A further evaluation of the clausius equation of state in thermodynamic property calculations

The Chemical Engineering Journal, 15 (1978) 121 - 129 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

A Further Evaluation of the Clausius Equation of State in Thermodynamic Property Calculations SALAH E. M. HAMAM* and BENJAMIN C.-Y. LU Department

of Chemical Engineering,

University

of Ottawa,

(Received 12 May 1977; in final form 21 September

Ottawa,

Ontario

(Canada)

1977)

analytical expressions have yet been developed which can describe without restrictions the volumetric behavior of both liquid and gas phases (the most important ones from the viewpoint of chemical engineering) over a wide range of temperature and pressure, including the critical region, for both pure components and mixtures. Recently, many attempts have been made in the literature to improve the existing equations of state, especially the simpler ones such as the Redlich-Kwong (RK) equation [l] and the Clausius equation [ 21. In these modifications, the parameters of the equations are considered to be temperature dependent as well as substance dependent [see, for example, refs. 3-81. The two-parameter RK equation proves to be a very powerful one for generating the desired physical properties when the modified procedure for evaluating the parameters is followed. However, from the theoretical and mathematical points of view, a suitable equation of state should contain a minimum of three parameters [9, lo]. The calculation of vapor-liquid equilibria for pure components by means of an equation of state must satisfy the following three requirements at saturation conditions [ 91:

A bstrac t A new modified procedure isproposed for the evaluation of the parameters of the Clausius equation in which the parameter fi2, is considered to be constant. The other two purecomponent parameters a2, and &, are considered to be tempemture dependent and are evaluated from vapor pressures and saturated liquid volumes. The values obtained are used for calculating PVT data, heats of mixing HE, volumes of mixing VE and vapor-liquid equilibrium (VLE) properties. The representation of PVT data for liquid n-heptane by the proposed modification is not satisfactory. When the calculated HE and VE values are compared with those obtained by the previous modification in which all three pammeters are considered to be tempemture dependent, the results obtained in VE are compamble but the proposed method yields better HE values. However, the VLE values calculated by the proposed method do not provide significant improvement over the modified Redlich-Kwong equation. The systems studied include ethane-carbon dioxide, propane-carbon dioxide, hydrogen sulfide-n-butane, nitrogen-argon, nitrogenoxygen, argon-oxygen, propane-ethanecarbon dioxide and nitrogen-argon-oxygen. The findings of this investigation may provide useful information for future development of new equations of state.

vQ.calc = vQ,ex,

(1)

V V.Calc

(2)

=

Vv.ex*

f v.calc = fQ.eale

(3)

Martin [lo] states that a linear transformation in the volume coordinate by the addition of a third parameter to a two-parameter equation may possibly provide the needed modification because it has no effect on the two pressure-volume derivatives of the critical isotherm which must vanish at the critical point. This is what Clausius [2] did in his

1. INTRODUCTION

In spite of the tremendous efforts devoted to searching for a suitable equation of state, no *Present address: Escuela De Ingenieria Quimica, Facultad De Ingenieria, Universidad Del Zulia, Maracaibo, Venezuela. 121

122

modification of the van der Waals equation. Perhaps Clausius’ effort was not fully appreciated because he offered a kinetic explanation that was only partially justified. In an earlier article [ 71 the Clausius equation was successfully applied to the calculation of vapor-liquid equilibrium (VLE) data by means of a modified procedure. The three parameters of the equation were considered to be temperature dependent and were determined from the vapor pressure p and the saturated liquid and vapor densities VQ and V,. When the modified Clausius equation was applied to the prediction of other thermodynamic properties, such as HE and VE, the results obtained [ll] indicated that the derivatives of the parameters, with respect to temperature, were too sensitive for the calculation, of HE values. The quantity c changed rapidly with temperature and did not have a special trend; some negative values were obtained. It should be mentioned that Martin’s mathematical explanation of the need for a third parameter implies that it should have a positive value to take care of the linear shift of the critical isotherm to the left on a P-V diagram. In this investigation, it was intended to modify the Clausius equation by a somewhat different procedure, and to evaluate the capability of the modified equation for calculating thermodynamic properties. 2. PJtOPOSEDMODIFICATION

