide with carbon dioxide: compositions C0„ (12%) - N„0 (88%) and

C0„ (6%) - N„0 (0.94%), respectively. The results are tables 7-10. Since

carbon dioxide is so similar to nitrous oxide, our accuracy assessments for

pure nitrous oxide are probably valid for mixtures. No data are available for

comparisons.

7. CONCLUSION

We have predicted the transport coefficients of nitrous oxide, and of

nitrous oxide/carbon dioxide mixtures using a significant modification of the

corresponding states procedure. Tabular values have been listed. Data available

for comparisons are scanty. More, and more reliable, data will be required if

the accuracy of the tables is to be improved.

8 . ACKNOWLEDGMENTS

We are very grateful to R. D. McCarty for his help with the PVT calculations,

to P. M. Holland for valuable discussions on the statistical mechanical

calculations reported in the Appendix, and to Karen Bowie for her help in

preparing the manuscript.

This work was supported by the Air Force Weapons Laboratory, Kirtland

Air Force Base, New Mexico, Contract No. 76-258. Also (in part) by the Office

of Standard Reference Data.

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47
9. REFERENCES

I] Hanley, H. J. M. , Cryogenics 16, 643 (1976).

2] Fisher, W. J., Phys. Rev. 28^ 73 (1909).

3] Smith, C. J., Proc. Phys. Soc. (London) 34_, 155 (1922).

4] Ishida, Y., Phys. Rev. 21, 550 (1923).

5] Gregory, H. and Archer, C. T., Proc. Roy. Soc. A121, 285 (1928).

6] Trautz, M. and Kurtz, F., Ann. Phys. 9_, 981 (1931).

7] Johnston, H. L. and McClosky, K. E. , J. Phys. Chem. 44, 1038 (1940).

8] Ellis, C. P. and Raw, C. J. G. , J. Chem. Phys. 20, 574 (1959).

9] Winkelman, A., Ann. Phys. 156, 497 (1875).

10] Eucken, A., Physik. Z. 14, 324 (1913).

II] Bruche E. and Littman, W. , Z. Phys. 67, 362 (1931).

12] Dickins, B. G. , Proc. Roy. Soc. A143 , 517 (1934).

13] Kannuluick, W. G. and Martin L. H. , Proc. Roy. Soc. A144 , 496 (1934).

14] Johnston, H. L. and Grilly, E. R. , J. Chem. Phys. 14, 233 (1946).

15] Kannuluick, W. G. and Donald, H. B., Aust. J. Sci. Res. A3, 417 (1950).

16] Bromley, L. A., Calif. Univ. Radiation Lab. (Berkeley) Rept. No. 1852 (1952)

17] Frank, E. V., Z. Phys. Chem. 201, 16 (1952).

18] Keys, F. G. , Trans. ASME 76, 809 (1954).

19] Richter, G. N. and Sage, B. H. , J. Chem. Eng. Data 8, 221 (1963).

20] Pereira, A. N. G. and Raw, C. J. G., Phys. Fluids 6^, 1091 (1963).

21] Mukhopadhyay, P. and Barua, A. K. , Trans. Fara. Soc. 63, 2379 (1967).

22] Maczek, A. 0. S. and Gray, P., Trans. Fara. Soc. 66, 127 (1970).

23] Bilyk, A. A., Gladkii, N. F., Kotelevskii, Yu. G. and Timofeev, B. D.,

Inv. Akad. Nauk. Belarusk SSR 1, 102 (1973).

24] Hanley, H. J. M. , McCarty, R. D. and Haynes, W. M. , J. Phys. Chem. Ref.

Data 2> 979 (1974): In press (1976).

25] Mo, K. C. and Gubbins, K. E., Chem. Eng. Comm. 1_, 281 (1974).

26] Leland, T. W. , Chappelear, P. S. and Gamson, B. W. , A.I.Ch.E. Journal 8^, 482 (1962).

27] Mollerup, J. and Rowlinson, J. S., Chem. Eng. Sci. _29, 1373 (1974).

28] Hanley, H. J. M. , McCarty, R. D. and Cohen, E. G. D. , Physica 6JD, 322 (1972)

Hanley, H. J. M. and Cohen, E. G. D. , Physica 83A, 215 (1976).

48
[29] Goodwin, R. D. , Nat. Bur. Stand. (U.S.), Tech. Note 653 (1974).

[30] Couch, E. J. and Kobe, K. A., J. Chem. Eng. Data 6, 229 (1961): Hirth, L. J. and Kobe, K. A., J. Chem. Eng. Data 6, 233 (1961).

