Chapter 8. The Statistical Associating Fluid Theory (SAFT)
Problem 1. Monomer fraction
Equation 8.4 gives the general expression for the fraction of non-bonded association sites A on
a component i.
i. Derive the pure-component version of this equation in the limit of a pure, self associating
fluid, X A .
ii. Derive the X A -expressions for the schemes 1, 3B and 4C shown in table 8.11. These
association schemes are often used for organic acids, alcohols and water, respectively.
Problem 2. Combining rules for cross-associating mixtures
Equations 8.49, 8.50 and 8.51 present combining rules used in SAFT for cross-associating
mixtures.
i. Show that the combining rule of equation 8.50:

Ai B j

 AB 
i i
2
Aj B j
and 
Ai B j
 
Ai Bi

Aj B j
  i j

  ij





3
8.50
can be derived from Elliott’s rule (equation 8.51;  i j   Ai Bi  j j ) when certain
assumptions are made. Are the assumptions reasonable?
ii. How different is the combining rule of equations 8.50 or 8.51 from the CR-1 rule (equation
AB
 Ai Bi   j j
Ai B j
AB
AB

8.49; 
and  i j   Ai Bi  j j )? The Elliott and CR-1 rules are also
2
extensively used in the CPA equation of state (see Chapter 9). Are these combining rules
identical in the CPA and SAFT models?
AB
AB
Problem 3. Comparison of methods for estimating parameters for associating fluids
Different estimation methods have been proposed for obtaining PC-SAFT parameters for
associating fluids from Grenner et al. 86,87 and from Gross and Sadowski77 .
i.
Which are the similarities and the differences of the two approaches?
ii.
Estimate the association parameters of some alcohols using the Grenner et al.
approach (equation 8.44b) and compare the values to those presented by Gross and
Sadowski77.
iii.
Can the method of Grenner et al. 86,87 be extended to other families of associating
fluids beyond alcohols and glycols ? How can this be accomplished?
Problem 4. The original and simplified PC-SAFT equations of state
Table 8.10 presents the differences between the original and the simplified PC-SAFT equations
of state.
i.
Explain the similarities and differences between the two variants of PC-SAFT.
ii.
Starting from the expressions of the radial distribution function and hard-sphere
Helmholtz energy from PC-SAFT (equations 8.9 and 8.18), derive the expressions
for the simplified PC-SAFT using equation 8.41 i.e.    3 .
1
Problem 5. Comparison of GC PC-SAFT equations of state.
Tamouza et al.60,61 and Tihic et al.66 have proposed group-contribution methods for estimating
the parameters of the PC-SAFT equation of state.
i.
Discuss the similarities and differences of the two approaches with emphasis on
their current range of applicability and the types of compounds the two approaches
can be employed for.
ii.
Estimate with the two methods the three PC-SAFT non-associating parameters for
selected compounds and compare the results to the parameters of PC-SAFT as
estimated from vapor pressures and liquid densities (Appendix A, book’s website).
Problem 6. Comparison of estimation methods for PC-SAFT parameters
Two alternative approaches to estimate the three parameters of PC-SAFT for non-associating
mixtures are:
i.
Via the generalized correlations with the molecular weight e.g. those shown in the
caption of figure 8.2 for n-alkanes, while more correlations of this type for other
homologous series are presented in Table 8 of the manuscript by Tihic et al.19.
ii.
Via the DTU-group contribution method (Tihic et al.66), presented in equations 8.53
and the parameter tables of Appendix 8.B.
Select a few compounds belonging to different families e.g. one alkane, one ketone, one ester,
etc. and compare the two approaches. Include some compounds for which both first- and
second-order groups are available. Compare the results to the “experimental” data i.e. the PCSAFT parameters estimated directly from vapour pressure and liquid density data (see
Appendix A; book’s website). Discuss the accuracy and limitations of the two approaches.
Which method performs best for the compounds you have selected?
Problem 7. Comparison of estimation methods for PC-SAFT interaction parameters
The purpose of this problem is to compare various methods for estimating the interaction
parameters kij of the PC-SAFT equation of state. Select for example some of the binary
mixtures for which interaction parameters are presented in this chapter (as well as in Chapters
13 and 14) fitted to experimental data in order to use them in your assessment of the
capabilities of the various methods.
Compare equations 8.48a and 8.48b using PC-SAFT values for the segment diameters but
various possibilities for the ionization potential e.g.
i.
experimental values e.g. from Pesuit (1978, Ind. Eng. Chem. Fundam., 17, 235).
ii.
from correlations as a function of the segment diameter such as the one shown in
equation 3.28 (or you may develop your own).
iii.
as estimated from quantum chemistry calculations, see equation 3.31.
Compare the interaction parameters also to the values which would be obtained using the
London parameters and the geometric mean rule i.e. C12  C1C2 (1  k12 ) [see also Chapters 2
and 3]
Discuss the results and the accuracy of the three prediction methods mentioned in this problem
(i, ii and iii –using 8.48a and 8.48b). What is the effect of the ionization potential values ?
2