# Thermodynamic Equation Of State Biology Essay

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Anthropogenic emissions of CO2 to the atmosphere have being identified as the major compound affecting the stability of earth climate [IPCC, 2001]. Underground geologic storage of CO2 is, amongst other options, a viable economic route to achieving the United Nations Framework Convention on Climate change (UNFCC) stabilisation target while still maintaining the use of fossil fuel. The large storage capacity and common occurrence of saline aquifers interest investigators as a potential storage site. An accurate equation of state (EoS) is vital in reactive transport models that predict the ultimate fate of stored CO2. A new EoS, for the mixtures of CO2/water/salt fluids, is proposed to be developed using the recent modification of the statistical association fluid theory (SAFT), to aqueous mixtures of electrolyte solution (SAFT-VRE), [Galindo, A. et al., 1999] and molecular dynamics simulation (using DL_POLY program). Also improvement to thermodynamic description of the fluids mixtures at the critical point is aimed. This is to be achieved by improving the parameterisation of already existing literature potentials and incorporation of news terms (1) to the residual free energy of ion-ion interaction to account for ion-ion dispersion effect and (2) to the monomer-monomer interaction contribution via a third-order perturbation in powers of the square-well depth. Obtained binary interaction parameters (from sub-binary systems) will be validated against experiment and/or simulation result (where no experimental results are available).

## 2.2 Background: Introduction

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Global anthropogenic emissions of greenhouse gases (GHG), mainly CO2, from fossil fuel combustion to the atmosphere have being identified as affecting the stability of the earthââ‚¬â„¢s climate. A general consensus by the Inter-governmental Panel on Climate Change (IPCC) is that the emissions and relative causes must be mitigated [IPCC, 2001]. Also, meeting the United Nations Framework Convention on Climate change (UNFCC) stabilization target, large reductions in GHG emissions is required, particularly CO2 emissions. Underground geologic storage of CO2 (from stationery emission sources) is viewed as a viable economic strategy of achieving this reduction as well as increasing the flexibility in developing alternative energy sources [Czernichowski-Lauriol et al., 2002]. CO2 injection in geologic reservoirs is employed by the petroleum industry to improve recovery rates of oil and gas in declining oil and gas fields, a process known as enhanced oil recovery (EOR) [NETL, 2010b]. The large volume of saline aquifers (20% to 500% of projected CO2 emissions to 2050, Davidson et al., 2001), common occurrence and non-potential source for potable water makes storage in saline aquifers an option considered for geologic sequestration of CO2. Storage in saline aquifers can be achieved by either physical trapping (buoyant supercritical CO2), solubility trapping (Dissolution in brine), ionic (dissolved bicarbonate ion) and mineral (solid carbonate precipitate) trapping mechanisms [Czernichowski-Lauriol et al., 2002]. However, dissolution of CO2 in saline waters (solubility trapping) is considered the most important long-term retention state [Bickle, M. et al., 2007]. A key aspect of CO2 sequestration is the need to accurately predict CO2 solubility in aqueous solution at high pressures (associated with deep depth injection), over a geologic period of time. Hence a reliable equation of state is an essential ingredient for transport modelling which predicts the ultimate fate of stored CO2. Our idea is to develop a robust equation of state using the statistical associating fluid theory (SAFT) approach plus molecular dynamic simulations (using DL_POLY program) to accurately describe the thermodynamic properties and vapour-liquid equilibrium of CO2, water and salt mixtures, as this will aid in describing the solubility of CO2 in saline aquifers.

## 2.3 Background: Statistical Associating Fluid Theory (SAFT).

Carbon dioxide is usually injected into saline aquifers as a supercritical fluid. Interactions between CO2, water and salt (NaCl) are a combination of associating and non-associating chain fluid mixtures. While water is a highly associating fluid [Gill-Villegas. A., et al. 1997], CO2 is considered a non-associating chain molecule and strong electrolyte solution of aqueous sodium chloride (NaCl) have being considered non-associating [Robinson, R. A., 1965]. However this only applies to ambient conditions. It has being shown that ion-ion association (ion pairing) occur in aqueous electrolyte solution as temperatures approach the critical point for water (due to the decrease in dielectric constant of water) [Pitzer, K. S. J., 1993]. Readily employed engineering equation of state, such as Peng-Robinson, Soave-Redlich-Kwong and Benedict-Webb-Rubin, are improvements on the hard sphere contribution and/or mean field term of the van der Waals equation. Their empirical approach can accurately describe the thermodynamic behaviours of simple, nearly-spherical low molecular mass hydrocarbon and simple inorganic (e.g. N2, CO, O2 etc). However the reference rapidly becomes inappropriate in predicting fluids mixtures of highly associating and non-spherical chain molecules [Economou. G.I., 2002]. This is because, for such fluids new intermolecular forces such as columbic forces, complexing forces and forces due to association comes into play which are not explicitly taking into consideration by such approach [Economou. G.I., 2002]. A more appropriate reference should incorporate the contribution of molecular shape and association as they certainly affect the fluid structure. It is in this light that Chapman et al. (in 1990) proposed an equation of state for associating chain molecules based on a series of Wertheim first-order thermodynamic perturbation expansion of Helmholtz energy [Wertheim, M. S., 1984a; 1984b; 1986a; 1986b; 1986c] called statistical associating fluid theory (SAFT).

