Transport Properties
CO2-enhanced coal-bed methane recovery involvesmotion of a uid under the
in uence of gradients of pressure, concentration and temperature. In addition
to the equation of state, the viscosity, di�usivities and thermal conductivityof the uid can be expected to play a major role in the characterization of
this process. Calculation of these transport properties can be approached
either theoretically (mainly using the non-equilibrium kinetic theory formal-
ism developed by Chapman and Enskog), or semi-empirically (by application
of various corresponding-states principles). Both approaches have proved to
be useful. Thus, for low-density uids, the kinetic theory approach has the
advantage of greater rigor and generality, but the disadvantage of requiring
knowledge of the intermolecular potentials, which might not be available.
On the other hand, the corresponding-states-based correlations have provedparticularly valuable in the extrapolation to high densities.
Pure Gas Viscosity at Low Pressure
The viscosity of gases at low pressure can be determined from nonequilibriumkinetic theory according to a procedure devised by Chapman and Enskog.Molecular interaction e�ects are manifested in the values of the so-called`collision integrals', which involve the pairwise interaction potential, and ap-
pear as temperature-dependent correction factors to the elementary kinetic-
theory expressions for the transport properties. For all realistic models ofintermolecular potentials, these integrals require numerical evaluation, and
for the important particular case of the Lennard-Jones potential function,viz.,
(r) = 4�
"��
r
�12
���
r
�6#;
the values of the collision integral can be calculated by an empirical approxi-mant developed by Neufeld et al. [P.D. Neufeld, A.R. Janzen, and R.A. Aziz,
Journal of Chemical Physics, 57: 1100 (1972)], as a function of dimensionlesstemperature T � = kT=�. This is
v = A=T �B + C exp(�DT �) + E exp(�FT �);
where A = 1:16145, B = 0:14874, C = 0:52487, D = 0:77320, E = 2:16178,and F = 2:43787. The expression for the viscosity (in micropoise, 10�7 Pa�s)
1
is
� =5
16
p�MRT
��2v
= 26:69
pMT
��2v
;
where M is the molar mass (g�mol�1), and the length-scale parameter � is
expressed in �Angstr�om units (10�10 m). Since expressions for the viscosityof multi-component gas mixtures involve the viscosities of the pure gases,
this formula is best implemented by supplying an array of the molar masses
and Lennard-Jones parameters �, �=k [which are tabulated in Appendix C of
the monograph \The Properties of Gases and Liquids", by R.C. Reid, J.M.
Prausnitz, and T.K. Sherwood, 3ed., New York: McGraw-Hill (1977)] for the
components, as follows:
SUBROUTINE ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)
C This subroutine returns an array of low-pressure gas viscosities
C calculated from the Chapman-Enskog formula, in which the collision
C integral is estimated from the Lennard-Jones parameters
C $\epsilon/k$ in EPSK(1:NC) and $\sigma$ in SIGMA(1:NC) according
C to the empirical correlation devised by P.D. Neufeld, A.R. Janzen,
C and R.A. Aziz, J. Chem. Phys., 57: 1100 (1972).
C
C Reference: R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The
C Properties of Gases and Liquids", 3rd edition, New York: McGraw-Hill
C (1977), pp. 395-399.
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(NC) !Lennard-Jones parameters $\epsilon/k$, K
DIMENSION ETA(NC) !Viscosity, micropoise
DIMENSION RM(NC) !Molar masses, g/mol
DIMENSION SIGMA(NC) !Lennard-Jones diameters $\sigma$, Angstrom
C
C Parameters in the correlation for OMEGAV as a function of
C dimensionless temperature TSTAR:
C
C \[
C \Omega_v = A/T^{*B}+C\exp(-D T^{*})+E\exp(-F T^{*})
C \]
C
PARAMETER(RGAS=8.31451)
PARAMETER(A=1.16145)
PARAMETER(B=0.14874)
PARAMETER(C=0.52487)
PARAMETER(D=0.77320)
PARAMETER(E=2.16178)
2
PARAMETER(F=2.43787)
CONST=26.69
DO 1 I=1,NC
TSTAR=TK/EPSK(I)
SIGMSQ=SIGMA(I)*SIGMA(I)
OMEGAV=A/TSTAR**B+C*DEXP(-D*TSTAR)+E*DEXP(-F*TSTAR)
ETA(I)=CONST*DSQRT(RM(I)*TK)/OMEGAV/SIGMA(I)/SIGMA(I)
1 CONTINUE
RETURN
END
Mixed Gas Viscosity at Low Pressure
The kinetic theory of gases can also be used to obtain an expression for the
low-pressure viscosity of a gas mixture. According to the discussion given byReid et al. (1977), this is of the form
�m =NXi=1
yi�iPN
j=1�ijyj
;
where the array of mixture coeÆcients �ij can be determined according toa variety of approximations. The simplest of these, due to Herning and
Zipperer (1936) [F. Herning and L. Zipperer,Gas Wasserfach, 79: 49 (1936)],is
�ij =
sMj
Mi
;
while according to Wilke (1950) [C.R. Wilke, Journal of Chemical Physics,18: 517 (1950)],
�ij =[1 + (�i=�j)
1=2(Mj=Mi)1=4]2
[8(1 +Mi=Mj)]1=2:
These expressions work best for mixtures of nonpolar gases, characterized by
relatively weak intereractions, and as such seem to be suitable for applicationto the study of carbon dioxide-methane mixtures. Brokaw [R.S. Brokaw,Industrial and Engineering Chemistry Process Design and Development, 8:
240 (1969)] devised a more elaborate model which is applicable to mixtures
containing polar components:
�ij = AijSij
s�i
�j
3
where
Aij =mijqMij
"1 +
Mij �M0:45
ij
2(1 +Mij) + (1 +M0:45
ij)m�0:5
ij=(1 +mij)
#;
Mij =Mi
Mj
; mij =
"4
(1 +Mij)(1 + 1=Mij)
#0:25
;
and the polar correction factor Sij is set equal to 1 if the species are non-
polar. The viscosity of the low-pressure mixture is evaluated according to
these models by the function ETALPM:
FUNCTION ETALPM(MODEL,NC,ETA,RM,Y)
C This subroutine determines the low-pressure viscosity of a gas
C mixture containing NC components with individual viscosities
C ETA(1:NC), molar masses RM(1:NC), and mole fractions Y(1:NC)
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
PARAMETER(MAXC=10) !Maximum number of components
C
C Adjustable arrays
C
DIMENSION ETA(NC) !Pure-component viscosities, micropoise
DIMENSION RM(NC) !Molar masses, g/mol
DIMENSION Y(NC) !Mole fractions
C
C Fixed arrays
C
DIMENSION PHI(MAXC*MAXC) !Mixing coefficients, phi
DIMENSION PHIY(MAXC) !Elements of PHI*Y
C
C Identification of invalid inputs
C
IF(NC.GT.MAXC) THEN
WRITE(6,*) 'Error return from ETALPM: Too many components'
WRITE(6,*) NC,MAXC
STOP
ENDIF
C
C Construction of the matrix of parameters PHI
C
DO 2 I=1,NC
DO 1 J=1,NC
IJ=NC*(I-1)+J
4
ETAIJ=ETA(I)/ETA(J)
RMIJ=RM(I)/RM(J)
IF(MODEL.EQ.1) THEN
C
C In model 1, the mixing coefficients are obtained as described by
C C.R. Wilke, J. Chem. Phys., 18: 517 (1950):
C
XUPPER=1.+DSQRT(ETAIJ/DSQRT(RMIJ))
XUPPER=XUPPER*XUPPER
XLOWER=DSQRT(8.*(1.+RMIJ))
PHI(IJ)=XUPPER/XLOWER
ELSEIF(MODEL.EQ.2) THEN
C
C In model 2, the mixing coefficients are obtained as described by
C F. Herning and L. Zipperer, Gas Wasserfach, 79: 49 (1936):
C
PHI(IJ)=1./DSQRT(RMIJ)
ELSEIF(MODEL.EQ.3) THEN
C
C In model 3, the mixing coefficients are obtained as described by
C R.S. Brokaw, Ind. Eng. Chem. Proc. Des. Dev., 8: 240 (1969), for
C mixtures of non-polar gases:
C
SIJ=1. !Polar correction factor = 1
RMMIJ=(4./(1.+RMIJ)/(1.+1./RMIJ))**0.25
F1=RMIJ-RMIJ**0.45
F2=2.*(1.+RMIJ)
F3=(1.+RMIJ**0.45)/DSQRT(RMMIJ)/(1.+RMMIJ)
AIJ=(1.+F1/(F2+F3))*RMMIJ/DSQRT(RMIJ)
PHI(IJ)=DSQRT(ETAIJ)*SIJ*AIJ
ELSE
WRITE(6,*) 'Error return from ETALPM:'
WRITE(6,*) 'Unknown option ',MODEL
ENDIF
1 CONTINUE
2 CONTINUE
C
C Product of the arrays PHI and Y
C
DO 4 I=1,NC
PHIY(I)=0.
