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The development of new processes and modifications of existing ones in many industries such as chemical, manufacturing, oil and gas, pharmaceuticals etc. is an important part of the business. Important as well is the controlling and monitoring of those processes which involves the manipulation of the behaviour of the fluids involved. Therefore, the properties of the participating fluids in these industries are very critical in the maximization of the production yield and ultimately the profit of the company in general. The properties of mixtures are always a subject of discussion from the design of a project in these industries to the operation of the project. However, the use of the characteristics of these fluids continues even as the operation is in progress. In order to achieve the process objectives and the business goal at large, the thermodynamic properties of the fluids need to be more accurately described by the model used.
The accurate prediction of these thermodynamic characteristics of the fluid is very vital in the design and operation of the process and equipment used. For example, the properties of these mixtures are applied in the design and operation of distillation columns, reactors, compressors, boilers, fans, blowers, furnaces, pumps, dryers, analyzers, absorbers, adsorbers, furnaces, extruders etc. The design and operation of the equipment require a good estimation of the performance of the fluids connected to them. Over the past decades, scientists have been progressively developing models to accurately forecast the behaviour of fluid in the equipment.
The first model used to estimate these properties is the ideal gas equation which was clearly for gases in an ideal condition alone. The Van der Waals equation, the second model, brought better predictive figures than the former. Since then, a lot of models have been developed to give a better correlation of these properties. These models are : Redlich-Kwong, Soave-Redlich-Kwong (SRK), Peng-Robinson, Wilson, Non-Random Two Liquid (NRTL), Universal Quasi Chemical (UNIQUAC), group contribution methods: Predictive Soave-Redlich-Kwong (PSRK), Analytical Solution of Groups (ASOG), Universal Functional Activity Coefficient (UNIFAC), thermodynamic perturbation theory (TPT) and a host of others. However the formulations of a large number of those engineering codes are based on the principle of chemical (Chapman et al., 1990). The search for a more accurate predictive formula for the thermodynamic characteristics has given rise to several other methods.
Several researches have shown that complex mixtures require equations of state that incorporate several contributory properties of the fluids (Wei & Sadus, 2000). The activity coefficient models (Wilson, NRTL, UNIQAC) in a way to take care of the non-ideality of these complex mixtures employs activity coefficients to describe the real behaviour in the liquid and fugacity coefficients for the vapour phase (Prof. Dr. J. Gmehling, Industrial Chemistry). Despite the adjustment made in their equations, some deviations were observed for complex fluids (Prof. Dr. J. Gmehling, Industrial Chemistry). A completely different mode of correlating fluids properties is the group contribution approach and the perturbation theory which opens a wide field of different technique. Several models have been developed from the later theorem which includes the Wertheim perturbation theory (Wertheim, 1984) and to the statistical associating fluid theory (SAFT) that this work is based on.
Before the use of SAFT, the traditional cubic equations of state were commonly used to determine the nature of fluids. The SAFT Equation of state approach was developed to incorporate the terms that account for the behaviour of real molecules of complex mixtures (Economou, 2002). The SAFT model takes into account the fact that molecules of fluids are made up of spheres and these spheres interact with one another. These interactions gave rise to intermolecular forces, formation of chains of monomers and cause an association of the segments in the molecules (Chapman, 1989). The addition of the terms depicting these interactions has given a better prediction to the actual property of a mixture than the other conventional equation of states (Gil-Villegas et al., 1997) .
The structure of this work will be such that, the theory of the SAFT and of the various versions showing the contributory terms from various interactions shall be presented in the section one. However, we will concentrate on the theory and model of the particular SAFT that will be use in this project called statistical associating fluid theory with variable range Mie potential (SAFT-VR Mie). Furthermore in the same section one, we will illustrate some areas that SAFT-like equations of state have been successfully applied to forecast the behaviour of fluids. And how this SAFT-VR Mie equation of state (EOS) will be use to estimate the properties of petroleum mixtures will be discussed in section three. Section four will treat the parameter estimation part of this review where we will use experimental data to fit the parameters.