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SAFT relates the thermodynamic properties of a fluid to its intermolecular forces. In the SAFT approach, the molecular contribution to the macroscopic behaviour of the fluid is via a sum of terms which include the separate effect of the molecular shape (chain length), dispersion interaction and molecular association [Galindo, A. et al., 1998]. This consideration makes SAFT suitable for a broad range of molecules, from non-associating near-spherical and non-spherical molecules, to associating near-spherical and non-spherical molecules [Chapman, G. W. et al., 1990]. SAFT essentially considers complex molecules to be built-up of tangentially touching spherical monomers. The general equation for SAFT Helmholtz free energy for associating chain molecules is given by

(1)

Where is the ideal free energy, is the excess Helmholtz energy of the free monomers, is the Helmholtz free energy change on connecting the monomers into chains and is the contribution to the free energy due to intermolecular association. are the number of molecules, Boltzmann constant and temperature (K) respectively [Gill-Villegas, A. et al., 1997]. Several modification of the original Lennard-Jones (LJ) segment (used by Chapman et al.) has being made to improve the description of the monomer-monomer contribution [Banaszak, M. et al., 1993; Ghonasgi, D. & Chapman, G. J., 1994; Tavares, F. W. et al., 1995].

In the modified version of SAFT (SAFT-VR) developed by Gill-Villegas and co-workers, an arbitrary potential of variable range is used to used to describe the chain molecules of hard-core segment. An additional derived parameter, the range (, allows for treatment of highly non-conformal fluid mixtures [Gill-Villegas, A. et al., 1997]. In the SAFT-VR approach, the monomer-monomer dispersion interactions are represented by second-order high-temperature perturbation expansion using a compact expression for the first order perturbation term, (mean attractive energy). The derived second-perturbation term, describes fluctuation of the attractive energy due to the fluid compression effect of [Gill-Villegas, A. et al., 1997]. This effect correlates to macroscopic thermodynamic compression described by local density variation of the fluid. The expression is derived from Barker and Henderson perturbation theory, and given as

(2)

Where and are the first and second perturbation term associated with the attractive energy of the variable range. is the Helmholtz free energy for a mixture of hard spheres. is the inverse of temperature (T in Kelvin), and is Boltzmann constant. When applied to mixtures SAFT-VR is simply straightforward. The mean value theorem (MVT) for pure component is still applicable in evaluating for monomer mixtures. Also the contact value and radial distribution function for pure components in mixtures can be combined obtaining similar expressions to that of pure component [Gill-Villegas, A. et al., 1997]. The equation was tested for a square well potential (SW), a Yukawa (Y) potential and a Sutherland (S) potential. Excellent representation of the vapour-liquid equilibrium (VLE) for binary mixtures of water with non-electrolytes was observed below the critical region, when vapour-liquid coexistence densities were correlated with simulated results. SAFT-VR was however inadequate in describing the thermodynamic behaviour at the critical region. Galindo, A. et al., 1998, applied several mixing rule to account for the binary mixtures of non-conformal fluids using SAFT-VR, but the approach failed to adequately describe phase behaviours at the critical region.

Our aim is to extend the order of thermodynamic perturbation of the monomer-monomer attractive energy term to describe the phase behaviour at the critical region.

Having obtained a good description of the thermodynamic properties of water, Galindo, A. et al., 1999, extended the SAFT approach to mixtures of strong electrolyte solution (SAFT-VRE) using SW potential. Here, water molecules are modelled as hard sphere with four associating short range sites describing the hydrogen-bonding association and electrolyte molecules are modelled as two hard spheres (cation and anion) of different sizes. The mean-spherical-approximation (MSA) for the restricted primitive model was used to account for the long-range columbic ion-ion interaction. The long range water-water and ion-water attractive interaction were modelled as second-order high temperature perturbation expansion as with the SAFT-VR approach [Galindo, A. et al., 1999]. The general expression for the SAFT-VRE approach takes into consideration contributions from the ion-ion interactions and is given as

(3)

Here is the contribution to the Helmholtz free energy from ion-ion interactions. All other terms are the same as those in equation (1). The SAFT-VRE approach can be easily extended to solutions of mixed salts as the potential parameters used are determined in terms of ions. For all studied salts (including NaCl) in a temperature range of 273-373K, the SAFT-VRE calculated vapour pressure reproduced the experimental data well. However, saturated liquid densities are slightly overestimated [Galindo, A. et al., 1999].