DO 3 J=1,NC
IJ=NC*(I-1)+J
PHIY(I)=PHIY(I)+PHI(IJ)*Y(J)
3 CONTINUE
5
4 CONTINUE
C
C Application of the mixing rule
C
ETALPM=0.
DO 5 I=1,NC
ETALPM=ETALPM+Y(I)*ETA(I)/PHIY(I)
5 CONTINUE
RETURN
END
The operation of both of these codes can be illustrated by Example 9.5 of
Reid et al. (1977, page 414), which involves calculating the viscosity of a mix-
ture of methane (1) and butane, with y1=0.697, at 20ÆC. FromAppendix C of
Reid et al. (1977), the Lennard-Jones parameters are �=k=148.6 K, �=3.758�A for methane, and �=k=531.4 K, �=4.687 �A for butane. The choice be-tween the three approximations for the mixture coeÆcients is expressed bythe value of the parameter MODEL. For the Wilke expression (MODEL=1),the pure-component viscosities are 109.68 �P for methane and 72.92 �P for
butane, which compare favorably with the experimental values 109.4 and72.74 obtained by Kestin and Yata [J. Kestin and J. Yata, Journal of Chem-
ical Physics, 49: 4780 (1968)]. The calculated mixture value is 92.48, whichis in reasonable agreement with the experimental value 93.35 obtained bythe same authors; the discrepancy is a little greater than 1%. An example
of a driver program is as follows:
C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12
C Calculation of viscosities of gaseous mixtures of methane (1)
C and butane (2)
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(2),ETA(2),RM(2),SIGMA(2),Y(2)
NC=2
TK=293.15
C
C Lennard-Jones parameters and molar masses
C
EPSK(1)=148.6
SIGMA(1)=3.758
EPSK(2)=531.4
SIGMA(2)=4.687
RM(1)=16.043
RM(2)=58.124
6
C
C Pure-component viscosities
C
CALL ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 1 viscosity = ',ETA(1),' micropoise'
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 2 viscosity = ',ETA(2),' micropoise'
C
C Comparison of models for mixture viscosity
C
Y(1)=0.697
Y(2)=0.303
DO 1 MODEL=1,3
TEST=ETALPM(MODEL,NC,ETA,RM,Y)
WRITE(6,'(T2,A6,I1,A21,F6.2,A11)')
, 'Model ',MODEL,' Mixture viscosity = ',TEST,' micropoise'
1 CONTINUE
END
This produces the output:
Component 1 viscosity = 109.68 micropoise
Component 2 viscosity = 72.92 micropoise
Model 1 Mixture viscosity = 92.48 micropoise
Model 2 Mixture viscosity = 93.04 micropoise
Model 3 Mixture viscosity = 95.11 micropoise
Gas Viscosities at High Pressure and Density
The method recommended by Reid et al. for calculating the e�ect of pressure
on the viscosity of a gas mixture is that of Dean and Stiel [D.E. Dean andL.I. Stiel, AIChE Journal, 11: 526 (1965)], which is based on the use of a
corresponding-states principle to determine a residual viscosity, or additive
correction to the low-pressure viscosity. This in turn requires an accurate
equation of state, together with a suitable method for estimating the pseudo-critical parameters for the mixture. The formula is
(�m � �Æm)�m = 1:08[exp(1:439�rm)� exp(�1:11�1:858
rm)];
where the superscript Æ is used to distinguish the low-pressure viscosity from
that at the pressure of interest. The reduced density of the mixture is ob-tained by dividing the actual density by the pseudocritical density (obtained
7
from the Prausnitz-Gunn mixture rules), and the parameter � is
�m =T 1=6
rm
M1=2
m p2=3
rm
where the mixture reduced pressure prm and temperature Trm are similarly
obtained from the pseudocritical parameters, but the mixture molar mass
Mm is a simple mole-fraction average. The Prausnitz-Gunn mixture rules
are implemented by the following subroutine PGRULE:
SUBROUTINE PGRULE(NC,TC,PC,RHOC,RM,TPC,RHOPC,PPC,RMAV,Y)
C This subroutine implements the Prausnitz-Gunn mixture rules for
C the pseudocritical temperature TPC, density RHOPC, and pressure
C PPC, of an NC-component fluid mixture characterized by mole
C fractions Y(1:NC), molar masses RM(1:NC), critical pressures
C PC(1:NC), and critical densities RHOC(1:NC).
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
PARAMETER(RGAS=8.31451) !Gas constant, J/K/mol
DIMENSION PC(NC) !Critical pressures, MPa
DIMENSION RHOC(NC) !Critical densities, g/cm^3
DIMENSION RM(NC) !Molar masses, g/mol
DIMENSION TC(NC) !Critical temperatures, K
DIMENSION Y(NC) !Mole fractions
RMAV=0.
TPC=0.
ZPC=0.
VPC=0.