In the previously proposed procedure, values of V, are required in addition to VQ and p in the evaluation of the parameters. The uncertainty of the literature V, values is reflected in the numerical values of the parameters, which is perhaps the reason for the negative c values and for some difficulties in obtaining convergent solutions when the original values of the parameters were used as the initial guesses in the calculation procedure. It is therefore proposed that only two parameters be treated as temperature dependent whilst a positive value is assigned to the third parameter to modify the cohesive pressure term. The Clausius equation is given by p=--

RT V-b

a T( V + c)~

which may be rearranged to give 1

PV _=z=-RT

Ah

l-h

B(1+fQ2

(5)

where a = a S2V2P

T

(6)

b = s2;V&lc1S;

(7)

c = slcvc(3s

(3)

1 S z-z& b h z-z_ v

- 8)

RT,

(9)

PCVC BP

(10)

z

(11) A-a-a R2T3

B?!_ RT

C2 S2V2P =” T R2T3

(12) (13)

The original values of the quantities S2o, a b and a2, reported by Clausius are 0.4219, 0.2500 and 0.1250, respectively. In this work, the quantities a a and Slb are considered to be temperature dependent while 52, has an assigned value equal to the original value of 0.1250. This value is consistent with the two conditions at the critical point, (15) and is responsible for the linear shift of the critical isotherm to the left on a P-V diagram. The pure-component parameters SZ2,and fl b are evaluated from the saturated properties V, and p to satisfy the conditions of eqn. (1) and

Equation (16) is equivalent to eqn. (3) because 4 = f/P. The fugacity coefficients for the pure component i are obtained from the equations [ 71

123

hQ

ln @iQ= --1nZQ 1

-ln(l-hQ)-

_h

Q

(

-

1

l---Q

hv

In @iv = 1 -h, -

-zQ

1

(2+HQ)

(17)

-In&-ln(l-hh,)1

( 1-h”

-2,

1

(2 +H”)

TABLE 1 Valuesof Q, and !&,in the Clausiusequation (!& =0.1250)

(18)

Combining eqns. (16) - (18) gives ln

bivzQ

-A

pounds was based on the proposed modified procedure at the temperatures of interest. The values obtained for ethane, propane, carbon dioxide and n-heptane are listed in Table 1. The temperature-dependent characteristics of

Compound

T W

Ethane

224.04 233.15 241.45 243.15 244.26 255.35 263.15 266.48 269.25 283.15

0.43687 0.43712 0.44536 0.44148 0.44223 0.44747 0.44134 0.44114 0.44415 0.43587

0.46628 0.46492 0.44211 0.42558 0.42373 0.41357 0.37713 0.36832 0.36965 0.32518

Propane

231.10 240.00 244.66 250.00 260.00 266.48 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 368.00 368.50 369.00 369.82'

0.44734 0.45159 0.45344 0.45564 0.45878 0.46034 0.46094 0.46217 0.46235 0.46146 0.45948 0.45630 0.45168 0.44558 0.43780 0.43328 0.40539 0.41837 0.41921 0.47457

0.64779 0.63586 0.63005 0.62127 0.60502 0.59369 0.58681 0.56653 0.54379 0.51831 0.48985 0.45802 0.42209 0.38160 0.33597 0.31128 0.29263 0.23766 0.24003 0.45174

Heptane

303.15 323.15 373.15 423.15 473.15

0.51433 0.52068 0.52431 0.51508 0.48932

1.67656 1.63218 1.49387 1.28266 0.95958

Carbondioxide 222.04 233.15 241.45 243.15 244.26 255.35 263.15 266.48 269.25 283.15

