Thermodynamic frameworks

 

Based on the research background of this Team, we provide calculation tools that allow for phase equilibrium modelling using Thermodynamic models such as excess Gibbs energy models and equations of state.


i)                    GE-models: NRTL and UNIQUAC

 

Using this framework (built in MATLAB language), the user can describe solid-liquid equilibrium data for binary mixtures in a very convenient way. The solid liquid equilibrium condition is required:

 

                                                                (1)

 

where xi is the solubility (in mole fraction) and ΔHiS-L, TiS-L , and ΔCp,iS-L are the melting enthalpy, melting temperature, and heat-capacity difference during melting of pure solute (considering a hypothetical liquid state). These pure solute’s melting properties are input variables that the user must supply in the input excel file. Besides, the SLE condition also requires the calculation of activity coefficients of the solute in the liquid phase. This thermodynamic function is calculated in this framework through GE-models such as NRTL1 or UNIQUAC2.

    The program reads the input file supplied by the user (containing the solubility data and the initial values for the parameters), and uses a minimization routine (fminsearch) which fits the model’s binary parameters to the experimental data. Afterwards the SLE condition is applied again together with the thermodynamic model to calculate solubilities. The program outputs are the optimized parameters, the calculated solubilities and the models performance (through average relative deviations, ARD). This thermodynamic framework is a very convenient tool that can be used to represent in a single run, a large number of experimental data sets. Figure 1 shows the working structure.

 

Figure 1. Block structure of the thermodynamic modelling program


Examples of results using this framework


ii)                    PC-SAFT equation of state


We also provide another way to model phase equilibrium using a very robust equation of state based on statistical thermodynamics, the PC –SAFT3. This model is able to capture the volumetric properties and phase behaviour of pure compounds and mixtures exhibiting very complex interactions such as association effects. The model also accounts for non-spherical shape of the molecules. The general form of this equation of state calculates the reduced residual Helmholtz energy of a system (ares) as a sum of the reference-chain contribution (ahc), dispersive interactions (adisp), and association (aassoc) effects:

 

                                                                                                                   (2)

The molecules are portrayed as chains of spherical segments which have association sites assigned (see Figure 2).


Figure 2.  PC-SAFT molecular description of a given compound

 

In general, the model has five pure component parameters (three for non-associating compounds):


mseg      number of segments

σi            segment diameter

ui/kB      dispersion energy 

εAiBi/kB  association-energy

κAiBi       association-volume

 

Thermophysical properties and phase equilibria can be obtained from an equation of state by applying standard thermodynamic relations that allow the calculation of density and fugacity coefficients4. Our PC-SAFT framework (Property calculator), developed in Fortran language, is able to calculate volumetric properties and several phase equilibrium phenomena in mixtures up to three components:

           a) Density calculations

           b) Osmotic coefficients

c) Infinite dilution activity coefficients

d) Solid-liquid equilibrium

 

In addition, we have available a parameter estimator (built in Fortran language) program that allows to obtain the PC-SAFT pure component parameters combining several sets of experimental data:

 

    a)      Pure component density

    b)      Binary mixture density

    c)       Solid-liquid equilibria

    d)      Osmotic coefficients

    e)      Infinite dilution activity coefficients

 

Property calculator I/O structure



Figure 3. Block structure of the PC-SAFT (property calculator) modelling program



Parameter estimator I/O structrure

 

 Figure 4. Block structure of the PC-SAFT (parameter estimator) modelling program

 


Examples of results using this framework


    iii)                    Correlation of osmotic coefficients

 

For the correlation of the experimental osmotic coefficients, the ion-interaction Extended Pitzer model of Archer, and the local composition Modified Non-Random Two Liquid (MNRTL) equation are used.


a)      Ion-interaction model: Extended Pitzer model of Archer


In this model, the osmotic coefficients (f) are expressed as:

                                        (3)                              

where,

                                                 (4)                                                                  

                 (5)                           (6)          

                                          (7)


In these equations, the ion interaction parameters of the extended Pitzer model of Archer are b (0), b (1), b (2), C (0) and  C (1), dependent on temperature and pressure, Af is the Debye-Hückel constant in the molal scale, and  a1, a2, a3, and b can be adjustable parameters or kept fixed at constant values. The term I is the ionic strength in molality. Figure 5 shows the structure of the framework.



Figure 5. Block structure of the Extended Pitzer model of Archer modelling program

 

b)      MNRTL equation


The activity coefficient is described by two terms: a long-range contribution (LR) and a short-range contribution (SR):

                                                                                                                                                                (8)

For this model, the long-range term is the Pitzer-Debye-Hückel (PDH) equation on a mole fraction scale:

                                                                                                                     (9)

in which the term I is the ionic strength on a mole fraction basis, I = 1/2Σxizi2. The short-range contribution for the activity coefficient of the solvent is:

                                   (10)

  where τca,m and τm,ca are the parameters of the model, Xi is the effective mole fraction (Xi = xi·Ki, Ki = zi for ions and Ki = 1 for solvent), and it is assumed:

                                                                                                                                                                   (11) 

where ωca,m and ωm,ca are the adjustable parameters, and α is the nonrandomness factor.



Figure 6. Block structure of the MNRTL modelling program

Example of results with these frameworks


If you are interested in our frameworks for the description of phase equilibrium of your experimental data we will be pleased to help you. Do not hesitate to contact us for additional Information: [email protected].



References:

 

1.       Renon, H., Prausnitz, J. M. 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE Journal, 14, 135-144.

 

2.     Abrams, D. S., Prausnitz, J. M. 1975. Statistical thermodynamics of liquid-mixtures - new expression for excess Gibbs energy of partly or completely miscible systems. AIChE Journal, 21, 116-128.

 

3.   Gross, J., Sadowski, G. 2001. Perturbed-Chain SAFT: An equation of state based on a perturbation theory for chain molecules. Industrial & Engineering Chemistry Research, 40, 1244-1260.

 

4.       Prausnitz, J. M., Lichtenthaler, R. N.,Azevedo, E. G. 1999. Molecular thermodynamics of fluid-phase equilibria New Jersey, Prentice Hall.