# Application of LNET Models to Study Thermodiffusion

## Abstract

Linear nonequilibrium thermodynamic formulations to study thermodiffusion in liquid mixtures were introduced in Sect. 2.2 of Chap. 2. In this chapter, application of some of these formulations to thermodiffusion in different types of fluid mixtures and the outcomes are presented.

## Keywords

Thermoelectric Power Hydrocarbon Mixture Partial Molar Enthalpy Thermodiffusion Coefficient Binary Molten## 3.1 Liquid Hydrocarbon Mixtures

In this section, the application of two thermodiffusion models to hydrocarbon mixtures will be illustrated, viz., the Firoozabadi model and the Kempers model. These models are described in Sect. 2.2 of Chap. 2. In the ensuing paragraphs, the mixtures considered are at atmospheric pressure.

### 3.1.1 Binary Hydrocarbon Mixtures

*Q*

^{ ∗ }in terms of the partial molar energy of the components, the Kempers model formulates

*Q*

^{ ∗ }using the partial molar enthalpy.

### 3.1.2 Ternary Hydrocarbon Mixtures

_{12}–isobutylbenzene (IBB)–tetralin (THN) and nC

_{8}–nC

_{10}–methylnaphthalene (MN) evaluated with these two models are shown in Fig. 3.2. For each mixture, all three thermodiffusion coefficients are calculated. Both models work well for the nC

_{12}-IBB-THN mixture in which the mass fraction of all three components are 1 ∕ 3. However, the Kempers model predicts a wrong sign for the first component in the nC

_{12}–IBB–THN mixture. Noting that the thermodiffusion coefficient is very small for this component, an inability of the Kempers model to predict this is perhaps an indicator that it is not very sensitive for very small values of the thermodiffusion coefficients.

In the nC_{8}–nC_{10}–MN mixture, a similar result is obtained when the mass fraction of each component is 1 ∕ 3. However, as the composition changes, there are more discrepancies in the calculations. Specifically, the errors are much larger for the first two thermodiffusion coefficients. In fact, for the second thermodiffusion coefficient of this last mixture, both models predict the wrong sign, i.e., a wrong direction of separation. It must be noted that in applying these models, there might be differences in the values of the calculated coefficients even with the same model. This is because of the use of slightly different values of the model parameters that are not uniform in the literature.

## 3.2 Liquid Associating Mixtures

*PC-SAFT*equation of state, that is well suited for associating mixtures, for ease of comparison. The results of these models for two mixtures, viz., water–ethanol and water–methanol of various mole fractions of water are presented in Fig. 3.3a, b.

As expected, the errors are the largest in the Haase model, whereas the Firoozabadi model is somewhat more accurate. The accuracy of the latter is attributed to the fact that the model depends upon several tuning parameters that can be slightly tweaked to enhance the performance of the model. At present, the most accurate model seems to be the one by Eslamian and Saghir that is at least able to qualitatively predict the sign change in both mixtures in these figures. Specifically, in close agreement with the experiments, the model predicts the sign change at low concentration of the alcohol. Of course, there are still large errors in the accuracy of the magnitude of α in very dilute limits of the compositions. This is because of the complex interactions between the molecules due to additional forces like hydrogen bonding, which is still not very accurately represented in the model.

## 3.3 Dilute Polymer Mixtures

| | ||
---|---|---|---|

Solvent | [m | [m | |

Cyclohexanone | 3. 58 ± 0. 77 | 1.33 | |

THF | 12. 27 ± 1. 05 | 7.81 | |

Toluene | 12 ± 4. 97 | 6.84 | |

Ethyl acetate | 11. 57 ± 1. 87 | 9 | |

MEK | 22. 86 ± 1. 87 | 10.05 |

*D*

_{T}of polystyrene in two solvents, viz., ethyl acetate and tetrahydrofuran (THF) at 295 K are shown in Fig. 3.4 for various molecular weights of polystyrene. The experimental data reported in [6, 7] are also shown for comparison. In both mixtures, the model is able to capture the logarithmic trend in the variation of

*D*

_{ T }. In the polystyrene–ethyl acetate mixtures, the model predictions are in close agreement with the experimental data.