DO 1 I=1,NC
RMAV=RMAV+Y(I)*RM(I)
TPC=TPC+Y(I)*TC(I)
VPC=VPC+Y(I)*RM(I)/RHOC(I)
ZPC=ZPC+Y(I)*PC(I)*RM(I)/(RGAS*TC(I)*RHOC(I))
1 CONTINUE
PPC=ZPC*RGAS*TPC/VPC
RETURN
END
From the account of the Dean-Stiel method given by Reid et al. (1977), it
is evident that the crucial input is not in fact the pressure, but the densityof the mixture - and, of course, its composition, which appears not only inthe low-pressure viscosity but also in the pseudocritical constants through
the Prausnitz-Gunn mixture rules. This suggests that a subprogram based
8
on this model should accept as arguments the temperature, the number of
components, and adjustable arrays containing the mole fractions, critical
parameters, and molar masses of the components. Within this subprogram,
the �rst step would be to calculate the low-pressure mixture viscosity, the
second step would be to calculate the mixture pseudocritical parameters, and
�nally the mixture viscosity at the pressure of interest would be calculated.
The existing function ETALPM that evaluates the low-pressure viscosity
of gas mixtures consists of the integer parameters MODEL, NC, and the NC-
dimensional arrays ETA, RM, and Y; the pure-gas low-pressure viscosities
ETA are determined separately by calling the subroutine ETACEN. The cal-culation of pressure/density e�ects is more easily performed by a subroutine
ETAMIX that accepts the additional inputs arrays of critical parameters PC,
RHOC, TC, for the components, as well as the temperature TK and mixture
density RHOGC3 (g�cm�3). This arrangement allows for maximum exibil-
ity, in that the required density can be derived from a call to any suitable
equation of state. It is also useful to include not only the low- (or zero-)pressure viscosity ETA0 in addition to the viscosity ETARHO correspondingto the density RHOGC3 returned by the equation of state, since this makesit easier to identify the pressures at which appreciable deviations from the
low-pressure values are expected.In the example given by Reid et al. (op. cit., p. 434), the required density
was in fact obtained from the corresponding-states correlation developed byPitzer, but use of a speci�c equation of state would almost certainly produceresults of superior accuracy. Since volumetric data for the ternary system
methane-carbon dioxide-nitrogen have not been reported in suÆcient numberto give reliable estimates of the mixture second and third virial coeÆcients,
this mode of application can be illustrated instead by use of the three-termvirial equation of state
z = 1 +B�+ C�2 z3 � z2 � �z � = 0;
for the ternary system methane (1)-ethane (2)-carbon dioxide (3), where
B =3Xi=1
3Xj=1
Bijxixj; C =3Xi=1
3Xj=1
3Xk=1
Cijkxixjxk:
Values of the mixture virial coeÆcients Bij , Cijk have been derived by Houet al. [H. Hou, J.C. Holste, K.R. Hall, K.N. Marsh, and B.E. Gammon,
9
Journal of Chemical and Engineering Data, 41: 344-353 (1996)] from Burnet-
isochoric measurements at 300 and 320 K. Instead of using the exact solution
of the equivalent cubic equation for z,
z3 � z2 � �z � = 0;
where � � Bp=RT , � C(p=RT )2, the density for speci�ed pressure can be
obtained more expeditiously by Newton-Raphson iteration for the equivalent
equation
f(z) = 1 +�
z+
z2� z; f 0(z) = �
�
z2�
2
z3� 1;
which avoids the need for a call to another subroutine to solve the equation.
The FORTRAN code for this subroutine, and a driver program that tests it
at the state point p=6.36283 MPa, T=320 K, x1=0.24756, x2=0.56013 (forwhich the experimental value of the compressibility factor is z=0.71454) isas follows:
C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12
C Equation of state for mixtures of methane (1), ethane (2), and
C carbon dioxide (3)
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
PARAMETER(RMW1=16.0428,RMW2=30.06964,RMW3=44.0098)
PMPA=6.36283
TK=320.
X1=0.24756
X2=0.56013
CALL VEOS23(PMPA,RHOGC3,TK,X1,X2)
END
SUBROUTINE VEOS23(PMPA,RHOGC3,TK,X1,X2)
C Given temperature TK (K), pressure PMPA (MPa), and mole fractions
C X1, X2 of methane and ethane, respectively, this subroutine
C returns the mass density RHOGC3 (mol cm^-3) of a ternary mixture
C of methane (1), ethane (2), and carbon dioxide (3)
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
PARAMETER(RGAS=8.31451)
PARAMETER(ITMAX=10,TOL=1.D-15)
DIMENSION B(3,3),C(3,3,3),RM(3),X(3)
C
C Molar masses, g mol^-1
C
10
DATA RM(1)/16.0428/
DATA RM(2)/30.0696/
DATA RM(3)/44.0098/
C
C Mixture 2nd virial coefficients (cm^3 mol^-1) reported by H. Hou,
C J.C. Holste, K.R. Hall, K.N. Marsh, and B.E. Gammon, J. Chem.
C Eng. Data, 41: 344-353 (1996)
C
DATA B(1,1)/ -35.17/
DATA B(1,2)/ -76.71/
DATA B(1,3)/ -54.02/
DATA B(2,1)/ -76.71/
DATA B(2,2)/-159.42/
DATA B(2,3)/-105.65/
DATA B(3,1)/ -54.02/
DATA B(3,2)/-105.65/
DATA B(3,3)/-104.54/
C
C Mixture 3rd virial coefficients reported by Hou et al. (1996),
C cm^6 mol^-2
C
DATA C(1,1,1)/2229./
DATA C(2,2,2)/9692./
DATA C(3,3,3)/4411./
DATA C(1,1,2)/3594./
DATA C(1,2,1)/3594./
DATA C(2,1,1)/3594./
DATA C(1,1,3)/2641./
DATA C(1,3,1)/2641./
DATA C(3,1,1)/2641./
DATA C(1,2,2)/5902./
DATA C(2,1,2)/5902./
DATA C(2,2,1)/5902./
DATA C(1,3,3)/3195./
DATA C(3,1,3)/3195./
DATA C(3,3,1)/3195./
DATA C(2,2,3)/6963./
DATA C(2,3,2)/6963./
DATA C(3,2,2)/6963./
DATA C(2,3,3)/5075./
DATA C(3,2,3)/5075./
DATA C(3,3,2)/5075./
DATA C(1,2,3)/4204./
DATA C(2,3,1)/4204./
DATA C(3,1,2)/4204./
11
DATA C(1,3,2)/4204./
DATA C(2,1,3)/4204./
DATA C(3,2,1)/4204./
C
C Second and third virial coefficients of the mixed fluid;
C average molar mass
C
X(1)=X1
X(2)=X2
X(3)=1.-X1-X2
BMIX=0.
CMIX=0.
RMAV=0.
DO 3 I=1,3
RMAV=RMAV+X(I)*RM(I)
DO 2 J=1,3
BMIX=BMIX+B(I,J)*X(I)*X(J)
DO 1 K=1,3
CMIX=CMIX+C(I,J,K)*X(I)*X(J)*X(K)
1 CONTINUE
2 CONTINUE
3 CONTINUE
C
C Newton-Raphson iteration for the compressibility factor
C
PRT=PMPA/RGAS/TK
BETA=PRT*BMIX
GAMMA=PRT*PRT*CMIX
Z=1.