[31] McCarty, R. D. , Nat. Bur. Stand. (U.S.) Internal Report (1977) (in review).

[32] Hanley, H. J. M. , J. Phys. Chem. Ref. Data 2, 619 (1973).

[33] Hanley, H. J. M. and Ely, J. F., J. Phys. Chem. Ref. Data 2, 735 (1973).

[34] Ely, J. F. and Hanley, H. J. M. , Mol. Phys. 30, 565 (1975).

[35] Hanley, H. J. M. , Sengers, J. V. and Ely, J. F. , Proc. 14th Thermal

Conductivity Conference (Plenum Press, 1976), Eds. Klemens, P. G. and

Chu, T. K. , page 383.

[36] Klein, M. , Hanley, H. J. M. , Smith, F. J. and Holland, P., NSRDS-NBS

Monograph No. 47 (1974).

[37] Pople, J. A., Proc. Roy. Soc. A221 , 498, 508 (1954).

[38] Buckingham, A. D., Disch, R. L. and Dunmur, D. A., J. Am. Chem. Soc. 90 , 3107 (1968).

[39] Stroud, A. H. , Calculation of Multiple Integrals (Prentice-Hall, New

York, 1972).

[40] Sweet, J. R. and Steele, W. A., J. Chem. Phys. 50, 668 (1969).

49
APPENDIX. Application of Statistical Mechanics

The dilute gas coefficients, n and X , have been correlated by the empirical

equation (31). An alternative, however, is to consider the kinetic theory,

statistical mechanical, expressions such as

(TTm kT) 1/2

° - — —

n " 16 2 n (2,2)* (1A)

to 9.

where m is the molecular mass, k is Boltzmann's constant and a is a length

parameter, characteristic of the intermolecular pair potential function, (J). (2 2 ) * The collision integral, 0, , is a function of 4> and the temperature. Clearly

one requires an expression for c}) to use equation (1A) in practice.

The potential for a nonspherical molecule such as nitrous oxide can be

written as the sum of a spherical contribution, if> (i.e., 4> depends on the

intermolecular separation between molecules but not on their relative orientations)

,

and a nonspherical contribution, (J) , which incorporates the appropriate

ns electrostatic moments and molecular polarizabilities . Hence

) + (2A)

s ns

In previous work we have represented the spherical part by the m-6-8 potential

function [ 32-34]

:

s

cj, /e = 6±§X UX d- _ P-Yfa-8) M d

6 /a V d

8

s m-6 \r

J m-6 \r

J \rJ

where d = r . /a, with a and r . being defined by the conditions (J) (o) = and

min mm s

<J> (r ) = - e, respectively. The parameter m represents the 'hardness' of the

repulsive term while Y represents the 'strength' of the inverse eighth attraction

term.

50
It turns out that the viscosity coefficient is only weakly dependent on

the nonspherical contribution so a fit of viscosity data with equation (1A)

will determine the potential parameters of (J) . Possible errors in making this

simplication are discussed in reference [34].

Nitrous Oxide . Viscosity data, [6,7], of nitrous oxide was fitted to equation

(1A) using the m-6-8 collision integrals from reference [36], The parameters

thus determined are

m = 11 Y

= 3

a = 3.65 x 10~10 m r = 4.068 x 10

_1

°m (4A) m

e/k = 288.0 K

A deviation curve comparing calculated with experimental viscosity coefficients,

as represented by the function (31), is shown as figure 3a. The deviation is

systematic above 300 K, but well within the experimental uncertainty of the

data.

The thermal conductivity coefficient was also determined from the corresponding

kinetic theory expression. The result agreeded with experiment to within the

estimated error of 6%. Details will not be given in this report.

The Second Virial Coefficient of Nitrous Oxide . It is often instructive to

calculate the second virial coefficient, B, of a gas using the potential

found suitable for the viscosity. The calculation is worthwhile in itself.

Further, however, the second virial statistical mechanical expression and the

second virial data are independent of the viscosity. The calculation is thus

a valuable check on the potential, and on the viscosity measurements which led

to the potential parameters.

For a polyatomic molecule, the nonspherical contribution to equation (1),

(b , has a significant effect on the second virial coefficient and cannot be

ns neglected. In general one has

(J) = (j) (permanent) + (J>(induced) (5A)

51
where 4> (permanent) represents interactions between the permanent moments e.g.

dipole(u)-dipole, quadrupole(0)-quadrupole, and dipole-quadrupole interactions —

while (J) (induced) represents the contributions caused by the induction effects

of the permanent moments.