And since the SAFT equations of state (EOS) are highly nonlinear and implicit algebraic functions of our unknown parameters, as will be seen later, the implicit least squares parameter estimation will be discussed. More so the Maximum likelihood parameter estimation method will be presented as well to evaluate the weights that will be used in the objective function created by the least squares method. And lastly, the solution of the objective problem will be validated using an average absolute deviation (AAD %) and a confidence interval test.
As a complement to the works that have been done on SAFT-VR Mie EOS, this project work will focus on the use of the equation of state to accurately predict the behaviour of pure components in petroleum mixtures. In order to achieve this, we shall use experimental data value of vapour pressure, saturated liquid density, speed of sound and the isobaric heat capacity of the various individual components to correlate the parameters in the model. The computation of the constants in the model equation will be carried out by a formulation of an objective function using least squares parameter estimation technique. However, in using this method, the weights in the objective function shall be determined by the maximum likelihood principle as mentioned earlier. And finally, the values of the parameters calculated shall be investigated by evaluating their respective average absolute deviation (AAD %) and confidence interval.
Section one: Theory of SAFT-VR Mie
Overview of SAFT
Before the use of SAFT EOS, several fluid packages have been developed to correlate the properties of mixtures. The common models used are: Soave-Redlich-Kwong, UNIQUAC, UNIFAC, Wilson, NRTL, Peng-Robinson, PRSV, Van laar and so many others. These models were developed to take care for the non-idealities in fluids. However, SAFT take care of the difficulties encountered by the former models (Muller & Gubbins, 2001). SAFT is an equation of state used in correlating and predicting the properties of complex fluids (Muller & Gubbins, 2001). It has a better predicting ability than the other molecular based equation of state due to the generation of the EOS from the molecular level (Muller & Gubbins, 2001). The theory of SAFT model and its derivatives are based on Wertheim papers on perturbation theory which were the first to describe the thermodynamic perturbation theory (TPT) (Chapman, 1989).
It is well known that SAFT was developed by Chapman, Gubbins, Radosz, and co-workers at cornell university and Exxon research based on the Wertheim work on thermodynamic perturbation theory (TPT) (Economou, 2002). SAFT is formed from the theory of micro molecules interactions rather than on the traditional approach of assuming that the non-ideality of mixtures can be explained by the chemical reactions existing between the species (known as the chemical theory)(Muller & Gubbins, 2001). In the chemical theory, the associations of the molecules are taken as a chemical reaction whose new compound properties are summation of the chemical properties of the reactant components (Muller & Gubbins, 2001). In addition, the theory assumes that the monomers of the mixtures are made up of spherical segments (Muller & Gubbins, 2001).
More so, the SAFT approach takes into account the existence of intermolecular forces between the spherical segments in the molecules of mixtures consisting of hard spheres (Muller & Gubbins, 2001). SAFT also incorporates the forces resulting from the bonding in the chains of monomers of the mixture for example hydrogen bonding and takes care of the electrostatic attraction between molecules in the mixture by introducing an association term (Muller & Gubbins, 2001). The improvements in the values produced by applying SAFT can largely be credited to the fact that SAFT incorporates the molecular characteristics of components into its model equations while considering the molecules to be made up of segments (Galliero et al., 2007). These segments could be spherical or highly non-spherical segments nonetheless; it is incorporated into the equation (Galliero et al., 2007). SAFT therefore blend into its model the shape, size and structure of the segments that make up the compound. A typical representation of these segments and their interactions in the different stages is shown in figure 1 below.
Figure 1: representation of SAFT molecules showing their interactions (Kwon Cheong Hoon).
In a clear term, the segments of the chain molecules shown in figure 1 are actually united atom models hence the number of segments in the chain does not represent the number of carbon atoms but instead, the parameter m, provides an indication of the non-sphericity of the molecule (Gil-Villegas et al., 1997).