DO 4 ITER=1,ITMAX
F=(GAMMA/Z+BETA)/Z+1.-Z
FPRIME=-1.-BETA/Z/Z-2.*GAMMA/Z/Z/Z
DZ=-F/FPRIME
RELERR=DABS(DZ/Z)
C WRITE(6,*) ITER,Z,RELERR
IF(RELERR.LT.TOL) GOTO 5
Z=Z+DZ
4 CONTINUE
WRITE(6,*) 'Warning: no convergence in ',ITMAX,' iterations'
5 RHOGC3=RMAV*PRT/Z
RETURN
END
It produces the output
1 1.00000000000000 0.266590971881958
12
2 0.733409028118042 2.574907815503465D-02
3 0.714524421733423 2.085336958233519D-04
4 0.714375419315002 1.338264719687369D-08
5 0.714375409754768 1.152368903920541D-16
which indicates that the iteration from z=1 gives satisfactory convergence.
There is good agreement between this value of z and the experimental value,
which indicates that the values assigned to the virial coeÆcients are correct.
Listings of subroutine ETAMIX and its driver program are as follows,
C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12
C Program to investigate pressure effects on the viscosity of
C multicomponent gas mixtures
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(3),ETA(3),RM(3),SIGMA(3),Y(3)
DIMENSION PC(3),RHOC(3),TC(3)
C
C Lennard-Jones parameters and molar masses for methane (1),
C ethane (2), and carbon dioxide (3)
C
DATA EPSK(1),SIGMA(1),RM(1),Y(1)/148.6,3.758,16.0428,0.24756/
DATA EPSK(2),SIGMA(2),RM(2),Y(2)/215.7,4.443,30.0696,0.56013/
DATA EPSK(3),SIGMA(3),RM(3),Y(3)/195.2,3.941,44.0098,0.19231/
C
C Critical constants for mixture components
C
DATA PC(1),RHOC(1),TC(1)/4.54,0.1621,190.4/
DATA PC(2),RHOC(2),TC(2)/4.82,0.2032,305.2/
DATA PC(3),RHOC(3),TC(3)/7.28,0.4682,304.2/
NC=3
TK=320.
PMPA=6.36283
X1=Y(1)
X2=Y(2)
C
C Pure-component low-pressure viscosities
C
CALL ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 1 viscosity = ',ETA(1),' micropoise'
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 2 viscosity = ',ETA(2),' micropoise'
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 3 viscosity = ',ETA(3),' micropoise'
13
C
C Mass density of gas mixture
C
CALL VEOS23(PMPA,RHOGC3,TK,X1,X2)
WRITE(6,'(T2,A10,E10.4,A8)') 'Density = ',RHOGC3,' g cm^-3'
C
C Comparison of models for low-pressure mixture viscosity
C
WRITE(6,'(T2,A5,2A20)') 'Model','eta0/micropoise','eta/micropoise'
DO 1 MODEL=1,3
CALL ETAMIX(MODEL,NC,ETA,ETA0,ETARHO,PC,RHOC,RHOGC3,RM,TC,Y)
WRITE(6,'(T2,I5,2F20.1)') MODEL,ETA0,ETARHO
1 CONTINUE
END
SUBROUTINE ETAMIX(MODEL,NC,ETA,ETA0,ETARHO,PC,RHOC,RHOGC3,RM,TC,Y)
C This subroutine determines the viscosity of a gas of mass density
C RHOGC3 (g cm^-3) containing NC components with individual
C viscosities ETA(1:NC), molar masses RM(1:NC), and mole fractions
C Y(1:NC)
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
PARAMETER(MAXC=10) !Maximum number of components
C
C Adjustable arrays
C
DIMENSION ETA(NC) !Pure-component viscosities, micropoise
DIMENSION PC(NC) !Critical pressures, MPa
DIMENSION RHOC(NC) !Critical densities, g cm^-3
DIMENSION RM(NC) !Molar masses, g mol^-1
DIMENSION TC(NC) !Critical temperatures, K
DIMENSION Y(NC) !Mole fractions
C
C Fixed arrays
C
DIMENSION PHI(MAXC*MAXC) !Mixing coefficients, phi
DIMENSION PHIY(MAXC) !Elements of PHI*Y
C
C Identification of invalid inputs
C
IF(NC.GT.MAXC) THEN
WRITE(6,*) 'Error return from ETAMIX: Too many components'
WRITE(6,*) NC,MAXC
STOP
ENDIF
C
14
C Construction of the matrix of parameters PHI
C
DO 2 I=1,NC
DO 1 J=1,NC
IJ=NC*(I-1)+J
ETAIJ=ETA(I)/ETA(J)
RMIJ=RM(I)/RM(J)
IF(MODEL.EQ.1) THEN
C
C In model 1, the mixing coefficients are obtained as described by
C C.R. Wilke, J. Chem. Phys., 18: 517 (1950):
C
XUPPER=1.+DSQRT(ETAIJ/DSQRT(RMIJ))
XUPPER=XUPPER*XUPPER
XLOWER=DSQRT(8.*(1.+RMIJ))
PHI(IJ)=XUPPER/XLOWER
ELSEIF(MODEL.EQ.2) THEN
C
C In model 2, the mixing coefficients are obtained as described by
C F. Herning and L. Zipperer, Gas Wasserfach, 79: 49 (1936):
C
PHI(IJ)=1./DSQRT(RMIJ)
ELSEIF(MODEL.EQ.3) THEN
C
C In model 3, the mixing coefficients are obtained as described by
C R.S. Brokaw, Ind. Eng. Chem. Proc. Des. Dev., 8: 240 (1969), for
C mixtures of non-polar gases:
C
SIJ=1. !Polar correction factor = 1
RMMIJ=(4./(1.+RMIJ)/(1.+1./RMIJ))**0.25
F1=RMIJ-RMIJ**0.45
F2=2.*(1.+RMIJ)
F3=(1.+RMIJ**0.45)/DSQRT(RMMIJ)/(1.+RMMIJ)
AIJ=(1.+F1/(F2+F3))*RMMIJ/DSQRT(RMIJ)
PHI(IJ)=DSQRT(ETAIJ)*SIJ*AIJ
ELSE
WRITE(6,*) 'Error return from ETAMIX:'
WRITE(6,*) 'Unknown option ',MODEL
ENDIF
1 CONTINUE
2 CONTINUE
C
C Product of the arrays PHI and Y
C
DO 4 I=1,NC
15
PHIY(I)=0.
DO 3 J=1,NC
IJ=NC*(I-1)+J
PHIY(I)=PHIY(I)+PHI(IJ)*Y(J)
3 CONTINUE
4 CONTINUE
C
C Application of the mixing rule
C
ETA0=0.
DO 5 I=1,NC
ETA0=ETA0+Y(I)*ETA(I)/PHIY(I)
5 CONTINUE
C
C Calculation of the pseudocritical constants and pseudoreduced
C density
C
CALL PGRULE(NC,TC,PC,RHOC,RM,TPC,RHOPC,PPC,RMAV,Y)
XI=TPC**(1./6.)/DSQRT(RMAV)/(PPC/0.101325)**(2./3.)