Nitrous oxide is somewhat unusual in that it possesses a small dipole

but a large quadrupole. In the brief discussion following, we will consider

these moments but will exclude the effects of (J) (induced) . The potential for

nitrous oxide is thus

) + <J>(uu) + <K00) + <KuO) (6A)

where (j) is the potential (3A) with the parameters of equation (4A) ; (J)(uu), 4>(00)

and 4>(u©) are the dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole

terms, respectively.

Writing the equation of state as

= 1 + Bp + ..

.

(7A) pNkT

where N is Avogadro's number, the second virial coefficient is given formally

by

B = | if [exp(-e<f>)-l] dj^1 d^

2 (8A)

where 3 = 1/kT and R1 and R„ denote the position and angular configuration of

molecule 1 and 2, respectively.

Several techniques have been proposed to integrate equation (8A) of which

the most common is the expansion procedure of Pople [37]. One considers the

molecules to have point moments and to interact according to the coordinate

system shown in figure 4.

52
I X

Figure 4.

Equation (2A) for (j) is inserted into equation (8A) and the (J) term expanded

in powers of 3- The integrations over angles are then carried out to yield;

B*(T*) = B(kT/£)/b = B* „ + B +B^ + B^ (9A)

o m-6-8 y y0

3 Here b = 2ttN(J /3 and B* , n is the conventional reduced second virial coefficient

o m-6-8

for the spherical potential, as tabulated in reference [36]. For the model shown

in figure 4 one has for the dipole-dipole potential;

2

(J) = - ^r- [2 Cose, Cos0 o T - SinG, Sin6. Cos(j>] (10A)

yu 3 12 1 2 r

For (j) 00

:

2

j) Q0

= | ~ [i - 5 cos

2

e 1 - 5 cos

2

e 2 - 15 cos

2

e 1 cos

2

e

2 r (HA)

+ 2 (Sine Sine Costj) - 4 Cos6 Cos6

2

) ]

And for 6r ,-.: y0

y0 [(Cos6 - Cose, ) (3 Cos6. Cos6 - 2 Sin6 1 Sin6 Cost)))

y0 „ 4

M IVWB"2 l /w 1 2 1 2 2r

- Cos6 + Cos6 ]

(12A)

53
Hence expressions for B , B~ and B „ are

U y0

192 E

j = l

n 2

(2j + 1)!

*4i 2 U

J K. I 2(j-3)

6j T *2j (13A)

*2 2 3 where y = u /ca and

E (2k)!/(k!)

k=o

(14A)

21 _©

5 *2 10

216

245

*6

*3

X 15 + (15A)

*2 2,5 where =0 /ea and finally

5 y0

*? *2

3 u

" *2

*2

I„ - 11

72

35

*2 4

y 0^

m *3 X 13 + (16A)

Note that equations (13A) , (15A) and (16A) are expansions for which only the first

few terms are considered. The index, j, of equation (13A) is set equal to 4.

The above equations contain integrals I given by

exp [-B4>

s (r )] (r ) X

dr (17A)

where r = r/a.

The Pople expansion technique has the advantage that the integration over angles only has to be carried out once. Hence the calculation of the second

virial using a particular spherical potential reduces to the straightforward

integrations of equation (17A) ; the results of which can be tabulated for a given spherical potential.

54
f Returning to nitrous oxide: values for the moments are [38]

= 3.0 x 10

26 e.s.u.

(18A)

—1 8 u = 0. 166 x 10 e. s.u.

Insertion of these parameters into the expression for B , equation (9A) , led

to the graph shown as figure 3b. Also included in this figure are the data

points of Hirth and Kobe [30].

It can be seen that our prediction of the second virial is not unsatisfactory but the predicted values are consistently somewhat higher. Possible

J. reasons for the discrepancy are 1) neglect of the induction contribution to B , 2) lack of convergence of the expansions since the quadrupole moment of nitrous

oxide is relatively large and 3) uncertainty in the m-6-8 parameters. Unfortunately,

we cannot correct this last factor. We have shown that the m-6-8 parameters

should be selected from viscosity data taken over a wide temperature range (see

the discussion in reference [32]) but the data is limited for nitrous oxide.

We did, however, investigate 2) as remarked in the next section.

Direct Calculation of the Second Virial Coefficient . There are some drawbacks

in the expansion (9A) . For example, all I integrals have to be redetermined if

the spherical potential is modified and, of interest here, the arbitrary

truncation of the expansions can introduce convergence problems. Several

authors have investigated alternative methods to the expansion of the exponential

term in equation (8A) . The most direct is to carry the full integration

numerically using, for example, the Monte-carlo approach. This and other

integration schemes are discussed by Stroud [39].