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Examples of our workOur aim is to improve the VLE prediction over an increased temperature range by the addition of new terms to account for the ion-ion dispersion interaction effect. We will run the DL_POLY molecular dynamics simulation package on one and two component systems using the literature potential parameters for pure CO2 molecules and aqueous NaCL salt solution mixtures respectively and validate result against experiment. Once validated, simulation results can be used in addition to experimental data to validate the SAFT-VR expressions. We will also further run SAFT programme for CO2 as non-associating chain molecule using equation (1) and SAFT-VRE for mixtures of associating water molecule and two (cation and anion) associating ions using equations (3).

## 2.4 Research Hypothesis and Objectives

We propose developing a robust thermodynamic equation of state for CO2/water/salt mixtures by using the SAFT approach and Molecular dynamics (DL_POLY) simulations. This entails improving the parameterisation of the model and including new terms to improve the thermodynamic descriptions at the critical point of the mixtures. The project takes advantage of the recent advances in modelling highly non-conformal associating chain mixtures [Gill-Villegas, A. et al., 1997], strong electrolytic mixtures [Galindo, A. et al., 1999] and simulation representation of complex fluid mixtures [Koneshan, S. et al., 2000; Lopez-Rendon, R. et al., 2008], making it feasible. Never before has the quaternary system of CO2/water/salt mixtures being modelled using a SAFT approach and the increasing need to mitigate GHG emissions (especially CO2) makes the project quite timely. The hypothesis and objectives of the various work packages (WP) are:

WP1: the accurate prediction of the thermodynamics and structural properties of CO2 and aqueous solution mixtures will depend on the interaction potential model. Our objectives are to validate molecular dynamic simulation results (DL_POLY) for the various component mixtures utilising literature potentials validated against experiment. Where good matches are obtained, this will serve as a reference to improve the quality of SAFT parameters (where experimental data are lacking).

WP2: the mean spherical approximation (MSA) used for the restricted primitive model (RPM) in the SAFT-VRE approach accounts exclusively for the ion-ion coulombic interaction for which electrolyte ions are assumed to be immersed in a uniform dielectric medium. Dispersive interactions between the ions, however, are lacking. Our objective is to include a new term to account for the ion-ion dispersion effect in the salt solution.

WP3: in the SAFT-VR approach the monomer properties are obtained from a second-order high temperature perturbation expansion of the SW variable range potential. Our objective is to add a third-order perturbation term in powers of the square well depth () of to improve the description of the VLE at the critical point.

## 2.5 Programme and Methodology

WP1: we will first run DL_POLY with one component system for pure water and CO2 molecules using literature interaction potentials [Lopez-Rendon, R. et al., 2008] at a given temperature and pressure range. Water will be represented as an extended simple point charge (SPC/E), as this model takes into account the polarization of water in an approximate way. Simulated results will be validated against experimental result to verify the usability of the chosen force field. Validated results for the pure components of CO2 and water will be used to run DL_POLY with binary mixture system for water-CO2, water-NaCL and water-water mixtures, utilising literature interaction potential for NaCl [Koneshan, S. et al., 2000]. Simulated results will again be tested against experimental results. If convergences occur between the vapour pressure and liquid densities the simulated result will serve as a reference to correlate the SAFT model. A molecular simulation tracks the motion of individual molecules in time and can be used to interpret experimental results or serve as substitute where no experimental data are available.

Finally, we will run SAFT with one component and with binary mixture system for pure molecules of water and CO2 respectively, using literature parameters for the SAFT-VR approach (i.e. square well range, , depth, and segment diameter, , association energy, and association volume, ) [Galindo, A. et al., 1999; Alain, V. et al., 2004]. Where and , are the association energy and volume due to short range attraction between a hydrogen site and oxygen electron on two different molecules. Water is modelled as a hard-sphere with four short-range non-central associating sites representing hydrogen bonding () while CO2 is modelled as tangential touching spherical dimer chain molecule (without associating sites,). These parameters are validated against experimental and/or simulation result and optimised if not well-fitted using the simplex method. This is important in describing the thermodynamic properties of real substances.