The segments interact with each other, repelling and attracting each other with the same strength in all directions as seen in fig 1 above. They form chains within the molecule and with segments of other molecules in the case of a mixture (Galindo et al., 1998). Therefore, a molecule is described by the number of segment present in its chain m as shown in fig 2 below. The diameter of each of the segments is denoted by Ïƒ, so that the chain length is mÏƒ. In other words they form bonds for example hydrogen bonding, and so have association sites within its structure (Galindo et al., 1998). These association sites shown in fig.2 as the blue and white small spheres represent a lone pair of electrons on an electronegative atom or a hydrogen atom bonded to a very electronegative atom (Process Systems Enterprise Limited) have made the model very verse in estimating the behaviour of fluids.
SAFT has been shown to be a versatile model in handling complex fluid and materials like: polar solvents (CO2 and refrigerants), hydrogen bonded mixtures (carboxylic acid, Hydrogen Fluoride and water), polymers and electrolytes (inorganic salts, charged surfactants (Muller & Gubbins, 2001). These compound and mixtures have electrostatic, polar and other association forces that the traditional equations of states did not include in their models. The SAFT equation has been successfully applied to describe the thermodynamics and phase behaviour of numerous fluid mixtures such as n-alkanes, polymers, perfluoroalkane, carbon dioxide, replacement refrigerants, petroleum fluids and electrolytes (Sun, 2007). This equation with these terms have been successfully applied to predicting the properties of fluid and mixtures including aqueous mixtures and electrolytes, liquid-liquid immiscible systems amphiphilic systems, liquid crystals, polymers, petroleum fluids, and high-pressure phase equilibria (Muller & Gubbins, 2001). Particularly, Economou shown in his work the application of SAFT to thermodynamic and phase equilibrium properties of aqueous and polymer systems (Economou, 2002).
Figure 2: A typical spherical segment in a SAFT molecule (Process Systems Enterprise Limited).
Overview of various versions of SAFT
This section will give a brief description of the some SAFT-like approaches that have been developed from the original SAFT. Over times the original SAFT have been improved upon with the improvement reflecting on the ability of the new method to give a better prediction on the thermodynamic properties of a wide variety of complex fluids. The various SAFT that have been developed are: perturbed chain statistical associating fluid theory (PC-SAFT), SAFT-VR for square well potentials (SAFT-VR SW), SAFT-VR for Lennard-Jones chains (SAFT-VR LJC), SAFT-VR with crossover technique (SAFT-VRX),statistical associating fluid theory with variable range Mie potential (SAFT-VR Mie) and group-contribution statistical associating fluid theory (SAFT-Î³).
Speaking individually, SAFT-VR SW, in addition to the variable range nature of the repulsive and attractive forces, is model using the square well potential for its intermolecular force of attraction and repulsion (Muller & Gubbins, 2001). In another approach, SAFT-VR LJC, have it that the range varies as well but the estimation of the intermolecular potential is carried out using the Lennard-jones' model where the ranges are fixed at 6 and 12 for the attraction and repulsion respectively (Chen, 1998). The square-well potential and the Lennard-jones potentials are illustrated in figure 3 with the square well potential drawn in solid line while the Lennard-jones potential is shown in broken lines.
However, unlike the other SAFT-like equations of state that take their reference system as the hard spheres, PC-SAFT EOS takes its reference system as the chain molecules (Tan, 2008). In addition, the application of the Baker and Henderson perturbation theory leads to a complex formation of the first two perturbation terms in the PC-SAFT EOS(Tan, 2008). Another model is the SAFT-VRX which is a combination of the original SAFT-VR with the crossover technique developed by Kiselev in order to be able to predict properties of mixtures around the critical and sub-critical points (McCabe, 2004). We also have the SAFT-VR Mie potential, which will be discussed later in preceding section. It is a special case of the SAFT-VR LJC in which the potential ranges are varied (Lafitte, 2006). Lastly, we have the SAFT-Ï’ which incorporates the group contribution method into the original SAFT-VR to correlate the properties of fluids that consist of different types of atoms (Lymperiadis et al., 2007).