0.48066 0.47742 0.47579 0.47434 0.47325 0.47036 0.46704 0.46482 0.46339 0.44648

0.62965 0.62330 0.57204 0.56485 0.55969 0.52057 0.48996 0.47457 0.46172 0.37764

+lIl(l-hQ)+ Q

(2+HQ)=0

(19)

These equations were used to evaluate the two parameters a ,, and a b. The calculation procedure is similar to that reported earlier [7] but simpler because the value of. a, is fixed. A brief description of the procedure is as follows. (1) Assume initial values for n,, and 52b (e.g. a,, = 0.4219 and 52b = 0.2500). (2) For a given value of zQ,exp,, assume a value of @iv and calculate hQ by means of eqn. (19). (3) Calculate B from eqn. (10) and A from eqn. (5). (4) Substitute the values of A and B obtained from step (3) into eqns. (5) and (10) and solve simultaneously for 2, and h,. (5) Calculate @iv from eqn. (18). (6) Obtain a new hQ value from eqn. (19) and compare it with the value used in step (2). If the two values disagree, repeat steps (2) - (6) using the new @iv value obtained from step (5). Iteration is continued until the change in hQ is less than the specified tolerance. (The tolerance used in this investigation is 0.00005 .) When the Clausius equation is applied to the calculation of properties of mixtures, the mixing rules used in this work are identical to those reported previously [ 71. 3. RESULTS ANDDISCUSSION The evaluation of the temperaturedependent parameters !Z?=and a, for the pure com-

*Critical point.

124

-

nc

0.180 Ob 0.125

0.42

0.120

0.41

-

0.40 0.8

/

I

1

I

I

9.7

0.75

0.2

0.S

1.0

Tr Fig. 1. Temperature dependence,characteristics (a, = 0.1250).

of S&, and !& obtained from the Clausius equation for propane

these two parameters are depicted in Fig. 1 using the values obtained for propane. The physical properties required in the calculations for the four compounds listed in Table 1 were taken from refs. 12 - 16. The same procedure was used for evaluating the two parameters at the critical temperature, using the critical volume and pressure as the saturated properties. 3.1. Calculation of PVT values In this investigation, PVT data for liquid nheptane [17] at five temperatures were arbitrarily selected for testing the applicability of the proposed modification for PVT calculations. The results are summarized in Table 2 under the heading Clausius II. In the same table, the results obtained from the RK equation when both parameters are considered to be temperature dependent [ 41 and those from the Clausius equation with all three parameters considered to be temperature dependent (Clausius I) [ 71 are included for comparison purposes. Several interesting features are brought out by this comparison. The results obtained by

TABLE 2 A summarized comparison of calculated 2 values for liquid n-heptane (pressure range 50 - 5000 bar) Temperature (E)

n

303.15 323.15 373.15 423.15 473.15

9 9 9 9 9

IAz/ZI x 100% RK

Clausius I

Clausius II

2.19 2.12 2.23 2.49 3.55

4.54 2.60 5.03 10.80 17.89

5.91 9.04 14.88 23.24 39.20

the proposed method are worse than those obtained by the other two methods, indicating that the linear transformation is carried too far to the left on the pressure-volume diagram. This behavior was also observed by Sugie and Lu [ 181 in their attempt to modify the RK equation. The RK equation with only two temperature-dependent parameters yields better results than the Clausius equation with three temperaturedependent parameters over the complete pressure range investigated (50 5000 bar), indicating the superiority and the capability of the RK equation for representing

125

liquid PVT data. Although the calculated results are not reported in detail, the deviations between the calculated and experimental values in all cases increase with increase in pressure. However, all three methods should give reasonable values for the saturated properties (vapor pressure and saturated liquid volume) because these properties are used in the evaluation of their parameters. 3.2. Vapor-liquid equilibria calculations The Clausius equation with all three parameters considered to be temperature dependent is suitable for VLE calculations [7]. The proposed simplification was used to calculate VLE data for six binary and two ternary systems. Some of the systems selected were identical to those tested previously [7]. The results are listed in Tables 3 and 4, respectively, for the binary and ternary systems. The deviations obtained between the calcu-