Steele and Sweet [40] discuss a procedure in which the potential function,

<Kr, 9, 9„ (j)) , and the radial distribution' function, g(r, 6 9„ cf>) , are expande

in spherical harmonics and the integrations over angles performed numerically.

1 e.s.u. = 3.33564 x 10

U C m

2

55
The virial coefficient follows from a straightforward integration over r intermolecular separation.

the

Our work with nitrous oxide here, and previously with carbon dioxide [34],

suggested another integration technique which avoids the somewhat complex and

time consuming calculations alluded to above, but which can eliminate the

convergence problems of the expansion method. An outline of our procedure is as follows.

Consider the potential of equations (3A), (10A)-(12A) and the coordinate

system of figure 4. The two molecules are set at fixed r apart at some initial

relative angular orientation, e.g., 6 = 6 = (J) = 0. The angles are then

varied incrementally (say in increments of tt/4, tt/8, etc.) over their range of

tt c}) = 2tt) and the numerical value of the angle dependent

part of the potential recorded for each increment. Due, however, to the

symmetry of the model, many of the values are identical so only the different, e, values and their weight, wt, (their fraction of the total) are recorded. The

second virial coefficient for a given initial configuration follows after

integrations over r according to the expression

integration (0 ,

2ttN

e

Z i=l

(sswt)

1 1J o

(1

-4).(r, e 1

, e 2

, 4>) kT

9 r dr

Z (sswt)

i=l

(19A)

where

(sswt) (Sin©., Sin0 n wt) . 1 2 l

i = 1, e (20A)

Details of the calculation will be published elsewhere. Values of B for two

initial configurations at two temperatures were determined and are shown as triangles in figure 3b. One can conclude that the expansion procedure, shown

by the curve, did indeed underestimate slightly the virial.

56
It should be stressed that the calculations of the second virial coefficient

are predictions : one could get better agreement with experiment, without

affecting significantly the viscosity fit, by adjusting the potential parameters.

Finally, we conclude from the calculation of the second virial that our assessment of about + 3% accuracy in the viscosity data is reasonable.

57
Figure 3(a). Viscosity coefficient of dilute N„0: comparison between

kinetic theory values and equation (31). 3(b). Plot of the second virial,

B, versus temperature. Data, filled circles, from reference [30]. The curve

is from equation (9A) , while the triangles are results discussed in the text.

58
200 300 400 500

T, K

59
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RHPORTNO.

NBS-TN-693

2. Gov't Accession

No.

4. TITLE AND SUBTITLE

Predicted Values of the Viscosity and Thermal Conductivity

Coefficients of Nitrous Oxide

3. Recipient's Accession No.

5. Publication Hate March 1977

6. Performing Organization Code

275.02

7. AUTHOR(S)

Howard J. M. Hanley

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2750124

11. Contract /Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as i tern 9-

13. Type of Report & Period Covered

Fi nal

14. Sponsoring Agency Codt

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 2 00- word or less (actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The viscosity and thermal conductivity coefficients of nitrous oxide are calculated for

temperatures between 180 and 900 K (330 to 1600°R) for pressures to 23 MPa (y 3500 psi)

Tables of values are presented. Two mixtures with carbon dioxide are also discussed.

These transport coefficients (for the pure fluid and for the mixtures) were predicted

from thermodynamic data. Details of the prediction procedure are presented. Estimates

of the accuracy of the tabular values are + 6% for the viscosity and + 8% for the

thermal conductivity.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Carbon d i ox i de; corresponding states; mixtures; nitrous oxide;

prediction; thermal conductivity; transport property; viscosity.

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purpose of the standards is to establish nationally rec- ognized requirements for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS

administers this program as a supplement to the activities of the private sector standardizing organizations.

Consumer Information Series—Practical information,

based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable language and illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent

of Documents, Government Printing Office, Washington,

D.C. 20402. Order following NBS publications—NBSIR's and FIPS

from the National Technical Information Services,

Springfield, Va. 22161. Federal Information Processing Standards Publications (FIPS PUBS)—Publications in this series collectively constitute the Federal Information Processing Standards Register. Register serves as the official source of information in the Federal Government regarding stand- ards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NBS Interagency Reports (NBSIR) —A special series of interim or final reports on work performed by NBS for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service. A

literature survey issued biweekly. Annual subscription: Domestic, $20.00; Foreign, $25.00.

Liquified Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey issued quarterly. Annual subscription: $20.00.

Send subscription orders and remittances for the pre- ceding bibliographic services to National Bureau of Standards, Cryogenic Data Center (275.02) Boulder,

Colorado 80302.
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National Bureau of Standards

Washington, DC. 20234

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