RHOR=RHOGC3/RHOPC
C
C Calculation of the pressure effects according to the correlation
C of D.E. Dean and L.I. Stiel, AIChE Journal, 11: 526 (1965)
C
ETARHO=ETA0+(DEXP(1.439*RHOR)-DEXP(-1.111*(RHOR**1.858)))*1.08/XI
RETURN
END
The results
Component 1 viscosity = 118.07 micropoise
Component 2 viscosity = 100.47 micropoise
Component 3 viscosity = 161.01 micropoise
Density = 0.9801E-01 g cm^-3
Model eta0/micropoise eta/micropoise
1 116.9 149.1
2 118.2 150.4
3 116.5 148.8
show that the additive correction to the low-pressure viscosity is of the rightorder, in comparison to the example given by Reid et al. and, more impor-tantly, that the e�ect of pressure (or density) on viscosity is not negligible.
This pressure e�ect introduces a further nonlinearity into the equation of
motion of the uid through the porous medium.
16
Comparison with Experiment: Viscosity
Several general observations can be made concerning the agreement be-
tween experimental measurements and the various models and correlations
described in the preceding sections.
The most fundamental of these is that the spherically-symmetric Lennard-
Jones potential, upon which the Neufeld correlation for the Chapman-Enskogcollision integrals is based, is not necessarily appropriate for all uids. In
uids containing molecules of irregular shape, the multipole moments associ-
ated with the unsymmetrical distribution of electronic charge density make
signi�cant contributions to the intermolecular potential. Inclusion of these
e�ects in the calculation of transport coeÆcients is in general complicated
by the tensorial character of multipolar interactions, and by the need for
explicit inclusion of the rotational coordinates in the integrations. Interac-
tions between molecules containing permanent dipole moments can describedby the Stockmayer potential function; as a practical matter, the use of theStockmayer potential or some more general function involving higher-ordermultipole tensors can be expected to a�ect not only the values of the col-lision integrals, but also their temperature-dependence. Thus, a degree of
skepticism is appropriate when using the Lennard-Jones potential, by way ofthe Neufeld correlation or some similar result, as the basis for predicting thetemperature dependence of the viscosity of molecular uids. In this regardit is signi�cant that Reid et al. give no details of the viscosity data fromwhich their compilation of Lennard-Jones parameters were obtained, or of
the data-reduction procedure used for this purpose, particularly in regard tothe weighting scheme used.
A second point is that the predictions of the low-pressure viscosity for
mixtures depend not only on theoretical justi�cation of the various mix-
ture models, but are limited ultimately by the method used to estimatethe required end-member viscosities. Thus, a more accurate result might
be obtained if the pure-component viscosities were experimentally measuredrather than obtained from a correlation. This e�ect could be signi�cant if the
temperature of interest were outside the temperature range represented in
the viscosity data from which the Lennard-Jones parameters were obtained.The accuracy of the pure- uid viscosity values obtained from ETACEN are
ultimately determined not only by the range of conditions in the originaldata, but also by the precision, and the regression procedure used to infer
the Lennard-Jones parameters.
17
By extension, the calculation of viscosities at high pressures is likewise
limited by the mixture-model used to determine the low-pressure viscosity, in
addition to the theoretical justi�cation of the corresponding-states relation-
ship (including the method used to estimate the pseudocritical parameters)
used to extrapolate to the conditions of interest, and on the accuracy of the
equation of state.
In the light of the foregoing observations, it is of interest to compare the
predictions of the three mixture models with low-pressure viscosity measure-
ments by Jackson [W.M. Jackson, \Viscosities of the binary gas mixtures,
methane-carbon dioxide and ethylene-argon", Journal of Physical Chemistry,60: 789 (1956)] for CH4-CO2 mixtures:
C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12
C Calculation of viscosities of gaseous mixtures of methane (1)
C and carbon dioxide (2)
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(2),ETA(2),RM(2),SIGMA(2),Y(2)
DIMENSION RESID(3),SSR(3),TEST(3)
C
C Lennard-Jones parameters and molar masses
C
DATA EPSK(1),SIGMA(1),RM(1)/148.6,3.758,16.0428/
DATA EPSK(2),SIGMA(2),RM(2)/195.2,3.941,44.0098/
C
C Experimental data:
C [W.M. Jackson, Journal of Physical Chemistry, 60:789 (1956)]
C
DIMENSION XCH4(14),ETAEXP(14)
DATA XCH4( 1),ETAEXP( 1)/0.000,151.0/
DATA XCH4( 2),ETAEXP( 2)/0.022,150.9/
DATA XCH4( 3),ETAEXP( 3)/0.103,149.4/
DATA XCH4( 4),ETAEXP( 4)/0.183,147.7/
DATA XCH4( 5),ETAEXP( 5)/0.297,145.3/
DATA XCH4( 6),ETAEXP( 6)/0.421,141.4/
DATA XCH4( 7),ETAEXP( 7)/0.537,137.3/
DATA XCH4( 8),ETAEXP( 8)/0.651,132.8/
DATA XCH4( 9),ETAEXP( 9)/0.730,129.1/
DATA XCH4(10),ETAEXP(10)/0.789,124.8/
DATA XCH4(11),ETAEXP(11)/0.850,122.6/
DATA XCH4(12),ETAEXP(12)/0.905,119.2/
DATA XCH4(13),ETAEXP(13)/0.933,117.4/
DATA XCH4(14),ETAEXP(14)/1.000,111.4/
NC=2
18
TK=298.15
NDATA=14
C
C Pure-component viscosities
C
CALL ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 1 viscosity = ',ETA(1),' micropoise'
WRITE(6,'(T2,A24,F6.2,A11)')
,'Component 2 viscosity = ',ETA(2),' micropoise'
C
C Comparison of models for mixture viscosity
C
SSR(1)=0.
SSR(2)=0.
SSR(3)=0.
WRITE(6,'(T2,A5,8A8)') 'XCH4','eta/uP','Model 1','Res.',
,'Model 2','Res.','Model 3','Res.'
DO 2 I=1,NDATA
Y(1)=XCH4(I)
Y(2)=1.-XCH4(I)
DO 1 MODEL=1,3
TEST(MODEL)=ETALPM(MODEL,NC,ETA,RM,Y)
RESID(MODEL)=TEST(MODEL)-ETAEXP(I)
SSR(MODEL)=SSR(MODEL)+RESID(MODEL)*RESID(MODEL)
1 CONTINUE
WRITE(6,'(T2,F5.3,8F8.1)') XCH4(I),ETAEXP(I),
, (TEST(K),RESID(K),K=1,3)
2 CONTINUE
DO 3 MODEL=1,3
SDR=DSQRT(SSR(MODEL)/DBLE(NDATA-1))
WRITE(6,'(T2,A6,I1,A31,F4.2)')
, 'Model ',MODEL,' Residual standard deviation = ',SDR
3 CONTINUE
END
From the output
Component 1 viscosity = 111.27 micropoise
Component 2 viscosity = 150.97 micropoise
XCH4 eta/uP Model 1 Res. Model 2 Res. Model 3 Res.