In the polystyrene–tetrahydrofuran mixtures, the model is able to predict the sign change, as is observed in the experiments. However, the model predictions level off much higher than the experimental data, resulting in larger errors at higher molecular weights. This is perhaps due to the role of other physical and chemical properties of the polymer in the thermodiffusion process.

## 3.4 Molten Metal Mixtures

_{T}in this model is:

*e*,

*z*,

*z*

_{1},

*S*,

*S*

_{1}, and

*N*are electron charge, valence of the ions in the mixture, valence of the ions of component 1, thermoelectric power of the mixture, thermoelectric power of component 1 and Avogadro number, respectively. All other notations are as explained in Chap. 2.

α_{T} calculated using the Haase, Kempers, Drickamer, and Eslamian–Saghir models, compared with the experimental data of equimolar molten metal mixtures studied by Winter and Drickamer [9]

Mixture | Expt. | Haase | Kempers | Drickamer | Eslamian | |
---|---|---|---|---|---|---|

Sn-Bi | − 0. 10 | − 0. 01 | 0.002 | − 0. 518 | − 0. 396 | |

Sn-Cd | − 0. 35 | 0.141 | 0.026 | 0.140 | − 0. 263 | |

Sn-Zn | − 4. 10 | 0.709 | 0.152 | − 1. 918 | − 3. 085 | |

Sn-Pb | − 1. 90 | − 0. 146 | 0.065 | − 1. 331 | − 1. 143 | |

Sn-Pb | − 0. 83 | − 0. 146 | 0.061 | − 0. 778 | − 0. 685 | |

Sn-Ga | − 0. 18 | 0.101 | − 0. 028 | 0.789 | 0.374 | |

Bi-Pb | − 1. 13 | − 0. 004 | 0 | 0.005 | − 0. 072 |

## 3.5 Effect of the Equation of State

*v-PR*equation of state and the

*PC-SAFT*equation of state. The thermodiffusion coefficients of six binary and three ternary hydrocarbon mixtures are presented in Fig. 3.5a, b.

In both, binary and ternary mixtures, the Firoozabadi model coupled with the *v-PR* equation of state is the most accurate and is able to predict the thermodiffusion coefficients fairly accurately. Further, for this model, the largest error are in the nC_{8}–nC_{10}–MN mixture with a composition of 16.7–16.7–66.6 wt%. On the other hand, the Kempers model with the *PC-SAFT* equation of state is the most error prone combination to be employed.

The good performance of the Firoozabadi model coupled with *v-PR* equation of state is because of the following: (1) the matching parameters of this model can be tuned to make the model suitable for a certain mixture. Generally, via an initial tuning using some experimental data, the parameters can be adjusted to obtain good results. Subsequently, using these tuning values to study other mixtures does not produce very large errors. (2) the density predictions by the *v-PR* equation of state is very accurate. In fact *PC-SAFT* is notorious in overpredicting the density that has a negative impact on the thermodiffusion coefficients [5]. Also, *PC-SAFT* is primarily designed for associating mixtures. It uses just the partial molar enthalpy in calculating the gradient of chemical potential and eventually underpredicts the value. This means that the thermodiffusion coefficient is inflated in magnitude [5].

In summary, the application of the different thermodiffusion models presented in Chap. 2 to study thermodiffusion in various types of mixtures has been demonstrated in this chapter. Each model performs differently and is good in predicting the thermodiffusion parameters in some mixtures, whereas its performance can be very poor in others. In evaluating the sign change effects in the mixtures, it is seen that some of these models are able to predict the sign change qualitatively. Absolute accuracy of the models for a wide variety of mixtures is still lacking. This is because of the lack of accurate representation of the chemical effects, inter-particle interaction forces such as hydrogen bonding, etc.

Finally, the equation of state is also an important aspect of these models, which goes a long way in determining the accuracy of the models. For instance, a thermodiffusion model coupled with *v-PR* equation of state is more suited for hydrocarbon mixtures than using the *PC-SAFT* equation of state. The latter is more appropriate for associating mixtures.

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