WP2: we will use the optimized intermolecular potentials for pure water component (from WP1) to run SAFT with ternary mixture system for mixtures of water in strong electrolytic solution of NaCl, utilised the extended version of SAFT-VR for strong electrolyte solution (SAFT-VRE) [Galindo, A. et al., 1999]. Water (solvent) is modelled in the same spirit as SAFT-VR and contributions to Helmholtz free energy are given by equation (3). Solvent-solvent, solvent-ion and ion-ion interaction contribution will be considered [Galindo, A. et al., 1999]. MSA assume RPM will account for the long-range coulombic interaction. However the assumption of a zero long range attractive square well ion-ion interaction will be relaxed. This has being assumed in previously modelled water-NaCl mixtures [Galindo, A. et al., 1999; Gill-Villegas, A. et al., 2000] for which equally-sized ionic molecules are solvated in a uniform dielectric solvent medium at ambient conditions. This approach has however failed to describe, accurately, thermodynamic properties at the critical point. We will be taking into account the molality of saline aquifers to relax this assumption and add a new term accounting for the ion-ion dispersion effect of the coulombic contribution to Helmholtz free energy. This is so because the effect of strong electrolytic ions on properties of highly associating polar solvent such as water can alter the critical constant of water within the critical point, leading to ion-ion association [Koneshan, S. et al., 2000]. NaCl intermolecular parameters will be taking from literature [Galindo, A. et al., 1999].

Finally, determined intermolecular parameters of the ternary mixture will be fitted against experimental and/or simulated result and optimised using the simplex method.

WP3: lastly, in the SAFT-VR approach (basis for formalism in the SAFT-VRE), contribution to the Helmholtz free energy due to long-range dispersion forces is obtained via a second order high-temperature perturbation expansion of the variable range [Gill-Villegas, A. et al., 1997]. This level of approximation has excellently described thermodynamic behaviours below the critical point but fails as temperatures approach the critical point. It has being suggested that incorporating a new term, due to a third order perturbation in the powers of the attractive square-well depth (), to the monomer-monomer segment contribution will significantly improve thermodynamic description at the critical point (recent personal communication of Masters with Galindo). This we would evaluate for binary mixtures of optimised ternary intermolecular SW potential parameters of water with aqueous NaCl solution (WP2) and previously optimised CO2 dimer intermolecular parameters(WP1). It should be noted that never before has CO2/water/salt mixtures being modelled using SAFT approach and so no theoretical results are available. However the SAFT-VRE approach allows for such complex mixtures in its formalism using relatively straight forward combinations with mixing rules [Galindo, A. et al., 1999]. We will run CO2/water/salt mixtures using SAFT with quaternary system. New interaction to be considered will be CO2-water interactions in coexisting phases. Salt will be restricted to the liquid phase (as it is assumed to be non-volatile even at high temperatures) [Parisod, C. J., 1981].

Finally, modelled results will be compared with experimental results. Knowing the Helmholtz free energy all other macroscopic thermodynamic parameters at VLE can be evaluated, hence the solubility of CO2 in saline aquifer determined. Being able to accurately predict the solubility of CO2 in saline aquifers is essential for long-term sequestration of injected CO2.

Milestones of proposed research: these are

M1.1: New simulation results using DL_POLY one component and binary mixture systems for CO2/water/salt mixtures.

M1.2: Improved parameterisation of literature intermolecular potential for pure fluids of water and CO2 using SAFT one component and binary mixture system, respectively.

M2.1: Reformulation of the ion-ion coulombic interaction contribution to Helmholtz free energy to incorporate a dispersion effect between ions.

M2.2: Improved description of thermodynamic properties of water at critical point.

M3.1: Modification of SAFT-VR formalism in the monomer-monomer segment contribution to overall Helmholtz free energy to enhance predictive capability of approach within the critical point of mixtures.

M3.2: New intermolecular parameters for CO2/water/salt mixtures using SAFT with quaternary system approach validated against experimental results.

M3.3: Better VLE representation of theoretical model with experiment at the critical point of fluid mixtures.

## 2.6 Relevance to Academic Beneficiaries

One key benefit obtainable from this project is the development of a robust equation of state using statistical mechanics for CO2/water/salt fluid mixtures. Once this improvement is attained faster and more accurate description of the mixture will be developed enhancing the prediction of CO2 solubility in saline water using theoretical models. A positive outcome will certainly interest the research community, and this will correlate directly to industries (such as the petroleum industries) performing, or intending to explore the option of, CO2 sequestration in saline aquifers. Better prediction of the thermodynamic properties of electrolytic solution at the critical region using a statistical mechanics approach, is one of great interest to applied physical sciences, molecular physics and the engineering community.