Figure 3: illustration of square well potential (solid line) and Lennard-jones potential (broken line), (Iacovella, 2007).
Theory and Model of SAFT-VR Mie
This projec is based on SAFT-VR Mie potential and its use to determine the thermodynamic properties of mixtures relevant to the oil and gas industries. SAFT-VR Mie and SAFT-VR LJC share the same principle with a slight disparity. The only difference is in the replacement of the constant range values of 6 and 12 in the Lennard-Jones potential with variable ranges of m and n, given rise to variable range Mie potential (Lafitte, 2006). The variability of the intermolecular range incorporates the non-conformal properties of the range into the SAFT equations. In the work of Lafitte et al, they have shown that SAFT-VR Mie EOS was able to predict accurately the derivative properties such as the isothermal compressibility and the speed of sound in the condensed liquid phase of long-chain n-alkanes than other SAFT like methods (Lafitte, 2006). In this project, the SAFT-VR Mie equations will be used to obtain the thermodynamic properties of pure components in hydrocarbon mixtures.
The equation will be developed in the form of Helmholtz free energy since from Helmholtz free energy of a fluid; every other properties of the fluid can be obtained.
The Helmholtz energy for associating chain molecules can be represented with all the contributing parts in the SAFT approach as
In the equation 1 above, N is the number of molecules, T is the temperature, and k is the Boltzmann constant (Lafitte, 2006). The different contributions from the ideal state, the monomer segments (M), the formation of chains, and the existence of intermolecular association are respectively Aideal , Amonomer, Achain and Aassociation.
It is worthy of note to mention that in order to give room to soft sphere property of fluids, the repulsive part of the intermolecular potential was allowed to vary in the Mie potential and not the usual constant value Î»2=12 as commonly used (Lafitte, 2006).
The intermolecular potential
The Mie potentials (Î»2, Î»1) are given by
(Lafitte, 2006) 3
According to Baker and Henderson perturbation theory the soft core interactions, which is the basis for the inclusion of Mie potential in the SAFT-VR could be represented by an equivalent hard-core temperature-dependent diameter given as:
Where ÏƒBH(T) and its derivatives are evaluated using numerical integration for example Simpson's rule and trapezoidal rule (Lafitte, 2006).
The intermolecular potential for the Mie potentials is then given by
A diagrammatic representation of this potential is shown below in figure 4.
Figure 4: illustration of Mie potential (Lafitte, 2006)
The ideal Helmholtz free energy
The Helmholtz energy contribution in the ideal case is given by;
Where in the equation above Ïs is the molecular density and is equal to Ns/V in which case, Ns is the number of molecules of the component, V is the volume of the fluid and Î› is the thermal de Broglie wavelength of the component under investigation (Lafitte, 2006).
The Monomer contribution
The monomer free energy contribution for a component with the Mie potential is given by the equation;
Where ms is the number of spherical segments in the chain, and Ns is the total number of segments. The monomer free energy per segment of the mixture aM =A/(NskT) is obtained from the Barker Henderson high temperature expansion to the second term:
Where aHS is the free energy for a mixture of hard spheres of diameter ÏƒBH , Î²=1/kT, a1Mie and a2Mie are the first two perturbation expansion terms associated with the attractive term of the potential (Lafitte, 2006). Since we will be dealing with pure component, the Carnahan and Starling expression is used for the hard-sphere residual free energy as oppose to Boublik and Mansoori expression for mixtures (Galindo et al., 1998).
