Shape factors in hard convex body equations of state

 

Binay Praksh Akhouri

Department of Physics, Birsa College, Khunti 835210, Jharkhand, India

*Corresponding Author E-mail: binayakhouri@yahoo.in

 

ABSTRACT:

The analysis using the Carnahan Müller shape factor approach for the original chain-SAFT equation of state, which is based on the Carnahan–Starling hard-sphere reference equation of state, has been performed. The performance of the second order approximation for shape factor of every proposed equation of state of hard convex fluid has been found to be of comparable accuracy when compared to their respective original equation of state.

 

KEYWORDS:  Shape factors, compressibility factors, equations of state

 

 

INTRODUCTION:

Carnahan- Müller (CM) approach1-3 is one of the most effective way to obtain an equation of state for hard convex bodies. According to CM approach, the contribution of anisotropic shape of the bodies is shifted into a shape factor. Here, y is the packing fraction, defined as the ratio of the volume occupied by the spheres with respect to the system volume. The shape factor   may be related to the equation of state as:

Fig.1(a) Comparison of shape factor of different orders with the residual compressibility factor calculated with SAFT equation of state((eqn.( 8)) taking m=3.

 

 

Fig.1 compares the residual compressibility factor calculated with the SAFT equation of state [equ.(8)] and the  corresponding shape-factorized for different orders. For plotting the graphs the shape factors eqns. (11a), (11b) and (11c) are combined with eqn.(9) according to eqn.(4). The considered three values of for analyzing shape factor orders are shown in figures 1(a), 1(b) and 1(c) respectively. These diagrams show the comparison of different shape factor orders with the original equation of state: (i) the zeroth order shape factor is valid only for low value of packing fraction, and high values of segments, m  (ii) the first order shape factor has very good agreement with the eqn.(8) for all range of   and m (iii)  the second order shape factor is also valid for low value of packing fraction,   and high value of segments, m.

 

Fig.1(b) Comparison of shape factor of different orders with the residual compressibility factor calculated with SAFT equation of state((eqn.( 8)) taking m=9.

 

 

Fig.1(c) Comparison of shape factor of different orders with the residual compressibility factor calculated with SAFT equation of state((eqn.( 8)) taking m=12.

 

 

(B) SAFT equation for hard convex body

The shape factor can be performed using equations of state developed for the hard convex body  fluid. The derivation is based on the EoS of ISPT12, the proposed EoS of Nezbeda15, the EoS  by Boubli'k11 and the EoS given by Song and Mason13. All these equations include the ideal gas term. We start our analysis with ISPT equation of state.

 

 

Fig.2 Comparison of the different order shape factor equations for convex body fluid and the original ISPT EoS taking

α=2.0, 1.25, 1.005.  

 

 

Fig .3 Comparison of the different order shape factor equations for convex body fluid and the original Boubli'k EoS taking

α=2.0, 1.25, 1.005. 

 

Fig. 4 Comparison of the different order shape factor equations for convex body fluid and the original Nezbeda EoS taking

α=2.0, 1.25, 1.005. 

 

Fig.5 Comparison of the different order shape factor equations for convex body fluid and the original Song and Mason EoS  taking α=2.0, 1.25, 1.005. 

 

RESULT AND DISCUSSION:

In all the above derived second order shape factor equations (14c, 17c, 20c and 23c) one can observe that the third term in these equations are dissimilar . Also, the second term of the first order shaft factor of Nezbeda EoS is also different from others. In Fig.1-4,  it can be observed that zeroth order approximation is a reasonable approximation but the second order approximation corrects the first order approximation and gives a better agreement to the original equation of state. Therefore the second order approximation is a good approximation for hard convex body fluid and can therefore be regarded as a fortuitous cancellation of the zeroth and first order term.

 

CONCLUSION:

An equation of state based on shape factor for fluids composed of hard bodies of non-spherical shape works very well for a variety of hard bodies. The performances of the proposed equations are compared to the hard-sphere SAFT approach and found to agree well in accuracy. In all cases, the equation proved to be accurate and simple in use. The shape factor concept is not limited to the use of hard non-spherical or spherical reference fluid only but any suitable reference fluid for which there are accurate data can be used. The method also applies to mixtures.

 

REFERENCES:

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Received on 07.01.2017         Modified on 15.01.2017

Accepted on 27.01.2017         © AJRC All right reserved

Asian J. Research Chem. 2017; 10(1):36-40.

DOI:  10.5958/0974-4150.2017.00007.4