Incorporating critical divergence of isochoric heat capacity into the soft‐SAFT equation of state

Abstract

In the critical region, widely used equations of state, including molecular‐based statistical associating fluid theory (SAFT) equations, fail to quantitatively describe the critical anomalies affected by the diverging fluctuations of density due to their classical mean‐field approximation. Description of some properties in the critical region, in particular vapor‐liquid coexistence and isothermal compressibility, can be improved by fitting (multi) parameters of an equation of state to the properties obtained in accurate computer simulations. However, some anomalies in the critical region, which are caused exclusively by critical fluctuations, cannot be readily generated by this procedure. An example of a critical anomaly, completely absent in the mean‐field approximation, is the divergence of the isochoric heat capacity. A simplified methodology for modifying the original soft‐SAFT equation of state to take into account these anomalies is explored. Possible procedures for further improvement are mentioned. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3073–3080, 2015

A New Predictive Group-Contribution Ideal-Heat-Capacity Model and Its Influence on Second-Derivative Properties Calculated Using a Free-Energy Equation of State

“…Whereas the ideal contribution to c p approaches zero at the critical point, as one would expect (since the total heat capacity diverges whereas, of course, the ideal heat capacity is unaffected), the ideal contribution to the isochoric heat capacity at the critical point is ∼95%; there is no evidence of any sort of divergence of the calculated isochoric heat capacity at the critical point. This absence is actually expected using an engineering equation of state such as SAFT (or a cubic equation), as has been discussed by Llovell et al 82 The divergence is due exclusively to critical fluctuations, which are not accounted for in the equation of state (in order to capture the divergence, one would require a special treatment such as that introduced by Llovell et al 82 MPa) the experimental isochoric heat capacity is still (slightly) influenced by the vicinity of the critical point, as evidenced by the maximum in the gradient of the curve (traced by the red open-diamond symbols) at a temperature a little below 500 K. In this region, the calculated isochoric heat capacity represents a significant underestimate, and one sees no evidence of a maximum in the calculated curve. From Figure 7a we can discern that the ideal contribution here is ∼90%.…”

“…Whereas the ideal contribution to c p approaches zero at the critical point, as one would expect (since the total heat capacity diverges whereas, of course, the ideal heat capacity is unaffected), the ideal contribution to the isochoric heat capacity at the critical point is ∼95%; there is no evidence of any sort of divergence of the calculated isochoric heat capacity at the critical point. This absence is actually expected using an engineering equation of state such as SAFT (or a cubic equation), as has been discussed by Llovell et al The divergence is due exclusively to critical fluctuations, which are not accounted for in the equation of state (in order to capture the divergence, one would require a special treatment such as that introduced by Llovell et al or McCabe and Kiselev). It is interesting to review Figure panels a and b with this new insight.…”

Comparison of Two Types of Crossover Soave–Redlich–Kwong Equations of State for Derivative Properties of n-Alkanes

“…, and the universal value of is equal to approximately 0.11 [25]. On the other hand, the mean-field approximation predicts a value equal to 0 for this exponent.…”

“…This is an additive renormalization term given in Eq. ( 6), responsible for the divergent behavior of the isochoric heat capacity [25,31]. As seen in Eq.…”

“…19) indicates that the second approach yields similar results at the critical point, but is inferior to the other RG method for the description of the property in subcritical conditions. According to the summary of the main results of obtained with the crossover EoSs and the classical SRK EoS reported in Table 5, the importance of the Kernel term 31 to incorporate the short wavelength fluctuations not only at the divergence but also close to the critical point can be reflected [25]. The calculation of the critical exponent is an important test to evaluate the behavior of the EoS, as the mean-field models are unable to predict the correct values of the universal critical exponents.…”