The Chain Contribution
From the figure 1, it is seen that monomers of segments get bonded together as they get close to each other to form chains within the molecule. The number of segments in a chain is described as ms. The bonds formed in this case are typical bonds for example hydrogen bonding. The hydrogen bonds certainly have an impact on the behaviour of the molecules. This effect is packed as the chain contribution term of the Helmholtz free energy. The chain contribution is given as
The cavity function yb is obtained analytically using the Gil-Villegas approach where the Clausius virial theorem is combined with density derivative to give the following expression
(Lafitte, 2006). 11
The Association Contribution
The ions present in the molecule create some kind of electrostatic behaviour in the segments. The chains of segments formed as described in the previous section tend to interact with one another via these electrostatic sites. This results to the chains attraction sites attracting each other. However, this attraction produces some kind of energy that needs to be accounted for. The figure 3 below shows the interaction between associating sites:
Fig. 5: A typical representation showing m segments of a molecule with association sites i and j. (Muller & Gubbins, 2001).
According to Gil-Villegas et al (Gil-Villegas et al., 1997), the Helmholtz free energy contributing term for this association sites s between site a of a molecule and site b of another molecule is expressed as;
The fractional value of the molecules not bonded at site a, Xa can be calculated as
Section two: Previous application of SAFT-VR
n-Alkanes and n-Perfluoroalkanes
Gil-Villegas et al(Gil-Villegas et al., 1997) applied SAFT-VR EOS to estimate the properties of n-Alkanes and n-Perfluoroalkanes. In their work, the parameters Ïƒ, Î», É›/k and ms were estimated by fitting some objective function. The optimised objective function was obtained by fitting the model vapour pressure and saturated liquid densities to the experimental data for the pure components. Gil-Villegas et al shown that the number of segments ms for the homologous group of n-Alkanes and n-Perfluoroalkanes is related to the number of carbon atoms, C in the component and that the relationship is expressed by the empirical equations: ms=1+(C-1)/3 and ms=1+(C-1)/0.37 respectively (Gil-Villegas et al., 1997). From the equations, the computation of the constants of other members of the same homologous series can easily be computed without fitting the parameters of the individual component to the experimental data over again (Gil-Villegas et al., 1997).
Lafitte et al (Lafitte, 2006) built on the work of Gil-Villegas et al to correlate the second-derivative properties: isothermal compressibility, speed of sound, isobaric heat capacity, isentropic compressibility etc. Lafitte et al (Lafitte, 2006) used the SAFT-VR Mie EOS to predict the second-derivatives behaviours of n-Alkanes by including the speed of sound experimental data to the fitting method. They noted that the inclusion of the speed of sound data in the fitting procedure aids in the correlation of isothermal compressibility and other second-derivative properties in the condensed liquid phase. The speed of sound C was used because it incorporates in it the isothermal compressibility Î²T and the isobaric heat capacity Cp which is given by the Maxwell's equation:, where and(Lafitte, 2006).
With their technique, the following three parameters: Ïƒ, Î»2 and É›/k were fitted to the data with the number of segment ms obtained by applying the appropriate empirical relationship between ms and the number of carbon atoms in the component (Lafitte, 2006). The SAFT-VR Mie EOS results for the n-Alkanes were shown to be of better accuracy than the values calculated using SAFT-VR SW approach. Table 3 and 4 shown below, illustrate this comparison.
Table 1: Comparison of the prediction between the second-derivative properties (speed of sound C) using SAFT-VR Mie EOS and SAFT-VR EOS (Lafitte, 2006)
Table 2: Comparison of the prediction between the second-derivative property (isobaric heat capacity CP and its residual Î”CP) using SAFT-VR Mie EOS and SAFT-VR EOS (Lafitte, 2006)
From previous works on pure components, the parameters of petroleum mixtures can be determined by applying an appropriate mixing rule as treated by Galindo (Galindo et al., 1998). However, Lixin et al (Sun, 2007) has used SAFT-VR EOS in combination with a semi-continuous thermodynamic approach to predict the phase equilibria of non-associating gas condensates and light petroleum fractions. Hence, the association term of the SAFT-VR EOS was not included. More so, a square-well potential was use to describe the dispersive interaction of the molecules with the attractive and repulsive ranges of the intermolecular potential fixed at Î»1=6, Î»2=12 respectively (Sun, 2007). However, this work will use SAFT-VR Mie EOS to generate the parameters of pure petroleum components where the repulsive range will be taken as one of the unknown parameters
Section Three: Properties of Oil and Gas mixtures
Composition of oil and gas mixtures
The actual composition of the petroleum mixture has not been clearly understood, it can be classified into the group of paraffinic, naphthenic and aromatic compounds (Sun, 2007). There are also some compounds that accompany the organic compounds which are: water, sulphuric compounds, some traces of nitrogen compounds and other inert substances. In this work, we shall concentrate on fitting the properties of the individual components of the different group using the SAFT-VR Mie methodology. With the parameter of a member of a homologue series estimated that of the other members of the group can be extracted by the relationshi between the parameters and the number of carbon atoms in the component (Gil-Villegas et al., 1997).
