새로운 DDLC 혼합법칙을 이용한 고체-유체, 기체-액체 상평형계산

배순규, 김영삼*, 김선욱*

현대중공업, 울산대학교 화학공학과*

Phase Equilibrium Calculations for Solid-Fluid, Vapor-Liquid Systems

Using New DDLC Mixing Rule

Soon Gyu Bae, Young-Sam Kim*, Sunwook Kim*

Hyundai Heavy Ind., Dept. of Chem. Eng., Univ. of Ulsan*

Introduction

Local composition concept and DDLC mixing rules have been used extensively to describe the phase behaviors of highly nonideal fluid mixtures; however, as pointed out by Mathias and Copeman[1], the equation of state which adopts DDLC mixing rule possesses a critical problem to overpredict nonrandom effects even for the nonpolar mixtures. Several research groups proposed refined models to control nonrandom contribution mainly by adjusting the attractive part of chemical potential[2-4].  An interesting founding for the local composition or the first peak of the radial distribution function is that the size and energy differences of molecules show an adverse effect on the local compositions[5,6]. Therefore, it is believed that the size difference will diminish the nonrandom(local composition) contribution, which may be another route to cure the overprediction problem. In this work new local composition model which accounts correct size differences has been derived and utilized to calculate the solubilities of solid compounds in supercritical fluid mixtures.

Theoretical Derivation

The concept of local composition may be derived using radial distribution function[7] and the number of j molecules around central i molecule is given by,

[eqn: rm n sub ji = x sub j ~ rho Int from sigma sub ji to L sub ji ~ g sub ji (T,V,N sub 1 ,□□□,r)~ d under r]                              (1)

where Lji is the radius of coordination volume. The local composition xji may be represented by nji/□nji. One of the ambiguities for the local composition concept has been caused by the empirical definition of Lji and it becomes worse when constituent molecules show large size differences.  Moreover, the results by various researchers manifest that the short range force and local structure are mainly determined by the first peak height of radial distribution function(RDF).  Therefore, it might be better to define the local number of molecules with the first peak height of RDF, which is the same definition used by Gierycz and Nakanishi[6] for their molecular dynamic simulation of local composition. The number of j molecules around i molecule on the first peak of RDF, rji*, may be written using potential of mean force Wji.

[eqn: rm n sub ji (r sub ji sup * ) ~=&~ x sub j ~ rho ~ exp[ - W sub ji (T,V,N sub 1 ,□□□,r sub ji sup * )/~ kT]]                (2)

The potential of mean force is complex function changing with density, intermolecular interaction energy and size of the molecules. In this work two effects, i.e. packing effect and attractive effect, are considered to be the major contributions to determine the potential of mean force. The packing effect are mainly caused by the size of the molecules and the number of molecules at the first peak of RDF may be represented by following equations.

[eqn: rm n sub ji (r sub ji sup * )~=& x sub j ~ rho ~ exp[ - W sub ji (T,V,N sub 1 ,□□□,r sub ji sup * )/~ kT] # = rm &~ x sub j ~ rho~(packing~ effect) ~(attractive ~ effect) # ~= rm &~ x sub j ~ rho~ F(sigma sub ji , sigma sub ii , sigma sub jj , rho )~ exp [alpha(epsilon sub ji - epsilon sub ji sup o )~/~kT]]            (3)

And the local composition, xji, is given by

[eqn: rm x sub ji ~=~ {~ x sub j ~ F(sigma sub ji , sigma sub ii , sigma sub jj , rho )~ exp[alpha ( epsilon sub ji - epsilon sub ji sup o )/kT] } over { sum from k x sub k ~F(sigma sub ki , sigma sub ii , sigma sub kk , rho )~ exp[alpha (epsilon sub ki - epsilon sub ki sup o )/kT] }]              (4)

where the packing factor, F, is functions of density and size of the molecules.  Utilizing newly developed local composition model(equation 4) various forms of mixing rules may also be derived for the equation of states.

Calculation Results

1. Local Composition Modeling

The molecular dynamics LC data[6] have been used to test new LC theory and the results are compared with those from previous investigators[3,8,9]. A typical calculation results for LB-2-□2 mixtures are illustrated in Fig. 1.

The average absolute deviation for the calculated results are summarized in Table 1, and it can be found that this work shows excellent results compared to the other models, which strongly support the soundness of present approach for the size and energy asymmetries of molecules.

Fig. 1  Comparison of LC Theories for LB-2-□2 Mixtures

Table 1. Deviations from Molecular Simulation Data(%)

□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□

LB-2-1          LB-2-□2          LB-3-□3

□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□

X21/X11  X12/X22    X21/X11  X12/X22    X21/X11  X12/X22

□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□

Wilson      21.21    24.86     23.79    32.05      38.56    47.29

HLP         3.18    20.07     81.55    35.91      209.74   45.81

L & S       4.74     6.54     70.11    35.79      164.28   50.75

L & C       3.90     6.67     75.23    44.15      183.10   64.27

This Work   6.51     9.78      1.85     9.40        8.68   26.15

□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□

2. Solid-Fluid Equilibria

The new DDLC mixing rule for cubic EOS, which simply insert LC in the mixing rule of energy parameter, has been used to calculate the solubilities of pure solid or solid mixtures in various supercritical fluids with and without entrainers. Using new mixing rule we can show that this method possesses excellent capabilities to calculate the solubilities at supercritical fluid conditions as shown in Fig. 2.

Fig. 2. Solubilities of Naphthalene in Supercritical CO2

References

1. Mathias, P.M. and Copeman, T.W.: Fluid Phase Equilibria, 13, 91(1983).

2. Mollerup, J.: Fluid Phase Equilibria, 15, 189(1983).

3. Hu, Y., Azevedo, E.G. and Prausnitz, J.M.: Fluid Phase Equilibria, 13, 351,

(1983).

4. Johnston, K.P., Kim, S. and Wong, J.M.: Fluid Phase Equilibria, 38, 39,

(1987).

5. Troop, G.J. and Bearman R.J.: J. Chem. Phys., 44, 1423(1966).

6. Gierycz, P. and Nakanishi, K.: Fluid Phase Equilibria, 16, 256(1984).

7. Lee, L.L., Chung, T.H. and Starling, K.E.: Fluid Phase Equilibria, 12, 105,

(1983).

8. Lee, R.J. and Chao, K.C.: Fluid Phase Equilibria, 37, 327(1987).

9. Sandler, S.I. and Leee, K.H.: Fluid Phase Equilibria, 30, 135(1986).