0.000 151.0 151.0 0.0 151.0 0.0 151.0 0.0
0.022 150.9 150.6 -0.3 150.4 -0.5 150.5 -0.4
0.103 149.4 149.1 -0.3 148.4 -1.0 148.5 -0.9
0.183 147.7 147.4 -0.3 146.2 -1.5 146.3 -1.4
19
0.297 145.3 144.6 -0.7 142.9 -2.4 143.0 -2.3
0.421 141.4 141.0 -0.4 138.9 -2.5 138.9 -2.5
0.537 137.3 136.9 -0.4 134.6 -2.7 134.6 -2.7
0.651 132.8 132.2 -0.6 129.9 -2.9 129.9 -2.9
0.730 129.1 128.4 -0.7 126.4 -2.7 126.3 -2.8
0.789 124.8 125.3 0.5 123.5 -1.3 123.3 -1.5
0.850 122.6 121.7 -0.9 120.2 -2.4 120.1 -2.5
0.905 119.2 118.2 -1.0 117.2 -2.0 117.0 -2.2
0.933 117.4 116.2 -1.2 115.5 -1.9 115.4 -2.0
1.000 111.4 111.3 -0.1 111.3 -0.1 111.3 -0.1
Model 1 Residual standard deviation = 0.64
Model 2 Residual standard deviation = 2.03
Model 3 Residual standard deviation = 2.05
the measured pure-component viscosities are seen to be in excellent agree-
ment with those produced by ETACEN, apart from the discrepancy of 0.1�P for the viscosity of methane. This is less than 0.1%, and well within theprecision of 0.2% claimed by the author. (Incidentally, the Orsat gas analysisprocedure was identi�ed as the dominant contribution to the experimentalerror.) The deviations from experiment - as expressed quantitatively by the
standard deviations - can therefore be attributed almost entirely to the mix-ture models. The best overall predictions are seen to be those from the Wilkecorrelation (Model 1), which, somewhat surprisingly, are signi�cantly betterthan the predictions of the more elaborate model of Brokaw (Model 3). It isalso noteworthy that all three models underestimate the mixture viscosity.
The performance of the Dean-Stiel correlation can be compared withmeasured viscosities from DeWitt and Thodos [K. J. De Witt and G. Thodos,\Viscosities of Binary Mixtures in the Dense Gaseous State: The Methane-Carbon Dioxide System," Canadian Journal of Chemical Engineering, 44:148-151 (1966)] at pressures up to about 680 atm. The Hou et al. three-
term virial equation of state cannot be used to provide the required densityhere, since the upper pressure limit is far higher than the pressure range
over which that equation was parameterized, but fortunately, DeWitt and
Thodos recorded the uid density, as well as the pressure, corresponding totheir viscosity measurements. The viscosities of a mixture containing 75.7mol% carbon dioxide up to 683 atm and 200ÆC, are in the following program,
in which the low-pressure viscosity is estimated from the Wilke correlation
(MODEL=1):
C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12
C Program to investigate pressure effects on the viscosity of
20
C multicomponent gas mixtures
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(2),ETA(2),RM(2),SIGMA(2),Y(2)
DIMENSION PC(2),RHOC(2),TC(2)
C
C Low-Temperature viscosity of methane-carbon dioxide mixtures
C K.J. DeWitt and G. Thodos, Canadian Journal of Chemical
C Engineering, 44: 148 (1966)
C
DIMENSION TCELS(4),ETAEX0(4),PATM(4,11),RHO(4,11),ETAEXP(4,11)
DATA TCELS(1),ETAEX0(1)/50.1,155.1/
DATA PATM(1, 1),RHO(1, 1),ETAEXP(1, 1)/ 33.41,0.0543,162.6/
DATA PATM(1, 2),RHO(1, 2),ETAEXP(1, 2)/ 68.66,0.1254,180.1/
DATA PATM(1, 3),RHO(1, 3),ETAEXP(1, 3)/135.55,0.3370,279.1/
DATA PATM(1, 4),RHO(1, 4),ETAEXP(1, 4)/203.39,0.5126,419.8/
DATA PATM(1, 5),RHO(1, 5),ETAEXP(1, 5)/273.07,0.6055,532.3/
DATA PATM(1, 6),RHO(1, 6),ETAEXP(1, 6)/340.77,0.6609,613.6/
DATA PATM(1, 7),RHO(1, 7),ETAEXP(1, 7)/410.18,0.7000,675.8/
DATA PATM(1, 8),RHO(1, 8),ETAEXP(1, 8)/477.89,0.7298,726.1/
DATA PATM(1, 9),RHO(1, 9),ETAEXP(1, 9)/546.61,0.7562,781.1/
DATA PATM(1,10),RHO(1,10),ETAEXP(1,10)/613.64,0.7763,821.6/
DATA PATM(1,11),RHO(1,11),ETAEXP(1,11)/682.02,0.7908,855.6/
DATA TCELS(2),ETAEX0(2)/100.4,176.4/
DATA PATM(2, 1),RHO(2, 1),ETAEXP(2, 1)/ 34.64,0.0447,182.4/
DATA PATM(2, 2),RHO(2, 2),ETAEXP(2, 2)/ 67.50,0.0930,192.2/
DATA PATM(2, 3),RHO(2, 3),ETAEXP(2, 3)/137.04,0.2145,236.7/
DATA PATM(2, 4),RHO(2, 4),ETAEXP(2, 4)/205.63,0.3416,303.0/
DATA PATM(2, 5),RHO(2, 5),ETAEXP(2, 5)/273.41,0.4436,375.5/
DATA PATM(2, 6),RHO(2, 6),ETAEXP(2, 6)/341.11,0.5173,444.7/
DATA PATM(2, 7),RHO(2, 7),ETAEXP(2, 7)/409.84,0.5731,509.5/
DATA PATM(2, 8),RHO(2, 8),ETAEXP(2, 8)/478.57,0.6152,565.9/
DATA PATM(2, 9),RHO(2, 9),ETAEXP(2, 9)/545.59,0.6483,614.9/
DATA PATM(2,10),RHO(2,10),ETAEXP(2,10)/613.30,0.6788,662.5/
DATA PATM(2,11),RHO(2,11),ETAEXP(2,11)/683.38,0.7022,701.3/
DATA TCELS(3),ETAEX0(3)/150.7,197.4/
DATA PATM(3, 1),RHO(3, 1),ETAEXP(3, 1)/ 34.16,0.0409,202.7/
DATA PATM(3, 2),RHO(3, 2),ETAEXP(3, 2)/ 68.79,0.0789,209.7/
DATA PATM(3, 3),RHO(3, 3),ETAEXP(3, 3)/138.00,0.1686,237.7/
DATA PATM(3, 4),RHO(3, 4),ETAEXP(3, 4)/205.29,0.2586,277.7/
DATA PATM(3, 5),RHO(3, 5),ETAEXP(3, 5)/273.07,0.3424,322.5/
DATA PATM(3, 6),RHO(3, 6),ETAEXP(3, 6)/340.77,0.4127,370.4/
DATA PATM(3, 7),RHO(3, 7),ETAEXP(3, 7)/408.48,0.4715,419.0/
DATA PATM(3, 8),RHO(3, 8),ETAEXP(3, 8)/478.23,0.5188,465.9/