Thermodynamic Properties of the mixtures using various EOS and SAFT-VR Mie models
The classical cubic equation of state is based on the assumption that molecules of fluids are made up of a single sphere (monomeric), an assumption that is not realistically true (Prof. Dr. J. Gmehling, Industrial Chemistry). The engineering models like the local composition model: WILSON, UNIQAC, NRTL and the group contribution models; UNIFAC, assumed that the local compositions around individual molecules are independent (Abrams, 1975). Lafitte et al (Lafitte, 2006) demonstrated the reliability of SAFT-VR Mie by carrying out the estimation of the properties of n-Alkanes and comparing its results with that of other models. For example Table 5 below illustrates a comparison between the characteristics forecast by SAFT-VR Mie and SAFT-VR SW (Lafitte, 2006). In addition and worth mentioning is the fact that SAFT-VR Mie gives an inaccurate correlation of fluid behaviour at temperatures close to the critical point: critical temperature and pressure (Lafitte, 2006).
Table 3: The comparison of the computational strength between SAFT-VR Mie EOS and SAFT-VR EOS (Lafitte, 2006)
Section Four: Estimation of SAFT-VR Mie parameters
Experimental data for the Parameter Fitting
The parameters to be determined are the number of segments in the chain, ms, the diameter of each segment in the chain, Ïƒ, the depth of the potential well, É›/k, the attractive and repulsive part of the range of potential, Î»1 and Î»2 respectively. However, while the attractive range Î»1, is fixed at Î»1=6, the repulsive part is allowed to vary to incorporate the non-conformal properties of the repulsive range into the model (Galindo et al., 1998). In order to fit the parameters in the model above, the experimental data of the vapour pressure and saturated liquid density of the condensed liquid phase of the component is used (Lafitte, 2006). In addition, for the purpose of obtaining accurate predictions of the second derivative-properties, the speed of sound and heat capacity data should be used for the fitting procedure (Lafitte, 2006).
Parameter Estimation Problem Formulation and Solution
Since the model equations are highly non-linear and consist of an implicit function of the dependent variable, a non-linear implicit least squares estimation method is used as the fitting procedure (Swaminathan, 2005). Swaminathan and Visco (Swaminathan, 2005) used simplex algorithm to minimize the objective function in order to correlates SAFT-VR EOS parameters from experimental vapour, saturated liquid and vapour volume. In their work, a least square objective function (OBJ) given in equation (17) was minimized to calculate the parameters. In the same vein, Gil-Villegas et al (Gil-Villegas et al., 1997) used the same simplex optimisation approach to fit the calculated vapour-pressure curve and saturated liquid densities to the given experimental data of n-Alkanes and n-Perfluoroalkanes for SAFT-VR EOS (Swaminathan, 2005).
Where in the equation above, N is the number of experimental data.
On the other hand, Lafitte et al (Lafitte, 2006) computed the parameters of SAFT-VR Mie EOS by minimising an objective function given by equation 18(a), (b), and (c) below: 18a
In order to obtain accurately the values of the second-derivative properties of fluids, Lafitte et al (Lafitte, 2006) proposed the addition of the speed of sound and compressed liquid phase data. Then equation (16a) became:
Where, , , are the non-constant weights specified by the experimenter in all the cases mentioned above.
