Energy Additivity Approaches to QSPR Modeling in Estimation of Dynamic Viscosity of Fatty Acid Methyl Ester and Biodiesel

Abstract

Viscosity is an important physical property of fatty acid methyl esters (FAME) and biodiesel (mixture of FAMEs). In this work, quantitative structure–property relationship (QSPR) for estimation of dynamic viscosity of FAMEs and biodiesel is approached via the Gibbs energy additivity method. The Gibbs energy of dynamic viscous flow is simply derived from the sum of the Gibbs energy of kinematic viscous flow and Gibbs energy of volumetric expansion. The derived model can be used for estimation of dynamic viscosity of saturated and unsaturated FAMEs commonly found in nature. Also, the proposed model can be extended to a mixture of FAMEs or biodiesel as well as biodiesel blends. Thus, the dynamic viscosity of FAMEs as well as neat and blended biodiesels can be estimated by the same equation from the carbon number (z) and number of double bonds (n _d) at different temperature (T). The average absolute deviation (AAD) values for saturated, unsaturated FAMEs, biodiesels, and biodiesel blends (at 20–100 °C) are approximately the same as the original model for estimation of kinematic viscosity.

Appendix

Gibbs energy additivity for biodiesel (mixture of FAME),

$$\Delta G\; = \;x_{1} \Delta G_{1} \; + \;x_{2} \Delta G_{2} \; + \;x_{3} \Delta G_{3} \ldots x_{m} \Delta G_{m}$$

(A1)

$$\Delta G\; = \;\sum\limits_{i = 1}^{m} {x_{i} \Delta G_{i} }$$

(A2)

Equation A2 is similar to Eq. 3, where x _i is mole or mass fraction and ΔG _1…m are the Gibbs energy contribution from each FAME.

Expanding Eq. A2 to its enthalpy and entropy forms,

$$\begin{aligned} \Delta G\; = \;x_{1} \left( {s_{0,\rho } \; + \;s_{1,\rho } z_{1} \; + \;s_{2,\rho } n_{{{\text{d}}1}} \; + \;\frac{{h_{0,\rho } \; + \;h_{1,\rho } z_{1} \; + \;h_{2,\rho } n_{{{\text{d}}1}} }}{T}} \right)\; + \;x_{2} \left( {s_{0,\rho } \; + \;s_{1,\rho } z_{2} \; + \;s_{2,\rho } n_{{{\text{d}}2}} \; + \;\frac{{h_{0,\rho } \; + \;h_{1,\rho } z_{2} \; + \;h_{2,\rho } n_{{{\text{d}}2}} }}{T}} \right) + \hfill \\ \, x_{3} \left( {s_{0,\rho } \; + \;s_{1,\rho } z_{3} \; + \;s_{2,\rho } n_{{{\text{d}}3}} \; + \;\frac{{h_{0,\rho } \; + \;h_{1,\rho } z_{3} \; + \;h_{2,\rho } n_{{{\text{d}}3}} }}{T}} \right)\; + \; \ldots x_{m} \left( {s_{0,\rho } \; + \;s_{1,\rho } z_{m} \; + \;s_{2,\rho } n_{\text{dm}} \; + \;\frac{{h_{0,\rho } \; + \;h_{1,\rho } z_{m} \; + \;h_{2,\rho } n_{\text{dm}} }}{T}} \right), \hfill \\ \end{aligned}$$

(A3)

Rearranging,

$$\begin{aligned} \Delta G\; = \;s_{0,\rho } \left( {x_{1} \; + \;x_{2} \; + \;x_{3} \; + \cdots x_{m} } \right)\; + \;s_{1,\rho } \left( {x_{1} z_{1} \; + \;x_{2} z_{2} \; + \;x_{3} z_{3} \; + \cdots x_{m} z_{m} } \right) + \hfill \\ \, s_{2,\rho } \left( { \, x_{1} n_{{{\text{d}}1}} \; + \;x_{2} n_{{{\text{d}}2}} + x_{3} n_{{{\text{d}}3}} + \cdots x_{m} n_{\text{dm}} } \right)\; + \cdots \frac{{h_{0,\rho } }}{T}\left( {x_{1} \; + \;x_{2} \; + \;x_{3} + \cdots x_{m} } \right) + \hfill \\ \, \frac{{h_{1,\rho } }}{T}\left( {x_{1} z_{1} \; + \;x_{2} z_{2} \; + \;x_{3} z_{3} \; + \cdots x_{m} z_{m} } \right) + \frac{{h_{2,\rho } }}{T}\left( { \, x_{1} n_{{{\text{d}}1}} + x_{2} n_{{{\text{d}}2}} + x_{3} n_{{{\text{d}}3}} + \cdots x_{m} n_{\text{dm}} } \right) \hfill \\ \end{aligned}$$

(A4)

Averaging,

$$\Delta G\; = \;s_{0,\rho } \; + \;s_{1,\rho } \frac{{\sum\limits_{i = 1}^{m} {x_{i} z_{i} } }}{{\sum\limits_{i = 1}^{m} {x_{i} } }}\; + \;s_{2,\rho } \frac{{\sum\limits_{i = 1}^{m} {x_{i} n_{di} } }}{{\sum\limits_{i = 1}^{m} {x_{i} } }}\; + \;\frac{{h_{0,\rho } }}{T}\; + \;\frac{{h_{1,\rho } }}{T}\frac{{\sum\limits_{i = 1}^{m} {x_{i} z_{i} } }}{{\sum\limits_{i = 1}^{m} {x_{i} } }}\; + \;\frac{{h_{2,\rho } }}{T}\frac{{\sum\limits_{i = 1}^{m} {x_{i} n_{di} } }}{{\sum\limits_{i = 1}^{m} {x_{i} } }}$$

(A5)

And $z_{\text{ave}} \; = \;\frac{{\sum\limits_{i = 1}^{m} {x_{i} z_{i} } }}{{\sum\limits_{i = 1}^{m} {x_{i} } }}$; $n_{\text{d(ave)}} \; = \;\frac{{\sum\limits_{i = 1}^{m} {x_{i} n_{di} } }}{{\sum\limits_{i = 1}^{m} {x_{i} } }}.$