lated values and the values reported in the literature are comparable with those obtained previously [7]. The calculated results for the binary system propanecarbon dioxide [22] are depicted in Fig. 2. It appears that as long as saturated liquid volumes and vapor pressures are used in the evaluation of the temperature-dependent parameters, little difference can be detected with regard to the deviations in VLE calculations between the two modifications of the Clausius equation. This is also true when the two parameters of the RK equation are similarly considered and evaluated. The results obtained from the modified RK equations are included in Tables 3 and 4. As far as VLE calculation is concerned, the third parameter of the Clausius equation does not improve its capability significantly over the RK equation. Furthermore, the poor results obtained by the proposed method in the

TABLE3 Comparisonof calculated binary vapor-liquid equilibrium data IAPIPI,, x 100%

IAY I,

RK

This work

RK

This work

RK

This work

12 9

0.54 0.92

0.80 0.91 0.82 0.46

-

0.108 0.101 0.102 0.097 0.098 0.097

0.092

12 9 15 13

0.0213 0.0153 0.0272 -

0.0183 0.0075 0.0213

-25°F 0°F 25°F

0.35 0.89 0.74 0.77 0.77 0.47

0.086 0.090 0.084 0.085 0.088

19 19 19 20 20 20

223.15K 243.15K 263.15K 283.15K

11 12 11 13

1.01 0.72 0.74 0.35

0.80 0.70 0.66 0.56

0.0089 0.0068 0.0096 0.0082

0.0067 0.0047 0.0039 0.0052

0.099 0.098 0.099 0.101

0.085 0.084 0.087 0.083

21 21 21 21

Propane-carbon

dioxide

-20°F 20 "F

10 11

2.02

2.64 2.06

0.0062 0.0069

0.0067 0.0122

0.099 0.102

0.086 0.093

22 22

10

2.34

3.64

0.0121

0.0107

0.060

0.051 .

23

14

0.28

0.21

0.0019

0.0039

0.006

0.003

24

0.68

0.37

0.0040

0.0048

0.000

0.000

24

0.43

0.38

0.0006

0.0012

0.009

0.007

25

Tempemture

Ethane-carbon -60°F -20 “F 20 "F

Hydrogen

n

Data source

klz

dioxide

2.26

sulfide-butane

100°F Nitrogen-argon

110°C Nitrogkn-oxygen 110.05 “C

8

Argon-oxygen

110°C

12

126 TABLE

4

Comparison Temperature

of calculated n

ternary

IAWi,v RK

X 100%

This work

Propane(l)-ethane(2)-carbon 20 “F -20 “F

30 18

2.68 2.92

vapor-liquid

equilibrium

BY Iav

Data source

kij This work

RK

data

RK

This work

km

kl3

k23

klz

‘w

km

dioxide(3) 2.52 2.59

0.0066 0.0079

0.0090 0.0100

0.000 0.000

0.102 0.101

0.102 0.099

0.000 0.000

0.090 0.086

0.093 0.086

26 26

0.0146

0.0143

0.0006

0.000

0.009

0.003

0.000

0.007

24

Nitrogen(l)-argon(l)-owygen(3) 110 “C

27

1.54

0.73

very slight variation of k12 values with temperature is indicated in Table 3. 3.3. Calculation of VE The volume of mixing VE at constant T and P is defined by AV = VE = C ~i( Vi - Vi) i

1

(20)

where Vi is .the molar volume of the pure component i and Vi is the partial volume of the same _component in the mixture. The quantity Vi is defined by

In terms of the Clausius equation, the quantity Vi is represented by

1.0

Fig. 2. Comparison of calculated and experimental diVLE data for the binary system propane-carbon calculated (Clausius equation); A, 20 “F; oxide: .,-20 “F. ’

above PVT calculations are not reflected in the VLE calculations. Tks may be attributed to the fact that saturated properties are used in the evaluation of ,the parameters, hence making the procedure suitable for VLE calculation . It should be mentioned that the k12 values used in the VLE calculations were obtained by minimizing the sum of IAPI,. This procedure is recommended by Kato et al. [27]. A