Furthermore, the least square estimation method which involves the minimization of an objective function as stated above is based on the assumption that the residuals (the difference between the experimental value and the model value) are normally distributed and that the model is an adequate representation of the problem (Englezos, 2001). On this note, the weight can be randomly specified by the user but, since this approach does not clearly state how the weights should be determined, the maximum likelihood estimation method is used to calculate the weights (Englezos and Kalogerakis, 2001).
Maximum Likelihood Estimation
In some cases, the model might not be an adequate representation of the problem or the experimental data might not be reliable, hence, it is the importance to determine the weights in the objective function using the maximum likelihood method. For the condition where it is assumed that the weights do not change from data point to data point, the calculations of the weights in the objective function in the previous section as stated by Englezos and Kalogerakis (Englezos, 2001) are obtained by the following equations:
However, Englezos and Kalogerakis (Englezos, 2001) stated that if the weights change significantly, it is better to use a non-constant diagonal weighting matrix, so that for wi,VLE we have:
And similar equations can as well be written for the other weights.
Therefore the least squares objective function can be restructured to be:
And similarly the objective function can also be restructured for the case where the non-constant diagonal weighting matrix is introduced before it is minimized.
Solution and Validation of results
The model equations are implemented using a suitable Fortran code (Lymperiadis et al., 2007)and the general process modelling software (gPROMS) (Process Systems Enterprise Limited); to execute the optimisation problem.
For each of the different cases, the results can be validated and compared using the average absolute deviation (ADD %) using the formula as stated by Lymperiadis et al (Lymperiadis et al., 2007):
And to further evaluate the reliability of the values of our parameters Kicalc, the intervals between which we have a certain percentage confidence that the true values of our parameters lies between such intervals can be calculated using the joint confidence interval form given by Englezos and Kalogerakis (Englezos, 2001):
Where in the equation above, Nm is the number of experimental measurement, p is the number of parameters, so that Nm-p is the degree of freedom, Î± is the selected probability level in the Fisher's F-distribution, (Acalc)-1 is the ratio of the covariance matrix to the scaling factor and F Î±p,Nm-p is obtained from F-distribution tables where we will take v1=p and v2= Nm-p. The joint confidence interval equation above is an approximation for a non-linear model defined by analogy to the linear model (Englezos, 2001). Therefore to obtain non-approximate values for our parameters, we used the formula obtained from the likelihood ratio test as stated by Rooney and Biegler (Rooney, 2001), developed for non-linear models using log-likelihood function LR given as:
Where Î· is the Bartlett correction and X1-Î±,p2 describes the confidence region.
Section Five: Conclusion and Plan of work
In order to meet the objectives and ultimately the goal this project, we shall use experimental data values of vapour pressure, saturated liquid density, speed of sound and the isobaric heat capacity of the various individual components to correlate the constants in the model. The computation of the constants in the model equation will be carried out by a formulation of an objective function using least squares parameter estimation technique. In using this method, the weights in the objective function shall be computed by the maximum likelihood principle as described in section three. Then we shall employ the used of Fortran code and gPROMS to solve the objective function. And lastly, the values of the parameters calculated shall be investigated by evaluating their respective average absolute deviation (AAD %) and joint confidence interval.
Finally the plan of the project work will follow the sequence:
Learning the code for SAFT-VR Mie
Acquisition of data on oil and gas systems
Running simple parameter estimation problem
Computing confidence interval for the parameters estimated
Application of code to estimate the parameters of various pure components in oil systems
Present thesis to Supervisor
Submission of project.
The graphical illustration of the plan of the project work is shown below.
PROJECT WORK PLAN
1. Reviewing of Literature
Learning Code for SAFT-VR Mie
2.Acquisition of data on oil and gas systems
3.Running simple parameter estimation problem
4.Computing confidence interval for the simple parameters estimated
5.Application of code to estimate the parameters of various pure components in oil systems
7.Present thesis to Supervisor
8.Submission of project