Practical equation of state for non-spherical and asymmetric systems

Du Rand, Marlie (2004-12)

Thesis (PhD)--University of Stellenbosch, 2004.

Thesis

ENGLISH ABSTRACT: In this study an equation of state has been developed for the specific purpose of representing systems of simple non-polar spherical and chain-like components and their mixtures for practical applications. To be applied in engineering calculations, the model has to be accurate, be able to represent mixtures with large size asymmetry without the use large binary interaction parameters, and be mathematically simple enough to ensure rapid computations. The model is developed through a sequential evaluation of the statistical mechanical theory of particles and the various approaches available to extend it to real fluid systems. The equation of state developed in this work models the real fluid systems as interacting with a highly simplified two step potential model. The repulsive interactions are represented by a newly developed simplified form of the hard sphere equation of state, capable of representing the known hard sphere virial coefficients and phase behaviour to a high degree of accuracy. This equation has a realistic closest packed limiting density in between the idealised hard sphere fluid random and crystal structure limits. The attractive interactions between the particles are incorporated into the model through a perturbation expansion represented in the form of a double summation perturbation approximation. The perturbation matrix was optimised to have the lowest order in density necessary to still be able to accurately represent real fluid properties. In a novel approach to obtain simple mixing rules that result in the theoretically correct second virial coefficient composition dependence, the perturbation matrix is constrained in such a manner that only the first perturbation term has a term that is first order in density. From a detailed evaluation of the various methods available to represent chain-like non-spherical systems it was finally concluded that the Perturbed Hard Chain Theory provided an ideal compromise between model simplicity and accuracy, and this method is used to extend the equation to chain-like systems. Finally the model is extended to fluid mixtures by uniquely developed mixing rules resulting in the correct mixture composition dependence both at low and high system densities. The newly developed equation of state is shown to be capable of representing the pure component systems to a comparable degree of accuracy as the generally applied equations of state for non-spherical systems found in the literature. The proposed equation is furthermore also shown equal or improve on the predictive ability of these models in the representation offluid mixtures consisting out of similar chainlike or size and energetic asymmetric components. Finally, the computational time required to model the behaviour of large multi-component fluid mixtures using the new equation of state is significantly shorter that that of the other semi-empirical equations of state currently available in the literature.

AFRIKAANSE OPSOMMING: Hierdie werkstuk behels die ontwikkeling van ‘n toestandsvergelyking wat spesifiek gerig is op toepassings in alledaagse, praktiese ingenieurstipe berekeninge en daartoe instaat is om sisteme bestaande uit nie-polêre spferiese- en ketting-tipe komponente en hulle mengsels teKettingteorie (PHCT) die mees geskikde metode is vir hierdie doel en is op die vergelyking toegepas. As ‘n laaste stap in die toestandsvergelykingontwikkelling is daar mengreëls ontwikkel vir die vergelyking wat die korrekte samestellingsafhanklikheid toon vir beide die lae en hoë digtheidskondisies. Die model wat in hierdie studie ontwikkel is, is met verskeie ander bekende toestandsvergelykings, wat daartoe instaat is om nie-spferiese sisteme te modelleer, vergelyk en daar is gevind dat die nuwe model daartoe instaat is om suiwer sisteme net so goed as die bestaande vergelykings te modelleer. Verder is daar ook gevind dat die nuwe vergelyking die modellering van verskeie mensels van kettingtipe komponente en komponente van uiteenlopende groottes of interaksie energieë kan ewenaar of verbeter. Laastens is daar ook gevind dat die tyd nodig vir die modellering van die termodinamiese gedrag van mengsels van ‘n groot hoeveelheid komponente aansienlik korter is vir die nuwe model as die ander bekende semi-empiriese vergelykings. kan beskyf. Om aan hierdie vereistes te voldoen moet die toestandsvergelyking die relevante sisteme akkuraat kan modelleer, slegs klein interaksie parameters benodig om mengsels van komponente met groot verskille in molekulêre groottes akkuraat voor te stel en steeds wiskundig eenvoudig genoeg wees om vinnige berekeninge te verseker. Die vergelyking is ontwikkel deur ‘n sistematiese evaluering van die statisitiese meganiese teorie van partikels en die verskillende metodes om hierdie teorië op werklike sisteme toe te pas. Die toestandsvergelyking beskryf die intermolekulêre interaksie tussen die verskillende komponente met ‘n hoogs vereenvoudigde twee-stap interaksie potensiaal model. Die afstotende kragte tussen die komponente word in ag geneem deur ‘n nuwe vergelyking wat ontwikkel is om die gedrag van ‘n ideale harde spfeer sisteem te modelleer. Hierdie hardespfeermodel is daartoe instaat om die viriale koeffisiënte en die fase gedrag van teoretiese harde spfeer sisteme akkuraat te modelleer, en het ‘n maksimum digtheidslimiet wat tussen teoretiese waardes van ‘n perfek geordende en nie-geordende harde spheer sisteem lê. Die aantrekkinskragte tussen die partikels word beskou as ‘n perturbasie van die harde-spheer vergelyking. ‘n Term bestaande uit ‘n dubbelle sommasiefunksie word gebruik om hierdie perturbasie uitbreiding voor te stel. Die sommasie term is geoptimiseer sodat die finale toestandsvergelyking die laagste digtheidsgraad het wat steeds tot ‘n akkurate voorstelling van die termodinamiese gedrag van werklike sisteme lei. Die sommasiefunksie is so gespesifiseer dat die eerste term van die perturbasie uitbreiding slegs ‘n eerste graadse orde in digtheid het in ‘n unieke benadering om te verseker dat die mengreëls van die toestandsvergelyking die teoreties korrekte samestellingafhanklikheid van die mengselvirialekoeffisiente tot gevolg het. ‘n Deeglike ondersoek van die verskillende metodes om die toepassing van die toestandsvergelyking uit te brei tot die moddellering van nie-spheriese ketting-tipe molekules is gedoen en daar is uiteindelik tot die gevolgtrekking gekom dat die Geperturbeerde Harde

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