2

2 xjaII - tX!i/( V + C) f

-

X

T( V + c)~

'

1

RT

(V-ZQ2

2a -T(V+C)~

-1

(22)

In this investigation, the proposed method was used to calculate VE values for three binary systems: nitrogen-argon [28] at 83.78 K, argon-oxygen [29,30] at 83.82 and 90 K, and nitrogen-oxygen [29] at 83.82 K. The average absolute deviations obtained by the proposed method are about the same as those obtained when all three parameters are

127

considered to be temperature dependent, although the shape of the VE uersgusx curves is somewhat different. In contrast, the results obtained for the three systems by the modified RK equation (with both parameters considered to be temperature dependent) indicate a somewhat smaller average absolute deviation - a difference of about 0.001 ml (g mol)-‘. In all of these calculations, the k,s values were determined by minimizing the sum of IAVEI,. 3.4. Calculation of HE The heat of mixing HE at constant T and P is defined by AH=HE

=

x.Xi(Hf i

-Hi)-(H*

-H’)

(23)

where (H * -H)* is the enthalpy departure from ideal gas behavior, and can be obtained from the following thermodynamic relationship:

W* -HIT

=RT-Z-‘V+iiP-T(;),t

dV

Using the Clausius equation of state and the mixing rules mentioned above, we obtain Hf -Hi

W-8)dT

RT --

H l+H

S2V2P c cT c dS20 CR2T2 %!i’

.H

C, Zi XiXi h,/dT

dac

(25)

and H* -H .RT

-1+H

CW

CR2T2

where daij -= dT

aij

d(1 - kiJ (IdT

kij)-l + ‘2

+ ‘3)

(&

+ a,)-1

(27)

dbi dabi dT = Vc(4--S)dT dci dT = Vc(3s-88)

ds

(29)

The quantity kij is assumed to be independent of temperature, composition and pressure of the solution. Hence, eqn. (27) reduces to (atzi + aaj)-l

(30)

These equations were used in this work to calculate HE values for the binary systems argonnitrogen [29] at 84 K, argon-oxygen [29] at 84 K and nitrogen-oxygen [31] at 77 K. The

128 overallaverage absolute deviation in terms of

IAHEIobtained from the proposed method is about 0.7 cal (g mol)-‘, which is less than that obtained by the previously proposed method [7] (1.7 cal (g mol)-l). The HE values for these three systems are small. If the comparison of the calculated results is made in terms of average absolute percentage deviations, the improvement is more dramatic (7% uersus 20%).

pressure gas constant 11% temperature molar volume liquid mole fraction compressibility factor

P R S

T V x z

Greek symbols difference fugacity coefficient parameters of the Clausius equation of state acentric factor

4. CONCLUSIONS 0

The Clausius equation of state is suitable for VLE calculations regardless of whether two or three parameters are considered to be temperature dependent. With the choice of constant 52=, the proposed method improves the calculated HE values obtained from the previously proposed approach in which all three parameters are considered to be temperature dependent. When the same two saturated properties (such as saturated liquid volume and vapor pressure) are used in the evaluation of the two temperature-dependent parameters, it is not evident that the third parameter of the Clausius equation gives improvements over the RK equation in calculating thermodynamic properties. The poor representation of PVT data by the Clausius equation indicates that we may need a four-constant equation of state to obtain better results than those obtained from the modified RK equation.

Subscripts k i L V

Superscripts E *

REFERENCES

The authors are indebted to the National Research Council of Canada for financial support. 9

a, b, c

A,&

f h, H kii

n

C

parameters of the Clausius equation of state fugacity dummy parameters or enthalpy 1 - Tad( Tci TJ)‘*‘, binary interaction constant number of data points

excess property vapor phase ideal state partial property

V

ACKNOWLEDGMENT

NOMENCLATURE

critical property component identification liquid phase vapor phase

C

10

11 12

13 14

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