DATA PATM(3, 9),RHO(3, 9),ETAEXP(3, 9)/545.59,0.5582,511.2/
21
DATA PATM(3,10),RHO(3,10),ETAEXP(3,10)/614.66,0.5926,556.0/
DATA PATM(3,11),RHO(3,11),ETAEXP(3,11)/681.68,0.6219,596.4/
DATA TCELS(4),ETAEX0(4)/200.4,216.6/
DATA PATM(4, 1),RHO(4, 1),ETAEXP(4, 1)/ 33.41,0.0328,221.0/
DATA PATM(4, 2),RHO(4, 2),ETAEXP(4, 2)/ 69.81,0.0700,227.0/
DATA PATM(4, 3),RHO(4, 3),ETAEXP(4, 3)/138.75,0.1425,247.4/
DATA PATM(4, 4),RHO(4, 4),ETAEXP(4, 4)/205.84,0.2140,276.1/
DATA PATM(4, 5),RHO(4, 5),ETAEXP(4, 5)/273.41,0.2828,308.1/
DATA PATM(4, 6),RHO(4, 6),ETAEXP(4, 6)/341.79,0.3465,343.8/
DATA PATM(4, 7),RHO(4, 7),ETAEXP(4, 7)/409.50,0.3993,379.2/
DATA PATM(4, 8),RHO(4, 8),ETAEXP(4, 8)/478.91,0.4432,413.7/
DATA PATM(4, 9),RHO(4, 9),ETAEXP(4, 9)/545.25,0.4830,449.4/
DATA PATM(4,10),RHO(4,10),ETAEXP(4,10)/614.32,0.5199,485.9/
DATA PATM(4,11),RHO(4,11),ETAEXP(4,11)/682.70,0.5508,521.0/
C
C Lennard-Jones parameters and molar masses for methane (1),
C ethane (2), and carbon dioxide (3)
C
DATA EPSK(1),SIGMA(1),RM(1)/148.6,3.758,16.0428/
DATA EPSK(2),SIGMA(2),RM(2)/195.2,3.941,44.0098/
C
C Critical constants for mixture components
C
DATA PC(1),RHOC(1),TC(1)/4.54,0.1621,190.4/
DATA PC(2),RHOC(2),TC(2)/7.28,0.4682,304.2/
NC=2
XCO2=0.757
Y(2)=XCO2
Y(1)=1.-XCO2
MODEL=1
WRITE(6,'(T2,A7,F5.3)') 'xCO2 = ',XCO2
DO 2 ITEMP=1,4
TK=273.15+TCELS(ITEMP)
C
C Pure-component low-pressure viscosities
C
CALL ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)
WRITE(6,'(T2,A2,F5.1,A10)') 'T=',TCELS(ITEMP),' degrees C'
WRITE(6,'(T2,A24,F6.2,A11)')
, 'Component 1 viscosity = ',ETA(1),' micropoise'
WRITE(6,'(T2,A24,F6.2,A11)')
, 'Component 2 viscosity = ',ETA(2),' micropoise'
WRITE(6,'(T2,A24,F6.2,A11)')
, 'Low-pressure viscosity = ',ETAEX0(ITEMP),' micropoise'
WRITE(6,'(T2,A6,4A14)') 'p/atm','d/g cm^-3','eta(calc.)/uP',
22
, 'eta(expt.)/uP','Residual/uP'
DO 1 I=1,11
PMPA=PATM(ITEMP,I)*0.101325
RHOGC3=RHO(ITEMP,I)
CALL ETAMIX(MODEL,NC,ETA,ETA0,ETARHO,PC,RHOC,
, RHOGC3,RM,TC,Y)
RESID=ETARHO-ETAEXP(ITEMP,I)
WRITE(6,'(T2,F6.2,F14.4,3F14.2)') PATM(ITEMP,I),
, RHO(ITEMP,I),ETARHO,ETAEXP(ITEMP,I),RESID
1 CONTINUE
WRITE(6,*)
2 CONTINUE
END
The output
xCO2 = 0.757
T= 50.1 degrees C
Component 1 viscosity =119.06 micropoise
Component 2 viscosity =162.47 micropoise
Low-pressure viscosity =155.10 micropoise
p/atm d/g cm^-3 eta(calc.)/uP eta(expt.)/uP Residual/uP
33.41 0.0543 167.37 162.60 4.77
68.66 0.1254 186.75 180.10 6.65
135.55 0.3370 283.61 279.10 4.51
203.39 0.5126 426.43 419.80 6.63
273.07 0.6055 542.27 532.30 9.97
340.77 0.6609 631.44 613.60 17.84
410.18 0.7000 705.86 675.80 30.06
477.89 0.7298 770.06 726.10 43.96
546.61 0.7562 833.02 781.10 51.92
613.64 0.7763 885.18 821.60 63.58
682.02 0.7908 925.26 855.60 69.66
T=100.4 degrees C
Component 1 viscosity =133.91 micropoise
Component 2 viscosity =184.47 micropoise
Low-pressure viscosity =176.40 micropoise
p/atm d/g cm^-3 eta(calc.)/uP eta(expt.)/uP Residual/uP
34.64 0.0447 186.23 182.40 3.83
67.50 0.0930 198.11 192.20 5.91
137.04 0.2145 241.02 236.70 4.32
205.63 0.3416 307.45 303.00 4.45
273.41 0.4436 381.86 375.50 6.36
341.11 0.5173 452.44 444.70 7.74
23
409.84 0.5731 518.60 509.50 9.10
478.57 0.6152 577.63 565.90 11.73
545.59 0.6483 630.57 614.90 15.67
613.30 0.6788 685.20 662.50 22.70
683.38 0.7022 731.35 701.30 30.05
T=150.7 degrees C
Component 1 viscosity =147.90 micropoise
Component 2 viscosity =205.21 micropoise
Low-pressure viscosity =197.40 micropoise
p/atm d/g cm^-3 eta(calc.)/uP eta(expt.)/uP Residual/uP
34.16 0.0409 205.20 202.70 2.50
68.79 0.0789 214.12 209.70 4.42
138.00 0.1686 242.39 237.70 4.69
205.29 0.2586 281.12 277.70 3.42
273.07 0.3424 327.73 322.50 5.23
340.77 0.4127 376.63 370.40 6.23
408.48 0.4715 426.40 419.00 7.40
478.23 0.5188 473.84 465.90 7.94
545.59 0.5582 519.47 511.20 8.27
614.66 0.5926 564.65 556.00 8.65
681.68 0.6219 607.63 596.40 11.23
T=200.4 degrees C
Component 1 viscosity =161.00 micropoise
Component 2 viscosity =224.65 micropoise
Low-pressure viscosity =216.60 micropoise
p/atm d/g cm^-3 eta(calc.)/uP eta(expt.)/uP Residual/uP
33.41 0.0328 222.08 221.00 1.08
69.81 0.0700 230.41 227.00 3.41
138.75 0.1425 251.67 247.40 4.27
205.84 0.2140 279.13 276.10 3.03
273.41 0.2828 311.99 308.10 3.89
341.79 0.3465 348.86 343.80 5.06
409.50 0.3993 385.04 379.20 5.84
478.91 0.4432 419.85 413.70 6.15
545.25 0.4830 455.82 449.40 6.42
614.32 0.5199 493.58 485.90 7.68
682.70 0.5508 528.97 521.00 7.97
shows that the correlation consistently overestimates the viscosity by an ex-
tent that increases with increasing density (as one would expect), but de-
creases with increasing temperature. Over the range of conditions (up to
100 atm) relevant to coal-bed methane technology, the deviations are about3% at the lowest temperature (50ÆC) in the measurements of DeWitt and
24
Thodos; following this trend, the deviations can be expected to be somewhat
larger at lower temperatures.
The tentative conclusions that can be drawn from the calculations pre-
sented here are that (1) the Wilke correlation gives the best results for the
low-pressure mixture viscosity, and (2) the Dean-Stiel method gives results
of satisfactory accuracy for pressures up to about 100 atm.
Di�usion CoeÆcients
The starting point of the discussion of di�usion coeÆcients presented by Reid
et al. is the Chapman-Enskog formula
DAB =3
16
sM1 +M2
M1M2
2�kT1
n��2AB
D
fD =3
16
s2�kT
�
1
n��2AB
D
fD
where m1, m1 are the molecular masses, D is the collision integral appro-priate for di�usivity, and fD is a dimensionless parameter with a value close
to 1. The subscript applied to D signi�es di�usion of A in B, or vice versa.If the molar density is assumed to be obtained from the ideal gas equationof state in the form p = NkT=V = nkT , this becomes
DAB =3
16
sM1 +M2
M1M2
2�kT1
n��2AB
D
fD =3
16
s2�kT
�
kT
p��2AB
D
fD
Expressing the reduced mass � in molar units requires inclusion of a factorof Avogadro's number NA in the numerator of the fraction under the square
root, and if pressure is expressed in atmospheres and the length-scale of theinteraction potential function in �Angstr�om units, the conversion factor is
3
16
s2000�kNA
�
1020k
101325�= 1:858670 � 10�7m2 � s�1
�(g �mol�1)1=2(atm)(�A)2(K)�3=2
The working equation for the di�usivity is therefore
DAB=m2 � s�1 = [1:858670 � 10�7(g �mol�1)1=2(atm)(�A)2(K)�3=2]
�T 3=2fD
p��2AB
D
sM1 +M2
M1M2
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whereM1,M1 are the molar masses (g�mol�1). This is in agreement with the
formula 11.3-2 given by Reid et al., apart from the trivial factor of 1� 104
required to convert m2s�1 to cm2s�1.
The di�usion collision integral can be also be calculated, as a function of
the Lennard-Jones potential parameters, from an empirical approximation
developed by Neufeld et al., viz.,
d = A=T �B + C exp(�DT �) + E exp(�FT �) +G exp(�HT �):
Here, the reduced temperature T � for the pair AB is calculated from a char-
acteristic temperature obtained from a geometric mean mixing rule, and a
length scale obtained from an arithmetic mean:
�AB
k=
p�A�B
k�AB =
�A + �B
2:
Subroutine DCCEN0, which returns a matrix of di�usivities calculated inthis manner, is as follows:
SUBROUTINE DCCEN0(NC,DCOEFF,EPSK,PBAR,RM,SIGMA,TK)
C This subroutine returns an array of low-pressure gas diffusivities
C calculated from the Chapman-Enskog formula, in which the collision
C integral is estimated from the Lennard-Jones parameters
C $\epsilon/k$ in EPSK(1:NC) and $\sigma$ in SIGMA(1:NC) according
C to the empirical correlation devised by P.D. Neufeld, A.R. Janzen,
C and R.A. Aziz, J. Chem. Phys., 57: 1100 (1972).
C
C Reference: R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The
C Properties of Gases and Liquids", New York: McGraw-Hill (1977),
C 3ed., pp. 549-550.
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(NC) !Lennard-Jones parameters $\epsilon/k$, K
DIMENSION DCOEFF(NC*NC)!Diffusivity, cm^2 s^-1
DIMENSION RM(NC) !Molar masses, g/mol
DIMENSION SIGMA(NC) !Lennard-Jones diameters $\sigma$, Angstrom
C
C Parameters in the correlation for OMEGAV as a function of
C dimensionless temperature TSTAR:
C
C \[
C \Omega_d = A/T^{*B}+C\exp(-D T^{*})+E\exp(-F T^{*})
C +G\exp(-H T^{*})
C \]
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C
PARAMETER(PIE=3.14159265358979324)
PARAMETER(A=1.06036)
PARAMETER(B=0.15610)
PARAMETER(C=0.19300)
PARAMETER(D=0.47635)
PARAMETER(E=1.03587)
PARAMETER(F=1.52996)
PARAMETER(G=1.76474)
PARAMETER(H=3.89411)
PARAMETER(CONST=1.858670D-03)
PARAMETER(FD=1.D+00)
PATM=PBAR/1.01325
DO 2 I=1,NC
DO 1 J=1,NC
IJ=NC*(I-1)+J
EPSKIJ=DSQRT(EPSK(I)*EPSK(J))
SIGMIJ=(SIGMA(I)+SIGMA(J))/2.
RMUIJ=RM(I)*RM(J)/(RM(I)+RM(J))
TSTAR=TK/EPSKIJ
OMEGAD=A/TSTAR**B+C*DEXP(-D*TSTAR)+E*DEXP(-F*TSTAR)
, +G*DEXP(-H*TSTAR)
DCOEFF(IJ)=FD*CONST*(TK**1.5)/DSQRT(RMUIJ)
, /(PATM*SIGMIJ*SIGMIJ*OMEGAD)
1 CONTINUE
2 CONTINUE
RETURN
END
The driver program
C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12
C Calculation of diffusivities of gaseous mixtures
C
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION EPSK(2),RM(2),SIGMA(2),DCOEFF(4)
NC=2
TK=590.
C
C Lennard-Jones parameters and molar masses
C
SIGMA(1)=3.941
SIGMA(2)=3.798
EPSK(1)=195.2
EPSK(2)=71.4
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RM(1)=28.013
RM(2)=44.010
PBAR=1.01325
CALL DCCEN0(NC,DCOEFF,EPSK,PBAR,RM,SIGMA,TK)
DO 2 I=1,NC
DO 1 J=1,NC
IJ=NC*(I-1)+J
WRITE(6,*) DCOEFF(IJ)
1 CONTINUE
2 CONTINUE
END
calculates the binary di�usion coeÆcient of nitrogen and carbon dioxide,
and produces an array containing the self-di�usion coeÆcients of the pure
endmembers as well:
1 1 0.483
1 2 0.510
2 1 0.510
2 2 0.514
The value of the o�-diagonal element is in good agreement with that obtainedin Example 11-1 from Reid et al.
28